共查询到20条相似文献,搜索用时 31 毫秒
1.
Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T k (V) denote the set of all 2 k n k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log2 n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that
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$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$
2.
S. A. Nazarov 《Siberian Mathematical Journal》2018,59(1):85-101
We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter ε > 0. We describe the effects of almost complete reflection or transmission of waves related to the classical Vainstein anomaly and the presence of almost standing waves for the threshold value Λ k of the spectral parameter λ in continuous spectrum. The greatest attention is paid to analyzing the range λ ε = Λ k + ε2μ2 of the spectral parameter with μ ≤ μ0, which generates scattering coefficients depending on μ > 0 and presents the greatest difficulties in constructing and justifying the asymptotics. Almost complete reflection and transmission correspond to the cases of going away from the threshold (as μ → +∞) and approaching it (as μ → +0) characterized by simpler asymptotics. 相似文献
3.
Let X be a real normed space and let f: ? → X be a continuous mapping. Let T f (t 0) be the contingent of the graph G(f) at a point (t 0, f(t 0)) and let S + ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 X ). We show that
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- 1.If dimX < ∞ and the dilation D(f, t 0) of f at t 0 is finite then T f (t 0) ∩ S + is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
- 2.If dimX = ∞, then, given any compact set F ? S +, there exists a Lipschitz mapping f: ? → X such that T f (t 0) ∩ S + = F.
- 3.But if a closed set F ? S + has cardinality greater than that of the continuum then the relation T f (t 0) ∩ S + = F does not hold for any Lipschitz f: ? → X.
4.
In L 2(?3;?3), we consider a self-adjoint operator ? ε , ε > 0, generated by the differential expression curl η(x/ε)?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? ε 1/2 ) and ? ε ?1/2 sin(τ? ε 1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?0)1/2) and (?0)?1/2 sin(τ(?0)1/2), respectively, in the norm of the operators acting from the Sobolev space H s (with a suitable s) to ?2. Here ?0 is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? τ 2 v ε = ?? ε v ε , div v ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε). 相似文献
5.
N. N. Senik 《Functional Analysis and Its Applications》2016,50(1):71-75
We consider an operator Aε on L2(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d1 is positive, while d2 can be zero) given by Aε = ?div A(ε?1x1,x2)?, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (Aε ? μ)?1 and ?(Aε ? μ)?1 (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order. 相似文献
6.
Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ 0 with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem We prove that for any k ∈ ?, there exists ε k > 0 such that for all ε ∈ (0, ε k ) the problem (P ?? ) has a symmetric solution u ε , which looks like the superposition of k positive bubbles centered at the point ξ 0 as ε → 0. In particular, ξ 0 is a towering blow-up point.
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$$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}}\ \text { in }\ (M,g) . $$
7.
Marvin I. Knopp 《Journal of Number Theory》1980,12(1):2-9
If h, k ∈ Z, k > 0, the Dedekind sum is given by , with , . The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) , where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime. 相似文献
8.
Eungil Ko 《Integral Equations and Operator Theory》1997,28(4):492-499
In this paper we shall prove that if an operatorT∈L(⊕ 1 2 H) is an operator matrix of the form $$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$ whereT 1 is hyponormal andT 3 k =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar. 相似文献
9.
S. V. Kislyakov 《Functional Analysis and Its Applications》2009,43(3):236-238
Let m be a bounded function on ?+ whose p-variations on the intervals [2 k , 2 k+1], k ∈ ?, are uniformly bounded for some p < 2. Then the operator T, $ \widehat{Tf} = m\hat f $ , is of weak type (1, 1) on the space H 1(?). 相似文献
10.
In this paper it is shown that if T∈L(H) satisfies
- (i)
- T is a pure hyponormal operator;
- (ii)
- [T∗,T] is of rank two; and
- (iii)
- ker[T∗,T] is invariant for T,
11.
The problem concerned in this paper is the set-valued equation 0 ∈T(z) where T is a maximal monotone operator. For given xk and βk > 0, some existing approximate proximal point algorithms take x~(k+1) = xk such thatwhere {ηk} is a non-negative summable sequence. Instead of xk+1 = xk , the new iterate of the proposing method is given bywhere Ω is the domain of T and PΩ(·) denotes the projection on Ω. The convergence is proved under a significantly relaxed restriction supk>0 ηk<1. 相似文献
12.
We consider the Neumann problem outside a small neighborhood of a planar disk in the three-dimensional space. The surface of this neighborhood is assumed to be smooth, and its thickness is characterized by a small parameter ε. A uniform asymptotic expansion of the solution of this problem with respect to ε is constructed by the matching method. Since the problem turned out to be bisingular, an additional inner asymptotic expansion in the so-called stretched variables is constructed near the edge of the disk. A physical interpretation of the solution of this boundary value problem is the velocity potential of a laminar flow of an ideal fluid around a thin body, which is the neighborhood of the disk. It is assumed that this flow has unit velocity at a large distance from the disk, which is equivalent to the following condition for the potential: u(x1, x2, x3, ε) = x3+O(r?2) as r → ∞, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: ?u/?n = 0 at the boundary. After subtracting x3 from the solution u(x1, x2, x3, ε), we get a boundary value problem for the potential ?(x1, x2, x3, ε) of the perturbed motion. Since the integral of the function ??/?n over the surface of the body is zero, we have ?(x1, x2, x3, ε) = O(r?2) as r → ∞. Hence, all the coefficients of the outer asymptotic expansion with respect to ε have the same behavior at infinity. However, these coefficients have growing singularities at the approach to the edge of the disk, which implies the bisingularity of the problem. 相似文献
13.
Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes Bpqsm(Tk) with respect to the multiple trigonometric system T(k) in the metric of Lr(Tk) for a number of relations between the parameters s, p, q, r, and m (s = (s1,..., sn) ∈ R+n, 1 ≤ p, q, r ≤ ∞, m = (m1,..., mn) ∈ Nn, k = m1 +... + mn). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates. 相似文献
14.
Kassem Mustapha Maher Nour Bernardo Cockburn 《Advances in Computational Mathematics》2016,42(2):377-393
We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L ∞(0, T; L 2(Ω))-norm and to ?u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed. 相似文献
15.
In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V 1,…V t such that for all i the subtournament T[V i] induced on T by V i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k, t)=O(k 7 t 4) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists aninteger h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)=O(t 5) suffices. 相似文献
16.
Zhanmin Zhu 《Czechoslovak Mathematical Journal》2018,68(2):455-474
Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules such that F0,..., Fn?1 are finitely generated free and Kn?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext n R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor R n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.
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$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$
17.
Yutian Lei 《Journal of Mathematical Analysis and Applications》2007,334(2):1341-1362
This paper is concerned with the asymptotic behavior of the regularized minimizer uε=(uε1,uε2,…,uεn+1) of an energy functional
18.
We analyze the perturbations T?+?B of a selfadjoint operator T in a Hilbert space H with discrete spectrum ${\{ t_k\}, T \phi_k = t_k \phi_k}$ . In particular, if t k+1 ? t k ?? ck ?? - 1, ?? > 1/2 and ${\| B \phi_k \| = o(k^{\alpha - 1})}$ then the system of root vectors of T?+?B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions ${t_{k+p} - t_k \geq d > 0, \forall k}$ (with d and p fixed) and ${\| B \phi_k \| \rightarrow 0}$ a Riesz system {P k } of projections on invariant subspaces of T?+?B, Rank P k ?? p, is constructed (Theorem 3). 相似文献
19.
Stevo Stevi? 《Applied mathematics and computation》2010,216(1):179-5018
Suppose r∈(0,1],m∈N and 1?k1<k2<?<k2m+1, and let S2m+1={1,2,…,2m+1}. We show that every positive solution to the difference equation
20.
S. A. Gritsenko 《Proceedings of the Steklov Institute of Mathematics》2017,296(1):65-87
Let N 0(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ i T]. It is proved that N 0(T) ? T (ln T)1/2+1/16?ε, where ε is an arbitrarily small positive number. 相似文献