Publicaţii

a) Reviste indexate în baze de date internaţionale

[1] Paul Popescu, Marcela Popescu,

           Affine Hamiltonians in higher order geometry,             International Journal of Theoretical Physics 46 (10) (2007), pp. 2531-2549. (ISI: 0.389 -2006, Scopus, Ebsco, Scimago)

[2] Paul Popescu, Marcela Popescu,

          Lagrangians adapted to Submersions and Foliations, Differential Geometry and its Applications ) (accepted). (ISI: 0.457 - 2007, Scopus, Ebsco, Scimago)

[3] Paul Popescu, Marcela Popescu,

          On associated quasi-connections, Periodica Mathematica Hungarica, vol.31(1), 1995, 45-52.[Zbl 0857.53025][MR 96i:53023] (ISI since 2008, Scopus, Ebsco, Scimago)   

[4] Marcela Popescu, Paul Popescu,

          On the structure equations of the induced linear connections, Balkan Journal of Geometry and its Applications, Vol. 3, Nr.1, 1998,  103-109. [0949.53020, 0928.53012] [MR1664464 (99k:53038)] (ISI since 2008, inexed A by CNCSIS, Scopus, Ebsco, Numdam)

[5] Paul Popescu, Marcela Popescu,

          Geometrical objects on subbundles (II), Balkan Journal of Geometry and its Applications, Vol. 2, No.2, 1997, 105-111. [Zbl 0903.57015] [MR1662042 (2000f:53104)] (ISI since 2008, indexed A by CNCSIS, Scopus, Ebsco, Numdam)

[6] Marcela Popescu, Paul Popescu,

          On Lagrange epimorphisms and Lagrange submersions, Balkan Journal of Geometry and its Applications, 5, 2, 2000, 97-101.   [Zbl 0966.37024] [MR1811896  (2002g:37066)] (ISI since 2008, indexed A by CNCSIS, Scopus, Ebsco, Numdam)

[7] Paul Popescu, Marcela Popescu,

          On Hamiltonian submanifolds (I), Balkan Journal of Geometry and its Applications, 7, 2, 2002, 79-86. [Zbl 1034.53062] [MR1941111 (2003m:53130)] (ISI since 2008, indexed A by CNCSIS, Scopus, Ebsco, Numdam)

[8] Paul Popescu, Marcela Popescu,

          A general background of higher order geometry and induced objects on subspaces, Balkan Journal of Geometry and its Applications, 7, 1, 2002, 79-90. [Zbl pre01858856] [MR1940568 (2003j:53029)] (ISI since 2008, indexed A by CNCSIS, Scopus, Ebsco, Numdam)

[9] Marcela Popescu,

          On certaines differentials defined by pseudo- differential complexes, Analele Stiintifice Univ. Ovidius Constanta, III, fasc. 2, 1995, 103-106. (The second International Workshop on Differential Geometry and its Applications, Universitatea Ovidius din Constanţa, 25-28 Septembrie 1995.) [Zbl 0909.58045] [MR1439991 (97m:58217)] (ISI since 2008, indexed A by CNCSIS)

[10] Constantin Udriste, Marcela Popescu, Paul Popescu,

          Generalized Multitime Lagrangians and Hamiltonians, WSEAS Transactions on Mathematics, Volume 7, 2008, 66-72. (Scopus, Scimago) ISSN: 1109-2769

[11] V. Cruceanu, Marcela Popescu, Paul Popescu,

          On projectable objects on fibred manifolds, Archivum Mathematicum, 3, 2001, 185-296. [MR1860183 (2002f:53027)] (Scopus, Scimago)

b) Alte reviste internationale

[12] Marcela Popescu, Paul Popescu,

         On high order structures on vector bundles, Algebras, Groups and Geometries, Hadronic Press, 1(16), 1999, 91-103. [Zbl 1004.58001] [MR1704092 (2000f:58008)]

[13] Paul Popescu, Marcela Popescu,

          On graded algebra bundles, Novi Sad Journal of Mathematics, 3(29), 1999, 257-266.[Zbl 0999.58012][ MR1771003 (2001k:58032)]

[14] Marcela Popescu, Paul Popescu,

          Projectable nonlinear connections, Novi Sad Journal of Mathematics, 3(29), 1999, 249-256.[Zbl 1011.53052][MR1771002 (2001h:53102)]

c) Reviste nationale recunoscute CNCSIS

[15] Marcela Popescu, On induced non-linear connections, Analele Universităţii Al.I.Cuza din Iaşi, seria mat., T.XLII,f.1., 1996, 119-126. 

[16] Marcela Popescu, About some properties of fibre subbundles, Analele Universităţii din Craiova, seria mat.-inf., vol.XVIII, 1990, 50-56.

[17] Marcela Popescu, Induced connections on fibred submanifolds, Analele Universităţii din Craiova, seria mat.-inf.,vol.XXII, 1995, 43-50.

[18] Marcela Popescu, Paul Popescu,

          The structure equations of induced non-linear connectionons,Analele Universităţii din Timişoara, Seria matem., Vol. XXXIV, fasc. 2, 1996, 255-260. [MR1659360 (99k:53134)] (B)

[19] Marcela Popescu, Paul Popescu,

          On almost holonomic structures, Analele Universităţii din Craiova, seria mat.-inf., vol.XIX, 1991-92, 93-97. [Zbl 0851.53016] [MR1650137] (B)

[20] Marcela Popescu, Paul Popescu,

         The Darboux's property and the P()-derivability, Analele Universităţii din Craiova, seria mat.-inf., vol.XXII, 1995, 51-58. [MR1654212 (99m:26005)] (B)

[21] Marcela Popescu, Paul Popescu,

         d-linear connections on fibred manifolds, Analele Universităţii din Craiova, vol.XXII, 1996, 52-59. [MR1654908 (99k:53037)] (B)

[22] Marcela Popescu, Paul Popescu,

         On some derivable real functions, Analele Universităţii din Oradea, tomul II, fascicola matematică, 1992, 14-117. (B+)

[23] Marcela Popescu, Paul Popescu,

          On the p-derivable functions, Analele Universităţii din Oradea, tomul II, fascicola matematică, 1992, 118-124.(B+)

 

d) Volume ale unor Conferinţe ISI

 [24] Marcela Popescu, Paul Popescu,

          Lagrangians and Hamiltonians on affine bundles and higher order geometry, Proc. Vol. VII International Conference GEOMETRY AND TOPOLOGY OF MANIFOLDS, The Mathematical Legacy of Charles Ehresmann on the occasion of the hundredth anniversary of his birthday, Będlewo, Poland, May 8-15, 2005, Banach Center Publications, Inst. of Mathematics, Polish Acad. of Sc., Warsawa 2007, 451-469. (ISI Conference).

 [25] Paul Popescu, Marcela Popescu,

          The derived Lie pseudoalgebra of an anchored module, IV Conference on Geometry and Topology of Manifolds, Krynica  2002, Universitatis Iagellonicae Acta Mathematica, fasc.II 1278(42), 185-198 (2004), ISSN 0862-9544 (ISI Conference).

 [26] Paul Popescu, Marcela Popescu,

         On modules with differential, Proc. Vol. 8th International Conference on Diff. Geom. and Its Appl., Opava, 2001, Univ. Opava, Cehia, 2002. [Zbl pre01944442] [MR1978769 (2004e:16003)] (ISI Conference)  (ISSN 0926-2245)

e) Volume ale unor Conferinţe internaţionale

[27] Marcela Popescu,

          Induced semisprays and connections on submanifolds, vol. New Developements in Differential  Geometry, Budapest 1996, Kluwer Academic Publ., 1998, 317-328.

[28] Marcela Popescu, On vectorial Finsler connections, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, 13-16 decembrie 1995, Salonic, Grecia, Balkan Press, Bucharest, 1997, 61-64.

[29] Marcela Popescu, On the supplementary vector subbundles, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, 13-16 decembrie 1995, Salonic, Grecia, Balkan Press, Bucharest, 1997, 65-70.

[30] Paul Popescu, Marcela Popescu,

          Anchored vector bundles and Lie algebroids, Vol. Lie Algebroids, Banach Center Publications, Vol. 54, Inst. of Mathematics, Polish Acad. of Sc., 2001, 51-69.[ Zbl 1021.22001][ MR1881649 (2003a:58028)]

[31] Marcela Popescu, Paul Popescu,

          Geometric objects defined by Almost Lie Structures, Vol. Lie Algebroids, Banach Center Publications, Vol. 54, Inst. of Mathematics, Polish Acad. of Sc., 2001, 217-233. [Zbl 1001.53011] [MR1881656 (2003j:53036)]

[32] Marcela Popescu, Paul Popescu,

          Projectable nonlinear connections and foliations, Proceedings of Summer School in Differential Geometry, 3-7 Septembrie 1999,  Coimbra (Portugalia), CIM Publications, 159-165. [Zbl 0970.53016][ MR1859952 (2002j:53099)]

[33] Paul Popescu, Marcela Popescu, Embeddings of Almost Lie Structures,

          Proceedings of Summer School in Differential Geometry, 3-7 Septembrie 1999, Coimbra (Portugalia), CIM Publications, 149-157. [Zbl 01554190][ MR1859951 (2003d:58007)]

[34] Marcela Popescu, Paul Popescu,

          The derived Lie algebra of a module, Proc. Vol. I Colloquium on Lie Theory and Applications, Univ. Vigo, Spania, 2001, 151-162. [Zbl pre01836513][ MR1954771 (2004d:17040)]

[35] Paul Popescu, Marcela Popescu,

          Linear connections on modules with differentials, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, 21-26 octombrie, 1996, Salonic, Grecia, Balkan Press, Bucharest, 1997, 70-77. [Zbl 1004.58504][MR1710851 (2001b:58014)]

[36] Paul Popescu, Marcela Popescu,

         Geometrical objects on subbundles (I), Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, 21-26 octombrie, 1996, Salonic, Grecia, Balkan Press, Bucharest, 1997, 62-69. [Zbl 0938.57018]

[37] Marcela Popescu, Paul Popescu,

          Reductions associated with vector subbundles, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, 1994, Salonic, Grecia, Balkan Press, Bucharest, 1998, 56-60. [Zbl 0944.53043] [MR1648018 (99k:53135)]

[38] Paul Popescu, Marcela Popescu,

          Generalized algebroids and second order geometry, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, iulie 1997, Salonic, Grecia, Balkan Press, Bucharest,  2000, 69-78. [Zbl 0990.58015] [MR1896903 (2004e:58032)]

[39] Marcela Popescu, Paul Popescu,

          Non-linear d-connections, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, iulie 1998, Salonic, Grecia, Balkan Press, Bucharest, 1999, 83-90.[ Zbl 0970.53017]

[40] Paul Popescu, Marcela Popescu,

          Generalized algebroids and strongly homotopy algebra, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebras, iulie 1998, Salonic, Grecia, Balkan Press, Bucharest, 1999, 91-99.     [Zbl 0970.17013]

[41] Paul Popescu, Marcela Popescu,

          On non-commutative modules with differentials, Proccedings of Conference of Geometry and Its Applications in Technology and The Workshop on Global Analysis, Differential Geometry and Lie Algebras, iunie 1999, Salonic, Grecia, Balkan Press, Bucharest, 2001, 158-164. [Zbl 1016.16024]

[42] Marcela Popescu, Paul Popescu,

          Projectable nonlinear connections on submanifolds and geometric distributions on manifolds, Proccedings of Conference of Geometry and Its Applications in Technology and The Workshop on Global Analysis, Differential Geometry and Lie  Algebras, iulie 2000, Salonic, Grecia, Balkan Press, Bucharest, 2003, 104-108.

[43] Marcela Popescu, Paul Popescu,

          Higher order geometry and induced objects, Proceedings of The Conference of Applied Differential Geometry - General Relativity and The Workshop on Global Analysis, Differential Geometry and Lie Algebras, 2001, Geometry Balkan Press, 2004,  124-137.

[44] Marcela Popescu, Paul Popescu,

          On Finsler bundles and Finsler connections, Proceedings of The Conference of Applied Differential Geometry - General Relativity and The Workshop on Global Analysis, Differential Geometry and Lie Algebras, 2002, Geometry Balkan Press, 2004, 110-116.

[45] Paul Popescu, Marcela Popescu,

          On Hamiltonian submanifolds (II), Ann. of Univ. of Craiova, Physics AUC, Vol. 10, II, 2000, 199-210. (Proc. of the 2nd International Spring School and Workshop on Quantum Field theory and Hamilton Spaces, Călimăneşti, 2-7 Mai 2000) (B)

[46] Marcela Popescu, Paul Popescu,

          On Lagrange Submersions and Foliations, Ann. of Univ. of Craiova, Physics AUC, Vol. 12, III, 2002, 184-190. (Proc. of the 3rd International Spring School and Workshop on Quantum Field theory and Hamilton Spaces, Călimăneşti, 6-12 Mai 2002) (B)

[47] Paul Popescu, Marcela Popescu,

          On Non-Hamiltonian System, Ann. of Univ. of Craiova, Physics AUC, Vol. 16, II, 2006, 194-199. (Proc. of the 5th International Spring School and Workshop on Quantum Field theory and Hamilton Spaces, Călimăneşti, Mai 2006). (B)

[48] Paul Popescu, Marcela Popescu,

          Lifting and descending procedures for Lagrangians of higher order, Ann. of Univ. of Craiova, Physics AUC, Vol. 18, 2008, 136-140 (Proc. of the 6th International Spring School and Workshop on Quantum Field theory and Hamilton Spaces, Călimăneşti, Mai 2008). (B)

[49] Marcela Popescu, Paul Popescu,

          On algebra bundles and their pseudo-morphisms, General Mathematics, V.5, 1997, 331-338. (Third International Workshop on Diff. Geom. and its Appl. and the First German-Romanian Seminar on Geometry, Universitatea din Sibiu, 18-23 Septembrie, 1997.) [MR1723624 (2000h:58039)] (B)

[50] Marcela Popescu, Paul Popescu,

          On the derived bundle of a relative tangent space, Studii şi Cercetări Ştiinţifice, Seria Matematică, Nr. 10, 2000, 215-222. (Proc. of the Conference on Finsler and Lagrange Spaces, Bacău, Februarie 2000.) [MR1886297 (2003e:58005)]  (B+)

[51] Marcela Popescu, Paul Popescu,

         On affine hamiltonians and lagrangians of higher order and their subspaces,  BSG Procedings 13, The 5-th Conference of  Balkan Society of Geometers, Balkan Press, 2006, Editor in chief C.Udriste, Vol. Ed. V.Balan, 123-139, ISBN 973-8381-11-8.

[52] Marcela Popescu, Paul Popescu,

          From hamiltonians and lagrangians to legendrians, BSG Procedings 13, The 5-th Conference of  Balkan Society of Geometers, Balkan Press, 2006, Editor in chief C.Udriste, Vol. Ed. V.Balan,      115-122, ISBN 973-8381-11-8.

[53] Marcela Popescu, Paul Popescu, Dumitru Nicolae,

          Some aspects concerning non-Hamiltonians systems, Differential Geometry and Dynamical Systems-2007, BSG Proceedings 15, pp.163-167. (ISSN 1843-2654 - Printed version, ISSN 1843-2859 - Electronic version).

[54] Marcela Popescu, Paul Popescu,

          An intrinsic link between scalar and volume-valued Lagrangians, Differential Geometry and Dynamical Systems-2007, BSG Proceedings 15, pp. 168-175. (ISSN 1843-2654 · Printed version, ISSN 1843-2859 - Electronic version).

f) Volume ale unor Conferinţe naţionale

[55] Marcela Popescu, Paul Popescu,

          About some definitions of differentiability, Volumul A XIII-a Conferinţă Naţională de Geometrie şi Topologie, Universitatea Babeş-Bolyai, Cluj, 1982, 87-88 (recenzat în Referativnîi Jurnal, 136, Analiza, 1985).

[56] Marcela Popescu, Paul Popescu,

         Some constructions in the categories of anchored modules and preinfinitesimal modules, Lucrările celei de-a III-a Conferinţe Anuale a SSMR, Craiova,  26-29 mai 1999, Editura Reprograph, vol.I, Comunicări Ştiinţifice, 193-199.

[57] Marcela Popescu, Paul Popescu,

          Induced structures on anchored vector subbundles, în Vol. Lucr. Confer. Anuale SSM, Braşov, 2001, Ed.Univ.Transilvania, Vol. I, 2002, 239-248.

[58] Marcela Popescu, Nabi Toporan, Probleme de afectare. Consideraţii metodice. Aplicaţii pe calculator, în Vol. Confer. Anuale SSM, Braşov, 2001.

[59] Marcela Popescu, Paul Popescu, Nabi Toporan,

          Probleme de tip transport. Consideraţii metodice. Aplicaţii pe calculator, în Vol. Confer. Anuale SSM, Braşov, 2001.

g) În rezumat

[60] Marcela Popescu, Paul Popescu,

          An extension of the osculator bundle, Abstracts of the 3rd International Conference on Geometry, 1-6 Septembrie, 1997, Varna, Bulgaria, 52-53.

[61] Marcela Popescu, An extension of Gauss and Codazzi equations, Abstracts of The second Asian Congress on Mathematics, 17-20 octombrie 1995, Suranaree University of Technology, Nackhon-Ratchasima, Thailand, 135.

[62] Marcela Popescu, Semisprays induced on submanifolds, Abstracts on Conference on Differential Geometry, Budapest, July 27-30, 1996, Supplement, 7.

PARTICIPĂRI LA CONFERINŢE

a) Conferinţe Naţionale

1. Conferintele Nationale de Geometrie si Topologie: mai multe participari cu lucrari incepand cu anul 1982.

2. Seminarul National de Spatii Finsler, Brasov, Februarie, 1996; lucrarea a aparut in Analele Univ, din Timisoara.

3. The X-th National Conference on Finsler and Lagrange Geometry and Applications, Craiova, februarie 1998.

4. A III-a Conferinta Anuala a S.S.M., Craiova, mai 1999.

5. A V-a Conferinta Anuala a S.S.M., Brasov, mai 2001.

b) Conferinţe internaţionale din ţară

6. The second International Workshop on Differential Geometry and its Applications, 25-28 Septembrie 1995, Constanta.

7. First Conference of Balkan Society of Geometers, Universitatea Politehnică Bucureşti, 23-27 Septembrie 1996.

8. Third International Workshop on Diff. Geom. and its Appl. and the First German-Romanian Seminar on Geometry, Sibiu, september 18-23, 1997.

9. A treia conferinta BSG, Bucuresti, august 2000.

10. 2nd International Spring School and Workshop on Quantum Field theory and Hamilton Spaces, Călimăneşti, Mai 2000.

11. The 5-th Conference of Balkan Society of Geometers, August 29 - September 2, 2005 Mangalia, Romania.

12. International Conference on Quantum Field Theory, General Relativity, Gravitation and Non-linear Dynamics,  Calimanesti-Caciulata, May 2006.

13. The 5-th Conference of Balkan Society of Geometers August 29 - September 2, 2005 Mangalia, ROMANIA. (Member of Scientific Commettee)

c) Conferinţe internaţionale din străinătate

14. Colloquium on Differential Geometry, Debrecen, Hungary, July, 1994.

15. The Workshop on Global Analysis, Differential Geometry, Lie Algebras, December 1994, Thessaloniki, Greece.

16. The Second Asian Congress on Mathematics, 17-20 October 1995, Suranaree University of Technology, Nackhon-Ratchasima, Thailand.

17. The Workshop on Global Analysis, Differential Geometry, Lie Algebras, 13-16 December 1995, Thessaloniki, Greece.

18. Conference on Differential Geometry, July 27-30 , 1996, Budapest, Hungary,

19. First Conference of Balkan Society of Geometers, Universitatea Politehnica Bucuresti, 23-27 Septembrie 1996.

20. The Fourth International Congress of Differential Geometry, may 1996, Thessaloniki, Greece (invited speaker).

21. The Workshop on Global Analysis, Differential Geometry, Lie Algebras, october 21-26, 1996, Thessaloniki, Greece (invited speaker).

 22. The Workshop on Global Analysis, Differential Geometry, Lie Algebras, july 23-26, 1997, Thessaloniki, Greece (invited speaker).

23. XVIIth Nevanlinna Colloquium, Lausanne, EPFL, (Elvetia) 14-20 august, 1997.

24. International Conference on Geometry, 1-5 Septembrie 1997, Varna (Bulgaria).

25. The Conference of Geometry and its Applications in Technology, Thessaloniki (Greece), June 23-27 1999 (invited speaker).

26. Summer School in Differential Geometry, Coimbra (Portugal), september 3-7 1999.

27. Simposion on Geometry, Dortmund, July 2000.

28. Conference on Applied Differential Geometry, Aristotle University of Thessaloniki Greece, 27 august-3 sept. 2000 (invited speaker).

29. Lie Algebroids and Related Topics in Differential Geometry, Stefan Banach International Mathematical Center,Warsaw, June, 2000 (invited speaker).

30. Conference on Geometry and Topology of Manifolds, Krynica 29.04-5.05.2001 (Poland).    

31. 8th Iternational Conference on Differential Geometry and its Applications, August 27 - 31, 2001, Opava, Czech Republic.

32. The Conference of Applied Differential Geometry - General Relativity and The Workshop on Global Analysis, Differential Geometry and Lie Algebras, Aristotle University of Thessaloniki, Greece, July 2001 (invited speaker).

33. 4th International Conference of the Balkan Society of Geometers and Conference on Aplied Differential Geometry, Aristotle University of Thessaloniki, Greece, June 26-30,  2002 (invited speaker).

34. XIV Geometric Seminar, Technical Faculty ,,Mihajlo Pupin'', Zrenjanin (Serbia), October 2-5, 2003.

35. 7th Conference on Geometry and Topology of Manifolds. The Mathematical Legacy  of Charles Ehresmann. Bedlewo (Poland) 8.05.2005-15.05.2005.

36. 8th Conference on Geometry and Topology of Manifolds (Luxembourg- Poland- Ukraine conference, Lie algebroids, dynamical systems and applications) Przemysl (Poland) - L'viv (Ukraine) 30.04.07 - 6.05.2007.

 

CONTRACTE DE CERCETARE

1)      Grantul CNCSIS de tip A, 81/2005 (Universitatea din Craiova) - director.

2)     Grantul CNCSIS de tip A, 196/1999 (Universitatea din Craiova).

3)     Grantul CNCSIS de tip A, 159/1999, Contract nr.32576/1999, Tema nr.54, Comisia 1 (la   Universitatea din Iasi).

4)     Grantul CNCSIS de tip A, 34967(182)/2001-2004 (Universitatea Politehnică Bucureşti).

5)     Contract 7C23/2004, Univ Cv. 50C/26.11.2004. Proiectul a obtinut in anul 2006 Premiul pentru Excelenta in cercetare la Sectiunea Proiecte de Dezvoltare Tehnologica.

6)     Contract CEEX 100/13.10.2005, Subcontract 251/2005. 5 faze: 1/2005; 2/2006; 3/2007; 4/2007/ 5/2008.

7)     Contract pentru Platforma Interdisciplinara: TEHNOPLAT OLTENIA.

8)     Laborator de cercetare si modelare in matematici aplicate, Proiect Impact cu nr. 434, aprobat in sesiunea ianuarie 2007.

 

ALTE ACTIVITĂŢI

 - Referent la Zentralblatt fűr Math (reviwer number 9609).

- Membru în Editorial Board of BSG Proceedings.

     Participarea la editarea volumelor cu lucrările Workshop-urilor GLOBAL  ANALYSIS, DIFFERENTIAL GEOMETRY AND LIE ALGEBRAS din anii 1994 si 1995, care au apărut în editura BALKAN PRESS, Editor Prof. Gr. Tsagas şi   Editor asociat la volumele cu lucrările Workshop-urilor GLOBAL ANALYSIS, DIFFERENTIAL GEOMETRY AND LIE ALGEBRAS din anii 1996, 1997, 1998, 1999 2000, 2001 şi 2002 care au apărut în editura BALKAN PRESS, Editor Prof. Gr. Tsagas.

 
Educatie
Activitate didactica
Cercetare
Contacte
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

               

 DMA          Membri