Backwards time

Thompson K and Makri N 1999 Influence functionals with semiclassical propagators in combined fonA/ard-backward time J. Chem. Phys. 110 1343... 

The second and third ternns on the right-hand side of Equation (3.81) are approximated using known values at a backward time step as... 

So far, the Lagrangian density for a homogenous problem (no sink or source term in the diffusion equation) has been considered, subject to the requirement that the approximate trial function, ip, can be forced to satisfy the boundary conditions. In this sub-section, these limitations are removed and the Lagrangian density for the Green s function developed. The Green s functions for the forward and backward time process satisfy the equations... 

Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwarder backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller [19, 33, 35, 48]. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Reciprocally, the trajectories that approach the... 

For that purpose a secure timeserver is introduced. In this role an agent running within the platform is acting. The time agent supports two modes of operation as described in [1] forward-time hash function and backward-time hash function. The forward-time constructions permit key generation only after a given time. And the backward-time constructions permit key generation only before it. 

The backward-time hash function construction has the following steps (symbols as in the previous example) [1] ... 

Similarly, in order to use backward-time hash function construction there are additional steps required in the protocol. However, this time application first queries the Time Agent for the target time s secret and only then queries the Platform Manager for target container s secret. In the backward-time hash function construction, as described in the previous paragraph, the target... 

Time Agent is an agent that implements forward- and backward-time hash function constructions. There is a one instance of the Time Agent running in the agent platform. 

The forward- and backward time hash functions allows the creation of mobile agents that are able to display the content simultaneously on many target devices after certain point in time or the agents carrying the content with expiration date set. 

Backward Time Central Space (BTCS) Implicit Scheme... 

Because of the off-diagonal nature of the electronic Hamiltonian in the di-abatic basis, transitions among different electronic states can, and in general will, occur during the forward and backward time evolutions. As a first step towards a computationally convenient expression for the non-adiabatic correlation function, we shall make use of the mapping Hamiltonian formalism to account for these transitions [28-31]. 

Finally when a >0 (Figure 5.1.5e), x becomes unstable, due to the exponential growth in the x-direction. Most trajectories veer away from x and head out to infinity. An exception occurs if the trajectory starts on the y-axis then it walks a tightrope to the origin. In forward time, the trajectories are asymptotic to the x-axis in backward time, to the y-axis. Here x =0 is called a saddle point. The y-axis is called the stable manifold of the saddle point x, defined as the set of initial conditions Xg such that x(z) x as t -> o . Likewise, the unstable manifold of X is the set of initial conditions such that x(z) x as z. Here the unstable manifold is the x-axis. Note that a typical trajectory asymptotically approaches the unstable manifold as z —> o , and approaches the stable manifold as z —> -oo. This sounds backwards, but it s right ... 

The electric conductance of a molecular junction is calculated by recasting the Keldysh formalism in Liouville space. Dyson equations for non-equilibrium many-body Green functions (NEGF) are derived directly in real (physical) time. The various NEGFs appear naturally in the theory as time-ordered products of superoperators, while the Keldysh forward/backward time loop is avoided. 

Despite truncation errors after 1,000 time steps, the last tabulation-plot in Figure 21-3b shows that we have recaptured the step initial condition in tlmee ways we (1) obtained the exact left-to-right concentration values of 10% and 90%, (2) correctly imaged the transition boundary between the x = 5 to 6 ft nodes, and (3) extracted the two solutions just quoted using exactly the same number of backward time steps as we did forward time steps. In time lapse analysis, the... 

The data that we use, have TTF values between 1 and 36 months, since the product has 3 year warranty period. In our model, we do analysis for forward and backward time windows. While forward analysis is a widely used method in warranty analysis, investigating TTF values from backward is a new method that we introduce for accurate determination of the change point x and the hazard functions h i) and h i). In our method, and are used as the number of months that constitute the boundaries of time windows. In Figure 5, an explicit demonstration of the time windows are shown. 

Figure 5. Demonstration of forward and backward time windows.

In order to completely characterize the dynamics of a system, we must introduce a system propagator U(/) that bears information about the time-reversal properties of the system by including both its forward and backward time evolution. The itsual propagator exp(-Z,/) is insuflacient when the transition operator L =iC possesses broken time-reversal symmetry, i.e., when -Z, where Z is the time-reversed form of L. Of cotuse, the usttal propagator exp(-Z) is sufficient when we are deahng with a reversible system, i.e., when the symmetry relation L = -L holds. 

On the basis of time-reversal argrrments, one can assert that the backward time evolution of a system is given by 0(-f)exp(Zf). Hence, the propagator 0(f) assitmes the form... 

It follows from the above theorem that a rough system on the plane may possess only rough equilibrium states (nodes, foci and saddles) and rough limit cycles. As for separatrices of saddles, they either tend asymptotically to a node, a focus, or a limit cycle in forward or backward time, or leave the region G after a finite interval of time. 

E) there exists no separatrix which tends to a homoclinic loop of a saddle (in forward or backward time), as depicted in Fig. 8.1.5. 

As /i —0, the stable manifold of L fx) approaches the component of W 0) W 0) which contains F. The characteristic form of in the small neighborhood U of F is sketched in Fig. 13.4.11. Here, the local stable manifold can be continued along F in backward time so that it returns to the small neighborhood of O. Since A 0, the manifold is transverse to and therefore it is limited to the manifold as t —> — oo. This geometry of is a direct consequence of the non-vanishing of the value of A. We remark, however, that in some cases, even if A = 0, the manifold may be still limited to in backward time. 

Fig. 13.4.11. Behavior of the stable manifold Wf in backward time near a homoclinic loop to the saddle with real Ai, provided the non-degeneracy assumptions hold.

Fig. 13.4.14. The stable manifold of the saddle-focus continued along the homoclinic loop in backward time has a helicoid form.

The coupled set of flow and solid equations (Eqs. (3.1)-(3.8), (3.11)) were solved simultaneously. A finite volume scheme was adopted for the spatial discretization of the flow equations and solution was obtained with a SIMPLER method for the pressure-velocity field [8]. In the case of transient simulation, the 1-D transient solid energy equation was solved with a second order accurate, fully implicit scheme by using a quadratic backward time discretization [9]. The coupled flow and solid phases were solved iteratively and convergence was achieved at each time step when the solid temperamre did not vary at any position along the wall by more than 10 K. 

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