Extensions of Variational Methods, III

A Survey of Direct Methods for Solving Variational Problems

This study presents a comparative survey of direct methods for solving Variational Problems. Thisproblems can be used to solve various differential equations in physics and chemistry like RateEquation for a chemical reaction. There are procedures that any type of a differential equation isconvertible to a variational problem. Therefore finding the solution of a differential equation isequivalen...

Variational Formulation of Problems and Variational Methods

Variational Extensions to EM and Multinomial PCA

Several authors in recent years have proposed discrete analogues to principle component analysis intended to handle discrete or positive only data, for instance suited to analyzing sets of documents. Methods include non-negative matrix factorization, probabilistic latent semantic analysis, and latent Dirichlet allocation. This paperbegins with a review of the basic theory of the variational ext...

Generalization of matching extensions in graphs (III)

Proposed as a general framework, Liu and Yu [6] introduced (n, k, d)-graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let n, k and d be non-negative integers such that n+2k+d+2 6 |V (G)| and |V (G)| − n− d is even. If deleting any n vertices from G, the remaining subgraph H of G contains a k-matching and each k-matching can ...

Nonlocal Extensions of Similarity Methods

Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful local symmetry or local conservation law. It is shown how to construct systematically trees of ...