The fundamental group of the complement of the branch curve of ℂℙ1 × T

Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.     

The canonical embedding of an unramified morphism in an ��tale morphism

We show that every unramified morphism ${X\to Y}$ has a canonical and universal factorization ${X\hookrightarrow E_{X/Y}\to Y}$ where the first morphism is a closed embedding and the second is étale (but not separated).     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000     

How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and any class of fields. By we denote the minimal number of realizations of G as a Galois group over some field from the class . For G abelian and the class of algebraic extensions of ℚ we give an explicit formula for . Similarly we treat the case of an abelian p-group G and the class which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results. Received: 27 October 1998 / Revised version: 3 February 1999     

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

In this paper, we consider the combined quasineutral and low Mach number limit of compressible Euler–Poisson system coupled to a magnetic field. We prove that, as the Debye length and the Mach number tend to zero simultaneously in some way, the solution of compressible Euler–Poisson system coupled to a magnetic field will converge to that of ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

The fundamental theorem of calculus: visually?

The paper wants to show how it is possible to develop based on an adequate basic idea (so-called “Grundvorstellung”) of the derivative a visual understanding of the (first) Fundamental theorem of Calculus.     

The asymptotic σ-algebra of a recurrent random walk on a locally compact group

Let μ be a probability measure on a locally compact second countable groupG defining a recurrent (but not necessarily Harris) random walk. Denote byG ^∞ the space of paths and byB ^(a)the asymptotic σ-algebra. Let the starting measure be equivalent to the Haar measure and writeQ for the corresponding Markov measure onG ^∞. We prove thatL ^∞(G^∞, B^(a), Q) is in a canonical way isomorphic toL ^∞(G/N) whereN is the smallest closed normal subgroup ofG such that μ(zN)=1 for somez∈G. The groupG/N is either a finite cyclic group with generatorzN or a compact abelian group having the cyclic group as a dense subgroup. As a corollary we obtain that the set of all φ∈L ¹(G) such that coincides with the kernel of the canonical mapping ofL ¹(G)ontoL ¹(G/N). In particular, when μ is aperiodic, i.e.,G=N, then the random walk is mixing: for every φ∈L ¹(G) with ∝ φ=0.     

On exact solutions of Stokes second problem for a Burgers�� fluid, II. The cases �� =?��2/4 and �� > ��2/4

In this study, the exact solutions of the Stokes second problem for a Burgers?? fluid are presented when the relaxation time satisfies the conditions ?? =???²/4 and ?? >???²/4. The velocity field and the associated tangential stress, when only one initial condition is necessary for velocity, are determined by means of the Laplace transform. The physical interpretation for the emerging parameters is discussed with the help of graphical illustrations. The similar solutions for the Stokes?? first problem are obtained as the limiting cases of our solutions.     

What does a group algebra of a free group “know” about the group?

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra $K(F)$ when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in $K(F)$. Therefore $K(F)$ “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups.     

The Colin de Verdière number and sphere representations of a graph

Colin de Vedière introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G)2 if and only ifG is outerplanar, and (G)3 if and only ifG is planar. We prove that if the complement of a graphG onn nodes is outerplanar, then (G)n–4, and if it is planar, then (G)n–5. We give a full characterization of maximal planar graphs whose complementsG have (G)=n–5. In the opposite direction we show that ifG does not have twin nodes, then (G)n–3 implies that the complement ofG is outerplanar, and (G)n–4 implies that the complement ofG is planar.Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.     

The cohomology group of weak entwining structure

In this paper, we reveal that a weak entwining structure admits a rich cohomology theory. As an application we compute the cohomology of a weak entwining structure associated to a weak coalgebra-Galois extension.     

The full automorphism group of T‾

Let $\stackrel{￣}{G}$ be the wonderful compactification of a simple affine algebraic group G of adjoint type defined over $\mathbb{C}$. Let $\stackrel{￣}{T}?\stackrel{￣}{G}$ be the closure of a maximal torus $T?G$. We prove that the group of all automorphisms of the variety $\stackrel{￣}{T}$ is the semi-direct product $N_G(T)?D$, where $N_G(T)$ is the normalizer of T in G and D is the group of all automorphisms of the Dynkin diagram, if $G\ne \mathrm{PSL}(2,\mathbb{C})$. Note that if $G=\mathrm{PSL}(2,\mathbb{C})$, then $\stackrel{￣}{T}=\mathbb{C}{\mathbb{P}}^1$ and so in this case $\text{Aut}(\stackrel{￣}{T})=\mathrm{PSL}(2,\mathbb{C})$.     

The asymptotic behavior of the variance of the “surface area” of a random field

We study the asymptotics of the second moment of a geometric functional of surface area type of a realization of a homogeneous isotropic Gaussian random field over a disk of radius r as the latter is increased.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 106–110.     

The σ-algebra of pasts of a random walk over the orbits of a Bernoulli action of the group Z d

In the present paper, we study the σ-algebra of pasts Ξ = {ξ _n } _n of a random walk T over the orbits of a Bernoulli action of the group Z^d. We calculate the proper scaling and the scaling entropy of this sequence of partitions. We show that the proper scaling entropy of the σ-algebra of pasts is equal to . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 103–112.     

The algebra of Calderón-Zygmund kernels on a homogeneous group is inverse-closed

We show that the subalgebra of convolution operators with Calderón-Zygmund kernels on a homogeneous group G is inverse-closed in the algebra of all bounded linear operators on the Hilbert space L ²(G). The main tool used is a symbolic calculus, where the convolution of distributions on the group is translated via the abelian Fourier transform into a “twisted product” of symbols on the dual to the Lie algebra g of G.