THIS symposium commences with a collection of papers which developthe arithmetic of a practical concept of stochastic processes. Thesetheoretical performs day again to the many years 1953 and 1954 when thetheory of stochastic procedures experienced been created in two major instructions:1. the theory of the initial two moments, i.e. correlation theoryand the theory of Gaussian processes2. the concept of Markov processes.At that time non-linear transformations of random capabilities werebeing investigated by methods which have been intently allied to thesetheories.But while the first theory, which is primarily based on linear algebra,was very satisfactory theoretically and hassle-free in observe in thelinear transformation of stochastic procedures, it was patently inadequatein the investigation of non-linear transformations. This appliedespecially to the non-linear transformation of non-Gaussian processesand delayed non-linear transformations.The authors as a result set themselves the activity of investigating themathematics of a general theory of random features not restrictedto the 1st two moments (which had been insufficient for a entire descriptionof arbitrary processes) or sure by the Markov problems. Suchwas the origin of the apparatus of “characterizing functions”. Thoughless tasteful than the two earlier theories, it is the only just one that canbe employed in specific cases and for want of a much better theory of thesame degree of generality its complexity need to be tolerated. The idea rests on the subsequent major propositions: one. An arbitrary stochastic method can be explained by an infinite set of characterizing features with an rising sequence of arguments, a established of correlation capabilities, a set of moment features or a established of quasi-second capabilities. These capabilities are “cumulants” (semi-invariants), times or quasi-times (coefficients of the growth of a likelihood density into a multi-dimensional Edgeworth sequence), which can be regarded as features of diverse instants in time. 2. The part played by significant order functions is systematically lowered with boost of buy so that the standard infinite expansions of the idea may be made finite. three. Guidelines are formulated for modifying from just one set of features to one more. four. Formulae are proposed whereby characteristic features can be written in terms of characterizing features, and multi-dimensional chance densities in terms of quasi-minute capabilities. This gives rules for the linear transformation of quasi-second capabilities in the non-linear transformation of stochastic processes. Mathematically, no part at all is played by the ordinarily investigated homes of beneficial-definite minute capabilities etcetera., which are linked to non-negative likelihood and have for that reason been omitted. The equipment which is explained is adequate for an arbitrary stochastic method. Evidence of this is furnished by comparing it with the principle of correlated random details (6) and the formulae linking these two theories. This kind of a comparison shows that the mathematical equipment is basically symmetric and uniform. The continuation of the principle in the quantum discipline is also of fascination, exactly where the elementary associations for the instant features, distribution features and attribute functionals keep their validity and are expressed in the language of operators (see R. L. Stratonovich, Relating to distributions in consultant house, (O raspredeleniyakh v izobrazhayushchem prostrantsve). Zhur. eksj). teor. fiz., 31, 1012, 1956). The equipment of characterizing functions is seldom utilised in connexion with non-linear transformations of random features as examined in the afterwards chapters of the symposium. It is standard of the idea of non-linear transformation of fluctuations that there is no one common strategy which is identified to have positive aspects about all others. Numerous procedures might be applied for fast benefits in unique specific situations dependent on the interactions amongst the parameters of the issue. Consequently in some cases the linearization technique may be utilized to let the non-linear transformation to be diminished approximately to a linear transformation in one or an additional perception. In other circumstances, when the non-linear transformation is effected by a delay method in which the time constants noticeably exceed the correlation time of the fluctuations, the problem could be solved by the Fokker-Planck equation in certain, and by the Markov approximation in common. The asymptotic applicability of the Fokker Planck equation is viewed as in short article five in reference to regular correlated random time collection in radio engineering. The Fokker-Planck equation is utilised to analyse the result of sounds on a detector (see post ten) and a valve oscillator (see content articles sixteen, 21), and to look into computerized section manage (24, twenty five). Ultimately, it is possible in other cases to lower a non-linear delay transformation to a delay-absolutely free transformation by considering a quasi-static approximation. This proved feasible in the investigation of the outcome of a slender-band approach on an exponential detector stage (see content 8 and nine). These procedures are of training course not the only kinds to be utilised in this symposium. The articles or blog posts in the second chapter offer with the outcome of sounds on detector stages and similar non-linear units. The 3rd chapter is a assortment of papers which examine self-oscillations and parametric oscillations in the presence of random fluctuations. Two articles or blog posts (six, 7) are devoted to pulse-sort random time sequence which are at the moment of exclusive desire in connexion with the use of discrete methods. The fourth chapter is established apart for the investigation of random perform excursions and the calculation of the distribution of excursions in excess of the length, a difficulty posed by S. O. Rice in 1945. The first paper in chapter 4 provides a complete and rigorous solution of the challenge, but one which regrettably are unable to be realised in observe without problems. In specific, it follows from the results which are received that the distribution density about the period diminishes exponentially for excursions of excellent length. Some results relating to Gaussian fluctuations are provided in the other papers of this chapter (29-33). The fifth and final chapter features the very first of a new sequence of articles dealing with optimum programs. Whereas the previous articles or blog posts (except four of post No. two) are devoted to the analysis of techniques subject to electrical fluctuations, the papers in this chapter think about the synthesis of systems which will execute their features in the the best possible method. I t goes with no expressing that only a handful of these kinds of problems can be regarded as below and that several other folks await solution. After two mathematical papers (36, 37), which offer with the principle of conditional Markov processes, many papers observe which are centered on this idea and which resolve a amount cf difficulties in ideal filtration. The principle is a continuation of Wiener-Kolmogorov’s theory of linear filtering and its non-stationary generalisations.