Continuous rational maps into the unit 2-sphere

Investigated are continuous rational maps from a compact nonsingular real algebraic set into unit spheres. Special attention is devoted to such maps with values in the unit 2-sphere.     

Strong surjectivity of mappings of some 3-complexes into $$ M_{Q_8 } $$

Let K be a CW-complex of dimension 3 such that H ³(K;ℤ) = 0 and the orbit space of the 3-sphere with respect to the action of the quaternion group Q ₈ determined by the inclusion Q ₈ ⊆ . Given a point a ∈ , we show that there is no map f:K → which is strongly surjective, i.e., such that MR[f,a]=min{#(g ⁻¹(a))|g ∈ [f]} ≠ 0.      

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

On the surjectivity of weighted Gaussian maps

We study the surjectivity of suitable weighted Gaussian maps $\gamma _{a,b}(X,L)$ which provide a natural generalization of the standard Gaussian maps and encode the local geometry of the locus $\mathfrak{Th }^r_{g,h}\subset \mathcal M _g$ of curves endowed with an $h$ -th root $L$ of the canonical bundle satisfying $h^0(L)\ge r+1$ . In particular, we get a bound on the dimension of its Zariski tangent space, hich turns out to be sharp in the special case $r=0$ . Finally, we describe this locus in the case of complete intersection curves.     

On the surjectivity of some trace maps

LetK be a commutative ring with a unit element 1. Let Γ be a finite group acting onK via a mapt: Γ→Aut(K). For every subgroupH≤Γ define tr _H :K→K ^H by tr _h (x)=Σ_σ∈H σ(x). We proveTheorem: tr_Γ is surjective onto K ^Γ if and only if tr _P is surjective onto K ^P for every (cyclic) prime order subgroup P of Γ. This is false for certain non-commutative ringsK.     

Dynamical systems method and surjectivity of nonlinear maps

If F : H → H is a $C_{\mathrm{loc}}^2$-map in a real Hilbert space, sup_{u ∈ B(u0,R)}∥[F′(u)]⁻¹∥ ⩽ m(R), and ${\mathrm{limsup}}_{R>0}\frac{R}{m(R)}=\infty$, then F is surjective.     

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators

Let B(H) be the algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space H. For every T∈B(H), let m(T) and q(T) denote the minimum modulus and surjectivity modulus of T respectively. Let ?:B(H)→B(H) be a surjective linear map. In this paper, we prove that the following assertions are equivalent:

(i)

m(T)=m(?(T)) for all T∈B(H),

(ii)

q(T)=q(?(T)) for all T∈B(H),

(iii)

there exist two unitary operators U,V∈B(H) such that ?(T)=UTV for all T∈B(H).

This generalizes the result of Mbekhta [7, Theorem 3.1] to the non-unital case.     

Geodesic nets on the 2-sphere

In this paper we introduce the concept of a geodesic net, an idea which plays the role among graphs that geodesics play among simple closed curves. We establish the existence of specific geodesic nets on the 2-sphere in certain cases.

     

Dissecting the 2-sphere by immersions

The self intersection of an immersion dissects S ² into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular, for every n we construct an immersion with 2n triple points, for which all pieces are discs.      

Rational holomorphic maps from B2 into BN with degree 2

Rational proper holomorphic maps from the unit ball in C2 into the unit ball CN with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

The 2-homeotoxal tilings of the plane and the 2-sphere

A complete enumeration is obtained of the finite tilings of the 2-sphere and of the infinite tilings of the plane which are normal and admit two transitivity classes of edges under topological automorphisms of the tiling. Every type is found to be representable by a tiling in which the edges form two transitivity classes under isometric symmetries of the tiling.     

On the planarity of infinite 2-complexes

This paper is devoted to the problem of characterizing infinite planar polyhedra. We give two topological characterizations of planarity, and two others of proper planarity (embeddings with no accumulation points). We also give a combinatorial characterization of planarity, and proper planarity. In the case of compact polyhedra both results provide a weaker condition than that given by Gross and Rosen in [6].     

On the partition of the 2-sphere by geodesic nets

The main result of the paper is that for every natural number there exists a geodesic net with vertices of degree 3 or 4 partitioning the round 2-sphere into regions.