We present explicit models for non-elliptic genus one Shimura curves $X_0(D,N)$ with ${\mathrm{\Gamma }}_0\left(N\right)$-level structure arising from an indefinite quaternion algebra of reduced discriminant $D$, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan's work [10, Ch. III] on points with complex multiplication on Shimura curves.

Z.-J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper, we extend this work to the case of discrete quantum groups. That is, we show that a discrete quantum group, where we do not assume its unimodularity, has an invariant mean if and only if it is strongly Voiculescu amenable.

We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space andshow that totally umbilic hypersurfaces with vanishing curvatures are only horospheres. We also show that the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space.

We study families of submanifolds in symmetric spaces of compact type arising as exponential images of $s$-orbits of variable radii. If the $s$-orbit is symmetric such submanifolds are the most important examples of adapted submanifolds, i.e. of submanifolds of symmetric spaces with curvature invariant tangent and normal spaces.

We establish a relation between the gamma matrices of the functional equations satisfied by zeta functions associated with prehomogeneous vector spaces and certain integrals related to the intertwining operator of degenerate principal series representations of general linear groups.

The classification of almost split real forms of symmetrizable Kac-Moody Lie algebras is a rather straightforward infinite-dimensional generalization of the classification of real semi-simple Lie algebras in terms of the Tits index [J. Algebra, 171, 43--96 (1995)]. We study here the conjugate classes of their Cartan subalgebras under the adjoint groups or the full automorphism groups. Maximally split Cartan subalgebras of an almost split real Kac-Moody Lie algebra are mutually conjugate and one can generalize the Sugiura classification (given for real semi-simple Lie algebras) by comparing any Cartan subalgebra to a standard maximally split one. As in the classical case, we prove that the number of conjugate classes of Cartan subalgebras is always finite.

In this paper, we give an easy description of the elementary divisors of the Cartan matrices for symmetric groups in terms of the lengths of $p$-regular partitions and their Glaisher correspondents. Moreover, for $p=2$, it is done block wise. There we use certain kinds of cores and weights, which are similar to but different from the usual ones.

We give sufficient conditions for hypoellipticity of a second order operator with real-valued infinitely differentiable coefficients whose principal part is the product of a real-valued infinitely differentiable function $\phi \left(x\right)$ and the sum of squares of first order operators $X_1,\dots ,X_r$. These conditions are related to the way in which $\phi \left(x\right)$ changes its sign, and the rank of the Lie algebra generated by $\phi X_1,\dots ,\phi X_r$ and $X_0$ where $X_0$ is the first order term of the operator. Our result is an extension of that of [4], and it includes some cases not treated in [1], [5] and [8].

We construct spaces of initial conditions of Garnier system and its degenerate systems in two variables and describe them as symplectic manifolds.These systems are expressed as polynomial Hamiltonian systems on all affine charts.

Let $L$ be a very ample line bundle on a smooth complex projective variety $X$ of dimension $\ge 7$. We classify the polarized manifolds $(X,L)$ such that there exists a smooth member $A$ of $\left|L\right|$ endowed with a branched covering of degree five $\pi :A\to {\mathbf{P}}^n$. The cases of $\mathrm{deg}\pi =2$ and $3$ are already studied by Lanteri-Palleschi-Sommese.

We give a complete derived equivalence classification of all symmetric algebras of domestic representation type over an algebraically closed field. This completes previous work by R. Bocian and the authors,where in this paper we solve the crucial problem of distinguishing standard and nonstandard algebras up to derived equivalence. Our main tool are generalized Reynolds ideals, introduced by B. Külshammer for symmetric algebras in positive characteristic, and recently shown by A. Zimmermann to be invariants under derived equivalences.

The validity of Freedman's disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a large special class of 4-manifolds with free nonabelian fundamental groups. The goal of this paper is to show that this also holds for other fundamental groups which are not known to be good, and that it is best understood using controlled surgery theory of Pedersen-Quinn-Ranicki. We consider some examples of 4-manifolds which have the fundamental group either of a closed aspherical surface or of a 3-dimensional knot space. A more general theorem is stated in the appendix.

We study the homotopy groups of spaces of continuous maps between real projective spaces and we generalize the results given in [5], [8] and [12]. In particular, we determine the rational homotopy types of these spaces and compute their fundamental groups explicitly.

We give a necessary and sufficient condition for a discrete variety in a pseudoconvex open set $\mathrm{\Omega }$ of ${\mathbf{C}}^n$ to be an interpolating variety for Hörmander's weighted algebras of holomorphic functions in $\mathrm{\Omega }$.

We extend, in the context of real semisimple Lie group, a result of T. Yamamoto which asserts that ${\lim }_{m\to \mathrm{\infty }}\left[s_i\right(X^m){]}^{1/m}=|{\lambda }_i\left(X\right)|$, $i=1,\dots ,n$, where $s_1\left(X\right)\ge \cdots \ge s_n\left(X\right)$ are the singular values, and ${\lambda }_1\left(X\right),\dots ,{\lambda }_n\left(X\right)$ are the eigenvalues of the $n\times n$ matrix $X$, in which $\left|{\lambda }_1\right(X\left)\right|\ge \cdots \ge \left|{\lambda }_n\right(X\left)\right|$.

Some further results about special Vietoris continuous selections and totally disconnected spaces are obtained, also several applications are demonstrated. In particular, it is demonstrated that a homogeneous separable metrizable space has a continuous selection for its Vietoris hyperspace if and only if it is discrete,or a discrete sum of copies of the Cantor set, or is the irrational numbers.

We study $p$-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak $(1,p)$-Poincaré inequality, . We establish the barrier classification of regular boundary points from which it also follows that regularity is a local property of the boundary. We also prove boundary regularity at the fixed (given) boundary for solutions of the one-sided obstacle problem on bounded open sets. Regularity is further characterized in several other ways.

Our results apply also to Cheeger $p$-harmonic functions and in the Euclidean setting to $A$-harmonic functions, with the usual assumptions on $A$.