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In this paper, we introduce a frame diagonalization of matrices in $ M_n({\mathbb {C}}) $ Mn(C), called QR-frame diagonalization, by using QR-factorization. Furthermore, we show that every matrix A in $ M_n({\mathbb {C}}) $ Mn(C) is QR-frame diagonalizable. Also we introduce another frame diagonalization via Hadamard matrices in $ M_n({\mathbb {R}}) $ Mn(R), called Hadamard frame diagonalization, by using Hadamard matrices for n = 2, 4 or multiple of 4. We show that every matrix A in $ M_n({\mathbb {R}}) $ Mn(R) which n = 2, 4 or multiple of 4 is Hadamard frame diagonalizable. In this frame diagonalization, we can find entries on main diagonal Δ by using inner product. Moreover we introduce frame diagonalization of matrices on left quaternionic Hilbert spaces $ M_n({\mathbb {H}}) $ Mn(H). [ABSTRACT FROM AUTHOR]

Matrices are built and designed by applying procedures from lower order matrices. Matrix tensor products, direct sums or multiplication of matrices are such procedures and a matrix built from these is said to be a separable matrix. A non-separable matrix is a matrix which is not separable and is often referred to as an entangled 'matrix. The matrices built may retain properties of the lower order matrices or may also acquire new desired properties not inherent in the constituents. Here design methods for non-separable matrices of required types are derived. These can retain properties of lower order matrices or have new desirable properties. Infinite series of required type non-separable matrices are constructible by the general methods. Non-separable matrices of required types are required for applications and other uses; they can capture the structure in a unique way and thus perform much better than separable matrices. General new methods are developed with which to construct multidimensional entangled paraunitary matrices; these have applications for wavelet and filter bank design. The constructions are used to design new systems of non-separable unitary matrices; these have applications in quantum information theory. Some consequences include the design of full diversity constellations of unitary matrices, which are used in MIMO systems, and methods to design infinite series of special types of Hadamard matrices. [ABSTRACT FROM AUTHOR]

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size N\ge 4 has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras. [ABSTRACT FROM AUTHOR]

Unsupervised hashing for cross-modal retrieval has received much attention in the data mining area. Recent methods rely on image-text paired data to conduct unsupervised cross-modal hashing in batch samples. There are two main limitations for existing models: (1) learning of cross-modal representations is restricted to batches; (2) semantically similar samples may be wrongly treated as negative. In this paper, we propose a novel category-level contrastive learning for unsupervised cross-modal hashing, which alleviates the above problems and improves cross-modal query accuracy. To break the limitation of learning in small batches, a selected memory module is first proposed to take global relations into account. Then, we obtain pseudo labels through clustering and combine the labels with the Hadamard Matrix for category-centered learning. To reduce wrong negatives, we further propose a memory bank to store clusters of samples and construct negatives by selecting samples from different categories for contrastive learning. Extensive experiments show the significant superiority of our approach over the state-of-the-art models on MIRFLICKR-25K and NUS-WIDE datasets. [ABSTRACT FROM AUTHOR]

For secure image communications, the Internet must be shielded against unauthorized acquisition or malevolent access. The need for image encryption methods with enough capacity and good efficiency is growing due to the current situation. The purpose of this research work is to solve the problems in today's communication security using a proposed image encryption algorithm. The proposed image encryption system includes the combination of a chaotic system, compressive sensing and Hyper optimization algorithm. The initial values of chaos are extracted using the SHA512. Discrete Wavelet Transform (DWT) is applied to sparse image pixels. The image is shuffled using a FAN transform with a circular map. Next, the image is divided into blocks to facilitate the application of block compressive sensing that utilized the Hadamard measurement matrix. These blocks are masked as one part to quantify the pixels. The logistic map is improved by a hybrid transform which is the combination of Discrete Cosine Transform, Arnold Transform and Discrete Wavelet Transform (CAW). The image is finally scrambled using a Circular Map with a Hyper Optimization that combines two meta-heuristics algorithms namely GWO and COOT. According to simulation experiment findings and security assessments, the algorithm was extremely resilient to differential, statistical, and interference assaults. Based on the experimental results, it was found that the average rate of PSNR was 33.8189, the rate of entropy was 7.99454, the average rate of SSIM was 0.96144, the rate of UACI was 99.62892 and the average rate of NPCI was 33.50304. The results showed that it is an effective method of encryption and strong enough against various types of attacks. [ABSTRACT FROM AUTHOR]

A quaternionic Hadamard matrix (QHM) of order n is an n × n matrix H with non-zero entries in the quaternions such that H H ∗ = n I n , where I n and H ∗ denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements. [ABSTRACT FROM AUTHOR]

Let R = ϱ I n and D = 0 , e 1 , ... , e n , where ϱ > 1 and e i is the i-th coordinate vector in R n . The spectral properties of the n - dimensional Sierpinski measure μ R , D has been studied over two decades. In this paper, a special type of spectrum called a typical spectrum for μ R , D is considered. We show that ϱ ∈ (n + 1) N is necessary and sufficient for μ R , D to admit a typical spectrum if n + 1 is prime. And some necessary conditions for μ R , D to admit a typical spectrum are provided when n + 1 is not a prime number. Furthermore, under the condition on real Hadamard matrix, we prove that μ R , D admits a quasi-typical spectrum if and only if ϱ ∈ 2 N . These results show that the spectral properties of the Sierpinski measure μ R , D are really different between n + 1 is prime and non-prime. As a corollary, we prove that ϱ ∈ 2 N are the only integers such that μ R , D becomes a spectral measure when n = 3 . [ABSTRACT FROM AUTHOR]

We present a preliminary study of Schur norms $ \|M\|_{\textrm{S}}=\max \{ \|M\circ C\|: \|C\|=1\} $ ‖ M ‖ S = max { ‖ M ∘ C ‖ : ‖ C ‖ = 1 } , where M is a matrix whose entries are $ \pm 1 $ ± 1 , and $ \circ $ ∘ denotes the entrywise (i.e. Schur or Hadamard) product of the matrices. We recover a result of Johnsen that says that, if such a matrix M is $ n\times n $ n × n , then its Schur norm is bounded by $ \sqrt {n} $ n , and equality holds if and only if it is a Hadamard matrix. We develop a numerically efficient method of computing Schur norms, and as an application of our results we present several almost Hadamard matrices that are better than were previously known. [ABSTRACT FROM AUTHOR]

In this paper, we classified Hadamard 2-(51, 25, 12) designs having a non-abelian automorphism group of order 10, i.e., Frob10 ≌ Z5: Z2, where a subgroup isomorphic to Z2 fixed setwise all the Z5-orbits of the sets of points and blocks. Furthermore, we classified 2-(59, 29, 14) designs having a non-abelian automorphism group of order 14, i.e., Frob14 ≌ Z7: Z2, where a subgroup isomorphic to Z2 fixed setwise all the Z7-orbits of the sets of points and blocks. We also showed that there was no Hadamard 2-(59, 29, 14) design with automorphism group Frob21 ≌ Z7: Z3, where a subgroup of order 3 fixed setwise all the Z7-orbits of points and blocks. Additionally, we used Hadamard 2-(59, 29, 14) designs obtained to construct ternary linear codes and linear codes over the finite field of order 5. We constructed ternary self-dual codes and self-dual codes over the finite field of order 5 from the corresponding Hadamard matrices of order 60. [ABSTRACT FROM AUTHOR]

Single-pixel imaging (SPI) is an alternative method for obtaining images using a single photodetector, which has numerous advantages over the traditional matrix-based approach. However, most experimental SPI realizations provide relatively low resolution compared to matrix-based imaging systems. Here, we show a simple yet effective experimental method to scale up the resolution of SPI. Our imaging system utilizes patterns based on Hadamard matrices, which, when reshaped to a variable aspect ratio, allow us to improve resolution along one of the axes, while sweeping of patterns improves resolution along the second axis. This work paves the way towards novel imaging systems that retain the advantages of SPI and obtain resolution comparable to matrix-based systems. [ABSTRACT FROM AUTHOR]

In this work we define and study binary codes C q , k and C q , k ¯ obtained from neighborhood designs of Paley-type bipartite graphs P(q, k) and their complements, respectively for q an odd prime. We prove that for some values of q and k the codes C q , k are self-dual and the codes C q , k ¯ are self-orthogonal. Most of these codes tend to be with optimal or near optimal parameters. Next, we extend the codes C q , k to get doubly even self dual codes and find that most of these codes are extremal. [ABSTRACT FROM AUTHOR]

Characterizing the 6 × 6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the 6 × 6 CHMs except the H 2 -reducible matrices and the Tao matrix as the non- H 2 -reducible matrices. As far as we know, no non- H 2 -reducible matrices with analytic form have been found. In this paper, we find some non- H 2 -reducible matrices with analytic form. We also characterize some special 6 × 6 CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in C 6 consists of an MUB trio and the identity). Our results may lead to the complete classification of 6 × 6 CHMs and the solution of MUB problem in C 6 . [ABSTRACT FROM AUTHOR]

Maximin distance designs and orthogonal designs are two attractive classes of space-filling designs for computer experiments, but their theoretical constructions are challenging, especially the construction of optimal designs in terms of both the maximin distance and orthogonality criteria. This paper presents a systematic method for constructing orthogonal maximin distance designs with flexible numbers of runs and factors. The method is carried out by rotating the subarrays of a saturated two-level regular design in Yates order or its circular shifting version. The principal objective is to construct high-level designs from two-level designs, and the method is effective because the performance of high-level designs is determined by that of two-level designs under both the maximin distance and orthogonality criteria. The proposed method is also generalized by rotating the subarrays of a saturated two-level nonregular design such that the resulting designs have flexible run sizes. Comparison results reveal that the resulting orthogonal designs are well worthy of recommendation under the maximin distance criterion. An illustrative example is provided to show that the proposed designs have a good two-dimensional stratification property. An application is given to present the effectiveness of the proposed designs in building statistical surrogate models. [ABSTRACT FROM AUTHOR]

We introduce the Hadamard decomposition problem in the context of data analysis. The problem is to represent exactly or approximately a given matrix as the Hadamard (or element-wise) product of two or more low-rank matrices. The motivation for this problem comes from situations where the input matrix has a multiplicative structure. The Hadamard decomposition has potential for giving more succint but equally accurate representations of matrices when compared with the gold-standard of singular value decomposition (svd). Namely, the Hadamard product of two rank- h matrices can have rank as high as h 2 . We study the computational properties of the Hadamard decomposition problem and give gradient-based algorithms for solving it approximately. We also introduce a mixed model that combines svd and Hadamard decomposition. We present extensive empirical results comparing the approximation accuracy of the Hadamard decomposition with that of the svd using the same number of basis vectors. The results demonstrate that the Hadamard decomposition is competitive with the svd and, for some datasets, it yields a clearly higher approximation accuracy, indicating the presence of multiplicative structure in the data. [ABSTRACT FROM AUTHOR]

In a [Formula: see text]-factorial experiment with limited resources, when practitioners can identify the non-negligible effects and interactions beforehand, it is common to run an experiment with a saturated design that ensures the unbiased estimation of the non-negligible parameters of interest. We propose a method for the construction of D-optimal saturated designs for the mean, the main effects, and the second-order interactions of one factor with the remaining factors. In the process, we show the problem is just as hard as the Hadamard determinant problem.

The determinants of { ± 1 } -matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family. [ABSTRACT FROM AUTHOR]

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labellings of graphs, Hadamard matrices, and equiangular lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labellings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labellings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are 2 × 2 matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix. [ABSTRACT FROM AUTHOR]

Block cipher is a cryptographic field that is now widely applied in various domains. Besides its security, deployment issues, implementation costs, and flexibility across different platforms are also crucial in practice. From an efficiency perspective, the linear layer is often the slowest transformation and requires significant implementation costs in block ciphers. Many current works employ lookup table techniques for linear layers, but they are quite costly and do not save memory storage space for the lookup tables. In this paper, we propose a novel lookup table technique to reduce memory storage when executing software. This technique is applied to the linear layer of block ciphers with recursive Maximum Distance Separable (MDS) matrices, Hadamard MDS matrices, and circulant MDS matrices of considerable sizes (e.g. sizes of 16, 32, 64, and so on). The proposed lookup table technique leverages the recursive property of linear matrices and the similarity in elements of Hadamard or circulant MDS matrices, allowing the construction of a lookup table for a submatrix instead of the entire linear matrix. The proposed lookup table technique enables the execution of the diffusion layer with unchanged computational complexity (number of XOR operations and memory accesses) compared to conventional lookup table implementations but allows a substantial reduction in memory storage for the pre-computed tables, potentially reducing the storage needed by 4 or 8 times or more. The memory storage will be reduced even more as the size of the MDS matrix increases. For instance, analysis shows that when the matrix size is 64, the memory storage ratio with the proposed lookup table technique decreases by 87.5% compared to the conventional lookup table technique. This method also allows for more flexible software implementations of large-sized linear layers across different environments. [ABSTRACT FROM AUTHOR]

We prove that the stack-number of the strong product of three n-vertex paths is Θ (n 1 / 3) . The best previously known upper bound was O(n). No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest Δ 0 such that there exists a graph family with unbounded stack-number, bounded queue-number and maximum degree Δ 0 . We show that Δ 0 ∈ { 6 , 7 } . [ABSTRACT FROM AUTHOR]

The purpose of this study is to establish recent inequalities based on double integrals of mappings whose higher-order partial derivatives in absolute value are convex on the co-ordinates on rectangle from the plane. Also, some special cases of results improved in this study are examined. [ABSTRACT FROM AUTHOR]

This study investigates the existence of solutions to integral equations in the form of quadratic Urysohn type with Hadamard fractional variable order integral operator. Due to the lack of semigroup properties in variable-order fractional integrals, it becomes challenging to get the existence and uniqueness of corresponding integral equations, hence the problem is examined by employing the concepts of piecewise constant functions and generalized intervals to address this issue. In this context, the problem is reformulated as integral equations with constant orders to obtain the main results. Both the Schauder and Banach fixed point theorems are employed to prove the uniqueness results. In addition, an illustration is included in order to verify those results in the final step. [ABSTRACT FROM AUTHOR]

Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted. Finally, a multicomplex polar representation is provided. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of 2 n Radio Frequency (RF) signals can be represented as the real part of a single multicomplex signal modulated by several frequencies. The signal RFs are related through a Hadamard matrix to the modulating frequencies adopted in the multicomplex baseband representation. Moreover, an orthogonal decomposition is provided for the obtained multicomplex baseband signal as a function of the complex baseband representations of the input RF signals. [ABSTRACT FROM AUTHOR]

Nowadays, many image watermarking algorithms based on matrix decomposition have been proposed. In this paper, the properties of polar decomposition in the field of image watermarking is found, that is, embedding the watermark information into diagonal elements of the semi-positive definite matrix can improve the invisibility of the watermark. However, the diagonal elements have poor robustness. Based on this, lifting wavelet transform, which is the second generation wavelet transform, and Hadamard transform are used to improve the robustness of watermarking. In addition, image geometric correction algorithms are also combined to resist geometric attacks. The robustness and comparison experiments show that the proposed algorithm is superior to some state-of-the-art algorithms in some attacks. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of binary sequences with two-valued non-zero periodic autocorrelation sum and good periodic crosscorrelation sum as well as balanced properties. We make use of the sequences obtained in (No, J. et al., IEEE Trans. Inform. Theory 44(3), 1278-1282 2001) and adopt the extraction method similar to (Lüke, H. IEEE Trans. Inform. Theory 43(1) 1997). The new sequences are proven to be balanced or almost balanced. Based on these correlation and balanced properties, an important application is to construct Hadamard matrices of order p + 1 for p ≡ 3 ( mod 4) and 2 p + 2 for p ≡ 1 ( mod 4). Some examples are shown to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate the viability of Multi-Carrier Modulation (MCM) systems based on the Fast Walsh Hadamard Transform (FWHT). In addition, a nonlinear Joint Low-Complexity Optimized Zero Forcing Successive Interference Cancellation (JLCOZF-SIC) equalizer is proposed. To that end, general equations for the number of flops of the proposed equalizer and various other equalizers are given. This article discusses the use of Banded Matrix Approximation (BMA) as a technique for reducing complexity. The proposed equalizer uses BMA to accomplish both equalization and co-Carrier Frequency Offset (co-CFO) corrections. In addition, three cases involving the proposed equalizer were investigated. In the first case, diagonal compensation is used. In the second case, BMA compensation is used. In the third case, complete matrix compensation is used. In the presence of frequency offset, noise, and frequency-selective Rayleigh fading environments, analysis and simulation results show that the OFDM-FWHT system with the proposed equalizer outperforms the conventional OFDM system with various linear and nonlinear equalizers. [ABSTRACT FROM AUTHOR]

This article presents a technique to carry out fault classification using an equal-angle integer-period array convolutional neural network (EAIP-CNN) to process the electrostatic signal of working roller bearings. Firstly, electrostatic signals were collected using uniform angle sampling to ensure the angle intervals between two adjacent data points stayed the same and the signal length was fixed to a pre-determined number of rotation cycles. Then, this one-dimensional signal was transformed into a two-dimensional matrix, where the component of each row was the signal in one period, and the ordinate value of each row represented the corresponding rotation period. Therefore, the row and column indexes of the matrix had a specific meaning instead of simply splitting and stacking the data. Finally, the matrixes were utilized to train the CNN network and test the classification performance. The results show that the classification rate using this technique reaches 95.6%, which is higher than that of 2D CNNs without equal-angle integer-period arrays. [ABSTRACT FROM AUTHOR]

The XOR operator is a simple yet crucial computation in computer science, especially in cryptography. In symmetric cryptographic schemes, particularly in block ciphers, the AddRoundKey transformation is commonly used to XOR an internal state with a round key. One method to enhance the security of block ciphers is to diversify this transformation. In this paper, we propose some straightforward yet highly effective techniques for generating t-bit random XOR tables. One approach is based on the Hadamard matrix, while another draws inspiration from the popular intellectual game Sudoku. Additionally, we introduce algorithms to animate the XOR transformation for generalized block ciphers. Specifically, we apply our findings to the AES encryption standard to present the key-dependent AES algorithm. Furthermore, we conduct a security analysis and assess the randomness of the proposed key-dependent AES algorithm using NIST SP 800-22, Shannon entropy based on the ENT tool, and min-entropy based on NIST SP 800-90B. Thanks to the key-dependent random XOR tables, the key-dependent AES algorithm have become much more secure than AES, and they also achieve better results in some statistical standards than AES. [ABSTRACT FROM AUTHOR]

In this paper, we give a method for the constructions of Hadamard matrices of composite orders by using suitable T-matrices and known Hadamard matrices. We establish a formula for Tmatrices and Hadamard matrices and discuss under what condition we can get T-matrices from the known Hadamard matrices. [ABSTRACT FROM AUTHOR]

Many genetic models (including models for epistatic effects as well as genetic-by-environment) involve covariance structures that are Hadamard products of lower rank matrices. Implementing these models requires factorizing large Hadamard product matrices. The available algorithms for factorization do not scale well for big data, making the use of some of these models not feasible with large sample sizes. Here, based on properties of Hadamard products and (related) Kronecker products, we propose an algorithm that produces an approximate decomposition that is orders of magnitude faster than the standard eigenvalue decomposition. In this article, we describe the algorithm, show how it can be used to factorize large Hadamard product matrices, present benchmarks, and illustrate the use of the method by presenting an analysis of data from the northern testing locations of the G × E project from the Genomes to Fields Initiative (n ∼ 60,000). We implemented the proposed algorithm in the open-source "tensorEVD" R package. [ABSTRACT FROM AUTHOR]

The rising use of digital imaging applications has recently boosted the demand for different image compressing algorithms. Picture compression is used to reduce unnecessary information from an image. We can store the vital information of a picture via image compression, reducing storage space, time, and transmission bandwidth. Although the results of lossy compression reconstruction are not similar to the original image the compression technique is essential to save a large amount of memory and increase the speed of transmission, especially when dealing with images. In this research a database of different five images were considered namely; woman, car, Lenna, peppers, and house with sizes of 33, 47, 40, 44, and 51 Kb respectively. The compression was fulfilled by Walsh transform with four compression ratios 5%, 10%, 15%, and 20%. The Walsh transform performed well and gave the highest average PSNR of 29.1904 during a 10% of compression ratio. [ABSTRACT FROM AUTHOR]

The so-called matrix properties of finitely complete categories are a special type of exactness properties that can be encoded as extended matrices of integers. The relation of entailment of matrix properties gives an interesting preorder on the set of such matrices, which can be investigated independently of the category-theoretic considerations. In this paper, we conduct such investigation and obtain several results dealing with binary matrices, i.e., when the only integer entries in the matrix are 0 or 1. Central to these results is a complete description of the preorder in the case of a special type of binary matrices, which we call diagonal matrices, and which include matrices that define Mal'tsev, majority and arithmetical categories. [ABSTRACT FROM AUTHOR]

A novel hybrid image-adaptive watermarking technique utilizing Discrete Wavelet Transform (DWT) and Fast Walsh Hadamard Transform (FWHT) is presented in this paper for the safety of proprietary rights. The host image is partitioned into blocks of dimension 8 × 8 for entropy calculation. DWT is used to determine the low frequency coefficients from the high entropy blocks, which are further transformed by FWHT. These transformed frequency coefficients are modeled by Gaussian distribution. The watermark is added into the standard image using a modifiable Hybrid Strength Factor (HSF), which is computed using mean, standard deviation, entropy and kurtosis of the transformed coefficients of high entropy blocks. Four HSF's are used for embedding the watermarks with the help of limited side information consisting of locations of high entropy blocks and parameters of Gaussian distribution. Two novel contributions, firstly the amalgamation of DWT and FWHT to intensify the imperceptibility and robustness, and secondly the use of a statistical approach for watermark extraction have been presented in the paper. The efficacy of the technique is proved by better results using the performance metrics of PSNR (Peak Signal to Noise Ratio), SSIM (Structural Similarity Index Measure), NCC (Normalized Cross Correlation), GMSD (Gradient Mean Structural Deviation) and UIQI (Universal Image Quality Index) There was no compromise in the security and imperceptibility of the watermark when the scheme is put through various attacks. Also, a comparison with latest approaches too proves high invisibility and robustness of the suggested technique. [ABSTRACT FROM AUTHOR]

In this paper we show that inequalities for functions convex on the coordinates can be derived from inequalities for convex functions defined on real intervals, and essentially only this method works. As applications, we show how our result works for the Jensen's and Hermite- Hadamard inequalities for functions convex on the coordinates. Finally, we extend the classical notion of f -divergence functional to functions convex on the coordinates, and as a further application of our main result, we study the refinement of a basic inequality corresponding to the new divergence. [ABSTRACT FROM AUTHOR]

An |$n \times n$| matrix with |$\pm 1$| entries that acts on |${\mathbb {R}}^{n}$| as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools, we construct matrices with |$\pm 1$| entries that act as approximate scaled isometries in |${\mathbb {R}}^{n}$| for all |$n \in {\mathbb {N}}$|⁠. More precisely, the matrices we construct have condition numbers bounded by a constant independent of |$n$|⁠. Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in |${\mathbb {R}}^{n}$| formed by |$N$| vectors with independent identically distributed coordinate having a nondegenerate symmetric distribution contains many Riesz bases with high probability provided that |$N \ge \exp (Cn)$|⁠. On the other hand, we prove that if the entries are sub-Gaussian, then a random frame fails to contain a Riesz basis with probability close to |$1$| whenever |$N \le \exp (cn)$|⁠ , where |$c

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group's structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields. [ABSTRACT FROM AUTHOR]

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given. [ABSTRACT FROM AUTHOR]

The Orthogonal Frequency Division Multiplexing (OFDM) method provides effective bandwidth utilization and straightforward Frequency Domain Equalization (FDE). The OFDM system can be applied without a group of modulators and demodulators by using the Inverse Fast Fourier Transform (IFFT) and Fast Fourier Transform (FFT). The Fast Walsh Hadamard Transform (FWHT) and Inverse Fast Walsh Hadamard Transform (IFWHT) can also be used to build the OFDM system. In this work, we employ the IFWHT and FWHT to build the OFDM. This is followed by a proposal for a Joint Low-Complexity Regularized Zero Forcing-Successive Interference Cancellation (JLCRZF-SIC) equalizer with respect to various linear and nonlinear equalizers at various antenna configurations. A closed form for arithmetic operation count for the proposed equalizer is proposed. The proposed equalizer executes the equalization process more accurately than other schemes with low-complexity deployments. The analysis results show that the proposed equalizer outperforms the conventional one. [ABSTRACT FROM AUTHOR]

This article proposes a video watermarking technique by concealing the watermark in the low frequency bands of phase using Conjugate Symmetric Sequency Complex Hadamard Transform (CS-SCHT) and Speeded Up Robust Feature (SURF). The CS-SCHT is effective in dealing with attacks such as H.264 compression, Gaussian noise, Salt and pepper noise and speckle noise. However, the watermark in CS-SCHT based video watermarking scheme is erased when rotation, scaling and translation attack is applied. So in this work CS-SCHT is combined with a Speeded Up Robust Feature (SURF) algorithm. This watermarking scheme allows correct watermarked detection even if it is distorted by scaling, rotation, shearing, illumination change and translation. Image registration has been performed on watermarked image and SURF extracted affine transform for these attacks. The affine transform is inverse transformed to reposition the watermarked video to original, before the watermark extraction. The quality of signed video is analyzed by Peak signal-to-noise ratio (PSNR) and Structural Similarity index Measure (SSIM). Simulation results prove the proposed video watermarking scheme shows superiority in the results of imperceptibility and robustness. [ABSTRACT FROM AUTHOR]

For h > 0, α ∈ [0, h) and μ ∈ R denote by SDh(μ, α) a class of absolutely convergent in the half-plane Π0 = {s: Re s < 0} Dirichlet series F(s) = esh + P∞ k=1 fk exp{sλk} such that Re n (μ-1)F′(s)-μF′′(s)/h (μ-1)F(s)-μF′(s)/h o > α for all s ∈ Π0, and let ΣDh(μ, α) be a class of absolutely convergent in half-plane Π0 Dirichlet series F(s) = e-sh + P∞ k=1 fk exp{sλk} such that Re n (μ-1)F′(s)+μF′′(s)/h (μ-1)F(s)+μF′(s)/h o < -α for all s ∈ Π0. Then SDh(0, α) consists of pseudostarlike functions of order α and SDh(1, α) consists of pseudoconvex functions of order α. For functions from the classes SDh(μ, α) and ΣDh(μ, α), estimates for the coefficients and growth estimates are obtained. In particular, it is proved the following statements: 1) In order that function F(s) = esh + P∞ k=1 fk exp{sλk} belongs to SDh(μ, α), it is sufficient, and in the case when fk(μλk/h - μ + 1) ≤ 0 for all k ≥ 1, it is necessary that ∞P k=1 fk 􀀀 μλk h - μ + 1 (λk - α) ≤ h - α, where h > 0, α ∈ [0, h) (Theorem 1). 2) In order that function F(s) = e-sh+ P∞ k=1 fk exp{sλk} belongs to ΣDh(μ, α), it is sufficient, and in the case when fk(μλk/h + μ - 1) ≤ 0 for all k ≥ 1, it is necessary that ∞P k=1 fk 􀀀 μλk h + μ - 1 (λk + α) ≤ h - α, where h > 0, α ∈ [0, h) (Theorem 2). Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus m were also studied. [ABSTRACT FROM AUTHOR]

Let I = {1;: :: ;N} be a finite set of indices and K = IN the set of all sequences of indices equipped with the product measure and the product topology. Melo, da Cruz Neto, and de Brito [Strong convergence of alternating projections, J. Optim. Theory Appl. 194 (2022), 306-324] defined a family of sequences N0 ⊆ K so that whenever one iterates distance minimizing projections on N closed and convex subsets of an Hadamard space, the sequence of projections converges, provided it has at least one accumulation point. They proved that N0 has full measure, and in the sense of measure almost all iterates of projections converge. We observe that N0 is meager. The question, which almost all iterates converge in the topological sense, remains open. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section. [ABSTRACT FROM AUTHOR]

A Butson Hadamard matrix of order n over the kth root of unity is a square matrix H which entries are some complex kth root of unity such that HH* = nIn, where H* is the complex conjugate of H. A set of Butson Hadamard matrices of order n over the kth root of unity is denoted by BH(n,k). It is well-known that a Butson Hadamard matrices is a generalization of a Hadamard matrix. In this paper, we give some properties of Butson Hadamard matrices of order 4 which implies to the upper and the lower bounds of the number of its equivalence classes. We also showed that the entries of Butson Hadamard matrices of order 4 is 2k-th root of unity for some integer k. Furthermore, we describe the equivalence classes of Butson Hadamard matrices of order 4 by constructing the representative of the class. [ABSTRACT FROM AUTHOR]

Single-pixel imaging (SPI) is an alternative method for obtaining images using a single photodetector, which has numerous advantages over the traditional matrix-based approach. However, most experimental SPI realizations provide relatively low resolution compared to matrix-based imaging systems. Here, we show a simple yet effective experimental method to scale up the resolution of SPI. Our imaging system utilizes patterns based on Hadamard matrices, which, when reshaped to a variable aspect ratio, allow us to improve resolution along one of the axes, while sweeping of patterns improves resolution along the second axis. This work paves the way towards novel imaging systems that retain the advantages of SPI and obtain resolution comparable to matrix-based systems.

The investigations on short-circuit fault protection applications have shown that the majority of research are toward transmission networks. In the case of distribution networks, however, most studies have been focused on identifying the fault and its type or detecting the faulty bus, rather than locating the fault directly. This study presents two hardware-based applications, the first known prototype device attempts to directly predict fault location in branched distribution networks, which operate in real-time and off-line respectively. In the real-time application, a circuit is designed in Laboratory Virtual Instrument Engineering Workbench (LabVIEW) to transform 3 phase current-voltage modeling data into analog signals. The signals are then processed by a digital signal processor and the fault location is determined by the embedded algorithm in real time. The off-line application obtains results using the MATLAB Gui-based interface, which is integrated into the LattePanda development board. The algorithms inside both applications use Random Forest (RF) along with Fast Walsh Hadamard Transform (FWHT) feature to estimate the fault location. The performance of both systems is validated on a real 111-bus distribution feeder located in Turkey. [ABSTRACT FROM AUTHOR]

The integration of metasurfaces and optical waveguides is gradually attracting the attention of researchers because it allows for more efficient manipulation and guidance of light. However, most of the existing studies focus on passive devices, which lack dynamic modulation. This work utilizes the meta‐waveguides with liquid crystal(LC) to modulate the on‐chip spectrum, which is the first experimentally verified, to the authors' knowledge. By applying a voltage, the refractive index of the liquid crystal surrounding the meta‐waveguides can be tuned, resulting in a blue shift of the spectrum. The simulation shows that the 18.4 dB switching ratio can be achieved at 1550 nm. The meta‐waveguides are prepared using electron beam lithography (EBL), and the improved transmittance of the spectrum in the short wavelength is experimentally verified, which is consistent with the simulation trend. At 1551.64 nm wavelength, the device achieves a switching ratio of ≈16 dB with an active area of 8 µm × 0.4 µm. Based on this device, an optoelectronic computing architecture for the Hadamard matrix product and a novel wavelength selection switch are proposed. This work offers a promising avenue for on‐chip dynamic modulation in integrated photonics, which has the advantage of a compact active area, fast response time, and low energy consumption compared to conventional thermal‐light modulation. [ABSTRACT FROM AUTHOR]