Polar decomposition

Of these diacyl peroxides the ones that generate the most stable radicals (R ) are the most unstable diacyl peroxides. Most other diacyl peroxides decompose by competing free-radical and polar decomposition, ie, carboxy iaversion (188). Carboxy iaversion occurs to a much greater extent with certain diacyl peroxides having unsymmetrical diacyl peroxide stmctures (52,187,188,199) ... 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

The polar decomposition (A. 13) implies that the deformation may be viewed as two successive deformations, the first being a pure stretch from the reference configuration into an unrotated configuration, and the second being a... 

The decomposition of aliphatic peroxides produces oxygen radicals too unstable for paramagnetic measurement. These radicals initiate the polymerization of olefins and give the complex mixtures of decomposition products associated with radical mechanisms. On the other hand, aliphatic peroxides are also capable of polar decomposition reactions, a subject to be taken up in Chapter VIII. The characteristic reactions of the less stable oxygen free radicals are /3-cleavage to form... 

This decomposition is somewhat analogous to the polar decomposition of suitably unsymmetrical diacyl peroxides. However, it has not been possible to force the complementary mode of decomposition of a hydroxamic acid in which the nitrogen moiety departs with the electrons and leaves the oxygen electron-deficient.816... 

Although benzoyl peroxide will initiate the polymerization (by a radical chain reaction) of either styrene or acrylonitrile, -methoxy- -nitrobenzoyl peroxide will not initiate polymerization efficiently in the latter monomer because it is too rapidly destroyed by the polar decomposition. Acrylonitrile, but not styrene, causes the polar decomposition to predominate, and the intermediates of the polar decomposition are not catalysts for the polymerization of acrylonitrile. 

There is considerable overlap in the effective range of the initiators, but this is less troublesome than it might seem since the various mechanisms can be expected to differ in their response to inhibitors. And if the alternatives are free radical and ion-pair with one of the possible ions not very reactive, decision is easy. For example, the polar decomposition of >-methoxy-/> -nitro benzoyl peroxide in acrylonitrile initiates... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The kinetic and activation parameters for the decomposition of dimethylphenylsilyl hydrotrioxide involve large negative activation entropies, a significant substituent effect on the decomposition in ethyl acetate, dependence of the decomposition rate on the solvent polarity (acetone-rfe > methyl acetate > dimethyl ether) and no measurable effect of the radical inhibitor on the rate of decomposition. These features indicate the importance of polar decomposition pathways. Some of the mechanistic possibilities involving solvated dimeric 71 and/or polymeric hydrogen-bonded forms of the hydrotrioxide are shown in Scheme 18. 

Equation (5-9) can be regarded as a canonical (Lowdin) orthonormalisation of the set of vectors 0 , or equivalently as a polar decomposition of the operator 3l (J0rgensen 9) Thus the Schrodinger equation for the n-electron Hamiltonian, H, Eq. (2-2), can always be formally transformed to the eigenvalue problem (5-10 a) for the effective Hamiltonian,, acting in the subspace S sparmed by a finite set of orthonormal vectors 0 the ligand field Hamiltonian (1-5) must therefore be an approximation to this object. 

Polar decomposition of the field amplitude, as in (36), which is trivial for classical fields becomes far from being trivial for quantum fields because of the problems with proper definition of the Hermitian phase operator. It was quite natural to associate the photon number operator with the intensity of the field and somehow construct the phase operator conjugate to the number operator. The latter task, however, turned out not to be easy. Pegg and Barnett [11-13] introduced the Hermitian phase formalism, which is based on the observation that in a finite-dimensional state space, the states with well-defined phase exist [14]. Thus, they restrict the state space to a finite (cr + l)-dimensional Hilbert space H-+ spanned by the number states 0), 1),. .., a). In this space they define a complete orthonormal set of phase states by... 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

It is well known that the angular momentum of a quantum mechanical system is specified by a representation of the SU(2) algebra. If the corresponding enveloping algebra contains a uniquely defined scalar (the Casimir operator), the polar decomposition of the angular momentum can be obtained [51]. This polar decomposition determines a dual representation of the SU(2) algebra expressed in terms of so-called phase states [51], In particular, the Hermitian operator of the SU(2) quantum phase can be constructed [51],... 

This expression indicates that the motion of the rotor can no longer be described by a simple quantum movement driven by the preparation of the rotor in a pure rotation state. Several authors have shown that the interaction of a quantum system with a reservoir destroys the phase coherence of any wave packet prepared to profit from more semi-classical-like behavior [23,24]. To demonstrate this below, we use the Madelung polar decomposition [25] of the wave function, Eq. (7). For these purposes, the density matrix technique is usually preferred [26, 27], however, the alternate route practiced here has the advantage of direct extraction of the equations of motion of the rotor alone. 

Another approach to the analysis of Jones and Mueller-Jones matrix exploits the polar decomposition theorem [18]. This approach was first suggested in [19] and was explored in [20,21]. The polar decomposition of a Jones matrix J can be represented as ... 

Let A be the single positive self-adjoint operator and Jthe unique antiuni-tary operator in the polar decomposition of S ... 

Equations (4.11) and (4.12) are called polar decomposition of the deformation gradient. Particularly, U is called the right stretch tensor and V is called the left stretch tensor. 

Figure 4.4 shows a schematic of mapping from an undeformed body to a deformed body, as well as the decomposition of the deformation gradient to the plastic deformation gradient, which leads to an intermediate relaxed body with pure plastic deformation and is obtained by conceptually unloading the deformed body elastically to zero stress, and the elastic deformation gradient, which leads the intermediate body to the deformed body. If the total deformation gradient is F, the plastic component is F, and the elastic component is F, similar to polar decomposition, the multiplicative decomposition leads to... 

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