nonlocal symmetries related to backlund transformation and their applications starting from nonlocal symmetries related to backlund transformation bt many interesting results can be obtained taking the well known potential kdv pkdv equation as an example a new type of nonlocal symmetry in an elegant and compact form which comes from bt is presented and used to perform research works in two main subjects the nonlocal symmetry is localized by introducing suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions some other models both in finite and infinite dimensions are generated under new nonlocal symmetry the finite dimensional models are completely integrable in liouville sense which are shown equivalent to the results given through the nonlinearization method for lax pair

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

infinite hierarchies of t independent and t dependent conserved functionals of the federbush model the construction of four infinite hierarchies of t independent and t dependent conserved functionals for the federbush model is given a formal proof of the existence of these infinite hierarchies is given in appendix b

on modified factorizations for large scale linearly constrained optimization we consider the algebraic issues concerning the solution of general large scale linearly constrained nonlinear optimization problems particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods the main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null space of the constraints while neither adversely affecting the convergence of newton s method nor incurring a significant computational overhead numerical evidence to support the theoretical developments is provided

an infinite number of infinite hierarchies of conserved quantities of the federbush model the construction of two lie backlund transformations is given which are hamiltonian vector fields leading to an infinite number of hierarchies of conserved functionals and associated lie backlund transformations

explicit solutions from eigenfunction symmetry of the korteweg de vries equation in nonlinear science it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and painlevwaves actually even if for the most well known prototypical models such as the kortewet de vries kdv equation and the kadomtsev petviashvili kp equation this kind of problem has not yet been solved in this paper the explicit analytic interaction solutions betweensolitarywavesandcnoidalwavesareobtainedthroughthelocalizationprocedureofnonlocalsymmetries which are related to darboux transformation for the well known kdv equation the same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves rational solutions bessel function solutions and or general painlev e ii solutions

robust fixed size confidence procedures for a restricted parameter space on obtient des procedures robustes basees sur la resolution d un probleme de decision minimax robuste associe

on nonparametric confidence intervals an inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions as a corollary attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed length intervals and still maintain uniform coverage probability however adaptive confidence intervals can attain improved rates of convergence in some other classes of densities such as those satisfying a shape restriction

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

nonlocal symmetries and a darboux transformation for the camassa holm equation we announce two new structures associated with the camassa holm ch equation a lie algebra of nonlocal symmetries and a darboux transformation for this important equation which we construct using only our symmetries we also extend our results to the associated camassa holm equation introduced by j schiff 1998 physica d 121 24 43

new nonlocal symmetries with pseudopotentials the concept of nonlocal lie backlund symmetries can be generalized by including pseudopotentials for the kdv hd and akns equations the authors calculate generalized symmetries of such a kind the solitary wave solutions are obtained through the transformation of trivial solutions

non local symmetries and conservation laws for one dimensional gas dynamics equations the theory on the generation of conservation laws symmetry conditions and construction of conservation laws for differential equations preprint for systems of partial differential equations by the use of lie backlund symmetries is generalised to include the use of non local symmetries it is shown how the action of a symmetry on the appropriate conservation laws can be utilised to extend the symmetry generator with respect to a covering system systems of differential equations that describe one dimensional gas flow are used to demonstrate the direct calculation of new non local symmetries as well as the generation of new conservation laws also a solution that is invariant under a non local symmetry is found

minimax risk bounds in extreme value theory asymptotic minimax risk bounds for estimators of a positive extreme value index under zero one loss are investigated in the classical i i d setup to this end we prove the weak convergence of suitable local experiments with pareto distributions as center of localization to a white noise model which was previously studied in the context of nonparametric local density estimation and regression from this result we derive upper and lower bounds on the asymptotic minimax risk in the local and in certain global models as well finally the implications for fixed length confidence intervals are discussed in particular asymptotic confidence intervals with almost minimal length are constructed while the popular hill estimator is shown to yield a little longer confidence intervals

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

interior point methods for nonconvex nonlinear programming orderings and higher order methods the paper extends prior work by the authors on loqo an interior point algorithm for nonconvex nonlinear programming the specific topics covered include primal versus dual orderings and higher order methods which attempt to use each factorization of the hessian matrix more than once to improve computational efficiency results show that unlike linear and convex quadratic programming higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming but that a variant of mehrotra s predictor corrector algorithm can definitely improve performance

on nonlocal symmetries of some shallow water equations a recent construction of nonlocal symmetries for the korteweg de vries camassa holm and hunter saxton equations is reviewed and it is pointed out that in the camassa holm and hunter saxton case these symmetries can be considered as nonlocal symmetries of integro differential equations

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

asymptotic statistical equivalence for scalar ergodic diffusions for scalar diffusion models with unknown drift function asymptotic equivalence in the sense of le cam s deficiency between statistical experiments is considered under long time asymptotics a local asymptotic equivalence result is established with an accompanying sequence of simple gaussian shift experiments corresponding globally asymptotically equivalent experiments are obtained as compound experiments the results are extended in several directions including time discretisation an explicit transformation of decision functions from the gaussian to the diffusion experiment is constructed

optimal rates of convergence for estimates of the extreme value index hall and welsh established the best attainable rate of convergence for estimates of a positive extreme value index y under a certain second order condition implying that the distribution function of the maximum of n random variables converges at an algebraic rate to the pertaining extreme value distribution as a first generalization we obtain a surprisingly sharp bound on the estimation error if y is still assumed to be positive but the rate of convergence of the maximum may be nonalgebraic this result allows a more accurate evaluation of the asymptotic performance of an estimator for y than the hall and welsh theorem for example it is proved that the hill and the pickands estimators achieve the optimal rate but only the hill estimator attains the sharp bound finally an analogous result is derived for a general not necessarily positive extreme value index in this situation it turns out that location invariant estimators show the best performance 1 introduction consider n i i d random variables r v s whose distribution function d f f belongs to the weak domain of attraction of an extreme value d f

higher order predictor corrector interior point methods with application to quadratic objectives in this paper the authors explore the full utility of mehrotra s predictor corrector method in the context of linear and convex quadratic programs they describe a procedure for doing multiple corrections at each iteration and implement it within the framework of ob1 computational results are provided for the multiple correcting procedure using several strategies for determining the number of corrections in a given iteration the results indicate that iteration counts can be significantly reduced by allowing higher order corrections but at the the cost of extra work per iteration the procedure is shown to be a level m composite newton interior point method where m is the number of corrections performed in an iteration

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

infinite dimensional estabrook wahlquist prolongations for the sine gordon equation we are looking for the universal covering algebra for all symmetries of a given partial differential equation pde using the sine gordon equation as a typical example for a nonevolution equation for nonevolution equations estabrook wahlquist prolongation structures for nonlocal symmetries depend on the choice of a specific subideal of the contact module to define the pde for each inequivalent such choice we determine the most general solution of the prolongation equations as subalgebras of the infinite dimensional algebra of all vector fields over the space of nonlocal variables associated with the pde in the style of vinogradov covering spaces we show explicitly how previously known prolongation structures known to lie within the kac moody algebra a 1 1 are special cases of these general solutions although we are unable to identify the most general solutions with previously studied algebras we show the existence of gauge transformations between prolongation structures viewed as determinin

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

non local symmetry and generating solutions for harry dym type equations non local symmetries of the equations u0 f u u111 w0 g w11 w111 are investigated equations which admit non local linearization are described and formulas for generating solutions derived non lie ansatz u h x phi omega f x phi omega g x is used for reduction of some nonlinear equations

asymptotic equivalence of nonparametric autoregression and nonparametric regression it is proved that nonparametric autoregression is asymptotically equivalent in the sense of le cam s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design

conservation laws and symmetries of semilinear radial wave equations abstract classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions a radial hyperbolic equation a radial schrodinger equation and its derivative variant and two proposed radial generalizations of modified korteweg de vries equations as well as hamiltonian variants the mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives including any that exist only for special powers or dimensions all such cases for which these wave equations admit in particular dilational energies or conformal energies and inversion symmetries are determined in addition potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations as illustrative applications a discussion is given of energy norms conserved h s norms critical powers for blow up solutions and one dimensional optimal symmetry groups for invariant solutions

renormalization and white noise approximation for nonparametric functional estimation problems where wt is brownian motion in section 2 we show how invariance ideas can often clarify rates of convergence results as n x cc for a variety of parameter spaces f extensions and generalisations of these results can be found in donoho and low 1990a and low 1991 millar 1979 has illustrated the power of looking at nonparametric problems from the viewpoint of le cam s theory of experiments with the parameter space indexed by an infinite dimensional hilbert space this highly successful approach led to a simple unified theory for estimating distribution functions under for example sup norm loss the power of hilbert space parametrizations has not however been exploited in pointwise estimation problems arising for example from density estimation nonparametric regression or estimation of a variable intensity function from a poisson process in fact it is only recently that le cam s theory of experiments has even been brought to bear on the problem of density estimation and then only through a sequence of one dimensional parametrizations donoho and liu 1991 however did pick these one dimensional parametrizations in an optimal way and showed the power of le cam s methodology romano 1988 has also applied le cam s theory to sequences of one dimensional experiments naturally arising from estimation of the mode

the residual symmetry and exact solutions of the davey stewartson iii equation abstract through the truncated painleve expansion the residual symmetry of the davey stewartson iii system is derived this symmetry is localized to a properly prolonged system and the corresponding backlund transformation is obtained based on the transformation some exact solutions are generated furthermore several types of interaction wave solutions are obtained by using the consistent tanh expansion method

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

nonlocal symmetries of the hirota satsuma coupled korteweg de vries system and their applications exact interaction solutions and integrable hierarchy the nonlocal symmetry is derived from the known darboux transformation dt of the hirota satsuma coupled korteweg de vries hs ckdv system and infinitely many nonlocal symmetries are given by introducing the internal parameters by extending the hs ckdv system to an auxiliary system with five dependent variables the prolongation is found to localize the so called seed nonlocal symmetry related to the dt by applying the general lie point symmetry method to this enlarged system we obtain two main results a new type of finite symmetry transformation is derived which is different from the initial dt and can generate new solutions from old ones some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and painleve waves are computed through similarity reductions in addition two kinds of new integrable models are proposed from the obtained nonlocal symmetry the negative hs ckdv hierarchy by introducing the internal parameters the integrable models both in lower and higher dimensions by restricting the symmetry constraints

nonlocal symmetry darboux transformation and soliton cnoidal wave interaction solution for the shallow water wave equation abstract in classical shallow water wave sww theory there exist two integrable one dimensional sww equation hirota satsuma hs type and ablowitz kaup newell segur akns type in the boussinesq approximation in this paper we mainly focus on the integrable sww equation of akns type the nonlocal symmetry in form of square spectral function is derived starting from its lax pair infinitely many nonlocal symmetries are presented by introducing the arbitrary spectrum parameter these nonlocal symmetries can be localized and the sww equation is extended to enlarged system with auxiliary dependent variables then darboux transformation for the prolonged system is found by solving the initial value problem similarity reductions related to the nonlocal symmetry and explicit group invariant solutions are obtained it is shown that the soliton cnoidal wave interaction solution which represents soliton lying on a cnoidal periodic wave background can be obtained analytically interesting characteristics of the interaction solution between soliton and cnoidal periodic wave are displayed graphically

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

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

estimating tails of probability distributions we study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold a key idea is the use of pickands generalised pareto distribution and its fitting in most cases by the method of maximum likelihood the results cover all three limiting types of extreme value theory we propose a new estimator for an index of regular variation and show that it often performs better than hill s estimator we give new results for estimating the endpoint of a distribution extending earlier work by hall and by smith and weissman finally we give detailed results for the domain of attraction of exp e x and show that in most cases our proposed estimator is more efficient than two others one based on the exponential distribution and the other due to davis and resnick we also touch briefly on the problem of large deviations from a statistical point of view the results make extensive use of existing work on rates of convergence

extended soliton solutions for the kaup kupershmidt equation we extend the n soliton solutions of the kaup kupershmidt equation on a nonzero background decreasing as x 1 a 2 these new solutions describe the interaction of n solitary waves with a static bell shaped wave we give the conditions so that the backlund transformation relating those solutions and the n solitons of the sawada kotera equation will be satisfied

a moment estimator for the index of an extreme value distribution on generalise l estimateur bien connu de hill de l indice d une fonction de reparatition avec queue de variation reguliere a une estimation de l indice d une loi de valeurs extremes on demontre la convergence et la normalite asymptotique on utilise l estimateur pour certaines estimations comme celle d une quantile elevee et d un point d extremite

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem