The purpose of this paper is to put forward the basics results of complex fuzzy sets (CFSs) such ... more The purpose of this paper is to put forward the basics results of complex fuzzy sets (CFSs) such as union, intersection, complement, product into complex neutrosophic sets because as the CFSs and complex intuitionistics sets does give the erroneous and inconvenient information about uncertainty and periodicity and also there are results related to different norms. Moreover we give some results about the distance measures of complex neutrosophic sets and define some notions.
Uni-Soft Commutative Ideals with Thresholds in BCK=Bci-Algebras
The notion of a uni-soft commutative ideal with thresholds is introduced, and related properties ... more The notion of a uni-soft commutative ideal with thresholds is introduced, and related properties are investigated. Relations between a uni-soft ideal with thresholds and a uni-soft commutative ideal with thresholds are discussed. Conditions for a uni- soft ideal with thresholds to be a uni-soft commutative ideal with the same thresholds are provided. Characterizations of a uni-soft commutative ideal with thresholds are established
Extreme Preservers of Rank Inequalities of Boolean Matrix Sums
Journal of applied mathematics & informatics, 2008
We construct the sets of Boolean matrix pairs, which are naturally occurred at the extreme cases ... more We construct the sets of Boolean matrix pairs, which are naturally occurred at the extreme cases for the Boolean rank inequalities relative to the sums and difference of two Boolean matrices or compared between their Boolean ranks and their real ranks. For these sets, we consider the linear operators that preserve them. We characterize those linear operators as T(X) = PXQ or with appropriate invertible Boolean matrices P and Q.
Let $\Omega_n$ be the polyhedron of $n \times n$ doubly stochastic matrices, that is, nonnegative... more Let $\Omega_n$ be the polyhedron of $n \times n$ doubly stochastic matrices, that is, nonnegative matrices whose row and column sums are all equal to 1. The permanent of a $n \times n$ matrix $A = [a_{ij}]$ is defined by $$ per(A) = \sum_{\sigma}^ a_{1\sigma(a)} \cdots a_{n\sigma(n)} $$ where $\sigma$ runs over all permutations of ${1, 2, \ldots, n}$.
On the Minimum Permanents Related with Certain Barycentric Matrices
Journal of The Korean Mathematical Society, 1997
The permanent function on certain faces of the polytope of doubly stochastic matrices are studied... more The permanent function on certain faces of the polytope of doubly stochastic matrices are studied. These faces are shown to be barycentric, and the minimum values of permanent are determined.
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2018
In this paper, we studied the action of Picard modular group P SL(2, Z[i]) denoted by on the biqu... more In this paper, we studied the action of Picard modular group P SL(2, Z[i]) denoted by on the biquadratic field Q i, √ 3. We found patern of algebraic integers formed by this action. To prove results we used coset diagrams for the action of on Q i, √ 3 , proponded by Graham Higman.
We give the definition of fuzzy BCK-filter induced by a fuzzy set in a bounded BCK-algebra. We ve... more We give the definition of fuzzy BCK-filter induced by a fuzzy set in a bounded BCK-algebra. We verify that the family of fuzzy BCK-filters is a completely distributive lattice. Using the BCK-filter U (µ; α) for a given fuzzy set µ, we construct the fuzzy BCK-filter induced by µ.
Linear operators that strongly preserves the sign-central matrices
Bulletin of The Korean Mathematical Society, 1997
Let $M_{m,n}$ be the set of all $m \times n$ real matrices. For a matrix $A = [a_{ij}] \in M_{m,n... more Let $M_{m,n}$ be the set of all $m \times n$ real matrices. For a matrix $A = [a_{ij}] \in M_{m,n}$, the sign of $a_{ij}$ is defined by $$ sgn a_{ij} = { 0 if a_{ij} = 0, { +1 if a_{ij} > 0, { -1 if a_{ij}
The notions of hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set ... more The notions of hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set on BCK/BCIalgebras are introduced, and related properties are investigated. We prove that every hesitant fuzzy translation of a hesitant fuzzy subalgebra (ideal) is a hesitant fuzzy subalgebra (ideal). Conditions for a hesitant fuzzy set to be a hesitant fuzzy subalgebra (ideal) are provided. We show that if a hesitant fuzzy set is a hesitant fuzzy subalgebra (ideal), then its support is a subalgebra (ideal), and also prove that if the support of a hesitant fuzzy set is a subalgebra (ideal), then its hesitant fuzzy translation is a hesitant fuzzy subalgebra (ideal).
Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r ... more Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form b i , j b i , l b k , j b k , l with nonzero entries. We know that the isolation number of B is not greater than the rank of it. Thus, we investigate the upper bound of the rank of B and the rank of its support for the given matrix B with isolation number h over antinegative semirings.
A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface o... more A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k−1. A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and that maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus not k to graphs of genus not k. We show that such linear operators are necessarily vertex permutations.
Abstract Let M n ( B ) denote the set of n × n ( 0 , 1 ) -matrices with Boolean arithmetic. The s... more Abstract Let M n ( B ) denote the set of n × n ( 0 , 1 ) -matrices with Boolean arithmetic. The set of primitive matrices of exponent k, denoted E k , is the set of matrices such that A k has all nonzero entries and A j has zero entries for all j k . For 3 ≤ k ≤ n , we characterize those linear operators that map E k to E k and E k − 1 to E k − 1 . We also characterize those linear operators that strongly preserve E k for 3 ≤ k ≤ n , that is, that map E k to E k and the complement of E k to the complement of E k .
A criterion is presented in order to decide whether a given integer is a prime power or not. The ... more A criterion is presented in order to decide whether a given integer is a prime power or not. The criterion associates to each positive integer m a finite set of integers S(m) , each of them < m , and the properties of this set are studied. The notion of complementary pairs in S(m) is introduced and it is proved that if one is able to determine a complementary pair n, n ′ , then a partial factorization of the odd integer m can be obtained in polynomial time. Some particular cases and examples of these results are given.
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