On the factorization of a nonsingular rational matrix

Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions

Peripheral exercise blood pressure and resting central blood pressure are considered more relevant to cardiovascular health than resting peripheral blood pressure. Central exercise blood pressure may well be an even more useful measure, but there is no simple non-invasive means of determining it. The aim of the present study was to establish whether the estimation of central blood pressure from peripheral blood pressure using a transfer function derived at rest, would hold after aerobic exercise. Thirty healthy young men were studied before and immediately (< 1 min) and 10 min after 15 min bicycle exercise at 65-70% of maximum heart rate. Simultaneous carotid and radial artery waveforms were recorded, and radial-to-carotid generalized transfer functions (GTF) were calculated using Fourier analysis for rest and immediately postexercise. Central systolic blood pressure (SBP) and augmentation index (AIx) were calculated for measured and derived waves. The resting GTF underestimated central SBP and AIx immediately (-5.8 +/- 2.1 mmHg, P = 0.01; -8.3 +/- 2.9%, P = 0.008) and 10 min after (-2.0 +/- 0.7 mmHg, P = 0.008; -7.0 +/- 2.1%, P = 0.003) exercise. No significant bias was found between measured and derived (using resting GTF) carotid values at rest. The use of an exercise-specific GTF resulted in no specific bias immediately or 10 min after exercise, although it overestimated blood pressure and AIx at rest (2.5 +/- 1.0 mmHg, P = 0.02; 11.3 +/- 3.0%, P = 0.001). A peripheral-to-central arterial GTF derived at rest significantly underestimates key measures of central arterial pressure immediately after exercise, and pressure estimations may be improved by the use of an exercise-specific GTF.

Introduction: Central aortic pressure waveform analyses can provide clinically relevant information beyond conventional brachial blood pressure (BP) assessment. This waveform can be reproduced noninvasively through application of a generalized transfer function (GTF) on a peripheral waveform, as conventionally performed by applanation tonometry. Photoplethysmography (PPG) is an alternate approach; however, differences in measurement site and modality demand the use of a transfer function (TF) specific for those differences. This study aimed to compare central aortic waveform features generated from radial tonometry (reference) using a proprietary GTF with a central aortic waveform and its features generated from a simultaneous fingertip PPG measurement using a selective method where one of three different TFs is chosen based on the input signal harmonic profile. Methods: Brachial BP was measured in triplicate under resting conditions and was used for subsequent calibration. Multiple simultaneous radial tonometry (SphygmoCor CVMS) and fingertip PPG measurements were then performed in individual participants (n = 21, 10 females, age: 39 ± 16 years). Measurements were converted into central aortic waveforms with their respective TFs. Twenty central aortic pressure waveform parameters were compared through correlation analysis, Bland-Altman plots, and a repeated measure mixed-effects ANOVA model. Central aortic waveform shape was compared using the root-mean-squared error (RMSE). Results: Correlation (r) of PPG-derived parameters with radially tonometry-derived central aortic parameters was high ranging from 0.79 to 0.99. Mean differences of pressure-related parameters were within 1.3 mm Hg, and differences of time-related parameters ranged from −2.2 to 3.4%. While some parameters were statistically different, these differences are not physiologically meaningful. Central aortic waveform shape had an average RMSE of 1.8 ± 0.9%. Conclusion: Fingertip PPG-derived central aortic waveform parameters using a novel selective TF were comparable to central aortic waveform features derived from radial tonometry using a previously validated GTF.

When one column of a square nonsingular matrix, say $B^0 $, is replaced by another, the Bartels–Golub update of the LU factors of $B^0 $ forms a product form representation that is suitable for the efficient solution of associated systems of linear equations. However, it does not update the factors so as to obtain (even implicitly) the LU factors of the new matrix. In this note an exceedingly simple modification of the Bartels–Golub technique that yields the true LU factors is described, and it is shown to be mathematically equivalent to the update (specifically the partial pivoting version) recently proposed by Fletcher and Matthews [Math. Programming, 30 (1984), pp. 267–284]. The other commonly used update, Forrest–Tomlin, is quite evidently a mathematical variant of the Bartels–Golub technique, so this result enables a common ground for the three major updating techniques of linear programming to be established.

Gravity is a major recovery mechanism of naturally fractured reservoirs, where fracture gas drains matrix oil until equilibrium is reached with the capillary forces (wrt fluid densities, matrix gas capillary pressure and block height). The challenge of modelling gravity drainage in dual-medium simulation is to match the final maximum recovery, using integrated pseudo-capillary pressure curve, and the correct recovery kinetics. The paper suggests an approach to improve the simulation of the recovery kinetics in gravity drainage by dividing the matrix block for the fluid transfer function in two specific parts: saturation front part (SFP) and initial state part (ISP). As the invading gas enters the matrix the SFP and ISP areas increase and decrease respectively, until the final recovery is reached at equilibrium point. The contributions from each part are summed up to equal a mass conservation equation at each time step for each matrix cell. Properties of SFP depend on the invading fluid saturation and ISP hold the initial state properties, hence its name. This SubFace formulation can be implemented in flow simulator for reservoirs exhibiting a dual-medium behaviour. Our SubFace Transfer Function approach (SF), performs well versus not only conventional transfer functions (Kazemi, Gilman), but also versus two improved ones: Quandalle-Sabathier, and Lu-Blunt (non-Warren-Root General Transfer Function) in matching the results of fine-grid single-medium models under various parameters (capillary pressure, matrix shape and mobility). We also tested SF in mixed-wet water-oil system to assess its capability of modelling gravity and capillary imbibition. This new formulation improves dual-medium simulations of fractured reservoirs with an accurate representation of matrix-fracture exchanges, and better reserves assessment and reservoir management.

A method to derive approximate explicit solutions for structural mechanics problems

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

We consider n-input n-output convolution feedback systems characterized by y = G * e and e = u - y, where the open-loop transfer function ĝ contains a finite number of unstable poles and the expansion ĝa(s) + ∑i=0 ĝ Gi exp(-sti). The latter is such that, i) ĝa is the Laplace transform of L1 functions, ii) t0 = 0, iii) the delay-times ti, for i = 1, 2, ..., are positive and iv) the coefficient matrices Gi form an absolutely convergent series, i.e. ∑i=0 ĝ |Gi| < ∞. Thus the subsystem represented by G is an open-loop unstable distributed parameter system. One theorem is obtained. It gives necessary and sufficient conditions for stability. A basic device is the following: the sum of the principal parts of the Laurent expansions of ĝ at the unstable poles is factored as a ratio of two right coprime polynomial matrices. There are two necessary and sufficient conditions, the first is the usual infimum one, i.e. infRe s ĝ 0 |det[I+ ĝ(s)]| > 0, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition has a nice interpretation in terms of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate. The same is true if, instead of unity feedback, constant nonunity feedback represented by a nonsingular matrix is used. This allows us, through the use of the small gain theorem or the contraction mapping theorem, to consider non-linear systems, derived from the original one by introducing a non-linear time-varying gain in the forward loop.

Multiple LU factorizations of a singular matrix

In this letter, the Pascal matrix is used for transforming the normalized analog transfer function H(s) from the lowpass to the lowpass and highpass discrete transfer functions H(z). This algorithm is very simple; therefore, the transfer function H(s) can be easily transformed to the z domain using an appropriate calculator. The inverse Pascal matrix can be obtained without computing the determinant of the system, and then it is very easy to obtain the associated analog transfer function H(s) if the discrete transfer function H(z) is known.

This paper deals with real-time implementation of visual-motor control of a 7 degree of freedom (DOF) robot manipulator using self-organized map (SOM) based learning approach. The robot manipulator considered here is a 7 DOF PowerCube manipulator from Amtec Robotics. The primary objective is to reach a target point in the task space using only a single step movement from any arbitrary initial configuration of the robot manipulator. A new clustering algorithm using Kohonen SOM lattice has been proposed that maintains the fidelity of training data. Two different approaches have been proposed to find an inverse kinematic solution without using any orientation feedback. In the first approach, the inverse Jacobian matrices are learnt from the training data using function decomposition. It is shown that function decomposition leads to significant improvement in accuracy of inverse kinematic solution. In the second approach, a concept called sub-clustering in configuration space is suggested to provide multiple solutions for the inverse kinematic problem. Redundancy is resolved at position level using several criteria. A redundant manipulator is dexterous owing to the availability of multiple configurations for a given end-effector position. However, existing visual motor coordination schemes provide only one inverse kinematic solution for every target position even when the manipulator is kinematically redundant. Thus, the second approach provides a learning architecture that can capture redundancy from the training data. The training data are generated using explicit kinematic model of the combined robot manipulator and camera configuration. The training is carried out off-line and the trained network is used on-line to compute the joint angle vector to reach a target position in a single step only. The accuracy attained is better than the current state of art.

A novel method for factorizing the transfer function matrix of a multivariable system described by the triple (A, B, C) is presented. The factorization consists of a product of two rational function matrices, one of which is the inverse of a rational function matrix. Various properties of the factorization are given and the factorization procedure is illustrated by an example.

In this paper, we discuss the extension of carrier-assisted differential detection (CADD) from using a conventional interferometer to broad ranges of transfer functions. We show that by using a transfer function based on micro-ring resonator, the receiver sensitivity of CADD can be greatly improved.

High‐intensity interval exercise (HIIE) is recommended because it provides favourable haemodynamic stimulation, but excessive haemodynamic fluctuations may be an adverse impact on the brain. We tested whether the cerebral vasculature is protected against systemic blood flow fluctuation during HIIE. Fourteen healthy men (age 24 ± 2 years) underwent four 4‐min exercises at 80–90% of maximal workload (Wmax) interspaced by 3‐min active rest at 50–60% Wmax. Transcranial Doppler measured middle cerebral artery blood velocity (CBV). Systemic haemodynamics (Modelflow) and aortic pressure (AoP, general transfer function) were estimated from an invasively recorded brachial arterial pressure waveform. Using transfer function analysis, gain and phase between AoP and CBV (0.39–10.0 Hz) were calculated. Stroke volume, aortic pulse pressure and pulsatile CBV increased during exercise (time effect: P < 0.0001 for all), but a time‐domain index of aortic–cerebral pulsatile transition (pulsatile CBV/pulsatile AoP) decreased throughout the exercise bouts (time effect: P < 0.0001). Furthermore, transfer function gain reduced, and phase increased throughout the exercise bouts (time effect: P < 0.0001 for both), suggesting the attenuation and delay of pulsatile transition. The cerebral vascular conductance index (mean CBV/mean arterial pressure; time effect: P = 0.296), an inverse index of cerebral vascular tone, did not change even though systemic vascular conductance increased during exercise (time effect: P < 0.0001). The arterial system to the cerebral vasculature may attenuate pulsatile transition during HIIE as a defence mechanism against pulsatile fluctuation for the cerebral vasculature.

Run-to-run (R2R) control plays a vital role in monitoring or adjusting the manufacturing process of integrated circuits. In this article we propose a generalized quasi-MMSE controller for a process whose Input-Output (I-O) model follows a general Transfer Function (TF) model with ARIMA disturbance and analytically derive the long-term stability conditions and their limiting distribution. Furthermore, we use a comprehensive simulation study to compare the control performances among several potential controllers when the process I-O model follows a TF model of order (2, 2, 0) with ARIMA disturbance of order (2, 1, 2). The results demonstrate that using improper controllers may seriously affect the control performance in terms of the long-term stability conditions and the short-term total mean squared error. Supplementary materials are available for this article. Go to this article’s online edition of IIE Transactions for Appendices.