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Wavelets for Subdivision
Published in Charles Chui, Johan de Villiers, Wavelet Subdivision Methods, 2010
Charles Chui, Johan de Villiers
Let the Laurent polynomial A be defined by (9.5.9) for an arbitrary Laurent polynomial T. Then, by (9.5.2), we have P(z)A(z)+P(−z)A(−z)=P(z)[1+zT(z2)P(−z)]+P(−z)[1−zT(z2)P(z)]=P(z)+P(−z)=1,
Thermal stress around a smooth cavity in a plate subjected to uniform heat flux
Published in Journal of Thermal Stresses, 2021
Zhaohang Lee, Yu Tang, Wennan Zou
It can be seen from Figure 7 that in general the TNS varies sharply with the curvature of the cavity contour around the tips, but the TNSs around different tips have different variation characteristics and these characteristics might be affected by the direction of heat flux, even zero value of TNS appears at some tips. The tip with zero TNS depending on heat flux direction can be applied as the most ideal tip position for industrial design and material properties. It is worth mentioning that the curvature at the tips of pentagram presents bimodal characteristics, and the TNS is also different at the bimodal points (this phenomenon is caused by the platform at the tip). With the increase of terms in the Laurent polynomial of pentagram, the size of the platform at the tip will gradually decrease, and become infinitesimal when the pentagram is characterized by the Laurent series, and then the TNSs at the bimodal points will gradually approach to the same. For the case of bimodal tip, when the failure stress is reached, the crack may expand in both directions. According to Figure 7, for three different shapes of cavities, it looks like that the maximum TNSs are obtained at the maximum curvature, and their values depends on the values of maximum curvature, so the maximum TNS are pentagram (−99.073)> triangle (−55.75331) > square (−34.43897).
Control systems analysis for the Fornasini-Marchesini 2D systems model – progress after four decades
Published in International Journal of Control, 2018
Krzysztof Galkowski, Eric Rogers
As described in Section 2.4, in the behavioural systems setting for the representation and analysis of linear systems, the central object of study is the behaviour , i.e. the set of trajectories that satisfy the laws of the system, the properties of , and how such properties are reflected in properties of the representations of . Linear difference behaviours are defined as the solution set of a system of linear partial difference equations. Equivalently, a linear difference behaviour can be interpreted as the kernel of a polynomial partial difference operator represented by a polynomial matrix. In the case , where R is a Laurent polynomial matrix in the indeterminates This setting for the discrete case also extends to continuous-time systems.
Mitigation of ocular artifacts for EEG signal using improved earth worm optimization-based neural network and lifting wavelet transform
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2021
Devulapalli Shyam Prasad, Srinivasa Rao Chanamallu, Kodati Satya Prasad
In the above equations, the filter coefficients are denoted as and the length of the filter is denoted as and the respective Laurent polynomial degree is indicated by By dividing the filter coefficients into odd and even parts, those filters are represented in Eqs. (36) and (37).