In this paper, we study the Hayman T directions and the precise Borel directions of maximal kind of meromorphic solutions f(z) of the Schröder equations f(sz)=R(f(z)), where |s|<1 and R(w) is a rational function with $\deg [R]\geq 2$. We will show that, if $\operatorname {arg}[s]/2\pi \notin Q$, then f(z) has any direction as Hayman T direction and maximus Borel direction as well. This is a continue work of [Ishizaki, K. and Yanaihara, N., Borel and Julia directions of meromorphic Schröder functions, Math. Proc. Camb. Phil. Soc. 139 (2005), 139-147.] and [Yuan, W.J., Qi, J.M. and Seiki Mori. Singular directions of meromorphic solutions of some non-autonomous Schröder equations, Complex Analysis and its Applications Proceedings of the 15th ICFIDCAA held in Osaka (Japan), July 30-August 3, 2007].

In this note, we define a notion of finite-type for invariants of curves on surfaces as an analogue of the notion of finite-type for invariants of knots and 3-manifolds (Section 3). We also present a systematic construction for a large family of finite-type invariants SCI_n for curves on surfaces (Section 5). Arnold's invariants of plane isotopy classes of plane curves occur as invariants of order 1. Our theory of finite-type invariants of curves on surfaces is developed using the topological theory of words.

We consider an extremal problem for polynomials, which is dual to the well-known Smale mean value problem. We give a rough estimate depending only on the degree.

When we analyze the reflection phenomenon for the elastic wave, the one of the most complicated and interesting problems is to study the mode conversion case. For the elastic wave, there are waves of different modes and a remarkable phenomenon called "mode-conversion'' which causes serious difficulties. In this paper, by considering the non back-scattering case, we examine the singularities of the scattering kernel for the elastic wave equation with transverse incident waves and derive a new result about the singularities of the scattering kernel.

Let p be an odd prime greater than seven and A the mod p Steenrod algebra. In this paper we prove that in the cohomology of A the product $h_1 h_n \tilde \delta _{s + 4}\in {\rm Ext}_A^{s + 6, t(s,n) + s} ({\bf Z}_p , {\bf Z}_p)$ is nontrivial for $n \geq 5$, and trivial for $n=3, 4$, where $ \tilde \delta _{s + 4}$ is actually $\tilde \alpha _{s+4}^{(4)}$ described by X. Wang and Q. Zheng, $0 \leq s < p - 4$, $t(s,n) = 2(p-1)[(s + 1) + (s + 3)p + (s + 3)p^2 + (s + 4)p^3 + p^n ].$ We show our results by explicit combinatorial analysis of the (modified) May spectral sequence. The method of proof is very elementary.

In this paper, we make structural elucidation of some interesting arithmetical identities in the context of the Parseval identity.

In the continuous case, following Romanoff [R] and Wintner [Wi], we study the Hilbert space of square-integrable functions L₂(0,1) and provide a new complete orthonormal basis-the Clausen system-, which gives rise to a large number of intriguing arithmetical identities as manifestations of the Parseval identity. Especially, we shall refer to the identity of Mikolás-Mordell.

Secondly, we give a new look at enormous number of elementary mean square identities in number theory, including H. Walum's identity [Wa] and Mikolás' identity (1.16). We show that some of them may be viewed as the Parseval identity. Especially, the mean square formula for the Dirichlet L-function at 1 is nothing but the Parseval identity with respect to an orthonormal basis constructed by Y. Yamamoto [Y] for the linear space of all complex-valued periodic functions.