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A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators {fk}k=1K?L2(Rd) are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators fk must satisfy Rd|x||fk(x)|2dx=, namely, fk??H1/2(Rd). Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space Hd/2+?(Rd); our results provide an absolutely sharp improvement with H1/2(Rd). Our results are sharp in the sense that H1/2(Rd) cannot be replaced by Hs(Rd) for any s<1/2.  相似文献   

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Let N+(k)=2k/2k3/2f(k) and N?(k)=2k/2k1/2g(k) where f(k) and g(k)0 arbitrarily slowly as k. We show that the probability of a random 2-coloring of {1,2,,N+(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1,2,,N?(k)} containing a monochromatic k-term arithmetic progression approaches 0, as k. This improves an upper bound due to Brown, who had established an analogous result for N+(k)=2klogkf(k).  相似文献   

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Let q be a positive integer. Recently, Niu and Liu proved that, if nmax?{q,1198?q}, then the product (13+q3)(23+q3)?(n3+q3) is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and nmax?{q,11?q}, the product (1?+q?)(2?+q?)?(n?+q?) is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer Nq,? such that, for any positive integer nNq,?, the product (1?+q?)(2?+q?)?(n?+q?) is not a powerful number.  相似文献   

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This paper deals with the following nonlinear elliptic equation
?Δu+V(|y|,y)u=uN+2N?2,u>0,uH1(RN),
where (y,y)R2×RN?2, V(|y|,y) is a bounded non-negative function in R+×RN?2. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if N5 and r2V(r,y) has a stable critical point (r0,y0) with r0>0 and V(r0,y0)>0, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.  相似文献   

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We define a family KV(g,n+1) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 boundary components. The problem KV(0,3) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV(g,n+1) for arbitrary g and n. The key point is the solution to KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra g(g,n+1). In more detail, we show that every solution to KV(g,n+1) induces a Lie bialgebra isomorphism between g(g,n+1) and its associated graded grg(g,n+1). For g=0, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g1, n=0, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.  相似文献   

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We study LpLr restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the L(2d+2)/(d+3)L2 Stein–Tomas restriction result can be improved to the L(2d+4)/(d+4)L2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured LpL2 restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in (d+1) dimensions.  相似文献   

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Let Ω?RN be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)?ζ=Gζ,ζ|Γ=0, vanishes if GL1(Ω;R(N×N)×N) and ζW1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with GL1L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that:???:C°(Ω,Γ;R3)[0,),u?6sym(?uP?1)6L2(Ω) is a norm if PL(Ω;R3×3) with CurlPLp(Ω;R3×3), CurlP?1Lq(Ω;R3×3) for some p,q>1 with 1/p+1/q=1 as well as detP?c+>0. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ΦH1(Ω;R3), Ω?R3, satisfy sym(?Φ??Ψ)=0 for some ΨW1,(Ω;R3)H2(Ω;R3) with det?Ψ?c+>0. Then there exists a constant translation vector aR3 and a constant skew-symmetric matrix Aso(3), such that Φ=AΨ+a.  相似文献   

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This paper investigates the existence and asymptotic behavior of nodal solutions to the following gauged nonlinear Schrödinger equation
{?Δu+ωu+(h2(|x|)|x|2+|x|+h(s)su2(s)ds)u=λ|u|p?2u,xR2,u(x)=u(|x|)H1(R2),
where ω,λ>0, p>6 and
h(s)=120sru2(r)dr
is the so-called Chern–Simons term. We prove that for any positive integer k, the problem has a sign-changing solution uλk which changes sign exactly k times. Moreover, the energy of ukλ is strictly increasing in k, and for any sequence {λn}+(n), there exists a subsequence {λns}, such that (λns)1p?2ukλns converges in H1(R2) to wk as s, where wk also changes sign exactly k times and solves the following equation
?Δu+ωu=|u|p?2u,uH1(R2).
  相似文献   

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Under the assumption that VL2([0,π];dx), we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators ?d2/dx2+V in L2([0,π];dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators ?d2/dx2+V in Lp([0,π];dx), p(1,).  相似文献   

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For every real numbers a?1, b?1 with (a,b)(1,1), the curve parametrized by θR valued in C2?R4
γ:θ?(x(θ)+?1y(θ),u(θ)+?1v(θ))
with components:
x(θ):=a?1a(ab?1)cos?θ,y(θ):=b(a?1)ab?1sin?θ,u(θ):=b?1b(ab?1)sin?θ,v(θ):=?a(b?1)ab?1cos?θ,
has image contained in the CR-umbilical locus:
γ(R)?UmbCR(Ea,b)?Ea,b
of the ellipsoid Ea,b?C2 of equation ax2+y2+bu2+v2=1, where the CR-umbilical locus of a Levi nondegenerate hypersurface M3?C2 is the set of points at which the Cartan curvature of M vanishes.  相似文献   

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In the present paper we perform the homogenization of the semilinear elliptic problem
{uε0inΩε,?divA(x)Duε=F(x,uε)inΩε,uε=0on?Ωε.
In this problem F(x,s) is a Carathéodory function such that 0F(x,s)h(x)/Γ(s) a.e. xΩ for every s>0, with h in some Lr(Ω) and Γ a C1([0,+[) function such that Γ(0)=0 and Γ(s)>0 for every s>0. On the other hand the open sets Ωε are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” μu0 appears in the limit equation in the case where the function F(x,s) depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function F(x,s) satisfies 0F(x,s)h(x)(1s+1). In this case the solution uε to the problem belongs to H01(Ωε) and its definition is a “natural” and rather usual one.In the general case where F(x,s) exhibits a “strong singularity” at u=0, which is the purpose of the present paper, the solution uε to the problem only belongs to Hloc1(Ωε) but in general does not belong to H01(Ωε) anymore, even if uε vanishes on ?Ωε in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” μu0 still appears in the left-hand side while the source term F(x,u0) is not modified in the right-hand side.  相似文献   

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