Unitary condition

The excitation operator does not have to adhere to the unitary condition, as is the case for orthogonal orbitals. Each Brillouin matrix element (Eq. (4)) represents the stationary condition for the mixing of orbitals iffj and iffj according to Wi -> Vi + Vj- The wavefunction consisting of Vo and all singly excited states... 

Once we have identified the elements of the transition matrix and normalized this matrix to satisfy the unitary condition, the dispersion relation for a full multilayer is immediately obtained. Then the Nth power of this matrix is... 

The above relation is the unitary condition for nonabsorbing particles. In terms of the transition matrix, this condition is... 

Several years ago Baer proposed the use of a mahix A, that hansforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial diffei ential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

In general, air conditioning equipment can be classified into two broad types central (sometimes called field erected) and unitary (or packaged). 

Air Conditioning and Refrigeration Institute. (1999). ARI Unitary Directory. < http /Avww.ari. org >. 

The existence of a unitary transformation U(a,A) which relates the field operators in the two frames imposes certain conditions on the operators themselves they must satisfy certain commutation rules with P and Muv. Consider first the case of a space-time translation... 

As well known, this condition is not a restriction whenever the wavefunction is invariant under an unitary transformation [2],... 

For example, according to Hamermesh (see Ref. [11]), the number of real conditions to uniquely determine an (N x N) (complex) unitary matrix is N2, while the number of real conditions to uniquely fix a (real) orthogonal matrix of same dimensions is not N2/2 but N(N + l)/2. 

However, to determine the number of real pieces of information required to fix the projection from an M-dimensional space onto an /V-dimensional subspace spanned, not by the particular (occupied) basis in which P is diagonal, but by any basis of the subspace, it is necessary to subtract the numberof real parameters required to fix a particular basis in the /V-dimensional subspace from the total Kcy, such a number corresponds to the N2 real conditions that are necessary to fix a unitary transformation [11] in the subspace. But, as the phases ofthe eigenstates, < , are arbitrary as far as the physical state is concerned [4, 12], this latter number is reduced by N, the number of eigenstates belonging to the projection space. Hence, the number of independent real parameters in the unitary transformation which fixes the basis spanning the... 

Pecora noticed that the phase information of C is lost in the original constraints [i.e. P2 = P Tr P = N], but found it not at all clear . Here, we showed in which way one might take into account the loss of the phase information in C when calculating the number of conditions to uniquely determine C one has to impose, over and above the constraints arising from fixing the projector, the conditions to determine a particular unitary transformation in the TV-dimensional subspace, apart from the phases of the basis functions which are physically meaningless in the context of Quantum Mechanics. 

A vector space L defined over a field F is further called an inner-product space or unitary space if its elements satisfy one more condition ... 

Since the scalars are complex, this last condition is equivalent to two conditions. The number of parameters is therefore reduced from 8 to 4. The most general unitary matrix of order two involving four real parameters can be expressed in the form... 

It is known as the unitary unimodular group, or the special unitary group denoted by SU(2). Because of the extra condition on the determinant, SU(2) is a three-parameter group. 

When studying the border of universality, we always need to consider the limit of large graphs, that is, ue —> oo. This limit is in general not well defined, but may often be obvious from the examples considered. We will thus define the semiclassical limit loosely via a family of unitary-stochastic transition matrices Tn and associate USE s and take ue —> oo. The leading term in (13) then gives a condition for a family to show deviations from RMT statistics in terms of the spectrum of T the diagonal term must obey... 

Combining one of these equations with the condition for a unitary vector... 

The double trace term is due to the absence of the condition for the vanishing of the trace for the broken generator X5. It emerges naturally in the non linear realization framework at the same order in derivative expansion with respect to the single trace term. In the unitary gauge these two terms correspond to the five gluon masses [431. 

The unitary5 chemical agents in U.S. stockpiles are nerve agents (GB and VX)6 and three related forms of blister (mustard) agent (H, HD, and HT). The original stockpile contained 31,496 tons of unitary agents, which were stored in a variety of munitions and containers under ambient conditions, where they exist primarily as liquids. Because the stockpile at Johnston Atoll in the Pacific has been completely destroyed by incineration and the stockpile at Deseret Chemical Depot, in Utah, has been partially incinerated, as of October 3, 2001, the amount in storage was approximately 24,098 tons. 

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