results for au:Nien_C in:math

- Sep 01 2017 math.DS arXiv:1708.09506v1In recent work [Nien et al. 2016], the authors enumerated a classification of quadratic maps of the plane according to their critical sets and images. It is straightforward to show that quadratic maps which are affinely map equivalent are also equivalent in the critical set classification. The question remained whether maps that are equivalent in the critical set classification are also affinely map equivalent. This paper establishes a complete enumeration of the affine map equivalence classes. As a consequence, the relationship between affine map equivalence and critical set equivalence is established. In short, there are eighteen affine map equivalence classes. Three pairs of those classes have critical sets and images that match, but each pair has some other geometric property, preserved by affine map equivalence, that does not match. The other twelve affine map equivalence classes and the critical set equivalences are in one-to-one correspondence.
- Jul 13 2015 math.DS arXiv:1507.02732v1We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose critical sets are either ellipses or hyperbolas, was published in Delgado, et al. (2013). This work completes the classification by including all the nongeneric cases: the empty set, a single point, a single line, a parabola, two parallel lines, two intersecting lines, or the whole plane. We describe all possible images for each critical set case and illustrate the geometry of representative maps for each case.
- The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.