Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

The state of the second part of Hilbert's fifth problem


Author: J. Aczél
Journal: Bull. Amer. Math. Soc. 20 (1989), 153-163
MSC (1985): Primary 39B30; Secondary 01A55, 01A60, 01A65, 39B10, 39B20, 39B40, 39B70
DOI: https://doi.org/10.1090/S0273-0979-1989-15749-2
MathSciNet review: 981872
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • N. H. Abel (1823), Méthode générale pour trouver des fonctions d'une seule quantité variable lorsqu'une propriété des fonctions est exprimée par une équation entre deux variables, (Norwegian). Mag. Naturvidenskab. 1 (1823), No. 1, 1-10 (Oeuvres complètes, tome I. Grøndahl & Søn, Christiania, 1881, pp. 1-10).
  • N. H. Abel (1824), Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable, Manuscript, Christiania, c. 1824. (Oeuvres complètes, tome II. Grøndahl & Søn, Christiania, 1881, pp. 36-39).
  • N. H. Abel (1826A), Untersuchung der Functionen zweier unabhängigen veränderlichen Grössen x and y, wie f(x, y), welche die Eigenschaft haben, daβ f[z, f(x, y)] eine symmetrische Function von x, y und z ist, J. Reine Angew. Math. 1 (1826), 11-15. (Oeuvres complètes, tome I. Grøndahl & Søn, Christiania, 1881, pp. 61-65).
  • N. H. Abel (1826B), Unterschungen über die Reihe $ 1 + \frac m{1}x + \frac{m \cdot(m-1)}{1\cdot 2}\cdot x^2 + \frac{m \cdot (m-1)\cdot (m-2)}{1\cdot 2 \cdot 3}\cdot x^3 + \cdot\cdot\cdot\cdot\cdot\cdot$ , J. Reine Angew. Math. 1 (1826), 311-339. (Oeuvres complètes, tome I. Grøndahl & Søn, Christiania, 1881, pp. 219-250).
  • N. H. Abel (1826C), Note sur la fonction $\psi \chi = \chi + \frac x^2 {2^2] + \cdot\cdot\cdot + \frac x^n {n^2} + \cdot\cdot\cdot$, Manuscript, Freiberg, 1826. (Oeuvres complètes, tome II. Grøndahl & Søn, Christiania, 1881, pp. 189-193).
  • N. H. Abel (1827), Über die Functionen, die der Gleichung $\varphi x+\varphi y=\psi(xfy+yfx)$ genugthun, J. Reine Angew. Math. 2 (1827), 386-394. (Oeuvres complètes, tome I. Grøndahl & Søn, Christiania, 1881, pp. 389-398).
  • N. H. Abel (1828), Manuscript, Christiania, c. 1828. (Oeuvres complètes, tome II. Grøndahl & Søn, Christiania, 1881, pp. 287, 318-319).
  • J. Aczél (1949), Sur les opérations définies pour des nombres réels, Bull. Soc. Math. France 76 (1949), 59-64. MR 29946
  • J. Aczél (1961), Über die Differenzierbarkeit der integrierbaren Lösungen gewisser Funktionalgleichungen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3-4 (1960-1961), 5-8. MR 152770
  • J. Aczél (1966), Lectures on functional equations and their applications, Academic Press, New York-London, 1966. MR 208210
  • J. Aczél (1987), A short course on functional equations, based upon recent applications to the social and behavioral sciences, Reidel, Dordrecht-Boston-Lancaster-Tokyo, 1987. MR 875412
  • J. Aczél and J. K. Chung (1982), Integrable solutions of functional equations of a general type, Studia Sci. Math. Hungar. 17 (1982), 51-67. MR 761524
  • J. Aczél and J. Dhombres (1989), Functional equations in several variables with applications to mathematics, information theory and to the natural and social sciences, Cambridge Univ. Press, Cambridge-New York-New Rochelle-Melbourne-Sydney, 1989. MR 1004465
  • J. Aczél and A. D. Wallace (1967), A note on generalizations of transivitive systems of transformations, Colloq. Math. 17 (1967), 29-34. MR 217778
  • J. Andrade (1900), Sur l'équation fonctionnelle de Poisson, Bull. Soc. Math. France 28, (1900), 58-63.
  • W. Benz (1987), Ein Beitrag zu einem Problem von Herrn Fenyö, Abh. Math. Sem. Univ. Hamburg 57 (1987), 21-25. MR 927160
  • A. L. Cauchy (1821), Analyse algébrique, Paris, 1821. (Oeuvres, Sér. 2, tome 3. Gauthier-Villars, Paris, 1897).
  • J. K. Chung, PI. Kannappan and C. T. Ng (1985), A generalization of the cosine-sine functional equation on groups, Linear Algebra Appl. 66 (1985), 259-277. MR 781306
  • Z. Daróczy and H. Kiesewetter (1973), Eine Funktionalgleichung von Abel und die Grundgleichung der Information, Period. Math. Hungar. 4 (1973), 25-28. MR 329766
  • L. Fuchs (1963), Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris, 1963. MR 171864
  • H. Haruki (1965), Studies on certain functional equations from the standpoint of analytic function theory, Sci. Rep. Osaka Univ. 14 (1965), 1-40.
  • D. Hilbert (1902), Mathematical problems, Lecture delivered before the International Congress of Mathematicians at Paris in 1900, Bull. Amer. Math. Soc. 8 (1902), 437-479.
  • A. Járai (1986), On regular solutions of functional equations, Aequationes Math. 30 (1986), 21-54. MR 837038
  • M. Kac (1937), Une remarque sur les équations fonctionnelles, Comment. Math. Helv. 9 (1937), 170-171.
  • PI. Kannappan (1977, 1978), On Shannon's entropy and a functional equation of Abel, Period. Math. Hungar. 8 (1977), 41-44; Addendum, Period. Math. Hungar. 9 (1978), 297-298.
  • H. Kiesewetter (1957), Eine Bemerkung über partielle Differentiationen bei N. H. Abel, Publ. Math. Debrecen 5 (1957), 265-268. MR 118784
  • H. Kiesewetter (1961), Struktur linearer Funktionalgleichungen in Zusammenhang mit dem Abelschen Theorem, J. Reine Angew. Math. 206 (1961), 113-171. MR 131082
  • H. Kiesewetter (1965), Über die arc tan - Funktionalgleichung, ihre mehrdeutigen, stetigen Lösungen und eine nichtstetige Gruppe. Wiss. Z. Friedrich-Schiller-Univ. Jena Math.-Natur. Reihe 14 (1965), 417-421. MR 234149
  • J. F. C. Kingman (1966), On the algebra of queues, J. Appl. Prob. 3 (1966), 285-326. MR 203831
  • M. Kuczma (1968), Functional equations in a single variable, Polish Scientific Publishers, Warszawa, 1968. MR 228862
  • M. Kuczma, B. Choczewski and R. Ger (1989), Iterative functional equations, Cambridge Univ. Press, Cambridge-New York-New Rochelle-Melbourne-Sydney, 1989. MR 1067720
  • K. Lajkó (1987), Remark on the Hosszú functional equation, Wyz. Szkola Ped. Krakow, Rocznik Nauk-Dydakt. Prace Mat. 12 (1987), 192-193.
  • P. M. Lee (1964), On the axioms of information theory, Ann. Math. Statist. 35 (1964), 415-418. MR 158767
  • L. Losonczi (1985), Remark 32: The general solution of the arc tan equation, Proc. Twenty-third Internat. Sympos. on Functional Equations (Gargnano, Italy, June 2-11, 1985). Univ of Waterloo, Centre for Information Theory, Waterloo, Ont., 1985, pp. 74-76.
  • J. B. Miller (1969), Baxter operators and endomorphisms on Banach algebras, J. Math. Anal. Appl. 25 (1969), 503-520. MR 239458
  • Zs. Páles and R. W. Craigen (1989), The associativity equation revisited, Aequationes Math, (to appear). MR 1004503
  • F. Radó and J. A. Baker, (1987), Pexider's equation and aggregation of allocations, Aequationes Math. 32 (1987), 227-239. MR 900703
  • M. Sablik (1989), The continuous solutions of a functional equation of Abel, (Contribution to Hilbert's Fifth Problem), Aequationes Math, (submitted). MR 1044162
  • P. Volkmann and H. Weigel (1984), Über ein Problem von Fenyö, Aequationes Math. 27 (1984), 135-149. MR 758864
  • M. C. Zdun (1977), On integrable solutions of Abel's functional equation, Glas. Mat. Ser. III 12(32) (1977), 49-59. MR 470547

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 39B30, 01A55, 01A60, 01A65, 39B10, 39B20, 39B40, 39B70

Retrieve articles in all journals with MSC (1985): 39B30, 01A55, 01A60, 01A65, 39B10, 39B20, 39B40, 39B70


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1989-15749-2

American Mathematical Society