Though integer factorization and discrete logarithm problem are both practically and theoretically important, the computational complexity of these problems remained unknown. By comparing integer factorization problem with a problem in P and NP-complete problems, I show that the decision problem version of integer factorization problem is neither in P nor NP-complete. In addition, integer facto...

V -order is a total order on strings that determines an instance of Unique Maximal Factorization Families (UMFFs), generalization Lyndon words. The fundamental -comparison can be done in linear time and constant space. has been proposed as alternative to lexicographic (lexorder) the computation suffix arrays suffix-sorting induced by Burrows-Wheeler transform (BWT). In line with recent interest...

Factorization machines are a generic framework which allows to mimic many factorization models simply by feature engineering. In this way, they combine the high predictive accuracy of factorization models with the flexibility of feature engineering. Unfortunately, factorization machines involve a non-convex optimization problem and are thus subject to bad local minima. In this paper, we propose...

An ordering method and accompanying factorization for the direct solution of saddlepoint matrices is presented. A simple constraint on ordering together with an assumption on the rank of parts of the matrix are sufficient to guarantee the existence of the LDLT factorization, stability concerns aside. In fact, D may be taken to be a diagonal matrix with ±1 along the diagonal, and be fully determ...

This paper describes the design and implementation of three core factorization routines—LU, QR, and Cholesky—included in the out-of-core extension of ScaLAPACK. These routines allow the factorization and solution of a dense system that is too large to fit entirely in physical memory. The full matrix is stored on disk and the factorization routines transfer sub-matrice panels into memory. The ‘l...

For n even, a factorization of a complete graph Kn is a partition of the edges into n−1 perfect matchings, called the factors of the factorization. With respect to a factorization, a path is called rainbow if its edges are from distinct factors. A rainbow Hamiltonian path takes exactly one edge from each factor and is called orthogonal to the factorization. It is known that not all factorizatio...

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Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that is Lyndon words with respect to the inverse lexicographic order. We prove that any nonempty word w admits a canonical inverse Lyndon factorization, named IC...

Because of the limitations of matrix factorization, such as losing spatial structure information, the concept of low-rank tensor factorization (LRTF) has been applied for the recovery of a low dimensional subspace from high dimensional visual data. The low-rank tensor recovery is generally achieved by minimizing the loss function between the observed data and the factorization representation. T...

Factorization of linear partial differential operators (LPDOs) is a very wellstudied problem and a lot of pure existence theorems are known. The only known constructive factorization algorithm Beals-Kartashova (BK) factorization is presented in [1]). Its comparison with Hensel descent which is sometimes regarded as constructive, is given in [2], where the idea to use BKfactorization for approxi...