共查询到20条相似文献,搜索用时 561 毫秒
1.
Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T)
and a set
A ? XA\in\mathcal{X}
of positive measure, the set of integers n such that
A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon
is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tnx)f(T2nx)? f(Tknx) dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}
, where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d*(E)>0 and for all
{n ? \mathbbZ\colon d*(E?(E+n)?(E+2n)?(E+3n)) > d*(E)4-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\} 相似文献
2.
For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò02p s( \fracjn,eiq) e-i(j-k)q dq, j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s](n+1)E[s] \text as n?¥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear. 相似文献
3.
In this paper, we study the planar Hamiltonian system = J (A(θ)x + ▽f(x, θ)), θ = ω, x ∈ R2 , θ∈ Td , where f is real analytic in x and θ, A(θ) is a 2 × 2 real analytic symmetric matrix, J = (1-1 ) and ω is a Diophantine vector. Under the assumption that the unperturbed system = JA(θ)x, θ = ω is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system. 相似文献
4.
We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain
W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and
ò01 s-1 (meas\nolimits {x ? W: |a(x)| £ s })q ds < ¥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}. 相似文献
5.
The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q
s
and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Qs{q\in Q_{s}}. 相似文献
6.
Anders Bj?rner 《Combinatorica》2011,31(2):151-164
Let L be a finite distributive lattice and μ: L → ℝ+ a log-supermodular function. For functions k: L → ℝ+ let
Em (k;q)def ?x ? L k(x)m(x)qrank(x) ? \mathbbR+ [q] .E_\mu (k;q)^{\underline{\underline {def}} } \sum\limits_{x \in L} {k(x)\mu (x)q^{rank(x)} \in \mathbb{R}^ + [q]} . 相似文献
7.
A. Lemenant 《Journal of Mathematical Sciences》2012,181(6):820-836
We talk about the following minimization problem:
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