We formulate a theorem which provides a sufficient condition under which we can construct new approximate inverse systems from old. The result is at the heart of many constructions in the theory of approximate inverse systems and offers a unified approach to several important results in Topology such as Brown's approximation theorem, McCord's embedding theorem and results on expansion of $\Pi$-like spaces into inverse limits of spaces from $\Pi$.

Let $D$ be a Dirac operator on a compact oriented Riemannian manifold $M$ of dimension $2m$. The operator $D$ can be one of the four classical elliptic operators that arise from geometry, or one of the twisted operators of these four operators. Let $\lambda_{k}^{2}$ be the k-th nonzero eigenvalue of the operator $D^{2}$ counting with multiplicity. We show that \[ \lambda_{k}^{2}\leq c(2m)\max\{(\frac{N(a)}{V(M)})^{1/m},(\frac{2^{m-1}(m_{0}+k-1)-2^{-m}m_{0}+1}{|k_{0}|V(M)})^{1/m}} \], where $N(a)$ is an integer determined by the geometry of $M$,$m_{0}$ the dimension of the kernel of $D^{2}$ and $k_{0}$ an integer defined by the operator $D$. These results, in case $M$ being a surface, give a partial answer to a conjecture of Yau.

We show which inner automorphisms of $(SU(2), g)$ with an arbitrary left invariant metric $g$ into itself are affine transformations, and obtain affine transformations of $(SU(2), g)$ which are not harmonic, and study geodesics of $(SU(2), g)$ with some conditions.

Two theorems are proven: 1) For a topological space $X$ the pair $(\exp(X), \exp_{\omega}(X))$ of hyperspaces of compact and finite subsets of $X$ is homeomorphic to $(Q\times R^{\infty},\sigma\times R^{\infty})$ if and only if $X$ is a direct limit of a tower $X_{1}\subset X_{2}\subset\cdots$ of strongly countable-dimensional Peano continua such that each $X_{n}$ is nowhere dense in $X_{n+1}$; 2) The triple $(\mathscr{H}^{c}(R), \mathscr{H}_{L^{C}IP}(R),\mathscr{H}_{P^{C}L}(R))$ of homeomorphism groups of the line, endowed with the Whitney topology, is homeomorphic to $(s\times R^{\infty}, \Sigma\times R^{\infty}, \sigma\times R^{\infty})$.

We expand the generalized tempered distributions in terms of wavelets of ordinary functions and show the convergence of the wavelet expansions of the generalized tempered distributions.

Extending the famous Bowen-Franks-Misiurewicz's theorem concerning the topological entropy of continuous maps of an interval we prove that continuous maps of a class of continua have zero topological entropy if and only if the periods of all periodic points are powers of 2.

We consider a new conformal gauge invariant functional which is a natural curvature functional on the space of Weyl structures. We derive the first variation formula of its functional and characterize its critical points.