On Meromorphic Solutions of a Certain Type of Nonlinear Differential Equations

We consider transcendental meromorphic solutions with N(r, f)=S(r, f) of the following type of nonlinear differential equations:#br#f^{n}+P_{n-2}(f)=p_{1}(z)e^{α1(z)}+p_{2}(z)e^{α2(z)},#br#where n ≥ 2 is an integer, P_{n-2}(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p_{1}(z), p_{2}(z) are nonzero small functions of f, and α_{1}(z), α_{2}(z) are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.

Sub-additive Topological and Measure-theoretic Tail Pressures

In this paper, we give some definitions of the topological tail pressures for sub-additive potentials and prove that they are equivalent if the potentials are continuous. Under some assumptions, we get a variational principle which exhibits the relationship between topological tail pressure and measure-theoretic tail entropy. Finally, we define a new measure-theoretic tail pressure for sub-additive potentials and some interesting properties of it are obtained.

On Partial Regularity of Suitable Weak Solutions to the Stationary Fractional Navier-Stokes Equations in Dimension Four and Five

In this paper, we investigate the partial regularity of suitable weak solutions to the multidimensional stationary Navier-Stokes equations with fractional power of the Laplacian (-Δ)^{α} (n/6 ≤ α < 1 and α ≠1/2). It is shown that the n + 2 -6α (3 ≤ n ≤ 5) dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu[19] to four and five dimension. Moreover, the pressure in ε-regularity criteria is an improvement of corresponding results in[1, 13, 18, 20].

Mapping Properties of Certain Oscillatory Integrals on Modulation Spaces

Suppose β_{1} > α_{1} ≥ 0, β_{2} > α_{2} ≥ 0 and (k, j) ∈ R^{2}. In this paper, we mainly investigate the mapping properties of the operator#br#T_{α,β}f(x, y, z)=∫_{Q2}f(x -t, y -s, z -t^{k}s^{j})e^{-2πit-β1s-β2}t^{-1-α1s-1-α2}dtds#br#on modulation spaces, where Q^{2}=[0, 1] ×[0, 1] is the unit square in two dimensions.

Strong 3-Commutativity Preserving Maps on Standard Operator Algebras

Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ:A → A is said to be strong 3-commutativity preserving if[Φ(A), Φ(B)]_{3}=[A, B]_{3} for all A, B ∈ A, where[A, B]_{3} is the 3-commutator of A, B defined by[A, B]_{3}=[[[A, B], B], B] with[A, B]=AB -BA. The main result in this paper is shown that, if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h:A → F and a scalar λ ∈ F with λ^{4}=1 such that Φ(A)=λA + h(A)I for all A ∈ A.

Similarity and Parameterizations of Dilations of Pairs of Dual Group Frames in Hilbert Spaces

In this paper, firstly, in order to establish our main techniques we give a direct proof for the existence of the dilations for pairs of dual group frames. Then we focus on proving the uniqueness of such dilations in certain sense of similarity and giving an operator parameterization of the dilations of all pairs of dual group frames for a given group frame. We show that the operators which transform different dilations are of special structured lower triangular.

Wandering Subspaces and Quasi-wandering Subspaces in the Hardy-Sobolev Spaces

In this paper, we prove that for -(1/2) ≤ β ≤ 0, suppose M is an invariant subspaces of the Hardy-Sobolev spaces H_{β}^{2}(D) for T_{z}^{β}, then M⊖zM is a generating wandering subspace of M, that is, M=[M⊖zM]_{Tzβ}. Moreover, any non-trivial invariant subspace M of H_{β}2(D) is also generated by the quasi-wandering subspace P_{M}T_{z}^{β}M^{⊥}, that is, M=[P_{M}T_{z}^{β}M^{⊥}]_{Tzβ}.

Cluster Structures in 2-Calabi-Yau Triangulated Categories of Dynkin Type with Maximal Rigid Objects

In this paper, we consider two kinds of 2-Calabi-Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called A_{n,t}=D^{b}(KA_{(2t+1)(n+1)-3)}/τ^{t(n+1)-1}[1], where n, t ≥ 1, and D_{n,t}=D^{b}(KD_{2t(n+1)})/τ^{(n+1)}φ^{n}, where n, t ≥ 1, and φ is induced by an automorphism of D_{2t(n+1)} of order 2. Except the categories A_{n,1}, they all contain non-zero maximal rigid objects which are not cluster tilting. A_{n,1} contain cluster tilting objects. We define the cluster complex of A_{n,t} (resp. D_{n,t}) by using the geometric description of cluster categories of type A (resp. type D). We show that there is an isomorphism from the cluster complex of A_{n,t} (resp. D_{n,t}) to the cluster complex of root system of type B_{n}. In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan-Palu-Reiten:Let R_{An,t}, resp. R_{Dn,t}, be the full subcategory of A_{n,t}, resp. D_{n,t}, generated by the rigid objects. Then R_{An,t} ≃ R_{An,1} and R_{Dn,t} ≃ R_{An,1} as additive categories, for all t ≥ 1.