Direct Methods in Crystallography by M. M. Woolfson

Direct Limit of Krasner (m, n)-Hyperrings

The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings. In this regards we introduce and study direct limit of a direct system in category (m, n)-hyperrings. Also, we consider fundamental relation , as the smallest equivalence relation on an (m, n)-hyperring R such that the quotient space is an (m, n)-ring, to introdu...

ABS-Type Methods for Solving $m$ Linear Equations in $frac{m}{k}$ Steps for $k=1,2,cdots,m$

‎The ABS methods‎, ‎introduced by Abaffy‎, ‎Broyden and Spedicato‎, ‎are‎‎direct iteration methods for solving a linear system where the‎‎$i$-th iteration satisfies the first $i$ equations‎, ‎therefore a‎ ‎system of $m$ equations is solved in at most $m$ steps‎. ‎In this‎‎paper‎, ‎we introduce a class of ABS-type methods for solving a full row‎‎rank linear equations‎, ‎w...

M-FUZZIFYING MATROIDS INDUCED BY M-FUZZIFYING CLOSURE OPERATORS

In this paper, the notion of closure operators of matroids  is generalized to fuzzy setting  which is called $M$-fuzzifying closure operators, and some properties of $M$-fuzzifying closure operators are discussed. The $M$-fuzzifying matroid induced by an $M$-fuzzifying closure operator can induce an $M$-fuzzifying closure operator. Finally, the characterizations of $M$-fuzzifying acyclic matroi...

Direct methods for surface crystallography

In the most developed branches of crystallography, structure solution proceeds by two distinct steps: first, an approximate model of the structure is deduced directly from the measured data by an algorithm which assumes no preconceived model; second, a process of structure refinement simulates the experimental data for systematic variations of the parameters of such a model and determines a fin...

Two modified Jacobi methods for M-matrices