Time-dependent Operator Research Articles

On Subordination Chains with Normalization Given by a Time-dependent Linear Operator

In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball Bn in \({\mathbb{C}^n}\). The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases of this result have been recently obtained for subordination chains with normalization \({Df(0,t)=e^tI_n}\) or Df(0, t) = etA, t ≥ 0, where \({A\in L(\mathbb{C}^n,\mathbb{C}^n)}\). We also determine the form of the standard solutions to the Loewner differential equation associated with generalized spirallike mappings. In the last section we obtain the form of the solution in the presence of coefficient bounds.

A generalization of Chernoff's product formula for time-dependent operators

In this article we provide a set of sufficient conditions that allow a natural extension of Chernoff's product formula to the case of certain one-parameter family of functions taking values in the algebra L ( B ) of all bounded linear operators defined on a complex Banach space B . Those functions need not be contraction-valued and are intimately related to certain evolution operators U ( t , s ) 0 ⩽ s ⩽ t ⩽ T on B . The most direct consequences of our main result are new formulae of the Trotter–Kato type which involve either semigroups with time-dependent generators, or the resolvent operators associated with these generators. In the general case we can apply such formulae to evolution problems of parabolic type, as well as to Schrödinger evolution equations albeit in some very special cases. The formulae we prove may also be relevant to the numerical analysis of non-autonomous ordinary and partial differential equations.

Some statistical properties for the interaction between SU(1, 1) and SU(2) quantum systems

In this paper we introduce a Hamiltonian model which represents the interaction between SU(1, 1) and SU(2) quantum systems. The Heisenberg equations of motion are invoked to derive the exact time-dependent dynamical operators. The phenomena of the collapse and revival are observed for the intermediate states, however, for a large value of the excitation number m. It is also shown that the phenomenon of squeezing is observed in the intermediate state. Moreover, it is sensitive to the variation in the coupling parameter λ, the detuning parameter Δ, as well as in the Bargmann index k. The examination of the correlation function shows that the system displays anti-bunching for all periods of time except for a large value of the excitation number when k = 3/4. Our discussion for the variance squeezing also shows that the phenomenon of squeezing is pronounced in the quadrature variances for the odd parity case when the detuning parameter is large.

Development of a predictive model for long-term survival after lung transplantation and implications for the lung allocation score

Improving long-term survival after lung transplantation can be facilitated by identifying patient characteristics that are predictors of positive long-term outcomes. Validated survival modeling is important for guiding clinical decision-making, case-mix adjustment in comparative effectiveness research and refinement of the lung allocation system (LAS). We used the registry of the International Society for Heart and Lung Transplantation (ISHLT) to develop and validate a predictive model of 5-year survival after lung transplantation. A total of 18,072 eligible cases were randomly split into development and validation datasets. Pre-transplant recipient variables considered included age, gender, diagnosis, body mass index, serum creatinine, hemodynamic variables, pulmonary function variables, viral status and comorbidities. Predictors were considered in a stepwise approach with the Akaike Information Criteria (AIC). Time-dependent receiver operator characteristic (ROC) curves assessed predictive ability. A 1-year conditional model and three models for disease subgroups were considered. ROC methods were used to characterize the predictive potential of the LAS post-transplant model at 1 and 5 years. The baseline model included age, diagnosis, creatinine, bilirubin, oxygen requirement, cardiac output, Epstein-Barr virus status, transfusion history and diabetes history. Prediction of long-term survival was poor (area under the curve [AUC] = 0.582). Neither the 1-year conditional model (AUC = 0.573) nor models designed for separate diseases (AUC = 0.553 to 0.591) improved survival prediction. The predictive ability of the LAS post-transplant parameters was similar to that of our model (1-year AUC = 0.580 and 5-year AUC = 0.566). Models developed from pre-transplant characteristics poorly predict long-term survival. Models for separate diseases and 1-year conditional models did not improve prediction. Better databases and approaches to predict survival are needed to improve lung allocation.

Multi-Product Expansion with Suzuki’s Method: Generalization

In this paper we discuss the extension to exponential splitting methods with respect to time-dependent operators. We concentrate on the Suzuki’s method, which incorporates ideas to the time-ordered exponential of [3,11,12,34]. We formulate the methods with respect to higher order by using kernels for an extrapolation scheme. The advantages include more accurate and less computational intensive schemes to special time-dependent harmonic oscillator problems. The benefits of the higher order kernels are given on different numerical examples.

Nonlinear Stability of Time-Periodic Viscous Shocks

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.

Wave-driven dynamo action in spherical magnetohydrodynamic systems

Hydrodynamic and magnetohydrodynamic numerical studies of a mechanically forced two-vortex flow inside a sphere are reported. The simulations are performed in the intermediate regime between the laminar flow and developed turbulence, where a hydrodynamic instability is found to generate internal waves with a characteristic m=2 zonal wave number. It is shown that this time-periodic flow acts as a dynamo, although snapshots of the flow as well as the mean flow are not dynamos. The magnetic fields' growth rate exhibits resonance effects depending on the wave frequency. Furthermore, a cyclic self-killing and self-recovering dynamo based on the relative alignment of the velocity and magnetic fields is presented. The phenomena are explained in terms of a mixing of nonorthogonal eigenstates of the time-dependent linear operator of the magnetic induction equation. The potential relevance of this mechanism to dynamo experiments is discussed.

Driving quantum-walk spreading with the coin operator

We generalize the discrete quantum walk on the line using a time dependent unitary coin operator. We find an analytical relation between the long-time behaviors of the standard deviation and the coin operator. Selecting the coin time sequence allows to obtain a variety of predetermined asymptotic wave-function spreadings: ballistic, sub-ballistic, diffusive, sub-diffusive and localized.

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.

Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator

Stiy accurate implicit Runge{Kutta methods are studied for the time discretisation of nonlinear rst-order evolution equations. The equation is supposed to be governed by a time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fullls a certain growth condi- tion. It is proven that the piecewise constant as well as the piecewise linear interpolant of the time-discrete solution converge towards the exact weak solu- tion, provided the Runge{Kutta method is consistent and satises a stability criterion that implies algebraic stability; examples are the Radau IIA and Lo- batto IIIC methods. The convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part.

Diagonalization of Weyl Transforms and Heat Equations for Time-Dependent Hermite Operators

Weyl transforms with radial symbols are diagonalized in terms of explicit formulas for the eigenvalues with respect to the Hermite basis for \({L^2(\mathbb{R})}\) . The exact solutions of heat equations governed by time-dependent Hermite operators are analyzed in detail. Formulas for the heat kernels of these time-dependent Hermite operators are derived.

An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner

The projection method is a widely used fractional-step algorithm for solving the incompressible Navier–Stokes equations. Despite numerous improvements to the methodology, however, imposing physical boundary conditions with projection-based fluid solvers remains difficult, and obtaining high-order accuracy may not be possible for some choices of boundary conditions. In this work, we present an unsplit, linearly-implicit discretization of the incompressible Navier–Stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations. Since our scheme is not a fractional-step algorithm, it is straightforward to specify general physical boundary conditions accurately; however, this capability comes at the price of having to solve the time-dependent incompressible Stokes equations at each timestep. To solve this linear system efficiently, we employ a Krylov subspace method preconditioned by the projection method. In our implementation, the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning. The accuracy of the scheme is demonstrated for several problems, including forced and unforced analytic test cases and lid-driven cavity flows. These tests consider a variety of physical boundary conditions with Reynolds numbers ranging from 1 to 30000. The effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the Schur complement for the time-dependent incompressible Stokes operator. The projection method is found to be a more efficient preconditioner in most cases considered in the present work.

Variable time-step ϑ-scheme for nonlinear evolution equations governed by a monotone operator

The single-step ϑ-scheme on a variable time grid is employed for the approximate solution of the initial-value problem for a nonlinear first-order evolution equation. The evolution equation is supposed to be governed by a possibly time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. A piecewise constant as well as piecewise linear prolongation of the time-discrete solution is shown to converge towards the exact solution if ϑ≥1/2 (including the Crank-Nicolson scheme). In the appearance of a strongly continuous perturbation of the monotone main part, the method is still convergent if ϑ>1/2 and if the ratio of adjacent step sizes is bounded from above by a power of ϑ/(1−ϑ). Besides convergence also well-posedness of the time-discrete problem as well as a priori error estimates are studied.

A quantum statistical estimation of the amount of energy dissipated by molecular relaxation processes in liquids

The physical conditions concerning the far-infrared (FIR) absorption process in molecular liquids are discussed. It is shown that the well-known first-order perturbational treatment of the fluctuation-dissipation theorem (FDT) cannot be justified within quantum statistics. The basic physical concepts concerning the derivation and validity of the generalized FDT as revealed in an earlier article are discussed. It is shown that dissipation of irradiation energy is mainly due to the coupling V of the molecular system with the thermal bath, as proposed by van Vliet. The following cases are treated: (i) The coupling V leads to a time-dependent density operator. It turns out that nonvanishing dissipation is due to dynamical fluctuations of the density operator. (ii) The operator m(t) which represents the coupling of the system with an external field obeys, for V ≠ 0, a non-Markovian equation of motion. The irradiation energy dissipation is a direct consequence of memory effects concerning m(t). The “anomalous” temperature dependence of the FIR absorption bands of CH3CN is presented and explained by means of the generalized FDT.

Time-dependent Maxwell field operators and field energy density for an atom near a conducting wall

We consider the time evolution of the electric and magnetic field operators for a two-level atom, interacting with the electromagnetic field, placed near an infinite perfectly conducting wall. We solve iteratively the Heisenberg equations for the field operators and obtain the electric and magnetic energy density operators around the atom (valid for any initial state). Then we explicitly evaluate them for an initial state with the atom in its bare ground state and the field in the vacuum state. We show that the results can be physically interpreted as the superposition of the fields propagating directly from the atom and the fields reflected on the wall. Relativistic causality in the field propagation is discussed. Finally we apply these results to the calculation of the dynamical Casimir-Polder interaction energy in the far zone between two atoms when a boundary condition such as a conducting wall is present. Magnetic contributions to the interatomic Casimir-Polder interaction in the presence of the wall are also considered. We show that, in the limit of large times, the known results of the stationary case are recovered.

Mosco convergence of integral functionals and its applications

Questions relating to the Mosco convergence of integral functionals defined on the space of square integrable functions taking values in a Hilbert space are investigated. The integrands of these functionals are time-dependent proper, convex, lower semicontinuous functions on the Hilbert space. The results obtained are applied to the analysis of the dependence on the parameter of solutions of evolution equations involving time-dependent subdifferential operators. For example a parabolic inclusion is considered, where the right-hand side contains a sum of the -Laplacian and the subdifferential of the indicator function of a time-dependent closed convex set. The convergence as of solutions of this inclusion is investigated.Bibliography: 20 titles.

Time-dependent approach to the continuum shell model

The continuum shell model represents a merger of the traditional shell model, the tool for understanding nuclear structure, with the physics of reactions. In this work a new time-dependent approach to the continuum shell model is presented, where construction and application of the time-dependent evolution operator culminate in an effective and successful strategy for tackling the nonstationary many-body dynamics. Details behind the technique and methods to overcome general problems associated with quantum many-body physics on the verge of stability are discussed. Topics presented include the construction of the time-dependent Green's function, the full propagator from the exact solution of Dyson's equation, a treatment of decays and virtual self-energy terms, the explicit time dependence and survival probability of states, the strength function and collective features of unstable systems, the center-of-mass problem, computation of the cross section and its Blatt-Biedenharn angular decomposition, Coulomb amplitudes, and interference. An extensive comparison with the $R$-matrix approach is offered. Realistic examples are used to demonstrate the techniques.

Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.

The dynamic prediction of criminal recidivism: A three-wave prospective study.

A three-wave, prospective panel design was used to assess the extent to which static and dynamic risk factors could predict criminal recidivism in a sample of 136 adult male offenders released from Canadian federal prisons. Static measures were assessed only once, prior to release while dynamic measures were assessed on three separate occasions: pre-release, 1 month, and 3 months post-release. Recidivism was coded during an average of 10.2-month follow-up period (SD=19.2). A series of Cox regression survival analyses with time-dependent covariates and Receiver Operator Characteristic (ROC) analyses were conducted to assess predictive validity. Although the combined static and time-dependent dynamic model (AUC=.89, CI=.81-.93) significantly (p<.01) outperformed the pure static model (AUC=.81, CI=.73-.87) the confidence intervals did overlap to some extent. Implications for dynamic risk assessment and management are discussed.

Nyström discretization of parabolic boundary integral equations

A Nyström method for the discretization of thermal layer potentials is proposed and analyzed. The method is based on considering the potentials as generalized Abel integral operators in time, where the kernel is a time dependent surface integral operator. The time discretization is the trapezoidal rule with a corrected weight at the endpoint to compensate for singularities of the integrand. The spatial discretization is a standard quadrature rule for surface integrals of smooth functions. We will discuss stability and convergence results of this discretization scheme for second-kind boundary integral equations of the heat equation. The method is explicit, does not require the computation of influence coefficients, and can be combined easily with recently developed fast heat solvers.