Uniqueness of difference polynomials

<abstract><p>Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.</p></abstract>

Value distribution of meromorphic functions concerning rational functions and differences

Let c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

Einstein-Weyl structures on trans-Sasakian manifolds

In this article we study Einstein-Weyl structures on a 3-dimensional trans-Sasakian manifold M of type (α, β). First, we prove that a 3-dimensional trans-Sasakian manifold admitting both Einstein-Weyl structures W± = (g, ±θ) is Einstein, or is homothetic to a Sasakian manifold if α ≠ 0. Next for β ≠ 0 it is proved that M is Einstein, or is homothetic to an f-Kenmotsu manifold if it admits an Einstein-Weyl structure W = (g, κη) for some nonzero constant κ. Finally, a classification is obtained when a trans-Sasakian manifold admits a closed Einstein-Weyl structure. Further, if M is compact we also obtain two corollaries.

On Homogeneous Polynomials Determined by their Partial Derivatives

We prove that a generic homogeneous polynomial of degree $d$ is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order $k$ for $k\leqslant \frac{d}{2}-1$.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Sarason’s Conjecture of Toeplitz Operators on Fock-Sobolev Type Spaces

In this note, we will solve Sarason’s conjecture on the Fock-Sobolev type spaces and give a well solution that if Toeplitz product TuTv¯, with entire symbols u and v, is bounded if and only if u=eq, v=Ce-q, where q is a linear complex polynomial and C is a nonzero constant.