In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$-type groups. We first prove that, on $H$-type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$-type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3)95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc.142 (2014), 2101–2118].

We determine, up to multiplicative constants, the number of integers $n\leq x$ that have a divisor in $(y,2y]$ and no prime factor $\leq w$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table, which are free of prime factors $\leq w$, and the number of distinct fractions of the form $(a_{1}a_{2})/(b_{1}b_{2})$ with $1\leq a_{1}\leq b_{1}\leq N$ and $1\leq a_{2}\leq b_{2}\leq N$.

If $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ is a covering map between connected graphs, and $H$ is the subgroup of $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)/H$, where $N(H)$ is the normalizer of $H$ in $\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$. We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\unicode[STIX]{x1D6E4},v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ may be extended to a cover $g:\tilde{\unicode[STIX]{x1D6E5}}\rightarrow \unicode[STIX]{x1D6E4}$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.

In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$-algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$-Drazin invertible elements of a $C^{\ast }$-algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$-rings).

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$, was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$, it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve.

Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$, where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$-free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$, $(a,q)=1$, uniformly on $q$. When $j\geq 2$, we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.