Columns Of Matrix Research Articles

Formation of a dislocation dipole in the consecutive interfaces of two nanowires

The formation of a dipole of edge dislocations in the consecutive interfaces of two neighboring square-shaped nanowires has been investigated from an energy-based calculation. Assuming the dipole is nucleated in the center of the matrix row between the two nanowires, the energy variation induced by the climbing of the dislocations in the interfaces (one by interface) has been first determined. The energy barrier and unstable equilibrium positions of the dipole have been thus characterized. Considering then the dislocations can also glide into the interfaces, the resulting supplementary gliding energy variation has been considered and the dislocation stable equilibrium positions in the interfaces have been determined, assuming they are distributed symmetrically with respect to the central point of the structure. The effect of the width of the matrix row between the two nanowires on the critical misfit strain required for the introduction of the dislocations in the nanowire interfaces has been finally analyzed.

Pairwise comparison matrices with uniformly ordered efficient vectors

Our primary interest is understanding reciprocal matrices all of whose efficient vectors are ordinally the same, i.e., there is only one efficient order (we call these matrices uniformly ordered, UO). These are reciprocal matrices for which no efficient vector produces strict order reversals. A reciprocal matrix is called column ordered (CO) if each column is ordinally the same. Efficient vectors for a CO matrix with the same order of the columns always exist. For example, the entry-wise geometric mean of some or all columns of a reciprocal matrix is efficient and, if the matrix is CO, has the same order of the columns. A necessary, but not sufficient, condition for UO is that the matrix be CO and then the only efficient order should be satisfied by the columns (possibly weakly). In the case n=3, CO is necessary and sufficient for UO, but not for n>3. We characterize the 4-by-4 UO matrices and identify the three possible alternate orders when the matrix is CO (and give entry-wise conditions for their occurrence). We also describe the simple perturbed consistent matrices that are UO. Some of the technology developed for this purpose is of independent interest.

Clustering of high-dimensional observations

We present a novel clustering method for high-dimensional, low sample size (HDLSS) data. The method is distance-based, takes advantage of the distance concentration phenomenon and the limiting values of the dissimilarity indices to construct clusters. We describe an algorithm that orders each row of the dissimilarity matrix to estimate the change points, which define cluster boundaries. We construct an agreement matrix of the Rand indices of the row clusters. The minimum of the row sum of the agreement matrix provides us with the best clusters. We prove that the new method achieves perfect clustering as the number of features diverges for a fixed sample size. Several examples are presented to illustrate the proposed method. We compare the new method with four other clustering techniques, including high-dimensional k-means, minimal spanning tree and Hierarchical Scan. The clustering methods are applied to the Lymphoma data set.

DOCSIC: A Mean-Field Method for Orbital-by-Orbital Self-Interaction Correction.

We introduce a new method to remove the one-electron self-interaction error in approximate density functional calculations on an orbital-by-orbital basis, as originally proposed by Perdew and Zunger [Phys. Rev. B 1981, 23, 5048]. This method is motivated by a recent proposal by Pederson et al. [J. Chem. Phys. 2014, 140, 121103] to remove self-interaction that employs orbitals derived from the real-space density matrix, known as FLOSIC (Fermi Löwdin orbitals self-interaction correction). However, instead of Fermi Löwdin orbitals, our scheme utilizes columns of the density matrix to determine localized orbitals, like the localization procedure proposed by Fuemmeler et al. [J. Chem. Theory Comput. 2023, 19, 8572]. The new method, dubbed DOCSIC for density matrix as orbital coefficients self-interaction correction, contrasts with traditional Perdew-Zunger or FLOSIC in that it does not incorporate additional optimization parameters, and, unlike the average density self-interaction correction of Ciofini et al. [Chem. Phys. Lett. 2003, 380, 12], it makes use of localized orbitals. Another advantage of DOCSIC is that it can be implemented as a mean-field formalism. We show details of the self-consistent generalized Kohn-Sham implementation, some illustrative results, and we finally highlight its advantages and limitations.

Errorless robust JPEG steganography using steganographic polar codes

Recently, a robust steganographic algorithm that achieves errorless robustness against JPEG recompression has been proposed. The method employs a lattice embedding scheme and utilizes the syndrome-trellis code (STC) for practical embedding. However, we have noticed that errorless robust embedding with STC may encounter failures due to modifications on wet coefficients, especially when a high quality factor is used by the compression channel. To solve this problem, we have discovered that using steganographic polar code (SPC) for embedding has better performance in avoiding modifications on wet coefficients. In this paper, we conduct theoretical analysis to prove the better performance of SPC in wet paper embedding. We establish the condition of avoiding modifications on wet coefficients, followed by presenting a recursive calculation method for determining the distribution of columns in the generator matrix of SPC. The findings reveal that SPC can avoid modifications on wet coefficients under a larger number of wet coefficients compared with STC, and therefore we propose a better errorless robust embedding method employing SPC. The experimental results demonstrate that under close security performance, the proposed method achieves a higher success rate compared with embedding with STC. Specifically, when the quality factor of the compressor is 95 and the payload size is 0.4 bpnzac, our method achieves a success rate of 99.85%, surpassing the 91.95% success rate of the embedding with STC.

Selecting the number of factors in approximate factor models using group variable regularization

. We propose a novel method for the estimation of the number of factors in approximate factor models. The model is based on a penalized maximum likelihood approach incorporating an adaptive hierarchical lasso penalty function that enables setting entire columns of the factor loadings matrix to zero, which corresponds to removing uninformative factors. Additionally, the model is capable of estimating weak factors by allowing for sparsity in the non zero columns. We prove that the proposed estimator consistently estimates the true number of factors. Simulation experiments reveal superior selection accuracies for our method in finite samples over existing approaches, especially for data with cross-sectional and serial correlations. In an empirical application on a large macroeconomic dataset, we show that the mean squared forecast errors of an approximate factor model are lower if the number of included factors is selected by our method.

Strong consistency of an estimator by the truncated singular value decomposition for an errors-in-variables regression model with collinearity

In this paper, we prove strong consistency of an estimator by the truncated singular value decomposition for a multivariate errors-in-variables linear regression model with collinearity. This result is an extension of Gleser's proof of the strong consistency of total least squares solutions to the case with modern rank constraints. While the usual discussion of consistency in the absence of solution uniqueness deals with the minimal norm solution, the contribution of this study is to develop a theory that shows the strong consistency of a set of solutions. The proof is based on properties of orthogonal projections, specifically properties of the Rayleigh-Ritz procedure for computing eigenvalues. This makes it suitable for targeting problems where some row vectors of the matrices do not contain noise. Therefore, this paper gives a proof for the regression model with the above condition on the row vectors, resulting in a natural generalization of the strong consistency for the standard TLS estimator.

Structural design and surface networking analysis of a truss antenna based on hexagonal frustum deployable mechanism units

With the development of space technology, deployable mechanism has been widely applied in large aperture space antennas. In this paper, a hexagonal frustum deployable antenna unit and an innovative modular assembling concept are proposed for large parabolic truss antenna. Screw topology diagram and screw constraint matrix are established to solve degree of freedom (DOF) of the antenna mechanism. By separating column vectors of screw constraint matrix, whole actuations of antenna mechanism are obtained according to unique solution condition. A new hexagon antenna network method based on hexagon frustum units and rotation transformation matrix is proposed, and correctness and efficiency of the modeling principle are proved by simulation. Finally, a prototype of the proposed hexagonal frustum truss deployable antenna mechanism is constructed, and a deployment experiment is conducted, which demonstrate the mobility and deployment performance of the mechanism. The mechanism proposed in this paper provides a theoretical reference for the parabolic truss deployable antenna mechanism, which is potentially valuable in the application of large aperture deployable antennas.

Clustering Longitudinal Data for Growth Curve Modelling by Gibbs Sampler and Information Criterion

Clustering longitudinal data for growth curve modelling is considered in this paper, where we aim to optimally estimate the underpinning unknown group partition matrix. Instead of following the conventional soft clustering approach, which assumes the columns of the partition matrix to have i.i.d. multinomial or categorical prior distributions and uses a regression model with the response following a finite mixture distribution to estimate the posterior distribution of the partition matrix, we propose an iterative partition and regression procedure to find the best partition matrix and the associated best growth curve regression model for each identified cluster. We show that the best partition matrix is the one minimizing a recently developed empirical Bayes information criterion (eBIC), which, due to the involved combinatorial explosion, is difficult to compute via enumerating all candidate partition matrices. Thus, we develop a Gibbs sampling method to generate a Markov chain of candidate partition matrices that has its equilibrium probability distribution equal the one induced from eBIC. We further show that the best partition matrix, given a priori the number of latent clusters, can be consistently estimated and is computationally scalable based on this Markov chain. The number of latent clusters is also best estimated by minimizing eBIC. The proposed iterative clustering and regression method is assessed by a comprehensive simulation study before being applied to two real-world growth curve modelling examples involving longitudinal data clustering.

Discovering the signal subgraph: An iterative screening approach on graphs

Supervised learning on graphs is a challenging task due to the high dimensionality and inherent structural dependencies in the data, where each edge depends on a pair of vertices. Existing conventional methods are designed for standard Euclidean data and do not account for the structural information inherent in graphs. In this paper, we propose an iterative vertex screening method to achieve dimension reduction across multiple graph datasets with matched vertex sets and associated graph attributes. Our method aims to identify a signal subgraph to provide a more concise representation of the full graphs, potentially benefiting subsequent vertex classification tasks. The method screens the rows and columns of the adjacency matrix concurrently and stops when the resulting distance correlation is maximized. We establish the theoretical foundation of our method by proving that it estimates the true signal subgraph with high probability. Additionally, we establish the convergence rate of classification error under the Erdos-Renyi random graph model and prove that the subsequent classification can be asymptotically optimal, outperforming the entire graph under high-dimensional conditions. Our method is evaluated on various simulated datasets and real-world human and murine graphs derived from functional and structural magnetic resonance images. The results demonstrate its excellent performance in estimating the ground-truth signal subgraph and achieving superior classification accuracy.

Pengembangan Board Game Catur Matematika Sebagai Media Pembelajaran Pada Materi Matriks Serta Barisan Dan Deret

Nowadays, technological developments are very fast and sophisticated. This development has an impact on interactive learning in schools. Therefore, learning media is needed that is in line with the development of this technology. The goal is to make students more enthusiastic again. The aims of this research are: (1) to determine students' activeness in teaching and learning activities in mathematics for class XI students at SMAS Budaya Jakarta. (2) can produce interactive learning through this board game learning media with matrix material and rows and rows for class XI students at SMAS Budaya Jakarta. This development research method uses the ADDIE model analysis steps (Analysis, Design, Development, Implementation and Evaluation) because this step has an innovative and systematic system, thereby enabling researchers to achieve research objectives.

Sequentially extending space-filling experimental designs by optimally permuting and stacking columns of the design matrix

Researchers make available computationally expensive designs for computer experiments for widespread use by cataloging them and providing online links. This paper presents an algorithm that augments space-filling designs (SFDs) by optimally permuting and stacking columns of the design matrix to minimize the maximum absolute pairwise correlation among columns in the new extended design. The algorithm enables researchers to augment an SFD sequentially with batches of additional design points, which improves column orthogonality and adds more degrees of freedom for fitting metamodels. We show this method improves the correlation and space-filling properties of the resulting designs and allows us to extend some classes of designs to higher dimensions that are not easily obtainable. Moreover, the resulting extended designs compare well with many leading software-generated SFDs created from scratch in the extended design space.

A study on partial pivot ACA boundary element method for elasticity problems

The boundary element method (BEM) employing the partial pivot adaptive cross approximation (PACA) algorithm has been observed to experience convergence failures and reduced solution accuracy when solving elasticity problems, especially at large scales. To address this issue, this paper proposes an improved algorithm for both 2D and 3D elasticity problems.Our investigations revealed that when the normal of an element is parallel to the coordinate axes and the element is in the same plane as the source point, zero entries with regular distributions will appear in the coefficient matrix. This results in the premature satisfaction of the convergence criterion of the PACA algorithm, leading to the omission of some rows and columns of the coefficient matrix. To address this issue, this paper proposes improvements to the PACA algorithm through the Coordinate Rotation Method (CRM) and the Component Traversal Method (CTM), and validates the effectiveness of these two improved methods.Furthermore, we investigate reasons behind the decrease in accuracy when employing the CTM to solve large-scale problems. We attribute this to the oscillatory nature of the estimated relative error and propose the modified PACA (M-PACA) algorithm. M-PACA not only maintains the advantages of the PACA algorithm in reducing storage space and accelerating coefficient matrix generation but also addresses its limitations, ensuring more accurate and reliable solutions for elasticity problems.

Explainable graph clustering via expanders in the massively parallel computation model

Explainable clustering provides human-understandable reasons for decisions in black-box learning models. In a previous work, a decision tree built on the set of dimensions was used to define ranges of values for k-means clusters. For explainable graph clustering, we use expander graphs instead of dense subgraphs since powering an expander graph is guaranteed to result in a clique after at most a logarithmic number of steps.Consider a set of multi-dimensional points labeled with k labels. We introduce the heat map sorting problem as reordering the rows and columns of an input matrix (each point is a column and each row is a dimension) such that the labels of the entries of the matrix form connected components (clusters). A cluster is preserved if it remains connected, i.e., if it is not split into several clusters and no two clusters are merged. In the massively parallel computation model (MPC), each machine has a sublinear memory and the total memory of the machines is linear.We prove the problem is NP-hard. We give a fixed-parameter algorithm in MPC and an approximation algorithm based on expander decomposition. We empirically compare our algorithm with explainable k-means on several graphs of email and computer networks.

Secure Stego Method to Protect SMS Transmission

A novel method of message steganography will be proposed. The method will use a digital speech file as a covering media. The 32 binary bits of each speech sample will be used for data hiding, using these bits will increase the hiding capacity, and changing them will not much affect the sample value, thus the quality of the stego file will be always high. The proposed method will use a private key to protect the hidden message, the protection will be applied by shuffling the rows of the speech binary matrix using the contents of the generated indices key, The induces key will be generated by running a chaotic logistic map model using the values of the chaotic parameters included in the private key. The proposed method will be tested and implemented using various messages, the quality, sensitivity and speed of the proposed method will be analyzed to prove the enhancements provided by the proposed method.

Conditions for The Saturation of a Matrix Inequality

Matrix inequalities play a vital role in the study of linear algebra and information theory. In this paper, the author investigates the condition when a proven matrix inequality is saturated, and Pi’s are linearly independent matrices of the same size, and Si’s are also linearly independent matrices of the same size. The author constructs the conditions when Pi is a column vector or a 2 2 matrix, and Si is an arbitrary square matrix. The author finds a very potential result when investigating the column situation. The author also gets massive result in the 2 2 situation. In particular, the inequality is not saturated when Pi is a 2 2 matrix and K=2 or 3 and it could be saturated under the situation when K=4 if only if two certain matrices can enable the product of and form two established matrices. At the end of the essay, the author tries some more complex cases such as 2 3 matrices Pi when K=2.

The Algorithm of Gu and Eisenstat and D-Optimal Design of Experiments

This paper addresses the following problem: given m potential observations to determine n parameters, m>n, what is the best choice of n observations. The problem can be formulated as finding the n×n submatrix of the complete m×n observation matrix that has maximum determinant. An algorithm by Gu and Eisenstat for a determining a strongly rank-revealing QR factorisation of a matrix can be adapted to address this latter formulation. The algorithm starts with an initial selection of n rows of the observation matrix and then performs a sequence of row interchanges, with the determinant of the current submatrix strictly increasing at each step until no further improvement can be made. The algorithm implements rank-one updating strategies, which leads to a compact and efficient algorithm. The algorithm does not necessarily determine the global optimum but provides a practical approach to designing an effective measurement strategy. In this paper, we describe how the Gu–Eisenstat algorithm can be adapted to address the problem of optimal experimental design and used with the QR algorithm with column pivoting to provide effective designs. We also describe implementations of sequential algorithms to add further measurements that optimise the information gain at each step. We illustrate performance on several metrology examples.

The discrete empirical interpolation method in class identification and data summarization

AbstractThe discrete empirical interpolation method (DEIM) is well established as a means of performing model order reduction in approximating solutions to differential equations, but it has also more recently demonstrated potential in performing data class detection through subset selection. Leveraging the singular value decomposition (SVD) for dimension reduction, DEIM uses interpolatory projection to identify the representative rows and/or columns of a data matrix. This approach has been adapted to develop additional algorithms, including a CUR matrix factorization for performing dimension reduction while preserving the interpretability of the data. DEIM‐oversampling techniques have also been developed expressly for the purpose of index selection in identifying more DEIM representatives than would typically be allowed by the matrix rank. Even with these developments, there is still a relatively large gap in the literature regarding the use of DEIM in performing unsupervised learning tasks to analyze large datasets. Known examples of DEIM's demonstrated applicability include contexts such as physics‐based modeling/monitoring, electrocardiogram data summarization and classification, and document term subset selection. This overview presents a description of DEIM and some DEIM‐related algorithms, discusses existing results from the literature with an emphasis on more statistical‐learning‐based tasks, and identifies areas for further exploration moving forward.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification

Digital model reconstruction and geometric error analysis of additive manufacturing blisk enabled by 3D scanner

Additive manufacturing has advantages in manufacturing blisk. However, additive manufacturing blisk can deviate in size and shape from their original digital models. In this paper, the shape of a complex shaped blisk manufactured by additive manufacturing is measured by using a 3D scanner, and the geometric errors compared to the digital model are calculated by using a column vector row index-based K-Neighborhood search algorithm. The computed range for geometric errors falls within (0.0075 mm, 6.9177 mm). Approximately 94% of the data points have errors within 3 mm. Larger geometric errors are primarily concentrated at the edges of the blades, aligning with the characteristic features of additive manufacturing processes.

On the factor ambiguity of MCR problems for blockwise incomplete data sets

Multivariate curve resolution (MCR) methods are sometimes faced with missing or erroneous data, e.g., due to sensor saturation. In some cases, an estimation of the missing data is possible, but often MCR works with the largest submatrix without missing entries. This ignores all rows and columns of the data matrix that contain missing values. A successful approach to deal with incomplete data multisets has been proposed by Alier and Tauler (2013), but it does not include a factor ambiguity analysis. Here, the missing data problem is addressed in combination with a factor ambiguity analysis. An approach is presented that minimizes the factor ambiguity by extracting a maximum of spectral information even from incomplete rows and columns of the spectral data matrix. The method requires a high signal-to-noise ratio. Applications are presented for UV/Vis and HSI data.