Sentence examples for presented in explicit forms from inspiring English sources
According to the explicit out-of-plane displacement field solution, the analytical results of mechanical and electric fields are also presented in explicit forms.
By using the Mellin transform technique in conjunction with the image method, the two-dimensional full-field solutions of dissimilar isotropic composite annular wedges subjected to anti-plane concentrated forces and screw dislocations are presented in explicit forms.
By making use of the resolvent-kernel technique, the governing equation is solved and the solution to the original problem is presented in explicit form.
Panel matrices, which are presented in an explicit form, may be used to analyze structures by using standard finite element procedures.
Furthermore, the fundamental nuclei for doubly-curved structures are presented in their explicit form for the first time by the authors.
The bi-orthogonality conditions in more advanced cases, specifically, for non-axisymmetric waves in an elastic cylindrical shell and for waves in an elastic helical spring, are derived, but not presented in an explicit form via wavenumbers.
Secondly, by using Lyapunov stability theory and linear matrix inequality (LMI) techniques, the sufficient conditions which can guarantee the stability of augmented filtering error system are obtained and the parameters of the designed filter can be presented in an explicit form.
The aim of this paper is to derive some mathematical properties of X in explicit forms.
The necessary and sufficient conditions for linearization were provided in explicit forms.
Firstly by applying the Maximum Principle developed in this paper, the necessary and sufficient solvability condition of the finite horizon mean-field optimal control problem is presented in an explicit expression form (in analytical form) for the first time, which is completely different from the operator-type conditions obtained in previous works.
Thus, the proposed control laws become available in explicit form.
