In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses ( ), square brackets [ ], curly brackets { }, and angle brackets < >. In the typical use, a mathematical expression is enclosed between an "opening bracket" and a matching "closing bracket". Generally such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Additionally, there are several specific uses and meanings for the various brackets.

Historically, other notations, such as the vinculum, were similarly used for grouping; in present-day use, these notations have all specific meanings.

In the Z formal specification language, curly braces denote a set and angle brackets denote a sequence.

Symbols for representing angle brackets

A variety of different symbols are used to represent angle brackets. In e-mail and other ASCII text it is common to use the less-than (<) and greater-than (>) signs to represent angle brackets. Unicode has three pairs of dedicated characters: U+2329 (〈) and U+232A (〉) (left/right-pointing angle bracket), U+27E8 (⟨) and U+27E9 (⟩) (mathematical left/right angle bracket), and U+3008 (〈) and U+3009 (〉) (left/right angle bracket). In LaTeX the markup is \langle and \rangle: .

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Sun Oct 18 02:12:46 2009

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vectors and linear functionals in pure mathematics It is so called because the inner product of two states is denoted by a bracket consisting of a left part called the bra and a right part called the ket The notation was invented by Paul Dirac and is also known as Dirac notation

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mathematics It is so called because the inner product of two states is denoted by a bracket consisting of a left part called the bra and a right part called the ket The notation was invented by Paul Dirac and is also known as Dirac notation

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of invariants is in fact generated by only traces Since trace functions are invariant by all inner automorphisms the Culler Shalen construction essentially assumes that we are acting by on even if For instance for a free group of rank 2 and G = SO 2 the conjugation action is trivial and the G character variety is the torus But the trace algebra is a strictly small

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