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

solvable vector nonlinear riemann problems exact implicit solutions of dispersionless pdes and wave breaking we have recently solved the inverse spectral problem for integrable partial differential equations pdes in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter the associated inverse problem in particular can be formulated as a nonlinear riemann hilbert nrh problem on a given contour of the complex plane the most distinguished examples of integrable pdes of this type like the dispersionless kadomtsev petviashivili dkp the heavenly and the two dimensional dispersionless toda equations are real pdes associated with hamiltonian vector fields the corresponding nrh data satisfy suitable reality and symplectic constraints in this paper generalizing the examples of solvable nrh problems illustrated in manakov and santini 2009 j phys a math theor 42 095203 2008 j phys a math theor 41 055204 2009 j phys a math theor 42 404013 we present a general procedure to construct solvable nrh problems for integrable real pdes associated with hamiltonian vector fields allowing one to construct exact implicit solutions of such pdes parametrized by an arbitrary number of real functions of a single variable then we illustrate this theory on few distinguished examples for the dkp and heavenly equations for the dkp case we characterize a class of similarity solutions of solutions constant on their parabolic wave front and breaking simultaneously on it of localized solutions whose breaking point travels with constant speed along the wave front and of localized solutions breaking in a point of the x y plane for the heavenly equation we characterize two classes of symmetry reductions

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

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

anti self dual four manifolds with a parallel real spinor antiselfdual metrics in the signature that admit a covariantly constant real spinor are studied it is shown that finding such metrics reduces to solving a fourthorder integrable partial differ

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

the inverse spectral transform for the dunajski hierarchy and some of its reductions i cauchy problem and longtime behavior of solutions in this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations pdes arising from the commutation condition of pairs of one parameter families of vector fields recently developed by s v manakov and one of the authors to one distinguished class of equations the so called dunajski hierarchy we concentrate for concreteness i on the system of pdes characterizing a general anti self dual conformal structure in neutral signature ii on its first commuting flow and iii on some of their basic and novel reductions we formally solve their cauchy problem and we use it to construct the longtime behavior of solutions showing in particular that unlike the case of soliton pdes different dispersionless pdes belonging to the same hierarchy of commuting flows evolve in time in very different ways exhibiting either a smooth dynamics or a gradient catastrophe at finite time

stable isomorphism of modules over one dimensional rings abstract two r modules m and n are said to be stably isomorphic provided m r n n r n for some n 1 in this paper r is always a commutative reduced one dimensional noetherian ring with finite normalization r and m and n are torsionfree all modules are assumed to be finitely generated there is a natural action of r c the group of units of r c where c is the conductor of r in r on torsionfree r modules and the main theorem of this paper is that the orbits of the induced action of r c are exactly the stable isomorphism classes as applications we show that stably isomorphic modules are actually isomorphic if r is a domain finitely generated as an r algebra with at most one singular real maximal ideal we give several examples of modules that are stably isomorphic but not isomorphic illustrating that these hypotheses cannot be significantly weakened we are grateful to l s levy and the referee for several suggestions that have improved the exposition of this work

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

on a class of reductions of the manakov santini hierarchy connected with the interpolating system using the lax sato formulation of the manakov santini hierarchy we introduce a class of reductions such that the zero order reduction of this class corresponds to the dkp hierarchy and the first order reduction gives the hierarchy associated with the interpolating system introduced by dunajski we present the lax sato form of a reduced hierarchy for the interpolating system and also for the reduction of arbitrary order similar to the dkp hierarchy the lax sato equations for l the lax function split from the lax sato equations for m the orlov function due to the reduction and the reduced hierarchy for an arbitrary order of reduction is defined by lax sato equations for l only a characterization of the class of reductions in terms of the dressing data is given we also consider a waterbag reduction of the interpolating system hierarchy which defines 1 1 dimensional systems of hydrodynamic type

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

hodograph solutions for the manakov santini equation we investigate the integrable 2 1 dimensional manakov santini equation from the lax sato form several particular two and three component reductions are considered so that the manakov santini equation can be reduced to systems of hydrodynamic type then one can construct infinitely many exact solutions of the equation by the hodograph method

non hamiltonian generalizations of the dispersionless 2dtl hierarchy we consider two component integrable generalizations of the dispersionless two dimensional toda lattice 2dtl hierarchy connected with non hamiltonian vector fields similar to the manakov santini hierarchy generalizing the dkp hierarchy they form a one parametric family connected by hodograph type transformations generating equations and lax sato equations are introduced and a dressing scheme based on the vector nonlinear riemann problem is formulated the simplest two component generalization of the dispersionless 2dtl equation is derived and its differential reduction analogous to the dunajski interpolating system is presented a symmetric two component generalization of the dispersionless elliptic 2dtl equation is also constructed

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

torsion free modules over reduced witt rings abstract we compute the genus class group of a torsion free module over a reduced witt ring of finite stability index this is applied to modules locally isomorphic to odd degree extensions of formally real fields

primary ideals in witt rings the basic motivation for the study or primary ideals is a classical result in commutative algebra for a noetherian ring every ideal is a finite intersection of primary ideals called a primary decomposition of the ideal for witt rings a stronger result holds every ideal of wf has a primary decomposition if and only if f has only finitely many orderings and finite height this includes many non noetherian witt rings such as the witt rings of global fields in the last two sections of this paper we study the consequences of having a primary decomposition for every ideal in wf we find conditions equivalent to unique factorization of odd dimensional forms an ideal

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

hardness of subgraph and supergraph problems in c tournaments problems like the directed feedback vertex set problem have much better algorithms in tournaments when compared to general graphs this motivates us to study a natural generalization of tournaments named c tournaments and see if the structural properties of these graphs are helpful in obtaining similar algorithms c tournaments are simple digraphs which have directed paths of length at most c 1 between all pairs of vertices we study the complexity of feedback vertex set and r dominating set in c tournaments we also present hardness results on some graph editing problems involving c tournaments

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