scholarly journals Upper bounds for the Neumann eigenvalues on a bounded domain in euclidean space

1992 ◽  
Vol 106 (2) ◽  
pp. 353-357 ◽  
Author(s):  
P Kröger
Keyword(s):  
Euclidean Space ◽  
Bounded Domain ◽  
Upper Bounds ◽  
Neumann Eigenvalues

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

scholarly journals Weighted faces of Poisson hyperplane tessellations

2009 ◽  
Vol 41 (03) ◽  
pp. 682-694 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Upper Bounds ◽  
Dimensional Euclidean Space ◽  
Lower And Upper Bounds ◽  
Intrinsic Volumes ◽  
Vertex Number ◽  
Zero Cell ◽  
Lower Dimensional

We study lower-dimensional volume-weighted typical faces of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. After showing how their distribution can be derived from that of the zero cell, we obtain sharp lower and upper bounds for the expected vertex number of the volume-weighted typical k-face (k=2,…,d). The bounds are respectively attained by parallel mosaics and by isotropic tessellations. We conclude with a remark on expected face numbers and expected intrinsic volumes of the zero cell.

Generalization of Schwarz-Pick Lemma to Invariant Volume

1969 ◽  
Vol 21 ◽  
pp. 669-674
Author(s):  
K. T. Hahn ◽  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Explicit Form ◽  
Bounded Domain ◽  
Kähler Manifold ◽  
Siegel Domain ◽  
Homogeneous Domain ◽  
Kahler Manifold ◽  
Invariant Volume ◽  
Complex Euclidean Space ◽  
Bounded Homogeneous Domain

In this paper we give an extension of (6, Theorem 1), using a similar method of proof, to every homogeneous Siegel domain of second kind which can be mapped biholomorphically into a Kâhler manifold of a certain class (Theorem 1). Then by a well-known result of Vinberg, Gindikin, and Pjateckiï-Sapiro (10) that every bounded homogeneous domain D,contained in a complex euclidean space CN,can be mapped biholomorphically onto an affinely homogeneous Siegel domain of second kind, the theorem follows for D(Theorem 2). (6, Theorem 1) is a generalization of the Ahlfors version of the Schwarz-Pick lemma in C1(1) to invariant volume for a star-like homogeneous bounded domain in CN;see also (4). In § 3 we give the inequality for a special non-symmetric Siegel domain of second kind using an explicit form of TD(z, )due to Lu (7).

scholarly journals Lower bounds for fractional Laplacian eigenvalues

2014 ◽  
Vol 16 (06) ◽  
pp. 1450032
Author(s):  
Guoxin Wei ◽  
He-Jun Sun ◽  
Lingzhong Zeng
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Bounded Domain ◽  
Lower Bounds ◽  
Fractional Laplacian ◽  
Dimensional Euclidean Space ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of The Laplacian ◽  
Eigenvalue Inequality

In this paper, we investigate eigenvalues of fractional Laplacian (–Δ)α/2|D, where α ∈ (0, 2], on a bounded domain in an n-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which improves some results due to Yildirim Yolcu and Yolcu in [Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math. 15(3) (2013) 1250048]. In particular, for the case of Laplacian, we obtain a sharper eigenvalue inequality, which gives an improvement of the result due to Melas in [A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003) 631–636].

Universal inequalities for eigenvalues of a system of elliptic equations

2009 ◽  
Vol 139 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Hong-Cang Yang
Keyword(s):  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Lower Order ◽  
Elliptic Equations ◽  
Elastic Vibration ◽  
Dimensional Euclidean Space ◽  
Explicit Method ◽  
Universal Inequalities ◽  
Image Position

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume thatare eigenvalues of an eigenvalue problem of a system of n elliptic equations:In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

scholarly journals Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

2021 ◽  
Author(s):  
Léonard Tschanz
Keyword(s):  
Growth Rate ◽  
Cayley Graph ◽  
Bounded Domain ◽  
Polynomial Growth ◽  
Cayley Graphs ◽  
Upper Bounds ◽  
Comparison Theorems ◽  
Steklov Problem ◽  
Steklov Spectrum ◽  
Steklov Eigenvalues

AbstractWe study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$ ( Ω l , B l ) l = 1 ∞ a sequence of subgraphs of $$\Gamma$$ Γ such that $$|\Omega _l| \longrightarrow \infty$$ | Ω l | ⟶ ∞ , we prove that for each $$k \in {\mathbb {N}}$$ k ∈ N , the kth eigenvalue tends to 0 proportionally to $$1/|B|^{\frac{1}{d-1}}$$ 1 / | B | 1 d - 1 , where d represents the growth rate of $$\Gamma$$ Γ . The method consists in associating a manifold M to $$\Gamma$$ Γ and a bounded domain $$N \subset M$$ N ⊂ M to a subgraph $$(\Omega , B)$$ ( Ω , B ) of $$\Gamma$$ Γ . We find upper bounds for the Steklov spectrum of N and transfer these bounds to $$(\Omega , B)$$ ( Ω , B ) by discretizing N and using comparison theorems.

On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain

2018 ◽  
Vol 25 (2) ◽  
pp. 187-194 ◽  
Author(s):  
Grigor Barsegian ◽  
Wenjun Yuan
Keyword(s):  
Differential Equations ◽  
Bounded Domain ◽  
Meromorphic Functions ◽  
Upper Bounds ◽  
Higher Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Complex Differential Equations ◽  
Similar Solutions

Abstract The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.

Weighted faces of Poisson hyperplane tessellations

2009 ◽  
Vol 41 (3) ◽  
pp. 682-694 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Upper Bounds ◽  
Dimensional Euclidean Space ◽  
Lower And Upper Bounds ◽  
Intrinsic Volumes ◽  
Vertex Number ◽  
Zero Cell ◽  
Lower Dimensional

We study lower-dimensional volume-weighted typical faces of a stationary Poisson hyperplane tessellation in d-dimensional Euclidean space. After showing how their distribution can be derived from that of the zero cell, we obtain sharp lower and upper bounds for the expected vertex number of the volume-weighted typical k-face (k=2,…,d). The bounds are respectively attained by parallel mosaics and by isotropic tessellations. We conclude with a remark on expected face numbers and expected intrinsic volumes of the zero cell.

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