Linear Maps

Here u = Pu due to u G K. Application of the linear mapping I to this equation gives... 

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

First we define the linear map that produces the densities from N-particle states. It is a map from the space of A -particle Trace Class operators into the space of complex valued absolute integrable functions of space-spin variables... 

Forina M, Armanino C (1982) Eigenvector projection and simplified non-linear mapping of fatty acid content of Italian olive oils. Ann Chim [Rome] 72 127... 

Fig. 6. Terminal capping and lateral bulging of globular domains in the //-solenoid of the hemoglobin protease from E. coli (Otto et al., 2005). The //-solenoid domains are shown in blue and the remaining regions in dark yellow. (A) Ribbon diagram of the 3D structure and (B) linear map of the domain distribution within the amino acid sequence.

In our first implementation [13, 120, 121, 131] of this idea, we took the transition frequency to be a linear function of this electric field. We determined the coefficients of this linear function by fitting to the ab initio frequencies from water clusters (and in this case the clusters were not surrounded by point charges from the other molecules in the simulation). In the liquid simulation we simply calculate this electric field at every time step and then use this linear map (in this case the electric field was the full Ewald field from the simulation) to determine the frequency. In our later implementation [6, 98] we took the... 

Calculation of scores as described by Equations 2.20 and 2.21 can be geometrically considered as an orthogonal projection (a linear mapping) of a vector x on to a straight line defined by the loading vector b (Figure 2.15). For n objects, a score vector u is obtained containing the scores for the objects (the values of the linear latent variable for all objects). 

Further processing is also usually performed to transform the reflectance image cube to its logic (UR) form, which is effectively the sample absorbance . This results in chemical images in which brightness linearly maps to the analyte concentration, and is generally more useful for comparative as well as quantitative purposes. Note that for NIR measurements of undiluted solids the use of more rigorous functions such as those described by Kubelka and Munk are usually not required. ... 

As in Eqs.(14) and (15) for the 2-positive metric matrices, the 3-positive metric matrices are connected by linear mappings, which can be derived by rearranging the second-quantized operators. A 2-RDM is defined to be 3-positive if it arises from the contraction of a 3-positive 3-RDM ... 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

Because the hole and particle perspectives offer equivalent physical descriptions, the p-RDMs and p-HRDMs are related by a linear mapping [52, 53]. Thus if one of them is known, the other one is easily determined. The same linear mapping relates the p-particle and p-hole reduced Hamiltonian matrices ( K and K). An explicit form for the mapping may readily be determined by using the fermion anticommutation relation to convert the p-HRDM in Eq. (18) to the corresponding p-RDM. Eor p = 1 the result is simply... 

Domine, D., Devillers, J., Chastrette, M., and Karcher, W. (1993). Non-linear mapping for structure-activity and structure-property modeling. J. Chemometrics 7, 227-242. 

How is dimension reduction of chemical spaces achieved There are a number of different concepts and mathematical procedures to reduce the dimensionality of descriptor spaces with respect to a molecular dataset under investigation. These techniques include, for example, linear mapping, multidimensional scaling, factor analysis, or principal component analysis (PCA), as reviewed in ref. 8. Essentially, these techniques either try to identify those descriptors among the initially chosen ones that are most important to capture the chemical information encoded in a molecular dataset or, alternatively, attempt to construct new variables from original descriptor contributions. A representative example will be discussed below in more detail. 

Sammon, J.W. Jr., A non-linear mapping for data structure analysis, IEEE Trans. Comp., C-18,401-409,1969. 

If So is the spectrum of a perfect field k then classical Dieudonne theory gives us a con-travariant functor G M(G) of C(l)s0 into the category of finite dimensional fc-vectorspaces. Since there is a canonical isomorphism M(GM) = M(G) P the module M(G) is endowed with the action of Frobenius F and Verschiebung V. The functor G (Af(G),F,V) is an anti equivalence of C(l)s0 with the category of triples (Af, F, C ), where M is a finite dimensional fc-vectors pace and F — Af, resp. V M are linear maps such that F o V — 0... 

To extend this result to projective space of arbitrary finite dimension we will need the technical proposition below. Since addition does not descend to projective space, it makes no sense to talk of linear maps from one projective space to another. Yet something of linearity survives in projective space subspaces, as we saw in Proposition 10.1. The next step toward our classification is to show that physical symmetries preserve finite-dimensional linear subspaces and their dimensions. 

Proof. We shall use the description of (C2) in terms of matrices given in Theorem 1.14. Suppose Z is a T-invariant O-dimensional subscheme in (C2), and corresponds to a triple of matrices (Bi, B2, i). Recall that it is given as follows Define a iV-dimensional vector space V as H°(Oz), and a 1-dimensional vector space W. Then the multiplications of coordinate functions z, z2 6 C define endomorphisms Bi, B2. The natural map Oc2 —> Oz defines a linear map i W V. Prom this construction, V is a T-module, and W is the trivial T-module. The pair (Bi,B2) is T-equivariant, if it is considered as an element in Hom(V, Q V), where Q is 2-dimensional representation given by the inclusion T C SU(2). (This follows from that (Zi,z2) is an element in Q.) And i is also a T-equi variant homomorphism W —> V. 

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