1. The statement of the problem. Let B denote the boundary of a smooth, closed and bounded surface in £ , and V the exterior of B. Erect a spherical polar coordinate system with origin interior to B; and denote by r a point (r, 0, ) in V and by TB a point (r#, 6B, 4>B) on B. Further let v be a scalar wave function for the exterior V, i.e., (a) v(r) is of class C, r £ 7 = VVJB, (b) ( V 2 +^ 2...

any language foremost is composed of words and words are one of the most important elements in conveying meaning and they have different diversities and levels in conveying this meaning. quranic words are not the exception; therefore, to discover and to understand the different levels of conveying meaning (exterior and inner) by words, the quran reader can be more aware of the meaning and the p...

This is a very brief review of some results from Refs. [2] and [3]. In holographic renormalization, we studied the RG flow of a 2d N = (4, 4) CFT perturbed by a relevant operator, flowing to a conformal fixed point in the IR. Here, the supergravity dual is displayed, and the computation of correlators is discussed. The sample stress-energy correlator given here provides an opportunity to explic...

It is well-known that Birkhoff’s theorem is no longer valid in theories with more than four dimensions. Thus, in these theories the effective four-dimensional picture allows the existence of different possible, non-Schwarzschild, scenarios for the description of the spacetime outside of a spherical star, contrary to four-dimensional general relativity. We investigate the exterior spacetime of a...

and Applied Analysis 3 We denote by ∧l ∧l R the space of l-covectors in R and the direct sum ∧ R n ⊕ l 0 ∧l R 2.1 is a graded algebra with respect to the wedge product ∧. We will make use of the exterior derivative: d : C∞ ( Ω,∧l ) −→ C∞ ( Ω,∧l 1 ) 2.2 and its formal adjoint operator d∗ −1 nl 1 ∗ d∗ : C∞ ( Ω,∧l 1 ) −→ C∞ ( Ω,∧l ) , 2.3 known as the Hodge codifferential, where the symbol ∗ denot...

In this paper we show that the equivalence of the Apollonian metric and its inner metric remains unchanged by the removal of a point from the domain. For this we need to assume that the complement of the domain is not contained in a hyperplane. This improves a result of the authors wherein the same conclusion was reached under the stronger assumption that the domain contains an exterior point.

Let Ω ⊆ Rn be a compact convex set and x be a point in the exterior of Ω. The aperture angle of x relative to Ω is defined as the maximal angle of the smallest closed convex cone that contains Ω − x. This note provides an explicit formula, based on eigenvalues of symmetric matrices, for the aperture angle of a point relative to an ellipsoid.

Let Ω ⊆ Rn be a compact convex set and x be a point in the exterior of Ω. The aperture angle of x relative to Ω is defined as the maximal angle of the smallest closed convex cone that contains Ω − x. This note provides an explicit formula, based on eigenvalues of symmetric matrices, for the aperture angle of a point relative to an ellipsoid.