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.

We establish results related to homogeneous derivations, a concept introduced by Kanunnikov (2018). First, we prove the existence of a non-trivial family of derivations that are not homogeneous on graded rings. Furthermore, based on homogeneous derivations, we extend certain existing significant results in the context of prime (resp. semiprime) rings to gr-prime (resp. gr-semiprime) rings, such as Posner's and Herstein's theorems.

We prove that $\operatorname{vol}(S^{d})/8$ is the highest volume of a pair of $d$-dimensional isospectral and non-isometric spherical orbifolds for any $d\geq5$. Furthermore, we show that $\operatorname{vol}(S^{2n-1})/11$ is the highest volume of a pair of $(2n-1)$-dimensional isospectral and non-isometric spherical space forms if either $n\geq11$ and $n\equiv 1\pmod 5$, or $n\geq7$ and $n\equiv 2\pmod 5$, or $n\geq3$ and $n\equiv 3\pmod 5$.

We investigate the homothety conjecture for convex bodies of flotation of planar domains close to the unit disk $B$. We show that for every density $\mathcal{D}\in (0,\frac{1}{2})$, there exists $\gamma=\gamma(\mathcal{D})>0$ such that if $(1-\gamma)B\subset K\subset (1+\gamma)B$ and the convex body of flotation $K^\mathcal{D}$ of an origin-symmetric body $K$ of density $\mathcal{D}$ is homothetic to $K$, then $K$ is an ellipse. On the other hand, we also show that if the symmetry assumption is dropped, then there is an infinite set of densities accumulating at $\frac{1}{2}$ for which there is a body $K$ different from an ellipse with the property that $K^{\mathcal{D}}$ is homothetic to $K$.

Given any ring $R$ with unity $1$ and any $a, b, c \in R$, $a$ is called $(b,c)$-polar if there exist two idempotents $p, q \in R$ such that $p \in bRca$, $q \in abRc$, $pb = b$, $cq = c$, $cap = ca$ and $qab = ab$. These $p$ and $q$ are shown to be unique whenever they exist. The existence of $a^{\|(b,c)}$, the $(b,c)$-inverse of $a$, is shown to be equivalent to $a$ being $(b,c)$-polar, and hence $a^{\|(b,c)}$ is itself unique and expressed in terms of $p$ and $q$. Generalizing results of Koliha–Patrício and Song–Zhu–Mosić, further connections between the $(b,c)$-polar and $(b,c)$-invertible properties are found. Applying these results to bounded linear operators on a Banach space, we also generalize some known results in this setting.

This paper investigates a way to define left and right versions of the class of $G$-Drazin inverses for complex rectangular matrices. More precisely, the concepts of $W$-weighted left and right $G$-Drazin inverses are introduced and characterized by means of a simultaneous core-nilpotent decomposition as well as by a certain system of matrix equations. Then new partial orders associated with these weighted generalized inverses are presented and studied.

We prove that every nontrivial additive relation between algebraic conjugates of degree $d$ over $\mathbb{Q}$ has a corresponding multiplicative relation. The proof is constructive. The reverse statement is not true. These findings supplement the research of Smyth, Dixon, Girstmair and others. In addition, following a result of Kitaoka we show that all additive relations between four distinct algebraic conjugates of degree 4 over $\mathbb{Q}$ can be described as $\mathbb{Z}$-linear combinations of several basic nontrivial relations. On the other hand, we prove that an analogous result no longer holds for algebraic conjugates of degree 6 over $\mathbb{Q}$.

The aim of this paper is to show that the ultrapower $^\ast\mathbb{R}$ of the real line $\mathbb{R}$ with respect to a selective ultrafilter on the natural numbers (Choquet's absolute ultrafilter) can be naturally embedded in the prime spectrum of the usual topology on $\mathbb{R}$, viewed as a distributive lattice. Moreover, the topology induced on $^\ast\mathbb{R} \setminus \mathbb{R}$ through this embedding is separated (Hausdorff).

Let $q,p,k, m$ be positive integers with $m\geq k+p +1$, and let $A$ be a Weil algebra with $k$ generators. The (not necessarily regular) natural operators lifting $q$-forms on an $m$-dimensional manifold $M$ to $p$-forms on the Weil bundle $T^AM$ are completely described by means of the so-called excellent maps. As a consequence, we show that any natural operator lifting $q$-forms on an $m$-dimensional manifold $M$ to $p$-forms on $T^AM$ is regular, i.e., it sends smoothly parametrized families into smoothly parametrized families. We apply our general results to the case where $T^A$ is the $r$th- order tangent bundle $T^rM=J^r_0(\mathbf{R},M)$ (in particular, the tangent bundle).

We introduce the notion of the $m$-weak group inverse of a complex square matrix relative to an invertible Hermitian weight $E$. Several equivalent algebraic characterizations of this new type of generalized inverse are shown. In addition, various properties, representations, and characterizations involving the column space, null space, rank equalities, and projectors are presented. Finally, we use this new generalized inverse to solve a linear equation.