Unsupervised learning is one type of machine with an exceptionally high number applications in various fields. The most popular and best-known group unsupervised methods clustering methods. main goal to find hidden relationships between individual observations. There great interest different density estimation methods, especially when there are outliers the data. Density also can be applied dat...

We first give a new proof and also a new formulation for the Abhyankar-Gujar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gujar formula to give a more straightforward proof for the equivalence of the Jacobian conjecture with a special case of the vanishing conjecture of (homogeneous) quadratic differential operators with constant coefficients.

The Krĕın-Langer integral representation of a matrix-valued generalized Nevanlinna function arises in problems of spectral theory and interpolation. A version of this formula which is suitable for such problems, and a corresponding Stieltjes inversion formula, are derived. Some classes of generalized Nevanlinna functions which are defined in terms of behavior at infinity are characterized in te...

In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.

We give a survey of the status of some of the fundamental problems in harmonic analysis on semisimple symmetric spaces, including the description of the discrete series, the deenition of the Fourier transform, the inversion formula, the Plancherel formula and the Paley{ Wiener theorem.

A formula is given for the propagation of errors during matrix inversion. An explicit calculation for a 2 × 2 matrix using both the formula and a Monte Carlo calculation are compared. A prescription is given to determine when a matrix with uncertain elements is sufficiently nonsingular for the calculation of the covariances of the inverted matrix elements to be reliable.