Matrices inward matrix product

Carbo-Dorca, R. (2001) Inward matrix products extensions and applications to quantum mechanical foundations of QSAR. J. Mol. Struct. (Theochem), 537, 41—54. 

K.D.Sen and R.Carb6-Dorca, Inward Matrix Products, Generalised Density Functions and Rayleigh-Schrodinger Perturbation Theory , Institute of Computational Chemistry Technical Report IT-IQC-98-42. (1998)... 

This section of the chapter deals with the structure and properties of VSS and its mathematical framework. Because VSS is an interesting conceptual formalism, which supports inward matrix product (IMP) algebra,the applications of such a product in quantum chemistry, particularly in QS theory, will be discussed. 

Applied Sciences and Engineering (ECCOMAS 2000), CDROM edited by Facultat d ln-formatica de Barcelona (FIB)— Universitat Politecnica de Catalunya (UPC)—International Centre for Numerical Methods in Engineering (CIMNE) Barcelona, 2000, Computational Chemistry Section, Chapter 12. Quantum Quantitative Structure-Activity Relationships (QQSAR) A Comprehensive Discussion Based on Inward Matrix Products, Employed as a Tool to Find Approximate Solutions of Strictly Positive Linear Systems and Providing QSAR-Quantum Similarity Measures Connections. 

R. Carbo-Dorca,/. Mol. Struct. (THEOCHEM), 537, 41 (2001). Inward Matrix Products ... 

Whose elements are coincident with the diagonal terms of the d-type GTO s. In Eq. (13) there has been used an inward matrix product. 

Inward matrix products and complete sums are well documented in several previous places [21,61-65] and in CQS3. 

Manipulation of the TDQSH by means of the constraction of CSI has been also performed throughout devoted sections of CQS4. Generalized scalar products based on inward matrix products have been profusely employed for this purpose. The results show a wide variety of possible CSI definitions, which can be employed into MQOS description and ordering. 

For describing a general DF form, a definition of a new matrix product, the inward product, is needed, and it can be found in Appendix D along its properties. 

The interest in defining such a matrix product stems from the possibility to attach to it the most usual features of a multiplication composition rule. The following properties can be attached to the inward product ... 

These features and definitions permit to express density functions in a compact and extremely general form [92]. Such previous work may be referred to defining a matrix W as bearing a convex coefficient (hyper-)matrix of arbitrary dimensions Dim W)= [S]. In general, this can be obtained by choosing a complex matrix X, with adequate dimension Dim(X)= such that the inward product holds W = X X. Then, symbolically it can be assured that W = eW—>3 eX co = % % j special choice can be... 

It is a straightforward matter to generalise the inward product for applications within hyper-matrix spaces. 

Carburisation is observed in industrial processes where Fe-Ni-Cr alloys are used in carbonaceous atmospheres at high temperature, especially in the steam cracking of hydrocarbons for ethylene production. Carbon is transferred from the atmosphere into the metal matrix, diffuses inward and causes... 

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