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The paper considers implementations of some randomized algorithms in connection with obtaining a random $n^2 \times n^2$ Sudoku matrix with programming language C++. For this purpose we describes the set $\Pi_n$ of all $(2n) \times n$ matrices, consisting of elements of the set $\mathbb{Z}_n =\{ 1,2,\ldots ,n\}$, such that every row is a permutation. We emphasize the relationship between these matrices and the $n^2 \times n^2$ Sudoku matrices. An algorithm to obtain random $\Pi_n$ matrices is presented in this paper. Several auxiliary algorithms that are related to the underlying problem have been described. We evaluated all algorithms according to two criteria - probability evaluation, and time for generation of random objects and checking of belonging to a specific set. This evaluations are interesting from both theoretical and practical point of view because they are particularly useful in the analysis of computer programs., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1312.0192

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 to directly operate with every bit of an arbitrary variable situated in the computers 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., Comment: arXiv admin note: text overlap with arXiv:1312.0138

The work considers an equivalence relation in the set of all $n\times m$ matrices with entries in the set $[p]=\{ 0,1,\ldots , p-1 \}$. 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., Comment: arXiv admin note: text overlap with arXiv:1604.02714

The e-learning is an advanced version of traditional education. It is 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 who 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 students readiness for e-learning in Yemeni universities.

Context-free languages are widely used to describe the syntax of programming languages and natural languages. Usually, we describe a context-free language mathematically with the help of context-free grammar (for generation) or pushdown automata (for recognition). The purpose of this study is to describe some unconventional methods of description of context-free languages, namely a representation with the help of finite digraphs and with automata - generators of context-free languages. We will mainly focus on the mathematical models of these representations.

The main thrust of the article is to provide interesting example, useful for students of using bitwise operations in the programming languages C ++ and Java. As an example, we describe an algorithm for obtaining a Latin square of arbitrary order. We will outline some techniques for the use of bitwise operations.

The current paper is dedicated to the problem of finding the number of mutually non isomorphic bipartite graphs of the type $g=\langle R_g ,C_g ,E_g \rangle$ at given $n=|R_g |$ and $m=|C_g |$, where $R_g$ and $C_g$ are the two disjoint parts of the vertices of the graphs $g$, and $E_g$ is the set of edges, $Eg \subseteq R_g \times C_g$. 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 $n \times m$ canonical binary matrices and consequently for finding all with exactness up to isomorphism binary matrices with cardinality of each part equal to $n$ and $m$.

This work examines the concept of S-permutation matrices, namely $n^2 \times n^2$ permutation matrices containing a single 1 in each canonical $n \times n$ subsquare (block). The article suggests a formula for counting mutually disjoint pairs of $n^2 \times n^2$ S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of $n \times n$ binary matrices. The paper describe an algorithm that solves the main problem. To do that, every $n\times n$ binary matrix is represented uniquely as a n-tuple of integers., Comment: arXiv admin note: substantial text overlap with arXiv:1501.03395; text overlap with arXiv:1604.02691

This work examines the problem to describe an efficient algorithm for obtaining $n^2 \times n^2$ Sudoku matrices. For this purpose, we define the concepts of $n\times n$ $\Pi_n$-matrix and disjoint $\Pi_n$-matrices. The article, using the set-theoretical approach, describes an algorithm for obtaining $n^2$-tuples of $n\times n$ mutually disjoint $\Pi_n$ matrices. We show that in input $n^2$ mutually disjoint $\Pi_n$ matrices, it is not difficult to receive a Sudoku matrix.

This paper offers a glimpse of the major contributions made by Arabs to mathematics in middle ages history period. Its purpose is to stimulate interest in an object based on mutual respect and understanding. We give a short list of the most popular Arabic mathematicians and we comment some of their studies for using them in the high school teaching.

This article discusses some difficulties in the implementation of combinatorial algorithms associated with the choice of all elements with certain properties among the elements of a set with great cardinality.The problem has been resolved by using multidimensional arrays. Illustration of the method is a solution of the problem of obtaining one representative from each equivalence class with respect to the described in the article equivalence relation in the set of all $m\sim n$ binary matrices. This equivalence relation has an application in the mathematical modeling in the textile industry.

In this paper, we define the concepts of semi-canonical and canonical binary matrix. Strictly mathematical, we prove the correctness of these definitions. We describe and we implement an algorithm for finding all semi-canonical binary matrices taking into account the number of 1 in each of them. This problem relates to the combinatorial problem of finding all pairs of disjoint S-permutation matrices. In the described algorithm, the bit-wise operations are substantially used., Comment: Sixth International Scientific Conference - FMNS2015, South-West University, Blagoevgrad, Bulgaria, 113 - 124

The concept of S-permutation matrix is considered. A general formula for counting all disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the positive integer $n$ is formulated and proven in this paper. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of $n^2 \times n^2$ S-permutation matrices, it is sufficient to obtain some numerical characteristics of all $n\times n$ bipartite graphs., Comment: arXiv admin note: text overlap with arXiv:1211.1628

Let $n$ be a positive integer, $\sigma$ be an element of the symmetric group $\mathcal{S}_n$ and let $\sigma$ be a cycle of length $n$. The elements $\alpha ,\beta \in \mathcal{S}_n$ are $\sigma$-equivalent, if there are natural numbers $k$ and $l$, such that $\sigma^k \alpha =\beta\sigma^l$, which is the same as the condition to exist natural numbers $k_1$ and $l_1$, such that $\alpha = \sigma^{k_1} \beta\sigma^{l_1}$. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph $\Gamma_n$ 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.

The present work has been designed for students in secondary school and their teachers in mathematics. We will show how with the help of our knowledge of number systems we can solve problems from other fields of mathematics for example in combinatorial analysis and most of all when proving some combinatorial identities. To demonstrate discussed in this article method we have chosen several suitable mathematical tasks.

Different points of view on the nature and content of the assertiveness are followed in this paper. The main purpose is to study the assertive profile of Bulgarian students in computer science and computer engineering by analyzing the components of assertiveness. Research was performed using testing methods. It was found that the level of expressivity of this personal quality among subjects were above-average level.

In the work are defined the concepts semi-canonical and canonical binary matrix. What is described is an algorithm solving the combinatorial problem for finding the semi-canonical matrices in the set \Lambda_n^k consisting of all n\times n binary matrices having exactly k 1's in every row and every column without perambulating all elements. In the described algorithm bitwise operations are substantially used. In this way it becomes easier to find the solution to the problem for receiving one representative from every equivalence class regarding the introduced in the article equivalence relation in the set \Lambda_n^k . The last problem is equivalent to the problem for finding all canonical matrices in \Lambda_n^k .

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 $g=\langle R_g \cup C_g, E_g \rangle$ is formulated and proved, where $V=R_g \cup C_g$ is the set of vertices, $E_g$ is the set of edges of the graph $g$, $ |R_g |=m\ge 1$, $|C_g |= n\ge 1$, $|E_g |=k\ge 0$, $m,n$ and $k$ are integers.

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.

Some randomized algorithms, used to obtain a random $n^2 \times n^2$ Sudoku matrix, where $n$ is a natural number, is reviewed in this study. Below is described the set $\Pi_n$ of all $(2n) \times n$ matrices, consisting of elements of the set $\mathbb{Z}_n =\{ 1,2,\ldots ,n\}$, such that every row is a permutation. It is proved that such matrices would be particularly useful in developing efficient algorithms in generating Sudoku matrices. An algorithm to obtain random $\Pi_n$ matrices is presented in this paper. The algorithms are evaluated according to two criteria - probability evaluation, and time evaluation. This type of criteria is interesting from both theoretical and practical point of view because they are particularly useful in the analysis of computer programs.

An algorithm for obtaining all n\times 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.

Let $\Sigma = X\cup X^{-1} = \{ x_1 ,x_2 ,..., x_m ,x_1^{-1} ,x_2^{-1} ,..., x_m^{-1} \}$ and let $G$ be a group with set of generators $\Sigma$. Let $\mathfrak{L} (G) =\left\{ \left. \omega \in \Sigma^* \; \right\vert \;\omega \equiv e \; (\textrm{mod} \; G) \right\} \subseteq \Sigma^*$ be the group language representing $G$, where $\Sigma^*$ is a free monoid over $\Sigma$ and $e$ is the identity in $G$. The problem of determining whether a context-free language is subset of a group language is discussed. Polynomial algorithms are presented for testing whether a regular language, or a linear language is included in a group language. A few finite sets are built, such that each of them is included in the group language $\mathfrak{L} (G)$ if and only if the respective context-free language is included in $\mathfrak{L} (G)$., Comment: 14 pages

In the work we discuss the benefit of using bitwise operations in programming. Some interesting examples in this respect have been shown. What is described in detail is an algorithm for sorting an integer array with the substantial use of the bitwise operations. Besides its correctness we strictly prove that the described algorithm works in time O(n). In the work during the realisation of each of the examined algorithms we use the apparatus of the object-oriented programming with the syntax and the semantics of the programming language C++.

For any context-free grammar, we build a transition diagram, that is, a finite directed graph with labeled arcs, which describes the work of the grammar. This approach is new, and it is different from previously known graph models. We define the concept of proper walk in this transition diagram and we prove that a word belongs to a given context-free language if and only if this word can be obtained with the help of a proper walk., Comment: 4 figures

We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of integer sequences related to the discussed problem. The obtained sequences are new and they are not described in the On-Line Encyclopedia of Integer Sequences (OEIS). We show a relationship between some particular values of the parameters and the Fibonacci sequence.

Some techniques for the use of bitwise operations are described in the article. As an example, an open problem of isomorphism-free generations of combinatorial objects is discussed. An equivalence relation on the set of square binary matrices having the same number of units in each row and each column is defined. Each binary matrix is represented using ordered n-tuples of natural numbers. It is shown how by using the bitwise operations can be implemented an algorithm that gets canonical representatives which are extremal elements of equivalence classes relative to a double order on the set of considered objects., Comment: 9 pages, 2 tables

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 $n^2 \times n^2$ 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 $n^2 \times n^2$ S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type $g=$, where $V=R_g \cup C_g$ is the set of vertices, and $E_g$ is the set of edges of the graph $g$, $R_g \cap C_g =\emptyset$, $|R_g|=|C_g|=n$., Comment: 14 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1202.0401

The concept of S-permutation matrix is considered in this paper. It defines when two binary matrices are disjoint. For an arbitrary $n^2 \times n^2$ S-permutation matrix, a lower band of the number of all disjoint with it S-permutation matrices is found. A formula for counting a lower band of the number of all disjoint pairs of $n^2 \times n^2$ S-permutation matrices is formulated and proven. As a consequence, a lower band of the probability of two randomly generated S-permutation matrices to be disjoint is found. In particular, a different proof of a known assertion is obtained in the work. The cases when $n=2$ and $n=3$ are discussed in detail.

Computer administering of a psychological investigation is the computer representation of the entire procedure of psychological assessments - test construction, test implementation, results evaluation, storage and maintenance of the developed database, its statistical processing, analysis and interpretation. A mathematical description of psychological assessment with the aid of personality tests is discussed in this article. The set theory and the relational algebra are used in this description. A relational model of data, needed to design a computer system for automation of certain psychological assessments is given. Some finite sets and relation on them, which are necessary for creating a personality psychological test, are described. The described model could be used to develop real software for computer administering of any psychological test and there is full automation of the whole process: test construction, test implementation, result evaluation, storage of the developed database, statistical implementation, analysis and interpretation. A software project for computer administering personality psychological tests is suggested.

Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general $n^2 \times n^2$ case are discussed in this paper. All bipartite graphs of the type $g=$, where $|R_g |=|C_g |=2$ or $|R_g |=|C_g |=3$ are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the $4 \times 4$ and $9 \times 9$ cases are calculated., Comment: 18 pages, 13 figures. arXiv admin note: text overlap with arXiv:1211.1628

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. Key words: Education in programming, programming languages, data structures, set, Sudoku matrix, Sudoku puzzle.

The present study is meant to fill in some information gaps occurring in the most widespread and well-known educational and reference literature about programming. The stress is laid on a very useful instrument - the bitwise operations, topic which is, unfortunately, seldom dealt with in most of the well-known books on programming. In addition, the research is very useful as regards the topic of overloading operators in any Object-oriented programming course. Given some appropriate examples, with the emphasis being laid on some particular and data structures language constructions, the results are quite interesting. The algorithm of solving the popular Sudoku puzzle is one such entertaining example., Comment: Proceedings of the Fourth International Scientific Conference - FMNS2011, 8 - 11 June, 2011

An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described., Comment: Proceedings of the Fourth International Scientific Conference -- FMNS2011, 8 -- 11 June 2011

In this article we discuss the presentation of a random binary matrix using sequence of whole nonnegative numbers. We examine some advantages and disadvantages of this presentation as an alternative of the standard presentation using two-dimensional array. It is shown that the presentation of binary matrices using ordered n-tuples of natural numbers makes the algorithms faster and saves a lot of memory. In this work we use object-oriented programming using the syntax and the semantic of C++ programming language., Comment: Proceedings of the Thirty Ninth Spring Conference of the Union of Bulgarian Mathematicians, Albena, April 6-10, 2010

This piece of work presents a meaningful example for the advantages of using bitwise operations for creating effective algorithms in programming. A task connected with mathematical modeling in weaving industry is examined and computed., Comment: Proceedings of the Thirty Eighth Spring Conference of the Union of Bulgarian Mathematicians, Borovetz, April 1-?5, 2009

An equivalence relation in the set of all square binary matrices is described in this work. It is discussed a combinatoric problem about finding the cardinal number and the elements of the factor set according to this relation. We examine the possibility to get some special elements of this factor set. We propose an algorithm, which solves these problems. The results we have received are used to describe the topology of the different weaving structures., Comment: Proceedings of the Thirty Ninth Spring Conference of the Union of Bulgarian Mathematicians, Albena, April 6-10, 2010

The paper studies the set of all square binary matrices containing an exact number of 1's in each rows and in each column. A connection is established between the cardinal number of this set and the cardinal number of its subset of matrices containing 1 in the lower right corner. With the help of this result a new proof is advanced of the I. Good and J. Grook theorem. In connection with the firs result a classification has also been made of square binary matrices containing three 1's in each row and column and 1 in the lower right corner., Comment: 7 pages

Some difficulties regarding the application of the well-known sieve method are considered in the case when a practical (program) realization of selecting elements, having a particular property among the elements of a set with a sufficiently great cardinal number(cardinality). In this paper the problem has been resolved by using a modified version of the method, utilizing multidimensional arrays. As a theoretical illustration of the method of the multidimensional sieve, the problem of obtaining a single representative of each equivalence class with respect to a given relation of equivalence and obtaining the cardinality of the respective factor set is considered with relevant mathematical proofs.

A formal-linguistic approach for solving an entertaining task is made in this paper. The well-known task of the Hanoi towers is discussed in relation to some concepts of discrete mathematics. A context-free grammar which generate an algorithm for solving this task is described. A deterministic pushdown automation which in its work imitates the work of monks in solving the task of the Hanoi towers is built.