Bum-in time

Sputtering After ignition of the discharge a bum-in time is required to achieve a sputtering equilibrium. The surface layers have to be removed first, but once equilibrium is reached the discharge penetrates with constant velocity into the sample. Preburning times are usually up to 30 s for metals (at 90 W in argon, zinc 6 s brass ... 

After we have let the chain run a long time, the state the chain is in does not depend on the initial state of the chain. This length of time is called the burn-in period. A draw from the chain after the bum-in time is approximately a random draw from the posterior. However, the sequence of draws from the chain after that time is not a random sample from the posterior, rather it is a dependent sample. In Chapter 3, we saw how we could do inference on the parameters using a random sample from the posterior. In Section 7.3 we will continue with that approach to using the Markov chain Monte Carlo sample from the posterior. We will have to thin the sample so that we can consider it to be approximately a random sample. A chain with good mixing properties will require a shorter burn-in period and less thinning. 

The bum-in time is the number of steps needed for the draw from the chain to be considered a draw from the long-mn distribution. In other words, after the bum-in time the draw from the chain would be the same, no matter where the chain started from. Since we have set up the Markov chain to have the posterior as its long-run distribution, the draw is like a random draw from the posterior. However, consecutive draws from the sample continuing after the bum-in time will not be a random sample from the posterior. Subsequent values will be dependent on the first value drawn after bum-in. 

Independently simulate m independent realizations of the Markov chain, each started independently from an overdispersed starting distribution. They suggest finding all the modes of the target distribution, and letting the starting distribution be approximated by a mixture of Student s t distributions, one centered at each node. Each realization of the chain should be 2n steps. We discard the first n steps of each chain as the proposed bum-in time. 

The Gelman-Rubin statistic indicates the amount of improvement towards convergence that is possible if we allowed n to increase. Values of /A that are less than 1.10 show acceptable convergence. It should be noted that the value of the Gelman-Rubin statistics depends on the actual draws that occurred in the chains. It would give a different value if we repeated the process, so it is only indicative of what the bum-in time and thinning should be. 

It is hard to determine the adequate bum-in time. Examination of the trace plots and examination of the sample autocorrelation functions can be helpful. However, when there are widely separated modal regions, the chain may have... 

Coupling with the past is another method that can be used to determine the bum-in time. It starts a series of parallel chains from overdispersed starting points and uses the same random input values until all chains have conveiged. The number of steps this takes is adequate bum-in time as it shows the state of the chain is independent of the starting point. 

As part of the quality assurance system, overload tests may be performed specifically with the intent of weeding out defective items, and most electrical components are subjected to an initial bum-in time before delivery. 

Now consider the hypothetical problem of trying to teach the physics of space flight during the period in time between the formulation of Kepler s laws and the publication of Newton s laws. Such a course would introduce Kepler s laws to explain why all spacecraft proceed on elliptical orbits around a nearby heavenly body with the center of mass of that heavenly body in one of the focal points. It would further introduce a second principle to describe course corrections, and define the orbital jump to go from one ellipse to another. It would present a table for each type of known spacecraft with the bum time for its rockets to go from one tabulated course to another reachable tabulated course. Students completing this course could run mission control, but they would be confused about what is going on during the orbital jump and how it follows from Kepler s laws. 

The terms rate, speed, and velocity are all synonymous in chemical kinetics, though this is not so in mechanics. It takes different periods of time to complete different reactions. The neutralization reaction between acids and bases, mentioned earlier as an example of homogeneous reactions, takes place almost instantaneously at room temperature and under atmospheric pressure. However, it takes many days for iron to rust under these conditions. Thus, the rates of reactions that may take place under the same conditions of temperature and pressure may differ very significantly. When carbon or sulfur or phosphorus bums in... 

For comparative purposes it is usual to consider the effect when unit weight m of the propellant bums in unit time t. As momentum is equal to impulse in a flow system, then Ft = mV or, as t and m are both equal to 1, F -- V. Thus for the propellant the specific impulse Isp is given by... 

As with slow coke, the fast coke was shown to bum at a rate that is first order in coke and approximately first order in oxygen. However, the activation energy is approximately half that of the slow coke and the frequency factor is 1.0 x 10 sec" in contrast to 3.7 x 10 sec" for slow coke. At 950°F (783 K), the fast coke bums 17 times as fast as the slow coke. 

However, not all pairs of stars will meet such a cataclysmic end. The proof stems from the existence of novas. These also arise from the tempestuous love affairs between a white dwarf and a healthy star. The difference is that their love bums in a more reasonable manner, ejecting only a small portion of then-envelope at a time (roughly 10 " Mq). These ejecta are nevertheless loaded like galleons with radioactive isotopes such as beryllium-7 and sodium-22. One day it is hoped that their signature will be detected by the great INTEGRAL observatory. 

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