Labeled floor diagrams for plane curves

Sergey Fomin; Grigory Mikhalkin

Displaying similar documents to “Labeled floor diagrams for plane curves”

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

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For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision

Jeremy Yirmeyahu Kaminski, Michael Fryers, Mina Teicher (2005)

Journal of the European Mathematical Society

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We study some geometric configurations related to projections of an irreducible algebraic curve embedded in $ℂ ℙ^{3}$ onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve $X$ embedded in $ℂ ℙ^{3}$ , of degree $d$ and genus $g$ , can be recovered using its projections from points onto embedded projective planes. The embeddings are unknown. The only input is the defining...

Turán's problem and Ramsey numbers for trees

Zhi-Hong Sun, Lin-Lin Wang, Yi-Li Wu (2015)

Colloquium Mathematicae

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Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with $V = v ₀, v ₁, . . ., v_{n - 1}$ , $E ₁ = v ₀ v ₁, . . ., v ₀ v_{n - 3}, v_{n - 4} v_{n - 2}, v_{n - 3} v_{n - 1}$ and $E ₂ = v ₀ v ₁, . . ., v ₀ v_{n - 3}, v_{n - 3} v_{n - 2}, v_{n - 3} v_{n - 1}$ . For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for $r (T ₘ, T ₙ^{i})$ , where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.

Trees and the dynamics of polynomials

Laura G. DeMarco, Curtis T. McMullen (2008)

Annales scientifiques de l'École Normale Supérieure

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In this paper we study branched coverings of metrized, simplicial trees $F : T \to T$ which arise from polynomial maps $f : ℂ \to ℂ$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $ℙ T_{D}$ compactifying the moduli space of polynomials of degree $D$ ; that $F$ records the asymptotic behavior of the multipliers of $f$ ; and that any meromorphic family of polynomials over $Δ^{*}$ can be completed by a unique tree at its central fiber. In the cubic case we give a...

Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants ${𝒮𝒪}_{5}$ and ${𝒮𝒪}_{6}$

Wei Gao (2024)

Czechoslovak Mathematical Journal

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I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by ${𝒮𝒪}_{1}, {𝒮𝒪}_{2}, \dots, {𝒮𝒪}_{6}$ . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal...

The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

Jan J. Dijkstra, Kirsten I. S. Valkenburg (2010)

Fundamenta Mathematicae

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One way to generalize complete Erdős space $_{c}$ is to consider uncountable products of zero-dimensional $G_{δ}$ -subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $ℚ$ . The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$ , and therefore the set of rational points is finite.

A note on the cubical dimension of new classes of binary trees

Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena (2015)

Czechoslovak Mathematical Journal

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The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^{n}$ vertices, $n \geq 1$ , is $n$ . The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice...

Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

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The topology and combinatorial structure of the Mandelbrot set $ℳ^{d}$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in $ℳ^{d}$ . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^{d}$ . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

Complete pluripolar curves and graphs

Tomas Edlund (2004)

Annales Polonici Mathematici

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It is shown that there exist $C^{\infty}$ functions on the boundary of the unit disk whose graphs are complete pluripolar. Moreover, for any natural number k, such functions are dense in the space of $C^{k}$ functions on the boundary of the unit disk. We show that this result implies that the complete pluripolar closed $C^{\infty}$ curves are dense in the space of closed $C^{k}$ curves in ℂⁿ. We also show that on each closed subset of the complex plane there is a continuous function whose graph is complete pluripolar. ...

Quasi-tree graphs with the minimal Sombor indices

Yibo Li, Huiqing Liu, Ruiting Zhang (2022)

Czechoslovak Mathematical Journal

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The Sombor index $S O (G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)}$ of all edges $u v$ of $G$ , where $d_{G} (u)$ denotes the degree of the vertex $u$ in $G$ . A connected graph $G = (V, E)$ is called a quasi-tree if there exists $u \in V (G)$ such that $G - u$ is a tree. Denote $𝒬 (n, k) = {G : G$ is a quasi-tree graph of order $n$ with $G - u$ being a tree and $d_{G} (u) = k}$ . We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬 (n, k)$ . Furthermore, we characterized the corresponding extremal graphs, respectively.

A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

Yaping Mao, Christopher Melekian, Eddie Cheng (2023)

Czechoslovak Mathematical Journal

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For a connected graph $G = (V, E)$ and a set $S \subseteq V (G)$ with at least two vertices, an $S$ -Steiner tree is a subgraph $T = (V^{'}, E^{'})$ of $G$ that is a tree with $S \subseteq V^{'}$ . If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$ -Steiner tree. Two $S$ -Steiner trees are if they share no vertices other than $S$ and have no edges in common. For $S \subseteq V (G)$ and $| S | \geq 2$ , the pendant tree-connectivity $τ_{G} (S)$ is the maximum number of internally disjoint pendant $S$ -Steiner trees in $G$ , and for $k \geq 2$ , the $k$ -pendant tree-connectivity $τ_{k} (G)$ is the...