ISSN 1439-8516 CN 11-2039/O1
This paper studies the weighted Hardy inequalities on the discrete intervals with four different kinds of boundary conditions. The main result is the uniform expression of the basic estimate of the optimal constant with the corresponding boundary condition. Firstly, one-side boundary condition is considered, which means that the sequences vanish at the right endpoint (ND-case). Based on the dual method, it can be translated into the case vanishing at left endpoint (DN-case). Secondly, the condition is the case that the sequences vanish at two endpoints (DD-case). The third type of condition is the generality of the mean zero condition (NN-case), which is motivated from probability theory. To deal with the second and the third kinds of inequalities, the splitting technique is presented. Finally, as typical applications, some examples are included.
We conjecture that a Willmore torus having Willmore functional between 2π2 and 2π2√3 is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri's torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in Sn by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S5 attains the minimum 2π2√3, which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S5. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S3. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.
We consider viscous Burgers equations in one dimension of space and derive their solutions from stochastic variational principles on the corresponding group of homeomorphisms. The metrics considered on this group are Lp metrics. The velocity corresponds to the drift of some stochastic Lagrangian processes. Existence of minima is proved in some cases by direct methods. We also give a representation of the solutions of viscous Burgers equations in terms of stochastic forward-backward systems.
r (r∈[2,∞)) is bounded from Hardy space H1(μ) into L1(μ). Moreover, the authors show that, if f ∈ RBMO(μ), then[r(f)]r is either infinite everywhere or finite almost everywhere, and in the latter case,[r(f)]r belongs to RBLO(μ) with the norm no more than ‖f‖RBMO(μ) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of r from RBMO(μ) into RBLO(μ). The vector valued Calderón-Zygmund theory over (χ, d, μ) is also established with details in this paper.In this note, the exact value of the James constant for the l3-l1 space is obtained, J(l3-l1)=1.5573…. This result improves the known inequality, J(l3-l1)≤4/3√10, which was given by Dhompongsa, Piraisangjun and Saejung.
Let p be an odd prime. For the Stiefel manifold Wm+k,k=SU(m+k)/SU(m), we obtain an upper bound of its p-primary homotopy exponent in the stable range k≤m with k≤(p-1)2+1.
To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants, we define the genus expanded cut-and-join operators. Moreover all normalized the genus expanded cut-and-join operators with same degree form a differential algebra, which is isomorphic to the central subalgebra of the symmetric group algebra. As an application, we get some differential equations for the generating functions of the generalized Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.
In this paper, we characterize the symbols for (semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space. We show that for φ,ψ∈W1,∞, SφSψ=SφSψ on (Dh)⊥ if and only if φ and ψ satisfy one of the following conditions:(1) Both φ and ψ are harmonic functions; (2) There exist complex constants α and β, not both 0, such that φ=αψ+β.
In this note we study the general facility location problem with connectivity. We present an O(np2)-time algorithm for the general facility location problem with connectivity on trees. Furthermore, we present an O(np)-time algorithm for the general facility location problem with connectivity on equivalent binary trees.
Let H be a Hilbert space with dim H≥2 and Z∈ß(H) be an arbitrary but fixed operator. In this paper we show that an additive map Φ:ß(H) → ß(H) satisfies Φ(AB)=Φ(A)B=AΦ(B) for any A,B∈ß(H) with AB=Z if and only if Φ(AB)=Φ(A)B=AΦ(B), ∀A, B∈ß(H), that is, Φ is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Φ(A2)=AΦ(A)=Φ(A)A for any A∈ ß(H) with AA2=0 if and only if Φ(A)=AΦ(I)=Φ(I)A, ∀A∈ß(H), and generalize main results in Linear Algebra and its Application, 450, 243-249 (2014) to infinite dimensional case. New equivalent characterization of centralizers on ß(H) is obtained.
