Search results
Results from the WOW.Com Content Network
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
Two elements , are conjugate if there exists an element such that =, in which case is called a conjugate of and is called a conjugate of . In the case of the general linear group GL ( n ) {\displaystyle \operatorname {GL} (n)} of invertible matrices , the conjugacy relation is called matrix similarity .
One use of conjugate acids and bases lies in buffering systems, which include a buffer solution. In a buffer, a weak acid and its conjugate base (in the form of a salt), or a weak base and its conjugate acid, are used in order to limit the pH change during a titration process. Buffers have both organic and non-organic chemical applications.
The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e., (†) † =. Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the ...
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
In solution chemistry, it is common to use H + as an abbreviation for the solvated hydrogen ion, regardless of the solvent. In aqueous solution H + denotes a solvated hydronium ion rather than a proton. [9] [10] The designation of an acid or base as "conjugate" depends on the context. The conjugate acid BH + of a base B dissociates according to
A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute value at most 1, then α is a root of unity. There are quantitative forms of this, stating more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an ...
In abstract algebra, the conjugacy problem for a group G with a given presentation is the decision problem of determining, given two words x and y in G, whether or not they represent conjugate elements of G. That is, the problem is to determine whether there exists an element z of G such that =.