Equivalence transformation

No other synthetic method Is known that achieves the equivalent transformation. Rather elaborate procedures using an allylic anion type of the homoenolate "equivalents" or homoenolate radicals have been reported, but their tolerance to the structure of the enone acceptor is much narrower. 

It is hard to envisage a ligand, which generally requires a double equivalent transformation, being reduced in preference to the central atom. Even if Lr is formed as intermediate by direct interaction with Y, it is likely to act as a reductant, and the equilibrium M+a + L = M+a 1 + L will be shifted to the right. This is in fact the mechanism of ligand-bridged redox reactions as demonstrated in the case of Co(III)L-Cr(II) reactions (86). 

Another possibility is an atom transfer from the ligand to the central atom (a double equivalent transformation), which has been discussed above. In that case, M+a, in a low state of oxidation, is ligated to a ligand L, which is capable of oxidative atom transfer. This mechanism does not require, however, any interaction between a reducing agent, Y, and the system, ML+a. 

Effect of Equivalence Transformations on Extended Poincare Algebra... 

A useful aspect of the mercury(II) hydride method is that it can be directly coupled with the many standard techniques for heteromercuration of alkenes and cyclopropanes. The resulting overall transformation adds a heteroatom and a carbon atom across the carbon-carbon double bond of an alkene or the carbon-carbon single bond of a cyclopropane. This is a difficult transformation to conduct by standard ionic techniques. An alkene thus becomes an equivalent of synthon (12) and a cyclopropane of synthon (13 Scheme 34). Many equivalent transformations (like haloetherification and phenylselenolactoniza-tion) are available to make precursors for tin hydride mediated additions. 

Since the original proposition has been equivalently transformed to the proposition... 

Both the inhibition of this effect by CN ions, which form complexes with the enzyme metal cations, and the change in the distribution of the degradation intermediates at equivalent transformation rates of the organic pollutant, with or without an enzyme, suggest that the observed phenomenon really stemmed from the catalytic activity of SOD. These results emphasize the importance of superoxide. 

As a rule, more than two dimensionless numbers are necessary to describe a phys-ico-technological problem they cannot be produced as shown in the first three examples. The classical method to approach this problem involved a solution of a system of linear algebraic equations. They were formed separately for each of the base dimensions by exponents with which they appeared in the physical quantities. J. Pawlowski [5] replaced this relatively awkward and involved method by a simple and transparent matrix transformation ( equivalence transformation ) which will be presented in detail in the next example. 

The diffusion model can usually be used for the description of many stochastic distorted models. The equivalent transformation of a stochastic model to its associated diffusion model is fashioned by means of some limit theorems. The first class of limit theorems show the asymptotic transformation of stochastic models based on polystochastic chains the second class is oriented for the transformation of stochastic models based on a polystochastic process and the third class is carried out for models based on differential stochastic equations. 

From the structure of the continued fraction (2.1), we see that its value remains unchanged if a , b , a + (for all n 1) are multiphed by the same constant (other than zero). Operations of this kind do not aflfect the sequence of approximants and are called equivalence transformations. 

By use of equivalence transformations we can always transform a continue fraction K(a /b ) into an equivalent one whose partial numerators equal unity, or into an equivalent one whose partial denominators equal unity. We have, in fact. 

If we consider the Stieltjes fraction (3.1), write x= /z, and perform an equivalence transformation by multiplying by z the composing fractions, we obtain the S-fraction of the type... 

Only the rank r of the matrix is determined, using equivalent transformations these are linear combinations of rows (or columns). For this we form linear combinations of rows until the diagonal of a sub-matrix only contains ones, with the neighbouring diagonals all zeros. The first row is indicated by Z, the second by /A and so on, forming a new matrix, the rows of which will be indicated by dashes. So Z = Z. Z[. = Z3 4- Z4, Z) = —Z-2 + 3Z1. The new matrix looks like... 

The left hand sub-matrix now contains, as demanded, only ones in the main diagonal, in the neighbouring diagonals only zeros. The rank of the matrix is r = 4, as there are four rows that are linearly independent of each other By equivalent transformation no more rows can be converted into another. According to Buckingham [1.20], the number m of 7r-quantities m = n — r, where n is the number of original variables, in our case is n = 7. There are... 

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by cin equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representations in unitary form then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as... 

Cycloaddition reactions have considerable value in organic synthesis for a number of reasons, not the least of which are that two bonds are formed in one operation and that the reactions often exhibit high stereoselectivities. Even if this huge field were limited only to examples that fall into the category of asymmetric synthesis, it would take several volumes to completely do it justice. In this section, only selected [2+1]- and [4+2]-cycloadditions (and equivalent transformations) are covered, and the discussion is not limited to concerted processes. 

Suppose a main switchboard feeds load through transformers of different ratings and impedances. For the purpose of estimating fault current at an early stage in a project it is reasonable to combine all the distribution transformers into one equivalent transformer. The equivalent rating of all the... 

Another application of the graph model is to derive equivalence for certain parts of the model. In analysis and simulation of the model, it is often desirable to combine several components to an equivalent single component. The overall reduction of a model to a minimal system will be of great value not only for computations but also for theoretical study of essential characteristics of the system and comparison to the traditional compartment modeling. The graph model will facilitate the development of such equivalence transformations. 

A transformer model was devised for the coupling-loss mechanism associated with multifilament superconductors or cable conductors. The model is based on the fact that the loss is induced by an external magnetic field. A physical understanding of such induction phenomena can be obtained by analyzing an equivalent transformer circuit where the conductor is visualized as the secondary circuit with a time constant tq = L2/R2, and the exciting field is modeled as the primary circuit with a current, ii. For the filamentary superconductor, the time constant tq was chosen to be the same as that defined in field theory [% i.e.. 

We begin by transforming the problem to the fixed cr-interval [0,1], as was done in Section 7.1.3.1 (p. 187). The equivalent transformed problem on this interval is to find the controls and the time period that minimize... 

In mBuild, this equivalence transform is used to force four points of one compound to overlap with four points of another. Achieving this generally, requires that the same arrangement of four non-coplanar points must be added to any compound intended to make use of the equivalence transform. 

Any two Ports can be forced to overlap using the equivalence transform. Listing 2 demonstrates how this functionality can be leveraged via the simple yet common use case of creating an alkane polymer chain which will be used for screening— in this example, a CH2 group with the ports up and down delined. 

For a continuous signal, we can move from the time domain to the frequency domain via the Fourier transform, but as we are now dealing with digital signals, this formula cannot be used directly. We will now derive an equivalent transform for digital signals. Starting with the previously defined Fourier transform (Equation 10.21)... 

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