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1.
To a given immersion i:Mn? \mathbb Sn+1{i:M^n\to \mathbb S^{n+1}} with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|2. We prove the existence of a constant C n (R) depending on R and n so that R ≥ 1 and sup |A|2 = C n (R) imply that the hypersurface is a H(r)-torus \mathbb S1(?{1-r2})×\mathbb Sn-1 (r){\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C n (R) there is a complete hypersurface in \mathbb Sn+1{\mathbb S^{n+1}} with constant scalar curvature R and sup |A|2 = C, answering questions raised by Q. M. Cheng.  相似文献   

2.
The composition operators on weighted Bloch space   总被引:9,自引:0,他引:9  
We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B log = { f ? H(D): supz ? D (1-| z|2) ( log\frac21-| z|2 )| f¢(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +¥} +\infty \} , where H(D) be the class of all analytic functions on D.  相似文献   

3.
Given an isotropic random vector X with log-concave density in Euclidean space \mathbbRn{\mathbb{R}^n} , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that
l \mathbbP( | |X| - ?n | 3 t?n ) £ C  exp ( -cn1/2 min(t3, t) )   "t 3 0, \begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array}  相似文献   

4.
We give a short proof that the largest component C 1 of the random graph G(n, 1/n) is of size approximately n 2/3. The proof gives explicit bounds for the probability that the ratio is very large or very small. In particular, the probability that n −2/3|C 1| exceeds A is at most e - cA3{e^{ - c{A^3}}} for some c > 0.  相似文献   

5.
6.
Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t (v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t (v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t (v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t (v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if
lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ¥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}  相似文献   

7.
The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with |S| = W(nlog32)|S| = \Omega ({n^{{{\log }_3}2}}) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is
|S| = W( [(n)/(22?2 ?{log2n} ·log1/4n)] ).|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }} \cdot {{\log }^{1/4}}n}}} \right).  相似文献   

8.
Let X1,…, Xn be i.i.d. random variables symmetric about zero. Let Ri(t) be the rank of |Xitn−1/2| among |X1tn−1/2|,…, |Xntn−1/2| and Tn(t) = Σi = 1nφ((n + 1)−1Ri(t))sign(Xitn−1/2). We show that there exists a sequence of random variables Vn such that sup0 ≤ t ≤ 1 |Tn(t) − Tn(0) − tVn| → 0 in probability, as n → ∞. Vn is asymptotically normal.  相似文献   

9.
Let M n , n = 1, 2, ..., be a supercritical branching random walk in which the number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale W n related to M n is regular and W is a limit random variable. Let a(x) be a nonnegative function regularly varying at infinity with index greater than −1. We present sufficient conditions for the almost-sure convergence of the series . We also establish criteria for the finiteness of EW ln+ Wa(ln+ W) and E ln+|Z |a(ln+|Z |), where and (M n , Q n ) are independent identically distributed random vectors not necessarily related to M n . __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 326–342, March, 2006.  相似文献   

10.
Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rn 0 \in {\Bbb R}^n such that f-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.  相似文献   

11.
For the Dirichlet series F(s) = ?n = 1 anexp{ sln } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ a =0, we establish conditions for (λ n ) and (a n ) under which lnM( s, F ) = TR( 1 + o(1) )exp{ rR
/ | s| } \ln M\left( {\sigma, F} \right) = {T_R}\left( {1 + o(1)} \right)\exp \left\{ {{{{{\varrho_R}}} \left/ {{\left| \sigma \right|}} \right.}} \right\} for σ ↑ 0, where M( s, F ) = sup{ | F( s+ it ) |:t ? \mathbbR } M\left( {\sigma, F} \right) = \sup \left\{ {\left| {F\left( {\sigma + it} \right)} \right|:t \in \mathbb{R}} \right\} and T R and ϱ R are positive constants.  相似文献   

12.
We study the the following question in Random Graphs. We are given two disjoint sets L,R with |L| = n and |R| = m. We construct a random graph G by allowing each xL to choose d random neighbours in R. The question discussed is as to the size μ(G) of the largest matching in G. When considered in the context of Cuckoo Hashing, one key question is as to when is μ(G) = n whp? We answer this question exactly when d is at least three. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012  相似文献   

13.
The local irregularity of a digraph D is defined as il(D) = max {|d+ (x) − d (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V1, V2, …, Vc} be a partition of V(T) such that |V1| ≥ |V2| ≥ … ≥ |Vc|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2il(T) + 2|V1| + 2|V2| − 2, il(T) + 3|V1| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., il(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999  相似文献   

14.
A model for cleaning a graph with brushes was recently introduced. Let α = (v 1, v 2, . . . , v n ) be a permutation of the vertices of G; for each vertex v i let ${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}} and N-(vi)={j: vj vi ? E and j <  i}{N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}} ; finally let ba(G)=?i=1n max{|N+(vi)|-|N-(vi)|,0}{b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}. The Broom number is given by B(G) =  max α b α (G). We consider the Broom number of d-regular graphs, focusing on the asymptotic number for random d-regular graphs. Various lower and upper bounds are proposed. To get an asymptotically almost sure lower bound we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method (for fixed d). We further show that for any d-regular graph on n vertices there is a cleaning sequence such at least n(d + 1)/4 brushes are needed to clean a graph using this sequence. For an asymptotically almost sure upper bound, the pairing model is used to show that at most n(d+2?{d ln2})/4{n(d+2\sqrt{d \ln 2})/4} brushes can be used when a random d-regular graph is cleaned. This implies that for fixed large d, the Broom number of a random d-regular graph on n vertices is asymptotically almost surely \fracn4(d+Q(?d)){\frac{n}{4}(d+\Theta(\sqrt{d}))}.  相似文献   

15.
To a given immersion ${i:M^n\to \mathbb S^{n+1}}$ with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|2. We prove the existence of a constant C n (R) depending on R and n so that R ≥ 1 and sup |A|2 = C n (R) imply that the hypersurface is a H(r)-torus ${\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}$ . For R > (n ? 2)/n we use rotation hypersurfaces to show that for each value C > C n (R) there is a complete hypersurface in ${\mathbb S^{n+1}}$ with constant scalar curvature R and sup |A|2 = C, answering questions raised by Q. M. Cheng.  相似文献   

16.
For a connected cubic graph G of order n, we prove the existence of two disjoint dominating sets D 1 and D 2 with |D1|+|D2| £ \frac157198n+\frac89{|D_1|+|D_2|\leq \frac{157}{198}n+\frac{8}{9}}.  相似文献   

17.
LetW(D) denote the set of functionsf(z)=Σ n=0 A n Z n a nzn for which Σn=0 |a n |<+∞. Given any finite set lcub;f i (z)rcub; i=1 n inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f 1(z)z kn ,f 2(z)z kn+1, …,f n (z)z (k+1)n−1rcub; k=0 is a basis forW(D) which is equivalent to the basis lcub;z m rcub; m=0 . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e 2πiti /n.  相似文献   

18.
A basic integral equation of random fields estimation theory by the criterion of minimum of variance of the estimation error is of the form Rh = f, where and R(x, y) is a covariance function.The singular perturbation problem we study consists of finding the asymptotic behavior of the solution to the equation as 0.$$" align="middle" border="0"> The domain D can be an interval or a domain in Rn, n > 1. The class of operators R is defined by the class of their kernels R(x,y) which solve the equation Q(x, Dx)R(x, y) = P(x, Dx)δ(xy), where Q(x, Dx) and Px, Dx) are elliptic differential operators.  相似文献   

19.
Let ${\mathcal{P}_{d,n}}Let Pd,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRn{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? Pd,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that
supP ? Pd,n| p.v\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| £ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),  相似文献   

20.
We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form xy, chosen uniformly at random from among all 2‐clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n→α, the problem is known to have a phase transition at αc=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite‐size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α?(n,δ), α+(n,δ)) such that the probability of satisfiability is greater than 1?δ for α<α? and is less than δ for α>α+. We show that W(n,δ)=(1?Θ(n?1/3), 1+Θ(n?1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(?Θ(nε3)) above the window, and goes to one like 1?Θ(n?1|ε|?3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2‐SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001  相似文献   

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