nonlocal symmetries and formulae for generation of solutions for a class of diffusion convection equations abstract new formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation we study in what particular ways additional lie symmetries of diffusion convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations the formulae derived are used for the construction of exact solutions

group classification of variable coefficient generalized kawahara equations an exhaustive group classification of variable coefficient generalized kawahara equations is carried out as a result we derive new variable coefficient nonlinear models admitting lie symmetry extensions all inequivalent lie reductions of these equations to ordinary differential equations are performed we also present some examples on the construction of exact and numerical solutions

similarity reduction nonlocal and master symmetries of sixth order korteweg devries equation a systematic investigation to derive the lie point symmetries nonlocal and master symmetries of sixth order korteweg de vries equation kdv6 is presented using the obtained point symmetries similarity reductions are derived and constructed their particular solutions wherever possible it is shown that kdv6 admits infinitely many nonlocal and master symmetries the existence of infinitely many master symmetries ensures that kdv6 is completely integrable

on similarity solutions of zabolotskaya khokhlov equation abstract the new closed form solutions of the 2 1 dimensional zabolotskaya khokhlov equation are constructed by using the similarity transformations method via lie group theory the zabolotskaya khokhlov equation has been reduced into a new partial differential equation with smaller number of independent variables further using the similarity transformations method the new partial differential equation is reduced into an ordinary differential equation the ordinary differential equation provides semi analytical solutions elastic behaviour of multisolitons and dromion annihilation of solutions are analyzed physically through their evolutional profiles

nonlocal symmetries and recursion operators partial differential and differential difference equations abstract a systematic method to derive the nonlocal symmetries for partial differential and differential difference equations with two independent variables is presented and shown that the korteweg de vries kdv and burger s equations volterra and relativistic toda rt lattice equations admit a sequence of nonlocal symmetries an algorithm exploiting the obtained nonlocal symmetries is proposed to derive recursion operators involving nonlocal variables and illustrated it for the kdv and burger s equations volterra and rt lattice equations and shown that the former three equations admit factorisable recursion operators while the rt lattice equation possesses 2 2 matrix factorisable recursion operator the existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential difference equations does not always determine their mathematical structures for example bi hamiltonian representation

maximum likelihood estimation of rician distribution parameters the problem of parameter estimation from rician distributed data e g magnitude magnetic resonance images is addressed the properties of conventional estimation methods are discussed and compared to maximum likelihood ml estimation which is known to yield optimal results asymptotically in contrast to previously proposed methods ml estimation is demonstrated to be unbiased for high signal to noise ratio snr and to yield physical relevant results for low snr

linearisation and potential symmetries of certain systems of diffusion equations abstract we consider systems of two pure one dimensional diffusion equations that have considerable interest in soil science and mathematical biology we construct non local symmetries for these systems these are determined by expressing the equations in a partially and wholly conserved form and then by performing a potential symmetry analysis on those systems that can be linearised we give several examples of such systems and in a specific case we show how linearising and hodograph type mappings can lead to new solutions of the diffusion system

soliton solutions of blmp equation by lie symmetry approach abstract shallow water wave equations are usually described by korteweg de vries kdv type equations in this paper we have used lie transformation group theory to solve 2 1 dimensional boiti leon manna pempinelli blmp equation we have obtained some exact solutions of blmp equation in the explicit form through similarity reduction all the reported results are expressed in closed form and analysed physically through their evolution profiles the physical analysis reveals that the nature of solutions is parabolic quasi periodic multisoliton and asymptotic

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

new explicit solutions of einstein equations in the framework of gap theory the presented paper develops a new method for solving einstein equations explicitly not only the solutions are new but also the mathematical framework of their construction which is given by a nonstandard function theory built over nonstandard algebras in the following denoted as gaps and will be called here as gap theory it is shown that the gap theory succeeds in regions where group theory fails the group structure is too small to solve einstein equations explicitly

invariant discretization schemes for the shallow water equations invariant discretization schemes are derived for the one and two dimensional shallow water equations with periodic boundary conditions while originally designed for constructing invariant finite difference schemes we extend the use of difference invariants to allow construction of invariant finite volume methods as well it is found that the classical invariant schemes converge to the lagrangian formulation of the shallow water equations these schemes require redistribution of the grid points according to the physical fluid velocity i e the mesh cannot remain fixed in the course of the numerical integration invariant eulerian discretization schemes are proposed for the shallow water equations in computational coordinates instead of using the fluid velocity as the grid velocity an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level the numerical conservation of energy mass and momentum is evaluated for both the invariant

partial symmetry of initial value problems abstract it is generally believed that the symmetry of initial value problems ivp must leave both the partial differential equations pdes and initial conditions invariant in this paper we propose partial symmetry of ivp which needs less restrictive conditions on the pdes and initial conditions and only leaves the ivp invariant on some non empty subset of the whole solution set of the governing pdes thus some symmetries which either only leave the pdes invariant or only leave initial conditions invariant are partial symmetries of ivp then considering from two different starting points we define two types of partial symmetry and give the corresponding algorithms where one starts with the pdes and the other is from the initial conditions applications to boussinesq equation and a class of nonlinear heat equation are performed

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

symmetries of the fisher kolmogorov petrovskii piskunov equation with a nonlocal nonlinearity in a semiclassical approximation abstract the classical group analysis approach used to study the symmetries of integro differential equations in a semiclassical approximation is considered for a class of nearly linear integro differential equations in a semiclassical approximation an original integro differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis the approach is illustrated by the calculation of the lie symmetries in explicit form for a special case of the one dimensional nonlocal fisher kolmogorov petrovskii piskunov population equation

an equation admitting infinite true contact transformations abstract the linear wave equation is shown to possess the unique property that if w n is a true contact transformation admitted by the wave equation i e w n is not linear in the first derivatives of the dependent variable then so is n w n we comment of the physical implications

comparison of approximate symmetry and approximate homotopy symmetry to the cahn hilliard equation complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the cahn hilliard equation are performed and the reductions are constructed by an optimal system of one dimensional subalgebras zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations the relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one

generalized symmetries for the ideal mhd equations abstract for the nonlinear ideal mhd equations we give potential and local scaling symmetries which are two different generalizations of the classical lie point symmetries they transform given similarity solutions to new exact solutions some physical applications are discussed

quantum fluid dynamics description of a multiterm potential nonlinear schrodinger langevin equation via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear schrodinger langevin equation with a multiterm potential for the description of interactions in non conservative systems the nonlinear complete set of equations forming the basis of the model are derived the corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid particle behaviour

an exact solution of the schrodinger equation for a multiterm potential the authors show that the schrodinger equation with the potential bx cx2 dx3 ex4 is exactly solvable on the half line x or 0 provided two simple relations between b c d and e hold some remarks concerning the b d 0 case are made

similarity solutions of an equation describing ice sheet dynamics this paper focuses upon the derivation of the similarity solutions of a nonlinear equation associated with a free boundary problem arising in glaciology we present a potential symmetry analysis of this second order nonlinear degenerate parabolic equation related to non newtonian ice sheet dynamics in the isothermal case a full classical and also a non classical symmetry analysis are presented after obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation without any unilateral formulation a particular example of a similarity solution to a problem formulated with cauchy boundary conditions is described this allows us to obtain several qualitative properties of the free moving boundary in the presence of an accumulation ablation function with realistic physical properties

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

on the weak symmetry groups of partial differential equations abstract we define an algorithm to characterize all weak symmetry groups of a partial differential equation new invariant solutions to the fokker planck equation are found

impact of an improved combination of signals from array coils in diffusion tensor imaging an improved method for the combination of signals from array coils is presented as a way to reduce the influence of the noise floor on the estimation of diffusion tensor imaging dti parameters by an optimized combination of signals from the array channels and complex averaging of measurements this method leads to a significant reduction of the noise bias this combination algorithm allows computation of accurate tensors by using the simple two point method and is shown to provide results similar to the ones obtained using the standard signal combination and a nonlinear regression method with noise parameter estimation in many applications the use of this combination method would result in a scan time reduction in comparison to the current standard the effects of the improved combination on diffusion decay curves fractional anisotropy maps and apparent diffusion coefficient adc profiles are demonstrated

exact solutions for anharmonic oscillators completes the author s investigation see phys lett a vol 81 p 17 8 1981 and presents exact solutions and eigenvalues for anharmonic oscillators the solutions being given in the form of definite integrals the conditions for the validity of these results are investigated

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

derivation of conservation laws from nonlocal symmetries of differential equations an identity is derived which yields a correspondence between symmetries and conservation laws for self adjoint differential equations this identity does not rely on use of a lagrangian as needed to obtain conservation laws by noether s theorem moreover unlike noether s theorem which can only generate conservation laws from local symmetries the derived identity generates conservation laws from nonlocal as well as local symmetries it is explicitly shown how noether s theorem is extended by the identity conservation laws arising from nonlocal symmetries are obtained for a class of scalar wave equations with variable wave speeds the constants of motion resulting from these nonlocal conservation laws are shown to be linearly independent of all constants of motion resulting from local conservation laws

clinical dt mri estimation smoothing and fiber tracking with log euclidean metrics diffusion tensor magnetic resonance imaging dt mri or dti is an imaging modality that is gaining importance in clinical applications however in a clinical environment data have to be acquired rapidly often at the expense of the image quality this often results in dti datasets that are not suitable for complex postprocessing like fiber tracking we propose a new variational framework to improve the estimation of dt mri in this clinical context most of the existing estimation methods rely on a log gaussian noise gaussian noise on the image logarithms or a gaussian noise that do not reflect the rician nature of the noise in mr images with a low signal to noise ratio snr with these methods the rician noise induces a shrinking effect the tensor volume is underestimated when other noise models are used for the estimation in this paper we propose a maximum likelihood strategy that fully exploits the assumption of a rician noise to further reduce the influence of the noise we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself which results in a maximum a posteriori estimation optimizing such a nonlinear criterion requires adapted tools for tensor computing we show that riemannian metrics for tensors and more specifically the log euclidean metrics are a good candidate and that this criterion can be efficiently optimized experiments on synthetic data show that our method correctly handles the shrinking effect even with very low snr and that the positive definiteness of tensors is always ensured results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord

nonlocal symmetries of the kdv equation a loop algebra of nonlocal isovectors of the korteweg de vries kdv equation is introduced which is derived from the bi hamiltonian structure of that equation by inverting the usual recursion operator these symmetries are lifted to the modified kdv mkdv equation and two related evolution equations via coordinate transformations the effect on the algebraic structure of the isovector algebra is considered possible applications of these symmetries include recovering miura type transformations

nonlocal symmetries and the kaup kupershmidt equation computational and geometric aspects of nonlocal infinitesimal symmetries of nonlinear partial differential equations are considered in particular the relation of nonlocal symmetries with classical generalized and internal symmetries is briefly discussed a nonlocal symmetry for the kaup kupershmidt equation is introduced and studied in some detail some explicit particular solutions are found with its help and a darboux like transformation is also obtained

the existence of contact transformations for evolution type equations we investigate the existence of a one parameter group of contact transformations for evolution type equations ut f t x u ux uxx u n subscripts denote differentiation unless otherwise indicated where u n is the nth derivative of u with respect to x we prove that contact transformations of evolution equations where f is expandable as a power series in terms of all derivatives of order higher than one are just extended lie point transformations this result is extended to the case with m independent space variables as a consequence we obtain an ansatz for determining lie point transformations for nth order evolution equations with m independent space variables examples are given to verify the results obtained as well as to show how lie point transformations of these evolution type partial differential equations can be calculated from this ansatz

anharmonic oscillator with general polynomial potential this paper derives simple analytical formulas for the energy eigenvalues en of one dimensional anharmonic oscillators characterized by the potentials 2x2 m 2 x2 for doubly anharmonic oscillators over a wide range of n and these energy values agree well with the numerical values calculated by earlier workers

exact solutions for wave equations of two layered media with smooth transition the wave equation c2 x uxx utt 0 is solved for wave speeds c x corresponding to two layered media with smooth transition from layer to layer the wave speed c x has four free parameters to fit a given medium solutions are constructed from invariant solutions of a related system of first order partial differential equations that admit a four parameter symmetry group these solutions are superposed to solve general initial value problems for data with compact support the computation of the superposition coefficients uses elementary fourier analysis solutions are illustrated for various initial conditions

some relations between symmetries of nonlocally related systems the paper is concerned with relations between local symmetries of systems of partial differential equations pde within trees of nonlocally related pde systems it is shown that potential systems arising from a given system through linearly independent conservation laws are nonlocally related to each other further a theorem is proven stating that for a pde system which has precisely n linearly independent local conservation laws any local symmetry of the pde system is a projection of some local symmetry of the n plet potential system moreover a criterion is presented to determine whether or not a specific local symmetry of a given pde system is a projection of some local symmetry of a specific potential system examples are considered finally a formula for a symmetry of a given pde system in terms of a local symmetry of a nonlocally related subsystem is given the formula can be used to determine whether a symmetry of the subsystem yields a local or a nonlocal symmetry of the given system without the need to undertake a full symmetry classification and comparison between the given system and the subsystem

a constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex dwi in this paper we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion weighted images dwi the constrained variational principle involves the minimization of a regularization term of l sup p norms subject to a nonlinear inequality constraint on the data the data term we employ is the original stejskal tanner equation instead of the linearized version usually employed in literature the complex valued nonlinear form leads to a more accurate when compared to the linearized version estimate of the tensor field the inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor finally in order to accommodate the positive definite constraint on the diffusion tensor it is expressed in terms of cholesky factors and estimated the constrained variational principle is solved using the augmented lagrangian technique in conjunction with the limited memory quasi newton method experiments with complex valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm

potential symmetries to systems of nonlinear diffusion equations in this paper the potential symmetry method is developed to study systems of nonlinear diffusion equations potential variables of the systems are introduced through conservation laws such conservation laws yield equivalent systems auxiliary systems of pdes with the given dependent and potential variables as new dependent variables lie point symmetries of the auxiliary systems which cannot be projected to the vector fields of the given dependent and independent variables yield potential symmetries of the systems classification for systems of nonlinear diffusion equations with two and three components is performed symmetry reductions associated with the potential symmetries are presented

invariant solutions and conservation laws to nonconservative fp equation we generate conservation laws for the one dimensional nonconservative fokker planck fp equation also known as the kolmogorov forward equation which describes the time evolution of the probability density function of position and velocity of a particle and associate these where possible with lie symmetry group generators we determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied as the fokker planck equation is evolution type no recourse to a lagrangian formulation is made moreover we obtain invariant solutions for the fp equation via potential symmetries

classical and potential symmetries for a generalized fisher equation in this work we consider a generalized fisher equation and we have considered this equation from the point of view of the theory of symmetry reductions in partial differential equations generalizations of the fisher equation are needed to more accurately model complex diffusion and reaction effects found in many biological systems the reductions to ordinary differential equations are derived from the optimal system of subalgebras and new exact solutions are obtained the potential system has been achieved from the complete list of the conservation laws potential symmetries which are not local symmetries are carried out for the generalized fisher equation these symmetries lead to the linearization of the equation by non invertible mappings

on locally and nonlocally related potential systems for any partial differential equation pde system a local conservation law yields potential equations in terms of some potential variable which normally is a nonlocal variable the current paper examines situations when such a potential variable is a local variable i e is a function of the independent and dependent variables of a given pde system and their derivatives in the case of two independent variables a simple necessary and sufficient condition is presented for the locality of such a potential variable and this is illustrated by several examples as a particular example two dimensional reductions of equilibrium equations for fluid and plasma dynamics are considered it is shown that such reductions with respect to helical axial and translational symmetries have conservation laws which yield local potential variables this leads to showing that the well known johnson frieman kruskal oberman jfko and bragg hawthorne grad shafranov equations are locally related to the corresponding he

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

a class of fokker planck equations with logarithmic factors in diffusion and drift terms the x2 ln x dependence of the diffusion coefficient in the fokker planck equation is retrieved by means of symmetry arguments exact solutions of the equation with logarithmic factors in coefficients are presented algebraic and log algebraic solutions are found for some values of exponents they seem to be analogues on an interval of log normal distributions

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

nonlocal symmetry generators and explicit solutions of some partial differential equations the nonlocal symmetry of a partial differential equation is studied in this paper the partial differential equation written as a conservation law can be transformed into an equivalent system by introducing a suitable potential the nonlocal symmetry group generators of original partial differential equations can be obtained through their equivalent system further new explicit solutions can be constructed from the newly obtained symmetry generators the burgers equation is chosen as an example many new valuable explicit solutions and nonlocal symmetry generators are presented

weighted linear least squares estimation of diffusion mri parameters strengths limitations and pitfalls abstract purpose linear least squares estimators are widely used in diffusion mri for the estimation of diffusion parameters although adding proper weights is necessary to increase the precision of these linear estimators there is no consensus on how to practically define them in this study the impact of the commonly used weighting strategies on the accuracy and precision of linear diffusion parameter estimators is evaluated and compared with the nonlinear least squares estimation approach methods simulation and real data experiments were done to study the performance of the weighted linear least squares estimators with weights defined by a the squares of the respective noisy diffusion weighted signals and b the squares of the predicted signals which are reconstructed from a previous estimate of the diffusion model parameters results the negative effect of weighting strategy a on the accuracy of the estimator was surprisingly high multi step weighting strategies yield better performance and in some cases even outperformed the nonlinear least squares estimator conclusion if proper weighting strategies are applied the weighted linear least squares approach shows high performance characteristics in terms of accuracy precision and may even be preferred over nonlinear estimation methods

resolution of complex tissue microarchitecture using the diffusion orientation transform dot this article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion weighted magnetic resonance mr data this novel approach utilizes the fourier transform relationship between the water displacement probabilities and diffusion attenuated mr signal expressed in spherical coordinates the radial part of the fourier integral is evaluated analytically under the assumption that mr signal attenuates exponentially the values of the resulting functions are evaluated at a fixed distance away from the origin the spherical harmonic transform of these functions yields the laplace series coefficients of the probabilities on a sphere of fixed radius alternatively probability values can be computed nonparametrically using legendre polynomials orientation maps calculated from excised rat nervous tissue data demonstrate this technique s ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm this proposed methodology has a trivial extension to multiexponential diffusion weighted signal decay the developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system

least squares for diffusion tensor estimation revisited propagation of uncertainty with rician and non rician signals abstract least squares ls and its minimum variance counterpart weighted least squares wls have become very popular when estimating the diffusion tensor dt to the point that they are the standard in most of the existing software for diffusion mri they are based on the linearization of the stejskal tanner equation by means of the logarithmic compression of the diffusion signal due to the rician nature of noise in traditional systems a certain bias in the estimation is known to exist this artifact has been made patent through some experimental set ups but it is not clear how the distortion translates in the reconstructed dt and how important it is when compared to the other source of error contributing to the mean squared error mse in the estimate i e the variance in this paper we propose the analytical characterization of log rician noise and its propagation to the components of the dt through power series expansions we conclude that even in highly noisy scenarios the bias for log rician signals remains moderate when compared to the corresponding variance yet with the advent of parallel imaging pmri the rician model is not always valid we make our analysis extensive to a number of modern acquisition techniques through the study of a more general non central chi nc model since wls techniques were initially designed bearing in mind rician noise it is not clear whether or not they still apply to pmri an important finding in our work is that the common implementation of wls is nearly optimal when nc noise is considered unfortunately the bias in the estimation becomes far more important in this case to the point that it may nearly overwhelm the variance in given situations furthermore we evidence that such bias cannot be removed by increasing the number of acquired gradient directions a number of experiments have been conducted that corroborate our analytical findings while in vivo data have been used to test the actual relevance of the bias in the estimation

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

all solutions of standard symmetric linear partial differential equations have classical lie symmetry abstract it is proven that every solution of any linear partial differential equation with an independent variable deforming classical lie point symmetry is invariant under some classical lie point symmetry this is true for any number of independent variables and for equations of any order higher than one although this result makes use of the infinite dimensional component of the lie symmetry algebra due to linear superposition it is shown that new similarity solutions previously thought not to be classical can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra this result applies to all constant coefficient equations and to many variable coefficient equations such as fokker planck equations

framework for potential systems and nonlocal symmetries algorithmic approach an algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations des each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree useful conservation laws play an essential role in the construction of an extended tree a useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system nonlocal symmetries for a given system of des can arise from any system in its extended tree we construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations and in both cases obtain new nonlocal symmetries more importantly due to the equivalence of solution sets any coordinate independent method of analysis qualitative numerical perturbation etc can be applied to any system within the tree and may yield simpler computations and or results that cannot be obtained when the method is directly applied to the given system

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation we consider a class of inhomogeneous nonlinear diffusion equations inde that arise in solute transport theory hidden nonlocal symmetries that seem not to be recorded in the literature are systematically determined by considering an integrated equation obtained using the general integral variable rather than a system of first order partial differential equations pdes associated with the concentration and flux of a conservation law reductions for the inde to ordinary differential equations odes are performed and some invariant solutions are constructed

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

some more similarity solutions of the 2 1 dimensional blp system some more similarity solutions of the coupled boiti leon pempinelli system have been obtained by using similarity transformations method via lie group theory the solutions are attained under the restrictions imposed on the functions involved in infinitesimals then solutions are investigated through the graphical representation in different cases the cases depend upon some suitable choice of arbitrary functions involved there i e in the solutions to get physically meaningful structures analytical solutions are supplemented by numerical simulations with appropriate choice of arbitrary functions and constants eventually profiles show single soliton elastic multisolitons quasi periodic parabolic and asymptotic nature of solutions

exact solutions of semilinear radial wave equations in n dimensions abstract exact solutions are derived for an n dimensional radial wave equation with a general power nonlinearity the method which is applicable more generally to other nonlinear pdes involves an ansatz technique to solve a first order pde system of group invariant variables given by group foliations of the wave equation using the one dimensional admitted point symmetry groups these groups comprise scalings and time translations admitted for any nonlinearity power in addition to space time inversions admitted for a particular conformal nonlinearity power this is shown to yield not only group invariant solutions as derived by standard symmetry reduction but also other exact solutions of a more general form in particular solutions with interesting analytical behavior connected with blow ups as well as static monopoles are obtained

maximum a posteriori estimation of diffusion tensor parameters using a rician noise model why how and but the diffusion tensor is a commonly used model for diffusion weighted mr image data the parameters are typically estimated by ordinary or weighted least squares on log transformed data assuming normal or log normal distribution of measurement errors respectively this may not be adequate when using high b values and or performing high resolution scans resulting in poor snr in which case the difference between the assumed and the true rician noise model becomes important as a consequence the estimated diffusion parameters will be biased underestimating the true diffusion in this paper a computational framework is presented where parameters pertaining to a spectral decomposition of the diffusion tensor are estimated using a rician noise model the parameters are estimated using a fisher scoring scheme which gives robust and rapid convergence it is demonstrated how the fisher information matrix can be used as a generic tool for designing optimal experiments it is shown that the rician noise model leads to significantly less biased estimates for a large range of b values and snr but that the rician estimates have a poorer precision compared to the gaussian model for very low snr by pooling the rician estimates of uncertainty over neighbouring voxel estimates with higher precision but still not as high as with a gaussian model can be obtained we suggest the use of a rician estimator when it is important with truly quantitative values and when comparing different predictive models the higher precision of the gaussian estimates may be more important when the objective is to compare diffusion related parameters over time or across groups

estimation of fiber orientation probability density functions in high angular resolution diffusion imaging abstract an estimator of the orientation probability density function opdf of fiber tracts in the white matter of the brain from high angular resolution diffusion data is presented unlike q balls which use the funk radon transform to estimate the radial projection of the 3d probability density function the jacobian of the spherical coordinates is included in the funk radon approximation to the radial integral thus true angular marginalizations are computed which allows a strict probabilistic interpretation extensive experiments with both synthetic and real data show the better capability of our method to characterize complex micro architectures compared to other related approaches q balls and diffusion orientation transform especially for low values of the diffusion weighting parameter

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

nonlocal higher order symmetries for the federbush model by the introduction of nonlocal variables four first order nonlocal symmetries are obtained for the federbush model moreover the lie algebraic structure of the nonlocal symmetries is discussed

on invariance properties of the wave equation a complete group classification is given of both the wave equation c2 x uxx utt 0 i and its equivalent system vt ux c2 x vx ut ii when the wave speed c x const equations i and ii admit either a two or four parameter group for the exceptional case c x ax b 2 equation i admits an infinite group equations i and ii do not always admit the same group for a given c x the group for i can have more parameters or fewer parameters than that for ii moreover the groups can be different with the same number of parameters separately for i and ii all possible c x that admit a four parameter group are found explicitly the corresponding invariant similarity solutions are considered some of these wave speeds have realistic physical properties c x varies monotonically from one positive constant to another positive constant as x goes from to

conservation laws and hierarchies of potential symmetries for certain diffusion equations abstract we show that the so called hidden potential symmetries considered in a recent paper m l gandarias new potential symmetries for some evolution equations physica a 387 2008 2234 2242 are ordinary potential symmetries that can be obtained using the method introduced by bluman and collaborators g w bluman s kumei symmetries and differential equations springer new york 1989 g w bluman g j reid s kumei new classes of symmetries for partial differential equations j math phys 29 1988 806 811 in fact these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics furthermore we classify the conservation laws for classes of porous medium equations and then using the corresponding conserved potential systems we search for potential symmetries this is the approach one needs to adopt in order to determine the complete list of potential symmetries the provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones in particular in this way the potential systems potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described infinite series of infinite dimensional algebras of potential symmetries are constructed for such equations

on relation between potential and contact symmetries of evolution equations we prove that any evolution equation that can be written in the form of a conservation law reduces to an evolution equation admitting the group of contact symmetries leaving the temporal variable invariant and vice versa one of the consequences of this fact is that any evolution equation admitting the potential symmetries can be transformed into another evolution equation so that the potential symmetries map into contact symmetries of the latter based on this fact is out group approach to classification of evolution equations possessing nonlocal symmetries we present several examples of classifications of second order evolution equations admitting potential symmetries

a new methodology for the estimation of fiber populations in the white matter of the brain with the funk radon transform the funk radon transform frt is a powerful tool for the estimation of fiber populations with high angular resolution diffusion imaging hardi it is used in q ball imaging qbi and other hardi techniques such as the recent orientation probability density transform opdt to estimate fiber populations with very few restrictions on the diffusion model the frt consists in the integration of the attenuation signal sampled by the mri scanner on the unit sphere along equators orthogonal to the directions of interest it is easily proved that this calculation is equivalent to the integration of the diffusion propagator along such directions although a characteristic blurring with a bessel kernel is introduced under a different point of view the frt can be seen as an efficient way to compute the angular part of the integral of the attenuation signal in the plane orthogonal to each direction of the diffusion propagator in this paper stoke s theorem is used to prove that the frt can in fact be used to compute accurate estimates of the true integrals defining the functions of interest in hardi keeping the diffusion model as little restrictive as possible varying the assumptions on the attenuation signal we derive new estimators of fiber orientations generalizing both q balls and the opdt extensive experiments with both synthetic and real data have been intended to show that the new techniques improve existing ones in many situations

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions