Skip to main content
SpringerLink
Log in
Book cover

Value Distribution Theory and Related Topics pp 1–73Cite as

A New Program of Investigations in Analysis: Gamma-Lines Approaches

  • Chapter

Part of the book series: Advances in Complex Analysis and Its Applications ((ACAA,volume 3))

Abstract

A new program of mathematical studies primarily based on the theory of Gamma-lines and ideas of the Nevanlinna value distribution theory is presented. This program establishes new connections between a variety of mathematical fields: real and complex analysis, ordinary, partial and complex differential equations, differential geometry, real and complex algebraic geometry, and Hilbert’s topological problem 16. Preliminary results, related to some of the problems posed, are given. In addition, the usefulness of this program in applications will be discussed.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ablowitz M. J., Clarkson P. A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. xii+516 pp.

    Google Scholar 

  2. Ahlfors L., Zur Theorie Uberlagerungsflachen, Acta Math. 65 (1935), 157–194.

    MATH  Google Scholar 

  3. Barsegjan G., Defect values and the structure of covering surfaces, Izv, Akad. Nauk Armjan. SSR Ser. Mat. 12 (1977), 46–53. (Russian)

    MATH  MathSciNet  Google Scholar 

  4. Barsegjan G., Distribution of sums of α-points of meromorphic functions, Dokl. Akad. Nauk SSSR 232 (1977), 741–744. (Russian)

    MATH  MathSciNet  Google Scholar 

  5. Barsegjan G., A geometric approach to the problem of the branching of Riemann surfaces, Dokl. Akad. Nauk SSSR 237 (1977), 761–763. (Russian)

    MATH  MathSciNet  Google Scholar 

  6. Barsegjan G., New results in the theory of meromorphic functions, Dokl. Akad. Nauk SSSR 238 (1978), 777–780. (Russian)

    MATH  MathSciNet  Google Scholar 

  7. Barsegjan G., Geometry of meromorphic functions, Mat. Sb. (N.S.) 114(156) (1981), 179–225. (Russian)

    MATH  MathSciNet  Google Scholar 

  8. Barsegyan G., The property of closeness of α-points of meromorphic functions, Mat. Sb. (N.S.) 120(162) (1983), 42–67, 143. (Russian)

    MATH  MathSciNet  Google Scholar 

  9. Barsegyan G., The proximity property of α-points of meromorphic functions and the structure of univalent domains of Riemann surfaces, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 20 (1985), 375–400.

    MATH  MathSciNet  Google Scholar 

  10. Barsegyan, G., The property of closeness of α-points of meromorphic functions and the structure of univalent domains of Riemann surfaces, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 20 (1985), 407–425. (Russian)

    MATH  MathSciNet  Google Scholar 

  11. Barsegian G., Estimates of derivatives of meromorphic functions on sets of α-points, J. London Math. Soc. 34 (1986), 534–540.

    MATH  MathSciNet  Google Scholar 

  12. Barsegyan G., The property of the comparability of derivatives of meromorphic functions and of the distances between α-points, Izv. Akad. Nauk Armenii Mat. 26 (1991), 52–70. (Russian)

    MATH  MathSciNet  Google Scholar 

  13. Barsegian G., Principle of partitioning of meromorphic functions, Math. Montisnigri 5 (1995), 19–25.

    MATH  MathSciNet  Google Scholar 

  14. Barsegian G., Principle of closeness of sufficiently large sets of α-points of meromorphic functions in the unit disk, Hokkaido Math. J. 26 (1997), 451–456.

    MATH  MathSciNet  Google Scholar 

  15. Barsegyan G., Applications of the principle of partitioning of meromorphic functions, I, The comparability property, Izv. Akad. Nauk Armenii 32 (1997), 5–50. (Russian)

    MATH  Google Scholar 

  16. Barsegian G., Gamma-lines: on the geometry of real and complex functions, Taylor and Francis, London-New York, 2002.

    Google Scholar 

  17. Barsegian G. and Begehr H., Lines of catastrophe: problems, examples of solutions, connections with Nevanlinna theory and Gamma-lines, to appear in Proc. 3rd ISAAC Congress, Berlin 2001.

    Google Scholar 

  18. Barsegian G. and Begehr H., A note on some simple and general inequalities in analysis, Preprint.

    Google Scholar 

  19. Barsegian G. and Begehr H., To applied topics with ideas and methods of value distribution and G-lines, Abstract in the 3rd ISAAC Congress, Berlin, 2001.

    Google Scholar 

  20. Barsegian G., Begehr H. and Laine I., Stability phenomenon for generalizations of algebraic differential equations, Z. Anal. Anwendungen 21 (2002), 495–503.

    MathSciNet  Google Scholar 

  21. Barsegian G., Csordas G. and Yang C.C., Interpolations by complex functions with a prescribed geometry of interpolating points from the point of view of the principle of closeness of α-points, Preprint.

    Google Scholar 

  22. Barsegian G., Laine I. and Yang C.C., Stability phenomenon and problems for complex differential equations with relations to shared values, Mat. Stud. 13 (2000), 224–228.

    MathSciNet  Google Scholar 

  23. Barsegian G., Laine I. and Yang C.C., Locations of values of meromorphic solutions of algebraic differential equations, Complex Variables Theory Appl. 46 (2001), 323–336.

    MathSciNet  Google Scholar 

  24. Barsegian G., Laine I. and Yang C.C., On a method of estimating derivatives in complex differential equations, J. Math. Soc. Japan 54 (2002), 923–935.

    MathSciNet  Google Scholar 

  25. Barsegian G. and Lê D.T., Gamma-lines and proximity property in real algebraic geometry, Preprint.

    Google Scholar 

  26. Barsegian G. and Lê D.T., On a topological property of multi-valued solutions of some general classes of complex differential equations, Preprint.

    Google Scholar 

  27. Barsegian G., Sarkisian A. and Yang C.C., A new trend in complex differential equations: quasimeromorphic solutions, Preprint ICTP 46 (2002).

    Google Scholar 

  28. Barsegian G. and Sukiasian G., Applications of the principle of partitioning of meromorphic functions. II. Accumulation of α-points and the Liouville property, Izv. Nats. Akad. Nauk Armenii Mat. 32 (1997), 5–11. (Russian)

    Google Scholar 

  29. Barsegian G. and Sukiasyan G., Methods for study level sets of enough smooth functions, I, Preprint.

    Google Scholar 

  30. Barsegian G. and Sukiasyan G., Methods for study level sets of enough smooth functions, II, Preprint.

    Google Scholar 

  31. Barsegian G. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications. I. On the magnitudes of functions on the sets of α-points of derivatives, Complex Variables Theory Appl. 41 (2000), 293–313.

    MathSciNet  Google Scholar 

  32. Barsegian G. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications. II. On the derivatives of meromorphic functions and on inverse meromorphic derivatives, Complex Variables Theory Appl. 44 (2001), 13–27.

    MathSciNet  Google Scholar 

  33. Beardon A. and Yang C.C. (Ed.), Unsolved problems from the International Workshop on Value Distribution Theory and its Applications, J. Contemp. Math. Anal. 32, No. 4 (1997), 9–36.

    Google Scholar 

  34. Bechenbach E., Cootz T., The second fundamental theorem for meromorphic minimal surfaces, Bull. Amer. Math. Soc. 76 (1976), 711–716.

    Google Scholar 

  35. Begehr H., Complex analytic methods for partial differential equations, World Scientific, Singapore-New Jersey-London-Hong Kong, 1994. x+273 pp.

    Google Scholar 

  36. Borwein P., The arc length of the lemniscate ¦p(z)¦=1, Proc. Amer. Math. Soc. 123 (1995), 797–799.

    MATH  MathSciNet  Google Scholar 

  37. Browder F. (Ed.), Mathematical developments arising from Hilbert problems, Proc. Symp. Pure Math. 28, Amer. Math Soc., Providence, R. I., 1976. xii+628 pp.

    Google Scholar 

  38. Coolidge J.L., A treatise on algebraic plane curves, Dover Publications, New York, 1959. xxiv+513 pp.

    Google Scholar 

  39. Daniel M. and Kruskal M.D., Lakshmanan M. and Nakamura K., Singularity structure analysis of the continuum Heisenberg spin chain with anisotropy and transverse field: nonintegrability and chaos, J. Math. Phys. 33 (1992), 771–776.

    Article  MathSciNet  Google Scholar 

  40. Dubrovin B. and Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55–147.

    MathSciNet  Google Scholar 

  41. Erdös P., Herzog F. and Piranian G., Metric properties of polynomials, J. Analyse Math. 6, (1958), 125–148.

    MathSciNet  Google Scholar 

  42. Eremenko A. and Hayman W., On the length of lemniscates, Michigan Math. J. 46 (1999), 409–415.

    MathSciNet  Google Scholar 

  43. Gackstatter F., Methoden der Werteverteilungslere bei Minimalflächen im R3, Rend. Girc. Mat. Palermo (2) 20 (1971), 277–291.

    MathSciNet  Google Scholar 

  44. Gackstatter F., Topics on minimal surfaces, Lecture Notes, University of Sao Paulo.

    Google Scholar 

  45. Goldberg A.A., On one-valued integrals of differential equations of the first order, Ukrain. Math. Ž. 8 (1956), 254–261. (Russian)

    MATH  Google Scholar 

  46. Golubev V.V., Lectures on the Analytic Theory of Differential Equations, Gosudarst. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950, 436 pp. (Russian)

    Google Scholar 

  47. Harnack A., Über die Vieltheiligkeit der ebenen algebraischen Curven, Clebsch Ann. 10 (1876), 189–199.

    MATH  MathSciNet  Google Scholar 

  48. Hayman W. and Wu J.M., Level sets of univalent functions, Comment. Math. Helv. 56 (1981), 366–403.

    MathSciNet  Google Scholar 

  49. Hilbert D., Mathematische Probleme, Arch. d. Math. u. Phys. 1 (1901), 44–63, 213–237.

    MATH  Google Scholar 

  50. Hilbert D., Gesammelte Abhandlungen, Band III, Springer-Verlag, Berlin-New York, 1970. vii+435 pp.

    Google Scholar 

  51. Katajamäki K., Algebroid solutions of binomial and linear differential equations, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 90 (1993), 48 pp.

    Google Scholar 

  52. Laine I., Nevanlinna theory and complex differential equation, Walter de Gruyter, Berlin, 1993. viii+341 pp.

    Google Scholar 

  53. Lê D.T., Depth and perversity, Algebraic geometry and analytic geometry (Tokyo, 1990), 111–125, ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991.

    Google Scholar 

  54. Lê D.T., Seade J. and Verjovsky A., Conies and involutions, to appear.

    Google Scholar 

  55. Levin B. Ja., Distribution of zeros of entire functions, Translations of Mathematical Monographs 5, Amer. Math. Soc., Providence, R.I., 1980. xii+523 pp.

    Google Scholar 

  56. Lewis J. and Wu J.-M., On conjectures of Arakelyan and Littlewood, J. Analyse Math. 50 (1988), 259–283.

    MathSciNet  Google Scholar 

  57. Littlewood J.E., On some conjectural inequalities, with applications to the theory of integral functions, J. London Math. Soc. 27 (1952), 387–393.

    MATH  MathSciNet  Google Scholar 

  58. Nevanlinna R., Eindeutige analytische Funktionen, Springer, Berlin, 1936. viii+353 pp.

    Google Scholar 

  59. Osserman R., A survey on minimal surfaces, Van Nostrand, New York-London-Melbourne, 1969. iv+159 pp.

    Google Scholar 

  60. Poincaré H., Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré, Palermo Rend. 5 (1891), 161–191.

    MATH  Google Scholar 

  61. Poincaré H., Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré, Palermo Rend. 11 (1897), 193–239.

    MATH  Google Scholar 

  62. Pommerenke Ch., Einige Sätze über die Kapazität ebener Mengen, Math. Ann. 141 (1960), 143–152.

    Article  MATH  MathSciNet  Google Scholar 

  63. Pommerenke Ch., On metric properties of complex polynomials, Michigan Math. J. 8 (1961), 97–115.

    MATH  MathSciNet  Google Scholar 

  64. Selberg H., Algebroide Funktionen und Umkehrfunktionen Abelscher Integrale, Avh. Norske Vid. Akad. Oslo, 1934, No. 8, 1–72.

    Google Scholar 

  65. Steinmetz N., Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes, J. Reine Angew. Math. 368 (1986), 134–141.

    MATH  MathSciNet  Google Scholar 

  66. Valiron G., Directions de Borel des fonctions méromorphes, Mem. Sci. Math. 89, Gauthier-Villars, Paris, 1938, 70 pp.

    Google Scholar 

  67. Valiron G., Fonctions analytiques, Presses Universitaires de France, Paris, 1954, 236 pp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics of National Academy of Sciences of Armenia, 24-b Marshal Bagramian ave., Yerevan, 375019, Armenia

    G. Barsegian

Authors
  1. G. Barsegian
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. National Academy of Sciences of Armenia, Yerevan, Armenia

    G. Barsegian

  2. University of Joensuu, Joensuu, Finland

    I. Laine

  3. Hong Kong University of Science and Technology, Hong Kong, China

    C. C. Yang

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Kluwer Academic Publishers

About this chapter

Cite this chapter

Barsegian, G. (2004). A New Program of Investigations in Analysis: Gamma-Lines Approaches. In: Barsegian, G., Laine, I., Yang, C.C. (eds) Value Distribution Theory and Related Topics. Advances in Complex Analysis and Its Applications, vol 3. Springer, Boston, MA. https://doi.org/10.1007/1-4020-7951-6_1

Download citation

  • DOI: https://doi.org/10.1007/1-4020-7951-6_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4020-7950-4

  • Online ISBN: 978-1-4020-7951-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation