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1.
The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ k for almost all (in induced measure) points x onM according as the sum Σ r = 1/∞ Ψ(r)r k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r t . τ >k − 1.  相似文献   

2.
We consider the solutions of the First Painlevé Differential Equationω″=z+6w 2, commonly known as First Painlevé Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z −1 w(z 2), respectively:w#(z)=O(|z|3/4) andf#(z)=O(|z|3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painlevé’s Equations II and IV.  相似文献   

3.
Suppose thatAR n is a bounded set of diameter 1 and that:f:Al 2 is a map satisfying the nearisometry condition |xy|−ɛ≤|fxfy|≤|xy|+ɛ withɛ≤1. Then there is an isometryS:Al 2 such that |Sxfx|≤c nɛ for allx inA. IfA satisfies a thickness condition and iff:AR n , then there is an isometryS:R n R n with |Sxfx|≤c nɛ/q, whereq is a thickness parameter.  相似文献   

4.
Let CM be the bundle of connections of a principal G-bundle PM over a pseudo-Riemannian manifold (M,g) of signature (n+, n) and let EM be the associated bundle with P under a linear representation of G on a finite-dimensional vector space. For an arbitrary Lie group G, the O(n+, n) × G-invariant quadratic Lagrangians on J1(C × M E) are characterized. In particular, for a simple Lie group the Yang–Mills and Yang–Mills–Higgs Lagrangians are characterized, up to an scalar factor, to be the only O(n+, n) × G-invariant quadratic Lagrangians. These results are also analyzed on several examples of interest in gauge theory. Submitted: May 19, 2005; Accepted: April 25, 2006  相似文献   

5.
Letu be a function harmonic in the unit disc or in the plane, and letu(z)M(|z|) for a majorantM. We formulate conditions onM that guarantee thatu(z)≥−(1+o(1))M(|z|) for |z|→1 in the disc and for |z|→∞ in the plane.  相似文献   

6.
By some moment inequalities for α-mixing random variables, we prove the Bahadur representation of sample quantiles under very weak α-mixing coefficients. As application, the uniformly asymptotic normality is derived, the rate of which is near to n −1/6 under the given conditions.  相似文献   

7.
In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rn+1. If Mn−1 is a compact hypersurface of the sphere Sn(1) we represent by C(M)ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for nslant 7 there is an ε for which the truncate cone C(M)ε is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mn−1 of the sphere with zero scalar curvature and S3 different from zero, for which all truncated cones based on M are stable. Mathematics Subject Classifications (2000): 53C42, 53C40, 49F10, 57R70.  相似文献   

8.
Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n + 1)-dimensional Riemannian manifold N^n+1 whose (n - 1)th Ricci curvature satisfying Ric^N(n-1) (n - 1)c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If |φ|^2 - (n- 2)√n(n- 1)|H||φ|+ n(2n - 1)(H^2+ c) 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H^2 〉 0, then M must have only one end.  相似文献   

9.
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( n n+x ). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general SP n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension ( n /n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each PS and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).  相似文献   

10.
We obtain the upper bound O(214n/15 n−1/5) on the number of distinct values of all possible correlation functions between M-sequences of order n .  相似文献   

11.
Since the definition of attribute reduction based on classic discernibility matrix differs from that of it based on positive region, a new simplified discernibility matrix and the corresponding definition of attribute reduction were proposed. At the same time, it proves that the proposed definition is identical to its definition based on positive region. For computing simplified discernibility matrix, the indiscernibility relation, which is also called equivalence relation, should usually be calculated at first, so a new algorithm for computing equivalence relation was designed with radix sorting, whose temporal complexity is O(|C||U|). Furthermore, an efficient attribute reduction algorithm is proposed, whose temporal complexity and spatial complexity are cut down to max(O(|C|2|U pos ||U′|,O(|U||C|)) and max(O(|C||U pos ||U′|,O(|U|)) respectively. At last, an example is used to illustrate efficiency of the new algorithms.  相似文献   

12.
Sparse Dynamic Programming has emerged as an essential tool for the design of efficient algorithms for optimization problems coming from such diverse areas as computer science, computational biology, and speech recognition. We provide a new sparse dynamic programming technique that extends the Hunt–Szymanski paradigm for the computation of the longest common subsequence (LCS) and apply it to solve the LCS from Fragments problem: given a pair of strings X and Y (of length n and m, respectively) and a set M of matching substrings of X and Y, find the longest common subsequence based only on the symbol correspondences induced by the substrings. This problem arises in an application to analysis of software systems. Our algorithm solves the problem in O(|M| log |M|) time using balanced trees, or O(|M| log log min(|M|, nm/|M|)) time using Johnson's version of Flat Trees. These bounds apply for two cost measures. The algorithm can also be adapted to finding the usual LCS in O((m + n) log |Σ| + |M| log |M|) time using balanced trees or O((m + n) log |Σ| + |M| log log min (|M|, nm/|M|)) time using Johnson's version of Flat Trees, where M is the set of maximal matches between substrings of X and Y and Σ is the alphabet. These bounds improve on those of the original Hunt–Szymanski algorithm while retaining the overall approach.  相似文献   

13.
Let {ξ j ; j ∈ ℤ+ d be a centered stationary Gaussian random field, where ℤ+ d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝd, having nonnegative integer coordinates. For each j = (j 1 , ..., jd) in ℤ+ d , we denote |j| = j 1 ... j d and for m, n ∈ ℤ+ d , define S(m, n] = Σ m<j≤n ζ j , σ2(|nm|) = ES 2 (m, n], S n = S(0, n] and S 0 = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes. Research supported by NSERC Canada grants at Carleton University, Ottawa  相似文献   

14.
LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK H≥4λ and antiholomorphic Ricci curvatureρ A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P n−1(λ) andM=C P n(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.  相似文献   

15.
For fixed 1≦p<∞ theL p-semi-norms onR n are identified with positive linear functionals on the closed linear subspace ofC(R n ) spanned by the functions |<ξ, ·>| p , ξ∈R n . For every positive linear functional σ, on that space, the function Φσ:R n R given by Φσ is anL p-semi-norm and the mapping σ→Φσ is 1-1 and onto. The closed linear span of |<ξ, ·>| p , ξ∈R n is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in anyL p unlessp=2. Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.  相似文献   

16.
Let (M n ,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ 0 of its universal covering satisfies λ 0≤(n−1)2/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. The author was partially supported by NSF Grant 0505645.  相似文献   

17.
A generalization of the Blaschke product is constructed. This product enables one to factor out the zeros of the members of certain non-Nevanlinna classes of functions analytic in the unit disc, so that the remaining (non-vanishing) functions still belong to the same class. This is done for the classesA −n (0<n<∞) andB −n (0<n<2) defined as follows:fA −n iff |f(z)|≦C f (1−|z|)n ,fB n iff |f(z)|≦exp {C f (1−|z|)n }, whereC f depends onf.  相似文献   

18.
Conditions are obtained for (*)E|S T |γ<∞, γ>2 whereT is a stopping time and {S n=∑ 1 n ,X j n ,n⩾1} is a martingale and these ensure when (**)X n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET γ/2<∞, sup n≥1 E|X n |γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S k,T |γ<∞ and of the second moment equationES k,T 2 =σ 2 EΣ j=k T S k−1,j−1 2 where and {X n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X n , n≥1} withEX=0,EX 2=1 that the a.s. limit set of {(n log logn)k/2 S k,n ,n≥k} is [0,2 k/2/k!] or [−2 k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .  相似文献   

19.
In the linear model Xn × 1 = Cn × pθp × 1 + En × 1, Huber's theory of robust estimation of the regression vector θp × 1 is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector En × 1. In the first model considered, the restriction of F to a set [−a0, b0] is a standard normal distribution contaminated, with probability , by an unknown distribution symmetric about 0. In the second model, the restriction of F to [−a0, b0] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [−a0, b0] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θp × 1 are constructed, under mild regularity conditions on the sequence of design matrices {Cn × p}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets.  相似文献   

20.
Let X1, X2, . . . be i.i.d. random variables, and set Sn=X1+ . . . +Xn. Several authors proved convergence of series of the type f(ɛ)=∑ncnP(|Sn|>ɛan),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑ncnP(|Sn|>ɛan),ɛ>α, is equivalent to the convergence of the double sum ∑kncnP(|Sn|>kan). Two exceptional series required deriving necessary and sufficient conditions for E[supn|Sn|(logn)η/n]<∞,0≤η≤1.  相似文献   

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