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On one of Murthy-Ashbacher's conjectures related to Euler's totient function
K. T. Atanassov,M. V. Vassilev-Missana 장전수학회 2006 Proceedings of the Jangjeon mathematical society Vol.9 No.1
A. Murthy and C. Ashbacher formulated a lot of open problems and conjectures in the area of elementary number theory. One of them is: For every positive integer k there exists a number n such that n '(n) = k: In the paper it is shown that this conjecture is not valid. It is proved that the only values of k for which the Diophantine equation k:'(n) = n is valid, are k = 1; 2; 3, where ' is Euler's totient function.
Correction : Proc. Jangjeon Math. Soc. 8(2005), No. 2, pp. 123-130
P. M. Vassilev,M. V. Vassilev-Missana,K. T. Atanassov 장전수학회 2006 Proceedings of the Jangjeon mathematical society Vol.9 No.2
In the above paper the following corrections should be taken into account. Everywhere in the text the expression \an integer k satisfying (1)" should be read as \an integer k satisfying (2)". In the Proof of Lemma 3 variable s should be changed to k. On page 123 instead of [4] should be read [2] and in the last row instead of [3] should be read [2]; on page 129 in section \Final remarks" instead of [2] should be read [4]. In the text there are some other misprints, e.g., it is written that there are four instead of ¯ve assertions, but we hope that the benevolent reader will not hold this against us.
Optimization of a neural network of self-organizing maps type with time-limits by a generalized net
K. Atanassov,S. Sotirov 장전수학회 2006 Advanced Studies in Contemporary Mathematics Vol.13 No.2
The neural networks Self-Organizing Maps have got a multitude of applications. The introduced generalized net gives a possibility for optimization of the neural network allowing determination of its optimum architecture based on time limits for its learning.
On some Pascal's like triangles. Part 5
K. T. Atanassov 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.3
A series of Pascal's like triangles with dierent forms are described and some of their propertiesa are given.
Some properties of the operators from one type of intuitionistic fuzzy modal operators
K. T. Atanassov 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.15 No.1
The intuitionistic fuzzy modal operators belong to several types. During the last year one of these types of modal operators was extended independently and in different directions by Gokhan Cuvalcioglu and by the author. Here the relations between both extensions are discussed and some properties of the new operators are studied.
Generalized net model for parallel optimization of feed-forward neural network
K. T. Atanassov,S. Sotirov,A. Antonov 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.15 No.1
The used generalized net wil give us a possibility for parallel optimization of fed-forward neural network based on assigned training pairs and time limits.
On index matrices Part 5: Three dimensional index matrices
K. T. Atanassov 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.4
Continuing the research over index matrices, in this paper, the concept of a three dimensional index matrix is introduced. Over itm several operations and relations are defined.
Index matrix representations of the Pascal's like triangles
K. T. Atanassov 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.4
An arithmetic function is defined, and survey of its basic properties and some new results are studied.
A DIGITAL ARITHMETICAL FUNCTION AND SOME OF ITS APPLICATIONS
K. T. Atanassov 장전수학회 2015 Proceedings of the Jangjeon mathematical society Vol.18 No.4
An arithmetical function is dened and survey of its basic properties and some new results are studied.
On intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets
K. T. Atanassov,P. Rangasam 장전수학회 2009 Advanced Studies in Contemporary Mathematics Vol.18 No.1
A new operator de¯ned over an interval valued intuitionistic fuzzy sets is defined. It gives in a result an intutionistic fuzzy set. Some properties of the new operator are studied.