Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,

In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word "simplex" simply means any finite set of vertices.

Simplex

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The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincar, writing about algebraic topology in 1900, called them "generalized tetrahedra".In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").[3]

The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as tag_hash_120n, the other two being the cross-polytope family, labeled as tag_hash_122n, and the hypercubes, labeled as tag_hash_124n. A fourth family, the tessellation of n-dimensional space by infinitely many hypercubes, he labeled as tag_hash_126n.[4]

An n-simplex is the polytope with the fewest vertices that requires n dimensions. Consider a line segment AB as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point E somewhere outside the 3-space. The new shape ABCDE, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space.

The coefficients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.

A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.

Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given ( p i ) i {\displaystyle (p_{i})_{i}} with possibly negative entries, the closest point ( t i ) i {\displaystyle \left(t_{i}\right)_{i}} on the simplex has coordinates

This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.

The above regular n-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:

A highly symmetric way to construct a regular n-simplex is to use a representation of the cyclic group Zn+1 by orthogonal matrices. This is an n n orthogonal matrix Q such that Qn+1 = I is the identity matrix, but no lower power of Q is. Applying powers of this matrix to an appropriate vector v will produce the vertices of a regular n-simplex. To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix

where each tag_hash_142i is an integer between zero and n inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices Qi form a basis for the non-trivial irreducible real representations of Zn+1, and the vector being rotated is not stabilized by any of them.

If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping.[12] In particular, the volume of such a simplex is

The Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the (n + 1)-hypercube's edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.

In probability theory, the points of the standard n-simplex in (n + 1)-space form the space of possible probability distributions on a finite set consisting of n + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (n + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the kth vertex of the simplex is assigned to have the kth probability of the (n + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.

A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.

Herpes is an infection that is caused by a herpes simplex virus (HSV). Oral herpes causes cold sores around the mouth or face. Genital herpes affects the genitals, buttocks or anal area. Genital herpes is a sexually transmitted disease (STD). It affects the genitals, buttocks or anal area. Other herpes infections can affect the eyes, skin, or other parts of the body. The virus can be dangerous in newborn babies or in people with weak immune systems.

The Simons Simplex Collection (SSC) is a core project and resource of the Simons Foundation Autism Research Initiative (SFARI). The SSC achieved its primary goal to establish a permanent repository of genetic samples from 2,600 simplex families, each of which has one child affected with an autism spectrum disorder, and unaffected parents and siblings.

Tremendous advances have occurred over the past 30 years in the diagnosis and management of neonatal herpes simplex virus (HSV) disease. Mortality in patients with disseminated disease has decreased from 85 to 29%, and that in patients with central nervous system (CNS) disease has decreased from 50 to 4%. Morbidity has been improved more modestly: the proportion of patients with disseminated disease who are developing normally at 1 year has increased from 50 to 83%. While the proportion of patients with neurologic morbidity following CNS disease has remained essentially unchanged over the past three decades, the total number of patients who are developing normally following HSV CNS disease has increased due to the improved survival. Although additional therapeutic advances in the future are possible, more immediate methods for further improvements in outcome for patients with this potentially devastating disease lie in an enhanced awareness of neonatal HSV infection and disease. A thorough understanding of the biology and natural history of HSV in the gravid woman and the neonate provides the basis for such an index of suspicion and is provided in this article.

Results: Herpes simplex virus type 1 genomes were detected in 11 of 14 patients (79%) with Bell palsy but not in patients with the Ramsay-Hunt syndrome or in other controls. The nucleotide sequences of the PCR fragments were identical to those of the HSV-1 genome.

Isolation of herpes simplex virus (HSV) from brain tissue after biopsy has been considered the reference standard for the diagnosis of herpes simplex encephalitis (HSE). During the evaluation of antiviral treatment of HSE, cerebrospinal fluid (CSF) was obtained from patients with clinical disease indicative of HSE who underwent diagnostic brain biopsy. HSV DNA was detected by polymerase chain reaction (PCR) in CSF of 53 (98%) of 54 patients with biopsy-proven HSE and was detected in all 18 CSF specimens obtained before brain biopsy from patients with proven HSE. Four of 19 CSF specimens were positive after 2 weeks of antiviral therapy. Positive results were found in 3 (6%) of 47 patients whose brain tissue was culture-negative. Detection of HSV DNA in the CSF correlated significantly with age and focal radiographic findings. Thus, PCR detection of HSV DNA should be the standard for diagnosis of HSE. e24fc04721