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An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the ...
Central pattern generators (CPGs) are self-organizing biological neural circuits [1] [2] that produce rhythmic outputs in the absence of rhythmic input. [3] [4] [5] They are the source of the tightly-coupled patterns of neural activity that drive rhythmic and stereotyped motor behaviors like walking, swimming, breathing, or chewing.
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.
The generation effect is typically achieved in cognitive psychology experiments by asking participants to generate words from word fragments. [2] This effect has also been demonstrated using a variety of other materials, such as when generating a word after being presented with its antonym, [3] synonym, [1] picture, [4] arithmetic problems, [2] [5] or keyword in a paragraph. [6]
For example, GPT-3, and its precursor GPT-2, [11] are auto-regressive neural language models that contain billions of parameters, BigGAN [12] and VQ-VAE [13] which are used for image generation that can have hundreds of millions of parameters, and Jukebox is a very large generative model for musical audio that contains billions of parameters. [14]
Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration ...
The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used. It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style ...
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