canonical variables and analysis on so n 2 the approach of berezin to the quantization of so n 2 via generalized coherent states is considered in detail a family of n commuting observables is found in which the basis for an associated fock type representation space is expressed an interesting feature is that computations can be done by explicit matrix calculations in a particular basis the basic technical tool is the leibniz function the inner product of coherent states

linear dynamical systems coherent state manifolds flows and matrix riccati equation the classical motion and the quantum evolution determined by linear hamiltonians on coherent state manifolds of compact and noncompact hermitian symmetric space structure are considered a matrix riccati equation with opposite signs of the quadratic terms in the compact and noncompact cases determines the classical motion and the quantum evolution the geometric meaning of equations of motion as flow on a complex grassmann manifold and his noncompact dual is emphasized possibilities of generalizing the results to flag manifolds are pointed out

the numerical range and the spectrum of a product of two orthogonal projections the aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum several improvements removing the closure of the numerical range of the operator using a parametrization after its eigenvalues are possible under additional assumptions an estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the friedrichs angle applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed

on the construction of perfect morse functions on compact manifolds of coherent states perfect morse functions on the manifold of coherent states are effectively constructed the case of a compact connected simply connected lie group of symmetry having the same rank as the stationary group of the manifold of coherent states such that the manifold of coherent states is a kahlerian c space is considered it is proved that the set of perfect morse functions is dense in the set of energy functions for linear hamiltonians in the elements of the cartan algebra of the lie algebra of the representation of the group considered it is proved that the maximum number of orthogonal vectors on a coherent vector manifold is equal to the euler poincare characteristic of the manifold

on pairs of projections in a hilbert space we give an algebraic derivation of the canonical form of a generic pair of projections the result is used to determine the spectral shift of a pair of projections and various properties of fredholm pairs

generalized coherent states and berry s phase the quantum dynamics of time dependent hamiltonians of the kind h t u t h 0 u t is studied in the framework of generalized coherent states it is shown that the quantum and the classical dynamics are isomorphic and that the phase of t is simply the classical action in adiabatic approximation it is easy to extract berry s phase from this a discussion of generalized coherent states cs s necessary to deal with excited states is given in particular for su 2 su 1 1 and e 2 their structure is completely characterized showing that the deep connection with geometric quantization extends to these generalized cs s this technique may be employed to estimate higher order correction to the adiabatic approximation

high order finite difference methods for a second order dual phase lagging models of microscale heat transfer two adi solvers are given for 2d and 3d heat conduct equations at microscale they are uniquely solvable it is shown that they are of order 2 in time and order 4 in space numerical results exhibit the efficiency of the solvers in this paper a compact alternating direction implicit adi method which combines the fourth order compact difference for the approximations of the second spatial derivatives and the approximation factorizations of difference operators is firstly presented for solving two dimensional 2d second order dual phase lagging models of microscale heat transfer by the discrete energy method it is shown that it is second order accurate in time and fourth order accurate in space with respect to l2 norms additionally the compact adi method is successfully generalized to solve corresponding three dimensional 3d problem also the convergence result of the solver for 3d case is given rigorously finally numerical examples are carried out to testify the computational efficiency of the algorithms and exhibit the correctness of theoretical results

convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the paper deals with the convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the optimal convergence orders are obtained for the semidiscrete and full discrete backward euler methods respectively both the discrete solutions are proved to be asymptotically stable under the condition that the analytical solution is asymptotically stable

hopf rinow theorem in the sato grassmannian abstract let u 2 h be the banach lie group of unitary operators in the hilbert space h which are hilbert schmidt perturbations of the identity 1 in this paper we study the geometry of the unitary orbit u p u u u 2 h of an infinite projection p in h this orbit coincides with the connected component of p in the hilbert schmidt restricted grassmannian gr res p also known in the literature as the sato grassmannian corresponding to the polarization h p h p h it is known that the components of gr res p are differentiable manifolds here we give a simple proof of the fact that gr res 0 p is a smooth submanifold of the affine hilbert space p b 2 h where b 2 h denotes the space of hilbert schmidt operators of h also we show that gr res 0 p is a homogeneous reductive space we introduce a natural metric which consists in endowing every tangent space with the trace inner product and consider its levi civita connection this connection has been considered before for instance its sectional curvature has been computed we show that the levi civita connection coincides with a linear connection induced by the reductive structure a fact which allows for the easy computation of the geodesic curves we prove that the geodesics of the connection which are of the form t e t z p e t z for z a p co diagonal anti hermitic element of b 2 h have minimal length provided that z 2 note that the condition is given in terms of the usual operator norm a fact which implies that there exist minimal geodesics of arbitrary length also we show that any two points p 1 p 2 gr res 0 p are joined by a minimal geodesic if moreover p 1 p 2 1 the minimal geodesic is unique finally we replace the 2 norm by the k schatten norm k 2 and prove that the geodesics are also minimal for these norms up to a critical value of t which is estimated also in terms of the usual operator norm in the process minimality results in the k norms are also obtained for the group u 2 h

dynamics for a type of general reaction diffusion model abstract in this paper we discuss the following reaction diffusion model which is a general form of many population models u t x t u t x u t x f u t x we study the oscillatory behavior of solutions about the positive equilibrium k of system with neumann boundary conditions sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of k these results improve and complement existing results for system without diffusion moreover when these results are applied to the diffusive nicholson s blowflies model and the diffusive model of hematopoiesis some new results are obtained for the latter

the high order multistep adi solver for two dimensional nonlinear delayed reaction diffusion equations with variable coefficients abstract in this paper a high order compact multistep alternating direction implicit adi difference scheme is constructed and analyzed for two dimensional 2d nonlinear delayed reaction diffusion equations ndres with variable coefficients by the discrete energy method it is presented that it achieves the convergence rate of o 2 h x 4 h y 4 with respect to h 1 and l 2 norms in non constrained temporal grids and is unconditionally stable finally numerical results demonstrate the effectiveness of our adi solver and exhibit the correctness of theoretical results

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

phase portrait of the matrix riccati equation the matrix riccati equation which arises from optimal control and filtering problems is a quadratic differential equation on the space of real symmetric n times n matrices it is closely related via compactification of the phase space to the differential equations on the grassmann manifold and the lagrange grassmann manifold whose flows are generated by the action of one parameter subgroups of the general linear group and of the symplectic group respectively we determine the complete phase portraits of the riccati equations on all three spaces the asymptotic behavior of every solution is described the phase portraits are characterized topologically as well as set theoretically although the riccati equation is not generally a morse smale vector field we are able to show that it possesses suitable generalizations of many of the important properties of morse smale vector fields in particular the riccati equation satisfies a generalized version of the morse inequalities for a morse smale dynamica

a spectral galerkin method for nonlinear delay convection diffusion reaction equations this paper deals with the numerical approximation of a class of nonlinear delay convection diffusion reaction equations in order to derive an efficient numerical scheme to solve the equations we first convert the original equation into an equivalent reaction diffusion problem with an exponential transformation then we propose a fully discrete scheme by combining the crank nicolson method and the legendre spectral galerkin method the analytical and numerical stability criteria are obtained in l 2 norm it is proven under the suitable conditions that the method is convergent of second order in time and of exponential order in space finally several numerical experiments are given to illustrate the computational efficiency and the theoretical results

anti selfadjoint operators as commutators of projections abstract let a be an anti selfadjoint operator on a hilbert space h with a 1 2 we give a sufficient and necessary condition for a to be a commutator of a pair of orthogonal projections and establish the general representation of all pairs p q of orthogonal projections such that a p q q p then we discuss the path components of the set c a p q a p q q p we prove that the action of unitary group u a is transitive in each path component of c a when a is in generic position moreover we characterize the von neumann algebra generated by all projections in c a as an application we obtain that the set of all commutators of pairs of orthogonal projections is connected

on equations of motion on compact hermitian symmetric spaces the classical equations of motion on compact coherent state manifolds that have hermitian symmetric space structure are considered in the case of hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states the classical motion and the quantum evolution are both described by the same first order second degree differential equation the explicit forms of the classical equations of motion as matrix riccati equations on the grassmann manifold ga cn and the coset manifold so 2n u n attached to the hartree fock and respectively hartree fock bogoliubov problem are obtained

liberation of projections abstract we study the liberation process for projections p q p t q u t p u t q where u t is a free unitary brownian motion freely independent from p q its action on the operator valued angle q p t q between the projections induces a flow on the corresponding spectral measures t we prove that the cauchy transform of the measure satisfies a holomorphic pde we develop a theory of subordination for the boundary values of this pde and use it to show that the spectral measure t possesses a piecewise analytic density for any t 0 and any initial projections of trace 1 2 we us this to prove the unification conjecture for free entropy and information in this trace 1 2 setting

surjective isometries on grassmann spaces let h be a complex hilbert space n a given positive integer and let pn h be the set of all projections on h with rank n under the condition dimh 4n we describe the surjective isometries of pn h with respect to the gap metric the metric induced by the operator norm

exceptional lie algebras in terms of fermionic creation annihilation operators from the embedding e8 d4 d4 defining fermionic operators which transform respectively as the vectorial and spinorial representations of d4 the algebra of e8 is constructed in terms of a bilinear in the fermionic operators then by embeddings the adjoint and fundamental representations of the other exceptional algebras are explicitly written

ldg method for reaction diffusion dynamical systems with time delay abstract in this paper we introduce a local discontinuous galerkin method to solve nonlinear reaction diffusion dynamical systems with time delay stability and convergence of the schemes are obtained finally numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method

the compact and crank nicolson adi schemes for two dimensional semilinear multidelay parabolic equations this paper deals with numerical solutions of initial boundary value problems of the two dimensional semilinear multidelay parabolic equations two types of alternating direction implicit adi schemes are suggested the unique solvability convergence and unconditional stability of the schemes are analyzed and hence the corresponding criteria are established especially by using the discrete energy method it is proven that the compact adi scheme can attain second order accuracy in time and fourth order accuracy in space and the crank nicolson adi scheme has second order accuracy in both time and space numerical experiments are performed to verify the efficiency and accuracy of the both schemes

general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections abstract in this paper based on the block operator technique and operator spectral theory general explicit descriptions for intertwining operators and direct rotations of two orthogonal projections are established as a consequence kato s result kato 1996 14 is improved so are j avron r seiler and b simon s theorem 2 3 avron et al 1994 6 and c davis w m kahan s result davis and kahan 1970 11

instanton like solutions in chiral models abstract general two dimensional euclidean chiral models of field theory are considered in detail it is shown that in the case when the field takes its values in an arbitrary kahler manifold the duality equations reduce to the cauchy riemann equations on this manifold for homogeneous manifolds the solutions of these equations do exist and are given by rational functions

angles between infinite dimensional subspaces with applications to the rayleigh ritz and alternating projectors methods we define angles for infinite dimensional subspaces of hilbert spaces inspired by the work of e j hannan 1961 1962 the angles of dixmier and friedrichs and the gaps are characterized we establish connections between the angles corresponding to orthogonal complements the sensitivity of angles with respect to subspaces is estimated we show that the squared cosines of the angles from one subspace to another can be interpreted as ritz values in the rayleigh ritz method the hausdorff distance between the ritz values corresponding to different trial subspaces is shown to be bounded by a constant times the gap between the subspaces we prove a similar eigenvalue perturbation bound that involves the gap squared an ultimate acceleration of the classical alternating projectors method is proposed its convergence rate is estimated in terms of the angles we illustrate the acceleration for a domain decomposition method with a small overlap for the 1d diffusion equation

unitaries permuting two orthogonal projections let p and q be two orthogonal projections on a separable hilbert space h wang du and dou proved that there exists a unitary u with upu 1 q uqu 1 p if and only if dim ker p ker 1 q dim ker q ker 1 p both may be infinite we provide a new proof using the supersymmetric machinery of avron seiler and simon

high order compact adi methods for parabolic equations in this paper we develop a sixth order compact scheme coupled with alternating direction implicit adi methods and apply it to parabolic equations in both 2 d and 3 d unconditional stability is proved for linear diffusion problems with periodic boundary conditions numerical examples supporting our theoretical analysis are provided

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

the index of a pair of projections we discuss pairs of self adjoint projections with p q j 2n 1 the trace ideal and prove that for m n tr p q 2m 1 tr p q 2n 1 dim ker q ran p dim ker p ran q is an integer we also prove that there exists a unitary v interchanging p and q if and only if this integer is 0

homogeneous kahler manifolds paving the way towards new supersymmetric sigma models homogeneous kahler manifolds give rise to a broad class of supersymmetric sigma models containing as a rather special subclass the more familiar supersymmetric sigma models based on hermitian symmetric spaces in this article all homogeneous kahler manifolds with semisimple symmetry groupg are constructed and are classified in terms of dynkin diagrams explicit expressions for the complex structure and the kahler structure are given in terms of the lie algebra g ofg it is shown that for compactg one can always find an einstein kahler structure which is unique up to a constant multiple and for which the kahler potential takes a simple form