Showing papers on "Magic square published in 2022"

OUP accepted manuscript

Abstract: In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic K\"ahler symmetric spaces, which are mostly associated with the Freudenthal-Tits magic square. Motivated by the intriguing singularity structure previously found in such F-theory models with a gauge group $SU(6)$,$SO(12)$ or $E_7$, we investigate, as the final magical example, an F-theory on an elliptic fibration over a Hirzebruch surface of the non-split $I_6$ type, in which the unbroken gauge symmetry is supposed to be $Sp(3)$. We find significant qualitative differences between the previous F-theory models associated with the magic square and the present case. We argue that the relevant half-hypermultiplets arise at the $E_6$ points, where half-hypermultiplets ${\bf 20}$ of $SU(6)$ would have appeared in the split model. We also consider the problem on the non-local matter generation near the $D_6$ point. After stating what the problem is, we explain why this is so by using the recent result that a split/non-split transition can be regarded as a conifold transition.

Magic squares: Latin, semiclassical, and quantum

Abstract: Quantum magic squares have recently been introduced as a “magical” combination of quantum measurements. In contrast to quantum measurements, they cannot be purified (i.e., dilated to a quantum permutation matrix)—only the so-called semiclassical ones can. Purifying establishes a relation to an ideal world of fundamental theoretical and practical importance; the opposite of purifying is described by the matrix convex hull. In this paper, we prove that semiclassical magic squares can be purified to quantum Latin squares, which are “magical” combinations of orthonormal bases. Conversely, we prove that the matrix convex hull of quantum Latin squares is larger compared to the semiclassical ones. This tension is resolved by our third result: we prove that the quantum Latin squares that are semiclassical are precisely those constructed from a classical Latin square. Our work sheds light on the internal structure of quantum magic squares, on how this is affected by the matrix convex hull, and, more generally, on the nature of the “magical” composition rule, both at the semiclassical and at the quantum level.

An Efficient Medical Image Encryption Using Magic Square and PSO

Abstract: Encryption is essential for protecting sensitive data, especially images, against unauthorized access and exploitation. The goal of this work is to develop a more secure image encryption technique for image-based communication. The approach uses particle swarm optimization, chaotic map and magic square to offer an ideal encryption effect. This work introduces a novel encryption algorithm based on magic square. The image is first broken down into single-byte blocks, which are then replaced with the value of the magic square. The encrypted images are then utilized as particles and a starting assembly for the PSO optimization process. The correlation coefficient applied to neighboring pixels is used to define the ideal encrypted image as a fitness function. The results of the experiments reveal that the proposed approach can effectively encrypt images with various secret keys and has a decent encryption effect. As a result of the proposed work improves the public key method's security while simultaneously increasing memory economy.

Magic square and three-zero textures for Dirac neutrinos

Abstract: We show a matrix decomposition of flavor mass matrix for Dirac neutrinos [Formula: see text] by sum as [Formula: see text] where [Formula: see text] obeys the feature of the magic square and [Formula: see text] is three-zero texture. The favorable three-zero textures in the context of magic square are explored. It turned out that the so-called normal ordering of neutrino masses is favored over the inverted ordering in the context of magic square.

Personal and historical magic squares

Abstract: Parametrizations of 4 × 4 squares which allow to generate individual examples, using birthdays or other personally preferred numbers are developed. This will be done for magic squares that are delightful, perfect, skew symmetric, most perfect and pandiagonal (also called diabolic). Furthermore, the parametrizations explain the construction of famous historical magic squares. Also an idea for an artwork containing mathematics is given, called MathArt. 1 Historical Introduction Magic squares are always of interest to people irrespective of age and their acquaintance with mathematics. Earlier, magic squares appeared often on temples, in paintings and on mythological objects. Magic squares first appeared in ancient China, before they became an active subject westwards. They played a remarkable role in India, later in the Arabic world, in medieval Islam and finally in Europe and America. Legend has it that the first magic square is over 4000 years old. It is said that 12 Albert Fässler and Alagu S. Somasundaram the mystical Emperor Yu in China discovered small black and white circles on the shell of a turtle that had emerged from the Lo river. The arrangement of the circles representing the numbers 1 to 9 were structured in a special 3 × 3 square (see [2]). Here is the modern design of the so-called Lo Shu magic square

Construction of multiple odd magic squares

Abstract: A technique of constructing multiple odd magic square matrices is studied in this work. A specific rule of establishing odd magic to magic squares derived from the odd algebraic Latin squares is proposed. Magic squares are practically important from the properties of their equality in the sum of rows, columns and diagonals. An n*n odd magic square is an array containing the positive integers from 1 to n2 , arranged so that each of the rows, columns, and the two principal diagonals have the same sum.

An estimator of moment of inertia of magic square array of masses

Abstract: The classical definition of magic square of order is a square array of consecutive natural numbers from 1 to such that the raw, column and diagonal sum add up to the same number. When the magic square entries are considered as an array of masses of a rigid body, it is established that its moment of inertia is a function of This work consider a more general magic array of masses of a rigid body to establish its inertial moment as a function of the magic sum and the central entry of the array. The paper further discusses the advantage of this development.

An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

Abstract: In this paper, we discuss and mathematically compute the eigenvalues and the characteristic polynomials of special even square matrices of orders 4x4 and 8x8. Also, we introduce two 8th order compound magic squares. The computed values are verified using Maple software. First, for the 4th order square matrix, the characteristic polynomial was derived to be: λ(λ-2s)( λ²+4Θ) with the eigenvalues: 0,2 s, and two other conjugates. In further analysis, we performed numerical classification of the squares for the matrices of order 4. Second, for the 8th order magic square, the characteristic polynomial was obtained in the form: λ3(λ-4s)(λ4+Ωλ2+θ) where Ω,Θ are constants; the eigenvalues are 0,4 s, ∓√λ1, ∓√λ2; where λ1, λ2 are the roots of the quadratic equation: λ2+Ωλ+Θ=0. Third, for the franklin square, we obtained the eigenvalues 0,4 s, and the roots of the equation: λ2+aλ+b. Finally, we suggested a hybrid image encryption technique based on Franklin magic square matrices and improved substitution technique. The proposed a grayscale image encryption/ decryption algorithm uses circular rotation of bits and Franklin magic squares’ properties in conjunction with substitution techniques to obtain a very secure algorithm against attacks.

Magic square and three-zero textures for Dirac neutrinos

Abstract: We show a matrix decomposition of flavor mass matrix for Dirac neutrinos $M$ by sum as $M=M'+M^0$ where $M'$ obeys the feature of the magic square and $M^0$ is three-zero texture. The favorable three-zero textures in the context of magic square are explored. It turned out that so-called the normal ordering of neutrino masses is favored over the inverted ordering in the context of magic square.

Lattice path enumeration for semi-magic squares by Latin rectangles

Abstract: Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this poset, and we completely describe the orbits, covering data, and maximal chains for squares of size 4, 5, and 6. The last item gives the number of Latin squares in these cases. To calculate efficiently for size 6, we in turn identify orbits with certain equivalence classes of hypergraphs.

Moir\'e Landau fans and magic zeros

Abstract: We study the energy spectrum of moir\'e systems under a uniform magnetic field. The superlattice potential generally broadens Landau levels into Chern bands with finite bandwidth. However, we find that these Chern bands become flat at a discrete set of magnetic fields which we dub "magic zeros". The flat band subspace is generally different from the Landau level subspace in the absence of the moir\'e superlattice. By developing a semiclassical quantization method and taking account of superlattice induced Bragg reflection, we prove that magic zeros arise from the simultaneous quantization of two distinct $k$-space orbits. The flat bands at magic zeros provide a new setting for exploring crystalline fractional quantum Hall physics.

An octonionic construction of $E_8$ and the Lie algebra magic square

Abstract: We give a new construction of the Lie algebra of type $E_8$, in terms of $3\times3$ matrices, such that the Lie bracket has a natural description as the matrix commutator. This leads to a new interpretation of the Freudenthal-Tits magic square of Lie algebras, acting on themselves by commutation.