on casimir operators of conformal galilei algebras in a previous work f alshammari p s isaac and i marquette j phys a math theor 51 065206 2018 we introduced an algorithm that utilises differential operator realisations to find polynomial casimir operators of lie algebras in this article we build on this work by applying the algorithm to several classes of finite dimensional conformal galilei algebras with central extension in these cases we highlight the utility of an algebra anti automorphism and give relevant details through key examples

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

continuous subgroups of the generalized schrodinger groups some general results on the subalgebras of the lie algebra asch n of the generalized schrodinger group sch n and on the subalgebras of the lie algebra asch n of the generalized extended schrodinger group sch n have been obtained the subalgebra structure of the algebras asch n and asch n are studied with respect to inner automorphisms of the groups sch n and sch n respectively the maximal abelian subalgebras and the one dimensional subalgebras of the algebras asch n and asch n have been explicitly found the full classification of the subalgebras of the algebras asch 3 asch 3 which are nonconjugate to the subalgebras of asch 2 asch 2 respectively has been carried out

the exotic conformal galilei algebra and nonlinear partial differential equations abstract the conformal galilei algebra cga and the exotic conformal galilei algebra ecga are applied to construct partial differential equations pdes and systems of pdes which admit these algebras we show that there are no single second order pdes invariant under the cga but systems of pdes can admit this algebra moreover a wide class of nonlinear pdes exists which are conditionally invariant under cga it is further shown that there are systems of nonlinear pdes admitting ecga with the realisation obtained very recently in d martelli y tachikawa comments on galilei conformal field theories and their geometric realisation preprint arxiv 0903 5184v2 hep th 2009 moreover wide classes of nonlinear systems invariant under two different 10 dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented

nonlinear systems of the burgers type equations lie and q conditional symmetries ansatze and solutions abstract a class of nonlinear diffusion convection systems containing two burgers type equations is considered a complete description of lie symmetries is obtained for these systems new results of finding q conditional symmetries are also presented moreover a variety of lie and non lie ansatze and exact solutions of a diffusion convection system invariant under the generalized galilei algebra are constructed

nonlinear equations invariant under the poincare similitude and conformal groups in two dimensional space time all realizations of the lie algebras p 1 1 sim 1 1 and conf 1 1 are classified under the action of the group of local diffeomorphisms of r3 the result is used to obtain all second order scalar differential equations invariant under the corresponding poincare similitude and conformal groups the invariant equations are in general nonlinear and the requirement of linearity turns out to be very restrictive group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation

maximum principle of state constrained optimal control governed by fluid dynamic systems this work deals with maximum principle of optimal control governed by some fluid dynamic systems which include three dimensional navier stokes equations some type of state constraint is involved

a differential operator realisation approach for constructing casimir operators of non semisimple lie algebras we introduce a search algorithm that utilises differential operator realisations to find polynomial casimir operators of lie algebras to demonstrate the algorithm we look at two classes of examples 1 the model filiform lie algebras and 2 the schrodinger lie algebras we find that an abstract form of dimensional analysis assists us in our algorithm and greatly reduces the complexity of the problem

equivalence groupoids and group classification of multidimensional nonlinear schrodinger equations abstract we study admissible and equivalence point transformations between generalized multidimensional nonlinear schrodinger equations and classify lie symmetries of such equations we begin with a wide superclass of schrodinger type equations which includes all the other classes considered in the paper showing that this superclass is not normalized we partition it into two disjoint normalized subclasses which are not related by point transformations further constraining the arbitrary elements of the superclass we construct a hierarchy of normalized classes of schrodinger type equations this gives us an appropriate normalized superclass for the non normalized class of multidimensional nonlinear schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses after a preliminary study of lie symmetries of nonlinear schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension we exhaustively solve the group classification problem for such equations in space dimension two

formal linearization of nonlinear massive representations of the connected poincare group let u1 be an arbitrary finite direct sum of unitary irreducible representations each of positive mass 2 of the connected poincare group p0 r4 sl 2 c it is proved that each nonlinear representation of p0 with linear part u1 on the space of differentiable vectors of u1 is formally linearizable further it is remarked that commonly used nonlinear massless representations of r2 so0 1 1 are nonlinearizable and that the corresponding evolution equations do not have covariant wave operators

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

logarithmic correlators in nonrelativistic conformal field theory we show how logarithmic terms may arise in the correlators of fields which belong to the representation of the schrodinger virasoro algebra or the affine galilean conformal algebra gca we show that in gca only scaling operator can have a jordan form and rapidity cannot we observe that in both algebras logarithmic dependence appears along the time direction alone

meta conformal invariance and the boundedness of two point correlation functions the covariant two point functions derived from ward identities in direct space can be affected by consistency problems and can become unbounded for large time or space separations this difficulty arises for several extensions of dynamical scaling for example schrodinger invariance conformal galilei invariance or meta conformal invariance but not for standard ortho conformal invariance for meta conformal invariance in dimensions which acts as a dynamical symmetry of a simple advection equation these difficulties can be cured by going over to a dual space and an extension of these dynamical symmetries through the construction of a new generator in the cartan sub algebra this provides a canonical interpretation of meta conformally covariant two point functions as correlators galilei conformal correlators can be obtained from meta conformal invariance through a simple contraction in contrast by an analogus construction schrodinger covariant two point functions are causal response functions all these two point functions are bounded at large separations for sufficiently positive values of the scaling exponents

remark on higher derivative mechanics with l conformal galilei symmetry based on the results in a galajinsky and i masterov nucl phys b 866 212 2013 we consider a way to construct a higher derivative mechanical model which possesses the l conformal galilei symmetry the dynamical system describes generalized pais uhlenbeck oscillator in the presence of an effective external field

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

z2 z2 generalizations of n 1 superconformal galilei algebras and their representations we introduce two classes of novel color superalgebras of z2 z2 grading this is done by realizing members of each class within the universal enveloping algebra of the n 1 supersymmetric extension of the conformal galilei algebra this allows us to upgrade any representation of the super conformal galilei algebras to a representation of the z2 z2 graded algebra as an example boson fermion fock space representation of one class is given we also provide a vector field realization of members of the other class by using a generalization of the grassmann calculus to z2 z2 graded setting we introduce two classes of novel color superalgebras of z2 z2 grading this is done by realizing members of each class within the universal enveloping algebra of the n 1 supersymmetric extension of the conformal galilei algebra this allows us to upgrade any representation of the super conformal galilei algebras to a representation of the z2 z2 graded algebra as an example boson fermion fock space representation of one class is given we also provide a vector field realization of members of the other class by using a generalization of the grassmann calculus to z2 z2 graded setting

on covariant realizations of the euclid group we classify realizations of the lie algebras of the rotation o 3 and euclid e 3 groups within the class of first order differential operators in arbitrary finite dimensions it is established that there are only two distinct realizations of the lie algebra of the group o 3 which are inequivalent within the action of a diffeomorphism group using this result we describe a special subclass of realizations of the euclid algebra which are called covariant ones by analogy to similar objects considered in classical representation theory furthermore we give an exhaustive description of realizations of the lie algebra of the group o 4 and construct covariant realizations of the lie algebra of the generalized euclid group e 4

pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint over all y u y mad 0 t u subject to the semilinear equation y t ay t by t du t f t a e t 0 t 1 1 with state constraint y 0 y t s 1 2 the following hypotheses will be e9ective throughout this paper h1 v and h are real hilbert spaces with norms and moreover v h v algebraically and topologically with compact injections where v denotes the dual of

pontryagin maximum principle for optimal control of variational inequalities in this paper we investigate optimal control problems governed by variational inequalities we present a method for deriving optimality conditions in the form of pontryagin s principle the main tools used are the ekeland s variational principle combined with penalization and spike variation techniques

on the simplification of the form of lie transformation groups admitted by systems of evolution differential equations abstract for a system of two partial differential equations in two independent variables the lie point symmetry generator has the form t x u v t t x u v x t x u v u t x u v v we consider a system of evolution equations of general class and we provide restrictions on the functional forms of the coefficient functions of the lie generator these restrictions reduce the amount of calculations in determining the lie symmetries of the system under study we show that t except in one special case and we present further restrictions on the remaining coefficient functions

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

prime ideals of the enveloping algebra of the euclidean algebra and a classification of its simple weight modules a classification of the simple weight modules is given for the 6 dimensional euclidean lie algebra x1d522 3 x1d530 x1d529 2 v3 as an intermediate step a classification of all simple modules is given for the centralizer c of the cartan element h in the universal enveloping algebra x1d4b0 u x1d522 3 generators and defining relations for the algebra c are found there are three quadratic relations and one cubic relation the algebra c is a noetherian domain of gelfand kirillov dimension 5 classifications of prime primitive completely prime and maximal ideals are given for the algebra u

non local representations of the ageing algebra in higher dimensions the ageing lie algebra and especially its local representations for a dynamical exponent z 2 have played an important role in the description of systems undergoing simple ageing after a quench from a disordered state to the low temperature phase here the construction of representations of for generic values of z is described for any space dimension d 1 generalizing upon earlier results for d 1 the mechanism for the closure of the lie algebra is explained the lie algebra generators contain higher order differential operators or the riesz fractional derivative co variant two time response functions are derived some simple applications to the exactly solvable models of phase separation or interface growth with conserved dynamics are discussed

nonrelativistic conformal groups ii further developments and physical applications the finite dimensional conformal groups associated with the galilei and oscillating or expanding newton hooke space time manifolds was characterized by the present authors in a recent work three isomorphic group families one for each nonrelativistic kinematics were obtained whose members are labeled by a half integer number l since the action of these groups on their corresponding space time manifolds is only local a linearization is introduced here such that the corresponding action is well defined everywhere in particular the l 1 conformal cases that can be obtained by contraction from the well known minkowskian conformal group are treated in more detail as an application of physical interest the conformal invariance of the galilean electromagnetism is studied in order to achieve it the pertinent local representations of the galilean conformal algebras are derived

lie symmetries of a generalised non linear schrodinger equation ii exact solutions for pt i see ibid vol 21 p 1493 1988 the authors obtain group invariant solutions of the non linear equation i psi t delta psi a0 psi a1 psi mod psi mod 2 a2 psi mod psi mod 4 for which the symmetry group was previously shown to be the extended galilei group for a1a2 not 0 and the extended galilei similitude group for a1 0 or a2 0 they use the symmetry subgroups to reduce the equation to ordinary differential equations which are solved whenever possible with the help of a singularity analysis solutions are obtained in terms of elementary functions jacobi elliptic functions and painleve transcendents

threshold θ 2 contact processes on homogeneous trees we study the threshold 2 contact process on a homogeneous tree mathbb t b of degree b 1 with infection parameter 0 and started from a product measure with density p the corresponding mean field model displays a discontinuous transition at a critical point lambda rm c rm mf kappa theta and for lambda geq lambda rm c rm mf kappa theta it survives iff p geq p rm c rm mf kappa theta lambda where this critical density satisfies 0 0 we also show that lim b to infty b lambda rm c mathbb t b theta phi theta where 1 2 3 lim theta to infty phi theta infty and 0 lim inf b to infty b theta theta 1 p rm c mathbb t b theta infty leq lim sup b to infty b theta theta 1 p rm c mathbb t b theta infty infty

conformal carroll groups conformal extensions of levy leblond s carroll group based on geometric properties analogous to those of newton cartan space time are proposed the extensions are labeled by an integer k this framework includes and extends our recent study of the bondi metzner sachs bms and newman unti nu groups the relation to conformal galilei groups is clarified conformal carroll symmetry is illustrated by carrollian photons motion both in the newton cartan and carroll spaces may be related to that of strings in the bargmann space

fluid limit of threshold voter models on tori in this paper we are concerned with threshold voter models on tori assuming that the initial distribution of the process is product measure with density p we obtain a fluid limit of the proportion of vertices in state 1 as the dimension of the torus grows to infinity the fluid limit performs a phase transition phenomenon from p 1 2

convergence rates for subcritical threshold one contact processes on lattices in this paper we are concerned with threshold one contact processes on lattices we show that the probability that the origin is infected converges to 0 at an exponential rate i in the subcritical case furthermore we give a limit theorem for i as the degree of the lattice grows to infinity our results also hold for classic contact processes on lattices

optimal control of parabolic differential equations with two point boundary state constraints this paper deals with the necessary conditions of optimality for some optimal control problems governed by some parabolic differential equations involving monotone graphs the two point boundary time variable state constraints are discussed some analyses on the state equations are given

affine extension of galilean conformal algebra in 2 1 dimensions we show that a class of nonrelativistic algebras including noncentrally extended schrodinger algebra and galilean conformal algebra gca has an affine extension in 2 1 hitherto unknown this extension arises out of the conformal symmetries of the two dimensional complex plane we suggest that this affine form may be the symmetry that explains the relaxation of some classical phenomena toward their critical point this affine algebra admits a central extension and maybe realized in the bulk the bulk realization suggests that this algebra may be derived by looking at the asymptotic symmetry of an anti de sitter ads theory this suggests that ads cft conformal field theory duality may take on a special form in four dimensions

critical density points for threshold voter models on homogeneous trees in this paper we study the threshold voter models on homogeneous trees we give the definition of the critical density of the models and obtain an estimation of it furthermore we study the exponential rate at which the process converges to an absorbed state 0 in the subcritical case at the end of the paper we propose two conjectures about the phase transition phenomenons of our models

the survival of large dimensional threshold contact processes we study the threshold theta contact process on z d with infection parameter lambda we show that the critical point lambda c defined as the threshold for survival starting from every site occupied vanishes as d infinity this implies that the threshold theta voter model on z d has a nondegenerate extremal invariant measure when d is large

the complete convergence theorem for coexistent threshold voter models we consider the d dimensional threshold voter model it is known that except in the one dimensional nearest neighbor case coexistence occurs nontrivial invariant measures exist in fact there is a nontrivial limit 1 2 obtained by starting from the product measure with density 1 2 we show that in these coexistent cases t 0 1 1 1 2 as t where p 0 p 1 0 and 1 are the first hitting times of the all zero and all one configurations respectively and denotes weak convergence

necessary and sufficient conditions for optimal controls in viscous flow problems a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation characterising the feedback problem the maximum principle is established by two quite different methods

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

on irreducible representations of the exotic conformal galilei algebra we investigate the representations of the exotic conformal galilei algebra this is done by explicitly constructing all singular vectors within the verma modules and then deducing irreducibility of the associated highest weight quotient modules a resulting classification of infinite dimensional irreducible modules is presented

realizations of galilei algebras all inequivalent realizations of the galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by shirokov the varieties of the deformed galilei algebras are discussed and families of one parametric deformations are presented in explicit form it is also shown that a number of well known and physically interesting equations and systems are invariant with respect to the considered galilei algebras or their deformations

quantum structure of the motion groups of the two dimensional cayley klein geometries a simultaneous and global scheme of quantum deformation is defined for the set of algebras corresponding to the groups of motions of the two dimensional cayley klein geometries their central extensions are also considered under this unified pattern in both cases some fundamental properties characterizing the classical ck geometries as the existence of a set of commuting involutions contractions and dualities relationships remain in the quantum version

nonrelativistic conformal groups in this work a systematic study of finite dimensional nonrelativistic conformal groups is carried out under two complementary points of view first the conformal killing equation is solved to obtain a whole family of finite dimensional conformal algebras corresponding to each of the galilei and newton hooke kinematical groups some of their algebraic and geometrical properties are studied in a second step among the groups included in these families one can identify for example the contraction of the minkowski conformal group the analog for a nonrelativistic de sitter space or the nonextended schrodinger group

optimal control of the primitive equations of the ocean with state constraints abstract we investigate in this article pontryagin s maximum principle for a class of control problems associated with the primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in wang 2002 7 for the three dimensional navier stokes ns equations the main difference between this work and wang 2002 7 is that the nonlinearity in the pes is stronger than in the three dimensional ns systems

the time optimal control of navier stokes equations the maximum principle for the time optimal control of the navier stokes equations in 2 d is established

nonlinear galilei invariant pdes with infinite dimensional lie symmetry a class of nonlinear galilei invariant generalizations of the heat equation admitting infinite dimensional lie symmetry is presented lie symmetries and examples of exact solutions are found for some nonlinear equations generalizing the heat equation full sets of the absolute differential invariants for two non equivalent representations of the galilei algebra are constructed

aspects of infinite dimensional l super galilean conformal algebra in this work we construct an infinite dimensional l super galilean conformal algebra which is a generalization of the l 1 algebra found in the literature we give a classification of central extensions the vector field representation the coadjoint representation and the operator product expansion of the infinite dimensional l super galilean conformal algebra keeping possible applications in physics and mathematics in mind

c integrable nonlinear partial differentiation equations i a technique to generate c integrable nonlinear partial differentiation equations pdes i e nonlinear pdes linearizable by an appropriate change of variables is reported and several examples of such pdes are exhibited

pontryagin s maximum principle for optimal control of the stationary primitive equations of the ocean abstract we study in this article pontryagin s maximum principle for a class of control problems associated with the stationary primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in g wang pontryagin s maximum principle for optimal control of the stationary navier stokes equations nonlinear anal 52 2003 1853 1866 in the case of the three dimensional navier stokes ns equations the main difference between this work and the one cited above is that the nonlinearity in pes is stronger than that in the three dimensional ns equations

the modulation of weakly non linear dispersive waves near the marginal state of instability the modulation of a one dimensional weakly non linear purely dispersive quasi monochromatic wave the carrier is usually governed by the non linear schrodinger ns equation the critical wavenumber for which the carrier is marginally modulationally unstable is determined by the condition that the product of the coefficients of the non linear and dispersive terms in the ns equation is zero however near this marginal state the assumptions that lead to the ns equation are invalid and a modified form of the ns equation that involves higher order non linearities is appropriate this modified ns equation is here derived formally for a general system involving a single dependent variable and a revised form of the instability criterion is obtained the results are illustrated by considering a particular system described by a generalised korteweg de vries equation

on integrable doebner goldin equations we integrate sub families of a family of nonlinear schrodinger equations proposed by doebner and goldin in the 1 1 dimensional case which have exceptional lie symmetries since the method of integration involves non local transformations of dependent and independent variables general solutions obtained include implicitly determined functions by properly specifying one of the arbitrary functions contained in these solutions we obtain broad classes of explicit square integrable solutions the physical significance and some analytical properties of the solutions obtained are briefly discussed

optimal controls of boussinesq equations with state constraints abstract this work is concerned with the maximum principle for an optimal control problem governed by boussinesq equations some integral type state constraints are considered

state constrained optimal control problems governed by coupled nonlinear wave equations with memory this work considers the optimal control problems governed by coupled nonlinear wave equations with memory where the state constraint is a type of integral first we investigate the existence of the optimal solution for the optimal control problem and then we deduce the maximum principle for the optimal solution with state constraint moreover we give a concrete example which is covered by the main result

on non linear partial differential equations with an infinite dimensional conditional symmetry abstract the invariance of non linear partial differential equations under a certain infinite dimensional lie algebra a n z in n spatial dimensions is studied the special case a 1 2 was introduced in j stat phys 75 1994 1023 and contains the schrodinger lie algebra sch 1 as a lie subalgebra it is shown that there is no second order equation which is invariant under the massless realizations of a n z however a large class of strongly non linear partial differential equations is found which are conditionally invariant with respect to the massless realization of a n z such that the well known monge ampere equation is the required additional condition new exact solutions are found for some representatives of this class

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

necessary conditions for optimal control of distributed parameter systems the ekeland variational principle is used to prove a maximum principle for the optimal controls of a general semilinear evolutionary control system in a banach space with strictly convex dual the boundary constraint is of general type which includes the optimal periodic control problem as a special case

intertwining operators for l conformal galilei algebras and hierarchy of invariant equations the l conformal galilei algebra denoted by is a non semisimple lie algebra specified by a pair of parameters d l the algebra is regarded as a nonrelativistic analogue of the conformal algebra we derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry this is done by developing a representation theory such as verma modules singular vectors of and vector field representations for d 1 2

evolution equations invariant under two dimensional space time schrodinger group the most general second order evolution equation t f x t x x xx xx 0 invariant under the galilei galilei similitude and schrodinger groups in two dimensions is constructed a preliminary step is a classification of all possible realizations of the corresponding lie algebras of vector fields in r2 c parametrized by x t and applications of this study include the investigation of nonlinear alternatives to quantum mechanics and nonrelativistic classical field theories among the schrodinger invariant equations in particular are found integrable equations linearizable by contact transformations

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

on new galilei invariant equations in two dimensional spacetime we study realizations of galilei groups acting as transformation lie groups in the space of two independent variables and one dependent variable classification of realizations of the lie algebras and within the class of lie vector fields is carried out utilizing the classification results we have constructed the full sets of second order scalar differential equations in two dimensional spacetime invariant under the lie algebras and

principal agent problem in a fuzzy environment this paper studies implementation of a classical principal agent problem in which the agents characteristics are supposed to be unknown to the principal it is reasonable for us to employ fuzzy variables to denote these hidden characteristics since generally they are fuzzy by nature a fuzzy expected value model which aims to maximize the utility function of the principal under the incentive constraints is developed from the principal s point of view pontryagin maximum principle is adopted in this paper to find the optimal solutions to the fuzzy expected value model

invariant partial differential equations with two dimensional exotic centrally extended conformal galilei symmetry conformal galilei algebras cgas labeled by d l where d is the number of space dimensions and l denotes a spin l representation w r t the x1d530 x1d529 2 subalgebra admit two types of central extensions the ordinary one for any d and half integer l and the exotic central extension which only exists for d 2 and l for both types of central extensions invariant second order partial differential equations pdes with continuous spectrum were constructed by aizawa et al j phys a 46 405204 2013 it was later proved by aizawa et al j math phys 3 031701 2015 that the ordinary central extensions also lead to oscillator like pdes with discrete spectrum we close in this paper the existing gap constructing a new class of second order invariant pdes for the exotic centrally extended cgas they admit a discrete and bounded spectrum when applied to a lowest weight representation these pdes are markedly different with respect to their ordinary counterparts the l 1 case which is the prototype

coexistence in threshold voter models the threshold voter models considered in this paper are special cases of the nonlinear voter models which were introduced recently by cox and durrett they are spin systems on z d with transition rates begin equation c x eta begin cases 1 text if there is a y text with x y leq n text and eta x neq eta y 0 text otherwise end cases end equation this system is known to cluster if n d 1 and to coexist if n geq 4 in one dimension and if n is reasonably large in other dimensions cox and durrett conjectured that it coexists in all cases except n d 1 in this paper we prove this conjecture the proof is based on comparisons with threshold contact processes the hard part of the proof consists of showing that the second nearest neighbor threshold contact process in one dimension with parameter 1 survives the proof of this result is modeled after the proof by holley and liggett that the critical value of the basic contact process in one dimension is at most 2 by comparison with that proof however the fact that the interaction is not of nearest neighbor type presents substantial additional difficulties in fact part of the proof is computer aided

partially invariant solutions of a class of nonlinear schrodinger equations partially invariant solutions of a general class of nonlinear schrodinger equations involving four arbitrary functions of the modulus rho of the solution and its derivative rho x are obtained the modulus rho xi is assumed to depend on a symmetry variable xi whereas the phase omega x t depends on both independent variables both rho and omega are obtained explicitly as are the conditions on the coefficients in the equation necessary for such solutions to exist