We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-$2$ submanifolds with contact number ${\geq}3$. We also study surfaces in $\mathbb{E}^6$ with contact number ${\geq}4$. As an immediate consequence, we obtain the first explicit examples of non-spherical pseudo-umbilical surfaces in Euclidean spaces.

AMS 2000 Mathematics subject classification: Primary 53C40; 53A10. Secondary 53B25; 53C42

Let $w(z)$ be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in $|z|\le r$, it is shown that $n(r,w)\gg r^2$ (respectively, $n(r,w)\gg r^{3/2}$), from which the growth estimate $T(r,w)\gg r^2$ (respectively, $T(r,w)\gg r^{3/2}$) immediately follows.

AMS 2000 Mathematics subject classification: Primary 34M55; 34M10

We construct certain elements in the motivic cohomology group $H^3_{\mathcal{M}}(E\times E',\mathbb{Q}(2))$, where $E$ and $E'$ are elliptic curves over $\mathbb{Q}$. When $E$ is not isogenous to $E'$ these elements are analogous to circular units in real quadratic fields, as they come from modular parametrizations of the elliptic curves. We then find an analogue of the class-number formula for real quadratic fields, which specializes to the usual quadratic class-number formula when $E$ and $E'$ are quadratic twists.

AMS 2000 Mathematics subject classification: Primary 11F67; 14G35. Secondary 11F11; 11E45; 14G10

Let $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr. 186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

We study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

It is well known that $\varOmega^2S^{2n+1}$ is approximated by $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, the space of based holomorphic maps of degree $k$ from $S^2$ to $\mathbb{C}P^{n}$. In this paper we construct a space $G_{k}^{n}$ which is an analogue of $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, and prove that under the natural map $j_k:G_{k}^{n}\to\varOmega^2S^{2n}$, $G_{k}^{n}$ approximates $\varOmega^2S^{2n}$.

AMS 2000 Mathematics subject classification: Primary 55P35

Let $p^m$ be a power of a prime number $p$, $\mathbb{Dacute;_{p^m}$ be the dihedral group of order $2p^m$ and $k$ be a field where $p$ is invertible and containing a primitive $2p^m$-th root of unity. The aim of this paper is computing the Brauer group $BM(k,\mathbb{D}_{p^m},R_z)$ of the group Hopf algebra of $\mathbb{D}_{p^m}$ with respect to the quasi-triangular structure $R_z$ arising from the group Hopf algebra of the cyclic group $\mathbb{Z}_{p^m}$ of order $p^m,$ for $z$ coprime with $p$. The main result states that $BM(k,\mathbb{D}_{p^m},R_z)\cong \mathbb{Z}_2 \times k^{\cdot}/k^{\cdot 2} \times Br(k)$ when $p$ is odd and when $p=2,$$BM(k,\mathbb{D}_{2^m},R_z) \cong \mathbb{Z}_2\times \mathbb{Z}_2 \times k^{\cdot}/k^{\cdot 2} \times k^{\cdot}/k^{\cdot 2} \times Br(k).$

We define and study some properties of spectral maximal projections of a bounded operator on a complex Banach space. Then we apply these results to the new concepts of weakly projection-relative decomposable operators and projection-relative decomposable operators in the spirit of the works of C. Foias [6], A. Jafarian [7], I. Erdelyi and R. Lange [5].

Suppose that $A$ and $B$ are ‘isoloid’ operators acting on a complex Banach space, that is, every isolated point of their spectra is an eigenvalue. In this note it is shown that if Weyl's theorem holds for both $A$ and $B$ then it holds for $A\otimes B$.

Given $X$ a complex Banach space, $L$ a complex nilpotent finite dimensional Lie algebra, and $\rho\,{\colon}\, L\to L(X)$, a representation of $L$ in $X$ such that $\rho (l\,)\,{\in}\, K(X)$ for all $l\,{\in}\, L$, the Taylor, the Słodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $\rho$ are computed.

We characterize norm closed subspaces $B$ of $\linf (\partial D)$ such that $C(\partial D) B \subset B$ and maximal ones in the family of proper closed subspaces $B$ of $L^\infty(\partial D)$ such that $A(D) B \subset B$, where $A(D)$ is the disk algebra. Analogously, we characterize closed subspaces of $H^\infty$ that are simultaneously invariant under $S$ and $S^\ast$, the forward and the backward shift operators, and maximal invariant subspaces of $H^\infty$.

In 1989, Alladi, Erdös and Vaaler confirmed their own conjecture about a class of multiplicative functions by means of a deep result of Baranyai on hypergraphs. In this note we give a simple direct proof of the result which is derived in their proof as a consequence of the above mentioned graph theoretic result.

In this article the following are proved: 1. Let $G$ be an infinite $p$-group of cardinality either ${\bf {\mathbb N}_{0}}$ or greater than $2^{\bf {\mathbb N}_{0}}$. If $G$ is center-by-finite and non-$\skew5\check{C}$ernikov, then it is non-co-Hopfian; that is, $G$ is isomorphic to a proper subgroup of itself. 2. Let $G$ be a nilpotent $p$-group of class $2$ with $G/G'$ a non-$\skew5\check{C}$ernikov group of cardinality ${\bf {\mathbb N}_{0}}$ or greater than $2^{{\bf {\mathbb N}_{0}}}$. If $G'$ is of order $p$, then $G$ is non-co-Hopfian.

In this note we prove that a locally graded group $G$ in which all proper subgroups are (nilpotent of class not exceeding $n$)-by-Černikov, is itself (nilpotent of class not exceeding $n$)-by-Černikov.

As a preparatory result that is used for the proof of the former statement in the case of a periodic group, we also prove that a group $G$, containing a nilpotent of class $n$ subgroup of finite index, also contains a characteristic subgroup of finite index that is nilpotent of class not exceeding $n$.

We consider a polynomial $f \,{:}\, \mathbb{R}^n \rightarrow \mathbb{R}$ with isolated critical points and we relate $\chi(f^{-1}(0))$ and $\chi(\{f \ge 0\})-\chi(\{f \le 0\})$ to the topological degrees of polynomial maps defined in terms of $f$.

It is known that the concept of Moufang loops, Moufang 3-nets and groups with triality are strongly related. Due to S. Doro, a group with a splitting automorphism of order 3 can lead to a group with triality. This construction naturally appears in the classification of simple Moufang loops. In this paper, we consider groups with triality related to groups with splitting automorphism. We give a classification of Moufang loops corresponding to this construction.

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

In this paper we give a full asymptotic expansion for the number of closed geodesics in homology classes. Especially, we obtain formulae about the coefficients of error terms which depend on the homology class.

We explicitly classify helicoidal and translational constant mean curvature surfaces in ${\mathbb H}^2\times {\mathbb R}$.