linear dependence of a function set of m variables with vanishing generalized wronskians abstract necessary and sufficient conditions are established for a set of n functions i em e1 which together with their partial derivatives of order at least n 1 are continuous to be linearly dependent one of these conditions is that all generalized wronskians vanish identically

persistence of a continuous stochastic process with discrete time sampling non markov processes we consider the problem of discrete time persistence which deals with the zero crossings of a continuous stochastic process x t measured at discrete times t ndeltat for a gaussian stationary process the persistence no crossing probability decays as exp theta d t rho a n for large n where a exp deltat 2 and the discrete persistence exponent theta d is given by theta d ln rho 2 ln a using the independent interval approximation we show how theta d varies with deltat for small deltat and conclude that experimental measurements of persistence for smooth processes such as diffusion are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker we extend the matrix method developed by us previously phys rev e 64 015101 r 2001 to determine rho a for a two dimensional random walk and the one dimensional random acceleration problem we also consider alternating persistence which corresponds to a 0 and calculate rho a for this case

oscillator representations of the 2d conformal algebra and superalgebra for the virasoro algebra and for the neveu schwarz ramond nsr superalgebra oscillator representations are found by localising the miura and the super miura maps from the theory of the korteweg de vries kdv and the super kdv equations respectively

deformations of current lie algebras i small algebras in characteristic 2 abstract we compute low degree cohomology of current lie algebras extended over the 3 dimensional simple algebra compute deformations of related semisimple lie algebras and apply these results to classification of simple lie algebras of absolute toral rank 2 and having a cartan subalgebra of toral rank one everything is in characteristic 2

transmutations for darboux transformed operators with applications we solve the following problem let q1 be a continuous complex valued potential of a stationary schr dinger operator defined on a segment a a and q2 be the potential of a darboux transformed schr dinger operator that is where f is a nonvanishing solution of the equation suppose a transmutation operator t1 is known such that for any u c2 a a find an analogous transmutation operator for it is well known that the transmutation operators can be realized in the form of volterra integral operators with continuously differentiable kernels given a kernel k1 of the transmutation operator t1 we find the kernel k2 of t2 in a closed form in terms of k1 as a corollary interesting commutation relations between t1 and t2 are obtained which then are used in order to construct the transmutation operator for the one dimensional dirac system with a scalar potential

statistics of zero crossings in rough interfaces with fractional elasticity we study numerically the distribution of zero crossings in one dimensional elastic interfaces described by an overdamped langevin dynamics with periodic boundary conditions we model the elastic forces with a riesz feller fractional laplacian of order z 1 2 such that the interfaces spontaneously relax with a dynamical exponent z to a self affine geometry with roughness exponent by continuously increasing from 1 2 macroscopically flat interface described by independent ornstein uhlenbeck processes phys rev 36 823 1930 phrvao0031 899x10 1103 physrev 36 823 to 3 2 super rough mullins herring interface three different regimes are identified i 1 2 1 the effect on p l of short scale smoothening is also analyzed numerically and analytically a tight relation between the mean interval the mean width of the interface and the density of zeros is also reported the results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints along with discretization effects are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis

the generalized confluent supersymmetry algorithm representations and integral formulas we extend the confluent version of the higher order supersymmetry susy formalism to general linear differential equations of second order closed form representations of transformation functions their wronskians and of the general solutions to susy transformed equations are derived we use these results to construct formulas for resolving multiple integrals of special functions in terms of wronskians

simple lie algebras of small characteristic ii exceptional roots abstract letlbe a finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp 3 andta 2 dimensional torus in the semisimplep envelope ofl suppose thatlis not isomorphic to a melikian algebra it is proved in this paper that for every root l t the subalgebrak generated by i f pki whereki x li x l i 0 acts triangulably onl in particular this implies that in the terminology of r e block and r l wilson 1988 j algebra114 115 259 all roots of l t are nonexceptional

simple lie algebras of small characteristic iv solvable and classical roots abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 and t a torus of maximal dimension in the p envelope of l in der l in this paper we describe the t semisimple quotients of the 2 sections of l relative to t and prove that if all 1 sections of l relative to t are compositionally classical or solvable then l is either classical or a block algebra or a filtered lie algebra of type s

arbitrary order jordan chains associated with quantum mechanical hamiltonians representations and integral formulas we consider jordan chains of arbitrary order that are associated with quantum mechanical hamiltonians an explicit formula for the connection between integral and differential representations of the jordan chains is given as a byproduct we obtain a method that allows to resolve certain multiple integrals of special functions

korteweg de vries modified korteweg de vries systems and darboux transforms in 1 1 and 2 1 dimensions the jaulent miodek hierarchy m jaulent and i miodek lett math phys 1 243 1976 and its two dimensional analog are considered from the point of view of darboux transforms structures a new form of those transforms as well as old fashioned forms are presented the determinant form of solitons is obtained the combination of different transforms in the 2 1 case allows the buildup of localized solutions in both spatial dimensions including the two dimensional korteweg de vries kdv equation

a modular analogue of morozov s theorem on maximal subalgebras of simple lie algebras abstract let g be a simple algebraic group over an algebraically closed field of characteristic p 0 and suppose that p is a very good prime for g in this paper we prove that any maximal lie subalgebra m of g lie g with rad m 0 has the form m lie p for some maximal parabolic subgroup p of g this means that morozov s theorem on maximal subalgebras is valid under mild assumptions on g we show that such assumptions are necessary by providing a counterexample to morozov s theorem for groups of type e 8 over fields of characteristic 5 our proof relies on the main results and methods of the classification theory of finite dimensional simple lie algebras over fields of prime characteristic

simple lie algebras of small characteristic iii the toral rank 2 case abstract it is proved that every finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristic p 3 is of classical or cartan type or a melikian algebra

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

on integral and differential representations of jordan chains and the confluent supersymmetry algorithm we construct a relationship between integral and differential representation of second order jordan chains conditions to obtain regular potentials through the confluent supersymmetry algorithm when working with the differential representation are obtained using this relationship furthermore it is used to find normalization constants of wave functions of quantum systems that feature energy dependent potentials additionally this relationship is used to express certain integrals involving functions that are solution of schrodinger equations through derivatives

generalized schrodinger equations with quadratical energy dependence in the potential darboux transformations and application to the heun class we construct darboux transformations of arbitrary order for generalized schrodinger type equations the potentials of which are second degree polynomials in the energy our equations are allowed to contain first derivative terms with arbitrary coefficients such as they occur for example in position dependent mass scenarios our darboux transformations are shown to be applicable to equations from the heun class

persistence of non markovian gaussian stationary processes in discrete time the persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval although persistence is a simple concept to understand it is in gener

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

higher order darboux transformations and wronskian representations for schrodinger equations with quadratically energy dependent potentials we construct higher order darboux transformations for schrodinger equations with quadratically energy dependent potentials similar to the case of conventional darboux also known as supersymmetric transformations both the solutions and the potentials of transformed equations are expressed through wronskians we discuss properties of our darboux transformations and provide an application

a condition equivalent to linear dependence for functions with vanishing wronskian abstract it is well known that though the vanishing of the wronskian w of a set of functions on an interval i is a necessary condition for it to be linearly dependent it is not a sufficient one since peano in 1989 expressed an interest in finding classes of functions for which w 0 is sufficient for dependence and offered one such example himself others m b cher d r curtiss w hurewicz and g h meisters have provided related results here the author gives a final answer to the question by first generalizing peano s result using the order of a critical point thereby obtaining a dense set of intervals of dependence he then shows that w 0 together with a condition that the intersection of certain subspaces of en is nontrivial is equivalent to the linear dependence of on i the above results are used to establish the dynamical theorem that motion of a particle under the action of a central force field is planar so long as the particle is restricted from the origin the author provides a counterexample for the case in which the particle passes through the origin

a hierarchy of coupled korteweg de vries equations and the normalisation conditions of the hilbert riemann problem the authors discuss the normalisation conditions of the hilbert riemann problem which is of importance for the construction of the ladder of soliton solutions for a given spectral problem they provide the prescription for constructing the appropriate normalisation condition all this is applied to a new hierarchy of coupled korteweg de vries equations to find the soliton solution which displays some peculiar phenomena

generalized survival in equilibrium step fluctuations we investigate the dynamics of a generalized survival probability s t r defined with respect to an arbitrary reference level r rather than the average in equilibrium step fluctuations the exponential decay at large time scales of the generalized survival probability is numerically analyzed s t r is shown to exhibit simple scaling behavior as a function of system size l sampling time ensuremath delta t and the reference level r the generalized survival time scale ensuremath tau s r associated with s t r is shown to decay exponentially as a function of r

classification of the restricted simple lie algebras soit l une algebre de lie simple restreinte de dimension finie sur un corps algebriquement clos f de caracteristique p 7 alors l est de type classique ou de cartan

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

exact occupation time distribution in a non markovian sequence and its relation to spin glass models we compute exactly the distribution of the occupation time in a discrete non markovian toy sequence that appears in various physical contexts such as the diffusion processes and ising spin glass chains the non markovian property makes the results nontrivial even for this toy sequence the distribution is shown to have non gaussian tails characterized by a nontrivial large deviation function that is computed explicitly an exact mapping of this sequence to an ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model at a given temperature what is the distribution over disorder of the thermally averaged number of spins that are aligned to their local fields we show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the sherrington kirkpatrick model with gaussian disorder

on simple lie algebras over a field of characteristic 2 abstract we prove that the simple lie algebras constructed by g jurman 2004 in 2 are isomorphic to hamiltonian algebras as a corollary we answer all questions formulated in g jurman 2004 2 about isomorphisms of these algebras

c sections of lie algebras let m be a maximal subalgebra of a lie algebra l and a b a chief factor of l such that b m and a 6 m we call the factor algebra m a b a c section of m all such c sections are isomorphic and this concept is related those of c ideals and ideal index previously introduced by the author properties of c sections are studied and some new characterizations of solvable lie algebras are obtained mathematics subject classification 2000 17b05 17b20 17b30 17b50

toral rank one lie algebras let l be a finite dimensional lie algebra over an algebraically closed field f of characteristic p 0 assume that h is a cartan subalgebra of l we say that l has toral rank m relative to h if dim pa m where a is the set of roots with respect to h p is the prime field and pa is the p vector space spanned by a in this work we study lie algebras of toral rank one the motivation to investigate this class of algebras comes from the problem of classifying simple lie algebras if a is a finite dimensional simple lie algebra with cartan subalgebra h and if ci is a root of h then the subalgebra a ciep a has toral rank one relative to h in 12 133 wilson has shown that if a simple algebra a has characteristic p 7 then the restricted closure w of h in der a has the form r t z where i is the nil radical of b and t is the maximal torus which is central in j it follows that ad r on each root space a can be simultaneously upper triangularized indeed i is a lie algebra of nilpotent transformations on a and thus by 8 theorem 3 31 it annihilates a nonzero vector the space of such vectors is t invariant and so t i has a common eigenvector u on a one can proceed inductively on ajfu to argue that each root space a has a basis relative to which ad r is upper triangular with that in mind we say in what follows that a cartan subalgebra his triangularizable if there is a basis for each root space relative to which ad h is upper triangular we show in section 1 that if l is a lie algebra having a toral rank one cartan subalgebra h and if h is triangularizable then l modulo its maximal solvable ideal either is 0 or consists of a simple toral rank one lie algebra s together with derivations of s in h in the first case we say that the core of l is 0 and in the second case that the core of l is s wilson 141 has classified the simple toral rank one lie algebras using

linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized wronskians abstract we study linear dependence in the case of quotients of analytic functions in several variables real or complex we identify the least subcollection of generalized wronskians whose identical vanishing is sufficient for linear dependence our proof admits a straight forward algebraic generalization and also constitutes an alternative proof of the previously known result that the identical vanishing of the whole collection of generalized wronskians implies linear dependence motivated by the structure of this proof we introduce a method for calculating the space of linear relations we conclude with some reflections about this method that may be promising from a computational point of view

persistence exponents and the statistics of crossings and occupation times for gaussian stationary processes we consider the persistence probability the occupation time distribution and the distribution of the number of zero crossings for discrete or equivalently discretely sampled gaussian stationary processes gsps of zero mean we first consider the ornstein uhlenbeck process finding expressions for the mean and variance of the number of crossings and the partial survival probability we then elaborate on the correlator expansion developed in an earlier paper g c m a ehrhardt and a j bray phys rev lett 88 070602 2002 to calculate discretely sampled persistence exponents of gsps of known correlator by means of a series expansion in the correlator we apply this method to the processes d n x dt n ensuremath eta t with ng 3 incorporating an extrapolation of the series to the limit of continuous sampling we then extend the correlator method to calculate the occupation time and crossing number distributions as well as their partial survival distributions and the means and variances of the occupation time and number of crossings we apply these general methods to the d n x dt n ensuremath eta t processes for n 1 random walk n 2 random acceleration and larger n and to simple diffusion from random initial conditions in one to three dimensions the results for discrete sampling are extrapolated to the continuum limit where possible

on a new hierarchy of hamiltonian soliton equations a method is suggested for studying the hamiltonian structure of the nonlinear partial differential equations that can be solved by the use of the spectral transform soliton equations the method is applied to a new hierarchy of n 1 coupled partial differential equations related to a schrodinger like spectral problem it is shown that these soliton equations are integrable hamiltonian equations with commuting flows for n 1 and n 2 a miura like transformation is computed and the corresponding modified equations are explicitly given

elementary and binary darboux transformations at rings for a given idempotent p and some element from a differential associative ring we introduce gauge transformation p with the spectral parameter that leaves some linear operators form invariant the explicit form of the is derived for the generalized zakharov shabat problem the maps that factorize darboux transformations are referred to as elementary ones binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set we show an infinitesimal version of iterated transforms the spectral operator and soliton theories applications are outlined the nonabelian n wave interaction equation modification with additional linear terms its zero curvature representation soliton solutions and their stability with respect to infinitesimal deformations are studied

the classification of the simple modular lie algebras iii solution of the classical case we investigate the structure of simple modular lie algebras over an algebraically closed field of characteristic p 7 if the distinguished subalgebra q l t with respect to an optimal torus t in some p envelope lp of l equals l then either l has a two section of type h 2 1 z or l is of classical type

derivatives of any order of the confluent hypergeometric function f11 a b z with respect to the parameter a or b the derivatives to any order of the confluent hypergeometric kummer function f f11 a b z with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable kampe de feriet functions various properties reduction formulas recurrence relations particular cases and series and integral representations of the defined hypergeometric functions are given finally an application to the two body coulomb problem is presented the derivatives of f with respect to a are used to write the scattering wave function as a power series of the sommerfeld parameter

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis

recursive representation of wronskians in confluent supersymmetric quantum mechanics a recursive form of arbitrary order wronskian associated with transformation functions in the confluent algorithm of supersymmetric quantum mechanics susy is constructed with this recursive form regularity conditions for the generated potentials can be analyzed moreover as byproducts we obtain new representations of solutions to schrodinger equations that underwent a confluent susy transformation