The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

On two upper bounds for hypersurfaces involving a Thas’ invariant

Let $X^n$ be a hypersurface in ${\mathbb{P}}^{n+1}$ with $n\ge 1$ defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of ${\mathbb{F}}_q$-points on $X^n$ which involve a Thas’ invariant.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

Factorization of a class of polynomials over finite fields

We study the factorization of polynomials of the form $F_r(x)=b{x}^{q^r+1}?a{x}^{q^r}+dx?c$ over the finite field ${\mathbb{F}}_q$. We show that these polynomials are closely related to a natural action of the projective linear group $\mathrm{PGL}(2,q)$ on non-linear irreducible polynomials over ${\mathbb{F}}_q$. Namely, irreducible factors of $F_r(x)$ are exactly those polynomials that are invariant under the action of some non-trivial element $[A]\in \mathrm{PGL}(2,q)$. This connection enables us to enumerate irreducibles which are invariant under $[A]$. Since the class of polynomials $F_r(x)$ includes some interesting polynomials like $x^{q^r}?x$ or $x^{q^r+1}?1$, our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over ${\mathbb{F}}_q$. At the same time, we generalize recent results about certain invariant polynomials over the binary field ${\mathbb{F}}_2$.