Online first articles

1. Adaptive Density Estimation Using Projection Kernels and Penalized Comparison to Overfitting Van Ha Hoang , Tien Dat Nguyen , Thi Mong Ngoc Nguyen
   ▸ Abstract

   In this work, we study wavelet projection estimators for density estimation, based on compactly supported $\mathcal {S}$-regular scaling functions. The main issue is the choice of the resolution level, which determines the bias–variance trade-off. We select this level by a Penalized Comparison to Overfitting (PCO) criterion: each candidate estimator is compared in $\mathbb {L}^2(\mathbb {R})$ with a single overfitting reference, and the additional stochastic fluctuation is corrected by an explicit penalty. For the selected estimator, we prove a high-probability oracle inequality and a risk oracle inequality in expectation. Over Besov balls of densities satisfying a common $\mathbb {L}^\infty$ bound, the risk inequality yields the rate uniformly over the class, with a sufficient penalty threshold that can be chosen uniformly over the class. This rate has the classical minimax order, and the procedure adapts to the unknown Besov regularity. Numerical experiments on several density shapes illustrate that the data-driven level is close to the oracle one and adapts to the structure of the target.

2. Regularity Index of Embedded Fat Points Phan Van Thien , Phan Quang Nhu Anh
   ▸ Abstract

   Let $Z=m_1P_1+\cdots +m_sP_s$ be a set of fat points in $\mathbb{P}^n$. Let $\phi$ be a linear embedding of $\mathbb{P}^n$ in $\mathbb{P}^m,$ $ n < m$. We prove that the regularity index of $Z$ and the regularity index of $\phi(Z)$ are the same.

3. Generalized Kloosterman Sheaves Nicholas M. Katz , Pham Huu Tiep
   ▸ Abstract

   We determine in many (but not all) cases the geometric monodromy group $G_\textrm{geom}$ of the one-parameter local system $\mathcal {G}(f,a,\chi )$ on $\mathbb {G}_m$ in characteristic $p >0$, whose trace function (at $t \in L^\times$ for large enough finite extensions L of the field $\mathbb {F}_p$) is $$\begin{aligned} t \in L^\times \mapsto -\sum _{x \in L^\times }\psi (f(x)+t/x^a)\chi (x), \end{aligned}$$ with $\psi$ a (nontrivial) additive character, f(x) a polynomial of degree $A \ge 1$, $a \ge 1$, and $\chi$ a (possibly trivial) multiplicative character. These local systems are generalizations of Kloosterman sheaves.

4. Lê Modules and Hypersurfaces with One-dimensional Singular Sets David B. Massey
   ▸ Abstract

   By using our previous results on Lê modules and an upper-bound on the betti numbers which we proved with Lê, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces with one-dimensional singular sets and small Lê numbers.