Bernoullis Interpretation

We see that there are several ways of interpreting the term ef. From the Bernoulli equation, it represents the lost (i.e., dissipated) energy... 

The Suter-Flory model was successfully used to interpret the results of the epimerization reaction carried out on propylene oligomers (204) and on polypropylene itself (106, 205). In both cases a slight prevalence of the r dyad over the m (52/48) is observed. The epimerized polypropylene has a microstmcmre almost coincident with a Bernoulli distribution and represents the polymer sample closest to an ideal atactic polymer so far obtained. 

The same type of addition—as shown by X-ray analysis—occurs in the cationic polymerization of alkenyl ethers R—CH=CH—OR and of 8-chlorovinyl ethers (395). However, NMR analysis showed the presence of some configurational disorder (396). The stereochemistry of acrylate polymerization, determined by the use of deuterated monomers, was found to be strongly dependent on the reaction environment and, in particular, on the solvation of the growing-chain-catalyst system at both the a and jS carbon atoms (390, 397-399). Non-solvated contact ion pairs such as those existing in the presence of lithium catalysts in toluene at low temperature, are responsible for the formation of threo isotactic sequences from cis monomers and, therefore, involve a trans addition in contrast, solvent separated ion pairs (fluorenyllithium in THF) give rise to a predominantly syndiotactic polymer. Finally, in mixed ether-hydrocarbon solvents where there are probably peripherally solvated ion pairs, a predominantly isotactic polymer with nonconstant stereochemistry in the jS position is obtained. It seems evident fiom this complexity of situations that the micro-tacticity of anionic poly(methyl methacrylate) cannot be interpreted by a simple Bernoulli distribution, as has already been discussed in Sect. III-A. 

When considering the bending of a beam and attempting to extract a modulus value one must make several assumptions, the most important being that the modulus in tension is the same as in compression, and is independent of strain (at least for the range of strain involved). The simple Bernoulli-Euler theory is usually used to interpret the data. When performing resonance tests it is particularly useful to find a set of resonances and compare the measured frequency ratios with the theoretical ones given in the previous chapter. 

One of the earliest attempts of interpreting properties of systems based on states of atoms or molecules constituting the system, was made by Bernoulli in 1738. He considered an ideal gas within a cylinder and a piston similar to what is presented in Fig. 4.1. He considered the force exerted by the piston to be balanced by the pressure of the gas and he considered the pressure to be generated by the impact of gas molecules on piston and the consequent momentum change. He reasoned that when volume decreases, the average distance between molecules would decrease as the cube root of volume ratio, and the number of impact on the piston on this account would increase by the same ratio. On the other hand the number of particles adjacent to the piston will increase as per the area ratio which can be expressed as... 

It is therefore remarkable that 100 years or so before the laws of thermodynamics were formulated, Daniel Bernoulli developed a billiard ball model of a gas that gave a molecular interpretation to pressure and was later extended to give an understanding of temperature. This is truly a wonderful thing, because all it starts with is the assumption that the atoms or molecules of a gas can be treated as if they behave like perfectly elastic hard spheres—minute and perfect billiard balls. Then Newton s laws of motion are applied and all the gas laws follow, together with a molecular interpretation of temperature and absolute zero. You have no doubt... 

The case of the Reynolds number discussed above sho vs that the physical interpretation of one dimensionless group is not unique. Generally, the interpretation of dimensionless groups used in the flo v area in terms of different energies involved in the process, can be obtained starting vith the Bernoulli flovr equation. The relationship existing between the terms of this equation introduces one dimensionless group. 

Bayes demonstrated his theorem by inferring a posterior distribution for the parameter p of Eq. (4.3-2) from the observed number k of successes in n Bernoulli trials. His distribution formula expresses the probability, given k and n, that p lies somewhere between any two degrees of probability that can be named. The subtlety of the treatment delayed its impact until the middle of the twentieth century, though Gauss (1809) and Laplace (1810) used related methods. Stigler (1982, 1986) gives lucid discussions of Bayes classic paper and its various interpretations by famous statisticians. 

A considerable amount of data are available relating to the effect of substituents upon the rate of decarboxylation. Unfortunately, rates or activation parameters are usually not given individually for the free acid and the anion. As indicated previously, caution must be used in the interpretation of these overall rate coefficients. A number of years ago, Bernoulli et reported an extensive study of the... 

For steady, fully developed one-dimensional flow in a uniform pipe, the engineering Bernoulli and momentum equations provide equivalent interpretations of the friction loss ... 

We can now interpret the slide problem as an example of a sequence of independent, repeated Bernoulli trials with probability P of success. Here a success occurs when a person makes it to the bottom of the slide. If we let the random variable N represent the number of unsuccessful trials (attempts) required before the first success for that person (that is, success is achieved on trial N+1), then N has a geometric density defined by the equation P N) = P( - P). Here, P N) is the probability that N unsuccessful trials occur before the successful trial. This is the exponential decay law observed in Figure 10.1. Steve Czamecki notes that the expected value of N (the average number of attempts required before the successful trial) will be E N)- -P)/P. For the example with 10 holes, 1,023 unsuccessful attempts are required, on average, before the successful attempt. Thus, our intuition that the person will achieve one success out of 1,024 attempts, on the average, is correct. 

Interpretation. Consider a dichotomous experiment that has only two possible outcomes (alternative events) like a coin toss. One of the alternative outcomes (e.g., heads ) is generally called success while the other (e.g., tails ) failure. The probability of success is denoted by p, that of failure by q = I — p. Now, the Bernoulli variable X is defined as follows ... 

Interpretation. Consider a dichotomous game. Let p denote the probability of success in a single trial. Suppose that a series of n Bernoulli trials have been performed. Let Xi,X2,...,X denote the independent Bernoulli variables belonging to the respective trials. Then the random variable X = Xi- ------- -X has a B(n, p) binomial distribution. 

Note that X means the number of successful outcomes in the series of n Bernoulli trials. Note also that the above interpretation justifies the use of the symbol B 1, p) for the Bernoulli distribution. 

Despite the dominance of Newton s view, some people followed the kinetic interpretation. In 1738, Daniel Bernoulli, a Swiss mathematician and physicist, gave a quantitative explanation of Boyle s law using the kinetic interpretation. He even suggested that molecules move faster at high temperatures, in order to explain Amontons s experiments on the temperature dependence of gas volume and pressure. However, Bernoulli s paper attracted little notice. A similar kinetic interpretation of gases was submitted for publication to the Royal Society of London in 1848 by John James Waterston. His paper was rejected as nothing but nonsense. ... 

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