We first prove the consistency of: there is a universal graph of power ?1<2?0 = 2?1=?2. The consistency of the non-existence of a universal graph of power ?1 is trivial. Add ?2 Cohen generic reals. We then show that we can have 2?0=?2<2?1, and get similar results for other cardinals. 相似文献
In this paper we consider the following m-point fractional boundary value problem with p-Laplacian operator on infinite interval where 0<????1, 2<????3, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, ??p(s)=|s|p?2s,p>1, (??p)?1=??q, $\frac{1}{p}+\frac{1}{q}=1$. 0<??1<??2<?<??m?2<+??, ??i??0, i=1,2,??,m?2 satisfies $0 <\sum_{i=1}^{m-2}\beta_{i}\xi_{i}^{\alpha-1} < \Gamma(\alpha)$. We establish solvability of the above fractional boundary value problems by means of the properties of the Green function and some fixed-point theorems. 相似文献
In this paper we study equivalent formulations of the DP?Pp (1 < p < ∞). We show that X has the DP?Pp if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?π Y, the dual (X ??Y)? of their injective tensor product, and the bidual (X ?π Y)?? of their projective tensor product, do not have the DP Pp, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP?Pp, 1 < p < ∞. 相似文献
X is a nonnegative random variable such that EXt < ∞ for 0≤ t < λ ≤ ∞. The (l??) quantile of the distribution of X is bounded above by [??1 EXt]1?t. We show that there exist positive ?1 ≥ ?2 such that for all 0 <?≤?1 the function g(t) = [?-1EXt]1?t is log-convex in [0, c] and such that for all 0 < ? ≤ ?2 the function log g(t) is nonincreasing in [0, c]. 相似文献
Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ?3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C?r?1+ε |t|?ε/2 for ? T0 ≤ t < 0 and 0 < C? < ∞ allowed to be large. We prove that v is regular at time zero. 相似文献
Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0?<?p?<?∞ , Hp(W) denotes a weighted Hardy space on the unit circle. When W?≡?1, Hp(W) is the usual Hardy space Hp. We are interested in Hp( W)+ the set of all nonnegative functions in Hp( W). If p?≥?1/2, Hp+ consists of constant functions. However Hp( W)+ contains a nonconstant nonnegative function for some weight W. In this paper, if p?≥?1/2 we determine W and describe Hp( W)+ when the linear span of Hp( W)+ is of finite dimension. Moreover we show that the linear span of Hp(W)+ is of infinite dimension for arbitrary weight W when 0?<?p?<?1/2. 相似文献
We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ?1-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ?1-dense subsets of are isomorphic, is consistent. We e.g. prove Con(BA+(2?0>?2)). Let <KH,<> be the set of order types of ?1-dense homogeneous subsets of with the relation of embeddability. We prove that for every finite model <L, <->: Con(MA+ <KH, <-> ? <L, <->) iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ?1, and {U0,\h.;,Un?1} is a cover of D(X)XxX-}<x,x>¦x?X} consisting of symmetric open sets, then X can be partitioned into {Xi \brvbar; i ? ω} such that for every i ? ω there is l<n such that D(Xi)?Ul”. We also prove that MA+OCA [xrArr] 2 ?0 = ?2. 相似文献
In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φp(s)=|s|p?2?s,p>1;ξi∈(0,1) with 0<ξ1<ξ2<???<ξn?2<1 and αi,βi satisfy αi,βi∈[0,∞),0≤∑i=1n?2αi<1 and 0≤∑i=1n?2βi<1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. 相似文献
Let ø(x) be a truncated normal pdf over the interval [a,b], that is, assume ø(x)=exp[-(x–μ)2/2σ2]/∝baexp[-(x–μ)2/2σ2]dx for - ∞<a?x?b?< + ∞ and zero elsewhere. Suppose that X1,X2,…,Xn is a random sample of size n from this truncated distribution. Using known properties of exponential families of distributions and the system of Legendre polynomials over the interval [-1,1], we examine the maximum likelihood estimation of the parameters μ and σ2. 相似文献
It is known that for all monotone functions f : {0, 1}n → {0, 1}, if x ∈ {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n?α, then P[f(x) ≠ f(y)] < cn?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[fn(x) ≠ fn(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[fn(x) ≠ fn(y)] ≥ δ, with:
? = c(δ)n?α for any α < 1/2, using the recursive majority function with arity k = k(α);
? = c(δ)n?1/2logtn for t = log2 = .3257 …, using an explicit recursive majority function with increasing arities; and
? = c(δ)n?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.
In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α0 > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H1(?2)-solution u? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u? is also established as ? tends to zero. 相似文献
This paper is concerned with the following fourth-order m-point nonhomogeneous boundary value problem $$\begin{array}{l}u^{(4)}(t)=f(t,u(t),u^{\prime \prime }(t)),\quad 0<t<1,\\[3pt]u(0)=u(1)=u^{\prime \prime }(0)=0,\\[3pt]u^{\prime \prime }(1)-\displaystyle\sum_{i=1}^{m-2}a_{i}u^{\prime\prime }(\xi _{i})=-\lambda ,\end{array}$$ where ai≥0 (i=1,2,…,m?2), 0<ξ1<ξ2<???<ξm?2<1 and ∑i=1m?2aiξi<1, and λ>0 is a parameter. The existence and nonexistence of positive solution are discussed for suitable λ>0 when f is superlinear or sublinear. The main tool used is the well-known Guo-Krasnoselskii fixed point theorem. 相似文献
We study the Cauchy problem for the nonlinear heat equation ut-?u=|u|p-1u in RN. The initial data is of the form u0=λ?, where ?∈C0(RN) is fixed and λ>0. We first take 1<p<pf, where pf is the Fujita critical exponent, and ?∈C0(RN)∩L1(RN) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<ps, where ps is the Sobolev critical exponent, and ?(x) decaying as |x|-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial. 相似文献
Let ξ1, ξ2, ξ3,... be a sequence of independent random variables, such that μj?E[ξj], 0<α?Var[ξj] andE[|ξj?μj|2+δ] for some δ, 0<δ?1, and everyj?1. IfU and ξ0 are two random variables such thatE[ξ02]<∞ andE[|U|ξ02]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξj:j?1}, then under appropriate regularity conditions $$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$ whereSn?ξ1+ξ2+?+ξn,μj?E[ξj],sn2?Var[Sn], andcn=O(sn). 相似文献
Moser's C?-version of Kolmogorov's theorem on the persistence of maximal quasi-periodic solutions for nearly-integrable Hamiltonian system is extended to the persistence of non-maximal quasi-periodic solutions corresponding to lower-dimensional elliptic tori of any dimension n between one and the number of degrees of freedom. The theorem is proved for Hamiltonian functions of class C? for any ?>6n+5 and the quasi-periodic solutions are proved to be of class Cp for any p with 2<p<p* for a suitable p*=p*(n,?)>2 (which tends to infinity when ?→∞). 相似文献
We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = ?d2/dx2 + v(x), x ∈ L1([0, π], with Hper?1-potential and the free operator L0 = ?d2/dx2, subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $\left\| {S_N - S_N^0 :L^a \to L^b } \right\| \to 0if1 < a \leqslant b < \infty ,1/a - 1/b < 1/2,$, where SN and SN0 are the N-th partial sums of the spectral decompositions of L and L0. Moreover, if v ∈ H?α with 1/2 < α < 1 and $\frac{1} {a} = \frac{3} {2} - \alpha $, then we obtain the uniform equiconvergence ‖SN?SN0: La → L∞‖ → 0 as N → ∞. 相似文献
We introduce a bound M of f, ‖f‖∞?M?2‖f‖∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule. 相似文献
We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables one to construct embeddings of ?pn, p > 0, into various types of Banach or quasi-Banach spaces. In particular, for 0 < r < p < 2 with r ≤ 1, we construct a family of operators that embed ?pn into $\ell _r^{(1 + \eta )n}$, with sharp polynomial bounds in η > 0. 相似文献
Assume that W=e?Q where I:=(a,b), ?∞≦a<0<b≦∞, and Q:?I→[0,∞) is continuous and increasing. Let 0<p<∞, a<tr<tr?1<?<t1<b, pi>?1/p, i=1,2,…,r, and $U(x)=\prod_{i=1}^{r} {|x-t_{i}|}^{p_{i}}$. We give the Lp Christoffel functions for the Jacobi-exponential weight WU. In addition, we obtain restricted range inequalities. 相似文献