Projection onto subspace

Thus by summing the normalized eigenvalues of the first 5 dimensions one can judge the quality of a projection onto that subspace. 

One way to solve the problem of unphysically short atomic distances is to project onto the Rpm subspace only those grid points included in a selected strip (gray area), with width of a (cos a + sin a) in the A per subspace. The slope of RPai shown in Fig. 1 is 0.618..., an irrational number related to the golden mean [( /5 + l)/2 = 1.618...]. As a result, the projected ID structure contains two segments (denoted as L and S), and their distribution follows a ID quasiperiodic Fibonacci sequence [2] (c.f. Table 1). From another viewpoint, the ID quasiperiodic structure on the par subspace can be conversely decomposed into periodic components (square lattice) in a (higher) 2D space. The same strip/projection scheme holds for icosahedral QCs, which are truly 3D objects but apparently need a more complex and abstract 6D... 

Therefore, we want to decide which direction, among all possible choices, is common to all sample subspaces, or, at least, which direction represents the best zone of the sample subspaces in a least-square sense. Since a direction can be completely described by its unit vector, we can restrict the solution set to the surface of the unit sphere centered at the origin. Let us call y the solution of unitary modulus and its projection onto the fcth sample subspace (k = 1,..., s) represented by the matrix Ak. It is a simple matter to show that... 

Finding the least-square solution reduces to minimizing the sum S of squared deviations yk — j) between the estimated source solution and its projection onto each sample subspace. Thus, finding the minimum of... 

For each covariant rank 2 tensor S v (with two subscripted indices) we define a projection onto the soft subspace... 

For each rank 2 contravariant Riemannian tensors T (with two raised indices) we define a. K x K projection onto the hard subspace... 

Proof. We will prove the first conclusion of this proposition by induction on the dimension of VF. We start with the subspace of dimension zero, i.e., the trivial subspace 0. It is easy to check that the linear transformation taking every vector of V to the zero vector is an orthogonal projection onto 0. ... 

Next we must prove the inductive step. Fix any natural number n. Suppose that there exists an orthogonal projection onto any subspace of dimension n. Consider a subspace VF of dimension +1. Fix an element w e W such that w,w) = 1. Let W denote the subspace of VF perpendicular to w. Then W has dimension n, so there is a well-defined orthogonal projection flyy onto... 

Recall from Proposition 5.4 that orthogonal projection onto an invariant subspace of a unitary representation is a homomorphism of representations. Hence for any we have... 

After a particle in a state [u] is subjected to the measurement A and yields a value A, the particle will be in the state [flu], where fl is the orthogonal projection onto the subspace W, as in Assumption 2... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

The coupled Schrodinger equations can be projected onto the fa fa subspace by Feshbach partitioning, giving an equation for the coefficient function Xd(q) in the component faxdiq) of the total wave function. The effective Hamiltonian in this equation is tn + Vd(q) + Vopt, which contains an optical potential that is nonlocal in the <7-space. This operator is defined by its kernel in the fa - fa subspace,... 

Orthogonal outliers have a large orthogonal distance, but a small score distance, as, for example, case 5. They cannot be distinguished from the regular observations once they are projected onto the PCA subspace, but they lie far from this subspace. Consequently, it would be dangerous to replace that sample with its projected value, as its outlyingness would not be visible anymore. 

Alternatively, the essence of the theory can be expressed succinctly in mathematical terms. If the transition state reaction coordinate is defined by a 3N- dimensional vector in mass normalised space, which is projected onto the 3-dimensional mass normalised subspace of the product separation coordinate, the proportion q of the reverse critical energy appearing as relative translational energy of products is... 

In OCT the conditional probabilities determining the molecular communication channel in the basis-function resolution follow from the quantum-mechanical superposition principle [51] supplemented by the "physical" projection onto the subspace of the system-occupied MOs, which determines... 

Projection onto a limited basis, or subspace, entails a concommitant replacement of the conventional hamiltonian with an effective operator, some parts of which are necessarily energy dependent and also all other operators for the system must be replaced by effective operators chosen so that their matrix elements in the projected basis are exact. The first stage of our treatment - the primitive parameterization - explicitly neglects... 

It is not always feasible to directly measure the ancilla independently from the information system in other words, it is sometimes impossible to perform a projection onto disentangled subspaces of H of the form 7T/0Span o ) in some cases, as for the example proposed in Sec. 3, one can only project onto entangled subspaces of the total Hilbert space H. In such a case the information initially stored in the vector ipi) = J2i=i r< Iu<) G Hi must be transferred into an entangled state of X and A of the form ip) = i r> H) where the I vectors 0) (i = 1.. .., I) which form an orthonormal basis of the information-carrying subspace C, are generally not factorized as earlier but entangled states. Nevertheless the same method as before can be used in that case to protect information, albeit in a different subspace C. 

In the last step the erroneous state vector is projected onto the subspace C to recover the initial information. Projection is a non-unitary process which cannot be achieved through a Hamiltonian process, but requires the introduction of irreversibility. To this end, we make use of a path which is symmetric with the pumping step, and consists in two stimulated and one spontaneous emissions. To be more explicit, we apply two left circularly polarized lasers slightly detuned from the transitions (60/ <—> 5d, j = 3/2) and (5d, j = 3/2 <—> 5p, j = 3/2). Due to these laser fields, the atom is likely to fall towards the ground state and emit two stimulated and one spontaneous photons. 

Figure 8. Invariant cone for the matrices A and B corresponding to Poincare maps for the network of Eq. (8), projected onto the unit sphere, indicated by solid lines. Projections of the invariant subspaces through eigenvectors are indicated by dashed lines.

Using either an Arnoldi or a Lanzcos method we computed from Eq. (42) the eigensolutions of interest. These two methods are both iterative methods based on Krylov subspace projections [9]. The Arnoldi method is a generahzation of the Lanczos process and reduces to that method for real symmetric Hamiltonian matrices. The basic idea of Krylov subspace iterations is to approximate a subset of the eigensolutions of the large Hamiltonian matrix by a much smaller matrix, where this small matrix is an orthogonal projection onto a particular Krylov subspace. Our actual computations are based either on the spectrum transformed Lanczos code from T. Ericsson and A. Ruhe... 

---