2001, Vol. VIII, Núm. 1 http://hdl.handle.net/2099/2072 2019-11-22T00:29:37Z Generation of fuzzy mathematical morphologies http://hdl.handle.net/2099/3595 Generation of fuzzy mathematical morphologies Burillo López, Pedro; Frago Paños, Noé; Fuentes-González, Ramón Fuzzy Mathematical Morphology aims to extend the binary morphological operators to grey-level images. In order to define the basic morphological operations fuzzy erosion, dilation, opening and closing, we introduce a general method based upon fuzzy implication and inclusion grade operators, including as particular case, other ones existing in related literature In the definition of fuzzy erosion and dilation we use several fuzzy implications (Annexe A, Table of fuzzy implications), the paper includes a study on their practical effects on digital image processing. We also present some graphic examples of erosion and dilation with three different structuring elements $B(i, j)=1$, $B(i, j)=0.7$, $B(i, j)=0.4$, $i, j \in \{ 1,2, 3\}$ and various fuzzy implications. 2007-09-28T11:08:29Z Burillo López, Pedro Frago Paños, Noé Fuentes-González, Ramón Fuzzy Mathematical Morphology aims to extend the binary morphological operators to grey-level images. In order to define the basic morphological operations fuzzy erosion, dilation, opening and closing, we introduce a general method based upon fuzzy implication and inclusion grade operators, including as particular case, other ones existing in related literature In the definition of fuzzy erosion and dilation we use several fuzzy implications (Annexe A, Table of fuzzy implications), the paper includes a study on their practical effects on digital image processing. We also present some graphic examples of erosion and dilation with three different structuring elements $B(i, j)=1$, $B(i, j)=0.7$, $B(i, j)=0.4$, $i, j \in \{ 1,2, 3\}$ and various fuzzy implications. On four intuitionistic fuzzy topological operators http://hdl.handle.net/2099/3594 On four intuitionistic fuzzy topological operators Atanassov, Krassimir T. Four new operators, which are analogous of the topological operators "interior" and "closure", are defined. Some of their basic properties are studied. Their geometrical interpretations are given. 2007-09-28T10:55:43Z Atanassov, Krassimir T. Four new operators, which are analogous of the topological operators "interior" and "closure", are defined. Some of their basic properties are studied. Their geometrical interpretations are given. A theorem for basis operators over intuitionistic fuzzy sets http://hdl.handle.net/2099/3593 A theorem for basis operators over intuitionistic fuzzy sets Atanassov, Krassimir T. The concept of s-basis operators over intuitionistic fuzzy sets is introduced and all 2-, 2-, 4-, basis operators are listed. 2007-09-28T10:51:43Z Atanassov, Krassimir T. The concept of s-basis operators over intuitionistic fuzzy sets is introduced and all 2-, 2-, 4-, basis operators are listed. Topological automorphism groups of chains http://hdl.handle.net/2099/3592 Topological automorphism groups of chains Ovchinnikov, Sergei V. It is shown that any set--open topology on the automorphism group $A(X)$ of a chain $X$ coincides with the pointwise topology and that $A(X)$ is a topological group with respect to this topology. Topological properties of connectedness and compactness in $A(X)$ are investigated. In particular, it is shown that the automorphism group of a doubly homogeneous chain is generated by any neighborhood of the identity element. 2007-09-28T10:40:18Z Ovchinnikov, Sergei V. It is shown that any set--open topology on the automorphism group $A(X)$ of a chain $X$ coincides with the pointwise topology and that $A(X)$ is a topological group with respect to this topology. Topological properties of connectedness and compactness in $A(X)$ are investigated. In particular, it is shown that the automorphism group of a doubly homogeneous chain is generated by any neighborhood of the identity element. Application of Cauchy's equation in combinatorics and genetics http://hdl.handle.net/2099/3591 Application of Cauchy's equation in combinatorics and genetics Kannappan, Palaniappan We are familiar with the combinatorial formula $\left(\begin{array}{cc} n\\ r \end{array}\right) = \frac{n(n-1) \cdots (n - r + 1)}{r !} =$ number of possible ways of choosing $r$ objects out of $n$ objects\,. In section 1 of this paper we obtain $\left( \begin{array}{cc} n\\ 2\end{array}\right)$ and $\left( \begin{array}{cc} n\\ 3 \end{array}\right)$ by using a functional equation, {\it the additive Cauchy equation}. In genetics it is important to know the combinatorial function $g_{r}(n)=$ the number of possible ways of picking $r$ objects at a time from $n$ objects {\it allowing repetitions}, since this function describes the number of possibilities from a gene pool. Again we determine $g_2(n)$ and $g_3(n)$ with the help of the additive Cauchy equation in section 2. Functional equations are used increasingly in diverse fields. The method of finding $\left( \begin{array}{cc} n\\ 2 \end{array}\right), \left( \begin{array}{cc} n\\ 3 \end{array}\right), g_2 (n)$ and $g_3(n)$ (see Snow ) is similar to that of finding the well known sum of powers of integers $S_K(n) = 1^K + 2^K + \cdots + n^K$ (Acz\'{e}l , Snow ).\\ 2007-09-28T10:33:59Z Kannappan, Palaniappan We are familiar with the combinatorial formula $\left(\begin{array}{cc} n\\ r \end{array}\right) = \frac{n(n-1) \cdots (n - r + 1)}{r !} =$ number of possible ways of choosing $r$ objects out of $n$ objects\,. In section 1 of this paper we obtain $\left( \begin{array}{cc} n\\ 2\end{array}\right)$ and $\left( \begin{array}{cc} n\\ 3 \end{array}\right)$ by using a functional equation, {\it the additive Cauchy equation}. In genetics it is important to know the combinatorial function $g_{r}(n)=$ the number of possible ways of picking $r$ objects at a time from $n$ objects {\it allowing repetitions}, since this function describes the number of possibilities from a gene pool. Again we determine $g_2(n)$ and $g_3(n)$ with the help of the additive Cauchy equation in section 2. Functional equations are used increasingly in diverse fields. The method of finding $\left( \begin{array}{cc} n\\ 2 \end{array}\right), \left( \begin{array}{cc} n\\ 3 \end{array}\right), g_2 (n)$ and $g_3(n)$ (see Snow ) is similar to that of finding the well known sum of powers of integers $S_K(n) = 1^K + 2^K + \cdots + n^K$ (Acz\'{e}l , Snow ).\\ On the generalizations of Siegel's fixed point theorem http://hdl.handle.net/2099/3590 On the generalizations of Siegel's fixed point theorem Jung, J.S.; Chang, S.S.; Lee, B.S.; Cho, Y.J; Kang, S.M. In this paper, we establish a new version of Siegel's fixed point theorem in generating spaces of quasi-metric family. As consequences, we obtain general versions of the Downing-Kirk's fixed point and Caristi's fixed point theorem in the same spaces. Some applications of these results to fuzzy metric spaces and probabilistic metric spaces are presented. 2007-09-28T09:58:06Z Jung, J.S. Chang, S.S. Lee, B.S. Cho, Y.J Kang, S.M. In this paper, we establish a new version of Siegel's fixed point theorem in generating spaces of quasi-metric family. As consequences, we obtain general versions of the Downing-Kirk's fixed point and Caristi's fixed point theorem in the same spaces. Some applications of these results to fuzzy metric spaces and probabilistic metric spaces are presented.