Minimum squares

A sieve diameter is defined as the width of the minimum square aperture through which the particle will pass. A common sizing device associated with this definition is a series of sieves with square woven meshes. Two sieve standards, i.e., Tyler Standard and American... 

For this reason, in what follows no attempts have been made to use computer subroutines [such as XFIT used by, e.g. Alder et al. (1986)] to calculate minimum square atomic displacements for various pathways. The view has been taken, rather, that if it is not intuitively obvious that one pathway involves greater nuclear motion than another, then the answer one will get from displacement minimisation subroutines will depend on the function of the displacements minimised, and this is itself an intuitive choice. 

Assuming, for the moment, that the relationship is best represented by a straight line, the least-squares line is the regression giving the minimum squared deviation of the data from the line. The theoretical least-squares line can be represented mathematically as... 

Linear systems over a gaussian random vector. If x or X is a gaussian vector with mean value m and covariance (the minimum square error estimate for x is x and x = m, ) which is considered to be in a formal linear system completed with a zero-mean gaussian vector (v is N (0, R)) then we have ... 

The Kalman Filter Equations are here obtained from the formulation of the Kalman filter with the purpose of finding a recursive estimation procedure for the solution of a problem (estimation of state vector). Before detailing the procedure, we have to introduce other new notations. The minimum square estimate of for the given observations Yj= [yo>yi --yil defined by xy. Furthermore represents the error covariance associated with... 

The sieve diameter, for square mesh sieves, is the length of the minimum square aperture through which the particles can pass, though this definition needs modification for sieves which do not have square apertures. 

Legendre, A. M. Appendix. Sur la methode des moindres quaeres. In Nou-velles Methodes pour la Determination des Orbites des Cometes. pp. 72-75 Courcier Paris (1805). English translation Harvey, G. On the method of minimum squares, employed in the reduction of experiments, being a translation of the appendix to an essay of Legendre s entitled Nouvelles Methodes pour la Determination des Orbites des Cometes, with remarks. Edinburgh Philosophical Journal 7, 292-301 (1822). 

The width of the minimum square aperture through which the particle has passed... 

Finally, two words about new ideas. Let (X = C9/A,0) be p.p.a.v. Buser and Sarnak defined a metric invariant m X) = m(X,0) by setting it equal to the minimum square length, with respect to the Hermitian form, of a non-zero lattice vector. Then, on the one hand, they show that for a general p.p.a.v. X, ... 

The reservoir solids are most conveniently classified according to their size. Because the particles are of irregular shape, classification is on the basis of sieve opening of sieve screens. Sieve opening is the width of the minimum square aperture through which the particle can... 

The necessary data are usually obtained by means of external devices such as vision devices [3], the laser tracker [4] or the coordinate measuring machine. Then, the following step is to identify the kinematic parameters that provide the optimum value of an objective function. This function can be formulated in terms of a linear minimum-square. 

A standard sieve series usually consists of a set of sieves with apertures covering a wide range from microns to centimeters. The sieve size is defined as the minimum square aperture through which the particles can pass. Sieves are often referred to by their mesh size, that is, the number of wires per linear inch. Mesh size and the wire diameter determine the aperture size. The ratio of aperture of a given sieve to the aperture of the next one in a... 

This model, will be detailed in after-next section, yet worth noting from now how it presumes ihe knowledge of Ihe Si-atoms position in the unit cell, being therefore reliable only after the determination of their location, for example, from the rationalization of Ihe diffraction experiments. Nevertheless, a measure in estimating Ihe Iheoretical-to-experimental modeling can be provided by the procedure of Ihe least (minimum) squares fitting, based on the formula ... 

For parameter estimation of a given equation from experimental data, a common procedure is the minimum square error. By this, using the Solver, we will be looking to minimize the square of the difference between the actual experimental results and the results obtained by the proposed equation that is based on the parameters that need to be estimated. Figure 3.11 contains aU the cells we have included in bold the names we have given to the correspondent columns and cells as it will be indicated in order to follow the calculations. 

In fact plots with the relatively average values of the easy-hydrolysable nitrogen, mobile phosphorus and potassium content occupied more than half of the area, while the number of plots with bulk and minority of the content were noticeably less (Figures 14.1-14.3). Thus, 350 plots contained 101-250 mg/kg mobile P2O5 in their soil, while plots with lower and higher values of these indices occupied minimum square. Similar results characterized territorial distribution of sites with different easy-hydrolysable nitrogen and mobile potassium. 

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