double phase integral approximations a systematic simplification technique for wave equations with cutoffs and resonances the time independent wave equation d2 dz2 q2 z 0 where q2 z may have arbitrary order zeros and poles on or close to the real axis is transformed to a simpler wave equation of similar properties model approximate transformations leading from the original wave equation to the model are simply related to froman s higher order phase integrals but are nevertheless well defined at the pertinent zeros and poles of q2 z

energy eigenvalues for an arbitrary potential well with n minima for a general smooth one dimensional potential well with n minima containing n 1 barrier humps of arbitrary shape a phase integral quantization condition including terms is obtained from which the energy eigenvalues can be calculated with an accuracy relative to the chosen order of the froman phase integral approximation the quantization equation is derived from a condition for total transmission through a general system of potential barriers of arbitrary shape by means of a limiting procedure in which the two outermost barriers are made to grow large in a particular way finally the quantization condition is expressed by means of a concise recursion formula as an illustration the special case n 2 is treated in some detail for n 2 the quantization condition turns out to be identical to a previously well known phase integral quantization condition for the double minimum potential the accuracy of which has been examined in several papers by other authors their results concerning a symmetri

on the construction of infinitely many mutually noncongruent examples of minimal imbeddings of s 2n 1 into cp n n geqslant 2

study the stability of solutions of functional differential equations via fixed points in this paper we study the stability properties of solutions of a class of functional differential equations with variable delay by using the fixed point theory under an exponentially weighted metric we obtain some interesting sufficient conditions ensuring that the zero solution of the equations is stable and asymptotically stable

fixed points stability and exact linearization abstract we study the scalar equation x f t x x x b t g x t l 0 by means of contraction mappings conditions are obtained to ensure that each solution x t x t 0 0 as t the conditions allow f to grow as large as t but not as large as t 2 this is parallel to the classical result of smith quart j math oxford ser 12 2 1961 123 for the linear equation without a delay

phase integral approximation for coupled ordinary differential equations of the schrodinger type four generalizations of the phase integral approximation pia to sets of ordinary differential equations of schrodinger type uj x k 1nrjk x uk x 0 j 1 2 n are described the recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix r x rjk x either hermitian or non hermitian for hermitian and negative definite r x matrices a wronskian conserving pia theory is formulated which generalizes fulling s current conserving theory pertinent to positive definite r x matrices the idea of a modification of the pia which is well known for one equation u x r x u x 0 is generalized to sets a simplification of wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections if the pia is generated by a nondegenerate eigenvalue of the r x matrix the eliminated integration is the only one present in that case the simplified theory becomes fully algorit

transmission through cutoffs and resonances in the double phase integral approximation using the double phase integral approximation technique developed earlier for the wave equation d2 dz2 q2 z 0 we derive analytical formulas for the reflection r transmission t and absorption a coefficients they are valid to arbitrary order in the expansion parameter for functions q2 z having either two cutoffs or one cutoff and one resonance for two examples of this type the formulas for r t and a are checked against numerical results using approximations up to fifth order

the eigenvalues of the bounded λx2m oscillators the even and odd parity eigenvalues of the bounded pure x2m oscillators are obtained as the roots of two equations derived explicitly

on the stability of a second order retarded differential equation in this paper we study a second order retarded differential equation with variable delay applying the fixed point theory we obtain a necessary and sufficient condition ensuring that the zero solution of the equation is asymptotically stable our results improve and generalize the known ones in this direction

on the construction of codimension two minimal immersions of exotic spheres into euclidean spheres it is nowadays a well known fundamental fact in differential topology that there exist many smooth manifolds which are homeomorphic to but non diffeomorphic to spheres such manifolds were christened exotic spheres by their founder milnor 223 the structures of those exotic spheres which bound some parallelizable manifolds were systematically analyzed by the techniques of surgery in 20 for example it is not difficult to show that such exotic spheres can be embedded as codimension two submanifolds of the unit spheres in differential geometry and geometric variational theory the study of closed minimal submanifolds of the euclidean n sphere s 1 is directly related to that of the local structure of singularities of minimal subvarieties in the general riemannian setting therefore closed minimal submanifolds of s 1 are not only interesting nice geometric objects by themselves but are also of general theoretical importance in the study of geometric measure theory 16 18 24 among various problems on the closed minimal submanifolds of euclidean spheres the following two problems naturally distinguish themselves as especially interesting namely

fixed points and stability in differential equations with variable delays abstract in this paper we consider a linear scalar differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions an asymptotic stability theorem with a necessary and sufficient condition is proved

on the construction of infinitely many congruence classes of imbedded closed minimal hypersurfaces in s n 1 for all n geq 3

efficient integration of the one dimensional time independent wave equation for bound states and for wave propagation the wave equation is solved with the phase integral approximation in the asymptotic regions and integrated numerically elsewhere in this application the phase integral approximation is relatively simple in higher orders and its accuracy can be precisely controlled with the derived error estimates for the bound states the analytical formulas defining the termination error in the wave function are derived for the wave propagation an efficient formalism is given for calculating the reflection transmission and absorption coefficients

alternative phase integral approximations phase integral approximations of a new kind are obtained they are closely related to the phase integral approximations generated from an unspecified base function q z due to professor n froman and professor p o froman 1991 in the search for an optimal base function a sequence of successively improved base functions qn z are derived n 0 1 2 together with a concise recurrence formula for these functions that is suitable for computer calculations the function qn z is an approximate solution to the nonlinear q equation equivalent to the schrodinger equation the proposed approximation which is immediately obtained from qn z is expected to be somewhat more accurate for n or 2 than the corresponding phase integral approximation of order 2n 1 obtained from the function qn z given by qn z q z sigma n 0ny2n z a detailed comparison is made between qn z and qn z for n 0 1 2 and 3 the proposed approximations have the correct form required to combine them with the f matrix technique for solving connection problems

new stability conditions for mixed linear levin nohel integro differential equations for the mixed levin nohel integro differential equation we obtain new necessary and sufficient conditions of asymptotic stability these results improve those obtained by becker and burton stability fixed points and inverse of delays proc r soc edinburgh sect a 136 245 275 2006 10 1017 s0308210500004546 and jin and luo stability of an integro differential equation comput math appl 57 7 1080 1088 2009 10 1016 j camwa 2009 01 006 when b t 0 and supplement the 32 stability theorem when a t s 0 in addition the case of the equations with several delays is discussed as well

detailed behavior of the phase integral approximations at zeros and singularities of the square of the base function approximate solutions to the one dimensional time independent wave equation called the phase integral approximations are analyzed in the vicinity of characteristic points the approximations are of arbitrary order and are generated from an unspecified base function the general theory is illustrated by examples involving the power and or the exponential behavior of the square of the base function in these cases simple estimates are derived for the integrals which define the accuracy of the phase integral approximation and the optimum approximation order is determined

the energy levels and the corresponding normalized wave functions for a model of a compressed atom in the model of a compressed atom or ion considered in the present paper the boundary condition associated with the corresponding uncompressed atom i e the condition that the radial wave function must vanish at r is replaced by the boundary condition that the radial wave function must have a node at the finite distance r a the treatment of the problem of obtaining the energy shift due to the compression is based on the phase integral method developed by froman and froman an essential feature of which is that one can use exact formulas in the calculations and make all approximations in the final stage the treatment of the problem of obtaining the relative change of the wave function due to the compression is based on the rigorous evaluation of the normalization integral developed by furry phys rev 71 360 1947 and yngve j math phys 13 324 1972 in which one also uses exact formulas in the calculations and makes all approximations in the final stage since compression of an atom giv

transmission through a system of potential barriers i transmission coefficient the transmission coefficient for a one dimensional system of n potential barriers of general shapes unequal or equal is expressed exactly by means of the phase integral method of n froman and p o froman in terms of quantities characterizing the separate barriers and wells of the system the exact formula being of interest for further rigorous studies of transmission properties can for evaluation readily be converted into a useful approximate formula by insertion of available phase integral expressions of an arbitrary order for the characteristic quantities mentioned above error estimates are given the sub and superbarrier cases are treated in a unified way some key facts about the phase integral method are given in appendix a

transmission through a system of potential barriers ii necessary condition for complete transparency a maximum transmission problem a smooth one dimensional system of n potential barriers of arbitrary shapes unequal or equal is considered a general necessary condition for complete transparency is obtained that can be understood as a constraint on the reflection coefficients pertaining to the single barriers of the system a maximum transmission problem of a general kind is solved and the solution is used to give a physical interpretation of the necessary condition the sub and superbarrier cases are treated in a unified way the exact final formulas can readily be converted into accurate approximate ones by insertion of available phase integral expressions of an arbitrary order for certain characteristic quantities appearing in the formulas

a new method of dynamical stability i e fractional generalized hamiltonian method and its applications fractional generalized hamiltonian method of dynamical stability is presented six criterions for the new method of dynamical stability are given the new method s applications are studied five kinds of actual fractional dynamical models are constructed dynamical stability of the five actual fractional dynamical models is explored in the paper we present fractional generalized hamiltonian method of dynamical stability in terms of riesz riemann liouville derivative and study its applications for an actual dynamical system the fractional generalized hamiltonian method of constructing a fractional dynamical model is given and then the six criterions for fractional generalized hamiltonian method of dynamical stability are presented as applications by using the fractional generalized hamiltonian method we construct five kinds of actual fractional dynamical models which include a fractional euler poinsot model of rigid body that rotates with respect to a fixed point a fractional hojman urrutia model a fractional lorentz dirac model a fractional whittaker model and a fractional robbins lorenz model and we explore the dynamical stability of these models respectively this work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering

existence and uniqueness for a class of cauchy problems with characteristic initial manifolds on donne un theoreme d existence unicite pour une grande classe d equations d ordre 2 quasilineaires a donnees de cauchy sur une variete qui est caracteristique au sens ou l equation est singuliere d ordre 1 le long de la variete initiale