Block-diagonal Research Articles

New lower bounds on crossing numbers of Km,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{m,n}$$\\end{document} from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph Km,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{m,n}$$\\end{document}, extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr(K10,n)≥4.87057n2-10n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ extrm{cr}}\\limits (K_{10,n}) \\ge 4.87057 n^2 - 10n$$\\end{document}, cr(K11,n)≥5.99939n2-12.5n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ extrm{cr}}\\limits (K_{11,n}) \\ge 5.99939 n^2-12.5n$$\\end{document}, cr(K12,n)≥7.25579n2-15n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathop {\ extrm{cr}}\\limits (K_{12,n}) \\ge 7.25579 n^2 - 15n$$\\end{document}, cr(K13,n)≥8.65675n2-18n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ extrm{cr}}\\limits (K_{13,n}) \\ge 8.65675 n^2-18n$$\\end{document} for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

Block Diagonalization in the 5G SA Network

In this paper, we did programming regarding the Block diagonalization technology in the 5G standalone SA network, in this program, we have created a 5G site with 16 antennas(minimum of Massive MIMO) and 4 active users equipped of 4 antennas, this system is called Multi Users Massive MIMO system, the link that was chosen is the downlink,we have calculated the maximum throughput in the 5G downlink where we have obained a value of 1673864 b/ms, this value is divided by the number of Massive MIMO layers which worth 16 to get a transport block size of 104616 b/ms (no Cyclic redundancy check CRC). The Block Error rate BLER is null (no detection of errors in reception) because we are in the case of no crc and no channel coding (uncoded transmission), the signal of each user among 4 to be transmitted consists of 4 vectors, each vector has a length of 52308 that corresponds to the number of symbols which are the outputs of Quadrature Phase Shift Keying QPSK Mapping Operation. The received signal at each user equipment UE has a form which can be represented by the multiplication of preconding matrix of this UE with the channel matrix between this UE and the 5G site plus the noise received at the antennas of this UE. the results show that the product of channel gain between UE and the 5G site(known in emission) with the precoding matrix of the other UE gives a matrix which composes of imaginary elements each of which has a real part and imaginary part which both tend to zero(the inter users interferences IUI is canceled). The results show also that when the Signal to Noise Ratio SNR increases(several transmissions) the Bit Error Rate decreases.

The universal envelope of a polarized anti-Jordan triple system of block matrices

We investigate the universal associative envelope [Formula: see text] of the simple polarized anti-Jordan triple system [Formula: see text] of [Formula: see text] [Formula: see text] block matrices: [Formula: see text] with the triple product [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic 0. We prove that [Formula: see text], where [Formula: see text] is the ordinary associative algebra of all [Formula: see text] matrices over [Formula: see text]. From this result we classify up to equivalence all finite-dimensional irreducible representations of the anti-Jordan triple system [Formula: see text]. We conclude the paper with an open problem for further research.

Upper bounds of spectral radius of symmetric matrices and graphs

The spectral radius ρ(A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2×2 block matrix A=(0A12A21A22), then ρ(A)2≤ρ22+θ⁎, where θ⁎ is the largest root of the equation μ2=(x−ν)2(ρ22+x) and ρ2=ρ(A22), μ=ρ(A12A22A21), ν=ρ(A12A21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results.

Revisiting thread configuration of SpMV kernels on GPU: A machine learning based approach

Sparse matrix-vector multiplication (SpMV) optimization on GPUs has been challenging due to irregular memory accesses and unbalanced workloads. The majority of existing solutions assign a fixed number of threads to one or more rows of sparse matrices according to empirical formulas. However, this method does not give the optimal thread configuration and results in a significant performance loss. This paper proposes a new machine learning-based thread assignment strategy for SpMV on GPU, predicting the near-optimal thread configuration for matrices. Further, we partition irregular sparse matrices into blocks according to the distribution of non-zero elements and predict the optimal thread configuration for each block. A new SpMV kernel is designed to accelerate the execution of different blocks. Experimental results show that our machine learning-based approach can select the near-optimal thread configuration for most matrices. The efficiency of SpMV for irregular matrices is also improved by matrix partitioning and blockwise prediction. Finally, we dive into the trained model to find out the connection between the features of a sparse matrix and its optimal thread configuration.

Covariance matrix estimation method of FDA-MIMO radar with low snapshot size

Frequency diverse array and multi-input and multi-output (FDA-MIMO) radar have been applied to many fields due to its angle-range-dependent beampattern. A part of the related array processing algorithms can be successfully used on the basis of the known covariance matrix (CM). The estimation accuracy of the CM directly influences the algorithm performance. In particular, the estimation performance of the sample covariance matrix (SCM) will be degraded shapely once the sample size is less than the channel number. Aiming to improve the estimation accuracy of the SCM with FDA-MIMO radar, we propose a novel shrinkage-to-tapering (ST)-based method combined with the block Toeplitz rectification (STT). Firstly, each block matrix in the CM is processed by using the Toeplitz rectification method, and then, three different estimation matrix structures based on the ST method are proposed. Next, by plugging the unbiased estimators into the optimal shrinkage coefficient and normalized mean square error (MSE), their closed forms of their estimators can be given. The numerical simulation results demonstrate that the proposed three different STT approaches are superior to a number of existing methods in terms of CM estimation performance, and the estimation performance is related with the choice of tapering matrix.

Simulation of SMA-based engineering applications considering large displacement and rotation, thermomechanical coupling and partial phase transformation

Shape memory alloys (SMAs) undergo large recoverable deformation due to their inherent diffusionless martensitic phase transformation. Two factors need to be taken into account to incisively emulate the behaviour of SMA based structures: (i) consideration of large displacement and rotation (LDR) effect, and (ii) capturing the consequence of transformation induced material level coupling. In this study, both these effects are apprehended by extending the infinitesimal strain-based constitutive model of SMA as proposed by Lagoudas et al. (2012), assuming small elastic strain but with finite inelastic strain. To preclude any spurious stress generation out of large rotation, the Jaumann stress rate is used, and incremental objectivity for finite rotation between two successive time instants is conserved by rotating the strain increment and spin tensor to the rotation-independent mid-point configuration following Hughes–Winget (Hughes and Winget, 1980) algorithm. In addition, the contribution of thermoelastic and latent heat evolved during transformation is taken into consideration in the thermal equilibrium equation. Moreover, the effect of partial phase transformation yielding minor hysteresis loop has also been captured. The equilibrium equation is expressed in the current configuration following the Updated Lagrangian formulation. The mechanical and thermal equilibrium equations are solved concurrently in block matrix form using the Newton–Raphson (NR) iterative technique, considering the coupling terms resulting from the phase transformation. The efficacy and robustness of the developed finite element model are corroborated through various practical applications of SMA-based members, e.g., SMA ring, SMA-actuated beam, morphing of corrugated airfoil, orthodontic palatal expander, compliant gripper, undergoing LDR while subjected to different thermomechanical loading conditions. The combined effects of LDR along with thermomechanical coupling yield a stiffening behaviour during loading and sluggish response at the time of thermal recovery.

Autologous tooth bone graft block compared with advanced platelet-rich fibrin in alveolar ridge preservation: A clinico-radiographic study.

To determine the clinico-radiographic efficiency of partially demineralized dentin matrix block (PDDM block), a mixture of PDDM with advanced-platelet-rich fibrin+ (A-PRF+) and injectable platelet-rich fibrin versus A-PRF+ alone in alveolar socket preservation. Sixteen molar teeth indicated for extraction were randomly assigned into two groups. For the test group, sockets were packed with PDDM block and control group, with A-PRF+ plug alone. Clinical and radiographic cone-beam computed tomography methods were used to assess the horizontal and vertical ridge dimensional changes at baseline and 4 months. Clinically, the mid buccal and palatal crestal height (10.25 ± 0.86 and 9.75 ± 0.28 mm) and alveolar ridge width (11.37 ± 0.25 mm) were significantly higher in the test group as compared to the control group, 4 months after tooth extraction (P < 0.01). Radiographically, there was improved apposition and nonsignificant resorption for the test group in ridge height and width, whereas statistically significant higher resorption was seen in the control group at 4 months. The application of the PDDM block demonstrated efficacy in maintaining the dimensions of the extraction socket when compared to A-PRF+ alone. This autologous and immune-free regenerative biomaterial is widely obtainable, offering a glimpse into the potential of next-generation biofuels for regeneration.

The Block Landweber Iterative Method for Light Field Reconstruction from a Focal Stack

Light field imaging involves reconstructing a 4D light field from a 3D focal stack, which makes it challenging to reconstruct the light field from incomplete projection data. To address this problem, a linear projection system is established to model the focal stack imaging process using discrete refocusing equations. Based on this system, we propose the block Landweber iterative method to find the least-squares solution. This method computes the sparse matrix while iterating, which overcomes the problem of data storage. The 2-norm of the block matrix is utilized as the weighted matrix to normalize every block matrix on an identical scale, delivering an effective relaxation strategy under the convergence condition in the inconsistent case, which yields better reconstruction results and accelerates the convergence speed. The experimental results based on the image quality assessments of reference and non-reference images show that our method achieved better reconstruction results compared to other relevant common methods, even with fewer focal stacks and higher angle resolution.

Age Effects on Subtest and Composite Scores of the Wechsler Abbreviated Scale of Intelligence-Second Edition.

We evaluated age effects in the Wechsler Abbreviated Scale of Intelligence-Second Edition (WASI-II) standardization sample. This extends work completed using previous editions of the Wechsler Adult Intelligence Scales. Of the four subtests, Vocabulary (VC) and Similarities (SI) were most resistant to aging. VC showed minimal variation regardless of age; SI peaked at 30 to 54 years followed by a decline. Block Design (BD) and Matrix Reasoning (MR) showed substantial drops from the younger to older groups. BD peaked at 17 to 44 years and then declined; MR peaked at 20 to 29 years and then consistently deteriorated. The WASI-II Verbal Comprehension Index peak at 30 to 44 years was followed by a maximum drop at 85 to 90 years. The Perceptual Reasoning Index peaked at 20 to 29 years, with a marked decline by 65 to 69 years. The Full Scale IQ was average until age 65 years followed by a decline. Minor changes in points of peak performance and subsequent decline were seen as a function of Full Scale IQ level. Results were consistent with crystallized and fluid intelligence theory.

A between-cluster approach for clustering skew-symmetric data

In order to investigate exchanges between objects, a clustering model for skew-symmetric data is proposed, which relies on the between-cluster effects of the skew-symmetries that represent the imbalances of the observed exchanges between pairs of objects. The aim is to detect clusters of objects that share the same behaviour of exchange so that origin and destination clusters are identified. The proposed model is based on the decomposition of the skew-symmetric matrix pertaining to the imbalances between clusters into a sum of a number of off-diagonal block matrices. Each matrix can be approximated by a skew-symmetric matrix by using a truncated Singular Value Decomposition (SVD) which exploits the properties of the skew-symmetric matrices. The model is fitted in a least-squares framework and an efficient Alternating Least Squares algorithm is provided. Finally, in order to show the potentiality of the model and the features of the resulting clusters, an extensive simulation study and an illustrative application to real data are presented.

Covariance‐based soft clustering of functional data based on the Wasserstein–Procrustes metric

AbstractWe consider the problem of clustering functional data according to their covariance structure. We contribute a soft clustering methodology based on the Wasserstein–Procrustes distance, where the in‐between cluster variability is penalized by a term proportional to the entropy of the partition matrix. In this way, each covariance operator can be partially classified into more than one group. Such soft classification allows for clusters to overlap, and arises naturally in situations where the separation between all or some of the clusters is not well‐defined. We also discuss how to estimate the number of groups and to test for the presence of any cluster structure. The algorithm is illustrated using simulated and real data. An R implementation is available in the Appendix .

Response to "Comment on The novel diagonal suprascapular canal block for shoulder surgery analgesia: a comprehensive technical report".

Response to "Comment on The novel diagonal suprascapular canal block for shoulder surgery analgesia: a comprehensive technical report".

Theoretical analysis about model simplifications and boundary conditions on two-phase flow instability of parallel heated tube systems

Flow instability may happen in both subcritical two-phase flow boiling systems (such as U tube natural circulation steam generator of Pressurized Water Reactor, once through steam generator of High-Temperature Gas-Cooled Reactor, etc) and supercritical evaporation systems. In previous theoretical researches, the model of a single heated tube (SHT) with constant pressure drop boundary condition (ConPD) was always used to investigate the system stability. However, in real applications, there are usually multiple parallel heated tubes (MHT) and unheated connection regions before the branch heated tubes. The system is usually driven by a pump, and there is always a long unheated main pipe and throttling between the heated tubes (boiler or steam generator) and the pump. However, there lacks rigorous analysis about the effects of these model simplifications and boundary conditions. Take superheated two-phase flow boiling systems as examples, the theoretical models with different flow-driven boundary conditions (ConPD, constant mass flow rate (ConMF), and pump driven (PumpD)), different numbers of heat tubes (SHT, MHT), with or ignoring the unheated region (UHR), with or ignoring the bypass of a pump, with identical or un-identical geometry parameters and operation parameters are established. Linear ordinary differential equations describing the instability are obtained. Eigenvalues of the coefficient matrix of linear ordinary differential equations, and the property of the block matrix and the similarity matrix are used to analyze the effects of the above boundary conditions and model simplifications on the system stability. Two new dimensionless numbers, dimensionless pump number (Npump) and dimensionless bypass number (Nbypass), are proposed to describe the effect of pump and its bypass on stability quantitatively. The effect of the pump bypass on the stability can be attributed to the decrease of Npump. No matter ConPD or PumpD is adopted, SHT considering UHR is proved to be more stable than SHT ignoring UHR. No matter UHR is considered or ignored, SHT with PumpD is proved to be more stable than SHT with ConPD. No matter UHR of MHT at the main pipe is considered or ignored, the stability of MHT with ConPD or ConMF or PumpD is proved to be the same as that of SHT with ConPD ignoring UHR. When ConPD ignoring UHR is adopted, the stability of MHT with un-identical geometry and operation parameters is proved to be the same as that of its most unstable heated tube. The above conclusions about the effect of model simplification and boundary condition are obtained from superheated two-phase flow boiling systems, they also hold for saturated two-phase flow boiling systems and supercritical evaporation systems.

Positive semidefinite interval of matrix pencil and its applications to the generalized trust region subproblems

We are concerned with finding the set I⪰(A,B) of real values μ such that the matrix pencil A+μB is positive semidefinite. If A,B are not simultaneously diagonalizable via congruence (SDC), I⪰(A,B) either is empty or has only one value μ. When A,B are SDC, I⪰(A,B), if not empty, can be a singleton or an interval. Especially, if I⪰(A,B) is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval I≻(A,B). If A,B are both singular, then even I⪰(A,B) is an interval, its interior may not be I≻(A,B), but A,B are then decomposed as block diagonals of submatrices A1,B1 with B1 nonsingular such that I⪰(A,B)=I⪰(A1,B1). Applying I⪰(A,B), the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.

The Concavity of Conditional Maximum Likelihood Estimation for Logit Panel Data Models with Imputed Covariates

In estimating logistic regression models, convergence of the maximization algorithm is critical; however, this may fail. Numerous bias correction methods for maximum likelihood estimates of parameters have been conducted for cases of complete data sets, and also for longitudinal models. Balanced data sets yield consistent estimates from conditional logit estimators for binary response panel data models. When faced with a missing covariates problem, researchers adopt various imputation techniques to complete the data and without loss of generality; consistent estimates still suffice asymptotically. For maximum likelihood estimates of the parameters for logistic regression in cases of imputed covariates, the optimal choice of an imputation technique that yields the best estimates with minimum variance is still elusive. This paper aims to examine the behaviour of the Hessian matrix with optimal values of the imputed covariates vector, which will make the Newton–Raphson algorithm converge faster through a reduced absolute value of the product of the score function and the inverse fisher information component. We focus on a method used to modify the conditional likelihood function through the partitioning of the covariate matrix. We also confirm that the positive moduli of the Hessian for conditional estimators are sufficient for the concavity of the log-likelihood function, resulting in optimum parameter estimates. An increased Hessian modulus ensures the faster convergence of the parameter estimates. Simulation results reveal that model-based imputations perform better than classical imputation techniques, yielding estimates with smaller bias and higher precision for the conditional maximum likelihood estimation of nonlinear panel models.

Comment on "The novel diagonal suprascapular canal block for shoulder surgery analgesia: a comprehensive technical report".

Comment on "The novel diagonal suprascapular canal block for shoulder surgery analgesia: a comprehensive technical report".

A novel color image tampering detection and self-recovery based on fragile watermarking

In industrial applications, sensing devices' image data must be accurately identified and recovered to prevent economic losses. Nevertheless, most industrial image watermarking methods focus on grayscale images with difficulty in achieving superior performance in three parameters: watermarked image quality, tamper detection and content recovery quality. To resolve this issue, a color image tampering detection and self-recovery method based fragile watermarking scheme is proposed in this paper. In the proposed scheme, block-based regular marker with pixel-based continuous marker is used for watermark embedding to enhance the quality of watermarked images. Then, a feature extraction-based tampering detection scheme for diagonal blocks combined with three-layer authentication is developed to resist tampering attacks of diverse shapes and sizes. Additionally, the block pixel-level recovery mechanism along with the Smoothing Inpainting algorithm is designed to implement the high-quality recovery of tampered images. Compared with recent methods, experimental results demonstrate the superiority of the scheme on tamper detection and image self-recovery, in terms of the image recovery robustness, tamper localization and attack resistance. Our scheme has a PSNR value of 25.4921 dB and an SSIM value of 0.7728 for the recovered image at 90% tampering rate.

Relaxation strategy for the block Landweber method

The block Landweber iteration is a general method for the solution of linear systems which is widely applied for image reconstructions. The convergence behavior of the block Landweber iteration is of both theoretical and practical importance. When the linear system is consistent, we derive a neat iterative representation and find an optimal relaxation coefficient by minimizing the 2-norm of the newly derived iteration matrix in the range of the conjugate transpose of the corresponding block matrix. We also establish the accelerating convergence relaxation strategies based on the maximum eigenvalue of each block matrix. When the linear system is inconsistent, we normalize each block matrix by the 2-norms of the each weighted block matrix at a uniform scale to give the relaxation strategies. The numerical simulations show that the relaxation strategies proposed for the block Landweber iteration in both consistent and inconsistent cases in this paper have the better reconstruction performances. Compared with the Landweber iteration, the block Landweber iteration has superiority in reconstruction results.

Image encryption algorithm based on hyperchaotic system and dynamic DNA encoding

Since the existing DNA encryption algorithms only have fixed DNA coding and decoding rules and a single algorithm that cannot meet more complex and more secure encryption requirements, an image encryption algorithm based on hyperchaotic system and dynamic DNA coding is proposed. The algorithm proposes two new operation methods in the DNA operation rules, namely the equal-additive column transformation and the equal-subtractive column transformation. which combine the SHA-256 function and the external key to generate the initial value of the chaotic system to ensure that key and plaintext of algorithm are correlated. In the encryption process, the plaintext image is firstly converted into a two-dimensional matrix for rotation and scrambling. Then the chaotic sequence generated by the hyperchaotic Chen system is used to perform DNA dynamic encoding, decoding and operation on the scrambled matrix and block matrix generated by logistic chaotic sequence. Finally, the encrypted matrix is scrambled in one dimension. Simulation results show that the improved encryption algorithm has a larger key space, breaks the strong correlation between image pixel layers, and can effectively resist multiple attacks.