Volume 10, Issue 3

In this paper, we consider a Brück-type result for a homogeneous differential polynomial generated by a meromorphic function.

We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m -quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with (n-1){(n-1)}-dimensional Einstein fibers in suitable cases.

An explicit representation of the Green function of the Robin problem for the Poisson equation in the unit ball is given.

Moment functions play a basic role in probability theory. A natural generalization can be defined on hypergroups which leads to the concept of generalized moment function sequences. In a former paper we studied some function classes on hypergroup joins which play a basic role in spectral synthesis. Moment functions are also important basic blocks of spectral synthesis. All these functions can be characterized by well-known functional equations. In this paper we describe generalized moment function sequences on hypergroup joins.

The problem of how to find a sparse representation of a signal is an important one in applied and computational harmonic analysis. It is closely related to the problem of how to reconstruct a sparse vector from its projection in a much lower-dimensional vector space. This is the setting of compressed sensing, where the projection is given by a matrix with many more columns than rows. We introduce a class of random matrices that can be used to reconstruct sparse vectors in this paradigm. These matrices satisfy the restricted isometry property with overwhelming probability. We also discuss an application in dimensionality reduction where we initially discovered this class of matrices.

The purpose of this paper is to prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let ℒ⁢(X){\mathcal{L}(X)} be the algebra of all bounded linear operators of X into itself and let 𝒜⁢(X)⊂ℒ⁢(X){\mathcal{A}(X)\subset\mathcal{L}(X)} be a standard operator algebra. Suppose there exist linear mappings ℋ,𝒢:𝒜⁢(𝒳)→ℒ⁢(𝒳){\mathcal{H},\mathcal{G}\colon\mathcal{A(X)}\to\mathcal{L(X)}} satisfying the relations ℋ⁢(𝒜m+n)=ℋ⁢(𝒜m)⁢𝒜n+𝒜m⁢𝒢⁢(𝒜n),\displaystyle\mathcal{H}(\mathcal{A}^{m+n})=\mathcal{H}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{G}(\mathcal{A}^{n}),𝒢⁢(𝒜m+n)=𝒢⁢(𝒜m)⁢𝒜n+𝒜m⁢ℋ⁢(𝒜n)\displaystyle\mathcal{G}(\mathcal{A}^{m+n})=\mathcal{G}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{H}(\mathcal{A}^{n}) for all 𝒜∈𝒜⁢(𝒳){\mathcal{A}\in\mathcal{A(X)}} and some fixed integers m,n≥1{m,n\geq 1}. Then there exists ℬ∈ℒ⁢(𝒳){\mathcal{B}\in\mathcal{L(X)}}, such that ℋ⁢(𝒜)=𝒜⁢ℬ-ℬ⁢𝒜{\mathcal{H(A)}=\mathcal{AB}-\mathcal{BA}} for all 𝒜∈ℱ⁢(𝒳){\mathcal{A}\in\mathcal{F(X)}}, where ℱ⁢(𝒳){\mathcal{F(X)}} denotes the ideal of all finite rank operators in ℒ⁢(X){\mathcal{L}(X)}, and ℋ⁢(𝒜m)=𝒜m⁢ℬ-ℬ⁢𝒜m{\mathcal{H}(\mathcal{A}^{m})=\mathcal{A}^{m}\mathcal{B}-\mathcal{B}\mathcal{A% }^{m}} for all 𝒜∈𝒜⁢(𝒳){\mathcal{A}\in\mathcal{A(X)}}.

In this paper, we prove two theorems establishing the uniqueness of the derivatives of meromorphic functions f(k){f^{{(k)}\vphantom{k}}} and g(k){g^{{(k)}\vphantom{k}}} when they share two sets with least cardinalities. Our results improve some existing results of Fang–Lahiri, Banerjee–Bhattacharjee, Banerjee–Mallick and Banerjee–Chakraborty.

In this article, we introduce the notion of “metrically differentiable” for set-valued functions. By using this notion, it is shown that each Lipschitz set-valued function is differentiable almost everywhere. Its relationship to the differentiation in the sense of Hukuhara and its generalizations are also discussed.

In this paper, we consider the Cauchy problem for the relativistic Enskog equation with near vacuum data for a hard sphere gas in the Robertson–Walker space-time. We prove an existence and uniqueness result of the global (in time) mild solution in a suitable weighted space. We also study the asymptotic behavior of the solution as well as the L∞{L^{\infty}}-stability.