Packing Characteristics of Different Shaped Proppants for use with Hydrofracing - A Numerical Investigation using 3D FEMDEM

Imperial Users onl

Unlimited parity alternating partitions

We introduce a new type of partitions that consists of partitions whose different parts alternate in parity (e.g., $3+2+2+1+1$). Various properties of this partition function are studied. In particular, we obtain its asymptotic behavior by employing Ingham's Tauberian theorem

Note on sums involving the Euler function

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} \phi\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{\phi([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordell\`es et al. and Wu.Comment: Bull. Aust. Math. Soc., accepte