In this paper we give an answer to an open problem posed by M.Z. Lee et al. (2012) 2. More precisely, we prove that the classical regular magic square of odd order produced by the centroskew ...S-circulant matrix with the assignment aj=j−1, j=1,2,⋯,(n+1)/2 is always nonsingular. Moreover an explicit formula for computing the eigenvalues of classical regular magic squares of odd order is given.
Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to ...moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the
k
-fold divisor function.
As matrix representations of magic labelings of related hypergraphs, magic squares and their various variants have been applied to many domains. Among various subclasses, trimagic squares have been ...investigated for over a hundred years. The existence problem of trimagic squares with singly even orders and orders 16n has been solved completely. However, very little is known about the existence of trimagic squares with other even orders, except for only three examples and three families. We constructed normal trimagic squares by using product constructions, row–square magic rectangles, and trimagic pairs of orthogonal diagonal Latin squares. We gave a new product construction: for positive integers p, q, and r having the same parity, other than 1, 2, 3, or 6, if normal p × q and r × q row–square magic rectangles exist, then a normal trimagic square with order pqr exists. As its application, we constructed normal trimagic squares of orders 8q3 and 8pqr for all odd integers q not less than 7 and p, r ∈ {7, 11, 13, 17, 19, 23, 29, 31, 37}. Our construction can easily be extended to construct multimagic squares.
Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been ...exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.
Plethora of image encryption schemes exist in literature based on the construct of magic square for realizing the purpose of image obfuscation. This magic square carries out the scrambling project of ...the encryption. In these schemes, normally single and static magic square is implied. To render greater scrambling effects, this study proposes a novel image encryption scheme using all order-4 magic squares whose frequency reaches to the tune of 880. These magic squares have been dynamically selected to carry out the scrambling project. As the color image is input, it is broken into its gray scale red, green and blue components. These components are joined together to make a big gray scale image. Intertwining logistic map (ILM) has been used for the generation of random data. Besides, one more stream has been created through the arithmetic manipulation of the generated three streams. Streams generated by ILM has been used to realize the effects of confusion and diffusion. First and second streams out of the four streams randomly select the address from the big gray scale image to apply the randomly selected magic square by the third stream, in order to create the scrambling effects. The fourth and last stream of random numbers is used to create the diffusion effects in the scrambled image. Plaintext senstivity has been introduced by tempering the one initial value of the chaotic system through the usage of a characteristic of the given input color image. The experimentation and security analyses sections vividly demonstrate the strength, immunity from the diverse attacks and prospects for the real world application of the proposed image cipher. In particular, we got very promising stats of information entropy (7.9974) and computational time (0.9865 seconds). No doubt, they suggest the potential application of the proposed image cipher in some real world setting.
Using centroskew matrices, we provide a necessary and sufficient condition for a regular magic square to be nonsingular. Using latin squares and circulant matrices we describe a method of ...construction of nonsingular regular magic squares of order n where n is an odd prime power.
This paper presents improved even order magic square construction algorithms, including both single even order magic square and double even order magic square construction algorithms. Further, in ...order to show how the algorithms work, two specific magic squares are constructed. Moreover, the correctness of the algorithms is proved, and the complexity analysis of the algorithms is given. Finally, the improved even order magic square construction algorithms are applied in secure communication and authentication areas for multi-user shared electronic account in detail.
Hou et al. 4 have studied various spaces of magic squares over a field F and determined their dimensions. However, they left one open question unsolved, namely, if the characteristic of F is 2 or 3, ...exactly which n and k make Mn,k(1) nonempty, where Mn,k(1) denotes the set of all n×n matrices over F whose row sums, column sums, k diagonal sums, and k antidiagonal sums are all 1. We solve this completely.
A new parameterization for order-4 regular (or associative) magic square matrices leads to general formulas for their eigenvalues, eigenvectors, and singular value decomposition. Known ...transformations extend these results to order-4 pandiagonal and bent-diagonal magic squares. The effect of various transformations on the eigenvalues and singular values of these special magic squares is considered. Numerical examples are presented and numerical values are obtained from simple formulas for the eigenvalues and singular values of each of the 48 natural pandiagonal, regular, and bent-diagonal magic squares of order 4 and their reflections.
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