Stable Gauss maps from a global viewpoint

We associate weighted graphs to stable Gauss maps on orientable closed surfaces immersed in 3-space and prove that any bipartite weighted graph can be associated to some stable Gauss map.     

An algorithm for the determination of graphs associated to fold maps between closed surfaces

The aim of this paper is to introduce a computational tool that checks theoretical conditions in order to determine whether a weighted graph, as a topological invariant of stable maps, can be associated to stable maps without cusps (ie, fold maps) from closed surfaces to the projective plan.     

A branch and bound algorithm for minimizing the number of crossing arcs in bipartite graphs

Acyclic directed graphs are widely used in many fields of economic and social sciences. This has generated considerable interest in algorithms for drawing “good” maps of acyclic diagraphs. The most important criterion to obtain a readable map of an acyclic graph is that of minimizing the number of crossing arcs. In this paper, we present a branch and bound algorithm for solving the problem of minimizing the number of crossing arcs in a bipartite graph. Computational results are reported on a set of randomly generated test problems.     

A bijection for covered maps, or a shortcut between Harer-Zagier?s and Jackson?s formulas

We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked unicellular spanning submap (which can have any genus in {0,1,…,g}). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n+1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4].Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps.We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33]) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection.     

Characterization and algorithms of curve map graphs

A curve map is a planar map obtained by dividing the Euclidean plane into a finite number of regions by a finite set of two-way infinite Jordan curves (every one dividing the plane in two regions) such that no two curves intersect in more than one point. A line map is a curve map obtained by Jordan curves being all straight lines. A graph is called a curve map graph respectively a line map graph if it is the dual of a curve map respectively of a line map.In this paper we give a characterization of the curve map graphs and we describe a polynomial time algorithm for their recognition.     

First Order Semi-Local Invariants of Stable Maps of 3-Manifolds into the Plane

In the late 1980s, Vassiliev introduced new graded numericalinvariants of knots, which are now called Vassiliev invariantsor finite-type invariants. Since he made this definition, manypeople have been trying to construct Vassiliev type invariantsfor various mapping spaces. In the early 1990s, Arnold and Goryunovintroduced the notion of first order (local) invariants of stablemaps. In this paper, we define and study first order semi-localinvariants of stable maps and those of stable fold maps of aclosed orientable 3-dimensional manifold into the plane. Weshow that there are essentially eight first order semi-localinvariants. For a stable map, one of them is a constant invariant,six of them count the number of singular fibers of a given typewhich appear discretely (there are exactly six types of suchsingular fibers), and the last one is the Euler characteristicof the Stein factorization of this stable map. Besides theseinvariants, for stable fold maps, the Bennequin invariant ofthe singular value set corresponding to definite fold pointsis also a first order semi-local invariant. Our study of unstablefold maps with codimension 1 provides invariants for the connectedcomponents of the set of all fold maps. 2000 Mathematics SubjectClassification 57R45 (primary), 32S20, 58K15 (secondary).     

Minimizing singularities of generic plane disks with immersed boundaries

A cooriented circle immersion into the plane can be extended to a stable map of the disk which is an immersion in a neighborhood of the boundary and with outward normal vector field along the boundary equal to the given coorienting normal vector field. We express the minimal number of fold components of such a stable map as a function of its number of cusps and of the normal degree of its boundary. We also show that this minimum is attained for any cooriented circle immersion of normal degree not equal to one. The first author is a research fellow of the Royal Swedish Academy of Sciences sponsored by the Knut and Alice Wallenberg foundation.     

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines.Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.     

Forcing faces in plane bipartite graphs

Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G∈Ω is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.     

Characterization of curve map graphs

In this paper we provide a characterization of curve map graphs as defined by Gavril and Schönheim, and also give a recognition algorithm for them.A curve map graph is the dual of a map obtained by placing a finite number of two-way infinite Jordan curves in the Euclidean plane in such a way that each curve divides the plane into two regions, no two curves intersect in more than one point, and any two curves which intersect at a point cross at that point.Our method is based on Gavril and Schönheim's approach, but corrects several difficulties in their characterization.     

On the existence of maximal ω-limit sets for dendrite maps

For interval maps and also for graph maps, every ω-limit set is a subset of a maximal one. In this note we construct a continuous map on a dendrite with no maximal ω-limit set. Moreover, the set of branch points is nowhere dense, every ω-limit set of the map is nowhere dense, the set of periodic points and the set of recurrent points are equal and the set of ω-limit points is not closed (an example with the last property was constructed by the authors already in [Ko?an Z, Kornecká-Kurková V, Málek M. On the centre and the set of omega-limit points of continuous maps on dendrites. Topol Appl 2009;156:2923-2931]).     

Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

We consider a planar Riemann surface R made of a non‐compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in R caused by these points to be uniformly distributed in R in terms of Euclidean geometry. Then we associate a graph G with R by taking the Voronoi diagram for the uniformly distributed cusps and show that G represents certain geometric and analytic properties of R .     

The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees

A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.     

Strongly stable real infinitesimally symplectic mappings

We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.     

图象的自反与渐近自反

本文对B（X,Y）的自反子空间及渐近自反子空间上的映射（或映射组）,分别给出了一些判别它们的图象（或联合图象）仍为自反及渐近自反的方法。     

Stable Maps and Branch Divisors

We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.     

Counting unrooted maps on the plane

A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside (“infinite”) face. An unrooted map is an equivalence class of maps under orientation-preserving homeomorphism, and a rooted map is a map with a distinguished oriented edge. Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps. In this article, using the same technique we obtain closed formulae for counting unrooted plane maps of all these classes and their duals. The corresponding formulae for rooted maps are known to be all sum-free; the formulae that we obtain for unrooted maps contain only a sum over the divisors of n. We count also unrooted two-vertex plane maps.     

Lower bound on the spectrum of the two-dimensional Schrödinger operator with a <Emphasis Type="Italic">δ</Emphasis>-perturbation on a curve

We consider the two-dimensional Schrödinger operator with a δ-potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the considered operator that are expressed explicitly in terms of the interaction strength and a parameter characterizing the curve geometry. We estimate the bottom of the spectrum for a piecewise smooth curve using parameters characterizing the geometry of the separate pieces. As applications of the obtained results, we consider curves with a finite number of cusps and general “leaky” quantum graph as the support of the δ-potential.     

Set covering and packing formulations of graph coloring: Algorithms and first polyhedral results

We consider two (0, 1)-linear programming formulations of the graph (vertex-) coloring problem, in which variables are associated with stable sets of the input graph. The first one is a set covering formulation, where the set of vertices has to be covered by a minimum number of stable sets. The second is a set packing formulation, in which constraints express that two stable sets cannot have a common vertex, and large stable sets are preferred in the objective function. We identify facets with small coefficients for the polytopes associated with both formulations. We show by computational experiments that both formulations are about equally efficient when used in a branch-and-price algorithm. Next we propose some preprocessing, and show that it can substantially speed up the algorithm, if it is applied at each node of the enumeration tree. Finally we describe a cutting plane procedure for the set covering formulation, which often reduces the size of the enumeration tree.     

共变完全多正线性映射的共变投射表示

研究了C*-代数中的共变完全多正线性映射,证明了共变完全多正线性映射可以诱导Hilbert C*-模上的共变投射表示,并且给出了共变完全多正线性映射的KS- GNS(Kasparov,Stinespring,Gel’fand,Naimark,Segal)构造.