Monomial – Knowledge and References

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Micromechanical models for composite materials

Published in A.R. Bunsell, S. Joannès, A. Thionnet, Fundamentals of Fibre Reinforced Composite Materials, 2021

We consider an algebraic expression, i.e. a symbolic mathematical expression formed by monomials (themselves made up of numbers, letters, symbols) separated by an addition or subtraction operation (it is thus a polynomial expression). It is assumed that the letters and symbols present in the monomials can carry numeric and/or literal underscripts/upperscripts. In order to simplify the notations here, the word script means either an underscript or upperscript.

Control analysis and design via randomised coordinate polynomial minimisation

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Journal Information

Published in International Journal of Control, 2022

Giuseppe C. Calafiore, Carlo Novara, Corrado Possieri

First, note that given a monomial , it can be equivalently rewritten as , for each . Similarly, given , the ith coordinate-wise polynomial of f at is the univariate polynomial in with coefficients in that is obtained by considering all values in being fixed, and only the ith variate as variable, i.e. where . The ith coordinate-wise polynomial is useful when we update the estimate of the solution to the PMP (1) with : determining and essentially consists in computing m + 1 coordinate-wise polynomials. The next proposition, whose proof is given in Appendix A.5, provides an upper bound on the number of elementary operations (sum and products) that have to be carried out to determine such polynomials.