notation is used to denote the falling factorial, an n-th degree polynomial defined by Alternatively, the same notation may be encountered as representing the rising factorial, also called "Pochhammer symbol". Another notation for the same is . It can be defined by Quantum mechanics In quantum mechanics, angle brackets are also used as part of Dirac's formalism, bra–ket notation, to denote vectors from the dual spaces of the bra and the ket . In statistical mechanics, angle brackets denote ensemble or time average. Polynomial rings Square brackets are used to contain the variable(s) in polynomial rings. For example, is the ring of polynomials with real number coefficients and variable . Subring generated by an element or collection of elements If is a subring of a ring , and is an element of , then denotes the subring of generated by and . This subring consists of all the elements that can be obtained, starting from the elements of and , by repeated addition and multiplication; equivalently, it is the smallest subring of that contains and . For example, is the smallest subring of containing all the integers and ; it consists of all numbers of the form , where and are arbitrary integers. Another example: is the subring of consisting of all rational numbers whose denominator is a power of . More generally, if is a subring of a ring , and , then denotes the subring of generated by and . Even more generally, if is a subset of , then is the subring of generated by and . Lie bracket and commutator In group theory and ring theory, square brackets are used to denote the commutator. In group theory, the commutator [g,h] is commonly defined as g−1h−1gh. In ring theory, the commutator [a,b] is defined as ab − ba. Furthermore, braces may be used to denote the anticommutator: {a,b} is defined as ab + ba. The Lie bracket of a Lie algebra is a binary operation denoted by . By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. There are many different forms of Lie bracket, in particular the Lie derivative and the Jacobi–Lie bracket. Floor/ceiling functions and fractional part Square brackets, as in , are sometimes used to denote the floor function, which rounds a real number down to the next integer. Respectively, some authors use outwards pointing