It is shown that an application of the theory of regular variation (in the sense of Karamata) gives the possibility of determining the existence and precise asymptotic behavior of positive solutions of the third-order nonlinear differential equation $(|x''|^{\alpha-1}x'')' + q(t)|x|^\beta x = 0$, where $\alpha > \beta > 0$ are constants and $q:[a,\infty)\to(0,\infty)$ is a continuous regularly varying function.

Fintushel-Stern's knot surgery has given many exotic 4-manifolds. We show that if an elliptic fibration has two, parallel, oppositely-oriented vanishing cycles (for example $S^2\times S^2$ or Matsumoto's $S^4$), then the knot surgery does not change its differential structure. We also give a classification of link surgery of $S^2\times S^2$ and a generalization of Akbulut's celebrated result that Scharlemann's manifold is standard.

It is known that any 7-colorable knot in 3-space is presented by a diagram assigned by four of the seven colors. In this paper, we prove the existence of a 7-colorable 2-knot in 4-space such that any non-trivial 7-coloring requires at least six of the seven colors.

We give a method for studying of asymptotic behavior of solutions to a periodic continuous linear system. It is based on a representation of solutions given in the paper, which is a reformation of the variation of constants formula into the sum of a $\tau$-periodic function and an exponential-like function. By using such representations, the set of initial values is completely classified according to the asymptotic behavior of the solutions to the continuous system. In particular, the set of initial values of bounded solutions is precisely determined. To give the representation for the continuous system, we will establish translation formulae by comparing two representations of solutions to a discrete linear system. These two representations are deeply related to the binomial coefficients, the Bernoulli numbers and the Stirling numbers.