Triangular Matrix Research Articles

A Simple Preprocessing Method Enhances Machine Learning Application to EEG Data for Differential Diagnosis of Autism

A new pre-processing approach of EEG data to detect topological EEG features has been applied to a continuous segment of artifact-free EEG data lasting 10 minutes in ASCII format derived from 50 ASD children and 50 children with other Neuro-Psychiatric Disorders (NPD), matched for age and male/female ratios. Each EEG is transformed in a triangular matrix of 171 values expressing all reciprocal Manhattan distances among the 19 electrodes of to the international 10-20 system. From this matrix, the minimum spanning tree (MST) is calculated. Electrode identification serial codes sorted according to the decreasing number of links in MST, and the number of links in MST are taken as input vectors for machine learning systems. Machine learning systems have been applied to build up a predictive model to distinguish between the two diagnostic classes (autism vs NPD) following a rigorous validation protocol. The best machine learning system (KNN algorithm) obtained a global accuracy of 93.2% (92.37 % sensitivity and 94.03 % specificity) in differentiating ASD subjects from NPD subjects. The results obtained in this study suggest that the thank to the new pre-processing method introduced, there is the possibility to discriminate subjects with autism from subjects affected by other psychiatric disorders with a modest computational time reducing the information to 38 figures.

Dual modules, f-projective modules and Π-coherence over trivial ring extensions

Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M. We first describe explicitly the structure of a dual module over R ⋉ M . Then we characterize Mittag-Leffler modules and f-projective modules over R × M. Finally, we investigate when R ⋉ M is a left Π-coherent ring. In addition, some applications are given in formal triangular matrix rings.

A note on products of idempotents in the ring of upper triangular infinite matrices

It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for N × N matrices. It turns out that in this case the analogous theorem does not hold.

Computing expected moments of the Rényi parking problem on the circle

A highly accurate and efficient method to compute the expected values of the count, sum, and squared norm of the sum of the centre vectors of a random maximal sized collection of non-overlapping unit diameter disks touching a fixed unit-diameter disk is presented. This extends earlier work on Renyi's parking problem [Magyar Tud. Akad. Mat. Kutato Int. Kozl. 3 (1–2), 1958, pp. 109–127]. Underlying the method is a splitting of the the problem conditional on the value of the first disk. This splitting is proven and then used to derive integral equations for the expectations. These equations take a lower block triangular form. They are solved using substitution and approximation of the integrals to very high accuracy using a polynomial approximation within the blocks.

Image Watermarking Using Discrete Wavelet Transform and Singular Value Decomposition for Enhanced Imperceptibility and Robustness

Digital multimedia elements such as text, image, audio, and video can be easily manipulated because of the rapid rise of multimedia technology, making data protection a prime concern. Hence, copyright protection, content authentication, and integrity verification are today’s new challenging issues. To address these issues, digital image watermarking techniques have been proposed by several researchers. Image watermarking can be conducted through several transformations, such as discrete wavelet transform (DWT), singular value decomposition (SVD), orthogonal matrix Q and upper triangular matrix R (QR) decomposition, and non-subsampled contourlet transform (NSCT). However, a single transformation cannot simultaneously satisfy all the design requirements of image watermarking, which makes a platform to design a hybrid invisible image watermarking technique in this work. The proposed work combines four-level (4L) DWT and two-level (2L) SVD. The Arnold map initially encrypts the watermark image, and 2L SVD is applied to it to extract the s components of the watermark image. A 4L DWT is applied to the host image to extract the LL sub-band, and then 2L SVD is applied to extract s components that are embedded into the host image to generate the watermarked image. The dynamic-sized watermark maintains a balanced visual impact and non-blind watermarking preserves the quality and integrity of the host image. We have evaluated the performance after applying several intentional and unintentional attacks and found high imperceptibility and improved robustness with enhanced security to the system than existing state-of-the-art methods.

Distributed fiber optic sensing for micro- and macro-crack quantification: an interfacial-fracture-energy-based model

Clarifying the strain transfer between the host material with a crack opening displacement (COD) and the optical fiber after interfacial debonding remains a critical challenge for crack quantification. The interfacial debonding induced a triangular form strain profile around the crack due to the residual shear stresses at the interface. This study investigated the interfacial shear stress-slip relationship between the bare fiber and its coating layer through a plate splitting test. Based on that, we proposed an interfacial-fracture-energy-based analytical model to convert distributed fiber optic strains before and after interfacial debonding to CODs of micro- and macro-cracks. The accuracy of the model under a single crack was validated through plate splitting tests and the experimental data reported in the literature. Furthermore, experimental results from a four-point beam test were utilized to investigate the feasibility of the model under multiple cracks. The model, assisted by the strain superposition strategy, accurately quantified the CODs of multiple cracks before interfacial debonding but failed when interfacial debonding occurred. In addition, we proposed a simplified method based on the linear relationship between the debonded lengths and the corresponding CODs, which facilitates evaluation of the CODs after debonding.

Integrated triangular fuzzy KE-GRA-TOPSIS method for dynamic ranking of products of customers’ fuzzy Kansei preferences

As society evolves, companies produce more homogeneous products, shifting customers’ needs from functionality to emotions. Therefore, how quickly customers select products that meet their Kansei preferences has become a key concern. However, customer Kansei preferences vary from person to person and are ambiguous and uncertain, posing a challenge. To address this problem, this paper proposes a TF-KE-GRA-TOPSIS method that integrates triangular fuzzy Kansei engineering (TF-KE) with Grey Relational Analysis (GRA) and Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). Firstly, a Kansei evaluation system is constructed based on KE and fuzzy theory. A dynamic triangular fuzzy Kansei preference similarity decision matrix (TF-KPSDM) is defined to quantify customer satisfaction with fuzzy Kansei preferences. Secondly, dynamic objective weights are derived using Criteria Importance Though Intercrieria Correlation (CRITIC) and entropy, optimized through game theory to achieve superior combined weights. Thirdly, the GRA-TOPSIS method utilizes the TF-KPSDM and combined weights to rank products. Finally, taking the case of Kansei preference selection for electric bicycles, results indicate that the proposed method robustly avoids rank reversal and achieves greater accuracy than comparative models. This study can help companies dynamically recommend products to customers based on their Kansei preferences, increasing customer satisfaction and sales.

A Note on Singularity Categories and Triangular Matrix Algebras

A Note on Singularity Categories and Triangular Matrix Algebras

Medical image encryption algorithm based on a new five-dimensional multi-band multi-wing chaotic system and QR decomposition

In this study, we propose a medical image encryption algorithm based on a new five-dimensional (5D) multi-band multi-wing chaotic system and QR decomposition. First, we construct a new 5D multi-band multi-wing chaotic system through feedback control, which has a relatively large Lyapunov exponent. Second, we decompose the plaintext image matrix and chaotic sequence into an orthogonal matrix and upper triangular matrix using QR decomposition. We multiply the orthogonal matrix decomposed from the original image by the orthogonal matrix decomposed from the chaotic sequence. In this process, we use the chaotic sequence to control left and right multiplication. Simultaneously, we chaotically rearrange the elements in the upper triangular matrix using the improved Joseph loop and then multiply the two resulting matrices. Finally, we subject the product matrix to bit-level scrambling. From the theoretical analysis and simulation results, we observed that the key space of this method was relatively large, the key sensitivity was relatively strong, it resisted attacks of statistical analysis and gray value analysis well, and it had a good encryption effect for medical images.

Особый тип каменных топоров эпохи бронзы с реки Коротаиха (бассейн Баренцева моря)

Finds of thrust tools made of rocks and minerals are rare to find in the sites of the European North-East. This can be probably explained through a high mobility of the local hunters-gatherers and through the way such artefacts were left behind. Considering that there is an outstanding set of six axes and three workpieces found at Korotaikha 1979/3 site at the Korotaikha river (the Barentsev Sea basin). There they found not as a treasure; traces of their production were missing. The research aims to verify the assumption that the artefacts constitute a special type thrust tool series. To do so the study analysed the consequence of formation and layout of diverse macrotraces on the thrust tools surface. The study provides their morphography and classic morphometry. It is determined that the Korotaikha 1979/3 axes are made with the same technology. To start with, the shape of the future tool was formed with bifacial knapping. The final staged implied abrasive treatment (polishing) to form a straight sharp blade, level off and dull down sharp bifacial ribs along the longitudinal edges of the work pieces. The process preserved a large part of the surface formed by knapping. Thus, the Korotaikha 1979/3 axes are partly polished tools with a rectangular or nearly triangular form and lens-shaped cross- and longitudinal section. Macrotraces left by knocking upon solid materials, blade corrections and tool wear-out are evident of the axes use and repair. The nature of the work piece bilateral treatment conforms is harmonized with the bifacial knapping that the Korotaikha 1979/3 flint arrowheads and scrapers are made with. Along with homogeneous ceramics and evidence of bronze smelting production these items conform that these materials shall be attributed to records for studying the Korshak culture assigned to the Bronze Age and preliminarily dated back to the 2nd millennium B.C. Based on this data the study defined equivalent tools within sites located in a very short area at the Korotaikha river. However, the study failed to find equivalents outside this microregion. The production peculiarities, geography and attribution of these axes allow to define their special type of thrust tools of the Bronze Age European north-east. This type can be designated as the Korotaikha type axes. The location is found with an abnormal number of good-condition thrust tools left in one place. It may be associated with the functioning of a sanctuary or burial ground at this site.

Browder spectra of closed upper triangular operator matrices

<abstract><p>Let $ T_{B} = \left[\begin{array}{ll} A & B \\ 0 & D \end{array}\right] $ be an unbounded upper operator matrix with diagonal domain, acting in $ \mathcal H \oplus\mathcal K $, where $ \mathcal H $ and $ \mathcal K $ are Hilbert spaces. In this paper, some sufficient and necessary conditions are characterized under which $ T_{B} $ is Browder (resp., invertible) for some closable operator $ B $ with $ \mathcal D(B)\supset \mathcal D(D) $. Further, a sufficient and necessary condition is given under which the Browder spectrum (resp., spectrum) of $ T_{B} $ coincides with the union of the Browder spectrum (resp., spectrum) of its diagonal entries.</p></abstract>

A note on equivalent conditions for majorization

<abstract><p>In this paper, we introduced novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We used our new characterizations of majorization to derive an improved entropy inequality.</p></abstract>

On exact computational complexity of triangular factorization algorithms for general banded matrices

On exact computational complexity of triangular factorization algorithms for general banded matrices

On the Similarity over the Ring of Integers of Certain Nilpotent Matrices of Maximal Rank

This paper is devoted to the problem of matrix similarity recognition over the ring of integers for some families of matrices. Namely, nilpotent upper triangular matrices of maximal rank are considered such that only first and second superdiagonals of these matrices are non-zero. Several necessary conditions are obtained for similarity of such matrices to matrices of the form superdiag(a1,a2,…,an−1) with a single non-zero superdiagonal, that is a generalization of the Jordan cell Jn(0)=superdiag(1,1,…,1) . These conditions are formulated in simple terms of divisibility and greatest common divisors of matrix elements. The result is obtained by reducing the problem of similarity recognition to the problem of solving in integers a system of linear equations and applying the known necessary similarity conditions for arbitrary matrices. Under some additional conditions on the elements a1,a2,…,an−1 of the first superdiagonal of matrix A , it is proven that the matrix A is similar to matrix superdiag(a1,a2,…,an−1) regardless of the values of the elements of the second superdiagonal. Moreover, for the considered matrices of the third and the fourth orders, easily verifiable necessary and sufficient similarity conditions are obtained describing their similarity to a matrix of the form superdiag(a1,a2,…,an−1) .

A highly sensitive torsion sensor based on upright-inverted alternating triangular prism structure on the single-multi-single mode fiber

In this paper, a novel torsion sensor based on Mach-zehnder interferometer (MZI) is proposed and demonstrated. The proposed sensor is fabricated by the single-mode fiber (SMF) and multi-mode fiber (MMF) shear splicing, further employing CO2 laser to modulate the MMF surface into triangular prism form. And compare the number of different triangular prism pairs structure on torsion sensitivity. By increasing the number of triangular prism pairs effectively improve the torsion sensitivity. The experimental results show that in the range of 0–9.617 rad/m, the highest sensitivity fiber sensor can be obtained with a sensitivity of 3.3133 nm/(rad/m). Meanwhile, the temperature characteristic of the sensor is investigated, which presents a sensitivity of 21.67 pm/℃ in the range from 30 ℃ to 100 ℃. Therefore, the temperature crosstalk is 0.0065 (rad/m)/℃, which results in a low temperature crosstalk. In addition, the structure also has the advantages of low cost and simple manufacturing process.

New Banach Sequence Spaces Defined by Jordan Totient Function

In this study, a special lower triangular matrix derived by combining Riesz matrix and Jordan totient matrix is used to construct new Banach spaces. $\alpha-$,$\beta-$,$\gamma-$duals of the resulting spaces are obtained and some matrix operators are characterized.

MATRIX TRANSFORMS OF SPEED-SUMMABLE AND SPEED-BOUNDED SEQUENCES

First we recall the notions of $A^{\lambda}$-boundedness, $A^{\lambda}$-summability and the absolute $A^{\lambda}$-summability of sequences, and the notion of $\lambda$-reversibility of $A$, where $A$ is a matrix with real or complex entries and $\lambda$ is the speed of convergence, \ie; a monotonically increasing positive sequence. Let $B$ be a lower triangular matrix with real or complex entries, and $\mu=(\mu_{n})$ be another speed of convergence. We find necessary and sufficient conditions for a matrix $M$ (with real or complex entries) to map the set of all $A^{\lambda}$-bounded sequences (for a normal matrix $A$) into the set of all absolutely $B^{\mu}$-summable sequences, and the set of all $A^{\lambda}$-summable sequences (for a $\lambda$-reversible matrix $A$) into the set of all absolutely $B^{\mu}$-summable sequences.

Shape and quantum size effects on the electronic structure, oscillator strengths and state lifetimes of helium-like quantum dots

Transition energies, electric static polarizability, oscillator strengths and state lifetimes of helium-like quantum dots are determined as a function of their shape and size. A configuration interaction approach based on B-spline functions is used. We found that, the oscillator strengths present an extremum around a low value of the dot radius whatever the shape of the confining potential. This extremum is due to the pressure on the energy levels in such a way that for a particular value of the dot radius, the second electron in the excited one-electron state reaches the borders of the confinement potential. The extremum position depends on the impurity charge and the oscillator strength value at this point changes with the potential well shape. As well, the increase of the effective mass involves the shift of the position of the oscillator strength extremum towards weak radius values. The first excited state lifetime globally increases with the dot radius and the reduction of the nuclear charge but presents a shoulder for a certain low value of the dot radius mostly marked for a triangular form of the confining potential.

Observer-Based Adaptive NN Security Control for Switched Nonlinear Systems Against DoS Attacks: An ADT Approach.

In this article, a novel switched observer-based neural network (NN) adaptive control algorithm is established, which addresses the security control problem of switched nonlinear systems (SNSs) under denial-of-service (DoS) attacks. The considered SNSs are described in lower triangular form with external disturbances and unmodeled dynamics. Note that when an attack is launched in the sensor-controller channel, the controller will not receive any message, which makes the standard backstepping controller not workable. To tackle the challenge, a set of NN adaptive observers are designed under two different situations, which can switch adaptively depending on the DoS attack on/off. Further, an NN adaptive controller is constructed and the dynamic surface control method is borrowed to surmount the complexity explosion phenomenon. To eliminate double damage from DoS attacks and switches, a set of switching laws with average dwell time are designed via the multiple Lyapunov function method, which in combination with the proposed controllers, guarantees that all the signals in the closed-loop system are bounded. Finally, an illustrative example is offered to verify the availability of the proposed control algorithm.

Role of asymmetry and external noise in the development and synchronization of oscillations in the analog Hopfield neural networks with time delay.

The analog Hopfield neural network with time delay and random connections has been studied for its similarities in activity to human electroencephalogram and its usefulness in other areas of the applied sciences such as speech recognition, image analysis, and electrocardiogram modeling. Our goal here is to understand the mechanisms that affect the rhythmic activity in the neural network and how the addition of a Gaussian noise contributes to the network behavior. The neural network studied is composed of ten identical neurons. We investigated the excitatory and inhibitory networks with symmetric (square matrix) and asymmetric (triangular matrix) connections. The differential equations that model the network are solved numerically using the stochastic second-order Runge-Kutta method. Without noise, the neural networks with symmetric and asymmetric matrices possessed different synchronization properties: fully connected networks were synchronized both in time and in amplitude, while asymmetric networks were synchronized in time only. Saturation outputs of the excitatory neural networks do not depend on the time delay, whereas saturation oscillation amplitudes of inhibitory networks increase with the time delay until the steady state. The addition of the Gaussian noise is shown to significantly amplify small-amplitude oscillations, dramatically accelerates the rate of amplitude growth to saturation, and changes synchronization properties of the neural network outputs.