The modularity of the partition-generating function has many important consequences: for example, asymptotics and congruences for $p(n)$. In a pair of articles, Bringmann and Ono [11], [12] connected the rank, a partition statistic introduced by Dyson [18], to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in [14]. Here, we take a further step toward understanding how weak Maass forms arise from interesting partition statistics by placing certain $2$-marked Durfee symbols introduced by Andrews [1] into the framework of weak Maass forms. To do this, we construct a new class of functions that we call quasi-weak Maass forms because they have quasi-modular forms as components. As an application, we prove two conjectures of Andrews [1, Conjectures 11, 13]. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves and also their derivatives. As a side product, we introduce a new method that enables us to prove transformation laws for generating functions over incomplete lattices

The aim of this article is twofold. First, we show that the $C^0$-limit of a pair of commuting Hamiltonians commutes. This means, on the one hand, that if the limit of the Hamiltonians is smooth, the Poisson bracket of their limit still vanishes and, on the other hand, that we may define “commutation” for $C^0$-functions. The second part of this article deals with solving multi-time Hamilton-Jacobi equations using variational solutions. This extends the work of Barles and Tourin [BT] in the viscosity case to include the case of $C^0$-Hamiltonians, and it removes their convexity assumption, provided that we work in the setting of variational solutions

We prove a polynomial upper bound on the deviation of ergodic averages for almost all directional flows on every translation surface, in particular, for the generic directional flow of billiards in any Euclidean polygon with rational angles

Let ${\pi }_1$, ${\pi }_2$ be cuspidal automorphic representations of ${\mathrm{PGL}}_2(\mathbb{R})$ of conductor $1$ and Hecke eigenvalues ${\lambda }_{{\pi }_{1,2}}(n)$, and let $h>0$ be an integer. For any smooth compactly supported weight functions $W_{1,2}:{\mathbb{R}}^{\times }\to \mathbb{C}$ and any $Y>0$, a spectral decomposition of the shifted convolution sum ${\displaystyle \sum _{m\pm n=h}\frac{{\lambda }_{{\pi }_1}(|m|){\lambda }_{{\pi }_2}(|n|)}{\sqrt{\left|\mathrm{mn}\right|}}{W}_1(\frac{m}{Y})W_2(\frac{n}{Y})}$ is obtained. As an application, a spectral decomposition of the Dirichlet series ${\displaystyle \sum _{\begin{array}{l}m,n\ge 1\\ m-n=h\end{array}}\frac{{\lambda }_{{\pi }_1}(m){\lambda }_{{\pi }_2}(n)}{(m+n{)}^s}(\frac{\sqrt{\mathrm{mn}}}{m+n}{)}^{100}}$ is proved for $\mathfrak{R}s>1/2$ with polynomial growth on vertical lines in the $s$-aspect and uniformity in the $h$-aspect

Let $G^{+}$ be the group of real points of a possibly disconnected linear reductive algebraic group defined over $\mathbb{R}$ which is generated by the real points of a connected component $G^{\text{'}}$. Let $K$ be a maximal compact subgroup of the group of real points of the identity component of this algebraic group. We characterize the space of maps $\pi \mapsto \mathrm{tr}(\pi (f))$, where $\pi$ is an irreducible tempered representation of $G^{+}$ and $f$ varies over the space of smooth, compactly supported functions on $G^{\text{'}}$ which are left and right $K$-finite. This work is motivated by applications to the twisted Arthur-Selberg trace formula