reduced dynamics for entangled initial state of the full density operator starting from the schrodinger equation for a system and a reservoir we construct a quantum dynamical map that describes reduced dynamics we work with an arbitrary initial state for the system and reservoir and thus refrain from performing any factorizations the map is found to be nonlinear not completely positive and not unique to investigate the physical implications the bloch equations are constructed if the freedom the new map offers us is fully exploited then the classic inequality 2 for the damping coefficients must be replaced by a weaker condition

an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

an interpolation based parametric reduced order model combined with component mode synthesis abstract this paper presents an efficient interpolation based parametric reduced order model with component mode synthesis the off line sampling of a large scale structural dynamic system containing many parameters requires many computations to investigate the parameter dependency of a dynamical system therefore we introduce a dynamic substructuring scheme to execute off line sampling in the subdomain level in addition we suggest discriminating the interpolation of the subsystem depending on the characteristics of each substructural mode we synthesize the substructures and we then construct a two level semi parametrized rom by reducing the degrees of freedom of the interface in the on line stage we then demonstrate the efficiency and accuracy of the present method by computing the relative eigenvalue errors and the transient responses of the system with random values for the parameters and we conduct the design optimization of large scale systems under dynamic loading conditions a comparison of the present method with the full order model and a conventional reduced order model indicates that the present method is useful in carrying out an efficient analysis and design optimization for various large scale structural dynamic systems

on the existence of quantum subdynamics it is shown using only elementary operator algebra that an open quantum system coupled to its environment will have a subdynamics reduced dynamics as an exact consequence of the reversible dynamics of the composite system only when the states of system and environment are uncorrelated furthermore it is proved that for a finite temperature the kms condition for the lowest order correlation function cannot be reproduced by any type of linear subdynamics except the reversible hamiltonian one of a closed system the first statement can be seen as a particular case of a more general theorem of takesaki on the properties of conditional expectations in von neumann algebras the concept of subdynamics used here allows for memory effects no assumption is made of a markov property for dynamical systems based on commutative algebras of observables the subdynamics always exists as a stochastic process in the random variable defining the open subsystem

quantum theory of dissipative processes the markov approximation revisited adopting the standard mathematical framework for describing reduced dynamics we derive two formal identities for the density operator of an open quantum system each of these is equivalent to the old nakajima zwanzig equation the first identity is local in time it contains the inverse of the dynamical map which govern the evolution of the density operator we indicate a time interval on which this inverse exists the second identity constitutes a suitable starting point for going beyond the markov approximation in a controlled way on the basis of the bloch equations we argue once more that in studying quantum dissipation one has to pay attention to the von neumann conditions in the nakajima zwanzig equation we make the first born approximation the ensuing master equation possesses the correct weak coupling limit while proving this rather obvious but at the same time important statement we elucidate the mathematical methods which underlie the weak coupling limit moving to a two dimensional hilbert space we find that both for short and for long times our approximate master equation respects the von neumann conditions assuming exponential decay for correlation functions we propose a physical limit in which the solutions for the density operator become markovian in character we confirm the well known statement that as seen from a macroscopic standpoint the system starts from an effective initial condition the approach to equilibrium is exponential the accessory relaxation constants can differ from the usual bloch parameters and by more than 50

multiaccess quantum channels shared communication channels are subject to multiple access interference transmitter and receiver design techniques that explicitly deal with this interference have been shown to improve substantially the performance of communication systems over radio frequency and other classical channels quantum multiple access communication channels on the other hand have received comparatively little attention in this paper an input output model for multiple access quantum channels relevant to optical communications is proposed the model accounts for multiaccess interference signal attenuation and random noise and can be used in the analysis and design of communication systems using a result from optimization a perturbation method is developed to find the minimum achievable error probability in small interference channels it is shown that the quantum measurement that minimizes the error probability in a no interference channel is robust in the presence of small multiaccess interference the results are illustrated with numerical examples which show that optimal quantum detectors can significantly outperform conventional detectors even for moderate levels of crosstalk

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

conditional expectations in von neumann algebras and a theorem of takesaki conditional expectations play an important role in classical probability theory in the general context of von neumann algebras they were impliciteiy used by von neumann 41 chap ii and by dixmier 14 nakamura and turumaru 27 and umegaki 36 391 introduced an axiomatic definition of the concept of conditional expectation in the framework of von neumann or c algebras and established many properties of these objects especially in the context of von neumann algebras with a finite trace their starting point was the characterization given by moy 26 of the classical conditional expectations as operators on spaces of measurable functions tomiyama showed 33 that conditional expectations in the sense of the above mentioned authors can be characterized as norm one projection in c algebras the importance of norm one projection in the classification problem of von neumann algebras was recognized by hakeda and tomiyama 23 and subsequent research on this argument confirmed the usefulness of these objects this line of thought culminated in the fundamental work of connes 121 in which approximately finite von 245 0022 1236 82 020245 29 02 00 o

parametric reduced order models for predicting the vibration response of complex structures with component damage and uncertainties modeling and fast reanalysis techniques are proposed for predicting the dynamic response of complex structures with uncertainty represented by parameter variability in geometric and material properties at component level the novel models allow for accurate reanalyses and are useful in many applications where the model of the pristine structure may not capture the changes in the system level response due to component level parameter variations herein such models are obtained by using a novel approach based on a modified concept of component mode synthesis the novel models referred to as parametric reduced order models are developed for the general case of multiple substructures with parameter variabilities three types of parameteric variabilities are considered a geometric thickness variability b structural deformations dents and c cracks for the first case a novel parametrization of component level mass and stiffness matrices is employed to predict the system level response for the second case a novel approximate method based on static mode compensation is implemented for the third case cracks a generalized formulation for the bi linear frequency approximation is used the predicted vibration responses of complex structures are shown to agree very well with results obtained using a much more computationally expensive commercial tool

order reduction of large scale second order systems using krylov subspace methods in order reduction of large scale linear time invariant systems krylov subspace methods based on moment matching are among the best choices today however in many technical fields models typically consist of sets of second order differential equations and krylov subspace methods cannot directly be applied two methods for solving this problem are presented in this paper 1 an approach by su and craig is generalized and the number of matching moments is increased 2 a new approach via first order models is presented resulting in an even higher number of matching moments both solutions preserve the specific structure of the second order type model

weak hopf bialgebras in this chapter the r bialgebroids and hopf algebroids of the previous chapter are investigated further in the particular case when the base algebra r carries a so called separable frobenius structure separable frobenius structures on some algebra r are shown to correspond to separable frobenius structures on the forgetful functor from the category of r bimodules to the category of vector spaces based on that the bijection of chap 5 is refined to bijections between three structures for any algebra a first monoidal structures on the category of a modules together with separable frobenius structures on the forgetful functor to the category of vector spaces second bialgebroid structures on a over some base algebra r together with separable frobenius structures on r finally weak bialgebra structures on a the r bialgebroid a is a hopf algebroid if and only if the corresponding weak bialgebra a is a weak hopf algebra

the weak theory of monads abstract we construct a weak version em w k of lack and street s 2 category of monads in a 2 category k by replacing their compatibility constraint of 1 cells with the units of monads by an additional condition on the 2 cells a relation between monads in em w k and composite pre monads in k is discussed if k admits eilenberg moore constructions for monads we define two symmetrical notions of weak liftings for monads in k if moreover idempotent 2 cells in k split we describe both kinds of weak lifting via an appropriate pseudo functor em w k k weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses weak bialgebras are characterized as algebras and coalgebras such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads

evolution matrix in a coherence vector formulation for quantum markovian master equations of n level systems quantum markovian master equation in n dimensions are systematically transformed into vector form by using the lie algebra of the special unitary group su n density operators are represented through coherence vectors and the infinitesimal kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable in general a complete classification of the spectrum of the latter and its jordan canonical form together with all types of associated stationary states is given in detail the results of this analysis are particularly suited for computations in practical applications

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

a square root method for the density matrix and its applications to lindblad operators abstract the evolution of open systems subject to both hamiltonian and dissipative forces is studied by writing the nm element of the time t dependent density matrix in the form nm t 1 a 1 a n t m t the so called square root factors the t s are non square matrices and are averaged over a systems of the ensemble this square root description is exact evolution equations are then postulated for the t factors such as to reduce to the lindblad type evolution equations for the diagonal terms in the density matrix for the off diagonal terms they differ from the lindblad equations the square root factors t are not unique and the equations for the t s depend on the specific representation chosen two criteria can be suggested for fixing the choice of t s one is simplicity of the resulting equations and the other has to do with the reduction of the difference between the t formalism and the lindblad equations when the method is tested on cases which have been previously treated by other methods our results agree with them examples chosen are i molecular systems such that are either periodically driven near level degeneracies for which we calculate the decoherence occurring in multiple landau zener transitions or else when undergoing descent around conical intersections in the potential surfaces ii formal dissipative systems with lindblad type operators representing either a non markovian process or a two state system coupled to bosons attractive features of the present factorization method are complete positivity the no higher than linear increase of the implementation effort with the number of states involved and the introduction of randomness only at the start of the process

a method for interpolating on manifolds structural dynamics reduced order models a rigorous method for interpolating a set of parameterized linear structural dynamics reduced order models roms is presented by design this method does not operate on the underlying set of parameterized full order models hence it is amenable to an online real time implementation it is based on mapping appropriately the rom data onto a tangent space to the manifold of symmetric positive definite matrices interpolating the mapped data in this space and mapping back the result to the aforementioned manifold algorithms for computing the forward and backward mappings are offered for the case where the roms are derived from a general galerkin projection method and the case where they are constructed from modal reduction the proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input in all cases good accuracy is demonstrated at real time processing speeds copyright 2009 john wiley sons ltd

non markovian behavior in low temperature damping an application of the averaging method abstract we give a description of the reduced dynamics of an open quantum system whose initial state is in general correlated with the surroundings the averaging method whose mathematical theory was developed in another paper is used to construct a series expansion of the reduced dynamics in powers of the ratio 1 0 of the characteristic relaxation times 1 and 0 of the surroundings and of the system respectively to first order in 1 0 we obtain the well known markovian approximation which corresponds to the weak coupling limit starting from second order in 1 0 there appear various kinds of non markovian effects including initial slips which have attracted much attention in the recent literature the relevance of these higher order corrections stems from the fact that the characteristic time 1 cannot be much smaller than h kt for quantum surroundings at temperature t

weak hopf algebras and weak yang baxter operators abstract in this paper we introduce the notions of weak yang baxter operator and weak braided hopf algebra we prove that it is possible to obtain examples of these notions working with yetter drinfeld modules associated to a weak hopf algebra h with invertible antipode finally we complete the study of the structure of weak hopf algebras with a projection obtaining a categorical equivalence between the category of weak hopf algebra projections associated to h and the category of hopf algebras in the non strict braided monoidal category of left left yetter drinfeld modules over h

a constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex dwi in this paper we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion weighted images dwi the constrained variational principle involves the minimization of a regularization term of l sup p norms subject to a nonlinear inequality constraint on the data the data term we employ is the original stejskal tanner equation instead of the linearized version usually employed in literature the complex valued nonlinear form leads to a more accurate when compared to the linearized version estimate of the tensor field the inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor finally in order to accommodate the positive definite constraint on the diffusion tensor it is expressed in terms of cholesky factors and estimated the constrained variational principle is solved using the augmented lagrangian technique in conjunction with the limited memory quasi newton method experiments with complex valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

determinism and randomness in quantum dynamics a class of models is considered where a finite or infinite quantum dynamical system in a stationary state is probed by sequences of observations acting on a specified finite subsystem the whole set of such experiments is described by a time ordered causal and stationary quantum correlation kernel it is shown that any such kernel can be decomposed in a unique way into a convex combination of two kernels here called the regular and singular components a singular system has a strong deterministic property the predictability of the future from the knowledge of the past is limited only by the inevitable indeterminism of quantum measurements furthermore in this case the full set of correlation functions of arbitrary time order and hence the dynamical system itself is determined by the causal kernel finite systems and infinite systems satisfying the kms condition at finite temperature are of this type in the regular case the dynamics contains a shift there is a genuine asymptotic randomness and the dynamical system cannot be reconstructed in a unique way from the causal kernel non trivial quantum markov processes are shown to belong to this class

impact of positivity and complete positivity on accessibility of markovian dynamics we consider a two dimensional quantum control system evolving under an entropy increasing irreversible dynamics in the semigroup form considering a phenomenological approach to the dynamics we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive in particular we characterize the family of maps having different accessibilities and show the impact of that property on observable quantities by means of a simple physical model

frobenius monads and pseudomonoids six equivalent definitions of frobenius algebra in a monoidal category are provided in a monoidal bicategory a pseudoalgebra is frobenius if and only if it is star autonomous autonomous pseudoalgebras are also frobenius what it means for a morphism of a bicategory to be a projective equivalence is defined this concept is related to strongly separable frobenius algebras and weak monoidal morita equivalence wreath products of frobenius algebras are discussed

brownian motion as a problem of eliminating fast variables abstract the hamilton equations for a brownian particle involve its mass as a large parameter as a consequence its motion is relatively slow and the fast motion of the surrounding fluid molecules can be eliminated by the standard method of eliminating fast variables the result is equivalent to the known langevin equation plus additional higher orders all coefficients are expressed in terms of correlation functions of the microscopic force on the particle

reduced dynamics need not be completely positive the reduced dynamics of a quantum system in contact with a reservoir is generally thought to be completely positive this is certainly true if product initial conditions are used to define the dynamics we show that with correlated initial conditions it need not be so in this case the dynamics can properly be defined only on a subset of initial system states extension by linearity to all possible initial states is trivially possible but the extension may not be physically realizable and may not even be positive let alone completely positive

theory of spin relaxation processes abstract the relaxation properties of a spin system weakly coupled to lattice degrees of freedom are described using an equation of motion for the spin density matrix this equation is derived using a general weak coupling theory which has been previously developed to second order in the weak coupling parameter the results are in agreement with those obtained by bloch wangsness and redfield but the derivation does not make use of second order perturbation theory for short times the derivation can be extended beyond second order and ensures that the spin density matrix relaxes to its exact equilibrium form to the appropriate order in the weak coupling parameter

phenomenological criteria for the validity of quantum markovian equations we revisit the constraints on the coefficients of a quantum markovian equation recently obtained by dekker and valsakumar we extend these criteria for time dependent coefficients and establish further criteria where appropriate phenomenological behaviour is required such as dissipation and a boltzmann asymptotic equilibrium state

systematic elimination of fast variables in linear systems we consider linear systems in which some of the variables evolve on a much faster time scale than the remaining ones by means of a systematic perturbation theory in the time scale ratio we extract a reduced dynamics in terms of the slow variables only valid after an initial transient period the procedure provides a systematic extension of the usual adiabatic elimination scheme and gives corrections to it we also find an expression for the long time behavior of the correlation functions of the slow variables these asymptotic expressions do not extrapolate back towards the equal time correlations for t going to zero the reason for this initial slip is given and its magnitude calculated method and results are illustrated with a simple example of coupled oscillators

the fundamental theorem for weak braided bimonads abstract the theories of hopf bialgebras and weak hopf bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product as first generalisations these notions were formulated for monoidal categories with braidings if needed the present authors developed a theory of bimonads and hopf monads h on arbitrary categories a employing distributive laws allowing for a general form of the fundamental theorem for hopf algebras for bimonads h properties of braided hopf bialgebras were captured by requiring a yang baxter operator h h h h the purpose of this paper is to extend the features of weak hopf bialgebras to this general setting including an appropriate form of the fundamental theorem this subsumes the theory of braided hopf algebras based on weak yang baxter operators as considered by alonso alvarez and others

an interpolation scheme for the approximation of dynamical systems abstract the demand of many industries for highly accurate predictions of the structural response of dynamical systems leads to a rapidly growing complexity of the established numerical models due to the thereby arising large computational efforts the use of model reduction techniques becomes an essential task model reduction procedures in the context of deterministic structural analysis are already well developed and widely used such as guyan reduction and component mode synthesis techniques however the assumption of deterministically known structural properties is in most cases not realistic hence the rational propagation of uncertainties becomes an indispensable task in the robust prediction of the structural response the associated multiple simulation runs of complex models bring the topic of model reduction to the foreground of technical and scientific interest in this work a novel method which approximates eigenfrequencies and mode shape vectors needed for the evaluation of frequency response functions will be presented this meta model of structural matrices is established by employing interpolation between the matrices evaluated at some supporting points the technique has no difficulties with closely spaced eigenfrequencies and mode switching and it can be employed over the entire frequency range on the full model but also on the substructures and it is hence applicable together with deterministic model reduction techniques

solutions of generic bilinear master equations for a quantum oscillator positive and factorized conditions on stationary states abstract we obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the gaussian form the well behavedness and positive semidefiniteness of the stationary states could be characterized by a three dimensional minkowski vector by requiring the stationary states to satisfy a factorized condition we obtain a generic class of master equations that includes the well known ones and their generalizations some of which are completely positive a further subset of the master equations with the gibbs states as stationary states is also obtained for master equations with not completely positive generators an analysis on the stationary states for a given initial state suggests conditions on the coefficients of the master equations that generate positive evolution

hopf monads on monoidal categories we define hopf monads on an arbitrary monoidal category extending the definition given in bruguieres and virelizier 2007 5 for monoidal categories with duals a hopf monad is a bimonad or opmonoidal monad whose fusion operators are invertible this definition can be formulated in terms of hopf adjunctions which are comonoidal adjunctions with an invertibility condition on a monoidal category with internal homs a hopf monad is a bimonad admitting a left and a right antipode hopf monads generalize hopf algebras to the non braided setting they also generalize hopf algebroids which are linear hopf monads on a category of bimodules admitting a right adjoint we show that any finite tensor category is the category of finite dimensional modules over a hopf algebroid any hopf algebra in the center of a monoidal category c gives rise to a hopf monad on c the hopf monads so obtained are exactly the augmented hopf monads more generally if a hopf monad t is a retract of a hopf monad p then p is a cross product of t by a hopf algebra of the center of the category of t modules generalizing the radford majid bosonization of hopf algebras we show that the comonoidal comonad of a hopf adjunction is canonically represented by a cocommutative central coalgebra as a corollary we obtain an extension of sweedler s hopf module decomposition theorem to hopf monads in fact to the weaker notion of pre hopf monad

on the relationship betweent1 andt2 for stochastic relaxation models we consider the relaxation dynamics of two quantum levels coupled to a stochastic bath we emphasize that even if the matrix elements of the fluctuating hamiltonian are gaussian a second order cumulant truncation is not exact for various stochastic models including the case of a spin 1 2 particle in a fluctuating magnetic field we calculate 1 t1 the population relaxation rate and 1 t2 the phase relaxation rate up to fourth order in perturbation theory we show that unlike the commonly accepted second order result that 1 t2 1 2t1 when fourth order terms are included in some instances 1 t2 1 2t1

weak bimonads and weak hopf monads we define a weak bimonad as a monad t on a monoidal category m with the property that the eilenberg moore category mt is monoidal and the forgetful functor mt m is separable frobenius whenever m is also cauchy complete a simple set of axioms is provided that characterizes the monoidal structure of mt as a weak lifting of the monoidal structure of m the relation to bimonads and the relation to weak bimonoids in a braided monoidal category are revealed we also discuss antipodes obtaining the notion of weak hopf monad

reference state for arbitrary u consistent subspace the reduced dynamics of the system s interacting with the environment e is not given by a linear map in general however if it is given by a linear map then this map is also hermitian in order that the reduced dynamics of the system is given by a linear hermitian map there must be some restrictions on the set of possible initial states of the system environment or on the possible unitary evolutions of the whole se in this paper adding an ancillary reference space r we assign to each convex set of possible initial states of the system environment for which the reduced dynamics is hermitian a tripartite state which we call the reference state such that the set is given as the steered states from the reference state the set of possible initial states of the system is also given as the steered set from a bipartite reference state the relation between these two reference states is as where is the identity map on r and is a hermitian assignment map from s to se as an important consequence of introducing the reference state we generalize the result of buscemi 2014 phys rev lett 113 140502 we show that for a u consistent subspace the reduced dynamics of the system is completely positive for arbitrary unitary evolution of the whole system environment u if and only if the reference state is a markov state in addition we show that the evolution of the set of system environment system states is determined by the evolution of the reference state

virtues and limitations of markovian master equations with a time dependent generator a markovian master equation with time dependent generator is constructed that respects basic constraints of quantum mechanics in particular the von neumann conditions for the case of a two level system bloch equations with time dependent parameters are obtained necessary conditions on the latter are formulated by employing a time local counterpart of the nakajima zwanzig equation we establish a relation with unitary dynamics we also discuss the relation with the weak coupling limit on the basis of a uniqueness theorem a standard form for the generator of time local master equations is proposed the jaynes cummings model with atomic damping is solved the solution explicitly demonstrates that reduced dynamics can be described by time local master equations only on a finite time interval this limitation is caused by divergencies in the generator a limit of maximum entropy is presented that corroborates the foregoing statements a second limiting case demonstrates that divergencies may even occur for small perturbations of the weak coupling regime

positive maps positive polynomials and entanglement witnesses we link the study of positive quantum maps block positive operators and entanglement witnesses with problems related to multivariate polynomials for instance we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares although the general problem of describing the set of positive maps remains open in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity

complete positivity for time dependent qubit master equations it is shown that if the decoherence matrix corresponding to a qubit master equation has a block diagonal real part then the evolution is determined by a one dimensional oscillator equation further when the full decoherence matrix is block diagonal then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator hamiltonian or lagrangian when the solution of the oscillator equation is not known an explicit sufficient condition for complete positivity can still be obtained based on a hamiltonian lagrangian inequality a rotational form invariance property is used to characterize the evolution via a single first order nonlinear differential equation enabling some further exact results to be obtained a class of master equations is identified for which complete positivity reduces to the simpler condition of positivity

a reduced order matrices fitting scheme with log euclidean metrics for fast approximation of dynamic response of parametric structural systems abstract structural engineering practice often involves tasks such as design optimization or statistical analysis where iterative solutions of dynamical systems are required varying some parameter which affect the system matrices in a generally non linear way the approach of interpolating among matrices of reduced order models roms is not new in the literature and it is very promising in order to speed up the calculations in particular the method shows a great potential since it is applicable to any kind of fe model and of geometry with no restrictions regarding the element type constituting the full order model or the range of parameter variability yet dealing with symmetric positive definite spd full order matrices the spd nature of the reduced matrices must be preserved to this end a special mathematical framework must be used which allows to linearize the curved space of spd matrices in the literature of roms interpolation the common choice is to rely on the principles of riemannian geometry through the concept of the tangent plane to the spd manifold although mathematically robust this approach brings a marked distortion in the space of spd matrices also it shows numerical instabilities when dealing with strongly anisotropic matrices in this work a polynomial fitting of the reduced matrices is proposed with log euclidean metrics these metrics were obtained by giving the spd manifold the structure of a lie group while maintaining the already proposed approach of interpolating between a number of sampling roms the advantages of using log euclidean metrics are illustrated in details in particular a great improvement is shown in case of a unique reduction matrix whose columns span the solution of the parameter space of interest in an accurate enough way and in case of substructuring reduction techniques the reported examples of applications show excellent accuracy while maintaining the dramatic decrease in the computational time of the interpolatory scheme