August  2018, 11(4): 953-1009. doi: 10.3934/krm.2018038

On multi-dimensional hypocoercive BGK models

1. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, A-1090, Austria

2. 

Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstrasse 8-10, Vienna, A-1040, Austria

3. 

Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

* Corresponding author: A. Arnold

Received  November 2017 Revised  February 2018 Published  April 2018

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the $L^1$ distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.

Citation: Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038
References:
[1]

F. Achleitner, A. Arnold and E. A. Carlen, On linear hypocoercive BGK models, in From Particle Systems to Partial Differential Equations III, Springer Proceedings in Mathematics & Statistics, 162 (2016), 1–37. doi: 10.1007/978-3-319-32144-8_1.  Google Scholar

[2]

F. AchleitnerA. Arnold and D. Stürzer, Large-time behavior in non-symmetric Fokker-Planck equations, Riv. Mat. Univ. Parma, 6 (2015), 1-68.   Google Scholar

[3]

D. Aldous and P. Diaconis, Shuffling cards and stopping times, Amer. Math. Month., 93 (1986), 333-348.  doi: 10.1080/00029890.1986.11971821.  Google Scholar

[4]

A. ArnoldA. Einav and T. Wöhrer, On the rates of decay to equilibrium in degenerate and defective Fokker-Planck equations, J. Differential Equations, 264 (2018), 6843-6872.  doi: 10.1016/j.jde.2018.01.052.  Google Scholar

[5]

A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric FokkerPlanck equations with linear drift, preprint, arXiv: 1409.5425. Google Scholar

[6]

V. I. Arnold, Ordinary Differential Equations, MIT Press, Cambridge, Mass. -London, 1978.  Google Scholar

[7]

P. Bergmann and J. L. Lebowitz, New approach to nonequilibrium processes, Phys. Rev., 99 (1955), 578-587.  doi: 10.1103/PhysRev.99.578.  Google Scholar

[8]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.  Google Scholar

[9]

R. Bosi and M. J. Cáceres, The BGK model with external confining potential: Existence, long-time behaviour and time-periodic Maxwellian equilibria, J. Stat. Phys., 136 (2009), 297-330.  doi: 10.1007/s10955-009-9782-5.  Google Scholar

[10]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, preprint, arXiv: 1708.06180. Google Scholar

[11]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[12]

P. Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Nat. Acad. Sci. USA, 93 (1996), 1659-1664.  doi: 10.1073/pnas.93.4.1659.  Google Scholar

[13]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[14]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516.  doi: 10.1016/j.crma.2009.02.025.  Google Scholar

[15]

H. Grad, Note on N-dimensional Hermite polynomials, Comm. Pure Appl. Math., 2 (1949), 325-330.  doi: 10.1002/cpa.3160020402.  Google Scholar

[16]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.   Google Scholar

[17]

J. P. Hespanha, Linear Systems Theory, Princeton University Press, Princeton, 2009.  Google Scholar

[18]

M. HitrikK. Pravda-Starov and J. Viola, Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators, Bull. Sci. Math., 141 (2017), 615-675.   Google Scholar

[19]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013.  Google Scholar

[20]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Berlin-New York, 1976.  Google Scholar

[22]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194.  doi: 10.1017/S0308210500018308.  Google Scholar

[23]

B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.  Google Scholar

[24]

B. Perthame and M. Pulvirenti, Weighted $L^∞$ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295.  doi: 10.1007/BF00383223.  Google Scholar

[25]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden Day, San Francisco, 1964.  Google Scholar

[26]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, 2nd ed., Academic Press, New York, 1980.  Google Scholar

[27]

E. Ringeisen, Contributions a L'etude Mathématique des Équations Cinétiques, PhD thesis, Université Paris Diderot - Paris 7, 1991. Google Scholar

[28]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.  Google Scholar

[29]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.  Google Scholar

[30]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141 pp.  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

show all references

References:
[1]

F. Achleitner, A. Arnold and E. A. Carlen, On linear hypocoercive BGK models, in From Particle Systems to Partial Differential Equations III, Springer Proceedings in Mathematics & Statistics, 162 (2016), 1–37. doi: 10.1007/978-3-319-32144-8_1.  Google Scholar

[2]

F. AchleitnerA. Arnold and D. Stürzer, Large-time behavior in non-symmetric Fokker-Planck equations, Riv. Mat. Univ. Parma, 6 (2015), 1-68.   Google Scholar

[3]

D. Aldous and P. Diaconis, Shuffling cards and stopping times, Amer. Math. Month., 93 (1986), 333-348.  doi: 10.1080/00029890.1986.11971821.  Google Scholar

[4]

A. ArnoldA. Einav and T. Wöhrer, On the rates of decay to equilibrium in degenerate and defective Fokker-Planck equations, J. Differential Equations, 264 (2018), 6843-6872.  doi: 10.1016/j.jde.2018.01.052.  Google Scholar

[5]

A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric FokkerPlanck equations with linear drift, preprint, arXiv: 1409.5425. Google Scholar

[6]

V. I. Arnold, Ordinary Differential Equations, MIT Press, Cambridge, Mass. -London, 1978.  Google Scholar

[7]

P. Bergmann and J. L. Lebowitz, New approach to nonequilibrium processes, Phys. Rev., 99 (1955), 578-587.  doi: 10.1103/PhysRev.99.578.  Google Scholar

[8]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511.  Google Scholar

[9]

R. Bosi and M. J. Cáceres, The BGK model with external confining potential: Existence, long-time behaviour and time-periodic Maxwellian equilibria, J. Stat. Phys., 136 (2009), 297-330.  doi: 10.1007/s10955-009-9782-5.  Google Scholar

[10]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, preprint, arXiv: 1708.06180. Google Scholar

[11]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[12]

P. Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Nat. Acad. Sci. USA, 93 (1996), 1659-1664.  doi: 10.1073/pnas.93.4.1659.  Google Scholar

[13]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[14]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516.  doi: 10.1016/j.crma.2009.02.025.  Google Scholar

[15]

H. Grad, Note on N-dimensional Hermite polynomials, Comm. Pure Appl. Math., 2 (1949), 325-330.  doi: 10.1002/cpa.3160020402.  Google Scholar

[16]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.   Google Scholar

[17]

J. P. Hespanha, Linear Systems Theory, Princeton University Press, Princeton, 2009.  Google Scholar

[18]

M. HitrikK. Pravda-Starov and J. Viola, Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators, Bull. Sci. Math., 141 (2017), 615-675.   Google Scholar

[19]

R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013.  Google Scholar

[20]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Berlin-New York, 1976.  Google Scholar

[22]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194.  doi: 10.1017/S0308210500018308.  Google Scholar

[23]

B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205.  doi: 10.1016/0022-0396(89)90173-3.  Google Scholar

[24]

B. Perthame and M. Pulvirenti, Weighted $L^∞$ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295.  doi: 10.1007/BF00383223.  Google Scholar

[25]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden Day, San Francisco, 1964.  Google Scholar

[26]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, 2nd ed., Academic Press, New York, 1980.  Google Scholar

[27]

E. Ringeisen, Contributions a L'etude Mathématique des Équations Cinétiques, PhD thesis, Université Paris Diderot - Paris 7, 1991. Google Scholar

[28]

Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.  doi: 10.14492/hokmj/1381757663.  Google Scholar

[29]

T. UmedaS. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.  doi: 10.1007/BF03167068.  Google Scholar

[30]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141 pp.  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

Figure 1.  These two functions illustrate the time dependent decay estimate from (15). The values of $C_d,\,\lambda^d$ correspond to the 1D case with $L = 2\pi$, and we chose $\mathcal{E}^d(\tilde f^I) = 15$. We also show the two time scales of the BGK equation: $t_{{\rm init}}$ marks the intersection point of the two blue curves and it corresponds to the generic transport time. $t_2: = t_{{\rm init}}+\frac2\lambda$ marks the intersection point of the exponential curve with the value $2/e$, and $t_2-t_{{\rm init}}$ corresponds to the relaxation time scale. For larger values of $L$, $t_{{\rm init}}$ will be much larger
Figure 2.  For each cell length $L$ the constant $2\mu_*(L)$ obtained from Lemma 3.1 and Remark 9(a) yields a bound for the entropy decay rate in Theorem 1.1
Table 1.  We give a classification of Hermitian matrices ${\bf C}_1$, such that the associated matrix ${\bf C} = i{\bf C}_1 +{\rm diag}(0,0,c_2,\ldots,c_n)$ is hypocoercive. The restrictions on the coefficients of ${\bf C}_1$ are depicted as $0$ if zero, $\bullet$ if non-zero, and $\ast$ if there is no restriction. Furthermore, we give the corresponding two-parameter ansatz for the transformation matrix ${\bf P} = {\bf I}+{\bf A}$. The guideline to construct an admissible Hermitian perturbation matrix ${\bf A}$, is to put the parameters $\lambda_j$ at the positions of the (non-zero) coupling elements of ${\bf C}_1$. In case (2B2) this will be apparent after a suitable transformation, see the proof of Theorem 2.9
$ \underline{{\rm Case~ 2A:}}$
${\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\ast \\ \ast&\ast \end{matrix} &\begin{matrix}\ast&\bullet&\ast&\cdots&\ast \\ \bullet&\ast&\ast&\cdots&\ast \end{matrix}\\ \hline \begin{matrix}\ast&\bullet \\ \bullet&\ast \\ \ast&\ast \\ \vdots&\vdots \\ \ast&\ast \end{matrix} &\bf{*} \end{array}\right)\!,~~{\bf P}={\bf I} + \left(\begin{array}{c|c} \begin{matrix}0&0&0&\lambda_1 \\ 0&0&\lambda_2&0 \\ 0&\overline{\lambda_2}&0&0 \\ \overline{\lambda_1}&0&0&0 \end{matrix} &{\bf 0}\\ \hline {\bf 0}& {\bf 0} \end{array}\right)\!,~~ (40) $
where the upper right submatrix ${\bf C}_1^{ur}\in\mathbb{C}^{2\times(n-2)}$ has rank 2. Here, we assume w.l.o.g. that $|c_{1,4}c_{2,3}|\geq |c_{1,3}c_{2,4}|$ and $c_{1,4}\,c_{2,3} \ne c_{1,3}\,c_{2,4}$, such that $c_{2,3}\ne 0$ and $c_{1,4}\ne 0$.
$ \underline{{\rm Case~ 2B:}}$
${\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\ast \\ \ast&\ast \end{matrix}&\begin{matrix}\ast&\ast&\cdots&\ast \\ \bullet&\ast&\cdots&\ast \end{matrix}\\ \hline \begin{matrix}\ast&\bullet \\ \ast&\ast \\ \vdots&\vdots \\ \ast&\ast \end{matrix} &\bf * \end{array}\right)\!, ~~~~~ ~~ {\bf P}={\bf I} +{\bf U} \left(\begin{array}{c|c}\begin{matrix}0&\lambda_1&0 \\ \overline{\lambda_1}&0&\lambda_2 \\ 0&\overline{\lambda_2}&0 \end{matrix}&{\bf 0}\\ \hline {\bf 0}& {\bf 0} \end{array}\right) {\bf U}^* \!,~ (41) $
where the upper right submatrix ${\bf C}_1^{ur}\in\mathbb{C}^{2\times(n-2)}$ has rank 1. Again, we assume w.l.o.g. that $c_{2,3}\ne 0$. The right choice for the unitary matrix ${\bf U}$ depends on the structure of ${\bf C}_1$:
$ {\rm (2B1)}~~~~ {\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\bullet \\ \bullet&\ast \end{matrix}& \begin{matrix}0&0&\cdots&0 \\ \bullet&\ast&\cdots&\ast \end{matrix} \\ \hline \begin{matrix}0&\bullet \\ 0&\ast \\ \vdots&\vdots \\ 0&\ast \end{matrix} &\bf * \end{array}\right)\!, ~~~~~ {\bf U} ={\bf I}\,, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (42)$
${\rm (2B2)} ~~~~~ {\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\ast \\ \ast&\ast \end{matrix}& \begin{matrix}\bullet&\ast&\cdots&\ast \\ \bullet&\ast&\cdots&\ast \end{matrix} \\ \hline \begin{matrix}\bullet&\bullet \\ \ast&\ast \\ \vdots&\vdots \\ \ast&\ast \end{matrix} &\bf * \end{array}\right)\!,~~~~~{\bf U} = \left(\begin{array}{c|c} {\bf U}^{ul}&{\bf 0}\\ \hline {\bf 0}& {\bf I} \end{array}\right) \!,~~~~~~~~~ (43) $
with upper left submatrix ${\bf U}^{ul}=\tfrac1{\sqrt{|c_{1,3}|^2 +|c_{2,3}|^2}}\begin{pmatrix}\overline{c_{2,3}}&c_{1,3} \\ -\overline{c_{1,3}}&c_{2,3} \end{pmatrix}$.
$ \underline{{\rm Case~ 2A:}}$
${\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\ast \\ \ast&\ast \end{matrix} &\begin{matrix}\ast&\bullet&\ast&\cdots&\ast \\ \bullet&\ast&\ast&\cdots&\ast \end{matrix}\\ \hline \begin{matrix}\ast&\bullet \\ \bullet&\ast \\ \ast&\ast \\ \vdots&\vdots \\ \ast&\ast \end{matrix} &\bf{*} \end{array}\right)\!,~~{\bf P}={\bf I} + \left(\begin{array}{c|c} \begin{matrix}0&0&0&\lambda_1 \\ 0&0&\lambda_2&0 \\ 0&\overline{\lambda_2}&0&0 \\ \overline{\lambda_1}&0&0&0 \end{matrix} &{\bf 0}\\ \hline {\bf 0}& {\bf 0} \end{array}\right)\!,~~ (40) $
where the upper right submatrix ${\bf C}_1^{ur}\in\mathbb{C}^{2\times(n-2)}$ has rank 2. Here, we assume w.l.o.g. that $|c_{1,4}c_{2,3}|\geq |c_{1,3}c_{2,4}|$ and $c_{1,4}\,c_{2,3} \ne c_{1,3}\,c_{2,4}$, such that $c_{2,3}\ne 0$ and $c_{1,4}\ne 0$.
$ \underline{{\rm Case~ 2B:}}$
${\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\ast \\ \ast&\ast \end{matrix}&\begin{matrix}\ast&\ast&\cdots&\ast \\ \bullet&\ast&\cdots&\ast \end{matrix}\\ \hline \begin{matrix}\ast&\bullet \\ \ast&\ast \\ \vdots&\vdots \\ \ast&\ast \end{matrix} &\bf * \end{array}\right)\!, ~~~~~ ~~ {\bf P}={\bf I} +{\bf U} \left(\begin{array}{c|c}\begin{matrix}0&\lambda_1&0 \\ \overline{\lambda_1}&0&\lambda_2 \\ 0&\overline{\lambda_2}&0 \end{matrix}&{\bf 0}\\ \hline {\bf 0}& {\bf 0} \end{array}\right) {\bf U}^* \!,~ (41) $
where the upper right submatrix ${\bf C}_1^{ur}\in\mathbb{C}^{2\times(n-2)}$ has rank 1. Again, we assume w.l.o.g. that $c_{2,3}\ne 0$. The right choice for the unitary matrix ${\bf U}$ depends on the structure of ${\bf C}_1$:
$ {\rm (2B1)}~~~~ {\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\bullet \\ \bullet&\ast \end{matrix}& \begin{matrix}0&0&\cdots&0 \\ \bullet&\ast&\cdots&\ast \end{matrix} \\ \hline \begin{matrix}0&\bullet \\ 0&\ast \\ \vdots&\vdots \\ 0&\ast \end{matrix} &\bf * \end{array}\right)\!, ~~~~~ {\bf U} ={\bf I}\,, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (42)$
${\rm (2B2)} ~~~~~ {\bf C}_1 = \left(\begin{array}{c|c} \begin{matrix}\ast&\ast \\ \ast&\ast \end{matrix}& \begin{matrix}\bullet&\ast&\cdots&\ast \\ \bullet&\ast&\cdots&\ast \end{matrix} \\ \hline \begin{matrix}\bullet&\bullet \\ \ast&\ast \\ \vdots&\vdots \\ \ast&\ast \end{matrix} &\bf * \end{array}\right)\!,~~~~~{\bf U} = \left(\begin{array}{c|c} {\bf U}^{ul}&{\bf 0}\\ \hline {\bf 0}& {\bf I} \end{array}\right) \!,~~~~~~~~~ (43) $
with upper left submatrix ${\bf U}^{ul}=\tfrac1{\sqrt{|c_{1,3}|^2 +|c_{2,3}|^2}}\begin{pmatrix}\overline{c_{2,3}}&c_{1,3} \\ -\overline{c_{1,3}}&c_{2,3} \end{pmatrix}$.
Table 2.  Let $\delta_j(\kappa,\alpha,\beta,\gamma,\omega)$ denote the determinant of the upper left $j\times j$ submatrix of ${\bf D}_{\kappa,\alpha,\beta,\gamma,\omega}$ for integers $j = 1,2,\ldots,11$. For our choice $\beta = 2\alpha$, $\gamma = \alpha$ and $\omega = \sqrt{6}\alpha$, the minors $\delta_j(\kappa,\alpha) = \delta_j(\kappa,\alpha,2\alpha,\alpha,\sqrt{6}\alpha)$ are given in this table
$\delta_1(\kappa,\alpha)$ = $2\ell\alpha$
$\delta_2(\kappa,\alpha)$ = $4\ell^2\alpha^2$
$\delta_3(\kappa,\alpha)$ = $8\ell^3\alpha^3$
$\delta_4(\kappa,\alpha)$ = $44\ell^4 \alpha^4$
$\delta_5(\kappa,\alpha)$ = $22\ell^3 \alpha^4 (4\ell -4\ell^2 \alpha -{\alpha}/{\kappa^2})$
$\delta_6(\kappa,\alpha)$ = $\delta_5(\kappa,\alpha) p_6(\kappa,\alpha)/\ell$
    with $p_6(\kappa,\alpha) :=-\tfrac{54}{11} \ell^2 \alpha +2\ell -2\alpha /{\kappa^2}$.
$\delta_7(\kappa,\alpha)$ = $\frac{2}{11\ell^2} \delta_5(\kappa,\alpha) p_7(\kappa,\alpha)$
    with $p_{7}(\kappa,\alpha) = \big(p_{7,0}(\alpha) +p_{7,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{7,2}(\alpha)$,
    $p_{7,0}(\alpha) = 93\ell^2 \alpha^2 -34\ell\alpha$, $p_{7,1}(\alpha) = 12\alpha^2$,
    $p_{7,2}(\alpha) = 162\ell^4 \alpha^2 -120\ell^3 \alpha +22\ell^2$.
$\delta_8(\kappa,\alpha)$ = $44 \ell^3 \alpha^4 \frac{\delta_7(\kappa,\alpha)}{\delta_5(\kappa,\alpha)} p_8(\kappa,\alpha)$
    with $p_8(\kappa,\alpha) = 2\ell^3 \alpha^2 -6\ell^2\alpha +4\ell -{\alpha}/{\kappa^2}$.
$\delta_9(\kappa,\alpha)$ = $8 \ell \alpha^4 p_8(\kappa,\alpha) p_9(\kappa,\alpha)$
    with $p_{9}(\kappa,\alpha) = \big(p_{9,0}(\alpha) +p_{9,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{9,2}(\alpha)$,
    $p_{9,0}(\alpha) = -12\ell^3 \alpha^3 +198\ell^2 \alpha^2 -68\ell \alpha$, $p_{9,1}(\alpha) = 24\alpha^2$,
    $p_{9,2}(\alpha) = -81\ell^5 \alpha^3 +411\ell^4 \alpha^2 -262\ell^3 \alpha +44\ell^2$.
$\delta_{10}(\kappa,\alpha)$ = $2 \delta_9(\kappa,\alpha)$,
$\delta_{11}(\kappa,\alpha)$ = $64 \ell \alpha^4 p_8(\kappa,\alpha) p_{11}(\kappa,\alpha)$
    with $p_{11}(\kappa,\alpha) = \big(p_{11,0}(\alpha) +p_{11,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{11,2}(\alpha)$,
    $p_{11,0}(\alpha) = -72\ell^4 \alpha^4 -300\ell^3 \alpha^3 +294\ell^2 \alpha^2 -68\ell\alpha$, $p_{11,1}(\alpha) = 24\alpha^2$,
    $p_{11,2}(\alpha) = 162\ell^6 \alpha^4 -909\ell^5 \alpha^3 +963\ell^4 \alpha^2 -358\ell^3 \alpha +44\ell^2$.
$\delta_1(\kappa,\alpha)$ = $2\ell\alpha$
$\delta_2(\kappa,\alpha)$ = $4\ell^2\alpha^2$
$\delta_3(\kappa,\alpha)$ = $8\ell^3\alpha^3$
$\delta_4(\kappa,\alpha)$ = $44\ell^4 \alpha^4$
$\delta_5(\kappa,\alpha)$ = $22\ell^3 \alpha^4 (4\ell -4\ell^2 \alpha -{\alpha}/{\kappa^2})$
$\delta_6(\kappa,\alpha)$ = $\delta_5(\kappa,\alpha) p_6(\kappa,\alpha)/\ell$
    with $p_6(\kappa,\alpha) :=-\tfrac{54}{11} \ell^2 \alpha +2\ell -2\alpha /{\kappa^2}$.
$\delta_7(\kappa,\alpha)$ = $\frac{2}{11\ell^2} \delta_5(\kappa,\alpha) p_7(\kappa,\alpha)$
    with $p_{7}(\kappa,\alpha) = \big(p_{7,0}(\alpha) +p_{7,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{7,2}(\alpha)$,
    $p_{7,0}(\alpha) = 93\ell^2 \alpha^2 -34\ell\alpha$, $p_{7,1}(\alpha) = 12\alpha^2$,
    $p_{7,2}(\alpha) = 162\ell^4 \alpha^2 -120\ell^3 \alpha +22\ell^2$.
$\delta_8(\kappa,\alpha)$ = $44 \ell^3 \alpha^4 \frac{\delta_7(\kappa,\alpha)}{\delta_5(\kappa,\alpha)} p_8(\kappa,\alpha)$
    with $p_8(\kappa,\alpha) = 2\ell^3 \alpha^2 -6\ell^2\alpha +4\ell -{\alpha}/{\kappa^2}$.
$\delta_9(\kappa,\alpha)$ = $8 \ell \alpha^4 p_8(\kappa,\alpha) p_9(\kappa,\alpha)$
    with $p_{9}(\kappa,\alpha) = \big(p_{9,0}(\alpha) +p_{9,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{9,2}(\alpha)$,
    $p_{9,0}(\alpha) = -12\ell^3 \alpha^3 +198\ell^2 \alpha^2 -68\ell \alpha$, $p_{9,1}(\alpha) = 24\alpha^2$,
    $p_{9,2}(\alpha) = -81\ell^5 \alpha^3 +411\ell^4 \alpha^2 -262\ell^3 \alpha +44\ell^2$.
$\delta_{10}(\kappa,\alpha)$ = $2 \delta_9(\kappa,\alpha)$,
$\delta_{11}(\kappa,\alpha)$ = $64 \ell \alpha^4 p_8(\kappa,\alpha) p_{11}(\kappa,\alpha)$
    with $p_{11}(\kappa,\alpha) = \big(p_{11,0}(\alpha) +p_{11,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{11,2}(\alpha)$,
    $p_{11,0}(\alpha) = -72\ell^4 \alpha^4 -300\ell^3 \alpha^3 +294\ell^2 \alpha^2 -68\ell\alpha$, $p_{11,1}(\alpha) = 24\alpha^2$,
    $p_{11,2}(\alpha) = 162\ell^6 \alpha^4 -909\ell^5 \alpha^3 +963\ell^4 \alpha^2 -358\ell^3 \alpha +44\ell^2$.
Table 3.  Matrix ${\bf D}_{\kappa,\alpha,\beta,\gamma,\omega,\eta}$ with $\ell: = 2\pi/L>0$ and $A : = \tfrac{\ell}{\sqrt3} (\sqrt2 \alpha -\sqrt3 \beta)$, $B : = \tfrac2{\sqrt3} \ell (\sqrt2 \beta -\sqrt3 \alpha)$, $C : = \tfrac{\ell}{\sqrt3} (\sqrt2 \eta -\sqrt3 \beta)$, $D : = \tfrac{\ell}{\sqrt3} (\sqrt3 \eta -\sqrt2 \beta)$, $E : = 2-2\ell\omega$, and $F : = 2-2\tfrac{\sqrt2}{\sqrt3} \ell\beta$
$ \begin{pmatrix} 2\ell\alpha&0&0&0&\tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&A&0&\tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&0&0&0&0&0&0&0&0&0 \\ 0&B&0&0&0&0&0&- \frac{i}{\kappa} \beta&0&0&-C&0&0&\tfrac{3+\sqrt3}{6} \ell\beta&0&\tfrac{3-\sqrt3}{6} \ell\beta&0&0&0&0&0 \\ 0&0&2\ell \gamma&0&0&-\frac{i}{\kappa} \gamma&0&0&0&0&0&\sqrt2 \ell \gamma&0&0&0&0&0&0&0&0&0 \\ 0&0&0&2\ell\omega&0&0&-\frac{i}{\kappa} \omega&0&0&0&0&0&\sqrt2 \ell \omega&0&0&0&0&0&0&0&0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&0&2\ell\eta&0&0&D&0&\ell\eta&-\frac{i}{\kappa} \eta&0&0&0&0&0&0&0&0&0&2\ell\eta \\ 0&0&\frac{i}{\kappa} \gamma&0&0&2-2\ell\gamma&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&\frac{i}{\kappa} \omega&0&0&E&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ A&\frac{i}{\kappa} \beta&0&0&D&0&0&F&0&-\tfrac{\sqrt2}{\sqrt3} \ell\beta&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&0&0&0&0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&0&\ell\eta&0&0&-\tfrac{\sqrt2}{\sqrt3} \ell\beta&0&2&0&0&0&0&0&0&0&0&0&0&0 \\ 0&-C&0&0&\frac{i}{\kappa} \eta&0&0&0&0&0&2-2\ell\eta&0&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&0&0&0&0 \\ 0&0&\sqrt2 \ell\gamma&0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&0 \\ 0&0&0&\sqrt2 \ell\omega&0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&0 \\ 0&\tfrac{3+\sqrt3}{6} \ell\beta&0&0&0&0&0&0&0&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&0&2&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&0&0&0 \\ 0&\tfrac{3-\sqrt3}{6} \ell\beta&0&0&0&0&0&0&0&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&0&0&0&2&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0 \\ 0&0&0&0&2\ell\eta&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2 \\ \end{pmatrix}$
$ \begin{pmatrix} 2\ell\alpha&0&0&0&\tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&A&0&\tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&0&0&0&0&0&0&0&0&0 \\ 0&B&0&0&0&0&0&- \frac{i}{\kappa} \beta&0&0&-C&0&0&\tfrac{3+\sqrt3}{6} \ell\beta&0&\tfrac{3-\sqrt3}{6} \ell\beta&0&0&0&0&0 \\ 0&0&2\ell \gamma&0&0&-\frac{i}{\kappa} \gamma&0&0&0&0&0&\sqrt2 \ell \gamma&0&0&0&0&0&0&0&0&0 \\ 0&0&0&2\ell\omega&0&0&-\frac{i}{\kappa} \omega&0&0&0&0&0&\sqrt2 \ell \omega&0&0&0&0&0&0&0&0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&0&2\ell\eta&0&0&D&0&\ell\eta&-\frac{i}{\kappa} \eta&0&0&0&0&0&0&0&0&0&2\ell\eta \\ 0&0&\frac{i}{\kappa} \gamma&0&0&2-2\ell\gamma&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&\frac{i}{\kappa} \omega&0&0&E&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ A&\frac{i}{\kappa} \beta&0&0&D&0&0&F&0&-\tfrac{\sqrt2}{\sqrt3} \ell\beta&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&0&0&0&0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha&0&0&0&\ell\eta&0&0&-\tfrac{\sqrt2}{\sqrt3} \ell\beta&0&2&0&0&0&0&0&0&0&0&0&0&0 \\ 0&-C&0&0&\frac{i}{\kappa} \eta&0&0&0&0&0&2-2\ell\eta&0&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&0&0&0&0 \\ 0&0&\sqrt2 \ell\gamma&0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&0&0 \\ 0&0&0&\sqrt2 \ell\omega&0&0&0&0&0&0&0&0&2&0&0&0&0&0&0&0&0 \\ 0&\tfrac{3+\sqrt3}{6} \ell\beta&0&0&0&0&0&0&0&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&0&2&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&0&0&0 \\ 0&\tfrac{3-\sqrt3}{6} \ell\beta&0&0&0&0&0&0&0&0&-\tfrac{1}{\sqrt{3}} \ell\eta&0&0&0&0&2&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2&0 \\ 0&0&0&0&2\ell\eta&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&2 \\ \end{pmatrix}$
Table 4.  Matrix ${\bf D}_{\kappa,\alpha,\sqrt3 \alpha,\alpha,\alpha,\alpha}$ with $\ell: = 2\pi/L>0$ and $B: = 2 (\sqrt2 -1) \ell\alpha$, $E: = 2-2\ell\alpha$, $F: = 2-2\sqrt2 \ell \alpha$
$ \begin{pmatrix} 2\ell\alpha & 0 & 0 & 0 & \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & \tfrac{\sqrt2 -3}{\sqrt3} \ell \alpha & 0 & \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & B & 0 & 0 & 0 & 0 & 0 & - \frac{i}{\kappa} \sqrt{3}\alpha & 0 & 0 & \tfrac{3-\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & \tfrac{\sqrt3+1}{2} \ell\alpha & 0 & \tfrac{\sqrt3-1}{2} \ell\alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2\ell\alpha & 0 & 0 & -\frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & \sqrt2 \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2\ell\alpha & 0 & 0 & -\frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & \sqrt2 \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & 0 & 2\ell\alpha & 0 & 0 & -\frac{B}{2} & 0 & \ell\alpha & -\frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\ell\alpha \\ 0 & 0 & \frac{i}{\kappa} \alpha & 0 & 0 & E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{i}{\kappa} \alpha & 0 & 0 & E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{\sqrt2 -3}{\sqrt3} \ell\alpha & \frac{i}{\kappa} \sqrt{3}\alpha & 0 & 0 & -\frac{B}{2} & 0 & 0 & F & 0 & -\sqrt2 \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & 0 & \ell\alpha & 0 & 0 & -\sqrt2 \ell \alpha & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \tfrac{3-\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & \frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & E & 0 & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \sqrt2 \ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt2 \ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \tfrac{\sqrt3+1}{2} \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \tfrac{\sqrt3-1}{2} \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2\ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ \end{pmatrix} ~~~~(114)$
$ \begin{pmatrix} 2\ell\alpha & 0 & 0 & 0 & \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & \tfrac{\sqrt2 -3}{\sqrt3} \ell \alpha & 0 & \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & B & 0 & 0 & 0 & 0 & 0 & - \frac{i}{\kappa} \sqrt{3}\alpha & 0 & 0 & \tfrac{3-\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & \tfrac{\sqrt3+1}{2} \ell\alpha & 0 & \tfrac{\sqrt3-1}{2} \ell\alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2\ell\alpha & 0 & 0 & -\frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & \sqrt2 \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2\ell\alpha & 0 & 0 & -\frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & \sqrt2 \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & 0 & 2\ell\alpha & 0 & 0 & -\frac{B}{2} & 0 & \ell\alpha & -\frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\ell\alpha \\ 0 & 0 & \frac{i}{\kappa} \alpha & 0 & 0 & E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{i}{\kappa} \alpha & 0 & 0 & E & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{\sqrt2 -3}{\sqrt3} \ell\alpha & \frac{i}{\kappa} \sqrt{3}\alpha & 0 & 0 & -\frac{B}{2} & 0 & 0 & F & 0 & -\sqrt2 \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & 0 & \ell\alpha & 0 & 0 & -\sqrt2 \ell \alpha & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \tfrac{3-\sqrt2}{\sqrt3} \ell\alpha & 0 & 0 & \frac{i}{\kappa} \alpha & 0 & 0 & 0 & 0 & 0 & E & 0 & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \sqrt2 \ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt2 \ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \tfrac{\sqrt3+1}{2} \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \tfrac{\sqrt3-1}{2} \ell \alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\tfrac{1}{\sqrt{3}} \ell\alpha & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2\ell\alpha & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ \end{pmatrix} ~~~~(114)$
Table 5.  Let $\delta_j(\kappa,\alpha,\beta,\gamma,\omega,\eta)$ denote the determinant of the upper left $j\times j$ submatrix of ${\bf D}_{\kappa,\alpha,\beta,\gamma,\omega,\eta}$ for integers $j = 1,2,\ldots,21$. For our choice $\beta = \sqrt3 \alpha$, $\gamma = \alpha$, $\omega = \alpha$ and $\eta = \alpha$, the minors $\delta_j(\kappa,\alpha) = \delta_j(\kappa,\alpha,\sqrt3 \alpha,\alpha,\alpha,\alpha)$ for integers $j = 1,2,\ldots,13$, are given in this table
$\delta_1(\kappa,\alpha)$ = $2\ell\alpha$
$\delta_2(\kappa,\alpha)$ = $4(\sqrt2 -1)\ell^2\alpha^2$
$\delta_3(\kappa,\alpha)$ = $8(\sqrt2 -1)\ell^3\alpha^3$
$\delta_4(\kappa,\alpha)$ = $16(\sqrt2 -1)\ell^4 \alpha^4$
$\delta_5(\kappa,\alpha)$ = $\tfrac{80}{3} (\sqrt2 -1)\ell^5 \alpha^5$
$\delta_6(\kappa,\alpha)$ = $\tfrac{40}{3} (\sqrt2 -1)\ell^4 \alpha^5 p_6(\kappa,\alpha)$
     with $p_6(\kappa,\alpha) = -4\ell^2 \alpha -\frac{\alpha}{\kappa^2} +4\ell$.
$\delta_7(\kappa,\alpha)$ = $\tfrac{20}{3} (\sqrt2 -1)\ell^3 \alpha^5 p_6(\kappa,\alpha)^2$
$\delta_8(\kappa,\alpha)$ = $12 \ell^2\ \alpha^5\ p_6(\kappa,\alpha)^2\ p_8(\kappa,\alpha)$
    with $p_8(\kappa,\alpha) = \tfrac{2-3\sqrt2}{3} \ell^2 \alpha -\tfrac56 \frac{\alpha}{\kappa^2} +\tfrac{10}{9}(\sqrt2 -1)\ell$.
$\delta_9(\kappa,\alpha)$ = $2\ \delta_8(\kappa,\alpha)$
$\delta_{10}(\kappa,\alpha)$ = $\tfrac43 \ell^2\ \alpha^5\ p_6(\kappa,\alpha)^2\ p_{10}(\kappa,\alpha)$
     with $p_{10}(\kappa,\alpha) = 9((\sqrt2 -1)\ell^2 +\frac1{\kappa^2})\ell \alpha^2$
        $-6((8\sqrt2 -6)\ell^2 +\frac5{\kappa^2})\alpha +40(\sqrt2 -1)\ell$.
$\delta_{11}(\kappa,\alpha)$ = $\tfrac29 \ell\ \alpha^5\ p_6(\kappa,\alpha)^2\ p_{11}(\kappa,\alpha)$
     with $p_{11}(\kappa,\alpha) = \big(p_{11,0}(\alpha) +p_{11,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{11,2}(\alpha) \ell^2$,
     $p_{11,0}(\alpha) = (54\sqrt2 -144)\ell^3 \alpha^3 +(672 -72\sqrt2)\ell^2 \alpha^2 -(216 +144\sqrt2)\ell\alpha$,
     $p_{11,1}(\alpha) =18(6 -\ell\alpha) \alpha^2$,
     $p_{11,2}(\alpha) =(9 -54\sqrt2)\ell^3 \alpha^3 +(456\sqrt2 -24) \ell^2 \alpha^2$
       $+(472 -816\sqrt2) \ell \alpha + 480(\sqrt2 -1)$.
$\delta_{12}(\kappa,\alpha)$ = $\delta_{11}(\kappa,\alpha) \frac{p_{12}(\kappa,\alpha)}{p_6(\kappa,\alpha)}$ = $\tfrac29 \ell\ \alpha^5\ p_6(\kappa,\alpha)\ p_{11}(\kappa,\alpha)\ p_{12}(\kappa,\alpha)$
     with $p_{12}(\kappa,\alpha) = 4\ell^3 \alpha^2 -12\ell^2 \alpha +8\ell -\frac{2 \alpha}{\kappa^2}$.
$\delta_{13}(\kappa,\alpha)$ = $\delta_{12}(\kappa,\alpha) \frac{p_{12}(\kappa,\alpha)}{p_6(\kappa,\alpha)} = \delta_{11}(\kappa,\alpha) \Big(\frac{p_{12}(\kappa,\alpha)}{p_6(\kappa,\alpha)}\Big)^2 = \tfrac29 \ell\ \alpha^5\ p_{11}(\kappa,\alpha) p_{12}(\kappa,\alpha)^2$
$\delta_1(\kappa,\alpha)$ = $2\ell\alpha$
$\delta_2(\kappa,\alpha)$ = $4(\sqrt2 -1)\ell^2\alpha^2$
$\delta_3(\kappa,\alpha)$ = $8(\sqrt2 -1)\ell^3\alpha^3$
$\delta_4(\kappa,\alpha)$ = $16(\sqrt2 -1)\ell^4 \alpha^4$
$\delta_5(\kappa,\alpha)$ = $\tfrac{80}{3} (\sqrt2 -1)\ell^5 \alpha^5$
$\delta_6(\kappa,\alpha)$ = $\tfrac{40}{3} (\sqrt2 -1)\ell^4 \alpha^5 p_6(\kappa,\alpha)$
     with $p_6(\kappa,\alpha) = -4\ell^2 \alpha -\frac{\alpha}{\kappa^2} +4\ell$.
$\delta_7(\kappa,\alpha)$ = $\tfrac{20}{3} (\sqrt2 -1)\ell^3 \alpha^5 p_6(\kappa,\alpha)^2$
$\delta_8(\kappa,\alpha)$ = $12 \ell^2\ \alpha^5\ p_6(\kappa,\alpha)^2\ p_8(\kappa,\alpha)$
    with $p_8(\kappa,\alpha) = \tfrac{2-3\sqrt2}{3} \ell^2 \alpha -\tfrac56 \frac{\alpha}{\kappa^2} +\tfrac{10}{9}(\sqrt2 -1)\ell$.
$\delta_9(\kappa,\alpha)$ = $2\ \delta_8(\kappa,\alpha)$
$\delta_{10}(\kappa,\alpha)$ = $\tfrac43 \ell^2\ \alpha^5\ p_6(\kappa,\alpha)^2\ p_{10}(\kappa,\alpha)$
     with $p_{10}(\kappa,\alpha) = 9((\sqrt2 -1)\ell^2 +\frac1{\kappa^2})\ell \alpha^2$
        $-6((8\sqrt2 -6)\ell^2 +\frac5{\kappa^2})\alpha +40(\sqrt2 -1)\ell$.
$\delta_{11}(\kappa,\alpha)$ = $\tfrac29 \ell\ \alpha^5\ p_6(\kappa,\alpha)^2\ p_{11}(\kappa,\alpha)$
     with $p_{11}(\kappa,\alpha) = \big(p_{11,0}(\alpha) +p_{11,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} + p_{11,2}(\alpha) \ell^2$,
     $p_{11,0}(\alpha) = (54\sqrt2 -144)\ell^3 \alpha^3 +(672 -72\sqrt2)\ell^2 \alpha^2 -(216 +144\sqrt2)\ell\alpha$,
     $p_{11,1}(\alpha) =18(6 -\ell\alpha) \alpha^2$,
     $p_{11,2}(\alpha) =(9 -54\sqrt2)\ell^3 \alpha^3 +(456\sqrt2 -24) \ell^2 \alpha^2$
       $+(472 -816\sqrt2) \ell \alpha + 480(\sqrt2 -1)$.
$\delta_{12}(\kappa,\alpha)$ = $\delta_{11}(\kappa,\alpha) \frac{p_{12}(\kappa,\alpha)}{p_6(\kappa,\alpha)}$ = $\tfrac29 \ell\ \alpha^5\ p_6(\kappa,\alpha)\ p_{11}(\kappa,\alpha)\ p_{12}(\kappa,\alpha)$
     with $p_{12}(\kappa,\alpha) = 4\ell^3 \alpha^2 -12\ell^2 \alpha +8\ell -\frac{2 \alpha}{\kappa^2}$.
$\delta_{13}(\kappa,\alpha)$ = $\delta_{12}(\kappa,\alpha) \frac{p_{12}(\kappa,\alpha)}{p_6(\kappa,\alpha)} = \delta_{11}(\kappa,\alpha) \Big(\frac{p_{12}(\kappa,\alpha)}{p_6(\kappa,\alpha)}\Big)^2 = \tfrac29 \ell\ \alpha^5\ p_{11}(\kappa,\alpha) p_{12}(\kappa,\alpha)^2$
Table 6.  Let $\delta_j(\kappa,\alpha,\beta,\gamma,\omega,\eta)$ denote the determinant of the upper left $j\times j$ submatrix of ${\bf D}_{\kappa,\alpha,\beta,\gamma,\omega,\eta}$ for integers $j = 14,\ldots,21$. For our choice $\beta = \sqrt3 \alpha$, $\gamma = \alpha$, $\omega = \alpha$ and $\eta = \alpha$, the minors $\delta_j(\kappa,\alpha) = \delta_j(\kappa,\alpha,\sqrt3 \alpha,\alpha,\alpha,\alpha)$ are given in this table
$\delta_{14}(\kappa,\alpha)$ = $\tfrac1{9\ (1+\sqrt3)^2} \ell\ \alpha^5\ p_{12}(\kappa,\alpha)^2\ p_{14}(\kappa,\alpha)$
     with $p_{14}(\kappa,\alpha) = \big(p_{14,0}(\alpha) +p_{14,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} +\ell^2 p_{14,2}(\alpha)$,
     $p_{14,0}(\alpha) = (-108\sqrt6 -72\sqrt3 -180\sqrt2 -144) \ell^4 \alpha^4$
       $+(360\sqrt6 -1824\sqrt3 +720\sqrt2 -3396) \ell^3 \alpha^3$
       $+(-576\sqrt6 +5952\sqrt3 -1152\sqrt2 +11760) \ell^2 \alpha^2$
       $+(-1152\sqrt6 -1728\sqrt3 -2304\sqrt2 -3456) \ell \alpha$,
     $p_{14,1}(\alpha) = 144\ (\sqrt3 +2)\ (6 -\ell \alpha)\ \alpha^2$,
     $p_{14,2}(\alpha) = (1440 -180\sqrt6 +828\sqrt3 -324\sqrt2) \ell^4 \alpha^4$
       $-(9348 -336\sqrt6 -5400\sqrt3 -624\sqrt2) \ell^3 \alpha^3$
       $+(11056 +3424\sqrt6 +6368\sqrt3 +6864\sqrt2) \ell^2 \alpha^2$
       $+(4192 -6528\sqrt6 +1856\sqrt3 -13056\sqrt2) \ell \alpha$
       $+(3840\sqrt6 -3840\sqrt3 +7680\sqrt2 -7680)$.
$\delta_{15}(\kappa,\alpha)$ = $2\ \delta_{14}(\kappa,\alpha)$
$\delta_{16}(\kappa,\alpha)$ = $\tfrac89 \tfrac{2+\sqrt3}{(1+\sqrt3)^2} \ell\ \alpha^5\ p_{12}(\kappa,\alpha)^2\ p_{16}(\kappa,\alpha)$
     with $p_{16}(\kappa,\alpha) = \big(p_{16,0}(\alpha) +p_{16,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} +\ell^2 p_{16,2}(\alpha)$,
     $p_{16,0}(\alpha) = -36 (\sqrt2 +2) \ell^4 \alpha^4 +(144 \sqrt2 -744) \ell^3 \alpha^3$
       $+(-288 \sqrt2 +2976) \ell^2 \alpha^2 +(-576 \sqrt2 -864) \ell \alpha$,
     $p_{16,1}(\alpha) = 72 (6 -\alpha \ell) \alpha^2$,
     $p_{16,2}(\alpha) = 27 \ell^5 \alpha^5 +(-144 \sqrt2 +216) \ell^4 \alpha^4 +(-24 \sqrt2 -2412) \ell^3 \alpha^3$
       $+(1632 \sqrt2 +3104) \ell^2 \alpha^2 +(-3264 \sqrt2 +928) \ell \alpha +1920 (\sqrt2 -1)$.
$\delta_{17}(\kappa,\alpha)$ = $2\ \delta_{16}(\kappa,\alpha)$
$\delta_{18}(\kappa,\alpha)$ = $2^2\ \delta_{16}(\kappa,\alpha)$
$\delta_{19}(\kappa,\alpha)$ = $2^3\ \delta_{16}(\kappa,\alpha)$
$\delta_{20}(\kappa,\alpha)$ = $2^4\ \delta_{16}(\kappa,\alpha)$
$\delta_{21}(\kappa,\alpha)$ = $\tfrac{256 (\sqrt3 +2) (24\sqrt2 +61)}{23121 (\sqrt3 +1)^2} \ell\ \alpha^5\ p_{12}(\kappa,\alpha)^2\ p_{21}(\kappa,\alpha)$
     with $p_{21}(\kappa,\alpha) = \Big(p_{21,0}(\alpha) +p_{21,1}(\alpha) \frac1{\kappa^2}\Big) \frac1{\kappa^2} +\ell^2 p_{21,2}(\alpha)$,
     $p_{21,0}(\alpha) = (-1152 \sqrt2 +2928) \ell^5 \alpha^5 +(-468 \sqrt2 -2664) \ell^4 \alpha^4$
       $+(75024 \sqrt2 -175272) \ell^3 \alpha^3 +(-130464 \sqrt2 +300768) \ell^2 \alpha^2$
       $+(-14400 \sqrt2 -25056) \ell \alpha$,
     $p_{21,1}(\alpha) = (-1728 \sqrt2 +4392) (6 -\ell \alpha) \alpha^2$,
     $p_{21,2}(\alpha) = 7707 \ell^5 \alpha^5 +(-25248 \sqrt2 +95000) \ell^4 \alpha^4$
       $+(89448 \sqrt2 -353228) \ell^3 \alpha^3 +(158880 \sqrt2 +38048) \ell^2 \alpha^2$
       $+(-417216 \sqrt2 +464416) \ell \alpha +1920 (85\sqrt2 -109)$.
$\delta_{14}(\kappa,\alpha)$ = $\tfrac1{9\ (1+\sqrt3)^2} \ell\ \alpha^5\ p_{12}(\kappa,\alpha)^2\ p_{14}(\kappa,\alpha)$
     with $p_{14}(\kappa,\alpha) = \big(p_{14,0}(\alpha) +p_{14,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} +\ell^2 p_{14,2}(\alpha)$,
     $p_{14,0}(\alpha) = (-108\sqrt6 -72\sqrt3 -180\sqrt2 -144) \ell^4 \alpha^4$
       $+(360\sqrt6 -1824\sqrt3 +720\sqrt2 -3396) \ell^3 \alpha^3$
       $+(-576\sqrt6 +5952\sqrt3 -1152\sqrt2 +11760) \ell^2 \alpha^2$
       $+(-1152\sqrt6 -1728\sqrt3 -2304\sqrt2 -3456) \ell \alpha$,
     $p_{14,1}(\alpha) = 144\ (\sqrt3 +2)\ (6 -\ell \alpha)\ \alpha^2$,
     $p_{14,2}(\alpha) = (1440 -180\sqrt6 +828\sqrt3 -324\sqrt2) \ell^4 \alpha^4$
       $-(9348 -336\sqrt6 -5400\sqrt3 -624\sqrt2) \ell^3 \alpha^3$
       $+(11056 +3424\sqrt6 +6368\sqrt3 +6864\sqrt2) \ell^2 \alpha^2$
       $+(4192 -6528\sqrt6 +1856\sqrt3 -13056\sqrt2) \ell \alpha$
       $+(3840\sqrt6 -3840\sqrt3 +7680\sqrt2 -7680)$.
$\delta_{15}(\kappa,\alpha)$ = $2\ \delta_{14}(\kappa,\alpha)$
$\delta_{16}(\kappa,\alpha)$ = $\tfrac89 \tfrac{2+\sqrt3}{(1+\sqrt3)^2} \ell\ \alpha^5\ p_{12}(\kappa,\alpha)^2\ p_{16}(\kappa,\alpha)$
     with $p_{16}(\kappa,\alpha) = \big(p_{16,0}(\alpha) +p_{16,1}(\alpha) \frac1{\kappa^2}\big) \frac1{\kappa^2} +\ell^2 p_{16,2}(\alpha)$,
     $p_{16,0}(\alpha) = -36 (\sqrt2 +2) \ell^4 \alpha^4 +(144 \sqrt2 -744) \ell^3 \alpha^3$
       $+(-288 \sqrt2 +2976) \ell^2 \alpha^2 +(-576 \sqrt2 -864) \ell \alpha$,
     $p_{16,1}(\alpha) = 72 (6 -\alpha \ell) \alpha^2$,
     $p_{16,2}(\alpha) = 27 \ell^5 \alpha^5 +(-144 \sqrt2 +216) \ell^4 \alpha^4 +(-24 \sqrt2 -2412) \ell^3 \alpha^3$
       $+(1632 \sqrt2 +3104) \ell^2 \alpha^2 +(-3264 \sqrt2 +928) \ell \alpha +1920 (\sqrt2 -1)$.
$\delta_{17}(\kappa,\alpha)$ = $2\ \delta_{16}(\kappa,\alpha)$
$\delta_{18}(\kappa,\alpha)$ = $2^2\ \delta_{16}(\kappa,\alpha)$
$\delta_{19}(\kappa,\alpha)$ = $2^3\ \delta_{16}(\kappa,\alpha)$
$\delta_{20}(\kappa,\alpha)$ = $2^4\ \delta_{16}(\kappa,\alpha)$
$\delta_{21}(\kappa,\alpha)$ = $\tfrac{256 (\sqrt3 +2) (24\sqrt2 +61)}{23121 (\sqrt3 +1)^2} \ell\ \alpha^5\ p_{12}(\kappa,\alpha)^2\ p_{21}(\kappa,\alpha)$
     with $p_{21}(\kappa,\alpha) = \Big(p_{21,0}(\alpha) +p_{21,1}(\alpha) \frac1{\kappa^2}\Big) \frac1{\kappa^2} +\ell^2 p_{21,2}(\alpha)$,
     $p_{21,0}(\alpha) = (-1152 \sqrt2 +2928) \ell^5 \alpha^5 +(-468 \sqrt2 -2664) \ell^4 \alpha^4$
       $+(75024 \sqrt2 -175272) \ell^3 \alpha^3 +(-130464 \sqrt2 +300768) \ell^2 \alpha^2$
       $+(-14400 \sqrt2 -25056) \ell \alpha$,
     $p_{21,1}(\alpha) = (-1728 \sqrt2 +4392) (6 -\ell \alpha) \alpha^2$,
     $p_{21,2}(\alpha) = 7707 \ell^5 \alpha^5 +(-25248 \sqrt2 +95000) \ell^4 \alpha^4$
       $+(89448 \sqrt2 -353228) \ell^3 \alpha^3 +(158880 \sqrt2 +38048) \ell^2 \alpha^2$
       $+(-417216 \sqrt2 +464416) \ell \alpha +1920 (85\sqrt2 -109)$.
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