Main Diagonal Research Articles

Enhanced Technique of Constructing Multiple ODD Magic Square Matrices

An enhanced technique of constructing multiple odd magic square matrices is proposed in this work. A specific rule of establishing improved odd magic to magic squares derived from the odd algebraic Latin squares is studied and programmed here. Magic squares are practically important from the properties of their equality in the sum of rows, columns and diagonals. An n x n odd magic square is an array containing the positive integers from 1 to n2, arranged so that each of the rows, columns, and the two principal diagonals have the same sum.

A Comparative Study on Lattice Tower Angle Member Capacities as Per Indian, American, and BSEN Standards

The transmission lines and Telecommunication networks are normally supported by lattice towers. Therefore, the reliability of these essentials depends much on the reliability of the towers and their foundations. In Both telecom and power transmission line sectors, the towers are mass-produced and generally based on optimized tower weight and foundation volumes. The weight of a tower is influenced to a great extent by the selection of tower configuration, choice of bracing patterns, choice of steel grade, and profile type. The towers in general are lattice types consisting of main legs, diagonals, horizontals, cross-arm members, and peak members. The telecommunication and power transmission line tower members are generally made of steel equal-angle sections. These tower members are modeled and analyzed as pin-jointed 3-dimensional space truss models and the members are subjected to axial forces, either axial compression or axial tension in nature. Estimation of member compression capacity is the most vital parameter in design as per respective local standards and proving those estimations during full-scale model tower testing if carried out. This paper presents the differences in axial compression capacity of angle members as per Indian, American, and European standards viz., IS: 802 (Part 1/Sec 2)-2016, IS 800-2007, ASCE 10-15 and BS EN 1993-3-1:2006.

Generalized Sum-Asymmetry Model and Orthogonality of Test Statistic for Square Contingency Tables

For analyzing contingency tables, we are usually interested in whether or not the independence model holds. On the other hand, for the analysis of square contingency tables, we are usually interested in whether or not the model having the structure of symmetry or asymmetry with respect to the main diagonals cells holds. This study proposes a generalized sum-asymmetry model including the exponential and relative exponential sum-symmetry models. This generalized model indicates that the cumulative probability that the sum of classes for row and column variables is s within the upper right cell of the table, is exponentially higher than the cumulative probability that the sum of classes for row and column variables is s within the lower left cell. Additionally, this study gives a separation of the sum-symmetry model using the proposed model, and reveals that the new separation satisfies the asymptotic equivalence for the test statistic. The utilities of the proposed methods are demonstrated through the real data analysis.

Matrix representations of arbitrary bounded operators on Hilbert spaces

Abstract We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to interesting consequences, e.g., when the tuple consists of powers of a single operator. We also prove several variants of this result of independent interest. The paper substantially extends former research on matrix representations in infinite-dimensional spaces dealing mainly with prescribing the main diagonals.

Equalizer of Amplitude-Frequency Response of Acoustic Channels and its Adaptive Filter Based on Recursive Least Squares Algorithm with Diagonalized Correlation Matrix

The paper presents an equalizer of the amplitude-frequency response of the acoustic wave propagation channels in the rooms. The considered equalizer uses the so-called modified architecture based on the adaptive filter with the input signal filtering, which weights are calculated using the Recursive Least Squares (RLS) algorithm. The equalizer of the acoustic channels amplitude-frequency response usually requires the usage of the adaptive filters with the large number of weighs that means the large computational complexity of the adaptive filter. The adaptive filters based on the fast RLS algorithms are efficient in the terms of their complexity but often unstable if the number of adaptive filter weights is large. Therefore, the algorithms are poorly suited for usage in the amplitude-frequency response equalizers of the acoustic channels. Adaptive filters based on the RLS algorithm, which use the Matrix Inversion Lemma (MIL), are stable, but computationally complex. The equalizer presented in this paper uses a simplified adaptive filter based on the MIL RLS algorithm with a diagonalized correlation matrix of the filter input signal. In addition, in the filter, each of the square submatrices on the simplified correlation matrix diagonal is inverted using a computationally efficient version of the MIL that takes into account the symmetrical structure of the mentioned submatrices relatively their main diagonals. Computer simulation confirms the performance of the proposed equalizer and demonstrates its efficiency comparing to the unsimplified equalizer. The demo includes the graphs of the equalizer steady-state amplitude-frequency response, the acoustic channel amplitude-frequency responses before and after the equalization, and the power spectral density plots of the equalizer input signal and the signals passed through the equalizer and the acoustic medium. The equalizer can be used in the equipment for the high-quality voice and music reproduction in the conditions of the limited computing resources for the implementation of the equalizer.

On the possibility of implementing a simple shear in the cross-section of metal materials during caliber rolling

The article analyzes the stress-strain state in the zone of plastic deformation during special caliber rolling. The principle of caliber rolling technology is described, which makes it possible to combine shear and compression deformations in the cross-section of metals, alloys, and metal-matrix composites. The analysis results of stresses and strains during shear rolling in a diamond pass, which included compressive strains that had not been considered in previous studies, revealed that localization or point inversion of stresses and strains is observed in the plastic deformation zone. Stress and strain are localized along the minor and major diagonals of the diamond pass.

Robustness evaluation of a steel bridge in the district of Potenza (Italy)

In the current paper the Pastorella bridge in Lauria (district of Potenza) has been investigated. Firstly, considering the inaccessibility of the bridge with the exclusion of the main deck as a walkaway, an aerial photogrammetric survey with a drone has been done for the acquisition of geometric data. Secondly, a three-dimensional modelling of the bridge in FEM environment has been carried out. Based on the corrosion level detected in the bridge structure, forecasts about the progress of the corrosion damage during the whole lifetime of the bridge have also been made considering appropriate thickness reduction of structural elements based on literature provisions. Therefore, other than studying the intact bridge, four different corrosion scenarios have been considered: 1) corroded upper deck elements, 2) corroded lower chord elements; 3) corroded main diagonals and 4) corroded vertical members. On these FEM models push-down and linear dynamic analyses have been performed, so to know the health state of the bridge in terms of robustness for different corrosion levels.

A binomial expansion formula for weighted geometric means of unipotent matrices

Following the Kubo-Ando theory of operator means we consider the weighted geometric mean A # t B of n × n upper triangular matrices A and B whose main diagonals are all 1, named the upper unipotent matrices. We also present its binomial expansion A # t B = ∑ k = 0 n − 1 ( t k ) A ( A − 1 B − I ) k , t ∈ R . Showing that the weighted geometric mean is a geodesic of symmetry in the symmetric space equipped with point reflection, known as the Loos symmetric space, we derive several binomial identities on the Lie group of upper unipotent (resp. the Lie algebra of nilpotent) matrices.

Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences

In this paper, we will consider a class of locally $\Phi$-Beilinson-Green algebras, where $\Phi$ is an infinite admissible set of the integers, and show that symmetric approximation sequences in $n$-exangulated categories give rise to derived equivalences between quotient algebras of locally $\Phi$-Beilinson-Green algebras in the principal diagonals modulo some factorizable ghost and coghost ideals by the locally finite tilting family. Then we get a class of derived equivalent algebras that have not been obtained by using previous techniques. From higher exact sequences, we obtain derived equivalences between subalgebras of endomorphism algebras by constructing tilting complexes, which generalizes Chen and Xi's result for exact sequences. From a given derived equivalence, we get derived equivalences between locally $\Phi$-Beilinson-Green algebras of semi-Gorenstein modules. Finally, from given graded derived equivalences of group graded algebras, we get derived equivalences between associated Beilinson-Green algebras of group graded algebras.

Wetlands Classification Using Quad-Polarimetric Synthetic Aperture Radar through Convolutional Neural Networks Based on Polarimetric Features

Wetlands are the “kidneys” of the earth and are crucial to the ecological environment. In this study, we utilized GF-3 quad-polarimetric synthetic aperture radar (QP) images to classify the ground objects (nearshore water, seawater, spartina alterniflora, tamarix, reed, tidal flat, and suaeda salsa) in the Yellow River Delta through convolutional neural networks (CNNs) based on polarimetric features. In this case, four schemes were proposed based on the extracted polarimetric features from the polarization coherency matrix and reflection symmetry decomposition (RSD). Through the well-known CNNs: AlexNet and VGG16 as backbone networks to classify GF-3 QP images. After testing and analysis, 21 total polarimetric features from RSD and the polarization coherency matrix for QP image classification contributed to the highest overall accuracy (OA) of 96.54% and 94.93% on AlexNet and VGG16, respectively. The performance of the polarization coherency matrix and polarimetric power features was similar but better than just using three main diagonals of the polarization coherency matrix. We also conducted noise test experiments. The results indicated that OAs and kappa coefficients decreased in varying degrees after we added 1 to 3 channels of Gaussian random noise, which proved that the polarimetric features are helpful for classification. Thus, higher OAs and kappa coefficients can be acquired when more informative polarimetric features are input CNNs. In addition, the performance of RSD was slightly better than obtained using the polarimetric coherence matrix. Therefore, RSD can help improve the accuracy of polarimetric SAR image classification of wetland objects using CNNs.

New unitary equivalences for some operator matrices with applications

In this paper, first we consider cross-diagonal, in particular off-diagonal, operator matrices; those operator matrices such that all entries except those on the main and off diagonals are zero. We show that this operator matrix is unitarily equivalent to a block diagonal operator matrix whose diagonal blocks are all two-by-two, except at most one of them which is one-by-one. Then, using this unitary equivalence, we show that any left circulant operator matrix is unitarily equivalent to a direct sum of a one-by-one operator and an off-diagonal operator matrix. As an application, we give equalities for the numerical radius of some important operator matrices. In particular, for the following left circulant operator matrix, we show that where is the nth root of unity. Meanwhile, some inequalities for general operator matrices are obtained.

Constraints between concurrence and polarization for mixed states subjected to open system dynamics

Entanglement and polarization are mutually constrained by the relationship ${C}^{2}+{P}^{2}=1$, which engages the concurrence $C$ of a pure, two-qubit state and the degree of polarization $P$ of either of its two subsystems. How the above constraint generalizes for mixed states is an open question. We address mixed, two-qubit states of the $X$ type, i.e., those whose density matrix has nonzero elements only in the two main diagonals. We focus on mixed states that arise out of a pure, two-qubit state that is subjected to either the amplitude damping channel or the depolarizing channel. We derive alternative constraints for concurrence and polarization and test them experimentally with polarization-entangled photons. We argue that our theoretical results hold also for classical light, whenever two binary degrees of freedom can be entangled.

CONSTRUCTION OF MULTIPLE ODD MAGIC SQUARES

A technique of constructing multiple odd magic square matrices is studied in this work. A specific rule of establishing odd magic to magic squares derived from the odd algebraic Latin squares is proposed. Magic squares are practically important from the properties of their equality in the sum of rows, columns and diagonals. An n*n odd magic square is an array containing the positive integers from 1 to n2 , arranged so that each of the rows, columns, and the two principal diagonals have the same sum.

Time trends in social contacts before and during the COVID-19 pandemic: the CONNECT study

BackgroundSince the beginning of the COVID-19 pandemic, many countries, including Canada, have adopted unprecedented physical distancing measures such as closure of schools and non-essential businesses, and restrictions on gatherings and household visits. We described time trends in social contacts for the pre-pandemic and pandemic periods in Quebec, Canada.MethodsCONNECT is a population-based study of social contacts conducted shortly before (2018/2019) and during the COVID-19 pandemic (April 2020 – February 2021), using the same methodology for both periods. We recruited participants by random digit dialing and collected data by self-administered web-based questionnaires. Questionnaires documented socio-demographic characteristics and social contacts for two assigned days. A contact was defined as a two-way conversation at a distance ≤ 2 m or as a physical contact, irrespective of masking. We used weighted generalized linear models with a Poisson distribution and robust variance (taking possible overdispersion into account) to compare the mean number of social contacts over time and by socio-demographic characteristics.ResultsA total of 1291 and 5516 Quebecers completed the study before and during the pandemic, respectively. Contacts significantly decreased from a mean of 8 contacts/day prior to the pandemic to 3 contacts/day during the spring 2020 lockdown. Contacts remained lower than the pre-COVID period thereafter (lowest = 3 contacts/day during the Christmas 2020/2021 holidays, highest = 5 in September 2020). Contacts at work, during leisure activities/in other locations, and at home with visitors showed the greatest decreases since the beginning of the pandemic. All sociodemographic subgroups showed significant decreases of contacts since the beginning of the pandemic. The mixing matrices illustrated the impact of public health measures (e.g. school closure, gathering restrictions) with fewer contacts between children/teenagers and fewer contacts outside of the three main diagonals of contacts between same-age partners/siblings and between children and their parents.ConclusionPhysical distancing measures in Quebec significantly decreased social contacts, which most likely mitigated the spread of COVID-19.

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries tii are arranged in increasing order. To avoid calculations using the differences tii − tjj with close (including equal) tii and tjj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

On the interplay between operators, bases, and matrices

Given a bounded linear operator T on a separable Hilbert space, we develop an approach allowing one to construct a matrix representation for T having certain specified algebraic or asymptotic structure. We obtain matrix representations for T with preassigned bands of the main diagonals, with an upper bound for all of the matrix elements, and with entrywise rational-like lower and upper bounds for these elements. In particular, we substantially generalize and complement our results on diagonals of operators from [47] and other related results. Moreover, we obtain a vast generalization of a theorem by Stout (1981) [56], and (partially) answer his open question. Several of our results have no analogues in the literature.

An image encryption algorithm based on dynamic row scrambling and Zigzag transformation

This paper introduces two new one-dimensional chaotic systems. They can be considered as an improvement of Logistic map, Sine map and Tent map. One is the chaotic system composed of Sine and Tent maps (STCS), the other is Logistic-Logistic chaotic system (L-LCS). Based on the two, an image encryption algorithm using dynamic row scrambling and Zigzag transformation is proposed. Firstly, the image is traversed in different directions of odd and even rows, and the results are scrambled with the chaotic sequence generated by STCS. Secondly, based on standard Zigzag scrambling idea, a special traversal method is used, that is, for the different permutation of the lower triangle and the upper triangle of the matrix (each triangular matrix does not contain the values on the main and sub diagonals) in order and direction. Then, combined with the chaotic sequence generated by L-LCS, two different diffusion formulas are used to diffuse the segmented image. Finally, this encryption scheme can also be extended to color images. Experimental simulation and performance analysis show that the encryption algorithm has better security.

Striking patterns in natural magic squares’ associated electrostatic potentials: Matrices of the 4th and 5th order

A magic square is a square matrix whereby the sum of any row, column, or any one of the two principal diagonals is equal. A surrogate of this abstract mathematical construct, introduced in 2012 by Fahimi and Jaleh, is the “electrostatic potential (ESP)” that results from treating the matrix elements of the magic square as electric charges. The overarching idea is to characterize patterns associated with these matrices that can possibly be used, in the future, in reverse to generate these squares. This study focuses on squares of order 4 and 5 with 880 and 275,305,224 distinct (irreducible/unique) realizations, respectively. It is shown that characteristic patterns emerge from plots of the ESPs of the matrices representing the studied squares. The electrostatic potentials for natural magic squares exhibit a striking pattern of maxima and minima in all distinct 880 of the 4th order and all distinct 275,305,224 of the 5th order matrices. The minimum values of ESP of Dudeney groups are discussed. Equipotential points and certain constants are found among the ESP sums along horizontal and vertical lines on the square lattice. These findings may help to open a new perspective regarding magic squares unsolved problems. While mathematics often leads discovery in physics, the latter (physics) is used here to detect otherwise invisible patterns in a mathematical object such as magic squares.

Strongly self-dual polytopes and distance graphs in the unit sphere

Lovasz proved that the chromatic number of the graph formed by the principal diagonals of an $$n$$ -dimensional strongly self-dual polytope is greater than or equal to $$n+1$$ . There is equality if the length of the principal diagonals is greater than the Euclidean diameter of the monochromatic parts of that coloring of the unit sphere which is based on a partition of $$n+1$$ congruent spherical regular simplices. We determine this quantity for all $$n$$ and prove that in dimension three all such graphs can be colored by four colors.

Plane section recurrences in the Pascal pyramid

This work deals with the plane sections lying over the Pascal pyramid. Any plane section can be defined using two main diagonals with parameters (q1,r1) and (q2,r2). We study the main case r1=r2=1, which corresponds to plane sections crossing all parallel planes to the x axis and intersecting with the integer coordinates of the pyramid. We establish the recurrence relation satisfied by the sequence of numbers counting the sum of elements lying over each plane section.