second order pdes in four dimensions with half flat conformal structure we study second order partial differential equations pdes in four dimensions for which the conformal structure defined by the characteristic variety of the equation is half flat self dual or ant

linearly degenerate hierarchies of quasiclassical sdym type we demonstrate that sdym self dual yang mills equations for the lie algebra of one dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy we define the reduction in terms of wave functions and introduce a generating relation lax sato equations and the dressing scheme for the reduced hierarchy a multidimensional case is also discussed

new reductions of the kadomtsev petviashvili and two dimensional toda lattice hierarchies via symmetry constraints new types of reductions of the kadomtsev petviashvili kp hierarchy and the two dimensional toda lattice 2dtl hierarchy are considered on the basis of sato s approach within this approach these hierarchies are represented by infinite sets of equations for potentials u1 u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp and the 2dtl hierarchies are studied under constraints of the following type n 1n nsn u1 u2 u3 x where sn are symmetries for these hierarchies n are arbitrary constants and is an arbitrary linear functional of the quantity it is shown that for the kp hierarchy these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 u3 un many known systems and several new ones are among them symmetry reductions for the 2dtl hierarchy give rise both to finite dimensional dynamical systems and 1 1 dimensional discrete systems some few results for the modified kp hierarc

the domain of attraction and the stability region for stochastic partial differential equations with delays abstract in this paper a stochastic partial differential equation with delays is considered on the basis of the properties of nonnegative matrices stochastic convolution and the inequality technique sufficient conditions for determining the domain of p th moment attraction and the p th moment asymptotic stability region are obtained an example is also discussed to illustrate the efficiency of the results obtained

impulses induced p exponential input to state stability for a class of stochastic delayed partial differential equations abstractthis paper proposes a new stability concept called p exponential input to state stability piss for impulsive stochastic delayed partial differential equations isdpdes by employing the

symmetries of hyper kahler or poisson gauge field hierarchy symmetry properties of the space of complex or formal hyper kahler metrics are studied in the language of hyper kahler hierarchies the construction of finite symmetries is analogous to the theory of riemann hilbert transformations loop group elements now taking values in a pseudo group of canonical transformations of a simplectic manifold in spite of their highly nonlinear and involved nature infinitesimal expressions of these symmetries are shown to have a rather simple form these infinitesimal transformations are extended to the plebanski key functions to give rise to a nonlinear realization of a poisson loop algebra the poisson algebra structure turns out to originate in a contact structure behind a set of symplectic structures inherent in the hyper kahler hierarchy possible relations to membrane theory are briefly discussed

on the heavenly equation hierarchy and its reductions second heavenly equation hierarchy is considered using the framework of hyper kahler hierarchy developed by takasaki 1989 j math phys 30 1515 21 1989 j math phys 31 1877 88 generating equations for the hierarchy are introduced they are used to construct generating equations for reduced hierarchies general n reductions logarithmic reduction and rational reduction for one of the lax sato functions are discussed it is demonstrated that rational reduction is equivalent to the symmetry constraint

on the numerical solution of differential algebraic equations by pade series in this paper the solution for a differential algebraic equation dae can be expanded up to arbitrary order using maple computer algebra systems frank maple v crc press boca raton fl 1996 first we calculate power series of the given equations system then transform it into pade series form which gives an arbitrary order for solving dae numerically

lowast sdym fields and heavenly spaces i lowast sdym equations as an integrable system it is shown that the self dual yang mills sdym equations for the bracket lie algebra on a heavenly space can be reduced to one equation the master equation two hierarchies of conservation laws for this equation are constructed then the twistor transform and a solution to the riemann hilbert problem are given

twistor theory of hyper kahler metrics with hidden symmetries we review the hierarchy for the hyper kahler equations and define a notion of symmetry for solutions of this hierarchy a four dimensional hyper kahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry it is shown that a hyper kahler metric admits a hidden symmetry if it admits a certain killing spinor we show that if the hidden symmetry is tri holomorphic then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a twistor group action as introduced by bielawski twistor quotients of hyper kahler manifolds world scientific river edge nj 2001 this leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized legendre transform for the hyper kahler metric itself given by ivanov and rocek commun math phys 182 291 1996 we show that the ale spaces are examples of hyper kahler metrics admitting three tri holomorphic killing spinors these metrics are in this sense

symmetries of the self dual einstein equations i the infinite dimensional symmetry group and its low dimensional subgroups this is the first of two papers in which the authors give a complete classification of symmetry reduced solutions of plebanski s potential equation for self dual einstein spaces in this first part the infinite pseudogroup of symmetries of plebanski s equation is described and the conjugacy classes of all local subgroups of dimensions one two and three over both the real and complex numbers are classified then in the second paper this classification is used to obtain all symmetry reduced solutions

existence and uniqueness attraction for stochastic age structured population systems with diffusion and poisson jump in this paper the problems about the existence and uniqueness attraction for strong solution of stochastic age structured population systems with diffusion and poisson jump are considered under the non lipschitz condition with the lipschitz condition being considered as a special case the existence and uniqueness for such systems is firstly proved by using the burkholder davis gundy inequality b d g inequality and ito s formula and then by using a novel inequality technique some sufficient conditions ensuring the existence for the domain of attraction are established as another by product the exponential stability in mean square moment of strong solution for such systems can be also discussed

the ordinary successive approximations method and pade approximants for solving a differential equation with variant retarded argument the aim of this paper is to present an efficient numerical procedure for solving boundary value problems for a differential equation with retarded argument x t a t x t t t f t x t f t l 0 0 0

the exponential stability for stochastic delay partial differential equations abstract in this paper we consider by the energy equality the exponential stability of energy solutions to a non linear stochastic functional partial differential equation with finite delay r 0 in separable hilbert spaces our method is very simple and is powerful for concrete stochastic delay partial differential equations

the exponential behaviour and stabilizability of stochastic 2d navier stokes equations abstract some results on the pathwise exponential stability of the weak solutions to a stochastic 2d navier stokes equation are established the first ones are proved as a consequence of the exponential mean square stability of the solutions however some of them are improved by avoiding the previous mean square stability in some more particular and restrictive situations also some results and comments concerning the stabilizability and stabilization of these equations are stated

stochastic functional partial differential equations existence uniqueness and asymptotic decay property existence and uniqueness of strong solutions for a class of stochastic functional differential equations in hilbert spaces are established sufficient conditions which guarantee the transference of mean square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential equations are studied the stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics in particular as a direct consequence our main results improve some of those by mao shah in which it was proved that under certain conditions pathwise exponential stability is transferred from non delay equations to delay equations if the constant time lag appearing in the problem is sufficiently small while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions

stochastic functional evolution equations with monotone nonlinearity existence and stability of the mild solutions abstract in this paper we study a class of semilinear functional evolution equations in which the nonlinearity is demicontinuous and satisfies a semimonotone condition we prove the existence uniqueness and exponentially asymptotic stability of the mild solutions our approach is to apply a convenient version of burkholder inequality for convolution integrals and an iteration method based on the existence and measurability results for the functional integral equations in hilbert spaces an ito type inequality is the main tool to study the uniqueness p th moment and almost sure sample path asymptotic stability of the mild solutions we also give some examples to illustrate the applications of the theorems and meanwhile we compare the results obtained in this paper with some others appeared in the literature

three dimensional reductions of four dimensional quasilinear systems in this paper we show that four dimensional quasilinear systems of first order integrable by the method of two dimensional hydrodynamic reductions possess infinitely many three dimensional hydrodynamic reductions which are also integrable systems these three dimensional multi component integrable systems are irreducible to two dimensional hydrodynamic reductions in a generic case

on integrability in grassmann geometries integrable systems associated with fourfolds in gr 3 5 let gr d n be the grassmannian of d dimensional linear subspaces of an n dimensional vector space v n a submanifold x gr d n gives rise to a differential system x that governs d dimensional submanifolds of v n whose gaussian image is contained in x systems of the form x appear in numerous applications in continuum mechanics theory of integrable systems general relativity and differential geometry they include such wellknown examples as the dispersionless kadomtsev petviashvili equation the boyer finley equation plebansky s heavenly equations and so on in this paper we concentrate on the particularly interesting case of this construction where x is a fourfold in gr 3 5 our main goal is to investigate differential geometric and integrability aspects of the corresponding systems x we demonstrate the equivalence of several approaches to dispersionless integrability such as the method of hydrodynamic reductions the method of dispersionless lax pairs integrability on solutions based on the requirement that the characteristic variety of system x defines an einstein weyl geometry on every solution integrability on equation meaning integrability in twistor theoretic sense of the canonical gl 2 r structure induced on a fourfold x gr 3 5 all these seemingly different approaches lead to one and the same class of integrable systems x we prove that the moduli space of such systems is 6 dimensional we give a complete description of linearisable systems the corresponding fourfold x is a linear section of gr 3 5 and linearly degenerate systems the corresponding fourfold x is the image of a quadratic map p4 99k gr 3 5 the fourfolds corresponding to generic integrable systems are not algebraic and can be parametrised by generalised hypergeometric functions

hyper kahler hierarchies and their twistor theory a twistor construction of the hierarchy associated with the hyper kahler equations on a metric the anti self dual einstein vacuum equations asdve in four dimensions is given the recursion operator r is constructed and used to build an infinite dimensional symmetry algebra and in particular higher flows for the hyper kahler equations it is shown that r acts on the twistor data by multiplication with a rational function the structures are illustrated by the example of the sparling tod eguchi hansen solution

invariant set and attractor of nonautonomous functional differential systems the paper is devoted to the invariant set and periodic attractor for nonautonomous functional differential systems some sufficient criteria of the invariant set and periodic attractor are derived particularly we have provided an estimate on existence range of periodic attractor by using the properties of nonnegative matrices and differential inequality technique

existence uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in hilbert spaces abstract in this paper we shall consider the existence uniqueness and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r 0 dx t ax t f t x t dt g t x t dw t where we assume that a is a closed densely defined linear operator and the generator of a certain analytic semigroup f c h g c l 0 2 k h are two locally lipschitz continuous functions where c c r 0 d a l 0 2 k h are two proper infinite dimensional spaces 0 w t is a given k valued wiener process and both h and k are separable hilbert spaces

symmetries and tau function of higher dimensional dispersionless integrable hierarchies a higher dimensional analogue of the dispersionless kp hierarchy is introduced in addition to the two dimensional phase space variables k x of the dispersionless kp hierarchy this hierarchy has extra spatial dimensions compactified to a two or any even dimensional torus integrability of this hierarchy and the existence of an infinite dimensional of additional symmetries are ensured by an underlying twistor theoretical structure or a nonlinear riemann hilbert problem an analogue of the tau function whose logarithm gives the f function free energy or prepotential in the context of matrix models and topological conformal field theories is constructed the infinite dimensional symmetries can be extended to this tau or f function the extended symmetries just like those of the dispersionless kp hierarchy obey an anomalous commutation relations

stability of semilinear stochastic evolution equations stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of liapunov function sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved three examples are given to illustrate the theory

doubrov ferapontov general heavenly equation and the hyper kahler hierarchy we give a description of a recently introduced doubrov ferapontov general heavenly equation in terms of a closed differential plucker two form rationally depending on the spectral parameter we demonstrate that the general heavenly equation is an important generating equation in the context of takasaki hyper kahler hierarchy and it is also directly connected to hyper kahler geometry through the gindikin construction we develop a dressing scheme and introduce a formula for the potential satisfying of the general heavenly equation multidimensional generalization is also outlined

dunajski generalization of the second heavenly equation dressing method and the hierarchy dunajski generalization of the second heavenly equation is studied a dressing scheme applicable to the dunajski equation is developed an example of constructing solutions in terms of implicit functions is considered a dunajski equation hierarchy is described and its lax sato form is presented the dunajski equation hierarchy is characterized by the conservation of a three dimensional volume form in which a spectral variable is taken into account

dispersionless 3 1 dimensional integrable hierarchies in this paper we introduce a multi dimensional version of the r matrix approach to the construction of integrable hierarchies applying this method to the case of the lie algebra of functions with respect to the contact bracket we construct integrable hierarchies of 3 1 dimensional dispersionless systems of the type recently introduced in sergyeyev 2014 http arxiv org abs 1401 2122

spaces of complex null geodesics in complex riemannian geometry the notion of a conlplex rienlanmianl 11 mcanifold meaning a complex 1 manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point is briefly developed it is shown that provided ii 4 every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal bundle type in an associated space of comple x liull geodesics this relationship between local geometry and global complex analysis is stable under deformations introduction the present work provides a general framework analogous to but distinct from penrose s twistor correspondence penrose 15 atiyah et al 2 for the study of real analytic pseudo riemannian geometry on manifolds whose dimension exceeds three in which points of the original manifold correspond to compact complex submanifolds quadrics in an associated complex manifold the natural intermediate step in the construction is to consider complexification of the geometry leading to the notion of a complex riemannian manifold implicitly present for instance in the work of penrose in fact it is really the conformal structure that is of fundamental importance and so having briefly introduced holomorphic metrics in chapter i we proceed to establish the fundamental facts of holomorphic conformal geometry in chapter ii finally in chapter iii we develop our generalized twistor correspondence a brief historical note is in order a special case of this correspondence where the complex riemannian manifold is complex euclidean 4 space proved to be useful for the study of yang mills fields issenberg et at 7 witten 18 several years ago in the wake of this discovery the present author first sketched his ideas on the subject in 1979 in the informal twistor newsletter of the oxford mathematical institute a more fully developed version became the author s ph d thesis lebrun 10 which forms the basis of much of the present more refined paper several results are completely new however such as the theorems in 111 4 5 in dimension four some recent results have shown these notions to be of genuine physical interest within the last year a beguiling theory of the dirac manin 13 and wave lebrun 11 operators fitting neatly within this general framework has emerged for the present however we will examine only the big picture relegating this arguably more beautiful restricted case to treatment elsewhere received by the editors march 10 1982 1980 mathematics subject classification primary 32g10 53a30 secondary 32d15 53c15 53c22 83c99 d1983 american mathematical society 0002 9947 82 00001440 06 75

stability in distribution of mild solutions to stochastic partial differential delay equations with jumps the existence uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented the principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones as a consequence stability results of basak et al basak et al 1999 j math anal appl 202 604 622 and yuan et al yuan et al 2003 syst control lett 50 195 207 are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions in contrast to the almost sure exponential stability in ichikawa ichikawa 1982 j math anal appl 90 12 44 and luo liu luo liu 2008 stoch proc appl 118 864 895 and the moment exponential stability in luo liu we present a new result on the stability in distribution of mild solutions finally an example is given to demonstrate the applicability of our work

dispersionless integrable systems in 3d and einstein weyl geometry for several classes of second order dispersionless pdes we show that the symbols of their formal linearizations define conformal structures which must be einstein weyl in 3d or self dual in 4d if and only if the pde is integrable by the method of hydrodynamic reductions this demonstrates that the integrability of these dispersionless pdes can be seen from the geometry of their formal linearizations

domain of attraction of nonlinear discrete systems with delays abstract in this paper we obtain the methods to determine the domain of exponential attraction of a class of nonlinear discrete systems with delays examples are given to show the applicability of the proposed approach

fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays the fixed point theory is first used to consider the stability for stochastic partial differential equations with delays some conditions for the exponential stability in pth mean as well as in sample path of mild solutions are given these conditions do not require the monotone decreasing behavior of the delays which is necessary in t caraballo k liu exponential stability of mild solutions of stochastic partial differential equations with delays stoch anal appl 17 1999 743 763 ruhollan jahanipur stability of stochastic delay evolution equations with monotone nonlinearity stoch anal appl 21 2003 161 181 even in this special case our results also improve the results in t caraballo k liu exponential stability of mild solutions of stochastic partial differential equations with delays stoch anal appl 17 1999 743 763

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

a simple construction of recursion operators for multidimensional dispersionless integrable systems abstract we present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect to the latter new examples of recursion operators obtained using our technique include inter alia those for the general heavenly equation which describes a class of anti self dual solutions of the vacuum einstein equations and a six dimensional equation resulting from a system of ferapontov and khusnutdinova

a discrete generalized coupled dispersionless integrable system and its multisoliton solutions abstract an integrable discretization of generalized coupled dispersionless dgcd integrable system via lax pair is presented a lax pair for the dgcd system is defined a darboux transformation is used on the lax pair to obtain multi soliton solutions of the dgcd system the solutions are expressed in terms of quasideterminants explicit expressions of discrete one and two soliton solutions are obtained for the s u 2 case by using properties of quasideterminants we also study continuous analogue of the dgcd system by applying continuum limit

hierarchy structure in integrable systems of gauge fields and underlying lie algebras an improved version of nakamura s self dual yang mills hierarchy is presentd and its symmetry contents are studied the new hierarchy as well as the previous one represents a set of commuting dynamical flows in an infinite dimensional manifolds of loop type but includes a large set of dependent variables because of new degrees of freedom the theory acquires a more symmetric form with richer structures for example it allows a large symmetry algebra of riemann hilbert type which is actually a direct sum of two subalgebras left and right this phenomenon is basically the same as observed recently by avan and bellon on the case of principal chiral models in addition to these rather familiar symmeties a new type of symmetries referred to as coordinate transformation type are also introduced generators of the above dynamical flows are all included therein these two types of symmetries altogether form a big lie algebra which lead to more satisfactory understanding of symmetry properties of integrable systems of guage fields

integrability via geometry dispersionless differential equations in three and four dimensions we prove that the existence of a dispersionless lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation pde is equivalent to the canonical conformal structure defined by the symbol being einstein weyl on any solution in 3d and self dual on any solution in 4d the first main ingredient in the proof is a characteristic property for dispersionless lax pairs the second is the projective behaviour of the lax pair with respect to the spectral parameter both are established for nondegenerate determined systems of pdes of any order thus our main result applies more generally to any such pde system whose characteristic variety is a quadric hypersurface

evolution hyperheavenly equations the cauchy kovalevskaya and evolution forms of all hyperheavenly equations are presented some methods of integration of these equations are proposed

bi differential calculi and integrable models the existence of an infinite set of conserved currents in completely integrable classical models including chiral and toda models as well as the kp and self dual yang mills equations is traced back to a simple construction of an infinite chain of closed respectively covariantly constant 1 forms in a gauged bi differential calculus the latter consists of a differential algebra on which two differential maps act in a gauged bi differential calculus these maps are extended to flat covariant derivatives

an infinite number of hidden variables in hyper kahler metrics three types of hidden variables both independent and dependent are shown to underlie the hyper kahler geometry in a complexified setting the variables satisfy an infinite set of differential equations hierarchy just as in the case of most nonlinear integrable systems the notion of the plebanski key functions j math phys 16 2395 1975 is extended to this hierarchy to give an analog of the notion of the function two examples of special solutions which are reminiscent of several solution techniques in the theory of nonlinear integrable systems are presented for illustration

p attracting and p invariant sets for a class of impulsive stochastic functional differential equations in this paper a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered by establishing an l operator differential inequality and stochastic analysis technique we obtain the p attracting set and p invariant set of the impulsive stochastic functional differential equation an example is also discussed to illustrate the efficiency of the obtained results

the soliton correlation matrix and the reduction problem for integrable systems integrable 1 1 dimensional systems associated to linear first order matrix equations meromorphic in a complex parameter as formulated by zakharov mikhailov and shabat 1 3 zms are analyzed by a new method based upon the soliton correlation matrix m matrix the multi backlund transformation which is equivalent to the introduction of an arbitrary number of poles in the zms dressing matrix is expressed by a pair of matrix riccati equations for them matrix through a geometrical interpretation based upon group actions on grassman manifolds the solution of this system is explicitly determined in terms of the solutions to the zms linear system reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing them matrix suitably so as to preserve the class of invariant solutions

a connection between the einstein and yang mills equations it is our purpose here to show an unusual relationship between the einstein equations and the yang mills equations we give a correspondence between solutions of the self dual einstein vacuum equations and the self dual yang mills equations with a special choice of gauge group the extension of the argument to the full yang mills equations yields einstein s unifield equations we try to incorporate the full einstein vacuum equations but the approach is incomplete we first consider yang mills theory for an arbitrary lie algebra with the condition that the connection 1 form and curvature are constant on minkowski space this leads to a set of algebraic equations on the connection components we then specialize the lie algebra to be the infinite dimensional lie algebra of a group of diffeomorphisms of some manifold the algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act in the self dual case if we choose the connection components from the lie algebra of the volume preserving 4 dimensional diffeomorphism group the resulting equations are the same as those obtained by ashtekar jacobsen and smolin in their remarkable simplification of the self dual einstein vacuum equations an alternative derivation of the same equations begins with the self dual yang mills connection now depending only on the time then choosing the lie algebra as that of the volume preserving 3 dimensional diffeomorphisms when the reduced full yang mills equations are used in the same context we get einstein s equations for his unified theory based on absolute parallelism to incorporate the full einsteinvacuum equations we use as the lie group the semi direct product of the diffeomorphism group of a 4 dimensional manifold with the group of frame rotations of anso 1 3 bundle over the 4 manifold this last approach however yields equations more general than the vacuum equations

hydrodynamic reductions of multidimensional dispersionless pdes the test for integrability a d 1 dimensional dispersionless pde is said to be integrable if its n component hydrodynamic reductions are locally parametrized by d 1 n arbitrary functions of one variable the most important examples include the four dimensional heavenly equation descriptive of self dual ricci flat metrics and its six dimensional generalization arising in the context of sdiff 2 self dual yang mills equations given a multidimensional pde which does not pass the integrability test the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which when added to the equation make it an integrable system in fewer dimensions as an example of this phenomenon we discuss the second commuting flow of the dispersionless kp hierarchy considered separately this is a four dimensional pde which does not pass the integrability test however the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full 2 1 dimensional dis

uniform bounds from bounded lp functionals in reaction diffusion equations to prove global existence of classical or mild solutions of reaction diffusion equations a priori bounds in the uniform norm are needed but for interesting examples often one can only derive bounds for some lp norms using the structure of the reaction term they can be used to obtain uniform bounds two propositions are stated which give conditions for this procedure the proofs use the smoothing properties of analytic semigroups and the multiplicative gagliardo nirenberg inequality to illustrate the method we prove global existence of solutions for the brusselator and a volterra lotka system with one diffusing and one sedentary species

stability of stochastic partial differential equations with infinite delays in this paper we study the existence and the asymptotical stability in p th moment of mild solutions to stochastic partial differential equations with infinite delays dx t ax t f t x t t t dt g t x t d t dw t t 0 x 0 x d f 0 b m 0 0 h where t t t t d t with delays t t d t as t our method for investigating the stability of solutions is based on the fixed point theorem

on a class of integrable systems of monge ampere type we investigate a class of multi dimensional two component systems of monge ampere type that can be viewed as generalisations of heavenly type equations appearing in a self dual ricci flat geometry based on the jordan kronecker theory of the skew symmetric matrix pencils a classification of normal forms of such systems is obtained all two component systems of monge ampere type turn out to be integrable and can be represented as the commutativity conditions of parameter dependent vector fields geometrically systems of monge ampere type are associated with linear sections of the grassmannians this leads to an invariant differential geometric characterisation of the monge ampere property

asymptotic behavior of a class of reaction diffusion equations with delays abstract the authors analyze asymptotic behavior of the partial functional differential equations and the sufficient conditions on existence of global attractor of a class of reaction diffusion equations with delays are given

a numerical solution of the elasticity problem of settled of the wronkler ground with variable coefficients in this paper we have given numerical solution of the elasticity problem of settled on the wronkler ground with variable coefficient the approximation solution of boundary value problem which is pertinent to this has been converted to integral equations and then by using the successive approximation methods has been reached in addition to this the approximation solution of the problem was put into pade series form we applied these methods to an example which is the elasticity problem of unit length homogeny beam which is a special form of boundary value problem first we calculate the successive approximation of the given boundary value problem then transform it into pade series form which give an arbitrary order for solving differential equation numerically

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

an infinite hierarchy of conservation laws and nonlinear superposition principles for self dual einstein spaces self dual einstein spaces are shown to admit an infinite hierarchy of conservation laws and this hierarchy is then used to derive a formal version of penrose s twistor construction the set of formal holomorphic bundles of fiber dimension 2 over the riemann sphere p1 is shown to form a formal infinite group which is used to derive nonlinear superposition principles as an example of our methods a new self dual einstein space is obtained as the result of a collision of complex pp waves traveling in opposite directions

stabilization in probability and mean square of controlled stochastic dynamical system with state delay abstract although stochastic dynamical systems have received a great deal of attention in terms of stabilization studies so far there are few works on controlled stochastic dynamical systems with state delay in this paper a controlled stochastic dynamical system represented by a stochastic differential equation with state delay is considered condition under which the system is exponentially stable in mean square and in probability is examined