some properties of the discrete hamiltonian method abstract a method for approximating a large class of hamiltonian ordinary equations by area preserving discrete mapping equations is described the time steps can be quite large compared to conventional approaches it is shown that in general the mapping equations are implicit and the conditions under which a unique solution can be found are determined under these conditions newton s method is rapidly convergent numerical examples are studied and the appropriateness of these methods for use in plasma simulations is explored

the ito clifford integral abstract we present a theory of non commutative stochastic integration analogous to the ito theory it is shown that wick products of fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted operatorvalued square integrable integrands for square integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined and a doob meyer decomposition for supermartingales obtained

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

quantum risk sensitive estimation and robustness this paper studies a quantum risk sensitive estimation problem and investigates robustness properties of the filter this is a direct extension to the quantum case of analogous classical results all investigations are based on a discrete approximation model of the quantum system under consideration this allows us to study the problem in a simple mathematical setting we close the paper with some examples that demonstrate the robustness of the risk sensitive estimator

temperature states on loop groups boson fermion correspondence and a determinant identity the boson fermion correspondence for the representation of the car algebra over l2 s1 c defined by the kms state with chemical potential is considered and the non bijectivity shown using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of szego s theorem

two disjoint cycle cover bipancyclicity of balanced hypercubes abstract a bipartite graph g is two disjoint cycle cover r1 r2 bipancyclic if for any even integer l satisfying r1 l r2 there exist two vertex disjoint cycles c1 and c2 in g such that v c 1 l and v c 2 v g l where v g denotes the number of vertices in g in this paper we study the two disjoint cycle cover bipancyclicity of the n dimensional balanced hypercube bhn which is a hypercube variant network and is superior to hypercube due to having a smaller diameter as a consequence we show that bhn is two disjoint cycle cover 4 2 2 n 1 bipancyclic for n 2

on some new low storage implementations of time advancing runge kutta methods in this paper explicit runge kutta rk schemes with minimum storage requirements for systems with very large dimension that arise in the spatial discretization of some partial differential equations are considered a complete study of all four stage fourth order schemes of the minimum storage families of williamson 1980 2 van der houwen 1977 8 and ketcheson 2010 12 that require only two storage locations per variable is carried out it is found that whereas there exist no schemes of this type in the williamson and van der houwen families there are two isolated schemes and a one parameter family of fourth order schemes in four stage ketcheson s family this available parameter is used to obtain the optimal scheme taking into account the 2 norm of the coefficients of the leading error term in addition a new alternative minimum storage family to the s stage ketcheson that depends also on 3 s 3 free parameters is proposed this family contains both the williamson and van der houwen schemes but it is not included in ketcheson s family finally the results of some numerical experiments are presented to show the behavior of fourth order optimal schemes for some nonlinear problems

formal energy of symplectic scheme for hamiltonian systems and its applications ii abstract in this paper transforming the symplecticity of a difference scheme into the closedness of some differential 1 form and using the poincare lemma on closed differential forms the author deduces directly the existence of formal energy as a power series in stepsize of any symplectic scheme for hamiltonian systems theorem 1 and then proves its non conservativity of the original energy remark 1 according to theorem 1 the revertible symplectic schemes are characterized corollary 1 and 2 theorem 2 gives the formulae for calculation of the first three terms of formal energy then a few examples are displayed for the composition of two symplectic schemes the author obtains the valuable formulae of the first three terms of its formal energy theorem 3 finally as an application of theorem 3 two examples are given

two node disjoint paths in balanced hypercubes the balanced hypercube bhn proposed by wu and huang is a variation of the hypercube it has been proved that the balanced hypercube is a node transitive and bipartite graph assume that the nodes are divided into two bipartite node sets x and y u and x are two different nodes in x and v and y are two different nodes in y in this paper we prove that there exist two node disjoint paths p x y and r u v in bhn and v p x y v r u v v bhn where n 1 the hamiltonian laceability of bhn which was proved by xu et al is also obtained from the corollary of our result

path embedding in star graphs abstract the star graph interconnection network has been introduced as an attractive alternative to the hypercube network in this paper we consider the path embedding problem in star graphs assume that n 4 we prove that paths of all even lengths from d x y to n 2 can be embedded between two arbitrary vertices x and y from the same partite set in the n dimensional star graph in addition paths of all odd lengths from d x y to n 1 can be embedded between two arbitrary vertices x and y from different partite sets in the n dimensional star graph except that if x and y are adjacent there is no path of length 3 between them the result is optimal in the sense that paths of all possible lengths are found in star graphs

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

representations of lie superalgebras and generalized boson fermion equivalence in quantum stochastic calculus the boson fermion equivalence scheme of 5 can be generalized ton dimensions withr boson andn r fermion creation and annihilation fields the same stochastic integral prescription replaces then 2 generalized number processes ai j which form representations of the lie algebragl n by processes forming representations of the lie superalgebragl n r

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

the hamiltonian operator associated with some quantum stochastic evolutions we consider the hamiltonian operator associated to the quantum stochastic differential equation introduced by hudson and parthasarathy to describe a quantum mechanical evolution in the presence of a quantum noise we characterize such a hamiltonian in the case of arbitrary multiplicity and bounded coefficients we find an essentially self adjoint restriction of the operator and in particular we provide an explicit construction of a dense set of vectors belonging to its domain

splitting k symplectic methods for non canonical separable hamiltonian problems non canonical hamiltonian systems have k symplectic structures which are preserved by k symplectic numerical integrators there is no universal method to construct k symplectic integrators for arbitrary non canonical hamiltonian systems however in many cases of interest by using splitting we can construct explicit k symplectic methods for separable non canonical systems in this paper we identify situations where splitting k symplectic methods can be constructed comparative numerical experiments in three non canonical hamiltonian problems show that symmetric non symmetric splitting k symplectic methods applied to the non canonical systems are more efficient than the same order gauss methods non symmetric symplectic methods applied to the corresponding canonicalized systems for the non canonical lotka volterra model the splitting algorithms behave better in efficiency and energy conservation than the k symplectic method we construct via generating function technique in our numerical experiments the favorable energy conservation property of the splitting k symplectic methods is apparent

bipanconnectivity of balanced hypercubes the balanced hypercube proposed by wu and huang is a variant of the hypercube network in this paper paths of various lengths are embedded into balanced hypercubes a bipartite graph g is bipanconnected if for two arbitrary nodes x and y of g with distance d x y there exists a path of length l between x and y for every integer l with d x y l v g 1 and l d x y 0 mod 2 we prove that the n dimensional balanced hypercube bh n is bipanconnected for all n 1 this result is stronger than that obtained by xu et al which shows that the balanced hypercube is edge bipancyclic and hamiltonian laceable

edge pancyclicity and hamiltonian laceability of the balanced hypercubes abstract the balanced hypercube bh n is a variant of the hypercube q n huang and wu proved that bh n has better properties than q n with the same number of links and processors in particularly they showed that there exists a cycle of length 2 l in bh n for all l 2 l 2 n in this paper we improve this result by showing that bh n is edge pancyclic which means that for arbitrary edge e there exists a cycle of even length from 4 to 2 2 n containing e in bh n we also show that the balanced hypercubes are hamiltonian laceable

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

conditional diagnosability of alternating group graphs let an be the alternating group of degree n with n 3 set s 1 2 i 1 i 2 3 i n the alternating group graph denoted by agn is defined as the cayley graph on an with respect to s jwo et al networks 23 1993 315 326 introduced alternating group graph agn as an interconnection network topology for computing systems conditional diagnosability a new measure of diagnosability introduced by lai et al ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of agn is 4 for n 4 and 6n 18 for each n 5

an o 3 1 nonlinear sigma model and the ablowitz ladik hierarchy a solvable nonlinear 2d field model is proposed which is shown to be closely connected with the nonlinear schrodinger chain the conservation laws and soliton solutions for the corresponding field equations are obtained

meixner class of non commutative generalized stochastic processes with freely independent values ii the generating function let t be an underlying space with a non atomic measure on it in comm math phys 292 99 129 2009 the meixner class of non commutative generalized stochastic processes with freely independent values omega omega t t in t was characterized through the continuity of the corresponding orthogonal polynomials in this paper we derive a generating function for these orthogonal polynomials the first question we have to answer is what should serve as a generating function for a system of polynomials of infinitely many non commuting variables we construct a class of operator valued functions z z t t in t such that z t commutes with s for any s t in t then a generating function can be understood as g z omega sum n 0 infty int t n p n omega t 1 dots omega t n z t 1 dots z t n sigma dt 1 dots sigma dt n where p n omega t 1 dots omega t n is the kernel of the nth orthogonal polynomial we derive an explicit form of g z which has a resolvent form and resembles the generating function in the classical case albeit it involves integrals of non commuting operators we finally discuss a related problem of the action of the annihilation operators partial t t in t in contrast to the classical case we prove that the operators t related to the free gaussian and poisson processes have a property of globality this result is genuinely infinite dimensional since in one dimension one loses the notion of globality

fault tolerant cycle embedding in alternating group graphs abstract cycle embedding is an important issue in evaluating the efficiency of interconnection networks and is also an extension of the theoretical research on hamiltonicity in this paper we study the fault tolerant hamiltonicity of alternating group graphs which were recently proposed as interconnection topologies for parallel and distributed systems a cycle of length l is said to be an l cycle let f be a set of faulty vertices in an n dimensional alternating group graph ag n and let m be the number of vertices in ag n f a previous result in j m chang j s yang y l wang y cheng panconnectivity fault tolerant hamiltonicity and hamiltonian connectivity in alternating group graphs networks 44 2004 302 310 showed that for n 4 ag n f is hamiltonian i e ag n f possesses a hamiltonian cycle if f n 2 and ag n f is hamiltonian connected i e every two vertices of ag n f are connected by a hamiltonian path if f n 3 in this paper we show the following results i ag n f is pancyclic i e ag n f contains an l cycle for each l with 3 l m if f n 2 ii ag n f is vertex pancyclic i e every vertex of ag n f is contained in an l cycle for each l with 3 l m if f n 3 and iii ag n f is edge 4 pancyclic i e every edge of ag n f is contained in an l cycle for each l with 4 l m if f n 4 the first result is an improvement over the previous one and the last two results are new characterization of fault tolerant pancyclicity on ag n all the results we achieved are the best possible in the sense that the number of faulty vertices cannot be increased

a martingale characterization of quantum poisson processes we study the characterization problem of quantum poisson processes using the martingale property and some conditions on moments up to fourth order these conditions are inspired by an assumption on the quadratic variation that is equivalent in classical cases to the fact that the process has purely discontinuous sample paths with unit jumps

external finite element approximations of eigenfunctions in the case of multiple eigenvalues abstract the paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains h are not subdomains of the original domain r 2 the considerations are restricted to piecewise linear approximations special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues as yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h we avoid this disadvantage

the balanced hypercube a cube based system for fault tolerant applications in this paper we present an interconnection structure called the balanced hypercube which is a variation of the standard hypercube with desirable properties of strong connectivity regularity and symmetry the proposed structure is a special type of load balanced graph designed to tolerate processor failure in balanced hypercubes each processor has a backup matching processor that shares the same set of neighboring nodes therefore tasks that run on a faulty processor can be reactivated in the backup processor to provide efficient system reconfiguration other properties of balanced hypercubes are examined it is also shown that odd dimensional balanced hypercubes have smaller diameters than that of standard hypercubes as an application of balanced hypercubes we show a fault tolerant embedding of rings in balanced hypercubes

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

on martingale inequalities in non commutative stochastic analysis we develop a non commutativelpstochastic calculus for the clifford stochastic integral anl2theory of which has been developed by barnett streater and wilde the main results are certain non commutativelpinequalities relating clifford integrals and their integrands these results are applied to extend the domain of the clifford integral froml2tol1integrands and we give applications to optional stopping of clifford martingales proving an analog of a theorem of burkholder the stopped clifford processfthas zero expectation providedet in proving these results we establish a number of results relating the clifford integral to the differential calculus in the clifford algebra in particular we show that the clifford integral is given by the divergence operator and we prove an explicit martingale representation theorem both of these results correspond closely to basic results for stochastic analysis on wiener space thus furthering the analogy between the clifford process and brownian motion

finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions in this paper we use two symplectic schemes to simulate the ablowitz ladik model associated to the cubic nonlinear schrodinger equation and we compare them with nonsymplectic methods

conditional fault tolerant edge bipancyclicity of hypercubes with faulty vertices and edges let f be a faulty set in an n dimensional hypercube q n such that in q n f each vertex is incident to at least two edges and let f v f e be the numbers of faulty vertices and faulty edges in f respectively in this paper we consider the fault tolerant edge bipancyclicity of hypercubes it is shown that each edge in q n f for n 3 lies on a fault free cycle of any even length from 6 to 2 n 2 f v if f v f e 2 n 5 this gives an answer for a problem proposed by yang et al 2016 33 consider hypercubes with mixed faulty edges and faulty vertices under conditional fault model improve the previous results about the fault tolerant edge bipancyclicity of hypercubes give an affirmative answer for a problem proposed by yang et al 2016

fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs the arrangement graph a sub n k is a generalization of the star graph there are some results concerning fault hamiltonicity and fault hamiltonian connectivity of the arrangement graph however these results are restricted in some particular cases and thus are less completed we improve these results and obtain a stronger and simpler statement let n k spl ges 2 and f spl sube v a sub n k spl cup e a sub n k we prove that a sub n k f is hamiltonian if f spl les k n k 2 and a sub n k f is hamiltonian connected if f spl les k n k 3 these results are optimal

hamiltonian cycle embedding for fault tolerance in balanced hypercubes abstract the balanced hypercube bh n defined by wu and huang is a variant of the hypercube network yang proposed that fault tolerance of balanced hypercube bh n is an important issue in parallel computing which needs further study yang 2010 24 in this paper we prove that there exists a fault free hamiltonian path between any two adjacent vertices in bh n with 2 n 2 faulty edges as a corollary we derive that for any fault free edge e there exists a fault free hamiltonian cycle containing e in bh n with 2 n 2 faulty edges which is optimal in the sense of the number of faulty edges

a system of quantum stochastic differential equations in terms of non equilibrium thermo field dynamics within the framework of non equilibrium thermo field dynamics netfd quantum wiener processes at finite temperatures are constructed and its representation space is shown to be the thermal space after the introduction of the stochastic shrodinger equation a unified system of quantum stochastic differential equations including the quantum stochastic liouville equation and the quantum langevin equation is established within the quantum stochastic calculus

coherent quantum lqg control based on a recently developed notion of physical realizability for quantum linear stochastic systems we formulate a quantum lqg optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum such a control scheme is often referred to in the quantum control literature as coherent feedback control it distinguishes the present work from previous works on the quantum lqg problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom the difference in our formulation is the presence of additional non linear and linear constraints on the coefficients of the sought after controller rendering the problem as a type of constrained controller design problem due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard lqg problem we propose a numerical procedure for solving this problem based on an alternating projections algorithm and as an initial demonstration of the feasibility of this approach we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained for comparison we also consider the case of classical linear controllers that use direct or indirect measurements and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones

symmetric property and reliability of balanced hypercube huang and wu in ieee transactions on computers 46 1997 484 490 introduced the balanced hypercube bh n as an interconnection network topology for computing systems and they proved that bh n is vertex transitive however some other symmetric properties say edge transitivity and arc transitivity of bh n remained unknown in this paper we solve this problem and prove that bh n is an arc transitive cayley graph using this we also investigate some reliability measures including super connectivity cyclic connectivity etc in bh n first we prove that every minimum edge cut of bh n n ge 2 isolates a vertex and every minimum vertex cut of bh n n ge 3 isolates a vertex this is stronger than that obtained by wu and huang which shows the connectivity and edge connectivity of bh n are 2n second yang applied mathematics and computation 219 2012 970 975 proved that for n ge 2 the super connectivity of bh n is 4n 4 and the super edge connectivity of bh n is 4n 2 in this paper we proved that bh n n ge 2 is super lambda prime but not super kappa prime that is every minimum super edge cut of bh n n ge 2 isolates an edge but the minimum super vertex cut of bh n n ge 2 does not isolate an edge third we also obtain that for n ge 2 the cyclic connectivity of bh n is 4n 4 and the cyclic edge connectivity of bh n is 4 2n 2 that is to become a disconnected graph which has at least two components containing cycles we need to remove at least 4n 4 vertices resp 4 4n 2 edges from bh n n ge 2

a solvable model of microscopic frequency modulation ii rigorous treatment of the damping operator abstract the solvable model of quantum statistical mechanical frequency modulation which was proposed previously by the authors is solved exactly by rigorous treatment of the damping operator for the cases of boson and two state irrelevant subsystems to obtain the intensity distribution and the response function for a certain operator of the relevant subsystem the results are compared with the previous ones which were obtained within the conventional treatment of the damping operator

the series product and its application to quantum feedforward and feedback networks the purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks these methods build on and generalize existing methods for series or cascade connections by allowing for more general interfaces and by introducing an efficient algebraic tool the series product we also introduce another product which we call the concatenation product that is useful for assembling and representing systems without necessarily having connections we show how the concatenation and series products can be used to describe feedforward and feedback networks a selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology

some self adjoint quantum semimartingales it is proved that the quantum stochastic gauge integral preserves self adjointness of vacuum adapted processes this fact together with bounded perturbations and the link between the hudson parthasarathy calculus and vacuum adapted theory is used to produce many self adjoint quantum semimartingales

edge fault tolerant bipanconnectivity of hypercubes this paper shows that for any two distinct vertices u and v with distance d in the hypercube q n n 3 with at most 2n 5 faulty edges and each vertex incident with least two fault free edges there exist fault free uv paths of length in q n for every with d 4 2 n 1 and d 0 mod2 this result improves some known results on edge fault bipanconnectivity of hypercubes the proof is based on the recursive structure of q n

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

numerical methods for the simulation of trapped nonlinear schrodinger systems we propose analyze and compare the efficiency and accuracy of different numerical schemes for the solution of the nonlinear schrodinger equation with a trapping potential in particular we study schemes of finite difference pseudospectral and spectral types for the space discretization together with explicit symplectic multistep split step and standard variable step integrators to solve the time evolution all of these schemes are evaluated comparatively and some recommendations based on their accuracy and computational efficiency are made

noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations the notion of a unitary noncommutative stochastic process with independent and stationary increments is introduced and it is proved that such a process under a continuity assumption can be embedded into the solution of a quantum stochastic differential equation in the sense of hudson and parthasarathy 8

fermion ito s formula and stochastic evolutions an ito product formula is proved for stochastic integrals against fermion brownian motion and used to construct unitary processes satisfying stochastic differential equations as in the corresponding boson theory 10 11 these give rise to stochastic dilations of completely positive semigroups

mayer problem for quantum stochastic control this paper is concerned with the value function for optimal quantum stochastic control we established the lipschitz continuity of the value function which is inherent from the lipschitz property of the set valued map describing the quantum stochastic control by using a feedback multivalued map we also prove a characterization of optimal solutions relating to a noncommutative generalization of mayer problem

localization of quantum bernoulli noises the family k k k 0 of annihilation and creation operators acting on square integrable functionals of a bernoulli process z zk k 0 can be interpreted as quantum bernoulli noises in this note we consider the operator family lk lk k 0 where lk kek with ek being the conditional expectation operator given field zj 0 j k we show that lk resp lk is essentially a kind of localization of the annihilation operator k resp creation operator k we examine properties of the family lk lk k 0 and prove among other things that lk and lk satisfy a local canonical anti communication relation and lk k 0 forms a mutually orthogonal operator sequence although each lk is not a projection operator we find that the operator series k 0 lk xlk converges in the strong operator topology for each bounded operator x acting on square integrable functionals of z in particular we get an explicit sum of the operator series k 0 lk lk a useful norm estimate on k 0 lk xlk is also obtained final

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

edge fault tolerant edge bipancyclicity of balanced hypercubes abstract the balanced hypercube bh n is a variant of hypercube q n hao et al 2014 showed that there exists a fault free hamiltonian path between any two adjacent vertices in bh n with 2 n 2 faulty edges cheng et al 2015 proved that bh n is 6 edge bipancyclic after 2 n 3 faulty edges occur for all n 2 in this paper we improve these two results by demonstrating that bh n is 6 edge bipancyclic even when there exist 2 n 2 faulty edges for all n 2 our result is optimal with respect to the maximum number of tolerated edge faults

hamiltonian paths passing through prescribed edges in balanced hypercubes abstract the balanced hypercube was proposed as a novel interconnection network for large scale parallel systems it is known that the balanced hypercube is edge bipancyclic given a set of edges p with p n 1 and two vertices x and y in different bipartite sets we show that there exists a hamiltonian path from x to y passing through p in b h n if and only if p is a linear forest and neither x nor y is an internal vertex of p and x and y are not end vertices of a component of p simultaneously consequently the balanced hypercube contains a hamiltonian cycle passing through a set p of at most n prescribed edges if and only if p is a linear forest moreover our result improves some known results

su 1 1 lie algebraic approach to linear dissipative processes in quantum optics an su 1 1 lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems the liouville space formulation thermo field dynamics and the disentanglement theorem of su 1 1 lie algebra play essential roles in this formulation in the liouville space the time evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of su 1 1 lie algebra and the thermal state condition of thermo field dynamics the presented formulation is used for investigating a dissipative nonlinear oscillator the quantum mechanical model of phase modulation and the photon echo in the localized electron phonon system this algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems

relative number state representation and phase operator for physical systems a relative number state rns representation of a system composed of two distinguishable subsystems is proposed a phase variable as a quantum mechanical operator conjugate to a relative number operator is defined based on the rns representation space the phase operator is expressed as a unitary exponential operator the properties of the relative number operator the phase operator and their eigenstates are investigated in detail the phase variable has a maximum uncertainty in any stationary state also a time operator as a dynamical variable can be defined in the rns representation space the rns representation is closely related to the liouville space formulation and to thermofield dynamics the rns representation is shown to be a suitable method for investigating the josephson junction with ultrasmall capacitance a basic formulation of number phase quantization in the josephson junction is given in terms of the rns representation

matching preclusion for balanced hypercubes matching preclusion is a measure of robustness in the event of edge failure in interconnection networks the matching preclusion number of a graph g is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching and the conditional matching preclusion number of g is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or an almost perfect matching in this paper we consider balanced hypercubes we obtain that an n dimension balanced hypercube bh n has the matching preclusion number 2n and mainly prove that for the balanced hypercube bh n each matching preclusion set of cardinality 2n is trivial and the conditional matching preclusion number of balanced hypercube is 4n 2 whenever n 2

some results in lumped mass finite element approximation of eigenvalue problems using numerical quadrature formulas in this paper we establish the convergence and the rate of convergence for approximate eigenvalues and eigenfunctions of second order elliptic eigenvalue problems obtained by a lumped mass finite element approximation various aspects of lumped mass techniques have been discussed for such eigenvalue problems by fix 1972 ishihara 1977 strang and fix 1973 and tong et al 1971 among others in our approach the lumping of the mass matrix results from the use of a lobatto quadrature formula for the integrals over rectangular lagrange finite elements of degree k

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

quantum q white noise and a q central limit theorem we establish a connection between the azema martingales and certain quantum stochastic processes with increments satisfyingq commutation relations this leads to a theory ofq white noise onq bialgebras and to a generalization of the fock space representation theorem for white noise on bialgebras in particular quantum azema noise q interpolations between fermion and boson quantum brownian motion and unitary evolutions withq independent multiplicative increments are studied it follows from our results that the azema martingales and theq interpolations are central limits of sums ofq independent identically distributed quantum random variables

an investigation into the effect of product approximation in the numerical solution of the cubic nonlinear schro dinger equation abstract we study the effect of product approximation on the galerkin solutions of the one dimensional cubic schrodinger equation a crank nicolson scheme is used to discretize in time the paper describes two numerical experiments in the first we examine the approximation obtained by the standard galerkin method and discuss the possibility of enforcing discrete analogs of the conservation laws satisfied by the exact solution in the second experiment numerical results obtained by the product approximation version of the galerkin method are compared and the effectiveness of the method for different combinations of test and trial functions is also investigated

singular perturbation approximations for a class of linear quantum systems this technical note considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics the technical note presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction

stochastic liouville equation approach within non equilibrium thermo field dynamics abstract the development of non equilibrium thermo field dynamics called netfd in abbreviation was started in 1983 based on the damping theory in liouville space it has progressed considerably since then especially the structural similarity between the framework of netfd and the framework of the usual quantum field theory was revealed in this paper we will show that within the formalism of netfd the quantum langevin equation and the quantum stochastic liouville equation approach are also feasible for the construction of the quantum field theory for open systems

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

the symplecticity of multi step methods abstract for any linear multi step scheme feng kang defines the step transition operator characterizing it and defines the symplecticity of the method for hamiltonian systems through the operator in this paper the author gets a valuable expression of the step transition operator lemma 1 section 2 and proves a conjecture due to feng any linear multi step scheme is non sympletic theorem 1 section 3 similarly an interesting result in theorem 2 section 3 for a sort of generalized multi step schemes is obtained the results indicate that some novel approach is needed for the construction of sympletic multi step methods

embedding of cycles in arrangement graphs arrangement graphs have been proposed as an attractive interconnection topology for large multiprocessor systems the authors study these graphs by proving the existence of hamiltonian cycles in any arrangement graph they also prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph they show that an arrangement graph can be decomposed into node disjoint cycles in many different ways

symplectic methods for the ablowitz ladik discrete nonlinear schrodinger equation using several kinds of coordinate transformations we standardize the noncanonical symplectic structure of the ablowitz ladik model a l model of nonlinear schrodinger equation nlse then we employ some symplectic scheme to simulate the solitons motion and test the evolution of the discrete invariants of the a l model and also the conserved quantities of the original nlse in comparison with a higher order non symplectic scheme applied directly to the a l model we show the overwhelming superiorities of the symplectic method we also compare the implementation of the same symplectic scheme to different standardized hamiltonian systems resulting from different coordinate transformations and show that the symmetric coordinate transformation improves the numerical results obtained via the asymmetric one in preserving the invariants of the a l model and the original nlse

diagonal versions and quantum stochastic integrals on the symmetric fock space with nonadapted integrands we develop a quantum stochastic calculus for nonadapted processes and the three standard integrators on the symmetric fock space over an arbitrary polish space the integrable processes are restricted to a class of operators which possess some kernel description our approach is mainly based on the concept of diagonalized versions 16 in the case of the real line we consider connections to the usual stochastic and quantum stochastic calculi and derive ito formulae besides the standard integrals with two integrands and one integrator these formulae lead to a new type of quantum stochastic integrals which has two integrators and three integrands

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

continuous unital dilations of completely positive semigroups abstract since it began in the 1970s the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup several authors have proved the existence of dilations but have usually only achieved unital dilations of specific examples while general theorems have tended to produce non unital dilations a unique approach due to jean luc sauvageot leads to a unital dilation in a general setting but leaves unclear the continuity of the dilation semigroup the major purpose of this paper therefore is to further develop sauvageot s theory in order to prove the existence of continuous unital dilations the dilation is characterized by a combinatorial property akin to free independence this property is implicit in some of sauvageot s original calculations but a secondary goal of this paper is to present it as its own object of study which is done in section 4 this work is based on the author s ph d thesis

noncommutative brownian motion in monotone fock space an example of noncommutative brownian motion is constructed on the monotone fock space which is a kind of fock space generated by all the decreasing finite sequences of positive real numbers the probability distribution at time associated to this brownian motion is shown to be the arcsine law normalized to mean 0 and variance t

spectral families of quantum stochastic integrals we develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation conservation and annihilation processes in fock space these give a non commutative generalisation of classical stochastic integrals driven by poisson random measures a stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations as an application we construct stochastic flows on operator algebras driven by levy processes with finite levy measure

a martingale characterization of canonical commutation and anticommutation relations using a martingale condition and some restrictions on moments up to fourth order the characterisation problem of boson fermion and classical brownian motions is studied from a unified point of view entirely within the framework of elementary operator theory global commutation and anticommutation rules turn out to be consequences of corresponding commutation and anticommutation rules between past and future observables

quantum stochastic calculus with maximal operator domains quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains operator adaptedness conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of fock space together with sections of the adapted gradient operator free from exponential vector domains our stochastic integrals may be satisfactorily composed yielding quantum ito formulas for operator products as sums of stochastic integrals the calculus has seen two reformulations since its discovery one closely related to classical ito calculus the other to noncausal stochastic analysis and malliavin calculus our theory extends both of these approaches and may be viewed as a synthesis of the two the main application given here is existence and uniqueness for the attal meyer equations for implicit definition of quantum stochastic integrals

filtered random variables bialgebras and convolutions we introduce the filtered bialgebra which is a multivariate generalization of the unital bialgebra c x x p of polynomials in noncommuting variables x x x x and a projection p p p2 endowed with the coproduct x x 1 1 x x x p p x with p being group like we study the associated convolutions random walks and filtered random variables the gns representations of the limit states lead to filtered fundamental operators which are the ccr fundamental operators on the multiple symmetric fock space h over h l2 r g where g is a separable hilbert space multiplied by appropriate projections the importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main types of noncommutative random variables and fundamental operators respectively regardless of the type of noncommutative independence

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

a solvable model of microscopic frequency modulation i conventional treatment of the damping operator abstract a solvable model of quantum statistical mechanical frequency modulation is proposed and is solved for the cases of boson and two state irrelevant subsystems to obtain the intensity distribution for a certain operator of the relevant subsystem the damping theoretical approach which is based on the time independent projection operator method to the model is also investigated

solving quantum stochastic differential equations with unbounded coefficients we demonstrate a method for obtaining strong solutions to the right hudson parthasarathy quantum stochastic differential equation dut f utd t u0 1 where u is a contraction operator process and the matrix of coefficients f consists of unbounded operators this is achieved whenever there is a positive self adjoint reference operator c that behaves well with respect to the f allowing us to prove that domc1 2 is left invariant by the operators ut thereby giving rigorous meaning to the formal expression above we give conditions under which the solution u is an isometry or coisometry process and apply these results to construct unital homomorphic dilations of quantum markov semigroups arising in probability and physics