We study the distribution of closed geodesics in homology on a finite area hyperbolic surface. We obtain an estimate which is uniform as the homology class varies, refining an asymptotic formula due to C. Epstein.

In [19], R. Kadison proved that every surjective linear isometry $\Phi{:}\, {\C A} \to {\C B}$ between two unital C*-algebras has the form $$\Phi (x) = u T (x), \hbox{ $x\in {\C A}$,}$$ where $u$ is a unitary element in ${\C B}$ and $T$ is a Jordan *-isomorphism from${\C A}$ onto ${\C B}$. This result extends the classical Banach–Stone theorem [3, 32] obtained in the 1930s to non-abelian unital C*-algebras. A. L. Paterson and A. M. Sinclair extended Kadison's result to surjective isometries between C*-algebras by replacing the unitary element $u$ by a unitary element in the multiplier C*-algebra of the range algebra [28]. Thus, every surjective linear isometry between C*-algebras preserves the triple products as $$\J xyz \,{=}\, 2^{-1} ( x y^* z + z y^* x).$$

We prove results on the asymptotic behavior of the quantization error for self-similar distributions $P$ when $n$, the number of prototypes, goes to infinity. In particular, we show that the suitably scaled quantization error converges, if $P$ is nonarithmetic and that it is asymptotically periodic in the arithmetic case. The results hold also for distributions that are absolutely continuous with respect to a self-similar distribution.

By using Mawhin's continuation theorem of coincidence degree theory, we derive criteria for the existence of a positive periodic solution of a generalized prey-predator system with time delay.

Spherically symmetric space-times have been studied in general relativity ever since the beginning when as the first exact solution of the Einstein equation the Schwarzschild space-time was given. Later on an elaborate local theory of spherically symmetric space-times was worked out with such fundamental results as Birkhoff's theorem and essential results concerning their global geometry have been achieved. Yet it seems that a general global theory is lacking even now. Some fundamental facts concerning the global geometry of spherically symmetric space-times are presented below. The concept of axis and of transverse submanifold are introduced and applied to study and classification.

We consider the existence of unbounded solutions for the asymmetric oscillator $$(\vp_p(x'))'+(p-1)[\al \vp_p(x^+)-\be \vp_p(x^-)]=f(t)\eqno(1)$$where $\vp_p(u)=|u|^{p-2}u, p>1$, $\al$ and $\be$ are positive constants satisfying $$\al^{{-}\frac 1p}+\be^{{-}\frac 1p}=2m/n\eqno(2)$$ with ($m,n)=1,\,\, m, n\in N$ and $x^\pm=\max\{\pm x,0\}$, $f \in L^\infty[0,2\pi_p]$ is $2\pi_p$-periodic, $\pi_p={2\pi}/({p\sin (\pi/p)})$.

We study locally spatially homogeneous solutions of the Einstein–Vlasov system of all Bianchi types except IX with a positive cosmological constant. First the global existence of solutions of this system and the casual geodesic completeness are shown. Then the asymptotic behaviour of solutions in future time is investigated in various aspects.