Canonical vector representation

The canonical vector representation allows an easy as well as exact description of the ring conformations of these species see Table II. Each conformer is characterized by its projection on to idealized Chair (C), Boat (B), and Skew boat (S) conformations... 

Step 1. Norbornadiene C7H8 of symmetry C2V contains 8 CH single, 8 CC single and 2 jr-bonds, occupied by 36 electrons. (We disregard the inner carbon ls-orbitals). Accordingly, a SCF treatment yields 18 bonding canonical molecular orbitals (CMOs) irreducible representation Ai, 2 to A2, 4 to B and 5 to 82- We collect these 18 CMOs in a column vector... 

Denote by DE(G) the set of directed edges of Conn(G). Denote by V(G) the vector space with canonical basis feOdeOEto- For every cycle c of Conn(G), choose an orientation of it and denote by f(c) its representation in V(G). Denote... 

In the description of nature afforded by quantum mechanics, one classifies and characterizes the state of a total system in terms of the eigenvalues of a set of commuting observables acting on an element of the Hilbert space, the state vector. Molecular orbital theory in its canonical representation as originally... 

The embedding of encoded watermark bits from u, in the jth structure Mj is considered. A block diagram of the embedding scheme is depicted in Fig. 3. First, a canonic representation of the structure is obtained as described above. Next, the host data vector Xj is extracted (see also Section 4.7). Here, it is assumed that L j elements are extracted from the structure Mj, and the elements of are scaled such that the watermark w,- can be embedded with a variance = l. 

The upper part of the block diagram in Fig. 4 depicts the watermark detection scheme for one stmcture Mj. First, the data is transformed into its canonical representation. Next, the received vector rj is extracted. The extraction method must be identical to the host vector extraction used for watermark embedding. Thus, the length of rj is also Lx J. Second, tlie 64-bit hash of Mj is derived and the pseudo-random vectors t j. kj and ij are computed dependent on the copyright holders key K. After applying tlie spread transform, the demodulated soft watennark letters yj are derived from and kj as described in Section 2. The probability p dn,j = 1) of receiving a watennark letter dnj = 1 from the nth clement of is given by... 

Similar to —> MNA descriptors, signature descriptors are holographic vectors of count numbers of all the atomic signatures present in the data set molecules, the signature of an atom being a canonical representation of the atom environment up to a predefined depth, that is, a circular substructure descriptor [Visco, Pophale et al, 2002 Faulon, Visco et al, 2003]. 

Foldy representation for finite mass So far we have ignored the discrete operators P and T which generate the other 3 classes of It proves necessary to double the dimension of each irrep, so that for a representation (m,y) for mass m and spin y the dimension is 2(2y-l-1). Foldy [57] gives a canonical form in terms of a representation space of vector-valued functions P(ct,x) with inner product... 

The hyper band structure is, in fact, just another representation of the canonical structure matrix (on- and off-diagonal blocks) and it may be used to generate the fully hybridised energy-band structure for any specified vector P(E) = P (E),...,Pf(E) of potential functions through... 

We shall now set up a coupled-cluster optimization scheme that avoids the solution of the linear equations (13.4.5) in each iteration. In the canonical representation, we may invoke (13.2.44) to write the coupled-cluster vector function (13.4.3) and its Jacobian (13.4.4) of the nth iteration in the form... 

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