generalized quadratic mappings of r type in several variables let x y be vector spaces it is shown that if an even mapping f x y satisfies f 0 0 and 2cl 1d 2 cld 2 cl 2d 2 r2f j 1dxjr j 0 1 j 1d j lr2f j 1d 1 j xjr cld 1 cl 1d 1 2d 2cl 1 cld 2 cl 2d 2 j 1df xj for all x1 xd x then the even mapping f x y is quadratic furthermore we prove the cauchy rassias stability of the functional equation in banach spaces

on the hyers ulam rassias stability of approximately additive mappings abstract in this paper using an idea from the paper of p gvruta j math anal appl 184 1994 431 436 the results of th m rassias j math anal appl 158 1991 106 113 shall be generalized

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

hyers ulam stability of a generalized apollonius type quadratic mapping abstract let x y be linear spaces it is shown that if a mapping q x y satisfies the following functional equation 0 1 q i 1 n z i i 1 n x i q i 1 n z i i 1 n y i 1 2 q i 1 n x i i 1 n y i 2 q i 1 n z i i 1 n x i i 1 n y i 2 then the mapping q x y is quadratic we moreover prove the hyers ulam stability of the functional equation 0 1 in banach spaces

soliton structure versus singularity analysis third order completely intergrable nonlinear differential equations in 1 1 dimensions abstract backlund transformations between all known completely integrable third order differential equations in 1 1 dimensions are established and the corresponding transformations formulas for their hereditary operators and hamiltonian formulations are exhibited some of these backlund transformations are not injective therefore additional non commutative symmetry groups are found for some equations these non commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra a 1 1 new completely integrable third order integro differential equations some depending explicitly on x are given these new equations give rise to nonin equation connections between the singularity equations from the painleve analysis and the nonlinear equations for interacting solitons are established a common approach to singularity analysis and soliton structure is introduced the painleve analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine gordon and nonlinear schrodinger equation a method to recover the painleve series from its constant level term is exhibit the soliton singularity transform is recognized to be connected to the mobius group this gives rise to a darboux like result for the spectral properties of the recursion operator these connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton singularity transform this then leads to the result that only for scaling degrees 1 and 2 the usual painleve analysis can be carried out a new invariance principle connected to kernels of differential operators is discovered this new invariance for example connects the explicit solutions of the liouville equation with the miura transform simple methods are exhibited which allow to compute out of n soliton solutions of the kdv bargman potentials explicit solutions of equations like the harry dym equation certain solutions are plotted

on the hyers ulam stability of the functional equations that have the quadratic property abstract the hyers ulam stability of the quadratic functional equation 1 on a restricted domain shall be investigated and the result shall be applied to the study of an asymptotic behavior of that equation furthermore the hyers ulam stability problems of another quadratic equation 4 on a restricted domain shall also be treated under the approximately even or odd condition and some asymptotic behaviors of the quadratic mappings and the additive mappings shall be investigated

an extended harry dym hierarchy an extended harry dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the harry dym hierarchy associated with the pseudo differential operator l u u0 u1 1 the corresponding lax presentation possesses a self consistent source involving squared eigenfunctions the resulting extended harry dym hierarchy is reduced to the harry dym hierarchy with self consistent sources under the n reduction ln ln 2 and the k constrained harry dym hierarchy under the k constraint lk lk 2 ni 1qi 1ri 2 a few particular examples are computed together with their lax pairs

stability of functional equations on restricted domains in a group and their asymptotic behaviors in this paper we consider hyers ulam stability problems for the pexider equation the cauchy equation and the jensen equation in general restricted domains in a group the main purpose of this paper is to find restricted domains such that the functional inequality satisfied in those domains extends to the inequality for the whole domain and such that the hyers ulam stability theorem holds for the inequalities as it does when the inequality holds globally we also consider a distributional version of the hyers ulam stability of the pexider equation in restricted domains and its asymptotic behaviors

a fixed point method for perturbation of bimultipliers and jordan bimultipliers in c ternary algebras let a be a c ternary algebra a c bilinear t a a a is called a c ternary algebra bi multiplier if it satisfies t abc d t a b cd t a bcd abt c d for all a b c d a also the mapping t a a a is a called c ternary algebra jordan bimultiplier if it satisfies t aaa a t a a aa t a aaa aat a a for all a a using the fixed point method we investigate the generalized hyers ulam rassias stability of bimultipliers and jordan bimultipliers in c ternary algebras the concept of hyers ulam rassias stability originated from the th m rassias stability theorem that appeared in his paper th m rassias proc am math soc 72 297 1978

stability of functional equations of drygas in the space of schwartz distributions in this paper we reformulate and prove the stability theorems of s m jung and p k sahoo s m jung p k sahoo stability of a functional equation of drygas aequationes math 64 2002 263 273 in the spaces of generalized functions such as the schwartz distributions and tempered distributions

on a generalized nonlinear functional equation abstract the paper deals with the functional equation f x f f u x f v x under some special assumptions concerning the given functions u v and f our main result extends some results in the literature

vector valued invariant means revisited abstract we show that a banach space x is complemented in its ultraproducts if and only if for every amenable semigroup s the space of bounded x valued functions defined on s admits a an invariant average or b what we shall call an admissible assignment condition b still provides an equivalence for quasi banach spaces while condition a necessarily implies that the space is locally convex

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

generalized hyers ulam rassias stability of n sesquilinear quadratic mappings on banach modules over c algebras assume that x is a left banach module over a unital c algebra a it is shown that almost every n sesquilinear quadratic mapping h xxxxx n a is an n sesquilinear quadratic mapping when h rx y z 1 z n h x ry z 1 z n h x y rz 1 z 2 z n h x y z 1 z 2 rz n rh x y z 1 z 2 z n r 0 r 1 holds for all x y z 1 z email protected x moreover we prove the generalized hyers ulam rassias stability of an n sesquilinear quadratic mapping on a left banach module over a unital c algebra

on the local and nonlocal camassa holm hierarchies we construct the local and nonlocal conserved densities for the camassa holm equation by solving a suitable riccati equation we also define a kadomtsev petviashvili extension for the local camassa holm hierarchy

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

stability of exponential equations in schwartz distributions abstract we reformulate the superstability of exponential equation and cosine functional equation j a baker the stability of cosine equation proc amer math soc 80 1980 411 416 in some spaces of generalized functions such as the schwartz distributions sato hyperfunctions and gelfand generalized functions which completes the previous results of partial generalizations of the stability problems j chung a distributional version of functional equations and their stabilities nonlinear anal 62 2005 1037 1051 j chung s y chung d kim the stability of cauchy equations in the space of schwartz distributions j math anal appl 295 2004 107 114

on the cauchy rassias stability of a generalized additive functional equation let x and y be banach spaces and f x y an odd mapping we solve the following generalized additive functional equation rf j 1dxjr j 0 1 j 1d j lrf j 1d 1 j xjr cld 1 cl 1d 1 1 j 1df xj for all x1 xd x moreover we deal with the above functional equation in banach modules over a c algebra and obtain generalizations of the cauchy rassias stability the concept of cauchy rassias stability for the linear mapping was originated from th m rassias s stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

on the hyers ulam stability of linear mappings abstract let e1 be a real normed vector space and e2 a real banach space s m ulam posed the problem when does a linear mapping near an approximately linear mapping e1 e2 exist we give a new generalized solution to this problem an example illustrates when the answer to this question is negative the behaviour of bounded approximately additive mappings which do not satisfy hyers uiam stability is also investigated

on the stability of the linear mapping in banach modules abstract we prove the generalized hyers ulam rassias stability of the linear mapping in banach modules over a unital banach algebra

regular articlesquared eigenfunction symmetries for soliton equations part ii nonlinear integrable evolution equations in 1 1 dimensions arise from constraints of the 2 1 dimensional hierarchies associated with the kadomtsev petviashvili kp equation the modified kp equation and the dym equation respectively links of backlund type and of reciprocal type are known to exist between the 2 1 dimensional systems the corresponding links between the constrained flows are discussed in a general framework in particular squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered the links between the soliton hierarchies are extended to these symmetries

canonically modified nose hoover equation with explicit inclusion of the virial the nos e hoover equation has recently been introduced to simulate in a deterministic and reversible way the equilibrium properties of a system at constant temperature however for many one dimensional potentials such as the harmonic oscillator the nos e hoover scheme is not adequate since the dynamics is not sufficiently ergodic we present modifications of the nos e hoover equation in which the kinetic energy and the virial are treated in an equivalent manner and which explicitly include the virial within a canonical framework we show that these modifications can yield an adequate statistical description for one dimensional potentials such as the double well and harmonic oscillator

stability of c inner products abstract in this paper we investigate the stability of the inner products in hilbert c modules over unital c algebras

negative semi discrete kp and bkp hierarchies via nonlocal symmetries in this paper nonlocal symmetries for the bilinear semi discrete kadomtsev petviashvili kp and bilinear semi discrete b type kp bkp equations are derived by expanding these nonlocal symmetries in power series of each of two parameters we have derived bilinear negative semi discrete kp and bkp hierarchies an interesting observation is that bilinear positive and negative semi discrete kp hierarchies may be derived from the same nonlocal symmetry for the semi discrete kp equation

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

isomorphisms between c ternary algebras in this paper we prove the hyers ulam rassias stability of homomorphisms in c ternary algebras and of derivations on c ternary algebras for the following generalized cauchy jensen additive mapping 2f j 1pxj2 j 1dyj j 1pf xj 2 j 1df yj this is applied to investigate isomorphisms between c ternary algebras the concept of hyers ulam rassias stability originated from the rassias stability theorem that appeared in his paper on the stability of the linear mapping in banach spaces see proc amr math soc 72 297 300 1978

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

finite length scaling of collapsing directed walks we study the finite length scaling of self interacting partially directed self avoiding walks utilising enumeration data up to a total length of 6000 steps this facilitates the evaluation of the numerical techniques available for calculating exponents at the point and in the collapsed phase of walk type models another consequence is the conjecture of a new scaling theory for the collapsed region of the phase diagram and the suggestion that this should be applicable to the wider range of undirected problems including interacting saw we provide a phenomenological picture of the phase transition in terms of the condensation of droplets that allows us to understand the various length scales involved in the problem pacs numbers 05 50 q 05 70 fh 61 41 e email prel aleks brak tonyg mundoe maths mu oz au

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

ulam stability problem for quadratic mappings of euler lagrange abstract the purpose of this paper is to solve the stability problem of ulam for an approximate mapping f a b of the following quadratic functional equation of euler lagrange a b aq x a b bq y q ax by abq x y for all x y a with a and b linear spaces

the ghost symmetry of the bkp hierarchy in this paper we systematically develop the ghost symmetry of the kadomtsev petviashvili sub hierarchy of b type bkp through its actions on the lax operator l the eigenfunctions and the function in this process the spectral representation of the eigenfunctions and a new potential are introduced by using squared eigenfunction potential of the bkp hierarchy moreover the bilinear identity of the constrained bkp hierarchy and adler shiota van moerbeke formula of the bkp hierarchy are rederived compactly by means of the spectral representation and ghost symmetry

isomorphisms between c ternary algebras abstract in this paper we prove the hyers ulam rassias stability of homomorphisms in c ternary algebras and of derivations on c ternary algebras for the following cauchy jensen additive mappings 0 1 f x y 2 z f x y 2 z f x 2 f z 0 2 f x y 2 z f x y 2 z f y 0 3 2 f x y 2 z f x f y 2 f z these are applied to investigate isomorphisms between c ternary algebras the concept of hyers ulam rassias stability originated from the th m rassias stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

stability of a quadratic jensen type functional equation in the spaces of generalized functions abstract making use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic jensen type functional equation 9 f x y z 3 f x f y f z 4 f x y 2 f y z 2 f z x 2 in the spaces of schwartz tempered distributions and fourier hyperfunctions

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

conditional functional equations on restricted domains of measure zero let r02 r2 0 0 r 2 x y r2 x2 y2 and f r02 r g r 2 r in this paper we consider the ulam hyers stability of the functional equations f ux vy uy vx f x y f u v f ux vy uy vx f x y f u v g ux vy uy vx g x y g u v g ux vy uy vx g x y g u v for all x y u v where r4 is of 4 dimensional lebesgue measure zero the above functional equations are modified versions of the equations in 9 11 14 18 24 which arise from number theory and are in connection with characterizations of determinant and permanent of two by two matrices

nonlocal topological solitons of the sine gordon equation a special nonlocal symmetry related to the b cklund transformation is obtained for the 1 1 dimensional sine gordon sg equation in its polynomial form which can further generate infinitely many nonlocal symmetries of the sg equation due to the inclusion of an arbitrary spectrum parameter this nonlocal symmetry can be localized and hence turn into a point symmetry for the augmented system through the introduction of an auxiliary function consequently finite transformation similarity reductions and explicit invariant solutions related to the nonlocal symmetry are obtained it is known that the sg equation has topological solitons or kinks or antikinks based on the group invariant solutions a different kind of topological soliton which is a topological soliton situated on a cnoidal wave background is obtained analytically and graphically displayed

on approximation of approximately linear mappings by linear mappings abstract assume a is a normed linear space b is a banach space and f a b is a mapping approximately linear we solve the following ulam problem give conditions in order for a linear mapping near an approximately linear mapping to exist

on the generalized hyers ulam stability of module left derivations let a be a unital normed algebra and let m be a unitary banach left a module if f a m is an approximate module left derivation then f a m is a module left derivation moreover if m a is a semiprime unital banach algebra and f tx is continuous in t r for each fixed x in a then every approximately linear left derivation f a a is a linear derivation which maps a into the intersection of its center z a and its jacobson radical rad a in particular if a is semisimple then f is identically zero

hyers ulam rassias stability of homomorphisms in quasi banach algebras associated to the pexiderized cauchy functional equation abstract in this paper we prove the hyers ulam rassias stability of homomorphisms in quasi banach algebras associated to the pexiderized cauchy functional equation this is applied to investigate homomorphisms between quasi banach algebras the concept of hyers ulam rassias stability originated from th m rassias stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300