共查询到20条相似文献,搜索用时 875 毫秒
1.
Schnyder characterized planar graphs in terms of order dimension. Brightwell and Trotter proved that the dimension of the
vertex-edge-face poset P
M
of a planar map M is at most four. In this paper we investigate cases where dim(P
M
) ≤ 3 and also where dim(Q
M
) ≤ 3; here Q
M
denotes the vertex-face poset of M. We show:
• |
If M contains a K
4-subdivision, then dim(P
M
) = dim(Q
M
) = 4. 相似文献
2.
Karl-Theodor Sturm 《Acta Mathematica》2006,196(1):133-177
We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound
(introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as
the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to
and dim(M) ⩽ N.
The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact.
Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers
theorem. Moreover, it implies the doubling property and local, scale-invariant Poincaré inequalities on balls. In particular,
it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels. 相似文献
3.
Liangping Jiang 《Journal of Mathematical Sciences》2011,177(3):395-401
The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V(m + 1) \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10] 相似文献
4.
Kwang C. Shin 《Potential Analysis》2011,35(2):145-174
For integers m ≥ 3 and 1 ≤ ℓ ≤ m − 1, we study the eigenvalue problems − u
″(z) + [( − 1)ℓ(iz)
m
− P(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays
argz=-\fracp2±\frac(l+1)pm+2\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2} in the complex plane, where P is a polynomial of degree at most m − 1. We provide asymptotic expansions of the eigenvalues λ
n
. Then we show that if the eigenvalue problem is PT\mathcal{PT}-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when
gcd(m,l)=1\gcd(m,\ell)=1, the eigenvalue problem has infinitely many real eigenvalues if and only if one of its translations or itself is PT\mathcal{PT}-symmetric. Also, we will prove some other interesting direct and inverse spectral results. 相似文献
5.
Lorenzo Mazzieri 《Calculus of Variations and Partial Differential Equations》2009,34(4):453-473
In this paper we construct a family of new (topologically distinct) solutions to the Einstein constraint equations by performing
the generalized connected sum (or fiber sum) of two known compact m-dimensional constant mean curvature solutions (M
1, g
1, Π1) and (M
2, g
2,Π2) along a common isometrically embedded k-dimensional sub-manifold (K, g
K
). Away from the gluing locus the metric and the second fundamental form of the new solutions can be chosen as close as desired
to the ones of the original solutions. The proof is essentially based on the conformal method and the geometric construction
produces a polyneck between M
1 and M
2 whose metric is modeled fiber-wise (i. e. along the slices of the normal fiber bundle of K) around a Schwarzschild metric; for these reasons the codimension n : = m − k of K in M
1 and M
2 is required to be ≥ 3. In this sense our result is a generalization of the Isenberg–Mazzeo–Pollack gluing, which works
for connected sum at points and in dimension 3. The solutions we obtain for the Einstein constraint equations can be used
to produce new short time vacuum solutions of the Einstein system on a Lorentzian (m + 1)-dimensional manifold, as guaranteed by a well known result of Choquet-Bruhat. 相似文献
6.
Mitchel T. Keller Yi-Huang Shen Noah Streib Stephen J. Young 《Journal of Algebraic Combinatorics》2011,33(2):313-324
Let K be a field and S=K[x
1,…,x
n
]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in
the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze
squarefree Veronese ideals in S. In particular, if I
n,d
is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I
n,d
)=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I
n,d
)≤⌊(n−d)/(d+1)⌋+d. 相似文献
7.
Kathrin Bacher 《Potential Analysis》2010,32(1):1-15
In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces (M,d,m) denoted by BBL(K,N) for two numbers K,N ∈ ℝ with N ≥ 1. In the first part we prove that BBL(K,N) holds true on metric measure spaces satisfying a curvature-dimension condition CD(K,N) developed and studied by Lott and Villani in (Ann Math 169:903–991, 2007) as well as by Sturm in (Acta Math 196(1):133–177, 2006). The aim of the second part is to show that BBL(K,N) is stable under convergence of metric measure spaces with respect to the L
2-transportation distance. 相似文献
8.
Lin-Feng Wang 《Annals of Global Analysis and Geometry》2010,37(4):393-402
Let M be an n-dimensional complete non-compact Riemannian manifold, dμ = e
h
(x)dV(x) be the weighted measure and
\trianglem{\triangle_{\mu}} be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry–émery curvature is bounded from below by Ric
m
≥ −(m − 1)K, K ≥ 0, then the bottom of the Lm2{{\rm L}_{\mu}^2} spectrum λ1(M) is bounded by
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