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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]

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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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]

Supersaturated design (SSD) plays an important role in screening factors. E (f N O D) criterion is one of the most widely used criteria to evaluate multi-level and mixed-level SSDs. This paper provides some methods to construct multi-level E (f N O D) optimal SSDs with general run sizes, which can also be extended to construct mixed-level SSDs. The main idea of these methods is combining two processed generalized Hadamard matrices with the expansive replacement method. These proposed methods are easy to perform. Besides, the non-orthogonality between any two columns of the resulting SSDs is well controlled by that of the source designs. Some illustrative examples are given and several new SSDs are provided in this paper. • Convenient and flexible construction methods are provided. • Many new E (f N O D) optimal supersaturated designs are constructed. • Non-orthogonality of the new designs is well controlled by that of the source designs. [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]

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]

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 proposed spectral linear polarization imager and its realistic mathematical model. • The approach of loading the coded aperture and the necessity of merging micro-mirror units. • The measurements and reconstructions at different spectral resolutions. Spectral and polarization detections can increase the dimension of data acquisition and enhance the capabilities of target detection systems under complicated circumstances. A spectral linear polarization imager that combines double-disperser coded aperture imaging technology with a division of focal plane polarization detector is proposed in this paper to acquire spectral and polarization information simultaneously. First, the principles and mathematical model of the system are presented, where the proposed mathematical model considers the transformations among different coordinate systems and signal transmission processes, which is realistic and provides guidance for the calibration process. Then, targeting the detection and reconstruction methods based on the Hadamard transform, the loading approach of the coded aperture and the necessity of merging micro-mirror units are analyzed. Finally, according to the actual detection demands, the proposed system can achieve measurements and reconstructions at different spectral resolutions by loading 7th- and 15th-order cyclic S-matrix templates in real-time at the coded template. The results show the Pearson correlation coefficient and the root-mean-square error between the degree of linear polarization values acquired through proposed and reference systems are 0.9975 and 0.0238, respectively. Experimental results also demonstrate that although 15th-order cyclic S-matrix template can result a higher spectral resolution compared with 7th-order cyclic S-matrix template, it decreases the accuracy of the reconstructed results, indicating a trade-off between spectral resolution and accuracy in actual detections. [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]

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]

• Two methods to resist optical image forgery attack are proposed by combining hash function and image nonlinear correlation technology. • A raw hash value generated by a hash function is sent to the receiver over a secret channel. By comparing the original hash value with that of the decryption key, the receiver can determine whether the received key is a real key. • A sparse key map is generated by Hadamard function and sent to the receiver in a certain way. The receiver can judge the authenticity of the key according to the peak image presented. • Simulation results show that both of the two methods can effectively determine whether the keys of optical encryption system are tampered or not, thus effectively preventing forgery attacks. In forgery attacks, attackers will forge false ciphertexts or keys to deceive decrypters, so that people would lose their important information and even be threatened in terms of money or security. Therefore, the protection against forgery attack has become an important problem to be solved urgently in the field of information security. In this paper, two methods to resist optical image forgery attack are proposed by combining hash function and image nonlinear correlation technology. Firstly, a hash function is used to generate a raw hash value for the real key, and it is sent to the receiver over a secret channel. By comparing the original hash value with that of the decryption key, the receiver can determine whether the received key is a real key. In addition, a sparse key map is generated by Hadamard function and sent to the receiver in a certain way. The receiver can use this map to perform nonlinear correlation with the received key, and then judge the authenticity of the key according to the peak image presented. Simulation results show that both of the two methods can effectively determine whether the keys of optical encryption system are tampered or not, thus effectively preventing forgery attacks. [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]

Balanced Incomplete Block Designs (BIBD) are the designs obtained from the arrangement of symbols in blocks following established rules and satisfying relevant necessary conditions. BIBDs have broad applications in many fields.The entries in a Hadamard matrix H are all +1’s or -1’s, satisfying the property HHT = n In [2]. Rhotrix is a new concept in mathematics having applications in cryptography. In this paper, using SBIB designs the Hadamard Rhotrices are constructed. AMS Subject classification: 05B05, 05B20, 62K10.

In measuring a transmission matrix (TM) of a scattering medium, the normal way requires projecting a large number of patterns, resulting in significant measurement time. Here, we propose to do imaging through scattering media using a path-planning-based downsampling TM. The proposed method is verified by measuring TMs for imaging through scattering media when Walsh, Cake-cutting, and Russian doll Hadamard transforms (HTs) are adopted. Based on the spectral characteristics of different HTs of natural images, corresponding sampling paths are artificially planned, which guarantees the method to preferentially measure components in TM in relation to the more important spectra. This means that our method can reduce the number of patterns during measurement, thus improving the imaging speed while only sacrificing a little imaging quality. Experimental results demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]