Eigenvector map

When the scheme is not primal, the scheme maps each vertex into an edge. However, the unit row eigenvector as a stencil maps edges into vertices, and so the product of the mask with the unit row eigenvector maps vertices into vertices. 

Abbas, R., S.T. Holland, and I.J. Halvorsen, Novel minimum reflux calculation using eigenvector maps, in Distillation and Absorption 2010, G.K. de Haan, editor. 2010, Eindhoven, The Netherlands Eindhoven University of Technology. 

HyperChem offers a Reaction Map facility under the Setup menu. This is needed for the synchronous transit method to match reactants and products, and depending on X (a parameter having values between 0 and 1, determining how far away from reactants structures a transition structure can be expected) will connect atoms in reactants and products and give an estimated or expected transition structure. This procedure can also be used if the eigenvector following method is later chosen for a transition state search method, i.e., if you just want to get an estimate of the transition state geometry. 

Topological analysis of X-ray protein relative density maps utilizing the eigenvector following method... 

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... 

When the significant eigenvectors are more than 2 or 3, the information cannot be easily visualized by few eigenvector plots. In these cases the use of nonlinear mapping (NLM) can give a planar representation of the objects with greater fidelity to the structure of the information in the hyperspace of the variables... 

Proof. The main idea is to define an isomorphism by mapping the eigenvectors of pli) to the monomials in P". Choose any highest weight vector vo for the representation p. Because V is in irreducible representation of sm(2), the nontrivial subrepresentation VL constructed in Proposition 8.8 must be all of V. Hence we have an explicit basis for V, namely,... 

When the first direction, v has been found, the data are reflected such that the first eigenvector is mapped onto the first basis vector. Then the data are projected onto the orthogonal complement of the first eigenvector. This is simply done by omitting the first component of each (reflected) point. Doing so, the dimension of the projected data points can be reduced by 1 and, consequently, all the computations do not need to be done in the full r-dimensional space. 

Prove that an algebraic G is triangulable (9, Ex. 6) iff it has a unipotent normal closed subgroup U with G/U diagonalizable. [If G acts on V, then U acts trivially on a nonzero subspace V0. The map G - Aut(V0) factors through G/U, which will have an eigenvector.]... 

The semi-empirical molecular orbital calculation software MOPAC in the CAChe Work System for Windows ver. 6.01 (Fujitsu, Inc.) was used in all of calculations for optimization of geometry by the Eigenvector Following method, for search of potential energies of various geometries of intermediates by use of the program with Optimized map, for search of the reaction path from the reactants to the products via the transition state by calculation of the intrinsic reaction coordinate (IRC) [10]. 

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