Topic
Euler's factorization method
About: Euler's factorization method is a research topic. Over the lifetime, 534 publications have been published within this topic receiving 10572 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: The first-order differential-difference factorization method as mentioned in this paper is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists.
Abstract: The factorization method is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists. The underlying idea is to consider a pair of first-order differential-difference equations which are equivalent to a given second-order differential equation with boundary conditions. For a large class of such differential equations the method enables us to find immediately the eigenvalues and a manufacturing process for the normalized eigenfunctions. These results are obtained merely by consulting a table of the six possible factorization types.The manufacturing process is also used for the calculation of transition probabilities.The method is generalized so that it will handle perturbation problems.
1,763 citations
7 Aug 2005
TL;DR: A "direct" positive-preserving gradient descent algorithm and an alternating scheme based on repeated multiple rank-1 problems are derived and motivate the use of n-NTF in three areas of data analysis.
Abstract: We derive algorithms for finding a non-negative n-dimensional tensor factorization (n-NTF) which includes the non-negative matrix factorization (NMF) as a particular case when n = 2. We motivate the use of n-NTF in three areas of data analysis: (i) connection to latent class models in statistics, (ii) sparse image coding in computer vision, and (iii) model selection problems. We derive a "direct" positive-preserving gradient descent algorithm and an alternating scheme based on repeated multiple rank-1 problems.
651 citations
TL;DR: In this paper, the Euler equation has been considered by several authors including L. Lichtenstein (1925-30), J. Leray (1932-37), M. Wolibner (1938), T. Ebin and J. Marsden [2] have proved the existence of a local solution in the general case.
395 citations
TL;DR: This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly, and gives explicit formulee for several classes ofEuler sums in terms of harmonic numbers.
Abstract: This paper develops an approach to the evaluation of Euler sums that involve harmonic numbers, either linearly or nonlinearly. We give explicit formulee for several classes of Euler sums in terms o...
344 citations
TL;DR: The progress of the factorization method since the 1935 work of Dirac is briefly reviewed in this article, where the authors suggest that factorization seems an autonomous "driving force", offering substantial support to the present day Darboux and Backlund approaches.
Abstract: The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial support to the present day Darboux and Backlund approaches.
269 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 1 |
| 2018 | 4 |
| 2017 | 22 |
| 2016 | 13 |
| 2015 | 25 |