| 41. | Then choosing any smooth positive quadratic form on H gives a sub-Riemannian metric on the group.
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| 42. | In 1986, Margulis gave a complete resolution of the Oppenheim conjecture on quadratic forms and diophantine approximation.
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| 43. | Binary quadratic forms were considered already by Fermat, in particular, in the question of Eisenstein ).
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| 44. | That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.
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| 45. | For quadratic forms over a number field, there is a Hasse invariant ? for every finite place.
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| 46. | Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.
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| 47. | Then the Arf invariant of this quadratic form can be used to distinguish the two extra special groups.
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| 48. | With additional structure, this ?-symmetric form can be refined to an ?-quadratic form.
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| 49. | For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant.
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| 50. | :: : The quadratic form can be determined from the fixed points of the order three elements.
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