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.     

Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

The calculation of the L 2-norm of the index of a plane curve and related formulas

We provide formulas for calculating the L ²-norm of the index function of a rectifiable closed curve in the complex plane. Some applications to isoperimetric inequalities are given. The main tool used is the decomposition of a rectifiable closed curve into a sequence of Jordan curves, curves with null index functions, and an exceptional set.     

On Minimal Annuli with a Straight Line Boundary

Shiffman proved that if a minimal annulus A in a slab is bounded by two convex Jordan curves contained respectively in the two boundary planes P and Q of the slab, then A intersects all parallel planes between P and Q in strictly convex curves. We generalize Shiffman's result to the case that A is bounded by a strictly convex C² Jordan curve and a straight line. We show that in this case Shiffman's result is still true.     

Topology of dynamical systems in finite groups and number theory

We study and develop a very new object introduced by V.I. Arnold: a monad is a triple consisting of a finite set, a map from that finite set to itself and the monad graph which is the directed graph whose vertices are the elements of the finite set and whose arrows lead each vertex to its image (by the map). We consider the case in which the finite set entering in the monad definition is a finite group G and the map is the Frobenius map, for some k∈Z. We study the Frobenius dynamical system defined by the iteration of the monad fk, and also study the combinatorics and topology (i.e., the discrete invariants) of the monad graph. Our study provides useful information about several structures on the group associated to the monad graph. So, for example, several properties of the quadratic residues of finite commutative groups can be obtained in terms of the graph of the Frobenius monad .     

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.     

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.     

Relation-induced connectedness in the digital plane

We introduce and discuss a connectedness induced by n-ary relations (\(n>1\) an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer \(n>1\), we introduce one such n-ary relation on the digital plane \({\mathbb {Z}}^2\) and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For \(n=2\), such a structure coincides with the (specialization order of the) Khalimsky topology and, for \(n>2\), it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology.     

Gaussian maps of plane curves with nine singular points

We consider nonsingular curves which are the normalization of plane curves with nine ordinary singular points, viewing them as embedded in the blow-up X of the projective plane along their singular points. For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. The proof makes essential use of the geometry of X.     

On the curve diffusion flow of closed plane curves

In this paper, we consider the steepest descent H ^?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L ² oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.     

Bi-lipschitz extensions in the plane

We show that a bi-Lipschitz mapF from a subset of a line or a circle into the plane can be extended to a bi-Lipschitz map of the whole plane onto itself, with the bi-Lipschitz constant depending only on that ofF. This research was supported in part by NSF Grant DMS-9305792.     

A Note on the Complete Unitary Invariant for Some Subnormal Operators

A complete unitary invariant for an irreducible subnormal operators with m.n.e. N satisfying the condition that σ(N) consists of either a Jordan curve or two Jordan curves is given. Also the model of such subnormal operators is established.     

Segment representation of a subclass of co-planar graphs

A graph is a segment graph if its vertices can be mapped to line segments in the plane such that two vertices are adjacent if and only if their corresponding line segments intersect. Kratochvíl and Kuběna asked the question of whether the complements of planar graphs, called co-planar graphs, are segment graphs. We show here that the complements of all partial 2-trees are segment graphs.     

On edges crossing few other edges in simple topological complete graphs

We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graphT=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple if any two edges meet in at most one common point.Let h=h(n) be the smallest integer such that every simple topological complete graph on n vertices contains an edge crossing at most h other edges. We show that Ω(n^3/2)≤h(n)≤O(n²/log^1/4n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn².     

Orchards in elliptic curves over finite fields

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.     

The complete classification of fibres in pencils of curves of genus two

This article contains geometrical classification of all fibres in pencils of curves of genus two, which is essentially different from the numerical one given by Ogg ([11]) and Iitaka ([7]). Given a family π:X→D of curves of genus two which is smooth overD′=D?{0}, we define a multivalued holomorphic mapT _π fromD′ into the Siegel upper half plane of degree two, and three invariants called “monodromy”, “modulus point” and “degree”. We assert that the family π is completely determined byT _π, and its singular fibre by these three invariants. Hence all types of fibres are classified by these invariants and we list them up in a table, which is the main part of this article.     

A new proof of a theorem of Dembowski

It was shown by P.Dembowski [1;Satz 3] that any finite semiaffine plane(=FSAP) is of the types:

1. A finite affine plane,
2. A finite projective plane with one line and all its points except one deleted,
3. A finite projective plane with one point deleted,
4. A finite projective plane.

This was established by using the results obtained for natural parallelisms of incidence structures. The purpose of this note is to give a new proof based on purely combinatorial arguments.     

Fluctuations in the number of points on smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8].