Most probable equation

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

Under the proper conditions (said conditions being that the errors that prevent all the data relationships from being described by a single equation are normally [1, 2] distributed) it can be proven mathematically that the most probable equation is exactly the one that is the least square equation. While we have discussed this point... 

Schrodinger wave equation The fundamental equation of wave mechanics which relates energy to field. The equation which gives the most probable positions of any particle, when it is behaving in a wave form, in terms of the field. 

Equation (2.28), being statistical in nature, requires a large number of particles to be measured, especially if the spread of particle size is wide. The possibility of error from this source is stressed by Arnell and Henneberry who found that in a particular sample of finely ground quartz, two particles in a total of 335 had a diameter about twenty times the most probable diameter, and that if these were overlooked the calculated value of A would be nearly doubled. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

Tortuosity t is basically a correction factor applied to the Kozeny equation to account for the fact that in a real medium the pores are not straight (i.e., the length of the most probable flow path is longer than the overall length of the porous medium) ... 

The scheme of the interaction mechanism (Equation 88) testifies to an electro-affinity of MeFe" ions. In addition, MeFe" ions have a lower negative charge, smaller size and higher mobility compared to MeF6X(n+1> ions. The above arguments lead to the assumption that the reduction to metal form of niobium or tantalum from melts, both by electrolysis [368] and by alkali metals, most probably occurs due to interaction with MeF6 ions. The kinetics of the reduction processes are defined by flowing equilibriums between hexa-and heptacoordinated complexes. 

To calculate the most probable configuration, we start with equation (10.15) and take the logarithm of both sides to get... 

We will be able to use this equation to find the most probable energy distribution after we obtain appropriate values for a and 3. 

The addition of sulphinyl chlorides to trimethylsilyl enol ether 138 affording a-ketosulphoxides 139 (equation 76) represents an extension of the reaction of sulphinyl chlorides with ketones. This reaction has attracted attention only recently. Sergeev and coworkers192 reported that treatment of sulphinyl chlorides with acyclic enol ethers afforded a-ketosulphoxides 139 in good to excellent yields. Meanwell and Johnson193 observed that in the case of cyclic enol ethers the corresponding sulphoxides were formed only in very low yields. They found, however, that the introduction of an equivalent amount of a Lewis acid into the reaction mixture markedly promotes the desired reaction, whereas the use of catalytic amounts of a Lewis acid led to a substantial reduction in the yield. This is most probably due to the formation of a complex, between the a-ketosulphoxide and the Lewis acid. 

A closely related reaction of (—)-(S)-276 with the Grignard reagents obtained from a-acetylenic halides leads to the formation of mixtures of acetylenic sulphoxides 290 and allenic sulphoxides 291363 (equation 161). The latter compounds are most probably formed via transition state 292, which is analogous to 289. On the other hand, hex-l-ynyl p-tolyl sulphoxide 293 is smoothly prepared from hex-1 -ynylmagnesium bromide and (— )-(S)-276363 (equation 162). 

The softer, less basic potassium bromide and iodide did not react with the thiirene dioxide 19b. The latter was also inert towards potassium thiocyanate, selenocyanate or nitrile. It did react, however, with potassium thiophenoxide in DMF at room temperature to yield, most probably, the vinyl sulfmate 138 isolated as the corresponding sulfone39 (equation 56). 

The photolysis of various substituted thiete dioxides under similar conditions resulted in the formation of the unsaturated ketones (255)264, most probably via a vinyl sulfene intermediate followed by a loss of sulfur monoxide as shown in equation 96. The same results were obtained in the thermolysis of 6e (R1 = R3 = Ph R2 = R4 = H)231, which further demonstrates that similar mechanisms are operative in thermolyses and pho-tolyses of thietane dioxides and thiete dioxides. 

In a mechanistically related process, Fukamiya, Okano and Aratani have shown38 that the reaction of a sulphoxide with 3 mole equivalents of trifluoromethanesulphonyl chloride and iodine in cold, dry THF gives the corresponding sulphide in yields ranging from 52 to 96%. When compared to t-butyl bromide, this reagent is more expensive, and the products were isolated after chromatographic separation. The reaction pathway can most probably be described as depicted in equation (14) ... 

Meinwald and coworkers71 studied the chemistry of naphtho[l, 8-bc]thiete and its S-oxides. The reaction of the sulphone 2 with LAH (equation 29) is of particular and direct relevance to this section since it is different from the reductions that have been discussed thus far, because the major reaction pathway is now cleavage of an S—C bond, rather than a deoxygenation of the sulphur atom. The major product (equation 29) was isolated in 65% yield two minor products accounted for a further 15% yield. One of the minor products is 1-methylthionaphthalene and this was most probably produced by an initial reduction of the strained 1,8-naphthosulphone, 2, to the thiete, which was then cleaved to the thiol and subsequently methylated. Meinwald also showed71 that the thiete was subject to cleavage by LAH as well as that both molecules were susceptible to attack and cleavage by other nucleophiles, notably methyllithium. These reactions are in fact very useful in attempts to assess a probable mechanism for the reduction of sulphones by LAH and this will be discussed at the end of this section. 

The root mean square speed rrrm of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are i/mean = (8RT/-nM),a and... 

Figure 6. Various potential energy curves for the interaction between 02+-02 A = U(r) given by Equation 13 B = the inverse (12-6) power potential shown for comparison C = U(r) given by Equation 15 D = U(r) given by Equation 2. The most probable distance r and the potential at this distance U(r )...

The growing polymer chains have the most probable distribution defined by Equation (13.26). Typically, is large enough that PD 2 for the growing chains. It remains 2 when termination occurs by disproportionation. Example 13.5 shows that the polydispersity drops to 1.5 for termination by pure combination. The addition rules of Section 13.2.2 can be applied to determine 1.5 < PD < 2 for mixed-mode terminations, but disproportionation is the predominant form for commercial polymers. 

The other case which we consider is that of a most probable primary distribution. The molecular size distribution after random cross-linking must correspond exactly to that which would be obtained by random condensation of a mixture of bifunctional and tetrafunctional units. This follows as an extension of the correspondence between these two cases considered in the discussion of the critical condition given in the preceding section. The equations developed there are applicable to this case. 

The Gaussian function (7) is shown graphically in Fig. 77. The most probable location of one end of the chain relative to the other is at coincidence, i.e., at r = 0. The density, or probability, decreases monotonically with r, exactly as noted above for the one-dimensional case. Equation (7) likewise is unsatisfactory for values of r not much less than the full extension length nl. The extent of this limitation will be discussed presently. 

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