Diagonal Transformation Research Articles

Ergodic theory of diagonal orthogonal covariant quantum channels

We analyse the ergodic properties of quantum channels that are covariant with respect to diagonal orthogonal transformations. We prove that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix. This allows us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels. As an application of our analysis, we study dual unitary brickwork circuits which have recently been proposed as minimal models of quantum chaos in many-body systems. Upon imposing a local diagonal orthogonal invariance symmetry on these circuits, the long-term behaviour of spatio-temporal correlations between local observables in such circuits is completely determined by the ergodic properties of a channel that is covariant under diagonal orthogonal transformations. We utilize this fact to show that such symmetric dual unitary circuits exhibit a rich variety of ergodic behaviours, thus emphasizing their importance.

Enhancing Salp Swarm Optimization with Orthogonal Diagonalization Transformation for Damage Detection in Truss Bridge

Enhancing Salp Swarm Optimization with Orthogonal Diagonalization Transformation for Damage Detection in Truss Bridge

An algorithm for calculating spectral radius of $ s $-index weakly positive tensors

<abstract><p>In this paper, we introduced $ s $-index weakly positive tensors and discussed the calculation of the spectral radius of this kind of nonnegative tensors. Using the diagonal similarity transformation of tensor and Perron-Frobenius theory of nonnegative tensor, the calculation method of the maximum $ H $-eigenvalue of $ s $-index weakly positive tensors was given. A variable parameter was introduced in each iteration of the algorithm, which is equivalent to a translation transformation of the tensor in each iteration to improve the calculation speed. At the same time, it was proved that the algorithm is linearly convergent for the calculation of the spectral radius of $ s $-index weakly positive tensors. The final numerical example shows the effectiveness of the algorithm.</p></abstract>

Фермион с тремя массовыми параметрами во внешнем магнитном поле

Recently, models for a spin 1/2 particle with two (or three)
 mass parameters were developed. Specific features of
 these models are as follows. For corresponding two (or
 three) bispinors in absence of external fields, separate Diraclike
 equations are derived, they differ in masses. However,
 in presence of external electromagnetic or gravitational
 fields with non-vanishing Ricci scalar, the wave equation for
 bispinors does not split into separated equations but makes
 quite a definite mixing of two (or three) equations arises. In
 the present paper, the model of a fermion with three mass parameters
 is studied in presence of the external uniform magnetics
 field. After performing a diagonalizing transformation,
 three separate equations are obtained for particles with different
 anomalous magnetic moments. Their exact solutions
 and generalized energy spectra are found

Image encryption algorithm based on 2D hyper-chaotic system and central dogma of molecular biology

Abstract With the widespread use of images, image security has received much attention. Image security can be guaranteed by encrypting the plain image and transmitting the corresponding cipher image. This paper proposes an image encryption algorithm based on the novel two-dimensional (2D) hyper-chaotic system, bidirectional diagonal crossover transformations (BDCTs) and central dogma of molecular biology. Firstly, six chaotic sequences are generated using the proposed hyper-chaotic system, which are used in the permutation and diffusion processes. Secondly, the 5th–8th bit-planes of plain image are permuted using the designed BDCTs. Thirdly, the permuted image is dynamically encoded into a quaternary DNA sequence, and then the quaternary DNA sequence is transcribed into a quaternary RNA sequence. After that, the quaternary RNA sequence is diffused using quaternary RNA operations. Finally, the diffused RNA sequence is translated into a codon sequence, and then the designed codon-level multipoint crossover scheme is used to further improve the diffusion effect. Experiment results and security analyses demonstrate that our algorithm has high security and efficiency. In other words, our algorithm is quite suitable for real-time image cryptosystems.

On the phases of a semi-sectorial matrix and the essential phase of a Laplacian

In this paper, we extend the definition of phases of sectorial matrices to those of semi-sectorial matrices, which are possibly singular. Properties of the phases are also extended, including those of the Moore-Penrose generalized inverse, compressions and Schur complements, matrix sums and products. In particular, an interlacing relation is established between the phases of A+B and those of A and B combined. Also, a majorization relation is established between the phases of the nonzero eigenvalues of AB and the phases of the compressions of A and B, which leads to a generalized matrix small phase theorem. For the matrices which are not necessarily semi-sectorial, we define their (largest and smallest) essential phases via diagonal similarity transformation. An explicit expression for the essential phases of a Laplacian matrix of a directed graph is obtained.

Reductive functionalization of CO2 under mild reaction conditions for the catalytic synthesis of N-formamides by Mannich base based Cobalt (III) complex

Reductive functionalization of carbon dioxide is an important approach which not only reduces CO2 concentration but also functionalized it for the formation of value added chemicals. In this perspective, a hexadentate mononuclear cobalt (III) complex [CoIIIL] of Mannich base ligand (H4L) was synthesized and structurally characterized by single crystal X-ray diffraction. The electronic spectra of the complex showed two bands at 448 and 561 nm corresponds to the LMCT and d-d transitions respectively. The derived inexpensive [CoIIIL] complex was very much effective as catalyst in carbon dioxide functionalization reaction. The catalytic synthesis of mono selective N-formamide product from amines was produced through diagonal transformation of carbon dioxide in presence of polymethylhydrosiloxane (PMHS) as reducing agent and [CoIIIL] complex as catalyst. The effects of various reaction parameters were examined. Broad range of substrates (aromatic as well as aliphatic amines) smoothly produced good percentage of respective N-formamides product (72–94%) under mild reaction condition utilizing 1 atmospheric CO2. The high TON (1540–2000) and TOF value (192.5–400.0 h−1) for the catalytic reactions strongly depicted the productive nature of the [CoIII(L)] catalyst. The probable mechanestics studies clearly reveals that the [CoIIIL] complex activated the Si−H bond of PMHS to react with carbon dioxide in order to form the silyl formates intermediate for activation of N–H bonds in amines, thus leading to the excellent performance of the catalytic reaction.

A New Lyapunov-Based Event-Triggered Control of Linear Systems

This paper introduces a new Lyapunov sampling for event-triggered linear systems. Contrary to the majority of the existing literature, our proposed event-triggering mechanism (ETM) directly deals with the Lyapunov derivative rather than its upper bounds. Moreover, a new concept of ideality is introduced which serves as a measure of conservativity of the triggering mechanism during the stability regions of trajectories. To construct the ETM we employ a diagonal transformation to facilitate the design in a component-wise fashion. Moreover, the interpretation of stating ETM in the transformed coordinate is given by when the trajectories reach <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>-norm contour curves. A proof of Zeno behaviour exclusion is also given which is required to justify the efficiency of the presented method for practical implementation. The approach is illustrated by compelling example.

Augmenting the detection of Online Image theft using diagonal transformation and iris biometric watermarking

Augmenting the detection of Online Image theft using diagonal transformation and iris biometric watermarking

Evolving convolution neural network by optimal regularization random vector functional link for computational color constancy

To automatically eliminate the influence of external illumination on the acquired image, restore the true color information of the object, and provide true and stable color features for computer vision tasks, our work is based on the convolutional neural network (CNN) VGG19 for feature extraction of image information. Moreover, it proposes a model based on atom search optimization improved by chaotic-logistic maps to optimize the regularization random vector functional link (RRVFL). The optimized RRVFL replaces CNN’s regression layer to estimate the image illumination and then restore the image. First, we used chaotic-logistic maps to optimize the initial value of search agent of the atom search optimization (CASO) algorithm and then used the search agent segment mapping method proposed in our study to simultaneously optimize the number of nodes in hidden layer, regularization factors, input weight, and the bias of RRVFL hidden layer. This avoids the problem of serious fluctuation of prediction results and low accuracy caused by the randomness of the RRVFL parameters. After the predicted illumination information was obtained with the optimized RRVFL, the image was restored using the diagonal transformation method. Comparative experiments showed that the average angular errors of the illumination estimation of the V19-CASO-RRVFL algorithm proposed in our study were 1.73, 0.30, 0.3406, and 0.0976 lower than those of the gray-edge-2, GE-CASO-RRVFL, DS-Net, and color constancy on deep residual learning algorithms, respectively.

Power Function Method for Finding the Spectral Radius of Weakly Irreducible Nonnegative Tensors

Since the eigenvalue problem of real supersymmetric tensors was proposed, there have been many results regarding the numerical algorithms for computing the spectral radius of nonnegative tensors, among which the NQZ method is the most studied. However, the NQZ method is only suitable for calculating the spectral radius of a special weakly primitive tensor, or a weakly irreducible primitive tensor that satisfies certain conditions. In this paper, by means of diagonal similarrity transformation of tensors, we construct a numerical algorithm for computing the spectral radius of nonnegative tensors with the aid of power functions. This algorithm is suitable for the calculation of the spectral radius of all weakly irreducible nonnegative tensors. Furthermore, it is efficient and can be widely applied.

Exactness Conditions for Semidefinite Programming Relaxations of Generalization of the Extended Trust Region Subproblem

The extended trust region subproblem (ETRS) of minimizing a quadratic objective over the unit ball with additional linear constraints has attracted a lot of attention in the last few years because of its theoretical significance and wide spectra of applications. Several sufficient conditions to guarantee the exactness of its semidefinite programming (SDP) relaxation have been recently developed in the literature. In this paper, we consider a generalization of the extended trust region subproblem (GETRS), in which the unit ball constraint in the ETRS is replaced by a general, possibly nonconvex, quadratic constraint, and the linear constraints are replaced by a conic linear constraint. We derive sufficient conditions for guaranteeing the exactness of the SDP relaxation of the GETRS under mild assumptions. Our main tools are two classes of convex relaxations for the GETRS based on either a simultaneous diagonalization transformation of the quadratic forms or linear combinations of the quadratic forms. We also compare our results to the existing sufficient conditions in the literature. Finally, we extend our results to derive a new sufficient condition for the exactness of the SDP relaxation of general diagonal quadratically constrained quadratic programs, where each quadratic function is associated with a diagonal matrix. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11801087, 12171100, 72161160340]; Hong Kong Research Grants Council [Grants 14213716 and 14202017]; and Natural Science Foundation of Shanghai [Grant 22ZR1405100].

Interval estimator for uncertain descriptor systems: a block diagonal transformation‐based solution

AbstractThis paper investigates an interval observer (IO) design technique for a class of continuous‐time descriptor systems with bounded uncertainties. By adopting a block diagonal transformation, these constraints on the descriptor system cooperativity and the system sensor noise are well relaxed under a framework of the descriptor IO dynamic structure. Moreover, the sufficient existence condition of the descriptor IO for any descriptor system more than just the detectable or regular ones is formulated as the rank condition. By means of the elementary transformation, the rank condition is further transformed into a set of constrained matrix equations, and a valid constructive design algorithm is proposed under the stepwise parameter design process. Finally, the correctness and superiority of the obtained methods are illustrated through two simulation examples.

Diagonal nonlinear transformations preserve structure in covariance and precision matrices

For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision matrices reveal correlations and so-called partial correlations between variables, but these do not, in general, have any correspondence with respect to independence properties. In this paper, we prove that, for a certain class of non-Gaussian distributions, these correspondences still hold, exactly for the covariance and approximately for the precision. The distributions—sometimes referred to as “nonparanormal”—are given by diagonal transformations of multivariate normal random variables. We provide several analytic and numerical examples illustrating these results.

Augmenting the detection of online image theft using diagonal transformation and iris biometric watermarking

Augmenting the detection of online image theft using diagonal transformation and iris biometric watermarking

Illumination correction with optimized kernel extreme learning machine based on improved marine predators algorithm

AbstractAiming at the problem of poor image illumination correction accuracy, a kernel extreme learning machine optimized based on the differential evolution‐improved marine predators algorithm is proposed to estimate the scene illumination information and restore the image. First, in order to solve the problem of randomization of the initial population of the marine predators algorithm, differential evolutions is used to provide a set of suitable initial populations to the algorithm. Then, an improved algorithm is used to optimize the weight and bias of the kernel extreme learning machine to solve the problem of the randomness of the weight and bias, so that the learning machine converges to the global optimal value and avoids the unstable predictions. Finally, after obtaining the predicted illumination information, the image model is corrected to the image effect under standard illumination through diagonal transformation. It can be seen from the experimental results that the best predictive value of the differential evolution marine predator algorithm learning machine is 0.0135915, and the quasi‐bias is only 0.001687. Compared with traditional illumination estimation algorithms such as random vector functional link and extreme learning machine, the differential evolution marine predator algorithm learning machine algorithm proposed in this article has better stability, higher accuracy of calculated predicted values, and better image restoration effect.

Sliding Mode Control for Singularly Perturbed Systems Using Accurate Reduced Model

In order to deal with unmodeled dynamics in large vehicle systems, which have an ill condition of the state matrix, the use of model order reduction methods is a good approach. This article presents a new construction of the sliding mode controller for singularly perturbed systems. The controller design is based on a linear diagonal transformation of the singularly perturbed model. Furthermore, the use of a single sliding mode controller designed for the slow component of the diagonalized system is investigated. Simulation results indicate the performance improvement of the proposed controllers.

Techniques to Reduce π/4-Parity-Phase Circuits, Motivated by the ZX Calculus

To approximate arbitrary unitary transformations on one or more qubits, one must perform transformations which are outside of the Clifford group. The gate most commonly considered for this purpose is the T = diag(1, exp(i \pi/4)) gate. As T gates are computationally expensive to perform fault-tolerantly in the most promising error-correction technologies, minimising the "T-count" (the number of T gates) required to realise a given unitary in a Clifford+T circuit is of great interest. We describe techniques to find circuits with reduced T-count in unitary circuits, which develop on the ideas of Heyfron and Campbell [arXiv:1712.01557] with the help of the ZX calculus. Following [arXiv:1712.01557], we reduce the problem to that of minimising the T count of a CNOT+T circuit. The ZX calculus motivates a further reduction to simplifying a product of commuting "\pi/4-parity-phase" operations: diagonal unitary transformations which induce a relative phase of exp(i \pi/4) depending on the outcome of a parity computation on the standard basis (which motivated Kissinger and van de Wetering [1903.10477] to introduce "phase gadgets"). For a number of standard benchmark circuits, we show that these techniques -- in some cases supplemented by the TODD subroutine of Heyfron and Campbell [arXiv:1712.01557] -- yield T-counts comparable to or better than the best previously known results.

Accelerating the Sinkhorn–Knopp iteration by Arnoldi-type methods

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.

Generalized P and CP transformations in the three-Higgs-doublet model

We study generalised P and CP transformations in the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We find that there are two equivalence classes, with respect to flavour transformations, of generalised P transformations and there is only one class of CP transformations. We discuss the conditions the potential has to satisfy in order to be invariant under these transformations. We apply the method of bilinears which we briefly review. We discuss the relation to the conventional basis, where the potential is written in terms of scalar products of the doublet fields. In particular we reproduce the known result that a potential is invariant under CP transformations if and only if there is a conventional basis where all parameters are real. Eventually we study standard P and CP transformations in the $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDM the standard CP transformation corresponds to a diagonal linear transformation with only $\pm 1$ as diagonal elements. We give this matrix explicitly for arbitrary $n$.