Distribution Gaussian mathematical expression

In lack of analytical or numerical methods to obtain the spectra of complicated Hamiltonians, Wigner and Dyson analyzed ensembles of random matrices and were able to derive mathematical expressions. A Gaussian random matrix ensemble consists of square matrices with their matrix elements drawn from a Gaussian distribution... 

Kwong-Soave type). Epi is calculated from geometric considerations involving r, and H. Fa(H) can be related to the pore volume distribution fimction Fy(H) if the pore shape is known. In this work, we consider classical slit-like pores to describe the porous structure of the activated carbon and we take a bimodal gaussian for the mathematical expression of Fv(H). As a consequence ... 

The variation that is observed in experimental results can take many different forms or distributions. We consider here three of the best known that can be expressed in relatively straightforward mathematical terms the binomial distribution, the Poisson distribution and the Gaussian, or normal, distribution. These are all forms of parametric statistics which are based on the idea that the data are spread in a specific manner. Ideally, this should be demonstrated before a statistical analysis is carried out, but this is not often done. 

Since we are not able to find the true value for any parameter, we often make do with the average of all of the experimental data measured for that parameter, and consider this as the most probable value." Measured values, which necessarily contain experimental errors, should lie in a random manner on either side of this most probable value as expressed by the normal or Gaussian distribution. This distribution i. a bell-shaped curve that represents the number of measurements N that have a specific value x (which deviates from the mean or most probable value Xq by an amount x - Xo, representative of the error). Obviously the smaller the value of x - Xo, the higher the probability that the quantity being measured lies near the most likely value xq, which is at the top of the peak. A plot of N against x, shown in Figure 10.1, is called a Gaussian distribution or error curve, expressed mathematically as ... 

Solver, can provide parameters that fit a user-specified function to a set of data, but does not yield estimates of the precision of those parameters. Here we exploit the approach used in section 10.3 to compute the precision of the parameters obtained with Solver. We will make the usual assumptions that, in fitting a function F a ) to /Vexperimental data pairs x,y, all indeterminate uncertainties are restricted to y, and follow a single Gaussian distribution. Furthermore we will assume that Solver has already been used to find a solution ycaic based on a mathematical model expression of the type yta c = F (a(-), where at are the parameters Solver has adjusted. We can then use a second macro, called SolverAid, to estimate the standard deviations of those parameters a,. 

In a more general case of n and h being varied, the distribution with r(Wj) has a more sophisticated expression (according to Equation 90) (Yamzdkawa, 1971). In this connection, it seems reasonable to have such an approximation of r(flj) that would lead to the chief properties of the chain (Equations 97 and 100) with simpler mathematical operations. The Gaussian function... 

Finally, there is the possibility to simulate an experimental curve (spectrum) by a mathematical algorithm, e.g., by a polynomial, a Fourier transform expression, or the superposition of Gaussian or other suitable distribution curves (cf. Sec. 2.3.4, Eq. (2-41) (2-47)). In this case, one must keep in mind that for simulation of real spectra it is also necessary to add a noise function, produced by a random generator, to the PC-computed curve. Otherwise, it is not possible to transfer the results of the investigations to real signals produced by any apparatus. Of course, it is much easier to get useful derivatives from undisturbed curves than from real spectra containing noise. 

Physical theories often require mathematical approximations. When functions are expressed as polynomial series, approximations can be systematically improved by keeping terms of increasingly higher order. One of the most important expansions is the Taylor series, an expression of a function in terms of its derivatives. These methods show that a Gaussian distribution function is a second-order approximation to a binomial distribution near its peak. We will hnd this useful for random walks, which are used to interpret diffusion, thermal conduction, and polymer conformations. In the next chapter we develop additional mathematical tools. 

Precision an expression of the random error of a measurement series or, in other words, of the scattering of single values around the average value. The generally accepted way to express precision is with the standard deviation (STD). The latter is the mathematical term for the width of the Gaussian error distribution curve given in the form of o (the distance between the centre and the inflection point of the Gaussian curve). For practical purposes, instead of o, the estimated value 5 is determined from a finite population of... 

The mathematical description of the model is out of the scope of this paper. Briefly, in this model, each reactant beam density is fitted to gaussian radial and temporal distribution functions, the spread in relative translational energy is neglected and the densities are assumed to be constant within the probed volume, which is smaller than the reaction zone. These assumptions result in a simple analytic expression of the overlap integral. Calculations are carried out for each rovibrational state of the outcoming molecule and for extreme velocity vector orientations, i.e, forwards and backwards. An example of the correction function, F, obtained for the A1 + O2 reaction at = 0.49 eV is displayed on Fig. 1, together with the... 

The particular choice of a Gaussian charge distribution is made for mathematical convenience and cp is, of course, independent of the parameter n In Appendix K, we derive an explicit expression for C(q) using the ft-space representation and EWALD s transformation. The result is... 

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