Momentum bias

Momentum bias Carrying forward with an existing course of action despite incomplete or even contrary evidence. 

Momentum bias is the tendency to carry forward with an existing course of action despite incomplete or even contrary evidence. In culture change undertakings, this bias can play out as a lack of responsiveness and flexibility. Does the leader try to fix the problem using more of the same (i.e., using an old approach that has momentum in the organization but has proved insufficient in the past) Or does she discover a new approach ... 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

The objective of the method presented here is to develop a momentum distribution that will bias path dynamics along the slow manifold, permitting the efficient calculation of kinetic properties of infrequent reactions. 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

ARPES measurements (k/ = 0.215 0.01 A 1) [43,44]. This method, dubbed Fourier Transform STM (FT-STM) [48], has an energy resolution that depends on the bias voltage and temperature, and a momentum resolution that depends on the size of the image and the broadening of the Fermi contour due to the thermal decay of the standing waves. If both are selected properly this method can compete with the best ARUPS data available. 

The model contains the following elements r is the learning rate p is the momentum factor a, is the activation that accumulates in each hidden or output unit i A., is the activation that spreads from unit i to units above it co, is the weight associated with the connection between units i and / P, is the bias associated with unit i x, is the target level of output activation, externally set as 1 if the output unit i represents the correct situation or 0 if it does not and , is the error associated with unit i. 

Here rp(k) is the momentum relaxation time which is due to the electron-phonon and electron-impurity scattering,stands for the electron distribution functions of spin a, h(k) is the DP term which serves as an effective magnetic field and is composed of the Dresselhaus term [10] due to the bulk inversion asymmetry (BIA) and the Rashba term [11] due to the structure inversion asymmetry (SIA),... 

Figure 1. Isovalue surfaces of A2FI (p) for (a) tolane, 1, in the absence of an external field, (b) tolane thiolate, 2, in the absence of an external field, (c) tolane thiolate in an external field oriented in cooperation with the electron donation of sulfur (forward bias), and (d) tolane thiolate in an external field oriented in opposition to the electron donation of sulfur(reversebias). A2II (p) = -0.025 au everywhere on these surfaces, and A2n (p)< -0.025 au everywhere within these envelopes. The orientations of the molecules, and the momentum-space axes are as described in the "GettingOriented" section.

Figure 13. Total incorporated momentum per deposited atom in dependence on the substrate bias voltage during step 4. In addition the value for optimum cBN nucleation = —150 V 97% Ar/ 3% N2) is shown.

If the total momentum is chosen to be zero, then it will formally remain zero along trajectories (see [78, 254] for a discussion on the affect this has on canonical sampling). However, in practice, a small numerical artifact may be introduced. Moreover, under numerical discretization, bias due to finite stepsize may be present allowing a build up of total momentum in long simulations. 

It is desirable to construct formulations and numerical methods which exactly (i.e. up to rounding error) preserve the total momentum from step to step. One obvious approaeh to this problem is to simply project the momenta onto the linear momenrnm constraint at the end of each step (or after some number of steps). Such a projection introduces potential issues in terms of convergence order and would certainly complicate the analyses presented thus far in this book. Moreover, the optimal choice of projection is unclear and it is easy to define poor schemes (for example, modifying always the momentum of just the first particle in order to balance all the remaining components) which are likely to introduce artifacts (bias) in simulation. For this reason, there is interest in building in momentum conservation into the equations of motion (and indeed the integrator). Ideally this should be done in a localized and homogeneous way so that momentum is not transferred by a nonphysical mechanism between distant particles. 

When re-weighting the track spectrum in minimum bias events the dependence of the fake probability on the transverse momentum and the pseudorapidity is taken into account. The result is shown in Fig. 5.8. 

The muon trigger efficiency has been determined from data in minimum bias events. The statistical uncertainty on the trigger efficiency amounts to 3-5%, depending on the muon transverse momentum and pseudorapidity, and is taken as a systematic uncertainty. The muon reconstruction efficiency is known to a precision of 3%. The tracking efficiency for hadrons is known with a precision of 4%. This induces a systematic uncertainty of 2% on the number of events passing the event selection. The uncertainty in the tracking efficiency affects the b-fraction in the fit by about 1%. 

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