This paper is concerned with the construction of high order mass-lumping finite elements on simplexes and a program for computing mass-lumping finite elements on triangles and tetrahedra. The polynomial spaces for mass-lumping finite elements, as proposed in the literature, are presented and discussed. In particular, the unisolvence problem of symmetric point-sets for the polynomial spaces used in mass-lumping elements is addressed, and an interesting property of the unisolvent symmetric point-sets is observed and discussed. Though its theoretical proof is still lacking, this property seems to be true in general, and it can greatly reduce the number of cases to consider in the computations of mass-lumping elements. A program for computing mass-lumping finite elements on triangles and tetrahedra, derived from the code for computing numerical quadrature rules presented in [7], is introduced. New mass-lumping finite elements on triangles found using this program with higher orders, namely 7, 8 and 9, than those available in the literature are reported.

Basic features of a microcomputer package, BCDP, which is an auto-tutorial designed to teach the Simplex algorithm are described. The package may be used to augment lecture and text materials in introductory linear programming courses or as a review for advanced math programming courses. Preliminary evaluations of the effectiveness of the program to augment classroom instruction are very positive.

This paper presents the capabilities and limitations of a microcomputer linear programming package. The solution algorithm is a version of the revised simplex. Rapid problem entry, user ease of operation, sensitivity analyses on objective function and right hand sides are advantages. A problem size of 150 activities and 64 constraints can be solved in present form. Due to problem size, limitations and lack of parametric and integer programming routines, this package is thought to have the most value in teaching applications and research problems in the smaller size categories.

In the present paper, a coupled algorithm refining recursively the Hermite–Hadamard inequality on a simplex is investigated. Our approach allows us to express the integral mean value $M_{f}$ of a convex function $f$ on a simplex as both the limit of sequences and sum of series involving iterative lower and upper bounds of $M_{f}$. Two examples of interest are discussed.

We establish some inequalities for the second moment

$$\frac{1}{\left| K \right|}\,{{\int }_{K}}\left| x \right|_{2}^{2}dx$$

of a convex body $K$ under various assumptions on the position of $K$.

This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations.

We show that solvability of the abstract Dirichlet problem for Baire-two functions on a simplex X cannot be characterized by topological properties of the set of extreme points of X.

In this paper, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on the metric addition inequality of Alexander.

2000 Mathematics subject classification: primary 52A40.

A sequence of random triangles is constructed by choosing successively the three vertices of one triangle at random in the interior of its predecessor. A way is found to prove that the shapes of these triangles converge, almost surely, to collinear shapes, thus closing a gap in one of the central arguments of Mannion [5]. The new approach is based on a representation of the triangle process by a sequence of products of i.i.d. random matrices. We succeed in calculating the corresponding Lyapounov exponent.

Given the equations for the n + 1 hyperplanes that bound an n-simplex in Rn, simple formulae are derived for the contents of the n − r simplices (0 ≦ r < n) embedded in it. For example, when n = 3, the formulae include the volume of the tetrahedron, the areas of its faces and the lengths of its edges.

Buckminster Fuller has coined the name tetrahelix for a column of regular tetrahedra, each sharing two faces with neighbours, one 'below' and one 'above' [A. H. Boerdijk, Philips Research Reports 7 (1952), p. 309]. Such a column could well be employed in architecture, because it is both strong and attractive. The (n — 1)-dimensional analogue is based on a skew polygon such that every n consecutive vertices belong to a regular simplex. The generalized twist which shifts this polygon one step along itself is found to have the characteristic equation

(λ - 1)2{(n - 1)λn-2 + 2(n - 2)λn-3 + 3(n - 3)λn-4 + . . . + (n - 2)2λ + (n - 1)} = 0,

which can be derived from tan nθ = n tan θ by setting λ = exp (2θi).

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.