a hamiltonian and multi hamiltonian formulation of a rod model using quaternions abstract the geometrically exact model of an elastic rod formulated in 23 the simo reissner rod model has been proposed and investigated we present a constrained hamiltonian formulation of the elastic rod model as well as a constrained multi hamiltonian formulation of the model in both formulations quaternions are used to represent the group of rotations the resulting hamiltonian pde and multi hamiltonian formulation have simple looking formats involving constant structure matrices

objective energy momentum conserving integration for the constrained dynamics of geometrically exact beams in this paper the results in s leyendecker p betsch p steinmann energy conserving integration of constrained hamiltonian systems a comparison of approaches comput mech 33 2004 174 185 are extended to geometrically exact beams the finite element formulation for nonlinear beams in terms of directors providing a framework for the objective description of their dynamics is considered geometrically exact beams are analysed as hamiltonian systems subject to holonomic constraints with a hamiltonian being invariant under the action of so 3 the reparametrisation of the hamiltonian in terms of the invariants of so 3 is perfectly suited for a temporal discretisation which leads to energy momentum conserving integration in this connection the influence of alternative procedures the lagrange multiplier method the penalty method and the augmented lagrange method for the treatment of the constraints is investigated for the example of a beam with concentrated masses 2005 elsevier b v all rights reserved

a domain wall between single mode and bimodal states and its transition to dynamical behavior in inhomogeneous systems abstract we consider domain walls dws between single mode and bimodal states that occur in coupled nonlinear diffusion nld real ginzburg landau rgl and complex ginzburg landau cgl equations with a spatially dependent coupling coefficient group velocity terms are added to the nld and rgl equations which breaks the variational structure of these models in the simplest case of two coupled nld equations we reduce the description of stationary configurations to a single second order ordinary differential equation we demonstrate analytically that a necessary condition for existence of a stationary dw is that the group velocity must be below a certain threshold value above this threshold dynamical behavior sets in which we consider in detail in the cgl equations the dw may generate spatio temporal chaos depending on the nonlinear dispersion a spatially dependent coupling coefficient as considered in this paper can be realized at least in two different convection systems a rotating narrow annulus supporting two traveling wave wall modes and a large aspect ratio system with poor heat conductivity at the lateral boundaries where the two phases separated by the dw are rolls and square cells

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

midpoint rule for variational integrators on lie groups the midpoint rule provides a standard method to obtain symmetric symplectic and second order accurate variational integrators for mechanical systems whose configuration manifold is the vector space r n in this work we discuss how to extend this rule to a generic finite dimensional lie group g while retaining the same properties we show that the function k g g exp log g g e g plays a special role in the theory and for g so 3 we give a compact formula to compute it we also discuss sufficient conditions for the method to conserve momentum maps associated with left or right group actions as an example the variational integrator obtained from the midpoint rule is applied to simulating rigid body dynamics the resulting integrator is compared with state of the art symmetric and second order accurate integrators for rigid body motion

complementary inequalities to inequalities of jensen and ando based on the mond pecaric method abstract applying the mond pecaric method to unital positive linear maps we shall show several complementary inequalities to jensen s inequalities on positive linear maps and consequently obtain complementary inequalities to ando s inequalities associated with operator means we shall apply them to obtain complementary estimates for the results by ando aujla vasudeva and fujii on hadamard product and operator means

neural learning by geometric integration of reduced rigid body equations in previous contributions the second author of this paper presented a new class of algorithms for orthonormal learning of linear neural networks with p inputs and m outputs based on the equations describing the dynamics of a massive rigid frame on the stiefel manifold these algorithms exhibit good numerical stability strongly binding to the sub manifold of constraints and good controllability of the learning dynamics but are not completely satisfactory from a computational complexity point of view in the recent literature efficient methods of integration on the stiefel manifold have been proposed by various authors see for example phys d 156 2001 219 numer algorithms 32 2003 163 j numer anal 21 2001 463 numer math 83 1999 599 inspired by these approaches in this paper we propose a new and efficient representation of the mentioned learning equations and a new way to integrate them the numerical experiments show how the new formulation leads to significant computational savings especially when p m the effectiveness of the algorithms is substantiated by numerical experiments concerning principal subspace analysis and independent component analysis these experiments were carried out with both synthetic and real world data

matrix inequalities in statistical mechanics some matrix inequalities used in statistical mechanics are presented a straightforward proof of the thermodynamic inequality is given and its equivalence to the peierls bogoliubov inequality is shown

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

quantum control with piecewise constant control functions one of the long standing problems in the field of computational quantum control is to devise algorithms that converge to high yield controls with a desired structure particularly controls that can be reproduced by laboratory laser pulse shapers in this paper we propose an approach that is a considerable step forward toward the solution of this problem our treatment employs piecewise constant controls for these the concept of desired structure is addressed by including appropriate quadratic forms in the performance functionals thereafter correct extremal control equations are derived whereas the equations appearing in the literature are correct only to first or second order we illustrate numerically that our approach can give extremely high yield controls with a desired structure our results are valid not only in the context of quantum control but also for similiar bilinear systems arising in different areas in particular they are suitable for controlling linear flows with drift on unitary matrix groups

an extended lie trotter formula and its applications abstract in this paper we present a class of lie trotter formulae for hermitian operators including the formulae derived by hiai petz and furuta a lie trotter formula for weighted log euclidean geometric means of several positive definite operators is given in terms of sagae tanabe geometric and spectral geometric means

inequalities for quantum relative entropy some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log majorization result the thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the peierls bogoliubov inequality is obtained

two reverse inequalities associated with tsallis relative operator entropy via generalized kantorovich constant and their applications recently tsallis relative operator entropy tp a b and tsallis relative entropy dp a b are discussed by furuichi yanagi kuriyama we shall show two reverse inequalities involving tsallis relative operator entropy tp a b via generalized kantorovich constant k p as some applications of two reverse inequalities we shall show two trace reverse inequalities involving tr tp a b and dp a b and also a known reverse trace inequality involving the relative operator entropy s a b by fujii kamei and the umegaki relative entropy s a b is shown as a simple corollary we show the following result let a and b be strictly positive operators on a hilbert space h such that m1 i a m1 i 0 and m2 i b m2 i 0 put m m2m1 m m2m1 h mm m1m2m1m2 1 and p 0 1 let be normalized positive linear map on b h then the following inequalities hold i 1 k p p a p b tp a b tp a b tp a b and ii f p a tp a b tp a b tp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and k p 0 1 and f p mpphp hh 11 k p 1p 1 0 in addition let a and b be strictly positive definite matrices iii 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b and iv f p tr a dp a b tr tp a b dp a b in particular both iii and iv yield the following known result logs 1 tr a s a b tr s a b s a b where s 1 h1h 1elogh1h 1 is said to be the specht ratio and s 1 1

efficient time symmetric simulation of torqued rigid bodies using jacobi elliptic functions if the three moments of inertia are distinct the solution to the euler equations for the free rigid body is given in terms of jacobi elliptic functions using the arithmetic geometric mean algorithm abramowitz and stegun 1992 handbook of mathematical functions with formulas graphs and mathematical tables new york dover these functions can be calculated efficiently and accurately compared to standard numerical ode and lie poisson solvers the overall approach yields a faster and more accurate numerical solution to the euler equations this approach is designed for mass asymmetric rigid bodies in the case of symmetric bodies the exact solution is available in terms of trigonometric functions see dullweber et al 1997 j chem phys 107 5840 51 reich 1996 fields inst commun 10 181 91 and benettin et al 2001 siam j sci comp 23 1189 203 for details in this paper we consider the case of asymmetric rigid bodies subject to external forces we consider a strategy similar to the symplectic splitting method proposed in reich 1996 fields inst commun 10 181 91 and dullweber et al 1997 j chem phys 107 5840 51 the method proposed here is time symmetric we decompose the vector field of our problem into a free rigid body frb problem and another completely integrable vector field the frb problem consists of the euler equations and a differential equation for the 3 3 orientation matrix the euler equations are integrated exactly while the matrix equation is approximated using a truncated magnus series in our experiments we observe that the overall numerical solution benefits greatly from the very accurate solution of the euler equations we apply the method to the heavy top and the simulation of artificial satellite attitude dynamics

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

operator inequality implying generalized bebiano lemos providencia one a capital letter means n n matrix very recently fujii et al show for every a b 0 and p 1 a12as2bp sas21pa12p 1 s p s log a1 s2b1 sa1 s2 holds for any s 0 in fact yields bebiano lemos providencia inequality a12 as2bsas2 tsa12 log a1 t2bta1 t2 for s t 0 as an extension of we show the following result the following i and ii hold and they are equivalent i for every a 0 b 0 0 1 and each t 0 1 and any real number q 0 aq2a12ba12 aq2h log aq 1 t r 2a qr2a1 qt2ba1 qt2sa qr2 aq 1 t r 2 holds for s 1 and r t where 1 t r 1 t s r and h 1 t r s 1 t s r ii if a b 0 with a 0 then for t 0 1 and p 1 a1 t r ar2a t2bpa t2sar21 t r p t s r for s 1 and r t i implies in fact put t 0 s 1 qr 1 in i then aq2 a12ba12 aq2 1 q q log a1 q2b 1 q qa1 q2 holds and finally replace b by bp s a by as by 1p and also replace q by 1s then we have the desired

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

trace inequalities for logarithms and powers of j hermitian matrices abstract spectral inequalities are stated for the trace of the exponential or the logarithmic of certain j hermitian matrices j i r i n r 0 r n the obtained inequalities are established in the context of indefinite inner product spaces and they are known to be valid for hilbert space operators or operator algebras

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

composite zigzag structures in the 1d complex ginzburg landau equation we study the dynamics of the one dimensional complex ginzburg landau equation cgle in the regime where holes and defects organize themselves into composite superstructures which we call zigzags extensive numerical simulations of the cgle reveal a wide range of dynamical zigzag behavior which we summarize in a phase diagram we have performed a numerical linear stability and bifurcation analysis of regular zigzag structures which reveals that traveling zigzags bifurcate from stationary zigzags via a pitchfork bifurcation this bifurcation changes from supercritical forward to subcritical backward as a function of the cgle coefficients and we show the relevance of this for the phase diagram our findings indicate that in the zigzag parameter regime of the cgle the transition between defect rich and defect poor states is governed by bifurcations of the zigzag structures 2001 elsevier science b v all rights reserved

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

steady squares and hexagons on a subcritical ramp steady squares and hexagons on a subcritical ramp are studied both analytically and numerically within the framework of the lowest order amplitude equations on the subcritical ramp the external stress or control parameter varies continuously in space from subcritical to supercritical values at the subcritical end of the ramp pattern formation is suppressed and patterns fade away into the conduction solution it is shown that three dimensional patterns may change shape on a subcritical ramp a square pattern becomes a pattern of rolls as it fades with the roll axes aligned in the direction orthogonal to that in which the control parameter varies hexagons in systems with horizontal midplane symmetry become a pattern of rectangles before reaching the conduction solution there is a suggestion that hexagons in systems which lack this symmetry might fade away through a roll pattern numerical simulations are used to illustrate these phenomena

generalized shannon inequalities based on tsallis relative operator entropy abstract tsallis relative operator entropy is defined and then its properties are given shannon inequality and its reverse one in hilbert space operators derived by furuta linear algebra apll 381 2004 219 are extended in terms of the parameter of the tsallis relative operator entropy moreover the generalized tsallis relative operator entropy is introduced and then several operator inequalities are derived

convergence of lie group integrators we relate two notions of local error for integration schemes on riemannian homogeneous spaces and show how to derive global error estimates from such local bounds in doing so we prove for the first time that the lie butcher theory of lie group integrators leads to global error estimates

the exact computation of the free rigid body motion and its use in splitting methods this article investigates the use of the computation of the exact free rigid body motion as a component of splitting methods for rigid bodies subject to external forces we review various matrix and quaternion representations of the solution of the free rigid body equation which involve jacobi ellipic functions and elliptic integrals and are amenable to numerical computations we consider implementations which are exact i e computed to machine precision and semiexact i e approximated via quadrature formulas we perform a set of extensive numerical comparisons with state of the art geometrical integrators for rigid bodies such as the preprocessed discrete moser veselov method our numerical simulations indicate that these techniques combined with splitting methods can be favorably applied to the numerical integration of torqued rigid bodies

reverse inequalities involving two relative operator entropies and two relative entropies we shall discuss relation among tsallis relative operator entropy tp a b the relative operator entropy s a b by j i fujii kamei the tsallis relative entropy dp a b by furuichi yanagi kuriyama and the umegaki relative entropy s a b we show the following result let a and b be strictly positive definite matrices such that m1i a m1i 0 and m2i b m2i 0 put h m1m2m1m2 1 and p 0 1 then the following inequalities hold 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and the first inequality is the reverse one of the second known inequalty in particular logs 1 tr a s a b tr s a b s a b where s 1 is the specht ratio defined by s 1 h1h 1elogh1h 1 and the first inequality is the reverse one of the second known inequalty it is known that k p 0 1 for p 0 1 and s 1 1

tsallis entropies and matrix trace inequalities in quantum statistical mechanics extensions of trace inequalities arising in statistical mechanics are derived in a straightforward and unified way from a variational characterization of the tsallis entropy namely one parameter extension of the thermodynamic inequality is presented and its equivalence to a generalized peierls bogoliubov inequality is stated in the sense that one may be obtained from the other and vice versa

lie group variational integrators for the full body problem we develop the equations of motion for full body models that describe the dynamics of rigid bodies acting under their mutual gravity the equations are derived using a variational approach where variations are defined on the lie group of rigid body configurations both continuous equations of motion and variational integrators are developed in lagrangian and hamiltonian forms and the reduction from the inertial frame to a relative frame is also carried out the lie group variational integrators are shown to be symplectic to preserve conserved quantities and to guarantee exact evolution on the configuration space one of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies

integration methods based on canonical coordinates of the second kind we present a new class of integration methods for differential equations on manifolds in the framework of lie group actions canonical coordinates of the second kind is used for representing the lie group locally by means of its corresponding lie algebra the coordinate map itself can in many cases be computed inexpensively but the approach also involves the inversion of its differential a task that can be challenging to succeed it is necessary to consider carefully how to choose a basis for the lie algebra and the ordering of the basis is important as well for semisimple lie algebras one may take advantage of the root space decomposition to provide a basis with desirable properties the problem of ordering leads us to introduce the concept of an admissible ordered basis aob the existence of an aob is established for some of the most important lie algebras the computational cost analysis shows that the approach may lead to more efficient solvers for odes on manifolds than those based on canonical coordinates of the first kind presented by munthe kaas numerical experiments verify the derived properties of the new methods

preprocessed discrete moser veselov algorithm for the full dynamics of a rigid body the discrete moser veselov algorithm is an integrable discretization of the equations of motion for a free rigid body it is symplectic and time reversible and it conserves all first integrals of the system the only drawback is its low order we present a modification of this algorithm to arbitrarily high order which has negligible overhead but considerably improves the accuracy

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

an objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy momentum conserving scheme in dynamics we present in this paper a new finite element formulation of geometrically exact rod models in the three dimensional dynamic elastic range the proposed formulation leads to an objective or frame indifferent under superposed rigid body motions approximation of the strain measures of the rod involving finite rotations of the director frame in contrast with some existing formulations this goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod s cross section furthermore the proposed framework allows the development of time stepping algorithms that preserve the conservation laws of the underlying continuum hamiltonian system the conservation laws of linear and angular momenta are inherited by construction leading to an improved approximation of the rod s dynamics several numerical simulations are presented illustrating these properties copyright 2002 john wiley sons ltd

a note on the numerical integration of the kdv equation via isospectral deformations the main purpose of this paper is to test the performance of lie group integrators as applied to a semi discrete version of the kdv equation

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

a lower bound of quantum conditional mutual information in this paper a lower bound of quantum conditional mutual information is obtained by employing the peierls bogoliubov inequality and the golden thompson inequality comparison with the bounds obtained by other researchers indicates that our result is independent of any measurements it may give some new insights into squashed entanglement and perturbations of markov chain states

geometric direct search algorithms for image registration a widely used approach to image registration involves finding the general linear transformation that maximizes the mutual information between two images with the transformation being rigid body i e belonging to se 3 or volume preserving i e belonging to sl 3 in this paper we present coordinate invariant geometric versions of the nelder mead optimization algorithm on the groups sl 3 se 3 and their various subgroups that are applicable to a wide class of image registration problems because the algorithms respect the geometric structure of the underlying groups they are numerically more stable and exhibit better convergence properties than existing local coordinate based algorithms experimental results demonstrate the improved convergence properties of our geometric algorithms

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

matrix trace inequalities on tsallis relative entropy of negative order abstract in this paper we show matrix trace inequalities on the quantum tsallis relative entropy of negative order which includes the quasi geometric mean of positive definite matrices moreover we show an estimate of the difference between two tsallis relative entropies of negative order

lie group methods for rigid body dynamics and time integration on manifolds recently there has been an increasing interest in time integrators for ordinary differential equations which use lie group actions as a primitive in the design of the methods these methods are usually phrased in an abstract sense for arbitrary lie groups and actions we show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general an important part of the lie group methods involves the computation of a coordinate map and its derivative various options are available and they vary in cost accuracy and ability to approximately conserve invariants we discuss how the computation of these maps can be optimized for the rigid body case and we provide numerical experiments which give an idea of the performance of lie group methods compared to other known integration schemes

a class of intrinsic schemes for orthogonal integration numerical integration of odes on the orthogonal stiefel manifold is considered points on this manifold are represented as n k matrices with orthonormal columns of particular interest is the case when n gg k mainly two requirements are imposed on the integration schemes first they should have arithmetic complexity of order nk2 second they should be intrinsic in the sense that they require only the ode vector field to be defined on the stiefel manifold as opposed to for instance projection methods the design of the methods makes use of retractions maps two algorithms are proposed one where the retraction map is based on the qr decomposition of a matrix and one where it is based on the polar decomposition numerical experiments show that the new methods are superior to standard lie group methods with respect to arithmetic complexity and may be more reliable than projection methods owing to their intrinsic nature