actions of symbolic dynamical systems on c algebras ii simplicity of c symbolic crossed products and some examples a c symbolic dynamical system mathcal a rho sigma consists of a unital c algebra mathcal a and a finite family rho alpha alpha in sigma of endomorphisms of mathcal a indexed by symbols of satisfying some conditions the endomorphisms rho alpha alpha in sigma yield both a subshift and a c algebra of a hilbert c bimodule the obtained c algebra is regarded as a crossed product of mathcal a by the subshift we will study simplicity condition of these c algebras some examples such as irrational rotation cuntz krieger algebras will be studied

compact lie group action with the continuous rokhlin property abstract in this paper we study the continuous rokhlin property of c dynamical systems using techniques of equivariant kk theory and quantum group theory in particular we determine the kk equivalence class and give a classification of kirchberg g algebras when the g is a compact lie group with the hodgkin condition

trace spaces of counterexamples to naimark s problem abstract a counterexample to naimark s problem is a c algebra that is not isomorphic to the algebra of compact operators on some hilbert space yet still has only one irreducible representation up to unitary equivalence it is well known that such algebras must be nonseparable and in 2004 akemann and weaver used the diamond principle a set theoretic axiom independent from zfc to give the first counterexamples for any such counterexample a the unitary group u a acts transitively on the pure states which are the extreme points of the state space s a it is conceivable that this implies as happens for finite dimensional simplexes that the action of u a on s a has at most one fixed point i e a has at most one trace we give a strong negative answer here assuming diamond in particular we adapt the akemann weaver construction to show that the trace space of a counterexample to naimark s problem can be affinely homeomorphic to any metrizable choquet simplex and can also be nonseparable

consistency of a counterexample to naimark s problem we construct a c algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any hilbert space this answers an old question of naimark our construction uses a combinatorial statement called the diamond principle which is known to be consistent with but not provable from the standard axioms of set theory assuming that these axioms are consistent we prove that the statement there exists a counterexample to naimark s problem which is generated by elements is undecidable in standard set theory

purely infinite partial crossed products let a g be a partial dynamical system we show that there is a bijective correspondence between g invariant ideals of a and ideals in the partial crossed product a rg provided the action is exact and residually topologically free assuming in addition a technical condition automatic when a is abelian we show that a rg is purely infinite if and only if the positive nonzero elements in a are properly infinite in a rg as an application we verify pure infiniteness of various partial crossed products including realisations of the cuntz algebras on oa on and oz as partial crossed products

closed form expressions for the finite approximations of first and higher derivatives based on taylor series numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series in this paper we have presented closed form expressions of these approximations of arbitrary order for first and higher derivatives a comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses the comparison reveals that the central difference approximations can be used as digital differentiators because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator it is also observed that central difference approximations are in fact the same as maximally flat digital differentiators in the appendix a computer program written in mathematica is presented which can give the approximation of any order to the derivative of a function at a certain mesh point

spacelike convex surfaces with prescribed curvature in 2 1 minkowski space abstract we prove existence and uniqueness of solutions to the minkowski problem in any domain of dependence d in 2 1 dimensional minkowski space provided d is contained in the future cone over a point namely it is possible to find a smooth convex cauchy surface with prescribed curvature function on the image of the gauss map this is related to solutions of the monge ampere equation det d 2 u z 1 z 1 z 2 2 on the unit disc with the boundary condition u d for a smooth positive function and a bounded lower semicontinuous function we then prove that a domain of dependence d contains a convex cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function is in the zygmund class moreover in this case the surface of constant curvature k contained in d has bounded principal curvatures for every k 0 in this way we get a full classification of isometric immersions of the hyperbolic plane in minkowski space with bounded shape operator in terms of zygmund functions of d finally we prove that every domain of dependence as in the hypothesis of the minkowski problem is foliated by the surfaces of constant curvature k as k varies in 0

functionally fitted block methods for ordinary differential equations abstract functionally fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions in this paper we propose a new type of functionally fitted block methods for ordinary differential equations the basic theory for the proposed methods is established first we derive two sufficient conditions to ensure the existence of the functionally fitted block methods discuss their equivalence to collocation methods in a special case and independence on integration time for a set of separable basis functions we then obtain some basic characteristics of the methods by taylor series expansions and show that the order of accuracy of the r point functionally fitted block method is at least r for general ordinary differential equations experimental results are conducted to demonstrate the validity of the functionally fitted block methods

classification of homomorphisms and dynamical systems let a be a unital simple c algebra with tracial rank zero and let x be a compact metric space suppose that h 1 h 2 c x a are two unital monomorphisms we show that h 1 and h 2 are approximately unitarily equivalent if and only if h 1 h 2 in kl c x a and o h 1 f o h 2 f for every f c x and every trace r of a inspired by a theorem of tomiyama we introduce a notion of approximate conjugacy for minimal dynamical systems let x be a compact metric space and let x x be two minimal homeomorphisms using the above mentioned result we show that two dynamical systems are approximately conjugate in that sense if and only if a k theoretical condition is satisfied in the case that x is the cantor set this notion coincides with the strong orbit equivalence of giordano putnam and skau and the k theoretical condition is equivalent to saying that the associate crossed product c algebras are isomorphic another application of the above mentioned result is given for c dynamical systems related to a problem of kishimoto let a be a unital simple ah algebra with no dimension growth and with real rank zero and let a aut a we prove that if r fixes a large subgroup of k 0 a and has the tracial rokhlin property then ax z is again a unital simple ah algebra with no dimension growth and with real rank zero

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

a fourth algebraic order trigonometrically fitted predictor corrector scheme for ivps with oscillating solutions a scheme of trigonometrically fitted predictor corrector p c adams bashforth moulton methods is constructed in this paper our new p c method is based on the third order adams bashforth scheme as predictor and on the fourth order adams moulton scheme as corrector we tested the efficiency of our newly developed scheme against well known methods with excellent results the numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions

entire surfaces of constant curvature in minkowski 3 space this paper concerns the global theory of properly embedded spacelike surfaces in three dimensional minkowski space in relation to their gaussian curvature we prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant gaussian curvature this is a consequence of our classification of surfaces with bounded prescribed gaussian curvature sometimes called the minkowski problem for which partial results were obtained by li guan jian schoen and bonsante seppi some applications to minimal lagrangian self maps of the hyperbolic plane are obtained

simple nuclear c algebras not isomorphic to their opposites abstract we show that it is consistent with zermelo fraenkel set theory with the axiom of choice zfc that there is a simple nuclear nonseparable c c algebra which is not isomorphic to its opposite algebra we can furthermore guarantee that this example is an inductive limit of unital copies of the cuntz algebra o 2 o2 or of the canonical anticommutation relations car algebra

a parallel approach to the modified numerov like eigenvalue determination for sturm liouville problems abstract a modified numerov like eigenvalue algorithm previously introduced is parallelized an inplementation of this algorithm on a helios based parallel processing transputer system is discussed time savings with respect to a sequential approach are commented

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

homographic solutions of the curved 3 body problem abstract in the 2 dimensional curved 3 body problem we prove the existence of lagrangian and eulerian homographic orbits and provide their complete classification in the case of equal masses we also show that the only non homothetic hyperbolic eulerian solutions are the hyperbolic eulerian relative equilibria a result that proves their instability

purely infinite simple reduced c algebras of one relator separated graphs abstract given a separated graph e c there are two different c algebras associated to it the full graph c algebra c e c and the reduced one c red e c for a large class of separated graphs e c we prove that c red e c either is purely infinite simple or admits a faithful tracial state the main tool we use to show pure infiniteness of reduced graph c algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to dykema

new finite difference formulas for numerical differentiation conventional numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series and closed form expressions of these finite difference formulas have already been presented in this paper we present new finite difference formulas which are more accurate than the available ones especially for the oscillating functions having frequency components near the nyquist frequency closed form expressions of the new formulas are given for arbitrary order a comparison of the previously available three types of approximations is given with the presented formulas a computer program written in mathematica based on new formulas is given in the appendix for numerical differentiation of a function at a specified mesh point

equivariant kirchberg phillips type absorption for amenable group actions we show an equivariant kirchberg phillips type absorption theorem for pointwise outer actions of discrete amenable groups on kirchberg algebras with respect to natural model actions on the cuntz algebras mathcal o infty and mathcal o 2 this generalizes results known for finite groups and poly mathbb z groups the model actions are shown to be determined up to strong cocycle conjugacy by natural abstract properties which are verified for some examples of actions arising from tensorial shifts we also show the following homotopy rigidity result which may be understood as a precursor to a general kirchberg phillips type classification theory if two outer actions of an amenable group on a unital kirchberg algebra are equivariantly homotopy equivalent then they are conjugate this marks the first c dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups

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

two sided eigenvalue bounds for the spherically symmetric states of the schro dinger equation the eigenvalues of the radial schr6dinger equation are calculated very accurately by obtaining exact upper and lower bounds by truncating the usual unbounded domain 0 cx of the system to a finite interval of the form 0 g two auxiliary eigenvalue problems are defined it is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required moreover each auxiliary eigenvalue problem gives rise to an orthonormal set involving bessel functions the matrix representation of the hamiltonian is therefore derived by expanding the wave function into a fourier bessel series numerical results for single and doublewell polynomial oscillators as well as gaussian type non polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy whenever the boundary parameter g is in the neighborhood of some critical value denoted by ecr 1998 elsevier science b v all rights reserved

a simple separable exact c algebra not anti isomorphic to itself we give an example of an exact stably finite simple separable c algebra d which is not isomorphic to its opposite algebra moreover d has the following additional properties it is stably finite approximately divisible has real rank zero and stable rank one has a unique tracial state and the order on projections over d is determined by traces it also absorbs the jiang su algebra z and in fact absorbs the 3 uhf algebra we can also explicitly compute the k theory of d namely k 0 d cong mathbb z tfrac 1 3 with the standard order and k1 d 0 as well as the cuntz semigroup of d namely w d cong mathbb z tfrac 1 3 sqcup 0 infty

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

entire spacelike hypersurfaces of constant gaus curvature in minkowski space we prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed gaus curvature in minkowski space the proof is based on barrier constructions and local a priori estimates

pure infiniteness and ideal structure of c algebras associated to fell bundles abstract we investigate structural properties of the reduced cross sectional algebra c r b of a fell bundle b over a discrete group g conditions allowing one to determine the ideal structure of c r b are studied notions of aperiodicity paradoxicality and b infiniteness for the fell bundle b are introduced and investigated by themselves and in relation to the partial dynamical system dual to b several criteria of pure infiniteness of c r b are given it is shown that they generalize and unify corresponding results obtained in the context of crossed products by the following duos laca spielberg 34 jolissaint robertson 21 sierakowski rordam 47 giordano sierakowski 18 and ortega pardo 39 for exact separable fell bundles satisfying the residual intersection property primitive ideal space of c r b is determined the results of the paper are shown to be optimal when applied to graph c algebras applications to a class of exel larsen crossed products are presented

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

bimodules in crossed products of von neumann algebras abstract in this paper we study bimodules over a von neumann algebra m in the context of an inclusion m m g where g is a discrete group acting on a factor m by outer automorphisms we characterize the m bimodules x m g that are closed in the bures topology in terms of the subsets of g we show that this characterization also holds for w closed bimodules when g has the approximation property ap a class of groups that includes all amenable and weakly amenable ones as an application we prove a version of mercer s extension theorem for certain w continuous surjective isometric maps on x

z2 actions on kirchberg algebras abstract we classify a large class of z 2 actions on the kirchberg algebras employing the kasparov group kk 1 as the space of classification invariants

reduced free products of finite dimensional c algebras abstract we find a necessary and sufficient conditions for the simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional c algebras with specified traces on them

outer automorphisms and reduced crossed products of simple c algebras every outer automorphism of a separable simplec algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism the reduced crossed product of a simplec algebra by a discrete group of outer automorphisms is shown to be simple

a new perturbative technique for solving integro partial differential equations integro partial differential equations occur in many contexts in mathematical physics typical examples include time dependent diffusion equations containing a parameter e g the temperature that depends on integrals of the unknown distribution function the standard approach to solving the resulting nonlinear partial differential equation involves the use of predictor corrector algorithms which often require many iterations to achieve an acceptable level of convergence in this paper we present an alternative procedure that allows us to separate a family of integro partial differential equations into two related problems namely i a perturbation equation for the temperature and ii a linear partial differential equation for the distribution function we demonstrate that the variation of the temperature can be determined by solving the perturbation equation before solving for the distribution function convergent results for the temperature are obtained by recasting the divergent perturbation expa

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

phase fitted numerov type methods we present a hybrid two step method for the solution of second order initial value problem it costs only five function evaluations per step and attains sixth algebraic order the coefficients depend on the frequency of the problem and satisfy zero dissipation and phase fitted properties so the method is well suited for facing problems with oscillatory solutions numerical tests justify our effort

isomorphism and morita equivalence classes for crossed products of irrational rotation algebras by cyclic subgroups of sl2 z abstract let be irrational numbers and a b be matrices in s l 2 z of infinite order we compute the k theory of the crossed product a a z and show that a a z and a b z are isomorphic if and only if mod z and i a 1 is matrix equivalent to i b 1 combining this result and an explicit construction of equivariant bimodules we show that a a z and a b z are morita equivalent if and only if and are in the same g l 2 z orbit and i a 1 is matrix equivalent to i b 1 finally we determine the morita equivalence class of a f for any finite subgroup f of s l 2 z

on the morita equivalence of tensor algebras our objective is two fold first we want to develop a notion of morita equivalence for c correspondences that guarantees that if two c correspondences e and f are morita equivalent then the tensor algebras of e and f taoeu and taofu are strongly morita equivalent in the sense of 8 the toeplitz algebras toeu and tofu are strongly morita equivalent in the sense of rieffel 32 see 33 also and the cuntz pimsner algebras 28 ooeu and oofu are strongly morita equivalent in the same sense we were inspired by earlier work of curto williams and the first author in 15 and combes 14 these papers investigate conditions implying that the crossed product of two c dynamical systems are strongly morita equivalent given that the coefficient algebras are strongly morita equivalent abadie eilers and exel 1 were similarly inspired and developed a theory of morita equivalence for special c correspondences our analysis extends theirs but our arguments are different see example 2 5 and remarks 3 3 and 3 6 for a comparison of their results and ours second we want to investigate the converse implication that is we study the following question if taoeu and taofu are strongly morita equivalent in the sense of 8 when are e and f morita equivalent in the sense we define for this converse direction we focus on the tensor algebras rather than the toeplitz and cuntz pimsner algebras because very little can be said in the self adjoint setting here we were inspired by arveson s paper 2 in which it is proved that two ergodic measure preserving transformations are conjugate if and only if certain non self adjoint crossed products associated with the transformations are algebraically isomorphic see 3 also his analysis was further extended by k saito and the first author to cover non self adjoint crossed products built from automorphisms of general von neumann algebras they showed that non self adjoint crossed products built from properly outer automorphisms of von neumann algebras are isomorphic if and only if the automorphisms are conjugate see 22 and 23 our principal result in the converse direction theorem 7 2 asserts that if e and f are two aperiodic correspondences and if taoeu is strongly morita equivalent to taofu in the sense of 8 then e and f are morita equivalent as we shall see in 5 the notion of aperiodicity is an exact analogue for correspondences of the concept of free action in ergodic theory free action was a central hypothesis in arveson s

approximate equivalence for representations of c algebras and c dynamical systems abstract approximate equivalence unitary equivalence up to arbitrarily small compact perturbations is studied for representations of c algebras and covariant representations of c dynamical systems approximate equivalence for representations of c algebras is shown to be determined by a trace condition analogous to the trace condition for unitary equivalence of representations of finite groups for c dynamical systems with a compact abelian group acting approximate equivalence is shown to be determined by certain associated representations of the fixed point algebra of the action

the tracial rokhlin property for actions of finite groups on c algebras we define tracial analogs of the rokhlin property for actions of finite groups approximate representability of actions of finite abelian groups and approximate innerness we prove the following four analogs of related nontracial results bullet the crossed product of an infinite dimensional simple separable unital c algebra with tracial rank zero by an action of a finite group with the tracial rokhlin property again has tracial rank zero bullet an outer action of a finite abelian group on an infinite dimensional simple separable unital c algebra has the tracial rokhlin property if and only if its dual is tracially approximately representable and is tracially approximately representable if and only if its dual has the tracial rokhlin property bullet if a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital c algebra has the tracial rokhlin property then it is tracially approximately representable bullet an automorphism of an infinite dimensional simple separable unital c algebra a with tracial rank zero is tracially approximately inner if and only if it is the identity on k 0 a mod infinitesimals

on the structure of twisted group c algebras we first give general structural results for the twisted group algebras c g of a locally compact group g with large abelian subgroups in particular we use a theorem of williams to realise c g as the sections of a c bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras next we use a device of rosenberg to show that when is a discrete subgroup of a solvable lie group g the k groups k c are isomorphic to certain twisted k groups k g of the homogeneous space g and we discuss how the twisting class h 3 g z depends on the cocycle

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

on isometric immersions of the hyperbolic plane into the lorentz minkowski space and the monge ampere equation of a certain type

on a correction of numerov like eigenvalue approximations for sturm liouville problems for computing approximations for the eigenvalues of sturm liouville problems numerov s method and other finite difference methods are often used giving rise to an algebraic eigenvalue problem of which the smallest eigenvalues approximate the fundamental eigenvalue and the first few harmonics of the original problem recently the authors have derived a modified numerov method with step dependent coefficients which integrates exactly the functions 1 x x2 x3 sin kx and cos kx the parameter k is calculated by minimizing the error term associated to the above modified multistep method equation we shall show that this modified numerov method delivers much more accurate eigenvalues than the classical numerov method which is based on a same grid pattern several numerical experiments will be presented

an intrinsic approach in the curved n body problem the negative curvature case abstract we consider the motion of n point particles of positive masses that interact gravitationally on the 2 dimensional hyperbolic sphere which has negative constant gaussian curvature using the stereographic projection we derive the equations of motion of this curved n body problem in the poincare disk where we study the elliptic relative equilibria then we obtain the equations of motion in the poincare upper half plane in order to analyze the hyperbolic and parabolic relative equilibria using techniques of riemannian geometry we characterize each of the above classes of periodic orbits for n 2 and n 3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting

tracially z absorbing c algebras abstract we study a tracial notion of z absorption for simple unital c algebras we show that if a is a c algebra for which this property holds then a has almost unperforated cuntz semigroup and if in addition a is nuclear and separable we show this property is equivalent to having a a z we furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial rokhlin type property

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

semicrossed products of simple c algebras let mathcal a alpha and mathcal b beta be c dynamical systems and assume that mathcal a is a separable simple c algebra and that and are automorphisms then the semicrossed products mathcal a times alpha mathbb z and mathcal b times beta mathbb z are isometrically isomorphic if and only if the dynamical systems mathcal a alpha and mathcal b beta are outer conjugate

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

central sequence c algebras and tensorial absorption of the jiang su algebra we study properties of the central sequence algebra a a 0 of a c algebra a and we present an alternative approach to a recent result of matui and sato they prove that every unital separable simple nuclear c algebra whose trace simplex is nite dimensional tensorially absorbs the jiang su algebra if and only if it has the strict comparison property we extend their result to the case where the extreme boundary of the trace simplex is closed and of nite topological dimension we are also able to relax the assumption on the c algebra of having the strict com parison property to a weaker property that we call local weak comparison namely we prove that a unital separable simple nuclear c algebra whose trace simplex has nite dimensional closed extreme boundary tensorially absorbs the jiang su algebra if and only if it has the local weak comparison property we can also eliminate the nuclearity assumption if instead we assume the si property of matui and sato and moreover that each ii1 factor representation of the c algebra is a mcdu factor

numerical methods for solving radial schro dinger equations abstract an algorithm previously introduced by brown et al 1963 for solving radial schrodinger equations is revisited and implemented in a more accurate way the method is firstly applied to equations where potentials are present which are finite at the origin and which have an asymptotic behaviour v r 0 as r typical examples of potentials belonging to that class are the woods saxon and the morse potential furthermore the method is also used for coulomb like potentials such as the hulthen and the hellmann potential a comparison between the approximated numerical values and other available numerical and exact bound state energies is made