The ”massless” VECTOR field equation in de Sitter (dS) 4-dimensional space is gauge invariant under some special gauge transformations. It is also gauge invariant in ambient space notations in which the field equation is written in terms of the Casimir operators of dS group. In this paper the "massless" VECTOR field equation has been solved in the PHYSICAL case. It has been shown that the solution can be written as the multiplication of a generalized polarization VECTOR and a ”massless” conformally coupled scalar field in the ambient space notations. The PHYSICAL VECTOR twopoint FUNCTION has been calculated using ambient space formalism and its zero curvature limit has been considered. It is shown that the PHYSICAL VECTOR TWO-POINT can be written in terms of the conformally coupled massless scalar TWO-POINT FUNCTION in the ambient space notations. The TWO-POINT FUNCTION is expressed in terms of dS intrinsic coordinates from its ambient space counterpart, which is dS-invariant and is free from any divergences.

Let $X$ be a compact Hausdorff space, $A$ be a (commutative) Banach algebra and $\mathcal{A}$ be a Banach $A$-valued FUNCTION algebra on $X$. Let $\mathfrak{A}$ be the FUNCTION algebra on $X$, consisting of scalar-valued FUNCTIONs in $\mathcal{A}$. We study and characterize various amenability properties of the algebra $\mathcal{A}$ in terms of cohomological properties of $\mathfrak{A}$ and $A$. Containing some well-known examples, such as $C(X,A)$ and $Lip(X,A)$, the class of VECTOR-valued FUNCTION algebras also includes, in some sense, the tensor products $\mathfrak{A} \hat \otimes_\gamma A$. As consequences, some known results in this area are covered.

We give necessary and sufficient conditions for the boundedness and compactness of weighted composition operators between spaces of VECTOR- valued Lipschitz FUNCTIONs. We then show that a bounded separating linear operator between these spaces is indeed a weighted composition operator.

We continue the study begun by the third author of C*-Segal algebra-valued FUNCTION algebras with an emphasis on the order structure. Our main result is a characterization theorem for C*-Segal algebra-valued FUNCTION algebras with an order unitization. As an intermediate step, we establish a FUNCTION algebraic description of the multiplier module of arbitrary Segal algebra-valued FUNCTION algebras. We also consider the Gelfand representation of these algebras in the commutative case.

Fuzzification of support VECTOR machine has been utilized to deal with outlier and noise problem. This importance is achieved, by the means of fuzzy membership FUNCTION, which is generally built based on the distance of the points to the class centroid. The focus of this research is twofold. Firstly, by taking the advantage of robust statistics in the fuzzy SVM, more emphasis on reducing the impact of outliers on the generalizability of SVM has been placed. Moreover, the variety of membership FUNCTION for the elliptical data has been designated, based on the classic and robust Mahalanobis distance. Minimum covariance determinant and orthogonalised Gnanadesikan Kettenring estimators are employed in the structure of the robust– fuzzy SVM. By implementing the new membership FUNCTION, the disadvantages of the traditional fuzzy membership FUNCTION has been rectified. Simulated and real benchmarking data set confirm the effectiveness of the proposed methods. Compared with the traditional SVM and fuzzy SVM, these methods give a better performance on reducing the effects of outliers and significantly improves the classification accuracy and generalization.

We establish a sharp maximal FUNCTION estimate for some VECTOR-valued multilinear singular integral operators. As an application, we obtain the (Lp, Lq) -norm inequality for VECTOR-valued multilinear operators.