order, the Changhee numbers and polynomials of negative order, the Lah numbers, the Hermite polynomials, the central factorial numbers, the Bernoulli numbers of higher order, the Daehee numbers, the Bernstein b
For operators, σ(·) and tanh(·) refer to the element-wise sigmoid and hyperbolic tangent functions, and ∗ is the element-wise multiplication. Intuitively, ĉt corresponds to the new information one is going to store in the cell state. To derive the new cell state ct that is of...
In an acyclic SCM, the combination of noise distribution pϵ and functions fi uniquely determines a distribution of the system variables V1,…,Vn. This distribution is often referred to as the entailed distribution of the SCM. The entailed distribution p(⋅) of the original SCM (that is,...
Rogers-Ramanujan functionsG?llnitz-Gordon functionsRamanujan’s general theta functions11B6511F11In this paper, we find new proofs of modular relations for the Gllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schrter's formulas and some simple theta-...
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12 and 13 are devoted to the establishment of the relation between the functions \(\omega ({\bar{\xi }}^{\,\prime },\xi )\) and \({\widetilde{\omega }} ({\bar{\xi }}^{\,\prime },\xi )\) , which are the matrix elements of the operator \(a_{0}\) and of the ...
I am not sure whether there is a standard way of doing this? The formula returns the number of timeseach combination occurs. The formula is a 365 array formula because I have pretty much forgotten traditional methods even when they may be simpler (more often they are not)....
Contiguous relations are also used to imply some continued fraction expansions for hypergeometric functions as well as to make a correspondence between Lie algebras and special functions, such correspondence yields formulas of special functions [6]. In [7], several properties of coefficients of these ...
Mathematically, as long as a function is symmetric and the resulting kernel matrix is positive semi-definite, the function is a valid kernel function. Typical kernels include linear, polynomial, and radial basis functions, among others. Among many properties of kernel functions, an important one ...
it satisfies the heat equation. It is shown that all natural choices of this function (rational, hyperbolic, and elliptic) reproduce exactly the Painlevé list of equations. In this sense, the classical-quantum correspondence can be regarded as an alternative definition of the Painlevé equations. ...