An overview of curvature bounds and spectral theory of planar tessellations

Matthias Keller

Displaying similar documents to “An overview of curvature bounds and spectral theory of planar tessellations”

Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

Kinkar Ch. Das, Muhuo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $ω$ $(2 \leq ω \leq n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.

$Q$ -curvature and spectral invariants

Branson, Thomas

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Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes (2012)

Fundamenta Mathematicae

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We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature $ℳ_{p}^{α} (X) : = \int_{X} \int_{X} \int_{X} κ^{p} (x, y, z) d_{X}^{α} (x) d_{X}^{α} (y) d_{X}^{α} (z)$ , where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that $ℳ_{p}^{α} (X) < \infty$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant...

The resolution of the bounded $L^{2}$ curvature conjecture in general relativity

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

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This paper reports on the recent proof of the bounded $L^{2}$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^{2}$ -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

Curvature measures and fractals

Steffen Winter

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Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ’classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets $F \subseteq ℝ^{d}$ (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic...

Local spectrum and local spectral radius of an operator at a fixed vector

Janko Bračič, Vladimír Müller (2009)

Studia Mathematica

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Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with $σ_{T} (e) = σ_{δ} (T)$ , respectively $r_{T} (e) = r (T)$ , is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem

Guy Métivier (1981-1982)

Publications mathématiques et informatique de Rennes

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Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

On the total curvature of surfaces in $E^{4}$

Peter Wintgen (1978)

Colloquium Mathematicae

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A new characterization of the sphere in $R^{3}$

Thomas Hasanis (1980)

Annales Polonici Mathematici

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Let M be a closed connected surface in $R^{3}$ with positive Gaussian curvature K and let $K_{I} I$ be the curvature of its second fundamental form. It is shown that M is a sphere if $K_{I} I = c \sqrt H K^{r}$ , for some constants c and r, where H is the mean curvature of M.

On the joint spectral radius

Vladimír Müller (1997)

Annales Polonici Mathematici

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We prove the $_{p}$ -spectral radius formula for n-tuples of commuting Banach algebra elements

Strong spectral gaps for compact quotients of products of $PSL (2, ℝ)$ )

Dubi Kelmer, Peter Sarnak (2009)

Journal of the European Mathematical Society

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The existence of a strong spectral gap for quotients $Γ ∖ G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg conjectures. If $G$ has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral...

Formulae for joint spectral radii of sets of operators

Victor S. Shulman, Yuriĭ V. Turovskii (2002)

Studia Mathematica

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The formula $ϱ (M) = m a x ϱ_{χ} (M), r (M)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), $ϱ_{χ} (M)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

Mean curvature properties for $p$ -Laplace phase transitions

Berardino Sciunzi, Enrico Valdinoci (2005)

Journal of the European Mathematical Society

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This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of $p$ -Laplacian type and a double well potential $h_{0}$ with suitable growth conditions. We prove that level sets of solutions of $Δ_{p} u = h_{0}^{'} (u)$ possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature.

Gauss curvature estimates for minimal graphs

Maria Nowak, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We estimate the Gauss curvature of nonparametric minimal surfaces over the two-slit plane $ℂ ∖ ((- \infty, - 1] \cup [1, \infty))$ at points above the interval $(- 1, 1)$ .

Linear maps on Mₙ(ℂ) preserving the local spectral radius

Abdellatif Bourhim, Vivien G. Miller (2008)

Studia Mathematica

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Let x₀ be a nonzero vector in ℂⁿ. We show that a linear map Φ: Mₙ(ℂ) → Mₙ(ℂ) preserves the local spectral radius at x₀ if and only if there is α ∈ ℂ of modulus one and an invertible matrix A ∈ Mₙ(ℂ) such that Ax₀ = x₀ and $Φ (T) = α A T A^{- 1}$ for all T ∈ Mₙ(ℂ).

On the Spectral Characterizations of Graphs

Jing Huang, Shuchao Li (2017)

Discussiones Mathematicae Graph Theory

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Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let [...] p2q $𝒰_{p}^{2 q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] p2q $𝒱_{p}^{2 q}$ the subset of [...] p2q $𝒰_{p}^{2 q}$ with odd p...

On some properties of a Riemannian space $V_{4}$ with constant scalar curvature

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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Accurate Spectral Asymptotics for periodic operators

Victor Ivrii (1999)

Journées équations aux dérivées partielles

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Asymptotics with sharp remainder estimates are recovered for number $𝐍 (τ)$ of eigenvalues of operator $A (x, D) - t W (x, x)$ crossing level $E$ as $t$ runs from $0$ to $τ$ , $τ \to \infty$ . Here $A$ is periodic matrix operator, matrix $W$ is positive, periodic with respect to first copy of $x$ and decaying as second copy of $x$ goes to infinity, $E$ either belongs to a spectral gap of $A$ or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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Schur Lemma and the Spectral Mapping Formula

Antoni Wawrzyńczyk (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula $σ_{l} (S)$ = ${(λ (s))}_{s \in S} \in ℂ^{S} | {s - λ (s)}_{s \in S}$ generates a proper left ideal $.$ Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_{l} (S)$ has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.