Analysis and optimization of a class of non-linear staircase systems for random processes

1960 ◽  
Vol 1 (1) ◽  
pp. 447-457
Author(s):  
T. Prasad
Keyword(s):  
Random Processes ◽  
Non Linear

Characterization of Non-Linear Transformations Possessing Kernels

1970 ◽  
Vol 22 (3) ◽  
pp. 449-471 ◽  
Author(s):  
Victor J. Mizel
Keyword(s):  
Random Processes ◽  
Fading Memory ◽  
Linear Functionals ◽  
Linear Transformations ◽  
Main Motivation ◽  
Carathéodory Conditions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form(1.1)with a continuous “kernel” φ: R → R,or(1.2)with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see [8]) and to the theory of fading memory in continuum mechanics [3]. However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form(1.3)where the kernel φ: R × T → R satisfies “Carathéodory conditions”.

scholarly journals Two non-linear birth and death processes

1963 ◽  
Vol 3 (1) ◽  
pp. 104-116 ◽  
Author(s):  
J. A. Bather
Keyword(s):  
Random Processes ◽  
Conditional Probabilities ◽  
Birth And Death Processes ◽  
Non Linear ◽  
Image Position ◽  
Fixed Population

The random processes discussed here may be specified in the following way. A fixed population of N members is spilt into two distinct classes. Individuals move about randomly between the classes, and we are interested in the size of each class at any time, rather than in the behaviour of particular individuals. Let i(t) and N —i(t) be the numbers present in the repective classes at the time t. It is assumed that the process {i(t), t ≧ 0} is Markovian, and that transitions between the states j = 0, 1, … N, occur according to the conditional probabilities; and.

Illustrating Random Processes with Random Phase Modulation

1986 ◽  
Vol 23 (2) ◽  
pp. 151-158
Author(s):  
G. J. M. Aitken
Keyword(s):  
Phase Noise ◽  
Linear Relationship ◽  
Conditional Probability ◽  
Phase Modulation ◽  
Random Phase ◽  
Random Processes ◽  
Mathematical Framework ◽  
Synchronous Detection ◽  
Non Linear

Randomly phase-modulated cosines are a source of examples for illustrating the topics of variance, autocorrelation, conditional probability and filtering. Mathematical manipulations are neither difficult nor tedious despite the non-linear relationship between measured quantities and the phase noise. The basic mathematical framework is presented in the context of examples which include synchronous detection in the presence of phase perturbations.

A framework for causality under data assimilation

2020 ◽  
Author(s):  
Nachiketa Chakraborty ◽  
Peter Jan van Leeuwen ◽  
Michael de Caria ◽  
Manuel Pulido
Keyword(s):  
Causal Relation ◽  
Random Processes ◽  
Time Varying ◽  
Linear Dynamical Systems ◽  
Information Theoretic ◽  
Non Linear ◽  
Information Theoretic Measures ◽  
Non Gaussian ◽  
Causal Framework ◽  
Simple Systems

<p>Time varying processes in nature are often complex with non-linear and non-gaussian components. Complexity of environments and processes make it hard to disentangle different causal mechanisms which drives the observed time-series. It also makes it harder to make forecasts. The standard ways of studying causal relation in the geosciences which includes information theoretic measures of causation as well as predictive framework have deficiencies when applied to non-linear dynamical systems. Here we focus on investigating building a predictive causal framework that allows us to make predictions in simpler systems in a consistent way. We use a Bayesian framework to embed causal measures akin to mutual information from information theory to quantify relations between different random processes in this system. We examine causal relations in toy models and simple systems with a view to eventually applying to the interocean exchange problem in the Indian, the South Atlantic and the Southern Ocean. </p>

B. The Non-Linear Calculations for Pulsating Stars

1967 ◽  
Vol 28 ◽  
pp. 105-176
Author(s):  
Robert F. Christy
Keyword(s):  
Major Part ◽  
Afternoon Session ◽  
Discussion Session ◽  
Pulsating Stars ◽  
Fourth Symposium ◽  
Non Linear ◽  
Considerable Discussion ◽  
Informal Approach ◽  
Preceding Session

(Ed. note: The custom in these Symposia has been to have a summary-introductory presentation which lasts about 1 to 1.5 hours, during which discussion from the floor is minor and usually directed at technical clarification. The remainder of the session is then devoted to discussion of the whole subject, oriented around the summary-introduction. The preceding session, I-A, at Nice, followed this pattern. Christy suggested that we might experiment in his presentation with a much more informal approach, allowing considerable discussion of the points raised in the summary-introduction during its presentation, with perhaps the entire morning spent in this way, reserving the afternoon session for discussion only. At Varenna, in the Fourth Symposium, several of the summaryintroductory papers presented from the astronomical viewpoint had been so full of concepts unfamiliar to a number of the aerodynamicists-physicists present, that a major part of the following discussion session had been devoted to simply clarifying concepts and then repeating a considerable amount of what had been summarized. So, always looking for alternatives which help to increase the understanding between the different disciplines by introducing clarification of concept as expeditiously as possible, we tried Christy's suggestion. Thus you will find the pattern of the following different from that in session I-A. I am much indebted to Christy for extensive collaboration in editing the resulting combined presentation and discussion. As always, however, I have taken upon myself the responsibility for the final editing, and so all shortcomings are on my head.)

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