7. Several questions and a conjecture

Corollary 1

Corollary 2

Theorem 2

Proof

aii ¼ �

Corollary 3

Corollary 4

Corollary 5

Reð Þ λ<sup>i</sup> ≤ 0.

Vi dCi

This equation gives:

P<sup>Φ</sup>ij<sup>þ</sup>

PΦ<sup>p</sup> i

ith row looks like the disk in Figure 7.

ith row looks like the disk in Figure 6.

54 Matrix Theory-Applications and Theorems

and the ODE solutions are asymptotically stable.

dCi dt <sup>¼</sup> <sup>Φ</sup>ai Vi

Vi < 0, and also Ri ¼

Eq. (22) implies that the ith row of the MP-matrix has entries <sup>Φ</sup>ki

k ¼ i, and this equation does not contribute to the independent term.

In an open system, the Gershgorin disks look like those of Figure 8.

In an open system, being the ith tank an input one, the Gershgorin circle corresponding to the

If in an open system, all are input tanks, all the eigenvalues satisfy the condition Reð Þ λ<sup>i</sup> < 0,

In an open system, if the ith tank is not an input one, then the diagonal entry of the ith row is

aii < 0 and aii j j ¼ Ri being the sum of the non-diagonal entry modules of that row.

If Φai, Φbi, ⋯, Φni are the incoming fluxes from other tanks (a, b, ⋯, n) of the MP system,

fluxes from the ith tank to outside the system, then the corresponding ODE can be written as

Φni Vi Cn �

In this case a flux balance gives the following equation <sup>P</sup>Φki <sup>¼</sup> <sup>P</sup>Φij <sup>þ</sup> <sup>P</sup>Φ<sup>p</sup>

P<sup>Φ</sup>ij<sup>þ</sup>

In an open system, if the ith tank is not an input one, the Gershgorin circle corresponding to the

In an open system with input and non-input tanks, all the eigenvalues satisfy the condition

PΦ<sup>p</sup> i

� �Ci � <sup>Φ</sup><sup>1</sup>

<sup>P</sup>Φij <sup>þ</sup> <sup>P</sup>Φ<sup>p</sup>

Vi

i

<sup>i</sup> , Φ<sup>2</sup>

<sup>i</sup> Ci � … � <sup>Φ</sup><sup>s</sup>

Vi for k 6¼ i and �

Vi ¼ aii j j, and the theorem is proved.

<sup>i</sup> , ⋯, Φ<sup>s</sup>

Ci (23)

P<sup>Φ</sup>ij<sup>þ</sup>

PΦ<sup>p</sup> i Vi for

<sup>i</sup> , then

<sup>i</sup> are the

iCi (22)

ΦiA, ΦiB, ⋯, ΦiJ are the outgoing fluxes to other tanks (A, B, ⋯, J), and Φ<sup>1</sup>

dt <sup>¼</sup> <sup>Φ</sup>aiCa <sup>þ</sup> … <sup>þ</sup> <sup>Φ</sup>niCn � <sup>Φ</sup>iA <sup>þ</sup> … <sup>þ</sup> <sup>Φ</sup>iJ

Ca þ … þ

P<sup>Φ</sup>ki Vi ¼

As a consequence of the previous results, the following corollary can be stated.

In the previous section, some particular cases with λ<sup>i</sup> ¼ 0 and/or λ<sup>i</sup> ¼ a þ bi ∉ℜ (with a < 0, b 6¼ 0) were considered. A first question to analyze is if there exists an MP that satisfies any of these conditions. For this purpose, let us consider the closed MP of Figure 9, in which

ODE system can be written as , and the corresponding MP-matrix is �a 0 a 0 1

b �b 0 0 c �c B@ CA, being <sup>a</sup> <sup>¼</sup> <sup>Φ</sup> V<sup>1</sup> , <sup>b</sup> <sup>¼</sup> <sup>Φ</sup> V2 , and <sup>c</sup> <sup>¼</sup> <sup>Φ</sup> V3 . If Φ and Vi are chosen such that a ¼ 1,

on the answers to the questions and the conjecture presented in the last section, giving a

Square Matrices Associated to Mixing Problems ODE Systems

http://dx.doi.org/10.5772/intechopen.74437

57

The author wishes to thank Marjorie Chaves for her assistance and support in this work.

[1] Braun M. Differential Equations and their Applications. 3rd ed. New York: Springer;

[2] Groestch C. Inverse Problems: Activities for Undergraduates. Washington D.C.: Mathe-

[3] Martinez-Luaces V. Engaging secondary school and university teachers in modelling: Some experiences in South American countries. International Journal of Mathematical

[4] Martinez-Luaces V. Inverse-modelling problems in chemical engineering courses. In: Proceedings of the Southern Hemisphere Conference on Undergraduate Mathematics and Statistics Teaching and Learning. (Delta ´07); 26–30 November 2007; Calafate: Argentina.

[5] Martinez-Luaces V. Modelling and inverse-modelling: Experiences with O.D.E. Linear systems in engineering courses. International Journal of Mathematical Education in Sci-

[6] Martinez-Luaces V. Matrices in chemical problems: Characterization, properties and consequences about the stability of ODE Systems. In: Baswell A, editor. Advances in Mathe-

[7] Zill D. Differential Equations with Boundary-Value Problems. 9th ed. Boston: Cengage

challenging proposal for further research on this topic.

Address all correspondence to: victoreml@gmail.com Faculty of Engineering, UdelaR, Montevideo, Uruguay

2013. 546 p. DOI: 10.1007/978-1-4684-9229-3

matical Association of America; 1999. 222 p

ence and Technology. 2009;40(2):259-268

Education in Science and Technology. 2005;36(2–3):193-205

matics Research. New York: Nova Publishers; 2017. pp. 1-33

Acknowledgements

Author details

References

pp. 111-118

Learning; 2016. 630 p

Victor Martinez-Luaces

Figure 9. Three tanks with all the possible connections.

<sup>b</sup> <sup>¼</sup> 2, and <sup>c</sup> <sup>¼</sup> 3, it is easy to show that the eigenvalues are <sup>λ</sup><sup>1</sup> <sup>¼</sup> 0 and <sup>λ</sup>2,<sup>3</sup> ¼ �<sup>3</sup> � <sup>i</sup> ffiffiffi 2 <sup>p</sup> , which prove that null and/or complex eigenvalues are possible.

Other questions are not so simple like the previous one. The next two examples propose challenging problems that deserve to be studied:

Question 1:

Is it possible to find an MP-matrix with an eigenvalue λ<sup>i</sup> ¼ 0 such that AMð Þ0 > 1?

Question 2:

Is it possible to find an MP-matrix such that AMð Þ0 > GMð Þ0 ?

Question 3:

Is it possible to find an MP-matrix with complex eigenvalues in an open system?

Finally, it is interesting to observe that all cases analyzed here with λ<sup>i</sup> ¼ 0 correspond to closed systems. Moreover, in a previous book chapter [6], it was proved that Reð Þ λ<sup>i</sup> ≤ 0 , ∀i, in any MP open system with three tanks or less. Taking into account all these facts, it can be conjectured that in an open system, all the MP-matrix eigenvalues have negative real part and as a consequence, all the solutions are asymptotically stable.
