Variation problem, equivalent

Note 4.1 (Equivalence between the strong form and weak form Fundamental Lemma of the Variational Problem). Since the weak form (4.5) is derived from the strong form (4.1), the solution of the strong form is exactly the solution of the weak form. The converse is not always true. If the solution of the weak form can be regarded as sufficiently smooth, the converse is true, which is proved by the Fundamental Lemma of the Variational Problem. Note that the first term defined on the boundary of (4.5) can be considered separately from the rest of the terms defined in the domain, since v is arbitrary both on the boundary and in the... 

This variational problem is equivalent to the eigenvalues and eigenvectors problem for the matrix ... 

The FEA is conceptually quite different. Solution of the PDE is replaced with an equivalent variational problem find a function that minimizes the functional... 

Under these assumptions, the stochastic variational problem involving the random field o,m x, (o) has the following deterministic equivalent find u e Hq D) Lp(r), such that... 

Several parameters come into the relation between density and equivalence ratio. Generally, the variations act in the following sense a too-dense motor fuel results in too lean a mixture causing a potential unstable operation a motor fuel that is too light causes a rich mixture that generates greater pollution from unburned material. These problems are usually minimized by the widespread use of closed loop fuel-air ratio control systems installed on new vehicles with catalytic converters. 

Many variations of the Morgan Algorithm were introduced, because of problems finding the terminating condition of stage 1 (oscillating number of equivalent classes [80]) or special atoms with isospcctral points [81],... 

Herewith the problem of minimizing H over the set is equivalent to the variational inequality... 

In Section 3.1.3 a complete system of equations and inequalities holding on F, X (0,T) is found (i.e. boundary conditions on F, x (0,T) are found). Simultaneously, a relationship between two formulations of the problem is established, that is an equivalence of the variational inequality and the equations (3.3), (3.4) with appropriate boundary conditions is proved. 

Observe that variational inequality (3.106) is valid for every function X G 82- It means that a solution % to problem (3.106) with 9 G Si coincides with the unique solution to problem (3.100) with the same 9] i.e. problems (3.100) and (3.106) are equivalent. For small 5, we write down an extra variational inequality for which a solution exists, and demonstrate that the solution coincides with the solution of variational inequality (3.98). 

Then problem (3.114) can be written down in equivalent form by means of the following two variational inequalities ... 

Sufficiency. Given the example, the theory we employ to deduce the conditions must guarantee that the conditions on equivalence or dominance will be satisfied, since the example itself is simply an instance of the particular problem structure, and may not capture all the possible variations. The sufficient theory will thus end up being specialized by the example, but since we have asserted the need for validity of this process, it will still be guaranteed to satisfy the abstract necessary and sufficient conditions on dominance and equivalence. 

Variations of resistance with frequency can also be caused by electrode polarization. A conductance cell can be represented in a simplified way as resistance and capacitance in series, the latter being the double layer capacitance at the electrodes. Only if this capacitance is sufficiently large will the measured resistance be independent of frequency. To accomplish this, electrodes are often covered with platinum black 2>. This is generally unsuitable in nonaqueous solvent studies because of possible catalysis of chemical reactions and because of adsorption problems encountered with dilute solutions required for useful data. The equivalent circuit for a conductance cell is also complicated by impedances due to faradaic processes and the geometric capacity of the cell 2>3( . 

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