We determine the roots in F q 3 of the polynomial X 2q +1 + X + a for each positive integer and each a ∈ Fq, where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more explicit than the previous results in this area. Our results resolve an open problem and a conjecture of Zheng,

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A novel algorithm called orthogonal discriminant local tangent space alignment (O-DLTSA) is proposed for supervised feature extraction. Derived from local tangent space alignment (LTSA), O-DLTSA not only inherits the advantages of LTSA which uses local tangent space as a representation of the local geometry so as to preserve the local structure, but also makes full use of class information and orthogonal subspace to improve discriminant power. The experimental results of applying O-DLTSA to standard face databases demonstrate the effectiveness of the proposed method.

For each odd prime power $q$, we construct an infinite sequence of rational functions $f(X) \in{\mathbb{F}}_q(X)$, each of which is exceptional in the sense that for infinitely many $n$ the map $c \mapsto f(c)$ induces a bijection of ${\mathbb{P}}^1({\mathbb{F}}_{q^n})$. Moreover, each of our functions $f(X)$ is indecomposable in the sense that it cannot be written as the composition of lower-degree rational functions in ${\mathbb{F}}_q(X)$. These are the first known examples of wildly ramified indecomposable exceptional rational functions $f(X)$, other than linear changes of polynomials. In case $q$ is not a power of $3$, these are also the first known examples of indecomposable exceptional rational functions $f(X)$ over ${\mathbb{F}}_q$ which have non-solvable monodromy groups and have arbitrarily large degree.

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