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Algebraic Polynomial Identities
Published in Charles Chui, Johan de Villiers, Wavelet Subdivision Methods, 2010
Charles Chui, Johan de Villiers
To develop a unified theory that also applies to the construction of synthesis wavelets associated with the interpolatory scaling functions ϕmI, we will consider a more general polynomial G = Gd in this chapter, and study a class of algebraic polynomial identities generated by G. A fundamental theorem of existence and uniqueness is established, with a constructive proof that also yields an algorithm for constructing the minimum-degree polynomial solution HG of the above identity, with (1+z2)d replaced by the more general polynomial G(z).
Matchings in graphs
Published in V. K. Balakrishnan, Network Optimization, 2019
The stable marriage problem is the problem of finding a stable matching for an arbitrary preference matrix. The constructive proof of Theorem 5.7 gives an iterative algorithm to solve this problem. The matching which results from this ‘propose-and-reject’ algorithm is not only stable but also optimal.
Daubechies Wavelets
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
Hint: See Daubechies (1992). There is an analogous Bézout’s theorem for integers. A constructive proof of this later theorem is based upon the Euclidean algorithm for determining the greatest common divisor of two integers, and the extended Euclidean algorithm. Daubechies’ proof is similar to its number-theoretic analog.
On a model for the magnetoelastic vibrations of a perfectly conducting Mindlin–Timoshenko plate
Published in Applicable Analysis, 2021
(iii) The decay rate of for solutions of can probably be improved by using a different strategy, e.g. by providing a constructive proof rather than using a contradiction argument in the verification of the conditions of the B–T criterion. In this regard we refer to the work of Jorge da Silva et al. [21] who obtained a decay rate of for the classical Mindlin–Timoshenko model with Kelvin–Voigt damping in the system for the shear angles by establishing direct estimates. Our view that different approaches may produce different decay rates appears to be confirmed by recent work of Tebou [22]. This author in fact achieved a decay rate of order for solutions of the Mindlin–Timoshenko model, again with Dirichlet boundary conditions, but equipped with viscous damping in the system for the shear angles, by using the B–T criterion combined with the technique of energy multipliers, thus to provide a constructive proof instead of a proof by contradiction.
Stability of a schedule minimising the makespan for processing jobs on identical machines
Published in International Journal of Production Research, 2023
By using Theorem 3.1, one can also test whether the schedule Sk ∈ S(t) has a zero stability radius. If the strict equalities hold, the algorithm realises a constructive proof of Theorem 7.1.
The optimality box in uncertain data for minimising the sum of the weighted job completion times
Published in International Journal of Production Research, 2018
Tsung-Chyan Lai, Yuri N. Sotskov, Natalja G. Egorova, Frank Werner
The constructive proof of the following claim allows us to describe an -algorithm for constructing an optimal permutation for a special case of the problem .