Nondensity Property for ≤_T~h Reducibility

Let A, B are sets. We can define the reducibility ≤_T~h by A≤_T~hB iff there-exists a polynomial-bounded oracle Turing machine M such that(1) x∈A M~B accepts x.(2) There exists a polynomial q such that for all x, if M~B(x) queries B     

The Equivalence of Hypercontractivity and Logarithmic Sobolev Inequality for q ((1 ≤ q ≤ 1) -Ornstein-Uhlenbeck Semigroup

In this paper the author proves the equivalence of hypercontractivity and logarithmic Sobolev inequality for q-Ornstein-Uhlenbeck semigroup U_t~((q))= Γ_q(e~(-t)I)(-1 ≤q ≤ 1), where Γ_q is a q-Gaussian functor.     

sinx≤x≤tanx的简单运用

众所周知,sinx≤x≤tanx,z∈[0,π/2], (*)当且仅当x=0时等号成立．三角不等式竞赛题外形及结构较奇特,证明方法灵活多变,无章可循．如果能很好地利用重要结论(*),可收到出奇制胜的效果． 1、求证具体角的三角函数构成的不等式例1 证明:sin20°≤7/20．(第12届全俄数     

Gap theorems for Kähler-Ricci solitons

In this paper, we prove that a gradient shrinking compact K?hler-Ricci soliton cannot have too large Ricci curvature unless it is K?hler-Einstein. Received: 23 October 2007, Revised: 28 February 2008     

Bochner identities for Kählerian gradients

We discuss algebraic properties for the symbols of geometric first order differential operators on Kähler manifolds. Through a study of the universal enveloping algebra and higher Casimir elements, we know a lot of relations for the symbols, which induce Bochner identities for the operators. As applications, we have vanishing theorems, eigenvalue estimates, and so on.     

L^p-gradient Estimates of Symmetric Markov Semigroups for 1 〈 p ≤ 2

For 1 〈 p ≤2, an L^p-gradient estimate for a symmetric Markov semigroup is derived in a general framework, i.e. ‖Γ^/2（Ttf）‖p≤Cp/√t‖p, where F is a carre du champ operator. As a simple application we prove that F1/2（（I- L） ^-α） is a bounded operator from L^p to L^v provided that 1 〈 p 〈 2 and 1/2〈α〈1. For any 1 〈 p 〈 2, q 〉 2 and 1/2 〈α 〈 1, there exist two positive constants cq,α,Cp,α such that ‖Df‖p≤ Cp,α‖（I - L）^αf‖p,Cq,α（I-L）^（1-α）‖Df‖q＋‖f‖q, where D is the Malliavin gradient （[2]） and L the Ornstein-Uhlenbeck operator.     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

Regularity for weakly (K1, K2)-quasiregular mappings

In this paper, we first give the definition of weakly (K1, K2)-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Holder inequality, we obtain their regularity property: For any ql that satisfies 0 < K1n(n+4)/22n+1 × 100n2[23n/2(25n + 1)](n - q1) < 1, there exists p1 = p1(n, q1, K1, K2) > n, such that any (K1, K2)-quasiregular mapping f ∈W(loc)(1,q1)(Ω,Rn) is in fact in W(loc)(1,p1)(Ω,Rn). That is, f is (K1, K2)-quasiregular in the usual sense.     

Some comparison theorems for K?hler manifolds

In this work, we will verify some comparison results on K?hler manifolds. They are: complex Hessian comparison for the distance function from a closed complex submanifold of a K?hler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang (J Differ Geom 69(1):43–47, 2005).     

The Künneth formula for Hilbert complexes

A variant of the Künneth formula for tensor products of Fredholm complexes of Hilbert spaces is given.     

On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of K n

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by $\operatorname {cr}(P)$ , is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P). Similarly, a k-edge, 0??k??n/2?1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ??k-edges of P is denoted by E _??k (P). Let $\overline {\mathrm {cr}}(n)$ , h(n), and E _??k (n) denote the minimum of $\operatorname {cr}(P)$ , the maximum of h(P), and the minimum of E _??k (P), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E _??k (n) is tight for k<?(4n?2)/9? and improve it for all k???(4n?2)/9?. This in turn improves the lower bound on $\overline {\mathrm {cr}}(n)$ from $0.37968\binom{n}{4}+\varTheta (n^{3})$ to $\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})\geq 0.37997\binom{n}{4}+\varTheta (n^{3})$ . We also give the exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for all n??27. Exact values were known only for n??18 and odd n??21 for the crossing number, and for n??14 and odd n??21 for halving lines.     

A New Inequality for Weakly （K1, K2）-Quasiregular Mappings

We obtain a new inequality for weakly （K1,K2）-quasiregular mappings by using the McShane extension method. This inequality can be used to derive the self-improving regularity of （K1, K2）-Quasiregular Mappings.     

b≤2000,m≤10的广义Fermat素数

设b为偶数，本文基于作者的原有结果，进一步改进了算法，对于b≤2000,m≤10给出了所有广义Fermat素数F(b,m)=b^2m 1，其中最大的是1632^1024 1,有3290位。     

Duals of Hardy Spaces of B-valued Martingales for 0<r≤1

The dual of B-valued martingale Hardy space Hs(p)r(B) with small index 0 r ≤ 1,which is associated with the conditional p-variation of B-valued martingale,is characterized.In order to obtain the results,a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved.Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space.     

One-radius results for supermedian functions on {\mathbb R^d} , d ≤ 2

A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c ₀, where c ₀ = 1, if d = 2, whereas 2.50 < c ₀ < 2.51, if d = 1.