Universality among topological properties of electron density associated with the hydrogen–hydrogen nonbonding interactions

1992 ◽  
Vol 70 (2) ◽  
pp. 443-449 ◽  
Author(s):  
Jerzy Cioslowski ◽  
Stacey T. Mixon
Keyword(s):  
Electron Density ◽  
Functional Dependencies ◽  
Hydrogen Atoms ◽  
Topological Features ◽  
Second Derivatives ◽  
The Difference ◽  
Point Distance ◽  
Torsional Angles ◽  
Derivatives Of ◽  
Bond Ellipticity

Nonbonding repulsive interactions between hydrogen atoms, separated by less than ca. 2.18 Å and connected to two carbon atoms in the 1,4-positions, are associated with additional pairs of bond and ring critical points in the electron density, ρ(r), and the corresponding attractor interaction lines. Such topological features of ρ(r) are present in some planar benzenoid hydrocarbons, including chrysene, benzanthracene, and phenanthrene. They also appear in conformations of the biphenyl molecule with the torsional angles between the benzene rings lying within the range of 0°–27°. Properties such as the bond point – ring point distance, the difference in ρ(r) at the bond point and the ring point, and the bond ellipticity are found to follow universal functional dependencies with the hydrogen–hydrogen distance as the controlling variable. The same is true about the corresponding difference in the second derivatives of ρ(r) in the direction of the vector connecting the bond and ring points. Keywords: atoms in molecules, electron density, attractor interaction lines, steric repulsions.

scholarly journals On the difference method for approximation of second order derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 633-643
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya
Keyword(s):  
Difference Schemes ◽  
Rectangular Parallelepiped ◽  
Finite Difference Schemes ◽  
Compatibility Conditions ◽  
Averaging Operator ◽  
Second Derivatives ◽  
Boundary Functions ◽  
The Difference ◽  
Derivatives Of ◽  
Theoretical Results

We present and justify finite difference schemes with the 14-point averaging operator for the second derivatives of the solution of the Dirichlet problem for Laplace?s equations on a rectangular parallelepiped. The boundary functions ?j on the faces ?j,j = 1,2,..., 6 of the parallelepiped are supposed to have fifth derivatives belonging to the H?lder classes C5?, 0 < ? < 1. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. It is proved that the proposed difference schemes for the approximation of the pure and mixed second derivatives converge uniformly with order O(h3+?), 0 < ? < 1 and O(h3), respectively. Numerical experiments are illustrated to support the theoretical results.

On Jenkins-Strebel differentials for open Riemann surfaces

1980 ◽  
Vol 86 (3-4) ◽  
pp. 315-325
Author(s):  
Frederick P. Gardiner
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Infinite Family ◽  
Extremal Length ◽  
Extremum Problems ◽  
Second Derivatives ◽  
The Difference ◽  
Difference Quotients ◽  
Derivatives Of

SynopsisDual extremum problems associated with an infinite family of admissible loops on a Riemann surface are shown to be solvable by Jenkins-Strebel differentials. Then the inequalities associated with these problems are used to calculate the first derivatives of extremal length functionals on Teichmüller space and to estimate the difference quotients for the second derivatives of these functions.

scholarly journals WAVE PROCESSES IN A FLAT BODY WITH A SYMMETRIC LOCATED RECTANGULAR CUTOUT

2020 ◽  
Vol 5 (333) ◽  
pp. 10-18
Author(s):  
N. K. Ashirbayev ◽  
◽  
Zh.N. Ashirbayeva ◽  
M.T. Shomanbayeva ◽  
R.B. Bekmoldayeva ◽  
...  
Keyword(s):  
Numerical Technique ◽  
Dynamic Problems ◽  
Wave Processes ◽  
Local Character ◽  
Second Derivatives ◽  
Corner Points ◽  
The Difference ◽  
Propagation Of Waves ◽  
The Impact ◽  
Derivatives Of

The work is devoted to the generalization of the difference method of spatial characteristics to the case of the plane problem of the propagation of waves in a rectangular region of finite dimensions with a symmetrically-located rectangular cutout at the lateral boundaries. Based on the numerical technique developed in this work, the calculated finite - difference relations of dynamic problems are obtained at the corner points of a rectangular cutout, where the smoothness of functions that is “familiar” to dynamic problems is violated. At these corner points, the first and second derivatives of the desired functions suffer a discontinuity of the first kind. The results of the study are brought to a numerical solution. The effect of stress concentration in the vicinity of the cutout was studied and it was shown that the impact of the cutout on the particle velocity distribution, on the stress distribution has a local character.

Energy Approximation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Main Lemma ◽  
The Other ◽  
Coarse Scale ◽  
Continuous Solutions ◽  
Material Derivative ◽  
Energy Variation ◽  
Energy Approximation ◽  
The Difference ◽  
Derivatives Of

This chapter presents the equations and calculations for energy approximation. It establishes the estimates (261) and (262) of the Main Lemma (10.1) for continuous solutions; these estimates state that we are able to accurately prescribe the energy that the correction adds to the solution, as well as bound the difference between the time derivatives of these two quantities. The chapter also introduces the proposition for prescribing energy, followed by the relevant computations. Each integral contributing to the other term can be estimated. Another proposition for estimating control over the rate of energy variation is given. Finally, the coarse scale material derivative is considered.

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.

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.

A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

1990 ◽  
Vol 112 (1) ◽  
pp. 83-87 ◽  
Author(s):  
R. H. Fries ◽  
B. M. Coffey
Keyword(s):  
Numerical Simulation ◽  
White Noise ◽  
State Space ◽  
Spectral Characteristics ◽  
The State ◽  
Second Derivatives ◽  
Time Histories ◽  
Sample Interval ◽  
Rail Irregularities ◽  
Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

Effects of avertin versus xylazine-ketamine anesthesia on cardiac function in normal mice

2001 ◽  
Vol 281 (5) ◽  
pp. H1938-H1945 ◽  
Author(s):  
Chari Y. T. Hart ◽  
John C. Burnett ◽  
Margaret M. Redfield
Keyword(s):  
Cardiac Function ◽  
Diastolic Pressure ◽  
Systolic Function ◽  
Pressure Rise ◽  
Left Ventricular ◽  
Lv Function ◽  
Ketamine Anesthesia ◽  
The Difference ◽  
And Function ◽  
Derivatives Of

Anesthetic regimens commonly administered during studies that assess cardiac structure and function in mice are xylazine-ketamine (XK) and avertin (AV). While it is known that XK anesthesia produces more bradycardia in the mouse, the effects of XK and AV on cardiac function have not been compared. We anesthetized normal adult male Swiss Webster mice with XK or AV. Transthoracic echocardiography and closed-chest cardiac catheterization were performed to assess heart rate (HR), left ventricular (LV) dimensions at end diastole and end systole (LVDd and LVDs, respectively), fractional shortening (FS), LV end-diastolic pressure (LVEDP), the time constant of isovolumic relaxation (τ), and the first derivatives of LV pressure rise and fall (dP/d t max and dP/d t min, respectively). During echocardiography, HR was lower in XK than AV mice (250 ± 14 beats/min in XK vs. 453 ± 24 beats/min in AV, P < 0.05). Preload was increased in XK mice (LVDd: 4.1 ± 0.08 mm in XK vs. 3.8 ± 0.09 mm in AV, P < 0.05). FS, a load-dependent index of systolic function, was increased in XK mice (45 ± 1.2% in XK vs. 40 ± 0.8% in AV, P < 0.05). At LV catheterization, the difference in HR with AV (453 ± 24 beats/min) and XK (342 ± 30 beats/min, P < 0.05) anesthesia was more variable, and no significant differences in systolic or diastolic function were seen in the group as a whole. However, in XK mice with HR <300 beats/min, LVEDP was increased (28 ± 5 vs. 6.2 ± 2 mmHg in mice with HR >300 beats/min, P < 0.05), whereas systolic (LV dP/d t max: 4,402 ± 798 vs. 8,250 ± 415 mmHg/s in mice with HR >300 beats/min, P < 0.05) and diastolic (τ: 23 ± 2 vs. 14 ± 1 ms in mice with HR >300 beats/min, P < 0.05) function were impaired. Compared with AV, XK produces profound bradycardia with effects on loading conditions and ventricular function. The disparate findings at echocardiography and LV catheterization underscore the importance of comprehensive assessment of LV function in the mouse.

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