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

partner symmetries and non invariant solutions of four dimensional heavenly equations we extend our method of partner symmetries to the hyperbolic complex monge ampere equation and the second heavenly equation of plebanski we show the existence of partner symmetries and derive the relations between them for certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate legendre transformation the solutions of these linear equations are generically non invariant as a consequence we obtain explicitly new classes of heavenly metrics without killing vectors

discrete dynamical systems in l topological spaces in this paper we investigate iteration of zadeh image operators and their discrete dynamical systems in l topological spaces some basic concepts such as limit set invariant set periodic point nonwandering point and recurrent point are discussed in weakly induced topological spaces we also study topological mixing topological transitivity and minimality for such dynamical systems while comparing with those corresponding ones in crisp spaces

point set lattice theoretic topology this essay attempts to survey in a coherent way certain aspects of point set lattice theoretic or poslat topology by which we mean fuzzy topology grounded in notions of sets functions powersets and powerset operators the latter two being lattice theoretic in nature and examined from a lattice theoretic point of view using methods of category theory connections with other related issues and developments are frequently 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

partner symmetries of the complex monge ampere equation yield hyper kahler metrics without continuous symmetries we extend the mason newman lax pair for the elliptic complex monge ampere equation so that this equation itself emerges as an algebraic consequence we regard the function in the extended lax equations as a complex potential their differential compatibility condition coincides with the determining equation for the symmetries of the complex monge ampere equation we shall identify the real and imaginary parts of the potential which we call partner symmetries with the translational and dilatational symmetry characteristics respectively then we choose the dilatational symmetry characteristic as the new unknown replacing the kahler potential this directly leads to a legendre transformation studying the integrability conditions of the legendre transformed system we arrive at a set of linear equations satisfied by a single real potential this enables us to construct non invariant solutions of the legendre transform of the complex monge ampere equation using these solutions we obtained explicit legendre transformed hyper kahler metrics with a anti self dual riemann curvature 2 form that admit no killing vectors they satisfy the einstein field equations with euclidean signature we give the detailed derivation of the solution announced earlier and present a new solution with an added parameter we compare our method of partner symmetries for finding non invariant solutions to that of dunajski and mason who use hidden symmetries for the same purpose

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

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

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

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

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

bi hamiltonian structure of an asymmetric heavenly equation in the paper of sheftel and malykh 2009 j phys a math theor 42 395202 on the classification of second order pdes with four independent variables that possess partner symmetries an asymmetric heavenly equation appears as one of the canonical equations admitting partner symmetries here for the asymmetric heavenly equation formulated in a two component form we present the lax pair of olver ibragimov shabat type and obtain its multi hamiltonian structure therefore by magri s theorem it is a completely integrable bi hamiltonian system in four dimensions

neoclassical econometries the kernel the paper considers the problem of determining the parameters in static equality constrained neoclassical models frequently encountered in economic analysis these include the utility maximizing household subject to income and time constraints and the profit maximizing firm subject to a given technology the approach applies equally to the traditional regression based stochastic econometric model with additive normally distributed errors or to a deterministic analog where the distance between the actual data and model solutions for the agents decisions is minimized conventional econometric practice is extended in that closed form analytic solutions for the decision functions are not required and essentially arbitrary configurations of available data are permitted the computer is employed to jointly 1 solve the agents optimization problems 2 impute missing data and 3 determine whether there is sufficient information in the available data set to uniquely calibrate or estimate the model s parameters

invariance and stability of fuzzy systems the elaboration of a stability theory of fuzzy systems is fully justified by the use in a variety of practical situations of fuzzy controllers 11 27 28 however such a theory may be founded upon ideas that arise from a historical study of dynamical systems theory during the last quarter of the nineteenth century the study of ordinary differential equations underwent some rather radical changes prior to this period the major emphasis in the subject had been on the methods of solving equations it was during this period that peano 20 gave a rigorous proof of the existence theorem for solutions of ordinary differential equations during the same period lipschitz 151 and picard 22 showed that the method of successive approximations led to a proof of the existence and the uniqueness of solutions to the initial value problem these developments represent the dhuement of the attempts to solve differential equations as the works of peano lipschitz and picard were finishing one chapter in the book of differential equations another was being initiated with the research of liapunov 141 and poincare 23 this new chapter was based on an entirely different approach where one did not attempt to compute the solutions rather one tried to exploit their topological properties poincart in his famous m moire 23 was the first to look at an ordinary differential equation from the point of view of the geometry of the trajectories thereby creating the topological theory of ordinary differential equations almost at the same time the stability theory created by liapunov 141 although heavily quantitative in its methods stressed the importance of some topological features of ordinary differential equations during the 50 years following liapunov and poincare many important advances were made it is with the works of birkhoff 4 nemytskii and stepanov 191 and smale 26 that researchers realized that the essence of this theory was based on the notion of dynamical system based on liapunov s original works classical stability theory is concerned with the equilibrium points of the system and the dynamical

pointwise topology on completely distributive lattices abstract in this paper the author presents a kind of pointwise topology theory on complete completely distributive lattices which is both a generalization of fuzzy topology and a further generalization of the author s papers 14 15

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

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

imposing curvature restrictions on flexible functional forms abstract a general computational method for estimating the parameters of a flexible functional form subject to convexity quasi convexity concavity or quasi concavity at a point at several points or over a region is set forth and illustrated with an example

on cohen macaulay rings of invariants abstract we investigate the transfer of the cohen macaulay property from a commutative ring to a subring of invariants under the action of a finite group our point of view is ring theoretic and not a priori tailored to a particular type of group action as an illustration we discuss the special case of multiplicative actions that is actions on group algebras k z n via an action on z n

supersymmetric sigma models twistors and the atiyah hitchin metric the legendre transform and its generalizations originally found in supersymmetric models are techniques that can be used to givelocal constructions of hyperkahler metrics we give a twistor space interpretation to the generalizations of the legendre transform construction the atiyah hitchin metric on the moduli space of two monopoles is used as a detailed example

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

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

on the solutions of the dkp equation the nonlinear riemann hilbert problem longtime behaviour implicit solutions and wave breaking 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 in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation we showed in particular that the associated inverse problems can be expressed in terms of nonlinear riemann hilbert problems on the real axis in this paper we make use of the nonlinear riemann hilbert problem of dkp i to construct the longtime behaviour of the solutions of its cauchy problem ii to characterize a class of implicit solutions iii to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dkp at finite time as well as in the longtime regime and the corresponding universal behaviours near breaking

structure theorems over polynomial rings given a polynomial ring r over a field k and a finite group g we consider a finitely generated graded rg module s we regard s as a kg module and show that various conditions on s are equivalent such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of d karagueuzian p symonds the module structure of a group action on a polynomial ring a finiteness theorem preprint http www ma umist ac uk pas preprints

multi hamiltonian structure of plebanski s second heavenly equation we show that plebanski s second heavenly equation when written as a first order nonlinear evolutionary system admits multi hamiltonian structure therefore by magri s theorem it is a completely integrable system thus it is an example of a completely integrable system in four dimensions

group actions on rings and the cech complex abstract we have previously shown that when a finite group g acts on a polynomial ring s in n variables over a finite field k only finitely many isomorphism classes of indecomposable k g modules occur as summands of s we have also shown that the regularity of the invariant subring s g is at most zero which has various consequences for example s g is generated in degrees at most n g 1 provided n g 2 both of these theorems depend on the structure theorem of karagueuzian and the author which is proved by means of a long and complicated calculation the aim of this paper is to prove these results using a more conceptual method

some remarks on fuzzy powerset operators fuzzy powerset operators related to morphisms of underlying ground categories of categories of fuzzy topological spaces are considered and some properties of such operators related to isomorphisms fuzzy points and order reversing involutions are verified

self dual gravity is completely integrable we discover a multi hamiltonian structure of a complex monge ampere equation cma set in a real first order 2 component form therefore by magri s theorem this is a completely integrable system in four real dimensions we start with lagrangian and hamiltonian densities and obtain a symplectic form and the hamiltonian operator that determines the dirac bracket we have calculated all point symmetries of the 2 component cma system and hamiltonians of the symmetry flows we have found two new real recursion operators for symmetries which commute with the operator of a symmetry condition on solutions of the cma system these operators form two lax pairs for the 2 component system the recursion operators applied to the first hamiltonian operator generate infinitely many real hamiltonian structures we show how to construct an infinite hierarchy of higher commuting flows together with the corresponding infinite chain of their hamiltonians

chaos entropy and a generalized extension principle we show a relationship between a very simple criterion positive topological entropy and li yorke chaos a general definition of fuzzification and level set based on t norms conorms and their diagonal functions is introduced the chaos theorem of benhabib and day for set valued mappings is considerably strengthened and generalised

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

econometric methods for agricultural supply under uncertainty fertilizer use and crop response the paper develops general econometric methods for the treatment of modelling decision making under conditions of uncertainty and risk the problem of modelling fertilizer use and crop response in the context of irrigated and or rainfed water supply when there is uncertainty and risk with respect to weather conditions over the crop cycle and the subsequent price at marketing suggested the present approach as a by product the stochastic duality theory associated with the present primal problem is also derived

theory of fuzzy systems in this paper a theory of dynamical fuzzy systems is proposed it is shown that a fuzzy system can be represented by a flow describing the evolution of this fuzzy system in the space of fuzzy subsets of the state space the topological properties of the flow are then established

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

fuzzy dynamical systems a fuzzy dynamical system on an underlying complete locally compact metric state space x is defined axiomatically in terms of a fuzzy attainability set mapping on x this definition includes as special cases crisp single and multivalued dynamical systems on x it is shown that the support of such a fuzzy dynamical system on x is a crisp multivalued dynamical system on x and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of x in addition fuzzy trajectories are defined their existence established and various properties investigated

new hyperkahler metrics and new supermultiplets we present new constructions of hyperkahler metrics along with the three new classes ofn 4 supermultiplets that they derive from further we provide a general setting for understanding the constructions and give a detailed description of the multiplets inn 2 andn 4 superspace

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

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

a torelli type theorem for gravitational instantons on montre que chaque 4 variete hyper kahler asymptotiquement localement euclidienne et isometrique a un membre d une des familles obtenues dans un article precedent

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

neoclassical econometrics the agenda abstract a new approach to econometric modeling and parameter estimation which permits the construction of macro econometric models from general microeconomic foundations is proposed it provides inter alia full information calibration estimation methods which 1 a permits a statistical solution to the exact or consistent aggregation problem 2 b permits a new approach to the problem of parameter identification 3 c permits a systematic approach to the problem of model selection 4 d obviates the need for use of the duality theory of cost and profit functions by working solely on primal problems 5 e treats in principle the spectrum of perfectly competitive to monopolistic monopsonistic market structures 6 f applies in principle to both partial and general equilibrium models and 7 g obtains the structural form of limited dependent variable econometric estimation methods directly from neoclassical constrained optimization behavioral models all in the context of well behaved but essentially unrestricted functional forms subject to a general set of rationality postulates regarding the behavior of individual economic units

h1 topological molecular lattices abstract this paper is devoted to the study of h 1 topological molecular lattices a generalization of the hausdorff fuzzy topological spaces some primary properties of such topological molecular lattices are discussed for any complete and completely distributive lattice l the existence and uniqueness of minimal h 1 co topologies on l are investigated and the characteristic theorem of minimal h 1 co topologies on l is established

the symmetry structure of the anti self dual einstein hierarchy an important example of a multi dimensional integrable system is the anti self dual einstein equations by studying the symmetries of these equations a recursion operator is found and the associated hierarchy constructed owing to the properties of the recursion operator one may construct a hierarchy of symmetries and find the algebra generated by them in addition the lax pair for this hierarchy is constructed

hamiltonian structure of real monge ampere equations the variational principle for the real homogeneous monge ampere equation in two dimensions is shown to contain three arbitrary functions of four variables there exist two different specializations of this variational principle where the lagrangian is degenerate and furthermore contains an arbitrary function of two variables the hamiltonian formulation of these degenerate lagrangian systems requires the use of dirac s theory of constraints as in the case of most completely integrable systems the constraints are second class and dirac brackets directly yield the hamiltonian operators thus the real homogeneous monge ampere equation in two dimensions admits two classes of infinitely many hamiltonian operators namely a family of local as well as another family non local hamiltonian operators and symplectic 2 forms which depend on arbitrary functions of two variables the simplest non local hamiltonian operator corresponds to the kac moody algebra of vector fields and functions on the unit circle hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair in the case of real monge ampere equations with constant right hand side this compatible pair is the only pair of hamiltonian operators that survives then the complete integrability of all these real monge ampere equations follows by magri s theorem some of the remarkable properties we have obtained for the hamiltonian structure of the real homogeneous monge ampere equation in two dimensions turn out to be generic to the real homogeneous monge ampere equation and the geodesic flow for the complex homogeneous monge ampere equation in arbitrary number of dimensions hence among all integrable nonlinear evolution equations in one space and one time dimension the real homogeneous monge ampere equation is distinguished as one that retains its character as an integrable system in multiple dimensions

galois structure of homogeneous coordinate rings suppose g is a finite group acting on a projective scheme x over a commutative noetherian ring r we study the rg modules h 0 x f l n when n 0 and f and are coherent g sheaves on x such that l is an ample line bundle we show that the classes of these modules in the grothendieck group g 0 rg of all finitely generated rg modules lie in a finitely generated subgroup under various hypotheses we show that there is a finite set of indecomposable rg modules such that each h 0 x f l n is a direct sum of these indecomposables with multiplicities given by generalized hilbert polynomials for n 0

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

flexible functional forms and global curvature conditions empirically estimated flexible functional forms frequently fail to satisfy the appropriate theoretical curvature conditions lau and gallant and golub have worked out methods for imposing the appropriate curvature conditions locally but those local techniques frequently fail to yield satisfactory results we develop two methods for imposing curvature conditions globally in the context of cost function estimation the first method adopts lau s technique to a generalization of a functional form first proposed by mcfadden using this generalized mcfadden functional form it turns out that imposing the appropriate curvature conditions at one data point imposes the conditions globally the second method adopts a technique used by mcfadden and barnett which is based on the fact that a non negative sum of concave functions will be concave our various suggested techniques are illustrated using the u s manufacturing data utilized by berndt and khaled