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Topological data analysis and persistent homology have had impacts on Morse theory. [121] Morse theory has played a very important role in the theory of TDA, including on computation. Some work in persistent homology has extended results about Morse functions to tame functions or, even to continuous functions [citation needed]. A forgotten ...
Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy (a measure of average surprisal) of a random variable, to continuous probability distributions.
The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with probability density function f(x) with finite or infinite support on the real line is defined by analogy, using the above form of the entropy as an expectation: [11]: 224
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
The test functions used to evaluate the algorithms for MOP were taken from Deb, [4] Binh et al. [5] and Binh. [6] The software developed by Deb can be downloaded, [ 7 ] which implements the NSGA-II procedure with GAs, or the program posted on Internet, [ 8 ] which implements the NSGA-II procedure with ES.
If Z is a standard normal random variable, and V is a chi-squared distributed random variable with ν degrees of freedom that is independent of Z, then = + / is a noncentral t-distributed random variable with ν degrees of freedom and noncentrality parameter μ ≠ 0. Note that the noncentrality parameter may be negative.
Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. [7] Naturally also same function of one variable that depends on some complex parameter is a candidate.