共查询到20条相似文献,搜索用时 31 毫秒
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It is shown that AC(ℝ), the axiom of choice for families of non‐empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2 ” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 has power 2 , then the union has power 2 ” and (ii) “ℵ(2 ) ≠ ℵω” (ℵ(2 ) is Hartogs' aleph, the least ℵ not ≤ 2 ), is strictly weaker than the full axiom of choice AC. 相似文献
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《Nonlinear Analysis: Theory, Methods & Applications》2004,57(3):349-362
We show that in a smooth bounded domain , n⩾2, all global nonnegative solutions of ut−Δum=up with zero boundary data are uniformly bounded in by a constant depending on and τ but not on u0, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in depending on under the optimal condition 1<m<p<[(n+2)/(n−2)]m. 相似文献
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A fundamental problem in computer science is that of finding all the common zeros of m quadratic polynomials in n unknowns over F2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4log2n2n operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. We show that, under precise algebraic assumptions on the input system, the deterministic variant of our algorithm has complexity bounded by O(20.841n) when m=n, while a probabilistic variant of the Las Vegas type has expected complexity O(20.792n). Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to 1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as 200, and thus very relevant for cryptographic applications. 相似文献
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We give a general construction for Steiner triple systems on a countably infinite point set and show that it yields 2 nonisomorphic systems all of which are uniform and r‐sparse for all finite r?4. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 115–122, 2010 相似文献
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Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)c on a set of positive density. Letting Fk(x) denote the number of n ≤ x for which G(n) = k, it is shown that , where logrx denotes the r-fold iterated logarithm. 相似文献
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Satendra K Vaish 《Journal of Mathematical Analysis and Applications》1984,101(1):23-29
In this paper we obtain a growth relation for entire functions of qth order with respect to the distribution of its zeros. We also derive certain relations between the qth convergence exponents of two or more entire functions. The most striking result of the paper is: If f(z) has at least one zero, then , where n(r) is the number of zeros of f(z) in and . 相似文献
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A.M Odlyzko 《Advances in Mathematics》1982,44(2):180-205
It is shown that the coefficients an of the power series f(z) = ∑∞n=1anzn which satisfies the functional equation display periodic oscillations; as n → ∞, where and u(x) is a positive, nonconstant, continuous function which is periodic with period log(4 ? ø). Similar results are obtained for a wide class of power series that satisfy similar functional equations. Power series of these types are of interest in combinatorics and computer science since they often represent generating functions. For example, the nth coefficient of the power series satisfying (1) enumerates 2, 3-trees with n leaves. 相似文献
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Harry Dym 《Journal of Functional Analysis》1978,28(1):33-57
Let PT denote the orthogonal projection of L2(R1, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure , and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if is summable, if is locally summable, and if belongs to the span in L∞ of e?iyTH∞:T ? 0, in which h is chosen to be an outer function and h#(γ) agrees with the complex conjugate of h(γ) on the line, then exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections. 相似文献
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Elliptic operators , α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on or a quotient space are considered. It is shown that the Lp-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L1 radial convolution kernel. Some consequences are: A can be closed in all Lp (1 ? p ? ∞), and is essentially self-adjoint in L2 if it is symmetric; A generates an analytic semigroup e?tA in the right half plane, strongly Lp and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability , with φ analytic, is proved for , with convergence in Lp and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients. 相似文献
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We investigate those graphs Gn with the property that any tree on n vertices occurs as subgraph of Gn. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by and , respectively. 相似文献
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The usual Sobolev inequality in n, n ? 3, asserts that , with Sn being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? n. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with and with . (ii) If ? ≠ 0 on ?Ω, then with . Some further results and open problems in this area are also presented. 相似文献
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