On an invariant related to a linear inequality Academic Article uri icon

abstract

  • Let\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } = (\alpha _1 ,\alpha _2 , \ldots \alpha _m ) \in \mathbb{R}_{ > 0}^m \). Let\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } _{i,j} \) be the vector obtained from\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } \) on deleting the entries αi and αj. We investigate some invariants and near invariants related to the solutions\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \in } \in \{ \pm 1\} ^{m - 2} \) of the linear inequality $$\left| {\alpha _i - \alpha _j } \right|< \left\langle {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \in } ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } _{i,j} } \right\rangle< \alpha _i + \alpha _j ,$$ , where denotes the usual inner product. One of our methods relates, by the use of Rademacher functions, integrals involving products of trigonometric functions to these quantities.

publication date

  • January 1, 2002