e isometries in euclidean spaces abstract let 0 r r and a be a subset of the n dimensional euclidean space e n which is contained in b x 0 r and contains points x 0 x 0 re 1 x 0 re n where the vectors e i i 1 n are orthonormal we show that if f a e n is an e isometry then there is an affine isometry u such that f x u x n e r r for all x in a where n cn moreover if a is star shaped with respect to x 0 and contains b x 0 r then n c log n 1 both results are sharp

stability of banach spaces via nonlinear e isometries abstract in this paper we prove that the existence of an e isometry from a separable banach space x into y the james space or a reflexive space implies the existence of a linear isometry from x into y then we present a set valued mapping version lemma on non surjective e isometries of banach spaces using the above results we also discuss the rotundity and smoothness of banach spaces under the perturbation by e isometries

real trajectories in complex time method and barrier penetrationlike phenomena various applications of the real trajectories in the complex time method are discussed it is shown that the proper way of summation over semiclassical orbits removes existing doubts and leads to the more general formulas than the standard wkb type one in the energy eigenvalue problem this method is also used in the description of explicitly time dependent phenomena in the problem of escaping from a metastable state the estimation of a lifetime is in a good agreement with the inverse width of a false ground state energy

on an n dimensional mixed type additive and quadratic functional equation in this paper we investigate the generalized hyers ulam stability of the functional email protected k 2 k n 01fx email protected i 2n 1 k ix i 2 n 1f x 1 2 n email protected i 2nf x i f x i 0for integer values of n such that n 2 where f is a function from a normed space x to a banach space y the solutions of the equation are called additive quadratic mappings

a fixed point theorem and the hyers ulam stability in non archimedean spaces abstract we prove a fixed point theorem in complete non archimedean normed spaces and show its applications in the hyers ulam stability of some functional equations

decomposition of mappings approximately inner product preserving abstract for hilbert spaces e and f we consider the class of mappings f e f preserving approximately inner product in the following sense f x f y x y x y x y e for suitable function we show that the orthogonal projection of f onto some closed linear subspace h of f is a linear isometry while the projection onto h is bounded by x x several consequences of such a decomposition are given in particular stability and superstability of inner product preserving mappings are considered

quantum theory of anharmonic oscillators a variational and systematic general approximation method this is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1 2 omega 2q2 lambda q4 as a lowest order approximation an extremely simple formula for energy levels ei 0 i 1 2 1 4 3 alphai alphai is derived i being the quantum number of the energy level which covers any lambda i alphai is the real positive root of a cubic equation yi alphai3 alphai2 1 0 with yi 6lambda 2i2 2i 1 2i 1 this formula reproduces the exact energy levels within an error of about 1 for any lambda i the worst case is 2 for i 0 lambda to infinity systematically higher orders of this perturbation theory are developed which contains the usual perturbation theory for the limiting case of small lambda but this perturbation theory is valid for any i lambda the second order perturbation theory reduces the errors of our lowest order results by a factor of about 1 5 in general various ranges large intermediate small of i lambda are investigated and compared with the exact values obtained by the montroll group 1975 1978 for i 0 1 even the fourth order perturbation calculation can be elaborated explicitly which reduces the error to about 0 01 for any lambda for small lambda it gives correct numerical coefficients up to lambda4 terms as it should

isometric embeddings of compact spaces into banach spaces abstract we show the existence of a compact metric space k such that whenever k embeds isometrically into a banach space y then any separable banach space is linearly isometric to a subspace of y we also address the following related question if a banach space y contains an isometric copy of the unit ball or of some special compact subset of a separable banach space x does it necessarily contain a subspace isometric to x we answer positively this question when x is a polyhedral finite dimensional space c 0 or l 1

non surjective coarse isometries between lp spaces abstract in this paper we study stability and weak stability of coarse isometries of banach spaces as a result we show that if a coarse isometry f l p 1 1 1 l p 2 2 2 1 p is weakly stable at each point of a schauder basis then there is a linear isometry u l p 1 1 1 l p 2 2 2 where j j j j 1 2 are finite measure spaces furthermore if f is uniformly weakly stable then f x u x o x when x as an application we obtain that p f x u x o x is equivalent to f x u x o x as x where p y u x is a projection with p 1

introduction of neighborhood information in evidence theory and application to data fusion of radar and optical images with partial cloud cover abstract two ways of introducing spatial information in dempster shafer evidence theory are examined in the definition of the monosource mass functions and during data fusion in the latter case a neighborhood mass function is derived from the label image and combined with the radiometric masses according to the dempster orthogonal sum the main advantage of such a combination law is to adapt the importance of neighborhood information to the level of radiometric missing information the importance of introducing neighborhood information has been illustrated through the following application forest area detection using radar and optical images showing a partial cloud cover

properties of isometric mappings abstract some relations between isometry and linearity are examined in particular generalizations of the mazur ulam theorem and open problems are discussed

a universal theorem for stability of e isometries of banach spaces abstract let x y be two banach spaces and f x y be a standard e isometry for some e 0 in this paper we show the following sharp weak stability inequality of f for every x x there exists y with x r such that x x f x 2 e r for all x x it is not only a sharp quantitative extension of figiel s theorem but it also unifies generalizes and improves a series of known results about stability of e isometries for example if the mapping f satisfies c f co f x f x y then it is equivalent to the following sharp stability theorem there is a linear surjective operator t y x of norm one such that t f x x 2 e for all x x when the e isometry f is surjective it is equivalent to omladic semrl s theorem there is a surjective linear isometry u x y so that f x u x 2 e for all x x

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

quasilinearisation and three dimensional oscillators the anharmonic symmetric oscillator is a frequently used model for physical systems the recently developed quasilinearisation technique is applied to this model as well as to the simpler one without the quartic term in the potential

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

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

the algebraic method for two dimensional quantum atomic systems the authors propose an easy and effective variant of the realization of the dynamical symmetry of hydrogen like atoms in two dimensional space based on the relationship between the schrodinger equation for the isotropic harmonic oscillator in one dimensional complex space and the schrodinger equation for the one electron atom in two dimensional space that permits them to use the operator method to solve the schrodinger equation for the two dimensional atomic system for illustration of the algebraic method they consider the problem of the two dimensional hydrogenic donor in a magnetic field by the use of the path integral they also establish the relationship between the two dimensional coulomb green function and the green function for the isotropic harmonic oscillator in one dimensional complex space

the anharmonic oscillator perturbation series for cubic and quartic energy distortion twenty five terms of the perturbation series are calculated for the first five energy levels of an oscillator with lambda x4 energy distortion and a similar calculation is carried out for the first six levels for lambda x3 distortion for the positive quartic the alternating series is summed using the aitken delta 2 transformation for lambda from 0 1 to 100 an exponential integral model and a truncated binomial model are used to transform the series for negative distortion energies and the complex energy eigenvalues are calculated for lambda from 0 01 to 1

on stability of nonlinear non surjective e isometries of banach spaces abstract let x y be two banach spaces e 0 and let f x y be an e isometry with f 0 0 in this paper we show first that for every x x there exists y with x r such that f x x x 4 e r for all x x making use of it we prove that if y is reflexive and if e y the annihilator of the subspace f y consisting of all functionals bounded on co f x f x is complemented in y then there is a bounded linear operator t y x with t such that t f x x 4 e for all x x if in addition y is gateaux smooth strictly convex and admitting the kadec klee property in particular locally uniformly convex then we have the following sharp estimate t f x x 2 e for all x x

fixed point theorems and the ulam hyers stability in non archimedean cone metric spaces abstract let x p be a metric space with a k valued non archimedean metric p in this paper we prove the existence and approximation of a fixed point for operators f x x satisfying the contractive condition in the form p f x f y q p x y where q k k is an increasing operator then we study the generalized ulam hyers stability of fixed point equations we next obtain an extension of the krasnoselskii fixed point theorem for the sum of two operators finally an application to functional equations is given

phase isometries on real normed spaces abstract we say that a map f from a normed space x to another normed space y is a phase isometry if the equality f x f y f x f y x y x y holds for all x y x a normed space x is said to have the wigner property if for any normed space y and every surjective phase isometry f x y there exists a phase function e x 1 1 such that e f is a linear isometry we show that all smooth normed spaces l type spaces and l 1 spaces enjoy this property

an analytical description of some quantum systems in periodic external fields and quasistationary systems the peculiarities of the operator method in solving the schrodinger equation with periodic potentials are discussed approximate analytical solutions of the mathieu equation and schrodinger equations for a two level system are found on the basis of the operator method an analogous method is used for the analytical estimation of the quasistationary state energy and width

dempster shafer fusion of evidential pairwise markov chains hidden markov models have been extended in many directions leading to pairwise markov models triplet markov models or discriminative random fields all of which have been successfully applied in many fields covering signal and image processing the dempster shafer theory of evidence has also shown its interest in a wide range of situations involving reasoning under uncertainty and or information fusion there are however only few works dealing with both of these modeling tools simultaneously the aim of this paper which falls under this category of works is to propose a general evidential markov model offering wide modeling and processing possibilities regarding information imprecision sensor unreliability and data fusion the main interest of the proposed model relies in the possibility of achieving easily the dempster shafer fusion without destroying the markovianity

unified representation of sets of heterogeneous markov transition matrices markov chains are very efficient models and have been extensively applied in a wide range of fields covering queuing theory signal processing performance evaluation time series and finance for discrete finite first order markov chains which are among the most used models of this family the transition matrix can be seen as the model parameter since it encompasses the set of probabilities governing the system state estimating such a matrix is however not an easy task due to possible opposing expert reports or variability of conditions under which the estimation process is carried out in this paper we propose an original approach to infer a consensus transition matrix defined in accordance with the theory of evidence from a family of data samples or transition matrices to validate our method experiments are conducted on nonstationary label images and daily rainfall data the obtained results confirm the interest of the proposed evidential modeling with respect to the standard bayesian one

a fixed point approach to the stability of isometries in this paper we apply a fixed point theorem to the proof of hyers ulam rassias stability property for isometries from a normed space into a banach space in which the parallelogram law holds

isosceles orthogonality preserving property and its stability for real normed spaces we consider the class of linear operators preserving approximately the relation of isosceles orthogonality we show some general properties of such mappings next we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one in regard to this we generalize some results obtained earlier for inner product spaces with standard orthogonality relation

operator method for coupled anharmonic oscillators two coupled anharmonic oscillators are considered as a model for a nonperturbative description of the correlation and nonadiabatic effects which are typical for many dimensional quantum systems the eigenvalues and eigenfunctions for this model are found by means of the operator method modified for the case of degenerate solutions of the schr dinger equation it is shown that the zeroth approximation of the method allows one to find the analytical and uniformly suitable approximation for the energy levels and their splitting in the entire range of hamiltonian parameters and quantum numbers numerical calculations demonstrate the convergence of the successive approximations even for quasistationary states of the system the results are of interest for applied problems of spectroscopy and solid state physics

stability of the preservation of the equality of distance abstract we study the stability problem for mappings satisfying the equation f x y f x f y as a result we obtain the stability of functions preserving the equality of distance that is satisfying for all x y u v the condition x y u v f x f y f u f v

coarse version of the banach stone theorem abstract we show that if there exists a lipschitz homeomorphism t between the nets in the banach spaces c x and c y of continuous real valued functions on compact spaces x and y then the spaces x and y are homeomorphic provided l t l t 1 6 5 by l t and l t 1 we denote the lipschitz constants of the maps t and t 1 this improves the classical result of jarosz and the recent result of dutrieux and kalton where the constant obtained is 17 16 we also estimate the distance of the map t from the isometry of the spaces c x and c y

generalized fourier series for non linear systems abstract a simple generalization of the usual fourier series in which the generalized exponential and circular functions are used is presented several examples that correspond to the most usual series are given

on weak stability of e isometries on wedges and its applications abstract in this paper we study weak stability properties of an e isometry defined on a wedge w of a banach space x instead of the whole space x as a result we show that if f w y is an e isometry with f 0 0 for some banach space y then there exists a w compact absolutely convex set b b x satisfying that a p x sup x b x x x for all x w w and b for every x b there is b y so that f x x x 2 e for all x w this is a generalization of a recent result so called a universal theorem for stability of e isometries but the proof is more technical as its application we prove that if the e isometry f is defined on the positive cone w of a c k space or an abstract m space with a strong unit in particular l and l for a finite measure then we can choose the set b to be b x the closed unit ball of the dual x and further show that x is w to w continuously isometric to a subspace of y

nonsurjective nearisometries of banach spaces abstract we obtain sharp approximation results for into nearisometries between l p spaces and nearisometries into a hilbert space our main theorem is the optimal approximation result for nearsurjective nearisometries between general banach spaces

on approximate solutions of the linear functional equation of higher order abstract we show that under some assumptions every approximate solution of the linear functional equation of higher order in single variable generates a solution of the equation that is close to it we also give a description of a procedure that yields such a solution estimate the distance between those approximate and exact solutions to the equation and discuss the problem of uniqueness moreover as a consequence we obtain some results concerning the hyers ulam stability of the equation

symmetry transformations in indefinite metric spaces a generalization of wigner s theorem we formulate and prove a generalization to indefinite metric spaces of uhlhorn s version of wigner s theorem

on the αx2 βx4 interaction by applying the laplace transform method the authors obtain a class of exact solutions for the schrodinger equation with the potential v x alpha x2 beta x4 x 0 alpha or 0 beta 0 the corresponding energy spectrum being continuous a particular feature of these eigen solutions is that they may not be accessible to perturbation theory the properties of the solutions obtained are investigated and their relation to the case where x or 0 is included is discussed

two level system in a one mode quantum field numerical solution on the basis of the operator method accurate eigenvalues and eigenfunctions of a two level system interacting with a one mode quantum field are calculated numerically a special iteration procedure based on the operator method permits one to consider the solution within a wide range of the hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues characteristic features of the model are considered such as the level intersections the population of the field states and the chaotization in the system through the doubling of the frequencies

asymptotic properties of isometries abstract in this paper we prove two theorems on the local stability of isometries in connection with e p isometries these theorems reveal that a large class of e p isometries defined on various restricted domains are stable

fixed points of a mapping and hyers ulam stability abstract we show that many general results on hyers ulam stability of some functional equations in a single variable follow immediately from a simple fixed point theorem the theorem is formulated for self maps of some subsets of the space of functions from a nonempty set into the set of reals we also give some applications of that theorem e g in investigations of solutions of some difference equations and functional inequalities

rayleigh schrodinger perturbation theory with a strong perturbation anharmonic oscillators the bound state solutions of schrodinger s equation for the anharmonic oscillator potentials v x2 lambda x2k k 2 3 have been investigated using elementary techniques of low order variational perturbation theory for the quartic oscillator k 2 a scaled harmonic potential provides a remarkably accurate model for all lambda although this model is slightly less satisfactory for higher order anharmonicities k or 3 the perturbation procedures remain effective and can be applied successfully provided that higher order terms are calculated

on unitary representability of topological groups we prove that the additive group e k e of an mathcal l infty banach space e with the topology k e of uniform convergence on compact subsets of e is topologically isomorphic to a subgroup of the unitary group of some hilbert space is unitarily representable this is the same as proving that the topological group e k e is uniformly homeomorphic to a subset of ell 2 kappa for some as an immediate consequence preduals of commutative von neumann algebras or duals of commutative c algebras are unitarily representable in the topology of uniform convergence on compact subsets the unitary representability of free locally convex spaces and thus of free abelian topological groups on compact spaces follows as well the above facts cannot be extended to noncommutative von neumann algebras or general schwartz spaces

stability of e isometric embeddings into banach spaces of continuous functions abstract let x be a banach space t be a compact hausdorff space and c t be the real banach space of all continuous functions on t endowed with the supremum norm we show that if there exists a standard e isometric embedding f x c t then there are nonempty closed subset s t and a linearly isometric embedding g x span f x s c s defined as g u lim n f 2 n u s 2 n for each u x satisfying that f u s g u 4 e for all u x making use of this result and the well known simultaneous extension operator e c s c t we also prove that the existence of a standard e isometric embedding f x c t implies the existence of a linearly isometric embedding e g x span e f x s c t whenever t is metrizable these conclusions generalize several well known results for any compact hausdorff space resp compact metric space t we further obtain that if g x is complemented in span f x s resp e g x is complemented in span e f x s then the standard e isometric embedding f x c t is stable

on perturbed metric preserved mappings and their stability characterizations abstract let x z be two real banach spaces and e 0 in this paper we first show that if there is a mapping f x z with f 0 0 satisfying f x f y x y e for all x y x then we can define a linear surjective isometry u x z n for some closed subspace n of z by an invariant mean of x we then show that the subspace n plays a crucial role in this paper for example 1 u n x is a w to w continuous surjective isometry and in particular if y span f x is reflexive then the mapping f is stable if and only if n is complemented in y 2 if y is reflexive and n is complemented in y then for any projection p y n the operator t u p satisfies t f x x 4 e for all x x and 3 if in addition y is gateaux smooth and locally uniformly convex then t u p satisfies the sharp estimate t f x x 2 e for all x x we present similar results for such mappings on general banach spaces

on approximate isometries and application to stability of a functional equation abstract in this paper we show a generalized stability of isometries between banach spaces making use of this result we prove the corresponding stability of the functional equation f x y f x f y

maps preserving unitarily invariant norms of jordan product of matrices abstract surjective isometries in a given unitarily invariant norm on n by n complex matrices and with respect to jordan product are classified moreover the similar problem is considered for a much wider class of unitarily invariant functions

optimal choice of a parameter for the operator method of the solution of the schrodinger equation the rate of convergence of the operator method iterative series as a function of the variational parameter is investigated numerically the optimal method of calculation of this parameter is proposed

on linear isometries and e isometries between banach spaces abstract let x y be two banach spaces and f x y be a standard e isometry for some e 0 recently cheng et al showed that if co f x f x y then there exists a surjective linear operator t y x with t 1 such that the following sharp inequality holds t f x x 2 e for all x x making use of the above result we prove the following results suppose that co f x f x y then 1 if there is a linear isometry s x y such that t s id x then t s y t x is a w to w continuous linear projection with t s 1 2 if there exists a w to w continuous linear projection p y t x with p 1 then there is an unique linear isometry s p x y such that t s p id x and p t s p furthermore if p 1 p 2 are two w to w continuous linear projection from y onto t x with p 1 p 2 1 then s p 1 s p 2 we apply these results to provide an alternative proof of a recent theorem which gives an affirmative answer of a question proposed by vestfrid we also unify several known theorems concerning the stability of e isometries

generalized exponential circular and hyperbolic functions for nonlinear wave equations wave functions are presented in the form of generalized exponentials that are solutions of some of the most usual linear and nonlinear wave equations the solutions are given in terms of the elliptic functions of jacobi and presented in a form as similar as possible to the usual circular functions some simple theorems are demonstrated to present the solutions as the simplest possible extension of the usual exponentials

pseudospectral methods for solving an equation of hypergeometric type with a perturbation almost all regular or singular sturm liouville eigenvalue problems in the schrodinger form email protected x v x j x email protected x email protected email protected email protected r j x l 2 email protected email protected for a wide class of potentials v x may be transformed into the form s x y email protected x y q x email protected email protected a b r by means of intelligent transformations on both dependent and independent variables where s x and t x are polynomials of degrees at most 2 and 1 respectively and l is a parameter the last form is closely related to the equation of the hypergeometric type eht in which q x is identically zero it will be called here the equation of hypergeometric type with a perturbation ehtp the function q x may therefore be regarded as a perturbation it is well known that the eht has polynomial solutions of degree n for specific values of the parameter l i e l email protected n 0 n t 12 n 1 s which form a basis for the hilbert space l 2 a b of square integrable functions pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of ehtp and hence the energies e of the original schrodinger equation specimen computations are performed to support the convergence numerically

generalized stability of isometries abstract let x and y be real banach spaces and let e p 0 a mapping f x y is called an e p isometry if f x f y x y e x y p holds for all x y x a pair x y is p stable with respect to isometries if there exists a function 0 0 with lime 0 e 0 such that for every surjective e p isometry f x y there is a surjective isometry u x y satisfying the estimate f x u x e x p x x we show that every pair of banach spaces x y is p stable for 0 p r 2 r 2 is not 1 stable when p 1 a superstability phenomenon occurs for finite dimensional banach spaces

stability of almost surjective e isometries of banach spaces abstract we show that for every pair of banach spaces x and y and for every e isometry f x y with sup y s y lim inf t dist t y f x t 1 2 there exists an affine surjective isometry v y x such that f x v x 2 e for all x x

quantum anharmonic symmetrical oscillators using elliptic functions the authors study in the jwkb approximation the energy levels of the symmetric anharmonic oscillators v x ax2 bx4 for different signs and values of a and b comparisons are made with published results for specific cases and with numerical calculations they given an additional example of exact value to add to the very rare catalogue of known examples

on extension of isometries and approximate isometries between unit spheres article history received 10 april 2008 available online 19 november 2008 submitted by j bastero

the operator method of the approximate description of the quantum and classical systems systematic theory of the operator method om of the approximate solution of the schrodinger equation is considered the eigenfunctions and eigenvalues for various one and many dimensional systems are calculated and the arguments of the convergence of the om perturbation series are discussed it is also shown that om permits one to develop the absolute convergent approximation in the theory of nonlinear oscillations of the classical systems