Dual Representations

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

In Section 4.4 we saw how to build a representation from the action of a group on a set the new representation space is a space of functions. In this section, we apply this idea to linear functions on a vector space of a representation to define the dual representation. 

To define the dual representation we first must define dual vector spaces. 

The character of a dual representation is the complex conjugate of the character of the original. 

Proposition 5.11 Suppose (G, V, p) is a finite-dimensional unitary representation with character /, Then the character of the dual representation G, V, p ) is X - (Recall that x denotes the complex conjugate of the C-valuedfunction xf Fitt thermore, (G, V, p is a unitary representation with respect to the natural complex scalar product on V. ... 

In this section we have shown how a representation on a vector space determines a representation on the dual of the vector space. We will find the dual representation useful in Section 5,5, More generally, duality is an important theoretical concept in many mathematical settings. Physically, momentum space is dual to position space, so the name "momentum space in the physics literature often connotes duality. 

The edges, vertex circles and face circles of a primal-dual representation... 

For tori, we take their universal covers on the plane and use the primal-dual representation obtained from the program TorusDraw ([Dut04b]). For the projective plane F2, we take its universal cover, which is the sphere, and draw a circular frame,... 

To overcome the aforementioned difficulty we have to introduce the dual representation of V and i>(y). 

The dual representation of V will be invoked in terms of the intersection of a collection of regions that contain it, and it is described in the following theorem of Geoffrion (1972). 

Remark 5 The dual representation of the set V needs to be invoked so as to generate a collection of regions that contain it (i.e., system (6.7) corresponds to the set of constraints that have to be incorporated for the case of infeasible primal problems. 

Having introduced the dual representation of the set V, which corresponds to infeasible primal problems, we can now invoke the dual representation of o(y). 

The analysis presented for the primal problem (see section 6.3.3.1) remains the same. The analysis though for the master problem changes only in the dual representation of the projection of problem (6.2) (i.e., v(y)) on the y-space. In fact, theorem 3 is satisfied if in addition to the two conditions mentioned in C3 we have that... 

In its current formulation the ASEP/MD method introduces a dual representation of the solute molecule. At each cycle of the ASEP/MD calculation, the solute charge distribution is updated using quantum mechanics but during the molecular dynamics simulations the solute charge distribution is represented by a set of fixed point charges. The use of an inadequate set of charges in the solute description can introduce errors into the estimation of the solvent structure, and hence of the solute s properties... 

The significance of this finding is in establishing the Lanczos dual representation fn), Qn(u) that enables the following equivalent definitions ... 

Linear learning machines can be expressed in a dual representation, enabling expression of the hypotheses as a linear combination of the training point (xj) so that the decision rule can be evaluated by using just inner products between the test points (x) and the training points ... 

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],... 

For any atomic multipole transition, the excited state can be described in terms of the dual representation of corresponding SU(2) algebra, describing the azimuthal quantum phase of the angular momentum. In particular, the exponential of the phase operator and phase states can be constructed. The quantum phase variable has a discrete spectrum with (2j + 1) different eigenvalues. 

In Section III.B, we introduced the atomic quantum phase states through the use of the representation of the SU(2) algebra (37) and dual representation (48), corresponding to the angular momentum of the excited atomic state. The multipole radiation emitted by atoms carries the angular momentum of the excited atomic state and can also be specified by the angular momentum [2,26,27], The bare operators of the angular momentum of the electric dipole... 

We now turn to the construction of the dual representation of the photon operators, providing the field counterpart of the SU(2) phase representation of the atomic variables. It is easily seen that the atomic exponential of the SU(2) phase operator (41) takes [in the representation of dual states (46)] the following diagonal form... 

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