Abstract We demonstrate that the recently proposed soft gluon factorization (SGF) is equivalent to nonrelativistic QCD (NRQCD) for heavy quarkonium production or decay, which means that, any given process, these two theories are either both valid violated. use methods arrive at this conclusion. In first method, we apply physical process . Our explicit calculation s...

Pk -factorization of a complete bipartite graph for k, an even integer was studied by H. Wang [1]. Further, Beiling Du [2] extended the work of H.Wang, and studied the P2k-factorization of complete bipartite multigraph. For odd value of k the work on factorization was done by a number of researchers[3,4,5]. P3-factorization of complete bipartite graph was studied by K.Ushio [3]. P5-factorizatio...

A palindrome is a symmetric string, phrase, number, or other sequence of units sequence that reads the same forward and backward. We present an algorithm for maximal palindromic factorization of a finite string by adapting an Gusfield algorithm [15] for detecting all occurrences of maximal palindromes in a string in linear time to the length of the given string then using the breadth first sear...

A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reeective subcategories and factorization systems. The second part of the paper presents two important factorization structures...

I provide the details of the factorization of the Mersenne number 21061 − 1 by the Special Number Field Sieve. Although this factorization is easier than the completed factorization of RSA-768, it represents a new milestone for factorization using publicly available software.