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Invertibility
Published in Crista Arangala, Exploring Linear Algebra, 2019
A magic square is an arrangement of positive integers in a square grid, where the numbers in each row, in each column, and the numbers in the main diagonals, all add up to the same number. This sum is called the magic constant. An n × n magic square is called normal if it contains the number 1 through n2. n × n magic squares can be written as linear combinations of the permutation matrices of In.
Behavior of powers of odd ordered special circulant magic squares
Published in International Journal of Mathematical Education in Science and Technology, 2022
A magic square is an array of numbers, in which the sum of elements in each row, each column, and both the diagonals is the same number. If the condition for diagonals is ignored, then a magic square becomes a semi-magic square. The study of magic squares has a long history in India. Several magic squares exist in the work of Seer Garga. Nagarjuna, the Buddhist philosopher, has given a general form of a magic square of order four around 550 AD. Varahamihira in Brihatsamhita (6th century AD) described 4th order magic square. In the 12th century, the pan-diagonal magic square of order four was found at Jaina Temple at Khajuraho (Singh, 1986; Sridharan & Srinivas, 2012). The magic squares and cubes are thoroughly studied by Andrews (1960). Sreeranjini and Mallayya (2012) has given some special properties of magic squares. The algebraic study on magic squares is given in Teixeira (2018) and Ward (1980). The definitions and matrix properties of magic squares are quoted in Stephens and Nordgren (2012). Charles et al. (2010) has found the conditions under which properties of magic squares are not destroyed. Lucas (1891) has given a construction of third order magic square by using arithmetic progression. The magic squares are useful in graph theory, a system of linear equations, vibration analysis, music, game theory, cryptography, image processing, and many more. The nice thing about these magic squares is that their spectrum and eigenvalues can be computed explicitly.
A non-puzzle based interconnection scheme for energy savings and income generation from partially shaded photovoltaic modules
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2022
Namani Rakesh, Senthilkumar Subramaniam, Babu Natarajan, Malavya Udugula
Recently, various static reconfiguration methods have been developed to enhance power generation from partially shaded PV modules like SUDOKU method, which is implemented by distributing the PV modules column-wise to distribute the shade on PV modules using SUDOKU puzzle solving technique (Rani, Ilango, and Nagamani 2013). Furthermore, The Magic Square (MS) technique is implemented on magic square mathematical puzzle to get sum of numbers in each row, column, and diagonal equal to the magic constant. Hence, the shade on PV modules distributed over the entire array (Samikannu, Namani, and SenthilKumar 2016). But these methods involve more wiring line losses during the reconfiguration process. Hence, the researchers proposed Improvised Magic Technique and optimal SUDOKU method for reduction in mismatch losses and line losses in PV modules (Potnuru et al. 2015; Rakesh, Subramaniam, and Madhusudanan 2019). The switched PV modules technique is implemented on 2 kW solar rooftop power plant to improve the performance of the PV modules under unavoidable static shade conditions (Praveenkumar et al. 2017). Few researchers have proposed different techniques such as Latin square-TCT, Physical Relocation of the Modules with a Fixed Electrical Connection (PRM-FEC), etc., to reduce the mismatch losses and to find the best PV array configuration, which provides the highest performance (Rupendra et al. 2018; Sahu and Nayak 2016). The drawback of PRM-FEC array configuration is that the maximum power generation is not improved for all the types of shading patterns (Darmini and Sunitha 2017).
A Novel Magic Square Based Physical Reconfiguration for Power Enhancement in Larger Size Photovoltaic Array
Published in IETE Journal of Research, 2021
G. Harish Kumar Varma, Venugopal Reddy Barry, Rohit Kumar Jain
Very little work is done on the magic square puzzle. Generally, the magic square is a square grid filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal known as magic constant. The magic square is proposed for (4 × 4) PV array [30,32]. The proposed system does not use “n” distinct integers to obtain the magic square. The magic squares with repeated integers do not fall under this definition and are referred to as trivial. Further, the magic square unable to satisfy the property of the magic constant. The first column in the array remains constant. It means that if the shadow falls on the left side of the array, it will remain undistributed; this leads to a reduction in power output and causes multiple peaks in P-V characteristics [33]. Magic squares are implemented on (9 × 9) and (6 × 6) PV array and satisfy the property of the magic constant [34,35]. However, those methods analyze the PV characteristics only for long wide shading pattern and few other shading patterns, namely short wide, short-narrow and long- narrow shading pattern [34,35]. This paper proposed an oddly even magic square that can apply for any (4k + 2) size PV array and most of the shading conditions. The overall literature of various puzzle pattern schemes and optimization schemes for shade dispersion are presented in Table 1.