scholarly journals Some results on magic squares based on generating magic vectors and R-C similar transformations

2017 ◽  
Vol 5 (1) ◽  
pp. 82-96 ◽  
Author(s):  
Xiaoyang Ma ◽  
Kai-tai Fang ◽  
Yu hui Deng
Keyword(s):  
Transformation Method ◽  
New Method ◽  
Magic Squares ◽  
Magic Square ◽  
Similar Transformation

Abstract In this paper we propose a new method, based on R-C similar transformation method, to study classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging two rows and related two columns of a magic square. Many new results for classification of the magic squares of order 5 are obtained by the R-C similar transformation method. Relationships between basic forms and R-C similar magic squares are discussed. We also propose a so called GMV (generating magic vector) class set method for classification of magic squares of order 5, presenting 42 categories in total.

Magic squares of order four

1982 ◽  
Vol 306 (1495) ◽  
pp. 443-532 ◽  
Keyword(s):  
Analytical Methods ◽  
New Method ◽  
Magic Squares ◽  
Magic Square ◽  
All Solutions ◽  
History Of ◽  
Analytical Proof ◽  
Necessary And Sufficient ◽  
Order Four

A brief history of work on the 4 x 4 magic square is presented, with particular reference to Frenicle’s achievement over 300 years ago of establishing 880 as the number of essentially different squares by using the method of exhaustion (not convincingly repeated except by computer in 1976). He also established several central theorems. Our paper confirms the number 880 by a wholly new method of Frenicle quads and ‘part sums’, which leads to the classification of all solutions into, initially, six genera one of which has no members and thence to the enumeration of all possible solutions by analytical methods only. The working leads also to the first analytical proof independent of solutions that 12 and only 12 patterns formed by linking‘complementary’ numbers within a square are necessary and sufficient to describe all solutions - a fact which has been known since 1908, but not hitherto proved. A second method of construction and partial proof) greatly shortened by what has gone before, is also described. This yields a highly symmetrical list of the 880 magic squares. Together the two methods combine to explain many of the special characteristics and otherwise mysterious properties of these fascinating squares. The complete symmetrical list of squares ends the paper.

scholarly journals A new method for identifying the role of marital preferences at shaping marriage patterns

2021 ◽  
pp. 1-27
Author(s):  
Anna Naszodi ◽  
Francisco Mendonca
Keyword(s):  
Contingency Table ◽  
Transformation Method ◽  
New Method ◽  
Marriage Patterns ◽  
The Us ◽  
Analytical Properties ◽  
The Matrix ◽  
Transformation Methods

Abstract We develop a method which assumes that marital preferences are characterized either by the scalar-valued measure proposed by Liu and Lu, or by the matrix-valued generalized Liu–Lu measure. The new method transforms an observed contingency table into a counterfactual table while preserving its (generalized) Liu–Lu value. After exploring some analytical properties of the new method, we illustrate its application by decomposing changes in the prevalence of homogamy in the US between 1980 and 2010. We perform this decomposition with two alternative transformation methods as well where both methods capture preferences differently from Liu and Lu. Finally, we use survey evidence to support our claim that out of the three considered methods, the new transformation method is the most suitable for identifying the role of marital preferences at shaping marriage patterns. These data are also in favor of measuring assortativity in preferences à la Liu and Lu.

scholarly journals SEMANTIC NETWORK TRANSFORMATION METHOD FOR AUTOMATION OF PROGRAMMING PROBLEMS SOLUTIONS EVALUATION IN E-LEARNING

2020 ◽  
Vol 2020 (4) ◽  
pp. 7-17
Author(s):  
Alexander Fedorov ◽  
Alexey Nikolaevich Shikov
Keyword(s):  
Semantic Network ◽  
Single Layer ◽  
Transformation Method ◽  
Abstract Concepts ◽  
Program Code ◽  
Optimal Value ◽  
E Learning ◽  
Network Transformation ◽  
Cutoff Threshold

The article presents a semantic network transformation method for a programcode into an N-dimensional vector. The proposed method allows automating the quality assessment of solving programming problems in the process of e-learning. The method includes the authentic algorithms of building and converting the network. In order to determine the algorithm in the program code there is a template of this algorithm, presented in the form of a subgraph of abstract concepts of the language in the semantic network, built on the basis of this code. The search for the algorithm by comparing the subgraph of the network with the template network helped to identify the BFS algorithm with a given accuracy: the cutoff threshold for the perceptron outputs is 0.85, which is based on the calculation of accuracy of the single-layer perceptron in the classification of the MNIST base equal to 88%, which confirms the effectiveness of the developed method and requires further research using machine learning methods to find the optimal value of the coordinates of the nodes of the semantic network and templates of algorithms.

Wildcards – natural and artificial: the combination of a panel of experts and Fuzzy TOPSIS

foresight ◽  
2017 ◽  
Vol 19 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Mohsen Mohammadi ◽  
Mohammad Rahim Eivazi ◽  
Jafar Sajjadi
Keyword(s):  
Design Methodology ◽  
Fuzzy Topsis ◽  
New Method ◽  
Weak Signals ◽  
Content Type ◽  
Participatory Foresight ◽  
The Future ◽  
Similar Character ◽  
Topsis Model

Purpose The purpose of this paper is threefold: to classify wildcards into three particular types sharing similar characteristics; use the Fuzzy TOPSIS as a new method in foresight to turn qualitative ideas into quantitative ones; and apply a combination of Fuzzy TOPSIS and a panel of experts to prioritize weak signals. Design/methodology/approach In this paper, the authors classify wildcards into three particular types which share similar character: natural wildcards, artificial wildcards (Degree 1) and artificial wildcards (Degree 2). Wildcards point to unexpected and surprising events including important results that can form watershed in the development of a specific trend. In addition, the authors present a Fuzzy TOPSIS model which can be used in various cases to prioritize a number of weak signals and put them in order, so that the most important ones are likely to yield the wildcard in the future Findings The authors presented a classification of wildcards with the same characteristics being natural wildcards, artificial wildcards (Degree 1) and artificial wildcards (Degree 2). The authors also prioritized the weak signals to deal with the most important ones and take appropriate action in advance so as to minimize possible damages and maximize the benefits of potential wildcards in an uncertain environment. Originality/value In this paper, the authors report on the prioritizing of weak signals by applying Fuzzy TOPSIS and classify wildcards. This is significant because, by identifying the most important weak signals, appropriate actions can be taken in the future if necessary. The paper should be of interest to readers in the area of participatory foresight.

Magic squares of squares over a finite field

2021 ◽  
pp. 111-122
Author(s):  
Stewart Hengeveld ◽  
Giancarlo Labruna ◽  
Aihua Li
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Similar Question ◽  
Magic Squares ◽  
Content Type ◽  
Magic Square ◽  
Twin Primes ◽  
Quadratic Residues ◽  
Perfect Squares ◽  
Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

scholarly journals The Classification of 7- and 8-dimensional Naturally Reductive Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1246-1274 ◽  
Author(s):  
Reinier Storm
Keyword(s):  
New Method ◽  
Structure Theory ◽  
New Construction ◽  
Naturally Reductive

AbstractA new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.

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