Vertical bar

Summary

The vertical bar, , is a glyph with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally the word "or"), vbar, and others. Usage Mathematics The vertical bar is used as a mathematical symbol in numerous ways:

absolute value: |x|, read "the absolute value of x"
cardinality: |S|, read "the cardinality of the set S" or "the length of a string S".
conditional probability: P(X|Y), read "the probability of X given Y"
determinant: |A|, read "the determinant of the matrix A". When the matrix entries are written out, the determinant is denoted by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses of the matrix, as in \begin{vmatrix} a & b \ c & d\end{vmatrix}.
distance: P|ab, denoting the shortest distance between

Official source

https://en.wikipedia.org/wiki/Vertical_bar

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A generalized mountain-pass theorem: the existence of multiple critical points

Hans-Jörg Ruppen

We analyse the existence of multiple critical points for an even functional J : H -> R in the following context: the Hilbert space H can be split into an orthogonal sum H = Y circle plus Z in such a way that inf{J(u) : u is an element of Z and parallel to u parallel to = rho >= alpha > J(0) and that here exits a point b is an element of H with parallel to b parallel to > rho and with J(b)

Springer Heidelberg2016

Unique decomposition of homogeneous languages and application to isothetic regions

Nicolas René Jean Ninin

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019

Let 1 < p < infinity, let G and H be locally compact groups and let c) be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp (G) of the p-convolution operators on G into CVp (H) which extends the usual definition of the image of a bounded measure by omega. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let G(d) denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, vertical bar parallel to mu vertical bar parallel to CVp(G)

Cambridge University Press2008