lipschitz p integral operators and lipschitz p nuclear operators national natural science foundation of china 10526034 10701063 nsf dms 0800061 dms 1068838 dms 1200370

lipschitz operator ideals and the approximation property abstract we establish the basics of the theory of lipschitz operator ideals with the aim of recovering several classes of lipschitz maps related to absolute summability that have been introduced in the literature in the last years as an application we extend the notion and main results on the approximation property for banach spaces to the case of metric spaces

emerging notions of norm attainment for lipschitz maps between banach spaces abstract we classify several notions of norm attaining lipschitz maps which were introduced previously and present the relations among them in order to verify proper inclusions we also analyze some results for the sets of lipschitz maps satisfying each of these properties to be dense or not in lip 0 x y for instance we characterize a banach space y with the radon nikod m property in terms of the denseness of norm attaining lipschitz maps with values in y further we introduce a property called the local directional bishop phelps bollobas property for lipschitz compact maps which extends the one studied previously for scalar valued functions and provide some new positive results

lipschitz p r s integral operators and lipschitz p r s nuclear operators abstract we introduce the notions of strongly lipschitz p r s nuclear operators and strongly lipschitz p r s integral operators we develop a theory of strongly lipschitz p r s nuclear operators and strongly lipschitz p r s integral operators which closely parallels the theory for linear operators their close relationships with other lipschitz operator ideals are studied

a two step explicit p stable method of high phase lag order for linear periodic ivps in this paper we present a nonlinear two step explicit p stable method of fourth algebraic order and 12th phase lag order for solving one dimensional second order linear periodic initial value problems ivps of ordinary differential equations based on a special vector arithmetic with respect to an analytic function the method can be extended to be vector applicable for multi dimensional problems directly some numerical results are reported to illustrate the efficiency of the method

phase diagram in the generalized chiral clock models abstract using the results of the integrable chiral potts model we speculate about the possible phase diagram of the generalized chiral clock model in the integrable model it is found that the chiral fields are almost proportional to one another here we consider a chiral model with all the chiral fields exactly proportional to one another with fixed proportionality constants given by the integrable case this reduces the chiral field variables to just one denoted by the model thus defined includes the generalized clock model or z n model at 0 or 1 2 and the chiral clock models as special cases using the results of many authors on the z n model in the so called symmetric case with vertical and horizontal interactions equal and the behavior of the system at t 0 we can get a pretty good picture of the phase diagram of this chiral model the exact results show that the commonly held belief that the massless phase in the z n model for n 5 is in the same universality class as the classical xy model and the belief that the commensurate incommensurate phase transition is of the pokrovskii talapov type and the incommensurate fluid transition is of kosterlitz thouless type may not be correct as universality may not even hold on these phase boundaries

a remark on the approximation of p compact operators by finite rank operators abstract the banach operator ideal k p of p compact operators was introduced in p d sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 let p 1 and let x be a closed subspace of an l p space we show that x has the approximation property if and only if for every banach space y the linear space f y x of finite rank operators is k p norm dense in k p y x i e x has the k p approximation property

low temperature behaviour of the six state clock model with competing interactions the phase diagram of the six state clock model on a three dimensional lattice with first and second neighbour competing interactions along one direction is analysed using a systematic low temperature expansion carried to all orders where necessary a transfer matrix method simplifies the configurational analysis we find that an infinite number of commensurate phases are stable near a zero temperature multiphase point a richer hierarchy of mixed phases is found to be stable down to zero temperature than in the annni model the system corresponds to an xy model with infinite hexagonal anisotropy and may be relevant to rare earth multilayers

a two step explicit p stable method of high phase lag order for second order ivps this paper presents a two step explicit p stable method of high phase lag order for solving second order ordinary differential equations basing on a special vector operation with respect to a smooth function the method can be extended to be vector applicable in multi dimensional space

unconditional almost squareness and applications to spaces of lipschitz functions abstract we introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of almost square dual banach spaces we also take advantage of this notion to provide some criteria of non duality of some subspaces of scalar as well as vector valued lipschitz functions

two lipschitz operator ideals abstract we introduce and investigate the concept of two lipschitz operator ideal between pointed metric spaces and banach spaces we show the basics of this new theory and we give a procedure to create a two lipschitz operator ideal from a linear operator ideal we apply our result to the ideals of strongly p summing and compact linear operator to obtain their corresponding two lipschitz operator ideal also we establish a natural relation between two lipschitz and bilinear maps and show that the two lipschitz factorable p dominated operators are those which are associated to the well known p semi integral bilinear operators

a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

a series approach to wetting and layering transitions iii the chiral clock model for pt ii see ibid vol 21 p 159 171 1988 in the third of this triad of papers which study interfacial phase transitions using series expansions the authors treat the wetting transition of an interface in the three state chiral clock model previous work has shown that on a simple cubic lattice at low temperatures the interface wets through a large possibly infinite number of layering transitions they extend the low temperature series results to an arbitrary number of nearest neighbours and show that in the mean field limit of infinite coordination number only two layering transitions are seen this is in agreement with numerical solutions of mean field equations hence the mean field approximation in this case does not provide a correct description of interface behaviour in three dimensions

duality for lipschitz p summing operators abstract building upon the ideas of r arens and j eells 1956 1 we introduce the concept of spaces of banach space valued molecules whose duals can be naturally identified with spaces of operators between a metric space and a banach space on these spaces we define analogues of the tensor norms of chevet 1969 3 and saphar 1970 14 whose duals are spaces of lipschitz p summing operators in particular we identify the dual of the space of lipschitz p summing operators from a finite metric space to a banach space answering a question of j farmer and w b johnson 2009 6 and use it to give a new characterization of the non linear concept of lipschitz p summing operator between metric spaces in terms of linear operators between certain banach spaces more generally we define analogues of the norms of j t lapreste 1976 11 whose duals are analogues of a pietsch s p r s summing operators a pietsch 1980 12 as a special case we get a lipschitz version of q p dominated operators

a polynomial hutton theorem with applications abstract extending a classical linear result due to hutton to a nonlinear setting we prove that a continuous homogeneous polynomial between banach spaces can be approximated by finite rank polynomials if and only if its adjoint can be approximated by finite rank linear operators among other consequences we apply this result to generalize a classical result due to aron and schottenloher about the approximation property on spaces of polynomials and a recent result due to caliskan and rueda about the quasi approximation property on projective symmetric tensor products

an extended numerov type method for the numerical solution of the schrodinger equation abstract a new method which is based on the new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases 1 i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wavefunctions tend to zero for x 2 ii the general case for positive and negative eigenvalues and for the well known cases of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new method is more efficient compared with previously derived methods

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

four step methods for y f x y abstract a systematic investigation is undertaken here on the possible practical consequences of the fact that the linear four step methods for y f x y form a family with a certain structure the mentioned property is found to be surprisingly rich in consequences and three of them are of particular importance i drastic reduction in the number of iterations at each step ii increased flexibility with respect to the start of integration and to the modification of the step size during the integration process iii possibility of direct and simple appraisal of the local truncation error

weights of the exponential fitting multistep algorithms for first order odes abstract we describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first order differential equations the main novelties are that i we admit nonequidistant mesh points in the partition and ii the weights are determined on the basis of the exponential functions exp i x i 1 2 3 rather than on the power function set as it is done for the classical weights in this way the method allows computing not only the weights of the well established algorithms but also those of new ones another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms some relevant numerical illustrations are given

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

a p stable exponentially fitted method for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the resonance problem of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind

galois connection between lipschitz and linear operator ideals and minimal lipschitz operator ideals abstract we establish a relation between lipschitz operator ideals and linear operator ideals which fits in the framework of galois connection between lattices we use this relationship to give a criterion which allows us to recognize when a banach lipschitz operator ideal is of composition type or not also we introduce the concept of minimal banach lipschitz operator ideal which have analogous properties to minimal banach operator ideals we characterize minimal banach lipschitz operator ideals which are of composition type and present examples which are not of this class

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

a family of four step exponential fitted methods for the numerical integration of the radial schrodinger equation abstract a family of exponential four step methods is developed for the numerical integration of the one dimensional schrodinger equation the formula developed contains certain free parameters which allows it to be fitted automatically to exponential functions the new methods integrate more exponential functions and are very simple compared with the well known sixth algebraic order runge kutta type methods numerical results indicate that the new methods are much more accurate than other exponentially fitted methods

p stable exponentially fitted methods for the numerical integration of the schrodinger equation a p stable exponentially fitted method is developed in this paper for the numerical integration of the schrodinger equation an application to the bound states problem we solve the radial schrodinger equation in order to find eigenvalues for which the wavefunction and its derivative are continuous and the boundary conditions are satisfied and the resonance problem the point of a resonance is that phase changes rapidly through of the radial schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially fitted methods of the same kind the method can be applied to any problem of physics and chemistry which can be expressed as system of coupled second order differential equations which have oscillatory or periodic solutions this is because it has the property of the p stability i e the interval of periodic stability of the proposed method is equal to 0 which allow is to integrate successful problems with high oscillatory or periodic solution

a two step explicit p stable method for solving second order initial value problems the conceptions of periodicity interval and p stability were introduced by lambert and watson j inst math appl 18 1976 189 in connection with multistep methods for second order initial value problems this paper presents a new two step explicit p stable method of order two for solving initial value problems of second order ordinary differential equations based on a special vector operation the method can be extended to be vector applicable

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

a new variable step method for the numerical integration of the one dimensional schro dinger equation the numerical integration of the one dimensional schrodinger equation is considered a new variable step method is presented within the canonical functions scheme characterized namely by i the replacement of the schrodinger equation by the equivalent integral equation ii the representation of the wavefunction by a superposition of simple integrals iii the definition of taylor series like difference equations at each interval an estimate of the step size h is determined in order to obtain the desired local accuracy for the practical case of a given potential function with polynomial interpolations the step size h is directly given by a simple formula depending on i the potential function in the interval ii the initial values of the wavefunction at the origin of the interval this variable step method is applied to the morse potential function used for the recent raptis and cash method comput phys commun 36 113 1985 for this application the present method presents some advantages over the previous one

explicit numerov type methods with reduced number of stages we present in this paper a new approach for the derivation of hybrid explicit numerov type methods the new methodology does not require the intermediate use of high accuracy inter polatory nodes since we only need the taylor expansion of the internal points as a consequence a sixth order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now numerical results over some well known problems in physics and mechanics indicate the superiority of the new method 2003 elsevier science ltd all rights reserved

a finite difference method for the numerical solution of the schro dinger equation a new approach which is based on a new property of phase lag for computing eigenvalues of schrodinger equations with potentials is developed in this paper we investigate two cases i the specific case in which the potential v x is an even function with respect to x it is assumed also that the wave functions tend to zero for x ii the general case of the morse potential and of the woods saxon or optical potential numerical and theoretical results show that this new approach is more efficient compared to previously derived methods

layering transitions at an interface in the blume capel model an interface in the three dimensional blume capel model is studied using low temperature series and mean field theory by calculating the leading order diagrams in the low temperature series expansion to all orders using a transfer matrix technique the authors are able to show that the interface wets through an infinite sequence of layering transitions which become quasicontinuous as the bulk phase boundary is approached the mean field approximation agrees well with the low temperature series results

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

a series approach to wetting and layering transitions i potts models the authors show that series methods provide a complementary approach to mean field theories in the study of interface wetting and unbinding transitions they consider a q state potts model which contains an interface bound to a surface by a bulk field low temperature and high q series expansions are used to show that as the field tends to zero the interface unbinds from the surface through a sequence of first order layering transitions they discuss the various regions of the phase diagram where the different expansions are expected to be valid and compare their results with those obtained in the mean field approximation

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

approximation properties and schauder decompositions in lipschitz free spaces abstract we prove that the lipschitz free space over a doubling metric space has the bounded approximation property we also show that the lipschitz free spaces over l 1 n or l 1 have monotone finite dimensional schauder decompositions

lipschitz compact operators abstract we introduce the notion of lipschitz compact weakly compact finite rank approximable operators from a pointed metric space x into a banach space e we prove that every strongly lipschitz p nuclear operator is lipschitz compact and every strongly lipschitz p integral operator is lipschitz weakly compact a theory of lipschitz compact weakly compact finite rank operators which closely parallels the theory for linear operators is developed in terms of the lipschitz transpose map of a lipschitz operator we state lipschitz versions of schauder type theorems on the weak compactness of the adjoint of a weakly compact linear operator

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

some properties of lipschitz strongly p summing operators abstract we consider the space of molecules endowed with the transposed version of the chevet saphar norm and we identify its dual space with the space of lipschitz strongly p summing operators we also extend some old results to the category of lipschitz mappings and we give a factorization result of lipschitz p r s summing operators

the p approximation property in terms of density of finite rank operators abstract we characterize the p approximation property p ap introduced by sinha and karn d p sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 in terms of density of finite rank operators in the spaces of p compact and of adjoints of p summable operators as application the p ap of dual banach spaces is characterized via density of finite rank operators in the space of quasi p nuclear operators this relates the p ap to saphar s approximation property ap p as another application the p ap is characterized via a trace condition allowing to define the trace functional on certain subspaces of the space of nuclear operators

the bishop phelps bollobas property for lipschitz maps abstract in this paper we introduce and study a lipschitz version of the bishop phelps bollobas property lip bpb property this property deals with the possibility of making a uniformly simultaneous approximation of a lipschitz map f and a pair of points at which f almost attains its norm by a lipschitz map g and a pair of points such that g strongly attains its norm at the new pair of points we first show that if m is a finite pointed metric space and y is a finite dimensional banach space then the pair m y has the lip bpb property and that both finiteness assumptions are needed next we show that if m is a uniformly gromov concave pointed metric space i e the molecules of m form a set of uniformly strongly exposed points then m y has the lip bpb property for every banach space y we further prove that this is the case for finite concave metric spaces ultrametric spaces and holder metric spaces the extension of the lip bpb property from m r to some banach spaces y and some results for compact lipschitz maps are also discussed

layering transitions in the chiral clock model bethe approximation interface phenomena such as wetting and layering transitions of the three state chiral clock model in three dimensions are studied within the bethe approximation the bethe approximation yields the correct sequence of first order layering transitions this is in contrast to the mean field theory which is known to be qualitatively misleading for low temperatures at higher temperatures the discontinuities of the order parameter at the layering transitions become exceedingly small so that the magnetisation profile appears to vary continuously across the interface for finite values of the chiral field delta the width of the interfacial layer is found to remain finite at the first order bulk transition line critical wetting occurs only for delta 0 i e for the symmetric three state potts model

analysis of the multiphase region in the three state chiral clock model abstract the phase diagram of the axial three state chiral or asymmetric clock model on a d dimensional lattice is analyzed near its multiphase point for d 2 by using systematic low temperature expansions carried to all orders where necessary a matrix method simplifies the configurational analysis two infinitely long sequences of commensurate phases appear each terminating in a triple point at t 0 and having mean wavevectors q 2 j 3 2j 1 a with j 2 3 jmax where at fixed t j max 2 ln 1 2 exp 3j 2k b t so that jmax as t 0

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

criticality of the three state chiral clock model a renormalisation group study within a real space renormalisation group approach the authors study the phase diagram and the universality classes of the three state chiral clock model in a self dual planar hierarchical lattice they find that the chiral field delta is relevant at the pure potts critical point with a crossover exponent phi approximately 0 32 the critical line which separates the ferromagnetic phase from the non ferromagnetic ones is characterised by a multicritical lifshitz point located at k delta approximately 1 452 0 382

the banach ideal of a compact operators and related approximation properties abstract we use the notion of a compact sets determined by an operator ideal a introduced by carl and stephani 1984 to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework for banach operator ideals a we introduce a way to measure the size of a compact sets and use it to give a norm on k a the ideal of a compact operators then we study two types of approximation properties determined by a compact sets we focus our attention on an approximation property which makes use of the norm defined on k a this notion fits the definition of the a approximation property recently introduced by oja 2012 with k a instead of a we exemplify the power of the carl stephani theory and the geometric structure introduced here by appealing to some recent developments on p compactness