We consider Kolmogorov's problem of stationary flows in a thin layer with a bottom friction. Using the Lyapunov-Schmidt theory, we construct bifurcating solutions of this problem in the case where the linearized equations have simple eigenvalues. Compared with the previous paper [7], where we did not take account of the bottom friction, we find some interesting properties of bifurcation curves.

Let $M$ be a countable non-standard model of first order Peano arithmetic (PA) and $I$ a weakly definable proper initial segment that is closed under addition, multiplication and factorial. We show that there is another model $N$ of PA such that the structure of $I$-module of $M$ coincides with that of $N$ and the multiplication of $M$ coincides with that of $N$ on $I$ but does not coincide at some $(a, b)\not\in I^{2}$.

Let $M^{n}(n\geq 3)$ be an immersed hypersurface without umbilic points in the $(n+1)$-dimensional unit sphere $S^{n+1}$. Then $M^{n}$ is associated with a so-called Möbius form $\Phi$ and a Möbius metric $g$ which are invariants of $M^{n}$ under the Möbius transformation group of $S^{n+1}$. In this paper, we show that if $\Phi$ is identically zero and the Ricci curvature $Ric_{g}$ is pinched: $(n-1)(n-2)/n^{2}\leq Ric_{g}\leq$ $(n^{2}-2n+5)(n-2)/[n^{2}(n-1)]$, then it must be the case that $n=2p$ and $M^{n}$ is Möbius equivalent to $S^{p}(1/\sqrt{2})\times S^{p}(1/\sqrt{2})$.

We show that the rational functions associated with the lemniscate functions and primary numbers of Gaussian integers, defined by Eisenstein, has a symmetrical relation with respect to the zeros and the poles. We also consider some polynomials associated with the trigonometric functions and odd integers, and point out a similar symmetry of the polynomials.

In this paper we investigate the Hard Lefschetz property on certain compact symplectic solvmanifolds and constmct compact pseudo-Kähler solvmanifolds which do not have the Hard Lefschetz property. We also construct holomorphic symplectic structures, hypercomplex structures and pseudo-hyperkähler stmctures on certain compact solvmanifolds.

We prove that for real-analytic hypersurfaces $M$ in $C^{N}$, finite jet determination for biholomorphisms of $M$ holds under minimal assumptions on the geometry of $M$, in both the infinite type and in the finite type cases. In the finite type case, we extend from $N=2$ to arbitrary $N$ a result of Ebenfelt, Lamel and Zaitsev on 2-jet determination of such biholomorphisms.

The inhomogeneous Diophantine approximation algorithm of Nishioka et al., ($X$,$T_{2},$$c(x),$$d(x,$$y)$), was shown by Komatsu to be efficient for inhomogeneous Diophantine approximation, but lacks a properly founded natural extension and not all periodic points about the approximation are determined. A new algorithm, ($X$,$T,$$a(x),$$b(x,$$y)$), is proposed in this paper as a modification of ($X$,$T_{2},$$c(x),$$d(x,$$y)$), and is shown to be efficient for inhomogeneous Diophantine approximation similar to ($X$,$T_{2},$$c(x),$$d(x,$$y)$) but also to have a natural extension, which allows all periodic points about ($X$,$T,$$a(x),$$b(x,$$y)$) to be determined and gives $\lim\inf_{q\rightarrow\infty}q||q\alpha-\beta-p|$ for the periodic points $(\alpha,\beta)$.

We prove that for a subspace $A$ of a space $X$, the following statements are equivalent: (1) for any Fréchet space $Y$, every pointwise bounded equicontinuous subset of $C(A, Y)$ can be extended to a pointwise bounded equicontinuous subset of $C(X, Y)$; (2) every pointwise bounded equicontinuous subset of $C(A)$ can be extended to a pointwise bounded equicontinuous subset of $C(X)$; (3) for any Fréchet space $Y$, every function $f\in C(A, Y)$ can be extended to a function $g\in C(X, Y)$. This theorem and other results obtained in this paper generalize several known theorems due to Flood, Frantz and Heath-Lutzer-Zenor, etc.

In [2], B. Y. Chen proved that the Lagrangian $H$-umbilical submanifolds in complex Euclidean space C$^{n}$ are Lagrangian pseudospheres and complex extensors of the unit hypersphere of E$^{n}$, except the flat ones. Similar to this, we can define the Lagrangian $H$-umbilical submanifold in quatemion space forms. The main purpose of this paper is to classify the Lagrangian $H$-umbilical submanifolds in quatemion Euclidean space H$^{n}$.

In this paper, we describe the structure of a subalgebra $B_{s}^{k}(t)$ of a basic self-injective Nakayama algebra $B_{s}^{k}$, and we give a periodic projective bimodule resolution for $B_{s}^{k}(t)$.

We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field $k$ of characteristic zero and then apply them to Hopf algebras $H$ of dimension $p^{3}$ over $k$. There are 10 cases according to the group-like elements of $H$ and $H^{*}$. We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We also give a partial classification of the quasitriangular Hopf algebras of dimension $p^{3}$ over $k$, after studying extensions of a group algebra of order $p$ by a Taft algebra of dimension $p^{2}$. In particular, we prove that every ribbon Hopf algebra of dimension $p^{3}$ over $k$ is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some results from [1] and [4] on bounds for the dimension of the first term $H_{1}$ in the coradical filtration of $H$, we give the complete classification of the quasitriangular Hopf algebras of dimension 27.