traveling wave solutions of convection diffusion systems by center manifold reduction a convection di usion system in one space dimension is a partial di erential equation of the form ut a u ux b u ux x 1 1 here u rn and a u and b u are n n matrices that we shall assume are c2 functions of u if we ignore di usion we have the convection system ut a u ux 0 1 2 if a u df u for some 4ux function f rn rn then eq 1 2 becomes ut f u x 0 1 3 a system of conservation laws while eq 1 1 becomes ut f u x b u ux x 1 4 a system of viscous conservation laws the conservation law case occurs more often in applications and is far better studied see 5 and 12 however equations of the

center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

length two path centered surface areas of the n k star graph we discuss the notion of h centered surface area of a connected graph g where h is a subgraph of g i e the number of vertices in g at a certain distance from h and focus on the special case when h is a length two path in particular we derive explicit formulas for the length two path centered surface areas of the edge asymmetric n k star graph centered on two kinds of length two paths via a multivariate generating function approach

wei norman equations for a unitary evolution the wei norman technique allows one to express the solution of a system of linear non autonomous differential equations in terms of product of exponentials with time dependent exponents being solutions of a system of nonlinear differential equations we show that in the unitary case i e when the solution of the linear system is given by a unitary evolution operator the nonlinear system by an appropriate choice of ordering can be reduced to a hierarchy of matrix riccati equations to this end we consider a general linear non autonomous dynamical system on the special linear group the unitary case of particular significance for quantum optimal control problems is then obtained by restriction to anti hermitian generators we also point to the connections of the obtained results with the theory of the so called lie systems

distance formula and shortest paths for the n k star graphs the class of n k star graphs is a generalization of the class of star graphs thus a distance formula for the first class implies one for the second in this paper we show that the converse is also true another important concept is the number of shortest paths between two vertices this problem has been solved for the star graphs we will solve the corresponding problem for the n k star graphs

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

more explicit formulas for the matrix exponential abstract the matrix exponential plays an important role in the study of dynamical systems and linear systems an elementary method is discussed for computing eta for general a c n n it is shown that this method can be applied to generate explicit formulas for eta in various forms using this method some well known formulas are rederived and some new formulas are also derived as applications explicit formulas for n 3 or n 4 are given in each case a characterization of eta is also made based on the entries of a alone

constructing constrained invariant sets in multiscale continuum systems we present a method that we name the constrained invariant manifold method a visualization tool to construct stable and unstable invariant sets of a map or flow where the invariant sets are constrained to lie on a slow invariant manifold the construction of stable and unstable sets constrained to an unstable slow manifold is exemplified in a singularly perturbed model arising from a structural mechanical system consisting of a pendulum coupled to a viscoelastic rod additionally we extend the step and stagger method d sweet h nusse and j yorke phys rev lett 86 2261 2001 to calculate a delta pseudoorbit on a chaotic saddle constrained to the slow manifold in order to be able to compute the lyapunov exponents of the saddle

on the integrability of systems of nonlinear ordinary differential equations with superposition principles a new class of solvable nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations ode s describing each member of this class possess nonlinear superposition principles these systems of ode s are generally not derived from a hamiltonian and are classified by associated pairs of lie algebras of vector fields in this paper all such systems of n 3 ode s are integrated in a unified way by finding explicit integrals for them and relating them all to a pivotal member of their class the projective riccati equations moreover by perturbing two parametrically driven projective riccati equations thus making them nonsolvable in the above sense evidence is discovered of chaotic behavior on the poincare surface of section in the form of sensitive dependence on initial conditions near a boundary separating bounded from unbounded motion

adaptive resummation of markovian quantum dynamics in this thesis we derive a highly convergent nonperturbative expansion of markovian open quantum dynamics it is based on a splitting of the incoherent dynamics into periods of continuous evolution and abrupt jumps and attains its favorable convergence properties from an adaptive resummation of this so called jump expansion by means of the long standing problems of spatial particle detection and landau zener tunneling in the presence of dephasing we show that this adaptive resummation technique facilitates new highly accurate analytic approximations of markovian open systems the open landau zener model leads us to propose an efficient and robust incoherent control technique for the isomerization reaction of the visual pigment protein rhodopsin besides leading to approximate analytic descriptions of markovian open quantum dynamics the adaptive resummation of the jump expansion implies an efficient numerical simulation method we spell out the corresponding numerical algorithm by means of monte carlo integration of the relevant terms in the jump expansion and demonstrate it in a set of paradigmatic open quantum systems

a higher order godunov method for modeling finite deformation in elastic plastic solids in this paper we develop a first order system of conservation laws for finite deformation in solids describe its characteristic structure and use this analysis to develop a second order numerical method for problems involving finite deformation and plasticity the equations of mass momentum and energy conservation in lagrangian and eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient particular attention is paid to the influence of a curl constraint on the deformation gradient so that the characteristic speeds transform properly between the two frames of reference next we consider models in rate form for isotropic elastic plastic materials with work hardening and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion in spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric we still find that the characteristic speeds are always real for elastic behavior and essentially always real for plastic response these results allow us to construct a second order godunov method for the computation of three dimensional displacement in a one dimensional material viewed in the lagrangian frame of reference we also describe a technique for the approximate solution of riemann problems in order to determine numerical fluxes in this algorithm finally we present numerical examples of the results of the algorithm

conditional matching preclusion for the arrangement graphs the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex recently the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex it is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings in this paper we find this number and classify all optimal sets for the arrangement graphs one of the most popular interconnection networks

on the solution of the riccati equation by the taylor matrix method in this paper we suggest a matrix method to solve the riccati differential equation in terms of taylor polynomials this method is based on first taking the truncated taylor series of the function in equations and then substituting their matrix forms into the given equation hence the result matrix equation can be solved and the unknown taylor coefficients can be found approximately the solution is calculated in the form of a series with easily computable components the numerical results show the effectiveness of the method for this type of equation comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate

invariant manifolds for volterra integral equations of convolution type abstract in this paper we develop some elements of a qualitative theory for nonlinear volterra integral equations of convolution type our starting point is a local semiflow associated with the equation and acting on a space of compactly supported forcing functions within that framework we discuss the variation of constants formula the saddle point property the center manifold and hopf bifurcation some equations from population biology get special attention

generalized rankine hugoniot condition and shock solutions for quasilinear hyperbolic systems abstract for a quasilinear hyperbolic system we use the method of vanishing viscosity to construct shock solutions the solution consists of two regular regions separated by a free boundary shock we use melnikov s integral to obtain a system of differential algebraic equations that governs the motion of the shock for lax shocks in conservation laws these equations are equivalent to the rankine hugoniot condition for under compressive shocks in conservation laws or shocks in non conservation systems the melnikov type integral obtained in this paper generalizes the rankine hugoniot condition under some generic conditions we show that the initial value problem of shock solutions can be solved as a free boundary problem by the method of characteristics

fault tolerance of n k star networks this paper considers a refined measure s h for the fault tolerance of a network and for the generalized star network s n k determines s h s n k n h k 2 1 for 2 k n 1 and 0 h n k which implies that at least n h k 2 1 vertices of s n k have to be removed to get a disconnected graph without vertices of degree less than h this work generalizes some known results when the n k star graph is used to model the topological structure of a large scale parallel processing system this result can provide a more accurate measure for the fault tolerance of the system

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

on the surface area of the n k star graph we present an explicit formula for the surface area of the n k star graph i e the number of nodes at a certain distance from the identity node in the graph by identifying the unique cycle structures associated with the nodes in the graph deriving a distance expression in terms of such structures between the identity node of the graph and any other node and enumerating those cycle structures satisfying the distance restriction the above surface area derivation process can also be applied to some of the other node symmetric interconnection structures defined on the symmetric group when the aforementioned distance expression is available

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

singularity crossing phenomena in daes a two phase fluid flow application case study this paper analyzes the existence of smooth trajectories through singular points of differential algebraic equations or daes arising from traveling wave solutions of a degenerate convection diffusion model the dae system can be written in the quasilinear form a x x b x in this setting singularities are displayed when the matrix a x undergoes a rank change the singular hypersurface may be smoothly crossed by trajectories in a finite time if x is a geometric singularity satisfying certain directional conditions the basis of our analysis is a two phase fluid flow model in one spatial dimension with dissipative mechanism involved

algebraic solution of the lindblad equation for a collection of multilevel systems coupled to independent environments we consider the lindblad equation for a collection of multilevel systems coupled to independent environments the equation is symmetric under the exchange of the labels associated with each system and thus the open system dynamics takes place in the permutation symmetric subspace of the operator space the dimension of this space grows polynomially with the number of systems we construct a basis of this space and a set of superoperators whose action on this basis is easily specified for a given number of levels m these superoperators are written in terms of a bosonic realization of the generators of the lie algebra in some cases these results enable finding an analytic solution of the master equation using known lie algebraic methods to demonstrate this we obtain an analytic expression for the state operator of a collection of three level atoms coupled to independent radiation baths when analytic solutions are difficult to find the basis and the superoperators can be used to considerably reduce the computational resources required for simulations

robust procedures for converting among lindblad kraus and matrix representations of quantum dynamical semigroups given a quantum dynamical semigroup expressed as an exponential superoperator acting on a space of n dimensional density operators eigenvalue methods are presented by which canonical kraus and lindblad operator sum representations can be computed these methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography the statistical estimation of superoperators and their generators from a wide variety of experimental data theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the presence of random errors in the data

constructing vertex disjoint paths in n k star graphs this work describes a novel routing algorithm for constructing a container of width n 1 between a pair of vertices in an n k star graph with connectivity n 1 since lin et al t c lin d r duh h c cheng wide diameter of n k star networks in proceedings of the international conference on computing communications and control technologies vol 5 2004 pp 160 165 already calculated the wide diameters in n n 1 star and n 1 star graphs this study only considers an n k star with 2

nonclassical multidimensional viscous and inviscid shocks extending our earlier work on lax type shocks of systems of conservation laws we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general lax or undercompressive type shock waves of nonconservative hyperbolic systems with parabolic regularization the hyperbolic equations may be of variable multiplicity and the parabolic regularization may be of real or partially parabolic type we prove an existence result for inviscid nonconservative shocks that extends to multidimensional shocks a one dimensional result of x lin proved by quite different methods in addition we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero thereby justifying the small viscosity limit for multidimensional nonconservative shocks in our previous work on shocks we made use of conservative form especially in parts of the low frequency analysis thus most of the new analysis of this paper is concentrated in this area by adopting the more general nonconservative viewpoint we are able to shed new light on both the viscous and inviscid theory for example we can now provide a clearer geometric motivation for the low frequency analysis in the viscous case also we show that one may in the treatment of inviscid stability of nonclassical and or nonconservative shocks remove an apparently restrictive technical assumption made by mokrane and coulombel in their work on respectively shocktype nonconservative boundary problems and conservative undercompressive shocks another advantage of the nonconservative perspective is that lax and undercompressive shocks can be treated by exactly the same analysis

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given