Effect of an electric field on a methane molecule. II. Calculation of the degeneracy splitting of the ν3 band. Expression of the second derivatives of the CH4 dipole moment and evaluation of the second derivative of the C–H bond polarizability

1985 ◽  
Vol 83 (6) ◽  
pp. 2653-2660 ◽  
Author(s):  
R. Kahn ◽  
E. Cohen De Lara ◽  
K. D. Möller
Keyword(s):  
Dipole Moment ◽  
Electric Field ◽  
Methane Molecule ◽  
Second Derivative ◽  
Second Derivatives ◽  
Bond Polarizability ◽  
Derivatives Of

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

scholarly journals Advanced analysis in epidemiological modeling: Detection of wave

2021 ◽  
Author(s):  
Abdon Atangana ◽  
Seda IGRET ARAZ
Keyword(s):  
Simple Model ◽  
Reproductive Number ◽  
Second Derivative ◽  
Mathematical Concepts ◽  
Epidemiological Modeling ◽  
Derivative Analysis ◽  
Second Derivatives ◽  
Advanced Analysis ◽  
The Stability ◽  
Derivatives Of

Some mathematical concepts have been used in the last decades to predict the behavior of spread of infectious diseases. Among them, the reproductive number concept has been used in several published papers for study the stability of the spread. Some conditions were suggested to predict there would be either stability or instability. An analysis was also suggested to determine conditions under which infectious classes will increase or die out. Some authors pointed out limitations of the reproductive number, as they presented its inability to fairly help understand the spread patterns. The concept of strength number and analysis of second derivatives of the mathematical models were suggested as additional tools to help detect waves. In this paper, we aim at applying these additional analyses in a simple model to predict the future. Keywords: Strength number, second derivative analysis, waves, piecewise modeling.

Reanalysis Method for Second Derivatives of Static Displacement

2019 ◽  
Vol 17 (08) ◽  
pp. 1950056 ◽  
Author(s):  
Wenjie Zuo ◽  
Jiaxin Fang ◽  
Zengming Feng
Keyword(s):  
Numerical Results ◽  
Second Derivative ◽  
Static Displacement ◽  
Normalized Error ◽  
Second Derivatives ◽  
Algebraic Operations ◽  
Combined Approximations Method ◽  
Derivatives Of ◽  
Combined Approximations

The reanalysis method to obtain the second derivatives of static displacement is innovatively proposed in this paper. This method is based on the combined approximations method. The reanalysis formulations of the second derivative of static displacement are derived to provide a programmatic procedure of formulations construction. Besides, the normalized error and the number of algebraic operations are considered to evaluate the accuracy and efficiency, respectively. Finally, three typical numerical results verify the accuracy and robustness of the proposed method.

Traveltime and wavefront curvature calculations in three‐dimensional inhomogeneous layered media with curved interfaces

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1466-1494 ◽  
Author(s):  
H. Gjøystdal ◽  
J. E. Reinhardsen ◽  
B. Ursin
Keyword(s):  
Taylor Series ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Reference Source ◽  
Second Derivative ◽  
Second Derivatives ◽  
Derivative Matrix ◽  
Three Dimensional Models ◽  
Derivatives Of

The seismic rays and wavefront curvatures are determined by solving a system of nonlinear ordinary differential equations. For media with constant velocity and for media with constant velocity gradient, simplified solutions exist. In a general inhomogeneous medium these equations must be solved by numerical approximations. The integration of the ray‐tracing and wavefront curvature equations is then performed by a modified divided difference form of the Adams PECE (Predict‐Evaluate‐Correct‐Evaluate) formulas and local extrapolation. The interfaces between the layers are represented by bicubic splines. The changes in ray direction and wavefront curvature at the interfaces are computed using standard formulas. For three‐dimensional media, two quadratic traveltime approximations have been proposed. Both are based on a Taylor series expansion with reference to a ray from a reference source point to a reference receiver point. The first approximation corresponds to expanding the square of the traveltime in a Taylor series and taking the square root of the result. The second approximation corresponds to expanding the traveltime in a Taylor series. The two traveltime approximations may be expressed in source‐receiver coordinates or in midpoint‐half‐offset coordinates. Simplified expressions are obtained when the reference source and receiver coincide, giving zero‐offset approximations, for which the reference ray is a normal‐incidence ray. A new method is proposed for computing the second derivatives of the normal‐incidence traveltime with respect to the source‐receiver midpoint coordinates. By considering a beam of normal‐incidence rays it is shown that the second‐derivative matrix may be found by computing the wavefront curvature along a reference normal‐incidence ray starting at the reflection point with the wavefront curvature equal to the curvature of the reflecting interface. From this second‐derivative matrix the normal moveout velocity can be computed for any seismic line through the reference source‐receiver midpoint. It is also shown how a reverse wavefront curvature calculation may be used, in a time‐to‐depth migration scheme, to compute the curvature of the reflecting interface from the estimated second derivatives of the normal‐incidence traveltime. Numerical results for different three‐dimensional models indicate that the first traveltime approximation, based on an expansion of the square of the traveltime, is the most accurate for shallow reflectors and for simple models. For deeper reflectors the two approximations give comparable results, and for models with complicated velocity variations the second approximation may be slightly better than the first one, depending on the particular model chosen. A simplified traveltime approximation may be used in a three‐dimensional seismic velocity analysis. Instead of estimating the stacking velocity one must estimate three elements in a [Formula: see text] symmetric matrix. The accuracy and range of validity of the simplified traveltime approximation are investigated for different three‐dimensional models.

scholarly journals Polaron in Electric Field and Vibrational Spectrum of Polyacetylene

2019 ◽  
Vol 14 (1) ◽  
pp. 150-159 ◽  
Author(s):  
T.Yu. Astakhova ◽  
G.A. Vinogradov
Keyword(s):  
Electric Field ◽  
Vibrational Spectrum ◽  
Phase Velocity ◽  
Additional Condition ◽  
Fourier Spectrum ◽  
Optical Branch ◽  
System A ◽  
Second Derivatives ◽  
The Matrix ◽  
Derivatives Of

In the present paper, the evolution of a polaron in polyacetylene in electric field is studied in detail. It is shown that the dependence of the polaron velocity on the applied field is determined by the peculiarities of the vibrational spectrum of polyacetylene. The spectrum is calculated numerically using the Hessian, the matrix of the second derivatives of the potential energy of the system. A supersonic polaron moving in an electric field generates coherent vibrations of the optical branch of the spectrum of the polyacetylene. The phase velocity of these oscillations is equal to the polaron velocity. An additional condition for the excitation of a mode with a certain phase velocity is its correspondence to the Fourier spectrum of an impulse with polaron shape.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

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