In analogy to an inequality of Chen [Che93], Scharlach, Simon, Verstraelen and Vrancken [SSVV97] have found a new inequality for (equi-) affine spheres. This inequality is optimal and in this paper we classify those 3-dimensional elliptic affine spheres for which the corresponding equality is assumed. This is achieved through reducing the problem to the problem of classifying those 2-dimensional minimal surfaces in $S_{3}^{5}$ whose ellipses of curvature are circles. We end with the investigation of 2-dimensional minimal surfaces in $S_{3}^{5}$ with positive definite induced metric whose ellipses of curvature are circles.

We prove the three mutually related theorems: the theorem on the quantizability of canonical isomorphisms, the theorem on the quantizability of classical canonical commutation relations and asemiclassical version of von Neumann's theorem. Although some similar results can be obtained on the basis of the deformation theory (e.g. [16], [15], [10] ) , here we present the proofs which involve only elementary methods and notions. Moreover, in our approach we can easily compute the quantum corrections. Our deformation quantizations (semiclassical algebras) are additionally equipped with the deformation involutions and we study here the algebras of entire functions and of polynomials, instead of frequently used algebras of $C^\infty$ observables.

We show that the Isaacs character correspondence induces a block correspondence of finite groups and that the corresponding blocks have common invariants.

Global properties of all maximal solutions of the iterative functional differential equation $x''(t)=a[x(x(t))-x(t)]+1$ are considered. Using a correspondence among solutions of the above equation and those of the functional differential equation $y'(t)=ay(t- by(t)$), global properties of all maximal solutions of the last equation are described.

Let $G$ be a finite cyclic subgroup of $GL(2, C)$ of order $n$ which contains no reflections. Let ${\mathbf A}^{2}$ be the complex affine plane. We consider a certain subscheme $Hi1b^{G}({\mathbf A}^{2})$ of $Hi1b^{n}({\mathbf{}A}^{2})$ consisting of $G$-invariant zero-dimensional subschemes of length $n$. We describe the structure of $Hi1b^{G}({\mathbf{}A}^{2})$ and prove this is the minimal resolution of the quotient surface singularity ${\mathbf A}^{2}/G$ .

It is shown that any two Haar measures for a quasi Woronowicz algebra, a quantum group in the von Neumann algebra framework, are proportional.

This paper presents a trust-region algorithm for n-dimensional nonlinear optimization subject to m nonlinear inequality constraints. Equivalent KKT conditions are derived, being the basis for constructing the new algorithm. Global convergence of the algorithm to a first-order KKT point is established under mild conditions on the trial steps. Condition $m\leq n$ is required.

Let $D_{n}$, $Q_{n}$ and $SD_{n}$ be the dihedral group, the generalized quaternion group and the semidihedral group of order $2^{n+1}$, respectively. Let $C_{n}$ be the cyclic 2-group of order $2^{n}$. As is well-known these four kinds of 2-groups play an important role in character theory of 2-groups. Let $\phi$ be a faithful irreducible character of $H=D_{n}$, $Q_{n}$, $SD_{n}$ or $C_{n}$. In [3] we determined all the 2-groups $G$ such that $H$ is a normal subgroup of $G$ and the induced character $\phi^{G}$ is irreducible. There exist other nonabelian 2-groups $M_{n}$ with a cyclic subgroup of index 2. All the faithful irreducible characters of $M_{n}$ are algebraically conjugate to each other as in $H$. The purpose of the paper is to determine all the 2-groups $G$ with a no rmal subgroup isomorphic to $M_{n}$ such that $\phi^{G}$ is irreducible for a faithful irreducible characters $\phi$ of the no rmal subgroup.

We determine the structure of a $p$-vertical[vertical] subgroup $A$ of an arbitrary abelian group $G$ such that every neat hull of $A$ is $p$-purifiable[purifiable] in $G$.

This paper treats geodesic triangles on two-dimensional orientable Riemannian manifolds $M$. Fixing two vertices $A$ and $B$, we can consider the area and the interior angles of the geodesic triangle $\Delta PAB$ as smooth functions of $P$. Applying the Laplace operator to these functions, we obtain formulas for the area and interior angles of $\Delta APAB$. It is shown that if $M$ is of constant curvature, the area and interior angles of geodesic triangles are harmonic.

We investigate a necessary and sufficient condition for a family of $2^{r}$ Lagrangian and $2^{r}$ Legendrian submanifolds to be a symplectic and contact regular r-cubic configuration respectively.

We give sufficient conditions for the convergence of the series having the following form $$\sum_{k=1}^{\infty}k^{\delta}(\varphi(|a_{n_{k}}|)+\varphi(|b_{n_{k}}|)) ,$$ where $a_{k}$ and $b_{k}$ are Fourier coefficients, $\delta\geq 0$, $\varphi(u)(u\geq 0)$ is an increasing concave function, and $\{n_{k}\}$ is a certain increasing sequence of natural numbers.

The simple Lie groups with left invariant projectively flat affine connections are classified. This is accomplished by proving that some of the possibilities that appear in the previous work by Urakawa [U, Theorem C] do not actually occur.

Sufficient conditions are given by means of the best trigonometric approxi- mation in $L^{p}(1<p\leq 2)$ and structural properties of $f\in L^{p}$ for the convergence of the series $$\sum_{n=1}^{\infty}\omega_{n}(\varphi(|a_{n}|)+\varphi(|b_{n}|)) ,$$ where $a_{n}$ and $b_{n}$ are the Fourier coefficients of $f$, $\{\omega_{n}\}$ is a certain sequence of positive numbers, $\varphi(u)(u\geq 0)$ denotes an increasing concave function.