We define a Schwarz fractal derivative of order $n$ for a real-valued function $f(t)$ as the limit \[ (\mathrm{S}^{\alpha,\beta}_nf)(t_0)=\lim_{t\to t_0} \frac{\sum_{j=0}^n\binom{n}{j}(-1)^jf^\beta\bigl(t_0+\frac{n-2j}{2}(t-t_0)\bigr)}{(t^\alpha-t_0^\alpha)^n}, \] where $\alpha,\beta>0$ and $f^\beta:=f|f|^{\beta-1}$. This derivative naturally generalizes the one introduced by Riemann in 1854. We study its essential properties and its relationship with other fractal derivatives recently reported in the literature. We obtain certain analogues of the mean value and Rolle theorems, together with some of their most important consequences. Finally, we propose an extension of such derivatives to the several-variable setting.

We prove stability in weighted $W^{1,1}$ spaces for standard finite element approximations of the Poisson equation in convex polygonal or polyhedral domains, when the weight belongs to Muckenhoupt's class $A_1$ and the family of meshes is quasi-uniform.

We show that, contrary to the behavior of the higher-order Riesz transforms studied so far on the atomic Hardy space $\mathcal{H}^1(\mathbb{R}^n, \gamma)$ associated with the Ornstein–Uhlenbeck operator with respect to the $n$-dimensional Gaussian measure $\gamma$, the new Gaussian Riesz transforms are bounded from $\mathcal{H}^1(\mathbb{R}^n, \gamma)$ to $L^1(\mathbb{R}^n, \gamma)$, for any order and any dimension $n$. We also prove that the classical Gaussian Riesz transforms of higher order are bounded from an appropriate subspace of $\mathcal{H}^1(\mathbb{R}^n, \gamma)$ into $L^1(\mathbb{R}^n, \gamma)$, extending T. Bruno (2019) to the first-order case.

The zero forcing number, defined as the minimum size of a zero forcing set, serves as an upper bound for the graph nullity. An expanded path $P_{m_1, m_2, \ldots, m_k}$ (resp. expanded cycle $C_{m_1, m_2, \ldots, m_k}$) is obtained from the $k$-vertex path (resp. cycle) by replacing its $i$th vertex with an independent set of $m_i$ vertices. We show that the zero forcing number of $P_{m_1, m_2, \ldots, m_k}$ (resp. $C_{m_1, m_2, \ldots, m_k}$) belongs to $\{n-k, n-k+1\}$ (resp. $\{n-k+1, n-k+2\}$), where $n$ is the number of vertices, and determine when it equals $n-k+1$. As an application, we provide a new proof of a result of Liang, Li, and Xu characterizing triangle-free graphs with zero forcing number $n-3$. We also show that for any cycle-spliced graph (i.e., a connected graph all of whose blocks are cycles), the zero forcing number equals $c+1$, where $c$ is the cyclomatic number. This gives an upper bound for the nullity and extends a result of Wong, Zhou, and Tian for the bipartite case.