On the number of intersections of two polygons

Jakub Černý; Jan Kára; Daniel Kráľ; Pavel Podbrdský; Miroslava Sotáková; Robert Šámal

Displaying similar documents to “On the number of intersections of two polygons”

$𝔤$ -quasi-Frobenius Lie algebras

David N. Pham (2016)

Archivum Mathematicum

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A Lie version of Turaev’s $\bar{G}$ -Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$ -quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮, β)$ together with a left $𝔤$ -module structure which acts on $𝔮$ via derivations and for which $β$ is $𝔤$ -invariant. Geometrically, $𝔤$ -quasi-Frobenius Lie algebras are the Lie algebra structures associated to...

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano gli omomorfismi reticolari ( $\mathcal{L}$ -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$ -algebre di Lie non ammettono $\mathcal{L}$ -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « $\mathcal{L}$ -omomorfismo su una catena di lunghezza $n$ .

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras

Vladimir V. Bavula (2016)

Czechoslovak Mathematical Journal

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Let $L_{n} = K [x_{1}^{\pm 1}, ..., x_{n}^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_{n} : = {Der}_{K} (L_{n})$ the Lie algebra of $K$ -derivations of the algebra $L_{n}$ , the so-called Witt Lie algebra, and let $Vir$ be the Virasoro Lie algebra which is a $1$ -dimensional central extension of the Witt Lie algebra. The Lie algebras $W_{n}$ and $Vir$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${Aut}_{Lie} (Vir) ≃ {Aut}_{Lie} (W_{1}) ≃ {\pm 1} ⋉ K^{*}$ , and give a short proof that ${Aut}_{Lie} (W_{n}) ≃ {Aut}_{K - alg} (L_{n}) ≃ {GL}_{n} (ℤ) ⋉ K^{* n}$ .

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Subalgebra of $L_{1} (G)$ associated with Laplacian on a Lie group

A. Hulanicki (1974)

Colloquium Mathematicae

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Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

K. H. Neeb (2014)

Annales de l’institut Fourier

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A unitary representation $π$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i d π (x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$ . We classify all irreducible semibounded representations of the groups ${\hat{ℒ}}_{φ} (K)$ which are double extensions of the twisted loop group $ℒ_{φ} (K)$ , where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant)...

The covering semigroup of invariant control systems on Lie groups

Víctor Ayala, Eyüp Kizil (2016)

Kybernetika

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It is well known that the class of invariant control systems is really relevant both from theoretical and practical point of view. This work was an attempt to connect an invariant systems on a Lie group $G$ with its covering space. Furthermore, to obtain algebraic properties of this set. Let $G$ be a Lie group with identity $e$ and $Σ \subset 𝔤$ a cone in the Lie algebra $𝔤$ of $G$ that satisfies the Lie algebra rank condition. We use a formalism developed by Sussmann, to obtain an algebraic structure on...

When unit groups of continuous inverse algebras are regular Lie groups

Helge Glöckner, Karl-Hermann Neeb (2012)

Studia Mathematica

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It is a basic fact in infinite-dimensional Lie theory that the unit group $A^{\times}$ of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group $A^{\times}$ is regular in Milnor’s sense. Notably, $A^{\times}$ is regular if A is Mackey-complete and locally m-convex.

Correct solvability of a general differential equation of the first order in the space $L_{p} (ℝ)$

N. Chernyavskaya, L.A. Shuster (2015)

Archivum Mathematicum

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We consider the equation $- r (x) y^{'} (x) + q (x) y (x) = f (x), x \in ℝ$ where $f \in L_{p} (ℝ)$ , $p \in [1, \infty]$ ( $L_{\infty} (ℝ) : = C (ℝ)$ ) and $0 < r \in C^{} (ℝ), 0 \leq q \in L_{1}^{} (ℝ) .$ We obtain minimal requirements to the functions $r$ and $q$ , in addition to (), under which equation () is correctly solvable in $L_{p} (ℝ)$ , $p \in [1, \infty]$ .

Counting triangles that share their vertices with the unit $n$ -cube

Brandts, Jan, Cihangir, Apo

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This paper is about $0 / 1$ -triangles, which are the simplest nontrivial examples of $0 / 1$ -polytopes: convex hulls of a subset of vertices of the unit $n$ -cube $I^{n}$ . We consider the subclasses of right $0 / 1$ -triangles, and acute $0 / 1$ -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of $I^{n}$ .

The vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_{D} [S] = ⋃_{x, y \in S} I_{D} [x, y]$ . A set S of vertices is a detour convex set if $I_{D} [S] = S$ . The detour convex hull ${[S]}_{D}$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

Projection of Wreath Products of Lie Algebras

Alexander A. Lašhi (1980)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Sia $K$ un dominio a ideali principali, che non sia un campo e sia $L$ un'algebra di Lie su $K$ . Principale risultato: Se $L=A\,wr\,B$ , cioè se $L$ è prodotto intrecciato delle algebre di Lie precarie $A$ e $B$ , ogni isomorfismo reticolare normale di $L$ è indotto da un isomorfismo. Si prova anche, mediante un esempio, che per le algebre sopra un campo il teorema non è vero.

Sharp upper bounds for a singular perturbation problem related to micromagnetics

Arkady Poliakovsky (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We construct an upper bound for the following family of functionals ${E_{ε}}_{ε > 0}$ , which arises in the study of micromagnetics: $E_{ε} (u) = \int_{Ω} {ε | \nabla u |}^{2} + \frac{1}{ε} \int_{ℝ^{2}} {| H_{u} |}^{2} .$ Here $Ω$ is a bounded domain in $ℝ^{2}$ , $u \in H^{1} (Ω, S^{1})$ (corresponding to the magnetization) and $H_{u}$ , the demagnetizing field created by $u$ , is given by $\{\begin{matrix} div (\tilde{u} + H_{u}) = 0 & in ℝ^{2}, \\ curl H_{u} = 0 & in ℝ^{2}, \end{matrix}$ where $\tilde{u}$ is the extension of $u$ by $0$ in $ℝ^{2} ∖ Ω$ . Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.

The diffeomorphism group of a non-compact orbifold

A. Schmeding

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We endow the diffeomorphism group $D i f f_{O r b} (Q,)$ of a paracompact (reduced) orbifold with the structure of an infinite-dimensional Lie group modeled on the space of compactly supported sections of the tangent orbibundle. For a second countable orbifold, we prove that $D i f f_{O r b} (Q,)$ is C⁰-regular, and thus regular in the sense of Milnor. Furthermore, an explicit characterization of the Lie algebra associated to $D i f f_{O r b} (Q,)$ is given.

Stronger bounds for generalized degrees and Menger path systems

R.J. Faudree, Zs. Tuza (1995)

Discussiones Mathematicae Graph Theory

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For positive integers d and m, let $P_{d, m} (G)$ denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that $P_{d, m} (G)$ is satisfied have been investigated. In particular, it has been shown,...