Let $k$ be an algebraically closed field of prime characteristic $p$, $G$ a finite group and $P$ a $p$-subgroup of $G$. We investigate the relationship between the fusion system ${ℱ}_P(G)$ and the Brauer indecomposability of the Scott $kG$-module. We give a result which shows that there exists some relationship between $G$ and its local subgroups in terms of the Brauer indecomposability of Scott modules.

We study the large time asymptotics of small amplitude solutions to the Cauchy problem for the modified KdV equation with a fifth order dispersive term.

Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Nakayama et al. (2017) proposed a memoryless quasi-Newton method based on the spectral-scaling Broyden family and showed that the method satisfies the sufficient descent condition and converges globally. To relax the conditions on parameters in the method, we apply the modification technique by Kou and Dai (2015) to the method of Nakayama et al., and we give a hybrid method of the three-term conjugate gradient method and the memoryless quasi-Newton method based on the spectral-scaling Broyden family. We show that our method satisfies the sufficient descent condition, and we prove that the method converges globally. Furthermore, we give a concrete choice of parameters for our method. Finally, some numerical results are given.

For square contingency tables with the same row and column ordinal classifications, this paper shows that the diamond model holds if and only if the weighted covariance for the difference between the row and column classifications and the sum of them equals zero and the uniform association diamond model holds. An example is given.