조선시대 산학제도에서 훈도라는 하급관리가 중앙 그리고 지방 향교와 관아에도 있었다. 수군의 조직에도 있었던 훈도는 각종 계산을 담당했다는 것을 여러 가지 자료로부터 알 수 있... more 조선시대 산학제도에서 훈도라는 하급관리가 중앙 그리고 지방 향교와 관아에도 있었다. 수군의 조직에도 있었던 훈도는 각종 계산을 담당했다는 것을 여러 가지 자료로부터 알 수 있다. 또 조선의 산학서에 멀리 떨어진 바다에서 섬까지의 거리를 구하는 망해도술이 있었다. 이순신 장군이 임진왜란 때 학익진과 같은 새로운 전법으로 왜군을 물리쳤음을 알고 있다. 그런데 학익진을 펼쳐 판옥선과 왜선 사이의 거리를 정확히 측량하고 화포의 명중률을 높여 일시집중타로 전투를 승리로 이끈 밑바탕에 조선의 중인계급이었던 산학자가 있었다. 임진왜란 당시 이순신 장군은 뛰어난 전술은 물론 탁월한 지도통솔력에 의한 용병작전으로 왜군과의 해전에서 항상 승리했고, 그 뒷받침에는 조선의 특수한 수군제도와 거북선 및 화포와 같은 병기를 보유했다는 점이다. 아울러 학익진과 같은 새로운 전법을 성공적으로 이끌 수 있도록 보이지 않게 도운 조선의 산학자가 있었음을 알아야 한다.
Discussiones Mathematicae - General Algebra and Applications, 2000
The maximal column rank of an m by n matrix is the maximal number of the columns of A which are l... more The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.
Most who have heard of Sang-Seol Lee know him for his contribution to the Korean independence mov... more Most who have heard of Sang-Seol Lee know him for his contribution to the Korean independence movement nearly a hundred years ago. This paper, however, will discuss Lee's other great contribution to his country; that of being "The father of modern mathematical education in Korea". Lee passed the rigorous government officer examination with the highest honor and became a teacher for the royal prince. Later he became the president of Sunkyunkwan, a national institute of higher learning since 1398, and eventually a well-known university bearing the same name. Lee was also a highly regarded Confucian scholar and well versed in foreign languages. He wanted Korea to become a modern country and felt that the areas of science and engineering were studies that needed improving in order to achieve modernization. While researching Western textbooks on the subjects he realized that Western mathematics would be especially important for Korea. With that, it became his mission to int...
A real nonsingular n × n matrix A = (a ij) is called centrogonal if A −1 = (a n+1−i,n+1−j), it is... more A real nonsingular n × n matrix A = (a ij) is called centrogonal if A −1 = (a n+1−i,n+1−j), it is called principally centrogonal if all leading principal submatrices of A are centrogonal, and it is called inverse principally centrogonal if A −1 is principally centrogonal. We give a necessary and sufficient condition for a principally centrogonal matrix to be an inverse principally centrogonal matrix.
In this article, we give a comparative study on the last 300 years of USA and Korean tertiary mat... more In this article, we give a comparative study on the last 300 years of USA and Korean tertiary mathematics. The first mathematics classes in United States were offered before July, 1638, but the real founding of tertiary mathematics courses was in 1640 when Henry Dunster assumed the duties of the presidency at Harvard. President Dunster read arithmetics and geometry on Mondays and Tuesdays to the third year students during the first three quarters, and astronomy in the last quarter. So tertiary mathematics education in United States began at Harvard which is the oldest college in USA. After 230 years since then, Benjamin Peirce in 1870 made a major and first American contribution to mathematics and got an attention from European mathematicians. Major change on the role of Harvard mathematics from teaching to research made by G.D. Birkhoff when he joined as an assistant professor in 1912. Tertiary mathematics education in Korea started long before Chosun Dynasty. But it was given to o...
In this paper we establish some new identities involving Stirling numbers of both kinds. These id... more In this paper we establish some new identities involving Stirling numbers of both kinds. These identities are obtained via Riodan arrays with a variable x. Some well-known identities are obtained as special cases of the new identities for the specific number x.
The permanent of the m-by-n matrix A is the sum of all possible products of m elements from A wit... more The permanent of the m-by-n matrix A is the sum of all possible products of m elements from A with the property that the elements in each of the products lie on different lines of A. This scalar valued function of the matrix A occurs throughout the combinatorial literature in connection with various enumeration and extremal problems. In this note, we can construct a (0, 1)-matrix with a prescribed permanent, 1, 2, . . . , 2 n−1 . Also, we consider the distribution of (0, 1)-matrices with permanent less than or equal to n!.
In [4], it was shown that an n by n orthogonal matrix which has a row of nonzeros has at least ( ... more In [4], it was shown that an n by n orthogonal matrix which has a row of nonzeros has at least ( log2n + 3)n - log2n +1 nonzero entries. In this paper, the matrices achieving these bounds are constructed. The analogous sparsity problem for m by n row-orthogonal matrices which have a row of nonzeros in conjectured.
A real m n matrix A is semipositive (SP) if there is a vector x 0 such that Ax > 0, inequaliti... more A real m n matrix A is semipositive (SP) if there is a vector x 0 such that Ax > 0, inequalities being entrywise. A is minimally semipositive (MSP) i f A is semipositive and no column deleted submatrix of A is semipositive. We give a necessary and suucient condition for the sign pattern matrix with n positive e n tries to be minimally semipositive.
A partial matrix is a rectangular array in which some entries are specified, while the remaining ... more A partial matrix is a rectangular array in which some entries are specified, while the remaining unspecified entries are free to be chosen from an indicated set (such as the real numbers or complex numbers). A completion of a partial matrix is a specific choice of values for the unspecified entries resulting in a conventional ma- trix. Problems based on a variety of properties have been studied. But the positive definite completion problem has received the most attention, due, in part, to its role in several applications in probability and statistics, image enhancement, systems engineering, geophysics, etc. and to its relation with other completion problems including spectral norm contractions and Euclidean distance matrices which is im- portant for the molecular conformation problem in chemistry. Another view of the positive definite completion problem is as a mechanism for addressing a fundamen- tal problem in Euclidean geometry: which potential geometric configurations of vectors...
We say that a sign-pattern matrix of R requires the Perron property if every real matrix A in Q(B... more We say that a sign-pattern matrix of R requires the Perron property if every real matrix A in Q(B) satisfies the Perron property. We characterize the linear operators that strongly preserve matrices whose sign patterns require the Perron property. We show that any such linear operator T is nonsingular and can be written as the direct sum of two linear operators, the first acting on diagonal matrices and the second acting on matrices with diagonal entries all zero.
A Boolean matrix A is said to be r¡regular if each vertex in its digraph G(A) = G has outdegree a... more A Boolean matrix A is said to be r¡regular if each vertex in its digraph G(A) = G has outdegree and indegree exactly r: A is primitive if and only if there exist minimum integer k such that Ak > 0. For such a matrix, its digraph G is strongly connected and given any ordered pair of vertices x and y there is a directed walk from x to y of length k. Using the graph theory, we determine the lower bounds and upper bounds of r¡regular primitive matices in this paper. We also proved that the exponent of r¡regular primitive tournament T is just 3 and the exponent of r¡regular primitive symmetric matrix A satisfys exp(A) • 2(n ¡ r): 1. basic concepts
The affirmative feature which web based learning and problem based learning in the self directed ... more The affirmative feature which web based learning and problem based learning in the self directed learning having on-line and off-line study it amalgamates in connection Blended Learning method and specially in mathematics subject as a service course. We introduce a model of BL lecture of Mathematics and tried to find a new way for the quality improvement of university education.
We consider a 2-coloring of arcs on the primitive extremal tournament with the largest exponent o... more We consider a 2-coloring of arcs on the primitive extremal tournament with the largest exponent on n vertices. This 2-colored digraph is a 2-primitive tournament. Then, we consider the 2-exponent of a 2-primitive tournament. In this paper, we give an upper bound for the 2-exponent of the primitive extremal tournament.
조선시대 산학제도에서 훈도라는 하급관리가 중앙 그리고 지방 향교와 관아에도 있었다. 수군의 조직에도 있었던 훈도는 각종 계산을 담당했다는 것을 여러 가지 자료로부터 알 수 있... more 조선시대 산학제도에서 훈도라는 하급관리가 중앙 그리고 지방 향교와 관아에도 있었다. 수군의 조직에도 있었던 훈도는 각종 계산을 담당했다는 것을 여러 가지 자료로부터 알 수 있다. 또 조선의 산학서에 멀리 떨어진 바다에서 섬까지의 거리를 구하는 망해도술이 있었다. 이순신 장군이 임진왜란 때 학익진과 같은 새로운 전법으로 왜군을 물리쳤음을 알고 있다. 그런데 학익진을 펼쳐 판옥선과 왜선 사이의 거리를 정확히 측량하고 화포의 명중률을 높여 일시집중타로 전투를 승리로 이끈 밑바탕에 조선의 중인계급이었던 산학자가 있었다. 임진왜란 당시 이순신 장군은 뛰어난 전술은 물론 탁월한 지도통솔력에 의한 용병작전으로 왜군과의 해전에서 항상 승리했고, 그 뒷받침에는 조선의 특수한 수군제도와 거북선 및 화포와 같은 병기를 보유했다는 점이다. 아울러 학익진과 같은 새로운 전법을 성공적으로 이끌 수 있도록 보이지 않게 도운 조선의 산학자가 있었음을 알아야 한다.
Discussiones Mathematicae - General Algebra and Applications, 2000
The maximal column rank of an m by n matrix is the maximal number of the columns of A which are l... more The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.
Most who have heard of Sang-Seol Lee know him for his contribution to the Korean independence mov... more Most who have heard of Sang-Seol Lee know him for his contribution to the Korean independence movement nearly a hundred years ago. This paper, however, will discuss Lee's other great contribution to his country; that of being "The father of modern mathematical education in Korea". Lee passed the rigorous government officer examination with the highest honor and became a teacher for the royal prince. Later he became the president of Sunkyunkwan, a national institute of higher learning since 1398, and eventually a well-known university bearing the same name. Lee was also a highly regarded Confucian scholar and well versed in foreign languages. He wanted Korea to become a modern country and felt that the areas of science and engineering were studies that needed improving in order to achieve modernization. While researching Western textbooks on the subjects he realized that Western mathematics would be especially important for Korea. With that, it became his mission to int...
A real nonsingular n × n matrix A = (a ij) is called centrogonal if A −1 = (a n+1−i,n+1−j), it is... more A real nonsingular n × n matrix A = (a ij) is called centrogonal if A −1 = (a n+1−i,n+1−j), it is called principally centrogonal if all leading principal submatrices of A are centrogonal, and it is called inverse principally centrogonal if A −1 is principally centrogonal. We give a necessary and sufficient condition for a principally centrogonal matrix to be an inverse principally centrogonal matrix.
In this article, we give a comparative study on the last 300 years of USA and Korean tertiary mat... more In this article, we give a comparative study on the last 300 years of USA and Korean tertiary mathematics. The first mathematics classes in United States were offered before July, 1638, but the real founding of tertiary mathematics courses was in 1640 when Henry Dunster assumed the duties of the presidency at Harvard. President Dunster read arithmetics and geometry on Mondays and Tuesdays to the third year students during the first three quarters, and astronomy in the last quarter. So tertiary mathematics education in United States began at Harvard which is the oldest college in USA. After 230 years since then, Benjamin Peirce in 1870 made a major and first American contribution to mathematics and got an attention from European mathematicians. Major change on the role of Harvard mathematics from teaching to research made by G.D. Birkhoff when he joined as an assistant professor in 1912. Tertiary mathematics education in Korea started long before Chosun Dynasty. But it was given to o...
In this paper we establish some new identities involving Stirling numbers of both kinds. These id... more In this paper we establish some new identities involving Stirling numbers of both kinds. These identities are obtained via Riodan arrays with a variable x. Some well-known identities are obtained as special cases of the new identities for the specific number x.
The permanent of the m-by-n matrix A is the sum of all possible products of m elements from A wit... more The permanent of the m-by-n matrix A is the sum of all possible products of m elements from A with the property that the elements in each of the products lie on different lines of A. This scalar valued function of the matrix A occurs throughout the combinatorial literature in connection with various enumeration and extremal problems. In this note, we can construct a (0, 1)-matrix with a prescribed permanent, 1, 2, . . . , 2 n−1 . Also, we consider the distribution of (0, 1)-matrices with permanent less than or equal to n!.
In [4], it was shown that an n by n orthogonal matrix which has a row of nonzeros has at least ( ... more In [4], it was shown that an n by n orthogonal matrix which has a row of nonzeros has at least ( log2n + 3)n - log2n +1 nonzero entries. In this paper, the matrices achieving these bounds are constructed. The analogous sparsity problem for m by n row-orthogonal matrices which have a row of nonzeros in conjectured.
A real m n matrix A is semipositive (SP) if there is a vector x 0 such that Ax > 0, inequaliti... more A real m n matrix A is semipositive (SP) if there is a vector x 0 such that Ax > 0, inequalities being entrywise. A is minimally semipositive (MSP) i f A is semipositive and no column deleted submatrix of A is semipositive. We give a necessary and suucient condition for the sign pattern matrix with n positive e n tries to be minimally semipositive.
A partial matrix is a rectangular array in which some entries are specified, while the remaining ... more A partial matrix is a rectangular array in which some entries are specified, while the remaining unspecified entries are free to be chosen from an indicated set (such as the real numbers or complex numbers). A completion of a partial matrix is a specific choice of values for the unspecified entries resulting in a conventional ma- trix. Problems based on a variety of properties have been studied. But the positive definite completion problem has received the most attention, due, in part, to its role in several applications in probability and statistics, image enhancement, systems engineering, geophysics, etc. and to its relation with other completion problems including spectral norm contractions and Euclidean distance matrices which is im- portant for the molecular conformation problem in chemistry. Another view of the positive definite completion problem is as a mechanism for addressing a fundamen- tal problem in Euclidean geometry: which potential geometric configurations of vectors...
We say that a sign-pattern matrix of R requires the Perron property if every real matrix A in Q(B... more We say that a sign-pattern matrix of R requires the Perron property if every real matrix A in Q(B) satisfies the Perron property. We characterize the linear operators that strongly preserve matrices whose sign patterns require the Perron property. We show that any such linear operator T is nonsingular and can be written as the direct sum of two linear operators, the first acting on diagonal matrices and the second acting on matrices with diagonal entries all zero.
A Boolean matrix A is said to be r¡regular if each vertex in its digraph G(A) = G has outdegree a... more A Boolean matrix A is said to be r¡regular if each vertex in its digraph G(A) = G has outdegree and indegree exactly r: A is primitive if and only if there exist minimum integer k such that Ak > 0. For such a matrix, its digraph G is strongly connected and given any ordered pair of vertices x and y there is a directed walk from x to y of length k. Using the graph theory, we determine the lower bounds and upper bounds of r¡regular primitive matices in this paper. We also proved that the exponent of r¡regular primitive tournament T is just 3 and the exponent of r¡regular primitive symmetric matrix A satisfys exp(A) • 2(n ¡ r): 1. basic concepts
The affirmative feature which web based learning and problem based learning in the self directed ... more The affirmative feature which web based learning and problem based learning in the self directed learning having on-line and off-line study it amalgamates in connection Blended Learning method and specially in mathematics subject as a service course. We introduce a model of BL lecture of Mathematics and tried to find a new way for the quality improvement of university education.
We consider a 2-coloring of arcs on the primitive extremal tournament with the largest exponent o... more We consider a 2-coloring of arcs on the primitive extremal tournament with the largest exponent on n vertices. This 2-colored digraph is a 2-primitive tournament. Then, we consider the 2-exponent of a 2-primitive tournament. In this paper, we give an upper bound for the 2-exponent of the primitive extremal tournament.
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