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Neural network models, such as BP, LSTM, etc., support only numerical inputs, so data preprocessing needs to be carried out on the categorical variables to convert them into numerical data. For unordered multi-categorical variables, existing encoding methods may produce dimensional catastrophes and may also introduce additional order misrepresentation and distance bias in neural network computation. To solve the above problems, this paper proposes an unordered multi-categorical variable encoding method O-AE using orthogonal matrix for encoding and encoding representation learning and dimensionality reduction via an autoencoder. Bayesian optimization is used for hyperparameter optimization of the autoencoder. Finally, seven experiments were designed with the basic O-AE, Bayesian optimization of the hyperparameters of the autoencoder for O-AE, and other encoding methods to encode unordered multi-categorical variables in five datasets, and they were input into a BP neural network to carry out target prediction experiments. The results show that the experiments using O-AE and O-AE-b have better prediction results, proving that the method proposed in this paper is highly feasible and applicable and can be an optional method for the data processing of unordered multi-categorical variables. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green’s functions of the classical Dirichlet and Neumann boundary value problems, Green’s functions of the studied problems are constructed and integral representations of solutions to these problems are obtained.

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green's functions of the classical Dirichlet and Neumann boundary value problems, Green's functions of the studied problems are constructed and integral representations of solutions to these problems are obtained. [ABSTRACT FROM AUTHOR]

The technique of surrogate optimisation is to use a simpler function to approximate a complex function that is time-consuming to evaluate. We show that the maximum of a special type of surrogate function (Formula presented.) is at (Formula presented.), and that there is one and only one local maximum both in (Formula presented.) and (Formula presented.). This function (Formula presented.) has been found to be useful in various aspects of electronic structure theory, including proving the Carlson-Keller theorem, and localising orbitals. As one other example, we apply it here to optimise the ground state of molecules using the Generalised Valence Bond wavefunction.

Abstract Improved software for processing medical images has inspired tremendous interest in modern medicine in recent years. Modern healthcare equipment generates huge amounts of data, such as scanned medical images and computerized patient information, which must be secured for future use. Diversity in the healthcare industry, namely in the form of medical data, is one of the largest challenges for researchers. Cloud environment and the Block chain technology have both demonstrated their own use. The purpose of this study is to combine both technologies for safe and secure transaction. Storing or sending medical data through public clouds exposes information into potential eavesdropping, data breaches and unauthorized access. Encrypting data before transmission is crucial to mitigate these security risks. As a result, a Blockchain based Chaotic Arnold’s cat map Encryption Scheme (BCAES) is proposed in this paper. The BCAES first encrypts the image using Arnold’s cat map encryption scheme and then sends the encrypted image into Cloud Server and stores the signed document of plain image into blockchain. As blockchain is often considered more secure due to its distributed nature and consensus mechanism, data receiver will ensure data integrity and authenticity of image after decryption using signed document stored into the blockchain. Various analysis techniques have been used to examine the proposed scheme. The results of analysis like key sensitivity analysis, key space analysis, Information Entropy, histogram correlation of adjacent pixels, Number of Pixel Change Rate, Peak Signal Noise Ratio, Unified Average Changing Intensity, and similarity analysis like Mean Square Error, and Structural Similarity Index Measure illustrated that our proposed scheme is an efficient encryption scheme as compared to some recent literature. Our current achievements surpass all previous endeavors, setting a new standard of excellence.

Improved software for processing medical images has inspired tremendous interest in modern medicine in recent years. Modern healthcare equipment generates huge amounts of data, such as scanned medical images and computerized patient information, which must be secured for future use. Diversity in the healthcare industry, namely in the form of medical data, is one of the largest challenges for researchers. Cloud environment and the Block chain technology have both demonstrated their own use. The purpose of this study is to combine both technologies for safe and secure transaction. Storing or sending medical data through public clouds exposes information into potential eavesdropping, data breaches and unauthorized access. Encrypting data before transmission is crucial to mitigate these security risks. As a result, a Blockchain based Chaotic Arnold's cat map Encryption Scheme (BCAES) is proposed in this paper. The BCAES first encrypts the image using Arnold's cat map encryption scheme and then sends the encrypted image into Cloud Server and stores the signed document of plain image into blockchain. As blockchain is often considered more secure due to its distributed nature and consensus mechanism, data receiver will ensure data integrity and authenticity of image after decryption using signed document stored into the blockchain. Various analysis techniques have been used to examine the proposed scheme. The results of analysis like key sensitivity analysis, key space analysis, Information Entropy, histogram correlation of adjacent pixels, Number of Pixel Change Rate, Peak Signal Noise Ratio, Unified Average Changing Intensity, and similarity analysis like Mean Square Error, and Structural Similarity Index Measure illustrated that our proposed scheme is an efficient encryption scheme as compared to some recent literature. Our current achievements surpass all previous endeavors, setting a new standard of excellence. [ABSTRACT FROM AUTHOR]

Let O (3) denote the group of orthogonal 3 × 3 real matrices, and M the 5-dimensional real vector space of all 3 × 3 real symmetric matrices with trace zero. Let λ 1 (A) ≤ λ 2 (A) ≤ λ 3 (A) be the eigenvalues of A ∈ M and Ξ - = { A ∈ M ∣ λ 1 (A) = λ 2 (A) } . M is an inner product space with the inner product ⟨ A , B ⟩ = trace (A B) . Let G 3 (M) be the set of all 3-dimensional subspaces of M , a 6-dimensional Grassman manifold. O (3) acts on M on the left by conjugation via inner product preserving linear isomorphisms, which map any 3-dimensional subspace into another 3-dimensional subspace; thus G 3 (M) also has a left action of O (3) . G 3 (M) becomes a category, an action groupoid, with morphisms (V , M , W) ∈ G 3 (M) × O (3) × G 3 (M) , where W = M V M T . Composition of morphisms is (V 1 , N , V 2) ∘ (V 0 , M , V 1) = (V 0 , N M , V 2) . Let C be a category whose objects (V, S) consist of a real inner product space V and S ⊂ V , and whose arrows (V , S) → (W , T) consist of f : V → W , an inner product preserving real linear mapping such that f (S) ⊂ T . We have the functor V ↓ (V , M , W) W G 3 (M) ⟶ F - (V , Ξ - ∩ V) ↓ F - (M) (W , Ξ - ∩ W) C where F - (M) : V → W : A ↦ M A M T . Suppose further that S 3 denotes the group of permutations of { 1 , 2 , 3 } , and ρ : S 3 → O (3) denotes a group homomorphism which is isomorphic as a group representation to the natural representation of S 3 on R 3 (which permutes the coordinates). Let Obj (L S) denote the set of all V ∈ G 3 (M) whose isotropy subgroup contains S = ρ (S 3) as a subgroup. This paper completely describes the full subcategory L S of G 3 (M) with object set Obj (L S) , as well as the details of the above functor restricted to L S . Thus all the members V ∈ Obj (L S) are determined, as well as the smooth manifold structure on Obj (L S) ; it is embedded as a one-dimensional submanifold of G 3 (M) . The isotropy subgroups of all V ∈ Obj (L S) are computed and all pairs V , W ∈ Obj (L S) which are isomorphic via some M ∈ O (3) are determined. The sets Ξ - ∩ V are all determined, and the functorial mappings on morphism sets are computed. However, L S is not a Lie groupoid. The image of Obj (L S) under the functor π 1 F - is the collection of fibres of the smooth manifold ∐ V ∈ Obj (L S) V , which is the total space of the canonical vector bundle over the base manifold Obj (L S) . The bifurcation points of the family of subsets Ξ - ∩ V as V ranges over Obj (L S) (within this total space) are seen to be the points of Obj (L S) with infinite isotropy subgroups. We also show how this mathematical problem arises naturally from a problem in mathematical chemistry. Hence certain features of numerical calculations of energy eigenvalue intersection patterns of the simple chemical system H3 are rationalized through linearization about the triple intersection point. [ABSTRACT FROM AUTHOR]

Neural network models, such as BP, LSTM, etc., support only numerical inputs, so data preprocessing needs to be carried out on the categorical variables to convert them into numerical data. For unordered multi-categorical variables, existing encoding methods may produce dimensional catastrophes and may also introduce additional order misrepresentation and distance bias in neural network computation. To solve the above problems, this paper proposes an unordered multi-categorical variable encoding method O-AE using orthogonal matrix for encoding and encoding representation learning and dimensionality reduction via an autoencoder. Bayesian optimization is used for hyperparameter optimization of the autoencoder. Finally, seven experiments were designed with the basic O-AE, Bayesian optimization of the hyperparameters of the autoencoder for O-AE, and other encoding methods to encode unordered multi-categorical variables in five datasets, and they were input into a BP neural network to carry out target prediction experiments. The results show that the experiments using O-AE and O-AE-b have better prediction results, proving that the method proposed in this paper is highly feasible and applicable and can be an optional method for the data processing of unordered multi-categorical variables.

Given a proper cone K in the Euclidean space R n , a square matrix A is said to be K -semipositive if there exists an x ∈ K such that A x ∈ int (K) , the topological interior of K. The paper aims to study algebraic and geometrical properties of K -semipositive matrices with special emphasis on the self-dual proper Lorentz cone L + n = { x ∈ R n : x n ≥ 0 , ∑ i = 1 n − 1 x i 2 ≤ x n 2 }. More specifically, we discuss a few necessary and other sufficient algebraic conditions for L + n -semipositive matrices. Also, we provide algebraic characterizations for diagonal and orthogonal L + n -semipositive matrices. Furthermore, given a square matrix A and a proper cone K , geometric properties of the semipositive cone K A , K = { x ∈ K : A x ∈ K } and the cone of S A , K = { x : A x ∈ K } are discussed in terms of their extremals. As L + n is an ellipsoidal cone, at last we find results for the cones K A , L + n and S A , L + n to be ellipsoidal. [ABSTRACT FROM AUTHOR]

We introduce an approach based on the Givens representation for posteriorinference in statistical models with orthogonal matrix parameters, such asfactor models and probabilistic principal component analysis (PPCA). We showhow the Givens representation can be used to develop practical methods fortransforming densities over the Stiefel manifold into densities over subsets ofEuclidean space. We show how to deal with issues arising from the topology ofthe Stiefel manifold and how to inexpensively compute the change-of-measureterms. We introduce an auxiliary parameter approach that limits the impact oftopological issues. We provide both analysis of our methods and numericalexamples demonstrating the effectiveness of the approach. We also discuss howour Givens representation can be used to define general classes ofdistributions over the space of orthogonal matrices. We then givedemonstrations on several examples showing how the Givens approach performs inpractice in comparison with other methods.

We analyze two algorithms for computing the symplectic factorization A = LLT of a given symmetric positive definite symplectic matrix A. The first algorithm W1 is an implementation of the HHT factorization from Dopico and Johnson (SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009), see Theorem 5.2. The second one is a new algorithm W2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of a new class of nonlocal boundary value problems for the generalized Helmholtz equation in the unit ball. These problems are a generalization of the classical Dirichlet boundary value problem to the Helmholtz equation. For the considered problems, existence and uniqueness theorems are proved. Integral representation of solutions is established. Corresponding spectral questions are also investigated, namely, eigenfunctions and eigenvalues of the problem are found. [ABSTRACT FROM AUTHOR]

The recursive solution to the Procrustes problem -with or without constraints- is thoroughly investigated. Given known matrices A and B , the proposed solution minimizes the square of the Frobenius norm of the difference A X − B when rows or columns are added to A and B. The proposed method is based on efficient strategies which reduce the computational cost by utilizing previous computations when new data are acquired. This is particularly useful in the iterative solution of an unbalanced orthogonal Procrustes problem. The results show that the computational efficiency of the proposed recursive algorithms is more significant when the dimensions of the matrices are large. This demonstrates the usefulness of the proposed algorithms in the presence of high-dimensional data sets. The practicality of the new method is demonstrated through an application in machine learning, namely feature extraction for image processing. [ABSTRACT FROM AUTHOR]

O'Dorney has proved that for any orthogonal matrix U , there exists a signature matrix D (a diagonal matrix with diagonal entries ±1) by which | s i j | ≤ 1 , where S = (s i j) = Q (U D) is the Cayley transform of UD. In this paper, we prove that the upper bound can be reduced to 2 − 1 by multiplying by a certain signed permutation matrix, instead of a signature matrix. [ABSTRACT FROM AUTHOR]

In this paper, a novel secure image signal transmission scheme was proposed in wireless systems, in which the poly-polarization filtering and the orthogonal matrix (PPF-OM) were combined to protect the image signal and eliminate the polarization dependent loss (PDL) at the same time, which was caused by the non-ideal wireless channel. This scheme divided the image information sequence into two parts in order to modulate and reshape the results into symbol matrices with the same size. Then, two sets of polarization states (PSs) and orthogonal matrices (OMs) were designed to process the symbols in order to enhance information protection and eliminate the PDL. Legitimate users were able to apply the shared PSs and OMs, step by step, so the information could be recovered. However, for eavesdroppers, the received signals were random symbols that were difficult to demodulate. Then, the bit error rate and the secrecy rate were derived to evaluate the performance of the PPF-OM scheme. Finally, the simulations demonstrated the superior performance of the PPF-OM scheme for enhancing the information security and eliminating the PDL. [ABSTRACT FROM AUTHOR]

In this paper, using an orthogonal matrix in space , the notion of a nonlocal biharmonic operator is introduced. For the corresponding nonlocal biharmonic equation, solvability of boundary value problems with fractional conformable derivatives is studied. For the considered problems, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained and integral representations of solutions are presented. [ABSTRACT FROM AUTHOR]

This paper aims to describe the methods that were used in order to construct the Hadamard matrix and its basic properties and how it is applied in Cryptography. Moreover, this matrix was used to construct a new algorithm in cryptography.

The aim of this paper is to investigate the Legendre spectral method for solving the diffusion equation with a source term and mixed initial-boundary value problem in a finite rectangle Ω2, we use some techniques to convert the problem to a system of ordinary differential equations and by an analysis matrical we find a general term defines all ordinary differential equations of this system, we solve this general term we get the desired approximate solution, we also present the error estimate. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the Legendre spectral method for solving the diffusion equation with a source term and mixed initial-boundary value problem in a finite rectangle Ω2, we use some techniques to convert the problem to a system of ordinary differential equations and by an analysis matrical we find a general term defines all ordinary differential equations of this system, we solve this general term we get the desired approximate solution, we also present the error estimate. [ABSTRACT FROM AUTHOR]

In the last decade, the communication of images through the internet has increased. Due to the growing demands for data transfer through images, protection of data and safe communication is very important. For this purpose, many encryption techniques have been designed and developed. New and secured encryption schemes based on chaos theory have introduced methods for secure as well as fast communication. A modified image encryption process is proposed in this work with chaotic maps and orthogonal matrix in Hill cipher. Image encryption involves three phases. In the first phase, a chaotic Henon map is used for permuting the digital image. In the second phase, a Hill cipher is used whose encryption key is generated by an orthogonal matrix which further is produced from the equation of the plane. In the third phase, a sequence is generated by a chaotic tent map which is later XORed. Chaotic maps play an important role in the encryption process. To deal with the issues of fast and highly secured image processing, the prominent properties of non-periodical movement and non-convergence of chaotic theory play an important role. The proposed scheme is resistant to different attacks on the cipher image. Different tests have been applied to evaluate the proposed technique. The results of the tests such as key space analysis, key sensitivity analysis, and information entropy, histogram correlation of the adjacent pixels, number of pixel change rate (NPCR), peak signal to noise ratio (PSNR), and unified average changing intensity (UCAI) showed that our proposed scheme is an efficient encryption technique. The proposed approach is also compared with some state-of-the-art image encryption techniques. In the view of statistical analysis, we claim that our proposed encryption algorithm is secured. [ABSTRACT FROM AUTHOR]

The concept of polynomial orthogonal of type I matrices are introduced. We define the index of orthogonal of type I matrices and we extended some results of orthogonal of type I matrix to polynomial orthogonal of type I matrices. [ABSTRACT FROM AUTHOR]

The Euclidean hull dimension of a linear code is an important quantity to determine the parameters of an entanglement-assisted quantum error-correcting code (EAQECC) if the Euclidean construction is applied. In this paper, we study the Euclidean hull of a linear code by means of orthogonal matrices. We provide some methods to construct linear codes over F p m with hull of arbitrary dimensions. With existence of self-dual bases of F p m over F p , we determine a Gray map from F p m to F p m , and from a given linear code over F p m with one-dimensional hull, we construct, using such a Gray map, a linear code over F p with m-dimensional hull for all m when p is even and for all m odd when p is odd. Comparisons with classical constructions are made, and some good EAQECCs over F q , q = 2 , 3 , 4 , 5 , 9 , 13 , 17 , 49 are presented. [ABSTRACT FROM AUTHOR]

In this paper, we provide new methods and algorithms to construct Euclidean self-dual codes over large finite fields. With the existence of a dual basis, we study dual preserving linear maps, and as an application, we use them to construct self-orthogonal codes over small finite prime fields using the method of concatenation. Many new optimal self-orthogonal and self-dual codes are obtained. [ABSTRACT FROM AUTHOR]

Feature extraction (FE) plays an important role in machine learning. In order to handle the “dimensionality disaster”problem, the usual approach is to transform the original sample into a low-dimensional target space, in which the FE task is performed. However, the data in reality will always be corrupted by various noises or interfered by outliers, making the task of feature extraction extremely challenging. Thence, we propose a novel image FE method via approximate orthogonal low-rank embedding (AOLRE), which adopts an orthogonal matrix to reserve the major energy of the samples, and the introduction of the ℓ2,1-norm makes the features more compact, differentiated and interpretable. In addition, the weighted Schatten p-norm is adopted in this model for fully exploring the effects of different ranks while approaching the original hypothesis of low rank. Meanwhile, as a robust measure, the correntropy is applied in AOLRE. This can effectively suppress the adverse influences of contaminated data and enhance the robustness of the algorithm. Finally, the introduction of the classification loss item allows our model to effectively fit the supervised scene. Five common datasets are used to evaluate the performance of AOLRE. The results show that the recognition accuracy and robustness of AOLRE are significantly better than those of several advanced FE algorithms, and the improvement rate ranges from 2% to 15%.

In this paper, a novel secure image signal transmission scheme was proposed in wireless systems, in which the poly-polarization filtering and the orthogonal matrix (PPF-OM) were combined to protect the image signal and eliminate the polarization dependent loss (PDL) at the same time, which was caused by the non-ideal wireless channel. This scheme divided the image information sequence into two parts in order to modulate and reshape the results into symbol matrices with the same size. Then, two sets of polarization states (PSs) and orthogonal matrices (OMs) were designed to process the symbols in order to enhance information protection and eliminate the PDL. Legitimate users were able to apply the shared PSs and OMs, step by step, so the information could be recovered. However, for eavesdroppers, the received signals were random symbols that were difficult to demodulate. Then, the bit error rate and the secrecy rate were derived to evaluate the performance of the PPF-OM scheme. Finally, the simulations demonstrated the superior performance of the PPF-OM scheme for enhancing the information security and eliminating the PDL.

This paper aims to describe the methods that were used in order to construct the Hadamard matrix and its basic properties and how it is applied in Cryptography. Moreover, this matrix was used to construct a new algorithm in cryptography. [ABSTRACT FROM AUTHOR]

Motivated by applications to Bayesian inference for statistical models with orthogonal matrix parameters, we present polar expansion , a general approach to Monte Carlo simulation from probability distributions on the Stiefel manifold. To bypass many of the well-established challenges of simulating from the distribution of a random orthogonal matrix Q , we construct a distribution for an unconstrained random matrix X such that Q X , the orthogonal component of the polar decomposition of X , is equal in distribution to Q. The distribution of X is amenable to Markov chain Monte Carlo (MCMC) simulation using standard methods, and an approximation to the distribution of Q can be recovered from a Markov chain on the unconstrained space. When combined with modern MCMC software, polar expansion allows for routine and flexible posterior inference in models with orthogonal matrix parameters. We find that polar expansion with adaptive Hamiltonian Monte Carlo is an order of magnitude more efficient than competing MCMC approaches in a benchmark protein interaction network application. We also propose a new approach to Bayesian functional principal component analysis which we illustrate in a meteorological time series application. for this article are available online. [ABSTRACT FROM AUTHOR]

We develop an efficient algorithm for sampling the eigenvalues of random matrices distributed according to the Haar measure over the orthogonal or unitary group. Our technique samples directly a factorization of the Hessenberg form of such matrices, and then computes their eigenvalues with a tailored core-chasing algorithm. This approach requires a number of floating-point operations that is quadratic in the order of the matrix being sampled, and can be adapted to other matrix groups. In particular, we explain how it can be used to sample the Haar measure over the special orthogonal and unitary groups and the conditional probability distribution obtained by requiring the determinant of the sampled matrix be a given complex number on the complex unit circle. [ABSTRACT FROM AUTHOR]

A generalization of the well-known Helmert matrix was recently introduced in [Akkus, I., and G. Kizilaslan. 2019. Generalization of a statistical matrix and its factorization. Communications in Statistics - Theory and Methods. doi:10.1080/03610926.2019.1645854.]. We consider a modified form of this generalization and we obtain the LU decomposition (in closed form) of this modified generalized matrix. We also prove some special cases, that are all orthogonal, only using the Helmert matrix. [ABSTRACT FROM AUTHOR]

A Householder transformation, or Householder reflection, or Household matrix, is a reflection about a hyperplane with a unit normal vector. Not only have the Household matrices been used in QR decomposition efficiently but also implicitly and successfully applied in other areas. In the process of investigating a family of unitary filterbanks, a new family of Householder matrices are established. These matrices are produced when a matrix filter is required to preserve certain order of 2d digital polynomial signals. Naturally, they can be applied to image and signal processing among others. [ABSTRACT FROM AUTHOR]

Many statistical problems involve the estimation of a d × d orthogonal matrix Q . Such an estimation is often challenging due to the orthonormality constraints on Q . To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible d × d matrix as the product of a d × d permutation matrix P , a d × d unit lower triangular matrix L , and a d × d upper triangular matrix U . Thanks to the QR decomposition, we find the formulation of U when the PLU decomposition is applied to Q . We call the result as PLR decomposition; it produces a one-to-one correspondence between Q and the d d - 1 / 2 entries below the diagonal of L , which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to L . For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses. [ABSTRACT FROM AUTHOR]

In the last decade, the communication of images through the internet has increased. Due to the growing demands for data transfer through images, protection of data and safe communication is very important. For this purpose, many encryption techniques have been designed and developed. New and secured encryption schemes based on chaos theory have introduced methods for secure as well as fast communication. A modified image encryption process is proposed in this work with chaotic maps and orthogonal matrix in Hill cipher. Image encryption involves three phases. In the first phase, a chaotic Henon map is used for permuting the digital image. In the second phase, a Hill cipher is used whose encryption key is generated by an orthogonal matrix which further is produced from the equation of the plane. In the third phase, a sequence is generated by a chaotic tent map which is later XORed. Chaotic maps play an important role in the encryption process. To deal with the issues of fast and highly secured image processing, the prominent properties of non-periodical movement and non-convergence of chaotic theory play an important role. The proposed scheme is resistant to different attacks on the cipher image. Different tests have been applied to evaluate the proposed technique. The results of the tests such as key space analysis, key sensitivity analysis, and information entropy, histogram correlation of the adjacent pixels, number of pixel change rate (NPCR), peak signal to noise ratio (PSNR), and unified average changing intensity (UCAI) showed that our proposed scheme is an efficient encryption technique. The proposed approach is also compared with some state-of-the-art image encryption techniques. In the view of statistical analysis, we claim that our proposed encryption algorithm is secured.

In order to reveal the sectoral composition between the years (2003-2020) with the inflation-free (real) values of the Payments made by Debit Card, Credit Card and Internet Card, and to increase the efficiency of payment systems by identifying similar and differentiated sectors, Binary Loads (Orthogonal (Vertical) and Diagonal (Diagonal) matrices are created. The eigenvalues, eigenvectors (charges) and ordinary correlation results obtained from the matrices are combined in the plane of the binary load graph and the shapes of the columns are equalized to the eigenvalues using Mahalanobis distances, and without observation scaling so that the cosines of the angles between the vectors are equal to the correlations between the variables, the results are interpreted only in a constant proportionality. In the light of results such as eigenvalues, values, forward difference in eigenvalues, and the total variance ratio explained, it was determined that some of the sectors paid by card differ significantly from other sectors and clustered by carrying positive or negative loads (variances). [ABSTRACT FROM AUTHOR]

Background: Permanent magnet machines can be used for many applications, due to their high power density, high efficiency, observable torque‐to‐current ratio, and low maintenance cost. As such, the generator design in wind turbine systems is highly dependent on improvement of efficiency and generator performance. Thus, in order to improve the quality of manufactured electrical machines, a robust and precise optimization model should be developed to substitute the traditional deterministic models. Purpose: This paper investigated the optimization design process of an interior permanent magnet synchronous generator (IPMSG) for wind power systems using the finite element analysis (FEA) based on a prototype IPMSG. To that end, a multi‐objective optimization design of a permanent magnet generator (PMG) based on Taguchi method was proposed. Design/Method: The influence of the pole arc angle, magnet inset, magnet thickness, magnet width, and stator tooth width as well as slot depth were considered as the design parameters. The main characteristics of generator efficiency, total harmonic distortion (THD) and the amplitude of induction electromotive force (EMF) were taken as the optimization objectives, as well. The orthogonal matrix was established based on the number of the selected variables and the level factor of each variable, and finite element method (FEM) was, in turn, applied to solve the experimental matrix. As a result, an improved generator with higher maximum efficiency and EMF and lower THD was designed. Conclusions: Finally, the prototype IPMSG which was manufactured based on the analysis results and Taguchi method was tested. The experimental tests were then performed to confirm the validity of the proposed design process and the generator efficiency. The prototype experiments and the finite element simulation results verified the efficacy of the optimization method. [ABSTRACT FROM AUTHOR]

We obtain the two new variants of an explicit parametrization for the general Lorentz group. Formulas are given for the direct and inverse four-dimensional Lorentz transformations. These formulas use the orthogonal three- or four-dimensional matrices. We find the infinitesimal operators of the proper Lorentz group and the multiplication formulas (commutators) of the infinitesimal operators. The orthogonal three- and four-dimensional matrices are parameterized by lower triangular matrices containing three or six independent parameters. [ABSTRACT FROM AUTHOR]

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A set S ⊆ F 2 n is called degree-d zero-sum if the sum ∑ s ∈ S f (s) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n − d − 1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independent elements, and show relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices. We are particularly interested in the smallest of such sets and prove bounds on the minimum number of elements in a degree-d zero-sum set of rank n. The motivation for studying those objects comes from the fact that degree-d zero-sum sets of full rank can be used to build linear mappings that preserve special kinds of nonlinear invariants, similar to those obtained from orthogonal matrices and exploited by Todo, Leander and Sasaki for breaking the block ciphers Midori, Scream and iScream. [ABSTRACT FROM AUTHOR]

A new condition, the strong inner product property, is introduced and used to construct sign patterns of row orthogonal matrices. Using this property, infinite families of sign patterns allowing row orthogonality are found. These provide insight into the underlying combinatorial structure of row orthogonal matrices. Algorithmic techniques for verifying that a matrix has the strong inner product property are also presented. These techniques lead to a generalization of the strong inner product property and can be easily implemented using various software. [ABSTRACT FROM AUTHOR]