Correction to: Acta Mech https://doi.org/10.1007/s00707-023-03750-9


In the original publication, few errors were noted and are corrected in this correction.


In Section 6, under heading “Concluding remarks,” the last two paragraphs should be correctly read as:


Future works can include the damage detection of the rectangular plate with holes of arbitrary shapes with smooth boundaries. Also, some examples of computing the higher-order moments of inertia for the plates weakened by some holes can be presented using the experimental data. The results can be validated by available elasticity solutions.


To identify human faces in images or videos, or search for a face among an extensive collection of existing images, some facial features are identified and measured. Similarly, as we saw in this paper, there are some damage detector features, namely higher-order moments of inertia which are used for identifying the actual shape and location of a damage or a set of damages. Machine learning is a rapidly growing field and has been used for various tasks, such as facial recognition. Therefore, using machine learning algorithms to identify the shape and locations of damage or a set of damages in a plate via the measured higher-order moments of inertial is of particular interest for future work.

Equation 15 should be read as:

$$ \begin{aligned} M_{1,0}^{t} & = \frac{{x_{0} + x_{1}+ x_{2} }}{3}M_{0,0}^{1} + \frac{{x_{2}^{\prime} + x_{1}^{\prime} +x_{0}^{\prime} }}{3}M_{0,0}^{2} \\ M_{0,1}^{t} & = \frac{{y_{0} +y_{1} + y_{2} }}{3}M_{0,0}^{1} + \frac{{y_{2}^{\prime} +y_{1}^{\prime} + y_{0}^{\prime} }}{3}M_{0,0}^{2} \\ M_{2,0}^{t} & =\frac{{x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{1} x_{2} + x_{0} x_{2}+ x_{0} x_{1} }}{6}M_{0,0}^{1} \\ & \quad + \frac{{x_{0}^{{\prime2}}+ x_{1}^{{\prime2}} + x_{2}^{{\prime2}} + x_{1}^{\prime}x_{2}^{\prime} + x_{0}^{\prime} x_{2}^{\prime} + x_{0}^{\prime}x_{1}^{\prime} }}{6}M_{0,0}^{2} \\ M_{0,2}^{t} & = \frac{{y_{0}^{2} +y_{1}^{2} + y_{2}^{2} + y_{1} y_{2} + y_{0} y_{2} + y_{0} y_{1}}}{6}M_{0,0}^{1} \\ & \quad + \frac{{y_{0}^{{\prime2}} +y_{1}^{{\prime2}} + y_{2}^{{\prime2}} + y_{1}^{\prime} y_{2}^{\prime}+ y_{0}^{\prime} y_{2}^{\prime} + y_{0}^{\prime} y_{1}^{\prime}}}{6}M_{0,0}^{2} \\ M_{3,0}^{t} & = \frac{1}{10}\Big[x_{0}^{3} +x_{1}^{3} + x_{2}^{3} + x_{0}^{2} \left( {x_{1} + x_{2} } \right) +x_{1}^{2} \left( {x_{0} + x_{2} } \right) + x_{2}^{2} \left( { x_{0}+ x_{1} } \right) \\ & \quad + x_{0} x_{1} x_{2} \Big] {M_{00}^{1}+ \frac{1}{10}} \Big[x_{0}^{{\prime3}} + x_{1}^{{\prime3}} +x_{2}^{{\prime3}} + x_{0}^{{\prime2}} \left( {x_{1}^{\prime} +x_{2}^{\prime} } \right) + x_{1}^{{\prime2}} \left( {x_{0}^{\prime} +x_{2}^{\prime} } \right) \\ & \quad + x_{2}^{\prime2} \left({x_{0}^{\prime} + x_{1}^{\prime} } \right) + x_{0}^{\prime}x_{1}^{\prime} x_{2}^{\prime} \Big]M_{0,0}^{2} \\ M_{0,3}^{t} & =\frac{1}{10}\Big[y_{0}^{3} + y_{1}^{3} + y_{2}^{3} + y_{0}^{2} \left({y_{1} + y_{2} } \right) + y_{1}^{2} \left( {y_{0} + y_{2} } \right)+ y_{2}^{2} \left( { y_{0} + y_{1} } \right) \\ & \quad + y_{0}y_{1} y_{2} \Big] {M_{00}^{1} + \frac{1}{10}}\Big[y_{0}^{{\prime3}} + y_{1}^{{\prime3}} + y_{2}^{{\prime3}} +y_{0}^{{\prime2}} \left( {y_{1}^{\prime} + y_{2}^{\prime} } \right) +y_{1}^{{\prime2}} \left( {y_{0}^{\prime} + y_{2}^{\prime} } \right) \\& \quad + y_{2}^{{\prime2}} \left( { y_{0}^{\prime} + y_{1}^{\prime} }\right) + y_{0}^{\prime} y_{1}^{\prime} y_{2}^{\prime} \Big]M_{0,0}^{2} \\M_{4,0}^{t} & = \frac{1}{15}\Big[x_{0}^{4} + x_{1}^{4} + x_{2}^{4} +x_{0}^{3} \left( {x_{1} + x_{2} } \right) + x_{1}^{3} \left( { x_{0}+ x_{2} } \right) + x_{2}^{3} \left( {x_{0} + x_{1} } \right) \\ &\quad + x_{0} x_{1} x_{2} \left( {x_{0} + x_{1} + x_{2} } \right) +x_{0}^{2} x_{1}^{2} + x_{1}^{2} x_{2}^{2} + x_{0}^{2} x_{2}^{2}\Big]M_{0,0}^{1} \\ & \quad + \frac{1}{15}\Big[x_{0}^{{\prime4}} +x_{1}^{{\prime4}} + x_{2}^{{\prime4}} + x_{0}^{{\prime3}} \left({x_{1}^{\prime} + x_{2}^{\prime} } \right) + x_{1}^{{\prime3}} \left({x_{0}^{\prime} + x_{2}^{\prime} } \right) + x_{2}^{{\prime3}} \left({ x_{0}^{\prime} + x_{1}^{\prime} } \right) \\ & \quad + x_{0}^{\prime}x_{1}^{\prime} x_{2}^{\prime} \left( {x_{0}^{\prime} + x_{1}^{\prime} +x_{2}^{\prime} } \right) + x_{0}^{{\prime2}} x_{1}^{{\prime2}} +x_{1}^{{\prime2}} x_{2}^{{\prime2}} + x_{0}^{{\prime2}}x_{2}^{{\prime2}} \Big]M_{0,0}^{2} \\ M_{0,4}^{t} & =\frac{1}{15}\Big[y_{0}^{4} + y_{1}^{4} + y_{2}^{4} + y_{0}^{3} \left({y_{1} + y_{2} } \right) + y_{1}^{3} \left( { y_{0} + y_{2} }\right) + y_{2}^{3} \left( {y_{0} + y_{1} } \right) \\ & \quad +y_{0} y_{1} y_{2} \left( {y_{0} + y_{1} + y_{2} } \right) +y_{0}^{2} y_{1}^{2} + y_{1}^{2} y_{2}^{2} + y_{0}^{2} y_{2}^{2}\Big]M_{0,0}^{1} \\ & \quad + \frac{1}{15}\Big[y_{0}^{\prime4} +y_{1}^{\prime4} + y_{2}^{\prime4} + y_{0}^{\prime3} \left({y_{1}^{\prime} + y_{2}^{\prime} } \right) + y_{1}^{\prime3} \left({y_{0}^{\prime} + y_{2}^{\prime} } \right) + y_{2}^{\prime3} \left({ y_{0}^{\prime} + y_{1}^{\prime} } \right) \\ & \quad + y_{0}^{\prime}y_{1}^{\prime} y_{2}^{\prime} \left( {y_{0}^{\prime} + y_{1}^{\prime} +y_{2}^{\prime} } \right) + y_{0}^{\prime2} y_{1}^{\prime2} +y_{1}^{\prime2} y_{2}^{\prime2} + y_{0}^{\prime2}y_{2}^{\prime2} \Big]M_{0,0}^{2} \\ M_{{5,0}}^{t} & =\frac{1}{{21}}\Big[x_{0}^{5} + x_{1}^{5} + x_{2}^{5} + x_{0}^{4}\left( {x_{1} + x_{2} } \right) + x_{1}^{4} \left( {x_{0} + x_{2}} \right) + x_{2}^{4} \left( {x_{0} + x_{1} } \right) \\ &\quad + x_{0}^{3} \left( {x_{1}^{2} + x_{1} x_{2} + x_{2}^{2} }\right) + x_{1}^{3} \left( {x_{0}^{2} + x_{0} x_{2} + x_{2}^{2} }\right) + x_{2}^{3} \left( {x_{0}^{2} + x_{0} x_{1} + x_{1}^{2} }\right) \\ & \quad + x_{0} x_{1} x_{2} \left( {x_{1} x_{2} +x_{0} x_{2} + x_{0} x_{1} } \right)\Big] {M_{{0,0}}^{1} +\frac{1}{{21}}} \Big[x_{0}^{\prime5} + x_{1}^{\prime5} +x_{2}^{\prime5} + x_{0}^{\prime4} \left( {x_{1}^{\prime} +x_{2}^{\prime} } \right) \\ & \quad + x_{1}^{\prime4} \left({x_{0}^{\prime} + x_{2}^{\prime} } \right) + x_{2}^{\prime4} \left({x_{0}^{\prime} + x_{1}^{\prime} } \right) + x_{0}^{\prime3} \left({x_{1}^{\prime2} + x_{1}^{\prime} x_{2}^{\prime} + x_{2}^{\prime2}} \right) \\ & \quad + x_{1}^{\prime3} \left( {x_{0}^{\prime2}+ x_{0}^{\prime} x_{2}^{\prime} + x_{2}^{\prime2} } \right) +x_{2}^{\prime3} \left( {x_{0}^{\prime2} + x_{0}^{\prime}x_{1}^{\prime} + x_{1}^{\prime2} } \right) \\ & \quad +x_{0}^{\prime} x_{1}^{\prime} x_{2}^{\prime} \left( {x_{1}^{\prime}x_{2}^{\prime} + x_{0}^{\prime} x_{2}^{\prime} + x_{0}^{\prime}x_{1}^{\prime} } \right)\Big]M_{{0,0}}^{2} \\ M_{{0,5}}^{t} & =\frac{1}{{21}}\Big[y_{0}^{5} + y_{1}^{5} + y_{2}^{5} + y_{0}^{4}\left( {y_{1} + y_{2} } \right) + y_{1}^{4} \left( {y_{0} + y_{2}} \right) + y_{2}^{4} \left( {y_{0} + y_{1} } \right) \\ &\quad + y_{0}^{3} \left( {y_{1}^{2} + y_{1} y_{2} + y_{2}^{2} }\right) + y_{1}^{3} \left( {y_{0}^{2} + y_{0} y_{2} + y_{2}^{2} }\right) + y_{2}^{3} \left( {y_{0}^{2} + y_{0} y_{1} + y_{1}^{2} }\right) \\ & \quad + y_{0} y_{1} y_{2} \left( {y_{1} y_{2} +y_{0} y_{2} + y_{0} y_{1} } \right)\Big] {M_{{0,0}}^{1} +\frac{1}{{21}}} \Big[y_{0}^{\prime5} + y_{1}^{\prime5} +y_{2}^{\prime5} + y_{0}^{\prime4} \left( {y_{1}^{\prime} +y_{2}^{\prime} } \right) \\ & \quad + y_{1}^{\prime4} \left({y_{0}^{\prime} + y_{2}^{\prime} } \right) + y_{2}^{\prime4} \left({y_{0}^{\prime} + y_{1}^{\prime} } \right) + y_{0}^{\prime3} \left({y_{1}^{\prime2} + y_{1}^{\prime} y_{2}^{\prime} + y_{2}^{\prime2}} \right) \\ & + y_{1}^{\prime3} \left( {y_{0}^{\prime2} +y_{0}^{\prime} y_{2}^{\prime} + y_{2}^{\prime2} } \right) +y_{2}^{\prime3} \left( {y_{0}^{\prime2} + y_{0}^{\prime}y_{1}^{\prime} + y_{1}^{\prime2} } \right) \\ & +y_{0}^{\prime} y_{1}^{\prime} y_{2}^{\prime} \left( {y_{1}^{\prime}y_{2}^{\prime} + y_{0}^{\prime} y_{2}^{\prime} + y_{0}^{\prime}y_{1}^{\prime} } \right)\Big]M_{{0,0}}^{2} \\ M_{{6,0}}^{t} & =\frac{1}{{28}}\Big[x_{0}^{6} + x_{1}^{6} + x_{2}^{6} + x_{0}^{5}\left( {x_{1} + x_{2} } \right) + x_{1}^{5} \left( {x_{0} + x_{2}} \right) + x_{2}^{5} \left( {x_{0} + x_{1} } \right) \\ &\quad + x_{0} x_{1} x_{2} \left( {x_{0} + x_{1} + x_{2} }\right)\left( {x_{0} ^{2} + x_{1} ^{2} + x_{2} ^{2} } \right) +\left( {x_{0}^{2} + x_{1}^{2} } \right)x_{0}^{2} x_{1}^{2} \\& \quad + x_{0}^{2} x_{2}^{2} \left( {x_{0}^{2} + x_{2}^{2} }\right) + x_{1}^{2} x_{2}^{2} \left( {x_{1}^{2} + x_{2}^{2} }\right) + x_{0}^{3} x_{1}^{3} + x_{1}^{3} x_{2}^{3} + x_{0}^{3}x_{2}^{3} + x_{0}^{2} x_{1}^{2} x_{2}^{2} \Big]M_{{0,0}}^{1} \\ &\quad + \frac{1}{{28}}\Big[x_{0}^{\prime6} + x_{1}^{\prime6} +x_{2}^{\prime6} + x_{0}^{\prime5} \left( {x_{1}^{\prime} +x_{2}^{\prime} } \right) + x_{1}^{\prime5} \left( {x_{0}^{\prime} +x_{2}^{\prime} } \right) + x_{2}^{\prime5} \left( {x_{0}^{\prime} +x_{1}^{\prime} } \right) \\ & \quad + x_{0}^{\prime}x_{1}^{\prime} x_{2}^{\prime} \left( {x_{2}^{\prime} +x_{1}^{\prime} + x_{0}^{\prime} } \right)\left( {x_{0}^{\prime2} +x_{1}^{\prime2} + x_{2}^{\prime2} } \right) + \left({x_{0}^{\prime2} + x_{1}^{\prime2} } \right)x_{0}^{\prime2}x_{1}^{\prime2} \\ & + x_{0}^{\prime2} x_{2}^{\prime2} \left({x_{0}^{\prime2} + x_{2}^{\prime2} } \right) + x_{1}^{\prime2}x_{2}^{\prime2} \left( {x_{1}^{\prime2} + x_{2}^{\prime2} } \right)+ x_{0}^{\prime3} x_{1}^{\prime3} + x_{1}^{\prime3} x_{2}^{\prime3}+ x_{0}^{\prime3} x_{2}^{\prime3} \\ & \quad + x_{0}^{\prime2}x_{1}^{\prime2} x_{2}^{\prime2} \Big]M_{{0,0}}^{2} \\ M_{{0,6}}^{t}& = \frac{1}{{28}}\Big[y_{0}^{6} + y_{1}^{6} + y_{2}^{6} + y_{0}^{5}\left( {y_{1} + y_{2} } \right) + y_{1}^{5} \left( {y_{0} + y_{2}} \right) + y_{2}^{5} \left( {y_{0} + y_{1} } \right) \\ &\quad + y_{0} y_{1} y_{2} \left( {y_{0} + y_{1} + y_{2} }\right)\left( {y_{0} ^{2} + y_{1} ^{2} + y_{2} ^{2} } \right) +\left( {y_{0}^{2} + y_{1}^{2} } \right)y_{0}^{2} y_{1}^{2} \\& \quad + y_{0}^{2} y_{2}^{2} \left( {y_{0}^{2} + y_{2}^{2} }\right) + y_{1}^{2} y_{2}^{2} \left( {y_{1}^{2} + y_{2}^{2} }\right) + y_{0}^{3} y_{1}^{3} + y_{1}^{3} y_{2}^{3} + y_{0}^{3}y_{2}^{3} \\ & \quad + y_{0}^{2} y_{1}^{2} y_{2}^{2}\Big]M_{{0,0}}^{1} + \frac{1}{{28}}\Big[y_{0}^{\prime6} +y_{1}^{\prime6} + y_{2}^{\prime6} + y_{0}^{\prime5} \left({y_{1}^{\prime} + y_{2}^{\prime} } \right) + y_{1}^{\prime5} \left({y_{0}^{\prime} + y_{2}^{\prime} } \right) + y_{2}^{\prime5} \left({y_{0}^{\prime} + y_{1}^{\prime} } \right) \\ & \quad +y_{0}^{\prime} y_{1}^{\prime} y_{2}^{\prime} \left( {y_{2}^{\prime}+ y_{0}^{\prime} + y_{1}^{\prime} } \right)\left( {y_{0}^{\prime2}+ y_{1}^{\prime2} + y_{2}^{\prime2} } \right) + \left({y_{0}^{\prime2} + y_{1}^{\prime2} } \right)y_{0}^{\prime2}y_{1}^{\prime2} \\ & \quad + y_{0}^{\prime2} y_{2}^{\prime2}\left( {y_{0}^{\prime2} + y_{2}^{\prime2} } \right) +y_{1}^{\prime2} y_{2}^{\prime2} \left( {y_{1}^{\prime2} +y_{2}^{\prime2} } \right) + y_{0}^{\prime3} y_{1}^{\prime3} +y_{1}^{\prime3} y_{2}^{\prime3} + y_{0}^{\prime3} y_{2}^{\prime3}\\ & \quad + y_{0}^{\prime2} y_{1}^{\prime2} y_{2}^{\prime2}\Big]M_{{0,0}}^{2} \\ \end{aligned} $$
(15)

Equation 40 should be read as:

$$ \begin{aligned} &\mathop \sum \limits_{i = 0}^{m}\mathop \sum \limits_{j = 0}^{m - i} \{ a_{ij} \left( {r + 2}\right)\left( {i + 2} \right)M_{i + r + 2,j + s} \hfill \\ &\quad+ \nu \left( {j + 2} \right)\left[ {\left( {r + 2} \right)M_{i + r + 1,j +s + 1} - a\left( {r + 1} \right)M_{i + r,j + s + 1} } \right]b_{ij}\} \hfill \\ &\quad- \mathop \sum \limits_{i = 0}^{m} \mathop \sum \limits_{j = 0}^{m - i} \{ a\left[ {\left( {r + 1} \right)\left( {i + 2} \right) + \left( {r + 2} \right)\left( {i + 1} \right)}\right]a_{ij} \hfill \\ &\quad+ \left( {r + 2} \right)\nu bb_{ij} \left({j + 1} \right)\} M_{i + r + 1,j + s} \hfill \\ &\quad+ a\left( {r + 1}\right)\mathop \sum \limits_{i = 0}^{m} \mathop \sum \limits_{j =0}^{m - i} \left[ {aa_{ij} \left( {i + 1} \right) + \nu bb_{ij}\left( {j + 1} \right)} \right]M_{i + r,j + s} \hfill \\ &\quad-\frac{{\left( {\nu - 1} \right)}}{2}s \Bigg\{ \mathop \sum \limits_{i =0}^{m} \mathop \sum \limits_{j = 1}^{m - i} a_{ij} j\left( {M_{i + r + 4,j + s - 2} - 2aM_{i + r + 3,j + s - 2} + a^{2} M_{i + r + 2,j +s - 2} } \right) \hfill \\ & \quad + \mathop \sum \limits_{i =0}^{m} \mathop \sum \limits_{j = 1}^{m - i} b_{ij} i\left( {M_{i + r + 1,j + s + 1} - aM_{i + r, j + s + 1} -b M_{i + r + 1,j +s} } + ab M_{i + r + 1,j +s} \right) \Bigg\} \hfill \\ &\quad= W_{rs} ,0 \le r \le m,0 \le s \le m - r \hfill \\ &\mathop \sum \limits_{i = 0}^{m} \mathop \sum \limits_{j =0}^{m - i} \{ \nu a_{ij} \left( {i + 2} \right)\left[ {\left( {s +2} \right)M_{i + r + 1,j + s + 1} - b\left( {s + 1} \right)M_{i + r + 1,j + s} } \right] \hfill \\ &\quad+ \left( {s + 2} \right)\left( {j +2} \right)b_{ij} M_{i + r,j + s + 2} \} - \mathop \sum \limits_{i =0}^{m} \mathop \sum \limits_{j = 0}^{m - i} \{ \nu a\left( {s + 2}\right)\left( {i + 1} \right)a_{ij} \hfill \\ &\quad+\left[ {\left( {s +2} \right)\left( {j + 1} \right) + \left( {s + 1} \right)\left( {j +2} \right)} \right]bb_{ij} \} M_{i + r,j + s + 1} \hfill \\ &\quad+b\left( {s + 1} \right)\mathop \sum \limits_{i = 0}^{m} \mathop \sum \limits_{j = 0}^{m - i} \left[ {\nu aa_{ij} \left( {i + 1} \right) +bb_{ij} \left( {j + 1} \right)} \right]M_{i + r,j + s} \hfill \\ &\quad-\frac{{\left( {\nu - 1} \right)}}{2}r\Bigg\{ \mathop \sum \limits_{i =1}^{m} \mathop \sum \limits_{j = 0}^{m - i} b_{ij} i\left( {M_{i + r - 2,j + s + 4} - 2bM_{i + r - 2,j + s + 3} + b^{2} M_{i + r - 2,j +s + 2} } \right) \hfill \\ &\quad+ \mathop \sum \limits_{i = 0}^{m}\mathop \sum \limits_{j = 1}^{m - i} a_{ij} j\left( {M_{i + r + 1,j + s + 1} - aM_{i + r,j + s + 1} - bM_{i + r + 1,j + s} + abM_{i +r,j + s} } \right)\Bigg\} \hfill \\ &\quad= V_{rs} ,0 \le r \le m,0 \le s \le m - r \hfill \\ \end{aligned} $$
(40)

In Appendix C, \({C}_{13}\) should be read as:

$$\begin{aligned} C_{{30}} & = 2\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)^{2} , \\ C_{{22}} & = - 3\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} - 1} \right)\left( {m_{{0,0}}^{1} + 1} \right)^{2} , \\ C_{{21}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{2} m_{{1,0}}^{t} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} ~ + ~4m_{{0,0}}^{1} ~ + ~3} \right], \\ C_{{20}} & = - 9\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {4m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)m_{{2,0}}^{t} ~ - ~5m_{{0,0}}^{1} \left( {m_{{1,0}}^{t} } \right)^{2} - 3\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{14}} & = 3m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)^{2} \left( {\left( {m_{{0,0}}^{1} } \right)^{2} + 1} \right), \\ C_{{13}} & = - 12m_{{0,0}}^{1} m_{{1,0}}^{t} [\left( {m_{{0,0}}^{1} } \right)^{3} + ~3\left( {m_{{0,0}}^{1} } \right)^{2} ~ + ~5m_{{0,0}}^{1} + ~3] \\ C_{{12}} & = 6m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)\left[ {2m_{{0,0}}^{1} \left( {2m_{{0,0}}^{1} - 3} \right)m_{{2,0}}^{t} + 9\left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{11}} & = 36m_{{0,0}}^{1} m_{{1,0}}^{t} \left[ {2m_{{0,0}}^{1} \left( {2m_{{0,0}}^{1} + 3} \right)m_{{2,0}}^{t} - 9\left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{10}} & = 3m_{{0,0}}^{1} \left\{ { - ~10m_{{4,0}}^{t} \left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right) + ~40\left( {m_{{0,0}}^{1} } \right)^{3} m_{{3,0}}^{t} m_{{1,0}}^{t} } \right. \\ & \quad \left. { + 24\left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{2,0}}^{t} } \right)^{2} - 108m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} + 27\left( {2m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{4} } \right\}, \\ C_{{06}} & = - \left[ {\left( {m_{{0,0}}^{1} } \right)^{3} - ~1} \right]\left( {m_{{0,0}}^{1} + ~1} \right)^{2} , \\ C_{{05}} & = 6m_{{1,0}}^{t} \left[ {\left( {m_{{0,0}}^{1} } \right)^{4} + \left( {m_{{0,0}}^{1} } \right)^{3} - ~2\left( {m_{{0,0}}^{1} } \right)^{2} - ~5m_{{0,0}}^{1} - ~3} \right], \\ C_{{04}} & = 18\left( {m_{{1,0}}^{t} } \right)^{2} ~ - 3\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} ~ + ~1} \right]\left[ {3\left( {m_{{1,0}}^{t} } \right)^{2} + 4m_{{0,0}}^{1} m_{{2,0}}^{t} } \right] \\ & - 18\left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 1} \right]\left[ {m_{{0,0}}^{1} m_{{2,0}}^{t} - 3\left( {m_{{1,0}}^{t} } \right)^{2} } \right] - 12\left( {m_{{0,0}}^{1} + 1} \right)\left[ {2m_{{0,0}}^{1} m_{{2,0}}^{t} - 15\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ C_{{03}} & = 4\left( {m_{{0,0}}^{1} + 1} \right)\left[ { - 5m_{{3,0}}^{t} \left( {m_{{0,0}}^{1} } \right)^{3} + 54m_{{0,0}}^{1} m_{{2,0}}^{t} m_{{1,0}}^{t} - 135\left( {m_{{1,0}}^{t} } \right)^{3} } \right] \\ & \quad + 108\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {m_{{0,0}}^{1} m_{{2,0}}^{t} m_{{1,0}}^{t} - \left( {m_{{1,0}}^{t} } \right)^{3} } \right], \\ C_{{02}} & = 3\left\{ {5\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {1 - \left( {m_{{0,0}}^{1} } \right)^{2} } \right]m_{{4,0}}^{t} + 20\left( {m_{{0,0}}^{1} } \right)^{3} \left( {~m_{{0,0}}^{1} + 2} \right)m_{{3,0}}^{t} m_{{1,0}}^{t} } \right.~ \\ & \quad + 12\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 2m_{{0,0}}^{1} + ~3} \right]\left( {m_{{2,0}}^{t} } \right)^{2} - 54m_{{0,0}}^{1} \left[ {~\left( {m_{{0,0}}^{1} } \right)^{2} + 4m_{{0,0}}^{1} + 6} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} \\ & \quad \left. { + 27\left[ {~\left( {m_{{0,0}}^{1} } \right)^{2} + 10m_{{0,0}}^{1} ~ + 15} \right]\left( {m_{{1,0}}^{t} } \right)^{4} } \right\}, \\ C_{{01}} & = 6\left\{ {\left( {m_{{0,0}}^{1} } \right)^{4} \left[ {7\left( {m_{{0,0}}^{1} + 1} \right)m_{{5,0}}^{t} - ~40m_{{3,0}}^{t} m_{{2,0}}^{t} } \right]} \right. \\ & \quad - ~15\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {m_{{4,0}}^{t} m_{{1,0}}^{t} + 2\left( {m_{{1,0}}^{t} } \right)^{2} m_{{3,0}}^{t} } \right] + 36\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} - 3} \right)m_{{1,0}}^{t} \left( {m_{{2,0}}^{t} } \right)^{2} ~ \\ & \quad \left. { + 108m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{3} m_{{2,0}}^{t} ~ - ~81\left( {m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{5} } \right\}, \\ C_{{00}} & = - 9\left\{ {2~\left( {m_{{0,0}}^{1} } \right)^{4} \left( {7m_{{5,0}}^{t} m_{{1,0}}^{t} - 10m_{{4,0}}^{t} m_{{2,0}}^{t} } \right) + 3\left( {m_{{0,0}}^{1} } \right)^{3} \left[ { - 5m_{{4,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} + 16\left( {m_{{2,0}}^{t} } \right)^{3} } \right]} \right. \\ & \quad \left. { - ~108\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{1,0}}^{t} } \right)^{2} \left( {m_{{2,0}}^{t} } \right)^{2} + ~162m_{{0,0}}^{1} \left( {m_{{1,0}}^{t} } \right)^{4} m_{{2,0}}^{t} - ~81\left( {m_{{1,0}}^{t} } \right)^{6} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} D_{{40}} & = \left( {m_{{0,0}}^{1} } \right)^{4} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{3} , \\ D_{{32}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{2} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} ~ - 1} \right], \\ D_{{31}} & = - 12\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + ~1} \right)^{2} \left( {m_{{0,0}}^{1} + 3} \right)m_{{1,0}}^{1} , \\ D_{{30}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} ~ + ~1} \right)\left[ {4\left( {\left( {m_{{0,0}}^{1} } \right)^{2} + m_{{0,0}}^{1} } \right)m_{{2,0}}^{t} - 3\left( {5m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{2} } \right], \\ D_{{24}} & = 3\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} + ~1} \right)^{3} \left[ {\left( {m_{{0,0}}^{1} } \right)^{2} ~ - ~8m_{{0,0}}^{1} ~ + ~1} \right], \\ D_{{23}} & = 12\left( {m_{{0,0}}^{1} } \right)^{2} m_{{1,0}}^{t} \left( {m_{{0,0}}^{1} ~ + ~1} \right)^{2} \left[ {7\left( {m_{{0.0}}^{1} } \right)^{2} + ~14m_{{0,0}}^{1} - ~3} \right], \\ D_{{22}} & = - 6\left( {m_{{0,0}}^{1} } \right)^{2} \left( {m_{{0,0}}^{1} ~ + ~1} \right)\left\{ {3\left[ {25\left( {m_{{0,0}}^{1} } \right)^{2} + ~12m_{{0,0}}^{1} - 9} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 2m_{{0,0}}^{1} \left[ { - ~10\left( {m_{{0,0}}^{1} } \right)^{2} - m_{{0,0}}^{1} + 9} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{21}} & = 36\left( {m_{{0,0}}^{1} } \right)^{2} m_{{1,0}}^{t} \left\{ {3\left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - 6m_{{0,0}}^{1} - 3} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 2m_{{0.0}}^{1} \left[ {2\left( {m_{{0.0}}^{1} } \right)^{2} + 11m_{{0,0}}^{1} + 9} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{20}} & = 3\left( {m_{{0,0}}^{1} } \right)^{2} \left\{ {9\left[ {6\left( {m_{{0,0}}^{1} } \right)^{2} + 39m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t} } \right)^{4} + 24\left( {m_{{0,0}}^{1} } \right)^{2} [\left( {m_{{0,0}}^{1} } \right)^{2} + 4m_{{0,0}}^{1} } \right. \\ & \quad + 3]\left( {m_{{2,0}}^{t} } \right)^{2} - 36m_{{0,0}}^{1} \left[ {3\left( {m_{{0,0}}^{1} } \right)^{2} + 14m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} - 10\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)^{2} m_{{4,0}}^{t} \\ & \quad \left. { + 40\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)m_{{3,0}}^{t} m_{{1,0}}^{t} } \right\}, \\ D_{{16}} & = - 2m_{{0,0}}^{1} \left( {m_{{0,0}}^{1} + 1} \right)^{3} \left( {m_{{0,0}}^{1} - 1} \right)\left[ {\left( {m_{{0,0}}^{1} } \right)^{2} - ~5m_{{0,0}}^{1} ~ + ~1} \right], \\ D_{{15}} & = - 12m_{{0,0}}^{1} m_{{1,0}}^{t} \left( {m_{{00}}^{1} + 1} \right)^{2} \left[ {3\left( {m_{{0,0}}^{1} } \right)^{3} + 2\left( {m_{{0,0}}^{1} } \right)^{2} - 12m_{{0,0}}^{1} ~ + ~3} \right], \\ D_{{14}} & = - 6m_{{00}}^{1} \left( {m_{{00}}^{1} ~ + ~1} \right)\left\{ {3\left[ { - ~7\left( {m_{{0,0}}^{1} } \right)^{3} + 32\left( {m_{{0,0}}^{1} } \right)^{2} + 30m_{{0,0}}^{1} - 15} \right]\left( {m_{{1,0}}^{t} } \right)^{2} } \right. \\ & \quad \left. { + 2m_{{0,0}}^{1} \left[ {2\left( {m_{{0,0}}^{1} } \right)^{3} - 21\left( {m_{{0,0}}^{1} } \right)^{2} - 20m_{{0,0}}^{1} + 3} \right]m_{{2,0}}^{t} } \right\}, \\ D_{{13}} & = 8m_{{0,0}}^{1} \left\{ {10\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)^{2} m_{{3,0}}^{t} + 27\left[ {5\left( {m_{{0,0}}^{1} } \right)^{3} + 12\left( {m_{{0,0}}^{1} } \right)^{2} - 5} \right]\left( {m_{{1,0}}^{t} } \right)^{3} } \right. \\ & \quad \left. { - 18m_{{0,0}}^{1} \left[ {9\left( {m_{{0,0}}^{1} } \right)^{3} + 22\left( {m_{{0,0}}^{1} } \right)^{2} + 10m_{{0,0}}^{1} - 3} \right]m_{{2,0}}^{t} m_{{1,0}}^{t} } \right\}, \\ D_{{12}} & = - 6m_{{0,0}}^{1} \left\{ {25\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1} \right)\left[ {1 - \left( {m_{{0,0}}^{1} } \right)^{2} } \right]m_{{4,0}}^{t} } \right. \\ & \quad + 20\left( {m_{{0,0}}^{1} } \right)^{3} \left( {5m_{{0,0}}^{1} + 1} \right)\left( {m_{{0,0}}^{1} + 1} \right)m_{{3,0}}^{t} m_{{1,0}}^{t} \\ & \quad + 12\left( {m_{{0,0}}^{1} } \right)^{2} \left[ {11\left( {m_{{0,0}}^{1} } \right)^{2} + 5\left( {m_{{0,0}}^{1} } \right)^{3} - 3m_{{0,0}}^{1} - 9} \right]\left( {m_{{2,0}}^{t} } \right)^{2} \\ & \quad + 18m_{{0,0}}^{1} \left[ { - ~53\left( {m_{{0,0}}^{1} } \right)^{2} - 15\left( {m_{{0,0}}^{1} } \right)^{3} - 18m_{{0,0}}^{1} + 18} \right]\left( {m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} \\ & \quad \left. { + ~27\left[ {5\left( {m_{{0,0}}^{1} } \right)^{3} + 17\left( {m_{{0,0}}^{1} } \right)^{2} - 15m_{{0,0}}^{1} - ~15} \right]\left( {m_{{1,0}}^{t} } \right)^{4} } \right\}, \\ \end{aligned}$$
$$\begin{aligned} D_{{11}} & = - 36m_{{0,0}}^{1}m_{{1,0}}^{t} \{ 5\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {\left({m_{{0,0}}^{1} } \right)^{2} - 4m_{{0,0}}^{1} - 5}\right]m_{{4,0}}^{t} + 20\left( {m_{{0,0}}^{1} } \right)^{3} \left({2 - m_{{0,0}}^{1} } \right)m_{{3,0}}^{t} m_{{1,0}}^{t} \\ & \quad +12\left( {m_{{0,0}}^{1} } \right)^{2} \left[ { - \left({m_{{0,0}}^{1} } \right)^{2} + 6m_{{0,0}}^{1} + 9} \right]\left({m_{{2,0}}^{t} } \right)^{2} + 18m_{{0,0}}^{1} \left[ {3\left({m_{{0,0}}^{1} } \right)^{2} - 8m_{{0,0}}^{1} - 6} \right]\left({m_{{1,0}}^{t} } \right)^{2} m_{{2,0}}^{t} \\ & \quad + 9\left[ { -3\left( {m_{{0,0}}^{1} } \right)^{2} + 18m_{{0,0}}^{1} + 9}\right]\left( {m_{{1,0}}^{t} } \right)^{4} \} , \\ D_{{10}} & =18m_{{0,0}}^{1} \left\{ {20\left( {m_{{0,0}}^{1} } \right)^{4}\left[ {\left( {m_{{0,0}}^{1} + 1} \right)m_{{4,0}}^{t}m_{{2,0}}^{t} - 4m_{{1,0}}^{t} m_{{2,0}}^{t} m_{{3,0}}^{t} }\right]} \right. \\ & \quad + 324\left( {m_{{0,0}}^{1} }\right)^{2} \left( {m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t}} \right)^{2} \left( {m_{{2,0}}^{t} } \right)^{2} \\ & \quad +3\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {60m_{{3,0}}^{t} \left({m_{{1,0}}^{t} } \right)^{3} - 16\left( {m_{{0,0}}^{1} + 1}\right)\left( {m_{{2,0}}^{t} } \right)^{3} - 5\left( {3m_{{0,0}}^{1}+ 5} \right)m_{{4,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} }\right] \\ & \quad \left. { - 54m_{{0,0}}^{1} \left({11m_{{0,0}}^{1} + 3} \right)\left( {m_{{1,0}}^{t} } \right)^{4}m_{{2,0}}^{t} + 81\left( {3m_{{0,0}}^{1} + 1} \right)\left({m_{{1,0}}^{t} } \right)^{6} } \right\}, \\ D_{{08}} & = \left({m_{{0,0}}^{1} + 1} \right)^{3} \left[ {\left( {m_{{0,0}}^{1} }\right)^{2} - m_{{0,0}}^{1} + 1} \right]^{2} , \\ D_{{07}} & = -24m_{{1,0}}^{t} \left( {m_{{0,0}}^{1} + 1} \right)^{2} \left({\left( {m_{{0,0}}^{1} } \right)^{2} - m_{{0,0}}^{1} + 1} \right),\\ D_{{06}} & = 12\left( {m_{{0,0}}^{1} + 1} \right)\left\{ {\left[{9\left( {m_{{0,0}}^{1} } \right)^{3} + 21} \right]\left({m_{{1,0}}^{t} } \right)^{2} + m_{{0,0}}^{1} \left[ { - 5\left({m_{{0,0}}^{1} } \right)^{3} + 4m_{{0,0}}^{1} - 1}\right]m_{{2,0}}^{t} } \right\}, \\ D_{{05}} & = 72m_{{1,0}}^{t}\left( {m_{{0,0}}^{1} + 1} \right)\{ m_{{0,0}}^{1} \left[ {4\left({m_{{0,0}}^{1} } \right)^{3} + \left( {m_{{0,0}}^{1} } \right)^{2} -9m_{{0,0}}^{1} + 3} \right]m_{{2,0}}^{t} \\ & \quad \left. { -3\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 7} \right]\left({m_{{1,0}}^{t} } \right)^{2} } \right\}, \\ D_{{04}} & = \left\{{120\left( {m_{{0,0}}^{1} } \right)^{3} \left[ {\left({m_{{0,0}}^{1} } \right)^{3} + 1} \right]m_{{3,0}}^{t} m_{{1,0}}^{t}- 30\left( {m_{{0,0}}^{1} } \right)^{3} \left( {m_{{0,0}}^{1} + 1}\right)\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 1}\right]m_{{4,0}}^{t} } \right. \\ & \quad + 162\left( {m_{{0,0}}^{1}+ 1} \right)\left[ {\left( {m_{{0,0}}^{1} } \right)^{3} + 35}\right]\left( {m_{{1,0}}^{t} } \right)^{3} + 36\left( {m_{{0,0}}^{1}} \right)^{2} \left[2\left( {m_{{0,0}}^{1} } \right)^{4} + 2\left({m_{{0,0}}^{1} } \right)^{3} - 20\left( {m_{{0,0}}^{1} } \right)^{2}\right.\\ &\left. \quad - 17m_{{0,0}}^{1} + 3\right] {\left( {m_{{2,0}}^{t} }\right)^{2} + 108m_{{0,0}}^{1} } \left[24\left( {m_{{0,0}}^{1} }\right)^{2} - 5\left( {m_{{0,0}}^{1} } \right)^{3} - 15 +15m_{{0,0}}^{1}\right. \\ & \quad\left. \left. - 3\left( {m_{{0,0}}^{1}}\right)^{4} \right]\left( {m_{{1,0}}^{t} } \right)^{2} {m_{{2,0}}^{t} }\right\}, \\ D_{{03}} & = 24\left\{ { - 20\left( {m_{{0,0}}^{1} }\right)^{5} m_{{3,0}}^{t} m_{{2,0}}^{t} - 20\left( {m_{{0,0}}^{1} }\right)^{4} m_{{3,0}}^{t} m_{{2,0}}^{t} - 60\left( {m_{{0,0}}^{1} }\right)^{3} m_{{3,0}}^{t} \left( {m_{{1,0}}^{t} } \right)^{2} }\right. \\ & \quad + 15\left( {m_{{0,0}}^{1} } \right)^{3} \left({m_{{0,0}}^{1} + 1} \right)m_{{4,0}}^{t} m_{{1,0}}^{t} + 18\left({m_{{0,0}}^{1} } \right)^{2} \left[ {10\left( {m_{{0,0}}^{1} }\right)^{2} + 10m_{{0,0}}^{1} - 3} \right]m_{{1,0}}^{t} \left({m_{{2,0}}^{t} } \right)^{2} \\ & \quad \left. { + 54m_{{0,0}}^{1}\left[ { - 4\left( {m_{{0,0}}^{1} } \right)^{2} + 5} \right]\left({m_{{1,0}}^{t} } \right)^{3} m_{{2,0}}^{t} - 567\left({m_{{0,0}}^{1} + 1} \right)\left( {m_{{1,0}}^{t} } \right)^{5} }\right\}, \\ D_{{02}} & = 4\left\{ {\left( {m_{{0,0}}^{1} + 1}\right)[5103\left( {m_{{1,0}}^{t} } \right)^{6} - 3645m_{{0,0}}^{1}\left( {m_{{1,0}}^{t} } \right)^{4} m_{{2,0}}^{t} - 405\left({m_{{0,0}}^{1} } \right)^{3} m_{{4,0}}^{t} \left( {m_{{1,0}}^{t} }\right)^{2} } \right. \\ & \quad \left. { + 225\left( {m_{{0,0}}^{1}} \right)^{4} m_{{4,0}}^{t} m_{{2,0}}^{t} - 28\left({m_{{0,0}}^{1} } \right)^{5} \left( {m_{{0,0}}^{1} + 1}\right)m_{{6,0}}^{t} + 100\left( {m_{{0,0}}^{1} } \right)^{5} \left({m_{{3,0}}^{t} } \right)^{2} } \right] \\ & \quad + 972\left({m_{{0,0}}^{1} } \right)^{3} \left( {m_{{1,0}}^{t} } \right)^{4}m_{{2,0}}^{t} + 1620\left( {m_{{0,0}}^{1} } \right)^{3} \left({m_{{1,0}}^{t} } \right)^{3} m_{{3,0}}^{t} \\ & \quad + 162\left({m_{{0,0}}^{1} } \right)^{2} \left[ { - 10\left( {m_{{0,0}}^{1} }\right)^{2} - 9m_{{0,0}}^{1} + 9} \right]\left( {m_{{1,0}}^{t}m_{{2,0}}^{t} } \right)^{2} \\ & \quad \left. { + 108\left({m_{{0,0}}^{1} } \right)^{3} \left[ {2\left( {m_{{0,0}}^{1} }\right)^{2} - m_{{0,0}}^{1} - 3} \right]\left( {m_{{2,0}}^{t} }\right)^{3} + 180\left( {m_{{0,0}}^{1} } \right)^{4} \left( {1 +2m_{{0,0}}^{1} } \right)m_{{1,0}}^{t} m_{{2,0}}^{t} m_{{3,0}}^{t} }\right\}, \\ \end{aligned}$$
$$\begin{aligned} D_{01} & = 24m_{1,0}^{t} \left\{ {\left( {m_{0,0}^{1} + 1} \right)\left[ {28\left( {m_{0,0}^{1} } \right)^{5} m_{6,0}^{t} - 729\left( {m_{1,0}^{t} } \right)^{6} + 135\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} \left( {m_{1,0}^{t} } \right)^{2} } \right]} \right. \\ & \quad - 162\left( {4m_{0,0}^{1} + 3} \right)\left( {m_{0,0}^{1} } \right)^{2} \left( {m_{1,0}^{t} m_{2,0}^{t} } \right)^{2} - 45\left( {m_{0,0}^{1} } \right)^{4} \left( { 4m_{0,0}^{1} + 5} \right)m_{4,0}^{t} m_{2,0}^{t} \\ & \quad + 108\left( {2m_{0,0}^{1} + 3} \right)\left( {m_{0,0}^{1} } \right)^{3} \left( {m_{2,0}^{t} } \right)^{3} + 729m_{0,0}^{1} \left( {2m_{0,0}^{1} + 1} \right)\left( {m_{1,0}^{t} } \right)^{4} m_{2,0}^{t} \\ & \quad \left. { - 100\left( {m_{0,0}^{1} } \right)^{5} \left( {m_{3,0}^{t} } \right)^{2} + 360\left( {m_{0,0}^{1} } \right)^{4} m_{1,0}^{t} m_{2,0}^{t} m_{3,0}^{t} - 540\left( {m_{0,0}^{1} } \right)^{3} \left( {m_{1,0}^{t} } \right)^{3} m_{3,0}^{t} } \right\}, \\ D_{00} & = 9\left\{ {m_{0,0}^{1} [ - 27\left( {m_{1,0}^{t} } \right)^{4} + 5\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} - 4\left( {m_{0,0}^{1} } \right)^{2} \left[ {3\left( {m_{2,0}^{t} } \right)^{2} + 5m_{1,0}^{t} m_{3,0}^{t} } \right]} \right. \\ & \quad + 54m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} ]^{2} + [5\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} - 12\left( {m_{0,0}^{1} } \right)^{2} \left( {m_{2,0}^{t} } \right)^{2} + 54m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} \\ & \quad - 27\left( {m_{1,0}^{t} } \right)^{4} ]\left[ {5\left( {m_{0,0}^{1} } \right)^{3} m_{4,0}^{t} - 12\left( {m_{0,0}^{1} } \right)^{2} \left( {m_{2,0}^{t} } \right)^{2} + 126m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} - 27\left( {m_{1,0}^{t} } \right)^{4} } \right] \\ & \quad + 16m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} \{ 27m_{2,0}^{t} \left[ {2\left( {m_{00}^{1} } \right)^{2} \left( {m_{2,0}^{t} } \right)^{2} - 15m_{0,0}^{1} \left( {m_{1,0}^{t} } \right)^{2} m_{2,0}^{t} + 9\left( {m_{1,0}^{t} } \right)^{4} } \right] \left. { - 7\left( {m_{0,0}^{1} } \right)^{4} m_{6,0}^{t} } \right\} \\ \end{aligned}$$
(C.1)

The original article has been corrected.