A perturbed trapezoid inequality in terms of the fourth derivative

2002 ◽  
Vol 9 (1) ◽  
pp. 45-60 ◽  
Author(s):  
N. S. Barnett ◽  
S. S. Dragomir
Keyword(s):  
Trapezoid Inequality ◽  
Fourth Derivative

Nonlinear solution of the reaction–diffusion equation using a two-step third–fourth-derivative block method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Ali Aldalbahi ◽  
Jawad Raza ◽  
Zurni Omar ◽  
Mostafizur Rahaman ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Block Method ◽  
Wide Range ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Improved Accuracy

AbstractThe processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

OPTICAL SENSOR SYSTEM FOR THE NON-INVASIVE ASSESSMENT OF ARTERIAL STIFFNESS QUANTIFIED BY FOURTH DERIVATIVE OF PHOTOPLETHYSMOGRAM

2015 ◽  
Vol 27 (03) ◽  
pp. 1550021
Author(s):  
S. Mohanalakshmi ◽  
A. Sivasubramanian
Keyword(s):  
Arterial Stiffness ◽  
Elastic Properties ◽  
Healthy Subjects ◽  
Vascular System ◽  
Augmentation Index ◽  
Cardiac Risk ◽  
Contour Analysis ◽  
Non Invasive ◽  
Asymmetrical Shape ◽  
Fourth Derivative

Arterial stiffness, resulting in loss of the elastic properties of arteries walls, is an indicator of cardiovascular risk, though the presence of disease is not clinically evident. Augmentation index is an important biomarker of arterial stiffness by which the cardiac risk of the patient can be diagnosed. The current paper outlines the non-invasive assessment of arterial stiffness by analyzing the morphology or contour of PhotoPlethysmoGraph (PPG) signal. PPG pulse was optically acquired with the developed photometric measurement device and the desired features were extracted to determine PPG augmentation Index (PAI) through advanced signal processing implemented in MATLAB. PAI was quantified by the fourth derivative of the signal by enhancing the location of inflection point (augmentation point) after conditioning the signal by efficient pre-processing and filtering techniques. The results reveal that the statistical distribution of PAI for healthy subjects presents a very low value and a very tight distribution. On the contrary, patients have a higher value of PAI and a wide asymmetrical shape of distribution. This work also establishes the usefulness of PPG contour analysis in the investigation of changes in the elastic properties of the vascular system. In conclusion, PAI has revealed to be a non-invasive indicator for arterial stiffness assessments.

Velocity estimation in laterally varying media

Geophysics ◽  
1982 ◽  
Vol 47 (6) ◽  
pp. 884-897 ◽  
Author(s):  
Walter S. Lynn ◽  
Jon F. Claerbout
Keyword(s):  
Taylor Series Expansion ◽  
Velocity Estimation ◽  
Lateral Resolution ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Derivative Term ◽  
Fourth Derivative ◽  
Zero Offset ◽  
Lateral Variations

In areas of large lateral variations in velocity, stacking velocities computed on the basis of hyperbolic moveout can differ substantially from the actual root mean square (rms) velocities. This paper addresses the problem of obtaining rms or migration velocities from stacking velocities in such areas. The first‐order difference between the stacking and the vertical rms velocities due to lateral variations in velocity are shown to be related to the second lateral derivative of the rms slowness [Formula: see text]. Approximations leading to this relation are straight raypaths and that the vertical rms slowness to a given interface can be expressed as a second‐order Taylor series expansion in the midpoint direction. Under these approximations, the effect of the first lateral derivative of the slowness on the traveltime is negligible. The linearization of the equation relating the stacking and true velocities results in a set of equations whose inversion is unstable. Stability is achieved, however, by adding a nonphysical fourth derivative term which affects only the higher spatial wavenumbers, those beyond the lateral resolution of the lateral derivative method (LDM). Thus, given the stacking velocities and the zero‐offset traveltime to a given event as a function of midpoint, the LDM provides an estimate of the true vertical rms velocity to that event with a lateral resolution of about two mute zones or cable lengths. The LDM is applicable when lateral variations of velocity greater than 2 percent occur over the mute zone. At variations of 30 percent or greater, the internal assumptions of the LDM begin to break down. Synthetic models designed to test the LDM when the different assumptions are violated show that, in all cases, the results are not seriously affected. A test of the LDM on field data having a lateral velocity variation caused by sea floor topography gives a result which is supported by depth migration.

The renormalization group in perturbative quantum gravity

2021 ◽  
pp. 488-493
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Renormalization Group ◽  
Quantum Gravity ◽  
Minimal Subtraction Scheme ◽  
Loop Approximation ◽  
Renormalization Group Equations ◽  
Gauge Fixing ◽  
Perturbative Renormalization ◽  
Fourth Derivative ◽  
Perturbative Quantum ◽  
The One

This is a short chapter summarizing the main results concerning the renormalization group in models of pure quantum gravity, without matter fields. The chapter starts with a critical analysis of non-perturbative renormalization group approaches, such as the asymptotic safety hypothesis. After that, it presents solid one-loop results based on the minimal subtraction scheme in the one-loop approximation. The polynomial models that are briefly reviewed include the on-shell renormalization group in quantum general relativity, and renormalization group equations in fourth-derivative quantum gravity and superrenormalizable models. Special attention is paid to the gauge-fixing dependence of the renormalization group trajectories.

scholarly journals Gravitational waves and perspectives for quantum gravity

2014 ◽  
Vol 29 (30) ◽  
pp. 1430034 ◽  
Author(s):  
Ilya L. Shapiro ◽  
Ana M. Pelinson ◽  
Filipe de O. Salles
Keyword(s):  
Quantum Gravity ◽  
Gravitational Waves ◽  
Classical Solutions ◽  
Quantum Level ◽  
Quantum Matter ◽  
Higher Derivatives ◽  
Fourth Derivative ◽  
Matter Fields ◽  
Classical Background

Understanding the role of higher derivatives is probably one of the most relevant questions in quantum gravity theory. Already at the semiclassical level, when gravity is a classical background for quantum matter fields, the action of gravity should include fourth derivative terms to provide renormalizability in the vacuum sector. The same situation holds in the quantum theory of metric. At the same time, including the fourth derivative terms means the presence of massive ghosts, which are gauge-independent massive states with negative kinetic energy. At both classical and quantum level such ghosts violate stability and hence the theory becomes inconsistent. Several approaches to solve this contradiction were invented and we are proposing one more, which looks simpler than those what were considered before. We explore the dynamics of the gravitational waves on the background of classical solutions and give certain arguments that massive ghosts produce instability only when they are present as physical particles. At least on the cosmological background one can observe that if the initial frequency of the metric perturbations is much smaller than the mass of the ghost, no instabilities are present.

A Functional Generalization of Trapezoid Inequality

2015 ◽  
Vol 43 (4) ◽  
pp. 663-675
Author(s):  
Silvestru S. Dragomir
Keyword(s):  
Trapezoid Inequality

Fourth-derivative spectrophotometry of proteins

1984 ◽  
Vol 9 (12) ◽  
pp. 508-510 ◽  
Author(s):  
Esteve Padrós ◽  
Mireia Duñach ◽  
Antoni Morros ◽  
Manuel Sabés ◽  
Joan Mañosa
Keyword(s):  
Derivative Spectrophotometry ◽  
Fourth Derivative
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