Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 9
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Wu, Faen Xiong, Yueshan and Zhao, Xinnuan 2015. Classification of Quadratic Harmonic Maps of $$S^{7}$$ S 7 into $$S^{7}$$ S 7. The Journal of Geometric Analysis, Vol. 25, Issue. 3, p. 1992.

    Rodríguez-Ordóñez, Hugo 2007. A note on the fundamental theorem of algebra for the octonions. Expositiones Mathematicae, Vol. 25, Issue. 4, p. 355.

    Тиморин, Владлен Анатольевич Timorin, Vladlen Anatol'evich Тиморин, Владлен Анатольевич and Timorin, Vladlen Anatol'evich 2004. Окружности и алгебры Клиффорда. Функциональный анализ и его приложения, Vol. 38, Issue. 1, p. 56.

    TANG, ZIZHOU 2001. NEW CONSTRUCTIONS OF EIGENMAPS BETWEEN SPHERES. International Journal of Mathematics, Vol. 12, Issue. 03, p. 277.

    2000. Compositions of Quadratic Forms.

    Yiu, Paul 1994. Quadratic forms between euclidean spheres. Manuscripta Mathematica, Vol. 83, Issue. 1, p. 171.

    Alarcon, Ignacio and Yiu, Paul 1993. Compositions of hermitian forms. Linear and Multilinear Algebra, Vol. 36, Issue. 2, p. 141.

    Lam, K. Y. and Yiu, P. Y. H. 1989. Geometry of normed bilinear maps and the 16-square problem. Mathematische Annalen, Vol. 284, Issue. 3, p. 437.

    Tang, Zizhou 1989. Harmonic polynomial morphisms between Euclidean spaces. Science in China Series A: Mathematics, Vol. 42, Issue. 6, p. 570.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 3
  • November 1986, pp. 493-504

Quadratic forms between spheres and the non-existence of sums of squares formulae

  • Paul Y. H. Yiu (a1)
  • DOI:
  • Published online: 24 October 2008

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the type

where are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]J. F. Adams . On the nonexistence of elements of Hopf invariant one. Ann. of Math. 72 (1960), 20104.

[3]G. Al-Sabti and T. Bier . Elements in the stable homotopy groups of spheres which are not bilinearly representable, Bull. London Math. Soc. 10 (1978), 197200.

[5]H. Hopf . Ein topologischer Beitrag zur reellen Algebra. Comment. Math. Helv. 13 (1941), 219239.

[7]A. Hurwitz . Über die Komposition der quadratischen Formen. Math. Ann. 88 (1923), 125; reprinted in Math. Werke Bd. 2, pp. 641–666.

[8]K. Y. Lam . Construction of nonsingular bilinear maps. Topology 6 (1967), 423426.

[12]K. Y. Lam . Some new results in composition of quadratic forms. Invent. Math. 79 (1985), 467474.

[13]R. J. Milgram . Immersing protective spaces. Ann. Math. 85 (1967), 473482.

[15]G. F. Paechter . The groups . Quart. J. Math. Oxford7 (1956), 249268.

[16]J. Radon . Lineare Scharen orthogonalen Matrizen. Abh. Math. Sem. Univ. Hamburg1 (1922), 114.

[17]J. Roitberg . Dilatation phenomena in the homotopy groups of spheres. Advances in Math. 15 (1975), 198206.

[19]E. Stiefel . Üjer Richtungsfelder in den projektiven Raumen und einen Satz aus reellen Algebra. Comment. Math. Helv. 13 (1941), 201218.

[21]R. Wood . Polynomial maps from spheres to spheres. Invent. Math. 5 (1968), 163168.

[24]S. Yuzvinsky . A series of monomial pairings. Linear and Multilinear Algebra 15 (1984), 109119.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *