Let $X$ and $Y$ be Banach spaces and let $f\,:\,X\,\to \,Y$ be an odd mapping. For any rational number $r\,\ne \,2$ , C. Baak, D. H. Boo, and Th. M. Rassias proved the Hyers–Ulam stability of the functional equation

$$rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{x}_{j}}}}{r} \right)\,+\,\sum\limits_{\begin{smallmatrix}i\left( j \right)\,\in \left\{ 0,\,1 \right\} \\\sum\nolimits_{j=1}^{d}{i\left( j \right)=\ell } \end{smallmatrix}}{rf\left( \frac{\sum\nolimits_{j=1}^{d}{{{\left( -1 \right)}^{i\left( j \right)}}{{x}_{j}}}}{r} \right)}\,=\,\left( C_{d-1}^{\ell }\,-\,C_{d-1}^{\ell -1}\,+\,1 \right)\,\sum\limits_{j=1}^{d}{f\left( {{x}_{j}} \right),}$$

where $d$ and $\ell$ are positive integers so that $1\,<\,\ell \,<\,\frac{d}{2}$ , and $C_{q}^{p}\,:=\frac{q!}{\left( q-p \right)!p!},\,p,\,q\,\in \,\mathbb{N}$ with $p\,\le \,q$ .

In this note we solve this equation for arbitrary nonzero scalar $r$ and show that it is actuallyHyers–Ulam stable. We thus extend and generalize Baak et al.’s result. Other questions concerningthe $^{*}$ -homomorphisms and the multipliers between ${{C}^{*}}$ -algebras are also considered.

Let f(z) be an entire function of order less than 1/2. We consider an analogue of the Wiman-Valiron theory rewriting power series of f(z) into binomial series. As an application, it is shown that if a transcendental entire solution f(z) of a linear difference equation is of order χ < 1/2, then we have log M (r, f) = Lrχ(1 + o(1)) with a constant L > 0.

A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.

One-dimensional random packing, known as the car-parking problem, was first analyzed by Rényi (1958). A stochastic version of Kakutani's (1975) interval splitting is another typical model on a one-dimensional interval. We consider a generalized car-parking problem which contains the above two models as special cases. In the generalized model, one can park a car of length l, if there is a space not less than 1. We give the limiting packing density and the limiting distribution of the length of randomly selected gaps between cars. Our results bridge the two models of Rényi and Kakutani.