grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

graded contractions of representations of orthogonal and symplectic lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras o n c n or 5 and sp 2n c n or 2 with complex parameters are described for all maximal parabolic subalgebras all contractions which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras are found contractions of the irreducible representations for each parabolic contraction of the lie algebra are the main results of the article

graded contractions of casimir operators we describe graded contractions of casimir operators of lie algebras the formalism applies to any casimir operator given as a symmetric form we deal with the quadratic casimir operator in exhaustive detail and indicate the modifications for operators of higher degree

equivalence of diagonal contractions to generalized iw contractions with integer exponents abstract we present a simple and rigorous proof of the claim by weimar woods e weimar woods contractions generalized inonu wigner contractions and deformations of finite dimensional lie algebras rev math phys 12 2000 1505 1529 that any diagonal contraction e g a generalized inonu wigner contraction is equivalent to a generalized inonu wigner contraction with integer parameter powers

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

strange attractors in saddle node cycles prevalence and globality we consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle node cycles we show that they always exhibit henon like strange attractors for a set of parameter values with positive lebesgue density at the bifurcation value in open classes of such families the strange attractors are of global type their basins contain an a priori defined neighbourhood of the cycle furthermore the bifurcation parameter may also be a point of positive density of hyperbolic dynamics

a variety of exact travelling wave solutions for the 2 1 dimensional boiti leon pempinelli equation abstract in this paper the 2 1 dimensional boiti leon pempinelli blp equation will be studied the tanh coth method will be used to obtain exact travelling wave solutions for this equation the exp function method will also be applied to the blp equation to derive a new variety of travelling wave solutions with distinct physical structures

on the sensitivity of sectional anosov flows we study sectional anosov flows on compact 3 manifolds for which the maximal invariant and nonwandering sets coincide we prove that every vector field close to one of these flows is sensitive with respect to initial conditions

contractions from grading we note that large classes of contractions of algebras that arise in physics can be understood purely algebraically via identifying appropriate zm gradings and their generalizations on the parent algebra this includes various types of flat space carroll limits of finite and infinite dimensional a ds algebras as well as galilean and galilean conformal algebras our observations can be regarded as providing a natural context for the grassmann approach of krishnan et al j high energy phys 2014 3 36 we also introduce a related notion which we call partial grading that arises naturally in this context

a series solution of the foam drainage equation foaming occurs in many distillation and absorption processes the drainage of liquid foams involves the interplay of gravity surface tension and viscous forces in this paper we apply an analytic technique namely the homotopy analysis method ham to handle the foam drainage equation the powerful homotopy analysis method computes the solution in a rapidly convergent infinite series we also compare the numerical results obtained by hpm or adm and the same results obtained by using the analytic solutions obtained by ham

graded contractions of the gell mann graded sl 3 c the gell mann grading one of the four gradings of sl 3 c that cannot be further refined is considered as the initial grading for the graded contraction procedure using the symmetries of the gell mann grading the system of contraction equations is reduced and solved each non trivial solution of this system determines a lie algebra which is not isomorphic to the original algebra sl 3 c the resulting 53 contracted algebras are divided into two classes the first is represented by the algebras which are also continuous inonu wigner contractions the second is formed by the discrete graded contractions

graded contractions of lie algebras and central extensions we use the concept of grading of lie algebras to investigate the appearance of central charges during the contraction process as for the usual graded contractions one finds simultaneously the central extensions of classes of algebras rather than specific lie algebras to illustrate the method we consider in detail two physical applications the kinematical algebras of spacetime and the u n bosons limits of some lie algebras

a fixed point technique for investigating the stability of α β γ derivations on lie c algebras abstract we apply a fixed point theorem to investigate some new stability results for derivations on lie c algebras associated with the generalized cauchy jensen type additive functional equation i 1 n f x i 1 n 1 j 1 j i n x j 2 i 1 n f x i

subjoinings of affine kac moody algebras subjoinings among affine kac moody algebras is a relation generalising the algebra subalgebra relation it is analogous to subjoinings among semisimple finite lie algebras a list of maximal subjoinings of equal rank affine kac moody algebras is presented it is conjectured that there are no other maximal subjoinings

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra

the construction of solvable lie algebras from equidimensional nilpotent algebras abstract the study of gradings of solvable lie algebras l of finite dimensionover a field f of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras l uniquely determined by l up to f isomorphy conversely for any finite dimensional lie algebra l over f an algorithm is developed which yields in parametric form all solvable lie algebras l determining l as the corresponding nilpotent algebra the exposition is independent of lie grading theory it organizes in a novel way the classification of solvable lie algebras of given dimension around the same task for nilpotent algebras every isomorphy class of solvable f algebras is obtained in this way

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

solvable lie algebras of dimension 4 over perfect fields abstract the solvable lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame counted over finite fields of q elements and classified over the real and complex number fields the number dq n of nonisomorphic solvable lie algebras of dimension n over f q roughly speaking grows as a polynomial in n with qn 2 as highest term but there are smaller additional terms depending on the residue class of q modulo n 1 if n 4

graded contractions of representations of special linear lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras sl n c n or 3 are described for all maximal parabolic subalgebras all contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras contractions of the irreducible representations of sl n c for each parabolic contraction of the lie algebra are the main results of the article

new gradings of sl 3 c representations two new gradings of the lie algebra sl 3 c are the refined sl 2 c and o 3 gradings the grading in each case utilizes the sl 2 c or o 3 weight together with a new additive modular label a complete set of sl 3 c representation basis states labeled by each to the two sets of additive quantum numbers is found the newlabeling operator in both cases fails to commute with the cubic sl 3 c casimir operator and hence mixes states of the contragredient representations p q and q p

the tanh method and adomian decomposition method for solving the foam drainage equation foaming occurs in many distillation and absorption processes the drainage of liquid foams involves the interplay of gravity surface tension and viscous forces in this paper we use a semi analytic method the adomian decomposition method and an analytic method the tanh method to handle the foam drainage equation the powerful tanh method gives the solution in a closed form however adomian decomposition method computes the solution in a rapidly convergent infinite series the comparison between the two approaches is conducted to illustrate the performance of each method

lowest dimensional example on non universality of generalized inonu wigner contractions abstract we prove that there exists just one pair of complex four dimensional lie algebras such that a well defined contraction among them is not equivalent to a generalized iw contraction or to a one parametric subgroup degeneration in conventional algebraic terms over the field of real numbers this pair of algebras is split into two pairs with the same contracted algebra the example we constructed demonstrates that even in the dimension four generalized iw contractions are not sufficient for realizing all possible contractions and this is the lowest dimension in which generalized iw contractions are not universal moreover this is also the first example of nonexistence of generalized iw contraction for the case when the contracted algebra is not characteristically nilpotent and therefore admits nontrivial diagonal derivations the lower bound equal to three of nonnegative integer parameter exponents which are sufficient to realize all generalized iw contractions of four dimensional lie algebras is also found

exact travelling wave solutions of some nonlinear equations by g g expansion method abstract here we find exact solutions of some nonlinear evolution equations within the framework of the g g expansion method exact solutions of five nonlinear equations of physical importance viz the coupled schrodinger kdv equation the coupled nonlinear reaction diffusion equation the foam drainage equation the phi four equation and the dodd bullough mikhailov equation are obtained these general solutions can be reduced in some standard results derived by some other methods two and three dimensional plots of some of the results are also presented

graded contractions of the lie algebra e 2 1 the graded contractions of pseudo euclidean lie algebra are studied the non equivalent gradings of of type and a are extended to the entire lie algebra using the action of on the abelian ideal the translations the graded contractions embed into a large family of six dimensional lie algebras the family includes solvable nilpotent and nonsolvable lie algebras both decomposable and indecomposable ones the distinction between graded and inonu wigner contractions is analysed the physically most interesting lie algebras obtained by the contractions are the inhomogeneous galilei and pseudo galilei

graded contractions of sl 3 c the wigner inonu contraction is generalized to include gradings other than z2 and the formalism is applied to obtain all toroidal contractions of sl 3 c examples of contractions based on nontoroidal gradings are also given

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

the family of quaternionic quasi unitary lie algebras and their central extensions the family of quaternionic quasi unitary or quaternionic unitary cayley klein algebras is described in a unified setting this family includes the simple algebras sp n 1 and sp p q in the cartan series cn 1 as well as many non semisimple real lie algebras which can be obtained from these simple algebras by particular contractions the algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature this common framework allows one to perform the study of many properties for all these lie algebras simultaneously in this paper the central extensions for all quasi simple lie algebras of the quaternionic unitary cayley klein family are shown to be trivial no matter their dimension

on lie gradings iii gradings of the real forms of classical lie algebras abstract maximal abelian subgroups of diagonalizable automorphisms of lie algebra so called mad groups play a crucial role in the construction of fine gradings of lie algebra our aim is to give a description of mad groups for real forms of classical lie algebras we introduce four types of matrix subgroups of g l n c called out groups ad groups out groups and ad groups for each type of these subgroups we define a relation of equivalence the problem of classifying of all non conjugate mad groups on real forms of sl n c o n c or sp n c is transformed to the problem of classifying these equivalence classes the classification of these equivalence classes is presented here

graded contractions of affine lie algebras the method of graded contractions based on the preservation of automorphisms of finite order is applied to the affine lie algebras and their representations to yield a new class of infinite dimensional lie algebras and representations after the introduction of the horizontal and vertical gradings and the algorithm to find the horizontal toroidal gradings some general properties of the graded contractions are discussed and compared with the inonu wigner contractions the example of is discussed in detail

a comparison between adomian decomposition method and taylor series method in the series solutions in this work we will compare the performance of the adomian decomposition method and the taylor series method applied to the solution of linear and nonlinear ordinary differential equations the comparison shows that the decomposition method is reliable efficient and easy to use from a computational viewpoint although the two methods provide the solution in an infinite series the decomposition method provides a fast convergent series of easily computable components and eliminates cumbersome computational work needed by taylor series method

the modified decomposition method and pade approximants for solving the thomas fermi equation in this paper we propose a new approach to develop a nonperturbative approximate solution for the thomas fermi equation this approach rests on the recently developed modification of the adomian decomposition method the initial slope of the thomas fermi potential y 0 is computed by converting the obtained series solution into several diagonal pade approximants the proposed scheme is presented in a general way so that it can be used in applied sciences

graded contractions of bilinear invariant forms of lie algebra we introduce a new construction for bilinear invariant forms on lie algebras based on the method of graded contractions the general method is described and the z2 z3 and z2 x z2 contractions are found the results can be applied to all lie algebras and superalgebras finite or infinite dimensional which admit the chosen gradings we consider some examples contractions of the killing form toroidal contractions of sl 3 c and we briefly discuss the limit to new wess zumino witten actions

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

cayley klein algebras as graded contractions of so n 1 we study z2 x n graded contractions of the real compact simple lie algebra so n 1 and we identify within them the cayley klein algebras as a naturally distinguished subset

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

inonu wigner contraction of kac moody algebras inonu wigner contractions of affine kac moody algebras are discussed here the sugawara construction for the contracted affine algebra exists only for a fixed value of the level k which is determined in terms of the dimension of the uncontracted part of the starting lie algebra and the quadratic casimir in the adjoint representation further contractions of the g h coset spaces are discussed and an affine translation algebra is obtained which yields a virasoro algebra via a goddard kent olive gko construction with a central charge given by dim g h

generic family with robustly infinitely many sinks we show for every r d ge 0 or r d ge 2 the existence of a baire generic set of c d families of c r maps f a a in mathbb r k of a manifold m of dimension ge 2 so that for every a small the map f a has infinitely many sinks when the dimension of the manifold is ge 3 the generic set is formed by families of diffeomorphisms when m is the annulus this generic set is formed by local diffeomorphisms this is a counter example to a conjecture of pugh and shub

hopf superalgebra contractions and r matrix for fermions different contractions are applied to the quantum superalgebra ospq 1 mod 2 in the first of them the graded analogue s hq 1 of the one dimensional heisenberg quantum algebra is obtained and its r matrix explicitly calculated in the second contraction a z2 graded version of the q oscillator is proposed and finally the supersymmetric two dimensional euclidean quantum algebra s eq 2 is found

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

statistical stability and continuity of srb entropy for systems with gibbs markov structures we present conditions on families of diffeomorphisms that guarantee statistical stability and srb entropy continuity they rely on the existence of horseshoe like sets with infinitely many branches and variable return times as an application we consider the family of henon maps within the set of benedicks carleson parameters

solution of the basin problem for henon like attractors for a large class of non uniformly hyperbolic attractors of dissipative diffeomorphisms we prove that there are no holes in the basin of attraction stable manifolds of points in the attractor fill in a full lebesgue measure subset then lebesgue almost every point in the basin is generic for the srb sinai ruelle bowen measure of the attractor this solves a problem posed by sinai and by ruelle for this class of systems

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

fine group gradings of the real forms of sl 4 c sp 4 c and o 4 c we present an explicit description of the fine group gradings i e group gradings which cannot be further refined of the real forms of the semisimple lie algebras sl 4 c sp 4 c and o 4 c all together 12 real lie algebras are considered and a total of 44 of their fine group gradings are listed the inclusions sl 4 c sp 4 c o 4 c are an important tool in our presentation systematic use is made of the faithful representations of the three lie algebras by 4 4 matrices

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

prevalent dynamics at the first bifurcation of henon like families we study the dynamics of strongly dissipative henon like maps around the first bifurcation parameter a at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set we prove that a is a full lebesgue density point of the set of parameters for which lebesgue almost every initial point diverges to infinity under positive iteration a key ingredient is that a corresponds to the non recurrence of every critical point reminiscent of misiurewicz parameters in one dimensional dynamics adapting on the one hand benedicks carleson s parameter exclusion argument we construct a set of good parameters having a as a full density point adapting benedicks viana s volume control argument on the other we analyze lebesgue typical dynamics corresponding to these good parameters