Showing papers on "Magic square published in 2021"
TL;DR: It is shown that characteristic patterns emerge from plots of the ESPs of the matrices representing the studied squares, and these findings may help to open a new perspective regarding magic squares unsolved problems.
5 citations
14 Sep 2021
TL;DR: In this paper, the rows of a magic square of order 2k−3 were rearranged to avoid k-term monotone arithmetic progressions in [n] for k ≥ 3.
Abstract: In 1977, Davis et al. proposed a method to generate an arrangement of [n]={1,2,…,n} that avoids three-term monotone arithmetic progressions. Consequently, this arrangement avoids k-term monotone arithmetic progressions in [n] for k≥3. Hence, we are interested in finding an arrangement of [n] that avoids k-term monotone arithmetic progression, but allows k−1-term monotone arithmetic progression. In this paper, we propose a method to rearrange the rows of a magic square of order 2k−3 and show that this arrangement does not contain a k-term monotone arithmetic progression. Consequently, we show that there exists an arrangement of n consecutive integers such that it does not contain a k-term monotone arithmetic progression, but it contains a k−1-term monotone arithmetic progression.
4 citations
TL;DR: In this paper, the powers of odd ordered special circulant magic squares along with their magic constants are discussed and it is shown that we always obtain circulent semi-magic squares.
Abstract: This paper contains interesting facts regarding the powers of odd ordered special circulant magic squares along with their magic constants. It is shown that we always obtain circulant semi-magic sq...
2 citations
TL;DR: This paper is considered as a development of encryption algorithms based on Magic Square of Order Five, where both GF(P) and GF(28) are used to encode both images and text.
Abstract: This paper is considered as a development of encryption algorithms based on Magic Square of Order Five. Both GF(P) and GF(28) are used to encode both images and text. Where two different algorithms were used, the first using message length = 10 and the second message length = 14, and an unspecified number of rounds were added and a mask will be used in the even round will use the addition operation and in the odd round will used the multiplication operation so that the text resulting from the first round will be as input text for the next Round, and thus. The speed, complexity, NIST tests and histogram for the first ten rounds were calculated and compared with the results of the previous algorithm before the rounds were made, where the complexity in the first algorithm was = ((256)^ 15)^(r+1)× (256)^10 + or × (256)^25 and the complexity in the second algorithm = ((256)^11)^(r+1) ×(256)^14 + or × (256)^25 where r represents the number of round used.
2 citations
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TL;DR: In this paper, the Glued Magic Square game self-tests a convex combination of two or more inequivalent strategies, which is a generalisation of self-testing to multiple strategies.
Abstract: Self-testing results allow us to infer the underlying quantum mechanical description of states and measurements from classical outputs produced by non-communicating parties. The standard definition of self-testing does not apply in situations when there are two or more inequivalent optimal strategies. To address this, we introduce the notion of self-testing convex combinations of reference strategies, which is a generalisation of self-testing to multiple strategies. We show that the Glued Magic Square game [Quantum 4 (2020), p. 346] self-tests a convex combination of two inequivalent strategies. As a corollary, we obtain that the Glued Magic square game self-tests two EPR pairs thus answering an open question from [Quantum 4 (2020), p. 346]. Our self-test is robust and extends to natural generalisations of the Glued Magic Square game.
2 citations
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TL;DR: In this article, the authors illustrate how quantum information theory and free (i.e., noncommutative) semialgebraic geometry often study similar objects from different perspectives.
Abstract: We illustrate how quantum information theory and free (ie noncommutative) semialgebraic geometry often study similar objects from different perspectives We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field
1 citations
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TL;DR: In this paper, a general parameterization of an order-3 magic square derived by Lucas and a compound it to produce a parameterized order-9 magic square is presented. But the results are restricted to the Jordan canonical form and singular value decomposition of the compound Lucas magic square matrices.
Abstract: We review a general parameterization of an order-3 magic square derived by Lucas and we compound it to produce a parameterized order-9 magic square. Sequential compounding to higher order also is treated. Expressions are found for the matrices in the Jordan canonical form and the singular value decomposition of the compound Lucas magic square matrices. We develop a procedure for determining if an order-n magic square may be natural. This enables determination of numerical values for parameters in natural compound Lucas magic squares. Also, we find commuting pairs of compound Lucas matrices and formulas for matrix powers of order-3 and order-9 Lucas matrices. A parameterization due to Frierson is related to Lucas' parameterization and our results specialize to it, complementing previous results.
1 citations
14 May 2021
TL;DR: In this paper, the authors established infinite methods of building doubly even magic squares from n-order magic squares of n order (n > 20) which are formed by blocks of order four whose sums of elements of lines, columns and diagonals are all equal at 2n + 2.
Abstract: Here we have established infinite methods of building doubly even magic squares from doubly even magic squares of n order (n > 20) which are formed by blocks of order four whose sums of elements of lines, columns and diagonals are all equal at 2n + 2. Such a characteristic of these special magic squares causes a large production of other magic squares.
1 citations
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TL;DR: In this article, the authors used the characteristic map to give a new combinatorial proof of their result. And they also extended their computation to moments of traces of symmetric powers, where the same result holds but in a wider range.
Abstract: 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.
1 citations
TL;DR: A new method using magic squares has been proposed in this paper, where the magic square is constructed on constraints such as order of the matrix n and starting number a.
1 citations
14 Jul 2021
TL;DR: In this article, the eigenvalues of special matrices of orders: 3, 5, 7 and 9 were calculated using Maple simulation software, and the results were verified by using magic squares and logic operations.
Abstract: In this research, we mathematically calculated the eigenvalues of special matrices of orders: 3, 5, 7 and 9 and then verified the results using Maple simulation software. First, we found that for the matrix of order 3, the characteristic polynomial takes the form: (λ - 2s)(λ2 + 40) with the eigenvalues: 2s, and two conjugates. Second, for the magic matrix of order 5, we calculated the characteristic polynomial to be (λ - α) (λ4 + Lλ2 + K) with eigenvalues of the form $\alpha , \mp \sqrt {{\lambda _1}} , \mp \sqrt {{\lambda _2}} $, which are the roots of the quadratic equation: λ2 + Lλ + K = 0. Third, for the magic matrix of order 7, we computed the characteristic polynomial of the self- complementary magic. In this case, the eigenvalues have the form $\alpha , \mp \sqrt {{\lambda _1}} , \mp \sqrt {{\lambda _2}} ,, \mp \sqrt {{\lambda _3}} $, where λ 1 , λ 2 and λ 3 are the roots of a cubic equation. Finally, we utilized magic squares and logic operations to propose a robust reversible image encryption/decryption technique.
TL;DR: In this article, the super (a, d)-H-antimagic total labeling is closely related to graph labeling, graph decomposition, graph covering and magic square and it arouses more and more people's concerns.
Abstract: Super (a, d)-H-antimagic total labeling is closely related to graph labeling, graph decomposition, graph covering and magic square and it arouses more and more people’s concerns. In this paper we s...
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TL;DR: In this paper, the structure of the magic square C*-algebra $A(4)$ of size 4 is investigated, and it is shown that a certain twisted crossed product of the A(4)-algebra is isomorphic to the homogeneous C*algebra M_4(C(\mathbb{R} P^3)) by a certain action.
Abstract: In this paper, we investigate the structure of the magic square C*-algebra $A(4)$ of size 4. We show that a certain twisted crossed product of $A(4)$ is isomorphic to the homogeneous C*-algebra $M_4(C(\mathbb{R} P^3))$. Using this result, we show that $A(4)$ is isomorphic to the fixed point algebra of $M_4(C(\mathbb{R} P^3))$ by a certain action. From this concrete realization of $A(4)$, we compute the K-groups of $A(4)$ and their generators.
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TL;DR: In this article, the authors use the magic square game to obtain a self-test for Bell states where the one side needs only to measure single-qubit Pauli observables.
Abstract: Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. In this work we use the $3 \times n$ magic rectangle games (generalisations of the magic square game) to obtain a self-test for $n$ Bell states where the one side needs only to measure single-qubit Pauli observables. The protocol requires small input size (constant for Alice and $O(\log n)$ bits for Bob) and is robust with robustness $O(n^{5/2} \sqrt{\varepsilon})$, where $\varepsilon$ is the closeness of the observed correlations to the ideal. To achieve the desired self-test we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, generalise this strategy to the family of $3 \times n$ magic rectangle games, and supplement these nonlocal games with extra check rounds (of single and pairs of observables).
TL;DR: In this paper, the existence of normal bimagic squares of even order is investigated, the ideas of general row (column) Bimagic rectangles and magic pairs are introduced, which are applied to their construction, then they obtain the spectrum of the normal BIMG squares of order 2 u if and only if u ≥ 4.
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TL;DR: In this paper, the authors show how to construct Franklin squares of every order, as a superimposition of two squares, folded together like palms in the Indian greeting, Namaste.
Abstract: Narayana Pandita constructed magic squares as a superimposition of two squares, folded together like palms in the Indian greeting, Namaste. In this article, we show how to construct Franklin squares of every order, as a superimposition of two squares. We also explore the myriad of similarities in construction and properties of Franklin and Narayana squares.
TL;DR: In this article, the authors focus on the verses under the captions "gandhārṇava" and "kacchapuṭa" of the Bṛhatsaṁhitā and verify the claims made by Varāhamihira and the same amended by the commentator Bhaṭṭotpala for the number of perfumes.
Abstract: The sixth-century Indian scholar Varāhamihira elaborated on the processes for the preparation of perfumes in the ‘gandhayukti’ chapter of the Bṛhatsaṁhitā. He adapted the combinatorial analysis and magic square structure for arriving at the possible number of perfumes from the set of specified substances. This attempt is an excellent example of the multidisciplinary thought process of the Indian scholars in general and Varāhamihira in particular. This paper focuses on the verses under the captions ‘gandhārṇava’ and ‘kacchapuṭa’. In the first case, four substances are selected from sixteen, and the chosen four are permuted with the pre-defined proportions. In the second case, selected substances are placed in a 4 × 4 magic square. The claims by Varāhamihira and the same amended by the commentator Bhaṭṭotpala for the number of perfumes are verified here. The constraints imposed by Varāhamihira in the first case on the proportions of the two substances are considered. The correct number for the possible perfumes is determined mathematically in the first case and by the specially designed computer program in the second case. Valid lists of perfumes with ingredients obtained as the output are attached in the form of web links in the first case, and the same is placed in the Appendix for the second case.
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TL;DR: In this article, the authors give formulas for enumerating directed paths in the graded poset of semi-magic squares of size three, using Vandermonde convolution for finite graded posets and a direct method for deriving Regge symmetry formulas for un-normalized Clebsch-Gordan coefficients.
Abstract: We give formulas for enumerating directed paths in the graded poset of semi-magic squares of size three. We give two applications of these formulas: an advanced example of Vandermonde convolution for finite graded posets, and a direct method for deriving Regge symmetry formulas for un-normalized Clebsch-Gordan coefficients.
01 Sep 2021