Editors in Chief : Editorial Board

Editors in Chief :

Adelina Georgescu, Professor, Ilie Burdujan, Professor, Member of Academy of Romanian University of Agricultural Sciences and Veterinary Scientists, 54, Splaiul Medicine Independentei, 050094, Bucharest, “Ion Ionescu de la Brad” Iaşi ROMANIA Mihail Sadoveanu, 3, 700490 Iaşi, ROMANIA

Editorial Board:

Nuri Aksel, Professor, Dumitru Botnaru, Professor, Faculty of Applied Sciences, Chair of Superior Mathematics, Department of Applied Mechanics Technical University of Moldova, and Fluid Dynamics, Bv. Stefan cel Mare, 168, MD 2004, University of Bayreuth, D‐95440 Chişinău, REPUBLIC OF MOLDOVA Bayreuth, GERMANY

Tassos Bountis, Professor , Mitrofan Choban, Professor, Centre for Research and Applications of Nonlinear Member of Academy of Sciences of Moldova, Systems and Department of Mathematics, University of State University of Tiraspol, Iablocikin, 5, MD 2069, Patras, 26500, Patras, GREECE Chişinău, REPUBLIC OF MOLDOVA

Sanda Cleja‐Țigoiu, Professor, Constantin Fetecău, Professor, Faculty of Mathematics and Computer Science, Faculty of Mechanical Engineering, University of Bucharest, Academiei, 14, 010014, The Gh. Asachi Technical University of Iaşi, Bucharest, ROMANIA Bv. Dimitrie Mangeron, 61‐63, 700050, Iaşi, ROMANIA

Anca Veronica Ion, Senior Researcher Peter Knabner, Professor, Institute of Mathematical Statistics and Applied Chair for Applied Mathematics, Faculty of Science, Mathematics of The Romanian Academy, Friedrich Alexander University Calea 13 Septembrie, 13, 050711, Erlangen‐Nuremberg, Martensstr. 3, 91058, Erlangen, Bucharest, ROMANIA GERMANY

Boris V. Loginov, Professor, Nenad Mladenovici, Professor, Faculty of Natural Sciences, Ulyanovsk State Technical Mathematical Sciences, John Crank 210, University, Severny Venetz, 32, 432027, Brunel University, Uxbridge, UB8 3PH, Ulyanovsk, RUSSIA UNITED KINGDOM

Toader Morozan, Senior Researcher, Emilia Petrişor, Professor, Institute of Mathematics Simion Stoilow of The Romanian Department of Mathematics, Academy, Calea Griviței, 21, 010702, University Politehnica of Timişoara, Bucharest, ROMANIA Victoriei Square, 2, 300006, Timişoara, ROMANIA

Mihail Popa, Professor, Senior Researcher, Kumbakonam Rajagopal, Professor, Institute of Mathematics and Computer Science of The Department of Mathematics, Academy of Sciences of Moldova, Academiei, 5, MD 2028, Texas A&M University, Chişinău, REPUBLIC OF MOLDOVA Mailstop 3368, College Station, TX 77843‐3368, UNITED STATES OF AMERICA

Mefodie Rațiu, Professor, Senior Researcher, Mirela Ştefănescu, Professor, Institute of Mathematics and Computer Science of The Faculty of Mathematics and Computer Science, Academy of Sciences of Moldova, Academiei, 5, MD 2028, Ovidius University, Bv. Mamaia, 124, 900527, Chişinău, REPUBLIC OF MOLDOVA Constanța, ROMANIA

Nicolae Suciu, Senior Researcher, Kiyoyuki Tchizawa, Professor, Tiberiu Popoviciu Institute of Numerical Analysis, Kanrikogaku Kenkyusho, Ltd., P.O. Box 68‐1, 400110, 2‐2‐2 Sotokanda, Chiyoda‐ku, 101‐0021, Cluj‐Napoca, ROMANIA Tokyo, JAPAN

Vladilen A. Trenogin, Professor, Constantin Vârsan, Senior Researcher, Moscow Institute of Steel and Alloys, Institute of Mathematics Simion Stoilow of The Romanian B‐49, Leninsky Prospect, 4, 119049, Academy, Calea Griviței, 21, 010702, Moscow, RUSSIA Bucharest, ROMANIA

In memoriam Adelina Georgescu

April 25, 1942 – May 1, 2010

Am cunoscut pe marea noastră matematiciană Profesor Dr. Adelina Georgescu şi am admirat-o din inimă atât pentru importanta creaţie ştiinţifică şi frumoasa activitate didactică, cât şi pentru energia inepuizabilă cu care a luptat pentru neamul românesc. Cu negrăită emoţie îmi amintesc tot ce a reuşit pentru unitatea noastră, de la primele manifestări de după 1989, Sesiuni ştiinţifice organizate în 1991-1992 şi în continuare, pentru matematicienii români de la răsărit şi apus de Prut şi lupta pentru o cât mai strânsă colaborare între ei. Spiritul de iniţiativă şi talentul deosebit de organizare stau la temelia importantei Societăţi ROMAI, pe care a creat-o în 1992 şi a seriei de Conferinţe Internaţionale CAIM iniţiată în 1993, care a ajuns în 2010 la cea de a 18-a ediţie. Doresc ca această Societate să rodească mai departe pentru înflorirea matematicii în România şi peste hotare, mai ales în Republica Moldova. La aceste minunate realizări se adaugă cei doi fii Andrei şi Sergiu Moroianu, pe care i-a crescut în dragoste pentru matematică şi care sunt astăzi la rândul lor creatori de renume internaţional în această ştiinţă şi totodată întemeietori de rodnice familii. Li se adaugă doctorii în ştiinţe matematice precum şi alţi tineri călăuziţi de Doamna Profesor pe căile acestei ştiinţe. Retrăind amintiri despre Adelina, gândul mă duce la Doamna Oltea, mama lui Ştefan cel Mare, simbol al femeii române luptând pentru dreptatea şi înflorirea României. Fie ca tradiţia făurită de Doamna Profesor Adelina Georgescu să fie continuată şi dezvoltată strălucit şi mai departe! Prof. univ. Cabiria Andreian Cazacu

Cuvant tinut in cadrul CAIM 2010 la Iasi, la lansarea cartii Matematica si viata, scrisa de Adelina Georgescu si Lucia Dragotescu.

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I have known our great mathematician Professor Adelina Georgescu, and I admired her from all my heart both for her important scientific creation and for the beautiful didactical activity as well as for the inexhaustible energy with which she fought for the Romanian people. With unspeakable emotion I remember all she managed to do for our unity, from the first manifestations after 1989, the sessions organized in 1991-1992, and after, mainly for the Romanian mathematicians from the East and West of Prut and the fight for their close collaboration. Her vivid spirit and organizing talent lie at the base of the important Mathematical Society ROMAI, which she founded in 1992, and of the series of Conferences on Applied and Industrial Mathematics (CAIM) that she initiated in 1993, and that reached in 2010 the 18th edition. I wish that this Society to be also fruitful from now on, for the flourishing of the mathematics from Romania and abroad, especially from Republic of Moldova. To this wonderful achievements we must add her two sons, Andrei and Sergiu Moroianu, that she grew up in the spirit of love for mathematics and that are now, at their turn, well-known mathematicians and also heads of beautiful families. To them we must add the doctors in mathematics and many other young persons that were lead by Professor Adelina Georgescu on the ways of this science. While reminding Adelina, the thought takes me to Doamna Oltea, mother of Stephen the Great, the symbol of Romanian women fighting for justice and for the flourishing of our country, Romania. Let the tradition developed by Professor Adelina Georgescu be brightly continued and developed further!

Professor Cabiria Andreian Cazacu.

Speech held by Professor Cabiria Andreian Cazacu, at CAIM 2010, in Iasi, when the book Mathematics and Life, written by Adelina Georgescu and Lucia Dragotescu was launched.

A life dedicated to the whole "Knowledge" Liliana RESTUCCIA Department of Mathematics, University of Messina, Italy Considerate la vostra semenza: fatti non foste a viver come bruti, ma per seguir virtute e conoscenza. (Dante Alighieri, Divina Commedia, Inferno, Canto XXVI, versi 118-120)

This contribution contains the speech delivered by the author with the occasion of the Mini‐ symposium, organized in collaboration with D. Jou, M. S. Mongiovì and W. Muschik, as a part of the 2010 Conference of SIMAI (Italian Society of Applied and Industrial Mathematics), which was held between 21‐25 June in Cagliari, Sardegna, Italia. This Mini‐symposium, entitled "Recent Ideas in Non‐ Equilibrium Thermodynamics and Applications" was dedicated to the memory of Prof. Dr. Adelina Georgescu, co‐researcher and a great author’s friend.

We dedicate this Mini-symposium, regarding the latest ideas in Non- Equilibrium Thermodynamics and Applications to physical-mathematical models of complex materials, to Prof. Adelina Georgescu, a member of the Scientists’ Academy of Romania, a great master and an eminent scientist. Her death, on May 1st 2010, leaves behind a very great emptiness in the field of Applied Mathematics and not less in the lives of those who knew and worked with her. I am deeply grateful for her very precious advices in the organization of my research and the very enlightening discussions and remarks regarding a lot of mathematical concepts and tools, with their practical applications in different fields of science. She was my great and unforgettable friend, invaluable guide and scientific collaborator. Since 2002 I invited her as visiting professor at the University of Messina for giving lectures and seminars on: Non-linear Dynamics, Bifurcation theory, Variational problems in Mathematical-Physics, Thermodynamics of fluids, Asymptotic methods with applications to waves and shocks, Complex fluid flows, Modeling in Meteorology, Turbulence, Modeling in cancer dynamics. I always thanked her for her profound participation and for her very precious contribution in spreading her research methodologies (a scientific literature on foundations and recent progress in Mathematical-Physics due to the Romanian and Russian schools) and, especially for her publications (very useful due to their synthetic style and clarity in explaining all concepts) at the University of Messina. Some of her books and publications have been already translated from Romanian to Italian, in order to be known in Italy for their invaluable qualities. v

She also was a corresponding member of "Accademia Peloritana dei Pericolanti" of Messina; a special issue of the Academic Acts (on-line) of this institution, will be published, containing a collection of the lectures held by Prof. Adelina Georgescu during her visits at Messina. She was the founder and president of ROMAI (Romanian Society of Applied and Industrial Mathematics) since 1992. Furthermore, she founded the Institute of Applied Mathematics of Romanian Academy (IAM), which she led as director until 1995. In 1997 she became full professor at the University of Piteşti (Romania), where she founded the Chair of Applied Mathematics (the head of which she was for several years), where she created the group of research on Dynamical systems and Bifurcation. She was the editor in chief of ROMAI Journal since 2005. Since 1993, year by year she has organized the CAIM International Conferences, devoted to contributions in Applied and Industrial Mathematics, the Romanian analogous to SIMAI International Conferences in Italy. She delivered about 200 conferences at several universities and research units mostly from abroad. Moreover, she sustained short or plenary conferences at about 150 scientific meetings. She published (some are still in press) about 20 books (research monographies, three of them at Kluwer and Chapman & Hall and one at World Scientific, dictionaries, universitary texts), alone or in collaboration, and more than 200 papers in scientific journals most of them in ISI journals or refereed in MR and ZBM. She was awarded the Romanian Academy prize and many other distinctions. She was honored as a Doctor Honoris Causa of the State University of Tiraspol (Rep. of Moldova). She was a corresponding member of the Academy of Nonlinear Sciences (Moscow). The main topics of Prof. Adelina Georgescu's researches were in Hydrodynamics and applications to complex fluid flows, Turbulence, Meteorology, Perturbation theories for partial differential equations, Boundary layer theory, Hydrodynamics stability, Study of solutions of partial differential equations, Non- linear dynamics, Bifurcation theory, Variational problems in Mathematical-Physics, Modeling in cancer dynamics, Biomathematics, Synergetics. She dedicated to the progress of science her very generous and honest and highly moral and responsible life, studying very deeply mathematical and physical theories with applications and giving them new impulse. Her courageous ideas in a difficult politic framework had been well known by whole scientific world. She was a gifted and talented woman with a very great humanity and

Professors Adelina Georgescu and Liliana Restuccia, at University of Messina, 2007

very broad interests. Besides her basic research items like Physics and Mathematics (Applied Mathematics, in particular), she was interested in politics, history, geography, languages, religion, literature, medicine. At the University of Messina we will be open a Study Center "Adelina Georgescu" of Applied Mathematics, dedicated to her memory, for her work be known by the present generations and those to come. This Center has as objective to spread at international level her books and publications, her research methodologies, her ideas, in general, and in particular those in Synergetics (the new synthesis of the science) and in the directions to be followed in order to find the science truths. Furthermore, this Center has also the goal of organizing international conferences, meetings, study days, lectures and seminars for discussing issues as: foundations and recent progresses in different fields of Applied Mathematics, with the participation of speakers of high scientific level, researchers and students. I think that if there is a place where there is now Adelina Georgescu, well, then this is the Paradise, the Dante Alighieri’s Paradise of the whole "Knowledge", to which Adelina devoted her entire life.

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Published scientific works of Professor ADELINA GEORGESCU

A. BOOKS

1. Adelina Georgescu, Bifurcatia, fractali si haos determinist, chapter of Enciclopedia matematica, edited by M. Iosifescu, O. Stănăşilă, D. Ştefănoiu, Editura AGIR, 2010. 2. Adelina Georgescu, L. Palese, G. Raguso, Biomatematica. Modelli, dinamica e biforcazione, Cacucci Editore, Bari, 2009. 3. Adelina Georgescu, Lidia Palese, Stability criteria of fluid flows, Series on Advances in Mathematics for applied sciences 81, World Scientific, Singapore, 2009. 4. Adelina Georgescu, George-Valentin Cârlig, Cătălin-Liviu Bichir, Ramona Radoveneanu, Matematicieni români de pretutindeni, ed. a II-a, , Seria Mat. Apl. Ind. 24, Ed. Pamantul, Pitesti, 2006. 5. Adelina Georgescu, C.-L. Bichir, G.V. Cîrlig, Matematicieni români de pretutindeni, Seria de Mat. Apl. Ind. 18, Ed. Univ. Piteşti , Piteşti, 2004. 6. C. Rocşoreanu, Adelina Georgescu, N. Giurgiţeanu, FitzHugh-Nagumo model: bifurcation and dynamics, Kluwer, Dordrecht, 2000. 7. B.-N. Nicolescu, N. Popa, Adelina Georgescu, M.Boloşteanu, Mişcări ale fluidelor cavitante.Modelare şi soluţii, Seria Mat. Apl. Ind, 3, Ed. Univ. Piteşti, Piteşti, 1999. 8. N. Popa, B.-N. Nicolescu, Adelina Georgescu, M. Boloşteanu, Modelări matematice în teoria lubrificaţiei aplicate la etanşările frontale, Seria Mat. Apl. Ind., 2, Ed. Univ. Piteşti, Piteşti, 1999. 9. Adelina Georgescu, M. Moroianu, I. Oprea, Teoria bifurcaţiei. Principii şi aplicaţii, Seria Mat. Apl. Ind. 1, Ed. Univ. Piteşti, Piteşti, 1999. 10. Adelina Georgescu, Teoria stratului limită. Turbulenţă, Ed. Gh. Asachi, Iaşi, 1997. 11. Adelina Georgescu, Asymptotic treatment of differential equations, Applied Mathematics and Mathematical Computation, 9, Chapman and Hall, London, 1995. 12. Adelina Georgescu, I. Oprea, Bifurcation theory from application viewpoint, Tipografia Univ. Timişoara, 1994, (Monografii Matematice 51). 13. Adelina Georgescu, Sinergetica. Solitoni. Fractali. Haos determinist. Turbulenţă, Tipografia Univ. Timişoara, 1992. 14. H.Dumitrescu, Adelina Georgescu, V.Ceangă, GH.Ghiţă, J.Popovici, B.Nicolescu, Al.Dumitrache, Calculul elicei, Ed.Academiei, Bucureşti, 1990. 15. Adelina Georgescu, Aproximaţii asimptotice, Ed.Tehnică, Bucureşti, 1989.

16. Adelina Georgescu, Sinergetica-o nouă sinteză a ştiinţei, Ed.Tehnică, Bucureşti, 1987. 17. Adelina Georgescu, Hydrodynamic stability theory, Mechanics: Analysis, 9, Kluwer, Dordrecht, 1985. 18. St.N.Săvulescu, Adelina Georgescu, H.Dumitrescu, M.Bucur, Cercetări matematice în teoria modernă a stratului limită, Ed. Academiei, Bucureşti, 1981. 19. Adelina Georgescu, Teoria stabilităţii hidrodinamice, Ed. Ştiinţifică şi Enciclopedică, Bucureşti, 1976.

B. PAPERS

2010

1. Limit cycles by Finite Element Method for a one - parameter dynamical system associated to the Luo - Rudy I model, ROMAI J., this issue, 27-41. (with C. L. Bichir, B. Amuzescu, Gh. Nistor, et al.). 2. A linearization principle for the stability of the chemical equilibrium of a binary mixture, ROMAI J., this issue, 131-138 (with L. Palese). 3. Asymptotic waves as solutions of nonlinear PDEs deduced by double-scale method in viscoanelastic media with memory, ROMAI J., this issue, 139-154 (with L. Restuccia). 4. An application of double-scale method to the study of non-linear dissipative waves in Jeffreys media, submitted to Math. and Its Appl. (with L. Restuccia). 5. On the nonlinear stability of a binary mixture with chemical surface reactions, submitted to Math. and Its Appl. (with L. Palese). 6. Stability and self-sustained oscillations in a ventricular cardiomyocyte model, submitted to Interdisciplinary Sciences – Computational Life. (with C. L. Bichir, B. Amuzescu, M. Popescu, et al.). 7. Limit points and Hopf bifurcation points for a one - parameter dynamical system associated to the Luo - Rudy I model, to be published (with C. L. Bichir, B. Amuzescu, Gh. Nistor, et al.). 2009

8. Further results on approximate inertial manifolds for the FitzHugh-Nagumo model, Atti dell’ Accad. Peloritana dei Pericolanti, Scienze FMN, vol LXXXVII, nr. 2. (2009). (with C.-S. Nartea). 9. On the stability bounds in a problem of convection with uniform internal heat source, Atti dell’ Accad. Peloritana dei Pericolanti, Scienze FMN, acceptata. (with F.-I. Dragomirescu).

10. Degenerated Bogdanov-Takens bifurcations in an immuno-tumor model, Atti dell’ Accad. Peloritana dei Pericolanti, Scienze FMN, vol LXXXVII, nr. 1. (2009). (with M. Trifan). 11. Stability criteria for quasigeostrofic forced zonal flows ; I. Asymptotically vanishing linear perturbation energy, ROMAI J. 5, 1 (2009). 63-76 (with L. Palese). 12. Linear stability results in a magnetothermoconvection problem, An St. Univ. Ovidius Constanta, 17, 3 (2009), 119-129. (with F.-I. Dragomirescu). 13. Polynomial based methods for linear nonconstant coefficients eigenvalue problems, Proceedings of the Middle Volga Mathematical Society, 11, 2 (2009), 33-41. (with F.-I. Dragomirescu). 14. Thermodynamics of fluids, SIMAI e-Lecture Notes, Vol. 2 (2009), 1-26.

2008

15. A closed-form asymptotic solution of the FitzHugh-Nagumo model, Bul. Acad. St. Rep. Moldova, Seria Mat., 2 (2008), 24-34. (with Gh. Nistor, M.- N. Popescu, D. Popa). 16. A closed-form asymptotic solution of the FitzHugh-Nagumo model, Bul. Acad. St. Rep. Moldova, Seria Mat., 2 (2008), 24-34. (with Gh. Nistor, M.- N. Popescu, D. Popa). 17. Application of two spectral methods to a problem of convection with uniform internal heat source, Journal of Mathematics and Applications, 30 (2008), 43-52. (with F.-I. Dragomirescu). 18. A linear magnetic Bénard problem with Hall effect. Application of the Budiansky-DiPrima method, Trudy Srednevoljckogo Matematichescogo Obshchestva, Saransk, 10, 1 (2008), 294-306. (with L. Palese). 19. Determination of asymptotic waves in Maxwell media by double-scale method, Technische Mechanik, 28, 2 (2008), 140-151. (with L. Restuccia). 20. Approximate limit cycles for the Rayleigh model, ROMAI J. 4, 2 (2008), 73- 80. (with M. Sterpu, P. Băzăvan). 21. Lyapunov stability analysis in Taylor-Dean systems, ROMAI J. 4, 2 (2008), 81-98. (with F.-I. Dragomirescu).

2007

22. Analytical versus numerical results in a microconvection problem, Carpathian J. Math., 23, 1-2 (2007), 81-88. MR 2305839 (with F.-I. Dragomirescu) . 23. Relaxation oscillations and the “canard” phenomenon in the FitzHugh- Nagumo model, Cap. 4 in Recent Trends in Mechanics, 1, Ed. Academiei Romane, Bucuresti, 2007, 82-106 (with M.-N. Popescu, Gh. Nistor, D. Popa)

24. Dynamical approach in biomathematics, ROMAI J., 2, 2 (2007), 63-76. (with L. Palese, G. Raguso).

2006

25. Some results on dynamics generated by Bazykin model, Atti Accad. Peloritana dei Pericolanti, Scienze FMN, 84, C1A0601003 (2006), 1- 10.(with R. - M. Georgescu). 26. A linear magnetic Bénard problem with tensorial electrical conductivity, Bolletttino U.M.I. (8) 9-B (2006), 197-214. (Prima data publicata ca Rapp. Int. Dipto. Mat. Univ. Bari, 17 / 03.) (with L. Palese, A. Redaelli) 27. Further study of a microconvection model using the direct method, ROMAI J., 2, 1 (2006), 77-86. (with I. Dragomirescu). 28. Bifurcations and dynamics in the lymphocytes-tumor model, The 7th Congress of SIMAI 2004, Venezia, Italy, 2004. International Conference on Mathematical Models and Methods in Biology and Medicine -MMMBM 2005, Bedlewo, Poland, May 29-June 3, 2005, trimisa la Atti Accad. Peloritana dei Pericolanti, Scienze FMN. (with M. Trifan) . 29. Asymptotic waves from the point of view of double-scale method, Atti Accad. Peloritana dei Pericolanti, Scienze FMN, 84, C1A0601005 (2006), 1- 9. (with L. Restuccia) . 30. Linear stability bounds in a convection problem for variable gravity field, Bul. Acad. St. Rep. Moldova, Mat., 3(2006), 51-56. (with I. Dragomirescu) . 31. Neutral manifolds in a penetrative convection problem. I. Expansions in Fourier series of the solutions, Sci. Annals of UASVM Iasi, 49, 2 (2006), 19-32. MR 2300509 (with A. Labianca) . 32. Mathematical models in biodynamics, Sci. Annals of UASVM Iasi, 49, 2 (2006), 361-371.

2005

33. The static bifurcation in the Gray – Scott model, Rev. Roum. Sci. Tech. – Méc. Appl., 50, 1-3 (2005), 3-13. (with R. Curtu) . 34. A nonlinear hydromagnetic stability criterion derived by a generalized energy method, Bul. Acad. St. Rep. Moldova, Seria Mat., 1(47) (2005), 85- 91. (with C.-L. Bichir, L. Palese) . 35. A direct method and its application to a linear hydromagnetic stability problem, ROMAI J., 1, 1 (2005), 67-76. (with L. Palese, A. Redaelli) . 36. Sets governing the phase portrait (approximation of the asymptotic dynamics), ROMAI J., 1, 1 (2005), 83-94. (with S.-C. Ion).

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37. Application of the direct method to a microconvection model, Acta Universitatis Apulensis, Alba Iulia, Mathematics - Informatics, 10 (2005), 131-142. ZB 1113.65076, MR 2240129. (with F.- I. Dragomirescu). 38. Normal forms and unfoldings: a comparative study, Sci. Annals of UASVM Iaşi, 48, 2 (2005), 15-26 (with E. Codeci) . 39. Bifurcation of planar vector fields. I. Normal forms "at the point". One zero eigenvalue, Bul. Şt. Univ. Piteşti, Ser. Mec. Apl., 1 (12) (2005), 41-47. (with D. Sârbu ) . 40. Bifurcation of planar vector fields. II. Normal forms "at the point". Hopf and cups cases, Bul. Şt. Univ. Piteşti, Ser. Mec. Apl., 1 (12) (2005), 49-55. (with D. Sârbu ). 41. Continuity of characteristics of a thin layer flow driven by a surface tension gradient, ROMAI J., 1, 2 (2005), 11-16. (with E. Borsa). 2004

42. Bifurcation in the Goodwin model I, Rev. Roum. Sci. Tech. – Méc. Appl., 49, 1-6 (2004), 13-16. (with N. Giurgiţeanu, C. Rocşoreanu). 43. Curba valorilor de bifurcaţie Hopf pentru sisteme dinamice plane, Bul. St., Seria Mec. Apl., 10 (2004), 55-62. (with E. Codeci) . 44. Dynamic bifurcation diagrams for some models in economics and biology, Acta Universitatis Apulensis, Alba Iulia, Mathematics-Informatics, 8 (2004), 156-161. 45. On instability of the magnetic Bénard problem with Hall and ion-slip effects, Intern. J. Engng. Sci., 42 (2004), 1001-1012. (with L. Palese) . 46. Liapunov method applied to the anisotropic Bénard problem, Math. Sci. Res. J., 8, 7 (2004), 196-204. (with Lidia Palese) . 2003

47. A Lie algebra of a differential generalized FitzHugh – Nagumo system, Bul. Acad. Şt., Rep. Moldova, Seria Mat. 1 (41) (2003), 18-30. (with M. Popa, C. Rocşoreanu) 48. Approximation of pressure perturbations by FEM, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 9 (2003), 31-36. (with C. - L. Bichir) . 49. Global bifurcations for FitzHugh-Nagumo model, Dynamical Systems and Applications, Trends in Mathematics: Bifurcations, Symmetry and Patterns, Birkhäuser, Basel, 2003, 197-202. (with C. Rocşoreanu, N. Giurgiţeanu) . 50. Static and dynamic bifurcation of nonlinear oscillators, Bul. Şt. Univ. Piteşti, Seria Mec. Apl., 1, 7 (2003), 133-138. 51. A Lorenz-like model for the horizontal convection flow, Int. J. Non-Linear Mech., 38 (2003), 629-644. (with E. Bucchignani, D. Mansutti) . 52. Bifurcation in biodynamics, Sci. Annals of UASVM Iasi, 46, 2 (2003), 15- 34. 53. Numerical integration of the Orr-Sommerfeld equation by wavelet methods, Bul. St. Univ. Pitesti, Seria Mat.-Inf., 9 (2003), 25-30. (with L. Bichir). xiii

2002

54. Codimension - one bifurcations for a Rayleigh model, Bul. Acad. Şt. Rep. Moldova, Seria Mat., 1 (38), (2002), 69 – 76. (with M Sterpu). 55. Static bifurcation diagram for a microeconomic model, Bul. Acad. Şt. Rep. Moldova, 3 (40) (2002), 21-26. (with L. Ungureanu, M. Popescu). 56. Improved criteria in convection problems in the presence of thermodiffusive conductivity, Analele Univ. Timişoara, 40, 2 (2002), 49-66. (with L. Palese) . 57. Existence and regularity of the solution of a problem modelling the Bénard problem, Mathematical Reports, 4 (54), 1 (2002), 87-102. (with A. – V. Ion). 58. Degenerated Bogdanov – Takens points in an advertising model, Analele Universităţii de Vest, Seria Economie; Sudiile de Economie, Timişoara, (with L. Ungureanu). 59. Global bifurcations in an advertising model, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., (with Laura Ungureanu, Liviu Ungureanu) . 60. Stability criteria for quasigeostrophic forced zonal flows. I. Asymptotically vanishing linear perturbation energy, Magnetohydrodynamics: an International J. (with L. Palese). 61. Domains of attraction for a model in enzimology, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 8 (2002), 49-58. (with R. Curtu) . 62. Heteroclinic bifurcations for the FitzHugh – Nagumo system, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 8 (2002). (with C. Rocşoreanu, N. Giurgiţeanu). 63. Topological type of some nonhyperbolic equilibria in a problem of microeconomic dynamics, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 8 (2002). (with L. Ungureanu, M. Popescu). 64. On the Misra-Progogine-Courbage theory of irreversibility, Mathematica, 44 (67), 2 (2002), 215-231. (with N. Suciu). 65. Non – Newtonian solution and viscoelastic constitutive equations, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 5 (2000). (with C. Chiujdea). 66. Normal form for the degenerated Hopf bifurcation in an economic model, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., 5 (2000). (with L. Ungureanu). 67. k > 3 order degenerated Bautin bifurcation and Hopf bifurcation in a mathematical model of economical dynamics, Bul. Şt. Univ. Piteşti, Seria Mat. – Inf., (2002). (with L. Ungureanu, M. Popescu). 68. Static bifurcation diagram for a mathematical model governing the capital of a firm, Bul. Şt. Univ. Piteşti, Seria Mat.- Inf, 8 (2002), 177-181. (with L. Ungureanu). 69. Concavity of the limit cycles in the FitzHugh-Nagumo model, An. Şt. Univ. Al.I.Cuza, Iaşi, Mat. (N. S.), 47, 2 (2001), 287-298, (2002). (with C. Rocşoreanu, N. Giurgiţeanu).

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2001

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89. Set of attraction of certain initial data in a nonlinear diffusion problem, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 235-261 (with H. Schwarze, H. Vereecken, U. Jaekel). 90. Coincidence of the linear and nonlinear stability bounds in a horizontal thermal convection problem, Intern. J. Nonlin. Mech., 34, 4 (1999), 603-613. (with D. Mansutti). 91. New types of codimension-one and-two bifurcations in the plane, Inst. Matem. Acad. Rom, Preprint No. 12/1999. (with C. Rocşoreanu, N. Giurgiţeanu). 92. Regimes with two or three limit cycles in the FitzHugh-Nagumo system, ZAMM 79, Supplement 2 (1999). (with C. Rocşoreanu, N. Giurgiţeanu). 93. Asymptotic analysis of solute transport with linear nonequilibrium sorption in porous media, Transp. Porous Media, 36, 2 (1999), 189-210 (with H. Vereecken, U. Jaekel). 94. Symmetry of the solution of the nonlinear Reynolds equation describing mechanical face seals, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 333-343. (with B. Nicolescu, N. Popa). 95. Hopf bifurcation and canard phenomenon in the FitzHugh-Nagumo model, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 217-233. (with C. Rocşoreanu, N. Giurgiţeanu). 96. Convecţia termică cu efect Marangoni. Condiţii de echilibru, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 345-350. (with Gh. Nistor). 97. Applications of coarse-grained and stochastic averages to transport processes in porous media, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 435-445. (with N. Suciu, C. Vamos, U. Jaekel, H. Vereeken). 98. Modelul continuu multiplicator-accelerator. II. Cazul liniar pentru anumite valori negative ale parametrilor şi cazul neliniar, Bul. Şt. Univ. Piteşti, Seria Matematică şi Informatică, 3 (1999), 263-266. (with C. Georgescu).

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1992

136. Aspecte ale modelării stratului limită al atmosferei. I, Stud. Cerc. Mec. Apl., 51, 1 (1992) ), 25-41. 137. Studiul calitativ al ecuaţiilor diferenţiale, St. Cerc. Mec. Apl., 51, 3 (1992), 317-326. 138. Models of asymptotic approximation governing the atmospheric motion over a low obstacle, Stud. Cerc. Mat., 44, 3 (1992), 237-252. (with G. Marinoschi). 139. Linear stability of a turbulent flow of Maxwell fluids in pipes, Rev. Roum. Math. Pures Appl., 37, 7 (1992), 579-586 (with C. Chiujdea, R. Florea). 140. Bifurcation problems in linear stability of continua, Quaderni. Dipto. di Mat. Univ. Bari, 1 (1992). (with I. Oprea). 141. Aspecte ale modelării stratului limită al atmosferei. II, Stud. Cerc. Mec. Apl., 51, 2 (1992). xix

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146. Models of asymptotic approximation for synoptical flows, Zeitschrift für Meteorologie, 40, 1 (1990), 14-20. (with C. Vamoş). 147. Neutral stability curves for a thermal convection problem. II. The case of multiple solutions of the characteristic equation, Acta Mechanica, 81 (1990), 115-119. (with I. Oprea). 148. Metode numerice în teoria bifurcaţiei. III. Soluţii periodice, Stud. Cerc. Fiz., 42, 1 (1990), 117-125. (with I. Oprea). 149. Scenarii de turbulenţă în cadrul haosului determinist, Stud. Cerc. Mec. Apl., 49, 4 (1990), 413-417. (with C. Vamoş, N. Suciu). 150. Asimptote oblice din punctul de vedere al aproximaţiei asimptotice, Gaz. Mat. M, 2 (1990), 58-60. 151. On a hydrodynamic-social analogy, Rev. Roum. Philos. Logique, 34, 1-2 (1990), 100-102. 1989

152. Bifurcation manifolds in a multiparametric eigenvalue problem for linear hydromagnetic stability theory, Mathematica - Anal. Numér. Théor. Approx., 18, 2 (1989), 123-138 (with I. Oprea, C. Oprea). 153. Model de aproximaţie asimptotică de ordinul patru pentru ecuaţiile meteorologice primitive când numărul Rossby tinde la zero, St. Cerc. Meteorolgie, 3 (1989), 13-21 (with C. Vamoş). 154. Filtred equations as an asymptotic approximation model, Meteorology and Hydrology, 19, 2 (1989), 21-22 (with C. Vamoş). 155. Boundary layer separation I. Bubbles on leading edges, Rev. Roum. Sci. Tech.-Méc. Appl., 34, 5 (1989), 509-525, (with H. Dumitrescu, Al. Dumitrache). 156. Stabilitatea mişcării lichidelor pe un plan înclinat, Stud. Cerc. Mec. Apl., 48, 5 (1989). 157. Comparative study of the analytic methods used to solve problems in hydromagnetic stability theory, An. Univ. Bucureşti, Mat., 38, 1 (1989), 15- 20 (with A. Setelecan). 158. Lagrange and the calculus of variations, Noesis, 15 (1989), 29-35. xx

159. Fractalii şi unele aplicaţii ale lor, Stud. Cerc. Fiz., 41, 3 (1989), 269-288. 160. Suprafeţe neutrale bifurcate într-o problemă de inhibiţie a convecţiei termice datorită unui câmp magnetic, Stud. Cerc. Mec. Apl., 48, 3 (1989), 263-278. (with I. Oprea). 1988

161. Metode numerice în teoria bifurcaţiei. II. Soluţii staţionare în cazul infinit dimensional, Stud. Cerc. Fiz., 40, 1 (1988), 7-18. (with I. Oprea). 162. Bifurcation (catastrophe) surfaces in multiparametric eigenvalue problems in hydromagnetic stability theory, Bull. Inst. Politehn. Bucureşti, Ser. Construc. Maş, 50 (1988), 9-12 (with I. Oprea). 163. The bifurcation curve of characteristic equation provides the bifurcation point of the neutral curve of some elastic stability, Mathematica - Anal. Numér. Théor. Approx., 17, 2 (1988), 141-145 (with I. Oprea). 1987

164. Metode de rezolvare a unor probleme de valori proprii care apar în stabilitatea hidrodinamică liniară, St.Cerc.Fiz., 39, 1 (1987), 3-25 (with A. Setelecan). 165. Notă asupra unor probleme izoperimetrice în calculul elicei de randament maxim, St.Cerc.Mec.Apl., 46, 5 (1987), 478-482. 166. Exact solutions for some instability of Bénard type, Rev. Roum. Phys., 32, 4 (1987), 391-397. 1986

167. Proiectarea aerodinamică a elicei de randament maxim, St.Cerc. Mec.Apl., 45, 2 (1986), 129-141 (with H. Dumitrecsu, Al. Dumitrache). 168. Bifurcaţia stratului limită, BITNAV, 3 (1986), 148-149. 169. Metode numerice în teoria bifurcaţiei, Stud. Cerc. Fiz., 38, 10 (1986), 912- 924 (with I. Oprea). 1984

170. Echilibrul plasmei în sisteme toroidale şi stabilitatea sa macroscopică, St.Cerc.Fiz., 36, 1 (1984), 86-110. 1983

171. Neustanovivseesia ploscoe dvijenie tipa Puazeilia dlia jidkostei Rivlina- Eriksena, PMM, 47, 2 (1983), 342-344. (with S.S. Chetti). 172. Stabilitatea şi ramificarea în contextul sinergeticii, St.Cerc.Mec.Apl., 42, 2 (1983), 174-180. 1982

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174. Neutral stability curves for a thermal convection problem, Analele Univ. din Craiova, Secţia Mat. Fiz.-Chim., X (1982), 51-53 (with I. Oprea). 175. Characteristic equations for some eigenvalue problems in hydromagnetic stability theory, Mathematica, Cluj, 24 (47), 1-2 (1982), 31-41. 1981

176. On the nonexistence of regular solutions of a Blasius-like equation in the theory of the boundary layer of finite depth, Rev. Roum. Math. Pures Appl., 26, 6 (1981), 849-854 (with M. Moroianu). 177. . Recent results in fluid mechanics, Preprint 2, Univ. "Babeş - Bolyai", Fac. Mat., Cluj-Napoca, 1981. 178. On a Bénard convection in the presence of dielectrophoretic forces, J. Appl. Mech., 48 , 4 (1981), 980-981 (with O. Polotzka). 1980

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ROMAI J., 6, 2(2010), 1–14

ON SPECIAL DIFFERENTIAL SUBORDINATIONS USING MULTIPLIER TRANSFORMATION AND RUSCHEWEYH DERIVATIVE Alina Alb Lupa¸s Department of Mathematics and Computer Science, University of Oradea, Romania [email protected] Abstract In the present paper we define a new operator using the multiplier transformation and α α Ruscheweyh derivative. Denote by RIn,λ,l the operator given by RIn,λ,l : A → A, α n n RIn,λ,l f (z) = (1−α)R f (z)+αI (n, λ, l) f (z), for z ∈ U, where R f (z) denote the Ruscheweyh derivative, I (n, λ, l) f (z) is the multiplier transformation and An = { f ∈ H(U): f (z) = n+1 z + an+1z + ..., z ∈ U} is the class of normalized analytic functions with A1 = A. A certain subclass, denoted by RIn (δ, λ, l, α) , of analytic functions in the open unit disc is introduced by means of the new operator. By making use of the concept of differential subordination we will derive various properties and characteristics of the class RIn (δ, λ, l, α) . Also, several differential subordinations are established regardind the α operator RIn,λ,l. Keywords: differential subordination, convex function, best dominant, differential operator, Ruscheweyh derivative, multiplier transformations. 2000 MSC: 30C45, 30A20, 34A40. .

1. INTRODUCTION Denote by U the unit disc of the complex plane, U = {z ∈ C : |z| < 1} and H(U) the space of holomorphic functions in U. Let X∞ p j A (p, n) = { f ∈ H(U): f (z) = z + a jz , z ∈ U}, j=p+n with A (1, n) = An, A (1, 1) = A1 = A and

n n+1 H[a, n] = { f ∈ H(U): f (z) = a + anz + an+1z + ..., z ∈ U}, where p, n ∈ N, a ∈n C. o z f 00(z) Denote by K = f ∈ A : Re f 0(z) + 1 > 0, z ∈ U , the class of normalized convex functions in U. If f and g are analytic functions in U, we say that f is subordinate to g, written f ≺ g, if there is a function w analytic in U, with w(0) = 0, |w(z)| < 1, for all z ∈ U

1 2 Alina Alb Lupa¸s such that f (z) = g(w(z)) for all z ∈ U. If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (U) ⊆ g(U). Let ψ : C3 × U → C and h an univalent function in U. If p is analytic in U and satisﬁes the (second-order) diﬀerential subordination

ψ(p(z), zp0(z), z2 p00(z); z) ≺ h(z), for z ∈ U, (1) then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (1). A dominant eq that satisfies eq ≺ q for all dominants q of (1) is said to be the best dominant of (1). The best dominant is unique up to a rotation of U. Definition 1.1. (Ruscheweyh [3]) For f ∈ A, n ∈ N, the operator Rn is defined by Rn : A → A,

R0 f (z) = f (z) R1 f (z) = z f 0 (z) ... (n + 1) Rn+1 f (z) = z Rn f (z) 0 + nRn f (z) , for z ∈ U.

P∞ j n P∞ n j Remark 1.1. If f ∈ A, f (z) = z + j=2 a jz , then R f (z) = z + j=2 Cn+ j−1a jz , for z ∈ U. Definition 1.2. [6] For f ∈ A(p, n), p, n ∈ N, m ∈ N∪ {0}, λ, l ≥ 0, the operator Ip (m, λ, l) f (z) is defined by the following infinite series ! X∞ p + λ ( j − 1) + l m I (m, λ, l) f (z):= zp + a z j. p p + l j j=p+n

Remark 1.2. It follows from the above deﬁnition that

Ip (0, λ, l) f (z) = f (z),

0 (p + l) Ip (m + 1, λ, l) f (z) = p(1 − λ) + l Ip (m, λ, l) f (z) + λz Ip (m, λ, l) f (z) , for z ∈ U.

Remark 1.3. If p = 1, n = 1, we have A(1, 1) = A1 = A, I1 (m, λ, l) f (z) = I (m, λ, l) and

(l + 1) I (m + 1, λ, l) f (z) = [l + 1 − λ] I (m, λ, l) f (z) + λz (I (m, λ, l) f (z))0 ,

for z ∈ U. On special diﬀerential subordinations using multiplier transformation... 3

P∞ j Remark 1.4. If f ∈ A, f (z) = z + j=2 a jz , then ! X∞ 1 + λ ( j − 1) + l m I (m, λ, l) f (z) = z + a z j, l + 1 j j=2

for z ∈ U.

m Remark 1.5. For l = 0, λ ≥ 0, the operator Dλ = I (m, λ, 0) was introduced and studied by Al-Oboudi [5], which reduced to the S˘al˘agean diﬀerential operator S m = m I (m, 1, 0) [4] for λ = 1. The operator Il = I (m, 1, l) was studied recently by Cho and Srivastava [7] and Cho and Kim [8]. The operator Im = I (m, 1, 1) was studied δ by Uralegaddi and Somanatha [12], the operator Dλ = I (δ, λ, 0), with δ ∈ R, δ ≥ 0, was introduced by Acu and Owa [1]. Lemma 1.1. (Hallenbeck and Ruscheweyh [1, Th. 3.1.6, p. 71]) Let h be a convex function with h(0) = a, and let γ ∈ C\{0} be a complex number with Re γ ≥ 0. If p ∈ H[a, n] and 1 p(z) + zp0(z) ≺ h(z), for z ∈ U, γ then p(z) ≺ g(z) ≺ h(z), for z ∈ U, R γ z γ/n−1 where g(z) = nzγ/n 0 h(t)t dt, for z ∈ U. Lemma 1.2. (Miller and Mocanu [1]) Let g be a convex function in U and let h(z) = g(z) + nαzg0(z), for z ∈ U, where α > 0 and n is a positive integer. n n+1 If p(z) = g(0) + pnz + pn+1z + ..., for z ∈ U is holomorphic in U and

p(z) + αzp0(z) ≺ h(z), for z ∈ U, then p(z) ≺ g(z), for z ∈ U, and this result is sharp.

2. MAIN RESULTS α Deﬁnition 2.1. Let α, λ, l ≥ 0, n ∈ N. Denote by RIn,λ,l the operator given by α RIn,λ,l : A → A,

α n RIn,λ,l f (z) = (1 − α)R f (z) + αI (n, λ, l) f (z), for z ∈ U. P Remark 2.1. If f ∈ A, f (z) = z + ∞ a z j, then n j=2 j o α P∞ 1+λ( j−1)+l n n j RIn,λ,l f (z) = z + j=2 α l+1 + (1 − α) Cn+ j−1 a jz , for z ∈ U. 4 Alina Alb Lupa¸s

0 n Remark 2.2. For α = 0, RIn,λ,l f (z) = R f (z), where z ∈ U and for α = 1, 1 RIn,λ,l f (z) = I (n, λ, l) f (z), where z ∈ U, which was studied in [3]. α n For l = 0, we obtain RIn,λ,0 f (z) = RD1,α f (z) which was studied in [4] and for α n l = 0 and λ = 1, we obtain RIn,1,0 f (z) = Lα f (z) which was studied in [2]. α 0 0 For n = 0, RI0,λ,l f (z) = (1 − α) R f (z) + αI (0, λ, l) f (z) = f (z) = R f (z) = I (0, λ, l) f (z), where z ∈ U. Deﬁnition 2.2. Let δ ∈ [0, 1), α, λ, l ≥ 0 and n ∈ N. A function f ∈ A is said to be in the class RIn (δ, λ, l, α) if it satisﬁes the inequality α 0 Re RIn,λ,l f (z) > δ, for z ∈ U. (2)

Theorem 2.1. The set RIn (δ, λ, l, α) is convex. Proof. Let the functions X∞ j f j (z) = z + a jkz , for k = 1, 2, z ∈ U, j=2 be in the class RIn (δ, λ, l, α). It is suﬃcient to show that the function

h (z) = η1 f1 (z) + η2 f2 (z) is in the class RIn (δ, λ, l, α) with η1 andη2 nonnegative such that η1 + η2 = 1. P∞ j Since h (z) = z + j=2 η1a j1 + η2a j2 z , for z ∈ U, then ( ! ) X∞ 1 + λ ( j − 1) + l n RIα h (z) = z+ α + (1 − α) Cn η a + η a z j, for z ∈ U. n,λ,l l + 1 n+ j−1 1 j1 2 j2 j=2 (3) Diﬀerentiating (3) we obtain

∞ ( !n ) 0 X 1 + λ ( j − 1) + l RIα h (z) = 1+ α + (1 − α) Cn η a + η a jz j−1, n,λ,l l + 1 n+ j−1 1 j1 2 j2 j=2 for z ∈ U. Hence n o α 0 P∞ 1+λ( j−1)+l n n j−1 Re RIn,λ,lh (z) = 1 + Re η1 j=2 j α l+1 + (1 − α) Cn+ j−1 a j1z + n o P∞ 1+λ( j−1)+l n n j−1 +Re η2 j=2 j α l+1 + (1 − α) Cn+ j−1 a j2z . (4) Taking into account that f1, f2 ∈ RIn (δ, λ, l, α) we deduce  ( ! )   X∞ 1 + λ ( j − 1) + l n  Re η j α + (1 − α) Cn a z j−1 > η (δ − 1) , for k = 1, 2.  k l + 1 n+ j−1 jk  k j=2 (5) On special diﬀerential subordinations using multiplier transformation... 5

Using (5) we get from (4) α 0 Re RIn,λ,lh (z) > 1 + η1 (δ − 1) + η2 (δ − 1) = δ, for z ∈ U, which is equivalent that RIn (δ, λ, l, α) is convex. 1 0 Theorem 2.2. Let g be a convex function in U and let h (z) = g (z)+ c+2 zg (z) , where z ∈ U, c > 0. R z ( ) ( ) ( )( ) c+2 c ( ) If f ∈ RIn δ, λ, l, α and F z = Ic f z = zc+1 0 t f t dt, for z ∈ U, then α 0 RIn,λ,l f (z) ≺ h (z) , for z ∈ U, (6) implies α 0 RIn,λ,lF (z) ≺ g (z) , for z ∈ U, and this result is sharp. Proof. We obtain that Z z zc+1F (z) = (c + 2) tc f (t) dt. (7) 0 Diﬀerentiating (7), with respect to z, we have (c + 1) F (z) + zF0 (z) = (c + 2) f (z) and α α 0 α (c + 1) RIn,λ,lF (z) + z RIn,λ,lF (z) = (c + 2) RIn,λ,l f (z) , for z ∈ U. (8) Diﬀerentiating (8) we have

0 1 00 0 RIα F (z) + z RIα F (z) = RIα f (z) , for z ∈ U. (9) n,λ,l c + 2 n,λ,l n,λ,l Using (9), the diﬀerential subordination (6) becomes

0 1 00 1 RIα F (z) + z RIα F (z) ≺ g (z) + zg0 (z) . (10) n,λ,l c + 2 n,λ,l c + 2 If we denote α 0 p (z) = RIn,λ,lF (z) , for z ∈ U, (11) then p ∈ H [1, 1] . Replacing (11) in (10) we obtain 1 1 p (z) + zp0 (z) ≺ g (z) + zg0 (z) , for z ∈ U. c + 2 c + 2 Using Lemma 1.2 we have α 0 p (z) ≺ g (z) , for z ∈ U, i.e. RIn,λ,lF (z) ≺ g (z) , for z ∈ U, 6 Alina Alb Lupa¸s and g is the best dominant. Example 2.1. If f ∈ RD 1, 1, 0, 1 , then f 0 (z) + z f 00 (z) ≺ 3−2z implies F0 (z) + 1 2 3(1−z)2 R z 00 ( ) 1 ( ) 3 ( ) zF z ≺ 1−z , where F z = z2 0 t f t dt. 1+(2δ−1)z Theorem 2.3. Let h (z) = 1+z , where δ ∈ [0, 1) and c > 0. R z ( )( ) c+2 c ( ) If α, λ, l ≥ 0, n ∈ N and Ic f z = zc+1 0 t f t dt, for z ∈ U, then ∗ Ic [RIn (δ, λ, l, α)] ⊂ RIn δ , λ, l, α , (12)

R 1 c+1 ∗ ( )( ) t where δ = 2δ − 1 + c + 2 2 − 2δ 0 t+1 dt. Proof. The function h is convex and using the same steps as in the proof of Theorem 2.2 we get from the hypothesis of Theorem 2.3 that 1 p (z) + zp0 (z) ≺ h (z) , c + 2 where p (z) is deﬁned in (11). Using Lemma 1.1 we deduce that α 0 p (z) ≺ g (z) ≺ h (z) , i.e. RIn,λ,lF (z) ≺ g (z) ≺ h (z) , where Z Z c + 2 z 1 + (2δ − 1) t (c + 2)(2 − 2δ) z tc+1 g (z) = tc+1 dt = 2δ − 1 + dt. c+2 c+2 z 0 1 + t z 0 t + 1 Since g is convex and g (U) is symmetric with respect to the real axis, we deduce

Z 1 c+1 α 0 ∗ t Re RIn,λ,lF (z) ≥ minRe g (z) = Re g (1) = δ = 2δ−1+(c + 2)(2 − 2δ) dt. |z|=1 0 t + 1 (13) From (13) we deduce inclusion (12). Theorem 2.4. Let g be a convex function, g(0) = 1 and let h be the function h(z) = g(z) + zg0(z), for z ∈ U. If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination α 0 RIn,λ,l f (z) ≺ h(z), for z ∈ U, (14) then RIα f (z) n,λ,l ≺ g(z), for z ∈ U, z and this result is sharp. On special differential subordinations using multiplier transformation... 7

α Proof. By using the properties of operator RIn,λ,l, we have ( ! ) X∞ 1 + λ ( j − 1) + l n RIα f (z) = z + α + (1 − α) Cn a z j, for z ∈ U. n,λ,l l + 1 n+ j−1 j j=2 Consider X∞ ( !n ) 1 + λ ( j − 1) + l n j z + α + (1 − α) C a jz α l + 1 n+ j−1 RI f (z) j=2 p(z) = n,λ,l = = z z 2 = 1 + p1z + p2z + ..., for z ∈ U. We deduce that p ∈ H[1, 1]. α Let RIn,λ,l f (z) = zp(z), for z ∈ U. By diﬀerentiating we obtain α 0 0 RIn,λ,l f (z) = p(z) + zp (z), for z ∈ U. Then (14) becomes p(z) + zp0(z) ≺ h(z) = g(z) + zg0(z), for z ∈ U. By using Lemma 1.2, we have RIα f (z) p(z) ≺ g(z), for z ∈ U, i.e. n,λ,l ≺ g(z), for z ∈ U. z

Theorem 2.5. Let h be an holomorphic function which satisfies the inequality zh00(z) 1 Re 1 + h0(z) > − 2 , for z ∈ U, and h(0) = 1. If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination α 0 RIn,λ,l f (z) ≺ h(z), for z ∈ U, (15) then RIα f (z) n,λ,l ≺ q(z), for z ∈ U, z R 1 z where q(z) = z 0 h(t)dt. The function q is convex and it is the best dominant. Proof. Let n n o α P∞ 1+λ( j−1)+l n j RI f (z) z + j=2 α l+1 + (1 − α) Cn+ j−1 a jz p(z) = n,λ,l = z z ( ! ) X∞ 1 + λ ( j − 1) + l n X∞ = 1 + α + (1 − α) Cn a z j−1 = 1 + p z j−1, l + 1 n+ j−1 j j j=2 j=2 8 Alina Alb Lupa¸s for z ∈ U, p ∈ H[1, 1]. α 0 0 Differentiating, we obtain RIn,λ,l f (z) = p(z)+zp (z), for z ∈ U and (15) becomes p(z) + zp0(z) ≺ h(z), for z ∈ U.

Using Lemma 1.1, we have

α Z RI f (z) 1 z p(z) ≺ q(z), for z ∈ U, i.e. n,λ,l ≺ q(z) = h(t)dt, for z ∈ U, z z 0 and q is the best dominant.

Theorem 2.6. Let g be a convex function such that g (0) = 1 and let h be the function h (z) = g (z) + zg0 (z), for z ∈ U. If α, λ, l ≥ 0, n ∈ N, f ∈ A and the diﬀerential subordination   zRIα f (z)0  n+1,λ,l   α  ≺ h (z) , for z ∈ U (16) RIn,λ,l f (z) holds, then α RIn+1,λ,l f (z) α ≺ g (z) , for z ∈ U, RIn,λ,l f (z) and this result is sharp. P Proof. For f ∈ A, f (z) = z + ∞ a z j we have n j=2 j o α P∞ 1+λ( j−1)+l n n j RIn,λ,l f (z) = z + j=2 α l+1 + (1 − α) Cn+ j−1 a jz , for z ∈ U. Consider P∞ 1+λ( j−1)+l n+1 n+1 j α z + α + (1 − α) C a jz RIn+1,λ,l f (z) j=2 l+1 n+ j p(z) = α = P n n o . RI f (z) ∞ 1+λ( j−1)+l n j n,λ,l z + j=2 α l+1 + (1 − α) Cn+ j−1 a jz We have 0 0 RIα f (z) RIα f (z) 0 n+1,λ,l n,λ,l p (z) = α − p (z) · α RIn,λ,l f (z) RIn,λ,l f (z) and we obtain   zRIα f (z)0 0  n+1,λ,l  p (z) + z · p (z) =  α  RIn,λ,l f (z) . Relation (16) becomes

p(z) + zp0(z) ≺ h(z) = g(z) + zg0(z), for z ∈ U. On special diﬀerential subordinations using multiplier transformation... 9

By using Lemma 1.2, we have

α RIn+1,λ,l f (z) p(z) ≺ g(z), for z ∈ U, i.e. α ≺ g(z), for z ∈ U. RIn,λ,l f (z)

Theorem 2.7. Let g be a convex function such that g(0) = 0 and let h be the function h(z) = g(z) + zg0(z), for z ∈ U. If α, λ, l ≥ 0, n ∈ N, f ∈ A and the diﬀerential subordination

α α (n + 1) RIn+1,λ,l f (z) − (n − 1) RIn,λ,l f (z) − ! l + 1 α n + 1 − I (n + 1, λ, l) f (z) − I (n, λ, l) f (z) ≺ h(z), for z ∈ U (17) λ holds, then α RIn,λ,l f (z) ≺ g(z), for z ∈ U. This result is sharp. Proof. Let

α n p(z) = RIn,λ,α f (z) = (1 − α)R f (z) + αI (n, λ, l) f (z) = (18) ( ! ) X∞ 1 + λ ( j − 1) + l n = z + α + (1 − α) Cn a z j = p z + p z2 + .... l + 1 n+ j−1 j 1 2 j=2 We deduce that p ∈ H[0, 1]. α n By using the properties of operators RIn,λ,l, R and I (n, λ, l), after a short calculation, we obtain p (z) + zp0 (z) = (n + 1) RIα f (z) − (n − 1) RIα f (z) − n+1,λ,l n,λ,l l+1 −α n + 1 − λ [I (n + 1, λ, l) f (z) − I (n, λ, l) f (z)]. Using the notation in (18), the diﬀerential subordination becomes

p(z) + zp0(z) ≺ h(z) = g(z) + zg0(z).

By using Lemma 1.2, we have

α p(z) ≺ g(z), for z ∈ U, i.e. RIn,λ,l f (z) ≺ g(z), for z ∈ U, and this result is sharp.

1+(2β−1)z Theorem 2.8. Let h(z) = 1+z be a convex function in U, where 0 ≤ β < 1. 10 Alina Alb Lupa¸s

If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisﬁes the diﬀerential subordination

α α (n + 1) RIn+1,λ,l f (z) − (n − 1) RIn,λ,l f (z) − ! l + 1 −α n + 1 − I (n + 1, λ, l) f (z) − I (n, λ, l) f (z) ≺ h(z), for z ∈ U, (19) λ then α RIn,λ,l f (z) ≺ q(z), for z ∈ U, ln(1+z) where q is given by q(z) = 2β − 1 + 2(1 − β) z , for z ∈ U. The function q is convex and it is the best dominant. Proof. Following the same steps as in the proof of Theorem 2.7 and considering α p(z) = RIn,λ,l f (z), the diﬀerential subordination (19) becomes 1 + (2β − 1)z p(z) + zp0(z) ≺ h(z) = , for z ∈ U. 1 + z By using Lemma 1.1 for γ = 1 and n = 1, we have p(z) ≺ q(z), i.e.,

Z z Z z α 1 1 1 + (2β − 1)t 1 RIn,λ,l f (z) ≺ q(z) = h(t)dt = dt = 2β−1+2(1−β) ln(z+1), z 0 z 0 1 + t z for z ∈ U.

Theoremh 2.9.i Let h be an holomorphic function which satisfies the inequality zh00(z) 1 Re 1 + h0(z) > − 2 , for z ∈ U, and h (0) = 0. If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination

α α (n + 1) RIn+1,λ,l f (z) − (n − 1) RIn,λ,l f (z) − ! l + 1 α n + 1 − I (n + 1, λ, l) f (z) − I (n, λ, l) f (z) ≺ h(z), for z ∈ U, (20) λ then α RIn,λ,l f (z) ≺ q(z), for z ∈ U, R 1 z where q is given by q(z) = z 0 h(t)dt. The function q is convex and it is the best dominant.

α α Proof. Using the properties of operator RIn,λ,l and considering p (z) = RIn,λ,l f (z), we obtain 0 α α p(z) + zp (z) = (n + 1) RIn+1,λ,l f (z) − (n − 1) RIn,λ,l f (z) − ! l + 1 α n + 1 − I (n + 1, λ, l) f (z) − I (n, λ, l) f (z) , for z ∈ U. λ On special diﬀerential subordinations using multiplier transformation... 11

Then (20) becomes p(z) + zp0(z) ≺ h(z), for z ∈ U. Since p ∈ H[0, 1], using Lemma 1.1, we deduce Z z α 1 p(z) ≺ q(z), for z ∈ U, i.e. RIn,λ,l f (z) ≺ q(z) = h(t)dt, for z ∈ U, z 0 and q is the best dominant. Theorem 2.10. Let g be a convex function such that g(0) = 1 and let h be the function h(z) = g(z) + zg0(z), for z ∈ U. If α, λ, l ≥ 0, n ∈ N, f ∈ A and the diﬀerential subordination (n + 1)(n + 2) (n + 1)(2n + 1) n2 RIα f (z) − RIα f (z) + RIα f (z) + z n+2,λ,l z n+1,λ,l z n,λ,l " # α (l + 1)2 − (n + 1)(n + 2) I (n + 2, λ, l) f (z) − z λ2   ¯ α 2 (l + 1 − λ) l + 1   − (n + 1)(2n + 1) I (n + 1, λ, l) f (z) + z  λ2  " # α (l + 1 − λ)2 − n2 I (n, λ, l) f (z) ≺ h(z), for z ∈ U (21) z λ2 holds, then α 0 [RIn,λ,l f (z)] ≺ g(z), for z ∈ U. This result is sharp. Proof. Let α 0 n 0 0 p(z) = RIn,λ,l f (z) = (1 − α) R f (z) + α (I (n, λ, l) f (z)) (22) ( ! ) X∞ 1 + λ ( j − 1) + l n = 1 + α + (1 − α) Cn ja z j−1 = 1 + p z + p z2 + .... l + 1 n+ j−1 j 1 2 j=2 We deduce that p ∈ H[1, 1]. α n By using the properties of operators RIn,λ,l, R and I (n, λ, l), after a short calculation, we obtain

p (z) + zp0 (z) = (n+1)(n+2) RIα f (z) − (n+1)(2n+1) RIα f (z) + z n+2,λ,l z n+1,λ,l 2 2 + n RIα f (z) + α (l+1) − (n + 1)(n + 2) I (n + 2, λ, l) f (z) − z n,λ,l z λ2 2(l+1−λ)(l¯+1) − α − (n + 1)(2n + 1) I (n + 1, λ, l) f (z) + z λ2 2 + α (l+1−λ) − n2 I (n, λ, l) f (z) . z λ2 12 Alina Alb Lupa¸s

Using the notation in (22), the diﬀerential subordination becomes

p(z) + zp0(z) ≺ h(z) = g(z) + zg0(z).

By using Lemma 1.2, we have α 0 p(z) ≺ g(z), for z ∈ U, i.e. RIn,λ,l f (z) ≺ g(z), for z ∈ U, and this result is sharp.

Example 2.2. If n = 1, α = 1, λ = 1, l = 0, f ∈ A, we deduce that f 0(z) + 3z f 00(z) + z2 f 000(z) ≺ g(z) + zg0(z), which yields that f 0(z) + z f 00(z) ≺ g(z), for z ∈ U. 1+(2β−1)z Theorem 2.11. Let h(z) = 1+z be a convex function in U, where 0 ≤ β < 1. If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisﬁes the diﬀerential subordination

(n + 1)(n + 2) (n + 1)(2n + 1) n2 RIα f (z) − RIα f (z) + RIα f (z) + z n+2,λ,l z n+1,λ,l z n,λ,l " # α (l + 1)2 − (n + 1)(n + 2) I (n + 2, λ, l) f (z) − z λ2   ¯ α 2 (l + 1 − λ) l + 1   − (n + 1)(2n + 1) I (n + 1, λ, l) f (z) + z  λ2  " # α (l + 1 − λ)2 − n2 I (n, λ, l) f (z) ≺ h(z), for z ∈ U, (23) z λ2 then α 0 RIn,λ,l f (z) ≺ q(z), for z ∈ U, ln(1+z) where q is given by q(z) = 2β − 1 + 2(1 − β) z , for z ∈ U. The function q is convex and it is the best dominant.

Proof. Following the same steps as in the proof of Theorem 2.10 and considering α 0 p(z) = RIn,λ,l f (z) , the diﬀerential subordination (23) becomes 1 + (2β − 1)z p(z) + zp0(z) ≺ h(z) = , for z ∈ U. 1 + z By using Lemma 1.1 for γ = 1 and n = 1, we have p(z) ≺ q(z), i.e., Z z Z z α 0 1 1 1 + (2β − 1)t 1 RIn,λ,l f (z) ≺ q(z) = h(t)dt = dt = 2β−1+2(1−β) ln(z+1), z 0 z 0 1 + t z for z ∈ U. On special diﬀerential subordinations using multiplier transformation... 13

Theoremh 2.12.i Let h be an holomorphic function which satisfies the inequality zh00(z) 1 Re 1 + h0(z) > − 2 , for z ∈ U, and h (0) = 1. If α, λ, l ≥ 0, n ∈ N, f ∈ A and satisfies the differential subordination (n + 1)(n + 2) (n + 1)(2n + 1) n2 RIα f (z) − RIα f (z) + RIα f (z) + z n+2,λ,l z n+1,λ,l z n,λ,l " # α (l + 1)2 − (n + 1)(n + 2) I (n + 2, λ, l) f (z) − z λ2   ¯ α 2 (l + 1 − λ) l + 1   − (n + 1)(2n + 1) I (n + 1, λ, l) f (z) + z  λ2  " # α (l + 1 − λ)2 − n2 I (n, λ, l) f (z) ≺ h(z), for z ∈ U, (24) z λ2 then α 0 RIn,λ,l f (z) ≺ q(z), for z ∈ U, R 1 z where q is given by q(z) = z 0 h(t)dt. The function q is convex and it is the best dominant. α α 0 Proof. Using the properties of operator RIn,λ,l and considering p (z) = RIn,λ,l f (z) , we obtain (n + 1)(n + 2) (n + 1)(2n + 1) n2 p(z)+zp0(z) = RIα f (z)− RIα f (z)+ RIα f (z) + z n+2,λ,l z n+1,λ,l z n,λ,l " # α (l + 1)2 − (n + 1)(n + 2) I (n + 2, λ, l) f (z) − z λ2   ¯ α 2 (l + 1 − λ) l + 1   − (n + 1)(2n + 1) I (n + 1, λ, l) f (z) + z  λ2  " # α (l + 1 − λ)2 − n2 I (n, λ, l) f (z) , for z ∈ U. z λ2 Then (24) becomes

p(z) + zp0(z) ≺ h(z), for z ∈ U.

Since p ∈ H[1, 1], using Lemma 1.1, we deduce Z z α 0 1 p(z) ≺ q(z), for z ∈ U, i.e. RIn,λ,l f (z) ≺ q(z) = h(t)dt, for z ∈ U, z 0 and q is the best dominant. 14 Alina Alb Lupa¸s References

[1] M. Acu, S. Owa, Note on a class of starlike functions, RIMS, Kyoto, 2006. [2] A. Alb Lupas¸, On special differential subordinations using S˘al˘agean and Ruscheweyh operators, Math. Inequ. Appl., 12, 4(2009), 781-790. [3] A. Alb Lupas¸, A special comprehensive class of analytic functions defined by multiplier transformation, J. Comput. Anal. Appl., 12, 2(2010), 387-395. [4] A. Alb Lupas¸, On special differential subordinations using a generalized S˘al˘agean operator and Ruscheweyh derivative, J. Comput. Anal. Appl., 2011 (accepted). [5] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, Ind. J. Math. Math. Sci., 27(2004), 1429-1436. [6] A. Catas¸,˘ On certain class of p-valent functions defined by new multiplier transformations, Pro- ceedings Book of the International Symposium on Geometric Function Theory and Applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250. [7] N.E. Cho, H.M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling, 37, 1-2(2003), 39-49. [8] N.E. Cho, T.H. Kim, Multiplier transformations and strongly close-to-close functions, Bull. Ko- rean Math. Soc., 40, 3(2003), 399-410. [9] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Marcel Dekker Inc., New York, 2000. [10] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109- 115. [11] G. St. Sal˘ agean,˘ Subclasses of univalent functions, Lecture Notes in Math., Springer, Berlin, 1013(1983), 362-372. [12] B.A. Uralegaddi, C. Somanatha, Certain classes of univalent functions, Current topics in analytic function theory, World. Sci. Publishing, River Edge, N.Y., (1992), 371-374. ROMAI J., 6, 2(2010), 15–26

A PRODUCT FORMULA APPROACH TO AN INVERSE PROBLEM GOVERNED BY NONLINEAR PHASE-FIELD TRANSITION SYSTEM. CASE 1D Tommaso Benincasa1, Costic˘aMoro¸sanu2 1University of Bologna, Italy 2University ”Alexandru Ioan Cuza” of Ia¸si,Romania Abstract In this paper we study an inverse problem, in one space dimension case, connected with the industrial solidiﬁcation process called casting wire, as an optimal control problem governed by nonlinear phase-ﬁeld system with nonhomogeneous Cauchy-Neumann boundary conditions. We prove the convergence of an iterative scheme of fractional steps type for the optimal control problem. Moreover, necessary optimality conditions are established for the approximating process. The advantage of such approach leads to the idea of a numerical algorithm in order to approximate the original optimal control problem.

Keywords: nonlinear parabolic systems, phase-ﬁeld models, optimality conditions, applications, time- dependent initial-boundary value problems, fractional steps method, inverse problems. 2000 MSC: 35K55, 49N15, 62P30, 65M12, 65M32.

1. INTRODUCTION; SETTING OF THE PROBLEM Phase-ﬁeld models, strongly studied in recent years, describe the phase transitions between two diﬀerent phases in a pure material by a system of nonlinear parabolic equations. These models can be viewed as extensions of the classical Stefan problem in two phases. Consequently, the interface boundary between the phases can be constructed from the so-called phase function and, phenomena associated with surface tension and supercooling are incorporated into the model. The mathematical literature concerning the optimal control problems associated with such models is in a deep process of development, as the models are suitable for many modern applications. One of them is the subject of this paper.

1.1. PROBLEM FORMULATION

Denote by Ω = (0, b1) ⊂ R, 0 < b1 < +∞. Let T > 0 and we set: Q = (0, T) × Ω, Σ0 = {(t, x) ∈ Q, t = f (x)}, Σ = (0, T) × {b1},

15 16 Tommaso Benincasa, Costic˘aMoro¸sanu where t = f (x) is considered to be the equation of the moving boundary separating the liquid and solid phases, 0 = f (b0), 0 < b0 < b1 (see Figure 1.1). Consider the following nonlinear parabolic system in one space dimension:   `  ρc ut + ϕt = kuxx, in Q,  2  (1.1)  1  τϕ = ξ2ϕ + (ϕ − ϕ3) + 2u, in Q, t xx 2a subject to the non-homogeneous Cauchy-Neumann boundary conditions

ux + hu = w(t), ϕx = 0 on Σ, (1.2)

ux = 0, ϕx = 0 on Σ0, (1.3) and to the initial conditions

u(0, x) = u0(x), ϕ(0, x) = ϕ0(x) on Ω0 = [b0, b1], (1.4) where u is the reduced temperature distribution, ϕ is the phase function used to distinguish between the phase of Ω, u0, ϕ0 : Ω −→ R are given functions, w : [0, T] −→ R is the boundary control (the temperature surrounding at x = b1), w ∈ U where U = {v ∈ L∞([0, T]), −R ≤ v(t) ≤ 0 a.e. t ∈ [0, T]}; the positive parameters ρ, c, τ, ξ, `, k, h, a, have the following physical meaning: ρ - is the density, c - is the heat capacity, τ - is the relaxation time, ξ - is the length scale of the interface, ` - denotes the latent heat, k - the heat conductivity, h - the heat transfer coeﬃcient and a is an probabilistic measure on the individual atoms (a depends on ξ).

Figure 1.1.Geometrical image of the elements in inverse problem (Pinv). The mathematical model (1.1), introduced by Caginalp [3], has been established in the literature as an extension of the classical two phase Stefan problem to capture the eﬀects of surface tension, supercooling, and superheating. A product formula approach to an inverse problem... 17

As regards the existence, it is known that under appropriate conditions on u0, ϕ0 2,1 ∞ and w, the state system (1.1)-(1.4) has a unique solution u, ϕ ∈ W = Wp (Q)∩L (Q), p > 3/2 (see Proposition 2.1 in [6]). Given the positive numbers d1, d2, we deﬁne: the pure liquid region : {(t, x) ∈ Q, u(t, x) > d2 and ϕ(t, x) ≥ 1 + d1}, the pure solid region : {(t, x) ∈ Q, u(t, x) < −d2 and ϕ(t, x) ≤ −1 − d1}, the separating region : {(t, x) ∈ Q, |u(t, x)| < d2, |ϕ(t, x)| ≤ 1 + d1}. and we set Q0 = (t, x) ∈ Q, f (x) ≤ t ≤ T . Consider the following inverse problem: ∞ (Pinv) Given Σ0 f ind the boundary control w ∈ L ([0, T]) such that Q0 is in the liquid region, Q1 = Q \ Q0 is in the solid region and a neighbourhood of Σ0 is the separating region between the liquid and the solid region.

(Pinv) is in general ”ill posed” and a common way to treat this inverse problem is to reformulate it as an optimal control problem with an appropriate cost functional. Consequently, we will concern in the present paper with an optimal control problem associated to the inverse problemZ (Pinv), namely: β (P) Minimize L (w) = (u(t, x) − δ )+ 2 · χ dtdx+ 0 2 2 Q0 Q Z ZT 1 1 + (ϕ(t, x) − 1 − δ )2 · χ dtdx + w2(t) dt, 2 1 Q0 2 Q 0 on all (u, ϕ) solution of the system (1.1)-(1.4) and for all w ∈ U β > 0 is a given constant. In the above statement we denoted by u+ the positive part of u, i.e. ( u, if u > 0, u+ = 0, if u ≤ 0. We point out that problem (P) is an optimal problem with boundary control w(t) depending on time variable t ∈ [0, T], being dictated by the industrial solidiﬁcation process like casting wire.

1.2. APPROXIMATING PROCESS We associate to the nonlinear system (1.1)-(1.4) the following approximanting scheme (ε > 0):   `  ρcuε + ϕε = kuε  t 2 t xx  ε  in Q0 = (t, x) ∈ Q, ε ≤ t ≤ T , (1.5)  1  τϕε = ξ2ϕε + ϕε + 2uε, t xx 2a 18 Tommaso Benincasa, Costic˘aMoro¸sanu

ε ε ε ε ux + hu = w(t), ϕx = 0 on Σ = [ε, T] × {b1}, (1.6) ε ε ε ux = 0, ϕx = 0 on Σ0 = {(t, x) ∈ Q, ε ≤ t ≤ T}, (1.7) ε ε ε u (ε, x) = u0(x) ϕ+(ε, x) = z(ε, ϕ−(ε, x)) on Ωε. (1.8) ε where z(ε, ϕ−(ε, x)) is the solution of the Cauchy problem:   1  z0(s) + z3(s) = 0, s ∈ (0, ε),  2a  (1.9)   ε ε z(0) = ϕ−(ε, x), ϕ−(0, x) = ϕ0(x), ε ε ε ε and ϕ+(ε, x) = limt↓ε ϕ (t, x), ϕ−(ε, x) = limt↑ε ϕ (t, x). The convergence and weak stability of the approximating scheme (1.5)-(1.9), in a more general case (w(t, x) in place of w(t)), was studied in the paper [2]. Corresponding to the approximating scheme (1.5)-(1.9), we will consider the approximating optimal control problem: Z β (Pε) Minimize Lε(w) = (uε(t, x) − δ )+ 2 · χ dtdx+ 0 2 2 Q0 Q Z ZT 1 1 + (ϕε(t, x) − 1 − δ )2 · χ dtdx + w2(t) dt, 2 1 Q0 2 Q 0 on all (uε, ϕε) solution of (1.5)-(1.9) corresponding to w ∈ U. The main result of the present paper (Theorem 2.1) says that problem (P) can be approximated for ε → 0 by the sequence of optimal control problems (Pε) and so the computation of the approximate boundary control w(t) can be substituted by computation of an approximate control of (Pε). In Section 2 we prove the convergence results regarding the sequence of optimal control problem (Pε). Such a convergence scheme was studied (for an optimal control problem governed by nonlinear parabolic variational inequalities) by Barbu [1]. For other works in this context see [6] and references therein. Necessary optimality conditions for the approximating process (Pε) (Theorem 3.1) and, a conceptual algorithm of gradient type are established in the last Section.

2. THE CONVERGENCE OF PROBLEM (Pε) The main result of this paper is ∗ ε Theorem 2.1. Let {wε} be a sequence of optimal controllers for problem (P ). Then ε lim inf L (w) = inf {L0(w); w ∈ U} (2.1) ε→0 0 and ∗ lim L0(wε) = inf {L0(w); w ∈ U}. (2.2) ε→0 A product formula approach to an inverse problem... 19

∗ Moreover, every weak limit point of {wε} is an optimal controller for problem (P). Remark 2.1. Theorem 2.1 amounts to saying that (Pε) approximates problem (P) ∗ ε and, an optimal controller {wε} of (P ) is a suboptimal controller for problem (P). The main ingredient in the proof of the Theorem 2.1 is the following Lemma.

∗ ε Lemma 2.1. If {wε} is a sequence of optimal controllers for problems (P ) then there exists {εn} −→ 0 such that

∗ ∗ ∞ wεn −→ w weakly star in L (Σ), (2.3)

∗ ∗ 2 1 uεn −→ u strongly in L ((0, T); H (Ω)), (2.4) ∗ ∗ 2 1 ϕεn −→ ϕ strongly in L ((0, T); H (Ω)), (2.5) ∗ ∗ ∗ ∗ ∗ wεn wεn ∗ where (uεn , ϕεn , wεn ) = (uεn , ϕεn , wεn ) is the solution to (1.5)-(1.8) corresponding to ∗ ∗ ∗ ∗ w∗ w∗ ∗ w = wεn and (u , ϕ , w ) = (u , ϕ , w ) is the solution to (1.1)-(1.4) corresponding to w = w∗. Proof. Details on the demonstration of this Lemma can be found in the work [6, Lemma 3.1, pp. 11]. We omit them.

We can now give the proof of Theorem 2.1.

∗ ε ∗ ∗ ∗ Proof. Let {wε} be an optimal controller for problem (P ) and let (uε, ϕε, wε) be the ∗ corresponding solution of (1.5)-(1.8) with w = wε. Lemma 2.1 above allows us to ∗ ∞ conclude that there exist w ∈ L ([0, T]) and {εn} such that relations (2.3)-(2.5) are valid. Since: Z β u −→ (u(t, x) − δ )+ 2 · χ dtdx, 2 2 Q0 ZQ 1 ϕ −→ (ϕ(t, x) − 1 − δ )2 · χ dtdx, 2 1 Q0 Q ZT 1 w −→ w2(t) dt 2 0 are convex continuous functions, it follows that these are weakly lower semicontinu- ous functions. Hence ∗ εn ∗ L0(w ) ≤ lim inf L (w ). (2.6) n−→∞ 0 εn ∗ ∗ Letw ¯ be an optimal controller for problem (P). Since wεn is an optimal controller for problem (Pεn ) it follows that

Lεn (w∗ ) ≤ L (w ¯ ε). 0 εn 0 20 Tommaso Benincasa, Costic˘aMoro¸sanu

w¯ ∗ w¯ ∗ w¯ ∗ w¯ ∗ 2 1 But (see (2.4) and (2.5)) uεn −→ u ϕεn −→ ϕ strongly in L ((0, T); H (Ω)) and so, the latter inequalities implies

εn ε ε lim L (w ¯ ) ≤ L0(w ¯ ). (2.7) n−→∞ 0 From (2.6)-(2.7) we get

L (w∗) ≤ lim inf Lεn (w∗ ) ≤ lim sup Lεn (w∗ ) ≤ L (w ¯ ∗). 0 0 εn 0 εn 0 n−→∞ n−→∞ Hence lim inf Lεn (w∗ ) = L (w ¯ ∗) = inf{L (w), w ∈ U} 0 εn 0 0 εn−→0 and then (2.1) holds. ∗ ∗ wε wε ∗ To prove (2.2) we set:u ¯ε = u ,ϕ ¯ ε = ϕ (we recall that wε is chosen to be optimal ε in (P )). On a subsequence {εn} we have

∗ 0 ∞ wε −→ w weakly star in L ([0, T]), 2 1 u¯εn −→ u strongly in L ((0, T), H (Ω)), 2 1 ϕ¯ εn −→ ϕ strongly in L ((0, T), H (Ω)), where (u, ϕ, w0) satisfy (1.1)-(1.4), i.e., (u, ϕ) = (uw0 , ϕw0 ). Therefore, we derive

0 L0(w ) ≤ inf P and, because {εn} was chosen arbitrarily, (2.2) follows. ∗ ε Now, taking into account that wε is an optimal controller for problem (P ), it follows that ε ∗ ε L0(wε) ≤ L0(w) ∀w ∈ U. On the other part, on the basis of relation (2.6), we can put

∗ ε ∗ L0(w ) ≤ lim inf L (wε) ε→0 0 and thus, along with previous inequality, we may conclude that

∗ ε L0(w ) ≤ lim L (w) ∀w ∈ U. ε→0 0 Consequently ∗ L0(w ) ≤ L0(w) ∀w ∈ U i.e., the weak limit point w∗ is a suboptimal controller for problem (P). This completes the proof of Theorem 2.1. A product formula approach to an inverse problem... 21 3. NECESSARY OPTIMALITY CONDITIONS IN (Pε) Let (uε, ϕε, w) be the solution of (1.5)-(1.8) and letw ˜ ∈ L∞[0, T]) be arbitrary but ﬁxed and λ > 0. Set wλ = w + λw˜ and let (uλ,ε, ϕλ,ε) be the solution of (1.5)-(1.8) corresponding to wλ, that is:   `  ρcuλ,ε + ϕλ,ε = kuλ,ε,  t 2 t xx  ε  in Q0, (3.1)  1  τϕλ,ε = ξ2ϕλ,ε + ϕλ,ε + 2uλ,ε, t xx 2a subject to non-homogeneous Cauchy-Neumann boundary conditions:

λ,ε λ,ε λ λ,ε ε ux + hu = w , ϕx = 0 on Σ , (3.2)

λ,ε λ,ε ε ux = 0, ϕx = 0 on Σ0, (3.3) and initial conditions:

λ,ε λ,ε λ u (ε, x) = u0(x), ϕ+ (ε, x) = z (ε, ϕ0(x)) on Ωε, (3.4)

λ where z (ε, ϕ0(x)) is the solution of the Cauchy problem:   0 1 3  zλ(s) + zλ(s) = 0, s ∈ (0, ε),  2a  (3.5)   λ λ,ε λ,ε z (0) = ϕ˜ − (ε, x), ϕ˜ − (0, x) = ϕ0(x). Subtracting (1.5)-(1.8) from (3.1)-(3.4) and dividing by λ > 0, we get

 λ,ε ε λ,ε ε λ,ε ε  u − u ` ϕ − ϕ u − u  ρc + = k ,  λ t 2 λ t λ xx in Qε, (3.6)  0  ϕλ,ε − ϕε ϕλ,ε − ϕε 1 ϕλ,ε − ϕε uλ,ε − uε  τ = ξ2 + + 2 , λ t λ xx 2a λ λ uλ,ε − uε uλ,ε − uε wλ − w ϕλ,ε − ϕε + h = , = 0 on Σε, (3.7) λ x λ λ λ x uλ,ε − uε ϕλ,ε − ϕε = 0, = 0 on Σε, (3.8) λ x λ x 0

uλ,ε(ε, x) − uε(ε, x) ϕλ,ε(ε, x) − ϕε (ε, x) zλ(ε, ϕ (x)) − z(ε, ϕ (x)) = 0, + + = 0 0 on Ω . λ λ λ ε (3.9) 22 Tommaso Benincasa, Costic˘aMoro¸sanu

Letting λ tend to zero in (3.6)-(3.9) we get the system in variation (3.10)-(3.13) below   `  ρcu˜ε + ϕ˜ ε = ku˜ε ,  t 2 t xx  ε  in Q0, (3.10)  1  τϕ˜ ε = ξ2ϕ˜ ε + ϕ˜ ε + 2˜uε, t xx 2a ε ε ε ε u˜ x + hu˜ = w˜ , ϕ˜ x = 0 on Σ , (3.11) ε ε ε u˜ x = 0, ϕ˜ x = 0 on Σ0, (3.12) ε ε u˜ (ε, x) = 0, ϕ˜ +(ε, x) = η(ε, x) on Ωε. (3.13) uλ,ε − uε whereu ˜ε = lim , etc., and λ−→0 λ zλ(ε, ϕ (x)) − z(ε, ϕ (x)) η(ε, x) = lim 0 0 = λ−→0 λ

λ ε = z (ε, ϕ0(x)) · ϕ˜ −(ε, x) + z˜(ε, ϕ0(x)) = z˜(ε, ϕ0(x)) with η(·, x) the solution of the Cauchy problem   3  η0(s, x) + z2(s, x)η(s, x) = 0, s ∈ (0, ε),  2a  (3.14)   ε η(0, x) = ϕ˜ −(ε, x), that is Zε 3 η(s, x) = exp − z(t, ·)2dt ϕ˜ ε (ε, x). (3.15) 2a − 0 We now introduce the adjoint state system. For this, the system (3.10) can be written in the abstract form: ε! ε! u˜ u˜ ε ε = A ε in Q0 ϕ˜ t ϕ˜ where (here ∆ϕ = ϕxx)  ` ` 1   1 k∆− − ξ2∆+   ρc τ 2τρc 2a    A =   ,  2 1 1   ξ2∆+  τ τ 2a ( ) ∂ψ ∂γ D(A) = (ψ, γ) ∈ H2(Ω) × H2(Ω); + hψ ∈ L2(∂Ω), = 0 . ∂ν ∂ν A product formula approach to an inverse problem... 23

Then  ` 2   1 k∆−   ρc τ τ  ∗   A =   ,  ` 1 1 1   − ξ2∆+ ξ2∆+  2τρc 2a τ 2a

( ) ∂ψ ∂γ 2ρc ∂ψ D(A∗) = (ψ, γ) ∈ H2(Ω) × H2(Ω); + hψ = 0, = . ∂ν ∂ν ` ∂ν Thus, the adjoint state system is

ε! ε! ∂ ε ! p ∗ p ∂uε L0(w) ε = −A ε + ∂ ε q t q ∂ϕε L0(w) i.e.  k ` 2  pε + pε − pε + qε = β(uε − d )+ · χ , Qε,  t xx 2 Q0 0  ρc τρc τ  (3.16)  pε + hpε = 0, on Σε,  x   ε ε p−(ε, x) = 0, p−(T, x) = 0 x ∈ ΩT ,   2 2  ε `ξ ε ` ε ξ ε 1 ε ε ε  qt − pxx − p + qxx + q =(ϕ −1−d1) · χQ , in Q ,  2τρc 4aτρc τ 2aτ 0 0    `  qε = pε, on Σε,  x 2ρc x  (3.17)   ε ε  qx = 0, on Σ0,  ε  Z  ε 3 2 ε ε  q−(ε, x) = exp z (t, ·)dt q+(ε, x), q−(T, x) = 0, x ∈ ΩT .  2a 0 Let us introduce the cost functional 1 Lε(w) = Lε(w) + I (w) 1 0 2 U where, as usually, IU(w) is the indicator function of the set U. If w∗ is an optimal controller of problem (Pε), then Lε(w∗ + λw˜ ) − Lε(w∗) 1 1 ≥ 0 ∀λ > 0. λ that leads to (letting λ tend to zero) 24 Tommaso Benincasa, Costic˘aMoro¸sanu R Z ε ε + ε ε β u˜ (u − d2) · χQ0 dtdx + ϕ˜ (ϕ − 1 − d1) · χQ0 dtdx+ (3.18) Q Q ZT ∗ 0 ∗ ∗ + w w˜ dt + IU(w , w˜ ) ≥ 0 ∀w˜ ∈ TU(w ). 0 ε ε Multiplying (3.16)1 byu ˜ and (3.17)1 byϕ ˜ , using integration by parts and Green’s formula, we get R R R ε ε k ε ε ` ε ε pt u˜ dt dx + ρc p u˜ xxdt dx − τρc p u˜ dt dx+ Qε Qε Qε 0 R 0 R 0 + 2 qεu˜εdt dx + k pεu˜ε − pεu˜ε dt dγ = τ ρc x x (3.19) Qε Σε R0 ε + ε = β (u − d2) · χQ0 u˜ dtdx, ε Q0

R R R ε ε `ξ2 ε ε ` ε ε qt ϕ˜ dt dx − 2τρc p ϕ˜ xxdt dx − 4aτρc p ϕ˜ dt dx+ Qε Qε Qε 0 R 0 R 0 `ξ2 ε ε ε ε ξ2 ε ε ε ε + 2τρc pxϕ˜ − p ϕ˜ x dt dγ + τ q ϕ˜ x − ϕ˜ qx dtdγ+ (3.20) Σε Σε 2 R R R + ξ qεϕ˜ ε dt dx + 1 qεϕ˜ εdt dx = (ϕε − 1 − d ) · χ ϕ˜ ε dtdx. τ xx 2aτ 1 Q0 ε ε ε Q0 Q0 Q0 ε ε Now we multiply (3.11) by p , (3.16)2 byu ˜ , by subtraction we get

ε ε ε ε ε pxu˜ − u˜ x p = −p w˜ . (3.21)

Adding (3.19)-(3.20) and taking into account (3.11)2, (3.17)2, (3.21), we obtain R R R ε ε ε ε k ε pt u˜ dt dx + qt ϕ˜ dt dx + ρc p wdt˜ dγ + ε ε ε Q0 Q0 Σ R h 2 i ε k ε `ξ ε ` ε ` ε + p u˜ xx − ϕ˜ xx − ϕ˜ − u˜ dt dx + Qε ρc 2τρc 4aτρc τρc R0 h i ε ξ2 ε 1 ε 2 ε + q τ ϕ˜ xx + 2aτ ϕ˜ + τ u˜ dt dx = Qε 0 R R ε + ε ε ε = β (u − d2) · χQ0 u˜ dtdx + (ϕ − 1 − d1) · χQ0 ϕ˜ dtdx, ε ε Q0 Q0 i.e., making use of equations in (3.10), the last relation leads to R R ε ε ε ε ε ε ε ε k ε pt u˜ + p u˜t + qt ϕ˜ + q ϕ˜ t dt dx + ρc p wdt˜ dγ = Qε Σε 0 R R ε + ε ε ε = β (u − d2) · χQ0 u˜ dtdx + (ϕ − 1 − d1) · χQ0 ϕ˜ dtdx, ε ε Q0 Q0 A product formula approach to an inverse problem... 25

By Fubini’s theorem and deﬁnition of distributional derivative, the latter relation give us Z Z Z k pεwdt˜ dγ = β (uε − d )+ · χ u˜ε dtdx + (ϕε − 1 − d ) · χ ϕ˜ ε dtdx, ρc 2 Q0 1 Q0 ε ε ε Σ Q0 Q0 and then (3.18) becomes

Z ZT k pεw˜ dtdγ + w∗w˜ dt + I0 (w∗, w˜ ) ≥ 0 ∀w˜ ∈ T (w∗) ρc U U Σε 0 or ZT k pε(s, b ) + w∗(s) w˜ (s) ds + I0 (w∗, w˜ ) ≥ 0 ∀w˜ ∈ T (w∗). ρc 1 U U 0 The last inequality is equivalent with

∗ −r(t) ∈ ∂IU(w ) a.p.t. (t, x) ∈ [0, T],

k ε where r(t) = ρc p (t, b1) + w(t), and thus we can conclude that ( 0, if r(t) > 0, w∗(t) = (3.22) −R, i f r(t) < 0.

Summing up, we have proved the following maximum principle for problem (Pε) Theorem 3.1. Let (u∗,ε, ϕ∗,ε, w∗) be optimal in problem (Pε). Then the optimal control is given by (3.22) where (pε, qε) satisfy along with u∗,ε, ϕ∗,ε the dual system (3.16) − (3.17). Now we will present a numerical algorithm of gradient type in order to compute the approximating optimal control stated by Theorem 3.1. Algorithm InvPHT1D (Inverse PHase Transition case 1D)

P0. Choose w(0) ∈ U and set iter= 0; Choose ε > 0; P1. Compute z(ε, ·) from (1.9); P2. Compute (uε,iter, ϕε,iter) from (1.5)-(1.8); P3. Compute (pε,iter, qε,iter) from (3.16)-(3.17); P4. For t ∈ [0, T], compute k riter(t) = · pε,iter(t, b ) + witer; ρc 1 26 Tommaso Benincasa, Costic˘aMoro¸sanu

P5. Set ( 0, if riter(t) > 0, w˜ iter(t) = −R, if riter(t) < 0.

P6. Compute λiter ∈ [0, 1] solution of the minimization process:

ε iter iter min {L0(λw + (1 − λ)w ˜ , λ ∈ [0, 1]; iter+1 iter iter Set w = λiterw + (1 − λiter)w ˜ ; P7. If k witer+1 − witer k ≤ η /* the stopping criterion */ then STOP else iter:= iter+1; Go to P1. In the above, the variable iter represents the number of iterations after which the ε ε algorithm InvPHT1D found the optimal value of the cost functional L0(w) in (P ). ε iter+1 ε iter Remark 3.1. The stopping criterion in P7 could be k L0(w ) − L0(w ) k ≤ η, where η is a prescribed precision.

4. CONCLUSIONS The main novelty brought by this work is that the computation of the approximate solution corresponding to the nonlinear system (1.1) is replaced with calculation of the approximate solution for an ordinary equation and a linear system (compare step P1 in [5] with the steps P1-P2 in present paper). Numerical implementation of the conceptual algorithm InvPHT1D remain an open problem. We wish only to draw attention to the type of boundary condition considered here (see (1.2)) namely that they fully cover industrial application proposed by us for numerical simulations - a matter for further investigation.

References [1] V. Barbu, A product formula approach to nonlinear optimal control problems, SIAM J. Control Optim., 26(1988), 496-520. [2] T. Benincasa, C. Moros¸anu, Fractional steps scheme to approximate the phase-field transition system with nonhomogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. and Optimiz., 30, 3-4(2009), 199-213. [3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal., 92(1986), 205-245. [4] 4. M. Heinkenschloss, F. Troltzsch,¨ Analysis of the Lagrange-SQP- Newton Method for the Con- trol of a Phase Field Equation, Control & Cybernetics, 28, 2(1999), 177–211. [5] C. Moros¸anu, Numerical approach of an inverse problem in the phase field equations, An. S¸t. Univ. ”Al.I. Cuza” Ias¸i, T XXXIX, s. I-a, 4(1993), 419-436. [6] C. Moros¸anu, Boundary optimal control problem for the phase-field transition system using fractional steps method, Control & Cybernetics, 32, 1(2003), 05-32. ROMAI J., 6, 2(2010), 27–39

LIMIT CYCLES BY FINITE ELEMENT METHOD FOR A ONE - PARAMETER DYNAMICAL SYSTEM ASSOCIATED TO THE LUO - RUDY I MODEL C˘at˘alinLiviu Bichir1, Adelina Georgescu2, Bogdan Amuzescu 3, Gheorghe Nistor 4, Marin Popescu 5, Maria-Luiza Flonta 6, Alexandru Dan Corlan 7, Istvan Svab 8 1 Rostirea Maths Research, Galat¸i, Romania, 2 Academy of Romanian Scientists, Bucharest, Romania, 3, 6, 8 Faculty of Biology, University of Bucharest, Romania, 4, 5 University of Pite¸sti,Romania, 7 Bucharest University Emergency Hospital, Romania, [email protected], [email protected], [email protected], [email protected], ﬂ[email protected], [email protected], [email protected] Abstract A one - parameter dynamical system is associated to the mathematical problem governing the membrane excitability of a ventricular cardiomyocyte, according to the Luo- Rudy I model. Limit cycles are described by the solutions of an extended system. A ﬁnite element method time approximation (FEM) is used in order to formulate the approximate problem. Starting from a Hopf bifurcation point, approximate limit cycles are obtained, step by step, using an arc-length-continuation method and Newton’s method. Some numerical results are presented.

Keywords: limit cycle, ﬁnite element method time approximation, Luo-Rudy I model, arc-length- continuation method, Newton’s method. 2000 MSC: 37N25 37G15 37M20 65L60 90C53 37J25.

1. INTRODUCTION The well-known Hodgkin-Huxley model of the squid giant axon [16] represented a huge leap forward in comparison with the earlier models of excitable systems built from abstract sets of equations or from electrical circuits including non-linear components, e.g. [33]. The pioneering work of Denis Noble’s group made the transition from neuronal excitability models, characterized by Na+ and K+ conductances with fast gating kinetics, to cardiomyocyte electrophysiology models, a field expanding steadily for over five decades [23]. Nowadays, complex models accurately reproduc- ing transmembrane voltage changes as well as ion concentration dynamics between various subcellular compartments and buffering systems are incorporated into de- tailed anatomical models of the entire heart [24]. The Luo-Rudy I model of isolated guinea pig ventricular cardiomyocyte [21] was developed in the early 1990s starting

27 28 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al from the Beeler-Reuter model [1]. It includes more recent experimental data related to gating and permeation properties of several types of ion channels, obtained in the late 1980s with the advent of the patch-clamp technique [22]. The model comprises only three time and voltage-dependent ion currents (fast sodium current, slow inward current, time-dependent potassium current) plus three background currents (time-independent and plateau potassium current, background current), their dynamics being described by Hodgkin-Huxley type equations. This apparent simplicity, compared to more recent multicompartment models, renders it adequate for mathematical analysis using methods of linear stability and bifurcation theory. Nowadays, there exist numerous software packages for the numerical study of finite - dimensional dynamical systems, for example MATCONT, CL−MATCONT, CL−MATCONTM [7], [15], AUTO [8]. In [19], [8], [7], [15]; the periodic boundary value problems used to locate limit cycles are approximated using orthogonal col- location method. Finite differences method is also considered. In this paper, limit cycles are obtained for the dynamical system associated to the Luo-Rudy I model by using finite element method time approximation (FEM).

2. LUO-RUDY I MODEL The mathematical problem governing the membrane excitability of a ventricular cardiomyocyte, according to the Luo-Rudy I model [21], is a Cauchy problem

u(0) = u0 , (1) for the system of ﬁrst order ordinary diﬀerential equations du = F(η, u) , (2) dt where u = (u1,..., u8) = (V,[Ca]i, h, j, m, d, f , X), η = (η1, . . . , η13) = (Ist, Cm, gNa, 8 13 gsi, gKp, gb,[Na]0,[Na]i,[K]0,[K]i, PRNaK, Eb, T), M = R , F : R × M → M, F = (F1,..., F8),

1 3 F1(η, u) = − [Ist + η3u3u4u5(u1 − ENa(η7, η8, η13)) η2 +η4u6u7(u1 − c1 + c2 ln u2) +gK(η10)Xi(u1)(u1 − EK(η7, η8, η9, η10, η11, η13))u8

+gK1(η10)K1∞(η9, η10, η13, u1)(u1 − EK1(η9, η10, η13)) +η5Kp(u1)(u1 − EK p(η9, η10, η13)) + η6(u1 − η12)] , F2(η, u) = −c3η4u6u7(u1 − c1 + c2 ln u2) + c4(c5 − u2) ,

F`(η, u) = α`(u1) − (α`(u1) + β`(u1))u` , ` = 3,..., 8 .

For the deﬁnition of variables V,[Ca]i, h, j, m, d, f , X, parameters Ist, Cm, gNa, gsi, gK p, gb,[Na]0,[Na]i,[K]0,[K]i, PRNaK, Eb, T, constants c1,..., c5, functions Limit cycles by Finite Element Method for a one - parameter dynamical system... 29 gK, ENa, EK, EK1, EKp, K1∞, Xi, K p, α`, β`, default values of parameters and initial values of variables in the Luo-Rudy I model, the reader is referred to [21]. The reader is also referred to [20] for the continuity of the model, and to [4] for the treatment of the vector ﬁeld F singularities. F is of class C2 on the domain of interest.

3. THE ONE - PARAMETER DYNAMICAL SYSTEM ASSOCIATED TO THE LUO - RUDY I MODEL We performed the study of the dynamical system associated with the Cauchy problem (1), (2) by considering only the parameter η1 = Ist and ﬁxing the rest of parameters. Denote λ = η1 = Ist and η∗ the vector of the ﬁxed values of η2, . . . , η13. Let F : R × M → M, F(λ, u) = F(λ, η∗, u), F = (F1,..., F8). Consider the dynamical system associated with the Cauchy problem (1), (3), where du = F(λ, u) . (3) dt The equilibrium points of this problem are solutions of the equation

F(λ, u) = 0 . (4)

The existence of the solutions and the number were established by graphical representation in [4], for the domain of interest. The equilibrium curve (the bifurcation diagram) was obtained in [4], via an arc-length-continuation method [13] and Newton’s method [12], starting from a solution obtained by solving a nonlinear least-squares problem [13] for a value of λ for which the system has one solution. In [4], the results are obtained by reducing (4) to a system of two equations in (u1, u2) = (V, [Ca]i). Here, we used directly (4).

4. EXTENDED SYSTEM METHOD FOR LIMIT CYCLES The extended system in (λ, T, u)   du  − TF(λ, u) = 0,  dτ   u(0) − u(1) = 0, (5)    R1  dw(t)  < u(t), > dt = 0, 0 dt was introduced, in [19], [8], [7], in order to locate limit cycles of a general problem (1), (3). Here, T is the unknown period of the cycle, w is a component of a known 30 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al reference solution (λ,ˆ Tˆ, w) of (5), and, in our case, tuv X8 X8 2 8 < u, v >= uivi and kuk = ui for u, v ∈ R . i=1 i=1

The system (5) becomes determined in a continuation process. In order to approximate and solve it by FEM time approximation, let us obtain the weak form of (5) in the sequel. For this, we consider the function spaces: dx X = {x ∈ L2(0, 1; R8); ∈ L2(0, 1; R8), dt x = (x1,..., x8), xi(0) = xi(1), i = 1,..., 8} . dv V = {v ∈ L2(0, 1; R); ∈ L2(0, 1; R), v(0) = v(1)} . dt

The weak form of (5) is the problem in (λ, T, u) ∈ R × R × X   Z1 Z1  dv(τ)  u (τ) dτ + T F (λ, u(τ))v(τ)dτ = 0,  i dτ i   0 0 ∀v ∈ V, i = 1,..., 8, (6)   Z1  dw(t)  < u(t), > dt = 0 .  dt 0 5. ARC-LENGTH-CONTINUATION METHOD FOR (6) Following the usual practice ([17], [18], [7], [8], [12], [13], [14], [15], [19], [25], [27], [28], [29]), we also use an arc-length-continuation method in order to formulate an algorithm to solve (6) approximatively. Glowinski ([13], following Keller [17], [18]) and Doedel ([8], where Keller’s name is also cited) chose a continuation equation written in our case as

Z1 du(t) dT dλ k k2dt + ( )2 + ( )2 = 1 , (7) ds ds ds 0 where s is the curvilinear abscissa. Let (λ0, u0) be a Hopf bifurcation point, ±β0i a pair of purely imaginary eigenval- 0 0 0 0 0 ues of of the Jacobian matrix DuF(λ , u ), and a nonzero complex vector g = gr +igi . (λ0, u0) is located on the equilibrium curve during a continuation procedure using Limit cycles by Finite Element Method for a one - parameter dynamical system... 31

0 0 0 0 0 8 8 8 some test functions ([19], [14], [7]). (λ , β , u , gr , gi ) ∈ R × R × R × R × R is the solution of the extended system ([27], [28], [29])

 F(λ, u)     D F(λ, u)g + βg   u r i   D F(λ, u)g − βg   u i r  = 0 , (8)  g − 1   r,k  gi,k where k is a ﬁxed index of gr and of gi, 1 ≤ k ≤ 8. To solve (6), the extended system formed by (6) and (7), parametrized by s, was considered. Let 4s be an arc-length step and λn u(λ4s), T n T(n4s), un u(n4s). We have the algorithm (following the cases from [13], [8], [28], [29]): Algorithm 1. 1. take the Hopf bifurcation point (λ0, u0) and T 0 = 2π/β0; 0 0 retain gr , gi ; 2. for n = 0, (λ1, T 1, u1) ∈ R × R × X is obtained ([8], [29]) by (13) below

1 Z X8 dφ (t) u1(t) i dt = 0 , (9) i dt 0 i=1 and Z1 8 X 1 0 (ui (t) − ui (t))φi(t) dt = 4s , (10) 0 i=1 where 0 0 φ(t) = sin(2πt)gr + cos(2πt)gi , (11) by using Newton’s method with the initial iteration

(u1)0(t) = u0 + 4s φ(t) , (T 1)0 = T 0 , (λ1)0 = λ0 . (12)

3. for n ≥ 1, assuming that (λn−1,T n−1, un−1), (λn,T n, un) are known, (λn+1, T n+1, un+1) ∈ R × R × X is obtained from

1 1 Z dv(τ) Z un+1(τ) dτ + T n+1 F (λn+1, un+1(τ))v(τ)dτ = 0 , (13) i dτ i 0 0 ∀v ∈ V, i = 1,..., 8,

1 Z X8 dun(t) un+1(t) i dt = 0, (14) i dt 0 i=1 32 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al

Z1 X8 un(t) − un−1(t) (un+1(t) − un(t)) i i dt+ i i 4s 0 i=1 T n − T n−1 λn − λn−1 +(T n+1 − T n) + (λn+1 − λn) = 4s,, (15) 4s 4s by using Newton’s method with the initial iteration

((λn+1)0, (T n+1)0, (un+1)0) = (λn, T n, un). (16)

6. NEWTON’S METHOD FOR THE STEPS OF ALGORITHM 1 In (15) (n ≥ 1), let us denote λ∗ = λn, T ∗ = T n, u∗ = un, λ∗∗ = (λn − λn−1)/4s, T ∗∗ = (T n − T n−1)/4s, u∗∗ = (un − un−1)/4s. We write (13), (9), (10) (the iteration n = 0) in the same general form as (13), (14), (15). So denote u∗ = u0, u∗∗ = φ and consider λ∗ = λ0, T ∗ = T 0, λ∗∗ = 0, T ∗∗ = 0 in (15) and consider u∗ = u0 = φ in (14). Each step of Algorithm 1, given (λ∗, T ∗, u∗), (λ∗∗, T ∗∗, u∗∗), calculates (λn+1, T n+1, un+1) ∈ R × R × X, n ≥ 0, by (13),

Z1 X8 du∗(t) un+1(t) i dt = 0 , (17) i dt 0 i=1 and

Z1 X8 n+1 ∗ ∗∗ n+1 ∗ ∗∗ n+1 ∗ ∗∗ (ui (t) − ui (t))ui (t) dt + +(T − T )T + (λ − λ )λ = 4s . (18) 0 i=1 Newton’s method applied (13), (17) and (18), for n ≥ 0, leads to:

let ((λ1)0,(T 1)0,(u1)0), given by (12), be an initial iteration (m = 0), if n = 0;

let ((λn+1)0,(T n+1)0,(un+1)0), given by (16), be an initial iteration (m = 0), if n ≥ 1;

calculate (λn+1, T n+1, un+1) as the solution of the algorithm: for m ≥ 0, ((λn+1)m+1,(T n+1)m+1,(un+1)m+1) = (λm+1, T m+1, um+1) ∈ R × R × X is obtained by Limit cycles by Finite Element Method for a one - parameter dynamical system... 33

Z1 Z1 dv(τ) um+1(τ) dτ + T m+1 F (λm, um(τ))v(τ)dτ+ i dτ i 0 0 Z1 m m m m+1 m+1 +T DFi(λ , u (τ))(λ , u (τ))v(τ)dτ = (19) 0 Z1 m m m m m = T DFi(λ , u (τ))(λ , u (τ))v(τ)dτ , ∀v ∈ V, i = 1,..., 8. 0 Z1 X8 du∗(t) um+1(t) i dt = 0, (20) i dt 0 i=1

Z1 X8 m+1 ∗∗ m+1 ∗∗ m+1 ∗∗ ui (t)ui (t) dt + T T + λ λ = (21) 0 i=1 Z1 X8 ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ = ui (t)ui (t) dt + T T + λ λ + 4s . 0 i=1 7. APPROXIMATION OF PROBLEM (19), (20), (21) BY FINITE ELEMENT METHOD TIME APPROXIMATION In order to perform this approximation, let us divide the interval [0, 1] in N + 1 subintervals K = K j = [t j, t j+1], 0 ≤ j ≤ N, where 0 = t0 < t1 < . . . < tN+1 = 1. The sets K represent a triangulation Th of [0, 1]. Let us approximate the spaces V and X by the spaces

Vh = {v : [0, 1] → R; v ∈ C[0, 1], v(0) = v(1), v|K ∈ Pk(K), ∀K ∈ Th},

8 Xh = {x : [0, 1] → R ; x = (x1,..., x8), xi ∈ Vh, i = 1,..., 8} , respectively, where Pk(K) is the space of polynomials in t of degree less than or equal to k defined on K, k ≥ 2. Let k = 2. An element K ∈ Th has three nodal points. To obtain a function uh ∈ Xh reduces to obtain a function vh ∈ Vh. In order to obtain a function vh ∈ Vh, we use a basis of functions of Vh. Let JK = {1, 2, 3} be the local numeration for the nodes of K, where 1, 3 correspond to t j, t j+1 respectively and 2 corresponds to a node between t j and t j+1. Let {ψi, i ∈ JK} be the local quadratic basis of functions on K corresponding to the local nodes. Let J = {1,..., 2N + 1} be the global numeration 34 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al for the nodes of [0, 1]. The two numerations are related by a matrix L whose elements are the elements j ∈ J. Its rows are indexed by the elements K ∈ Th (by the number of the element K in a certain fixed numeration with elements from the set {1,..., N}) and its columns, by the local numeration i ∈ JK, that is j = L(K, i). A function vh ∈ Vh is defined by its values v j from the nodes j ∈ J, X X vh(t) = v j ψi(t) , (22) K∈ Th i∈ JK , j=L(K,i) and a function uh ∈ Xh is defined by its values u j from the nodes j ∈ J, X X uh(t) = u j ψi(t) . (23) K∈ Th i∈ JK , j=L(K,i)

So, an unknown function uh = ((uh)1, ...,(uh)8) is reduced to the unknowns u j, u j = ((u j)1, ...,(u j)8), j ∈ J. In (19), (20), (21), approximate (λm+1, T m+1, um+1) ∈ R × R × X by (λm+1, T m+1, m+1 m+1 uh ) ∈ R × R × Xh. Taking uh = uh, uh given by (23), and v = ψ`, for all ` ∈ JK, for all K ∈ Th, we obtain the discrete variant of problem (19), (20), (21) as the 8·(2N+1) following problem in (λ, T, u1, ..., u2N+1) ∈ R × R × R , written suitable for the assembly process,  X  X Z Z  m+1 ψ` (τ) m+1 m m  (u )n ψ (τ) dτ + T Fi(λ , u (τ)) ψ dτ +  j i dτ h ` K∈ Th i∈ JK , j=L(K,i) K K X Z m m+1 m m + T < u j , (DuFn(λ , uh (τ))ψi(τ)) ψ` dτ > + i∈ JK , j=L(K,i) K  Z  m+1 m m  +λ (DλFn(λ , u (τ))) ψ dτ  = (24) h `  K   X  Z Z  m  m m m m m m  = T  (DuFn(λ , u (τ))u (τ)) ψ dτ + (DλFn(λ , u (τ))λ ) ψ dτ  ,  h h ` h `  K∈ Th K K n = 1,..., 8, m+1 m+1 u0 = u2N+1 , (25) Z X X du∗(τ) < um+1, ψ (τ) h dτ > = 0 , (26) j i dτ K∈ Th i∈ JK , j=L(K,i) K Limit cycles by Finite Element Method for a one - parameter dynamical system... 35

X X Z m+1 ∗∗ m+1 ∗∗ m+1 ∗∗ < u j , ψi(τ) uh (τ) dτ > +T T + λ λ = (27) K∈ Th i∈ JK , j=L(K,i) K

X Z ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ = < uh(τ), uh (τ) > dτ + T T + λ λ + 4s , K∈ Th K

for all ` ∈ JK, for all K ∈ Th. 8. NUMERICAL RESULTS Based on [30] and on the computer programs for [2] and [3], relations (24), (25), (26), (27) and the algorithm at the end of section 5 furnished the numerical results of this section. Let (λ0, u0) be the Hopf bifurcation point located during the construction of the equilibrium curve by a continuation procedure in [5]. 0 0 0 0 0 The solution (λ , β , u , gr , gi ) of (8), calculated in [5], is λ0 = −1.0140472901 , β0 = 0.0162886062, u0 = (−24.3132508542, 0.0034641214, 0.0, 0.0, 0.9176777444, 0.5025242162, 4920204612, 0.5071561613), 0 gr = (1.0, 0.0000468233, 0.0, 0.0, 0.0093195354, 0.0198748652, −0.0072420216, 0.0001706577), 0 gi = (0.0, 0.0000029062, 0.0, 0.0, −0.0000171311, −0.0118192370, 0.0136415789, −0.0017802907). 0 0 The eigenvalues of the Jacobian matrix DuF(λ , u ), calculated by the QR algorithm, are ± 0.0162886062 i, −8.8611865338, −0.1026761869, −0.0647560667, −0.0024565181, −1.7398266947, −0.2049715178. These data are considered in the step 1 of the algorithm at the end of section 5. We took Cm = 1, gNa = 23, gsi = 0.09, gK = 0.282, gK1 = 0.6047, gK p = 0.0183, Gb = 0.03921, [Na]0 = 140, [Na]i = 18, [K]0 = 5.4, [K]i = 145, PRNaK = 0.01833, Eb = −59.87, T = 310. In order to solve (6) numerically by the algorithm at the end of section 5 and by (24), (25), (26), (27), we performedR calculations using 4s = 1.0 and 500 iterations in the continuation process. Integrals f (τ) dτ were calculated using Gauss integration K formula with three integration points. Figure 1 and 2 present some results obtained using 20 elements K (41 nodes) (N = 20, J = {1,..., 41} in section 7). The curves of the projections of the limit cycles, on the planes indicates in ﬁgure, are plots generated from values calculated in the nodes, corresponding to a ﬁxed value of the parameter. Two projections of some limit cycles and of a part of the equilibrium curve (marked by ”N”) are presented in Fig. 1. The Hopf bifurcation point is marked by ”•”. 36 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al

0.9

0.8

0.7 f 0.6

0.5

0.4

0 −20 −40 0 −60 −1 −0.5 −80 −2 −1.5 V I st

0.6

0.5

0.4

X 0.3

0.2

0.1

0 −20 −40 −60 0 −80 −2 −1.5 −1 −0.5 V I st

Fig. 1. Two projections of limit cycles and of a part of the equilibrium curve (marked by ”N”). The Hopf bifurcation point is marked by ”•”. Limit cycles by Finite Element Method for a one - parameter dynamical system... 37

−3 −3 x 10 x 10 5 5 i

4 h 0 [Ca] 3 −5 −30 −20 −10 0 −30 −20 −10 0 V V −3 x 10 1 5 0.9 m j 0 0.8

0.7 −5 −30 −20 −10 0 −30 −20 −10 0 V V

0.8 0.8 f d 0.6 0.6 0.4 0.4 0.2 −30 −20 −10 0 −30 −20 −10 0 V V

0.55

X 0.5

0.45 −30 −20 −10 0 V

Fig. 2. Projections of two limit cycles calculated for Ist = −1.2000465026 and for Ist = −1.2000183729 (marked by ”x”) (20 elements, 41 nodes). 38 C. L. Bichir, A. Georgescu, B. Amuzescu, Gh. Nistor, M. Popescu, et al

In Fig. 2, there are represented the projections of the plots of two limit cycles calculated for Ist = −1.2000465026 (iteration 148) and for Ist = −1.2000183729 (iteration 248, marked by ”x” in ﬁgure). The results obtained are relevant from a biological point of view, pointing to unstable electrical behavior of the modeled system in certain conditions, translated into oscillatory regimes such as early afterdepolarizations [32] or self-sustained oscillations [4], which may in turn synchronize, resulting in life-threatening arrhythmias: premature ventricular complexes or torsades-de-pointes, degenerating in rapid poly- morphic ventricular tacycardia or ﬁbrillation [26]. Acknowlegdements: This research was partially supported from grant PNCDI2 61- 010 to M-LF by the Romanian Ministry of Education, Research, and Innovation.

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[14] W.J.F. Govaerts, Numerical methods for Bifurcations of Dynamical Equilibria, SIAM, Philadel- phia, 2000. [15] W. Govaerts, Yu. A. Kuznetsov, R. Khoshsiar Ghaziani, H.G.E. Meijer, Cl MatContM: A toolbox for continuation and bifurcation of cycles of maps, 2008, http://www.matcont.ugent.be/doc−cl−matcontM.pdf [16] A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544. [17] H.B.Keller, Numerical Solution of Bifurcation Eigenvalue Problems, in Applications in Bifurca- tion Theory, ed. by P.Rabinowitz, Academic, New York, 1977. [18] H.B.Keller, Global Homotopies and Newton Methods, in Recent Advances in Numerical Meth- ods, ed. by C. de Boor, G.H.Golub, Academic, New York, 1978. [19] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998. [20] L. Livshitz, Y. Rudy, Uniqueness and stability of action potential models during rest, pacing, and conduction using problem - solving environment, Biophysical J., 97 (2009), 1265-1276. [21] C.H. Luo, Y. Rudy, A model of the ventricular cardiac action potential. Depolarization, repolar- ization, and their interaction, Circ. Res., 68 (1991), 1501-1526. [22] E. Neher, B. Sakmann, Single-channel currents recorded from membrane of denervated frog muscle fibres, Nature, 260 (1976), 799-802. [23] D. Noble, Modelling the heart: insights, failures and progress, Bioessays, 24 (2002), 1155-1163. [24] D. Noble, From the Hodgkin-Huxley axon to the virtual hear, J. Physiol., 580 (2007), 15-22. Epub 2006 Oct 2005. [25] T.S.Parker, L.O.Chua, Practical Numerical Algorithms for Chaotic Systems, Springer, New York, 1989. [26] D. Sato, L. H. Xie, A. A. Sovari, D. X. Tran, N. Morita, F. Xie, H. Karagueuzian, A. Garfinkel, J. N. Weiss, Z. Qu , Synchronization of chaotic early afterdepolarizations in the genesis of cardiac arrhythmias, Proc. Natl. Acad. Sci. USA, 106 (2009), 2983-2988. Epub 2009 Feb 2913. [27] R. Seydel, Numerical computation of branch points in nonlinear equations, Numer. Math., 33 (1979), 339-352. [28] R. Seydel, Nonlinear Computation, invited lecture and paper presented at the Distinguished Ple- nary Lecture session on Nonlinear Science in the 21st Century, 4th IEEE International Workshops on Cellular Neural Networks and Applications, and Nonlinear Dynamics of Electronic Systems, Sevilla, June, 26, 1996. [29] R. Seydel, Practical Bifurcation and Stability Analysis, Springer, New York, 2010. [30] C. Taylor, T.G. Hughes, Finite Element Programming of the Navier-Stokes Equations, Pineridge Press, Swansea, U.K., 1981. [31] R.Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam, 1979. [32] D. X, Tran, D. Sato, A. Yochelis, J. N. Weiss, A. Garfinkel, Z. Qu, Bifurcation and chaos in a model of cardiac early afterdepolarizations, Phys. Rev. Lett., 102:258103 (2009). Epub 252009 Jun 258125. [33] B. Van der Pol, J. Van der Mark, The heartbeat considered as a relaxation oscillation and an electrical model of the heart, Phil. Mag. (suppl.), 6 (1928), 763-775.

ROMAI J., 6, 2(2010), 41–53

THE FACTORIZATION OF THE RIGHT PRODUCT OF TWO SUBCATEGORIES Dumitru Botnaru, Alina T¸urcanu Technical University of Moldova, Chi¸sin˘au,Republic of Moldova [email protected], alina−[email protected] Abstract In the category of locally convex spaces the right product of two subcategories is a reflective subcategory. In the topological completely regular spaces a similar property is not always true. The factorization of this product according to a structure of factorization leads always to a reflective subcategory. Thus, some well known compactifications in the topology appear as this type of factorization.

Keywords: reﬂective and coreﬂective subcategories, the right product of two subcategories, τ-complete spaces. 2000 MSC: 18A20, 18 B30. INTRODUCTION

In the sequel we use the following notations: Eu (resp. Mu ) denotes the class of universal epi (resp. mono); e b Ep, (resp. Mp) the class of precise epi (resp. mono) : Ep= Mu, Mp= Eu; C2V - the category of Hausdorff locally convex topological vector space; Th - the category of Tikhonov spaces (the completely regular Hausdorff spaces); if K (resp. R) is coreflective (resp. reflective) subcategory, then k: C →R (r: C→R) is the coreflector (resp. reflector) functor; (Epi, M f ) - (the class of epimorphisms, the class of strict monomorphisms) = (the class of mappings with dense image, the class of topological inclusions with closed image); (E f , Mono) - (the class of strict epimorphisms, the class of monomorphisms) (Eu, Mp)-(the class of universal epimorphisms, the class of precise monomorphisms)=(the class of surjective mapping, the class of topological inclusions). For concepts from general topology see [7], from topology of locally convex spaces see [8], and for those related to factorization structures see also [8]. The right product of a coreflective and reflective subcategory was introduced and examined in the paper [5]. Necessary and sufficient conditions for the product to be a reflective subcategory were identified. In the category C2V of Hausdorff locally convex topological vector spaces many cases when this product is a reflective subcategory were found. The examination of the right product of two subcategories is requested by the following situations:

41 42 Dumitru Botnaru, Alina T¸urcanu

1. The right product appears in natural way when studying the semi-reflexive subcategories [2]. 2. The relative torsions theories, that are so frequent in the category C2V, can be performed as right product theories. 3. Many reflective subcategories can be explained as the right product of two subcategories of certain type [3]. In the category C2V, as well as in the category of Tikhonov spaces, there are examples when this product is not a reflective subcategory (Theorem 1.1). The properties of right product factorization are examined in this paper (Lemma 1.2). There are stipulated the conditions when this factorization defines a reflective subcategory (Theorem 1.3). In Section 2 it is proved that τ-complete spaces [6], [11] could be constructed as such factorizations (Theorem 2.3). In Section 3 it is shown how the subcategory of τ-complete spaces could be performed in two ways: either varying in the product the coreflective subcategory, or varying the factorization structure. In Section 4 there are formulated some issues for the category of Tikhonov spaces.

1. THE RIGHT PRODUCT OF TWO SUBCATEGORIES Let K be a coreﬂective subcategory, and R - a reﬂective subcategory of the category C. For any object X of the category C assume that rX : X → rX is R-replique of object X, and kX : kX → X and kkrX : krX → rX are the K-corepliques of respective objects. Then rXkX = krXg (1) for some morphism g. Since g = k(rX) we can write the preceding equality

rXkX = krXk(rX). (2)

We assume that in the category C pushout squares exist and we construct the pushout square on the morphisms kX and k(rX):

vXkX = gXk(rX). (3)

Then there exists a morphism uX so that

rX = uXvX, (4)

krX = uXgX. (5) The factorization of the right product of two subcategories 43

Figure 1.1

We denote by K ∗d R the full subcategory of all objects of category C, isomorphic with the objects of vX, X ∈| C | form.

Deﬁnition 1.1. The subcategory K ∗d R is called the d-product or the right product of subcategories K and R.

Theorem 1.1. Let C be a category with pull-back and pushout squares, R a monoreflective subcategory, K - a epireflective subcategory, and V = K ∗d R. Then, the following affirnations are equivalent: 1. V is a reflective subcategory of category C. 2. For any object X of category C the morphism vX is V-replique of object X. 3. For any object X of category C the morphism vX is an epi. 4. For any object X of category C the morphism uX is R-replique of object vX. 5. X ∈| V |⇔ rXkX is K-coreplique of object rX.

The proving is performed as in the case of category C2V ([5], Theorem 2.5). We mention that there are known various cases when subcategory V is reflective (see [5], Theorems 3.2.-3.4. and [3] Theorem 2.6.). Let us examine the right product of two subcategories in the category Th of Tikhonov spaces. Let K be a coreflective subcategory. Then it contains the subcategory D of the spaces with discreteT topology. It follows that it is a monoreflective subcategory, and then it is (Eu Mono) - coreflective, where Eu is class of universal epimorphisms (continued and surjective maps). It is obvious that in the category Th the reflective subcategory R is monoreflective iff when R includes the subcategory of compact spaces: Comp ⊂ R. Let Comp ⊂ R. We reffer to Figure 1.1. Let’s presume that V is a reflective subcategory. Then uX is Comp-replique of object vX. So, uX is a topological inclusion. And from the equality (5) it results that uX is a surjective application. Therefore, uX is an isomorphism. Thus, we proved:

Theorem 1.2. Let K be a coreflective subcategory, R - a reflective subcategory in the category Th and Comp ⊂ R. Then, two cases are possible: 1. K ∗d R = R. 2. K ∗d R is not a reflective subcategory of Th category. 44 Dumitru Botnaru, Alina T¸urcanu

But some examples of reflective subcategory of Th category show they can be obtained as a simple modification of the right product of two subcategories. In what follows we will describe this modification. Definition 1.2. The class of morphisms A of category C is called right-stable if, because g0 f = f 0g, it is a pushout square, and from f ∈ A it follows that f 0 ∈ A.

In the categories Th and C2V the pair (Eu, Mp) = (the class of surjective morphisms, the class of topological inclusions) is a factorization structure. Therefore the class Eu is right-stable. Lemma 1.1. Let C be a category with pull-back and pushout squares in which the class Eu is right-stable, K - a monoreflective subcategory, and R - a reflective subcategory. Then for any abject X of the category C, uX is a monomorphism. T Proof. Any monoreflective subcategory is (Eu Mono)-coreflective. We use the no- X tations from the beginning of the section. In the pushout square (3) k ∈ Eu and X rX according to the hypothesis that g ∈ Eu. In the equality (5) we have k ∈ Mono X X and g ∈ Eu. According to the Lemma 1.3 [5], we deduce that u ∈ Mono.

Corollary 1.1. In the categories Th and C2V for any two subcategories one coreflective and other reflective, uX is always a monomorphism. Lemma 1.2. Assume in the category C and in the subcategory K and R, for any object X, uX is a monomorphism. Then: 1. For any object X of category C the morphism gX is K-coreplique of object vX : gX = kvX. 2. The correspondence X 7−→ vX defines a covariant functor v : C → C. 3. R is a monoreflective subcategory of the category K ∗d R. X 4. For any object X of the subcategory K ∗d R the morphism v is sectionable. Proof. 1. Consider f : A → vX, where A ∈| K |. Then uX f = krXg for some morphism g.

Figure 1.2 The factorization of the right product of two subcategories 45

We have uX f = krXg = uXgXg i.e. uX f = uXgXg, and since uX is a mono, we deduce that f = gXg. The uniqueness of the morphism g which satisﬁes the preceding equality results from the fact that krX is K-coreplique of object rX. So, we proved that gX = kvX. 2. Let us deﬁne the functor v on morphisms. Let f : X → Y ∈ C.

Figure 1.3 Then Y X r f = f1r , (6) for some morphism f1, where f1 = r( f );

rX rY f1k = k f2, (7) for some morphism f2, where f2 = k( f1). We have Y Y X Y X X X u v f k = (from (4), for object Y) = r f k =(from (6))= f1r k =(from (2)) rX X rY X Y vY X = f1k k(r ) = (from (7)) = k f2k(r )(= from (5) for object Y) = u k f2k(r ), i.e.

Y Y X Y vY X u v f k = u k f2k(r ). (8) 46 Dumitru Botnaru, Alina T¸urcanu

Since uY is a mono, from equality (8) we obtain

Y X vY X (v f )k = (k f2)k(r ). (9) From equality (9) and the fact that (3) is an pushout square, we deduce that

Y X v f = f3v , (10)

vY vX k f2 = f3k . (11)

We deﬁne v( f ) = f3. 3. Let us get look again to Fig.1.1. Let X ∈| R |. Then rX, and with him and k(rX) X and v , are isomorphisms. Therefore R ⊂ K ∗d R. Further on, let be rvX : vX → rvX R replique of object vX. Then uX = t · rvX X vX for some morphism t. Since u is a mono of category C , we deduceT that r is the same. The only thing we can add is: the subcategory R is (Epi Mu)-reﬂective in the category K ∗d R. 4. Let’s complete the diagram from Fig. 1.1 with an analogous diagram built for the object vX.

Figure 1.4 We have uX = tXrvX (12) for some morphism tX. Then tXkrvX = krXhX (13) for some morphism hX = k(tX). We have The factorization of the right product of two subcategories 47 krXhXk(rvX)= (from (13)) =tXkrvXk(rvX)=(from (2) for object vX) = tXrvXkvX =(from (12))= uXkvX = (from (10) since gX = kvx) = krX, i.e. krXhXk(rvX) = krX. (14) Therefore hXk(rvX) = 1. (15) Then in the pushout square vvXkvX = kvvXk(rvX) (16) the morphism k(rvX) is sectional. Therefore the morphism vvX is the same. We can assert that rvXvX = lXrX (17) for some morphism lX. We have tXlXrX = (from (17)) = tXrvXvX = (from (1)) = uXvX = (from (4)) = rX i.e. tXlXrX = rX (18) or tXlX = 1. (19)

We assume that the conditions of preceding lemma are satisfied, and (P, I) is a factorization structure in the category C. Let L be the full subcategory of category C comprising I-subobjects of objects of subcategories K ∗d R. For any object X of category C let vX = iXlX (20) be (P, I)-factorization of morphism vX. Theorem 1.3. The correspondence X 7−→ (lX, lX) defines the category L as a P-reflective subcategory of the category C. Proof. Let be A ∈| L |, and f : X −→ A ∈ C. We show that the morphism f is extended through morphism lX. According to the hypothesis there exists an object B ∈| K ∗d R | and a morphism i : A −→ B ∈ I. Then

vB(i f ) = v(i f )vX, (21) or

(vBi) f = (v(i f )iX)lX. (22) Since vB is sectional we deduce that vBi ∈ I, and lX ∈ P. Therefore lX ⊥ vBi, i.e. f = glX, (23) 48 Dumitru Botnaru, Alina T¸urcanu

v(i f )iX = vBig, (24) for some morphism g. The uniqueness of morphism g, that satisﬁes equality (23), results from the fact that lX is an epi.

Figure 1.5

2. THE SUBCATEGORY OF τ-COMPLETE SPACES We examine some coreflective subcategories of the category Th of Tikhonov spaces. Definition 2.1. Let τ be an cardinal. 1. In a topological space the intersection of τ open sets is called Gτ -set. − − 2. Pτ -change of the space (X, t) is called the space (X, tτ ), where the basis of − topology tτ is formed by Gα-set, α < τ. 3. Pτ-change of space (X, t) is called the space (X, tτ), where the basis of topology tτ is formed by Gτ-sets. We note with K−(τ) (respectively K(τ)) the full subcategory of the category Th − comprising all spaces (X, t) for which t = tτ (respectively, t = tτ). We observe that K(τ) = K−(τ+), where τ+ is the first cardinal which follows τ. If τ is limiting cardinal, then K−(τ) , K−(λ), for any cardinal λ. Therefore, it is enough to examine only the subcategories K−(τ). It is easily checked that K−(τ) (similar by and K(τ)) are the coreflective subcategories of category Th with coreflective functors.

− − − − Pτ : Th −→ K (τ), Pτ (X, t) = (X, tτ ),

Pτ : Th −→ K(τ), Pτ(X, t) = (X, tτ).

We mention the following properties of the subcategories K−(τ): 1. K−(τ) = Th for τ ≤ ω. 2. Let α < β be. Then K−(α) ⊇ K−(β). 3. Let Disc be the subcategory of discrete spaces. Then ∩{K−(τ)/τ} = ∩{K(τ)/τ} = Disc. Therefore, we can conclude that Disc = K−(∞) = K(∞), considering that τ < ∞ for every cardinal τ. The factorization of the right product of two subcategories 49

Theorem 2.1. ([4], Theorem1.2). Consider ω ≤ α < β. Then:

K−(β) ⊂ K−(α) and K−(β) , K−(α).

Deﬁnition 2.2. Let τ be an cardinal. The Tikhonov space X is called Q(τ)-space (respectively Q−(τ)-space), if X is closed in K(τ)-coreplique (respectively K−(τ)- coreplique) of space βX, where βX is Comp-replique of spaces X. We note with Q(τ) (respectively, Q−(τ) ) - the full subcategory of all Q(τ)-spaces (respectively, Q−(τ) -spaces). The categories Q(τ) have been studied by A. Cigoghidze [6], and the Q−(τ) by H. Herrlich [11]. − Let (X, t) be a Tikhonov space, (Y, u) = β(X, t), and (Y, uτ) − K (τ) -coreplique of space (Y, u). Let X be the closure of the set X in the space (Y, uτ). On set X we − induce the topology u0 out of space (Y, u). The topology space (X, u0) we note vt X, − or vt (X, t).

Figure 2.1

Evidently, Q(τ) = Q−(τ+) for any limiting cardinals τ. But for limiting cardinals τ, the subcategories Q−(τ) are of other form that Q(τ). For a subcategory A of category Th we note with PrA the subcategory that contains the products of objects of category A, and with M f (A) the subcategory that contains M f -subobjects of objects of subcategory A. Theorem 2.2. ([4], Theorem 2.6). 1. The subcategory Q−(τ) is a monoreflective subcategory (therefore also epireflective) of category Th with reflector functor − − vt : Th −→ Q (τ). 2. Q−(ω) = Q(n) = Comp, n ∈ N. − 3. Q (ωI) = Q(ω) = Q - the subcategory of Hewitt spaces. − τ 4.Q (τ) = M f Pr(R(τ)), where R(τ) = [−1; 1] \ {−1; 1}. − 5. Q (τ) = M f Pr(E(τ)), where E(τ) = Πα<τR(α). 6. α < β and ω < β. Then Q−(α) ⊂ Q−(β) and Q−(α) , Q−(β) . Corollary 2.1. Let τ be a limiting cardinal. Then

Q(τ) = M f Pr(∪Q(λ): λ > τ). 50 Dumitru Botnaru, Alina T¸urcanu

Remark 2.1. In [11] is deﬁned the problem of existence of generators for subcate- − gories Q (τ), i.e. if there is a space Aτ, so that − Q (τ) = M f Pr(Aτ). Ignoring the case in [6], the problem is solved for subcategories Q(τ) and fully in the precedent theorem.

− Theorem 2.3. The reﬂective subcategory Q (τ) is (Epi, M f )-factorization of the − right product K (τ) ∗d Comp. Proof. For Tikhonov space (X, t) let (Y, u) = β(X, t) the Stone-Cech˘ compactiﬁcation, X Y − and pτ :(X, tτ) −→ (X, t) and pτ :(Y, uτ) −→ (Y, u) the Kτ -coreplique of respective objects.

Figure 2.2 On the set Y we examine the inductive topology m, which is not mandatory being X Y the Tikhonov topology, defined by applications β and pτ : X −1 Y −1 G ∈ m ⇐⇒ [(β ) (G) ∈ t and (pτt) (G) ∈ uτ]. The square X X gpτ(β ) = hpτ , (25) X X is the pushout square constructed on morphisms pτ and pτ(β( )) in the category of topological spaces. In the sequel we construct the pushout square on these morphisms in the category Th. Let F be the set of continued defined functions (Y, m) with values in the field of real numbers R: F = Hom((Y, m), R), and m(F) - the topology defined on the set Y of this set F: The factorization of the right product of two subcategories 51

m(F) = { f −1(G)| f ∈ F and G is open in R}. Let l be the canonic application. Then

X X X (lh)pτ = g pτ(β ) (26) X X is the pushout square constructed on morphisms pτ and pτ(β ) in the category Th. Let Z be the closure of set X in the space (Y, m(F)), and let the topology v be the one induced from space (Y, m(F)). Since lX is an epi in the category Th it follows that uXiX is the Stone-Cech compactification of space (Z, v). Thus we can consider that the topology v is the one induced from application uXiX = βZ on set Z from space β(X, t). The closure of set X in the spaces (Y, m). A set A is closed in the space (Y, m) iff −1 −1 the set h (A) is closed in the space (X, t) and g (A) is closed in the space (Y, uτ). If −1 X ⊂ A, then the set A is closed iff the set g (A) is closed in the space (Y, uτ).

The closure of set X in the spaces (Y, m) and (Y, uτ) coincides: clmX = cluτ X. The last set we denote as T : T = cluτ X.

Let us prove that the closure of set X in the space (Y, uτ) coincides with Z: cluτ X = clm(F)X. Firstly we mention that T ⊂ Z and will prove the reverse inclusion. Let be y ∈ Y\T. Then exists a set H ⊂ Y \ T, that comprise point y so that: 1. H is closed in the topology u. 2. H is a Gτ-set in the topology u. 3. H is closed and open in the topology uτ. 4. H remains open and closed in the topology m. 5. H remains open and closed in the topology m(F). The theorem is proved.

3. THE CASE OF THE SUBCATEGORY Q−(τ) Let be R and L two reﬂective subcategories of the category C and L ⊂ R. If C is local and colocal small with projective limits, then there exists a class L(R) of factorization structure in the category C with the following property. For any object X of category C let be rX : X −→ rX and lX : X −→ lX the respective replique. Then lX = vXrX (27) for some morphism vX. If the subcategory L is monoreﬂective, then in the written equality all morphisms are bimorphisms. We note U = {rX|X ∈ |C| }, V = {vX|X ∈ |C| }. Always U ⊥ V. 52 Dumitru Botnaru, Alina T¸urcanu

Assume P00(R) = P00 = Vq; I00(R) = I00 = Vqx; P0(R) = P0 = Uxq; I0(R) = Ux. Theorem 3.1. [1] Let C be local and colocal small with projective limits and the subcategory L is monoreﬂective. Then: 1. (P0, I0) and (P00, I00) are structures of factorization in the subcategory C. 2. Let (P, I) be a structure of factorization in the category C. The following aﬃr- mations are equivalent: a) for any object X of category C the equality (1) is (P, I)-factorization of morphism lX; b) P0 ⊂ P ⊂ P00. Relying on this theorem we assert

Theorem 3.2. Let τ be a cardinal, τ > ℵ0. 1. The reflective subcategory Q−(τ) could be obtained factorizing the right product X − (the morphism v ) K (τ) ∗d Comp following the structure of factorization (Epi, M f ). 2. In the category Th there ar the structures of factorization (P, I), P0 (Q−(τ)) ⊂ P ⊂ P00(Q−(τ)), so that the reflective subcategory Q−(τ) could be obtained doing the − (P, I)- factorization of right product K (ω) ∗d Comp. 4. PROBLEMS In paper [4] some classes of coreflective subcategories in the category Th are examined.

− Deﬁnition 4.1. Let τ be an cardinal. The topological space (X, t) is called kτ -space, if any function deﬁnite on set X and continue on every compact K ⊂ X, | K |< τ is continue on space (X, t).

− − Let C (τ) a full subcategory of all kτ -spaces. Theorem 4.1. The subcategory C−(τ) is a coreﬂective subcategory of category Th. Deﬁnition 4.2. The weight w(X, t) of the topological space (X, t) is called minimal of cardinals |B|, where B is basis of spaces (X, t).

− Deﬁnition 4.3. Let τ be an cardinal. The Tikhonov space is called bτ -space if every function deﬁnite on set X and continue on every compact K ⊂ X, w(K) < τ is continue on space (X, t).

− − Let B (τ) is a subcategory full of all bτ -spaces. Theorem 4.2. 1. The subcategory B−(τ) is a coreﬂective subcategory of category Th. 2. Let be τ ≤ ω. Then B−(τ) = Disc. 3. ∩B−(τ) = C - subcategory of k-functionals space. The factorization of the right product of two subcategories 53

4. Let ω ≤ α < β and α a regular cardinal and β ≥ α+. Then B−(α) ⊂ B−(β) and B−(α) , B−(β).

Problem 4.1. 1. Let us describe the subcategory R which is (Epi, M f )-factorization − of the right product C (τ) ∗d Comp. 2. R-replique of an arbitrary object X of category Th is also the closing of set X in the C−(τ)-coreplique of space βX (with induce topology from βX)? − 3. The same thing is valid for the right product B (τ) ∗d Comp. References

[1] Botnaru D., Structure bicategorielles complementaires, ROMAI J., 5, 2(2009), 5-27. [2] Botnaru D., Cerbu O., Semireflexive product of two subcategories, Proc.of the Sixth Congress of Romanian Mathematicians, Bucharest, 2007, V.1, p. 5-19. [3] Botnaru D., Cerbu O., Some properties of semireflexivity subcategories (submitted). [4] Botnaru D., Robu R., Some categorical aspects of Tikhonov spaces, Scientific Annals. of MSU, The series ”Physical-Mathematical Sciences”, Chis¸inau,˘ 2000, 87-90 (in Romanian). [5] Botnaru D., Turcanu A., Les produits de gauche et de droit de deux sous-categories, Acta et coment, Chis¸inau,˘ III(2003), 57-73. [6] Cigoghidze A.C., About properties close of compactness, Uspehi mat.nauk, 35, 6 (216)(1980), 177-178. [7] Engelking R., General topology, Warszawa, 1985. [8] Grothendieck A., Topological vector spaces, Gordon and Breach, New York, London, Paris, 1973. [9] Herrlich H., Fortsetzbarkeit stetiger abbildungen und kompactheitsgrad topologischer Raume, Math. Z., 96, 1(1967), 64-72.

ROMAI J., 6, 2(2010), 55–67

CLASSIFICATION OF A CLASS OF QUADRATIC DIFFERENTIAL SYSTEMS WITH DERIVATIONS Ilie Burdujan University of Agricultural Sciences and Veterinary Medicine ”Ion Ionescu de la Brad” Ia¸si, Romania burdujan [email protected] Abstract The classification up to a center-affinity of a class of homogeneous quadratic differential systems defined on R3, which have a semisimple derivation with a 2-dimensional kernel is achieved. It is proved that there exist eighteen classes of affinely nonequivalent such systems.

Keywords: homogeneous quadratic dynamical systems, nilpotent derivation. 2000 MSC: Primary 34G20, Secondary 34L30, 15A69.

1. INTRODUCTION The study of a homogeneous quadratic differential system (S ) (briefly, HQDS) on a B space A can be achieved by using suitable algebraic tools. This is possible because a commutative algebra A(·) is naturally associated with any such a system (S ). In general, algebra A(·) is not associative. There exists (see [11], [1]) a 1-to- 1 correspondence between the classes of affinely equivalent HQDSs on A and the classes of isomorphic commutative algebras on A. A strong connection between the qualitative properties of HQDSs and the properties - invariant up to an isomorphism - of the corresponding commutative algebras emerges from this correspondence. In particular, any derivation/automorphism of (S ) is a derivation/automorphism of its associated algebra A(·) (see [5], [9]). As it is well known (see, for example, [1]), if a derivation of A(·) having only real eigenvalues exists, then its semisimple and nilpotent parts are necessarily derivations, too. The commutative algebras on R3 having a nilpotent derivation were already classified (see [4], [6]). The problem of classification of 3-dimensional real commutative algebras having at least a semisimple derivation is an intricate task and it is still unsolved. In this paper, in order to save the space, we shall classify - up to a center-affinity - only a particular class of HQDSs on R3 which have a semisimple derivation. It is proved that this class of HQDSs is naturally partitioned in eighteen mutually disjoint subclasses consisting of nonequivalent systems, i.e., every system in one subclass is not equivalent to any system in another subclass; moreover, two systems in the same subclass that correspond to different values of parameters are nonequivalent, too.

55 56 Ilie Burdujan

Our results are based on the remark: the problem of classification up to an affine equivalence of HQDSs on Rn is equivalent with the problem of classification up to an isomorphism of all real n-dimensional commutative algebras. That is why we shall classify, up to an isomorphism, the class of real 3-dimensional commutative algebras that correspond to the analyzed class of HQDSs. The lattices of the subalgebras of these algebras are used as a main tool for solving the isomorphism problems. In a forthcoming paper we shall classify the set of all HQDSs on R3 which have a semisimple derivation.

2. REAL COMMUTATIVE ALGEBRAS ON R3 HAVING A SEMISIMPLE DERIVATION Let us consider a nontrivial real 3-dimensional commutative algebra A(·). Then, any basis in A allows to identify A with R3. Now, let us suppose that A(·) has a nonzero semisimple derivation. Then Der A necessarily contains a semisimple derivation D having its spectrum SpecD of the form SpecD = {1, λ, µ} (λ, µ ∈ R). Accordingly, there exists a basis B = (e1, e2, e3) such that D(e1) = e1, D(e2) = λe2, D(e3) = µe3. 2 2 2 Obviously, the case λ = µ = 1 is not possible. Since D(e1) = 2e1, then either e1 = 0 2 (if 2 < SpecD) or e1 , 0 (if 2 ∈ SpecD). More exactly, it follows   2  e1 = 0 i f 2 < SpecD  2  e1 = κe2 i f λ = 2 , µ  2  e1 = ωe3 i f µ = 2 , λ  2 e1 = κe2 + ωe3 i f λ = µ = 2. Each of the following equations can be exploited in a similar way: 2 2 D(e1e2) = (1 + λ)e1e2, D(e1e3) = (1 + µ)e1e3, D(e2) = 2λe2, 2 2 D(e2e3) = (λ + µ)e2e3, D(e3) = 2µe3. In fact, at least one of the elements of the set {2, 1 + λ, 1 + µ, 2λ, 2µ, λ + µ} belongs to SpecD. A simple inspection of papers [4], [6], [10] provides the following cases of the spectrum of D that have to be considered: {1, 0, 0}, {1, 1, 0}, {1, 1, 2}, {1, 2, 0}, {1, 2, 2}, {1, 2, 3}, {1, 2, 4}, {1, −1, 0}. In what follows we deal with a class of algebras which have a semisimple derivation D with SpecD = {1, 0, 0}, only. This is the case when dimR Ker D = 2.

3. CASE SPEC D = {1, 0, 0}

There exists a basis B = {e1, e2, e3} such that

D(e1) = e1, D(e2) = 0, D(e3) = 0. Classiﬁcation of a class of quadratic diﬀerential systems with derivations 57

Then, from the equations

2 2 D(e1) = 2e1, D(e1e2) = e1e2, D(e1e3) = e1e3, 2 2 D(e2) = D(e2e3) = D(e3) = 0 emerges the following multiplication table of algebra A(·)

2 2 e1 = 0 e2 = αe2 + βe3 e1e2 = κe1 e2e3 = γe2 + δe3 2 e1e3 = ωe1 e3 = εe2 + θe3. It results that A = Im D ⊕ Ker D where Re1 is an ideal and Ker D is a subalgebra of A(·). The algebra Ker D can have (see [11]) either a basis of nilpotent elements of index two or a single 1-subspace of such nilpotent elements, only. Among the algebras of the former kind, there exists one (see [11]) which has the following multiplication table: 2 2 e2 = 0, e2e3 = e3, e3 = γe2 + e3 (γ , 0). It suggests us to deal with algebras deﬁned by the following multiplication table:

2 2 Table T e1 = 0 e2 = 0 e1e2 = αe1 e2e3 = e3 2 e1e3 = βe1 e3 = γe2 + e3 with α, β, γ ∈ R, γ , 0. We shall denote by A(α, β, γ) any algebra on R3 with the multiplication table T in basis B.

Algebra A(α, β, γ) Any algebra A = A(α, β, γ)(γ , 0) is not simple because it contains the ideal Im D. Distinct triples of real numbers (α, β, γ)(γ , 0) determine non-isomorphic algebras. Indeed, the following result is proved by a straightforward checking. Theorem 3.1. The algebras A(α, β, γ)(γ , 0) and A(p, q, r)(r , 0) are isomorphic if and only if α = p, β = q, γ = r. In order to ﬁnd a partition for class A(α, β, γ)(γ , 0) into subclasses of noniso- morphic subalgebras we look for the family of its 1-dimensional subalgebras. To this end we need to ﬁnd its annihilator elements, nilpotent (of index two) elements and idempotent (of index two) elements. This problem is solved in the next three propositions. Proposition 3.1. The set Ann (A) of all annihilator elements of A = A(α, β, γ) is: ( {0} i f α2 + β2 , 0 Ann (A) = Im D i f α = β = 0. 58 Ilie Burdujan

Proposition 3.2. The set N(A) of all nilpotent elements (of index two) of A = A(α, β, γ) is: ( Re ∪ Re i f α , 0 N(A) = 1 2 SpanR{e1, e2} i f α = 0. Proposition 3.3. The set of idempotent elements (of index two) of any algebra A(α, β, γ) (γ , 0) is: ◦ 1 1 I(A) = ∅ if γ < − 8 , ◦ 1 1 2 I(A) = {− 2 e2 + 2e3} if γ = − 8 and α − 4β + 1 , 0, ◦ 1 1 3 I(A) = {xe1 − 2 e2 + 2e3 | x ∈ R} if γ = − 8 and α − 4β + 1 = 0, ◦ 1 4 I(A) = {y1e2 + z1e3, y2e2 + z2e3 | x ∈ R} if γ > − 8 , γ , 0, 2(αy1 + βz1) − 1 , 0 and 2(αy2 + βz2) − 1 , 0, ◦ 1 5 I(A) = {y1e2 + z1e3} ∪ {xe1 + y2e2 + z2e3 | x ∈ R} if γ > − 8 , γ , 0, 2(αy1 + βz1) − 1 , 0 and 2(αy2 + βz2) − 1 = 0, ◦ 1 6 I(A) = {xe1 + y1e2 + z1e3, | x ∈ R} ∪ {y2e2 + z2e3} if γ > − 8 , γ , 0, 2(αy1 + βz1) − 1 = 0 and 2(αy2 + βz2) − 1 , 0, ◦ 7 I(A) = {xe1 + y1e2 + z1e3 | x ∈ R} ∪ {xe1 + y2e2 + z2e3 | x ∈ R} if 1 γ > − 8 , γ = 0, 2(αy1 + βz1) − 1 = 0 and 2(αy2 + βz2) − 1 , 0. 1 Corollary 3.1. Every algebra A(α, β, γ)(γ < − 8 ) is not isomorphic to any algebra 1 1 A(p, q, 8 ) or A(p, q, r)(r > − 8 , r , 0). Proof. It is enough to notice that any two such classes of algebras have distinct families of idempotent elements.

Properties of algebra A(α, β, γ) In what follows, we list the properties of these eighteen subclasses of algebras that help us to solve easily the isomorphism problem. Moreover, note that the lattices of subalgebras of each subclass of algebras were obtained.

◦ 1 2 2 Properties of algebra A1 =A(α, β, γ)(α , 0, γ < − 8 , (α − 1) + (2β − 1) , 0) • AnnA = {0} • N(A) = Re1 ∪ Re2 • I(A) = ∅ • Subalgebras: Re1, Re2, SpanR{e1, e2}, SpanR{e2, e3} • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1 Classiﬁcation of a class of quadratic diﬀerential systems with derivations 59

◦ 1 1 Properties of algebra A2 =A(1, 2 , γ)(γ < − 8 ) • AnnA = {0} • N(A) = Re1 ∪ Re2 • I(A) = ∅ 2 2 • Subalgebras: Re1, Re2, SpanR{e2, ae1 + be3} (a + b , 0) • Ideals : Re1 2 • A = A   0 0 1    0 0 0  • Der A = RD ⊕ RD1 where [D1] =   and [D, D1] = D1 0 0 0   x 0 y    0 1 0  • Aut (A) = {T(x, y) | x, y ∈ R, x , 0} where T(x, y) =   . 0 0 1

◦ 1 Properties of algebra A3 =A(0, β, γ)(β , 0, γ < − 8 ) • AnnA = {0} • N(A) = SpanR{e1, e2} • I(A) = ∅ 2 2 • Subalgebras: R(pe1 + qe2)(p + q , 0), SpanR{e1, e2}, SpanR{e2, e3} • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A4 =A(α, β, − 8 )(α , 0, α − 4β + 1 , 0) • AnnA = {0} • N(A) = Re1 ∪ Re2 1 • I(A) = {− 2 e2 + 2e3} 1 • Subalgebras: Re1, Re2, R(− 2 e2 + 2e3), SpanR{e1, e2}, S panR{e2, e3}, 1 SpanR{e1, − 2 e2 + 2e3} • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1 60 Ilie Burdujan

◦ 1 1 Properties of algebra A5 =A(0, β, − 8 )(β , 0, β , 4 ) • AnnA = {0} • N(A) = SpanR{e1, e2} 1 • I(A) = {− 2 e2 + 2e3} 2 2 1 • Subalgebras: Re1, Re2, R(pe1+qe2)(p +q , 0), R(− 2 e2+2e3), SpanR{e1, e2}, 1 SpanR{e2, e3}, S panR{e1, − 2 e2 + 2e3} • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 1 1 Properties of algebra A6 =A(4β − 1, β, − 8 )(β < { 4 , 2 }) • AnnA = {0} • N(A) = Re1 ∪ Re2 1 • I(A) = {xe1 − 2 e2 + 2e3 | x ∈ R} 1 • Subalgebras: Re1, Re2, R(xe1 − 2 e2 + 2e3)(x ∈ R), SpanR{e1, e2}, 1 SpanR{e2, e3}, SpanR{e1, − 2 e2 + 2e3} • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 1 Properties of algebra A7 =A(1, 2 , − 8 ) • AnnA = {0} • N(A) = Re1 ∪ Re2 1 • I(A) = {xe1 − 2 e2 + 2e3 | x ∈ R} 1 • Subalgebras: Re1, Re2, R(xe1 − 2 e2 + 2e3)(x ∈ R), SpanR{e1, e2}, 1 SpanR{e2, e3}, S panR{e1, − 2 e2 + 2e3}, S panR{e2, ae1 + be3} (a, b ∈ R)}, 1 cx Span {xe − e + 2e , e + be + ce } (b, c ∈ R, 4b + c , 0) R 1 2 2 3 2 1 2 3 • Ideals : Re1 2 • A = A   0 0 1    0 0 0  • Der A = RD ⊕ RD1 where [D1] =   and [D, D1] = D1 0 0 0 Classiﬁcation of a class of quadratic diﬀerential systems with derivations 61   x 0 y    0 1 0  • Aut (A) = {T(x, y) | x, y ∈ R, x , 0} where T(x, y) =   . 0 0 1

◦ 1 1 Properties of algebra A8 =A(0, 4 , − 8 ) • AnnA = {0} • N(A) = SpanR{e1, e2} 1 • I(A) = {xe1 − 2 e2 + 2e3 | x ∈ R} 2 2 • Subalgebras: R(pe1 + qe2)(p + q , 0), SpanR{e1, e2}, SpanR{e2, e3}, 1 ∗ SpanR{e1, − 2 e2 + 2e3}, SpanR{pe1 + qe2, e3}(p, q ∈ R ), 1 SpanR{− 2 e2 + 2e3, ae1 + be2 + ce3} (a, b, c ∈ R, 4b + c , 0), 1 SpanR{xe1 − 2 e2 + 2e3, (1 − 2xq)e1 + qe2} (x, q ∈ R, xq , 0) • Ideals : Re1 2 • A = A   0 4 1    0 0 0  • Der A = RD ⊕ RD1 where [D1] =   and [D, D1] = D1 0 0 0   x 4y y    0 1 0  • Aut (A) = {T(x, y) | x, y ∈ R, x , 0} where T(x, y) =   . 0 0 1

◦ 1 Properties of algebra A9 =A(α, β, γ)(α , 0, γ , 0, γ > − 8 , 2(αy1 + βz1) − 1 , 0, 2(αy2 + βz2) − 1 , 0) • AnnA = {0} • N(A) = Re1 ∪ Re2 • I(A) = {y1e2 + z1e3, y2e2 + z2e3} • Subalgebras: Re1, Re2, R(y1e2 + z1e3), R(y2e2 + z2e3), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{y1e2 + z1e3, e1}, SpanR{y1e2 + z1e3, e2}, SpanR{y2e2 + z2e3, e1}, SpanR{y2e2 + z2e3, e2}, SpanR{y1e2 +z1e3, ae1 +b[γ(α−z1 −αz1 +2βz1 +1)e2 +2(β+γz1 +αγz1)e3]} (a, b ∈ R), SpanR{y2e2 +z2e3, ae1 +b[γ(α−z2 −αz2 +2βz2 +1)e2 +(β+γz2 +αγz2)e3]} (a, b ∈ R) • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A10 =A(0, β, γ)(γ , 0, γ > − 8 , 2βz1 − 1 , 0, 2βz2 − 1 , 0, β , 0) • AnnA = {0} 62 Ilie Burdujan

• N(A) = SpanR{e1, e2} • I(A) = {y1e2 + z1e3, y2e2 + z2e3} 2 2 • Subalgebras: R(pe1 + qe2)(p + q , 0), R(y1e2 + z1e3), R(y2e2 + z2e3), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{y1e2 + z1e3, e1}, SpanR{y2e2 + z2e3, e1}, SpanR{y1e2 + z1e3, ae1 + b[γ(1 − z1 + 2βz1)e2 + 2(β + γz1)e3]} (a, b ∈ R, γ , 1), SpanR{y2e2 + z2e3, ae1 + b[γ(1 − z2 + 2βz2)e2 + 2(β + γz2)e3]} (a, b ∈ R, γ , 1) There exists other subalgebras if α, β, γ are connected by some appropriate equations, namely: p 2 1 SpanR{pe1 +qe2, q (b+βc)e1 +be2 +ce3} i f γ = 2β −β, β , 4 (b, c, p, q ∈ R, q , 0) • Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A11 =A(α, β, γ)(α , 0, γ , 0, γ > − 8 , 2(αy1 + βz1) − 1 , 0, 2(αy2 + βz2) − 1 = 0) • AnnA = {0} • N(A) = Re1 ∪ Re2 • I(A) = {y1e2 + z1e3} ∪ {xe1 + y2e2 + z2e3 | x ∈ R} • Subalgebras: Re1, Re2, R(y1e2+z1e3), R(xe1+y2e2+z2e3)(x ∈ R), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{y1e2 + z1e3, e1}, SpanR{y2e2 + z2e3, e1}, If x , 0 a biparametric family of 2-dimensional algebras of type SpanR{xe1 + y2e2 + z2e3, ae1 + be2 + ce3} is obtained; they correspond to the following conditions: 2 2 4x(1 − α)b + 4γx(βz2 − 1)c + 2(z2 − 2)ab + (4γz2 + z2 − 1)ac+ +4x(γz2 + αγz2 − β + 1)bc = 0, 3 2 2 2 −γxc + 2xb c + xbc − 2(y2 − αy2 + βz2)abc + (−y2 + 2βy2 + γz2)ac − 2 −2αz2ab = 0.

• Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A12 = A(0, β, γ)(γ , 0, γ > − 8 , 2βz1 −1 , 0, 2βz2 −1 = 2 0, i.e. γ = 2β − β, β = −γz1, β , 0) • AnnA = {0} • N(A) =SpanR{e1, e2} • I(A) = {y1e2 + z1e3} ∪ {xe1 + y2e2 + z2e3 | x ∈ R} Classiﬁcation of a class of quadratic diﬀerential systems with derivations 63

• Subalgebras: R(pe1 + qe2), R(y1e2 + z1e3), R(xe1 + y2e2 + z2e3)(x ∈ R), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{y1e2 + z1e3, e1}, SpanR{y2e2 + z2e3, e1}, p Span{pe1 + qe2, q (b + βc)e1 + be2 + ce3} (b, c, p, q ∈ R, pq , 0), SpanR{y1e2 + z1e3, ae1 + b[γ(2βz1 − z1 + 1)e2 + 2(β + γz1)e3]} (a, b ∈ R, β , −1), 2β−1 SpanR{xe1 + y2e2 + z2e3, ae1 + b( 2 e2 + e3)} (x, a, b ∈ R), 4βa − a − 4β2 xb SpanR{xe1 + y2e2 + z2e3, ae1 + 4βx e2 + be3} (x, a, b ∈ R, x , 0) • Ideals : Re1 2 • A = A   0 1 −γz  1   0 0 0  • Der A = RD ⊕ RD1 where [D1] =   and [D, D1] = D1 0 0 0   x y −γz y  1  ∗  0 1 0  • Aut (A) = {T(x, y) | x ∈ R } where T(x) =   . 0 0 1 ◦ 1 Properties of algebra A13 =A(α, β, γ)(α , 0, γ , 0, γ > − 8 , 2(αy1 +βz1)−1 = 2 2 2 0 , 2(αy1 +βz2)−1 , 0, (α−1) +(2β−1) , 0, i.e. 2γ(1−α) +(2β−1)(α−2β) = 0) • AnnA = {0} • N(A) = Re1 ∪ Re2 • I(A) = {xe1 + y1e2 + z1e3 | x ∈ R} ∪ {y2e2 + z2e3} • Subalgebras: Re1, Re2, R(y1e2+z1e3), R(xe1+y2e2+z2e3)(x ∈ R), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{y1e2 + z1e3, e1}, SpanR{y2e2 + z2e3, e1}, If x , 0 a 2-parametric family of 2-dimensional algebras of type S panR{xe1 + y1e2 + z1e3, ae1 + be2 + ce3} is obtained; they correspond to the following conditions:

2 2 4x(1 − α)b + 4γx(βz1 − 1)c + 2(z1 − 2)ab + (4γz1 + z1 − 1)ac+ +4x(γz1 + αγz2 − β + 1)bc = 0, 2 2 2 2 −γxc + 2xb c + xbc − 2(y1 − αy1 + βz1)abc + (−y1 + 2βy1 + γz1)ac − 2 −2αz1ab = 0.

• Ideals : Re1 • A2 = A • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 1 Properties of algebra A14 =A(1, 2 , γ)(γ , 0, γ > − 8 ) • AnnA = {0} • N(A) = Re1 ∪ Re2 • I(A) = {xe1 + y1e2 + z1e3 | x ∈ R} ∪ {y2e2 + z2e3} 64 Ilie Burdujan

• Subalgebras: Re1, Re2, R(xe1+y1e2+z1e3)(x ∈ R), R(y2e2+z2e3), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{e1, y1e2 + z1e3}, SpanR{e1, y2e2 + z2e3}, SpanR{e2, ae1 + be3} (a, b ∈ R), SpanR{xe1 + y1e2 + z1e3, −4xe1 + z1(e2 − 4e3)}, ∗ SpanR{y2e2 + z2e3, ae1 + b(y1e2 + z1e3)(a, b ∈ R ) • Ideals : Re1 2 • A = A   0 0 1    0 0 0  • Der A = RD ⊕ RD1 where [D1] =   and [D, D1] = D1 0 0 0   x 0 y    0 1 0  • Aut (A) = {T(x, y) | x, y ∈ R, x , 0} where T(x, y) =   . 0 0 1 ◦ 1 Properties of algebra A15 =A(0, β, γ)(γ , 0, γ > − 8 , 2βz1 − 1 = 0, 2βz2 − 1 , 2 0, i.e. γ = 2β − β, β = −γz2) • AnnA = {0} • N(A) = S panR{e1, e2} • I(A) = {xe1 + y1e2 + z1e3 | x ∈ R} ∪ {y2e2 + z2e3} • Subalgebras: R(pe1 + qe2)(p, q ∈ R), R(xe1 + y1e2 + z1e3)(x ∈ R), R(y2e2 + z2e3), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{y1e2 + z1e3, e1}, p SpanR{y2e2 +z2e3, e1}, Span{pe1 +qe2, q (b+βc)e1+be2+ce3} (b, c, p, q ∈ R, pq , 0), SpanR{y2e2 + z2e3, ae1 + b[γ(2βz2 − z2 + 1)e2 + 2(β + γz2)e3]} (a, b ∈ R, β , −1), 2β−1 SpanR{xe1 + y2e2 + z2e3, ae1 + b( 2 e2 + e3)} (x, a, b ∈ R), 4βa − a − 4β2 xb SpanR{xe1 + y1e2 + z1e3, ae1 + 4βx e2 + be3} (x, a, b ∈ R, x , 0) • Ideals : Re1 2 • A = A   0 1 −γz  2   0 0 0  • Der A = RD ⊕ RD1 where [D1] =   and [D, D1] = D1 0 0 0   x y −γz y  2  ∗  0 1 0  • Aut (A) = {T(x, y) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A16 =A(0, 0, γ)(γ < − 8 )

• AnnA = Re1 • N(A) = S panR{e1, e2} • I(A) = ∅ 2 2 • Subalgebras: R(pe1 + qe2)(p + q , 0), SpanR{e1, e2}, SpanR{e2, e3} • Ideals : Re1, SpanR{e1, e2} • A2 = A Classiﬁcation of a class of quadratic diﬀerential systems with derivations 65

• Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A17 =A(0, 0, − 8 )

• AnnA = Re1 • N(A) = SpanR{e1, e2} 1 • I(A) = {− 2 e2 + 2e3} 2 2 1 • Subalgebras: R(pe1 + qe2)(p + q , 0), R(− 2 e2 + 2e3), SpanR{e1, e2}, 1 1 SpanR{e2, e3}, SpanR{e1, − 2 e2 + 2e3}, SpanR{e2, − 2 e2 + 2e3} • Ideals : Re1, SpanR{e1, e2} 2 1 • A = SpanR{− 8 e2 + e3, e3} • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1

◦ 1 Properties of algebra A18 =A(0, 0, γ)(γ , 0, γ > − 8 )

• AnnA = Re1 • N(A) = SpanR{e1, e2} • I(A) = {y1e2 + z1e3} ∪ {y2e2 + z2e3} 2 2 • Subalgebras: R(pe1 + qe2)(p + q , 0), R(y1e2 + z1e3), R(y2e2 + z2e3), SpanR{e1, e2}, SpanR{e2, e3}, SpanR{e1, y1e2 + z1e3}, SpanR{e1, y2e2 + z2e3} • Ideals : Re1, SpanR{e1, e2} 2 • A = SpanR{γe2 + e3, e3} • Der A = RD   x 0 0   ∗  0 1 0  • Aut (A) = {T(x) | x ∈ R } where T(x) =   . 0 0 1 These lists of properties of classes Ai◦ allow us to assert the following result.

Theorem 3.2. Every algebra of class Ai◦ is not isomorphic to any algebra of class A j◦ (i , j, i, j ∈ {1, 2, ..., 18}).

Accordingly, the class of homogeneous quadratic diﬀerential systems   x0 = 2αxy + 2βxz  (S ) y0 = γz2   z0 = 2yz + z2 66 Ilie Burdujan is naturally divided into eighteen classes of aﬃnely nonequivalent systems. The last two equations of (S) supply the prime integral F(y, z), namely   2y2 + yz − 2γz2 2γz − y  ln + p 1 arctg p i f γ < − 1  4 8  y γ |8γ + 1| 2γy |8γ + 1|  2  (4y + z) 8y 1 F(y, z) =  ln 4 − i f γ = − 8  4y z + 4y  ! 1 p  2 2 √  |2y + yz − 2γz | 2 8γ+1 2γz − (1 + 8γ + 1)y 1  4 p i f γ > − . y 2γz − (1 − 8γ + 1)y 8 Consequently, any trajectory of (S) lies on a cylinder F(y, z) = const.

4. THE LATTICE OF SUBALGEBRAS OF A(α, β, γ) It is well known that isomorphic algebras have necessarily isomorphic lattices of their subalgebras. Consequently, the lattices of all subalgebras of two algebras become a natural tool in proving whether these algebras are or are not isomorphic. Moreover, the lattice of subalgebras of any algebra A delivers a partition of its vector space in nonempty mutually disjoint cells. In its turn, this partition provides a partition of the set of all solutions of the associated homogeneous quadratic diﬀer- ential system. In the previous section we have obtained the lattice of all subalgebras of any algebra A(α, β, γ), even if, this ﬁnding is a customary task. For particular classes of algebras this problem can be easily solved, namely for algebras A with Der A = RD (i.e. algebras with fewer symmetries). For example, the lattice of subalgebras for algebra A1◦ is

{{0}, Re1, Re2, S panR{e1, e2}, S panR{e2, e3}, A}. These subalgebras define a partition of R3 by means of their ground subspaces Ox, Oy, xOy and yOz. Accordingly, any solution of the corresponding HQDS (S) has its orbit in one of the cells bounded by these subalgebras or it is contained in one of these subalgebras. Thus, some orbits lay either on the axis Ox or on Oy, other orbits are contained in the plane xOy or yOz (but have no intersection with Ox and Oy) and the other ones are contained inside of one of the four quadrants defined in R3 by planes xOy and yOz. The lattice of subalgebras for algebra A2◦ is 2 2 {{0}, Re1, Re2, S panR{e2, ae1 + be3} (a + b , 0), A}. A comparison of the two lattices allows us to decide that A1◦ and A2◦ are not isomorphic. On the other hand, each solution of the corresponding HQDS (S) is contained in a subalgebra, i.e. it is necessarily a null torsion curve. Indeed, a straightforward computation confirms us that the triple scalar product (r’, r”, r”’) = 0 where 2 2 r’ = (2xy + xz)e1 + (γz )e2 + (2yz + z )e3. Moreover, each trajectory of a HQDS associated to one of the algebras A16◦, A17◦, A18◦ has null torsion, too. Classification of a class of quadratic differential systems with derivations 67 5. CONCLUSIONS The analyzed class of commutative algebras is divided into eighteen classes of non-isomorphic algebras. Accordingly, the corresponding class of homogeneous quadratic differential systems is partitioned into eighteen classes of affinely nonequivalent systems. Since A is a vector direct sum of an ideal and a subalgebra, each such a system is decoupled into subsystems that can be solved. The most part of classes of algebras have Der A = RD, so that the corresponding orbits of solutions have just a few symmetries. The algebras in the classes A2◦, A7◦, A8◦, A12◦, A14◦, A15◦ are the only algebras that have two-dimensional derivation algebras.

References

[1] I. B, Quadratic differential systems, Publ. House PIM, 2008.(Romanian) [2] I. B, A classification of a class of homogeneous quadratic dynamical systems on R3 with derivations, Bull. I.P. Ias¸i, Sect. Matematica, Mecanica teoretica, Fizica, T.LIV (2008), 37–47. [3] I. B, Homogeneous quadratic dynamical systems on R3 having derivations with complex eigenvalues, Libertas Mathematica, XXVIII (2008), 69–92. [4] I. B, Classification of Quadratic Differential Systems on R3 having a nilpotent of Order 3 Derivation, Libertas Mathematica, XXIX (2009), 47-64. [5] I. B, Automorphisms and derivations of homogeneous quadratic differential systems, Ro- mai J., 6, (12010), 15-28. [6] I. B, Classification of Quadratic Differential Systems on R3 having a nilpotent of Order 2 Derivation (in press). [7] T. D, Classifications and Analysis of Two-dimensional Real Homogeneous Quadratic Differ- ential Equation Systems, J. Diff. Eqs., 32(1979), 311–334. [8] I. K, Algebras with many derivations, ”Aspects of Mathematics and its Applications” (ed. J.A. Barroso), Elsevier Science Publishers B.V., 1986, 431. [9] K. M. K , A. A. S, Quadratic Dynamical Systems and Algebras, J. of Diff. Eqs, 117 (1995), 67–127 . [10] K. M. K , A. A. S, Automorphisms ans Derivations of Differential Equations and Al- gebras, Rocky Mountain J. of Mathematics, 24, 1,(1994), 135–153. [11] L. M, Quadratic Differential Equations and Non-associative Algebras, in ”Contributions to the Theory of Nonlinear Oscillations” Annals of Mathematics Studies, no. 45, Princeton Uni- versity Press, Princeton, N. Y., 1960. [12] A.A. S, R. W, Introduction to Lie groups and Lie algebras, Academic Press, New York, 1973. [13] R. D. S, An Introduction to Nonassociative Algebras, Academic Press, New York, London, 1966. [14] N. I. V, K. S. S˘ı, Geometrical Classification of Quadratic differential systems (Rus- sian), Differentialnye Uravnenje, 13, 5(1977), 803–814.

ROMAI J., 6, 2(2010), 69–81

ON THE 3D-FLOW OF THE HEAVY AND VISCOUS LIQUID IN A SUCTION PUMP CHAMBER Mircea Dimitrie Cazacu1, Cabiria Andreian Cazacu2 1Polytechnic University of Bucharest, Romania 2”Simion Stoilow” Institute of Mathematics of the Romanian Academy, Bucharest, Romania [email protected], [email protected] Abstract We present the essence of the theoretical and experimental researches, developed in the field of the three-dimensional flow with free surface of a heavy and viscous liquid in the suction chamber of a pump, with or without air carrying away or appearance of the dangerous cavitational phenomenon, problem of a special importance for a faultless working of a pump plant, especially of these relevant to the nuclear electric power plants, which constituted the object of our former preoccupations. From the mathematical point of view we present the solving particularities of the two-dimensional boundary problems, specific to the three-dimensional domain limits of the pump suction chamber flow, as well as the possibility to obtain a stable numerical solution. From the practical point of view, we show the possibilities of laboratory modelling of this complex phenomenon, by simultaneously use of many similitude criterions, as well as the manner to determine the characteristic curves, specific to each designed pump suction chamber. Because the drawings of the pump station of Romanian nuclear power station Cer- navoda on Danube are classified, we shall present the theoretical and experimental researches performed on the model existent in the Laboratory of New Technologies of Energy Conversion and Magneto-Hydrodynamics, founded after the world energetic cri- sis from 1973 [1] in the POLITEHNICA University of Bucharest, Power Engineering Faculty, Hydraulics, Hydraulic Machines and Environment Engineering Department.

Keywords: liquid three-dimensional flow with free surface, three-dimensional flow in the pump suction chamber, complex modelling of liquid three-dimensional flow with free surface. 2000 MSC: 35 G 30, 35 J 25, 35 J 40, 35 Q 30, 65 N 12, 76 D 17.

1. INTRODUCTION The theoretical research concerning the three-dimensional flow of the real liquid in a suction chamber of a pump, [2], [3], is justified by the flow effect with or without appearance of whirls, with or without air training [4] or by the appearance of the cavitation destructive phenomenon at the liquid entry in a pump, as well as by their negative influence on the hydraulic and energetic pump efficiency, the quick cavitational erosions, the machine vibrations followed by the fast wear, eventually by the

69 70 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu pump suction less and by their pumping cease, extremely dangerous for the pumping station, for instance of a nuclear electric power plant [1]. These situations may come in the sight in special work conditions linked with the low level of the water in the river bed or in the pump suction chamber, or by an ineﬃcient design of this installation for diﬀerent cases of pumps working [5], [6].

2. NUMERICAL INTEGRATION OF THE HEAVY AND VISCOUS LIQUID THREE-DIMENSIONAL FLOW IN A PUMP SUCTION CHAMBER We present the mathematical specific boundary conditions [2], [3], which intervene in the problem of the three-dimensional viscous liquid steady flow into the suction chamber of a pump plant, represented in the Figure 1. Thus, for the uniform flow of the viscous and heavy liquid in the entrance and going out sections of the domain, we must consider the two-dimensional special boundary problems: - for the fluid entrance section we must solve a mixed Dirichlet-Neumann problem for a Poisson equation with an unknown a priori constant, depending of the slope angle α of the suction chamber bottom, which can be determined by a successive calculus cycle of the computer program, satisfying a normalization condition, which, for the given value of mean input velocity and entrance section dimensions, ensures the desired flow-rate, - for the fluid going out pipe section we must solve a Dirichlet problem for another 0 Poisson equation, having the pressure drop gradient pz in the suction pipe also as unknown a priori constant, which can be determined also by a successive calculus cycle satisfying a normalization condition and which establishes the pressure drop gradient on the pipe length for the same flow-rate, satisfying another normalization condition.

2.1. PARTIAL DIFFERENTIAL EQUATIONS OF A VISCOUS AND HEAVY LIQUID STEADY FLOW The three-dimensional motion equations in Cartesian trihedron (Fig.1) are

0 0 0 0 1 0 U + U U + U V + U W + P = ν(U” 2 + U” 2 + U” 2 ) + g sin α, (1) T X Y Z ρ X X Y Z

0 0 0 0 1 0 V + V U + V V = V W + P = ν(V” 2 + V” 2 + V” 2 ), (2) T X Y Z ρ Y X Y Z

0 0 0 0 1 W + W U + W V + W W + = ν(W” 2 + W” 2 + W” 2 ) + g cos α, (3) T X Y Z ρ X Y Z On the 3D-flow of the heavy and viscous liquid in a suction pump chamber 71 the mass conservation equation for an incompressible fluid being 0 0 0 UX + VY + WZ = 0, (4) where, as usual, (U, V, W) is the velocity of the fluid, P is its the pressure, ρ - the fluid density, and ν - its kinematical viscosity. T and X, Y, Z represent time and spatial independent variables, respectively.

2.2. THE DIMENSIONLESS FORM OF THE EQUATION SYSTEM For more generality of the numerical solving [3], [5] - [8] we shall work with dimensionless equations, by using the characteristic physical magnitudes of flow: - the suction chamber width B H, approximately equal with its height, - the liquid mean velocity of the entrance in suction chamber Um = Q/BH, - the air atmospheric pressure on its free surface P0, - and the period T0 of whirl appearance in the three dimensional flow. With the new dimensionless variables and functions: X Y Z T U V W P x = , y = , z = , t = , u = , v = , w = , p = , (5) H H H T0 Um Um Um P0 the partial differential equations (1) to (4) become in the dimensionless form:

0 0 0 0 0 1 sin α 0 Sh ut + uxu + uyv + uzw + Eu px = ∆ u + , (1 ) Re x,y,z Fr 72 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu

0 0 0 0 0 1 0 Sh vt + vxu + vyv + vzw + Eu py = ∆ v, (2 ) Re x,y,z

0 0 0 0 0 1 cos α 0 Sh wt + wxu + wyv + wzw + Eu pz = ∆ w + , (3 ) Re x,y,z Fr in which one puts into evidence the following criteria of hydrodynamic flow simi- 2 larity, numbers of: Strouhal Sh = H/T0Um, Euler Eu = P0/ρUm, Reynolds Re = 2 HUm/ν and Froude Fr = Um/gH; the mass conservation equation is an invariant 0 0 0 0 ux + vy + wz = 0. (4 ) 2.3. THE NUMERICAL SOLVING METHOD The numerical integration of partial differential equation system was performed by an iterative calculus of unknown functions u, v and w, given by the algebraic relations associated to the partial differential equations, by introducing their expressions deduced from the finite Taylor’s series developments [8] in a cubical grid, having the same step δx = δy = δz = χ:

0 f1,2,5 − f3,4,6 f1,2,5 − 2 f0 + f3,4,6 f = and f ”2 2 2 = . (6) x,y,z 2χ x ,y ,z χ2 2.4. THE BOUNDARY CONDITIONS FOR THE THREE-DIMENSIONAL STEADY FLOW - on the solid walls, due to the molecular adhesion condition, we consider

uSW = vSW = wSW = 0 (7) and consequently, from equation (3’) for α ≈ 0 the hydrostatic pressure distribution becomes γH pSW(z) = 1 + z = 1 + Ar z, (8) P0 in which we introduced the hydrostatic similarity number Ar = γH/P0, dedicated to the Archimedean lift discoverer, - on the free surface, considered to be a plane due to the liquid important weight and small ﬂow velocities, we have the conditions

pFS(x, y, 0) = 1 and wFS(x, y, 0) = 0, (9) excepting a null measure set of points constituted by the whirl centers on the free surface, in whose neighbourhood the ﬂuid has a descent motion (Fig. 1). Neglecting the liquid friction with the air, we shall cancel both shearing stress components, obtaining: 0 0 0 τzx |FS= uz + wx = uz |FS= 0, → u6 = u5 (10) 0 0 0 τxy |FS= vx + wy = vz |FS= 0, → v6 = v5. On the 3D-ﬂow of the heavy and viscous liquid in a suction pump chamber 73

The value w6 is calculated from the mass conservation equation (4’), written in finite differences (that is not unstable in this local appliance [5] [6]) u − u v − v w − w 1 3 + 2 4 + 5 6 = 0 → w | = w + u − u + v − v , (11) 2x 2x 2x 6 FS 5 1 3 2 4 - in the entrance section, we consider the uniformly and steady flow at the normal depth of the current parallel with the channel bottom slope i = sin α, which leads us to the condition v |ES= w |ES= 0 and to the hydrostatic pressure repartition, the distribution of u(y, z) velocity component being obtained from the first motion equation (1’), which in these boundary conditions become a Poisson type equation with the constant α a priori unknown, written also in finite differences

Re 1 X4 χ2 Re ∆ u + sin α = 0 → u0 = ui + sin α. (12) x,z Fr 4 4 Fr i=1 The numerical solving is possible by an iterative cycle on the computer, until to the obtaining of slope value i, necessary to the given Re and Fr numbers corresponding to the ﬂow velocity Um, when the heavy force component g sin α is balanced by the interior friction forces, acting by the liquid adhesion to the channel bottom and its solid walls. The boundary conditions for the Poisson equation (12) for solving in the frame of plane the mixed problem Dirichlet-Neumann

USW = 0 on the solid walls, 0 =zx|FS= µUz = 0 on the free surface, lead us to solve a mixed problem Dirichlet-Neumann, whose normalization condition is ZB ZH Ui(Y, Z)dYdZ = Q = BHUm, (13) 0 0 or, in the iterative numerical calculus for the dimensionless case,

j −1 j −1 j −1 kXM−1 XN χ2 XN χ2 XN χ2 kXM−1 χ2 u + u + u + u = q = 1, (130) k,j 2 kM,j 8 k=2,j 4 k,j=2 k=2 j=2 j=2 j=2 k=2 representing the ﬂow-rate which is brought by this velocity repartition in the entrance section, for the given values of Um, therefore Re and Fr numbers, as well the considered grid step χ. For the starting of the calculus it is necessary to introduce only a single velocity value in a point as a seed, for instance uS = 2 in the point j = 20 and k = 40 in the middle of the free surface, in the rest of the domain the initial arbitrary values being 74 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu equal with zero, or, by admittance of a paraboloidal repartition a the initial given arbitrary values concerning the velocity distribution in the domain, in the shape

2 1 1 2 1 2 uva(y, z) = 8(1 − z ) · ( − y) · ( + y) = 8(1 − z ) · ( − y ), 2 2 4 (14) 2 uva( j, k) = 4us · a ( j − 1) · (k − 1) · [2(k − 1) · a][2( j − 1) · a], because we have 1 y = ( j − 1)a − and y = 1 − (k − 1)a, (15) 2 - in the exit section, we shall consider that the streamlines are parallel with the vertical pipeline walls, uES = vES = 0, the velocity component w(x, y) distribution in the uniformly and steady ﬂow verifying also a Poisson type equation with an a priori 0 unknown constant pz, but this time due to the ignorance of the pressure drop on the 0 vertical pipeline pz, deduced from the third motion equation (3’) in approximation cos α ≈ 1, determined above by solving the entry section problem

X4 0 Re 1 2 1 0 Re Eu pz = cos α + ∆ w, → w0 = [ wi + χ Re( − Eu pz)]. (16) Fr x,y 4 Fr i=1 The boundary condition to solve the Poisson equation (16) in the frame of Dirichlet problem, is given by the liquid adhesion wES = 0 on the interior solid wall of suction pipe. The Poisson type equation solving, in which one does not know a priori the pres- 0 sure drop pz along the pipe, can be made by numerical way, also by an iterative 0 cycle on computer, yielding up to the arbitrary values pz & 0.1, which exceed a few the pressure distribution deduced from the hydrostatic repartition (8), until the velocity integration wES in the exit section (Fig. 2) shall equalize the unitary value of dimensionless ﬂow-rate, calculated in the case of non-symmetrical or symmetrical supposed velocity repartitions, using the forms of [6]   X20 X10 1 X X9 1 X20 X  q ≈ χ2 ·  w + · w + · w  = 1, nonsym  ij 8 ij 8 ij i=17 j=7 i∈{17, 2} j=8 i=18 j∈{7, 1} (17)

2 qsym ≈ χ · [9w17, 8 + 4(w18, 8 + w17, 7)] = 1. 2.5. CONCLUSIONS ON THE USED BOUNDARY SPECIFIC PROBLEMS IN THE 3D-FLOWS The Dirichlet problem, proposed by the German mathematician Peter Gustav Lejeune Dirichlet, being generally speciﬁc to the equations of elliptic type, corresponding to the repartition of scalar physic magnitudes given by harmonic functions On the 3D-ﬂow of the heavy and viscous liquid in a suction pump chamber 75

(for example of the temperature distribution in a homogeneous medium, without convection i.e. U = V = 0, and without interior heat sources in virtue of Laplace equation) has here a speciﬁc character. The Neumann problem formulated by the German physicist Carl Gottfried Neumann, consists in giving, on the boundary C of the domain D, the 0 values of the normal derivative of the function, fn.

To specify the unknown function values in the domain, it is necessary also in this case to fulﬁl the so-called normalization condition, giving the ﬂux through the frontier line

Z 0 φ = fndl. (18) C

In the three-dimensional case of mixed boundary problems, by combination of these two classical problems, to solve the Poisson equations with unknown constants, intervene the well-known normalization conditions, even for the Dirichlet problem only. Initially, the computational program has been made in Cobol [4]], due to the great number of the data, working successively to relax the three velocity components values in a plane, considering their values also in the two neighbouring planes, the graphical representation of the three-dimension flow being shown by the three velocity components as in Figure 2. 76 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu 2.6. THE IMPORTANCE OF THE COMPLEX PHENOMENA FROM A PUMP SUCTION CHAMBER FLOW The faultless working of the pump plant for the nuclear electric power stations, which must work even in the case of nuclear reaction stopping, yet 10 to 12 hours till the complete cooling of the reactor, is the main aim to ensure the nuclear security. In the second part of this paper we present the experimental research performed, regarding the complex modelling of the spatial flow and the specific phenomena, which can be produced in the pump suction chamber. The research has been started in the year 1977 together with my former student and then collaborator Dr. Eng. Costel Iancu [2] for the pump station design of the nuclear power station Cernavoda in Romania [1] and continued by the construction of the Model installation endowed with the measuring apparatus for a good working research of a pump suction (Fig. 1 to 3) in the Laboratory of new technologies of energy conversion and environment protection, realized in the year 1979 with the help of the former Eng. PhD student Dumitru Ispas [3]-[6].

3. THE COMPLEX MODELLING OF THE HEAVY AND VISCOUS LIQUID 3D- FLOW IN A PUMP SUCTION CHAMBER After doctoral Thesis defending in 1957, M.D.Cazacu published a work on the laboratory complex modelling of the complicated water-hammer phenomenon [9], considering not only the dimensionless motion equation, but also the mass conservation and state equation of the compressible liquid, as well as the pipe elastic deformation with inﬂuence upon the sound propagation velocity. He decided at that time to pay attention to this very important phenomenon, taking into consideration the high security level of a pump plant for a nuclear electric power plant, since the research on the laboratory model was not conclusive in the consulted technical literature in this ﬁeld.

3.1. THE EXPERIMENTAL INSTALLATION DESCRIPTION AND MEASURING APPARATUS The experimental installation is presented in Fig. 3, and it consists in 1-stained glass chamber, 2-supporting feet, 3-tightening cap, 4-air tap, 5-flow reassure, 6- cen- trifugal pump, 7-direct current electric motor, 8-pump pressing pipeline, 9-discharge control valve, 10-flow meter, 11-pump suction pipeline, 12-suction basket, 13-emptying tap, 14-air carrying away whirl, 15-whirl on the bottom, 16-blades against the whirls. On the 3D-flow of the heavy and viscous liquid in a suction pump chamber 77

The water level in the pump suction chamber may be changed through the tap 13, the water leaving the chamber through the pump suction pipe 12 producing two whirls without or with air carrying away 14, which due to the rotational flow ω caused by the pump suction flow rates, smaller then the nominal flow rate, change their positions: one remaining on the free surface, but at the other chamber side the second removing on the chamber bottom, where with the solid particles uncoupled from the concrete and concentrated in the whirl axis, due to the bigger flow velocities, leads at a strong erosion of the pump suction chamber.

For the measuring of different physical magnitudes we utilized the apparatus: - a mechanic tachometer contacted with the electric motor shaft end, to measure the pump number of rotations n, - a diaphragm to measure the pump flow rate Q, - a graduated rule from transparent plastic to measure the water level Z in the suction chamber, fixed at the suction chamber glass, - a manual chronometer to measure the frequency of whirl appearance, - a precision metallic mano-vacuum-meter or a mercury manometer tube to measure the air pressure at the suction chamber free surface.

3.2. THE FLOW COMPLEX MODELLING USING MORE SIMILARITY CRITERIONS The short history concerning these ineﬃcient foregoing modelling attempts began any decades before, when the researchers tried to utilize the very known Froude’s criterion for the modelling of this gravitational ﬂow in a pump suction chamber, but which gave on the Model, smaller then the Reality, built at the length scale 78 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu

λ = H/H0 = D/D0 > 1, lesser velocities at which the whirls could not to appear

V2 V02 V Q Fr = = Fr0 = → β = = λ0,5 ≥ 1 and k = = λ2β = λ2,5, (19) gH g0H0 V0 Q0 in which we noted with β the velocity ratio and with k the flow rate ratio. For this reason, in the last decade of that period, the researchers, without no justi- fication, have enlarged the velocities on the Model till the same values as the velocities of the Reality, introducing the so called equal velocity criterion, for which M. D. Cazacu gave a proof considering Weber’s criterion, thinking at the free surface whirls with air carrying away similarity σ σ0 We = = We0 = → β = 1 = λ0 and then k = λ2, (20) ρV2 ρ0V02 where σ is the air-water superficial tension. As a good expert in the recommendations of the pump manufacturers and pump station designers, that the water access to the pump suction baskets must be as possible direct and because the estimation of the separation of viscous fluid flow is given by Reynold’s number, he considered also this kind of modelling, supposing that the water cinematic viscosity is the same VD V0D0 V υ Re = = Re0 = → β = = λ−1 ≈ λ−1 and k = λ, (21) υ υ0 V0 υ0 in which the velocities on the Model are greater than those from the Reality. Another very important problem was the modelling of the influence of pump working, especially at lesser flow rates than that nominal. He introduced the pump modelling by its rapidity criterion nq

nQ1/2 n0Q01/2 n = = n0 = , (22) q H3/4 q H03/4 from which we can choose the pump number of rotations on the laboratory Model

k1/2 n0 = n = nλ1/4β1/2 (23) λ3/4 and determine the convenient geometric scale in the case of composed similarity 0 Froude and pump rapidity nq, in which case he gave in Table 1 the pressure P0 necessary at the water free surface in suction chamber, to beneﬁt also of the cavitation phenomenon modelling at pump or at the strong erosive whirl, which ends on a side wall or on the bottom of the suction chamber built from the concrete, in the aim to eliminate his appearance in the Reality [5]. On the 3D-ﬂow of the heavy and viscous liquid in a suction pump chamber 79

Table 1

n’[rot/min] 3.000 1.500 1.000 750 600 n = 500 λ = (n0/n)2 36 9 4 2, 25 1, 44 1 P0 − P0 (T 0) 0 v = P0 − Pv(T) = 1/β2 = 1/λ = σ/σ0 0, 0277 0, 111 0, 25 0, 444 0, 6944 1 0 P0(bar) 0, 261 0, 342 0, 43 0, 5778 0, 7678 1

In this table one sees the advantage of the length scale λ = 4, corresponding to the standardized number of rotations 1,000 rot/min of the pump model, inclusively under pressure that we must assure at the model free surface, as well as that which assures only the cavitational phenomenon similarity, for which we must have also accomplished the equality relation of the cavitational criterion

P − P (T) P0 − P0 (T 0) σ = 0 v = σ0 = 0 v . (24) ρν2/2 ρ0ν02/2

Finally, in the case of Strouhal’s similarity, concerning the whirl appearance frequency in time, we shall have the relation, from which we can deduce the frequency of whirl appearance in reality as function of that obtained on the model

D f D0 f 0 D0V Sh = = Sh0 = → f = f 0 . (25) V V0 V0D

The symmetrical position (Fig. 1) of the two whirls from the right part can be modified by the pump working at littler flow rates (Fig. 3), by ω rotation impressed to the chamber flow as in the left part, when one whirl arrive to us and the other being disposed vertically on the bottom chamber (Fig. 4), causing its accelerated wear.

3.3. CONCLUSIONS AND IMPORTANT REMARKS With this occasion we shall observe that the greater models become too expensive and the smaller models introduce forces of other nature, which modify completely the development of the hydrodynamic process. On the basis of this modelling method one sees, that we can study a complex hydro-aerodynamic similarity using 6 criterions, as: Froude, Weber, Reynolds, pump rapidity, cavitation criterion and Strouhal, not only individually but also simultaneously. 80 Mircea Dimitrie Cazacu, Cabiria Andreian Cazacu

In this way we can also determine the convenient technical-economically length scale of the Model, to have also a good modelling similarity. In the same time we can determine the whirl apparition frequency in reality, that is very important because the erosion of the pump suction chamber walls is however a problem of time and we can also determine the zones that can be covered with metallic plates and one can determine for any pump plant, working at diﬀerent exploitation situations its characteristic curves (Fig. 4), which determine the region of a good operation.

References

[1] M.D.Cazacu, C.Iancu, Studiu privind aspirat¸ia ¸siaduct¸iunea pompelor mari de circulat¸ie din termocentrale CNE ¸sia criteriilor de similitudine pentru verificarea pe model (Study regarding the suction and delivery of the great circulation pumps from the Electric Nuclear Power Sta- tions and the similarity criterions for the check on the model). Contr. nr.201/1977, Beneficiary − ICEMENERG − Bucharest. [2] M.D.Cazacu, Gh.Baran, A.Ciocanea, High education and research development concerning the recoverable, inexhaustible and new energy sources in the POLITEHNICA University of Bucharest. Internat. Conf. on Energy and Environment, 23-25 Oct.2003, Polytechnic Univ. of Bucharest, Sect. 5 - Energetic and Educational Politics, Ses.2 - Educational and Environmental protection Politics, 5, 120-125. [3] M.D.Cazacu, On the boundary conditions in three-dimensional viscous flows. The 5th Congress of Romanian Mathematicians, June 22-28, 2003, University of Pitesti, Romania, p. 24. [4] M.D.Cazacu, Boundary mathematical problems in viscous liquid three-dimensional flows. Inter- nat. Conf. on Theory and Applications of Mathematics and Informatics - ICTAMI 2007, Alba Iulia, Acta Universitatis Apulensis, Mathematics-Informatics, No. 15 / 2008, 281 - 289. [5] C.Iancu, Contribut¸ii teoretice ¸siexperimentale la studiul curgerii fluidelor cu mi¸scare de rotat¸ie ¸sivârtejlegat de suprafet¸e libere (Theoretical and experimental contributions at the study of fluid flows with rotation motion and whirl bound of free surfaces). Doctoral Thesis defended in 1970 at the Polytechnic Institute of Bucharest, Romania. On the 3D-flow of the heavy and viscous liquid in a suction pump chamber 81

[6] M.D.Cazacu, D.Ispas, Cercet˘ariteoretice ¸siexperimentale asupra curgerii lichidului vˆıscosˆın camera de aspirat¸ie a unei pompe (Theoretical and experimental research on the viscous liquid flow in a pump suction chamber), Conf. of Hydraulic Machines and Hydrodynamics, Timisoara, Romania, 18 - 19 Oct. 1985, vol. III, 67-74. [7] M.D.Cazacu, D.A.Ispas, M.Gh.Petcu, Pipe-channel suction flow researches. Proc. of XXth IAHR Congress (Subject B.a ), Moscow SSRU, September 5 - 9, 1983, VI, 214 - 221. [8] M.D.Cazacu, On the solution stability in the numerical integration of non-linear with partial differential equations. Proceedings of the Internat. Soc. of Analysis its Appl. and Computation, 17-21 Sept. 2002, Yerevan, Armenia. Complex analysis, Differential equations and Related topics. Publishing House Gitityun, 2004, Vol. III, 99-112. [9] D.Dumitrescu, M.D.Cazacu, Theoretische und experimentelle Betrachtungen überdie Strömung zäherFlüssigkeiten um eine Platte bei kleinen und mittleren Reynoldszahlen. Zeitschrift fur¨ Angewandte Mathematik und Mechanik, 1, 50, 1970, 257- 280. [10] M.D.Cazacu, La similitude hydrodynamique concernant le phénomènedu coup de bélier. Rev. Roum. de Sci. Techn. Mec.´ Appl., Rom. Acad. Publ. House, Tome 12, No. 2, 1967, Bucharest, 317- 327.

ROMAI J., 6, 2(2010), 83–91

FIXED POINT THEOREMS IN E - METRIC SPACES Mitrofan M. Choban, Laurent¸iu I. Calmut¸chi Department of Mathematics, Tiraspol State University, Chi¸sin˘au,Republic of Moldova [email protected] Abstract The aim of the present article is to give some general methods in the ﬁxed point theory for mappings of general topological spaces. Using the notions of the multi-metric space and of E-metric space, we prove the analogous of several theorems about common ﬁxed points of commutative semigroups of mappings.

Keywords: ﬁxed point, m-scale, semiﬁeld, multi-metric space, E-metric space, pseudo-metric. 2000 MSC: 54H25, 54E15, 54H13, 12J17, 54E40.

1. INTRODUCTION In the sequel, any space is considered to be Tychonoff and non-empty. We use the terminology from [6, 7]. Let R be the space of real numbers, ω = {0, 1, 2, ...} and N = {1, 2, ...}. In [1, 2] the metric spaces over topological semifields were introduced. The general concept of the metrizability of spaces is contained in [14]. The notion of a topological semifield may be generalized in the following way. We say that E is a metric scale or, briefly, an m-scale if: 1. E is a topological algebra over the field of reals, R; 2. E is a commutative ring with the unit 1 , 0; 3. E is a vector lattice and 0 ≤ xy provided 0 ≤ x and 0 ≤ y; 4. For any neighborhood U of 0 in E there exists a neighborhood V of 0 in E such that if x ∈ V and 0 ≤ y ≤ x, then y ∈ U. 5. For any non-empty upper bounded set A of E there exists the suppremum ∨A. (The set A is upper bounded if there exists b ∈ E such that x ≤ b for any x ∈ A.) If the set A is not upper bounded, then we put ∨A = ∞. From the condition 4 it follows: 6. If 0 ≤ yn ≤ xn and limn→∞ xn = 0, then limn→∞yn = 0 too. From the condition 5 it follows: 7. If the non-empty set A ⊆ E is lower bounded, then there exists the infimum ∧A. Any topological semifield is an m-scale. A topological product of m-scales is an m-scale.

83 84 Mitrofan M. Choban, Laurent¸iu I. Calmut¸chi

Let E be an m-scale. Denote by E−1 = {x : x · y = 1 for some y ∈ E} the set of all invertible elements of E. By N(0, E) we denote some base of the space E at the point 0. n We consider that 0 ≤ x 1 if 0 ≤ x < 1, 1 − x is invertible and limn→∞ x = 0. We put E(+,1) = {x ∈ E : 0 ≤ x 1}. If t ∈ R and 0 ≤ t < 1, then t · 1 ∈ E(+,1). We identify t with t · 1 ∈ E for each t ∈ R. A mapping d : X × X −→ E is called a pseudo-metric over m-scale E or a pseudo- E-metric if it satisﬁes the following axioms: d(x, x) = 0, (∀)x ∈ E; d(x, y) = d(y, x), (∀)x, y ∈ E; d(x, y) ≤ d(x, z) + d(z, y)(∀)x, y, z ∈ E.

Every pseudo-E-metric is non-negative, i.e. d(x, y) ≥ 0 for all x, y ∈ X. An E-pseudo-metric d is called an E-metric if it satisﬁes the following axiom:

d(x, y) = 0 if and only if x = y. The ordered triple (X, d, E) is called a metric space over m-scale E or an E-metric space if d is an E-metric on X. Let d be a pseudo-E-metric on X. If A ⊆ B, B ⊆ X, x ∈ X, r ∈ E and r ≥ 0, then d(A, B) = in f {d(x, y): x ∈ A, y ∈ B} and B(x, d, r) = {y ∈ X : d(x, y) < r}. If U ∈ N(0, E), then we put B(x, d, U) = {y ∈ X : d(x, y) ∈ U} for any x ∈ X. The family {B(x, d, U): x ∈ X, U ∈ N(0, E)} is the base of the topology T(d) of the pseudo-E-metric space (X, d, E). If d is an E-metric, then the space (X, T(d)) is a Tychonoff (completely regular and Hausdorff) space. The pseudo-E-metric d generates on X the canonical equivalence relation: x ∼ y iff d(x, y) = 0. Let X/d be the quotient set with the canonical projection πd : X −→ X/ρ ¯ −1 −1 and the metric d(u, v) = ρ(πd (u), πd (v)).

Deﬁnition 1.1. Let A be a non-empty set and {Eα : α ∈ A} be a family of m-scales. A multi-Eα-metric space is a pair (X, P), where X is a set and P = {dα : X × X −→ Eα : α ∈ A} is a non-empty family of pseudo-Eα-metrics on X satisfying the condition: x = y if and only if d(x, y) = 0 for each d ∈ P.

Remark 1.1. Let A be a non-empty set and {Eα : α ∈ A} be a family of m-scales. Then E = Π{Eα : α ∈ A} is an m-scale. If (X, P) is a multi-Eα-metric space, then: 1. d(x, y) = (dα(x, y): α ∈) is an E-metric on X and T(d) is the topology of the multi-Eα-metric space X. 2. Let ρα(x, y) = {bβ ∈ Eβ : β ∈ A}, where bα = dα(x, y) and bβ = 0 for all β , α. Then Q = {ρα : α ∈ A} is a family of pseudo-E-metrics on X and T(d) is the topology of the multi-E-metric space (X, Q). Fixed point theorems in E - metric spaces 85

Thus we may consider that E = Eα for any α ∈ A in the Definition 1.1. In particular, any multi-E-metric space is an E0-metric space. If E = R, then the pseudo-E-metric is called a pseudo-metric and the pseudo-E- metric space is called a pseudo-metric space. Fix a multi-E-metric space (X, P). A subset V ⊆ X is called P-open if for any x ∈ V there exists a set U ∈ N(0, E) and a finite set A = A(x, U) ⊆ P such that B(x, A, U) = ∩{B(x, ρ, U): ρ ∈ A} ⊆ V. The family T(P) of all P-open subsets is a completely regular Hausdorff topology on X. If T is a completely regular Hausdorff topology on X, then T = T(P) for some family P of pseudo-E-metrics on X (see [6]). If d(x, y) = (ρ(x, y): ρP), then d is an EP-metric on X. Let X be a space and F be a non-empty set of mappings f : X −→ X. If Fix( f ) , ∅ for each f ∈ F, then X is called a fixed point space relative to F. By Fix(F) = {x ∈ X : f (x) = x for any f ∈ F} we denote the set of all common fixed points of the mappings f ∈ F. If ϕ : X −→ X is a mapping, then by Coin(F, ϕ) = {x ∈ X : f (x) = ϕ(x) for any f ∈ F} we denote the set of all coincidence points of the mappings f ∈ F and mapping ϕ. The excellent book [7] contains the fixed point theory for metric spaces with the important applications in distinct domains. Several results for general topological spaces with interesting applications are contained in the surveys [7, 12]. Distinct generalizations of the Banach fixed point principle were proposed in [3, 7, 12]. Dis- tinct classes of metrizable spaces were examined in [8, 9, 14]. In [16], the following general problem arose it was arisen the following general problem: to find topological analogies of the Banach fixed point principle. Some solutions to this problem were proposed in [5, 17]. This article is a continuation of the works [5, 16, 17, 11]. For each family γ of subsets of a set X and any x ∈ X the set S t(x, γ) = ∪{U ∈ γ : x ∈ U} is the star of the point x relative to γ. We put S t(A, γ) = ∪{S t(x, γ): x ∈ A}.

2. COMPLETE MULTI-METRIC SPACES Fix an m-scale E and an E-metric space (X, d, E). A set {xµ : µ ∈ M} is a net or a generalized sequence if M is a directed set. A point x ∈ X is a limit of the net {xµ : µ ∈ M} and we put limµ∈M xµ = x if for any U ∈ N(0, E) there exists λ ∈ M such that d(x, xµ) ∈ U for any µ ≥ λ. A net {xµ : µ ∈ M} is called fundamental if for any U ∈ N(0, E) there exists λ ∈ M such that d(xµ, xη) ∈ U for any µ, η ≥ λ. Any convergent net is fundamental. The limit of a fundamental sequence is unique (if the limit exists). The space (X, d, E) is called complete if any fundamental net is convergent (see [6, 1, 2]).

Example 2.1. Let A be an uncountable set. We put Iα = I = [0, 1] for any α ∈ A. A Let Y = I = Π{Iα : α ∈ A}, 0A = (0α : α ∈ A) ∈ Y and X = Y \{0A}. The space E = RA is a topological ﬁeld and an m-scale. By construction, X ⊆ IA ⊆ E. We put d(x, y) = (|xα − yα| : α ∈ A) and dα(x, y) = |xα − yα| for any pair of points 86 Mitrofan M. Choban, Laurent¸iu I. Calmut¸chi

x = (xα : α ∈ A) ∈ Y and y = (yα : α ∈ A) ∈ Y. Let P = {dα : α ∈ A}. Then (X, d, E) is an E-metric space and (X, P) is a multi-metric space. Obviously, T(d) = T(P). The −n space Y is compact and the space X is pseudocompact. We put xn = (2α : α ∈ A). Then {xn : n ∈ N} is a fundamental sequence. We have limP xn = 0A. Thus the spaces (X, d, E) and (X, P) are not complete.

Remark 2.1. Let (X, d, E) be an E-metric space and the space E be ﬁrst countable, i.e. it is metrizable in the usual sense. Then the space X is metrizable in the usual sense too.

3. CLOSURE OPERATORS ON PSEUDO-E-METRIC SPACES Fix an m-scale E, a topological space X and a pseudo-E-metric d on a space X. Denote by exp(X) the set of all subsets of X. A family γ ⊆ exp(X) is centered if ∩η , ∅ for any ﬁnite subfamily η ⊆ γ. If A is a non-empty subset of X, then diam(A) = ∨{d(x, y): x, y ∈ A}. Consider that diam(∅) = 0. A closure operator on the pseudo-E-metric space (X, d, E) is a mapping C : exp(X) −→ exp(X) satisfying the conditions: 1. if A ⊆ B, then C(A) ⊆ C(B); 2. A ⊆ C(A) and diam(C(A)) = diam(A); 3. C(C(A)) = C(A). Let C be a closure operator on X. The set A ⊆ X is C-closed if C(A) = A. The space X is called C-compact if ∩γ , ∅ for any centred family γ of non-empty C- closed sets. If the space X is compact and C(A) = clX A is the closure of the set A, then C is a closure operator on X and X is a C-compact space.

4. NIEMYTZKI FIXED POINT THEOREM FOR MULTI-METRIC SPACES The next theorem for compact metric spaces due to V. Niemytzki (see [15, 12]) and L.P.Belluce and W.A.Kirk [4].

Theorem 4.1. Let f, ϕ : X −→ X be mappings of a countably compact multi-metric space (X, P, R) into itself, ϕ(X) = X and d(ϕ(x), ϕ(y)) ≥ d(x, y) for all points x, y ∈ X and d ∈ P. Assume that d( f (x), f (y)) < d(x, y) for any pair of distinct points x, y ∈ X and any d ∈ P. Then: 1. The set Fix( f ) = {x ∈ X : f (x) = x} is non-empty and it is a singleton. 2. The set Coin( f, ϕ) = {x ∈ X : f (x) = ϕ(x)} is non-empty and it is a singleton.

Theorem 4.2. Let f, ϕ : X −→ X be mappings of a countably compact multi-metric space (X, P, R) into itself, ϕ(X) = X and d(ϕ(x), ϕ(y)) > d(x, y) for any pair of distinct Fixed point theorems in E - metric spaces 87 points x, y ∈ X and any d ∈ P. Assume that d( f (x), f (y)) ≤ d(x, y) for all points x, y ∈ X and d ∈ P. Then: 1. the set Fix(ϕ) = {x ∈ X : ϕ(x) = x} is non-empty and it is a singleton. 2. the set Coin( f, ϕ) = {x ∈ X : f (x) = ϕ(x)} is non-empty and it is a singleton.

Proof. The mapping ϕ is one-to-one and the mappings f and ϕ−1 are uniformly continuous. Moreover, if g = ϕ−1 ◦ f , i.e. g(x) = ϕ−1( f (x)) for any x ∈ X, then d(g(x), g(y)) < d(x, y) for any pair of distinct points x, y ∈ X and any d ∈ P. Fix ρ ∈ P. Consider the projection πρ : X −→ X/ρ of (X, ρ) onto the metric space (X/ρ, ρ¯) and the mapping gρ : X/ρ −→ X/ρ, where g(πρ(x)) = πρ(gρ(x)) and ρ(x, y) = ρ¯(g(x), g(y)) for all x, y ∈ X. By construction, X/ρ is a compact metric space andρ ¯(g(x), g(y)) < ρ¯(x, y) for all x, y ∈ X/ρ. Suppose that for any n ∈ ω there exists xn, yn ∈ X such that −n ρ(g(xn), g(yn) ≥ (1 − 2 )ρ(x.y). Since X/ρ is a metrizable compact space, there exist an inﬁnite subset L ⊆ ω, an inﬁnite subset M ⊆ L and two points x, y ∈ X such that: −n - πρ(x) is the limit of the subsequence {πρ(xn): n ∈ L} and ρ(x, xn) < 2 for any n ∈ L; −n - πρ(y) is the limit of the subsequence {πρ(yn): n ∈ M} and ρ(y, yn) < 2 for any n ∈ M. There exists m ∈ ω such that 2−m+2 < ρ(x, y) − ρ(g(x), g(y)). By construction, ρ(g(x), g(y)) = lim ρ(g(xn), g(yn)) = lim ρ(xn, yn) = ρ(x, y), a contradiction. Thus there exists a number kρ < 1 such that ρ(g(x), g(y)) ≤ k · ρ(x, y) for all x, y ∈ X. Fix x = x0. We put xn = g(xn−1) for any n ∈ N. There exists b ∈ E such that b · (1 − k) = 1. Then d(xn, xn+1) = d(g(xn−1), g(xn)) ≤ k · d(xn−1, xn) for any n ∈ N. Hence n d(xn, xn+1) ≤ k · d(x0, x1) for any n ∈ N. Obviously, d(xn, xm) ≤ d(xn, xn+1 + n n+1 m−1 m ... + d(xm−1, xm) ≤ (k + k + ... + k + k ) · d(x0, x1). Therefore d(xn, xm) ≤ n n+m n (k − k ) · b · d(x0, x1) provided n, m ∈ N and n ≤ m. Since limn→∞k = 0, the sequence {xn : n ∈ ω} is fundamental in (X, ρ). Since the space X is countably compact and ρ is an arbitrary pseudo-metric, there exists a unique point b ∈ X such that b = lim xn. Then g(b) = lim g(xn) = lim xn+1 = b. Thus f (b) = ϕ(b). The assertion 2 is proved. The assertion 1 follows from the assertion 2. The proof is complete.

The assertion 1 for compact metric spaces was proved by V.Niemytzki (see [15, 12]) in 1936.

5. MAPPINGS OF E-METRIC SPACES WITH DIMINISHING ORBITAL DIAMETERS Fix an m-scale E. Let (X, d, E) be a pseudo-E-metric space and let X be endowed with the topology T(d) generated by the pseudo-E-metric d. Obviously, clA is a closure operator on X. 88 Mitrofan M. Choban, Laurent¸iu I. Calmut¸chi

Consider a commutative semigroup F of mappings f : X −→ X. We can assume that the identity mapping eX : X −→ X is an element of F. For any point x ∈ X the set F(x) = {(g(x)) : g ∈ F} is the F-orbit of x. Let r(x, F, d) = ∧{diam(F(g(x)) : g ∈ F}. Since F is a semigroup, the family F2(x) = {F(g(x)) : g ∈ F} is centered. The set L ⊆ X is F-invariant if g(L) ⊆ L for any g ∈ F. Let In(x, F, d) = ∩{H ⊆ X : x ∈ X, clH = H, H is F-invariant}. Obviously, F(x) ⊆ In(x, F, d). A semigroup F is said to have diminishing orbital diameters (d.o.d.) (see [4, 10, 13]) if for every point x ∈ X we have r(x, F) < diam(F(x)) provided diam(F(x)) > 0. Our deﬁnition is more general then that in [13]. The set VCoin(F) = {x ∈ X : d(x, f (x)) = 0 for any f ∈ F} is the set of all d-ﬁxed points of the family F. If d is a E-metric, then VCoin(F) = Coin(F).

Theorem 5.1. Let F : X −→ X be a commutative semigroup of continuous mappings with diminishing orbital diameters of the pseudo-E-metric space (X, d, E) and the space (X, T(d)) is compact. Then VCoin(F) , ∅. Moreover, clF(x) ∩ VCoin(F) , ∅ for any point x ∈ X.

Proof. We can assume that the identity mapping eX ∈ F. Since the mappings F are continuous, the closure of an invariant set is invariant. Fix a point b ∈ X. If diam(F(b)) = 0, then b ∈ F(b) ⊆ VCoin(F). Assume that diam(F(b) > 0. Since F has d.o.d. and the space (X, T(d)) is compact, there exists a minimal invariant d-closed subset H of X such that diam(H) < ∞ and H ⊆ clF(b). We aﬃrm that diam(H) = 0. Assume that diam(H) > 0. Fix c ∈ H. If diam(F(c)) = 0), then P = clF(c) ⊆ H, diam(P) = 0 and P is a d-closed invariant set, in contradiction with the minimality of the set H. Thus 0 < diam(F(c)) ≤ diam(H). Since F has d.o.d., there exists a point y ∈ F(c) such that diam(F(y)) < diam((F(c)). Then B = clF(y) ⊆ H, diam(B) < diam(H) and B is a d-closed invariant set, a contradiction with the minimality of the set H. Therefore diam(H) = 0. Then H ⊆ clF(x) ∩ VCoin(F).

Corollary 5.1. Let F : X −→ X be a commutative semigroup of mappings with diminishing orbital diameters of the compact E-metric space (X, d, E). Then Coin(F) , ∅. Moreover, clF(x) ∩ Coin(F) , ∅ for any point x ∈ X.

The Corollary 4.2 for the compact metric spaces was proved in [4, 10, 13].

Remark 5.1. If E = R and F : X −→ X is a commutative semigroup of continuous mappings with diminishing orbital diameters of the pseudo-E-metric space (X, d, E), then d( f (x), f (y)) ≤ d(x, y) for all x, y ∈ X and f ∈ F. In particular, the mappings F are continuous. Fixed point theorems in E - metric spaces 89

6. MAPPINGS OF MULTI-E-METRIC SPACES WITH DIMINISHING ORBITAL DIAMETERS Fix an m-scale E. Let (X, P) be a multi-E-metric space and let X be endowed with the topology T(P) generated by the pseudo-E-metrics P. Obviously, clA is a closure operator on X. For any A ⊆ X and d ∈ P the element diamd(A) = ∧{d(x, y): x, y ∈ A} is the d-diameter of the set A. We put diam(A) = {diamd(A): d ∈ P}. The element b = (bd : d ∈ P) is called an E-P-number if bd ∈ {∞} ∪ E for any d ∈ P. If b = (bd : d ∈ P) and c = (cd : d ∈ P) are two E-P-numbers, then b < c if and only if bd ≤ cd for all d ∈ P and bρ < cρ for some ρ ∈ P. By construction, diam(A) is an E-P-number. Consider a commutative semigroup F of mappings f : X −→ X. We can assume that the identity mapping eX : X −→ X is an element of F. For any point x ∈ X the set F(x) = {(g(x): g ∈ F} is the F-orbit of x. Let r(x, F, d) = ∧{diamd(F(g(x)) : g ∈ F} and r(x, F) = (r(x, F, d): d ∈ P). By construction, r(x, F) is an E-P-number. Let In(x, F) = ∩{H ⊆ X : x ∈ X, clH = H, H is F-invariant}. Obviously, F(x) ⊆ In(x, F). A semigroup F is said to have diminishing orbital diameters (d.o.d.) if for every point x ∈ X we have r(x, F) < diam(F(x)) provided diam(F(x)) > 0.

Theorem 6.1. Let F : X −→ X be a commutative semigroup of continuous mappings with diminishing orbital diameters of the multi-E-metric space (X, P) and the space (X, T(P)) is compact. Then Coin(F) , ∅. Moreover, clF(x) ∩ Coin(F) , ∅ for any point x ∈ X.

Proof. The proof is similar to the proof of Therem 4.1. We can assume that the identity mapping eX ∈ F. Since the mappings F are continuous, the closure of an invariant set is invariant. Fix a point b ∈ X. If diam(F(b)) = 0, then b ∈ F(b) ⊆ Coin(F). Assume that diam(F(b) > 0. Since F has d.o.d. and the space (X, T(d)) is compact, there exists a minimal invariant closed subset H of X such that diam(H) < diam(F(b)) and H ⊆ clF(b). We aﬃrm that diam(H) = 0. Assume that diam(H) > 0. Fix c ∈ H. If diam(F(c)) = 0), then P = clF(c) ⊆ H, diam(P) = 0 and P is a closed invariant set, a contradiction with the minimality of the set H. Thus 0 < diam(F(c)) ≤ diam(H). Since F has d.o.d., there exists a point y ∈ F(c) such that diam(F(y)) < diam((F(c)). Then B = clF(y) ⊆ H, diam(B) < diam(H) and B is a closed invariant set, in contradiction with the minimality of the set H. Therefore diam(H) = 0. Then H ⊆ clF(x) ∩ Coin(F).

By the virtue of the following example, the method of multi-metrics is more eﬀec- tive than the method of metrics. 90 Mitrofan M. Choban, Laurent¸iu I. Calmut¸chi

Example 5.2. Let Xn = [0, 1] and ρn(x, y) = (n + 1) · |x − y| for all x, y ∈ Xn and n ∈ ω. We put X = Π{Xn : n ∈ ω} and d(x, y) = (n + 1) · |xn − yn| for all x = (xn : n ∈ ω) ∈ X, y = (yn : n ∈ ω) ∈ X and n ∈ ω. If P = {dn : n ∈ ω}, −1 then (X, P) is a metrizable multi-metric space. Let f (x) = (2 xn : n ∈ ω) and −n−1 g(x) = (2 xn : n ∈ ω) for any x = (xn : n ∈ ω) ∈ X. Denote by F the semigroup with identity generated by the mappings f and g. Since f g = g f , the semigroup F is ω commutative. If t ∈ R, then t¯ = (xn = t : n ∈ ω) ∈ R . If 0¯ < x = (xn : n ∈ ω) ∈ X, then diam(F(g(x)) < diam(F(x) and diam(F(g(x)) < 1.¯ Moreover, r(x, F) = 0 for any x ∈ X. Thus the semigroup F has diminishing orbital diameters. By construction, −1 diam(F(t¯)) = (2 ·(n+1)·t : n ∈ ω) for any t ∈ [0, 1]. If t > 0, then sup{diamdn (F(t¯): n ∈ ω} = ∞.

7. ASYMPTOTICAL PROPERTIES OF MAPPINGS Fix an m-scale E. Let E(−,1) = E(+,1) ∩ E−1. If ∈ E(−,1) and 0 < t < 1, then t · , t · 1 ∈ E(−,1). Consider an E-metric space (X, d, E) and a non-empty commutative semigroup F : X −→ X of continuous mappings of X into itself. A semigroup F : X −→ X is called asymptotically non-expansive (see [10, 13]) if for any x, y ∈ X there exists g ∈ F such that d( f g(x), f g(y)) ≤ d(x, y) for all f ∈ F. If A ⊆ X, then the set AF = {z ∈ X : there exists x ∈ A such that for every f ∈ F and ∈ E(−,1) there exists g ∈ F with d( f g(x), z) < } is called the F-closure of the set A in X.

Proposition 7.1. Let F : X −→ X be a non-empty commutative semigroup of continuous asymptotically non-expansive mappings of an E-metric space (X, d, E) into itself. If z ∈ XF, then: 1. For all f ∈ F and ∈ E(−,1) there exists g ∈ F such that d( f g(z), z) < }. 2. For any f ∈ F the mapping f |F(z) is an isometrical embedding of F(z) into F(z). 3. diam(F(z)) = diam(F(g(z))) for all g ∈ F.

Proof. Is similar to the proof of Propositions 1 and 2 from [10].

Corollary 7.1. ([13], Proposition 1, for metric space). Let F : X −→ X be a non- empty commutative semigroup of continuous asymptotically non-expansive mappings of an E-metric space (X, d, E) into itself. If the semigroup F has diminishing orbital diameters and z ∈ XF, then F(z) = {z} and z ∈ Coin(F). References

[1] M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, An outline of the theory of topological semiﬁelds, Russian Math. Surveys, 21:4 (1966), 163-192. Fixed point theorems in E - metric spaces 91

[2] M. Ya. Antonovskii, V. G. Boltyanskii, T. A. Sarymsakov, Topological semifields and their applications to general topology, American Math. Society. Translations, Series 2, 106, 1977. [3] V. Berinde, M. Choban, Remarks on some completeness conditions involved in several common fixed point theorems, Creative Mathematics and Informatics, 19:1(2010), 3-12. [4] L. P. Belluce, W. A. Kirk, Fixed point theorems for certain classes of nonexpansive mappings, Proceed. Amer. Math. Soc., 20 (1969), 141-146. [5] L. Calmut¸chi, M. Choban, On mappings with fixed points, Buletin S¸tiint¸ific. Universitatea din Pites¸ti, Matematica s¸i Informatica,˘ 3 (1999), 91-96. [6] R. Engelking, General Topology, PWN. Warszawa, 1977. [7] A. Granas, J. Dugundji, Fixed point theory, Springer, New York, 2003. [8] G. Gruenhage, Generalized Metric spaces, In. Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan, eds., Elsevier, Amsterdam, 1984. [9] G. Gruenhage, Metrizable spaces and generalizations, In: Recent Progress in General Topology II, M. Husekˇ and J. van Mill, eds., Elsevier, Amsterdam, 2002, 201-225. [10] R.D.Molmes, P.P.Narayanaswami, On asymptotically nonexpansive semigroups of mappings, Canadian Math. Bull., 13 (1970), 209-214. [11] K. Iseki, On Banach theorem on contractive mappings, Proceed. Japan Academy, 41 :2 (1965), 145-146. [12] A. A. Ivanov, Fixed points of mappings of metric spaces, Journal of Soviet Mathematics 12:1 (1979), 1-64. [13] Mo Tak Kiang, Semigroups with diminishing orbital diameters, Pacific J. Math. 41:1 (1972), 143-151. [14] S. I. Nedev, M.M.Choban, General conception of the metrizability of topological spaces, Annales of the Sophia University, Mathematics, 65 (1973), 111-165. [15] V. Niemytzki, The method of fixed points in analysis, Uspekhi Matem. Nauk, 1 (1936), 141-174. [16] J. Stepfans, S. Watson, W. Just, A topological fixed point theorem for compact Hausdorff spaces, York University, Preprint, 1991. [17] I. V. Yashchenko, Fixed points of mappings and convergence of the iterations of the dual mappings, In: Obshchaya Topologia: Otobrajenia, Proizvedenia i Razmernosti Prostranstv, Moskva: Moskov. Gosud. Uni-t, 1995, 131-142.

ROMAI J., 6, 2(2010), 93–105

DISLOCATIONS AND DISCLINATIONS IN FINITE ELASTO-PLASTICITY Sanda Cleja-T¸igoiu Faculty of Mathematics and Computer Science, University of Bucharest, Romania [email protected] Abstract The paper deals with analytical description of the dislocation and disclination, which are lattice defects of the crystalline materials. The evolution equations for disclinations, having as source the screw dislocations are derived within the constitutive framework of second order plasticity developed by the author in the papers appeared in Int. J. Fracture (2007), (2010).

Keywords: dislocation, disclination, elastic and plastic distortions, micro and macro forces, balance equations, connections, evolution and constitutive equations. 2000 MSC: 74C99, 74A20.

1. INTRODUCTION The plastic deformability of metals, which are crystalline materials, is produced because of the existence of lattice defects inside the microstructure. The lattice defects, among which the dislocations, disclinations and point defects are mathematically modeled by the differential geometry concepts, as torsion, curvature and metric property of certain connection, see Kroner¨ [10], [11], de Wit [7], but without any elasto-plastic constitutive equations. The elastic models for crystal defects can be found in Teodosiu [14]. We are not dealing with curved space but with curved geometry in flat space as stated de Wit [7]. 1. The nature of the geometry is determined by the linear connection Γ, fixed by its coefficients, the curvature tensor R, the Cartan torsion or torsion tensor S; 2. the metric tensor C, to measure the distance; 3. the non-metricity measure Q, of the connection relative to the metric tensor, i.e. in terms of Γ and C. Geometry for which R, S, Q are non-vanishing is non-metric, non-Riemannian. We restrict ourself to the case Q = 0, which means that the geometry is metric. If R = 0 the geometry is called flat. If S = 0 the geometry is called symmetric. If S = 0 the geometry is called Riemannian. If R = 0, S = 0 geometry is called Euclidean. In this paper the dislocations and disclinations are lattice defects of interest. The dislocations are characterized by the Cartan torsion, or by the non-zero curl of plastic connection, which means that the plastic distorsion can not be derived from a certain potential. The disclinations are characterized by a non-vanishing curvature R.

93 94 Sanda Cleja-T¸igoiu

The mathematical description of the continuously distributed dislocations is given by Noll [13], and the differential geometry support within the context of continuum mechanics can be found also in [12]. In this paper a peculiar mathematical problem is analyzed: find the disclinations, which are solutions of the appropriate evolution equation in such a way that the micro balance equation are satisfied, when the distribution of the dislocations is given. To give the mathematical description of the problem, we precise the general constitutive framework which is able to capture the dislocations and disclinations. We mention here the direction developed by Clayton et al. [3] within a micropolar elasto- plastic model, in order to emphasize the translational (dislocation) and rotational (disclinaton) defects. The behaviour of elasto-plastic body is described within the constitutive framework of second order plasticity, see Cleja-T¸igoiu [5], [6], based on the decomposition rule of the motion connection into the elastic and plastic second order deformations, see Cleja-T¸igoiu [4], and on the existence of configurations with torsion. The so called configuration with torsion is denoted by K, and it is described through a pair, composed by Fp, an invertible second order tensor which is called plastic distorsion (p) and Γ, a third order tensor field which is called plastic connection. (p) The pair (Fp, Γ) defines the second order plastic deformation with respect to the reference configuration of the body B. The actual configuration of the body is associated with t he motion function, which is defined at every time t by χ(·, t): B −→ E, E being the Euclidean (flat space) with a three dimensional vector space V. The second order elastic deformation, as a consequence of the decomposition rule, is defined with respect to the so called configuration with torsion K, which is time dependent. Two type of forces, macro and micro forces, are considered in the model. They are viewed as pairs of stress (a second order tensorial field) and stress momentum (a third order tensorial field), and they obey their own balance laws. The micro forces satisfy the viscoplastic type constitutive equations, in Kt. The evolution equations for Kt, which means the plastic distortion and plastic connection, have to be given. They have been derived to be compatible with an appropriate dissipation postulate. The energetic arguments, like a virtual power principle, macro and micro balance equations, and especially a dissipation postulate (see energy imbalance principle, for instance in Gurtin [9]) in order to prove thermomechanics restrictions, see Gurtin [9], Cleja-T¸igoiu [5]. In our adopted formalism the measure of the dislocations is characterized by the non-vanishing plastic curl (see Bilby [1], Noll [13]), while the disclinations has been related to a certain second order tensor Λ which enters the expression of the plastic connection and generates a non-zero curvature, apart from de Wit [7], where a measure of disclination is considered to be a second order curvature tensor. The following notations, definitions and relationships will be used in the paper: Dislocations and disclinations in finite elasto-plasticity 95

u · v, u × v, u ⊗ v denote scalar, cross and tensorial products of vectors; a ⊗ b and a ⊗ b ⊗ c are a second order tensor and a third order tensor defined by (a ⊗ b)u = a(b · u), (a ⊗ b ⊗ c)u = (a ⊗ b)(c · u), for all vectors u. For A ∈ Lin (Lin- the space of second order tensors), we introduce: the tensorial product A ⊗ a for a ∈ V, is a third order tensor, with the property (A ⊗ a)v = A(a · v), ∀v ∈ V. I is the identity tensor in Lin, AT denotes the transpose of A ∈ Lin, ∇A is the derivative (or the gradient) of the field A in a coordinate system {xa} ∂Ai j (with respect to the reference configuration), ∇A = ei ⊗ e j ⊗ ek. ∂xk Definition 1.1. The curl operator is defined for any smooth second order field, say A, through (curlA)(u × v) = (∇A)u)v − ((∇A)v)u, ∀ u, v, z ∈ V. (1) Lin(V, Lin) = {N : V −→ Lin, linear}− defines the space of all third order i j k tensors and it is given by N = Ni jki ⊗ i ⊗ i . T The scalar product of two second order tensor A, B is A · B := tr(AB ) = Ai jBi j, and the scalar product of third order tensors is given by N · M = Ni jk Mi jk, in a Carte- sian coordinate system. AB denotes the composition of A, B ∈ Lin. The compositions of A ∈ Lin and N, a third order tensor, defines the appropriate third order tensorial i j i p fields, via the formulae NAu = Ni jkAkpupi ⊗ i , and ANu = Ai jN jpkuki ⊗ i , which are written in a Cartesian basis, for any vector u. The affine connection is defined in a coordinate system by its coordinate representation i m k Γ = Γmkei ⊗ e ⊗ e . (2) We introduce the third order tensor field Γ[F1, F2], which is generated by a third order field Γ together with the second order tensors F1, F2 through the formula

(Γ[F1, F2]u)v = (Γ(F1u)) F2v, ∀u, v ∈ V. (3)

For any Λ1, Λ2 ∈ Lin we deﬁne a third order tensor associated with them, denoted Λ1 × Λ2, by ((Λ1 × Λ2)u)v = (Λ1u) × (Λ2v), ∀u, v ∈ V. (4)

For A, a third order tensor, we deﬁne the vector ﬁeld tr(2)A through the relationship written for all vectors (tr(2)A) · u = tr(Au). (5)

Three types of second order tensors, A B, A r B and A l B will be associated with any pair A, B of third order tensors, following the rules written for all L ∈ Lin

(A B) · L = A[I, L] · B = AiskLsnBink (A r B) · L = A · (LB) = Ai jkLinBn jk (6) (A l B) · L = A · (BL) = Ai jkBi jnLkn. 96 Sanda Cleja-T¸igoiu 2. GEOMETRIC RELATIONSHIPS Let F(X, t) = ∇χ(X, t) be the deformation gradient at time t, X ∈ B, and Γ = F−1∇F be the motion connection or the material connection. ∇χF is a gradient in the actual conﬁguration, while the gradient in the conﬁguration with torsion K, ∇KF, is calculated by p −1 ∇KF := (∇F)(F ) . (7) Ax.1 The decomposition of the second order deformation, (F, Γ), associated with (e) (p) e p the motion of the body B, into elastic, (F , Γ K), and plastic, (F , Γ), second order deformations is given by F = FeFp, (p) (e) (8) p −1 p p −1 Γ =Γ +(F ) Γ K [F , F ], Γ = F ∇F.

(p) Here the plastic connection with respect to the configuration with torsion, Γ K, is (p) related with the plastic connection,Γ, previously defined with respect to the reference configuration, by the following relationships

(p) (p) p p −1 p −1 (9) Γ K= −F Γ [(F ) , (F ) ]. The plastic metric tensor Cp and strain gradient C are defined with respect to the reference configuration, while the elastic metric tensor Ce is defined in configuration with torsion by Cp := (Fp)T Fp, Ce = (Fp)−T C(Fp)−1, C = FT F. (10) Definition 2.1. The Bilby’s type plastic connection is defined in a coordinate system through (p) A:= (Fp)−1∇Fp. (11) Remark 2.1. The Cartan torsion that belongs to the Bilby’s connection is given by (Su)v = (Fp)−1 ((∇Fp)u)v − ((∇Fp)u)v , while the curvature tensor is vanishing. Moreover, if the second order torsion tensor N is defined by N(u × v) = (Su)v, then it has the representation N = (Fp)−1curlFp. Consequently, we can say that the torsion tensor S is equivalent to curlFp. Let us introduce the expression for the plastic connection with respect to the reference configuration built by Cleja-Tigoiu in [5], which has metric property with respect to Cp, and that allows a represented under the form

(p) (p) Γ=A +(Cp)−1(Λ × I), (12) Dislocations and disclinations in ﬁnite elasto-plasticity 97 where the third order tensor Λ × I, generated by the second order (covariant) tensor Λ is deﬁned by (4), namely ((Λ × I)u)v = Λu × v, ∀ u, v ∈ V. (13) Λ is called the disclination tensor.

3. SCREW DISLOCATION First the skew dislocation will be defined in connection with the definition of the Burgers vector. The Burgers vector can be defined in terms of the plastic distortion p F , by considering a closed curve (circuit) C0 in the reference configuration of the body and A0 a surface with normal N bounded by C0 Z Z Z p p b = F dX = (curl(F ))NdA = αKnKdAK, (14) C0 A0 AK where αK is Noll’s dislocation density in [13] 1 α ≡ (curl(Fp))(Fp)T . (15) K detFp The expression of the Burgers vector can be aproximate by the formula p b ' curl(F )N area(A0). (16) In crystal plasticity the presence of the defects inside the crystals is measured by non- vanishing Burgers vector. The integral representation (14) shows that non-zero curl of plastic distortion, supposed to be continuum and non-zero in a certain material neighborhood, leads to a non-vanishing Burgers vector. Definition 3.1. We say that the plastic distortion Fp characterizes a screw dislocation if the generated Burgers vector through a circuit with the appropriate normal N is collinear with the normal, i.e. b k N, in contrast with the edge dislocation when b ⊥ N. p Let us introduce a Cartesian basis (e1, e2, e3) and a plastic distortion F which defines a screw dislocation that correspond to a Burgers vector directed to e3, given by p F = I + e3 ⊗ τ, τ ⊥ e3, with (17) 2 p p p p τ : D ⊂ R −→ V, τ := F31e1 + F32e2, J := det(F ) = 1. The mathematical description of the problem related to the screw dislocation within the finite elasticity is performed by Cermelli and Gurtin [2]. If we consider that τ = τ(x1, x2), the curl of the plastic distortion, see (1), is given by p p ∂F ∂F curl Fp = 31 − 32 e ⊗ e , (18) ∂x2 ∂x1 3 3 98 Sanda Cleja-T¸igoiu and Bilby’s type plastic connection

(p) ∂τβ A:= (Fp)−1∇Fp = e ⊗ ∇τ, ∇τ = e ⊗ e , β, δ ∈ {1, 2}. (19) 3 ∂xδ β δ The representation given in (18) justiﬁes the name attributed to the plastic distortion introduced in (17), taking into account the expression for the Burgers vector, for a plane curve with the normal e3. (p) Let us remark its appropriate trace tr(2) A, deﬁned by the formula (5)

(p) (p) ∂τ3 tr A ·u := tr(A u) = tr((e ⊗ ∇τ)u) = e ∇τ(u) = uk = 0, (20) (2) 3 3 ∂xk is zero, since τ3 : τ · e3 = 0, τ being a vector function orthogonal on e3. The plastic metric tensor Cp has the representation

p C = I + τ ⊗ e3 + e3 ⊗ τ + τ ⊗ τ. (21) Let us introduce the disclination tensor Λ represented in terms of Frank vector ω, like in Cleja-T¸igoiu [5] Λ := ηω ⊗ ζ, (22) ζ the tangent vector line for the disclination field, and the scalar valued function η have to be defined. It follows that ∇Λ := ω ⊗ ζ ⊗ ∇η, Λ˙ := η˙ω ⊗ ζ, (23) if we take constant value for ζ. Hypothesis. We assume that Frank and Burgers vectors are orthogonal, ω · b = 0, here b = e3. This hypothesis corresponds to the physical meaning attributed to these types of lattice defects. See for instance [3]. Remark 3.1. In Fig.1 the Burgers vector produced by the screw dislocation and the rotation produced by the disclinations have been plotted, following the comments from [3]. 4. MACRO AND MICRO FORCES Within the constitutive framework developed in [6], we consider that the free energy in the reference configuration can be introduce in terms of the deformation fields listed below (p) (p) ψ = ψ(Ce, Fp, A, Λ, ∇Λ) ≡ ψev(C, Fp, A) + ψd(Λ, ∇Λ), (24) but in contrast with [6] we have no influence of the elastic connection. Concerning the free energy function we assume that the defect energy is given by κ ψd(∇Λ) = 2 β2∇Λ · ∇Λ, (25) 2 2 Dislocations and disclinations in finite elasto-plasticity 99

Fig. 1. Lattice defects: Screw dislocaton with Burgers vector b, disclination with Frank vector ω.

with β2 a length scale parameter. Ax.2 The macro forces are T− the non-symmetric Cauchy stress and macro stress momentum, µ, which is described by a third order tensor ﬁeld, satisfy the macro balance equation div T + ρb = 0, −2Ta = div µ, (26)

Ta is the skew symmetric part of the stress tensor. The equations (26) are similar to those proposed in Fleck et al. [8], see also Cleja-Tigoiu [4], [6] We denote by ρ, ρ,˜ ρ0 the mass densities in the actual configuration, in the configuration with torsion and in the reference one. When we pull back the macro stress µ to the configuration with torsion, the appropriate expression for the macro stress momentum µK is derived

1 e T 1 e −T e −T e p −1 µK := (F ) µ[(F ) , (F ) ], where F = F(F ) . (27) ρ˜0 eρ

Ax.3 The micro stress denoted by Υλ and micro momentum respectively, denoted by µλ, which are associated with the disclinations satisfy in K their own balance equation (see [6]):

λ λ λ p λ p λ p −T λ Υ = divK µ + ρ˜B ⇐⇒ J Υ = div J µ (F ) + ρ˜B . (28)

Hereρ ˜Bλ is mass density of the couple body force, J p =| detFp | . Ax.4 The micro stress Υp and micro stress momentum µp are associated with the plastic mechanism, and they satisfy their own balance equation in the conﬁguration 100 Sanda Cleja-T¸igoiu with torsion K : p p λ Υ = divK (µ − µK) + ρ˜B ⇐⇒ (29) p p p p p −T p J Υ = div J (µ − µK)(F ) + ρ˜B . Hereρ ˜Bp is appropriate mass density of the couple body force.The equation (29) could be found in a modiﬁed form in [5]. The constitutive restrictions imposed by the free energy imbalance, that have been obtained by Cleja-T¸igoiu in [6], are adapted to the proposed here model as it follows. The elastic type constitutive functions are derived from the free energy function, viewed like a potential for Cauchy stress 1 T = 2F(∂ ψ)FT , µ = ∂ ψ, (30) ρ C Γ but here µ = 0, as a consequence of the supposition made in (24) that the free energy function is not dependent on Γ. We postulate here an energetic (non-dissipative) constitutive equation for the micro stress momentum associated with plastic behaviour and disclinations, respectively

1 p µ0 = ∂(p)ψ ρ0 A (31) 1 λ d 2 µ0 = ∂∇Λψ (∇Λ) = κ2β2 ω ⊗ ζ ⊗ ∇η. ρ0 The last equality is a consequence of (25) together with (23). The expressions for the micro stress plastic momentum µp and the macro stress momentum µK both of them being considered with respect to K, pulled back to the reference conﬁguration cab be expressed under the form

1 p p T 1 p p −T p −T µ0 := (F ) µ [(F ) , (F ) ], ρ0 eρ (32) 1 p T 1 p −T p −T µ0 := (F ) µK[(F ) , (F ) ], ρ0 eρ as well as the micro stress momentum associated with the disclination in K can be λ expressed in terms of µ0 by

1 λ p −T 1 λ p T p T µ := (F ) µ0[(F ) , (F ) ]. (33) ρ˜ eρ0 The viscoplastic type, dissipative evolution equations for plastic distorsion Fp and Λ have been postulated in [6] to be compatible with the appropriate dissipation inequality. The viscoplastic evolution equation for plastic distortion, giving rise to the rate Dislocations and disclinations in finite elasto-plasticity 101 of plastic distortion with respect to the reference configuration is given by lp = −(Fp)−1F˙ p 1 p p T p (34) (Σ0 − Σ0 ) + (F ) ∂Fp ψ = ξ1 l . ρ0 Definition 4.1. The Mandel type stress measures in the reference configuration are associated with the appropriate stresses as it follows

1 p 1 p T p p −T 1 1 T −T Σ0 = (F ) Υ (F ) , Σ0 = F TF (35) ρ0 ρ˜ ρ0 ρ and 1 λ 1 p T λ p −T Σ0 = (F ) Υ (F ) . (36) ρ0 ρ˜ The viscoplastic type, dissipative evolution equations are postulated for disclination Λ (p) 1 λ 1 λ Σ0 − ∂Λ ψ + A µ0 − ρ0 ρ0 (37) (p) (p) 1 λ 1 λ ˙ − µ0 r A − µ0 tr(2)(A) = ξ3 Λ. ρ0 ρ0 Ax.5 The scalar constitutive functions ξ1, ξ3 are deﬁned in such a way to be compatible with the dissipation inequality p p ˙ ˙ ξ1 l · l + ξ3 Λ · Λ ≥ 0. (38) Note that the dissipation inequality is reduced to the expression written in the left hand side of (39) if the viscoplastic type equations (34) and (38) have been accepted. λ The Mandel type stress measure associated with the disclination mechanism, Σ0 , is λ λ related to the micro stress Υ via (37), while µ0 is expressed in (31). 5. DISCLINATIONS GENERATED BY A PLASTIC DISTORTION We suppose that the second order disclination tensor Λ is described in terms of Frank vector ω, with the scalar intensity function η and the disclination line ζ, say constant during the deformation process, have to be found. We are now able to solve the problem: Find the disclination tensor Λ, having the expression (22), to be solution of the evolution equation (38) with the micro stresses related by (37), and which is compatible with the micro balance equation associated with the disclination, (28)2. We take into account the physically motivated hypothesis that the Frank and Burg- ers vectors are (ﬁxed) orthogonal, as well as that the plastic distorsion (17) characterizes the screw dislocation, then

ω · e3 = 0, τ · e3 = 0, ∇(τ)u · e3 = 0, ∀ u ∈ V. (39) 102 Sanda Cleja-T¸igoiu

First of all we eliminate the micro stress from (38), via (28) together with (37). By deﬁnitions and the hypotheses, from (33) together with (31)2 we get λ p −T p −T p T µ (F ) = ρ0(F ) ∂∇Λψ[I, (F ) ]. (40) As a direct consequence of (24) together with (25) and (23) we can prove the following relationship

1 λ p −T 2 µ (F ) = κ2β2 {ω ⊗ ζ ⊗ ∇η + (ζ · e3)(ω ⊗ ζ ⊗ ∇η)}. (41) ρ0 Consequently, we apply the divergence operator to (42) and then

λ p −T 2 div µ (F ) = κ2β2ρ0 ∆η{ω ⊗ ζ + (ζ · e3) ω ⊗ τ}+ (42) 2 +κ2β2ρ0(ζ · e3) ω ⊗ ((∇ τ)∇η) holds. Here ∆η denotes the Laplacean of the scalar function η. Proposition 5.1. In the evolution equation for Λ, written in (38) the term

(p) 1 λ 2 A µ0 = κ2β2 e3 ⊗ ∇τ ω ⊗ ζ ⊗ ∇η , (43) ρ0 is vanishing, since its components are given by

(p) 1 λ 2 A µ0 sn = κ2β2 (es · ζ) en · (∇τ)∇η (e3 · ω) = 0, (44) ρ0 while the following expression

(p) 1 λ 2 µ0 r A = κ2β2 ζ · ∇τ ∇η e3 ⊗ ω. (45) ρ0 has to be introduced inside. In order to prove the above relationships we calculate the components of the second order tensor ﬁeld, starting from the deﬁnitions introduced in (6). The scalar product of the tensor written in the left hand side of (44) with (en ⊗ es) gives rise to the components sn of the tensor. We perform the tensorial operations and we arrive at 2 κ2(β2) (e3 ⊗ ∇τ) (ω ⊗ ζ ⊗ ∇η) · (en ⊗ es) =

2 = κ2(β2) (e3 ⊗ ∇τ)[I, en ⊗ es] · ω ⊗ ζ ⊗ ∇η = (46) 2 T = κ2(β2) (e3 ⊗ es ⊗ (∇τ) en) · ω ⊗ ζ ⊗ ∇η =

2 = κ2(β2) (e3 ⊗ ω)(es · ζ)en · (∇τ)∇η . Dislocations and disclinations in ﬁnite elasto-plasticity 103

Note the formula proved below via (3) has been used in the previously written expression, namely

(e3 ⊗ ∇τ)[I, en ⊗ es]u = (e3 ⊗ (∇τ)u)(en ⊗ es) = (47) = (e3 ⊗ es)(∇τ)u · en) = (e3 ⊗ es ⊗ (∇τ)en)u, hold for all u ∈ V, since ∇τ ∈ Lin. We proceed similarly to the calculus of the term written in the left hand side of (46). We take the scalar product with (en ⊗ es) and we use the deﬁnitions introduced in (6) 1 (p) µλ A · (e ⊗ e ) = ρ 0 r n s 0 (48) (p) 2 = κ2β2 ζ · ∇τ ∇η · (en ⊗ es) A, as well as the formula

(en ⊗ es)(e3 ⊗ ∇τ) = δ3sen ⊗ ∇τ. (49) We use the balance equation for micro forces associated with the dislocation (28) p T p −T composed on the left with (F ) = I + τ ⊗ e3 and on right with (F ) = I − τ ⊗ e3 in order to express the Mandel’s type stress tensor (37)

1 λ 2 Σ0 = κ2β2 {∆η ω ⊗ ζ + (ζ · e3)ω ⊗ τ + ρ0

2 (50) +(ζ · e3)ω ⊗ ((∇τ)∇η)} − κ2β2 ∆η(ζ · τ)(ω ⊗ e3)−

2 2 −κ2β2 ∆η(ζ · e3){∆η | τ | +((∇τ)∇η) · τ}ω ⊗ e3. Proposition 5.2. The evolution equation for Λ written in (38) becomes

1 λ 2 Σ0 − κ2β2 ζ · ∇τ ∇η e3 ⊗ ω = ξ3 η˙ω ⊗ ζ, (51) ρ0

(p) where the ﬁrst term is given by (51), since tr(2) A= 0. We investigate now the consequences that follows from (52) and (51). If ζ · e3 = 0, the projection on ζ of the equation (52) together with (51) leads to

2 ξ3η˙ = κ2β2∆η. (52) When we return to the equation (52), we conclude that the following conditions (ζ · τ)∆η = 0, ζ · ∇τ ∇η = 0 (53) 104 Sanda Cleja-T¸igoiu necessarily hold. If ζ · e3 , 0, as ω · e3 = 0, τ · e3 = 0, the projection of the evolution equation on the Burgers vector, here on e3, is reduced to

2 2 +κ2β2 − ∆η (ζ · τ)ω + (ζ · e3){(1− | τ | )∆η− (54) −((∇τ)∇η) · τ}ω = ξ3 η˙(ζ · e3)ω.

If we restrict to the condition ζ = e3, the equation (55) becomes

2 2 κ2β2 {∆η(1− | τ | ) − ((∇τ)∇η) · (τ)} = ξ3 η.˙ (55) When we return to the equation (51) together with (50) the compatibility condition

∆η τ + (∇τ)∇η = 0 (56) follows. We consider the scalar product of (57) with τ and we arrive at the equality

∆η | τ |2 +τ · ((∇τ)∇η) = 0. (57)

Note that the equation (56) together with (58) becomes (53). Thus we proved the following result.

Theorem 5.1. Let us assume that e3 · ω = 0.

1 If the disclination line ζ is orthogonal to the Burgers vector, namely e3 · ζ = 0 the evolution equation for the density of disclination is given by

2 ξ3η˙ = κ2β2∆η, (58) and the compatibility condition is reduced to ζ · τ = 0.

2 If the disclination line is collinear with Burgers vector, i.e. ζ k e3, the evolution equation for the disclination intensity is still given by (59), while the compatibility condition, namely between the plastic distortion expressed in terms of τ and η, is derived under the form

τ∆η + (∇τ)∇η = 0. (59)

6. CONCLUSIONS In Theorem 5.1 under the assumption that the plastic distortion is described by a screw dislocation we derived the non-local evolution for scalar disclination density. We assumed that the disclination is described in terms of Frank vector and disclination line. If the Frank and Burgers vectors are orthogonal, and moreover the dislocation and the disclination lines are either collinear, or perpendicular one to the another, then the evolution of scalar disclination density η is not inﬂuenced by the Dislocations and disclinations in ﬁnite elasto-plasticity 105 evolution of plastic distorsion. A non-local evolution equation for η is derived. The result is similar with those obtained in [6], for the case of plastic shear distortion, but there the directions of the Frank vector and the Burgers vector could be arbitrarily given. The same type of the analysis can be conducted for the edge dislocation.

Acknowledgments. The author acknowledges support from the Romanian Ministry of Education and Research through PN-II, IDEI program (Contract no. 576/2008).

References

[1] B.A. Bilby, Continuous distribution of dislocations, In: Sneddon IN, Hill R (eds) Progress in Solid Mechanics. North-Holland, Amsterdam, 1960. [2] P. Cermelli, M.E. Gurtin, The Motion of Screw Dislocations in Crystalline Materials Undergoing Antiplane Shear: Glide, Cross-Slip, Fine Cross-Slips, Arch. Rat. Mech. Anal., 148(1999), 3-52. [3] J.D. Clayton, D.L. McDowell, D.J. Bammann, Modeling dislocations and disclinations with finite micropolar elastoplasticity, Int. J. Plasticity, 22, 2(2006), 210-256. [4] S. Cleja-T¸igoiu, Couple stresses and non-Riemannian plastic connection in finite elasto- plasticity, ZAMP, 39(2002), 996-1013. [5] S. Cleja-T¸igoiu, Material forces in finite elasto-plasticity with continuously distributed dislocations, Int J Fracture, 147, 1-4(2007), 67-81. [6] S. Cleja-T¸igoiu, Elasto-plastic materials with lattice defects modeled by second order deformations with non-zero curvature, Int. J. Fracture, 166, 1-2(2010), 61-75. [7] R. de Wit, A view of the relation between the continuum theory of lattice defects and non- Euclidean geometry in the linear approximation, Int. J. Engng. Sci., 19(12)(1981), 1475-1506. [8] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metall. Mater., 42, 2(1994), 475-487. [9] M.E. Gurtin, On the plasticity of single crystal: free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids, 48, 5(2000), 989-1036. [10] E. Kroner,¨ The Differential geometry of Elementary Point and Line Defects in Bravais Crystals, Int. J. Theoretical Physics, 29, 11(1990), 1219-1237. [11] E. Kroner,¨ The internal mechanical state of solids with defects, Int. J. Solids Structures, 29, 14- 15(1992), 1849-1857. [12] J.E. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity, (1983) Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. [13] W. Noll, Materially Uniform Simple Bodies with Inhomogeneities, Arch. Rat. Mech. Anal., 1967, and in The Foundations of Mechanics and Thermodynamics, Selected papers (1974) Springer- Verlag/ Berlin Heidelberg New York, 1974. [14] C. Teodosiu, Elastic Models of Crystal Defects, Ed. Academiei, Springer-Verlag Berlin Heidel- berg New York, 1982.

ROMAI J., 6, 2(2010), 107–120

ANALYSIS AND NUMERICAL APPROACH TO UNIDIMENSIONAL ELASTO-PLASTIC PROBLEM WITH MIXED HARDENING Sanda Cleja-T¸igoiu1, Nadia Stoicut¸a2, Olimpiu Stoicut¸a2 1Faculty of Mathematics and Computer Science, University of Bucharest, Romania 2University of Petro¸sani,Romania [email protected] Abstract In this paper is determined by the ﬁnite element method combined with the Euler method and Newton method, the solution of the problem of elasto-plastic deformation of a one- dimensional bar using weak formulation of the problem with initial and boundary data.

Keywords: elasto-plastic model, constitutive equation, weak solution, ﬁnite element method, numerical simulation. 2000 MSC: 97U99.

1. INTRODUCTION The constitutive framework which describes the set of the constitutive and evolution equation in the mechanical problem can be found in the references [1], [5], and [11]. The initial and boundary value problem, which describes the dynamic behaviour of one-dimensional elasto-plastic material with isotropic and mixed hardening has been solved by J.C. Simo and T. J. Hughes in [10]. In order to solve the one- dimensional problem, the finite element method is applied together with the Newton method and the Return Mapping Algorithm. From the continuum model which describes the constitutive equation in the elasto-plastic model, through an elastic type constitutive equation together with the evolution equation for plastic strain and internal variables, the discrete algorithmic equations are obtained in [12] by applying an Euler type difference scheme. The Return Mapping Algorithm proposed in [12] involve the computation of the elastic trial stress together with the test for plastic loading. To solve the one-dimensional, initial and boundary value problem, we start from the quasi-static rate boundary value problem formulated at a generic moment of time and associated with the equilibrium equation. In order to determine the weak solution of the rate boundary value problem, which represents the time derivative of the displacement field u, i.e. the velocity at time t in the one-dimensional body, we apply the finite element method, those main results are resumed from [2], [8] and [11]. The method proposed here works for loading elsto-plastic process only. To up- date the current values of σ, εp, α, k, of the unknowns the Euler method and Newton

107 108 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a method are applied in order to integrate the differential equation system. The numerical algorithms combine the finite element method with the numerical methods related to the differential type equations, i.e. the Euler and Newton method, and the coupled problems are simultaneously solved.

2. THE UNIDIMENSIONAL MATHEMATICAL MODEL OF ELASTO-PLASTIC PROBLEM WITH MIXED HARDENING Let x ∈ Ω be a material point of the body Ω ⊂ R. Ω is an one dimensional body, which deforms in time, say on the interval I = [0, T). We denote by u : Ω × I → R - the displacement field, by σ : Ω × I → R - the Cauchy stress field and with ∂u ε : Ω × I → R, with ε(x, t) = (x, t) ≡ ε(u)(x, t) -the strain. We introduce also ∂x the hardening variables: α : Ω × I → R the kinematic hardening variable and k : Ω× I → R the isotropic hardening variable, with k > 0, which play different roles in describing the deformability of the surface of the plasticity in the stress space, during the irreversible deformation process. Following Cleja-Tigoiu and Cristescu [1985] (see also Chabauche [1989], Paraschiv-Munteanu and Cleja-Tigoiu [2004], we introduce the basic assumptions within the constitutive framework of the elasto- plastic model with mixed hardening. [see 2004]: 1. the rate of the strain tensorε ˙ can be decomposed into the rate of elastic part and that of plastic part, denoted byε ˙e andε ˙ p, respectively ε˙ = ε˙e + ε˙ p, (1) 2. the elastic type constitutive equation is given by

σ˙ = Eε˙e, (2) 3. the irreversible properties of the material are described in terms of the yield function F(σ, α, k), which dependent on the stress and hardening variables; the rate of the plastic strain tensor is described by the associated ﬂow rule through ∂F(σ, α, k) ε˙ p = λ , (3) ∂σ where λ, the so-called plastic factor is a function of the state of the material, and it is deﬁned through the Kuhn-Tucker and consistency conditions λ ≥ 0, F ≤ 0, λF = 0 (4) λF˙ = 0. (5) 4. the variation of the kinematic hardening variable α is given by Armstrong and Frederik [1996] law α˙ = Cε˙ p − γαk˙, (6) Analysis and numerical approach to unidimensional elasto-plastic problem... 109

C and γ are material parameters, while the isotropic hardening variable k is given by r 2 k˙ = ε˙ p · ε˙ p (7) 3 5. we add the initial condition

σ(0) = 0, ε(0) = 0, εp(0) = 0, α(0) = 0, k(0) = 0 (8) which correspond to a physical body undeformed and unstressed at time t = 0. In the considered one-dimensional model the yield function is given by

F(σ, α, k):= |σ − α| − F(k) (9) where F is a strictly increasing function, deﬁned through

F(k) = R(k) + κ, R(0) = 0, R0(k) > 0, k > 0 (10)

κ = σY represents the initial yield condition in the uniaxial test. The isotropic changing in the dimension of the yield surface is represented by Chaboche [1989]

R(k) = Q(1 − e−bk), Q, b > 0 (11)

Q and b are material constants. Taking into account the expression (10), the yield function is derived under the form h i −bk F(σ, α, k) = |σ − α| − Q(1 − e ) + σY (12) The plastic factor is calculated from (5) hβi λ = H(F) (13) h where the complementary plastic factor β and the hardening variable h are deﬁned through Eε(˙u)(σ − α) β = (14) −bk Q(1 − e ) + σY r r 2 σ − α 2 h = E + C − γα + Qbe−bk (15) −bk 3 Q(1 − e ) + σY 3 We introduce the supposition that h > 0 on F(σ, α, k) = 0. If h becomes zero in the process, then the solution cannot be deﬁned sinceε ˙ p → ∞. We can say that the material is damaged. In relation (14) ∂u˙(x, t) du˙(x, t) ε(˙u(x, t)) = ≡ , (15) ∂x dx 110 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a with the last notation introduced for the sake of simplicity. The Heaviside function, from the relationship (13) is ( 0, if F < 0 H(F) = (17) 1, if F ≥ 0

By eliminating the elastic part of the strain we arrive at the following result.

Proposition 2.1. The one-dimensional elasto-plastic model with mixed hardening is described by the following diﬀerential system hβi σ − α ε˙ p = H(F) −bk h Q(1 − e ) + σY r  hβi  σ − α 2  α˙ = C − γα H(F)  −bk  h Q(1 − e ) + σY 3 (17) hβi σ − α σ˙ = Eε(˙u) − E H(F) −bk q h Q(1 − e ) + σY hβi k˙ = 2 H(F) 3 h where β as a function on u˙ and h are given by the relations (14) together with (16).

The equilibrium equation in terms of one-dimensional Cauchy stress σ(x, t) has the form: ∂σ div σ(x, t) + b (x, t) = 0 in Ω × I, with div σ := (19) 1 ∂x where b1 ∈ R is the body force. The boundary conditions are formulated on the boundary ∂Ω = Γ of the body which is divided into two parts Γu and Γσ, such that Γu ∪ Γσ = Γ and Γu ∩ Γσ = ∅. They are given by

σ11(x, t)n = f1(x, t) in Γσ × I ; u(x, t) = g1(x, t) in Γu × I. (19)

Here n is the outward unit normal field on ∂Ω and n = 1 in the one-dimensional case. The functions f1(x, t) and g1(x, t) are given. Problem P: Given the functions b1(x, t), f1(x, t) and g1(x, t), find the real valued functions u, σ, ε, εp, α, k that are defined on Ω × [0, T) and satisfy (19) together with (20) and the differential type constitutive equations, listed in (18) together with (14) and (16). To solve it we start from the variational formulation of the equilibrium problem for elasto-plastic one-dimensional bar. Analysis and numerical approach to unidimensional elasto-plastic problem... 111 3. THE VARIATIONAL FORMULATION In what follows, we recall the main ideas from Johnson [1987] and Hughes [1987], which are appropriate to our description. Unlike Johnson and Hughes, which used a ∂ ∂v ∂u local momentum equation of the form σ + ρb = ρ in Ω × I where v = , in ∂x ∂t ∂t this paper the equilibrium equation is given by relation (19). We determine the weak solutions of the rate quasi-static boundary value problem associated at a generic moment t, which is derived by taking the time derivative of the equilibrium equation (19) together with the boundary condition (20) " # ∂σ(x, t) ∂b (x, t) div + 1 = 0 in Ω × I ∂t ∂t ∂σ(x, t) ∂ f (x, t) = 1 in Γ × I ∂t ∂t σ (20) ∂u(x, t) ∂g (x, t) = 1 in Γ × I ∂t ∂t u For the problem P, at a generic stage of the process the current values, i.e. at the time t, the current plastic domain is the form: p Ωt = { x ∈ Ω| F(σ(x, t), α(x, t), k(x, t)) = 0} . (22) The set of kinematically admissible velocity field is denoted by: ( ) ∂w V = w |w : Ω → R ; ∈ L2(Ω), w = g˙ ⊂ H1(Ω) (23) ad ∂x Γu 1 where    Z    2 2 2 L (Ω) = w w : Ω → R; w dx = kwkL2(Ω) < ∞ (24)   Ω is the space of the square integrable functions on Ω, and ( ) ∂w H1(Ω) = w ∈ L2(Ω) ∈ L2(Ω) (25) ∂x is the Sobolev space. Theorem 3.1. At every time t the rate of the displacement field, u˙, satisfy the following relationships: Z Z du˙(x, t) dw(x, t) db (x, t) Eep (x, t) dx = 1 w(x, t)dx+ dx dx dt Ω " # " Ω # (25) d f1(x, t) dσ(x, t) dg1(x, t) + w(x, t) + dt dt dt Γσ Γu 112 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a which hold for every admissible vector field w ∈ Vad. Proof. By multiplying the first equation of equation (21) with an admissible displacement w, integrating on a domain Ω and applying Green’s formula we get Z " # Z dσ(x, t) dw(x, t) dσ(x, t) db (x, t) dx = w(x, t) | + 1 w(x, t)dx. dt dx dt Γ dt (27) Ω Ω Further, we perform some transformation in the left hand side of the relation (27) by introducing the third relation from (18) and we have Z R σ(x, t) − α(x, t) dw(x, t) E du˙(x,t) dw(x,t) dx − Eλ HFdx = dx dx −bk(x,t) Ω Q(1 − e ) + σY dx Ω h i h i (28) R db (x,t) d f (x,t) dσ(x,t) dg (x,t) = 1 w(x, t)dx + 1 w(x, t) + 1 dt dt Γ dt dt Γ Ω σ u where the rate of strain ε(˙u(x, t)) is replaced from (16). The expression of β and h are replaced by the relations (14) and (16), but the positive part of the expression of β enters (28). We remark that only under the assumption that

Eε(˙u(x, t))(σ − α) > 0 (29) along the process, i.e. when no unloading is produced, (28) becomes R E (σ(x,t)−α(x,t)) 2 du˙(x,t) dw(x,t) E 1 − −bk(x,t) H(F) dx = h(x,t) Q(1−e )+σY dx dx Ω h i h i (29) R db (x,t) d f (x,t) dσ(x,t) dg (x,t) 1 w(x, t)dx + 1 w(x, t) + 1 . dt dt Γ dt dt Γ Ω σ u

In (30) following notation has been introduced, but only under the hypothesis formulated in (29),   E if H(F) = 0, ep  E (x, t) =  E σ(x,t)−α(x,t) 2 (31)  E 1 − −bk(x,t) if H(F) = 1, h(x,t) Q(1−e )+σY where Eep is the so-called elasto-plastic modulus. We introduced the supposition that h > 0 on F(σ, α, k) = 0. Consequently the equality (30) together with (31) becomes: Z Z du˙(x, t) dw(x, t) db (x, t) Eep (x, t) dx = 1 w(x, t)dx+ dx dx dt Ω " # " Ω # (32) d f1(x, t) dσ(x, t) dg1(x, t) + w(x, t) + . dt dt dt Γσ Γu Analysis and numerical approach to unidimensional elasto-plastic problem... 113

The relation (32) is in fact the variational representation of the solution.

Remark. In case when the displacement u(x, t) = 0 on the boundary Γu, i.e. g1(x, t) = 0, the variational formulation given above becomes: Z Z h i ep du˙(x, t) dw(x, t) E (x, t) dx = b˙1(x, t)w(x, t)dx + f˙1(x, t)w(x, t) , (33) dx dx Γσ Ω Ω written for all w ∈ Vad, which are vanishing on Γu. As a consequence of the above theorem, the following statement holds. Theorem 3.2. Find a displacement ﬁeld u (·, t), solution of the variational formulation a(˙u, w) = hL, wi ∀u ∈ Vad, w ∈ Vad (34) where a (·, ·) : Vad × Vad → R is the bilinear and symmetric form deﬁned by Z a(˙u, w) = Eep(x, t)ε(˙u(x, t))ε (w(x, t)) dx. (35) Ω Here L is a linear functional Z h i

hL, wi = b˙1(x, t)w(x, t)dx + f˙1(x, t)w(x, t) ; ∀t ∈ [0, T] (36) Γσ Ω where   E if H(F) = 0, ep  E (x, t) =  E σ(x,t)−α(x,t) 2 (37)  E 1 − −bk(x,t) if H(F) = 1, h(x,t) Q(1−e )+σY but under the hypothesis written in (29). Hypothesis. We assume that the material properties are given in such a way to ensure that a(˙,)˙ be a bilinear, symmetric, continuous and coercive form. Next,we replace the material particle x ∈ Ω with ξ ∈ Ω, to avoid misunderstand- ings. The problem to be solved is presented below Problem P1. Consider the following diﬀerential system d x(t) = f (x(t), u˙ (ξ, t)) ; x(t ) = x (38) dt 0 0 where the vector x(t) has the components

T x(t) = (x1(t) x2(t) x3(t) x4(t)) (39)

p x1 = ε , x2 = α, x3 = σ, x4 = k (40) 114 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a while the vector valued function which deﬁnes the system (38) has the form:

T f (x(t), u˙ (ξ, t)) = ( f1 (x, u˙) f2 (x, u˙) f3 (x, u˙) f4 (x, u˙)) (41)  x − x  f (x, u˙) = λ (x, u˙) 3 2  1 −bx  Q(1 − e 4 ) + σY    r    x − x 2   f (x, u˙) = λ (x, u˙) C 3 2 − γx   2  −bx 2  Q(1 − e 4 ) + σY 3  (42)    f3 (x, u˙) = E (ε (u˙) − f1 (x, u˙))   r   2  f4 (x, u˙) = λ (x, u˙) 3 hβ (x(t), u˙ (ξ, t))i λ (x(t), u˙ (ξ, t)) = H (F (x(t))) (43) h(x(t)) Eε (u˙ (ξ, t)) (x (t) − x (t)) β (x(t), u˙ (ξ, t)) = 3 2 (44) −bx (t) Q(1 − e 4 ) + σY r r 2 x3(t) − x2(t) 2 h(x(t)) = E + C − γx (t) + Qbe−bx4(t) (45) 2 −bx (t) 3 Q(1 − e 4 ) + σY 3 ( 0, if F(x(t)) < 0 H(F(x(t))) = (46) 1, if F(x(t)) ≥ 0 h i −bx4(t) F(x(t)) = |x3(t) − x2(t)| − Q(1 − e ) + σY . (47)

Determine the displacement u ∈ Vad ﬁeld, such thatu ˙(·, t) ∈ Vad, the plastic deformation εp, the internal variables α, k and the stress σ which satisfy at every time t the variational formulation

Z L Eep (x(t)) ε(˙u(ξ, t))ε(w(ξ, t))dξ = 0 (48) Z L h i

= b˙1(ξ, t)w(ξ, t)dξ + f˙1(ξ, t)w(ξ, t) 0 ξ=L   E ! if H(F) = 0 ep  2 E (x(t)) =  E x3(t)−x2(t) (49)  E 1 − −bx (t) if H(F) = 1  h(x(t)) Q(1−e 4 )+σY and having the time evolution for any fixed particle given by the above differential system. Analysis and numerical approach to unidimensional elasto-plastic problem... 115 4. DISCRETIZATION BY FINITE ELEMENT METHOD Finite element method is generally used to solve a variational problem, or the discretization form of certain variational formulation of the problem. Here we use the finite element method to find the weak solutions of the problem, which satisfy the variational equation (48) coupled with the system of the differential equations (38). The problem P1 is solved using the finite element method, and hence we briefly presents this method, following the papers and books given by Ferreira [8], Fish [9], Johnson [10]. We consider one-dimensional body, which is identified with an interval of the real SN axis, i.e. Ω = [0, L]. The time interval [0, T] is discretized by [0, T] = [tn, tn+1]. n=1 [0,L] is at its turnh discretizedi in ne network elements, where a network element has e e e the form Ωe = ξ1, ξ2, ξ3 , e = 1, ne. Let nN the total number of nodes used in the discretization. We apply the finite element method to the elasto-plastic problem formulated for the one-dimensional bar. Thus, we divide the bar into ne elements with nN nodes, each element having three nodes. Since we consider the one-dimensional case, the number of degrees of freedom ngl is equal to one for each node in the network. Concerning the boundary conditions: ξ = 0 (Γu) is considered to be the fixed end of the bar thus the displacement is zero, while at a traction boundary condition is applied at ξ = L (Γσ). Further we proceed to the effective implementation of finite element method proposed by Fish [9], Johnson [10]. The global approximation of the trial solution u(ξ, t) is Xne u(ξ, t) = Ne(ξ)ue(t) ≡ N(ξ)u(t). (50) e=1 The vectors N(ξ) and u(t) which enter the relation (50) have the form N(ξ) = N1(ξ) N2(ξ) N3(ξ) ... NnN (ξ) (51)

T u(t) = u1(t) u2(t) u3(t) ... unN (t) . (52) In the same way, the weight function w(ξ, t) is calculated using the relationship

Xne w(ξ, t) = Ne(ξ)we(t) ≡ N(ξ)w(t) (53) e=1 where the vector w(t) is given under the form

T w(t) = w1(t) w2(t) w3(t) ... wnN (t) . (54) 116 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a

Also, the displacements ue(ξ, t) and the weight function we(ξ, t) of the three nodes of each element can be calculated using the following relations: ue(ξ, t) = Ne(ξ)ue(t) ≡ Ne(ξ)Le u(t) (55) we(ξ, t) = Ne(ξ)we(t) ≡ Ne(ξ)Le w(t) (56) where the vectors ue(t) and we(t) are calculated using the relationships:

ue(t) = Leu(t) (57)

we(t) = Lew(t) (58) Here Le, called the selection matrix, is composed by a number of lines equal to the number of degrees of freedom per element and a number of columns equal to the total number of degrees of freedom in the network and is build with the help of the relationships ( 1, Ine(i, e) = j Le = δ = (59) i j Ine(i,e), j 0, Ine(i, e) , j. In the relation (59), Ine is the matrix of connection, that is a matrix that has a line for each item. The line e of the matrix Ine contains the nodes that make up the item. In the relations (55) and (56), the vector of the interpolation function of an element e, if this element has three nodes, is of the form h i e e e e Ni (ξ) = N1(ξ) N2(ξ) N3(ξ) (60) and the displacements vector for an element in the three nodes are h i e e e e u (t) = u1(t) u2(t) u3(t) (61) h i e e e e w (t) = w1(t) w2(t) w3(t) . (62) For the unidimensional case when the network element contains three nodes, the e shape functions Ni (ξ) given by the relation (60) are built as is follows: ξ − ξe ξ − ξe Ne(ξ) = 2 3 (63) 1 e e e e ξ1 − ξ2 ξ1 − ξ3 ξ − ξe ξ − ξe Ne(ξ) = 1 3 (64) 2 e e e e ξ2 − ξ1 ξ2 − ξ3 ξ − ξe ξ − ξe Ne(ξ) = 1 2 . (65) 3 e e e e ξ3 − ξ1 ξ3 − ξ2 Analysis and numerical approach to unidimensional elasto-plastic problem... 117

When we take the time derivate of the displacement written in (55), we have:

due(ξ, t) ≡ u˙e(ξ, t) = Ne(ξ)Le u˙(t) (66) dt Thus the rate of the strain tensor is calculated using the following relationship:

du˙e(ξ, t) dNe(ξ) ε(˙ue(ξ, t)) = = Leu˙(t) ≡ Be(ξ)Le u˙(t) (67) dξ dξ

dNe(ξ) h i where Be(ξ) = , and Be(ξ) = Be(ξ) Be(ξ) Be(ξ) . dξ 1 2 3 On the other hand, the represented of the strain measure is: ε we(ξ, t) = Be(ξ)Le w(t). (68)

In the weak form (48), the integral over (0, L) is viewed as a sum of integrals over individual element domain, Ωe. Using the notations (67) and (68) in the variational formulation (48) we have:

Pne R Be(ξ)Lew(t) T (Eep)e (x(t)) Be(ξ)Leu˙(t) dξ = e=1Ωe ne R ne h i (69) P e e T P e e T N (ξ)L w(t) b˙1(ξ, t)dξ + N (ξ)L w(t) f˙1(ξ, t) . Γe e=1Ωe e=1 σ Moreover (69) can be written under the form      Pne R   w(t)T  LeT  Be(ξ)T (Eep)e (x(t))(Be(ξ)) dΩe Le u˙(t) = e=1 Ωe  (70)  Pne R Pne h i  = w(t)T  LeT Ne(ξ)T b˙ (ξ, t)dΩe + LeT Ne(ξ)T f˙ (ξ, t)  .  1 1 Γe  e=1 Ωe e=1 σ We introduce the following notation relations  R  Ke(x(t)) = Be(ξ)T (Eep (x(t)))e (Be(ξ)) dΩe   R Ωe  Re(t) = Ne(ξ)T b˙ (ξ, t)dΩe  b 1 (71)  Ωhe i  e e T ˙  R (t) = N (ξ) f1(ξ, t) e f Γσ which allow us to rewrite (70) in the form:

 n   n n  Xe  Xe Xe  w(t)T  LeT KeLe u˙(t) −  LeT R e + LeT R e = 0. (72)    b f  e=1 e=1 e=1 118 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a

The matrices introduced in (71) have the meaning, Ke is the element stiﬀness e e matrix, Rb is the matrix of the external forces and R f is the matrix of the internal forces. If Xne K(x(t)) = LeT Ke(x(t))Le (73) e=1  Pne  eT e  Rb(t) = L Rb(t)  e=1 R(t) = Rb(t) + R f (t) →  ne (74)  P eT e  R f (t) = L R f (t) e=1 then the relation (72) can be written in a shorter form w(t)T [K(x(t))˙u(t) − R(t)] = 0 ; ∀ w(t) (75) As w(t) is arbitrarily given, relationship (75) becomes K(x(t))˙u(t) = R(t). (76) Thus, in view of the above, in the following we apply the Gauss quadrature formula to the integral from relationship (70), i.e.:

Xn−1 e ( ) e T ep ( ) e e Kmn x(t) = h2 AiBm(h1 + h2τi) E x(t) Bn(h1 + h2τi) (77) i=0

e where the matrix K is symmetric, and m = 1, nN and n = 1, nN. e The integral Rb(t) given by the relation from (71) can be similarly computed by using the Gauss quadrature formula, as follows:   n−1  X  Re (t) = h Ne (h + h τ )T b˙(h + h τ , t) , m = 1, n (78) b m  2 m 1 2 i 1 2 i  N i=0 b + a b − a where we used the notations h = , h = . We obtain the components 1 2 2 2 XnN Fm(x(t), u˙(t)) = [Kmn(x(t))˙un(t)] − Rm(t) (79) n=1 of the relation (76), that can be written in following form F(x(t), u˙(t)) := K(x(t))˙u(t) − R(t) = 0. (80) To solve the system of equations (80) relative to the unknownu ˙(t), we apply New- ton’s method. Thus we have: " #−1 ∂F (x(tn), u˙ (tn)) u˙ (tn+1) = u˙ (tn) − F (x(tn), u˙ (tn)) , n = 0, 1, ... . (81) ∂u˙(tn) Analysis and numerical approach to unidimensional elasto-plastic problem... 119

Simultaneously we apply the Euler method to the non-linear system of diﬀerential equations given by (38). The iterative formulae can be derived under the form:

x (tn+1) = x (tn) + ∆t f (x (tn) , N(ξ)˙u(tn)) , x (t0) = 0, n = 0, 1, 2, ... (82) as a consequence of (50), where ∆t = tn+1 − tn is the step of the method. We present now the main steps of the algorithm applied to the formulated problem:

u˙(t0) = 0; x (t0) = 0

For n = 0 to N

x (tn+1) = x (tn) + ∆t f (x (tn) , N(ξ)˙u(tn)) " #−1 (83) ∂F (x(t ), u˙ (t )) n n u˙ (tn+1) = u˙ (tn) − F (x(tn), u˙ (tn)) ∂u˙(tn) end

To solve the problem P1, the algorithm presented in the relationship (83) is run in every point of the network.

5. NUMERICAL APPLICATION Numerical application presented here, aims to highlight the numerical algorithm of solving the problem P1.

Fig. 1. The graph of the stress depending on the strain in the point 30 120 Sanda Cleja-T¸igoiu, Nadia Stoicut¸a, Olimpiu Stoicut¸a

In this sense, the material chosen is steel DP 600, whose parameters are given below (Broggiato [2008]): ( ) E = 182.000 [MPa]; σ = 349, 4 [MPa]; Q = 50, 1 [MPa] Y . (84) C = 17.400 [MPa]; b = 27, 5 [−]; γ = 125, 9 [−] As an example we consider an uni-dimensional bar, whit Ω = [0, 30]. The bar has length L = 30 [cm]. The bar is fixed in the node ξ = 0, and the traction force f1(ξ, t) = 450 sin t is located in the node ξ = 6. The body force is considered by 2 form b1 = 0, 5 t . We specify also that for the meshing, the bar was divided into ten elements, each element having three nodes, total number of nodes in the network being 21. Since, the bar is fixed in the node ξ = 0, the displacement in this node is zero. In the network, the elements have been divided into equal intervals. In these conditions, after running the program, which implements the numerical algorithm of the problem P1, we obtain the following information presented in the figure 1.

Acknowledgement: Sanda Cleja-Tigoiu acknowledges the support from the Ministery of Education Research and Inovation under CNSIS PN2 Programm Idei, PCCE, Contract No. 100/2009.

References [1] P. J. Armstrong, C. O. Frederick, A mathematical representation of the multiaxial Bauschinger effect, G.E.G.B. Report RD-B-N 731, 1996. [2] J. Bathe, Finite element procedure, Vols. I, II, III, Prentice Hall, Englewood Cliffs, New Jersey, 1996. [3] T. Belytschko, K. L. Wing, B. Moran, Nonlinear Finite Elements for Continua and Structures, British Library Cataloguing in Publication Data, Toronto, 2006. [4] G. B.Broggiato , F. Campana, L. Cortese The Chaboche nonlinear kinematic hardening model: calibration methodology and validation , Meccanica, 43, 2(2008), 115-124. [5] J. L. Chaboche, Constitutive equations for cyclic plasticity and cyclic visco-pasticity , Int. J. Plasticity, 1989. [6] S. Cleja-Tigoiu, N. Cristescu, Teoria plasticitatii cu aplicatii la prelucrarea materialelor, Ed. Univ. Bucuresti, 1985. [7] S. Cleja-Tigoiu, E. Soos, Elastoplastic model with relaxed configurations and internal state variables, Appl. Mech. Rev., 43, 7(1990), 131-151. [8] A. J. M. Ferreira, Matlab Codes for Finite Element Analysis Solid and Structures , Springer, Berlin, 2009. [9] J. Fish, T. Belytschko, A First Course in Finite Elements , John Wiley and Sons, Ltd., USA, 2007. [10] C. Johnson, Numerical solution of partial diferential equation by the finite element method , Cambridge University Press, Cambridge, 2002. [11] I. Paraschiv-Munteanu, S. Cleja-Tigoiu, E. Soos, Plasticitate cu aplicat¸ii ˆıngeomecanic˘a , Ed. Universitat¸ii˘ din Bucures¸ti, 2004. [12] J. C. Simo, T. J. R. Hughes, Computational Inelasticity , Springer, New-York,1998. ROMAI J., 6, 2(2010), 121–130

SOME NATURAL DIAGONAL STRUCTURES ON THE TANGENT BUNDLES AND ON THE TANGENT SPHERE BUNDLES Simona-Luiza Drut¸˘a-Romaniuc1, Vasile Oproiu2 1Department of Sciences, ”Al. I. Cuza” University of Ia¸si,Romania 2Faculty of Mathematics, ”Al. I. Cuza” University of Ia¸si,Romania [email protected], [email protected] Abstract We study the properties of some geometric structures defined on the tangent and tangent sphere bundles of a Riemannian manifold by using the natural lifts. These lifts are obtained from the Riemannian metric g of the base manifold. We get some almost Hermitian structures defined on the tangent bundle and find the conditions under which they are Kahlerian.¨ Then we study some specific properties of such structures (to have constant holomorphic sectional curvature, to be Einstein, etc). Similar problems are considered for the tangent sphere bundles Tr M endowed with the the induced almost contact structures (the quality of Tr M to be a Sasaki space form, or an η-Einstein manifold).

Keywords: tangent bundle, natural lift, Einstein structure, tangent sphere bundle, almost contact structure, Sasaki space form, η-Einstein manifold. 2000 MSC: 53C05, 53C15, 53C55.

1. INTRODUCTION Some new interesting geometric structures on the tangent bundle of a Riemannian manifold (M, g) may be obtained by lifting the metric g from the base manifold. In 1998, in a joint work with N. Papaghiuc ([16]), the second author introduced on the tangent bundle of a Riemannian manifold, a Riemannian metric deﬁned by a regular Lagrangian depending on the energy density only. The same author has studied some properties of a natural lift G, of diagonal type, of the Riemannian metric g (so that it is no longer obtained from a Lagrangian) and a natural almost complex structure J of diagonal type on TM (see [12]-[15]). The condition for (TM, G, J) to be a Kahler¨ Einstein manifold leads to the conditions for (M, g) to have constant sectional curvature, and for (TM, G, J) to have constant sectional holomorphic curvature or to be a locally symmetric space. In [12] and [15], the second author excluded some important cases which appeared, in a certain sense, as singular cases. The case where the Riemannian metric G is deﬁned by the Lagrangian L (excluded in [12]) was studied in [16]. Other singular cases excluded in [12] and [13] were studied by N. Papaghiuc in [18] and [19]. Other geometric structures obtained by considering natural lifts of the metric g from the base manifold to its tangent bundle have been studied in (see [1], [8], [17]).

121 122 Simona-Luiza Drut¸˘a-Romaniuc, Vasile Oproiu

In this paper we present a survey on the geometric structures defined by the natural diagonal lifts of the metric from the base manifold to the tangent bundle, as well as the structures defined by the restrictions of these lifts to the tangent sphere bundles of constant radius r, seen as hypersurfaces of the tangent bundles, consisting of the tangent vectors of norm equal to r only. It is known that every almost Hermitian structure from the tangent bundle induces an almost contact structure on the tangent sphere bundle of constant radius r. Impor- tant results in this direction may be found in the recent surveys [1] and [9], but also in the papers [2]-[4], [11], [20]. The most part of the authors who worked in this field considered unitary tangent sphere bundles, endowed with the induced Sasaki metric (see [21]), but O. Kowalski and M. Sekizawa showed that the geometry of the tangent sphere bundles depends on the radius. They determined in [9] the g-natural metrics of the mentioned type on the tangent sphere bundles of constant radii, having constant scalar curvature, and in [2] M. T. K. Abassi and O. Kowalski obtained the conditions under which the g-natural metrics on the unit tangent sphere bundle are Einstein. In the paper [5] we considered on the tangent bundle TM of a Riemannian manifold M a natural diagonal metric, obtained by the second author in [15], and denoted in the section 4 by Ge. We proved that the tangent sphere bundle Tr M endowed with the Riemannian metric induced from Ge is never a space form, then we found the conditions under which (Tr M, G) is an Einstein manifold. In [6] we obtained some almost contact metric structures (ϕ, ξ, η, G) on the tangent sphere bundles, induced by some almost Hermitian structures (Ge, J) of natural diagonal lift type on the tangent bundle of a Riemannian manifold (M, g). The above almost contact metric structures are not automatically contact metric structures. In order to get such structures we made some rescalings of the metric, of the fundamental vector field, and of the 1-form. Then we gave the characterization of the Sasakian structures of natural diagonal lift type on the tangent sphere bundles. In this case, the base manifold must be a space form. We studied in [7] the holomorphic φ-sectional curvature of the obtained Sasakian manifolds, and we proved that there are no natural diagonal tangent sphere bundles of constant holomorphic φ-sectional curvature. On the other hand, the study of the conditions under which the obtained Sasakian tangent sphere bundles (Tr M, ϕ, ξ, η, G) are η-Eistein manifolds, namely the corresponding Ricci tensor field has the form

Ric = ρG + ση ⊗ η, where ρ and σ are smooth real functions, led in [6] to two cases, which will be presented in the final section of this paper. The manifolds, tensor fields and other geometric objects we consider in this paper are assumed to be differentiable of class C∞ (i.e. smooth). The well known summation convention is used throughout this paper, the range of the indices h, i, j, k, l, r being always {1,..., n}. Some natural diagonal structures... 123 2. PRELIMINARY RESULTS Consider a smooth n-dimensional Riemannian manifold (M, g) and its tangent bundle τ : TM → M. The total space TM has a structure of a 2n-dimensional smooth manifold, induced from the smooth manifold structure of M. This structure is obtained by using local charts on TM induced from usual local charts on M. If (U, ϕ) = (U, x1,..., xn) is a local chart on M, then the corresponding induced local chart on TM is (τ−1(U), Φ) = (τ−1(U), x1,..., xn, y1,..., yn), where the local coordinates xi, y j, i, j = 1,..., n, are defined as follows. The first n local coordinates of a tangent vector y ∈ τ−1(U) are the local coordinates in the local chart (U, ϕ) of its base point, i.e. xi = xi ◦ τ, by an abuse of notation. The last n local coordinates y j, j = 1,..., n, of y ∈ τ−1(U) are the vector space coordinates of y with respect to the natural basis in Tτ(y)M defined by the local chart (U, ϕ). Due to this special structure of differentiable manifold for TM, it is possible to introduce the concept of M-tensor field on it (see [10]). Denote by ∇˙ the Levi Civita connection of the Riemannian metric g on M. Then we have the direct sum decomposition TTM = VTM ⊕ HTM (1) of the tangent bundle to TM into the vertical distribution VTM = Ker τ∗ and the horizontal distribution HTM defined by ∇˙ (see [22]). The vertical and horizontal lifts of a vector field X on M will be denoted by XV and XH respectively. The set of vector ∂ ∂ −1 fields { ∂y1 ,..., ∂yn } on τ (U) defines a local frame field for VTM, and for HTM we δ δ δ ∂ h ∂ h k h have the local frame field { δx1 ,..., δxn }, where δxi = ∂xi − Γ0i ∂yh , Γ0i = y Γki, and h Γki(x) are the Christoffel symbols of g. ∂ δ The set { ∂yi , δx j }i, j=1,n, denoted also by {∂i, δ j}i, j=1,n, defines a local frame on TM, adapted to the direct sum decomposition (1). Consider the energy density of the tangent vector y with respect to the Riemannian metric g 1 1 1 t = kyk2 = g (y, y) = g (x)yiyk, y ∈ τ−1(U). (2) 2 2 τ(y) 2 ik Obviously, we have t ∈ [0, ∞) for every y ∈ TM.

3. NATURAL DIAGONAL LIFTED STRUCTURES ON THE TANGENT BUNDLE The second author constructed in [15] an (1,1)-tensor ﬁeld J on the tangent bundle TM, obtained as natural 1-st order lift of the metric g from the base manifold to the tangent bundle TM:

Jδi = a1(t)∂i + b1(t)g0iC, J∂i = −a2(t)δi − b2(t)g0iCe, (3) h where a1, a2, b1, b2 are smooth functions of the energy density, C = y ∂h is the Liou- h ville vector ﬁeld and Ce = y δh is the geodesic spray. 124 Simona-Luiza Drut¸˘a-Romaniuc, Vasile Oproiu

The invariant expressions of the above (1,1)-tensor ﬁeld are

H V V JXy = a1(t)Xy + b1(t)gτ(y)(X, y)yy , 1 V H H ∀X ∈ T0(M), ∀y ∈ TM. (4) JXy = −a2(t)Xy − b2(t)gτ(y)(X, y)yy ,

Studying the conditions under which J2 = −I, it was easy to prove the following result. Proposition 3.1. ([15]) The (1, 1)-tensor field J given by the relations (3) or (4) defines an almost complex structure on the tangent bundle if and only if a2 = 1/a1, b2 = −b1/[a1(a1 + 2tb1)]. Then it was considered on TM a natural Riemannian metric of diagonal type G on TM, induced by g. The expression in local adapted frame are defined by the M-tensor fields (1) Gi j = G(∂i, ∂ j) = c1gi j + d1g0ig0 j, (2) (5) Gi j = G(δi, δ j) = c2gi j + d2g0ig0 j, G(∂i, δ j) = G(δ j, ∂i) = 0. where the coefficients c1, c2, d1, d2 are smooth functions depending on the energy density t ∈ [0.∞) and the conditions for G to be positive definite are given by c1 > 0, c2 > 0, c1 + 2td1 > 0, c2 + 2td2 > 0, for every t ≥ 0. The invariant expressions of the above metric are

H H G(Xy , Yy ) = c1(t)gτ(y)(X, Y) + d1(t)gτ(y)(X, y)gτ(y)(Y, y), V V G(Xy , Yy ) = c2(t)gτ(y)(X, Y) + d2(t)gτ(y)(X, y)gτ(y)(Y, y), (6) V H e H V G(Xy , Yy ) = G(Xy , Xy ) = 0,

1 ∀X, Y ∈ T0(TM), ∀y ∈ TM. Theorem 3.1. ([14]) Let J be a natural diagonal almost complex structure on TM defined by g and the functions a1, a2, b1, b2. Then the family of natural diagonal Riemannian metrics G, given by (5) or (6) is almost Hermitian with respect to J, i.e. G(JX, JY) = G(X, Y), if and only if c c c + 2td c + 2td 1 = 2 = λ, 1 1 = 2 2 = λ + 2tµ, (7) a1 a2 a1 + 2tb1 a2 + 2tb2 where λ > 0, µ > 0 are functions of t. In [14], the second author obtained some classes of natural almost Hermitian structures (G, J) on TM, these classes are obtained from the well known classification of the almost Hermitian structures in sixteen classes. The results concerning this classification are given by Theorems 4, 5, 6 and 7 in [14]. Some very important particular cases of the almost Hermitian structure obtained in Theorem 3.1 have been studied further. So, if we consider the case when c1(t) = Some natural diagonal structures... 125

1 a1(t) = u(t), b1(t) = d1(t) = v(t), b2(t) = d2(t) = w(t), a2(t) = c2(t) = u(t) , λ(t) = 1 v and µ(t) = 0, where w(t) = − u(u+2tv) , then all the conditions from Proposition 3.1 and Theorem 3.1 are satisfied, and we obtain the almost Hermitian structure defined and studied by the second author in [15]. Theorem 3.2. ([15]) Under the above conditions, the almost Hermitian manifold (TM, G, J) is an almost Kählermanifold. Moreover, the almost complex structure J on TM is integrable (i.e. (TM, G, J) is a Kählermanifold) if (M, g) has constant sectional curvature c and the function v is given by c − uu0 v = . 2tu0 − u In [15] the second author obtained a Kahler-Einstein¨ structure even in the case where (M, g) has positive constant sectional curvature, but only on a tube around the zero section in TM ([15, Theorem 4.2]). He also obtained on TM a structure of Kahler¨ manifold with constant holomorphic sectional curvature ([15, Theorem 5.1]). The particular case of the above situation, when the function u(t) = 1, for all t ∈ [0, ∞), has been studied by the same author in [12]. In this case he stated Theorem 3.3. ([12]) (TM, G, J) is an almost Kählerianmanifold and the almost complex structure J is integrable if and only if (M, g) has constant sectional curvature c and v = −c. If c < 0 is obtained a Kählerstructure on whole TM. In the case where the constant c is positive is obtained a Kählerstructure in the tube around the zero 1 section in TM defined by t < 2c . Moreover, in the same paper it was proved Theorem 3.4. If (M, g) has negative constant sectional curvature then its tangent bundle has a structure of KählerEinstein manifold. If (M, g) has positive constant sectional curvature then a tube around the zero section in TM has a structure of KählerEinstein manifold. Another important case which appear as a singular case is when v(t) = 0 for all t ∈ [0, ∞) which implies w(t) = 0. This case has been studied by N. Papaghiuc in [18], [19], and he obtained Theorem 3.5. ([18]) Assume that the function u(t) satisfies the condition u(0)u0(0) , 0. Then the almost complex structure J on TM or on a tube around the zero section in TM is integrable if and only if the base manifold (M, g) has constant sectional 0 curvature c√and the function u(t) satisfies the ordinary differential equation uu = c, i.e. u(t) = 2ct + A, where A is an arbitrary real constant. 126 Simona-Luiza Drut¸˘a-Romaniuc, Vasile Oproiu

Another result from [18] (see also [19]) is given by Theorem 3.6. If n , 2, then the Kählerianmanifold (TM, G, J) cannot be an Einstein manifold and cannot have constant holomorphic sectional curvature. If n = 2, then the Kählerianmanifold (TM, G, J) is Ricci flat. Remark that the singular case when v(t) = u0(t) was studied in [16].

4. NATURAL DIAGONAL STRUCTURES ON THE TANGENT SPHERE BUNDLES OF CONSTANT RADIUS R Let us recall some results from [5], concerning tangent sphere bundles endowed with a natural diagonal lifted metric. 2 We denote by Tr M = {y ∈ TM : gτ(y)(y, y) = r }, with r ∈ (0, ∞), and the projection τ : Tr M → M, τ = τ ◦ i, where i is the inclusion map. The horizontal lift of any vector field on M is tangent to Tr M, but the vertical lift is not always tangent to Tr M. The tangential lift of a vector X to (p, y) ∈ Tr M, used in some recent papers like [3], [9], [11], [20], is defined by 1 XT = XV − g (X, y)yV . y y r2 τ(y) y T A generator system for the tangent bundle to Tr M is given by δi and ∂ j = ∂ j − 1 k T r2 g0 jy ∂k, i, j, k = 1, n. Remark that the vector fields {∂ j } j=1,n satisfy the relation j T T y ∂ j = 0, so they are not independent. In any point of Tr M the vectors {∂i }i=1,...n span an (n − 1)-dimensional subspace of TTr M. i The vector field N = y ∂i is normal to Tr M in TM. From now on we shall denote by Ge the metric G defined by the relations (5) or 0 (6), and by G the metric on Tr M induced by the metric of TM. Remark that the functions c1, c2, d1, d2 become constant, since in the case of the tangent sphere r2 bundle of constant radius r, the energy density t becomes a constat equal to 2 . It follows 0 H H G (Xy , Yy ) = c1gτ(y)(X, Y) + d1gτ(y)(X, y)gτ(y)(Y, y), 0 T T 1 G (Xy , Yy ) = c2[gτ(y)(X, Y) − r2 gτ(y)(X, y)gτ(y)(Y, y)], (8) 0 H T 0 T H G (Xy , Yy ) = G (Yy , Xy ) = 0, 1 0 ∀X, Y ∈ T0(M), ∀y ∈ Tr M, where c1, d1, c2 are constants. The conditions for G to 2 be positive are c1 > 0, c2 > 0, c1 + r d1 > 0. The Levi-Civita connection ∇, associated to the Riemannian metric G0 on the tangent sphere bundle Tr M has the form  T h T T h T h ∇ T ∂ = A ∂ , ∇δ ∂ = Γ ∂ + B δh,  ∂i j i j h i j i j h ji  h h h T ∇ T δ j = B δh, ∇δ δ j = Γ δh + C ∂ , ∂i i j i i j i j h Some natural diagonal structures... 127 the expressions of the involved M-tensor fields being given in [5]. In [5] we obtained the horizontal and tangential components of the curvature tensor field K, denoted by sequences of H and T, to indicate horizontal, or tangential argument on a certain position. For example, we may write

T h h T K(δi, δ j)∂k = HHTHki jδh + HHTTki j∂h . Then we proved the following result

Theorem 4.1. ([5]) The tangent sphere bundle Tr M, with the Riemannian metric induced by the natural metric Ge of diagonal lift type on the tangent bundle TM, has never constant sectional curvature. 0 We computed the Ricci tensor ﬁeld of the manifold (Tr M, G ), and we obtained T T the components Ric(δ j, δk) = RicHH jk and Ric(∂ j , ∂k ) = RicTT jk:

2 2 2 d1(2c1+r d1) d1(2c1+r d1)[c1n+r d1(n−1)] RicHH jk = Ric jk − 2 g jk + 2 2 g0 jg0k 2c2(c1+r d1) 2r c1c2(c1+r d1) (9) c2 h l c2d1 h − R R + 2 R Rh0 j0 2c1 0kl jh0 2c1(c1+r d1) 0k0

4 2 2 2 r d1−2c1(c1+r d1)(n−2) 1 c2 hi RicTT jk = 2c r2(c +d r2) ( r2 g0 jg0k − g jk) + 2 Rhik0R j0 1 1 1 4c1 2 (10) c2d1 h − 2 2 R 0 j0Rh0k0. 2c1(c1+r d1) Analyzing the vanishing conditions of the diﬀerence Ric(X, Y) − ρG0(X, Y), for 1 every X, Y ∈ T0(Tr M) we stated

Theorem 4.2. ([5]) The tangent sphere bundle Tr M of an n-dimensional Riemannian (M, g) of constant sectional curvature c is Einstein with respect to the metric G0 induced by the natural diagonal lifted metric Ge deﬁned on TM, i.e. it exists a real 1 constant ρ such that Ric(X, Y) = ρG(X, Y), for every X, Y ∈ T0(Tr M), if and only if r2d n d n c(n − 1)2(n − 2) c = 1 , c = 1 , ρ = . 1 2 2 2 n − 2 c(n − 2) r d1n In the paper [6] we studied the almost contact metric structures (ϕ, ξ, η, G) on the tangent sphere bundles, induced by the natural diagonal almost Hermitian structures (Ge, J) on TM, characterized in Theorem 3.1, and we proved

Theorem 4.3. The almost contact metric structure (ϕ, ξ, η, G) on Tr M, with ϕ, ξ, η, and G given respectively by

H T T H a2 H ϕX = a1X , ϕX = −a2X + r2 g(X, y)y , 1 H T H 0 ξ = 2λr2α y , η(X ) = 0, η(X ) = 2αλg(X, y), G = αG , 1 for every tangent vector ﬁelds X, Y ∈ T0(M), and every tangent vector y ∈ Tr M, 2 where α = c1+r d1 and the metric G0 is given by (8), is a contact metric structure, 4r2λ2 128 Simona-Luiza Drut¸˘a-Romaniuc, Vasile Oproiu and it is Sasakian if and only if the base manifold has constant sectional curvature 2 a1 c = r2 .

The contact metric manifold (Tr M, ϕ, ξ, η, G) is η−Einstein if the corresponding Ricci tensor may be written as Ric = ρG + ση ⊗ η, (11) where ρ and σ are smooth real functions. Using the expressions (9) and (10), we wrote the relation (11) with respect to the T generator system {δi, ∂ j }i, j=1,n, and taking into account Theorem 4.3 we proved

Theorem 4.4. ([6]) The Sasakian manifold (Tr M, ϕ, ξ, η, G), characterized in Theo- rem 4.3, is η−Einstein if and only if d d n − cc (n − 2) Case I) c = cc r2, ρ = cc (2n − 3) − 1 , σ = λ2 1 2 ; 1 2 2 2 2 2 2 2cc2r 2cc2(cc2 + d1)r cc r2 − c (n − 2) n(n − 2)(c + cc r2) CaseII) d = 2 1 , ρ = 1 2 , 1 2 2 (n − 1)r 2c1c2(n − 1)r 2 2 2 2 2 4 2r cc1c2{n[n(n − 4) + 6] − 2} − n (n − 2)(c + c c r ) σ = λ2 1 2 , 2 2 2c1c2(c1 + cc2r ) 2 a1 where c = r2 is the constant sectional curvature of the base manifold M.

In [7] we studied the condition under which the Sasaki manifold (Tr M, ϕ, ξ, η, G) characterized in Theorem 4.3 is a Sasaki space form, i.e. it has constant holomorphic φ-sectional curvature. The tensor ﬁeld corresponding to the curvature of the Sasakian space form (Tr M, ϕ, ξ, η, G) is given by:

k+3 k−1 K0(X, Y)Z = 4 [G(Y, Z)X − G(X, Z)Y] + 4 [η(X)η(Z)Y − η(Y)η(Z)X +G(X, Z)η(Y)ξ − G(Y, Z)η(X)ξ + G(Z, ϕY)ϕX − G(Z, ϕX)ϕY + 2G(X, ϕY)ϕZ], where k is a constant. T With respect to the generator system {δi, ∂ j }i, j=1,...,n, we have six components of K0. We exemplify by the expressions of two of them: T T T h T T T h K0(∂i , ∂ j )∂k = TTTT0ki j∂h , K0(∂i , ∂ j )δk = TTHH0ki jδh, where the M-tensor ﬁelds involved as coeﬃcients have the expressions:

TTTT h = k+3 α(G0(2)δh − G0(2)δh), n 0hki j 4 k j i ki j i o h 2 k−1 0(1) 1 h l l h h l l h l h l h TTHH0ki j = a2 4 αG kl r2 g0i(δ j y − δ jy ) − g0 j(δi y − δiy ) + δ jδi − δiδ j , Studying the vanishing conditions of the difference between the curvature tensor field K of the Riemannian manifold (Tr M, G) and the curvature tensor field K0 corresponding to the Sasaki manifold (Tr M, ϕ, ξ, η, G), we obtained that some compo- h nents of this difference vanish under certain very restrictive conditions, but TTTTki j− Some natural diagonal structures... 129

h TTTT0ki j takes the form 1 h 1 1 i δh(g − g g ) − δh(g − g g ) , 0, r2 i jk r2 0 j 0k j ik r2 0i 0k which is never equal to zero, so we proved the ﬁnal theorem from [7].

Theorem 4.5. The Sasakian manifold (Tr M, ϕ, ξ, η, G), given by Theorem 4.3, is never a Sasaki space form.

Acknowledgements. The ﬁrst author was supported by the Program POSDRU/89/1.5/S/49944, ”AL. I. Cuza” University of Ias¸i, Romania.

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[18] N. Papaghiuc, Another Kaehler structure on the tangen bundle of a space form, Demonstratio Mathematica, 31, 4(1998), 855-866. [19] N. Papaghiuc, On an Einstein structure on the tangent bundle of a space form, Publ. Math. De- brecen, 55(1999), 349-361. [20] J. H. Park, K. Sekigawa, When are the tangent sphere bundles of a Riemannian manifold η- Einstein?, Ann. Global An. Geom, 36, 3(2009), 275-284. [21] S. Sasaki,On the diﬀerential geometry of the tangent bundle of Riemannian manifolds,Tohokuˆ Math. J., 10(1958), 238-354. [22] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, M. Dekker Inc., New York, 1973. ROMAI J., 6, 2(2010), 131–138

A LINEARIZATION PRINCIPLE FOR THE STABILITY OF THE CHEMICAL EQUILIBRIUM OF A BINARY MIXTURE Adelina Georgescu1, Lidia Palese2 1Academy of Romanian Scientists, Bucharest, Romania 2Dipartimento di Matematica, Universit`adi Bari, Italy [email protected] Abstract We consider the stability of a chemical equilibrium of a thermally conducting two component reactive viscous mixture which is situated in a horizontal layer heated from below and experiencing a catalyzed chemical reaction at the bottom plate. We provide a method for an easy derivation of the nonlinear stability bound, derived explicitly in terms of the involved physical parameters. It enables us to derive a linearization principle in a large sense, i.e. to prove that the linear and nonlinear stability bounds are equal.

Keywords: hydrodynamic stability, horizontal thermal convection, energy method. 2000 MSC: 76E15, 76E30.

1. INTRODUCTION The convective instability and the nonlinear stability of a homogeneous fluid in a gravitational field heated from below, the classical Benard´ problem, is a well known interesting problem in several fields of fluid mechanics [1], [2], [3], [4], [5]. The stability problem for mechanical equilibria of binary mixture in the case of competing effects (e.g., temperature, concentration, magnetic field) is of a big importance in astrophysics, geophysics, oceanography, meteorology. That is why it received a large attention, mostly in the Oberbeck-Boussinesq approximation [5], [6], [7]. The point of loss of linear stability is usually also a bifurcation point at which the convective motions set in [4], [8]. However, in certain situations the linear and non linear stability bounds are not equal. In particular, subcritical instability may occur explaining unusual phenomena. Whence a special interest for the study of the relationship between linear and nonlinear stability bounds and, thus, of the linearization principle [9], [10], [11], [12]. A very strong approach to deal with linearization principle (in the sense of the coincidence of linear and nonlinear stability bounds) in convection problem was settled in [2], where an energy, defined in terms of linear combinations of the concentration and temperature fields, was used.

131 132 Adelina Georgescu, Lidia Palese

In addition, a parametric differentiation was employed to get the best non linear bound. For reactive fluids of technological interest, chemical reactions such as the dissociation of nitrogen, oxygen or hydrogen gas near the gas-solid interface of a space vehicle when returning to the earth’s atmosphere [13], [14], [15], [16], can give temperature and concentration gradients which influence the transport process and can alter hydrodynamic stabilities. In [17] a nonlinear stability analysis of the conduction-diffusion Benard´ problem, with the upper surface stress free and the lower one experiencing a catalyzed chemical reaction is performed, obtaining a condition of nonlinear stability of the equilibrium solution, and showing that, in a limiting case, there is the possibility of subcritical bifurcation in a well-determined region of the parameters’ space. In the present paper we consider a fluid mixture in a horizontal layer heated from below, the bottom plate being catalytic. We evaluate the effects of heterogeneous surface catalyzed reactions on the hidrodynamic stability of the chemical equilibrium. The model adopted is that of Bdzil and Frisch. We consider a Newtonian fluid model and, by introducing some linear combinations of temperature and concentration, we derive a system equivalent to the perturbation evolution equations, generalizing the Joseph’s method of parametric differentiation [18], [19], [20], changing the given problem governing the evolution equations in order to obtain an optimum stability bound. The determination of the nonlinear stability bound is reduced to the solution of an algebraic system. With symmetrization arguments for the involved linear operators we can determine the nonlinear stability bound in terms of the physical parameters, in the case in which the Prandtl and Schmidt numbers are equal, in a region of the parameters’ space. In the last section the equality between linear and nonlinear stability bounds was proved, at least in the class of normal modes perturbations.

2. THE INITIAL/BOUNDARY VALUE PROBLEM FOR PERTURBATION

We consider the mixture composed of the dimer A2 and the monomer A, (A2, A) described by a Newtonian model, in the layer bounded by the surfaces z = 0 and z = 1, being the lower surface catalytic, that is, the interconversion (A2 A) occurs via the surface z = 0 [15]. However, the conditions that must be satisfied at the catalyzed boundary z = 0, are [15]: J~· ~k = 0, Q~ · ~k = 0 where J~ is the mass flux, ~k is the unit vector in the vertical upward direction, and Q~ is the heat flux. The chemical equilibrium S 0 is characterized by the temperature (T) and degree of dissociation (fraction of pure monomers) (C) fields [13], [15]:

T(z) = T1 + β(1 − z), C(z) = C1 + γ(1 − z), (1) A linearization principle for the stability of the chemical equilibrium of a binary mixture 133 where C1 and T1 are the values of C and T at z = 1 and the constants β and γ are given in [13], [15]. Let us now perturb S 0 ≡ (~0, P¯, T¯, C¯) up to a cellular motion (we denote with V a periodicity cell) characterized by a velocity ~u = ~0 + ~u, a pressure p = P¯ + p0 a temperature T = T¯ + θ and a concentration C = C¯ + γ fields, where ~u, p0, θ, γ are the corresponding perturbation fields. In the Boussinesq approximation the evolution of the perturbation fields is governed by the following equations, written in nondimensional coordinates [17], ∂ ~u + (~u · ∇)~u = −∇p0 + ∆~u + (Rθ + Cγ)~k,, (2) ∂t ∂ P ( θ + ~u · ∇θ) = ∆θ − Rw, (t, ~x) ∈ (0, ∞) × V (3) r ∂t ∂ S ( γ + ~u · ∇γ) = ∆γ + Cw, (4) c ∂t in a subset of L2(V), namely , n N = (~u, p, θ, γ) ∈ L2(V) | div~u = 0; ∂u = ∂v = w = 0 on ∂V , ∂z ∂z 2 o ∂θ ∂γ (5) ~u = 0 on ∂V1, θ = γ = 0 on ∂V2, ∂z = −sγ, ∂z = rγ on ∂V1 . where ~u = (u, v, w), ∂V is the boundary of V, ∂V1 = ∂V ∩{z = 0}, ∂V2 = ∂V ∩{z = 1}. The perturbation fields depend on the time t and space ~x = (x, y, z) and R2, C2, are the thermal and concentrational numbers of Rayleigh, while Pr and S c are Prandtl and Schmidt numbers, respectively. In addition, r, s > 0 are dimensionless surface reactions numbers. The basic state S 0 corresponding to the zero solution of the initial-boundary value problem for (2)- (5) in the class N is called non linearly stable if a Lyapunov function E(t), the energy, remains bounded when t → ∞ [2]. The stability or instability of S 0 depends on six physical parameters occurring in (2) - (5): Pr, S c, R, C, r and s. In the following section we apply the Joseph’s generalized method from [18] - [20] to derive the evolution equation for the energy E.

3. THE EVOLUTION EQUATIONS FOR THE PERTURBATION FIELDS

In order to derive the stability boundaries we perform the operations a{(3) + (4)g3} and b{(4) + (3)g2}, we obtain ∂θ ∂γ aP + ~u · ∇θ + ag S + ~u · ∇γ = a(Cg − R)w + a∆θ + ag ∆γ, (6) r ∂t 3 c ∂t 3 3 ∂γ ∂θ bS + ~u · ∇γ + bg P + ~u · ∇θ = b(C − g R)w + bg ∆θ + b∆γ, (7) c ∂t 2 r ∂t 2 2 134 Adelina Georgescu, Lidia Palese where a, b, g2, g3 are some positive constants. By multiplying (2) by ~u, (6) by θ , and (7) by γ, integrating them over V in N and ∂γ ∂θ adding the results, if we require that the coeﬃcients of θ and γ be equal, i.e. ∂t ∂t

bg2 = τag3, (8) where τ = S c/Pr, we obtain 1 d < |~u|2 + aP θ2 + bS γ2 + 2ag S θγ >= − < |∇~u|2 + a|∇θ|2 + b|∇γ|2+ 2 dt r c 3 c 1 α +(ag + bg )|∇θ · ∇γ| > +aR( + αg − 1) < θw > +bR( + α − g ) < γw > + 3 2 a 3 b 2 Z 2 + θγ(s − rg3)a + γ (sg2 − r)b dxdy, (9) ∂V R 1 where < f >= V f dxdydz, α = C/R. By choosing s r g = , g = , (10) 3 r 2 s in the case τ = 1, if we require that √ ag3 + bg2 = 2 ab, (11) then the terms in θ and γ in < · > from the right-hand side of (9) form a perfect square. From (11) it follows that s2 b = a . (12) r Let us introduce √ √ √ a φ = aθ + bγ ≡ (rθ + sγ). (13) 1 r Moreover, if we require that the coefficients of < θw > and < γw > satisfy the relation 1 √ s s α r r √ √ + a(α − 1) = √ ( )2 + (α − ) a , (14) a r r a s s it follows s α = . (15) r Then (9) is in the form of the evolution equation for energy E(t) dE 1 √ s = − < |∇~u|2 + |∇φ |2 > +R √ + a(α − 1) < φ w >, (16) dt 1 a r 1 where 2 2 E(t) =< |~u| + Prφ1 > /2. (17) A linearization principle for the stability of the chemical equilibrium of a binary mixture 135 4. THE NONLINEAR STABILITY BOUND AND ITS EQUALITY WITH THE LINEAR STABILITY BOUND In order to find the nonlinear stability bound we follow the Joseph’s generalized method of parametric differentiation [18]-[20]. Denoting 1 √ s 2A = R| √ + a(α − 1)| (18) a r relation (16) implies dE p ≤ −ξ2 1 − A/ R E(t) (19) dt a∗ where [4], [5], [21], [22], [23] 2 < |∇u|2 + |∇φ |2 > 1 2 < φ w > ξ2 = min 1 , √ = max 1 . (20) u,φ1 2 2 u,φ1 < |∇u|2 + |∇φ |2 > < |u| + Prφ1 > Ra∗ 1 If p A < Ra∗, (21) the basic state S 0 is (nonlinearly) stable because (16) implies dE p ≤ −ξ2(1 − A/ Ra )E. (22) dt ∗

Relations (18) and (21) show that S 0 is stable if p . 1 √ s R < RE ≡ 2 Ra∗ | √ + a(α − 1)|. (23) a r

Therefore, RE is a non linear√ stability√ bound. But RE is maximum if 1/ a + a(αs/r−1) is minimum. Thus we shall determine a with the aid of the Joseph’s parametric diﬀerentiation method [2], i.e. imposing d 1 √ s | √ + a(α − 1)| = 0. (24) da a r s If r α > 1, the solution 1 a = s (25) r α − 1 of (24) gives, in terms of the physical quantities, the non linear stability bound [23] r √ s −1 R = R ( )2 − 1 , (26) E a∗ r whence the following 136 Adelina Georgescu, Lidia Palese

Theorem 4.1. For physical parameters τ = 1, R, C = αR, s/r = α, (s/r)2 > 1, the zero solution of (2)-(5), corresponding to the basic conduction state, is non linearly asymptotically stable if R < RE, where RE is given by (26), or, equivalently, if 2 2 C − R < Ra∗ where Ra∗ is given by (20). Let us consider the steady problem, obtained by linearizing (2) - (4) about the trivial solution, ∆u + (Rθ + Cγ)k = ∇p, (27) −Ru · k + ∆θ = 0, (28) Cu · k + ∆γ = 0. (29) Elimination of u · k between (28) multiplied by r and (29) multiplied by s implies r s ∆φ + R ( )2 − 1 w = 0, (30) 1 r moreover (27), taking into account (15), is equivalent to r s ∆u + R ( )2 − 1 φ k = ∇p, (31) r 1 Then from (31)-(30) it follows ( ∆u + µφ k = ∇p, 1 (32) ∆φ1 + µw = 0, q s 2 where µ = R ( r ) − 1 . We proved that, at least in the class of normal mode perturbations, the smallest eigenvalues of the problems (27)-(29) and (32) are the same. The system (32) represents the Euler-Lagrange equations for the minimum of the 2 2 functional < |∇u| + |∇φ1| > /(2 < φ1w >), namely√ this minimum is µ. Taking into account (20)2 it follows that µ = Ra∗. As a consequence the chemical equilibrium has the linear stability bound RL which satisﬁes the relation s µ2 ≡ R2 ( )2 − 1 , (33) L r this implies, taking into account (26), RL = RE, whence the following result: Theorem 4.2. For physical parameters τ = 1, R, C = αR, s/r = α, (s/r)2 > 1, the zero solution of (2)-(5), corresponding to the basic conduction state is non linearly asymptotically stable if R < RE A linearization principle for the stability of the chemical equilibrium of a binary mixture 137 and, in the class of normal mode perturbations,

RL = RE, whence the linear and non linear stability bounds coincide.

5. CONCLUSIONS We applied a generalized Joseph’s approach to some convection problems. In the Joseph’s approach it is used a linear combination of temperature and concentration to deﬁne the perturbation energy. Our idea was to incorporate the Joseph’s relation for the rate of change of the product of temperature and concentration directly into the evolution equations obtaining equations with better symmetries. In this way, the term in velocity from the temperature equation contributed to the symmetric part of the obtained equations. Otherwise, if the initial evolution equations were used this contribution is null and, correspondingly, the stability criterion, weaker. This is due to the sign minus in the quoted term in the initial equation for temperature. In [23] we derive the evolution equation for the perturbation energy, and, in the same region of the parameters’ space, we determine a nonlinear stability bound in terms of the involved physical parameters. This limit is the same as in the present paper, because the evolution equation for the perturbation energy from [23] can be derived considering some more general evolution equations with better symmetries, which incorporate the given equations, such as (6), (7). Indeed in this paper the given problem governing the perturbation evolution was changed in order to obtain an optimum energy inequation. The initial equations (3)- (4) were replaced by (6)-(7) in which the initial evolution equations were present through the terms in g2 and g3. These terms drastically changed the linear part of the initial equations and allows us a much more advantageous symmetrization and an equivalent formulation of the linear stability problem that was nothing else but the Euler system associated to the maximum problem of the nonlinear stability, whence a linearization principle in the sense of the coincidence of both linear and nonlinear stability bounds.

References [1] S.Chandrasekhar, Hydrodynamic and Hydromagnetic stability, Oxford, Clarendon Press, 1968. [2] D.D. Joseph, Global Stability of the Conduction-Diffusion Solution, Arch. Rat. Mech. Anal., 36, 4(1970), 285-292. [3] D.D. Joseph, Stability of fluid motions I-II, Springer, Berlin, 1976. [4] A. Georgescu, Hydrodynamic stability theory, Kluwer, Dordrecht, 1985. [5] B. Straughan, The Energy Method, Stability and Nonlinear Convection, Springer, New York, 2003. [6] S. Rionero G. Mulone, On the nonlinear stability of a thermodiffusive fluid mixture in a mixed problem, J. Math. Anal. Appl., 124, (1987), 165-188. 138 Adelina Georgescu, Lidia Palese

[7] S. Rionero G. Mulone, A nonlinear stability analysis of the magnetic Be´ nard problem through the Lyapunov direct method,, Arch. Rational Mech. Anal., 103, 4 (1988), 347-368. [8] A. Georgescu, I. Oprea, Bifurcation theory from the applications point of view, Monografii Matematice. Universitatea de vest din Timis¸oara, 51, 1994. [9] G. Prodi, Teoremi di tipo locale per il sistema di Navier-Stokes e stabilita` delle soluzioni stazionarie, REND. SEM. MAT. Padova, 32, (1962), 374-397. [10] V. I. Yudovich, On the stability of stationary flow of viscous incompressible fluids, Dokl. Akad. Nauk. SSSR, 161, 5 (1965), 1037-1040. (Russian) (translated as Soviet Physics Dokl., 10, 4 (1965), 293-295.) [11] D. H. Sattinger, The mathematical problem of hydrodynamic stability, J. Math. Mech., 19, 9 (1970), 197-217. [12] O. A. Ladyzhenskaya, V. A. Solonnikov, The linearization principle and invariant manifolds for problems of magnetohydrodynamics, J. SOV. MATH., 4, 8 (1977) 384-422. [13] J. Bdzil, H.L. Frisch, Chemical Instabilities. II. Chemical Surface Reactions and Hydrodynamic Instability Phys. Fluids, 14, 3 (1971), 475-481. [14] J. Bdzil, H.L. Frisch, Chemical Instabilities. IV. Nonisothermal Chemical Surface Reactions and Hydrodynamic Instability, Phys. Fluids, 14, 6 (1971), 1077-1086. [15] C. L. Mc Taggart, B. Straughan, Chemical Surface Reactions and Nonlinear Stability by the Method of Energy, SIAM J. Math. Anal., 17, 2 (1986), 342-351. [16] D.E. Loper, P.H. Roberts, On The Motion of an Airon-Alloy Core Containing a Slurry, Geophys. Astrophys. Fluid Dynamics, 9, 1978, 289-331. [17] B. Straughan, Nonlinear Convective Stability by the Method of Energy: Recents Results, Atti delle Giornate di Lavoro su Onde e Stabilita` nei mezzi continui, Cosenza (Italy), 1983, 323-338. [18] A. Georgescu, L. Palese, Extension of a Joseph’s criterion to the non linear stability of mechanical equilibria in the presence of thermodiffusive conductivity, Theor. Comp. Fl. Mech., 8, 1996, 403-413. [19] A. Georgescu, L. Palese, A. Redaelli, On a New Method inHydrodynamic Stability Theory, Math. Sci. Res. Hot Line, 4, 7 (2000), 1-16. [20] A. Georgescu, L. Palese, A. Redaelli, The Complete Form for the Joseph Extended Criterion, Annali Universita` di Ferrera , Sez. VII, Sc. Mat. 48, (2001), 9-22. [21] A. Georgescu, L. Palese, Stability Criteria for Fluid Flows, Advances in Math. for Appl. Sc., 81, World Scient. Singapore, 2010. [22] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. [23] A. Georgescu, L. Palese, On the nonlinear stability of a binary mixture with chemical surface reactions, Preprint. ROMAI J., 6, 2(2010), 139–155

ASYMPTOTIC WAVES AS SOLUTIONS OF NONLINEAR PDES DEDUCED BY DOUBLE- SCALE METHOD IN VISCOANELASTIC MEDIA WITH MEMORY Adelina Georgescu1, Liliana Restuccia2 1Academy of Romanian Scientists, Bucharest, Romania 2Department of Mathematics, University of Messina, Italy, [email protected] Abstract In this paper the double scale method is applied to investigate non-linear dissipative waves in isotropic viscoanelastic media with shape and volumetric memory, that were studied by one of the authors (L. R.) in more classical way. The system of PDEs describing these media include terms containing second order derivatives multiplied by a small parameter. The solutions are looked for in the form of asymptotic expansions with respect to an asymptotic sequence of powers of the small parameter, related to the thickness of internal layers across which the solutions or/and some of their derivatives vary steeply. An one-dimensional application is worked out containing original results.

Keywords: Partial diﬀerential equations, double-scale method, non-linear dissipative waves, asymptotic methods, rheological media. 2000 MSC: 34E05, 34E10, 34E13, 73B20, 73B99.

1. INTRODUCTION In previous papers we sketched out the general use of the double-scale method to nonlinear hyperbolic partial differential equations (PDEs) in order to study the asymptotic waves [1] and as examples the models governing the motion of anelastic media (Maxwell media) [2] and viscoanelastic media (Jeffreys media) [3], both of them without shape and volumetric memory, were studied. In this paper the double scale method is applied to investigate asymptotic waves in viscoanelastic media with shape and bulk memory, in which a viscous flow phenomenon occurs, that were studied by one of the authors (L. R.) in a more classical way [4]. Since the closed-form solutions of nonlinear PDEs are rare, usually the solution is looked for in the form of an asymptotic series of powers of a small parameter, which is related to the thickness of internal layers, across which the solution or/and some of its derivatives varies steeply. This asymptotic expansion is called asymptotic solution of the system of PDEs. Correspondingly, a new independent (fast) variable ξ is defined that models just this fast variation across the internal layers situated near

139 140 Adelina Georgescu, Liliana Restuccia some surfaces S and the slow variation along S. The asymptotic method involving ξ is known as the double-scale method [1]. The multiple-scale method, and, in particular, the double-scale approach, is appropriate to phenomena which possess qualitatively distinct aspects at various scales. For instance, at some well-determined times or space coordinates, the characteristics of the motion vary steeply, while at larger scale the characteristics are slow and describe another type of motion. Furthermore, the scales are defined by some small parameters. The mathematical aspects involved into the study of asymptotic waves belong to singular perturbation theory, namely the double-scale method [9] - [16]. This approach was initiated in the papers of Poincaré, Krylov and Bogolinskow [9], [10]. The interest in nonlinear waves was manifest as early as the years ’50 and ’60 of the last century. Subsequently, a lot of applications to various equations from elasticity, fluid mechanics and thermodynamics were worked out [17]- [8]. The use of nonlinear asymptotic waves received a particular attention soon after the seminal work [6], especially from the thermodynamicists. In the contest of rheological media studies were carried out, some of them belonging to one of the authors (L. R.) (see [4], [29] and [30].) In this paper previous treatments performed in [4], using a method of G. Boillat [6], generalized by D. Fusco [31], are seen in the modern light of double-scale method, as it is presented in a monograph on asymptotic treatments of the other author A. G. [16]. In Section 2 we give a physical interpretation of the new (fast) variable ξ, related to the surface across which the solution or/and some of its derivatives vary steeply. In Section 3 the equations governing the motion of viscoanelastic media with memory in the framework of classical irreversible thermodynamics (TIP) with internal variables [32]-[38] are introduced and the mechanical relaxation equation valid for these media is described. In Section 4 we present the double-scale method, obtaining the first and second asymptotic approximation. The application of the double-scale method is standard: the solution U(xα, ξ) is written as an asymptotic power series of the small parameter, the coefficients Ui, (i = 1, 2, ...) being functions of xα and ξ. Introducing the series in the PDEs, after the matching of the series in the right and left-hand sides, the equations for Ui, with i ≥ 1 are obtained. They are called equations of order i and Ui are the asymptotic approximations of order i. A special meaning is given for U0. It is taken as the initial, unperturbed state, where no small parameter occurs. In Sections 5 and 6 the derivation of the first approximation of the wave front and of the first approximation of the solution U is discussed in details. In Section 7 an one-dimensional application, containing original results and revealing the influence of the internal variable on the mechanical relaxation of the medium, is considered. Asymptotic waves as solutions of nonlinear PDEs ... 141 2. APPLICATION OF DOUBLE-SCALE METHOD TO NONLINEAR PDES We deal with those smooth solutions U(xα), called asymptotic waves, which evolve as progressive waves, i.e. there exists a family of hypersurfaces S (defined by the equation ϕ(xα) = 0) moving in the Euclidean space E3+1, consisting of points of coordinates xα (α = 0,1,2,3) or, equivalently, of the time t = x0 and the space coordinates xi (i=1,2,3), ϕ(t, xi) = ξ¯ = const, (1) such that the solutions U or/and some of their derivatives vary steeply across S, while along S their variation is slow [1]. This means that around S there exist (asymptotic) internal layers, such that the order of magnitude (i.e. the scale) of the solutions U or/and some of their derivatives inside these layers and far away from them differ very much. Therefore, it is natural to introduce a new independent variable ξ, related to the hypersurfaces S, ξ = ωξ¯ = ωϕ(t, xi), (2) where ξ is asymptotically fixed , i.e. ξ = Ord(1) as ω−1 → 0. ω is a large parameter (ω 1) and usually its dimension is a frequency. Asymptotically, this means that both ”slow” (old) and ”fast” (new) variables are necessary to characterize completely the behaviour of the solution as some parameter tends to zero [16]. Each variable (slow or fast) has a characteristic scale. That is why, an approximate solution is looked for as depending on the old as well as on the new variables, where the new variables are thought of as independent of the old variables. When no interior layers are present, all characteristic quantities have the same scale. If only one new variable of a different scale is imposed, we say that the problem has a double scale. In this case, the appropriate method, used to derive solutions of asymptotic approximation is called the double-scale method. Usually, the new variable is a space variable xi or the time t multiplied by a function of the small parameter. Then, we have U = U(xα, ξ). Furthermore, taking into account that U is sufficiently smooth, hence it has sufficiently many bounded derivatives, it follows that, except for the terms containing ω, all other terms are asymptotically fixed and the computation can proceed formally. In this way, if xα = xα(s) are the parametric equations of a curve C in E3+1, we have dU ∂U ∂ϕ ∂U dxα = ω + , ds ∂ξ ∂s ∂xα ds where the dummy index convention is used. This relation shows that, indeed, along dϕ C, U does not vary too much if C belongs to the hypersurface S (in this case ds = 0) but it has a large variation if C is not situated on S. For these reasons, ξ is referred to as the fast variable. Once introduced the fast variable ξ, in order to apply the double- scale method we must define the equations to which it applies. Thus, let E3+1 be an Euclidean space, let P ∈ E3+1 be a current point, let U = U(P) be the unknown vector 142 Adelina Georgescu, Liliana Restuccia function solution of a system of PDEs written in the following matrix form " # ∂2U ∂2U Aα(U)U + ω−1 Hk + Hik = B(U), (3) α ∂t∂xk ∂xi∂xk (α = 0, 1, 2, 3; i, k = 1, 2, 3), where x0 = t (time), x1, x2, x3 are the space coordinates, U is the vector of the un- α ∂U α k ik known functions (which depend on x ), Uα = ∂xα , A , H , H are appropriate matrices, and α A (U)Uα = B(U), (4) is the associated system of nonlinear hyperbolic PDEs. In [4] it was shown that the motion of isotropic viscoanelastic media with memory, in the isothermal case, is described by a system of nonlinear PDEs having the matrix form (3) and including terms containing second order derivatives, multiplied by a very small parameter, that have a balancing effect on the non-linear steepening of waves. In [4] non-linear dissipative waves were worked out by one of the author (L. R.) using a asymptotic method developed by G. Boillat [6] and generalized by D. Fusco [31]. In this paper we study the asymptotic waves deduced in [4] from the point of view of double scale-method. Following A. Jeffrey in [24], the solution hypersurfaces of systems of PDEs are referred to as waves because they may be interpreted as representing propagating wavefronts. When physical problems are associated with such interpretation the solution on the side of the wavefront towards which propagation takes place may then be regarded as being the undisturbed solution ahead of the wavefront, whilst the solution on the other side may be regarded as a propagating disturbance wave which is entering a region occupied by the undisturbed solution. We conclude this Section reminding that the wavefront ϕ is an unknown function. In order to determine it, we recall its equation is ϕ(t, x1, x2, x3) = 0. This implies that dϕ ∂ϕ along the wavefront we have dt = 0, implying ∂t + v · gradϕ = 0 or, equivalently, ∂ϕ ∂t gradϕ |gradϕ| + v · |gradϕ| = 0. Obviously, gradϕ = n, (5) |gradϕ| such that the previous equality reads ∂ϕ ∂t + v · n = 0. (6) |gradϕ| Introducing the notation ∂ϕ λ = − ∂t , (7) |gradϕ| we have λ(U, n) = v · n, (8) where λ is called the velocity normal to the progressive wave. Asymptotic waves as solutions of nonlinear PDEs ... 143 3. GOVERNING EQUATIONS FOR VISCOANELASTIC MEDIA WITH MEMORY In [32] a general thermodynamic theory for mechanical phenomena in continuous media was developed by G. Kluitenberg, using the methods of classical irreversible thermodynamics with internal variables [33]-[36]. It was assumed that several microscopic phenomena occur which give rise to inelastic strains due to effects of lattice defects, like slip, dislocations, atom vacancies, disclinations. From the theory it follows that several types of macroscopic stress fields may occur in a medium: a stress (eq) (vi) field ταβ (α, β = 1, 2, 3) of thermoelastic nature, a stress field ταβ analogous to the (k) viscous stress in ordinary fluids and n stress fields τ(m)αβ(k = 1, 2, ..., n), in arbitrary number, connected with microscopic stress fields surrounding imperfections in the (eq) (vi) medium. The stress field ταβ + ταβ is the mechanical stress field which occurs in the equations of motion and in the first law of thermodynamics. The stress fields (eq) (k) ταβ +τ(m)αβ play the role of thermodynamic affinities in the phenomenological equations which are generalizations of Lévy’s law. Temperature effects were fully taken into account. Explicit mechanical relaxation equations, with constant phenomenological coefficients, were deduced, linearizing equations of state. For distortional phenomena in isotropic media the following linear relation among the deviators of the mechanical stress tensor, the first n derivatives with respect to time of this tensor, the tensor of total strain (the sum of the elastic and inelastic strains) and the first n + 1 derivatives with respect to time of this last tensor was deduced, where n is the number of phenomena that give rise to inelastic deformations

Xn−1 dm dn Xn+1 dm R(τ) τ˜ + R(τ) τ˜ + τ˜ = R() ˜ + R() ˜ (i, k = 1, 2, 3). (9) (d)0 ik (d)m dtm ik dtn ik (d)0 ik (d)m dtm ik m=1 m=1 d (τ) In (9) dt is the material time derivative, the quantities R(d)m (m = 0, 1, ..., n − 1), () R(d)m (m = 0, 1, ..., n + 1) are algebraic functions of the coeﬃcients occurring in the phenomenological equations and in the equations of state. In [37] and [38] it was shown that the relation between τ, the scalar part of the stress tensor, and , the scalar part of the strain tensor, is given by

Xn−1 dmτ0 dnτ0 Xn+1 dm R(τ) τ0 + R(τ) + = R() + R() (v)0 (v)m dtm dtn (v)0 (v)m dtm m=1 m=1 Xn dm +R(T) (T − T 0) + R(T) T, (10) (v)0 (v)m dtm m=1 τ0 = τ − τ0, where τ0 and T 0 are the scalar part of the stress tensor and the temperature, respectively, of the medium in a state of thermodynamic equilibrium. This equilibrium 144 Adelina Georgescu, Liliana Restuccia

(τ) state plays the role of a reference state. The quantities R(v)m (m = 0, 1, ..., n − 1), () (T) R(v)m (m = 0, 1, ..., n + 1) and R(v)m (m = 0, 1, ..., n) are algebraic functions of the coefficients occurring in the phenomenological equations and in the equations of state. The well-known Burgers equation is a special case of this relation for n = 2, i.e. when only two internal variables of mechanical origin are taken into consideration. The rheological relations for ordinary viscous fluids, for thermoelastic media and for Maxwell, Kelvin (Voigt), Jeffreys, Poyting-Thomson, Prandtl-Reuss, Bingham, Saint Venant and Hooke media are special cases of these more general mentioned above relations too (see also [37]- [41]). The total strain is given by elastic and inelastic (el) (in) deformations εαβ = εαβ + εαβ (α, β = 1, 2, 3). The inelastic strain is due to lattice (in) Pn (k) (k) defects and related phenomena: εαβ = k=1 εαβ, where εαβ is the k-th contribution to the inelastic strain of the k-th microscopic phenomenon and it is called partial inelas- (k) tic strain tensor. In this theory n − 1 partial inelastic strain tensors εαβ are introduced as internal variables in the thermodynamic state vector. Furthermore, it is assumed that the gradient of the displacement field is small. This implies that the deformations are supposed to be small from a kinematical (or geometrical point of view). However the translations and the velocity of the medium may be large [36]. Then, the strain 1 ∂ ∂ tensor εik is assumed to be small, i.e. εik = 2 ∂xk ui + ∂xi uk (i, k = 1, 2, 3), where ui is the i-th component of the displacement field u and xi is the i-th component of the position vector x in Eulerian coordinates in a Cartesian reference frame. It should be emphasized, however, that the same physical ideas which are developed in this theory can also be reformulated for the case where the deformations are large from a kinematical point of view [36]. Then, assume that only one microscopic phenomenon gives rise to inelastic strain. In the isothermal case, the mechanical relaxation equations describing the behaviour of isotropic viscoanelastic media of order one (n = 1), i.e. when only one internal variable of mechanical origin is taken into consideration, with shape and bulk memory, can be written in the following form [4] d d d2 R(τ) P˜ + P˜ + R() ˜ + R() ˜ + R() ˜ = 0 (i, k = 1, 2, 3), (11) (d)0 ik dt ik (d)0 ik (d)1 dt ik (d)2 dt2 ik d d d2 R(τ) P0 + P0 + R() + R() + R() = 0, (12) (v)0 dt (v)0 (v)1 dt (v)2 dt2 ˜ where Pik and ˜ik are the deviators of the mechanical pressure tensor Pik and of the strain tensor , respectively, and dεik = 1 ∂vi + ∂vk . We define P in terms of the ik dt 2 ∂xk ∂xi ik 0 0 0 symmetric Cauchy tensor Pik = −τik, P = P − P = −(τ − τ ) and, hence, we have 1 1 P˜ = P − P δ , P = P , P = trP, ik ik 3 ss ik 3 ss ss

Pik = P˜ik + Pδik, P˜ ss = 0, Asymptotic waves as solutions of nonlinear PDEs ... 145 where the scalar part P of the tensor Pik is the hydrostatic pressure. Analogous deﬁ- nitions are valid for ˜ik and . Moreover, in eqns. (11) and (12) the coeﬃcients satisfy the following relations

(τ) (1,1) (1,1) R(d)0 = a ηs ≥ 0, (13)

() (0,0) (1,1) (0,0) (1,1) R(d)0 = a (a − a )ηs ≥ 0, (14) () (0,0) (0,1) (1,1) (0,0) (1,1) (0,1) 2 R(d)1 = a 1 + 2ηs + a ηs ηs + ηs ≥ 0, (15)

() (0,0) (τ) (1,1) (1,1) R(d)2 = ηs ≥ 0, R(v)0 = b η(v) ≥ 0, (16) () (0,0) (1,1) (0,0) (1,1) () (0,0) R(v)0 = b (b − b )η(v) ≥ 0, R(v)2 = ηv ≥ 0, (17) () (0,0) (0,1) (1,1) (0,0) (1,1) (0,1) 2 R(v)1 = b 1 + 2ηv + b ηv ηv + ηv ≥ 0, (18)

() (τ) () () (τ) () () (τ) () R(d)1 − R(d)0R(d)2 ≥ 0, R(d)1R(d)0 − R(d)0 ≥ 0, R(v)1 − R(v)0R(v)2 ≥ 0 () (τ) () and R(v)1R(v)0 − R(v)0 ≥ 0. In eqns. (13) - (18) a(0,0), a(1,1), b(0,0) and b(1,1) are scalar constants which occur (0,0) (0,1) (1,1) (0,0) (0,1) (1,1) in the equations of state, while the coefficients ηs , ηs , ηs , ηv , ηv and ηv are called phenomenological coefficients. Indicating by vi the i-th component of the velocity field v, we have du ∂u ∂u v = i = i + v i , (19) i dt ∂t j ∂x j where the definition of material derivative is taken into consideration. The balance equations for the mass density and momentum in the case of isotropic viscoanelastic media with shape an bulk memory read ∂ρ ∂ + (ρvi) = 0 (i = 1, 2, 3), (20) ∂t ∂xi ! ∂ ∂ ∂ ∂ ρ v + v v + P˜ + P = 0. (21) i k k i k ik ∂t ∂x ∂x ∂xi 4. EQUATIONS OF FIRST AND SECOND ASYMPTOTIC APPROXIMATION Let us now apply the double-scale method to the system (3), written in matrix form, for viscoanelastic media with memory. To this aim all the quantities, depending on xα, are considered as depending on xα and ξ. Consequently, the derivatives α ∂ ∂ ∂ ∂ξ ∂ ∂ ∂ϕ with respect to x must be replaced by ∂xα = ∂xα + ∂ξ ∂xα = ∂xα + ω ∂ξ ∂xα . Then, let 146 Adelina Georgescu, Liliana Restuccia us choose the solution of the equations as an asymptotic series ofn powers of the smallo parameter, , namely with respect to the asymptotic sequence 1, εa+1, εa+2, ..., or 1 2 1, ε p , ε p , ..., , as ε → 0. In [4], [29] and [30] it is considered p = 1, and ε = ω−1, such that U(xα, ξ) is written as an asymptotic series with respect to the asymptotic sequence 1, ω−1, ω−2, ..., as ω−1 → 0 , the Ui (i = 1, 2, ...) being functions of xα and ξ,

U(xα, ξ) ∼ U0(xα, ξ) + ω−1U1(xα, ξ) + O(ω−2), as ω−1 → 0, (22) where U0(xα, ξ) is a known solution of

α 0 0 0 A (U )Uα(U ) = B(U ) (23) and it is taken constant and as the initial, unperturbed state, where no small parameters occur [31]. We recall that ξ = ωϕ(xα), ω 1 is a real parameter and ϕ(xα) is the unknown wavefront which is to be determined. Then, the derivative U = ∂U has the following form α ∂xα ∂U −1 ∂U1 ∂U1 ∂ϕ −1 ∂2U2 ∂ϕ −2 −1 ∂xα ∼ ω ∂xα + ω ∂ξ ∂xα + ω ∂ξ ∂xα + O(ω ), as ω → 0 and the following asymptotic expansions are deduced for Aα, Hk, Hik and B: ! 1 1 Aα(U) ∼ Aα(U0) + ∇Aα(U0)U1 + O , as ω−1 → 0, (24) ω ω2 ! 1 1 Hk(U) ∼ Hk(U0) + ∇Hk(U0)U1 + O , as ω−1 → 0 (k = 1, 2, 3), (25) ω ω2 ! 1 1 Hik(U) ∼ Hik(U0) + ∇Hik(U0)U1 + O , as ω−1 → 0 (i, k = 1, 2, 3), (26) ω ω2 ! 1 1 B(U) ∼ B(U0) + ∇B(U0)U1 + O , as ω−1 → 0, (27) ω ω2 ∂ where ∇ = ∂U . Then, we introduce the asymptotic expansions (24) - (27) into the equations of the system (3) in order to obtain the sets of equations of various order of asymptotic approximation. Each such set has as a solution one approximation Ui(i = 1, 2, ...) of U. For the ﬁrst and second asymptotic approximation we have

∂U1 (Aα) Φ = 0 (α = 0, 1, 2, 3), (28) 0 α ∂ξ

2 h 1 1 (Aα) Φ ∂U = − (Aα) ∂U + (∇Aα) U1 Φ ∂U 0 α ∂ξ 0 ∂xα 0 α ∂ξ 2 1 2 1 (29) + (Hk) Φ Φ ∂ U + (Hik) Φ Φ ∂ U − (∇B) U1 , 0 0 k ∂ξ2 0 i k ∂ξ2 0 Asymptotic waves as solutions of nonlinear PDEs ... 147

∂ϕ ∂ϕ where Φα = ∂xα , Φk = ∂xk (k = 1, 2, 3) and the symbol (...)0 indicates that the quantities are computed in U0. Equation (28) is linear in U1, while (29) is aﬃne in U2. Now, by introducing in eq. (28) the quantities (5) and (7), we obtain

∂U1 (A − λI) = 0, (30) 0n ∂ξ

i where (An)0 = A0n and An(U) = A ni. In the case where the eigenvalues are real and the eigenvectors of the matrix An are linearly independent, the system of PDEs (4) is hyperbolic (see [1] for the deﬁnition of hyperbolicity). Furthermore, in [6] it was shown that only for the waves propagating with a velocity λ, that does not satisfy the Lax - Boillat exceptionality condition, i.e. ∇λ · r , 0 (with r the right eigenvector of (An)0 corresponding to the eigenvalue λ), our results are valid. Eq. (30) shows that ∂U1 ∂ξ can be taken as equal to the right-eigenvector r of A0n, corresponding to some eigenvalue λ. By integration, it follows that U1(xα, ξ) has the form

U1(xα, ξ) = u(xα, ξ)r(U0, n) + ν1(xα), (31) where u is a scalar function to be determined and ν1 is an arbitrary vector of integration which can be taken as zero, without loss of generality. Consequently, in order to determine U1 we must determine u, which gives rise to a phenomenon of distortion of the signals and governs the first-order perturbation obeying a non-linear partial differential equation, that will be studied in Section 6. Of course, equations of asymptotic approximation of higher order can be written and they are affine, but their solution is very difficult.

5. FIRST APPROXIMATION OF WAVE FRONT In this section we show how the wave front ϕ(t, x1, x2, x3) = 0 can be determined (see [4]). Following the general theory [5], [6] we introduce the quantity

Ψ(U, Φα) = ϕt + |gradϕ|λ(U, n). (32) The characteristic equations for (32) are

α dx ∂Ψ dΦα ∂Ψ = , = − α (α = 0, 1, 2, 3), (33) dσ ∂Φα dσ ∂x where σ is the time along the rays. The i−th component of the radial velocity Λ is deﬁned by ! dxi ∂Ψ ∂λ ∂λ Λi(U, n) ≡ = = λni + − n · ni = λni + vi − (nkvk)ni (34) dσ ∂φi ∂ni ∂n 148 Adelina Georgescu, Liliana Restuccia

(i = 1, 2, 3). Hence, Λ(U, n) = v − (vn − λ)n. (35) Since we are considering the propagation into an uniform unperturbed state, it is known that the wave front ϕ satisﬁes the partial diﬀerential equation

0 0 0 0 Ψ(U , Φα) = ϕt + |gradϕ|λ(U , n ) = Ψ = 0, (36) where n0 is the constant value of n and represents the normal vector at the point (xi)0 deﬁned by ! gradϕ grad0ϕ0 ∂ n0 = = , with (grad0) ≡ (i = 1, 2, 3). (37) 0 0 i i 0 |gradϕ| t=0 |grad ϕ | ∂(x ) The characteristic equations for (36) are dxα ∂Ψ0 = (α = 0, 1, 2, 3), (38) dσ ∂Φα dΦ ∂Ψ0 α = − (α = 0, 1, 2, 3), (39) dσ ∂xα where σ is the time along the rays. By integration of eq.(38) one obtains

0 i i 0 0 0 0 i 0 i x = t = σ, x = (x ) + Λi (U , n )t, with (x ) = (x )t=0 (i = 1, 2, 3). (40) h i 0 0 i 0 If we denote by ϕ the given initial surface ϕ (x ) = (ϕ)t=0 and if the Jacobian of the transformation x → x|t=0

0 ∂Λk J = det δik + t , 0 (i, k = 1, 2, 3) ∂(xi)0

0 is non vanishing xi can be deduced from (40) and ϕ takes the form i 0 i 0 0 0 ϕ(t, x ) = ϕ x − Λi (U , n )t . (41)

6. FIRST APPROXIMATION OF U In [4] it is shown that, by utilizing (28) and (29) (see [6] and [31]), the following equation for u(xα, ξ) can be obtained: √ 2 ∂u ∂u 1 ∂ J 0 ∂ u 0 + (∇Ψ · r)0 u + √ u + µ = ν u, (42) ∂σ ∂ξ J ∂σ ∂ξ2 where 0 (∇Ψ · r)0 = (|gradϕ|) (∇λ · r)0 , (43) Asymptotic waves as solutions of nonlinear PDEs ... 149 h i l · Hk ∂ϕ ∂ϕ + Hik ∂ϕ ∂ϕ r ∂t ∂xk ∂xi ∂xk 0 µ0 = , (44) (l · r)0 (l · ∇B r) ν0 = 0 , (45) (l · r)0 with l the left eigenvector and r the right eigenvector corresponding to the eigenvalue λ, that does not satisfy the Lax-Boillat condition. By using the transformation of variables (see [31]) Z σ w Z σ z w e 0 u = √ e , κ = √ (∇Ψ · r)0 dσ, with w = ν dσ, (46) J 0 J 0 equation (42) can be reduced to an equation of the type √ ∂z ∂z ∂2z µ0 Je−w + z + µˆ 0 = 0, whereµ ˆ 0 = , (47) 2 ∂κ ∂ξ ∂ξ (∇Ψ · r)0 which is similar to Burger’s equation and is valid along the characteristic rays. Equa- tion (47)1 can be reduced to the semilinear heat equation [42] ∂h ∂2h h dµˆ 0 = µˆ 0 − hlog , (48) ∂κ ∂ξ µˆ 0 dκ that has been extensively studied and for which the solution is known, using the following Hopf transformation ∂ z(ξ, κ) = µˆ logh(ξ, κ). (49) 0 ∂ξ 7. ONE-DIMENSIONAL CASE In this Section the one dimensional case is studied, containing original results. Consider the system of equations (11), (12), (19), (20) and (21). Assume that v2 = v3 = 0, u2 = u3 = 0, x2 = x3 = 0 and that the involved physical quantities depend 0 only on x1, denoted by x. Denote v1(x, t) by v, u1(x, t) by u, P by P and the components of the deviator of the mechanical pressure tensor Pik by Dik, being Dik = Dki. Then, the system (11), (12), (19), (20) and (21) becomes ∂u ∂u v = + v , (50) ∂t ∂x ∂ρ ∂ρ ∂v + v + ρ = 0, (51) ∂t ∂x ∂x ∂v ∂v 1 ∂D 1 ∂P + v + 11 + = 0, (52) ∂t ∂x ρ ∂x ρ ∂x 150 Adelina Georgescu, Liliana Restuccia

∂D 21 = 0, (53) ∂x ∂D 31 = 0, (54) ∂x ∂D ∂D 2 ∂u 2 ∂v 11 + v 11 + R() + R() ∂t ∂x 3 (d)0 ∂x 3 (d)1 ∂x 2 ∂2v 2 ∂2v + R() + R() v + R(τ) D = 0, (55) 3 (d)2 ∂t∂x 3 (d)2 ∂x2 (d)0 11 ∂D ∂D 12 + v 12 + R(τ) D = 0, (56) ∂t ∂x (d)0 12 ∂D ∂D 13 + v 13 + R(τ) D = 0, (57) ∂t ∂x (d)0 13 ∂D ∂D 1 ∂u 1 ∂v 22 + v 22 − R() − R() ∂t ∂x 3 (d)0 ∂x 3 (d)1 ∂x 1 ∂2v 1 ∂2v − R() − R() v + R(τ) D = 0, (58) 3 (d)2 ∂t∂x 3 (d)2 ∂x2 (d)0 22 ∂D ∂D 23 + v 23 + R(τ) D = 0, (59) ∂t ∂x (d)0 23 ∂P ∂P 1 ∂u 1 ∂v 1 ∂2v 1 ∂2v +v + R() + R() + R() + R() v + R(τ) P = 0. (60) ∂t ∂x 3 (v)0 ∂x 3 (v)1 ∂x 3 (v)2 ∂t∂x 3 (v)2 ∂x2 (v)0

Eqns. (53) and (54) show that D21 = f (t), D31 = f1(t), where f and f1 are functions of t. Therefore, from eqns. (56) and (57) we obtain the following results

(τ) (τ) −R(d)0t 0 −R(d)0t 0 D12 = e + D12, D13 = e + D13, revealing that the medium presents a relaxation time for its mechanical properties, due to the presence of a tensorial internal variable. Then, the remained system of equ.s (50) - (52), (55) and (58) - (60) takes the following matrix form (3) " # ∂2U ∂2U U + AU + ω−1 H1 + H11 = B(U), (61) t x ∂t∂x ∂x2 having the following associated system of nonlinear hyperbolic PDEs

Ut + AUx = B(U), (62)

0 T where A (U) = I, U = (u, ρ, v, D11, D22, D23, P) , (τ) (τ) (τ) (τ) T B = (v, 0, 0, −R(d)0D11, −R(d)0D22, −R(d)0D23, −R(v)0P) , Asymptotic waves as solutions of nonlinear PDEs ... 151    v 0 0 0 0 0 0     0 v ρ 0 0 0 0   1 1   0 0 v 0 0   ρ ρ   2 R() 0 2 R() v 0 0 0  A =  3 (d)0 3 (d)1  , (63)  1 () 1 ()   − 3 R 0 − 3 R 0 v 0 0   (d)0 (d)1   0 0 0 0 0 v 0   1 () 1 ()  3 R(v)0 0 3 R(v)1 0 0 0 v    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 2 R0() 0 0 0 0    1  3 (d)2  11  2 0()  H =   , H =  0 3 R(d)2v 0 0 0 0 0  ,  0 0 − 1 R0() 0 0 0 0     3 (d)2   0 − 1 R0() v 0 0 0 0 0   0 0 0 0 0 0 0   3 (d)2     0 0 0 0 0 0 0   0 0 1 R0() 0 0 0 0     3 (v)2  1 0()   0 3 R(v)2v 0 0 0 0 0 (64) () −1 0() () −1 0() T with R(d)2 = ω R(d)2 and R(v)2 = ω R(v)2. The symbol (...) means that U and B are column vectors. We obtain the following expressions for the eigenvalues of the matrix A: • λ1 = v (of multiplicity equalr to 5); () () (±) 2R(d)1+R(v)1 • λ2 = v ± γ, with γ = 3ρ . (±) The eigenvalues λ2 are simple eigenvalues. The right and left eigenvectors corre- (±) sponding to λ2 can be taken, respectively, as    () T  2 R(d)1 2  r(±) = 0, ρ, −(v − λ(±)), R() , − , 0, (v − λ(±))2 − R()  , (65) 2  2 3 (d)1 3 2 3 (d)1

   () ()   2R + R 1 1 (±)  (d)0 (v)0 (±)  l2 =  q , 0, − v − λ2 , , 0, 0,  . (66)  () () ρ ρ 3ρ 2R(d)1 + R(v)1

(±) Only the eigenvalues λ2 do not satisfy the Lax - Boillat exceptionality condition (±) (±) because ∇λ2 · r2 , 0. Then, our results are valid for them. Now, let us consider only the longitudinal wave traveling in the right direction and the case where the propagation occurs into an uniform unperturbed state U0, i. e.

(+) 0 0 λ2 = v + γ and U = (0, ρ , 0, 0, 0, 0, 0), 152 Adelina Georgescu, Liliana Restuccia where ρ0 is constant. The characteristic rays, given by eq. (40), are

0 0 (+) 0 0 0 x = σ = t, x = (x) + λ2 (U )σ = (x) + γ t, (67) whence the wave front is h i ϕ(t, x) = ϕ0 x(t) − γ0t , (68) r 2R() +R() 0 (d)1 (v)1 implying ϕx = 1. Here, γ = 3ρ0 . Now, we compute the terms in (42). We have

(+) (+) (+) ∇Ψ · r2 = ϕx(∇λ2 · r2 ), ! ∂ ∂ ∂ ∂ ∂ ∂ ∂ with ∇ = , , , , , , . Hence, we have ∂u ∂ρ ∂v ∂D11 ∂D22 ∂D23 ∂P ! γ0 γ ∇λ(+) · r(+) = , being ∇λ(+) = 0, − , 1, 0, 0, 0, 0 . (69) 2 2 0 2 2 2ρ Furthermore, we obtain    () (τ) () () (τ) ()  2 R(d)1R(d)0 − R(d)0 + R(v)1R(v)0 − R(v)0  l(+) · ∇B r(+) = −   , (70) 2 2 0  3ρ  0    () ()  2 2R(d)1 + R(v)1  2 l(+) · r(+) =   = 2 γ0 , (71) 2 2 0  3ρ  0 () (τ) () () (τ) () 2 R(d)1R(d)0 − R(d)0 + R(v)1R(v)0 − R(v)0 ν0 = − , (72) () () 2 2R(d)1 + R(v)1 2 l(+) · H1 ∂ϕ ∂ϕ + H11 ∂ ϕ r(+) ∂ϕ 0() 0() 2 ∂t ∂x ∂x2 2 ∂t 2R(d)2 + R(v)2 µ0 = 0 = 0 , (73) (+) (+) q l · r 0 () () 2 2 0 2 3ρ 2R(d)1 + R(v)1 This example, in its simplicity, shows the inﬂuence of a tensorial internal variable on the motion of viscoanelastic media with shape and bulk memory.

Note of L.R. The present paper is one of a series of works, in the area of asymptotic waves in rheological media, where joint original results and previous treatments performed in classical way by one of the authors (L. R), are seen in the modern light of double-scale method, as it is presented in a monograph on asymptotic treatments of the other author A. G.. These works were started and planned since 2004 in occasion of a stay of A. G at the Department of Mathematics of the University of Messina, for Asymptotic waves as solutions of nonlinear PDEs ... 153 delivering a series of lectures on ”Asymptotic methods with applications to waves and shocks”. They were continued during subsequent visits of A.G. at Messina in 2005 and 2007 (for holding seminars and lectures on Applied Mathematics) and fin- ished in next visits of L.R. at Bucharest in 2007 and 2009. This paper was written in a final version in 2010. These studies gave rise from very enlightening remarks concerning some mathematical tools, together with their practical applications in different fields of science. The author L. R. is very grateful to Adelina Georgescu for her deep involvement in this work regarding a systematic formalization of known results, obtained in the field of rheological media, and of joint new results derived on the same subject.

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A NEW MODIFIED GALERKIN METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS Anca-Veronica Ion ”Gh. Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania [email protected] Abstract The nonlinear Galerkin method as well as the postprocessed Galerkin method were built upon the notion of approximate inertial manifold (a.i.m.). We conceived a modiﬁed Galerkin method that uses the so-called ”induced trajectories” deﬁned in [19], instead of the a.i.m.s. The paper presents this new method, that provides a sequence of functions approximating the solution of the Navier Stokes equations. The convergence of this sequence of functions to the exact solution is proved and some comments on the relationship between our method and the nonlinear postprocessed Galerkin method are given.

Keywords: Navier-Stokes equations, induced trajectories, nonlinear and post-processed Galerkin methods, approximate inertial manifolds. 2000 MSC: 35K55, 65M60.

1. INTRODUCTION In the study of dissipative dynamical systems generated by evolution partial differential equations, the notion of inertial manifold arose. Defined in [6], this notion was the object of a large number of works. An inertial manifold (i.m.) is a finite dimensional manifold in the phase space, having some smoothness properties (at least Lipschitz), that is invariant to the considered dissipative dynamical system and ex- ponentially attracts all the trajectories of the system. The restriction of the (infinite dimensional) dynamical system to an i.m. is a finite dimensional dynamical system that has the large time asymptotic properties of the infinite dynamical system. Since not all the dissipative dynamical systems generated by PDEs have i.m.s, and if they have one, it can not be effectively constructed otherwise than approximately, another notion was defined, that of approximate inertial manifold (a.i.m.) [5], [20], [7], [2], [13], to give only a few titles. An a.i.m. is a finite dimensional manifold that has the property that for each trajectory there is a certain moment of time starting from which the trajectory is contained in a narrow neighborhood of this manifold. This property was exploited in order to construct some modified Galerkin methods

155 156 Anca-Veronica Ion with the help of the a.i.m.s, namely the nonlinear (NL) Galerkin method [14], [3], the postprocessed (PP) Galerkin method [8] or some combinations of these two. In this paper we consider the Navier-Stokes equations for a two-dimensional flow, with periodic boundary conditions. For this problem we propose a new modified Galerkin method, that relies not on the a.i.m.s but on some more basic notion, that of induced trajectories, defined in [19]. In [19] a family of a.i.m.s is constructed having as starting point these induced trajectories. Hence, the use of induced trajectories in defining a modified Galerkin method will lead to some simplifications in the numerical algorithm, compared to the NL Galerkin (eventually postprocessed) method. Our work is organized as follows. The functional framework and the basic hypothesis are settled in Section 2. In Section 3 the projector used in the Galerkin method is constructed and some estimates of the ”higher modes” part of the solution are given. In Section 4 a family of induced trajectories similar to those defined in [19], is presented. Our modified Galerkin method is presented in Section 5. In Section 6 we give some bounds for the norms of the functions involved in our method and prove that the sequence of functions provided by our algorithm converges to the solution of the Navier Stokes equations. In Section 7 we compare our method to the NL and PP Galerkin method, that makes use of a high accuracy a.i.m., from the point of view of the amount of computations involved.

2. THE EQUATIONS, THE FUNCTIONAL FRAMEWORK The plane ﬂow of an incompressible Newtonian ﬂuid is modelled by the Navier- Stokes equations:

∂u − ν∆u + (u · ∇) u+∇p = f, (1) ∂t divu = 0, (2)

u (0, ·) = u0 (·) , (3) where u = u (t, x) is the fluid velocity, x ∈Ω ⊂ R2, u (., x) : [0, ∞) → R2, p (., x) : [0, ∞) → R is the pressure of the fluid, ν is the kinematic viscosity, and f is the volume force. We take here Ω = (0, l) × (0, l) and consider the case of periodic boundary conditions. h i 2 2 We assume that f ∈ Lper (Ω) is independent of time. Since we consider periodic boundary conditions, we assume that [17], [16], Z 1 f = f (x) dx = 0, (4) 2 l Ω that the pressure is a periodic function on Ω, and that the average of the velocity over the periodicity cell is zero. The velocity u thus belongs to the space A new modified Galerkin method for the two-dimensional Navier-Stokes equations 157 h i2 H = v; v ∈ L2 (Ω) , div v = 0, v = 0 . The scalar product in H is hu, vi = R per ( ) ( ) ( ) Ω u1v1 + u2v2 dx, (where u = u1, u2 , v = v1, v2 ). The induced norm is denoted by | |. h i 1 2 We also use the space V = u ∈ Hper (Ω) , div u = 0, u = 0 , with the scalar P product hhu, vii = 2 ∂ui , ∂vi , and the induced norm, denoted by k k . We denote i, j=1 ∂x j ∂x j A = −∆. A is defined on D(A) = H∩H2 (Ω) . The classical variational formulation of the Navier-Stokes equations [17] leads to the equation du + νAu + (u · ∇) u = f in V 0, (5) dt u (0) = u0, u0 ∈ H. (6) Consider the trilinear form b(u, v, w) = ((u · ∇) v, w) and the bilinear mapping B(u, v) defined by (B(u, v), w) = b(u, v, w). The notation B(u) = B(u, u) will be also used. For the bilinear mapping B(u, v) the following inequalities

1 1 |B (u, v)| ≤ c1 |u| 2 |∆u| 2 kvk , (∀) u ∈D(A), v ∈V, (7) " !# 1 |∆u|2 2 |B (u, v)| ≤ c kuk kvk 1 + ln , (∀) u ∈D(A), v ∈V, (8) 2 2 λ1 kuk hold [10], [17], [19]. We also remind the following properties of the trilinear form b(u, v, w) (valid for periodic boundary conditions [16]): b(u, v, w) = −b(u, w, v), (9) b(u, v, v) = 0, and the following inequalities [16]

1 1 1 1 |b(u, v, w)| ≤ c3 |u| 2 kuk 2 kvk |w| 2 kwk 2 , (∀) u, v, w ∈V, (10)

1 1 1 1 |b(u, v, w)| ≤ c4 |u| 2 kuk 2 kvk 2 |∆v| 2 |w| , (∀) u ∈V, v ∈ D (A) , w ∈H. (11) For the problem (5), (6) we have the classical existence and uniqueness results for the Navier-Stokes equations in R2, with periodic boundary conditions.

Theorem 2.1. [17]. a) If u0 ∈ H, f ∈ H, then the problem (5), (6) has an unique weak solutionu ∈ C0 ([0, T]; H) ∩ L2 (0, T; V) . b) If, in addition to the hypotheses in a), u0 ∈ V, then there is an unique strong solution u ∈ C0 ([0, T]; V)∩L2 (0, T; D(A)) . The solution is, in this latter case, analytic in time on the positive real axis.

The semi-dynamical system {S (t)}t≥0 generated by problem (5) is dissipative [18], [16]. More precisely, there is a ρ0 > 0 such that for every R > 0, there is a t0(R) > 0 158 Anca-Veronica Ion with the property that for every u0 ∈ H with |u0| ≤ R, we have |S (t) u0| ≤ ρ0 for t > t0(R). In addition, there are absorbing balls in V and D (A) for {S (t)}t≥0, i.e. there are ρ1 > 0, ρ2 > 0 and t1(R), t2(R) with t2(R) ≥ t1(R) ≥ t0(R) such that for every R > 0, |u0| ≤ R implies kS (t) u0k ≤ ρ1 for t > t1(R) and |AS (t) u0| ≤ ρ2 for t > t2(R). 3. THE DECOMPOSITION OF THE SPACE, THE PROJECTIONS OF THE EQUATIONS 4π2 2 2 2 The eigenvalues of A are λ j1, j2 = l2 j1 + j2 , j = ( j1, j2) ∈ N \ {(0, 0)} . The corresponding eigenfunctions are √ 2 ( j , ∓ j ) j x ± j x ws± = 2 1 sin 2π 1 1 2 2 , j1, j2 l |j| l √ 2 ( j , ∓ j ) j x ± j x wc± = 2 1 cos 2π 1 1 2 2 , j1, j2 l |j| l

1 2 2 2 where |j| = j1 + j2 [19]. For a ﬁxed m ∈ N we consider the set Γm of eigenvalues λ j1, j2 having 0 ≤ j1, j2 ≤ m. We deﬁne 4π2 λ : = λ = λ = , 1 1,0 0,1 l2 4π2 λ : = λ = λ = (m + 1)2 , m+1 m+1,0 0,m+1 l2 λ 1 δ = δ (m) := 1 = . 2 λm+1 (m + 1)

Obviously, λm+1 is the least eigenvalue not belonging to Γm. Let Hm be the ﬁnite dimensional space generated by the eigenfunctions corresponding to the eigenvalues of Γm. We denote by P the orthogonal projection operator on this subspace and by Q the orthogonal projection operator on the complementary subspace, and, for the solution u of (5), we set p = Pu, q = Qu. We project equation (5) by using P, and Q, to obtain dp + νAp + PB(p + q) = Pf, (12) dt dq + νAq + QB(p + q) = Qf. (13) dt A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 159

Estimates for the ”small” component of the solution

In [5] it is proved that for every R > 0, there is a moment t3 (R) ≥ t2(R) such that for every |u0| ≤ R,

1 1 1 |q (t)| ≤ K0L 2 δ, kq (t)k ≤ K1L 2 δ 2 , 1 1 (14) 0 0 2 2 |q (t)| ≤ K0L δ, |∆q (t)| ≤ K2L , t ≥ t3 (R) , 0 0 where q (t) is the time derivative of q(t), the coefficients K0, K0, K1, K2 depend of ν, | f | , λ, and, for the way we chose the projection subspaces, L = L(m) = 1+ln(2m2) (see also [19]). The constant L comes from the use of inequality (8) in the proof of (14). More specific ! ! |∆p|2 λ L = sup 1 + ln = max 1 + ln = 1 + ln(2m2). (15) 2 p∈PH λ1 kpk λ∈Γm λ1 In [12] we improved the estimates (14) by eliminating L (which tends to infinity with m) from the constants. We proved the following result. e e e0 e Theorem 3.1. There are some positive numbers C0, C1, C 0, C2, that depend only on ν, λ, |Qf| , (and not on n) and there is a moment t4(R) such that, for t ≥ t4(R) the following inequalities hold |q(t)| ≤ Ce0δ, (16)

1 kq(t)k ≤ Ce1δ 2 , (17)

0 e0 q (t) ≤ C 0δ, (18)

|∆q(t)| ≤ Ce2. (19) 4. INDUCED TRAJECTORIES FOR THE NAVIER- STOKES EQUATIONS + In [19] a family of functions, {ui}i∈N, ui : R → H that define the so-called induced trajectories, {ui(t); t ≥ 0}, is constructed. In [20] the definitions are somehow simplified. We use here functions that share properties of both families of functions from [19] and [20]. The functions have the form ui(t) = p(t) + qi(t), where, as above, p(t) = P(u(t)), and the functions qi are successively defined as:

−1 q0(t) = (νA) [Qf − QB (p(t))], (20) −1 q1(t) = (νA) [Qf − QB (p(t) + q0(t))], (21) −1 0 q j(t) = (νA) [Qf − QB p(t) + q j−1(t) − q j−2(t)]. (22) 160 Anca-Veronica Ion

For the sake of some later discussions (Section 5), we denote the right-hand sides of (22) by Ψ j(p, q j−1, q j−2), j ≥ 0, with the convention q−1 = q−2 = 0. For the functions q j the following inequalities are proved in [20]:

1+ j/2 3/2+ j/2 |q j − q| ≤ k jL δ , (23) 1+ j/2 1+ j/2 kq j − qk ≤ k jL δ , (24) 1+ j/2 1/2+ j/2 |A(q j − q)| ≤ k jL δ , (25) 0 1+ j/2 3/2+ j/2 |(q j − q) | ≤ k jL δ . (26) They hold for t large enough (i.e. for t larger than a certain value t(ν, | f |, λ)). These inequalities imply similar ones for the difference u − uj. The definitions (20) - (22) are the starting point for the construction, in [19], [20], of a family of a.i.m.s, {M j} j≥0. The first one of these a.i.m.s, M0, was defined in [5] independent of the notion of induced trajectories. The a.i.m.s of this family were used in the nonlinear Galerkin method [14], [3] as well as in the post-processed Galerkin method [15].

5. OUR MODIFIED GALERKIN METHOD

Let us take a T > t4. The interval [0, T] is the interval on which we seek the approximate solution. Obviously, inequalities (16) - (17) and (23) - (26) are valid for t ∈ [t4, T] . In the sequel we present our modified Galerkin method. It has several levels, of increasing accuracy. Level 0. We define the first step of this level as the classical Galerkin method. Let us consider the Cauchy problem

dp − ν∆p + PB (p) = Pf, (27) dt p(0) = Pu0.

This problem is the Galerkin approximation of our problem. We denote by p0 (t) its solution and deﬁne

−1 eq0(t) = (νA) [Qf − QB (p0(t))]. (28) At this level, the exact solution u is approximated by

eu0 (t) = p0 (t) + eq0(t). (29) This preliminary level diﬀers from the PP Galerkin method of [8] only in the postprocessing part, in the fact that we compute eq0(t) at any moment of time and not only at the end of the time interval on which (27) is integrated. A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 161

Level 1. Now we consider the problem dp − ν∆p + PB p+eq = Pf, (30) dt 0 p(0) = Pu0, with eq0(t) computed at the preceding level. Since eq0(t) is already known, this equation is not more difficult to integrate than the simple Galerkin equation attached to the Navier-Stokes equation. It is an adjusted Galerkin equation since the nonlinear term is adjusted by adding to p(t) the term eq0(t) that approximates q(t) better than 0 does. We denote by p1 (t) the solution of problem (30). Then we define −1 eq1(t) = (νA) [Qf − QB p1(t) + eq0(t) ] . The approximate solution will be defined at this level as

eu1 (t) = p1 (t) + eq1(t). (31)

This function is an approximation of the function u1 (that defines the second induced trajectory - see Section 4). Level k. Let us consider a k ∈ N. We assume that we already constructed eqk−2 and eqk−1. Now, we define pk as the solution of the problem dp + νAp + PB p + eq = Pf, dt k−1 (32) p(0) = Pu0, and then set eqk as " # deq (t) eq (t) = (νA)−1 Qf − QB p (t) + eq (t) − k−2 . (33) k k k−1 dt The corresponding approximate solution of (5)-(6) is defined by

euk (t) = pk (t) + eqk (t) .

Remarks. 1. The right hand side of (33) is equal to Ψk(pk (t) ,eqk−1 (t) ,eqk−2 (t)), (the functions Ψ j are deﬁned in Section 4) and, as can be proved, eqk is an approximation of qk . Our functions euk are approximations of the functions uk = p + qk that generate the induced trajectories. Our construction by-passes the construction of a.i.m.s. and is based directly upon that of the induced trajectories. We might name the sets euk(t); t ≥ 0 - approximate induced trajectories. 2. If k is the last level we construct, than we may compute eqk only at the moment of interest (T for example) as in the PP Galerkin method of [8]. We name our method the repeatedly adjusted and postprocessed Galerkin method (abreviated as R-APP Galerkin method). 162 Anca-Veronica Ion 6. CONVERGENCE OF THE METHOD

We now prove that the sequence of functions euk deﬁned in the previous section converges in the norm of V to the exact solution u of the Navier Stokes equations. We must ﬁrst give some estimates for the norms of the functions pk, eqk.

6.1. UPPER BOUNDS FOR pk, eqk

In the sequel we assume that the initial condition of the studied problem u0 belongs to V and is taken in a radius R ball of H, that is |u0| ≤ R.

Theorem 6.1. There is a t5 ≥ t4 and there is a natural number M0, such that, for the functions p j, eq j, deﬁned in the previous section, the following inequalities hold

|p j| ≤ η0, kp jk ≤ η1, |∆p j| ≤ η2, j ≥ 0,

1/2 0 0 |eq j| ≤ κ0δ, keq jk ≤ κ1δ , |∆eq j| ≤ κ2, eq j (t) ≤ κ0δ, j ≥ 0, if t ≥ t5 and m ≥ M0. 2|Qf| 2|Qf| The numbers η , η , η depend only on ν, λ1, f and κ0 = , κ1 = 1/2 , 0 1 2 νλ1 νλ1 2 κ2 = ν |Qf|.

Proof. For p0, by using the method for the full Navier-Stokes eqns., [18], the inequality

lim sup |p0(t)| ≤ η0,0 := |Pf|/(νλ1) t→∞ may be proved. Hence for a t0(R) (the same t0(R) from the proof of the existence of the absorbing ball in H for u), we have |p0| ≤ η0,0, for t > t0(R). Also the inequality kp0k ≤ η0,1, where η0,1 is obtained from ρ1 (given e.g. in [18] pg. 111) by replacing |f| with |Pf|. We can see that η0,0 ≤ ρ0, η0,1 ≤ ρ1. We must remark that, since the equations for p0 have the same form as the Navier Stokes equations, we can prove that p0 is analytic in time, as is proved for u in [17]. More than that, p0(t) is the restriction to the real axis of an analytic function (a D(A) valued function) of a complex variable. The domain of analyticity is a neighborhood of the positive real axis (in the complex plane). Now, as in [12], take m as an even number, put n = m/2, and consider the space Hn deﬁned similarly as Hm, Pn the projection operator on Hn, and Qn = P − Pn. We also set pp(t) = Pnp0(t), pq(t) = Qnp0(t) (hence p0(t) = pp(t) + pq(t)) and

2 2 δn = λ1,0/λn+1,0 = 1/(n + 1) , Ln = 1 + ln(2n ). A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 163

By using the method of proof of Theorem 1 given in [12], we may show e e 1/2 e that |∆pq| ≤ C2, kpqk ≤ C1δn , |pq| ≤ C0δn. For eq0 we have

|ν∆eq0| ≤ |Qf| + |QB (p0) | ≤ ≤ |Qf| + |QB pp, pq | + |QB pq, pp | + |QB pq, pq |, since QB pp, pp = 0 [12]. By using inequalities (7), it follows that (8),

1/2 1/2 1/2 |ν∆eq0| ≤ |Qf| + c2Ln kppk kpqk + c1|pq| |∆pq| kppk+ 1/2 1/2 +c1|pq| |∆pq| kpqk ≤ e 1/2 1/2 e1/2 e1/2 1/2 e1/2 e1/2 e ≤ |Qf| + c2C1Ln η0,1δn + c1C0 C2 η0,1δn + c1C0 C2 C1δn. 1/2 1/2 The function Ln δn is decreasing and tends to 0 when n → ∞ hence there is an n0 such that for n ≥ n0 the sum of the last three terms above is less than |Qf|. For such an n, 2 |∆eq | ≤ |Qf| := κ . 0 ν 2 1/2 As consequences, by using the inequalities |Aq| > λn+1kqk > λn+1|q|, that hold for every q ∈ QD(A), we obtain 2|Qf| 2|Qf| keq k ≤ δ1/2 := κ δ1/2 and |eq | ≤ δ := κ δ. 0 1/2 1 0 νλ 0 νλ1 1 0 In order to obtain an estimate for eq0(t), we observe that the analyticity of p0 implies, by (28), the analyticity of eq0. The domain of analyticity is the same as for p0. Then, we consider a point t ∈ [0, T], a closed curve in the domain of analyticity of eq0 that surrounds t, and, by using the Cauchy integral formula, we obtain an estimate of the form

0 eq0 (t) ≤ eκδ. Now we assume that for every 0 ≤ j ≤ k − 1, we have

1/2 0 k∆eq jk ≤ κ2, keq jk ≤ κ1δ , |eq j| ≤ κ0δ, eq j (t) ≤ eκδ.

The function pk is deﬁned as the solution of equation (32). We use the classical method for obtaining bounds for the norms of the solution of the Navier Stokes equation, as is presented in [18], that is, we take the scalar product of (32) with its solution, pk, and we obtain 1 d |p |2 + νkp k2 = hPf, p i − hPB(p + eq ), p i ≤ (34) 2 dt k k k k k−1 k 164 Anca-Veronica Ion

≤ |Pf| |pk| + |hPB(pk,eqk−1), pki| + |hPB(eqk−1,eqk−1), pki|.

Here we have used the equalities hPB(pk, pk), pki = hB(pk, pk), pki = 0, and hPB(eqk−1, pk), pki = hB(eqk−1, pk), pki = 0, due to (9). We estimate the terms in (34): ν 2c2L |hPB(p ,eq ), p i| ≤ c L1/2kp k keq k |p | ≤ kp k2 + 2 keq k2 |p |2, k k−1 k 2 k k−1 k 8 k ν k−1 k 1/2 1/2 |hPB(eqk−1,eqk−1), pki| ≤ c1|eqk−1| |∆eqk−1| keqk−1k |pk|. (34) becomes 1 d 7ν 2c2L |p |2 + kp k2 ≤ |Pf| |p | + 2 κ2δ |p |2 + c |eq |1/2|∆eq |1/2keq k |p |. 2 dt k 8 k k ν 1 k 1 k−1 k−1 k−1 k 7ν 2 In the LHS of the inequality above, we use λ1|pk| ≤ kpkk and replace 8 kpkk 7νλ1 2 with 8 |pk| . Moreover, since the function of m, Lδ, is decreasing with m and tends to 0 when m → ∞ we may choose m1 ≥ m0 := 2n0 such that 2c2κ2 νλ 2 1 L(m)δ(m) ≤ 1 , for m ≥ m . ν 8 1

Now, the relation for |pk| becomes 2 1 d 2 7νλ1 2 2 2 2c1 2 3νλ1 2 |pk| + |pk| ≤ |Pf| + |eqk−1| |∆eqk−1| keqk−1k + |pk| , (35) 2 dt 8 νλ1 νλ1 8 and, thus, 2 1 d 2 νλ1 2 2 2 2c1 2 |pk| + |pk| ≤ |Pf| + |eqk−1| |∆eqk−1| keqk−1k ≤ (36) 2 dt 2 νλ1 νλ1 2 2 2 2c1 2 2 ≤ |Pf| + κ0κ2κ1δ . νλ1 νλ1

For m ≥ m1 with m1 fixed above we have, by the definition of κ1, and considering that ν ≤ 1, ν4λ2 ν2λ2 L(m)δ(m) ≤ 1 ≤ 1 , 2 2 2 2 64c2|Qf| 32c2|Qf| hence 2 2 2 4 2c1 2 2 16c1 4 2 16c1|Qf| 2 2 κ0κ2κ δ = |Qf| δ(m) = L(m) δ(m) ≤ 1 3 3 2 νλ1 νλ1 νλ1L(m) A new modified Galerkin method for the two-dimensional Navier-Stokes equations 165

2 4 4 2 3 16c |Qf| (νλ ) c ν λ1 ≤ 1 1 ≤ 1 . 3 2 2 4 4 4 2 νλ1L(m) 32 c2|Qf| 64c2L(m) Finally we obtain

2 3 d 4 c ν λ1 |p |2 + νλ |p |2 ≤ |Pf|2 + 1 , (37) k 1 k 4 2 dt νλ1 32c2L(m) and 4 c2ν2 |p (t)|2 ≤ |p (0)|2e−νλ1t + |Pf|2 + 1 . k k 2 4 2 (νλ1) 32c2L(m)

νλ1 2 − 2 t 1 2 If we chose a et such that R e ≤ 2 |Pf| , for t > et then (νλ1) √ 5 c1ν |pk| ≤ |Pf| + 2 := η0. νλ1 8c2 A short look in the proof of the existence of an absorbing ball in H for the Navier Stokes eqns., given in [18] pg 109, shows that et may be taken as equal to t0(R). Inequalities kpkk ≤ η1 and k∆pkk ≤ η2 may be proved by methods similar to those used in [18], III.2 combined to those above. The fact that η1, η2 depend only on ν, λ1, f is a consequence of the uniform bounds of the various norms of eq j. The presentation of the proofs here would lengthen too much the paper. Now, in what concerns eqk, we start from its deﬁnition, (33), and see that

|ν∆eqk| ≤ |Qf| + |QB(pk, pk)| + |QB(pk, eqk−1)|+ (38)

0 +|QB(eqk−1, pk)| + |QB(eqk−1, eqk−1)| + |eqk−2|.

For pk we deﬁne pkp = Pnpk, pkq = Qnpk. For pkq, by the method of [12], we can prove inequalities of the form

1/2 |pkq| ≤ C0δn, kpkqk ≤ C1δn , |∆pkq| ≤ C2, with C1, C2, C3 independent of k, since the estimates for the various norms of eqk−1 do not depend on k. This and the fact that QB(pkp, pkp) = 0, enable us to write (by (8), (11))

|QB(pk, pk)| ≤ |QB(pkp, pkq)| + |QB(pkq, pkp)| + |QB(pkq, pkq)| ≤

1/2 1/2 1/2 1/2 3/4 1/2 1/2 1/2 1/2 ≤ c2Ln η1C1δn +c4C0 C1 δn η1 η2 +c4C0 C1C2 δn → 0, when n → ∞. 166 Anca-Veronica Ion

The following three terms in the RHS of (38) are of a smaller order than |QB(pk, pk)|, and, by the induction hypothesis

0 0 |eqk−2| ≤ κ0δ.

Now, it is obvious that we can choose an m2 ≥ m1 such that the sum after |Qf| in the RHS of (38) is smaller than |Qf|, for any m > m2 such that 2 k∆eq k ≤ |Qf|. k ν

From here, two other inequalities of interest for eqk follow: 2|Qf| 2|Qf| keq k ≤ δ1/2, |eq | ≤ δ. k 1/2 k νλ νλ1 1

0 To obtain an estimate for eqk, we have to remark that, as for p0, it can be proved that every function pk is analytic in time, and is the restriction to the real axis of an analytic function of complex variable deﬁned on a neighborhood of the real axis, in the complex plane. Actually, the domain of analyticity may be proved to be independent of k, since the norms of eq j, 0 ≤ j ≤ k are bounded uniformly (with upper bounds not depending on j). e0 0 By using the Cauchy integral formula, we will ﬁnd |q k| ≤ eκ δ. 0 0 We set κ0 := max eκ, eκ , and the relation

e0 0 |q k| ≤ κ0δ follow. It is very important to remark that, at step k + 1 (that is, for the functions pk+1, eqk+1), the same constants as at step k would be implied in the estimates. Some differences appear only between the first two steps (0 and 1) and the following, because of the different definitions of the functions in these different cases. Hence no further increase of m besides the last one (m > m2) is necessary. We set M0 = m2. Thus the induction hypothesis on eqk is confirmed and we proved also that the estimates for pk hold for any k ≥ 0, completing the proof.

6.2. THE CONVERGENCE RESULTS

In the proof of the main result of this section, we need the following result that is a direct consequence of Lemma 1 from [8]. A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 167

Lemma 6.1. Consider a function G : [0, ∞) 7→ H. For a s ≥ 0, we denote by Gbi (s) the coordinate of G(s) with respect to the eigenfunction wi , such that, j,l ! j,l P P4 bi i G(s) = G j,l (s) w j,l . If j,l i=1

bi i G j,l (s) ≤ c j,l, f or s ∈ [0, t], 0 ≤ j, l ≤ m, 1 ≤ i ≤ 4, then   1 2 2 Z t  X X4 i  1  c j,l  e−ν(t−s)APG(s)ds ≤   . (39) ν  2  0 j,k≤m i=1 λ j,l We state and prove our result concerning the convergence of the sequence of approximate solutions given by the R-APP Galerkin method.

Theorem 6.2. The sequence of functions euk (t) , k ≥ 0, converges in V uniformly with respect to t ∈ [t5, T] to the exact solution of the problem (5)- (6). More precisely, there is a natural number M such that for m > M, and t ∈ [t5, T], the inequalities :

1 1/4 5/4+k/4 u − euk (t) ≤ (1 + δ )δ , (40) λ1    1  e  1/2 1/4 3/4+k/4 k(u − uk)(t)k ≤ λ1 + 1/2 δ  δ , (41) λ1 hold for every k ≥ 0. Proof. We prove our assertion by induction. 1. We start with k = 0. h i2 2 Theorem 1 and Remark 1 from [9] (for f ∈ Lper(Ω) ) combined with our estimates (16)-(18), lead to

1/2 3/2 p − p0 (t) ≤ CL δ , (42) for t ∈ [t3, T], where C = C(T, ρ1). We must say that C is of the order of eT . Since the function of m, L1/2δ1/4, tends to 0 when m → ∞ and C does not depend on m, we can choose a m3 ≥ m2 (m2 was deﬁned in the proof of 1/2 1/4 Theorem 6.1) such that for m ≥ m3, the inequality CL δ < 1 is true, hence 5/4 |p − p0| ≤ δ , for m ≥ m3. 1/2 From here, by using the inequality kpk ≤ λm+1,0|p|, valid for every p ∈ Hm, 1/2 3/4 1/4 we obtain kp − p0k ≤ λ1 δ , |∆(p − p0)| ≤ λ1δ , for m ≥ m3. 168 Anca-Veronica Ion We estimate the various norms of q − eq0 (t): ν∆ q − eq = QB(p + q) − QB (p ) + dq 0 0 dt

1/2 1/2 1/2 1/2 5/8 1/4 3/8 1/2 1/2 QB p − p0, p ≤ c4|p − p0| kp − p0k kpk |∆p| ≤ c4δ λ1 δ ρ1 ρ2 = c λ1/4ρ1/2ρ1/2δ, 4 1 1 2 1/2 1/2 1/2 1/2 1/2 3/4 QB p0, p − p0 ≤ c1|p0| |∆p0| kp − p0k ≤ c1η0 η2 λ1 δ . With (7) and (17), the third term yields

1 1 1/2 1/2 2 2 e 1/2 |QB(p, q)| ≤ c1 |p| |∆p| kqk ≤ c1ρ0 ρ2 C1δ , and the fourth, with (11), (16), (17), 1/2 1/2 1/2 1/2 e1/2 e1/2 1/2 1/2 3/4 |QB(q, p)| ≤ c4|q| kqk kpk |∆p| ≤ c4C0 C1 ρ1 ρ2 δ . The ﬁfth term is smaller than the preceding one, and for the last term we use (18). Putting all the above inequalities together, we obtain 1 ∆ q (t) − eq (t) ≤ (c λ1/2ρ1/2ρ1/2δ1/2 + c η1/2η1/2λ1/2δ1/4+ 0 ν 4 1 1 2 1 0 2 1/2 1/2 e e1/2 e1/2 1/2 1/2 1/4 e0 1/2 1/2 +c1ρ0 ρ2 C1 + 2c4C0 C2 ρ1 ρ2 δ + C0δ )δ , 1/2 for t great enough. We denote by K0 the coeﬃcient of δ above. We thus 1/2 obtained ∆ q (t) − eq0 (t) ≤ K0δ , and, as consequences, K K q (t) − eq (t) ≤ 0 δ, q (t) − eq (t) ≤ 0 δ3/2. (43) 0 1/2 0 λ λ1 1

As we remarked in the proof of Theorem 6.1., eq0 is the restriction to the positive real axis of a complex variable analytic function, deﬁned on a neighborhood of this axis. On the other hand, q = Qu has the same property [17], but the domain of analyticity may be diﬀerent (being also a neighborhood of the positive real axis). We take the intersection of the two domains, and by using the Cauchy integral formula, we obtain an inequality of the form

0 0 0 3/2 q (t) − eq0 (t) ≤ K0δ . (44) A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 169

Inequality (42) and the second inequality of (43) imply

K0 1/4 5/4 u (t) − eu0 (t) ≤ (1 + δ )δ . (45) λ1

e 2. We now estimate |p − p1| and q − q1 . We have, by subtracting (27) from (12) and by adding and subtracting in the RHS the term PB p1 + eq0, p + q , d p − p1 = ν∆ (p − p1) − PB p + q − p1 + eq0 , p + q − dt −PB p1 + eq0, p + q − p1 + eq0 . From here, by using the semigroup of linear operators of inﬁnitesimal generator νA, we obtain

d νtA νtA e p − p1 (t) = e −PB p − p1, u − PB q − eq0, u − dt −PB p1 + eq0, q − eq0 − PB p1 + eq0, p − p1 , and, by integrating and using p(0) = p1(0),

Z t −ν(t−s)A p − p1 (t) = − e PB p − p1, u + PB p1 + eq0, p − p1 ds − 0 Z t −ν(t−s)A − e PB p1 + eq0, q − eq0 + PB q − eq0, u ds. 0 Inspired by [8] and [9], we use the inequalities [1] ( C A1−αu |v| ≤ C A1/2u |v| , A−αB (u, v) ≤ C |u| A1−αv ≤ C |u| A1/2v ,

α −νtA −α − νλ t valid for α ∈ (1/2, 1) and [11] A e ≤ Ct e 2 , and obtain

Z t νλ −α − 2 (t−s) p − p1 (t) ≤ C(ρ1, η1, κ1) (t − s) e p − p1 (s) ds+ 0

Z t −ν(t−s)A + e PB p1 + eq0, q − eq0 + PB q − eq0, p + q (s) ds . 0 A form of Gronwall inequality ([11], Lemma 7.1.1) implies 170 Anca-Veronica Ion

p − p1 (t) ≤ (46)

Z t −ν(t−s)A ≤ C(T, ρ1, η1, κ1)max e PB p1 + eq0, q − eq0 + PB q − eq0, p + q (s) ds . 0≤t≤T 0 T We must remark that C(T, ρ1, η1, κ1) above is of the order of e . By using the method of [8], we ﬁnd estimates for the coordinates of the several terms in the RHS of (46):

1 K0 K0 2 e[ e e 2 e B q0, q − q0 j,k ≤ q0 A q − q0 ≤ κ0δ 1/2 δ = κ0 1/2 δ , (47) λ1 λ1

1 K0 K0 [ 2 3/2 e 1/2 e 2 B q − eq0, q j,k ≤ q − eq0 A q ≤ δ C1δ = C1δ , (48) λ1 λ1 1 [ 2 B p1, q − eq0 j,k ≤ A q − eq0 I − Pm− j p + |(I − Pm−k) p| (49) K0 1 1 ≤ 1/2 δρ0 + , λ λm− j+1 λm−k+1 1   K  1 1  [ 0 3/2   B q − eq0, p j,k ≤ δ ρ1  1 + 1  , (50) λ1  2 2  λm− j+1 λm−k+1 where Pm− j represents the projection operator on the space spanned by the eigenfunctions corresponding to the eigenvalues in Γm− j and λ j = λ j,0. By using the inequalities (39), (47), (48) and X −2 e λ j,k ≤ C, j,k≤m it follows that

Z t −ν(t−s)A 1 K0 K0 2 e PB eq0, q − eq0 + PB q − eq0, q ds ≤ (κ0 + Ce1)δ . ν 1/2 λ 0 λ1 1

R t In order to estimate the term e−ν(t−s)APB p , q − eq ds we use (49) and 0 1 0 the inequality X 1 C 3/2 2 2 ≤ 3 = Cδ , j,k≤m λ j,kλm− j+1 (m + 1) proved in [8]. It follows

Z t −ν(t−s)A K0 1+ 3 e PB p , q − eq ds ≤ C ρ δ 4 . 1 0 1/2 0 0 νλ1 A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 171

R t A similar estimate can be proved for e−ν(t−s)A PB q − eq , p ds , hence 0 0 ﬁnally, by using (46), we obtain

1 K K K K p − p ≤ C(T, ρ , η , κ ) (κ 0 δ1/4 + 0 Ce δ1/4 + C 0 ρ + C 0 ρ )δ7/4. 1 1 1 1 ν 0 1/2 λ 1 1/2 0 λ 1 λ1 1 λ1 1 (51) Now, if K0 > 1, we deﬁne m4 such that, for m ≥ m4 the inequality

1 K K K K C(T, ρ , η , κ ) (κ 0 δ1/4 + 0 Ce δ1/4 + C 0 ρ + C 0 ρ )δ1/4 ≤ 1 (52) 1 1 1 ν 0 1/2 λ 1 1/2 0 λ 1 λ1 1 λ1 1 holds. If K0 ≤ 1, then m4 will be chosen such that

1 1 1 1 1 C(T, ρ , η , κ ) (κ δ1/4 + Ce δ1/4 + C ρ + C ρ )δ1/4 ≤ 1. (53) 1 1 1 ν 0 1/2 λ 1 1/2 0 λ 1 λ1 1 λ1 1

With this assumption we obtain, for m > m4 and t enough large,

3/2 p − p1 (t) ≤ δ . (54)

Now, we estimate the various norms of q − eq1 :

ν∆ q − eq1 ≤ |QB (p − p1, u)| + QB q − eq0, u + + |QB (p , p − p )| + QB p , q − eq + (55) 1 1 1 0 dq + QB eq0, p − p1 + QB eq0, q − eq0 + dt .

As we did for q − eq0 , we estimate one by one the terms from the right side. For the ﬁrst two, we use (11):

1/2 1/2 1/2 |QB (p − p1, u)| ≤ c4|p − p1| kp − p1k kuk |∆u| 3/4 1/4 1/2 1/2 1/4 1/2 5/4 ≤ c4δ λ1 δ ρ1 ρ2 = c4λ1 ρ1 ρ2δ ; 1 1 1 1 2 2 QB q − eq0, u ≤ c4 q − eq0 q − eq0 kuk 2 |∆u| 2

K0 1/2 1/2 5/4 = c4 3/4 ρ1 ρ2 δ . λ1 172 Anca-Veronica Ion

The following terms can be evaluated by using (7):

1/2 1/2 QB p1, p − p1 ≤ c1|p1| kp1k p − p1 1/2 1/2 1/2 ≤ c1η0 η1 λ1 δ, 1 1 2 2 QB p1, q − eq0 ≤ c1 |p1| ∆p1 q − eq0

1/2 1/2 K0 ≤ c1η0 η2 1/2 δ, λ1 1 1 2 2 QB eq0, p − p1 ≤ c1 eq0 ∆eq0 kp − p1k 1/2 1/2 1/2 3/2 ≤ c1κ0 κ2 λ1 δ , 1 1 2 2 QB eq0, q − eq0 ≤ c1 eq0 ∆eq0 q − eq0

1/2 1/2 K0 3/2 ≤ c1κ0 κ2 1/2 δ . λ1 With (18) we obtain   1  1/4 1/2 K0 1/2 1/2 1/2 1/2 1/2 ∆ q − eq ≤ c λ ρ ρ δ1/4 + c ρ ρ δ1/4 + c η η λ + 1 ν  4 1 1 2 4 3/4 1 2 1 0 1 1 λ1  K K  1/2 1/2 0 1/2 1/2 1/2 1/2 1/2 1/2 0 1/2 e0  +c1η0 η2 1/2 + c1κ0 κ2 λ1 δ + c1κ0 κ2 1/2 δ + C0 δ. (56) λ1 λ1

We denote the coeﬃcient of δ above by K1(= K1(m) since it depends on m). At this point we make a new assumption on m. That is, we consider a natural 1/4 1/4 number m5 such that K1(m5)δ(m5) < 1. It follows that K1(m)δ(m) < 1 for any m > m5, and for such an m, 1 1 ∆ q − eq ≤ δ3/4, q − eq ≤ δ5/4, q − eq ≤ δ7/4. (57) 1 1 1/2 1 λ λ1 1

The arguments used to state the analyticity in time of q0 remain valid for q1 and the following relation, that will be used later, follows

0 0 0 7/4 q − eq1 ≤ K1δ . By using (54) and (57) we now obtain

1 1/4 3/2 u − eu1 ≤ (1 + δ )δ . (58) λ1 A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 173

3. The induction step. We assume that, for every 0 ≤ j ≤ k + 1 the inequalities

p − p ≤ δ5/4+ j/4, j 1/2+ j/4 ∆(q − q j) ≤ δ , hold. We prove that similar inequalities hold also for j = k + 2 :

−νtA p − pk+2 (t) = e p − pk+2 (0) − Z t −ν(t−s)A − e PB p − pk+2, u + PB pk+2 + eqk+1, p − pk+2 ds− 0 Z t −ν(t−s)A − e PB pk+2 + eqk+1, q − eqk+1 + PB q − eqk+1, u ds. 0 As we did for p − p1 (t) , we obtain

Z t νλ −α − 2 (t−s) p − pk+2 (t) ≤ C(ρ1, η1, κ1) (t − s) e (p − pk+2)(s) ds+ 0

Z t −ν(t−s)A + e PB pk+2 + eqk+1, q − eqk+1 + PB q − eqk+1, p + q ds . 0 The already cited Gronwall-type Lemma of [11] implies p − pk+2 (t) ≤

Z t −ν(t−s)A ≤ C(T, ρ1, η1, κ1)max e PB pk+2 + eqk+1, q − eqk+1 + PB q − eqk+1, u ds . 0≤t≤T 0

Remark. The coeﬃcients C(ρ1, η1, κ1) and C(T, ρ1, η1, κ1) from the above inequalities are equal to the coeﬃcients from the similar inequalities concerning p − p1 (t) . We evaluate the coordinates of each term in the brackets after e−ν(t−s)A :

1 1 1+(k+1)/4 e [ e e 2 e B qk+1, q − qk+1 j,l ≤ qk+1 A q − qk+1 ≤ κ0δ 1/2 δ λ1 κ0 2+(k+1)/4 = 1/2 δ , λ1 174 Anca-Veronica Ion

1 1 1 [ 2 3/2+(k+1)/4 e 2 B q − eqk+1, q j,l ≤ q − eqk+1 A q δ C1δ λ1 1 = Ce δ2+(k+1)/4, λ 1 1 1 [ 2 B pk+2, q − eqk+1 j,l ≤ A q − eqk+1 I − Pm− j pk+2 + |(I − Pm−l) pk+2| 1 1+(k+1)/4 ≤ δ η0 1/λm− j+1 + 1/λm−l+1 , λ1/2 1 1 1 1 [ 3/2+(k+1)/4 2 2 B q − eqk+1, p j,l ≤ ρ1δ 1/λm− j+1 + 1/λm−l+1 . λ1 The arguments used for the terms involved in p − p1 (t) lead to

Z t 1 κ Ce −ν(t−s)A e e e 0 1 2+(k+1)/4 e PB qk+1, q − qk+1 + PB q − qk+1, q ds ≤ ( 1/2 + )δ , 0 ν λ λ1 1 Z t −ν(t−s)A C η0 ρ1 k/4+2 e PB pk+2, q − eqk+1 + PB q − eqk+1, p ds ≤ ( + )δ . ν 1/2 λ 0 λ1 1

By putting these results together, it follows

1 κ Ce Cη Cρ p − p (t) ≤ C(T, ρ , η , κ ) ( 0 δ1/4 + 1 δ1/4 + 0 + 1 )δ5/4+(k+3)/4. k+2 1 1 1 ν 1/2 λ 1/2 λ λ1 1 λ1 1 (59) By comparing the above inequality to (52), (53), we see that for m > m5, the inequality

1 κ Ce Cη Cρ C(T, ρ , η , κ ) ( 0 δ1/4 + 1 δ1/4 + 0 + 1 )δ1/4 < 1 1 1 1 ν 1/2 λ 1/2 λ λ1 1 λ1 1 holds. Thus

5/4+(k+2)/4 p − pk+2 (t) ≤ δ that conﬁrms our induction hypothesis in what concerns p . k Now, for ν∆ q − eqk+2 (t) we have

0 0 ν∆ q − eqk+2 = QB(p + q) − QB(pk+2 + eqk+1) + q − eqk, A new modiﬁed Galerkin method for the two-dimensional Navier-Stokes equations 175 thus ν∆ q − eqk+2 ≤ |QB (p − pk+2, u)| + QB q − eqk+1, u + + | QB (pk+2, p − pk+2)| + QB pk+2, q − eqk+1 + 0 0 + QB eqk+1, p − pk+2 + QB eqk+1, q − eqk+1 + q − eqk . (60) By using the induction hypotheses, we obtain

1/2 1/2 1/2 1/2 QB(p − pk+2, u) ≤ c4|p − pk+2| p − pk+2 ρ1 ρ2 ≤ 1/2 1/2 1/4 1+(k+2)/4 ≤ c4ρ1 ρ2 λ1 δ , 1 1 1 1 2 2 QB(q − eqk+1, u) ≤ c4 q − eqk+1 q − eqk+1 kuk 2 |∆u| 2

1 1/2 1/2 1+(k+2)/4 ≤ c4 3/4 ρ1 ρ2 δ , λ1

1/2 1/2 QB(pk+2, p − pk+2) ≤ c1|pk+2| |∆pk+2| p − pk+2 1/2 1/2 1/2 1+(k+1)/4 ≤ c1η0 η2 λ1 δ , 1 1 2 2 QB(pk+2, q − eqk+1) ≤ c1 |pk+2| ∆pk+2 q − eqk+1

1/2 1/2 1 1+(k+1)/4 ≤ c1η0 η2 1/2 δ , λ1 1 1 2 2 QB(eqk+1, p − pk+2) ≤ c1 eqk+1 ∆eqk+1 p − pk+2 1/2 1/2 1/2 3/2+(k+1)/4 ≤ c1κ0 κ2 λ1 δ , 1 1 2 2 QB eqk+1, q − eqk+1 ≤ c1 eqk+1 ∆eqk+1 q − eqk+1

1/2 1/2 1 2+(k+1)/4 ≤ c1κ0 κ2 1/2 δ . λ1

0 0 In what concerns |q − eq k|, we remind the observation from the proof of Theorem 6.1, that the complex domain of analyticity of p j must be the same for every j ≥ 2, since the norms of eq j have upper bounds that do not depend on j. Then, we obtain that the domain of analyticity of eq j must also be independent of j. Hence we may obtain estimates of the form

0 0 0 3/2+ j/4 |q − eq j| ≤ K δ , with K0 independent of j. 176 Anca-Veronica Ion

1 1 ∆ q − eq ≤ (c ρ1/2ρ1/2λ1/4δ1/4 + c ρ1/2ρ1/2δ1/4 + k+2 ν 4 1 2 1 4 3/4 1 2 λ1 1/2 1/2 1/2 1/2 1/2 1 1/2 1/2 1/2 1/2 + c1η0 η2 λ1 + c1η0 η2 1/2 + c1κ0 κ2 λ1 δ + λ1 1/2 1/2 1 0 1/4 3/4+(k+2)/4 + c1κ0 κ2 1/2 δ + K δ )δ . λ1 We denote by Ke the coeﬃcient of δ3/4+(k+2)/4 above. By comparing the RHS of the above inequality with that of (56), we ﬁnd that, for m > m5, the inequality Keδ1/4 < 1 holds, thus

1/2+(k+2)/4 ∆ q − eqk+2 ≤ δ , from where 1 1 q − eq ≤ δ1+(k+2)/4, q − eq ≤ δ3/2+(k+2)/4. k+2 1/2 k+2 λ λ1 1 0 0 0 3/2+(k+2)/4 This will imply also |q − eq k+2| ≤ K δ From (59) and the above estimates it follows that

1 1/4 5/4+(k+2)/4 u − euk+2 ≤ (1 + δ )δ . (61) λ1

The inequality above shows that the sequence of functions euk defined in Sec- tion 5 is strongly convergent in the norm of H to the exact solution u. We also have 1/2 −1/2 ku − eukk ≤ kp − pkk + kq − eqkk ≤ λn+1|p − pk| + λn+1 |∆(q − eqk|) 1/2 3/4+k/4 1 1+k/4 ≤ λ1 δ + 1/2 δ , λ1 that proves that euk → u when k → ∞, in the norm of V. By setting M := m5, the conclusions of the theorem follow. Remarks. 1. We could have obtained estimates of the form 5/4+k/2 e 3/2+k/2 p − pk (t) ≤ Ck(T)δ , q − eqk ≤ Ck(T)δ where Ck(T), Cek(T) depend increasingly on k and T. We had to ”sacrifice”, at every level, a δ1/4 factor in order to obtain estimates with coefficients that do not depend on k and on T. A new modified Galerkin method for the two-dimensional Navier-Stokes equations 177

2. The value of M in the above theorem might be very large. With the price of having an even smaller exponent of δ in the previous error estimates, we 1/4 1/2 3/4 could replace δ with δ or δ in the conditions that deﬁne m3, m4, m5. This will lead to smaller values of M, thus of the dimension of HM. 3. The fact that M is large is not convenient, since one advantage of the modiﬁed Galerkin methods is that the dimension of the space Hm is smaller than that used in the classical Galerkin method. However, the above theorem is just a result of convergence of the method, and its performances depend on the method of proof. Whether or not the R-APP Galerkin method works well for a small dimension projector P is to be proved by numerical experiments.

7. COMMENTS ON THE METHOD 1. We conceived the R-APP Galerkin method in order to bring simpliﬁca- tions to the NL Galerkin methods that use high-accuracy a.i.m.s. Hence, the method is meaningful only if we use at least three levels (k ≥ 2). The NL Galerkin method based on the use of high accuracy a.i.m.s [3], [15], applied to the Navier-Stokes problem and corresponding to our level k, k ≥ 2, consists in solving the ﬁnite dimensional problem

dep − ν∆ep+PB(ep + Φ ep ) = Pf, (61) dt k ep (0) = Pu (0) , for the approximation ep of p = Pu. Here Φk : PH → QH is the function defining an a.i.m. of high accuracy. The advantage of this method towards ours seems to be that the system of equations for ep is integrated only once, while in our method several integrations are necessary. But the recursive definition of Φk (requiring the definitions of all Φ j with j < k) [3] makes the total volume of computations in the NL Galerkin method to be greater than that of our method. The simplification occurs in our method from the fact 0 that qk−2 (from the definition of qk) may be approximated by the numerical derivative (qk−2(t) − qk−2(t − h)) /h (since we have already computed qk−2(t) 0 at every time step). This must be compared with the definitions of z j,m in [3] 1 or q j in [15]. This simplification is present in the postprocessing step of every level, hence also in the postprocessing of the last level, k. Thus, if we compare R-APP Galerkin method to the NL PP Galerkin method that uses high level of accuracy a.i.m.s [15], we see that, because of its simpler definition, our method comprises less computations than the other in the part that corre- 178 Anca-Veronica Ion sponds to the NL Galerkin method (that is from level 0 up to the determination of pk) as well as in the part that corresponds to the postprocessing method (that is the computation of eqk and its summing to pk). Also, the structure of our method, with iterative levels, makes the computations easier to program, since the programs for the numerical integration of the systems of ODEs for pk(t) should have the same structure for all k, only the coordinates of eqk−1(t) remaining to be replaced in the nonlinear term. Moreover, each level represents a certain approximation of the solution, so we can enjoy partial results. 2. The memory of the computer is better organized in our method, since at the beginning of the computations for the level k, we may erase from the memory the values of p j(t), j < k and of eql(t), l < k − 2, and keep only those 0 of eqk−1(t), eqk−2(t). This feature may be improved by taking eq k−1 instead of 0 eq k−2, in the definition of eqk. This will not affect the accuracy of the method and will allow us to keep in the memory of the computer, at level k, only the previously computed values of eqk−1 and erase all other intermediate functions. A similar procedure is not possible in the NL Galerkin method, where the definition of Φk requires at every step of the numerical integration the appeal of all Φ j, j < k [3], [15]. 3. As in all modified Galerkin methods, problems appear due to f. If this function has a infinity of nonzero coefficients in its Fourier series, it will generate a infinite number of non-zero coordinates in eq j(t). A truncation criterion must be applied and it will depend on f. Thus the number of coordinates of eq j(t) to be computed depends on j and on the given function f. References

[1] P. Constantin, C. Foias¸, Navier-Stokes Equations, Chicago Lectures in Math., Univ. of Chicago Press, IL, 1988; [2] A. Debussche, M. Marion, On the construction of families of approximate inertial manifolds, J. Diff. Eqns., 100(1992), 173-201; [3] C. Devulder, M. Marion, A class of numerical algorithms for large time integration: the nonlinear Galerkin methods, SIAM J. Numer. Anal., 29(1992), 462-483; [4] C. Devulder, M. Marion, E.S.Titi, On the rate of convergence of Nonlinear Galerkin methods, Math. Comp. 60(1993), 495-515; [5] C. Foias¸, O.Manley, R.Temam, Modelling of the interactions of the small and large eddies in two dimensional turbulent flows, Math. Modelling and Num. Anal., 22(1988), 93-114, ; [6] C. Foias¸, G. R. Sell, R. Temam, Variétésinertielles des équationsdifferentielles, C.R. Acad. Sci., Ser.I, 301(1985), 139-141; [7] C. Foias¸, G. R. Sell, E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, Journal of Differential Equations, 1(1989), 199-244; A new modified Galerkin method for the two-dimensional Navier-Stokes equations 179

[8] B. Garcia-Archilla, J. Novo, E.S. Titi, Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds, SIAM J. Numer. Anal., 35(1998), 941-972; [9] B. Garcia-Archilla, Julia Novo, E.S. Titi, An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equation, Mathematics of Computation, 68(1999), 893-911; [10] A. Georgescu, Hydrodynamic stability theory, Martinus Nijhoff Publ., Dordrecht, 1985; [11] D. Henry, Geometric theory of semilinear parabolic equations, Springer, Berlin, 1991; [12] A.-V. Ion, Improvement of an inequality for the solution of the two dimensional Navier-Stokes equations, ROMAI Journal, 4, 1(2008), 119-130. [13] M. Marion, Approximate inertial manifolds for reaction-diffusion equations in high space dimension, J. Dynamics Differential Equations, 1(1989), 245-267; [14] M. Marion, R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26(1989), 1139- 1157; [15] J. Novo, E.S. Titi, S. Wynne, Efficient methods using high accuracy approximate inertial manifolds, Numer. Math., 87(2001), 523-554; [16] J. C.Robinson, Infinite-dimensional dynamical systems; An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge University Press, 2001; [17] R. Temam, Navier-Stokes equations and nonlinear functional analysis, CBMS-NSF Reg. Conf. Ser. in Appl. Math., SIAM, 1995; [18] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer Appl. Math. Sci., 68, Springer, New-York, 1997; [19] R. Temam, Induced trajectories and approximate inertial manifolds, Math. Mod. Num. Anal., 23(1989), 541-561; [20] R. Temam, Attractors for the Navier-Stokes equations, localization and approximation, J. Fac. Sci. Univ. Tokyo, Sec. IA, Math., 36(1989), 629-647.

ROMAI J., 6, 2(2010), 181–184

FORCING A CONTROLLED DIFFUSION PROCESS TO LEAVE THROUGH THE RIGHT END OF AN INTERVAL Mario Lefebvre Department of Mathematics and Industrial Engineering, Ecole´ Polytechnique, Montréal,Canada [email protected] Abstract Let {X(t), t ≥ 0} be a one-dimensional controlled diffusion process evolving in the interval [c, d]. We consider the problem of finding the control that minimizes the mathematical expectation of a cost function with quadratic control costs on the way and a terminal cost function that is infinite if the process hits c before d. The optimal control is obtained explicitly and particular cases are presented.

Keywords: optimal stochastic control, LQG homing, Brownian motion, ﬁrst exit time, Kolmogorov backward equation. 2000 MSC: 93E20.

1. INTRODUCTION Let {X(t), t ≥ 0} be a one-dimensional controlled diffusion process defined by the stochastic differential equation

dX(t) = m[X(t)] dt + b[X(t)] u(t) dt + {v[X(t)]}1/2 dB(t), (1) where {B(t), t ≥ 0} is a standard Brownian motion, u(t) is the control variable and b(·) , 0. We deﬁne the random variable T(x) by

T(x) = inf{t > 0 : X(t) = c or d | X(0) = x}.

Our aim is to determine the value of the control u∗(t) that minimizes the expected value of the cost function Z T(x) 1 2 J(x) = 2 q(x) u (t) dt + K[X(T), T], 0 where q is a positive function and K is the termination cost function. Next, let {x(t), t ≥ 0} be the uncontrolled process obtained by setting u(t) ≡ 0 in (1), and let τ be the same as T, but for {x(t), t ≥ 0}. If the condition

P[τ(x) < ∞] = 1 (2)

181 182 Mario Lefebvre holds, and if the functions b, v and q are such that

b2(x) ≡ α > 0, (3) q(x) v(x) then making use of a result proved by Whittle (see [2], p. 289), we can state that the optimal control u∗ [= u∗(0)] is given by

v(x) G0(x) u∗ = , (4) b(x) G(x) where G(x):= E exp {−α K[x(τ), τ]} | x(0) = x . In Lefebvre [1], the author solved the problem of forcing the controlled process {X(t), t ≥ 0} to stay in the continuation region C := (−∞, d) until a ﬁxed time t0 by giving an inﬁnite penalty if T1 < t0, where

T1(x) = inf{t > 0 : X(t) = d | X(0) = x < d}.

In the present paper, we consider the controlled process {X(t), t ≥ 0} in the interval [c, d]. We want the process to leave the continuation region through its right end. We will take K[X(T), T] = K[X(T)], where the function K is such that K(c) = ∞ and K(d) ∈ R. That is, we give an infinite penalty if X(t) reaches c (before d). The constant d can be chosen as large as we want. The larger it is, the longer it will take X(t) to attain this value. By giving an infinite penalty if the final value of X(t) is equal to c, we force the process to avoid this boundary. We assume that there are no constraints on the control variable u(t). In the next section, we will obtain an explicit formula for the optimal control u∗, and we will present some particular cases.

2. OPTIMAL CONTROL Let πd(x):= P[x(τ) = d | x(0) = x].

The function πd satisﬁes the Kolmogorov backward equation v(x) π00(x) + m(x) π0 (x) = 0, 2 d d and is subject to the boundary conditions

πd(c) = 0 and πd(d) = 1. Forcing a controlled diﬀusion process to leave through the right end of an interval 183

We easily ﬁnd that n o R x R u 2 m(s) c exp c − v(s) ds du πd(x) = n o . (5) R d R u 2 m(s) c exp c − v(s) ds du We will prove the following proposition. Proposition 2.1. Assume that the conditions (2) and (3) are satisﬁed, and that the termination cost function is K[X(T), T] = K[X(T)], with K(c) = ∞ and K(d) ∈ R. Then, the optimal control is given by n o R x 2 m(u) v(x) exp − v(u) du u∗ = R nRc o for c < x < d. (6) b(x) x u 2 m(s) c exp c − v(s) ds du

Proof. We can write that P[x(τ) = c | x(0) = x] = 1 − πd(x). Hence, we deduce from Whittle’s result that u∗ is given by (4), with G(x) = E exp{−α K[x(τ)]} | x(0) = x −αK(c) −αK(d) = e [1 − πd(x)] + e πd(x). Since K(c) = ∞, we obtain that

−αK(d) G(x) = e πd(x), so that 0 −αK(d) 0 G (x) = e πd(x). Hence, the optimal solution (6) follows at once from (5). Remarks. i) Because the interval [c, d] is bounded, the condition (2) is not restrictive. Furthermore, when b, q and v are all constant functions, then the condition (3) is automatically fulﬁlled. ii) We see that the optimal control does not depend on the value of K(d). iii) In many applications, we would like to take c = 0. If the uncontrolled process {x(t), t ≥ 0} can attain the boundary at the origin, then we can indeed replace c by 0 in (6). Particular cases. I) First, if m(x) ≡ 0, then x − c π (x) = d d − c and v(x) 1 u∗ = for c < x < d. b(x) x − c Notice that this case includes the (controlled) standard Brownian motion, for which m(x) ≡ 0 and v(x) ≡ 1. 184 Mario Lefebvre

2 II) Next, assume that q(x) ∝ b (x). With m(x) ≡ m0 , 0 and v(x) ≡ v0 > 0, so that the uncontrolled process {x(t), t ≥ 0} is a Wiener process with drift coefficient m0 and diffusion coefficient v0, we find that 2 m 1 u∗ = 0 n o for c < x < d. b(x) exp 2 m0 (x − c) − 1 v0 III) Finally, if {x(t), t ≥ 0} is a geometric Brownian motion defined by x(t) = eB(t), then m(x) = x/2 and v(x) = x2. The origin being a natural boundary for the geometric Brownian motion, we must take c > 0. The optimal control takes the form x 1 u∗ = for 0 < c < x < d. b(x) ln(x/c)

Here, q(x) must be proportional to b2(x)/x2.

3. CONCLUSION Based on the work presented in Lefebvre [1], we have solved the problem of opti- mally controlling a general diﬀusion process so that it leaves the continuation region (c, d) through the right-hand side of the interval. The objective could have been to leave through the left-hand side instead. Moreover, we could assume that d = ∞. In that case, we could try to maximize the time spent by the controlled process in the interval (c, ∞). Finally, the same type of problem as the one solved here could be considered in two dimensions. For example, (X1(t), X2(t)) could be a controlled two-dimensional Brownian motion, and T be the ﬁrst time that X1(t) hits the boundary x1 = c. If the termination cost is a function of X2(T), then we would have to determine the distribution of this variable, which is a continuous rather than discrete random variable.

References

[1] M. Lefebvre, Forcing a stochastic process to stay in or to leave a given region, Ann. Appl. Probab., 1(1991), 167-172. [2] P. Whittle, Optimization over Time, Vol. I, Wiley, Chichester, 1982. ROMAI J., 6, 2(2010), 185–198

STUDIES AND APPLICATIONS OF ABSOLUTE STABILITY IN THE AUTOMATIC REGULATION CASE OF THE NONLINEAR DYNAMICAL SYSTEMS Mircea Lupu1, Olivia Florea2,Ciprian Lupu3 1Faculty of Mathematics and Computer Science, Transilvania University of Bra¸sov,Romania Academy of Romanian Scientists (Corresponding Member), Bucharest, Romania 2Faculty of Mathematics and Computer Science, Transilvania University of Bra¸sov,Romania 3Faculty of Automatics and Computer Science, Politehnica University of Bucharest, Romania [email protected], olivia.fl[email protected], [email protected] Abstract In this paper the methods of study of the automatic regulation of the absolute stability for some nonlinear dynamical systems are presented. Two methods for the absolute stability are specified: a) the Lurie method with the effective determination of the Lyapunov function; b) the frequencies method of the Romanian researcher V. M. Popov, that uses the transfer function in the critical cases. The applications envisaged here refer to metal cutting tools machine. Both analytical and numerical aspects are considered.

Keywords: : nonlinear systems, automatic control system, absolute stability, tools machine. 2000 MSC: 34D23.

1. INTRODUCTION The automatic regulation for the stability of dynamical systems occupies a fundamental position in science and technique, and it is applied to the optimization of the technological process of cutting tools, of robots, of vehicles (or some machines components) movement regime, of energetic radioactive regimes, chemical, electro- magnetic, thermal, hidroaerodynamic regimes, etc. The complex technical achievements lead to complex mathematical models for closed circuits with input - output, following for the automatic regulation the integration of some mechanisms and devices with inverse reaction of response for the control and the fast and eﬃcient elimination of the perturbations which can appears along these processes or dynamical regimes. Generally these dynamical regimes are nonlinear and some contributions and special achievements for automatic regulation, generating the automatic regulation of absolute stability (a.r.a.s.) for these classes of nonliniarities were necessary. We highlight two special methods of a.r.a.s.:

1 Lyapunov’s function method discovered by A.I. Lurie [10], [12], [17] and developed into a series of studies by M.A Aizerman, V.A. Iacubovici, F.R. Gantmaher, R.E. Kalman, D.R. Merkin [11] and others [1] [14];

185 186 Mircea Lupu, Olivia Florea, Ciprian Lupu

2 Frequency method developed by researcher V. M. Popov [15] generalizing Nyquist’s criterion, then developed in many studies [1], [2], [12]. We note the contributions of Romanian researchers recognized by the works and monographs on the stability and optimal control theory: C. Corduneanu, A. Halanay, V. Barbu, Vl. Rasvan, V. Ionescu, M.E. Popescu, S. Chiriacescu, A. Georgescu and also directly on a.r.a.s.: I. Dumitrache [4] D. Popescu 13], C. Belea [2], V. Rasvan [16], S. Chiriacescu [3] and other recent works [5] [6] [7] [8] [9]. The research has shown that both methods are equivalent, and studies can be performed qualitatively or numerically. In this paper we present the two methods and apply them to a concrete problem.

2. A.R.A.S. USING THE LYAPUNOV’S FUNCTION METHOD In this part we’ll present the Lurie’s ideas and the effective method for finding the Lyapunov’s function. [10] [11] [2] [16] Generally, the systems of automatic regulation (s.r.a.) consist of the controlled processor system, sensor elements of measurement, acquisition board, and the mechanism feedback controller. The regulator comprises all the sensors and the acquisition board, but the controller is included in the feedback mechanism. Parameters charac- terizing the object control system (to control the working mode) are measured by sensors, and their records with the sensor response mechanism ζ is transmitted to the acquisition board. This processes the command σ, which is mechanically transmitted to the controller which, on its turn, distributes the object state and interact simultaneously adjusting the response mechanism. We highlight below the dynamic system equations. We denote by x1, x2,..., xn the state parameters of the regime’s subject which must be controlled, i.e. the coordinates and the sensorial speeds. We recall that the variation of these parameters of the open circuit (excluding the controller) system is de- Pn scribed by linear differential equations with constant coefficients:x ˙k = ak j x j, k = j=1 1,..., n. If the system is with closed loop, then the variables x1, x2,..., xn will be under the influence of the regulation body, and we denote by ξ its state. In this case for the autonomous closed system we have the equations: Xn x˙k = ak j x j + bkξ, k = 1,..., n. (1) j=1 The relation between the output ζ and the input ξ is ζ = kξ. (2) The acquisition board collects the signals and transmits them to the input sensors in order to obtain the embedded system Studies and applications of absolute stability in the automatic regulation case ... 187

Xn σ = c j x j − rξ, (3) j=1 where c j, r are transfer numbers, r > 0 is the transfer coeﬃcient of the inverse rigid connection (the regulator characteristics) [10], [11], [12]. The connection between the output (linear) function σ of the controller and the nonlinear input ϕ in the case of automatic regulation is expressed by the relation

ξ˙ = ϕ(σ). (4) The characteristic function of the controller ϕ(σ), σ ∈ (−∞, +∞) is continuous and satisﬁes the conditions [11], [6], [7]:

a) ϕ(0) = 0, b) σ · ϕ(σ) > 0, ∀σ , 0, (5) R ±∞ c) 0 ϕ(σ)dσ = ∞. The graph of the function ϕ lies in the quarters I, III. The functions ϕ(σ) are named admissible. Moreover, we assume that the sector condition ϕ(σ) 0 < < k (6) σ is satisfied, where k is the amplification coefficient. Example 2.1. 1 ϕ(σ) = sgn(σ) · ln(σ2 + 1), k > 1; 2 ϕ(σ) = a(eσ − 1), k ≤ a.

We assume that the n × n square matrix A = ak j is nonsingular. The equations (1), (3), (4) model the perturbed system with the zeros x(0, 0,..., 0), ξ = 0 .   b  1   ···  c ... c By setting B =   , C = ( 1 n ), C’ the transpose matrix of C, this bn system becomes   x  1  ˙ ˙ 0  ···  X = AX + Bξ, ξ = ϕ(σ), σ = C X − rξ, X =   . (7) xn 3. THE CANONICAL FORM AND THE LYAPUNOV FUNCTION OF SYSTEM (7)

We assume that A with det A = ∆0 , 0 is Hurwitz, i.e. the characteristic polynomial P(λ) = (−1)n det(A − λE) (8) 188 Mircea Lupu, Olivia Florea, Ciprian Lupu has simple roots with Re(λk) < 0, k = 1,..., n. The system (7) is brought to the canonical form if the matrix A is brought to the  λ1 0   .  Jordan form J = diagA =  ..  . There is a matrix T = (tk j) such that   0 λn

T −1AT = J, (⇔ AT = TJ), det T , 0. (9)

With the linear transform:   y  1   ···  X = TY, Y =   , (10) yn from (7) we obtain: TY˙ = ATY+Bξ, ξ˙ = ϕ(σ), σ = C0TY − rξ , that implies:

˙ ˙ 0 −1 0 0 Y = JY + B1ξ, ξ = ϕ(σ), σ = C1Y − rξ, B1 = T B, C1 = C T. (11) Hence, with the linear transform:   z  1  Z = JY + B ξ, σ = C0 Y − rξ, Z =  ···  , (12) 1 1   zn the problem (1),(4) becomes ( Z˙ = JZ + B1ϕ(σ), 0 (13) σ˙ = C1Z − rϕ(σ).

The disturbed system (13) with the equilibrium solution ( zk = 0, σ = 0) is equivalent with system (7) with the equilibrium solution (xk = 0, ξ = 0) and the transform (12) is nondegenerate if the determinant ∆ is non-null:

JB1 0 −1 ∆ = 0 , 0 ⇔ r + C1J B1 , 0 (14) C1 −r −1 −1 −1 −1 0 0 Returning to J = T A T, B1 = T B, C1 = C T transforms we obtain from (13) the ﬁnal condition

r + C0A−1B , 0. (15) Lurie’s problem consists in establishing the conditions for the asymptotic stability of the the null solution (xk = 0, ξ = 0) of system (7) (equivalent with (13) ) with solution zk = 0, σ = 0 for the initial perturbations and for any admissible functions ϕ(σ) deﬁned in (5), (6). The conditions imposed on the matrix A and on the function ϕ(σ) imply the absolute stability (a.s) of the system. [1] [13] Studies and applications of absolute stability in the automatic regulation case ... 189

We remark that if ϕ(σ) is linear, since the matrix A is Hurwitz, the systems (7), respectively (13) are asymptotic stable. The simplicity of system (13) entails imme- diate techniques for determining the Lyapunov function V = V(z1,..., zn, σ) attached to the system (13). The function V(z, σ) of class C1 is a Lyapunov function for system (13) if V(z = 0, σ = 0) = 0, it is positive deﬁned (V(z, σ) > 0), it is radially unlimited to ∞, with ˙ dV dV the absolute derivative V = dt negative deﬁned ( dt < 0) for (z , 0, σ , 0) in a vicinity of the equilibrium point. For the case of automatic regulation we search the function V = V(z, σ) as the sum of a quadratic form zk corresponding to the linear block A and an integral term corresponding to the non linear part

Z σ Z σ 0 V(z, σ) = Z PZ + ϕ(σ)dσ = V1(z, σ) + ϕ(σ)dσ. (16) 0 0 From theory [1] [4] Z0PZ is the quadratic form deﬁned strictly positive if the matrix P is symmetric (P = P0) and we have A0P + PA = −Q where Q is symmetric and positive (with positive eigenvalues). The integral term from (15) is strictly positive because of the conditions (5). Ob- viously, V(z = 0, σ = 0) = 0. Next we impose the condition V˙ < 0 and obtain conditions for parameters ck, r for a.r.a.s.. From (16), by using (13) and

0 0 0 0 0 0 0 Q = Q , P = P , B1PZ + Z PB1 = B1PZ + (PB1) Z = 2(PB1) Z, for

dV(z, σ) = Z0(J0P + PJ)Z − rϕ2(σ) + ϕ(σ)(B0 PZ + Z0PB ) + ϕ(σ)C , dt 1 1 1 we obtain

!0 dV 1 = −Z0QZ − rϕ2(σ) + 2ϕ(σ) PB + C Z, (17) dt 1 2 1 V˙ (z = 0, σ = 0) = 0.

The connection between the matrices P(pi j), Q(qi j) can be established. From λi + 0 0 λ j , 0, i, j = 1,..., n, P = P , J = diagA and Q = Q , we have qi j = − λi pi j + λ j pi j that implies

qi j pi j = − . (18) λi + λ j

Remark 3.1. The matrix A is stable with λi + λ j , 0 if Q is a quadratic form positive deﬁned. 190 Mircea Lupu, Olivia Florea, Ciprian Lupu

Example 3.1. If we choose Q = E (E - the unit matrix) and P is obtained from (17), than the below observation is valid. We must have (−V˙ ) positive defined (since V˙ < 0 ). Apply to the RHS of (17) the Silvester criterion demanding that all diagonal minors of (17) to be positive. Because Q is positive as quadratic form, than the first n inequalities are satisfied and the last inequality is ! ! 1 / 1 r > PB + C Q−1 PB + C . (19) 1 2 1 1 2 1 If the regulator parameters verify the conditions (15), (19) there are sufficient conditions for the asymptotic stability of the solution (x = 0, ξ = 0) of the system (1), (3), (4) [10] [16].

Remark 3.2. A choice technique of the quadratic form V1(z) for pi j according to Lurie is:

Xs nX−2s Xn Xn ε akzka jz j V (z) = ε z z + z2 − 1 2k−1 2k 2 2s+k λ + λ k=1 k=1 k=1 j=1 k j where a1, a2,..., a2s are complex conjugated, a2s+1,..., an are real corresponding to roots λk determining the coeﬃcients ak . Remark 3.3. The two transforms for the diagonal system (1), (3), (4) to obtain (13) can be replaced directly by the transform [12]

Xn N (λ ) x = − k i z , (120) k D0(λ ) i i=1 i n n P where from (7) P(λ) = (−1) D(λ), Nk(λ) = biDik(λ) , Dik are the corresponding i=1 algebraic complements of (i, k) from D(λ) = A−λE. In this case the simpliﬁed system analogous to (13) is:

Xn 0 z˙k = λkzk + ϕ(λ), σ˙ = fizi − rϕ(σ), k = 1, ..., n (13 ) i=1 for which we will build easier V(z, ϕ) .

For the case when a root is null (P(0) = 0) and all the others! have z˜ Re(λk) < 0, k = 1, ..., n − 1, the system (13), with Z = , becomes: z1 0 00 z˜˙ = J˜Z˜ + B˜1ϕ, z˙1 = b0ϕ, ε˙ = C˜ 1Z˜ + C0z1 − rϕ, (13 ) Studies and applications of absolute stability in the automatic regulation case ... 191 where J˜ is a (n − 1) × (n − 1) matrix, Z˜, B˜1 are (n − 1, 1) (column) matrices, while C˜1 is a (1, n − 1) (row) matrix. In this case the Lyapunov function is searched in the form ( Z σ ) 2 0 0 V(˜z, z1, σ) = az1 + z˜ Pz˜ + ϕ(σ)dσ (16 ). 0 For proofs and recent applications we recommend [2][12][11].

4. THE FREQUENCY METHOD FOR A.R.A.S. This method obtained by V. M. Popov [15] is applied to the dynamical system with continuous nonlinearities. We present in this section the method and criteria given by Aizerman, Kalman, Jakubovici [16] [11]. Consider the dynamical, autonomous, non homogeneous system Pn x˙i = ail xl + biu, i = 1, ..., n, l=1 Pn (20) σ = cl xl, u = −ϕ(σ), l=1 where ail, bi, cl are real constants, u is the arbitrary function of input, continuous, nonlinear with ϕ(σ) and σ is the output function. d By using the Laplace transform, and by replacing the operator dt with s we obtain from (20): Xn Xn sxi = ail xl + biu, σ = cl xl, i = 1, ..., n. (21) l=1 l=1 Eliminating from (21) the characteristic parameters of the regulator we obtain σ = W(s)u, σ = W(s)(−ϕ), (22) where W(s) = Qm(s) is the transfer function and Q(s) are polynomials of degrees m, Qn(s) respectively n, with m < n [4] [5] [13]. The transfer function connects σ and ϕ; the function ϕ satisﬁes the conditions (5) ϕ(σ) and the sector condition (6) 0 < σ < k ≤ ∞ - the plot ϕ = ϕ(σ) in the plane (σ, ϕ) will be the sector 0 ≤ ϕ(σ) ≤ kσ. The sector condition and the nonlinearity of ϕ determine the system (σ, ϕ) with closed loop through the impulse function ϕ. We study the absolute stability of the null solution (x = 0, u = 0) of system (20). Because the system is closed and nonlinear we can’t apply directly the Nyquist criterion, [4] [5] [15]. If ϕ ≡ kσ then the system is linear and this criterion can be applied. P Since the block ail xl is linear and biu is nonlinear it results that the roots of the characteristic polynomial P(λ) = (−1)(A − λE) = 0, P(λi) = 0, the poles of W(s) and k will inﬂuence the determination of the absolute stability criteria. 192 Mircea Lupu, Olivia Florea, Ciprian Lupu √ From W(s = jω) = U(ω) + jV(ω), j = −1 we have the hodograph for the axes (U, V) [2] [4] [5] [6] [12]:

U = U(ω), V = V(ω), 0 ≤ ω ≤ ∞. (23)

If all poles of W(s) have Re(si) < 0 then the system is uncritical; if some of the poles of W(s) are null or on the imaginary axis and the rest have Re(si) < 0 then the system is in the critical case. We enunciate the criteria for absolute stability of automatic control a.r.a.s. by the frequency method.

Criterion 1. (the uncritical case). Assume that the following conditions are satisﬁed for system (20):

a) The function ϕ(σ) satisfy (5), (6),

b) All poles of W(s) have Re(si) < 0 ,

c) There is a q ∈ R such that for any ω ≥ 0 the condition 1 + Re (1 + jωq) W( jω) ≥ 0 (24) k holds. Then the system (20) is automatic regulated and absolute stable for the null solution (x = 0, u = 0). From (24) we obtain 1 + U(ω) − qωV(ω) ≥ 0. (240) k From a geometric point of view, criterion (24) shows that in the plane U1 = U, V1 = ωV there exists the line 1 + U − qV = 0 (2400) k 1 1 1 passing through − k , 0 such that the plot of the hodograph is under this line for ω ≥ 0, k > 0 .

Criterion 2. (the critical case when there is a simple null pole s0 = 0 ). Assume that the following conditions are satisﬁed: a) The function ϕ verify (5), (6), b) W(s) has a simple null pole, and the others poles si have Re(si) < 0, Studies and applications of absolute stability in the automatic regulation case ... 193

c) ρ = lim sW(s) > 0 and there is a q ∈ R such that for any ω ≥ 0 condition (24) s→0 holds. Then for the system (20) for the null solution we have a.r.a.s.

Criterion 3. (the critical case when s = 0 is a double pole). Assume that the following conditions are satisﬁed: a) The function ϕ(σ) verify (5), (6) and the sector condition for k = ∞, b) W(s) has a double pole in s=0 andh the otheri poles have Re(si) < 0, c) ρ = lim s2W(s) > 0 , µ = lim d s2W(s) > 0 and π(ω) = ωImW( jω) < 0 for s→0 s→0 ds ∀ω ≥ 0. Then for the system (20) we have a.r.a.s. for the null solution.

Remark 4.1. The form of these criteria (1, 2, 3) has an analytical character but their veriﬁcation for the construction of hodograph values of the coeﬃcients must be done numerically. For special cases we recommend the monographs [2] [4] [12] [16].

5. THE ABSOLUTE STABILITY IN THE AUTOMATIC REGULATION OF METAL CUTTING The high precision of the metal cutting tools implies an automatic regulation of the processes. In this section we present the original results for modeling and the for the study of the nonlinear dynamics of the cutting processes (CP) with tools into metal blocks, for composite materials, blocks or hardwood, practically defined in [3]. These (CP) are: CP of drilling, CP of milling, CP of grinding, screw machine, spindle bearing. Machine tool bar is provided with an inner elastic hard metal cutting, cutting inside to run the required geometric rotation and advancing to step slow. Because of the variation in hardness, density, coefficient of elasticity, material composition manufactured by the process disturbances will occur in work mode: transverse vibration due to shaft rotation or longitudinal vibrations to advance. The automatic controller is equipped with sensors, micrometers, tensiometers, rigid response mechanisms of signals output power amplifiers and accelerators. Their purpose is to adjust the characteristics to obtain asymptotic stability of the system work, resulting in high precision components. We apply the two methods described in the previous sections.

5.1. THE A.R.A.S. METHOD BY LYAPUNOV FUNCTION Consider the dynamic system modeled mathematically, brought to a canonical autonomous form, that features automatic adjustment for absolute stability of dynamic cutting machining processes. [3] [14] 194 Mircea Lupu, Olivia Florea, Ciprian Lupu   x˙1 = a11 x1 + b1ξ   x˙2 = a23 x3 ˙  σ = c2 x2 + c4 x4 − rξ, ξ = ϕ(σ) (25)  x˙3 = a31 x1 + a32 x2 + a33 x3  x˙4 = a44 x4 + b4ξ where ai j, bi, r, ξ are constants i, j = 1, 2, 3, 4, and

a = −m < 0, a = n > 0, a = −εn < 0, a = −p < 0, a = −l < 0 11 31 32 33 44 (26) a23 = 1, c2 = 1, c4 = c < 0, b1 = b > 0, b4 = d − r > 0, r > 0.

These represent mass inertia, elastic constants, strain or pressure coeﬃcients, and σ, r, ξ are the characteristics of the controller. We assume that the input function ϕ is generally nonlinear and the conditions (5) (6) hold. We observe that the linear response function σ of the controller evaluate the elements x2 - the speed of rotation of the cutting bar and x4 - the speed of advancing its material, [3]. We check the absolute stability of the zero solution of the system (x = 0, ξ = 0) . We have det A , 0. Moreover, Re(λi) < 0, i = 1, 2, 3, 4, as the following computations show:

a11 − λ 0 0 0 0 −λ a 0 P(λ) = D(λ) = 23 = a a a − λ 0 (27) 31 32 33 0 0 0 a44 − λ 2 = (a11 − λ)(a44 − λ)(λ − λa33 − a23a32) = 0,

λ1 = a11 = −m < 0, λ4 = a44 = −l < 0, (27) q ! 1 λ = −p ± p2 − 4εn < 0, λ ∈ R 2,3 2 i

In this case, following the diagonalization method (Section 3) with the formulas (9) - (13) or by directly choosing the alternative from Remark 3.3 we get the diagonal system in zi and σ (12’), (13’):

X4 z˙i = λizi + ϕ(σ);σ ˙ = fizi − rϕ(σ), i = 1, ..., 4, (29) i=1

b1a31 −b1a31 f1 = , f2 = (29) (λ1 − λ2)(λ1 − λ3) (λ1 − λ2)(λ2 − λ3) b1a31 f3 = , f4 = b4c4 < 0. (λ1 − λ3)(λ2 − λ3) Studies and applications of absolute stability in the automatic regulation case ... 195

We observe that f1 + f2 + f3 = 0 and whatever is the choice of order quantities, λ1, λ2, λ3 are strictly negative, and always two of the functions fi, i = 1, 2, 3 have the same sign and the third function takes opposite sign using relation (26). In this case, if f4 < 0 we can construct such Lyapunov function:

σ X3 2 2 X3 X3 Z 1 2 1 ai zi aia j V(z, σ) = − f4z4 − − ziz j + ϕ(σ)dσ (31) 2 2 λi λi + λ j i=1 i=1 j=1 0 where the real coefficients a1, a2, a3 will be determined. From λi < 0, λi +λ j < 0, V(0) = 0 , the terms after i = 1, 2, 3 determine a positive definite quadratic form and since the integral is positive, we have V(z, σ) > 0 in a neighborhood of the null solution. We calculate V˙ (z, σ) for the function V defined above:

  X3  2 X  a 2aia j  V˙ (z, σ) = − f λ z2 − (a z + a z + a z )2 − ϕ z  i + − f  . (32) 4 4 4 1 1 2 2 3 3 i  λ λ + λ i i=1 i j,i i j

We remark that V˙ (z = 0, σ = 0) = 0 and in order to have the strict negativity, all the parentheses from the sum that multiplies ϕ must be null, that is

2 X a 2aia j i + − f = 0, i = 1, 2, 3 (33) λ λ + λ i i j,i i j

The system (33) comprises three equations Fi(a1, a2, a3) = 0, i = 1, 2, 3 with three unknowns and the local existence of solutions is ensured by the condition on the system Jacobian: J = D(F1,F2,F3) , 0. If each equation from (33) is multiplied D(a1,a2,a3) respectively by 1 and all equations are summed, then condition (33) proves to be λi equivalent to   X3 X3 2 fi  ai  S := =   > 0. (34) λ  λ  i=1 i i=1 i This condition indicates that the known sum (S) is strictly positive and in the para- 3 √ P ai metric space (a1, a2, a3) the plane (π12) λ = ± S where exists a solution, does i=1 i not admit the null solution because fi , 0. The Jacobian J can be shown to be √ S (a + a + a )[a (λ + λ ) + a (λ + λ ) + a (λ + λ )] J = ±8 1 2 3 1 2 3 2 1 3 3 1 2 , 0. λ1λ2λ3(λ1 + λ2)(λ1 + λ3)(λ2 + λ3) 196 Mircea Lupu, Olivia Florea, Ciprian Lupu

The condition above shows that the solution is contained in the plane (π12), but not in the intersections of this plane with the planes a1 + a2 + a3 = 0 or a1(λ2 + λ3) + a2(λ1 + λ3) + a3(λ1 + λ2) = 0. Analyzing the system (33) by the sign of fi, we see that a1, a2, a3 can not have all the same sign. If this would happen, then fi > 0. We proved that there exist solutions of system (33) and, for these, V˙ < 0 ; it follows that the Lyapunov function provide the automatic regulation of the absolute stability. For this application suﬃcient conditions of type (15), (16) with numerical data, are also obtained.

5.2. THE FREQUENCY METHOD FOR A.R.A.S In the following study the frequency method presented in Section 4 will be applied to problem (25), (26). Because the system (25) is equivalent with (29) the function u = −ϕ(s) satisﬁes the sector conditions. By applying the Laplace transform, the transfer function W(s) is found. So, from (29) is obtained, for σ = W(s)(−ϕ)

X4 szi = λizi + ϕ; sσ = fizi − rϕ. (35) i=1

Eliminating zi from (35) we obtain the transfer function from σ = W(s)(−ϕ)    X4  1  λi fi  W(s) = r −  . (36) s  s − λ  i=1 i

Because the real roots si = λi satisfy Re(λi) < 0, the transfer function has a simple pole in s = 0 and the rest of real roots with Re(si) < 0. In this case we may use the Criterion 2 of critical singularity from Section 4 for a.r.a.s.. Here, the conditions (15), (19) and II a), b) were veriﬁed in the preceding subsection and only condition c) must be veriﬁed. P4 ρ = lim sW(s) = r + fi = r + f4 = r + b4c4 > 0 implies r + c(d − r) > 0 that means s→0 i=1 cd r > 1−c > 0, d < 0 . From W(s = jω) = U(ω) + jV(ω) , we have:

4 4 2 X f λ r 1 X λ fi U(ω) = i i , V(ω) = − + i . 2 2 ω ω 2 2 i=1 λi + ω i=1 λi + ω For given k > 0, from the condition 0 < ϕ(s) < kσ with ϕ(σ) specified, some q ∈ R verifying the condition (24’) may be determined. The parameters λi, fi are known from (28), (30), the nonlinear function ϕ is chosen with σ from (25) and for specified numerical k and, consequently, the delimitation of q are determined. The existence of these conditions can be checked hodographically for a.r.a.s. at this application. Studies and applications of absolute stability in the automatic regulation case ... 197 6. CONCLUSION In the paper, two methods for a.r.a.s. very useful in the fundamental and applicative research, are presented. Their use is exemplified in the application in Section 5. For other studies the published results of the researchers, the works [11] [12] [16] [17] are recommended.

7. ACKNOWLEDGEMENT This work was supported by CNCSIS - IDEI Program of Romanian Research, De- velopment and Integration National Plan II, Grant no. 1044/2007 and ”Automatics, Process Control and Computers” Research Center from University ”Politehnica” of Bucharest (U.P.B.-A.C.P.C.) projects.

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[1] E. A. Barbasin, Liapunov’s function, Nauka, Moscov, 1970 (in Russian) [2] C. Belea, The system theory - nonlinear system, Ed. Did. Ped., Bucures¸ti, 1985 (in Romanian) [3] S. Chiriacescu, Stability in the Dynamics of Metal Cutting, Ed. Elsevier, 1990; Dynamics of cutting machines, Ed. Tehnica, Bucures¸ti, 2004 (in Romanian) [4] I. Dumitrache, The engineering of automatic regulation, Ed. Politehnica Press, Bucures¸ti, 2005. [5] C. Lupu &co., Industrial process control systems, Ed. Printech, Bucuresti, 2004, (in Romanian). [6] C. Lupu &co., Practical solution for nonlinear process control, Ed. Politehnica Press, Bucuresti, 2010 (in Romanian) [7] M. Lupu, O. Florea, C. Lupu, Theoretical and practical methods regarding the absorbitors of oscillations and the multi-model automatic regulatin of systems, Proceedings of The 5th Interna- tional Conference Dynamical Systems and Applications, Ovidius University Annals Series: civil Engineering, Volume 1, Special Issue 11, pg. 63-72, 2009 [8] M. Lupu, F. Isaia, The study of some nonlinear dynamical systems medelled by a more general Reylegh - Van der Pol equation, J. Creative Math, 16(2007), Univ. Nord Baia Mare, 81-90. [9] M. Lupu, F. Isaia, The mathematical modeling and the stability study of some speed regulators for nonlinear oscilating systems, Analele Univ. Bucuresti, LV(2006), 203-212. [10] A. Y. Lurie, Nonlinear problems from the automatic control, Ed. Gostehizdat Moskow, 1951 (in Russian). [11] D. R. Merkin, Introduction in the movement stability theory, Ed. Nauka, Moskow, 1987 (in Rus- sian). [12] R. A. Nalepin &co., Exact methods for the snonlinear system control in case of automatic regulation, M.S. Moskva, 1971. [13] D. Popescu & co., Modelisation, Identiﬁcation et Commande des Systemes, Ed. Acad. Romane, Bucuresti, 2004. [14] E. P. Popov, Applied theory of control of nonlinear systems, Nauka, 1973, Verlag Technik, Berlin 1964. [15] V. M. Popov, The hypersensitivity of automatic systems, Ed. Acad Romane, 1966, Nauka 1970, Springer -Verlag, 1973. [16] Vl. Rasvan, Theory of Stability, Ed. St. Enciclopedica, Bucuresti, 1987, (in Romanian) 198 Mircea Lupu, Olivia Florea, Ciprian Lupu

[17] N. Rouche, P. Habets, M. Leloy, Stability theory by Lyapunov’s Direct Method, Springer Verlag, 1977. ROMAI J., 6, 2(2010), 199–220

SOLVING UNDISCOUNTED INFINITE HORIZON OPTIMIZATION PROBLEMS: A NONSTANDARD APPROACH Dapeng Cai1, Takashi Gyoshin Nitta2 1Institute for Advanced Research, Nagoya University, Nagoya, Japan 2Department of Mathematics, Faculty of Education, Mie University, Japan Abstract Undiscounted infinite horizon optimization problems are intrinsically difficult because (i) the objective functional may not converge; (ii) boundary conditions at the infinite terminal time cannot be rigorously expressed in the real number field. In this paper, by ex- tending real numbers to hyper-real numbers, we derive the optimal solution to an undiscounted infinite horizon optimization problem that has an infinite objective functional. We demonstrate that under a hyper-real terminal time, there exists a unique optimal solution in the hyper-real number field. We show that under fairly general conditions, the standard part of the hyper-real optimal path is the optimum among all feasible paths in the standard real number field, in the sense of two modified overtaking criteria. We also examine the applicability of our approach by considering two parametric examples.

Keywords: inﬁnite-horizon optimization, boundary condition, overtaking criterion. 2000 MSC: C61, E21, O41.

1. INTRODUCTION Most environmental issues, for example, global climate change, loss of biodiver- sity, require the evaluation and comparison of policies whose effects can be expected to spread out into the far distant future (Nordhaus, 1994; Weitzman, 1998; 1999). In economics, such issues have been considered in the framework of the infinite horizon optimization problems. However, with an infinite horizon, the objective functional in general may diverge and one ends up at comparing infinity with infinity, which has been impossible within the standard real number field. As a compromise, economists have been discounting future values, although they have long been scathing about its ethical dimensions (see, for example, Ramsey, 1928; Pigou, 1932; Harrod, 1948; Solow, 1974; Cline, 1992; Anand and Sen, 2000; and Stern, 2007). Discounting raises moral and logical difficulties because a positive discount rate generates a fundamental asymmetry between the treatments of the present and future generations (Heal, 1998). It is thus imperative to tackle these issues in a manner that is ethically and generationally equitable. This calls for the construction of the optimal solutions to the undiscounted infinite horizon optimization problems. The technical difficulties of not discounting lies in the fact that “there is not enough room in the set of real numbers to accommodate and label numerically all the different

199 200 Dapeng Cai, Takashi Gyoshin Nitta satisfaction levels that may occur in relation to consumption programs for an infinite future (Koopmans, 1960; p. 288)”, when there is “a preference for postponing satisfaction, or even neutrality toward timing (ibid.).” Moreover, to rigorously express the boundary conditions at the infinite terminal time, one needs to distinguish the state at the infinite terminal time with that at the time after the infinite terminal time, which has been also impossible in the real number field. To overcome these difficulties, one natural practice would be to extend the real number field R to the hyper-real number field, which has been applied to analyze economic and financial issues. An excellent survey on the economic applications of the non-standard analysis is available in An- derson (1991), which provides ”a careful development of non-standard methodology in sufficient detail to allow the reader to use it in diverse areas in mathematical economics” (p. 2147). Rubio has applied non-standard analysis to optimization theory (Rubio, 1994; 2000).However, by far such analyses have been relatively general and geometrical; they are not readily applicable to most problems in economic dynamics. On the other hand, Okumura et al. (2009a, b) present the generalized transversality condition for infinite horizon optimization problems that may have unbounded objective functionals in the real number field. This calls for the constructions of optimal solutions to undiscounted infinite horizon optimization problems. In this paper, we present an approach that explicitly tackles undiscounted infinite horizon optimization problems in the hyper real number field. We first consider a simple finite horizon undiscounted optimization problem. By resorting to the arguments in the nonstandard analysis, we extend real numbers into hyper-real numbers and present a mathematically correct infinite horizon extension of the finite horizon problem (especially to specify the relevant boundary condition at T = ∞). We then demonstrate that there exists a unique optimal solution to this infinite horizon problem in the hyper-real number field: the limit of the solution for the finite horizon problem is the unique solution to the infinite horizon problem. The conjecture that the limit of the solutions for the finite-horizon problem is the unique solution to the infinite-horizon problem has been around for a while (the case with a discount factor has been examined in for example, Stokey and Lucas (1989)). We show that “the limit of the solutions for the finite-horizon problem” is the unique “projection” of the unique hyper-real optimal solution on the real number field. We also show the conditions under which the unique “projection” on the real number field is the unique optimum to the infinite horizon problem, in the sense of two modified overtaking criteria, respectively. Of the two modified overtaking criteria that we use as our intertemporal optimality criteria, one is common in the literature, which examines the difference between the summation of the utility along the optimal path and that of any other attainable path when T → ∞. The other one, however, is new and takes the quotient form, with its focus on the quotient of the summation of utility along the optimal path divided by that of any other attainable path when T → ∞. We clarify the conditions under which the limits of the solutions for the finite horizon problems are optimal Solving undiscounted infinite horizon optimization problems: a nonstandard approach 201 among all attainable paths for the infinite horizon problem in the sense of the two modified overtaking criteria. In the two simple parametric examples we consider, we show that the modified overtaking criterion in the common form is only valid for one example. On the other hand, the quotient form modified overtaking criterion is apparently weaker, and is valid under both examples. These two examples suggest that the two overtaking criteria differ fundamentally. It can also be easily shown that when discounting is incorporated and the objective functional converges, the criteria are equivalent to each other. The difference emerges when the objective functional diverges.

2. THE MODEL SETTING 2.1. TWO PERPLEXITIES OF THE INFINITE HORIZON OPTIMIZATION PROBLEM We consider an economy that is composed of many identical households, each forming an immortal extended family. In each period, a representative household invests k(t) at the beginning of the period t to produce f(k(t)) amount of output. The production process is postulated as follows: f : Rm → Rm, f is continuous diﬀerentiable, with f(0) = 0. In each period, the output is divided between current consumption c(t) ≡ (ci(t)), with 1 6 i 6 m, and investment, k(t + 1) ≡ (ki(t + 1)), with 1 6 i 6 m. Furthermore, ci(t) > 0, and ki(t + 1) > 0. The problem we consider in this paper is the following: X∞ Maximize U (c (t) , t) (P) t=0

sub ject to : k (t + 1) = f (k (t)) − c (t) , given k (0) = k0, and to the condition that goods will not be wasted. The instantaneous utility function U : Rm+1 → R is assumed to be continuously differentiable. Notice that the objective functional of (P) is not necessarily finite. We first consider the finite horizon version of problem (P). Given the planning horizon T ∈ [0, ∞), the criterion for a social planner to judge the welfare of the representative household takes the form XT max U (c (t) , t), (P0) c(t) t=0 where the instantaneous utility function U : Rm+1 → R is continuous differentiable. The household chooses the path of c(t) that maximizes (P’), which is subject to

the budget constraint : k(t + 1) = f(k(t)) − c(t), (1)

the initial capital stock : k(0) ≡ k0 > 0, and 202 Dapeng Cai, Takashi Gyoshin Nitta

the boundary condition : k(T + 1) = 0. (2)

T An unique solution to problem (P’), subject to (2), {kT (t), cT (t)}t=0, can be found readily by applying the method of Lagrange, with the ﬁrst-order conditions given as (below the subscripts denote the length of the planning horizon):

∂U(cT (t), t) = λiT (t), (3) ∂ci

∂ fi(kT (t)) λiT (t) = λiT (t − 1). (4) ∂ki By substituting (1), (3), into (4) and by rearranging terms, we have

∂ f (k (t)) ∂U((f(k (t)) − k (t + 1)), t) ∂U((f(k (t − 1)) − k (t)), t) i T T T = T T . (5) ∂ki ∂ci ∂ci

We proceed to extend the planning horizon of this problem to infinity. P∞ Two perplexities arise immediately. First of all, the infinite series U(c(t), t) will t=0 in general diverge and the maximization of which may be meaningless in the real number field. Second, it would also be imperative to present an appropriate analogue of the boundary condition (2) for the infinite-horizon case, the na¨ıve extension of which would be k(∞+1) = 0. However, as ∞ has not been defined in R, k(∞+1) = 0 is indefinable, and we can only state that “goods will not be wasted” in the context of real number field, as in Stokey and Lucas (1989). It seems that the boundary condition can be stated instead as lim k(T + 1) = 0, however, which also implies T→∞ lim k(T) = 0. Since the capital stock k at the infinite time is not specified as zero, we T→∞ have arrived at a contradiction and it is inappropriate to state the boundary condition with the concept of limit in R.

2.2. A REFORMULATION OF THE PROBLEM Instead, we extend the field of real numbers so that different “levels of infinity” can be accommodated and we can distinguish “∞” and “∞+1”. Following Anderson [2] (1999, pp. 2150-2151), we extend real numbers. into hyper-real numbers. We define hyper-real number as ∗R ≡ RN u, where N is a set of natural numbers, u is a free ultrafilter that is a maximal filter containing the Fréchet filter. ∗ The elements of R are represented by sequences and are denoted as [< an >], where an ∈ R. Instead of T, we consider an infinite star finite number T˜ ≡ [< Tn >]. We extend c(t) to C(t) ≡ [< Cn(t) >], and k(t) to K(t) ≡ [< Kn(t) >]. Moreover, U : Rm+1 → R and f : Rm → Rm are extended to ∗U : ∗Rm+1 → ∗R and ∗f : ∗Rm → ∗Rm, respectively. Solving undiscounted infinite horizon optimization problems: a nonstandard approach 203

PT PT˜ Accordingly, U(c(t), t) is extended to∗ ∗U(C(t), t), with its elements rep- " t=0 # []=0 PTn resented by < U(Cn(t), t) > . Hence, the maximization problem can be reformu- t=0 lated as XT˜ max ∗ ∗U(C(t), t), ∗(P)

[]=0 ∗ ∗ sub ject to C([< tn >]) = f(K([< tn >])) − K([< tn >] + 1), (1) given K(0), and

the f act that goods will not be wasted : K(T˜ + 1) = 0. ∗(2) Next, we show how to solve the optimization problem in the hyper-real ﬁeld. ∗ m For [< λn >]: {1, 2,..., T˜} → R , we form the Lagrangean,

XT˜ ∗ ∗ ∗ L = ( U(C([< tn >]), t) − [< λn >] (C([< tn >])− []=0

∗ − f(K([< tn >])) + K([< tn >] + 1) ). The ﬁrst-order conditions are familiar:

∗ ∂U(C([< tn >]), t) ∗ = [< λin >]([< tn >]), (3) ∂ci

∗ ∂fi(K([< tn >])) ∗ [< λin >]([< tn >]) = [< λin >]([< tn >] − 1). (4) ∂ki Combining ∗(2), ∗(3), and ∗(4), we have ∂f (K([< t >])) ∂U(∗f(K([< t >])) − K([< t >] + 1), t) ∗ i n ∗ n n = ∂ki ∂ci ∂U(∗f(K([< t >] − 1)) − K([< t >]), t) = ∗ n n , ∗(5) ∂ci where t = 1, 2,..., T˜ . Equation ∗(5) is a second-order diﬀerence equation in K(t); hence it has a two- parameter family of solutions. The unique optimum of interest is the one solution in this family that in addition satisﬁes the two boundary conditions

given K(0), K(T˜ + 1) = 0

. 204 Dapeng Cai, Takashi Gyoshin Nitta

For each index n, equation ∗(5) is equivalent to

∂ f (K (t )) ∂U(∗ f (K (t )) − K (t + 1), t ) ∂U(∗ f (K (t − 1)) − K (t ), t ) ∗ i n n ∗ n n n n n = ∗ n n n n n , ∂ki ∂ci ∂ci ∗(50) where tn = 1, 2,..., Tn, and an unique solution (KT˜ (t), CT˜ (t)) can be obtained by applying the following boundary conditions:

given Kn(0), Kn(Tn + 1) = 0. All the variables in equations ∗(3), ∗(4), and ∗(5) depend on T˜. In particular, [< λn >]([< tn >]) stands for the shadow price for capital at time [< tn >], given the planning horizon T˜. In other words, the value of λ at each time changes with the length of the planning horizon. In what follows, we assume that there exists a unique T˜ ∗ ∗ ∗ optimal solution {K(t), C(t)}t=0 to problem (P), subject to (1) and (2). With a ﬁnite horizon T, because ci(t) > 0 and ki(t + 1) > 0, 0 6 ki(t + 1) 6 fi(k(t)), T and the set of sequences {k(t)}t=0 satisfying (1) and (2) is a closed, bounded, and convex subset of Rm. Moreover, the objective function (P’) is continuous and strictly concave. It is well-known that there is exactly one solution to (P’) (Stokey and Lucas, T˜ ∗ 1989). Hence, the set of sequences {K(t)}t=0 satisfying (2) is a *closed, *bounded, and *convex subset of ∗Rm, and the objective function *(P) is continuous and strictly T˜ ∗ concave. Therefore, there exists a unique solution {K(t), C(t)}t=0 to (P) in the hyper- real number ﬁeld.

2.3. THE LEGITIMACY OF THE LIMIT OF THE SOLUTIONS FOR THE FINITE-HORIZON PROBLEM Denote by est the standard mapping from ∗R → R ∪ {±∞}. As is widely known in the nonstandard analysis, if there exist lim kT (t) and lim cT (t), then T→∞ T→∞ lim kT (t) = est(KT˜ (t)), lim cT (t) = est(CT˜ (t)), where t ∈ R. T→∞ T→∞ Hence, from the nonstandard optimal conditions, we have speciﬁed a unique path 2m (est(KT˜ (t)),est(CT˜ (t))) in R , although we still need to verify whether such a path is indeed the optimal solution in R2m.

2.4. TWO OVERTAKING CRITERIA In what follows, we show that under fairly general conditions, the optimum obtained by applying Theorem 2.1 (below) is indeed the unique optimum among all feasible paths in R2m, under two modified overtaking criteria, modified to incorporate the boundary condition that goods are not to be wasted. One is common in the literature, which examines the difference between the summation of the utility along the optimal path and that of any other attainable path as T → ∞. The other one, Solving undiscounted infinite horizon optimization problems: a nonstandard approach 205 however, is new and takes the quotient form, with its focus on the quotient of the summation of utility along the optimal path divided by that of any other attainable path when T → ∞. For simplicity, we only consider reversible investment. For two attainable paths (k1, c1) and (k2, c2) that satisfy the following conditions: the budget constraint: k(t + 1) = f (k(t)) − c(t), the initial capital stock: k(0) ≡ k0, and the boundary condition that no goods should be wasted; we give the following definitions:

Definition 2.1. (Modified Overtaking Criterion I) (k1, c1) and (k2, c2) are two infinite horizon attainable paths that satisfy conditions (i)-(iii). For all t, (k2, c2) overtakes (k1, c1) if    TX−1 XT−1     lim  U (c2 (t) , t) + U (f (k2 (T)) , T) −  U (c1 (t) , t) + U (f (k1 (T)) , T) > 0. T→∞ t=0 t=0

Definition 2.2. (Modified Overtaking Criterion II) (k1, c1) and (k2, c2) are two infinite horizon attainable paths that satisfy conditions (i)-(iii). For U(c1(t)) > 0, U(c2(t)) > 0 (resp. U(c1(t)) < 0, U(c2(t)) < 0), all t, (k2, c2) overtakes (k1, c1) if TP−1 U (c2 (t) , t) + U (f (k2 (T)) , T) lim t=0 > 1 T−1 T→∞ P U (c1 (t) , t) + U (f (k1 (T)) , T) t=0

 T−1   P   U (c1 (t) , t) + U (f (k1 (T)) , T)   t=0  resp. lim < 1 .  T→∞ TP−1   U (c2 (t) , t) + U (f (k2 (T)) , T)  t=0 Based on the notion of weak maximality [14], the optimality criteria are deﬁned as follows:

Definition 2.3. For all t, an infinite horizon attainable path (k, c) is optimal if no other infinite horizon attainable path (k1, c1) overtakes it:    TX−1 XT−1     lim  U (c2 (t) , t) + U (f (k2 (T)) , T) −  U (c1 (t) , t) + U (f (k1 (T)) , T) 6 0. T→∞ t=0 t=0

Definition 2.4. For U(c(t)) > 0, U(c1(t)) > 0 (resp. U(c(t)) < 0, U(c1(t)) < 0), all t, an infinite horizon attainable path (k, c) is optimal if no other infinite horizon 206 Dapeng Cai, Takashi Gyoshin Nitta attainable path (k1, c1) overtakes it: TP−1 U (c1 (t) , t) + U (f (k1 (T)) , T) lim t=0 6 1 TP−1 T→∞ U (c (t) , t) + U (f (k (T)) , T) t=0

 T−1   P   U (c1 (t) , t) + U (f (k1 (T)) , T)   t=0  resp. lim > 1 .  T→∞ TP−1   U (c (t) , t) + U (f (k (T)) , T)  t=0

Both (k1, c1) and (k, c) are extended to hyper-real number ﬁeld and are reformulated to satisfy the boundary condition that no goods should be wasted. The resultant two paths are then compared by using the notion of weak maximality. According to the non-standard argument, the standard parts of the two derived paths are exactly their original paths when 0 < t < ∞, respectively. Notice that our optimal programs are not restricted to optimal stationary programs.

2.5. THE PROOF OF THE CONJECTURE To establish the legitimacy of the conjecture, we also need the following lemmas:

Lemma 2.1. Let aT and bT , T ∈ [0, ∞), be two sequences. If lim bT = 0, then T→∞

lim (aT + bT ) = lim aT + lim bT = lim aT T→∞ T→∞ T→∞ T→∞ lim (aT + bT ) = lim aT + lim bT = lim aT . T→∞ T→∞ T→∞ T→∞

Lemma 2.2. If lim a(t) > 0, lim b(t) > 0, then lim(ab)(t) = lim a(t) · lim b(t). t→∞ t→∞ t→∞ t→∞ t→∞

Proof. Let lim b(t) = β, then for an arbitrary ε > 0, there exists a T1 > 0, such that t→∞ if t > T1, β − ε < b(t) < β + ε. We denote inf{a(s) |s > t} as a(t). Since lim a(t) > 0, t→∞ there exists a T2, such that if t > T2, then a(t) > 0, that is, if s > T2, then a(s) > 0. Hence, if s > max(T1, T2), then a(s)(β − ε) < (ab)(s) < a(s)(β + ε). Assuming that t > max(T1, T2), we have

inf{a(s)(β − ε) |s > t} 6 inf{(ab)(s) |s > t} 6 inf{a(s)(β + ε) |s > t} , which implies a(t)(β − ε) 6 (ab)(t) 6 a(t)(β + ε). Solving undiscounted inﬁnite horizon optimization problems: a nonstandard approach 207

Hence, lim a(t)(β − ε) 6 lim(ab)(t) 6 lim a(t)(β + ε). t→∞ t→∞ t→∞ Since ε is arbitrarily chosen, lim ab(t) = lim a(t)β. t→∞ t→∞ Theorem 2.1. We consider: T˜−1 ˜ P Condition (i). For an arbitrary infinite number T, (U(CT˜ (t), t) − U(C(t), t)) + t=0 ˜ ˜ ˜ ˜ U(CT˜ (T), T) − U(f(K(T)), T) is infinitesimal. If Condition (i) is satisfied, then for an arbitrary (k1, c1), (k, c) overtakes (k1, c1) in the sense of the modified overtaking criterion as defined in Definition 2.3. ˜ Proof. For an arbitrary infinite number T, CT˜ is an optimal solution. Hence, for an arbitrary admissible path (k1, c1), we have   T˜ T˜−1  X X  U(C (t), t) −  U(C (t), t) + U(f(K (T˜)), T˜) > 0. T˜  1 1  t=0 t=0

0 From the Nonstandard Extension Theorem, there exists a! ﬁnite T such that if T < T , T 0 T 0−1 P P 0 0 then U(cT 0 (t), t) − U(c1(t), t) + U(f(k1(T )), T ) > 0. t=0 t=0 We then have ! TP−1 lim (U(c(t), t) − U(c1(t), t)) + U(f(k(T)), T) − U(f(k1(T)), T) = T→∞ t=0 TX−1 = lim ( (U(cT (t), t) − U(c1(t), t)) + U(f(kT (T)), T) − U(f(k1(T)), T) + T→∞ |t= 0 {z } (A) XT−1 + (U(c(t), t) − U(cT (t), t)) + U(f(k(T)), T) − U(f(kT (T)), T)) |t= 0 {z } (B) By Lemma 2.1, the above equation can be further restated as

TX−1 lim ( (U(cT (t), t) − U(c1(t), t)) + U(f(kT (T)), T) − U(f(k1(T)), T) + T→∞ |t= 0 {z } (A)

TX−1 + (U(c(t), t) − U(cT (t), t)) + U(f(k(T)), T) − U(f(kT (T)), T)) |t= 0 {z } (B) 208 Dapeng Cai, Takashi Gyoshin Nitta

TX−1 = lim ( (U(cT (t), t) − U(c1(t), t)) + U(f(kT (T)), T) − U(f(k1(T)), T))+ T→∞ |t= 0 {z } (A) TX−1 + lim ( (U(c(t), t) − U(cT (t), t)) + U(f(k(T)), T) − U(f(kT (T)), T)). T→∞ |t= 0 {z } (B) From Condition (i), we have   TX−1    lim  (U(c(t), t) − U(cT (t), t)) + U(f(k(T)), T) − U(f(kT (T)), T) = 0. T→∞   t=0 Hence, TX−1 lim ( (U(cT (t), t) − U(c1(t), t)) + U(f(kT (T)), T) − U(f(k1(T)), T) + T→∞ |t= 0 {z } (A) TX−1 + (U(c(t), t) − U(cT (t), t)) + U(f(k(T)), T) − U(f(kT (T)), T)) |t= 0 {z } (B) TX−1 = lim ( (U(cT (t), t) − U(c1(t), t)) + U(f(kT (T)), T) − U(f(k1(T)), T)) = T→∞ |t= 0 {z } (A) ! PT TP−1 = lim (U(cT (t), t) − (U(c1(t), t)) + U(f(k1(T)), T)) > 0. T→∞ t=0 t=0 q.e.d. Note that under this deﬁnition, we are able to compare all paths. In Theorems 2.2 and 2.4, we consider the case for U(c1(t)) < 0, U(c2(t)) < 0, all t, results for U(c1(t)) > 0, U(c2(t)) > 0 can be obtained similarly. Theorem 2.2. : We consider: Condition (ii). For an arbitrary inﬁnite number T˜,

T˜−1 P ˜ ˜ ˜ ˜ U(CT˜ (t), t) − U(C(t), t) + U(CT˜ (T), T) − U(f(K(T)), T) t=0 is in f initesimal. PT˜ U(CT˜ (t), t) t=0 Solving undiscounted inﬁnite horizon optimization problems: a nonstandard approach 209

If Condition (ii) is satisfied, then for an arbitrary (k1, c1), (k, c) overtakes (k1, c1) in the sense of the modified overtaking criterion as defined in Definition 2.4.

˜ Proof. For an arbitrary inﬁnite number T, CT˜ is an optimal solution. Hence, for an arbitrary admissible path (k1, c1), we have

˜ TP−1 ˜ ˜ U(C1(t), t) + U(f(K1(T˜)), T˜) t=0 > 1 PT˜ U(CT˜ (t), t) t=0 . From the Nonstandard Extension Theorem, there exists a ﬁnite T such that T < T 0 TP0−1  TP−1  0 0   U(c1(t),t)+U(f(k1(T )),T )  U(c1(t),t)+U(f(k1(T)),T)  implies t=0 > 1. Hence, lim  t=0  > 1. TP0  PT  U(cT0 (t),t) T→∞ U(cT (t),t) t=0 t=0 We then have PT˜ PT˜ U(CT˜ (t),t) U(CT˜ (t),t) t=0 = t=0 . T˜−1 T˜ T˜−1 T˜−1 P ˜ ˜ P P P ˜ ˜ ˜ ˜ U(C(t),t)+U(f(K(T)),T) U(CT˜ (t),t)−( U(CT˜ (t),t)− U(C(t),t))−U(CT˜ (T),T)+U(f(K(T)),T) t=0 t=0 t=0 t=0 PT˜ U(CT˜ (t),t) Condition (i) implies that the standard part of t=0 is 1. But, since T˜P−1 U(C(t),t)+U(f(K(T˜)),T˜)  t=0    TP−1  TP−1   PT  U(c1(t),t)+U(f(k1(T)),T)  U(c1(t),t)+U(f(k1(T)),T)   U(cT (t),t)  t=0 =  t=0  ·  t=0  , TP−1  PT   TP−1  U(c(t),t)+U(f(k(T)),T) U(cT (t),t) U(c(t),t)+U(f(k(T)),T) t=0 t=0 t=0 from Lemma 2.1, we have TP−1 U(c1(t),t)+U(f(k1(T)),T) t=0 lim TP−1 = T→∞ U(c(t),t)+U(f(k(T)),T) t=0     TP−1   PT   U(c1(t),t)+U(f(k1(T)),T)   U(cT (t),t)  = lim  t=0  · lim  t=0  > 1.  PT  T→∞  TP−1  T→∞ U(cT (t),t) U(c(t),t)+U(f(k(T)),T) t=0 t=0 q.e.d.

Hence we have shown that the path obtained is indeed optimal among all feasible paths in R, if the two conditions listed in Theorem 2.1 are simultaneously satisfied. If the maximand is finite, we can always resort to the standard Lagrange method to find the optimum. On the other hand, Condition (ii) requires that the loss accompanying the limit practice is infinitesimal as compared to the sum of the utility sequence to be maximized. 210 Dapeng Cai, Takashi Gyoshin Nitta

Finally, we explore the implications of these assumptions. For simplicity, we set m = 1. Our results can be easily extended to cases in which m takes other values. Note that when t = T, kT (T + 1) = 0, (5) is reduced to

0 0 0 f (kT (T)) U ( f (kT (T))) = U ( f (kT (T − 1)) − kT (T)) , (6) and we see that kT (T − 1) can be uniquely determined given kT (T). In other words, 1 1 kT (0) can be expressed as a function of kT (T). Letting g(x) ≡ U0(x) , ϕ(x) ≡ f 0(x) ,(??) can be restated as

ϕ (kT (T)) g ( f (kT (T))) = g ( f (kT (T − 1)) − kT (T)) . We then have

−1 f (kT (T − 1)) = kT (T) + g (ϕ (kT (T)) g ( f (kT (T)))) , implying −1 −1 kT (T − 1) = f kT (T) + g (ϕ (kT (T)) g ( f (kT (T)))) . (7) On the other hand, when t = T − 1, we have

−1 f (kT (T − 2)) = g (ϕ (kT (T − 1)) g ( f (kT (T − 1)) − kT (T))) + kT (T − 1) (8), implying −1 −1 kT (T − 2) = f g (ϕ (kT (T − 1)) g ( f (kT (T − 1)) − kT (T))) + kT (T − 1) .

−1 Letting ψT,T (k) ≡ g (ϕ (k) g ( f (k)))+k, we then have f (kT (T − 1)) = ψT,T (kT (T)), −1 i.e., kT (T) = ψT,T ( f (kT (T − 1))), and, by substituting it into (8), we have −1 −1 f (kT (T − 2)) = g ϕ (kT (T − 1)) g f (kT (T − 1)) − ψT,T ( f (kT (T − 1))) +kT (T − 1) .

The ﬁrst subscript T in ψ−1denotes the length of the planning horizon, whereas the second T is used to denote time. From (8), we see that

−1 f (kT (T−1)) − kT (T) = g (ϕ(kT (T )) g ( f (kT (T)))) = −1 −1 −1 = g ϕ ψT,T ( f (kT (T − 1))) g f ψT,T ( f (kT (T − 1))) . Hence f (kT (T − 2)) = = g−1 (ϕ (k (T − 1))) ϕ ψ−1 ( f (k (T − 1))) g f ψ−1 ( f (k (T − 1))) +k (T − 1) . T T,T T T,T T T −1 −1 −1 Letting ψT,T−1(k) ≡ g (ϕ(k)) ϕ ψT,T ( f (k)) g f ψT,T ( f (k)) + k, we see that −1 f (kT (T − 2)) = ψT,T−1 (kT (T − 1)) , i.e., kT (T − 1) = ψT,T−1 ( f (kT (T − 2))). Solving undiscounted inﬁnite horizon optimization problems: a nonstandard approach 211

In general, we have

−1 f (kT (t)) = g (ϕ (kT (t + 1))) g ( f (kT (t + 1))− kT (t + 2)) + kT (T + 1) = −1 −1 −1 = g (ϕ (kT (t + 1))) g ψT,t+2 ( f (kT (t + 1))) g f ψT,t+2 ( f (kT (t + 1))) + +kT (T + 1), implying −1 −1 −1 ψT,t+1 (k) = g (ϕ (k)) ϕ ψT,t+2 ( f (k)) g f ψT,t+2 ( f (k)) + k.

Hence, we deﬁne ψT,t (0 6 t 6 T) as −1 −1 −1 ψT,t (k) = g (ϕ (k)) ϕ ψT,t+1 ( f (k)) g f ψT,t+1 ( f (k)) + k, and we have −1 kT (t) = ψT,t ( f (kT (t − 1))) ,

−1 cT (t) = f (kT (t)) − ψT,t+1 ( f (kT (t))) ≡ φT,t (kT (t)) , In other words,

−1 −1 −1 −1 kT (t) = ψT,t ( f (kT (t − 1))) = ψT,t ◦ f ◦ ψT,t−1 ◦ f ◦ · · · ◦ ψT,0 ◦ f (k (0)) , −1 −1 −1 cT (t) = φT,t ψT,t ◦ f ◦ ψT,t−1 ◦ f ◦ · · · ◦ ψT,0 ◦ f (k (0)) .

As ψT,t is derived inductively from ψT,t+1, if there exists lim ψT,t, then lim ψT,t does T→∞ T→∞ not depends on t. We deﬁne ψ ≡ lim ψT,t. Similarly, we deﬁne φ ≡ lim φT,t, which T→∞ t→∞ also does not depend on t. We then have

k (t) = ψ−1 ( f (k (t − 1))) ,

−1 c (t) = f (k (t)) − ψ ( f (kT (t − 1))) ≡ φ (kT (t)) .

k (t) = ψ−1 ◦ f ◦ ψ−1 ◦ f ◦ · · · ◦ ψ−1 ◦ f (k (0)) , c (t) = φ ψ−1 ◦ f ◦ ψ−1 ◦ f ◦ · · · ◦ ψ−1 ◦ f (k (0)) . Therefore, we can restate the conditions in Theorem 2.1 and 2.2, using the above. The above analysis reveals that if there exists lim ψT,t, then k(t) and c(t) exist. T→∞ Most importantly, as ψ and φ do not depend on t, and k (t) = ψ−1 ( f (k (t − 1))), c (t) = φ (k (t)), hence we see that both the saving function ψ−1 and the consumption function φ does not depend on t. This result indicates that for the optimal path, the relationship between saving and consumption is unchanging over time. 212 Dapeng Cai, Takashi Gyoshin Nitta 3. EXAMPLES Example 1. We consider the following social planner’s problem, in which the social planner treats the future equally by assuming a zero rate of time preference. The planner’s objective is X∞ max (c (t))α, α > 0, c(t) t=0 subject to k (t + 1) = f (k (t)) − c (t) , where f (k (t)) = ak (t), a > 0, k (0) > 0 given, 0 < k (t) < 1, and the fact that goods shall not be wasted. We ﬁrst solve the ﬁnite horizon version of problem: PT max (c (t))α, where T ∈ [0, ∞), α > 0, c(t) t=0 subject to c (t) + k (t + 1) = f (k (t)) , where f (k (t)) = ak (t), k (0) > 0 given,

0 < k (t) < 1, k (T + 1) = 0.

We extend T → T˜, the ﬁnite horizon problem is reformulated as

XT˜ max ∗ ∗ (C (t))α, C(t) []=0 subject to C([< tn >]) + K([< tn >] + 1) = a(K([< tn >])), where 0 < α < 1, K(0) given,

0 < K([< tn >]) < 1, K(T˜ + 1) = 0. The paths of C(t) and K(t) are uniquely determined given k(0) > 0 and k(T + 1) = 0. The solution (KT˜ (t), CT˜ (t)) is given by T˜+1 t(α−1)−t −t−α a K (0) a α−1 − a α−1 t K ˜ (t) = a K (0) − , T (T˜+1)(α−1)−α −T˜−1−α a α−1 − a α−1

−α α−1 K (0) 1 − a t − α−1 CT˜ (t) = ˜ a . − α −αT−2α a α−1 − a α−1 Solving undiscounted inﬁnite horizon optimization problems: a nonstandard approach 213

−T˜−2α Obviously, a α−1 is inﬁnitesimal when (i) α > 1 and a > 1, or when (ii) α < 1 and a < 1. We have

t ◦ −1 − 1 K (t) = K (0) a a α−1 ,

t ◦ α − 1 C (t) = K (0) a α−1 − 1 a α−1 .

α α Because a α−1 > 1 under both case (i) and case (ii), when see that a α−1 > 1 and − 1 ◦ ◦ a α−1 < 1, indicating that C (t) is decreasing in t. On the other hand, K (t) is also decreasing in t. Moreover, ◦ α α C (t) − CT˜ (t) = !α !α! α α 1 1 1 t α − α−1 − α−1 = K (0) 1 − a α − a . − − α − T˜−1−α −T˜−1 a α−1 a α−1 − a α−1 Hence,

T˜P−1 α α α ◦ α ˜ ◦ ˜ CT˜ (t) − C (t) + CT˜ T − f K T t=0 ! ! ! ! α α T˜ α − α α 1 1 1−a− α−1 = K (0) 1 − a α−1 − α αT˜−2 − α − 1 − a α−1 1−a α−1 | a α − 1 − a 1 − {zα }  (I)       − α α   K (0) 1 − a α−1  α  − T˜  −1 − T˜  +  a α−1  − aK (0) a a α−1  .  − α − αT˜−2    a α−1 − a α−1 | {z } | {z } (III)  (II)

αT˜−2 − αT˜ − T˜ Because 0 < a < 1, a 1−α , a α−1 , and a α−1 are all infinitesimals, we see that (I), (II), T˜P−1 α α α ◦ α ˜ ◦ ˜ and (III) are infinitesimals. Hence, CT˜ (t) − C (t) + CT˜ T − f K T t=0 is infinitesimal and Condition (i) in Theorem 2.1 is satisfied. Example 2. We consider the following social planner’s problem, in which the social planner treats the future equally by assuming a zero rate of time preference. The planner’s objective is X∞ max ln(c(t)), c(t) t=0 subject to c(t) + k(t + 1) = f (k(t)), where f (k(t)) = k(t)α, 0 < α < 1, k(0) given, 0 < k(t) < 1, and the fact that goods shall not be wasted. 214 Dapeng Cai, Takashi Gyoshin Nitta

We ﬁrst solve the ﬁnite horizon version of problem: PT max ln(c(t)), where T ∈ [0, ∞), c(t) t=0 subject to c(t) + k(t + 1) = f (k(t)), where f (k(t)) = k(t)α, 0 < α < 1, k(0) given,

0 < k(t) < 1, k(T + 1) = 0. We extend T → T˜, the ﬁnite horizon problem is reformulated as

XT˜ max ∗ ∗ ln(C(t)), C(t) []=0 subject to α C([< tn >]) + K([< tn >] + 1) = (K([< tn >])) , where 0 < α < 1, K(0) given,

0 < K([< tn >]) < 1, K(T˜ + 1) = 0. As 0 < k(t) < 1, 0 < c(t) < 1. The paths of c(t) and k(t) are uniquely determined 1−αT−t α given k(0) > 0 and k(T + 1) = 0, with k(t + 1) = α 1−αT−t+1 k (t), t = 0, 1,..., T. Also, the solution (kT (t), cT (t)) is

t T−t+1 1−α αt 1 − α α 1−α k(0) kT (t) = t−1 , 1 − αT−t+2 1−α 1 − αT−t+3 α(1−α) ··· 1 − αT+1 α

t α(1−α ) αt (1 − α) α 1−α k(0) cT (t) = t−1 t . 1 − αT−t+1 1−α 1 − αT−t+2 α(1−α) ··· 1 − αT α (1−α) 1 − αT+1 α Next we extend T → T˜, where T˜ is an infinite star finite number. Accordingly, we extend t ∈ {0, 1, ··· , T} → t ∈ {0, 1, ··· , T˜}. We have α lim kT (t) = est(KT˜ (t + 1)) = α est KT˜ (t) , t ∈ [0, ∞). T→∞ Hence αt αt−1+···+1 αt 1/(1−α) k(0) est(K ˜ (t)) = (α) K = (α) , T 0 (α)1/(1−α) αt+1 α/(1−α) k(0) est(C ˜ (t)) = (1 − α)(α) , T (α)1/(1−α) −αt+1 −1 −α/(1−α) k(0) est(λ ˜ (t)) = (1 − α) (α) , T (α)1/(1−α) ˜ where λT˜ (t) is the Lagrange multiplier for time horizon T. It stands for the shadow price of capital at time t. Solving undiscounted infinite horizon optimization problems: a nonstandard approach 215

◦ ◦ ◦ Denote est(KT˜ (t)) ≡ K (t), est(CT˜ (t)) ≡ C (t), est(λT˜ (t)) ≡ λ (t). The following proposition shows that (k(t), c(t)) is optimal in R under Definition 2.3. ◦ ◦ Proposition 3.1. Let k(t) = K (t), c(t) = C (t). For an arbitrary (k1, c1), T−1 P α ln(c1(t)) + ln(k1(T)) lim t=0 > 1, TP−1 T→∞ ln(c(t)) + ln(k(T))α t=0 i.e. Condition (ii) in Theorem 2.2 is satisfied and (k(t), c(t)) is optimal in R under Definition 2.3. Proof. For an arbitrary infinite number T˜,   TP−1  T˜P−1  α  ∗ ∗ ˜ α  ln(c1(t)) + ln((k1(T)) )  ln( c1(t)) + ln(( k1(T)) ) t=0  t=0  lim = est   . T→∞ TP−1  T˜P−1  ln(c(t)) + ln((k(T))α)  ln(∗c(t)) + ln((∗k(T˜))α)  t=0 t=0 In what follows, we neglect the notation * if doing so does not cause confusion. We see that T˜P−1 T˜P−1 PT˜ ˜ α ˜ α ln(c1(t)) + ln((k1(T)) ) ln(c1(t)) + ln((k1(T)) ) ln(cT˜ (t)) t=0 = t=0 · t=0 . T˜P−1 PT˜ T˜P−1 ˜ α ˜ α ln(c(t)) + ln((k(T)) ) ln(cT˜ (t)) ln(c(t)) + ln((k(T)) ) t=0 t=0 t=0 (9)

Furthermore, for the sake of simplicity, we use lowercase letters cT˜ to stand for CT˜ . ˜ Here cT˜ is the optimal solution for t ∈ [0, T], hence we have

T˜−1 P ˜ α ln(c1(t)) + ln((k1(T˜)) ) t=0 > 1. PT˜ ln(cT˜ (t)) t=0 From the Nonstandard Extension Theorem, there exists a ﬁnite T such that if T < T 0, TP0−1  TP−1  0 α  α  ln(c1(t))+ln((k1(T )) )  ln(c1(t))+ln(k1(T) )  then t=0 > 1. Hence, lim  t=0  > 1. TP0  PT  ln(cT0 (t)) T→∞ ln(cT (t)) t=0 t=0 Next, we consider the diﬀerence between (c(t), k(t)) and (cT˜ (t), kT˜ (t)). First, XT˜−1 XT˜−1 XT˜−1 ln(cT˜ (t)) − ln c(t) = (ln(cT˜ (t)) − ln c(t)), t=0 t=0 t=0 216 Dapeng Cai, Takashi Gyoshin Nitta here

ln(cT˜ (t)) − ln c(t) = ˜ ˜ ˜ ˜ ˜ ˜ = −(1 − α)α−(T˜−t+1){αT˜−t+1 ln(1 − αT˜−t+1) + ··· + αT˜ ln(1 − αT˜ )} − αt ln(1 − αT˜+1). Let x = αs, T˜ − t + 1 6 s 6 T˜, the right-hand side of the above equality is

T˜ ˜ X ˜ −(1 − α)α−(T˜−t+1) (αs ln(1 − αs)) − αt ln(1 − αT˜+1), s=T˜−t+1 the absolute value of which is bounded by the absolute value of

Z T˜ ˜ ˜ −(1 − α)α−(T˜−t+1) αs ln(1 − αs)ds − αt ln(1 − αT˜+1) ≡ (∗). T˜−t+1 Let αs ≡ x, we see that ln α · αsds = dx, hence

Z T˜ ˜ dx ˜ (∗) = −(1 − α)α−(T˜−t+1) ln(1 − x) − αt ln(1 − αT˜+1). T˜−t+1 ln α Let −dx = dy,(∗) can be further rewritten using

˜ Z 1−αT˜ ˜ ˜ −α−(T˜−t+1) ln ydy − αt ln(1 − αT˜+1) ˜ 1−αT˜−t+1 ˜ ˜ ˜ ˜ = α− T−t+1 − 1 ln 1 − α T−t+1 + 1 − α− T−t+1 − αt−1 ln 1 − αT˜ | {z } | {z } (I) (II) ln α ˜ −αt−1 + αt ln 1 − αT˜+1 . |{z} 1 − α (III) | {z } (IV)

T˜P−1 We observe that (ln(cT˜ (t)) − ln c(t)) is bounded by the absolute value of t=0 T˜−1 T˜−1 1−α P P − ln α {(I) + (II) + (III) + (IV)} . Next, we show that (ln(cT˜ (t)) − ln c(t)) is t=0 t=0 bounded. First, for term (I), letting s = T˜ − t + 1, we can rewrite the sum of (I) as

X2 α−s − 1 ln 1 − αs + 1 . s=T˜+1 Solving undiscounted inﬁnite horizon optimization problems: a nonstandard approach 217

˜ Letting αs = x, we see 0 < αT˜+1 6 x 6 α2 < 1. The sum of (I) can be further rewritten as Z 2 Z 2 α n o dx 1 α n o x−1 − 1 ln (1 − x) + 1 = x−1 − 1 ln (1 − x) + 1 x−1dx. ˜ ˜ αT˜+1 ln α · x ln α αT˜+1 Letting y(x) = x−1 − 1 ln (1 − x) + 1 x−1, we see that

dy −3 −2 −2 −1 −1 dx = (−2x + x ) ln(1 − x) + (x − x ) 1−x = −2+x ln(1 − x) − 2 x3 x2 = −2+x −x − 1 x2 − 1 x3 − · · · − 2 x3 2 3 x2 1 1 3 2 4 3 5 n−2 n = x3 6 x + 4·3 x + 5·4 x + ··· + n(n−1) x + ··· > 0. Hence, we see that y(x) is monotonically increasing in x, and we have lim y(x) = α2 − 1 ln 1 − α2 + 1 α2 > 0. x→α2 Next, we consider lim y(x). It can be easily shown that x→0 y(x) = x−1 − 1 ln (1 − x) + 1 x−1 1 1 1 1 1 1 2 1 1 n−1 = 1 − 2 + 2 − 3 x + 3 − 4 x + ··· + n − n+1 x + ··· . Hence, we see that lim y(x) = 1 , which implies y(x) is bounded in (0, α2). x→0 2 R 2 1 α ˜ Therefore, we have ln α αT˜+1 y(x)dx < +∞. For term (II), we see that  ˜ ˜ ˜   α−(T˜−1) 1−αT˜ α−1 1−αT˜    T˜ the sum of (II) = −  1−α − 1−α  ln 1 − α

˜ α −T˜ T˜ α+α−1−αT˜−1 T˜ =− 1−α α ln 1 − α + 1−α ln 1 − α . ˜ Let αT˜ = x, since T˜ is an infinite number, x is also an infinitesimal. Obviously, the second term is also an infinitesimal. For the first term, as 0 0 −x ˜ ˜ ln (1 − x ) 0 est α−T˜ ln 1 − αT˜ = lim αT ln 1 − αT = lim = lim 1−x = −1, T→∞ x0→0 x0 x0→0 x0 α we see that the sum of (II) is bounded by 1−α . ˜ T˜−1 ˜ P t−1 α−1(1−αT ) For term (III), we see that −α = − 1−α , hence, the sum of (III) is t=0 α−1 bounded by − 1−α . On the other hand, for term (IV), we see that

T˜−1 ˜ X ln α ˜ ln α ˜ 1 − αT αt ln 1 − αT˜+1 = ln 1 − αT˜+1 , 1 − α 1 − α 1 − α t=0 218 Dapeng Cai, Takashi Gyoshin Nitta which is an inﬁnitesimal. T˜P−1 Hence, we have shown that (ln(cT˜ (t)) − ln c(t)) is bounded. t=0 ˜ ˜ α Next, we consider ln(cT˜ (T)) and ln(k(T) ). We see that ,

˜ ˜ α(1−αT ) ln(cT˜ (T)) = ln(1 − α) + 1−α ln α − (1 − α) ln(1 − α) 1−α 2 2 T˜+1 T˜+1 − α {α ln(1 − α ) + ··· + α ln(1 − α )} ˜ ˜ ˜ −αT˜+1 ln(1 − αT˜+1) + αT˜ ln(k(0)).

˜ T˜P+1 T˜R+1 As αt ln(1 − αt) is bounded by αx ln(1 − αx)dx, let αx = y, t=2 2

˜ ZT˜+1 ZαT˜+1 dy αx ln(1 − αx)dx = ln(1 − y) ln α 2 α2

−1 T˜+1 T˜+1 T˜+1 2 2 2 = ln α ((1 − α ) ln(1 − α ) − (1 − α ) − (1 − α ) ln(1 − α ) + (1 − α )), −1 2 2 2 which is bounded by ln α (−1 − (1 − α ) ln(1 − α ) + (1 − α )). !α T˜ ˜ α 1−α αt α On the other hand, ln(k(T˜) ) = ln α 1−α k(0) , which is bounded by ln(α 1−α ). We see that the last term of the right-hand side of equation (9) is equal to

PT˜ ln(cT˜ (t)) t=0 × T˜P−1 ln(c(t)) + ln((k(T˜))α) t=0

T˜−1 P ˜ ln(cT˜ (t)) + ln cT˜ (T) × t=0 . PT˜ T˜P−1 T˜P−1 ˜ ˜ α ln(cT˜ (t)) − ( ln(cT˜ (t)) − ln(c(t))) − ln cT˜ (T) + ln(k(T) ) t=0 t=0 t=0 ˜ ˜ As ln(cT˜ (T))is bounded, not 0 and not infinitesimal, and as ln(cT˜ (t)) (for all t ∈ [0, T]) PT˜ is always negative, we see that ln(cT˜ (t)) is negative infinite. t=0 ˜ In summary, since ln cT˜ (T) and TX˜−1 XT˜−1 ˜ ˜ α −( ln(cT˜ (t)) − ln(c(t))) − ln cT˜ (T) + ln(k(T) ) t=0 t=0 Solving undiscounted infinite horizon optimization problems: a nonstandard approach 219

PT˜ are both bounded, and ln(cT˜ (t)) is an infinite value, we see that the standard part t=0 of the last term in the right-hand side of equation (9) is 1. T˜P−1 Hence, for an arbitrary infinite number T˜, as ln(c(t)) is infinite and ln(k(T˜)α) is t=0  ˜     PT˜  PT  ln(c ˜ (t))   ln(cT (t))   T˜    bounded, est  t=0  = 1, that is, lim  t=0  = 1.  T˜P−1  T→∞  TP−1   ln(c(t))+ln(k(T˜)α)  ln(c(t))+ln(k(T)α) t=0 t=0 T−1 T P α P ln(c1(t))+ln(k1(T) ) ln(cT (t)) t=0 t=0 Let T ≡ a(T), T−1 ≡ b(T). P P α ln(cT (t)) ln(c(t))+ln(k(T) ) t=0 t=0 From Lemma 2.2, we have TP−1  TP−1   PT  α  α    ln(c1(t))+ln(k1(T) )  ln(c1(t))+ln(k1(T) )   ln(cT (t))  t=0  t=0   t=0  lim T−1 = lim  T  · lim  T−1  > 1. P α  P  T→∞  P α  T→∞ ln(c(t))+ln(k(T) ) T→∞ ln(cT (t)) ln(c(t))+ln(k(T) ) t=0 t=0 t=0 References

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THE FIRST EXAMPLE OF MAXIMAL ITERATIVE ALGEBRA OF THE FUNCTIONS OF TOPOLOGICAL BOOLEAN ALGEBRA OF ORDER 16 WITH 3 OPEN ELEMENTS Mefodie T. Rat¸iu Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Chi¸sinau, Republic of Moldova [email protected] Abstract A.I. Mal’tsev [6] proposed the problem to obtain the description of iterative algebras of functions in propositional logics. From functional point of view a special interest represent the separation and description of maximal iterative algebras. L. Esakia and V. Meshi [1] and independly L.Maksimova [5] discovered the existence of pre-tabular modal logic EM4. It is approximated by the logics of a series of topological Boolean i algebras ∆i(i = 1, 2,... ) of order 2 with 3 open elements. In this work, for ﬁrst time, a maximal subalgebra of iterative algebra of functions of topological Boolean algebra of order 16 with one open atom is built.

Keywords: iterative algebra, maximal algebra, topological boolean algebra. 2000 MSC: 03B45, 03B50.

1. INTRODUCTION The study of iterative algebras, initiated by E. Post [7] and A.I. Mal’tsev [6], is closely related to functional problems of the theory of multivalent logics and of logical calculi. Any iterative algebra is supported by a closed class of k-valent (k = 2, 3,... ) logic functions or by a functions of a logics defined by logical calculi. The superposition, in its diverse variations, plays an essential role as signature of these algebras. But let us remember that any superpositions may be expressed easily by a finite number of special superposition indicated, for example, in the work [6, p.25]. We can conclude that iterative algebras have finite signature. Therefore the research in terms of such algebras is equivalent to the investigation of their bases, which, in most cases, are closed classes of k-valent logic functions. The general problem of description of closed classes for k-valent logic, with k ≥ 3, is equivalent to the problem of the description of corresponding iterative algebras, but it is complicated in principle by the existence of subalgebras with infinite numerable bases as well as by the existence of subalgebras without base [9, 11]. Consequently, the general problem of the description of iterative algebras remains open. In more details was examined the problem of (functional) completeness in some iterative al-

221 222 Mefodie T. Rat¸iu gebra. It requires to be clarified necessary and sufficient conditions that an arbitrary system of its elements to generate full algebra. A proper subalgebra of a given algebra is called maximal if supplementing it with any element of the difference of these algebras generate the whole given algebra [6]. The maximal subalgebras play a very special role in the completeness problem of systems in the iterative algebras. Remem- ber that in any k-valent logic to any maximal subalgebra it corresponds respectively some pre-complete class of functions or formulas. In this work for first time it is built one maximal subalgebra of iterative algebra of topological Boolean algebra of order 16 with 3 open elements including 2 trivial items and one atom.

2. PRELIMINARIES By a topological Boolean algebra [8] we mean any universal algebra < M;&, ∨, ⊃ , ¬, >, such that the system, < M;&, ∨, ⊃, ¬ > is a Boolean algebra, and the symbol represents the operation of taken inside. In this work a signiﬁcant role will be played by the series of following topological boolean algebras: ∆1 with 2 n elements, ∆2 with 4 elements, . . . , ∆n with 2 elements, . . . , every of them having 2 trivial open elements and one open atom. This algebras can be represented by a series of diagrams, from which we present the following four (see the Figures 1 and 2): 1 1 u 1 u u \ cc c \ u cuu u \ c u # ,l # u\\ u c, c l# e , cu# l e u, luuu e c ## e c # u eu ccu# 0 0 0

Fig. 1. The diagrams of ∆1, ∆2 and ∆3 algebras

The elements 0 and 1 represent the smallest and respectively the greatest elements of algebra, while other elements of ∆i (i = 2, 3, 4) are denoted by small letters of Greek alphabet. The open elements of algebra are marked by (square). The conjunction of two elements of algebra is determined on the diagram by the lowest edge up and the disjunction - by the biggest bound down to those elements. The inside of any element α, i.e. α, is the greatest open element that is less than α. In [1, 5] it was remarked that the series of algebras ∆1, ∆2, . . . generates a variety of topological Boolean algebras. It is known [2] that the modal logics represent an important family of logics which, unlike k-valent logic, not every of them can be described by ﬁnite models. Between topological Boolean algebras and modal logics there is a closely The ﬁrst example of maximal iterative algebra... 223

u 1 ¨HH ¨¨ J H ¨ J H ¨ HH ¨¨ J ψ H ¨ ν J H β ρ u¨ P t Jau Hu #Q !!QP !! aa !!Q E Q ! PP! E a! D Q # !! !Q! P E ! a # !E Q ! Q P!P! aaD Q !! Q!! Q ! E PP D a Q #! E ! Q !!Q PP aaQ ##! ! QQ ! Q E PPDu aQ ε ua! τ E ua! ! u QEuδ D θ aQuω ca a γ l l ¤ a ¥ a ¥ c aa aa l l ¤ c a¥ a¥ l ¥ a ¥ a l ¤ c aa aal l cc¥ a ¥ al l ¤ α Hua ual uµ u¤ H ϕS ¨¨ σ H ¨¨ HH S ¨ H S ¨ H ¨¨ HHS u¨ 0

Fig. 2. The diagram of ∆4 algebra relation. Namely, for any modal logic L there exists a topological Boolean algebra whose logic coincides with L. Modal formulas are traditionally built from variables p, q, r,... (sometimes with indexes) with logical operators & (conjunction), ∨ (disjunction), ⊃ (implication), ¬ (negation), (necessity) and parentheses. Operator ¬¬ (possibility) is denoted by diamond . Let us notice that the logical operators and the operations of topological Boolean algebra signature are denoted by the same symbol, which facilitates the interpretation of formulas on algebras. Formulas (p & ¬p) and (p ⊃ p) are called constants and we note them by ciphers 0 and 1, respectively. The formulas usually we denote by capital letters of the Latin alphabet. The formula (A ⊃ B) &(B ⊃ A) is called the equivalence of formulas A and B and it is denoted by symbol A ∼ B. An arbitrary function which is realized by a formula F after its interpretation on a topological Boolean algebra ∆ will be called a function F of the algebra ∆. When interpreting modal formulas on a topological Boolean algebra ∆, the set of all modal formulas identically equal to 1 on ∆ constitutes a modal logic that is denoted by L∆. The best-known modal logics are the logic S 4 and the logic S 5 [2], the latter being simpler and it responding better to ancient thoughts about modalities. A modal logic is called tabular if there is a finite topological Boolean algebra ∆ such that L = L∆. A modal logic is called pre-tabular, if it is not tabular, but any of its proper extensions is already a tabular logic. The logic S 5 is a well known example [5] of pre-tabular modal logic. In the works [1, 5] one new pre-tabular modal logic - the logic of EM4 was presented. This logic is obtained by adding to S 4 the following three formulas as new axioms: p ⊃ p, T(Z), T(Y), 224 Mefodie T. Rat¸iu where Z = ((p ⊃ q) ∨ (q ⊃ p)), Y = ((¬¬p &((p ⊃ q) ⊃ q)) ⊃ q), and the operator T means prescribing square before of each subformula. This modal logic corresponds fully to the variety of topological Boolean algebras generated by the series of algebras ∆1, ∆2,... [1, 5]. Following A.V. Kuznetsov [3, 4] , a formula F is called expressible in a logic L by a system of formulas Σ, if F can be obtained from the variables and formulas of Σ with a finite number of applications of weak rule of substitution which allows the transition from two formulae by substitution of one of them into the another instead of all entries of a variable, and the rule replacement by an equivalent in the logic L which allows the passage of one formula to any another formula equivalent to it in L. A formula is called direct expressible by Σ if it can be obtained from the variables and formulas of Σ with a finite number of applications of weak rule of substitution. We will say that a modal formula F(p1,..., pn) preserves a predicate R(x1,..., xm) on topological Boolean algebra ∆ if, for any elements αi j ∈ ∆ (i = 1,..., m; j = 1,..., n), from the fact that are true the sentences

R(α11, . . . , αm1),..., R(α1n, . . . , αmn) it result that the statement

R(F[α11, . . . , α1n],..., F[αm1, . . . , αmn]) are true. If the predicate R is deﬁned on a ﬁnite algebra of a degree not too large, then R often can be given by its determining matrix

βi j (i = 1,..., m; j = 1,..., l) whose elements belong to the algebra ∆, so that following sentence is true

R(β1k, . . . , βmk) for k = 1,..., l.

For example, each of formulae ¬p, p and (p & q) preserves on the algebra ∆4 the following matrix that will be used below    0 αϕµετγρσδθωνψβ 1     0 αµϕτεγρσδωθψνβ 1   0 ϕαµεγτρσθδωνβψ 1    . (1)  0 ϕµαγετρσθωδβνψ 1     0 µαϕτγερσωδθψβν 1    0 µϕαγτερσωθδβψν 1

Taking into account the fact that disjunction and implication are expressible in classical logic by means of conjunction and negation, it follows that any formula preserves the matrix (1) on ∆4 algebra. The ﬁrst example of maximal iterative algebra... 225 3. THE MAIN RESULT

For any function f (p1,..., pn) of algebra ∆4 let us consider it n-dimensional (i.e. with n entries) table. This picture (i.e. the part where are located function values and not the argument values) is divided into sectors so as be satisﬁed that: two arbitrary cells with two sets of coordinates < α1, . . . , αn > and < β1, . . . , βn > belong to the same sector then and only then when it take place the equality

αi = βi (i = 1,..., n). For example, on the tables below of some functions (i.e. derivative operations) of algebra ∆4 the unique lines just divides the table in such sectors.

Table 1

p 0 αϕµετγρσδθωνψβ 1 p 0 0 0 0 0 0 0 0 σ σ σ σ σ σ σ 1 p 0 ρ ρ ρ ρ ρ ρ ρ 1 1 1 1 1 1 1 1 p 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 ¬p & p 0 γ τ ε µ ϕ α 0 ρ γ τ ε µ ϕ α 0 p ⊃ p 1 β ψ ν ω θ δ σ 1 β ψ ν ω θ δ 1 p ⊃ p 1 δ θ ω ν ψ β 1 σ δ θ ω ν ψ β 1

A cell of sector is called main, if all its coordinates are equal to 0 or 1. Obviously, any sector has a single main cell. If the coordinates of the main cell of a sector constitute the collection < α1, . . . , αn >, then we will denote this sector by symbol Q < α1, . . . , αn >.

Theorem 3.1. The iterative algebra of the functions of algebra ∆4, which preserves the following line matrix with 12 elements 0 α µ ε γ ρ σ δ ω ν β 1 (2) is maximal in the iterative algebra of all functions of ∆4 algebra.

Indeed, we denote by Γ the iterative algebra of all functions of algebra ∆4, which preserve the matrix (2). The fact that algebra Γ is proper subalgebra of iterative algebra of all functions of ∆4 algebra results from fact that formula p∨q does not preserve matrix (2), because there is equality ψ = δ ∨ ω which shows that the disjunction of 2 elements δ and ω of the matrix (2) is equal to the element ψ that does not belong to the matrix (2). Let us notice now that any element of the matrix (2) belongs to one of the following 2 isomorphic subalgebras of algebra ∆4: < 0, α, γ, ρ, σ, δ, β, 1; Ω >, < 0, µ, ε, ρ, σ, ω, ν, 1; Ω >, 226 Mefodie T. Rat¸iu

Table 2 p ∨ q

p\q 0 αϕµετγρσδθωνψβ 1

0 0 α ϕ µ ε τ γ ρ σ δ θ ω ν ψ β 1 α α α ε τ ε τ ρ ρ δ δ ν ψ ν ψ 1 1 ϕ ϕ ε ϕ γ ε ρ γ ρ θ ν θ β ν 1 β 1 µ µ τ γ µ ρ τ γ ρ ω ψ β ω 1 ψ β 1 ε ε ε ε ρ ε ρ ρ ρ ν ν ν 1 ν 1 1 1 τ τ τ ρ τ ρ τ ρ ρ ψ ψ 1 ψ 1 ψ 1 1 γ γ ρ γ γ ρ ρ γ ρ β 1 β β 1 1 β 1 ρ ρ ρ ρ ρ ρ ρ ρ ρ 1 1 1 1 1 1 1 1 σ σ δ σ ω ν ψ β 1 σ δ θ ω ν ψ β 1 δ δ δ ν ψ ν ψ 1 1 δ δ ν ψ ν ψ 1 1 θ θ ν θ β ν 1 β 1 θ ν θ β ν 1 β 1 ω ω ψ β ω 1 ψ β 1 ω ψ β ω 1 ψ β 1 ν ν ν ν 1 ν 1 1 1 ν ν ν 1 ν 1 1 1 ψ ψ ψ 1 ψ 1 ψ 1 1 ψ ψ 1 ψ 1 ψ 1 1 β β 1 β β 1 1 β 1 β 1 β β 1 1 β 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 where Ω = {&, ∨, ⊃, ¬, }. Therefore, by applying any unary formula to the elements of matrix (2), only elements of this matrix can be obtained. Thus, any unary formula belongs to subalgebra Γ. It can be easily checked that the formulas p & q and p∨q belong to subalgebra Γ. Let us denote by S 0(p, q) the following binary formula (p ∨ q) ∨ (p ⊃ q) ∨ (q ⊃ p) ∨ (¬p ∨ ¬q).

Notice that this formula belongs to algebra Γ because the formula S 0 can take only the values 0, 1 or σ which belong to the matrix (2). Let present the table of 2-dimensional function S 0. Next we need following formula:

T0(p, q) = (p ∨ q ∨ S 0[¬(p ⊃ p), ¬(q ⊃ q)])& &(p ∨ ¬q ∨ S 0[¬(p ⊃ p), q])& &(¬p ∨ q ∨ S 0[p, ¬(q ⊃ q)])&

&(¬p ∨ ¬q ∨ S 0(p, q)).

For use this formula we present below her 2-dimensional table on algebra ∆4. On the basis of the tables of p ∨ q and of T0 it is not diﬃcult to verify that the following lemma holds. The ﬁrst example of maximal iterative algebra... 227

Table 3 S 0

p\q 0 αϕµετγρσδθωνψβ 1

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ϕ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 µ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ε 1 1 1 1 1 σ σ 1 1 σ σ 1 1 1 1 1 τ 1 1 1 1 σ 1 σ 1 1 σ 1 σ 1 1 1 1 γ 1 1 1 1 σ σ 1 1 1 1 σ σ 1 1 1 1 ρ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 σ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 δ 1 1 1 1 σ σ 1 1 1 1 σ σ 1 1 1 1 θ 1 1 1 1 σ 1 σ 1 1 σ 1 σ 1 1 1 1 ω 1 1 1 1 1 σ σ 1 1 σ σ 1 1 1 1 1 ν 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ψ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 β 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Lemma 3.1. Formula D = (T0(p, q) ⊃ (p ∨ q)) belongs to subalgebra Γ.

The table of formula D on algebra ∆4 can be described by the following scheme   ρ in 6 cells in which the value of T is σ of the  0  sector Q < 0, 0 >, D(p, q) =  1 in the other 18 cells where the value of T is σ,  0  p ∨ q in remaining cells.

Next we need the following formula:

T1(p, q) = (p ∨ ¬q ∨ S 0(p, q))& &(p ∨ ¬q ∨ S 0[p, (q ⊃ q)])&

&(¬p ∨ q ∨ S 0[(p ⊃ p), q])& &(¬p ∨ ¬q ∨ S 0[(p ⊃ p), (q ⊃ q)]).

Lemma 3.2. Formula K = T1 ⊃ (p & q) belongs to subalgebra Γ. 228 Mefodie T. Rat¸iu

Table 4 T0

p\q 0 αϕµετγρσδθωνψβ 1

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 1 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1 ϕ 1 σ 1 σ 1 1 1 1 1 σ 1 σ 1 1 1 1 µ 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1 1 ε 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 τ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 γ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 σ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 δ 1 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1 θ 1 σ 1 σ 1 1 1 1 1 σ 1 σ 1 1 1 1 ω 1 σ σ 1 1 1 1 1 1 σ σ 1 1 1 1 1 ν 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ψ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 β 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

By using the Lemma 3.1 and Lemma 3.2 it is not diﬃcult to ﬁnish the proof of Theorem 3.1.

References

[1] Esakia L., Meshi V., Five critical modal systems, Theoria, 43(1977), 52–62. [2] Feys R., Modal Logics, Paris, 1965. [3] Kuznetsov A.V., On functional expressibility in the superintuitionistic Logics. Math. issled., Kishinev, 6, 4(1971), 75–122 (in Russian). [4] Kuznetsov A.V., On tools for the discovery of non-deducibility or non-expressibility, Logical deduction, Nauka, Moscow, 1979, 5–33 (in Russian). [5] Maksimova L., Pre-tabular extensions of Lewis’s Logic, Algebra and Logic, 14, 1(1975), 28–55 (in Russian). [6] Mal’tsev A.I., Post’s iterative algebras. Novosibirsk, 1976 (in Russian). [7] Post E.L., Two-valued iterative systems of mathematical logic, Princeton, 1941. [8] Rasiowa E., Sikorski R., The Mathematics of metamathematics, Warszawa, 1963. [9] Ratsa M. Iterative chain classes of pseudo-boolean functions, Kishinev, 1990 (in Russian). [10] Scroggs S. I., Extensions of the Lewis system S5, J.Simb.Logic, 6, 2 (1951), 112–120. [11] Yanov Y.I., Muchnik A.A., About the existence of k-valued closed classes having not ﬁnite bases, Doklady AN SSSR, 127, 1(1959), 44–46 (in Russian). ROMAI J., 6, 2(2010), 229–236

ON MIDDLE BOL LOOPS Parascovia Sˆarbu State University of Moldova, Chisinau, Republic of Moldova [email protected] Abstract A loop Q(·) is middle Bol if every its loop isotope has the anti-automorphic inverse property: (xy)−1 = y−1 x−1. Every middle Bol loop can be obtained using left (right) Bol loops ([2]). Connections between structure and properties of middle Bol loops and of the corresponding left Bol loops are studied in the present work.

Keywords: left Bol loop, middle Bol loop, isotopy, autotopism. 2000 MSC: 20N05.

1. INTRODUCTION Middle Bol loops were deﬁned by V. Belousov ([1]). A loop Q(·) is called middle Bol if the primitive loop Q(·, /, \) satisﬁes the identity

x(yz \ x) = (x/z)(y \ x). (1)

The identity (1) corresponds to the third closure condition B3 in algebraic nets so it is universal in loops, i.e. any loop isotope of a middle Bol loop is middle Bol ([1]). In particular, every Moufang loop is middle Bol. Note that the identity (1) is a generalization of the Moufang identity (x · yz)x = xy · zx. It was proved by Gwaramija in [2] that: if Q(·) is a left (right) Bol loop then the groupoid Q(◦), where x ◦ y = y(y−1x · y) (respectively, x ◦ y = (y · xy−1)y), ∀x, y ∈ Q, is a middle Bol loop and, conversely, if Q(◦) is a middle Bol loop then there exists a left (right) Bol loop Q(·) such that x ◦ y = y(y−1 x · y) (respectively, x ◦ y = (y · xy−1)y), ∀x, y ∈ Q. This result implies that if Q(·) is a left Bol loop and Q(◦) is the corresponding middle Bol loop then x ◦ y = x/y−1 and x · y = x//y−1, where ”/” (”//”) is the left division in Q(·) (resp. in Q(◦)). The class of left (right) Bol loops is intensively developed at present (see, for example [4]) Connections between structure and properties of middle Bol loops and of the corresponding left Bol loops are studied in the present work. It is noted that two middle Bol loops are isomorphic if and only if the corresponding left (right) Bol loops are isomorphic, and a general form of the autotopisms of middle Bol loops is deduced. Relations between diﬀerent sets of elements, such as nucleus, left (right, middle) nu- clei, the set of Moufang elements, the center, etc. of a middle Bol loop and of the corresponding left Bol loop are established.

229 230 Parascovia Sˆarbu 2. MIDDLE BOL LOOPS AND THE CORRESPONDING LEFT BOL LOOPS Proposition 2.1. Two middle Bol loops Q(◦) and Q(⊗) are isomorphic if and only if the corresponding left Bol loops are isomorphic. Proof. Let Q(·) and Q(∗) be two left Bol loops and let Q(◦) and Q(⊗) the corresponding middle Bol loops, respectively, i.e. x ◦ y = y(y−1x · y) and x ⊗ y = y ∗ ((y−1 ∗ x) ∗ y), ∀x, y ∈ Q. If Q(·) Q(∗) and if ϕ is an isomorphism from Q(·) to Q(∗), then for every x, y ∈ Q: x ◦ y = y(y−1 x · y) = ϕ−1(ϕ(y) ∗ ((ϕ(y−1) ∗ ϕ(x)) ∗ ϕ(y))) = ϕ−1(ϕ(x) ⊗ ϕ(y)), so Q(◦) Q(⊗). Conversely, if Q(◦) and Q(⊗) are isomorphic and if ϕ is an isomorphism from Q(◦) to Q(⊗), then ϕ(x ◦ y) = ϕ(x) ⊗ ϕ(y) ⇒ ϕ(x/y−1) = ϕ(x)//ϕ(y)−1, where ”/” (”//”)is the left division in Q(·) (resp., in Q(∗)). Denoting x/y−1 by z we get x = z · y−1, so the last equality implies: ϕ(z) = ϕ(z · y−1)//ϕ(y)−1 ⇒ ϕ(z · y−1) = ϕ(z) ∗ ϕ(y)−1 ⇒ ϕ(z · y−1) = ϕ(z) ∗ ϕ(y−1), ∀x, y ∈ Q, i.e. ϕ is an isomorphism from Q(·) to Q(∗). Remark 2.1. It is known ([2]) that:

a) The unit in a left Bol loop Q(·) and in the corresponding middle Bol loop Q(◦) is the same. Moreover, the left (right) inverse of an element x in Q(·) coincide with the left (right) inverse of x in Q(◦).

b) Since both left and middle Bol loops are power-associative, the equality x ◦ y = y(y−1 x · y) implies that each element x has the same order in Q(·) and in Q(◦). It is known ([3]) that the order of each element in a ﬁnite left Bol loop divides the order of the loop. So the same property is valid in ﬁnite middle Bol loops.

Let Q(·) be a loop and let consider: i. the set of all Moufang elements of Q(·): M(·) = {a ∈ Q| ax · ya = (a · xy)a, ∀x, y ∈ Q}, ii. the set of all Bol elements of Q(·): B(·) = {a ∈ Q| a(x · ay) = (a · xa)y, ∀x, y ∈ Q}, iii. the left nucleus of Q(·): (·) Nl = {a ∈ Q| a · xy = ax · y, ∀x, y ∈ Q}, iv. the right nucleus of Q(·): (·) Nr = {a ∈ Q| xy · a = x · ya, ∀x, y ∈ Q}, On middle Bol loops 231

v. the middle nucleus of Q(·): (·) Nm = {a ∈ Q| xa · y = x · ay, ∀x, y ∈ Q}. vi. the Moufang center of Q(·): C(·) = {a ∈ Q| a2 · xy = ax · ay, ∀x, y ∈ Q}

Lemma 2.1. Let Q(·) be a left Bol loop and let Q(◦) be the corresponding middle Bol 3 loop. Then T1 = (α, β, γ) ∈ S Q is an autotopism of Q(·) if and only if T2 = (γ, IβI, α) (where I(x) = x−1, ∀x ∈ Q) is an autotopism of Q(◦). Proof. As mentioned above, x · y = x//y−1 = x//I(y), ∀x, y ∈ Q, where ”//” is the 3 left division in Q(◦). If T1 = (α, β, γ) ∈ S Q is an autotopism of Q(·) then γ(xy) = α(x) · β(y), ∀x, y ∈ Q, i.e. γ(x//I(y)) = α(x)//Iβ(y), ∀x, y ∈ Q, which is equivalent to α(x) = γ(x//y) ◦ IβI(y), ∀x, y ∈ Q. Taking x//y = z in the last equality, we get α(z ◦ y) = γ(z) ◦ IβI(y), ∀x, y ∈ Q, i.e. (γ, IβI, α) is an autotopism of the loop Q(◦). Corollary 2.1. The group of autotopisms of a middle Bol loop is isomorphic to the group of autotopisms of the corresponding left Bol loop. Proof. Indeed, the mapping σ :(α, β, γ) → (γ, IβI, α) is an isomorphism between two groups. Proposition 2.2. Let Q(·) be a left Bol loop and let Q(◦) be the corresponding middle Bol loop. The following sentences are true: (◦) (◦) (◦) (·) (·) (◦) (·) (◦) (·) (◦) 1. Nl = Nr j M , 2. Nm = Nl = Nl , 3. Nr = Nm , 4. N = N , (·) (·) (◦) (·) (◦) (·) (◦) 5. M ∩ Nl ⊆ Nm , 6. C j B , 7. C = C . (◦) (◦) (◦) Proof. 1. The equality Nl = Nr is known (see, for example, [2]). If a ∈ Nl = (◦) Nr then T1 = (La, ε, La) and T2 = (ε, Ra, Ra) are autotopisms of Q(◦), so T1T2 = (◦) (La, Ra, RaLa) is an autotopism of Q(◦), i.e. a ∈ M . · (◦) −1 −1 (·) 2. If a ∈ Nm then La = a ◦ x = x(x a · x) = x(x · ax) = ax = La (x), ∀x, y ∈ Q, (◦) (·) (·) (·) (·) (·) so La = La . Hence a ∈ Nl = Nm if and only if T = (La , ε, La ) is an autotopism (◦) (◦) of Q(·) so, according to Lemma 2.1, if and only if T = (La , ε, La ) is an autotopism (◦) of Q(◦), i.e. if and only if a ∈ Nl . (·) (◦) −1 −1 (·) 3. If a ∈ Nr then Ra (x) = x ◦ a = a(a x · a) = a(a · xa) = x · a = Ra (x), (·) (◦) (·) (·) (·) ∀x ∈ Q, so Ra = Ra . According to Lemma 2.1, a ∈ Nr if and only if (ε, Ra , Ra ) (·) (·) (◦) (◦) is an autotopism of Q(·), i.e., if and only if (Ra , IRa I, ε) = (Ra , IRa I, ε) is an autotopism of Q(◦). But IR(◦)I(x) = IR(◦)(x−1) = I(x−1 ◦ a) = a−1 ◦ x = L(◦) (x), ∀x ∈ Q, so IR(◦)I = a a a−1 a L(◦) . Hence, if a ∈ N(·) then (R(◦), L(◦) , ε) is an autotopism of Q(◦), i.e. x ◦ y = a−1 r a a−1 (x◦a)◦(a−1 ◦y), ∀x, y ∈ Q. Taking x = e in the last equality, we have y = a◦(a−1 ◦y), ∀x, y ∈ Q so, for y = a ◦ z the same equality implies x ◦ (a ◦ z) = (x ◦ a) ◦ z, ∀x, z ∈ Q, (◦) (·) (◦) i.e. a ∈ Nm and then Nr j Nm . 232 Parascovia Sˆarbu

(◦) −1 On the other hand, a ∈ Nm ⇔ (x◦a)◦y = x◦(a◦y), ∀x, y ∈ Q. Taking y = a ◦z, we get (x◦a)◦(a−1◦z) = x◦(a◦(a−1◦z)) = x◦z, so (R(◦), L(◦) , ε) = (R(·), IR(·)I, ε) is an a a−1 a a (◦) (◦) autotopism of Q(◦). Now, according to Lemma 2.1, (ε, Ra , Ra ) is an autotopism of (◦) (◦) (◦) (·) Q(·), which implies Ra (x · y) = x · Ra (y) and, taking y = e we get: Ra (x) = Ra (x), (◦) (·) (·) (·) ∀x ∈ Q, so Ra = Ra , hence (ε, Ra , Ra ) is an autotopism of Q(·), which means that (·) (◦) (·) (◦) (·) a ∈ Nr and Nm j Nr . So, we proved that Nm = Nr .

4. Remind that N = Nl ∩ Nr ∩ Nm, so the proof follows from 1.-3. (·) (◦) −1 (·) 5. If a ∈ M then Ra (x) = x ◦ a = a(a x · a) = x · a = Ra (x), ∀x ∈ Q, so (◦) (·) Ra = Ra . (·) (·) (·) (·) (·) On the other hand, a ∈ M if and only if (La , Ra , La Ra ) is an autotopism of (·) (·) (·) (·) Q(·), hence (La Ra , IRa I, La ) is an autotopism of Q(◦). So as IR(·)I = IR(◦)I ⇒ IR(◦)I(x) = I(x−1 ◦ a) = a−1 ◦ x, ∀x ∈ Q, i.e. IR(◦)I = L(·) , a a a a a−1 we get that a ∈ M(·) if and only if,

T = (L(·)R(◦), L(◦) , L(·)) = (L(·), ε, L(·))(R(◦), L(◦) , ε) a a a−1 a a a a a−1 (·) (·) (·) (·) is an autotopism of Q(◦). If a ∈ M ∩ Nl then T and (La , ε, La ) are autotopisms of Q(◦) so (R(◦), L(◦) , ε) is an autotopism of Q(◦), which implies (x ◦ a) ◦ (a−1 ◦ y) = a a−1 x ◦ y, ∀x, y ∈ Q. Taking x = e in the last equality, it follows that L(◦) = L(◦)−1, so a−1 a (◦) (◦)−1 ◦ (Ra , La , ε), i.e. a ∈ Nm. 6. It follows from the deﬁnition of the Moufang center that

C(·) = {a ∈ Q|a2 · xy = ax · ay, ∀x, y ∈ Q} =

{a ∈ Q|a · (a · xy) = ax · ay, ∀x, y ∈ Q}. Let us remark now that if a ∈ C(·) than a · x = x · a, for every x ∈ Q. Indeed, taking y = e in the equality a · (a · xy) = ax · ay we get a · (a · x) = ax · a, ∀x ∈ Q and denoting ax by z: az = za, ∀z ∈ Q. Also, if a ∈ C(·) then, taking x → a−1x and y → a−1y in the equality a · (a · xy) = ax · ay, we have a[a · (a−1x · a−1y)] = xy, which implies a−1 x · a−1y = a−1(a−1 · xy), i.e. a−1 ∈ C(·). Here we used the equalities a−1 · ax = x and xa · a−1 which are true for all a ∈ C(·) and all x ∈ Q. (·) (·) (·) Now, using Lemma 2.1 and the equality La = Ra , we get that a ∈ C if and only (·) (·) (·)2 (◦)2 (◦) (◦) if (Ra , Ra , Ra ) is an autotopism of Q(·), i.e. if and only if (Ra , IRa I, Ra ) is an autotpism of Q(◦). So as IR(◦)I(x) = I(x−1 ◦ a) = a−1 ◦ x = L(◦) (x) we get that a ∈ C(·) if and only if a a−1 the identity (x ◦ y) ◦ a = [(x ◦ a) ◦ a] ◦ (a−1 ◦ y) (2) On middle Bol loops 233 holds for every x, y ∈ Q. Now, taking x = a−1 in (2), we have (a−1 ◦y)◦a = a◦(a−1 ◦y) so, denoting a−1 ◦ y by z, the last equality means z ◦ a = a ◦ z, for every z ∈ Q, i.e. (◦) (◦) −1 Ra = La . Also, taking y = e in (2), we get x ◦ a = [(x ◦ a) ◦ a] ◦ a so, denoting x ◦ a by z, it implies z = (z ◦ a) ◦ a−1, ∀z ∈ Q. Remark that if a ∈ C(·) then R(◦) = L(◦) so a−1 ◦ x = x ◦ a−1, ∀x ∈ Q. Hence, from a−1 a−1 the equality z = (z ◦ a) ◦ a−1 it follows z = a−1 ◦ (a ◦ z), ∀z ∈ Q. Now, taking y = c ◦ y in (2), we get a ◦ (x ◦ (a ◦ y)) = (a ◦ (x ◦ a)) ◦ y, ∀x, y ∈ Q, i.e. a ∈ B(◦), so C(·) j B(◦). 7. Using 6. and Lemma 2.1, we get that a ∈ C(·) if and only if the identity (2) holds. But the equality (2) is equivalent to a ◦ (x ◦ y) = (a ◦ (x ◦ a)) ◦ (a−1 ◦ y), so it is equivalent to the equality a ◦ (a ◦ (x ◦ y)) = a ◦ ((a ◦ (x ◦ a)) ◦ (a−1 ◦ y)). Now, using the properties of the elements of C(◦), deduced in 6., we have:

a ◦ ((a ◦ (x ◦ a)) ◦ (a−1 ◦ y)) = a ◦ (((x ◦ a) ◦ a) ◦ (a−1 ◦ y)) = (((x ◦ a) ◦ a) ◦ (a−1 ◦ y)) ◦ a = (x ◦ a) ◦ ((a ◦ (a−1 ◦ y)) ◦ a) = (x ◦ a) ◦ (y ◦ a) = (a ◦ x) ◦ (a ◦ y), ∀x, y ∈ Q, i.e. a ∈ C(◦). So, C(·) = C(◦). Proposition 2.3. The Moufang center of a left Bol loop is a subloop. Proof. Indeed, let Q(·) be a left Bol loop and let C(·) be its Moufang center. So as for every a, b ∈ C(·) we have x · a = b ⇒ x = b · a−1 and a · x = b ⇒ x = a−1 · b, C(·) is a subloop if and only if it contains the product of every two its elements. For a, b ∈ C(·) we have:

0 0 0 0 ab · (ab · xy) = ab · [ab · (ax · ay )] = ab · [ab · a(a · x y )] = 0 0 0 0 0 0 ab · a2[b(a · x y )] = a2[b · a(b · (a · x y ))] = a2 · [(b · ab)(a · x y )] = 0 0 0 0 0 0 a2 · [ab2 · (a · x y ))] = a2 · [a2 · (b2 · x y ))] = a4 · (b2 · x y ), where x0 = ax and y0 = ay. On the other hand,

0 0 0 0 (ab · x) · (ab · y) = (ab · ax ) · (ab · ay ) = (a2 · bx )(a2 · by ) =

0 0 0 0 a4(bx · by ) = a4(b2 · x y ), where x0 = ax and y0 = ay. Hence ab · (ab · xy) = (ab · x) · (ab · y), for every x, y ∈ Q, i.e. ab ∈ C(·), so C(·) is a subloop of Q(·). Proposition 2.4. If L(·) is a subloop of a left Bol loop Q(·) then L(◦) is a subloop of the corresponding middle Bol loop Q(◦). Proof. Indeed, if L(·) is a subloop of a left Bol loop Q(·) then for every x, y ∈ L, x ◦ y = y(y−1 x · y) ∈ L. More, if a, b ∈ L then each of the equations a ◦ x = b and 234 Parascovia Sˆarbu y ◦ a = b has its solution in L: a ◦ x = b ⇔ x = (b \ a)−1 ∈ L and, analogously, y ◦ a = b ⇔ y/a−1 = b ⇔ y = b · a−1 ∈ L. Corollary 2.2. From ﬁrst two remarks and the equality C(·) = C(◦) it follows that the Moufang center C(◦) is a subloop of the middle Bol loop Q(◦).

Let Q(·, /, \) be an arbitrary loop . An element a ∈ Q is called a middle Bol element if the equality a(yz \ a) = (a/z) · (z \ a), holds for every y, z ∈ Q ([2]). So, we can consider the mapping

Ia : Q → Q, Ia(x) = x \ a, −1 which is a bijection. From this deﬁnition it follows that Ia = a/x, ∀x ∈ Q, and that −1 a ∈ Q is a middle Bol element if and only if LaIa(xy) = Ia (y) · Ia(x), x, y ∈ Q, i.e. if −1 and only if (Ia , Ia, LaIa) is an anti-autotopism of Q(·). Lemma 2.2. If Q(·) is a middle Bol loop then, for every a ∈ Q, the triple T = −1 (Ia I, IaI, LaIaI) is an autotopism of Q(·). −1 Proof. Indeed, so as (Ia , Ia, LaIa) and (I, I, I) are anti-autotopisms of Q(·) it follows that T is an autotopism of Q(·). Remind that if Q(·) is a quasigroup then a bijection τ : Q → Q is called a left (right) pseudo-automorphism if there exists an element c ∈ Q such that c · τ(xy) = (c · τ(x))τ(y) (respectively, τ(xy) · c = τ(x)(τ(y) · c)), ∀x, y ∈ Q. In this case the element c ∈ Q is called a left (resp. right) companion of τ. So, it follows from the deﬁnition that τ is a left (right) pseudo-automorphism with the companion c if and only if (Lcτ, τ, Lcτ) (resp., (τ, Rcτ, Rcτ)) is an autotopism of Q(·).

Proposition 2.5. If Q(·) is a left Bol loop and c ∈ Q then a bijection τ ∈ S Q is a left pseudo-automorphism with the companion c if and only if IτI is a left pseudo- automorphism with the companion c of the corresponding middle Bol loop.

Proof. The mapping τ ∈ S Q is a left pseudo-automorphism of Q(·) with the compan- (·) (·) ion c if and only if (Lc τ, τ, Lc τ) is an autotopism of Q(·) so, according to Lemma (·) (·) 2.1, if and only if (Lc τ, IτI, Lc τ) is an autotopism of the corresponding middle Bol loop Q(◦), i.e. if and only if the equality

(·) (·) Lc τ(x ◦ y) = Lc τ(x) ◦ IτI(y), (·) (◦) holds for every x, y ∈ Q. Taking x = e in the last equality we get Lc τ(y) = Lc IτI(y), (◦) where Lc is the left translation in the loop Q(◦), i.e. (·) (·) (◦) (◦) (Lc τ, IτI, Lc τ) = (Lc IτI, IτI, Lc IτI), On middle Bol loops 235 which means that IτI is a left pseudo-automorphism of Q(◦) with the companion c. Corollary 2.3. The group of left pseudo-automorphisms of a left Bol loop Q(·) is isomorphic to the group of left pseudo-automorphisms of the corresponding middle Bol loop. Proof. Indeed, the mapping τ → IτI is an isomorphism between two groups.

It is known ([1]) that every autotopism T = (α, β, γ) of a left Bol loop Q(·) can be represented as follows:

(·) (·) (·)−1 (·)−1 (·)−1 T = (Lk τ, τ, Lk τ)(Rb Lb , Lb, Lb ), (3) where τ is a left pseudo-automorphism with the companion k of the loop Q(·) and b = β(e), e being the unit of Q(·).

Proposition 2.6. Every autotopism T = (α, β, γ) of a middle Bol loop Q(◦) can be represented in the form:

(◦) (◦) −1 (◦)−1 (Lk τ, τ, Lk τ)(IIa, IIa , IIaLa ), (4) where τ is a left pseudo-automorphism of Q(◦) with the companion k, a = β(e). Proof. If T = (α, β, γ) is an autotopism of a middle Bol loop Q(◦, //, \\) then, according to Lemma 2.1, T1 = (γ, IβI, α) is an autotopism of the corresponding left Bol loop Q(·, /, \). According to the general form (3), there exists a left pseudo-automorphism τ0 of Q(·) with the companion k and an element b = β(e) ∈ Q, where e is the unit of Q(·), such that

(·) 0 0 (·) 0 (·)−1 (·)−1 (·)−1 (γ, IβI, α) = (Lk τ , τ , Lk τ )(Rb Lb , Lb, Lb ). (5) −1 · −1 From the equality x·y = x//I(y) it follows x·I(y) = x//y = Ix (y), so LxI(y) = Ix (y), ∀y ∈ Q, i.e.

· −1 Lx = Ix I, (6) for every x ∈ Q. So as Q(◦) is a middle Bol loop, according to Lemma 2.2, the tuples −1 (◦) −1 −1 −1 (◦)−1 T = (Ia I, IaI, La IaI) and T = (IIa, IIa , IIa La ) are autotopisms of Q(◦) for every x ∈ Q. (·)−1 (·)−1 (·) (·)−1 On the other hand, if (Rb Lb , Lb , Lb ) is an autotopism of Q(·), then T = (·)−1 (·) (·)−1 (Lb , ILb I, Rb ) is an autotopism of Q(·). Now, taking b = a and using the equality (6), we get that

−1 (·)−1 T1 = (IIa, IIa , Ra IIa) 236 Parascovia Sˆarbu

−1 is an autotopism of Q(◦). So as the autotpisms T and T1 have the same two components, we get T −1 = T and

(·)−1 −1 (◦)−1 Ra IIa = IIa La (7) So, from (5), it follows

(·) 0 0 (·) 0 (·)−1 (·) (·)−1 (·)−1 (α, β, γ) = (Lk τ , Iτ I, Lk τ )(Lb , ILb I, Rb Lb ). (8) Remark now that the mapping Iτ0 I is a left pseudo-automorphism of Q(◦) with the (·) 0 0 (·) 0 companion k. Indeed, so as (Lk τ , Iτ I, Lk τ ) is an autotopism of Q(◦), we have (·) 0 (·) 0 0 (·) 0 Lk τ (x ◦ y) = Lk τ (x) ◦ Iτ I(y), ∀x, y ∈ Q, which implies for x = e: Lk τ (y) = (◦) 0 (·) 0 (◦) 0 Lk Iτ I(y), ∀y ∈ Q, i.e. Lk τ = Lk Iτ I. Hence

(·) 0 0 (·) 0 (◦) 0 0 (◦) 0 (Lk τ , Iτ I, Lk τ ) = (Lk Iτ I, Iτ I, Lk Iτ I) (9) is an autotopism of Q(◦), i.e. Iτ0 I is a pseudo-automorphism of Q(◦) with the companion k. Denoting Iτ0 I by τ, b by a and using (6), (7) and (9), from (8) we get:

(◦) (◦) −1 (◦)−1 (α, β, γ) = (Lk τ, τ, Lk τ)(IIa, IIa , IIaLa ), where a = β(e). Corollary 2.4. If T = (α, β, γ) is an autotopism of a middle Bol loop Q(◦) then there exists a left pseudo-automorphism τ of Q(◦) with a companion k such that

−1 −1 (◦)−1 T = (Ik τI, τ, IkτI)(IIa, IIa , IIaLa ) where a = β(e). Proof. Let Q(◦) be a middle Bol loop and let Q(·) be the corresponding left Bol loop −1 (·)−1 then, for a ∈ Q, we have Ia(x) = x \\a and Ia (x) = a//x, so Ia = ILa i.e. (·) −1 La = Ia I.

This work was partially supported by the CSCDT ASM institutional project No. 06.411.027F.

References

[1] V. Belousov, Foundations of the theory of quasigroups and loops, Nauka, Moscow, 1967.(Rus- sian) [2] A. Gwaramija, On a class of loops, Uch. Zapiski MGPI., 375, (1971), 25-34. (Russian) [3] P. Syrbu, Loops with universal elasticity, J. Quasigroups and Rel. Syst., 1, 1(1994), 57-65. [4] G. Nagy, A class of simple proper Bol loops, Manuscripta Math., 1, 127(2008), 81-88. ROMAI J., 6, 2(2010), 237–244

ALTMAN ORDERING PRINCIPLES AND DEPENDENT CHOICES Mihai Turinici ”A. Myller” Mathematical Seminar; ”A. I. Cuza” University; Ia¸si,Romania [email protected] Abstract Some ordering principles related to Altman’s [Nonlinear Analysis, 6 (1982), 157-165] are equivalent with the Dependent Choices Principle. This is also true for many inter- mediary statements, including the Szaz unboundedness criterion [Math. Moravica, 5 (2001), 1-6].

Keywords: Quasi-order, pseudometric, (ascending) Cauchy/asymptotic sequence, regularity, maximal element, sequential inductivity, monotone function, unboundedness. 2000 MSC: 49J53, 47J30.

1. INTRODUCTION Let M be some nonempty set; and (≤), some quasi-order (i.e.: reﬂexive and tran- sitive relation) over it. Denote M(x, ≤):= {t ∈ M; x ≤ t}, x ∈ M; the subsets M(x, ≥), x ∈ M, are introduced in a similar way, modulo (≥)(=the dual quasi-order). Further, take some application F : M × M → R with the properties (a01) F is (≤)-pseudometric: F(x, y) ≥ 0, when x ≤ y (a02) ∀y ∈ M, F(·, y) is decreasing on M(y, ≥).

Let ψF stand for the function (from M to R+ ∪ {∞})

ψF(x) = sup{F(x, y); y ∈ M(x, ≤)}, x ∈ M.

From (a02), ψF is (≤)-decreasing; for, if x1 ≤ x2, then (denoting for simplicity Mi = M(xi, ≤), i ∈ {1, 2}),

ψF(x1) = sup{F(x1, y); y ∈ M1} ≥ sup{F(x1, y); y ∈ M2} ≥ sup{F(x2, y); y ∈ M2} = ψF(x2).

On the other hand, the alternative ∞ ∈ ψF(M) cannot be avoided, in general. But, under a regularity condition like (a03) F(x, ·) is bounded above over M(x, ≤), ∀x ∈ M, this happens [in the sense: ψF(M) ⊆ R+].

237 238 Mihai Turinici

Call z ∈ M,(≤, F)-semi-maximal when: z ≤ w ∈ M =⇒ F(z, w) = 0; note that, in terms of the above introduced function, this amounts to: ψF(z) = 0. The following 1982 Altman’s ordering principle [1] (in short: AOP) is our starting point: Theorem 1.1. Let (a01)–(a03) hold, as well as (a04) (M, ≤) is sequentially inductive: each ascending sequence is bounded above (a05) (M, ≤, F) is semi regular: ((xn)=ascending) =⇒ lim infn F(xn, xn+1) = 0. Then, for each u ∈ M there exists a (≤, F)-semi-maximal v ∈ M with u ≤ v. [In fact, the original statement was formulated in terms of the dual quasi-order (≥) and the function Φ(x, y) = −F(y, x), x, y ∈ M. Moreover (in terms of this translation), (a02) and (a05) were taken – respectively – in their stronger form

(a06) ∀y ∈ M, F(·, y) is decreasing [x1 ≤ x2 =⇒ F(x1, y) ≥ F(x2, y)]

(a07) (M, ≤, F) is regular: [(xn)=ascending] =⇒ limn F(xn, xn+1) = 0. But, the author’s reasoning also works in this relaxed setting]. This result includes the 1976 Brezis-Browder ordering principle [2] (in short: BB). Moreover, as precise by the quoted authors, their statement includes Ekeland’s variational principle [5] (in short: EVP); hence, so does AOP. Now, BB and EVP found some basic applications to control and optimization, generalized differential calculus, critical point theory and global analysis; see the quoted papers for a survey of these. So, it must be not surprising that AOP was subjected to different extensions. For example, a pseudometric enlargement of this ordering principle was obtained by Turinici [11]. Further contributions in the area were provided by Kang and Park [7]; see also Zhu and Li [15]. The obtained facts are interesting from a technical viewpoint. However, we must emphasize that, whenever a maximality principle of this type is to be applied, a substitution of it by the Brezis-Browder’s is always possible. This raises the question of to what extent are these extensions of BB effective. As already shown in Turinici [12], the answer is negative for most of these. It is our aim in the following to provide a simpler proof of this fact, with respect to AOP; details will be given in Section 3. Note that, as a consequence, AOP is reducible to the De- pendent Choices Principle (in short: DC) due to Tarski [10]; for a direct verification we refer to Section 2. Now, in a close connection with Theorem 1.1, the following Szaz unboundedness criterion [9] (in short: S-U) is available: Theorem 1.2. Suppose that (in addition to (a01)+(a02))

(a08) each ascending sequence (xn) with supn F(x0, xn) < ∞ is bounded above and lim infn F(xn, xn+1) = 0 Altman ordering principles and dependent choices 239

(a09) ∀x ∈ M: ψF(x) > 0 [i.e.: ∃y ∈ M(x, ≤) with F(x, y) > 0]

Then, ψF(x) = ∞, for all x ∈ M. This statement is shown to imply a related one in Brezis and Browder [2]; which, as precise there, includes BB. Moreover [according to the author’s observation] his result includes AOP as well. It is our aim in the following to show (in Section 4) that the reciprocal inclusion is also true; i.e., that S-U is nothing but a logical equivalent of AOP. Further aspects will be delineated elsewhere.

2. DC IMPLIES AOP In the following, a direct proof of AOP is provided, via DC. (A) Let M be a nonempty set; and R ⊆ M × M stand for a relation on it. For each x ∈ M, denote M(x, R) = {y ∈ M; xRy} (the section of R through x). Proposition 2.1. (Dependent Choices Principle) Suppose that (b01) M(c, R) is nonempty, for each c ∈ M.

Then, for each a ∈ M there exists (xn) ⊆ M with x0 = a and xnRxn+1, for all n. This principle, due to Tarski [10] is deductible from the Axiom of Choice (in short: AC), but not conversely; cf. Wolk [14]. Moreover, it suﬃces for constructing a large part of the ”usual” mathematics; see Moore [8, Appendix 2, Table 4]. (B) We may now pass to the promised proof of AOP Proof. Let the premises of the quoted statement be in use. If, by absurd, its conclusion is not true, there must be some u ∈ M with (cf. Section 1)

(b02) ψF(x) > 0, for all x in U := M(u, ≤). Let γ be arbitrary ﬁxed in ]0, 1[. We introduce a relation R over U as:

(b03) xRy iﬀ x ≤ y, F(x, y) > γψF(x). From (b02), U(x, R) is nonempty, for each x ∈ U. So, by (DC), there must be a sequence (xn) in U with (x0 = u and) xnRxn+1, ∀n; i.e.,

xn ≤ xn+1 and F(xn, xn+1) > γψF(xn), for all n. Let v be an upper bound (in U) of this (ascending) sequence (assured by (a04)). As ψF is decreasing, we have [by the above relations] F(xn, xn+1) > γψF(v), ∀n. Passing to lim inf as n → ∞ yields ψF(v) = 0; in contradiction to (b02). Hence, this working assumption cannot be accepted; and the conclusion follows. Note that only the values of F on gr(≤):= {(x, y) ∈ M × M; x ≤ y} were considered here. So, we may substitute (a01) with its extended version 240 Mihai Turinici

(b04) F(x, y) ≥ 0, for all x, y ∈ M (i.e.: F(M × M) ⊆ R+). For, otherwise, the truncated function F∗(x, y) = max{0, F(x, y)}, x, y ∈ M, (fulfilling (b04)) has all properties of F; and, from the conclusion of Theorem 1.1 written for F∗ we derive the same fact (modulo F); because F∗ = F on gr(≤). Suppose that this extension was effectively performed. In this case, we claim that the boundedness condition (a03) may be dropped. For, the associated function G(x, y) = F(x, y)/(1 + F(x, y)), x, y ∈ M, fulfills such a property; as well as all remaining ones (involving F). This, along with the observation that the notions of (≤, F)-semi-maximal and (≤, G)-semi-maximal are identical, proves our claim. Further technical aspects may be found in Turinici [11]; see also Kang and Park [7].

3. AOP =⇒ (S-U) =⇒ BB Under these preliminary facts, we may now return to our initial setting. (A) Concerning the former inclusion, its original proof uses Altman’s ideas [1] for establishing Theorem 1.1. A slightly diﬀerent reasoning is that given below.

Proof. Assume that conclusion of Theorem 1.2 would be false:

(c01) ψF(x0) < ∞, for some x0 ∈ M.

(Here, ψF : M → R ∪ {∞} is that of Section 1). Denote for simplicity M0 = M(x0, ≤). Given the ascending sequence (yn) in M0, we have F(y0, yn) ≤ ψF(y0) ≤ ψF(x0) < ∞, ∀n; wherefrom (by (a08)), (yn) is bounded above in M (hence in M0) and lim infn F(yn, yn+1) = 0; so that, (a04)+(a05) hold over M0. Moreover, (a03) is clearly true (over M0) in view of (c01). Summing up, Theorem 1.1 applies to (M0, ≤) and F. It gives us, for the starting point x0 ∈ M0, some z ∈ M0 with the property: z ≤ w ∈ M0 =⇒ F(z, w) = 0; or, equivalently: ψF(z) = 0. But then, (a09) cannot hold; contradiction. Hence (c01) is false; and the conclusion follows.

(B) Let again (M, ≤) be a quasi-ordered structure; and ϕ : M → R, some function. By the construction (c02) F(x, y) = ϕ(x) − ϕ(y), x, y ∈ M, one gives, via (S-U), a basic result in Brezis and Browder [2, Theorem 1] (called: Brezis-Browder’s unboundedness criterion; in short: BB-U). Theorem 3.1. Assume that ϕ is (≤)-decreasing and

(c03) each ascending sequence (xn) with infn ϕ(xn) > −∞ is bounded above (c04) ∀x ∈ M, ∃y ∈ M(x, ≤) with ϕ(x) > ϕ(y).

Then, inf[ϕ(M(x, ≤))] = −∞, ∀x ∈ M. Altman ordering principles and dependent choices 241

A basic application of it is to be given under the lines below. Let the triplet (M, ≤ ; ϕ) be introduced as before. Call z ∈ M,(≤, ϕ)-maximal when: z ≤ w ∈ M imply ϕ(z) = ϕ(w). The following 1976 Brezis-Browder ordering principle [2] (in short: BB) about the existence of such points, is available. Theorem 3.2. Let (M, ≤) be sequentially inductive and ϕ be (≤)-decreasing, bounded from below. Then, for each u ∈ M there exists a (≤, ϕ)-maximal v ∈ M with u ≤ v. Proof. Assume this is not true; then, there must be some u ∈ M such that (c04) holds with Mu := M(u, ≤) in place of M. But then, the preceding statement yields inf[ϕ(Mu)] = −∞; in contradiction with the choice of ϕ. Note that the boundedness from below condition is not essential for the conclusion above; see Carjˆ a,˘ Necula and Vrabie [4, Ch 2, Sect 2.1] for details. Moreover (cf. Zhu and Li [15]), (R, ≥) may be substituted by a separable ordered structure (P, ≤) without altering the conclusion above; see also Turinici [13]. For the moment, BB follows from AOP, via (S-U). A direct deduction of this is available, by the construction (c02). In fact, (a01)+(a02) hold, from the monotonicity of ϕ. On the other hand, (a03) holds too, by the boundedness from below of ϕ; and, ﬁnally, (a05) is obtainable from both these. Summing up, Theorem 1.1 is applicable to our data. This, and the observation that (≤, ϕ)-maximal is identical with (≤, F)- semi-maximal (in the context of (c02)) ends the argument. Finally, note that BB (hence, a fortiori, AOP) includes EVP. Further aspects may be found in Hyers, Isac and Rassias [6, Ch 5].

4. (BB-LC) IMPLIES (DC) By the developments above, we have the (chain of) implications (DC) =⇒ (AOP) =⇒ (S-U) =⇒ (BB). So, it is natural asking whether these may be reversed. The natural setting for solving such a problem is (ZF)(=the Zermelo-Fraenkel system) without (AC); referred to in the following as the reduced Zermelo-Fraenkel system. Let X be a nonempty set; and (≤) be an order (i.e.: antisymmetric quasi-order) over it. We say that (≤) has the inf-lattice property, provided: x ∧ y := inf(x, y) exists, for all x, y ∈ X. Further, call z ∈ X,(≤)-maximal if X(z, ≤) = {z}; the class of all these points will be denoted as max(X, ≤). Accordingly, (≤) is called a Zorn order when max(X, ≤) is nonempty and coﬁnal in X [for each u ∈ X, there exists a (≤)-maximal v ∈ X with u ≤ v]. Now, the statement below is a particular case of BB: Theorem 4.1. Let the partially ordered structure (X, ≤) be such that (d01) (X, ≤) is sequentially complete (d02) (≤) has the inf-lattice property.

In addition, suppose that there exists a function ϕ : X → R+ with 242 Mihai Turinici

(d03) ϕ is strictly decreasing and ϕ(X) is countable. Then, (≤) is a Zorn order. We shall refer to it as: the lattice-countable version of BB (in short: BB-Lc). Clearly, (BB) =⇒ (BB-Lc); since (by (d03)) x ≤ y and ϕ(x) = ϕ(y) imply x = y. The remarkable fact to be added is that this last principle yields (DC); and so, it completes the circle between all these results. Proposition 4.1. We have (in the reduced Zermelo-Fraenkel system) (BB-Lc) =⇒ (DC). So (by the above) the maximal/unboundedness results (AOP), (S-U) and (BB) are all equivalent with (DC); hence, mutually equivalent. Proof. Let M be some nonempty set; and R stand for some relation over M with the property (b01). Fix in the following a ∈ M; as well as some b ∈ M(a, R). For each n ≥ 2 in N (= the set of natural numbers), let N(n, >):= {0, ..., n − 1} stand for the initial segment determined by n; and Xn denote the class of all ﬁnite sequences x : N(n, >) → M with: x(0) = a, x(1) = b and x(m)Rx(m + 1) for 0 ≤ m ≤ n − 2. In this case, N(n, >) is just Dom(x) (the domain of x); and n = card(N(n, >)) will be referred to as the order of x [denoted as ω(x)]. Put X = ∪{Xn; n ≥ 2}. Let stand for the partial order (on X)

(d04) x y iff Dom(x) ⊆ Dom(y) and x = y|Dom(x); and ≺ denote its associated strict order. All we have to prove is that (X, ) has strictly ascending infinite sequences. (A) Let x, y ∈ X be arbitrary fixed. Denote K(x, y):= {n ∈ Dom(x) ∩ Dom(y); x(n) , y(n)}. If x and y are comparable (i.e.: either x y or y x; written as: x <> y) then K(x, y) = ∅. Conversely, if K(x, y) = ∅, then x y if Dom(x) ⊆ Dom(y) and y x if Dom(y) ⊆ Dom(x); hence x <> y. Summing up, we have the characterization

(x, y ∈ X): x <> y if and only if K(x, y) = ∅.

The negation of this property means: x and y are not comparable (denoted as: x||y). By the property above, it means: K(x, y) , ∅. Note that, in such a case, k(x, y):= min(K(x, y)) is well defined; and N(k(x, y), >) it is the largest initial interval of Dom(x)∩ Dom(y) where x and y are identical. Lemma 4.1. The order () has the inf-lattice property. Moreover, the map x 7→ ω(x) is strictly increasing: x ≺ y implies ω(x) < ω(y). Proof. Let x, y ∈ X be arbitrary fixed. The case x <> y is clear; so, without loss, one may assume that x||y. Note that, by the remark above, K(x, y) , ∅ and k := k(x, y) exists. Let the finite sequence z ∈ Xk be introduced as z = x|N(k,>) = y|N(k,>). For the Altman ordering principles and dependent choices 243 moment z x and z y. Suppose that w ∈ Xh fulfills the same properties. Then, the restrictions of x and y to N(h, >) are identical; wherefrom (see above) h ≤ k and w z; which tells us that z = x ∧ y. The last part is obvious.

(B) Denote ϕ(x) = 3−ω(x), x ∈ X. Clearly, ϕ(X) = {3−n; n ≥ 2}; hence, ϕ has countably many strictly positive values. Moreover, if x, y ∈ X are such that x ≺ y, then (by Lemma 4.1) ϕ(x) > ϕ(y); which tells us that ϕ is strictly decreasing. (C) From (b01), max(X, ) = ∅; i.e.: for each x ∈ X there exists y ∈ X with x ≺ y. This, along with (BB-Lc), tells us that (X, ≤) is not sequentially complete: there exists at least one ascending sequence (xn) in X which is not bounded above. By this last property, B(k):= {n ∈ N(k, <); xk ≺ xn} is nonempty, for all k ∈ N. Deﬁne g(n) = min[B(n)], n ∈ N; hence n < g(n) and xn ≺ xg(n), for all n. The iterative process [p(0) = g(0), p(n + 1) = g(p(n)), n ≥ 0] is therefore constructible without any use of (DC); and gives a strictly ascending sequence of ranks (p(n)). So, (yn := xp(n); n ∈ N) is a subsequence of (xn) with yn ≺ ym (hence ω(yn) < ω(ym)) for n < m; moreover, as ω(y0) > 1, we must have ω(yn) > n + 1, for all n. But then, the sequence (cn = yn(n); n ∈ N) is well deﬁned (in M) with c0 = a and cnRcn+1, for all n. The proof is thereby complete.

In particular, when the speciﬁc assumptions (d02) and (d03) (the second half) are ignored in Theorem 4.1, Proposition 4.1 is comparable with a statement in Brunner [3]. Further aspects may be found in Turinici [13].

References

[1] M. Altman, A generalization of the Brezis-Browder principle on ordered sets, Nonlinear Analy- sis, 6 (1982), 157-165. [2] H. Brezis, F. E. Browder, A general principle on ordered sets in nonlinear functional analysis, Advances Math., 21 (1976), 355-364. [3] N. Brunner, Topologische Maximalprinzipien, Zeitschr. Math. Logik Grundl. Math., 33 (1987), 135-139. [4] O. Carjˆ a,˘ M. Necula, I. I. Vrabie, Viability, Invariance and Applications, North Holland Mathe- matics Studies vol. 207, Elsevier B. V., Amsterdam, 2007. [5] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (New Series), 1 (1979), 443-474. [6] D. H. Hyers, G. Isac, T. M. Rassias, Topics in Nonlinear Analysis and Applications, World Sci. Publ., Singapore, 1997. [7] B. G. Kang, S. Park, On generalized ordering principles in nonlinear analysis, Nonlinear Anal- ysis, 14 (1990), 159-165. [8] G. H. Moore, Zermelo’s Axiom of Choice: its Origin, Development and Inﬂuence, Springer, New York, 1982. [9] A. Szaz, An Altman type generalization of the Brezis-Browder ordering principle, Math. Morav- ica, 5 (2001), 1-6. 244 Mihai Turinici

[10] A. Tarski, Axiomatic and algebraic aspects of two theorems on sums of cardinals, Fund. Math., 35 (1948), 79-104. [11] M. Turinici, A generalization of Altman’s ordering principle, Proc. Amer. Math. Soc., 90 (1984), 128-132. [12] M. Turinici, Metric variants of the Brezis-Browder ordering principle, Demonstr. Math., 22 (1989), 213-228. [13] M. Turinici, Brezis-Browder Principles and Applications, in (P. M. Pardalos et al., eds.) ”Nonlin- ear Analysis and Variational Problems” (Springer Optimiz. Appl., vol. 35), pp. 153-197, Springer Science+Business Media, LLC, 2010. [14] E. S. Wolk, On the principle of dependent choices and some forms of Zorn’s lemma, Canad. Math. Bull., 26 (1983), 365-367. [15] J. Zhu, S. J. Li, Generalization of ordering principles and applications, J. Optim. Th. Appl., 132 (2007), 493-507. ROMAI J., 6, 2(2010), 245–245

NOTE

We were informed by Prof. Francesco J. Aragon´ Artacho from Univ. of Alicante, Spain, about the following unpleasant situation: the paper ”Proximal point methods for variational inequalities involving regular mappings” signed by Corina L. Chiriac in ROMAI Journal v. 6, 1(2010), 41-45, was also published in Analele Universitatii Oradea, facultatea Matematica, TXVII (2010), 65-69 under the title ”Convergence of the proximal point algorithm variational inequalities with singular mappings”. More- over, the paper has a consistent intersection with the paper ”Convergence of the proximal point method for metrically regular mappings”, by F. J. Aragon´ Artacho, A. L. Donchev and M. H. Geoﬀroy, published in ESAIM: Proceedings 17 (2007), 1-8, without even citing this work. Prof. F. J. Aragon´ Artacho concluded this to be a case of plagiarism. The Directory Council of ROMAI compared the three papers involved in the above assertions and found the conclusion formulated by Prof. Aragon Artacho as justiﬁed. As a consequence, we, the members of the Directory Council decide to exclude the paper signed by Corina L. Chiriac from the issue v. 6, 1(2010), posted on the website of ROMAI Journal http://rj.romai.ro/. We also consider this paper morally excluded from the printed issue (even if we have not the means to do this physically). We will inform the mathematical databases that are reviewing our journal about this decision. Corina L. Chiriac will not be allowed to publish again in ROMAI Journal.

Directory Council of ROMAI

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