**[1] Space vector PWM control of dual three-phase induction machine using vector space decomposition**

The technique of vector space decomposition control of voltage source inverter fed dual three-phase induction machines is presented in this paper. By vector space decomposition, the analytical modeling and control of the machine are accomplished in three two-dimensional orthogonal subspaces and the dynamics of the electromechanical energy conversion related and the nonelectromechanical energy conversion related machine variables are thereby totally decoupled. A space vector PWM technique is also developed based on the vector space decomposition to limit the 5th, 7th, 17th, 19th,… harmonic currents which in such a system would be otherwise difficult to control. The techniques developed in this paper can be generalized for the control of an induction machine with an arbitrary number of phases.

**[2] A critical analysis of vector space model for information retrieval**

Notations and definitions necessary to identify the concepts and relationships that are important in modelling information retrieval objects and processes in the context of vector spaces are presented. Earlier work on the use of vector model is evaluated in terms of the concepts introduced and certain problems and inconsistencies are identified. More importantly, this investigation should lead to a clear understanding of the issues and problems in using the vector space model in information retrieval. © 1986 John Wiley & Sons, Inc.

**[3] A vector space model for automatic indexing**

In a document retrieval, or other pattern matching environment where stored entities (documents) are compared with each other or with incoming patterns (search requests), it appears that the best indexing (property) space is one where each entity lies as far away from the others as possible; in these circumstances the value of an indexing system may be expressible as a function of the density of the object space; in particular, retrieval performance may correlate inversely with space density. An approach based on space density computations is used to choose an optimum indexing vocabulary for a collection of documents. Typical evaluation results are shown, demonstating the usefulness of the model.

**[4] Modelling Grammaticality-grading in Natural Language Systems Using a Vector Space Approach**

There exist several natural language processing systems that focus on checking the grammaticality (grammatical correctness or incorrectness) of natural language texts. Studies however showed that most existing systems do not assign specific scores to the grammaticality of the analysed text. Such scores would for instance prove very useful to second language learners and tutors, for judging the progress made in the learning process and assigning performance scores respectively. The current study was embarked upon to address this problem. A grammaticality grading model which comprised of 6 equations was developed using a vector space approach. The model was implemented in a natural language processing system. Correlation analysis showed that the grading (in %) performed using the developed model correlated at a coefficient of determination (R2) value of 0.9985 with the percentage of grammatical sentences in evaluated texts. The developed model is therefore deemed suitable for grammaticality grading in natural language texts.

**[5] Action of a Polynomial Matrix on a Vector of Power Series**

The adjoint of the right multiplication of a row vector by a fixed polynomial matrix gives a left operation of the polynomial matrix on column vectors of power series. This explain the polynomial matrix and vector of powers series “multiplication”, used to define discrete linear dynamical systems, according to Willems and Oberst theory.

**Reference**

**[1] **Zhao, Y. and Lipo, T.A., 1995. Space vector PWM control of dual three-phase induction machine using vector space decomposition. *IEEE Transactions on industry applications*, *31*(5), pp.1100-1109.

**[2] **Raghavan, V.V. and Wong, S.M., 1986. A critical analysis of vector space model for information retrieval. *Journal of the American Society for information Science*, *37*(5), pp.279-287.

**[3] **Salton, G., Wong, A. and Yang, C.S., 1975. A vector space model for automatic indexing. *Communications of the ACM*, *18*(11), pp.613-620.

**[4] **Aregbesola, M.K., Ganiyu, R.A., Olabiyisi, S.O., Omidiora, E.O. and Alo, O.O., 2017. Modelling Grammaticality-grading in Natural Language Systems Using a Vector Space Approach. *Journal of Advances in Mathematics and Computer Science*, pp.1-15.

**[5] **Andriamifidisoa, R., 2016. Action of a polynomial matrix on a vector of power series. *Journal of Advances in Mathematics and Computer Science*, pp.1-8.