finite dimensional lie algebras of order f f lie algebras are natural generalizations of lie algebras f 1 and lie superalgebras f 2 when f 2 not many finite dimensional examples are known in this article we construct finite dimensional f lie algebras f 2 by an inductive process starting from lie algebras and lie superalgebras matrix realizations of f lie algebras constructed in this way from su n sp 2n so n and sl n m osp 2 m are given we obtain nontrivial extensions of the poincare algebra by inonu wigner contraction of certain f lie algebras with f 2

involutive systems of differential equations einstein s strength versus cartan s degre d arbitraire three new theorems relating einstein s notions of strength and compatibility to the field of the initial value problem are presented these theorems result i in a first proof of matthews conjectures concerning this relation for a wider class of systems of partial differential equations ii in a new interpretation of einstein s compatibility condition and iii in the exact relation between einstein s strength and cartan s degre d arbitraire

revisit to sign changing solutions for the nonlinear schrodinger poisson system in r3 abstract in this paper we investigate the existence of solutions for the nonlinear schrodinger poisson system with zero mass by introducing some new ideas we prove via the constraint variational method and the quantitative deformation lemma that the system has a sign changing solution

ground state sign changing solutions for a schrodinger poisson system with a 3 linear growth nonlinearity abstract in this paper we investigate the existence and asymptotic behavior of ground state sign changing solutions to a class of schrodinger poisson systems u v x u u f x u u 2 u x r 3 u 2 x r 3 where v is a smooth function f is nonnegative 0 1 and 1 is the first eigenvalue of the problem u v x u f x u in h with the help of the sign changing nehari manifold we obtain that the schrodinger poisson system possesses at least one ground state sign changing solution u for all 0 and each 1 moreover we prove that its energy is strictly larger than twice that of ground state solutions besides we give a convergence property of u as 0 this paper can be regarded as the complementary work of shuai and wang 23 wang and zhou 24

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

on the schrodinger poisson slater system behavior of minimizers radial and nonradial cases this paper is motivated by the study of a version of the so called schrodinger poisson slater problem delta u omega u lambda left u 2 star frac 1 x right u u p 2 u where u in h 1 mathbb r 3 we are concerned mostly with p in 2 3 the behavior of radial minimizers motivates the study of the static case 0 among other things we obtain a general lower bound for the coulomb energy which could be useful in other frameworks the radial and nonradial cases turn out to yield essentially different situations

on the strength of maxwell s equations the strength of a set of field equations first defined by einstein as the number of taylor coefficients of field variables that could be chosen arbitrarily is used to show that the amount of initial data required by the electromagnetic formulation of maxwell s theory in free space is equal without approximation to that required by the potential formulation in each formulation the strength is interpreted in terms of the amount of initial data required to provide a solution of the cauchy initial value problem and in terms of the invariance properties of the formulation equality of the strengths of the two formulations of maxwell s theory is used to support the assertion that knowledge of the strengths of other established field theories provides a means for predicting the possible existence of unknown formulations of the theories

trefoil symmetry ii another clover extension we construct a novel graded extension of the poincare group with integer spin multiplets as novel generators the extension involves z4 z4 graded parameters and produce space time translations through the composition of novel symmetric vector and scalar multiplets

z2n graded extensions of supersymmetric quantum mechanics via clifford algebras it is shown that the n 1 supersymmetric quantum mechanics sqm can be extended to a z2n graded superalgebra this is done by presenting quantum mechanical models that realize with the aid of clifford gamma matrices the z2n graded poincare algebra in one dimensional spacetime reflecting the fact that the z2n graded poincare algebra has a number of central elements a sequence of models defining the z2n graded version of sqm is provided for a given value of n in a model of the sequence the central elements having the same z2n degree are realized as dependent or independent operators it is observed that as the clifford algebras of larger dimension are used more central elements are realized as independent operators

infinitely many positive solutions for a schrodinger poisson system abstract we are interested in the existence of infinitely many positive solutions of the schrodinger poisson system u u v x u u p 1 u x r 3 v x u 2 x r 3 where v x is a positive bounded function 1 p 5 and v r r x has the following decay property v r a r m o 1 r m with a 0 m 3 2 0 the solutions obtained are non radial

casimir elements of e lie algebras the classical framework for investigating the casimir elements of a lie algebra is generalized to the case of an e lie algebra l we construct the standard l module isomorphism of the e symmetric algebra of l onto its enveloping algebra and we introduce the harish chandra homomorphism in case the generators of l can be written in a canonical two index form we construct the associated standard sequence of casimir elements and derive a formula for their eigenvalues in an arbitrary highest weight module

exterior differential systems janet riquier theory and the riemann lanczos problems in two three and four dimensions we discuss the riemann lanczos problems in two three and four dimensions using the theory of exterior differential systems and janet riquier theory we show that the riemann lanczos problem in two dimensions is always a system in involution for each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result in three dimensions the riemann lanczos problem is not in involution as an identity occurs this does not prevent the existence of singular solutions and we give an example for the reduced godel space time a prolongation of this problem whereby an integrability condition is added leads to a prolonged system in involution the riemann lanczos problem in four dimensions is not in involution and needs to be prolongated as bampi and caviglia suggested but singular solutions of it can be found and we give examples for the godel kasner and debever hubaut space times

trefoil symmetry iii the full clover extension we aim to approach a trefoil symmetry involving z4 z4 q graded parameters multiplets of generators of integer spin 2 with naive dimensions in the interval 0 1 the resulting full clover extension involves as particular subalgebras the minimal clover extensions previously obtained we address the introduction of further scalar dimensionless multiplets and central charges they could provide connections among external and internal symmetries perhaps after some graded symmetry breaking

existence of solutions for a class of nonlinear schrodinger equations with potential vanishing at infinity abstract in this paper we investigate the existence of positive ground state solution for the following class of elliptic equations u v x u k x f u in r n where n 3 v k are nonnegative continuous functions and f is a continuous function with a quasicritical growth here we prove a hardy type inequality and use it together with variational method to get a ground state solution

chern simons terms in metric affine space time bianchi identities as euler lagrange equations the metric affine gauge theory of gravity encompasses a space time with the following geometrical fields coframe metric g and an independent linear connection within this geometrical framework all four forms b dc are constructed which qualify as boundary terms for a gauge lagrangian that is they are gl 4 r scalars as well as exact forms derived from chern simons type three forms c the result of our search is summarized in eq 4 20 the translational piece dctt is new the boundary terms effectively serve as lagrangians for the bianchi identities of nonmetricity torsion and curvature in the canonical formalism the normal parts of the chern simons three forms represent generating functions that are capable of generating new ashtekar type variables eventually the bach lanczos identity is generalized to the metric affine space time

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

i q graded lie algebraic extensions of the poincare algebra constraints on i and q the constraints on the index set i and on the function q of the i q graded lie algebras over k containing the poincare lie algebra are studied by using the single grading model particular choices for i and q consistent with the found constraints are determined for k c gradings are then found for which i i z2 z4n z4n gre with n n and gre an abelian group these gradings provide a way for algebraic extensions of the poincare lie algebra beyond the z2 gradings of supersymmetry and supergravity in these algebraic extensions each other commuting space time parameter can either commute or anticommute with the further parameters of the i q graded super manifold different field representations can have with each other generalized commutative behavior beyond commutativity and anticommutativity

sign changing solutions for a class of kirchhoff type problem in bounded domains abstract we are interested in the existence of least energy sign changing solutions for a class of kirchhoff type problem in bounded domains because the so called nonlocal term b u 2 d x u is involving in the equation the variational functional of the equation has totally different properties from the case of b 0 combining constraint variational method and quantitative deformation lemma we prove that the problem possesses one least energy sign changing solution u b moreover we show that the energy of u b is strictly larger than the ground state energy finally we regard b as a parameter and give a convergence property of u b as b 0

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

trefoil symmetries i clover extensions beyond coleman mandula theorem a graded minimal lie algebraic extension of the space time symmetry is constructed involving only spin 1 multiplets as novel generators the extension involves z4 z4 graded parameters and generators it provides a bosonic analog to supersymmetry since the composition of three symmetric vector charges produces a space time translation there arise three noncommutative four dimensional manifolds with pseudometric

multiple solutions for a class of schrodinger poisson system with indefinite nonlinearity in this article we investigate the existence and multiplicity of solutions of schrodinger poisson system u u l x u k x u 2u h x u x r3 l x u2 x r3 where the potential k x allows sign changing we obtain the existence and multiplicity of solutions for the system which can be regarded as complementary work of huang et al 2013 12

on the arbitrariness of the general solution of an involutive partial differential equation the relationship between the strength of a differential equation as introduced by einstein its cartan characters and its hilbert polynomial is studied using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems the problem of computing the number of arbitrary functions in the general solution is treated finally the effect of gauge symmetries is considered