simplex

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simplex

[′sim‚pleks]
(mathematics)
An n-dimensional simplex in a euclidean space consists of n + 1 linearly independent points p0, p1,…, pn together with all line segments a0 p0+ a1 p1+ ⋯ + an pn where the ai ≥ 0 and a0+ a1+ ⋯ + an = 1; a triangle with its interior and a tetrahedron with its interior are examples.
(quantum mechanics)
The eigenvalue of a nucleus or other object with an octupole (pear) shape under an operation consisting of rotation through 180° about an axis perpendicular to the symmetry axis, followed by inversion.

Simplex

 

a method of two-way communication wherein, at each communication station, transmission alternates with reception.


Simplex

 

in mathematics, the simplest convex polyhedron of some given dimension n. When n = 3, we have a three-dimensional simplex, which is a tetrahedron; the tetrahedron may be irregular. A two-dimensional simplex is a triangle, a one-dimensional simplex is a line segment, and a zero-dimensional simplex is a point.

An n-dimensional simplex has n + 1 vertices, which do not belong to any (n - 1)-dimensional subspace of the Euclidean space (of dimension at least n) in which the simplex lies. Conversely, any n + 1 points of a Euclidean m-dimensional space Rm, mn, that do not lie in a subspace of dimension less than n uniquely determine an n-dimensional simplex with vertices at the given points e0, e1, • • •, en. This simplex can be defined as the convex closure of the set of the given n + 1 points—that is, as the intersection of all convex polyhedra of Rm that contain the points.

If a system of Cartesian coordinates x1, x2, • • •, xm is defined in Rm such that the vertex ei, i = 0, 1, • • •, n, has the coordinates Simplex, then the simplex with the vertices e0, e1, • • •, en consists of all points of Rm whose coordinates are of the form

where μ(0), μ(1) •••, μ(n) are arbitrary nonnegative numbers whose sum is 1. By analogy with the case where n ≤ 3, we can say that all points of a simplex with given vertices are obtained if we place arbitrary nonnegative masses (not all of which are zero) at the vertices and determine the center of gravity of these masses. It should be noted that the requirement that the sum of the masses be equal to 1 eliminates only the case where all the masses are zero.

Any r + 1 vertices, 0 ≤ r ≤ n − 1, selected from the given n + 1 vertices of an n-dimensional simplex determine an r-dimensional simplex, which is called a face of the original simplex. The zero-dimensional faces of a simplex are its vertices; the one-dimensional faces are called its edges.

REFERENCES

Aleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Pontriagin, L. S. Osnovy kombinatornoi topologii. Moscow-Leningrad, 1947. Pages 23–31.

simplex

(communications)
Used to describe a communications channel that can only ever carry a signal in one direction, like a one-way street. Television is an example of (broadcast) simplex communication.

Opposite: duplex.

simplex

(algorithm)

simplex

One way transmission. Contrast with half-duplex and full-duplex.
References in periodicals archive ?
where K[W] denotes the subcomplex of K consisting of all simplices whose vertices belong to W.
In this section a new Lipschitz bound over simplices is proposed.
i]} I[member of]I is fat if all its simplices are [phi]-fat for some [phi]>0.
iii) the number of simplices per polytopal element is uniformly bounded.
that is, the union of interiors of all simplices having s as a vertex.
The incremental cavity algorithm (Watson 1981) deletes all n-dimensional simplices that are no longer empty, according the Delaunay circum-sphere criterion, after the insertion of the new point.
A (geometric) simplicial complex [DELTA] is a finite family of simplices such that (i) if F is in [DELTA] and G is a face of F, then G is also in [DELTA], and (ii) for any two elements F and H of [DELTA], F [intersection] H is a face of both F and H.
p] are not just piecewise-linear but actually linear (by which we really mean "affine"), and one might hope to base a proof of Theorem 1 on this; unfortunately, the images of these simplices under [[rho].
h] consisting of simplices or d-parallelotopes of the domain [OMEGA], where each element T [member of] [T.
locum inueniunt, perfectiones esse simpliciter simplices, nec, nisi cum limitationibus seu imperfectionibus, ullam inuoluere repugnantiam"; Knutzen, Systema Causarum efficientium, [sections]33, p.
d] (or, more generally, in a point lattice) we call a full-dimensional simplex unimodular when it is an affine lattice basis and we call a triangulation unimodular when all its full-dimensional simplices are.