Matrix Algebra Methods Research Articles

Matrix Formulation for Minimum Response of Undamped Structure

A useful method for the determination of frequencies that correspond to points of local minima in the frequency response of finite element structure is presented. The method is based on the application of linear matrix algebra methods in conjunction with the fundamental definitions for existence of local minima. The procedure was initially developed for determination of antiresonant frequencies, and is now extended for application to minimum response. The derivations of the procedures for antiresonance and minimum response are both included in this paper. A significant advantage of this method is that it allows for the formulation of sensitivity relationships similar to what is commonly used for resonant frequency. Thus, there is potential for expanded use of minimum response information for structural optimization, damage detection, and other related tasks. In addition, generality in the formulation is retained such that the response may be defined as any linear combination of the coordinate responses, and any arbitrary forcing function may be considered. A numerical example, with a simple one-dimensional spring-mass system, is presented to provide for illustration and validation of the approach.

Analysis of the Bacteriorhodopsin Photocycle by Singular Value Decomposition with Self-Modeling: A Critical Evaluation Using Realistic Simulated Data

Data matrices consisting of sample optical absorption as a function of wavelength and another variable, such as time, are decomposable using known matrix algebraic methods. The natural decomposition, based on the Beer−Lambert law, into the product of component absorbance and (time dependent) concentration matrices is usually not straightforward. Singular value decomposition yields orthonormal spectral and kinetic eigenvectors, with mathematical but not physical meaning. The connection of the two decompositions is explored with reference to the problem of the bacteriorhodopsin photocycle. The limitations and applicability of singular value decomposition with self-modeling is evaluated with known stoichiometric constraints on the intermediate kinetics and compared to other techniques applied to the same problem. The improved method of exponential fit assisted self-modeling is introduced and demonstrated on realistic simulated data.

A RIS treatment of the mean-square dipole moment of PMMA chains in consideration of the pendant ester group orientations

According to the six-state scheme proposed by Vacatello and Flory, the mean-square dipole moments 〈 M 2〉/ x ( x: number of monomeric units) of poly(methyl methacrylate) (PMMA) chains and their temperature dependence d ln 〈M 2〉/ dT were calculated by the matrix algebra method of rotational isomeric state theory. To take account of the orientation effect of polar ester groups, the statistical weight matrices, U′ and U″, were recast into an expanded form. A Bernoullian probability λ introduced to measure the preference between the two alternative orientations ( χ=0 and χ= π) of the side ester groups was found to give rise to a large effect on the mean-square dipole moment of polymers. When the skeletal bonds of a meso dyad are both in trans, an exceptionally strong dipole-dipole interaction may take place. If the two adjacent ester groups adopt the same orientation, the dipoles must be situated nearly parallel to each other in a close proximity. Such repulsive interactions may be alleviated by taking a staggered conformation, i.e. ( χ, χ′)=(0, π) or ( π,0). Such neighbor-dependent correlation inherent to the meso dyad is considered by assigning a parameter λ′ to the proper element of the U m ″ matrix. The values of 〈 M 2〉/ x observed at 30 °C were satisfactorily reproduced by adopting λ=0.270 and λ′=0.865. The thermal coefficients d ln 〈M 2〉/ dT were found to be reconciled with the available experimental data by allowing the orientation parameters λ and λ′ to be slightly temperature dependent. The stereochemical characteristics of the chain terminals were examined in order to extend the treatment to include the dipole moment of PMMA oligomers.

Effect of the Pendant Ester Group Orientations on the Dipole Moment of Poly(alkyl acrylate)s

The dipole moment analysis of vinyl polymers carrying polar side groups such as poly(methyl acrylate) (PMA), poly(methyl methacrylate) (PMMA), and poly(ethyl methyacrylate) (PEMA) has been performed by the expanded matrix algebra method with due consideration of the ester group orientations. The stereochemical characteristics of the pendant ester groups have been exactly treated by introducing a Bernoullian probability λ representing the occurrence of the cis conformation around the first articulated Cα−C* bond. Special care is required for the orientational correlation of the neighboring ester groups when the skeletal conformations around the two successive bonds are both in trans. Such a meso-tt effect has been accommodated by assigning probability λ‘ or (1 − λ‘) to the proper elements of the statistical weight matrix. Experimental observations were found to be satisfactorily reproduced with λ = 0.3−0.4, indicating that the trans conformation is more preferred for the ester group. The orientational corr...

Decomposition of spectra in EPR dosimetry using the matrix method

The matrix method of EPR spectra decomposition is developed and adapted for routine application in retrospective EPR dosimetry with teeth. According to this method, the initial EPR spectra are decomposed (using methods of matrix algebra) into several reference components (reference matrices) that are specific for each material. Proposed procedure has been tested on the example of tooth enamel. Reference spectra were a spectrum of an empty sample tube and three standard signals of enamel (two at g=2.0045, both for the native signal and one at g ⊥=2.0018, g ||=1.9973 for the dosimetric signal). Values of dosimetric signals obtained using the given method have been compared with data obtained by manual manipulation of spectra, and good coincidence was observed. This allows considering the proposed method as potent for application in routine EPR dosimetry.

Factor Analysis Using Subspace Factor Models: Some Theoretical Results and an Application to UK Inflation Forecasting

Recent work in the macroeconometric literature considers the problem of summarising efficiently a large set of variables and using this summary for a variety of purposes including forecasting. Work in this field has been carried out in a series of recent papers. This paper provides an alternative method for estimating factors derived from a factor state space model. This model has a clear dynamic interpretation. Further, the method does not require iterative estimation techniques and due to a modification introduced, can accommodate cases where the number of variables exceeds the number of observations. The computational cost and robustness of the method is comparable to that of principal component analysis because matrix algebraic methods are used.

Shear, principal, and equivalent strains in equal-channel angular deformation

The shear and principal strains involved in equal channel angular deformation (ECAD) were analyzed using a variety of methods. A general expression for the total shear strain calculated by integrating infinitesimal strain increments gave the same result as that from simple geometric considerations. The magnitude and direction of the accumulated principal strains were calculated based on a geometric and a matrix algebra method, respectively. For an intersecting angle of π/2, the maximum normal strain is 0.881 in the direction at π/8 (22.5 deg) from the longitudinal direction of the material in the exit channel. The direction of the maximum principal strain should be used as the direction of grain elongation. Since the principal direction of strain rotates during ECAD, the total shear strain and principal strains so calculated do not have the same meaning as those in a strain tensor. Consequently, the “equivalent” strain based on the second invariant of a strain tensor is no longer an invariant. Indeed, the equivalent strains calculated using the total shear strain and that using the total principal strains differed as the intensity of deformation increased. The method based on matrix algebra is potentially useful in mathematical analysis and computer calculation of ECAD.

Eigenvalue spectra and geometric transformation of polyhedra

Starting from one fullerene, the three geometric transformations, cap, face dual and edge dual produce series of carbon clusters and deltahedra. The geometric relation between these polyhedra implies that their topological matrices and eigenvalue spectra must be relative. We have developed a matrix algebra method to research some polyhedra with high symmetry and one kind, two or three kinds of equivalent vertices such as C 60( I h), resolve their exact eigenvalues, and proved this point.

A New Method For Mineral Quantification to Aid In Hydrocarbon Exploration And Exploitation

Abstract Quantitative information about the reservoir rock minerals is important for making technical and business decisions in hydro-carbon exploration and exploitation. Minerals are usually quantified using mineral properties available from published data and rock properties measured in the laboratory used cored samples or in the field using geochemical well logs. Despite considerable efforts by many researchers, the rapid quantification of minerals. with error estimates remains a challenge. The most widely used, method for rapid mineral quantification is the matrix algebra method that uses the least-squares principle. Although fast and easy to implement, the conventional matrix algebra method is computationally unstable leading to unrealistic values (negative or greater than one) for mineral fractions. In this paper, we present a computationally stable method that retains the speed of the conventional matrix algebra method while overcoming its limitation. The present method can be applied to both laboratory (core samples) and downhole (geo-chemical well logs) analyses. It is effective in handling over determined, determined, and under-determined systems. It also handles both fixed and variable mineral properties. Unlike the conventional matrix algebra method, the present method supplements the rock and mineral properties with several constraining equations that incorporate prior information about the mineral fractions. The prior information about the mineral fractions, and the measured rock properties are weighted by the reciprocals of their respective error variances. Involving only matrix operations, the resulting equation to obtain mineral fractions is easy to implement and fast to compute. Programmed as an Excel macro or in Visual Basic, the method has been successfully implemented in our laboratory since 1993 for quantifying minerals in core samples from diverse rock formations. Introduction Reservoir rocks are an assemblage of minerals such as quartz, carbonates, feldspars, kaolinite, illite, and smectite. Quantitative analysis of these minerals is used by exploration and production geologists as well as by reservoir and production engineers. Exploration geologists use the rock mineralogy to reduce the risk in discovering oil by determining the thermal and diagenetic history of a basin, defining the provenance (source area) and the depositional environment of the sediments, and correlating certain minerals with well logs. Exploitation geologists and reservoir engineers use the rock mineralogy to assess reservoir quality develop effective depletion strategies, and predict the effect of rock-fluid interactions. Production engineers use the rock mineralogy to design work-over and completion strategies such as selection of drilling fluids and proper stimulation methods (e.g., acidizing). Traditionally, rock mineralogy has been determined in the laboratory using cored rock samples(1–4). Over the past few years, there have been efforts to develop mineral logs by downhole geochemical welllogging(5–7). Laboratory and Downhole Mineral Quantification The methodologies for routine laboratory and downhole mineral quantification are similar. In both cases, the rock properties and mineral properties are used to estimate mineral fractions. In the laboratory, X-ray fluorescence is used to measure rock properties (elemental composition expressed as oxides). These properties can be supplemented by other rock properties such as loss on ignition (LOI) and cation-exchange capacity (CEC).

Approximation Model For Multi-Service Transmission

AbstractDisparate service times of different classes of traffic in multi-service transmission models cause numerical difficulties in exact analysis. Utilizing the difference in time scale, an approximation model is developed. The accumulation and depletion of low priority data packets are modelled as the service time and inter-arrival time processes of a PH/PH/1 queue. The PH/PH/1 model is related to the stochastic fluid flow model, and the eigenvalues of a key matrix determine the stationary distribution of the waiting time (and the idle time) of the PH/PH/1 queue and of the buffer content in the stochastic fluid flow model. Exploiting the common results of the two methods, numerical examples are presented for the moments of queue length of low priority data packets conditional on various levels of voice calls or active speakers present. A stochastic fluid flow interpretation is given of the coupling matrix employed in the matrix-algebraic method of PH/PH/1 queue and the connection with results obtained ...

Efficiency of the Lucas scintillation cell

The Lucas scintillation cell (LC) and its modifications are among the most sensitive instruments to detect low levels of radon. In this work we describe a comprehensive investigation of the counting efficiency of the LC. A known quantity of 222Rn was transferred into the LC and the decay and ingrowth of Rn and its daughters was followed in an automated measurement with counting intervals ranging from 5 s to 12 h. A total of 7014 measurements were taken in a period exceeding 5 months. The measurements taken during Rn transfer to the LC indicated that the separation was not ideal and that small amounts of daughters were also transferred to the LC. A numerical model was then developed and fitted to the data. The model includes kinetics of Rn transfer to the LC, a matrix-algebra method for description of decay and growth of Rn and its daughters, individual counting efficiencies, as well as several small corrections. Individual α counting efficiencies (counts/particle) were determined: 0.76(2) for 222Rn, 0.77(3) for 218Po, 0.91(1) for 214Po, and 0.8283(6) for 210Po. A contribution from electrons to the efficiency was found to be insignificant. The work is concluded by providing an overall counting efficiency of LC as a function of decay and counting times, which is of practical interest to radon measurements.

Solution of the Maxwell field equations in vacuum for arbitrary charge and current distributions using the methods of matrix algebra

A new matrix representation of classical electromagnetic theory is presented. The basis of this representation is a space-time, eight-by-eight differential matrix operator. This matrix operator is initially formulated from the differential form of the Maxwell field equations for vacuum. The resulting matrix formulation of Maxwell's equations allows simple and direct derivation of the electromagnetic wave and charge continuity equations, the Lorentz conditions and definition of the electromagnetic potentials, the Lorentz and Coulomb gauges, the electromagnetic potential wave equations, and Poynting's conservation of energy theorem. A four-dimensional Fourier transform of the matrix equations casts them into an eight-dimensional transfer theorem. The transfer function has an inverse, and this allows the equations to be inverted. This expresses the fields directly in terms of the charge and current source distributions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Calculation of the mean-square radius of gyration for polymer chains with side-groups

The mean-square radius of gyration of polymer chains with side-groups was formulated using the method of matrix algebra with the rotational isomeric state approximation. If the effect of the side-groups is neglected, the expression reduces to that reported by Flory. The molecular weight dependence of the root-mean-square radius of gyration for isotactic polypropylene was obtained as S 2 1 2 =0.34M 1 2 p] in fair agreement with experimental data from small-angle neutron scattering, reported by Ballard et al.

Frequency Window Method for Strongly Coupled and Multiply Connected Structural Systems: Multiple-Mode Windows

In a companion paper, a method is presented to analyze the dynamic behavior of a structural system consisting of a main structure and strongly coupled, multiply connected substructures. Lagrange’s equations are used to develop a characteristic equation for connected substructures in terms of substructure impedances and mobilities. A frequency window method is used to reduce the complexity of the problem by a decomposition of the impedance and mobility functions into dominant and higher-order rational expressions. From the reduced problem, simple expressions for the modal properties are developed using matrix algebraic methods, which provide insight into the resonance characteristics of the connected substructures. Onemode windows were discussed in detail and examples were presented. In the present paper the theory is extended to multiple-mode windows.

Frequency Window Method for Strongly Coupled and Multiply Connected Structural Systems: One-Mode Windows

A method is presented to analyze the dynamic behavior of a structural system consisting of a main structure and strongly coupled, multiply connected substructures. Lagrange’s equations are used to develop a characteristic equation for connected substructures in terms of substructure impedances and mobilities. Then, a frequency window method is used to reduce the complexity of the problem by a decomposition of the impedance and mobility functions into dominant and highorder rational expressions. From the reduced problem, simple expressions for the modal properties are developed using matrix algebraic methods, which provide insight into the resonance characteristics of the connected substructures. Higherorder terms, which become significant for strongly coupled substructures, are included in the eigenvalue analysis by using an iterative procedure. It is shown that the frequency window method developed in this paper, used as a numerical scheme, produces results which converge to exact results after only a few iterations.

Improved expression of mean-square radius of gyration. I. Vinyl polymers

Taking account of the effect of large side groups, the mean-square radius of gyration of vinyl polymer chains has been deduced by the method of matrix algebra. According to previous approaches to the computation of conformational energy, every substituent of vinyl polymer is confined to an orientation in which the conformational energy is minimum, i.e., its plane is approximately perpendicular to the one defined by the skeletal bonds flanking the Cα atom. This model facilitates the derivation of mean-square radius of gyration. Neglecting the effect of side groups, the expression for mean-square radius of gyration given in this work reduces to be that reported by Flory. Numerical calculations indicate that for tactic polypropylene, polystyrene, poly(methyl acrylate), and poly-1,2-butadiene the dependence of mean-square radius of gyration on molecular weight can be expressed as 〈S2〉=aMb, which is in agreement with the experimental data.

Application of projection matrices to the calculation of unperturbed dimensions of chain polymers

AbstractThe projection operator method of matrix algebra is applied to calculate the unperturbed dimensions of an independently hindered rotating bond model for linear chain molecules. It does not require eigenvectors of a given transformation matrix in order to evaluate products of that matrix. This is an advantage to usual matrix diagonalization methods. As a result, we obtained four different types of conformations depending upon the values of parameters adopted. One of them shows a typical dependence on the number of bonds N, which has also been given by the usual matrix diagonalization method. Two further conformations are rather extended compared with the first one. In addition, as the fourth one we obtained also a helix‐like conformation. These conformation types correspond to those previously obtained by continuous models for stiff‐chain polymers.

A simple matrix method for determining flux control coefficients in complex pathways

Flux control coefficients express in quantitative terms the extent to which the steady state flux through a metabolic pathway is controlled by a particular parameter. Enzyme flux control coefficients can be calculated using matrix algebra methods which express the control coefficients in terms of parameters which can be determined experimentally (enzyme elasticities, flux ratios, metabolite ratios). This paper describes an algorithm based on a ‘constraint’ matrix which enables expressions for enzyme control coefficients to be written for pathways of any complexity.

Contribution to the crystallography of the bainite transformation in a high strength low alloy steel 40 NSCD 7-7-3 (AISI 300M)

During continuous cooling performed from 950°C at a rate between 1 and 10°C s −1, the high strength low alloyed steel 300M acquires a mixed structure composed of plate martensite, lower bainite and retained austenite. The formation of lower bainite has been interpreted in terms of phenomenological theory developed in the case of martensite formation. Indeed, the habit plane of lower bainite experimentally determined well agrees with the one calculated from matrix algebra or graphical method, when the shear elements of the austenite lattice are as follows: (1̄11) γ, [211] γ and (1̄11) γ, [121̄] γ. The PITSCH's orientation relationship was evidenced between cementite and ferrite of lower bainite, while between ferrite of lower bainite and austenite, an orientation relationship closer to Nishiyama-Wassermann's than Kurdjumov-Sach's was found. For appropriate thermal treatments, upper bainite and austenite under form of “S” where also revealed. Moreover, silicon does not seem to induce the formation of silicocarbides and would not play a role in the mechanism of the bainite transformation; it only contributes to delay this transformation.

Quantitative genetics of allotetraploid and autotetraploid populations

Theoretical studies were conducted to compare quantitative genetic parameters in hypothetical allotetraploid and autotetraploid populations with the same genetic information. Concepts of factorial experiments and methods of direct product matrix algebra were used to express genetic effect contrasts in similar forms for each ploidy type. Quantitative genetic effects could be expressed as follows: (i) the population mean; (ii) four linear contrasts that could be equated to additive effects with either ploidy type; (iii) six contrasts resulting from interactions between two alleles, two of these equated to dominance effects and four to additive × additive effects in the allotetraploid and all six equated to digenic effects in the autotetraploid; (iv) four contrasts from interactions between three alleles, which equated to additive × dominance effects in the allotetraploid and to trigenic effects in the autotetraploid; and (v) one contrast that resulted from interactions between four alleles, which equated to the dominance × dominance effect in the allotetraploid and the quadrigenic effect in the autotetraploid. In hypothetical populations for which gene functions were the same and gene frequencies were equal, contrasts involving similar numbers of genes were identical for each ploidy type. The mean and total variance of the hypothetical populations would be identical. The parent–offspring covariance, variance among half-sib families, and the variance among full-sib families would be slightly larger in allotetraploid than in autotetraploid populations for traits affected by genes with nonadditive effects. We concluded that response to recurrent selection, general combining ability in diallel crosses, and the probability of finding a superior single cross hybrid would be larger in the allotetraploid than in the autotetraploid populations when nonadditive genetic effects are present. The relationship between ploidy types for specific combining ability in diallel crosses depended on the importance of digenic effects. Plant breeders working on species with autotetraploid inheritance should consider looking for allotetraploid forms before embarking on programs to produce hybrid cultivars.Key words: genetic variance; covariance, parent–offspring; breeding behavior; matrix algebra.