Let $G$ be a finite group and $X$ a conjugacy class of $G$. We denote $\mathrm{rank}(G:X)$ to be the minimum number of elements of $X$ generating $G$. In the present paper we investigate the ranks of the Conway group $\mathit{Co}_1$. Computations were carried with the aid of computer algebra system $\mathbf{GAP}$ [16].

We consider extremal holomorphic curves for the truncated defect relation when the number of vectors whose truncated defects are equal to 1 is large. Some applications to another defect are given.

In this paper, we give explicit formulas for the secondary Whittaker functions for $P_J$-principal series representations of $Sp(3,{\mathbf R})$, which are power series solutions of a holonomic system of rank 24.

We take sectors as real $2$-dimensional objects associated with trajectories for Kähler magnetic fields. As a sequel of [2] we make a bit more consideration on comparison theorem on mangetic Jacobi fields and study lengths of arcs for sectors.

Two seemingly unrelated problems are intimately connected. The first is the equsingularity problem in $\mathbf{R}^2$: For an analytic family $f_t:(\mathbf{R}^2,0)\to (\mathbf{R},0)$, when should it be called an ``equisingular deformation''? This amounts to finding a suitable trivialization condition (as strong as possible) and, of course, a criterion. The second is on the Morse stability. We define $\mathbf{R}_*$, which is $\mathbf{R}$ ``enriched'' with a class of infinitesimals. How to generalize the Morse Stability Theorem to polynomials over $\mathbf{R}_*$? The space $\mathbf{R}_*$ is much smaller than the space used in Non-standard Analysis. Our infinitesimals are analytic arcs, represented by fractional power series. In our Theorem II, (B) is a trivialization condition which can serve as a definition for equisingular deformation; (A), and (A$'$) in Addendum \refAdd1, are criteria, using the stability of ``critical points'' and the ``complete initial form''; (C) is the Morse stability (Remark 1.6).

We prove some new results on the triviality and the finiteness of the Galois and flat cohomology of unipotent algebraic group schemes over non-perfect (and in particular, also local and global function) fields. Some applications are also given to certain local - global principles.