Mathematical derivation of the Leontief inverse

2017 ◽  
pp. 83-83
Author(s):  
Keyword(s):  
Leontief Inverse ◽  
Mathematical Derivation

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

Nonlinear Axisymmetric Deformation Model for Structures of Revolution

2011 ◽  
Vol 3 (4) ◽  
pp. 420-447
Author(s):  
Ayman Mourad ◽  
Jawad Zaarour
Keyword(s):  
Coordinate System ◽  
Three Dimensional ◽  
Axisymmetric Deformation ◽  
Large Displacement ◽  
Deformation Model ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Cylindrical Coordinate ◽  
Mathematical Derivation ◽  
Displacement Effects

AbstractAn axisymmetric formulation for modeling three-dimensional deformation of structures of revolution is presented. The axisymmetric deformation model is described using the cylindrical coordinate system. Large displacement effects and material nonlinearities and anisotropy are accommodated by the formulation. Mathematical derivation of the formulation is given, and an example is presented to demonstrate the capabilities and efficiency of the technique compared to the full three-dimensional model.

The Maxslope Taxometric Procedure: Mathematical Derivation, Parameter Estimation, Consistency Tests

2004 ◽  
Vol 95 (2) ◽  
pp. 517-550 ◽  
Author(s):  
William M. Grove
Keyword(s):  
Parameter Estimation ◽  
Nonlinear Regression ◽  
Classification Accuracy ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Indicator Variable ◽  
Regression Slope ◽  
The Relationship ◽  
Mathematical Derivation ◽  
Made In

This article first explains concepts in taxometrics, including the meaning of “taxon” in relation to taxometric procedures. It then mathematically develops the MAXSLOPE procedure of Grove and Meehl which relies on nonlinear regression of one taxometric indicator variable on another. Sufficient conditions for MAXSLOPE's validity are set forth. The relationship between the point of maximum regression slope (MAXSLOPE point) and the HITMAX cut, i.e., the point on a variable which, if used as a diagnostic cut-off score, yields maximum classification accuracy, is analyzed. A sufficient condition is given for the MAXSLOPE point to equal the HITMAX cut; however, most distributions have different MAXSLOPE and HITMAX points. Equations and an algorithm are spelled out for making a graphical test for the existence of a taxon, estimating taxometric parameters, and conducting consistency tests; the latter serve as stringent checks on the validity of a taxonic conjecture. The plausibility of assumptions made, in deriving MAXSLOPE equations, is discussed, and the qualitative effects of violations of these assumptions are explained.

Determination of Singularity-Free Zones in the Workspace of Planar 3-P̱RR Parallel Mechanisms

2006 ◽  
Vol 129 (6) ◽  
pp. 649-652 ◽  
Author(s):  
Mehdi Tale Masouleh ◽  
Clément Gosselin
Keyword(s):  
Parallel Mechanisms ◽  
Numerical Examples ◽  
Mathematical Derivation

This paper presents an algorithm for the determination of singularity-free zones in the workspace of the planar 3-P̱RR mechanism. The mathematical derivation of the algorithm is first given. Numerical examples are then included to demonstrate the application of the proposed approach.

scholarly journals Exact Solution for the Torsional Vibration of an Elastic Pile in a Radially Inhomogeneous Saturated Soil

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Xibin Li ◽  
Zhiqing Zhang ◽  
Jianchao Sheng
Keyword(s):  
Exact Solution ◽  
Torsional Vibration ◽  
Saturated Soil ◽  
Exponential Form ◽  
Soil System ◽  
Time Harmonic ◽  
Radial Inhomogeneity ◽  
Inner Region ◽  
Radially Inhomogeneous ◽  
Mathematical Derivation

An exact solution is proposed to study the time-harmonic torsional vibration of an elastic pile embedded in a radially inhomogeneous saturated soil. The radially inhomogeneous saturated soil is composed of inner disturbed and outer semi-infinite undisturbed concentric annular regions, with the shear modulus of the inner region changing in an exponential form along the radial direction. The governing equation of each region of the saturated soil is solved through rigorous mathematical derivation and the soil torsional impedance is derived with an exact and explicit expression. Making use of the boundary and continuity conditions of the pile-soil system, the torsional complex stiffness at the pile top is obtained in an exact closed form in the frequency domain. Selected numerical results are presented to investigate the influence of the radial inhomogeneity of the surrounding soil on the vibration characteristics of the pile-soil system.

scholarly journals Reply to “Comments on ‘Incorporating the Effects of Moisture into a Dynamical Parameter: Moist Vorticity and Moist Divergence’”

2016 ◽  
Vol 31 (4) ◽  
pp. 1397-1405
Author(s):  
Weihong Qian ◽  
Ning Jiang ◽  
Jun Du
Keyword(s):  
Large Scale ◽  
Value Added ◽  
Heavy Rain ◽  
Heavy Precipitation ◽  
Small Scale ◽  
Precipitation Efficiency ◽  
Long Duration ◽  
Advection Term ◽  
Rain Events ◽  
Mathematical Derivation

Abstract Mathematical derivation, meteorological justification, and comparison to model direct precipitation forecasts are the three main concerns recently raised by Schultz and Spengler about moist divergence (MD) and moist vorticity (MV), which were introduced in earlier work by Qian et al. That previous work demonstrated that MD (MV) can in principle be derived mathematically with a value-added empirical modification. MD (MV) has a solid meteorological basis. It combines ascent motion and high moisture: the two elements necessary for rainfall. However, precipitation efficiency is not considered in MD (MV). Given the omission of an advection term in the mathematical derivation and the lack of precipitation efficiency, MD (MV) might be suitable mainly for heavy rain events with large areal coverage and long duration caused by large-scale quasi-stationary weather systems, but not for local intense heavy rain events caused by small-scale convection. In addition, MD (MV) is not capable of describing precipitation intensity. MD (MV) worked reasonably well in predicting heavy rain locations from short to medium ranges as compared with the ECMWF model precipitation forecasts. MD (MV) was generally worse than (though sometimes similar to) the model heavy rain forecast at shorter ranges (about a week) but became comparable or even better at longer ranges (around 10 days). It should be reiterated that MD (MV) is not intended to be a primary tool for predicting heavy rain areas, especially in the short range, but is a useful parameter for calibrating model heavy precipitation forecasts, as stated in the original paper.

Mathematical derivation of the minimally acceptable all-porcelain margin angle

2005 ◽  
Vol 93 (5) ◽  
pp. 467-472 ◽  
Author(s):  
Deborah L. Dalvit ◽  
R.A. Jackson ◽  
M. Chad Hawkins ◽  
M. Harry Parker
Keyword(s):  
Mathematical Derivation
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