Infinity history

Achilles in the Quantum Universe The Definitive History of Infinity... 

Random walks on square lattices with two or more dimensions are somewhat more complicated than in one dimension, but not essentially more difficult. One easily finds, for instance, that the mean square distance after r steps is again proportional to r. However, in several dimensions it is also possible to formulate the excluded volume problem, which is the random walk with the additional stipulation that no lattice point can be occupied more than once. This model is used as a simplified description of a polymer each carbon atom can have any position in space, given only the fixed length of the links and the fact that no two carbon atoms can overlap. This problem has been the subject of extensive approximate, numerical, and asymptotic studies. They indicate that the mean square distance between the end points of a polymer of r links is proportional to r6/5 for large r. A fully satisfactory solution of the problem, however, has not been found. The difficulty is that the model is essentially non-Markovian the probability distribution of the position of the next carbon atom depends not only on the previous one or two, but on all previous positions. It can formally be treated as a Markov process by adding an infinity of variables to take the whole history into account, but that does not help in solving the problem. 

There is a certain order which is observed in human knowledge that is impossible to invert, and on which depends all the success of our discoveries. It is thus that the celebrated doctrine of M. Rouelle on the different quantities of acid which can enter into the composition of the same salt must necessarily precede the history of gypsum and its varieties. This discovery so fecund, the greatest that has been made in theoretical chemistry since Stahl, will serve as basis for all the aetiologies one will find in this memoir. Furthermore, I do not doubt that it will serve in the future as the foundation of an infinity of others and that it will unveil for the posterity the most impenetrable mysteries of nature. ... 

This is the Markovian memory-less approximation to the Master Equation. In this approximation, the effective time evolution operator becomes independent of t and the integral may be extended to infinity. It is also consistent to assume that the system lost memory of the initial state of the reservoir, whatever this was. In the limit when Uq is calculated in perturbation theory and pq(0) = 0, we obtain the conventional Born-Markov time evolution which has a long and successful history. 

The use of the lower limit of minus infinity is a mathematical convenience it implies that to calculate the stress at time t, in the most general case, one must know the strain history infinitely far into the past, i.e., at all times t prior to time t. In practice this is not necessary. In general, an experiment is started at some time (f = 0) when the material is in a stress-free state. In this case, t(0) = 0, and... 

I think we become interested in the concept of infinity early in childhood. Perhaps our initial fascination starts when we hear about large numbers, or outer space, or death, or eternity, or God. When I was a boy, I often visited my father s library to examine his large collection of old books. The one that stimulated my early thoughts about infinity was not a mathematics book, nor a book on philosophy, nor one on religion. It was a history book published in 1921 titled The Story of Mankind. In the book, Hendrik Willem Van Loon starts with a little parable next to a sketch of a mountain ... 

Although the above formulation was presented for displacement time history, it can be easily modified for velocity or acceleration measurements. In this case the right hand side of Equations (3.4) or (3.17) can be modified with the corresponding expressions for velocity or acceleration. Of course, the case of relative acceleration with white noise excitation is not realistic since the response variance is infinity. However, the absolute acceleration measurements can be considered for ground excitation. Another choice is to utilize a band-limited excitation model. 

If the function G s) approaches zero as s approaches infinity (a condition which as we shall see corresponds to a viscoelastic liquid), there is an alternative formulationexpressed in terms of the history of the strain rather than that of the rate of strain ... 

The expression of p(x) is correct if the time duration T goes to infinity that is to say if the sample time history continues indefinitely. Measurement of the time... 

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