Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$ . Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.

Suppose G and H are bipartite graphs and $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph of H induced between $L(v)$ and $L(v')$ is a matching, whenever $vv'\in E(G)$ . We show for each $\varepsilon>0$ that if H has maximum degree D and $|L(v)| \ge (1+\varepsilon )D/\log D$ for all $v\in V(G)$ , then H admits an independent transversal with respect to L, provided D is sufficiently large. This bound on the part sizes is asymptotically sharp up to a factor $2$ . We also show some asymmetric variants of this result.

By a tournament Tn on n vertices, we shall mean a directed graph on n vertices for which every pair of distinct vertices form the endpoints of exactly one directed edge (e.g., see [5]). If x and y are vertices of Tn we say that x dominates y if the edge between x and y is directed from x to y. In 1962, K. Schütte [2] raised the following question: Given k > 0, is there a tournament Tn(k) such that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. (Such a tournament will be said to have property Pk.)

Romyar Sharifi has constructed a map $\varpi _M$ from the first homology of the modular curve $X_1(M)$ to the K-group $K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M+\zeta _M^{-1}, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[1/2]$, where $\zeta _M$ is a primitive Mth root of unity. Sharifi conjectured that $\varpi _M$ is annihilated by a certain Eisenstein ideal. Fukaya and Kato proved this conjecture after tensoring with $\operatorname {\mathrm {\mathbf {Z}}}_p$ for a prime $p\geq 5$ dividing M. More recently, Sharifi and Venkatesh proved the conjecture for Hecke operators away from M. In this note, we prove two main results. First, we give a relation between $\varpi _M$ and $\varpi _{M'}$ when $M' \mid M$. Our method relies on the techniques developed by Sharifi and Venkatesh. We then use this result in combination with results of Fukaya and Kato in order to get the Eisenstein property of $\varpi _M$ for Hecke operators of index dividing M.

Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$ , and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$ . In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$ , the Moran measure

$$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$

$(R,D)$

In a recent note in the Bulletin, Murphy [5] gave a short proof that for complex m×n matrices A and B, r(A+B)=r(A)+r(B) if the rows of A are orthogonal to the rows of B and the columns of A are orthogonal to the columns of B. His proof was elegant and simple, an improvement on an earlier proof of the same result by Meyer [4].

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.