Showing papers on "Magic square published in 2001"
Book•
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01 Jan 2001
TL;DR: The second edition of as discussed by the authors contains a new chapter on magic labeling of directed graphs and interesting counting arguments new research problems and exercises covering a range of difficulties a fully updated bibliography and index This concise, selfcontained exposition is unique in its focus on the theory of magic graphs/labelings.
Abstract: Magic squares are among the more popular mathematical recreations. Over the last 50 years, many generalizations of magic ideas have been applied to graphs. Recently there has been a resurgence of interest in magic labelings due to a number of results that have applications to the problem of decomposing graphs into trees. Key features of this second edition include: a new chapter on magic labeling of directed graphs applications of theorems from graph theory and interesting counting arguments new research problems and exercises covering a range of difficulties a fully updated bibliography and index This concise, self-contained exposition is unique in its focus on the theory of magic graphs/labelings. It may serve as a graduate or advanced undergraduate text for courses in mathematics or computer science, and as reference for the researcher.
266 citations
TL;DR: In this article, the authors connect algebraic geometry and representation theory associated with Freudenthal's magic square, and give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras.
108 citations
01 Jan 2001
TL;DR: In this paper, the authors illustrate the use of Partition Analysis and of the corresponding package Omega by focusing on three different aspects of combinatorial work: the construction of bisections (for the refined lecture hall Partition Theorem), exploitation of recursive patterns (for Cayley compositions), and finding nonnegative integer solutions of linear systems of diophantine equations (for magic squares of size 3).
Abstract: A significant portion of MacMahon’s famous book “Combinatory Analysis” is devoted to the development of “Partition Analysis” as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMahon’s ideas have not received due attention with the exception of work by Richard Stanley. A long range object of a series of articles is to change this situation by demonstrating the power of MacMahon’s method in current combinatorial and partition-theoretic research. The renaissance of MacMahon’s technique partly is due to the fact that it is ideally suited for being supplemented by modern computer algebra methods. In this paper we illustrate the use of Partition Analysis and of the corresponding package Omega by focusing on three different aspects of combinatorial work: the construction of bisections (for the Refined Lecture Hall Partition Theorem), exploitation of recursive patterns (for Cayley compositions), and finding nonnegative integer solutions of linear systems of diophantine equations (for magic squares of size 3).
39 citations
TL;DR: A significant algorithmic improvement of the Omega package is presented, which overcomes a problem related to the computational treatment of roots of unity and turns out to be superior to "The Method of Elliott" which is described in MacMahon's book.
Abstract: In his famous book "Combinatory Analysis" MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. By developing the Omega package we have shown that Partition Analysis is ideally suited for being supplemented by computer algebra methods. The object of this paper is to present a significant algorithmic improvement of this package. It overcomes a problem related to the computational treatment of roots of unity. Moreover, this new reduction strategy turns out to be superior to "The Method of Elliott" which is described in MacMahon's book. In order to make this article as self-contained as possible we give a brief introduction to Partition Analysis together with a variety of illustrative examples. For instance, the generating function of magic pentagrams is computed.
38 citations
Book•
09 Nov 2001
TL;DR: The luoshu, or magic squre of order three, has been a symbol of the Divine, a good luck charm, a cosmogram of the world order, and a template for fengshui-through the ages as mentioned in this paper.
Abstract: A symbol of the Divine, a good luck charm, a cosmogram of the world order, a template for fengshui-through the ages, the luoshu, or magic squre of order three, has fascinated people of many different cultures. In this riveting account of cultural detective work, renowned mathematics educator, Frank J. Swetz relates how he uncovered the previously h
19 citations
Book•
18 Jul 2001
TL;DR: The Opaque Cube Flip, the Psychic Robot Mathematics and Word Play Tiling the Bent Tromino, and Magic Tricks on a Computer
Abstract: The Opaque Cube Flip, the Psychic Robot Mathematics and Word Play Tiling the Bent Tromino Covering a Cube with Congruent Polygons Magic Tricks on a Computer Variations on the 12345679 Trick Cornering the King toroidal Currency Lewis Carroll's Pillow Problems Lewis Carroll's Word Ladders The Ant on the 1 X 1 X 2 Three-point Tiling Lucky Numbers and 2187 3 X 3 Magic Squares Primes in Arithmetic Progression Prime Magic Squares The Dominono Game The Growth of Recreational Mathematics
9 citations
TL;DR: In this article, it was shown that the inverse of a nonsingular semi-magic square has the same property as the singular semiauthority of a quadratic matrix, where rows and columns of which add up to an identical constant.
Abstract: A semi-magic square is a quadratic matrix, the rows and columns of which add up to an identical constant. It is well known that the inverse of a nonsingular semi-magic square has the same property. It is shown that this result can be extended to the Moore?Penrose inverse of a singular semi-magic square and also to a more general class of matrices. Finally, the Drazin inverse of such matrices is also considered.
8 citations
TL;DR: Ezra (Bud) Brown (Brown@math.vt.edu) has degrees from Rice and Louisiana State, and has been at Virginia Tech since the first Nixon Administration.
Abstract: Ezra (Bud) Brown (brown@math.vt.edu) has degrees from Rice and Louisiana State, and has been at Virginia Tech since the first Nixon Administration. His research interests include graph theory, the combinatorics of finite sets, and number theory?especially elliptic curves. In 1999, he received the MAA MD-DC-VA Section Award for Outstanding Teaching, and he loves to talk about mathematics and its history to anyone, especially students.
7 citations
Journal Article•
4 citations
TL;DR: In this paper, the authors investigated some properties of Smarandache sequences of the 2nd kind and demonstrated that these numbers are near prime numbers, and established that prime numbers can be computed from the similar analytical expressions, and that they can be used for constructing Magic squares 3x3 or Magic squares 9x9.
Abstract: In this paper we investigate some properties of Smarandache sequences of the 2nd kind and demonstrate that these numbers are near prime numbers. In particular, we establish that prime numbers and Smarandache numbers of the 2nd kind (a) may be computed from the similar analytical expressions, (b) may be used for constructing Magic squares 3x3 or Magic squares 9x9, consisted of 9 Magic squares 3x3.
Patent•
27 Jul 2001
TL;DR: In this paper, the authors proposed a method to solve problems that the utility range of a magic square is narrowed and the utility value and utility effect of the magic square can not be improved since a method for preparing all possible squares is not conventionally provided for magic squares equal to or greater than sixth-order square.
Abstract: PROBLEM TO BE SOLVED: To solve problems that the utility range of a magic square is narrowed and the utility value and utility effect of the magic square can not be improved since a method for preparing all possible squares is not conventionally provided for magic squares equal to or greater than sixth-order square. SOLUTION: According to the order of location, symbols are located at the respective positions of a matrix and when a symbol different from the symbol up to the moment can not be located at that position, a symbol, which is not equal with the symbol located up to the moment, is located backward to the preceding position according to the order of selection from the symbol of the next selection order located at that position. After mutually different symbols are completely located at all the positions of the matrix concerning the symbol, which is not the same, the respective sums of respective rows, respective columns, main diagonal elements and sub-diagonal elements in the matrix are calculated and it is discriminated whether or not each of the calculated results is equal with each of sums calculated in a step S1 (S3-S9, S12, S13, S10 and S11). The matrix, with which the equal discriminated results are provided, is adopted as a magic square (S14).
Posted Content•
TL;DR: In this paper, a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g $ ranges over the exceptional complex simple Lie algebras was given.
Abstract: We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g$ ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic chart. Our proofs use the triality model of the magic square which we review and present a simplified proof of its validity. We conclude with some general remarks about obtaining "series" of Lie algebras in the spirit of Deligne and Vogel.
01 Jan 2001
TL;DR: In this paper, the authors presented an analysis in English of the impact of knowledge in English-to-Thai translation. But they did not mention any references to the authorship.
Abstract: IN THAI......................................................................iv ABSTRACT IN ENGLISH................................................................v ACKNOWLEDGEMENTS................................................................vi CONTENTS......................................................................................vii CHAPTER
01 Jan 2001
TL;DR: In this article, the study combines magic squares in the ancient east with that in ancient west and lists the thought and cases about permutation and combination, especially in ancient India, and some chess methods and games in ancient East.
Abstract: From ancient magic squares,old examples on permutation and combination,chess methods and games the study combines magic squares in the ancient east with that in ancient west and lists the thought and cases about permutation and combination, especially in ancient India, and some chess methods and games in the ancient East. Those methods and games contain some profound mathematical knowledge, pointing out that the combinatorial thought came from the ancient east. Luoshu , the oldest 3 magic square in the world, is the predecessor of modern configuration and design.