ISSN 1439-8516 CN 11-2039/O1
In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Fréchet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*.
In this paper, we study the higher genus FJRW theory of Fermat cubic singularity with maximal group of diagonal symmetries using Givental formalism. As results, we prove the finite generation property and holomorphic anomaly equation for the associated FJRW theory. Via general LG-LG mirror theorem, our results also hold for the Saito-Givental theory of the Fermat cubic singularity.
We present some improvements of the Li-Yau heat kernel estimate on a Riemannian manifold with Ricci curvature bounded below. As a consequence we prove a gradient estimate to the heat kernel with an optimal leading term.
We study the Knieper measures of the geodesic flows on non-compact rank 1 manifolds of non-positive curvature. We construct the Busemann density on the ideal boundary, and prove that if there is a Knieper measure on T1M with finite total mass, then the Knieper measure is unique, up to a scalar multiple. Our result partially extends Paulin-Pollicott-Shapira's work on the uniqueness of finite Gibbs measure of geodesic flows on negatively curved non-compact manifolds to non-compact manifolds of non-positive curvature.
Assume that r is a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bilinear form, and ν:r → osp(p) is a homomorphism of Lie superalgebras. In this paper, we give a necessary and sufficient condition for r⊕p to be a quadratic Lie superalgebra. Then, we define the cubic Dirac operator D(g,r) on g and give a formula of (D(g,r))2. Finally, we get the Vogan's conjecture for quadratic Lie superalgebras by D(g,r).
By the new spectrum originated from the single-valued extension property, we give the necessary and sufficient conditions for a bounded linear operator defined on a Banach space for which property (ω) holds. Meanwhile, the relationship between hypercyclic property (or supercyclic property) and property (ω) is discussed.
Define the differential operators φn for n ∈ N inductively by φ1[f](z)=f(z) and φn+1[f](z)=f(z)φn[f](z) + d/dz φn[f](z). For a positive integer k ≥ 2 and a positive number δ, let ?? be the family of functions f meromorphic on domain D ⊂ C such that φk[f](z) ≠ 0 and|Res(f, a) -j| ≥ δ for all j ∈ {0, 1,..., k -1} and all simple poles a of f in D. Then ?? is quasi-normal on D of order 1.
In this paper, we prove that the weighted BMO space
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is independent of the scale p ∈ (0, ∞) in sense of norm when ω ∈ A1. Moreover, we can replace Lp(ω) by Lp,∞(ω). As an application, we characterize this space by the boundedness of the bilinear commutators[b, T]j (j=1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol b, from Lp1(ω)×Lp2(ω) to Lp(ω1-p) with 1 < p1, p2 < ∞ and 1/p=1/p1 + 1/p2. Thus we answer the open problem proposed by Chaffee affirmatively.
The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-k-coloring of a graph is a vertex coloring using k distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every d-degenerate graph with maximum degree at most Δ is equitably tree-k-colorable for every integer k ≥ (Δ + 1)/2 provided that Δ ≥ 9.818d, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.
Let G be a nonempty closed subset of a Banach space X. Let ??(X) be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and ??G(X)={A ∈ ??(X):A ∩ G=∅}, where the closure is taken in the metric space (??(X), H). For x ∈ X and F ∈ ??G(X), we denote the nearest point problem inf{||x -g||:g ∈ G} by min(x, G) and the mutually nearest point problem inf{||f -g||:f ∈ F, g ∈ G} by min(F, G). In this paper, parallel to well-posedness of the problems min(x, G) and min(F, G) which are defined by De Blasi et al., we further introduce the weak well-posedness of the problems min(x, G) and min(F, G). Under the assumption that the Banach space X has some geometric properties, we prove a series of results on weak well-posedness of min(x, G) and min(F, G). We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.
