Proceedings
of the
XXVI Congreso de Ecuaciones Diferenciales y Aplicaciones XVI Congreso de Matematica´ Aplicada
Gijon´ (Asturias), Spain
June 14-18, 2021
Editors: Rafael Gallego, Mariano Mateos Esta obra esta´ bajo una licencia Reconocimiento- No comercial- Sin Obra Derivada 3.0 Espana˜ de Creative Com- mons. Para ver una copia de esta licencia, visite http://creativecommons.org/licenses/by-nc-nd/3.0/es/ o envie una carta a Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA.
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Coordinadores: Rafael Gallego, Mariano Mateos (2021), Proceedings of the XXVI Congreso de Ecuaciones Diferenciales y Aplicaciones / XVI Congreso de Matematica´ Aplicada. Universidad de Oviedo. La autor´ıa de cualquier art´ıculo o texto utilizado del libro debera´ ser reconocida complementariamente.
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2 Foreword
It is with great pleasure that we present the Proceedings of the 26th Congress of Differential Equations and Appli- cations / 16th Congress of Applied Mathematics (XXVI CEDYA / XVI CMA), the biennial congress of the Spanish Society of Applied Mathematics SeMA, which is held in Gijon,´ Spain from June 14 to June 18, 2021. E In this volume we gather the short papers sent by some of the almost three hundred and twenty communications presented in the conference. Abstracts of all those communications can be found in the abstract book of the congress. Moreover, full papers by invited lecturers will shortly appear in a special issue of the SeMA Journal. E The first CEDYA was celebrated in 1978 in Madrid, and the first joint CEDYA / CMAalaga tookplaceinM´ in 1989. Our congress focuses on different fields of applied mathematics: Dynamical Systems and Ordinary Differ- ential Equations, Partial Differential Equations, Numerical Analysis and Simulation, Numerical Linear Algebra, Optimal Control and Inverse Problems and Applications of Mathematics to Industry, Social Sciences, and Biol- ogy. Communications in other related topics such as Scientific Computation, Approximation Theory, Discrete Mathematics and Mathematical Education are also common. For the last few editions, the congress has been structured in mini-symposia. In Gijon,´ we will have eighteen minis-symposia, proposed by different researchers and groups, and also five thematic sessions organized by the local organizing committee to distribute the individual contributions. We will also have a poster session and ten invited lectures. Among all the mini-symposia, we want to highlight the one dedicated to the memory of our colleague Francisco Javier “Pancho” Sayas, which gathers two plenary lectures, thirty-six talks, and more than forty invited people that have expressed their wish to pay tribute to his figure and work. This edition has been deeply marked by the COVID-19 pandemic. First scheduled for June 2020, we had to postpone it one year, and move to a hybrid format. Roughly half of the participants attended the conference online, while the other half came to Gijon.´ Taking a normal conference and moving to a hybrid format in one year has meant a lot of efforts from all the parties involved. Not only did we, as organizing committee, see how much of the work already done had to be undone and redone in a different way, but also the administration staff, the scientific committee, the mini-symposia organizers, and many of the contributors had to work overtime for the change. Just to name a few of the problems that all of us faced: some of the already accepted mini-symposia and contributed talks had to be withdrawn for different reasons (mainly because of the lack of flexibility of the funding agencies); it became quite clear since the very first moment that, no matter how well things evolved, it wouldbe nearly impossible for most international participants to come to Gijon;´ reservations with the hotels and contracts with the suppliers had to be cancelled; and there was a lot of uncertainty, and even anxiety could be said, until we were able to confirm that the face-to-face part of the congress could take place asplanned. On the other hand, in the new open call for scientific proposals, we had a nice surprise: many people that would have not been able to participate in the original congress were sending new ideas for mini-symposia, individual contributions and posters. This meant that the total number of communications was about twenty percent greater than the original one, with most of the new contributions sent by students. There were almost one hundred and twenty students registered for this CEDYA / CMA. The hybrid format allows students to participate at very low expense for their funding agencies, and this gives them the opportunity to attend different conferences and get more merits. But this, which can be seen as an advantage, makes it harder for them to obtain a full conference experience. Alfred´ Renyi´ said: “a mathematician is a device for turning coffee into theorems”. Experience has taught us that a congress is the best place for a mathematician to have a lot of coffee. And coffee cannot be served online.
In Gijon,´ June 4, 2021
The Local Organizing Committee from the Universidad de Oviedo
3 Scientific Committee
Juan Luis Vazquez,´ Universidad Autonoma´ de Madrid Mar´ıa Paz Calvo, Universidad de Valladolid Laura Grigori, INRIA Paris Jose´ Antonio Langa, Universidad de Sevilla Mikel Lezaun, Euskal Herriko Unibersitatea Peter Monk, University of Delaware Ira Neitzel, Universitat¨ Bonn Jose´ Angel´ Rodr´ıguez, Universidad de Oviedo Fernando Teran,´ Universidad Carlos III de Madrid Sponsors
Sociedad Espanola˜ de Matematica´ Aplicada Departamento de Matematicas´ de la Universidad de Oviedo Escuela Politecnica´ de Ingenier´ıa de Gijon´ Gijon´ Convention Bureau Ayuntamiento de Gijon´ Local Organizing Committee from the Universidad de Oviedo
Pedro Alonso Velazquez´ Rafael Gallego Mariano Mateos Omar Menendez´ Virginia Selgas Marisa Serrano Jesus´ Suarez´ Perez´ del R´ıo
4 Contents
On numerical approximations to diffuse-interface tumor growth models Acosta-Soba D., Guillen-Gonz´ alez´ F. and Rodr´ıguez-Galvan´ J.R...... 8
An optimized sixth-order explicit RKN method to solve oscillating systems Ahmed Demba M., Ramos H., Kumam P. and Watthayu W...... 15
The propagation of smallness property and its utility in controllability problems Apraiz J...... 23
Theoretical and numerical results for some inverse problems for PDEs Apraiz J., Doubova A., Fernandez-Cara´ E. and Yamamoto M...... 31
Pricing TARN options with a stochastic local volatility model Arregui I. and Rafales´ J...... 39
XVA for American options with two stochastic factors: modelling, mathematical analysis and numerical methods Arregui I., Salvador B., Sevˇ coviˇ cˇ D. and Vazquez´ C...... 44
A numerical method to solve Maxwell’s equations in 3D singular geometry Assous F. and Raichik I...... 51
Analysis of a SEIRS metapopulation model with fast migration Atienza P. and Sanz-Lorenzo L...... 58
Goal-oriented adaptive finite element methods with optimal computational complexity Becker R., Gantner G., Innerberger M. and Praetorius D...... 65
On volume constraint problems related to the fractional Laplacian Bellido J.C. and Ortega A...... 73
A semi-implicit Lagrange-projection-type finite volume scheme exactly well-balanced for 1D shallow-water system Caballero-Cardenas´ C., Castro M.J., Morales de Luna T. and Munoz-Ruiz˜ M.L...... 82
SEIRD model with nonlocal diffusion Calvo Pereira A.N...... 90
Two-sided methods for the nonlinear eigenvalue problem Campos C. and Roman J.E...... 97
Fractionary iterative methods for solving nonlinear problems Candelario G., Cordero A., Torregrosa J.R. and Vassileva M.P...... 105
Well posedness and numerical solution of kinetic models for angiogenesis Carpio A., Cebrian´ E. and Duro G...... 109
Variable time-step modal methods to integrate the time-dependent neutron diffusion equation Carreno˜ A., Vidal-Ferrandiz` A., Ginestar D. and Verdu´ G...... 114
5 CONTENTS
Homoclinic bifurcations in the unfolding of the nilpotent singularity of codimension 4 in R4 Casas P.S., Drubi F. and Ibanez´ S...... 122
Different approximations of the parameter for low-order iterative methods with memory Chicharro F.I., Garrido N., Sarr´ıa I. and Orcos L...... 130
Designing new derivative-free memory methods to solve nonlinear scalar problems Cordero A., Garrido N., Torregrosa J.R. and Triguero P...... 135
Iterative processes with arbitrary order of convergence for approximating generalized inverses Cordero A., Soto-Quiros´ P. and Torregrosa J.R...... 141
FCF formulation of Einstein equations: local uniqueness and numerical accuracy and stability Cordero-Carrion´ I., Santos-Perez´ S. and Cerda-Dur´ an´ P...... 148
New Galilean spacetimes to model an expanding universe De la Fuente D...... 155
Numerical approximation of dispersive shallow flows on spherical coordinates Escalante C. and Castro M.J...... 160
New contributions to the control of PDEs and their applications Fernandez-Cara´ E...... 167
Saddle-node bifurcation of canard limit cycles in piecewise linear systems Fernandez-Garc´ ´ıa S., Carmona V. and Teruel A.E...... 172
On the amplitudes of spherical harmonics of gravitational potencial and generalised products of inertia Flor´ıa L...... 177
Turing instability analysis of a singular cross-diffusion problem Galiano G. and Gonzalez-Tabernero´ V...... 184
Weakly nonlinear analysis of a system with nonlocal diffusion Galiano G. and Velasco J...... 192
What is the humanitarian aid required after tsunami? Gonzalez-Vida´ J.M., Ortega S., Mac´ıas J., Castro M.J., Michelini A. and Azzarone A...... 197
On Keller-Segel systems with fractional diffusion Granero-Belinchon´ R...... 201
An arbitrary high order ADER Discontinous Galerking (DG) numerical scheme for the multilayer shallow water model with variable density Guerrero Fernandez´ E., Castro D´ıaz M.J., Dumbser M. and Morales de Luna T...... 208
Picard-type iterations for solving Fredholm integral equations Gutierrez´ J.M. and Hernandez-Ver´ on´ M.A...... 216
High-order well-balanced methods for systems of balance laws based on collocation RK ODE solvers Gomez-Bueno´ I., Castro M.J., Pares´ C. and Russo G...... 220
An algorithm to create conservative Galerkin projection between meshes Gomez-Molina´ P., Sanz-Lorenzo L. and Carpio J...... 228
On iterative schemes for matrix equations Hernandez-Ver´ on´ M.A. and Romero N...... 236
A predictor-corrector iterative scheme for improving the accessibility of the Steffensen-type methods Hernandez-Ver´ on´ M.A., Magren˜an´ A.A., Mart´ınez E. and Sukhjit S...... 242
6 CONTENTS
Recent developments in modeling free-surface flows with vertically-resolved velocity profiles using moments Koellermeier J...... 247
Stability of a one degree of freedom Hamiltonian system in a case of zero quadratic and cubic terms Lanchares V. and Bardin B...... 253
Minimal complexity of subharmonics in a class of planar periodic predator-prey models Lopez-G´ omez´ J., Munoz-Hern˜ andez´ E. and Zanolin F...... 258
On a non-linear system of PDEs with application to tumor identification Maestre F. and Pedregal P...... 265
Fractional evolution equations in dicrete sequences spaces Miana P.J...... 271
KPZ equation approximated by a nonlocal equation Molino A...... 277
Symmetry analysis and conservation laws of a family of non-linear viscoelastic wave equations Marquez´ A. and Bruzon´ M...... 284
Flux-corrected methods for chemotaxis equations Navarro Izquierdo A.M., Redondo Neble M.V. and Rodr´ıguez Galvan´ J.R...... 289
Ejection-collision orbits in two degrees of freedom problems Olle´ M., Alvarez-Ram´ ´ırez M., Barrabes´ E. and Medina M...... 295
Teaching experience in the Differential Equations Semi-Virtual Method course of the Tecnologico´ de Costa Rica Oviedo N.G...... 300
Nonlinear analysis in lorentzian geometry: the maximal hypersurface equation in a generalized Robertson-Walker spacetime Pelegr´ın J.A.S...... 307
Well-balanced algorithms for relativistic fluids on a Schwarzschild background Pimentel-Garc´ıa E., Pares´ C. and LeFloch P.G...... 313
Asymptotic analysis of the behavior of a viscous fluid between two very close mobile surfaces Rodr´ıguez J.M. and Taboada-Vazquez´ R...... 321
Convergence rates for Galerkin approximation for magnetohydrodynamic type equations Rodr´ıguez-Bellido M.A., Rojas-Medar M.A. and Sepulveda-Cerda´ A...... 325
Asymptotic aspects of the logistic equation under diffusion Sabina de Lis J.C. and Segura de Leon´ S...... 332
Analysis of turbulence models for flow simulation in the aorta Santos S., Rojas J.M., Romero P., Lozano M., Conejero J.A. and Garc´ıa-Fernandez´ I...... 339
Overdetermined elliptic problems in onduloid-type domains with general nonlinearities Wu J...... 344
A method to construct irreducible totally nonnegative matrices with a given Jordan canonical form Canto´ B., Canto´ R. and Urbano A.M...... 352
7 XXVICONGRESODE ECUACIONES DIFERENCIALESY APLICACIONES XVICONGRESODE MATEMATICA´ APLICADA Gijon,´ 14-18 junio 2021 (pp. 352–358)
A method to construct irreducible totally nonnegative matrices with a given Jordan canonical form
Begoña Cantó1, Rafael Cantó1, Ana M. Urbano1 Institut de Matemática Multidisciplinar, Universitat Politècnica de València, 46071 València, Spain.
Abstract Let 퐴 R푛 푛 be an irreducible totally nonnegative matrix (ITN), that is, 퐴 is irreducible with all its minors ∈ × nonnegative. A triple 푛, 푟, 푝 is called realizable if there exists an ITN matrix 퐴 R푛 푛 with rank 퐴 = 푟 ( ) ∈ × ( ) and 푝-rank 퐴 = 푝 (recall that 푝-rank 퐴 is the size of the largest invertible principal submatrix of 퐴). Each ( ) ( ) ITN matrix 퐴 associated with a realizable triple 푛, 푟, 푝 has 푝 positive and distinct eigenvalues, and for the ( ) zero eigenvalue it is verified that 푛 푟 and 푛 푝 are the geometric and the algebraic multiplicity, respectively. − − Moreover, since rank 퐴푝 = 푝, 퐴 has 푛 푟 zero Jordan blocks whose sizes are given by the Segre characteristic, ( ) − 푆 = 푠1, 푠2, . . . , 푠푛 푟 , with 푠푖 푝, 푖 = 1, 2, . . . , 푛 푟. ( − ) ≤ − We know the number of zero Jordan canonical forms of ITN matrices associated with a realizable triple 푛, 푟, 푝 and all these zero Jordan canonical forms. The following important question that we present in this ( ) talk deals with how to construct an ITN matrix 퐴 associated with 푛, 푟, 푝 and exactly with one of these Segre ( ) characteristic 푆 corresponding to the zero eigenvalue.
1. Introduction A matrix 퐴 R푛 푛 is called totally nonnegative if all its minors are nonnegative and it is abbreviated as TN. The ∈ × wide study of these matrices is due to the large number of applications in different branches of science, see for instance [1, 7–18]. Now, we recall some basic concepts that we will use throughout the paper: 1. The rank of 퐴, denoted by rank 퐴 , is the size of the largest invertible square submatrix of 퐴. The principal ( ) rank of 퐴, denoted by 푝-rank 퐴 , is the size of the largest invertible principal submatrix of 퐴. It is clear that ( ) 0 푝-rank 퐴 rank 퐴 푛 ≤ ( ) ≤ ( ) ≤ 2. The characteristic polynomial of a matrix 퐴 is given by
푛 푛 푘 푛 푘 푞 퐴 휆 = det 휆퐼 퐴 = 휆 1 det 퐴 훼 휆 − ( ) ( − ) + (− ) ( [ ]) 푘=1 훼 (k,n) ∑︁ ©∀ ∈Q∑︁ ª ® where (k,n) denotes the set of all increasing sequences of 푘 natural numbers less than or equal to 푛, for Q 푘, 푛 N, 1 푘 푛, see [1]. If 훼 = 훼 , 훼 , . . . , 훼푘 « 푘,푛 and 훽 = 훽 , 훽 ¬, . . . , 훽푘 푘,푛, 퐴 훼 훽 ∈ ≤ ≤ ( 1 2 ) ∈ Q ( 1 2 ) ∈ Q [ | ] denotes the 푘 푘 submatrix of 퐴 lying in rows 훼푖 and columns 훽푖, 푖 = 1, 2, . . . , 푘. The principal submatrix × 퐴 훼 훼 is abbreviated as 퐴 훼 . [ | ] [ ] If 퐴 is TN and 푝-rank 퐴 = 푝, the minors of the same order have the same sign or are zero, then there are ( ) no cancelations in the summands and then,
푛 푝 푝 푝 푘 푝 푘 푞 퐴 휆 = 휆 − 휆 푘=1 1 훼 (k,n) det 퐴 훼 휆 − ( ) + (− ) ∀ ∈Q ( [ ]) 푛 푝 푝 푝 1 푝 = 휆 − 휆 푐 휆 − 1 푐 푝 . − Í1 + · · ·Í + (− ) Then, if 퐴 is a TN matrix with rank 퐴 = 푟 and 푝-rank 퐴 = 푝, has 푝 nonzero eigenvalues and the algebraic ( ) ( ) and geometric multiplicities of the zero eigenvalue are equal to 푛 푝 and 푛 푟, respectively. − − 푛 푛 3. A matrix 퐴 R × , with 푛 2, is an irreducible matrix if there is not a permutation matrix 푃 such that 퐵∈ 퐶 ≥ 푃퐴푃푇 = , where 푂 is an 푛 푟 푟 zero matrix (1 푟 푛 1 . If 푛 = 1, 퐴 = 푎 is irreducible 푂 퐷 ( − ) × ≤ ≤ − ) ( ) when 푎 ≠0.
Fallat, Gekhtman and Johnson in [8] characterize the irreducible TN matrices as follows: a TN matrix 푛 푛 퐴 = 푎푖 푗 R is irreducible if and only if 푎푖 푗 > 0 for all 푖, 푗 such that 푖 푗 1 and they represent this [ ] ∈ × | − | ≤ class of matrices by ITN.
CEDYA/CMA 352 ISBN 978-84-18482-21-2 B.CANTO´ ,R.CANTOAND´ A.M.URBANO
4. If there exists an ITN matrix 퐴 R푛 푛 with rank 퐴 = 푟 and 푝-rank 퐴 = 푝, then the triple 푛, 푟, 푝 is ∈ × ( ) ( ) ( ) called realizable [8, p. 709], and 퐴 is considered as an ITN matrix associated with the triple 푛, 푟, 푝 . In 푛 푝 ( ) order to a triple 푛, 푟, 푝 be realizable it is necessary that 푝 푟 푛 − . ( ) ≤ ≤ − 푝 5. If 퐴 is an associated matrix with a realizable triple 푛, 푟, 푝 then, its 푝 nonzero eigenvalues are positive and ( ) distinct ( [8, Theorem 3.3]). That is, if 휆1, . . . , 휆 푝, . . . 휆푛 are the eigenvalues of 퐴, we have
휆1 > 휆2 > . . . > 휆 푝 > 0, and 휆 푝 1 = 휆 푝 2 = ... = 휆푛 = 0, (1.1) + + Moreover, since the algebraic multiplicity of the zero eigenvalue is 푛 푝 and rank 퐴푝 = 푝, the size of the − ( ) zero Jordan blocks of 퐴 is at most 푝. Taking into account the above results, given a realizable triple 푛, 푟, 푝 the following questions arises in a natutal ( ) way: First question: How many different zero Jordan canonical forms are associated with a realizable triple 푛, 푟, 푝 ? ( ) As we have seen, the ITN matrices associated with a realizable triple 푛, 푟, 푝 verify that the algebraic multiplicity ( ) of the zero eigenvalue is 푛 푝, the geometric one is 푛 푟 and the size of the Jordan blocks is maximum 푝. Therefore, − − this problem is equivalent to the following in how many ways can we distribute 푛 푝 marbles in 푛 푟 bags, knowing − − that all bags must have at least one marble and that at most each bag will fit 푝 marbles. 푝 In [6], by using Number Theory, the authors calculated this number (represented by 푝푛( )푟 푛 푝 ) and they gived − ( − ) an algorithm to obtain it. For example, if we have the triple realizable 19, 14, 8 applying this algorithm we have 8 ( ) that 푝 ( ) 11 = 10. 5 ( ) Second question: Since we konw the number of different zero Jordan canonical forms associated with a realizable triple 푛, 푟, 푝 , then what are these zero Jordan forms? ( ) In [6], using properties and the full rank 퐿푈 factorization of ITN matrices and the Flanders Theorem the authors give and Procedure and the corresponding algorithm to compute the specific different zero Jordan canonical forms. For example, for the realizable triple 19, 14, 8 we have obtained that there are 10 different zero Jordan canonical ( ) forms and applying the new algorithm we obtain these specific zero Jordan structures, all of them have 5 zero Jordan blocks of differents sizes. These structures are, 7 1 1 1 1 6 2 1 1 1 5 3 1 1 1 5 2 2 1 1 4 4 1 1 1 4 3 2 1 1 4 2 2 2 1 3 3 3 1 1 3 3 2 2 1 3 2 2 2 2
Remark 1.1 The sizes of the zero Jordan blocks of a matrix 퐴 are known as the Segre characteristic of 퐴 relative to its zero eigenvalue. Given an ITN matrix 퐴 associated to a realizable triple 푛, 푟, 푝 , if we represent this Segre ( ) sequence by 푆 = 푠1, 푠2, . . . , 푠푛 푟 then, it is satisfied that ( − ) 1 푠 min 푟 푝 1, 푝 ( ) 1 ≤ { − + } 2 푠푖 푠푖 1, 푖 = 2, 3, . . . , 푛 푟 (1.2) ( ) ≤ − − 푛 푟 3 − 푠푖 = 푛 푝 ( ) 푖=1 − Associated to the Segre characteristic 푆 = 푠1Í, 푠2, . . . , 푠푛 푟 we have the Weyr characteristic of 퐴 relative to the ( − ) zero eigenvalue 푊 = 푤 , 푤 , . . . , 푤푠 , where 푤푖 = Car 푘 : 푠푘 푖 for 푖 = 1, 2, . . . , 푠 and ( 1 2 1 ) { ≥ } 1 1 푤 = dim Ker 퐴 = 푛 푟 ( ) 1 ( ) − 2 푤푖 푤푖 1, 푖 = 2, 3, . . . , 푠1 ( ) ≤ − 푖 푖 (1.3) 3 푤 푗 = dim Ker 퐴 ( ) 푗=1 ( ) 푠1 푠 4 푤 푗 = dim Ker 퐴 1 = 푛 푝 ( ) Í 푗=1 ( ) − Í 353 A METHOD TO CONSTRUCT IRREDUCIBLE TOTALLY NONNEGATIVE MATRICES
Thirth question: Finally, knowing the number of zero Jordan canonical forms and the specific structures associated with a realizable triple 푛, 푟, 푝 , the following question is the main goal of this work, how to construct an ITN ( ) matrix associated with a realizable triple 푛, 푟, 푝 and with 푛 푟 zero Jordan blocks whose sizes are given by the ( ) − Segre characteristic 푆 = 푠1, 푠2, . . . , 푠푛 푟 satisfying (1.2). ( − ) To answer this question, in the next section we first described a procedure that allow us to construct an upper block echelon matrix 푈 R푛 푛, with rank 푈 = 푟, 푝-rank 푈 = 푝 and 푛 푟 zero Jordan blocks whose sizes are ∈ × ( ) ( ) − given by the Segre characteristic 푆 satisfying (1.2). After that, from 푈 we will obtain the desired ITN matrix 퐴 associated with the realizable triple 푛, 푟, 푝 and with the same zero Jordan structure that 푈 as 퐴 = 퐿푈, where 퐿 is ( ) a lower triangular matrix with all its nonzero entries equal to 1.
2. Constructing an upper block echelon TN matrix 푈 with a zero Jordan canonical form In this section we describe a procedure to construct an upper block echelon matrix 푈 R푛 푛, with rank 푈 = 푟, ∈ × ( ) 푝-rank 푈 = 푝 and 푛 푟 zero Jordan blocks whose sizes are given by the Segre characteristic 푆 satisfaying (1.2). ( ) − We recall that a matrix is upper block echelon if each nonzero block, starting from the left, is to the right of the nonzero blocks below and the zero blocks are at the bottom. A matrix is a lower (block) echelon matrix if its transpose is an upper (block) echelon matrix. In the Procedure 1 we use the nonsingular ITN matrix 1 1 1 ... 1 1 1 2 2 ... 2 2 1 2 3 ... 3 3 푉푞 = . . . . . = min 푖, 푗 푞 푞 . . . . . [ { }] × 1 2 3 . . . 푞 1 푞 1 − − 1 2 3 . . . 푞 1 푞 − and the following MatLab notation: 퐴 푖, : denotes the 푖-th row of 퐴and 퐴 :, 푗 denotes its 푗-th column; ones 푛, 푚 ( ) ( ) ( ) denotes the 푛 푚 matrix of ones; triu ones 푛, 푚 denotes the upper triangular part of ones 푛, 푚 ; zeros 푛, 푚 × ( ( )) ( ) ( ) denotes the 푛 푚 zero matrix. × Note that if 푟 = 푝 the algebraic and geometric multiplicity of the zero eigenvalue Of 푈 is the same, therefore 푈 has 푛 푟 zero Jordan blocks of size 1 1. In this case is easy to see that the matrix 푈 can be the following − × triu(ones 푝, 푛 푈 = ( )) . zeros 푛 푝, 푛 ( − ) If 푝 < 푟 we construct a matrix 푈 by blocks as follows
푈11 푈12 푈13 푈14 . . . 푈1,푠 1 푈1,푠 푈1,푠 1 1− 1 1+ 푂 푂 푈23 푈24 . . . 푈2,푠 1 푈2,푠 푈2,푠 1 1− 1 1+ 푂 푂 푂 푈34 . . . 푈3,푠 1 푈3,푠 푈3,푠 1 1− 1 1+ ...... 푈 = ...... . 푂 푂 푂 푂 . . . 푂 푈푠 1,푠 푈푠 1,푠 1 1− 1 1− 1+ 푂 푂 푂 푂 . . . 푂 푂 푈푠 ,푠 1 1 1+ 푂 푂 푂 푂 . . . 푂 푂 푂 Each block and its size are given in the following procedure. Procedure 1. Given a realizable triple 푛, 푟, 푝 and the Segre characteristic 푆 = 푠1, 푠2, . . . , 푠푛 푟 satisfying (1.2), ( ) ( − ) this procedure obtains an upper block echelon matrix 푈 R푛 푛, with rank 푈 = 푟, 푝-rank 푈 = 푝 and 푛 푟 zero ∈ × ( ) ( ) − Jordan blocks whose sizes are given by 푆.
Step 1. Obtain the conjugated sequence of 푆, 푊 = 푤 , 푤 , . . . , 푤푠 and from 푊 define 푅 = 0, 푟 , . . . , 푟푠 , with ( 1 2 1 ) ( 2 1 ) 푟푖 = 푤푖, 푖 = 2, 3, . . . , 푠1. Step 2. Calculate 푛 = 푝 1 푠 and construct 1 + − 1 푈11 푈12 푈13 . . . 푈1,푠 푈1,푠 1 = triu(ones 푛1, 푛 1 1+ ( ))] Step 3. Construct 푈23 = 푉푟 1 and 2+
푂 푂 푈23 푈24 . . . 푈2,푠 푈2,푠 1 1 1+ = zeros 푟 1, 푛 푟 푈 푈 :, 푟 1 ones 1, 푛 푛 2푟 1 [ ( 2 + 1 + 2) 23 23 ( 2 + ) ∗ ( − 1 − 2 − )]
354 B.CANTO´ ,R.CANTOAND´ A.M.URBANO
Step 4. For 푖 = 3, 4, . . . , 푠1
4.1. If 푟푖 = 푟푖 1 construct 푈푖,푖 1 = 푉푟푖 1. − + +
푉푟푖 1 4.2. If 푟푖 < 푟푖 1 construct 푈푖,푖 1 = + − + ones 푟푖 1 푟푖, 1 푉푟푖 1 푟푖 1, : ( − − ) ∗ + ( + ) After, in both cases,
푂 . . . 푂 푈푖,푖 1 푈푖,푖 2 . . . 푈푖,푠1 푈푖,푠1 1 푖 1 + + + 푖 = zeros 푟푖 1 1, 푛1 푟2 푗− 푟 푗 1 푈푖,푖 1 푈푖,푖 1 :, 푟푖 1 ones 1, 푛 푛1 푟2 푗 푟 푗 1 [ ( − + + + =2 ( + )) + + ( + ) ∗ ( − − − =2 ( + ))] Í Í Step 5. Finally, the last block is equal to
푠 1 1− zeros 푛 푛1 푟2 1 푟푖 1 , 푛 − − ( + ) − ( + ) ! ∑︁푖=2 □ In the answer to the second question we have seen that the realizable triple 19, 14, 8 has associated 10 different ( ) zero Jordan canonical forms, being 4, 3, 2, 1, 1 one of them. In the following example we construct an upper ( ) block echelon TN matrix 푈 with this Jordan canonical form using Procedure 1.
Example 2.1 Obtain a 19 19 upper block echelon TN matrix 푈, with rank 푈 = 14, 푝-rank 푈 = 8 and with 5 × ( ) ( ) zero Jordan blocks of sizes 푆 = 4, 3, 2, 1, 1 . ( ) Since 푟 ≠ 푝 and 푠1 = 4, following Procedure 1 we construct an upper block TN matrix
푈11 푈12 푈13 푈14 푈15 푂 푂 푈23 푈24 푈25 푈 = 푂 푂 푂 푈 푈 . 34 35 푂 푂 푂 푂 푈 45 푂 푂 푂 푂 푂 Step 1. The conjugated sequence of 푆 is 푊= 5, 3, 2, 1 and then, 푅 = 0, 3, 2, 1 . ( ) ( ) Step 2. 푛 = 푝 1 푠 = 5 and 1 + − 1 푈 푈 푈 푈 푈 = triu(ones 5, 19 = 11 12 13 14 15 ( )) 1111111111111111111 0111111111111111111 0011111111111111111 . 0001111111111111111 0000111111111111111 Step 3. 푈23 = 푉4 and
푂 푂 푈 푈 푈 , = zeros 4, 8 푈 푈 :, 4 ones 1, 7 = 23 24 2 5 [ ( ) 23 23 ( ) ∗ ( )] 0000000011111111111 0000000012222222222 . 0000000012333333333 0000000012344444444 Step 4. For 푖 = 3, since 푟3 < 푟2, construct 1 1 1 푉 1 2 2 푈 = 3 = . 34 ones 1, 1 푉 3, : 1 2 3 ( ) ∗ 3 ( ) 1 2 3 355 A METHOD TO CONSTRUCT IRREDUCIBLE TOTALLY NONNEGATIVE MATRICES and 푂 푂 푂 푈34 푈35 = zeros 4, 12 푈34 푈 34 :, 3 ones 1, 4 = [ ( ) ( )( ) ∗ ( )] 0000000000001111111 0000000000001222222 . 0000000000001233333 0000000000001233333 Now, for 푖 = 4 since푟4 < 푟3, construct 1 1 푉 푈 = 2 = 1 2 45 ones 1, 1 푉 2, : ( ) ∗ 2 ( ) 1 2 and 푂 푂 푂 푂 푈45 = zeros 3, 15 푈45 푈45 :, 2 ones 1, 2 = [ ( ) ( )( ) ∗ ( )] 0000000000000001111 0000000000000001222 0000000000000001222 Step 5. The last zero block is 푠 1 1− zeros 푛 푛1 푟2 1 푟푖 1 , 푛 = zeros 3, 19 . " − − ( + ) − ( + ) !# [ ( )] ∑︁푖=2 Therefore, the matrix 푈 is
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 1 2 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 0 0 1 2 3 4 4 4 4 4 4 4 4 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 . 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The following result proves that the matrix 푈 constructed by Procedure 1 verifies the desired properties. Theorem 2.2 [6, Theorem 1] Consider the matrix 푈 constructed by Procedure 1. Then the following properties hold: 1. 푈 is a TN matrix with rank 푈 = 푟 and 푝-rank 푈 = 푝. ( ) ( ) 2. 푈 has 푛 푟 zeros Jordan blocks whose sizes are given by the sequence 푆 = 푠1, 푠2, . . . , 푠푛 푟 . − ( − )
356 B.CANTO´ ,R.CANTOAND´ A.M.URBANO
3. Construct an ITN matrix with a prescribed zero Jordan structure In this section we construct an ITN matrix 퐴 associated with the realizable triple 푛, 푟, 푝 and with a zero Jordan ( ) canonical form associated with this triple. For that, we use the procedure given in the previous section to construct an upper block echelon TN matrix 푈 of size 푛 푛, with rank 푈 = 푟, 푝-rank 푈 = 푝 and with a zero Jordan × ( ) ( ) canonical form associated with this triple. Now, we give the following procedure to compute the matrix 퐴.
Procedure 2. Given a realizable triple 푛, 푟, 푝 and the Segre characteristic 푆 = 푠1, 푠2, . . . , 푠푛 푟 satisfying (1.2), ( ) ( − ) this procedure obtains an ITN matrix 퐴 R푛 푛, associated with this triple and with 푛 푟 zero Jordan blocks whose ∈ × − sizes are given by 푆.
Step 1. Apply Procedure 1 to construct the upper block matrix 푈. Step 2. Construct the lower triangular TN matriz 퐿 = tril(ones 푛, 푛 . ( )) Step 3. Obtain 퐴 = 퐿 푈. ∗ □ The following result proves that the matrix 퐴 satisfies the prescribed conditions.
Theorem 3.1 [6, Proposition 1, Theorem 2] The matrix 퐴 constructed by Procedure 2 satisfies the following conditions: 1. 퐴 is a ITN matrix.
2. rank 퐴 = 푟. ( ) 3. 푝-rank 퐴 = 푝. ( ) 4. Matrices 퐴 and 푈 have the same zero Jordan structure.
Example 3.2 Construct a 19 19 ITN matrix 퐴, associated with the realizable triple 19, 14, 8 and with 5 zero × ( ) Jordan blocks of sizes 푆 = 4, 3, 2, 1, 1 . ( ) Using the matrix 푈 obtained in Example 2.1 and following Procedure 2, we have
퐴 = tril(ones 푛, 푛 푈 = ( )) ∗ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 2 3 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 2 3 4 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 1 2 3 4 5 5 5 5 7 8 8 8 8 8 8 8 8 8 8 1 2 3 4 5 5 5 5 8 10 11 11 11 11 11 11 11 11 11 1 2 3 4 5 5 5 5 9 12 14 15 15 15 15 15 15 15 15 1 2 3 4 5 5 5 5 9 12 14 15 16 16 16 16 16 16 16 1 2 3 4 5 5 5 5 9 12 14 15 17 18 18 18 18 18 18 1 2 3 4 5 5 5 5 9 12 14 15 18 20 21 21 21 21 21 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 24 24 24 24 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 25 25 25 25 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 26 27 27 27 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 27 29 29 29 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 27 29 29 29 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 27 29 29 29 1 2 3 4 5 5 5 5 9 12 14 15 19 22 24 27 29 29 29 Acknowledgements This work has been supported by the Ministerio de Economía y Competitividad under the Spanish DGI grant MTM2017-85669-P-AR.
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