A proper Euler’s magic matrix is an integer n×n matrix M∈Zn×n such that M⋅Mt=γ⋅I for some nonzero constant γ, the sum of the squares of the entries along each of the two main diagonals equals γ, and the squares of all entries in M are pairwise distinct. Euler constructed such matrices for n=4. In this work, we use multiplication matrices of the octonions to construct examples for n=8, and prove that no such matrix exists for n=3.

Complex and robust self-organization requires defined initial conditions and dynamic boundaries – neighboring tissues and extracellular matrix (ECM) that actively evolve to guide morphogenesis. A major challenge in tissue engineering is identifying material properties that mimic dynamic tissue boundaries but that are compatible with the engineering tools necessary for controlling the initial conditions of culture. Here we describe a highly tunable granular biomaterial, MAGIC matrix, that supports long-term bioprinting and gold-standard tissue self-organization. MAGIC matrix is designed for two temperature regimes: at 4 °C it exhibits reversible yield-stress behavior to support hours-long high-fidelity 3D printing without compromising cell viability; when transferred to cell culture at 37 °C, the material cross-links and exhibits viscoelasticity and stress relaxation that can be tuned to match numerous conditions, including that of reconstituted basement membrane matrices like Matrigel. We demonstrate that the timescale of stress relaxation and loss tangent are decoupled in MAGIC matrices, allowing us to test the role of stress relaxation rate and strain-dependence across formulations with identical storage and loss moduli. We find that fast absolute stress relaxation rates and large relative deformation magnitudes are required to optimize for morphogenesis. We apply optimized MAGIC matrices toward precise extrusion bioprinting of saturated cell suspensions directly into 3D culture. The ability to carefully control initial conditions for tissue growth yields dramatic increases in organoid reproducibility and complexity across multiple tissue types. We also fabricate perfusable 3D microphysiological systems that experience large strains in response to pressurization due to the compliant and dynamic tissue boundaries. Combined, our results both identify key parameters for optimal organoid morphogenesis in an engineered material and lay the foundation for fabricating more complex and reproducible tissue morphologies by canalizing their self-organization in both space and time.

A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.

At present, the Sudoku matrix, turtle shell matrix, and octagonal matrix have been put forward according to the magic matrix-based data hiding methods. Moreover, the magic matrices to be designed depend on the size of the embedding capacity. In addition, by determining the classification of points of pixel pairs after applying a magic matrix and by determining the traversal area, the average peak signal-to-noise ratio (PSNR) can be improved. Therefore, this topic intends to propose a data hiding method based on a 16 × 16 Sudoku matrix by using the 16 × 16 Sudoku matrix and extending it to a double-layer magic matrix. Low-cost data embedding methods are also studied, in order to improve the PSNR and maintain good image quality with the same embedding capacity.

Summary. It is known that if M is a 3 × 3 magic matrix then every positive, odd power of M is a magic matrix, while an even power need not be. Given any n × n magic matrix M, we find all polynomials f such that f(M) is a magic matrix.

Magic squares have long been popular in recreational mathematics. Their potential for introducing students to ideas in linear algebra was recognised over forty years ago in [1] and later in [2]. More recently they have proved to be a fascinating topic for undergraduate exploration, especially when students have access to a computer algebra package [3]. Some results on powers of magic square matrices can be found in [4], [5] and [6]. (Readers who google the title ‘Odd magic powers’ of Thompson’s paper [5] will be treated to a wide variety of non-mathematical exotica!)

Data hiding is a technology that generates meaningful stego media by embedding secret data in the cover media. In this paper, we propose a novel data hiding scheme using a mini-Sudoku matrix (MSM). A candidate block associated with the mapping value of a cover pixel pair on the MSM is used to indicate where the original pair would be modified to. According to the candidate block, each non-overlapping pixel pair can embed four bits of secret data. Among them, the first two bits are mapped from a specific element in the MSM while the other two are represented by the horizontal and vertical coordinates of this element with the modulus function. In addition, our MSM can be easily extended, and the proposed method can be used in other magic matrices to increase their embedding capacities. The experimental results indicate that our proposed scheme provides images with better visual quality than the previous methods. Furthermore, the security of the proposed scheme is verified by using pixel difference histogram (PDH) and RS steganalysis.

Data hiding is a technology that embeds data into a cover carrier in an imperceptible way while still allowing the hidden data to be extracted accurately from the stego-carrier, which is one important branch of computer science and has drawn attention of scholars in the last decade. Turtle shell-based (TSB) schemes have become popular in recent years due to their higher embedding capacity (EC) and better visual quality of the stego-image than most of the none magic matrices based (MMB) schemes. This paper proposes a two-layer turtle shell matrix-based (TTSMB) scheme for data hiding, in which an extra attribute presented by a 4-ary digit is assigned to each element of the turtle shell matrix with symmetrical distribution. Therefore, compared with the original TSB scheme, two more bits are embedded into each pixel pair to obtain a higher EC up to 2.5 bits per pixel (bpp). The experimental results reveal that under the condition of the same visual quality, the EC of the proposed scheme outperforms state-of-the-art data hiding schemes.

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We prove that a positive matrix with all permutation products equal is diagonally equivalent to J, the all-1s matrix. Then we give a simple proof of the rank inequality for diagonally magic matrices (J. Inequal. Appl. 2015:318, 2015).

We study a new class of matrices called diagonally magic matrices. We prove that such a matrix has rank at most 2 and that any square submatrix of a diagonally magic matrix is diagonally magic.