Krasimir Yordzhev - Academia.edu (original) (raw)
Papers by Krasimir Yordzhev
The book examines some combinatorial problems related to the number of Sudoku matrices. The probl... more The book examines some combinatorial problems related to the number of Sudoku matrices. The problem is reduced to the task of finding the number of mutually disjoint pairs of S-permutation matrices. We describe some algorithms that solve this problem. Essential role in the description of the corresponding formulas and algorithms play bipartite graphs and operations with binary matrices. Solving the main problem, we have discussed also other mathematical and algorithmic problems, which in itself are interesting.
arXiv (Cornell University), Jan 17, 2012
Some aspects of programming education are examined in this work. It is emphasised, based on the e... more Some aspects of programming education are examined in this work. It is emphasised, based on the entertainment value, the most appropriate examples are chosen to demonstrate the different language constructions and data structures. Such an example is the demonstrated algorithm for solving the widespread nowadays "Sudoku" puzzle. This is made, because of the connection with the term set and putting it into practice in the programming. Using the so built program there are solved some combinatorial problems, connected to the Sudoku matrices.
Applied Mathematics and Computation, Jun 1, 2014
ABSTRACT The article discusses the set of square n×nn×n binary matrices with the same number of 1... more ABSTRACT The article discusses the set of square n×nn×n binary matrices with the same number of 1’s in each row and each column. An equivalence relation on this set is introduced. Each binary matrix is represented using ordered n-tuples of natural numbers. We are looking for a formula which calculates the number of elements of each factor-set by the introduced equivalence relation. We show a relationship between some particular values of the parameters and the Fibonacci sequence.
Discrete Applied Mathematics, Dec 1, 2013
In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers... more In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the integer n naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type g = R g ∪ C g , E g , where V = R g ∪ C g is the set of vertices, and E g is the set of edges of the graph g, R g ∩ C g = ∅, |R g | = |C g | = n.
Asian Journal of Research in Computer Science, Sep 29, 2018
We contemplate this article to help the teachers of programming in his aspiration for giving some... more We contemplate this article to help the teachers of programming in his aspiration for giving some appropriate and interesting examples. The work will be especially useful for students-future programmers, and for their lecturers. Some of the strong sides of these programming languages C/C++ and Java are the possibilities of low-level programming. Some of the means for this possibility are the introduced standard bitwise operations, with the help of which, it is possible directly operate with every bit of an arbitrary variable situated in the computer's memory. In the current study, we are going to describe some methodical aspects for work with the bitwise operations and we will discuss the benefit of using bitwise operations in programming. The article shows some advantages of using bitwise operations, realizing various operations with sets.
arXiv (Cornell University), Apr 10, 2016
The current paper is dedicated to the problem of finding the number of mutually non isomorphic bi... more The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type g = Rg, Cg, Eg at given n = |Rg| and m = |Cg|, where Rg and Cg are the two disjoint parts of the vertices of the graphs g, and Eg is the set of edges, Eg ⊆ Rg × Cg. For this purpose, the concept of canonical binary matrix is introduced. The different canonical matrices unambiguously describe the different with exactness up to isomorphism bipartite graphs. We have found a necessary and sufficient condition an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all n × m canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to n and m.
Asian-european Journal of Mathematics, Jun 8, 2022
In addition to the deterministic models, the stochastic models are successfully used for mathemat... more In addition to the deterministic models, the stochastic models are successfully used for mathematical description of real-world processes and phenomena. They use knowledge from the mathematical theories of probability and statistics to analyze specific properties of complex living systems. This paper considers some stochastic models of various processes and phenomena in genetics. Basic knowledge used in classical stochastic models is considered and applied to real problems from medical genetics with their solutions.
International journal of education and management engineering, Jan 8, 2014
Asian-european Journal of Mathematics, Jun 1, 2014
Let n be a positive integer, σ be an element of the symmetric group Sn and let σ be a cycle of le... more Let n be a positive integer, σ be an element of the symmetric group Sn and let σ be a cycle of length n. The elements α, β ∈ Sn are σ-equivalent, if there are natural numbers k and l, such that σ k α = βσ l , which is the same as the condition to exist natural numbers k1 and l1, such that α = σ k 1 βσ l 1. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph Γn with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalence classes according to this relation.
American Journal of Applied Mathematics, 2013
Notes on Number Theory and Discrete Mathematics, Dec 1, 2018
The work considers an equivalence relation in the set of all n × m matrices with entries in the s... more The work considers an equivalence relation in the set of all n × m matrices with entries in the set [p] = {0, 1,. .. , p − 1}. In each element of the factor-set generated by this relation, we define the concept of canonical matrix, namely the minimal element with respect to the lexicographic order. We have found a necessary and sufficient condition for an arbitrary matrix with entries in the set [p] to be canonical. For this purpose, the matrices are uniquely represented by ordered n-tuples of integers.
Ìnformacìjnì tehnologìï v osvìtì, Jul 25, 2011
Some aspects of programming education are examined in this work. It is emphasised, based on the e... more Some aspects of programming education are examined in this work. It is emphasised, based on the entertainment value, the most appropriate examples are chosen to demonstrate the different language constructions and data structures. Such an example is the demonstrated algorithm for solving the widespread nowadays "Sudoku" puzzle. This is made, because of the connection with the term set and putting it into practice in the programming. Using the so built program there are solved some combinatorial problems, connected to the Sudoku matrices.
arXiv (Cornell University), Apr 25, 2014
The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some nume... more The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these characteristics of the set of all bipartite graphs of the type g = Rg ∪ Cg, Eg is formulated and proved, where V = Rg ∪ Cg is the set of vertices, Eg is the set of edges of the graph g, |Rg| = m ≥ 1, |Cg| = n ≥ 1, |Eg| = k ≥ 0, m, n and k are integers.
Some numerical characteristics of bipartite graphs in relation to the problem of finding all disj... more Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general n 2 × n 2 case are discussed in this paper. All bipartite graphs of the type g = Rg ∪ Cg, Eg , where |Rg| = |Cg| = 2 or |Rg| = |Cg| = 3 are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the 4 × 4 and 9 × 9 cases are calculated.
Journal of mathematical sciences and applications, Jan 23, 2013
An algorithm for obtaining all n n × binary matrices having exactly 2 units in every row and ever... more An algorithm for obtaining all n n × binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.
arXiv (Cornell University), Oct 9, 2020
The e-learning is an advanced version of the traditional education. It's defined as a way of lear... more The e-learning is an advanced version of the traditional education. It's defined as a way of learning by using the communication mechanisms of modern computer networks and multimedia, including voice, image, and graphics and mechanisms to search electronic libraries, as well as web portals, whether in the context of distance learning or in the classroom. The people engage in the transition to web-supported education are the administrative staff, the faculty, and the students. They all have their needs and they all should meet specific requirements in order to facilitate the transition. The article presents the results of questionnaire research of the student's readiness for e-learning in Yemeni universities.
The book examines some combinatorial problems related to the number of Sudoku matrices. The probl... more The book examines some combinatorial problems related to the number of Sudoku matrices. The problem is reduced to the task of finding the number of mutually disjoint pairs of S-permutation matrices. We describe some algorithms that solve this problem. Essential role in the description of the corresponding formulas and algorithms play bipartite graphs and operations with binary matrices. Solving the main problem, we have discussed also other mathematical and algorithmic problems, which in itself are interesting.
In this work, application of binary matrices in k-valued logic is given. Objects of our study are... more In this work, application of binary matrices in k-valued logic is given. Objects of our study are classes of k-valued functions and combinatorial problems related to them. We use proofs connected with finding the number of some classes of binary matrices. Some combinatorial identities and inequalities are obtained which would also be of independent interest.
arXiv (Cornell University), Dec 1, 2013
The study proves the existence of an algorithm to receive all elements of a class of binary matri... more The study proves the existence of an algorithm to receive all elements of a class of binary matrices without obtaining redundant elements, e. g. without obtaining binary matrices that do not belong to the class. This makes it possible to avoid checking whether each of the objects received possesses the necessary properties. This significantly improves the efficiency of the algorithm in terms of the criterion of time. Certain useful educational effects related to the analysis of such problems in programming classes are also pointed out.
arXiv (Cornell University), Dec 1, 2013
An algorithm for obtaining all n × n binary matrices having exactly 2 units in every row and ever... more An algorithm for obtaining all n × n binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.
The book examines some combinatorial problems related to the number of Sudoku matrices. The probl... more The book examines some combinatorial problems related to the number of Sudoku matrices. The problem is reduced to the task of finding the number of mutually disjoint pairs of S-permutation matrices. We describe some algorithms that solve this problem. Essential role in the description of the corresponding formulas and algorithms play bipartite graphs and operations with binary matrices. Solving the main problem, we have discussed also other mathematical and algorithmic problems, which in itself are interesting.
arXiv (Cornell University), Jan 17, 2012
Some aspects of programming education are examined in this work. It is emphasised, based on the e... more Some aspects of programming education are examined in this work. It is emphasised, based on the entertainment value, the most appropriate examples are chosen to demonstrate the different language constructions and data structures. Such an example is the demonstrated algorithm for solving the widespread nowadays "Sudoku" puzzle. This is made, because of the connection with the term set and putting it into practice in the programming. Using the so built program there are solved some combinatorial problems, connected to the Sudoku matrices.
Applied Mathematics and Computation, Jun 1, 2014
ABSTRACT The article discusses the set of square n×nn×n binary matrices with the same number of 1... more ABSTRACT The article discusses the set of square n×nn×n binary matrices with the same number of 1’s in each row and each column. An equivalence relation on this set is introduced. Each binary matrix is represented using ordered n-tuples of natural numbers. We are looking for a formula which calculates the number of elements of each factor-set by the introduced equivalence relation. We show a relationship between some particular values of the parameters and the Fibonacci sequence.
Discrete Applied Mathematics, Dec 1, 2013
In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers... more In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the integer n naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type g = R g ∪ C g , E g , where V = R g ∪ C g is the set of vertices, and E g is the set of edges of the graph g, R g ∩ C g = ∅, |R g | = |C g | = n.
Asian Journal of Research in Computer Science, Sep 29, 2018
We contemplate this article to help the teachers of programming in his aspiration for giving some... more We contemplate this article to help the teachers of programming in his aspiration for giving some appropriate and interesting examples. The work will be especially useful for students-future programmers, and for their lecturers. Some of the strong sides of these programming languages C/C++ and Java are the possibilities of low-level programming. Some of the means for this possibility are the introduced standard bitwise operations, with the help of which, it is possible directly operate with every bit of an arbitrary variable situated in the computer's memory. In the current study, we are going to describe some methodical aspects for work with the bitwise operations and we will discuss the benefit of using bitwise operations in programming. The article shows some advantages of using bitwise operations, realizing various operations with sets.
arXiv (Cornell University), Apr 10, 2016
The current paper is dedicated to the problem of finding the number of mutually non isomorphic bi... more The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type g = Rg, Cg, Eg at given n = |Rg| and m = |Cg|, where Rg and Cg are the two disjoint parts of the vertices of the graphs g, and Eg is the set of edges, Eg ⊆ Rg × Cg. For this purpose, the concept of canonical binary matrix is introduced. The different canonical matrices unambiguously describe the different with exactness up to isomorphism bipartite graphs. We have found a necessary and sufficient condition an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all n × m canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to n and m.
Asian-european Journal of Mathematics, Jun 8, 2022
In addition to the deterministic models, the stochastic models are successfully used for mathemat... more In addition to the deterministic models, the stochastic models are successfully used for mathematical description of real-world processes and phenomena. They use knowledge from the mathematical theories of probability and statistics to analyze specific properties of complex living systems. This paper considers some stochastic models of various processes and phenomena in genetics. Basic knowledge used in classical stochastic models is considered and applied to real problems from medical genetics with their solutions.
International journal of education and management engineering, Jan 8, 2014
Asian-european Journal of Mathematics, Jun 1, 2014
Let n be a positive integer, σ be an element of the symmetric group Sn and let σ be a cycle of le... more Let n be a positive integer, σ be an element of the symmetric group Sn and let σ be a cycle of length n. The elements α, β ∈ Sn are σ-equivalent, if there are natural numbers k and l, such that σ k α = βσ l , which is the same as the condition to exist natural numbers k1 and l1, such that α = σ k 1 βσ l 1. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph Γn with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalence classes according to this relation.
American Journal of Applied Mathematics, 2013
Notes on Number Theory and Discrete Mathematics, Dec 1, 2018
The work considers an equivalence relation in the set of all n × m matrices with entries in the s... more The work considers an equivalence relation in the set of all n × m matrices with entries in the set [p] = {0, 1,. .. , p − 1}. In each element of the factor-set generated by this relation, we define the concept of canonical matrix, namely the minimal element with respect to the lexicographic order. We have found a necessary and sufficient condition for an arbitrary matrix with entries in the set [p] to be canonical. For this purpose, the matrices are uniquely represented by ordered n-tuples of integers.
Ìnformacìjnì tehnologìï v osvìtì, Jul 25, 2011
Some aspects of programming education are examined in this work. It is emphasised, based on the e... more Some aspects of programming education are examined in this work. It is emphasised, based on the entertainment value, the most appropriate examples are chosen to demonstrate the different language constructions and data structures. Such an example is the demonstrated algorithm for solving the widespread nowadays "Sudoku" puzzle. This is made, because of the connection with the term set and putting it into practice in the programming. Using the so built program there are solved some combinatorial problems, connected to the Sudoku matrices.
arXiv (Cornell University), Apr 25, 2014
The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some nume... more The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these characteristics of the set of all bipartite graphs of the type g = Rg ∪ Cg, Eg is formulated and proved, where V = Rg ∪ Cg is the set of vertices, Eg is the set of edges of the graph g, |Rg| = m ≥ 1, |Cg| = n ≥ 1, |Eg| = k ≥ 0, m, n and k are integers.
Some numerical characteristics of bipartite graphs in relation to the problem of finding all disj... more Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general n 2 × n 2 case are discussed in this paper. All bipartite graphs of the type g = Rg ∪ Cg, Eg , where |Rg| = |Cg| = 2 or |Rg| = |Cg| = 3 are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the 4 × 4 and 9 × 9 cases are calculated.
Journal of mathematical sciences and applications, Jan 23, 2013
An algorithm for obtaining all n n × binary matrices having exactly 2 units in every row and ever... more An algorithm for obtaining all n n × binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.
arXiv (Cornell University), Oct 9, 2020
The e-learning is an advanced version of the traditional education. It's defined as a way of lear... more The e-learning is an advanced version of the traditional education. It's defined as a way of learning by using the communication mechanisms of modern computer networks and multimedia, including voice, image, and graphics and mechanisms to search electronic libraries, as well as web portals, whether in the context of distance learning or in the classroom. The people engage in the transition to web-supported education are the administrative staff, the faculty, and the students. They all have their needs and they all should meet specific requirements in order to facilitate the transition. The article presents the results of questionnaire research of the student's readiness for e-learning in Yemeni universities.
The book examines some combinatorial problems related to the number of Sudoku matrices. The probl... more The book examines some combinatorial problems related to the number of Sudoku matrices. The problem is reduced to the task of finding the number of mutually disjoint pairs of S-permutation matrices. We describe some algorithms that solve this problem. Essential role in the description of the corresponding formulas and algorithms play bipartite graphs and operations with binary matrices. Solving the main problem, we have discussed also other mathematical and algorithmic problems, which in itself are interesting.
In this work, application of binary matrices in k-valued logic is given. Objects of our study are... more In this work, application of binary matrices in k-valued logic is given. Objects of our study are classes of k-valued functions and combinatorial problems related to them. We use proofs connected with finding the number of some classes of binary matrices. Some combinatorial identities and inequalities are obtained which would also be of independent interest.
arXiv (Cornell University), Dec 1, 2013
The study proves the existence of an algorithm to receive all elements of a class of binary matri... more The study proves the existence of an algorithm to receive all elements of a class of binary matrices without obtaining redundant elements, e. g. without obtaining binary matrices that do not belong to the class. This makes it possible to avoid checking whether each of the objects received possesses the necessary properties. This significantly improves the efficiency of the algorithm in terms of the criterion of time. Certain useful educational effects related to the analysis of such problems in programming classes are also pointed out.
arXiv (Cornell University), Dec 1, 2013
An algorithm for obtaining all n × n binary matrices having exactly 2 units in every row and ever... more An algorithm for obtaining all n × n binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.