Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such exte...

Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of multipartite for four-qubit systems. Based on concurrence nine different classes states, with each class being closed under stochastic local operation and classical communication, we construct tetrahedron. Proper measure presented by using volume For non entangled pure classifies bi-sepa...

We present a method for the numerical evaluation of 6D and 5D singular integrals appearing in Volume Integral Equations. It is an extension Sauter-Schwab/Taylor-Duffy strategy triangle-triangle interaction to tetrahedron-tetrahedron triangle-tetrahedron integrals. The general advantages these kind quadrature that they allow use different kinds kernel basis functions. They also work on curviline...

The Ponzano-Regge quantum gravity model has been a mystery since the time of its construction [1]. Ponzano and Regge found an unexpected relation between the 6j-symbol and Regge’s discrete version of the action functional of (Euclidean) general relativity in 3-dimensions. The description of the relation follows. Consider a 3-dimensional simplex manifold, which consists of tetrahedra whose faces...

By pure analogy with a usual three-dimensional origami, we can fold a regular octahedral material along flat faces in 4-space. The octahedron comprises four congruent tetrahedra. We will show a procedure to fold a tetrahedron along bisectors of the dihedral angles. This procedure demonstrates that each two of four surfaces of the given tetrahedron coincide with each other and that the point of ...

A generalized tetrahedron group is defined to be a group admitting the following presentation: 〈x, y, z | x = y = z = W p 1 (x, y) = W q 2 (y, z) = W r 3 (x, z) = 1〉, 2 ≤ l,m, n, p, q, r, where each Wi(a, b) is a cyclically reduced word involving both a and b. These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized tri...

Let T be a tetrahedral mesh. We present a 3-D local refinement algorithm for T which is mainly based on an 8-subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron T ∈ T produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, η(Ti ) ≥ cη(T), wher...

The sum of external angles of a polygon is always constant, π 2 . There are several elementary proofs of this fact. In the similar way, there is an invariant in polyhedron that is π 4 . To see this, let us consider a regular tetrahedron as an example. Tetrahedron has four vertices. Three regular triangles gather at each vertex. Developing the tetrahedron around each vertex, there is an open ang...

The use of a recyclable phase transfer catalyst, SiO2-PEG-ImBr, is demonstrated in a simple and highly efficient synthesis of phenacyl derivatives by nucleophilic substitution reaction of phenacyl halides with different anions in water. Advantages of this system are easy work-up, moderate to good yields, and recyclable catalyst. The catalyst can be recycled and reused several times with no loss...