The statistics of stiff chain molecules I. The particle scattering factor

1970 ◽  
Vol 316 (1525) ◽  
pp. 185-199 ◽  
Keyword(s):  
Distribution Functions ◽  
Chain Model ◽  
Particle Scattering ◽  
Hindered Rotation ◽  
Chain Molecules ◽  
Chain Stiffness ◽  
Angle Scattering ◽  
Analytic Expressions ◽  
Scattering Factor ◽  
The Difference

Analytic expressions for the particle scattering factor of stiff chains have been derived, both for the wormlike and a discrete chain model with an axial symmetric potential of hindered rotation. The angular distribution functions agree well with the results of Monte Carlo calculations by Heine, Kratky & Roeppert (1962), if the chains are longer than five persistence lengths. The particle scattering factor of short chains can be well represented by the simple Guinier approximation. A transition point from the behaviour of a coil to that of the rod-like short chain sections has been determined by graphical extrapolation and appears at X a = 2.87 ± 0.05. The difference between the wormlike and the discrete chain models turned out to be smaller than 14% even for an alkane type chain with free rotation of the chain elements and decreases with increasing chain stiffness. The influence of the cross-section has been taken into account by representing the chain by a pearl necklace. Comparison with X -ray small angle scattering measurements of a cellulosetricarbanilate reveals close similarities between calculated and experimental curves.

Phenylalanyl-tRNA Synthetase from Baker'Yeast: Structural Organization of the Enzyme and Its Complex with tRNAPhe as Determined by X-Ray Small-Angle Scattering

1979 ◽  
Vol 34 (1-2) ◽  
pp. 20-26 ◽  
Author(s):  
Ingrid Pilz ◽  
Karin Goral ◽  
Friedrich v. d. Haar
Keyword(s):  
Molecular Weight ◽  
Small Angle ◽  
Quaternary Structure ◽  
Trna Synthetase ◽  
Distribution Functions ◽  
Maximum Diameter ◽  
Radius Of Gyration ◽  
X Ray ◽  
Angle Scattering ◽  
The Difference

Abstract The quaternary structure of the phenylalanyl-tRNA synthetase and its complex with tRNAPhe was studied in dilute solutions by small angle X-ray scattering. For the free synthetase the radius of gyration was determined as 5.5 nm, the volume 523 nm3, the maximum diameter 17.5 nm and the molecular weight as 260 000 using an isopotential specific volume of 0.735. The overall shape could be best approximated by a flat cylinder with dimensions 18.2 nm X 11.5 nm X 4 nm ; the loose structure was approximated by building up the cylinder by spheres (diameter 4.2 nm).The corresponding parameters of the enzyme tRNA complex were the following: radius of gyration 5.9 nm, volume 543 nm 3, maximum diameter 21 nm and molecular weight 290 000. These parameters suggest an 1:1 complex, whereby it must be assumed that the tRNA molecule is attached in the extension of the longer axis. From the difference in the distance distribution functions of the free enzyme and the complex it is evident that we have to assume a change of conformation (contraction) of the enzyme upon the binding of the specific tRNA.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

Extended two-dimensional belief function based on divergence measurement

2020 ◽  
pp. 1-8
Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu
Keyword(s):  
Belief Function ◽  
Distribution Functions ◽  
Divergence Measure ◽  
Uncertain Information ◽  
Belief Functions ◽  
Two Dimensional ◽  
Probability Distribution Functions ◽  
Basic Probability ◽  
The Difference ◽  
Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

scholarly journals Pentagon functions for scattering of five massless particles

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
D. Chicherin ◽  
V. Sotnikov
Keyword(s):  
Phase Space ◽  
Numerical Evaluation ◽  
Public Library ◽  
Basis Set ◽  
Particle Scattering ◽  
Minimal Basis ◽  
Feynman Integrals ◽  
Analytic Expressions ◽  
Transcendental Functions ◽  
Branch Cuts

Abstract We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.

Subgrid modelling of ion pitch-angle scattering for magnetosheath waves in a global hybrid-Vlasov simulation

2021 ◽  
Author(s):  
Maxime Dubart ◽  
Urs Ganse ◽  
Adnane Osmane ◽  
Andreas Johlander ◽  
Markus Battarbee ◽  
...  
Keyword(s):  
Velocity Distribution ◽  
Numerical Simulations ◽  
Spatial Resolution ◽  
Pitch Angle ◽  
Space Physics ◽  
Distribution Functions ◽  
Cyclotron Instability ◽  
Angle Scattering ◽  
Velocity Distribution Functions ◽  
Proton Cyclotron

<p>Numerical simulations are widely used in modern space physics and are an essential tool to understand or discover new phenomena which cannot be observed using spacecraft measurements. However, numerical simulations are limited by the space grid resolution of the system and the computational costs of having a high spatial resolution. Therefore, some physics may be unresolved in part of the system due to its low spatial resolution. We have previously identified, using Vlasiator, that the proton cyclotron instability is not resolved for grid cell sizes larger than four times the inertial length in the solar wind, for waves in the downstream of the quasi-perpendicular shock in the magnetosheath of a global hybrid-Vlasov simulation. This leads to unphysically high perpendicular temperature and a dominance of the mirror mode waves. In this study, we use high-resolution simulations to measure and quantify how the proton cyclotron instability diffuses and isotropizes the velocity distribution functions. We investigate the process of pitch-angle scattering during the development of the instability and propose a method for the sub-grid modelling of the diffusion process of the instability at low resolution. This allows us to model the isotropization of the velocity distribution functions and to reduce the temperature anisotropy in the plasma while saving computational resources.</p>

Calculations and Simulations

1997 ◽  
Author(s):  
Burak Erman ◽  
James E. Mark
Keyword(s):  
Gaussian Approximation ◽  
Rubber Elasticity ◽  
Chain Model ◽  
Chain Molecules ◽  
Rotational Isomeric State ◽  
End To End ◽  
State Models ◽  
Areas Of Interest ◽  
Segmental Orientation

The classical theories of rubber elasticity are based on the Gaussian chain model. The only molecular parameter that enters these theories is the mean-square end-to-end separation of the chains constituting the network. However, there are various areas of interest that require characterization of molecular quantities beyond the Gaussian description. Examples are segmental orientation, birefringence, rotational isomerization, and finite extensibility, and we will address these properties in the following chapters. One often needs a more realistic distribution function for the end-to-end vector, as well as for averages of the products of several vectorial quantities, as will be evident in these chapters. The foundations for such characterizations, and several examples of their applications, are given in this chapter. Several aspects of rubber elasticity (such as the dependence of the elastic free energy on network topology, number of effective junctions, and contributions from entanglements) are successfully explained by theories based on the freely jointed chain and the Gaussian approximation. Details of the real chemical structure are not required at the length scales describing these phenomena. On the other hand, studies of birefringence, thermoelasticity, rotational isomerization upon stretching, strain dichroism, local segmental orientation and mobility, and characterization of networks with short chains require the use of more realistic network chain models. In this section, properties of rotational isomeric state models for the chains are discussed. The notation is based largely on the Flory book, Statistical Mechanics of Chain Molecules. More recent information is readily found in the literature. Due to the simplicity of its structure, a polyethylene-like chain serves as a convenient model for discussing the statistical properties of real chains. This simplicity can be seen in figure 8.1, which shows the planar form of a small portion of a polyethylene chain. Bond lengths and bond angles may be regarded as fixed in the study of rubber elasticity because their rapid fluctuations are usually in the range of only ±0.05 A and ±5°, respectively. The chain changes its configuration only through torsional rotations about the backbone bonds, shown, for example, by the angle for the ith bond in figure 8.1.

scholarly journals Modeling of public transport waiting time indicator for the transport network of a large city

2018 ◽  
Vol 224 ◽  
pp. 04018 ◽  
Author(s):  
Olga Lebedeva ◽  
Marina Kripak
Keyword(s):  
Waiting Time ◽  
Large City ◽  
Distribution Functions ◽  
Weibull Function ◽  
Explanatory Variables ◽  
Transport Services ◽  
Time Period ◽  
Bus Stops ◽  
The Difference ◽  
Expected Waiting Time

The need to develop and improve public passenger transport in major cities was noted. It was reflected that waiting time at bus stops is one of the factors that have a big impact on the passenger quality assessment of transport services. The results of an empirical study of the actual and anticipated waiting time at bus stops were given. It was noted that the reliability functions were used in the field of ride duration modeling, traffic restoration time after an accident, and length of making the decision to travel. The waiting time distribution functions using the lognormal function and the Weibull function were chosen. The results of modeling were objective, the dependent variables in it were the expected waiting time of passengers and the difference between the anticipated and the actual waiting time. The explanatory variables were sex, age, time period, purpose of the trip and the actual waiting time. The results of the research showed that the age, purpose of the trip and the time period influence the waiting time perception, prolong it and lead to its reassessment.

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