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1.
Let S ⊂ ℝR n +1 be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ q = ψ|Λ(X)|q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov  相似文献   

2.
Let f ε Cn+1[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials qk associated with a distribution dα on [−1, 1]. It is shown that if qn+1/qn max(qn+1(1)/qn(1), −qn+1(−1)/qn(−1)), then fH[f] fn + 1 · qn+1/qn + 1(n + 1), where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)α (1 + t)β dt, α, β > −1, the condition on qn+1/qn is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.  相似文献   

3.
Let Xn, n , be i.i.d. with mean 0, variance 1, and EXn¦r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σi = 1n Xi t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n(r − 2)/2)), if the distances of the set B from the σ-fields σ(X1, …, Xn) are of order O(1/n(r − 2)/2)(lg n)β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n(r − 2)/2)) for r (O(lg lgn/n(r − 2)/2)) for r ).  相似文献   

4.
Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β 1, we construct inductively a series of rank n residually connected geometries Γ(n, β, Ω), n β, by putting Γ(β, β, Ω) = Ω and extending Γ(n - 1, β, Ω) with a partial geometry.  相似文献   

5.
Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (pm)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate of T(θ), based on a set of i.i.d. observations (X1, Y1), …, (Xn, Yn), that achieves the optimal nonparametric rate of convergence nr in Lq-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T(θ) that achieves the optimal rate (n/log n)r in L-norm restricted to compacts.  相似文献   

6.
Let {Xt} be a Gaussian ARMA process with spectral density fθ(λ), where θ is an unknown parameter. The problem considered is that of testing a simple hypothesis H:θ = θ0 against the alternative A:θ ≠ θ0. For this problem we propose a class of tests , which contains the likelihood ratio (LR), Wald (W), modified Wald (MW) and Rao (R) tests as special cases. Then we derive the χ2 type asymptotic expansion of the distribution of T up to order n−1, where n is the sample size. Also we derive the χ2 type asymptotic expansion of the distribution of T under the sequence of alternatives An: θ = θ0 + /√n, ε > 0. Then we compare the local powers of the LR, W, MW, and R tests on the basis of their asymptotic expansions.  相似文献   

7.
Let Bn( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials Bn( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {Bn( f,q;x)} to f(x) in the norm of C[0,1] has the order qn (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn( f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞.  相似文献   

8.
Let {pk(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation Dq(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {pk(1)(x;q)} be associated polynomials of the polynomials {pk(x; q)}. Explicit forms of the coefficients bn,k and cn,k in the expansions
are given in terms of basic hypergeometric functions. Here k(x) equals xk if σ+(0)=0, or (x;q)k if σ+(1)=0, where σ+(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by pn−1(1)(x;q) in a special form, recurrence relations (in k) for bn,k and cn,k are obtained in terms of σ and τ.  相似文献   

9.
We give a direct formulation of the invariant polynomials μGq(n)(, Δi,;, xi,i + 1,) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions Sλ known as Schur functions. To this end, we show after the change of variables Δi = γi − δi and xi, i + 1 = δi − δi + 1 thatμGq(n)(,Δi;, xi, i + 1,) becomes an integral linear combination of products of Schur functions Sα(, γi,) · Sβ(, δi,) in the variables {γ1,…, γn} and {δ1,…, δn}, respectively. That is, we give a direct proof that μGq(n)(,Δi,;, xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {γ1,…, γn} and {δ1,…, δn}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials μmGq(n)1,…, γn; δ1,…, δm). These new symmetries enable us to give an explicit formula for both μmG1(n)(γ; δ) and 1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for μmGq(n)(γ; δ).  相似文献   

10.
The combinatorial complexity of the Voronoi diagram ofnlines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to beO(n2α(n)log n), where α(n) is a slowly growing inverse of the Ackermann function. There are arrangements ofnlines where this complexity can be as large as Ω(n2α(n)).  相似文献   

11.
Let A = (aij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, aij = aji, i, J = 1,… ,n, ai = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if ai are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m3k(log2 k + log n) + m2k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.  相似文献   

12.
Let p and q be two permutations over {1, 2,…, n}. We denote by m(p, q) the number of integers i, 1 ≤ in, such that p(i) = q(i). For each fixed permutation p, a query is a permutation q of the same size and the answer a(q) to this query is m(p, q). We investigate the problem of finding the minimum number of queries required to identify an unknown permutation p. A polynomial-time algorithm that identifies a permutation of size n by O(n · log2n) queries is presented. The lower bound of this problem is also considered. It is proved that the problem of determining the size of the search space created by a given set of queries and answers is #P-complete. Since this counting problem is essential for the analysis of the lower bound, a complete analysis of the lower bound appears infeasible. We conjecture, based on some preliminary analysis, that the lower bound is Ω(n · log2n).  相似文献   

13.
The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W2(1). The target is either a subset of W2(1) a point in W2(1), a subset of the Euclidean space En or a point of En. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of Em and Ei respectively with q(t) = C(t) q′(t), − p(t) = B(t) p′(t), q′(t) ε Emp′(t) ε Er and B(t) is an n × r′-matrices and C(t) an n × m-matrix.  相似文献   

14.
Let nq(k, d) denote the smallest value of n for which there exists an [n, k, d; q]-code. It is known (cf. (J. Combin. Inform. Syst. Sci.18, 1993, 161–191)) that (1) n3(6, 195) {294, 295}, n3(6, 194) {293, 294}, n3(6, 193) {292, 293}, n3(6, 192) {290, 291}, n3(6, 191) {289, 290}, n3(6, 165) {250, 251} and (2) there is a one-to-one correspondence between the set of all nonequivalent [294, 6, 195; 3]-codes meeting the Griesmer bound and the set of all {v2 + 2v3 + v4, v1 + 2v2 + v3; 5, 3}-minihypers, where vi = (3i − 1)/(3 − 1) for any integer i ≥ 0. The purpose of this paper is to show that (1) n3(6, 195) = 294, n3(6, 194) = 293, n3(6, 193) = 292, n3(6, 192) = 290, n3(6, 191) = 289, n3(6, 165) = 250 and (2) a [294, 6, 195; 3]-code is unique up to equivalence using a characterization of the corresponding {v2 + 2v3 + v4, v1 + 2v2 + v3; 5, 3}-minihypers.  相似文献   

15.
Cheriyan and Hagerup developed a randomized algorithm to compute the maximum flow in a graph with n nodes and m edges in O(mn + n2 log2n) expected time. The randomization is used to efficiently play a certain combinatorial game that arises during the computation. We give a version of their algorithm where a general version of their game arises. Then we give a strategy for the game that yields a deterministic algorithm for computing the maximum flow in a directed graph with n nodes and m edges that runs in time O(mn(logm/n log nn)). Our algorithm gives an O(mn) deterministic algorithm for all m/n = Ω(nε) for any positive constant ε, and is currently the fastest deterministic algorithm for computing maximum flow as long as m/n = ω(log n).  相似文献   

16.
Consider the equation −ε2Δuε + q(x)uε = f(uε) in , u(∞) < ∞, ε = const > 0. Under what assumptions on q(x) and f(u) can one prove that the solution uε exists and limε→0uε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper.  相似文献   

17.
The n-widths of the unit ball Ap of the Hardy space Hp in Lq( −1, 1) are determined asymptotically. It is shown that for 1 ≤ q < p ≤∞ there exist constants k1 and k2 such that [formula]≤ dn(Ap, Lq(−1, 1)),dn(Ap, Lq(−1, 1)), δn(Ap, Lq(−1, 1))[formula]where dn, dn, and δn denote the Kolmogorov, Gel′fand and linear n-widths, respectively. This result is an improvement of estimates previously obtained by Burchard and Höllig and by the author.  相似文献   

18.
Given a set of values x1, x2,..., xn, of which k are nonzero, the compaction problem is the problem of moving the nonzero elements into the first k consecutive memory locations. The chaining problem asks that the nonzero elements be put into a linked list. One can in addition require that the elements remain in the same order, leading to the problems of ordered compaction and ordered chaining, respectively. This paper introduces a technique involving perfect hash functions that leads to a deterministic algorithm for ordered compaction running on a CRCW PRAM in time O(log k/log log n) using n processors. A matching lower bound for unordered compaction is given. The ordered chaining problem is shown to be solvable in time O(α(k)) with n processors (where α is a functional inverse of Ackermann′s function) and unordered chaining is shown to he solvable in constant time with n processors when k < n1/4− ε.  相似文献   

19.
LetN m (q) be the set of nonisotropic lines in the vector space of dimensionm over a finite field of orderq. In a paper by Bannai, Hao, Song and Wei, it was shown that the association scheme character tableP(Sp(2n, q),N 2n (q)), withn 3 andq odd, is controlled byP(Sp(4,q),N 4(q)) which is in turn controlled byP(O(3,q),O(3,q)/O + (2,q)). Our purpose in this paper is to compute the entries in the character tableP(O(3,q), O(3,q)/O + (2,q)) explicitly, which is left open in that paper.  相似文献   

20.
Let q be a prime power and suppose that e and n are integers satisfying 1 e n − 1. Then the Grassmann graph Γ(e, q, n) has as vertices the e-dimensional subspaces of a vector space of dimension n over the field Fq, where two vertices are adjacent iff they meet in a subspace of dimension e − 1. In this paper, a characterization of Γ(e, q, n) in terms of parameters is obtained provided that and ( if q ε {2, 3}) and if q = 3). As a consequence we can show that these Grassmann graphs are uniquely determined as distance-regular graphs by their intersection arrays.  相似文献   

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