Quadratic functions solution

The Galerldn finite element method results when the Galerldn method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerldn finite element equations. The element integrals are defined as... 

From an adsorption study of aliphatic compounds on an Hg electrode," it follows that AG° is a quadratic function of the charge density and applied potential. The value of AG° reaches minimum (the most negative value) at ffmax. i-e-, at the charge of maximum solute adsorption, which is usually in the vicinity of the potential of zero charge. 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

The above-mentioned 256 mixed solutions were measured with the multichannel taste sensor. Therefore, data on the output electrical potential pattern were taken for the 256 solutions. While the data on each channel output were dispersed discretely in the four-dimensional space constructed from four different concentrations, we approximated them by a quadratic function of the concentrations. As a result, eight quadratic functions were obtained. The data can be regarded as expressed by a set of eight different functions (corresponding to 8 channels) of concentrations of four taste substances. 

The solution to Eq. (1.6) for the electronic energy in, e.g., a diatomic molecule is well known. In this case there is only one internuclear coordinate and the electronic energy, Ei(R), is consequently represented by a curve as a function of the internuclear distance. For small displacements around the equilibrium bond length R = Ro, the curve can be represented by a quadratic function. Thus, when we expand to second order around a minimum at R = Ro,... 

Steepest descent is simple to implement and requires modest storage, O(k) however, progress toward a minimum may be very slow, especially near a solution. The convergence rate of SD when applied to a convex quadratic function, as in Eq. [22], is only linear. The associated convergence ratio is no greater than [(k - 1)/(k + l)]4 where k, the condition number, is the ratio of largest to smallest eigenvalues of A ... 

The quadratic effect of an externally applied field on the absorption coefficient is described by the imaginary part of the third-order susceptibility -o) a),0,0). influences the molar decadic absorption coefficient of the solute. The absorption coefficient in the presence of the field is a quadratic function of the applied field strength (118),... 

If, as is mostly the case in experiment, the analysing field E and polarizing field Ep are applied in the same direction, the electric permittivity variation tensor (12) possesses but one non-zero component, in the field direction, usually denoted by Ae,( p). This particular case is referred to as the effect of electric saturation in an dectric field. In molecular liquids Ae,( ) is in general a quadratic function of the field strength Ep, as proved r )eat y by Piekara and his co-workers. Lately, Davies has published a review on aspects of recent tfielectric studies, particularly dielectric saturation in liquids and molecular solutions, as well as in solutions of macromolecules where complete dielectric saturation has been observed. 

Ounkovsky and Volovai found that the viscosity-compositioii plots of binary mixtures of liquids having equal viscosities are not usually linear. Lautie found for dilute solutions of non-electrolytes that the relative viscosity 97/770 naay be represented as a linear or quadratic function of concentration. 

Seme and Muller (1987) describe attempts to hnd statistical empirical relations between experimental variables and the measured sorption ratios (R(js). Mucciardi and Orr (1977) and Mucciardi (1978) used linear (polynomial regression of first-order independent variables) and nonlinear (multinomial quadratic functions of paired independent variables, termed the Adaptive Learning Network) techniques to examine effects of several variables on sorption coefficients. The dependent variables considered included cation-exchange capacity (CEC) and surface area (S A) of the solid substrate, solution variables (Na, Ca, Cl, HCO3), time, pH, and Eh. Techniques such as these allow modelers to constmct a narrow probability density function for K s. 

It is important to underline the fact that, in the case where A is a linear operator, where D and M are Hilbert spaces, and where s(m) is a quadratic functional, the solution of the minimization problem (2.44) is unique. Note that the quadratic functional is a functional 17(111) with the property... 

Several features of the optimization problem are apparent in Figure 19.1(a). The model is nonlinear with respect to parameters nevertheless, the objective function is well behaved near the solution where it can be approximated by a quadratic function. The contours projected onto the base of the plot have an elliptical shape. The major axis of the ellipse does not lie along either axis. 

Table 3 shows results of recorded fluorescence emission intensity as a function of concentration of quinine sulphate in acidic solutions. These data are plotted in Figure 3 with regression lines calculated from least squares estimated lines for a linear model, a quadratic model and a cubic model. The correlation for each fitted model with the experimental data is also given. It is obvious by visual inspection that the straight line represents a poor estimate of the association between the data despite the apparently high value of the correlation coefficient. The observed lack of fit may be due to random errors in the measured dependent variable or due to the incorrect use of a linear model. The latter is the more likely cause of error in the present case. This is confirmed by examining the differences between the model values and the actual results. Figure 4. With the linear model, the residuals exhibit a distinct pattern as a function of concentration. They are not randomly distributed as would be the case if a more appropriate model was employed, e.g. the quadratic function. 

The best known Krylov subspace method is the method of Conjugate Gradients (CG) by Hestenes and Stiefel [70]. If A is S5mimetric positive definite, the solution of the problem Ax = b corresponds to determining a local minimum of the quadratic function ... 

Step 11 Go back and plot along the points (r, z) = (0,2) and (0.5,2) or on boundary 3 to obtain Figure 10.5. This figure shows the outlet velocity. The exact solution is a quadratic function of position, which is satisfied in Figure 10.5. However, you also expected a peak value of 2.0. The peak value is not 2.0 because of the slight error in the entry conditions. What is true is that the flow rate out equals the flow rate in, which is slightly less than you desired due to the bump in the inlet profile. This discrepancy decreases as you refine the mesh. 

Viscosities of dilute polymer solutions increase as the polymer concentration is increased (21). Several correlations between these two variables have been suggested. A widely used relationship is the Huggins equation which relates the viscosity to concentration in a quadratic functionality as follows ... 

Solution of Equation 14 is obtained by the finite element model RMA-4 for any stipulated flow field u x,t) and v y,t) over a continuum of elements, usually of triangular or rectangular shape (although isoparametric elements with quadratic functions defining the sides are allowed). Depths may be fixed or variable, both in time and space. RMA-4 has been applied successfully to a wide variety of practical problems (13). The version used here is a modification, RMA-4A, that is designed specifically to deal with the kinetics of copper as described. 

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