Pervasive liquid metal direct writing electronics with roller-ball pen

A roller-ball pen enabled direct writing electronics via room temperature liquid metal ink was proposed. With the rolling to print mechanism, the metallic inks were smoothly written on flexible polymer substrate to form conductive tracks and electronic devices. The contact angle analyzer and scanning electron microscope were implemented to probe the inner property of the obtained electronics. An ever high writing resolution with line width and thickness as 200{\mu}m and 80{\mu}m, respectively was realized. Further, with the administration of external writing pressure, GaIn24.5 droplets embody increasing wettability on polymer which demonstrates the pervasive adaptability of the roller-ball pen electronics

Combinatorial proofs of some properties of tangent and Genocchi numbers

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers