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commutator | A Wisdom Archive on commutator | | commutator A selection of articles related to commutator | |
| We recommend this article: commutator - 1, and also this: commutator - 2. |
| | commutator, Commutator, Commutator - Group theory, Commutator - Ring theory, Commutator - Identities | | | | Top | » Page 4 « Page 5 | |
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| ARTICLES RELATED TO commutator | | | | |  commutator: Encyclopedia II - Ternary logic - Dyadic two-argument functionsIn order to avoid confusion with the binary numeral system, functions accepting two inputs will be referred to as dyadic functions, following the convention introduced in The Art of Assembly Language by Randall Hyde.
Ternary logic - Commutativity.
As stated above, there are 19,683 dyadic ternary functions. However, only of these are commutative. Of the four functions defined above, OR, AND, and EQUIV are commutative, while IF/THEN is not. For com ...
See also:Ternary logic, Ternary logic - Formal definitions, Ternary logic - Representation of values, Ternary logic - Ternary operators, Ternary logic - Dyadic two-argument functions, Ternary logic - Commutativity, Ternary logic - Implementation Read more here: » Ternary logic: Encyclopedia II - Ternary logic - Dyadic two-argument functions |
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| | | | | | commutator: Encyclopedia II - Multiset - Examples One of the most natural and simple examples is the multiset of prime factors of a number n. Here the underlying set of elements is the set of prime divisors of n. For example the number 120 has the prime factorization
120 = 233151
which gives the multiset {2, 2, 2, 3, 5}.
Another is the multiset of solutions of an algebraic equation. Everyone learns in secondary school that a quadratic equation has two solutions, but in some cases th ...
See also:Multiset, Multiset - Formal definition, Multiset - Examples, Multiset - Operations, Multiset - Multiset coefficients, Multiset - Free commutative monoids Read more here: » Multiset: Encyclopedia II - Multiset - Examples |
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| | | | | commutator: Encyclopedia II - Multiset - Operations The usual set operations such as union, intersection and Cartesian product can be easily generalized for multisets.
Suppose (A, m) and (B, n) are multisets
The union can be defined as (A ∪ B, f) where f(x) = max{m(x), n(x)}.
The intersection can be defined as (A ∩ B, f) where f(x) = min{m(x), n(x)}.
The cartesian product can be defined as (A ...
See also:Multiset, Multiset - Formal definition, Multiset - Examples, Multiset - Operations, Multiset - Multiset coefficients, Multiset - Free commutative monoids Read more here: » Multiset: Encyclopedia II - Multiset - Operations |
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| | | | | | | commutator: Encyclopedia II - Addition - Properties
Addition - Commutativity.
Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same. Symbolically, if a and b are any two numbers, then
a + b = b + a.
The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to spe ...
See also:Addition, Addition - Notation and terminology, Addition - Interpretations, Addition - Combining sets, Addition - Extending a measure, Addition - Combining translations, Addition - Properties, Addition - Commutativity, Addition - Associativity, Addition - Zero and one, Addition - Units, Addition - Definitions and proofs for the real numbers, Addition - Naturals, Addition - Integers, Addition - Rationals, Addition - Reals, Addition - Generalizations, Addition - In algebra, Addition - Addition of sets, Addition - Related operations, Addition - Notes Read more here: » Addition: Encyclopedia II - Addition - Properties |
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| | | | | commutator: Encyclopedia II - Addition - Properties
Addition - Commutativity.
Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same. Symbolically, if a and b are any two numbers, then
a + b = b + a.
The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to spe ...
See also:Addition, Addition - Notation and terminology, Addition - Interpretations, Addition - Combining sets, Addition - Extending a measure, Addition - Combining translations, Addition - Properties, Addition - Commutativity, Addition - Associativity, Addition - Zero and one, Addition - Units, Addition - Performing addition, Addition - Definitions and proofs for the real numbers, Addition - Naturals, Addition - Integers, Addition - Rationals, Addition - Reals, Addition - Generalizations, Addition - In algebra, Addition - Addition of sets, Addition - Related operations, Addition - Arithmetic, Addition - Ordering, Addition - Other ways to add, Addition - In literature, Addition - Notes Read more here: » Addition: Encyclopedia II - Addition - Properties |
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| | | | | | commutator: Encyclopedia II - Logical disjunction - Union The union used in set theory is defined in terms of a logical disjunction: x ∈ A ∪ B if and only if (x ∈ A) ∨ (x ∈ B). Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws.
...
See also:Logical disjunction, Logical disjunction - Definition, Logical disjunction - Symbol, Logical disjunction - Associativity and Commutativity, Logical disjunction - Bitwise operation, Logical disjunction - Union, Logical disjunction - Note Read more here: » Logical disjunction: Encyclopedia II - Logical disjunction - Union |
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| | | | | commutator: Encyclopedia II - Stone–von Neumann theorem - The Heisenberg group The commutation relations for P, Q look very similar to the commutation relations that define the Lie algebra of general Heisenberg group Hn for n a positive integer. This is the Lie group of (n+2) × (n+2) square matrices of the form
In fact, using the Heisenberg group, we can formulate a far-reaching generalization of the Stone von Neumann theorem. Note that the ce ...
See also:Stone–von Neumann theorem, Stone–von Neumann theorem - Trying to represent the commutation relations, Stone–von Neumann theorem - Weyl form of the canonical commutation relations, Stone–von Neumann theorem - Another formulation, Stone–von Neumann theorem - The Heisenberg group, Stone–von Neumann theorem - Relation to the Fourier transform, Stone–von Neumann theorem - Representations of finite Heisenberg groups, Stone–von Neumann theorem - Generalizations Read more here: » Stone–von Neumann theorem: Encyclopedia II - Stone–von Neumann theorem - The Heisenberg group |
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| | | | | | commutator: Encyclopedia II - Convolution - Definition The convolution of f and g is written f * g. It is defined as the integral of the product of the two functions after one is reversed and shifted.
The integration range depends on the domain on which the functions are defined. While the symbol t is used above, it need not represent the time domain. In the case of a finite integration range, f and g are often considered to extend periodically in both directions ...
See also:Convolution, Convolution - Uses, Convolution - Definition, Convolution - Properties, Convolution - Commutativity, Convolution - Associativity, Convolution - Distributivity, Convolution - Associativity with scalar multiplication, Convolution - Convolution theorem, Convolution - Convolutions on groups Read more here: » Convolution: Encyclopedia II - Convolution - Definition |
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| | | | | | | | | | commutator: Encyclopedia II - Convolution - Convolutions on groups If G is a suitable group endowed with a measure m (for instance, a locally compact Hausdorff topological group with the Haar measure) and if f and g are real or complex valued m-integrable functions of G, then we can define their convolution by
In this case, it is also possible to give, for instance, a Convolution Theorem, however it is much more difficult to phrase and requires representation theory for these types of groups and the Peter-Weyl theorem of Harmonic analysis. It is very difficult to do these calculations without more structure, and Lie groups turn ...
See also:Convolution, Convolution - Uses, Convolution - Definition, Convolution - Properties, Convolution - Commutativity, Convolution - Associativity, Convolution - Distributivity, Convolution - Associativity with scalar multiplication, Convolution - Convolution theorem, Convolution - Convolutions on groups Read more here: » Convolution: Encyclopedia II - Convolution - Convolutions on groups |
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| | | | | | commutator: Encyclopedia II - Stone–von Neumann theorem - Relation to the Fourier transform For any non-zero h, the mapping
is an automorphism of Hn which is the identity on the center of Hn. In particular, the representations Uh and Uh α are unitarily equivalent. This means that there is a unitary operator W on L2(Rn) such that for any g in Hn,
Moreover, by irreducibility of the representations Uh, it foll ...
See also:Stone–von Neumann theorem, Stone–von Neumann theorem - Trying to represent the commutation relations, Stone–von Neumann theorem - Weyl form of the canonical commutation relations, Stone–von Neumann theorem - Another formulation, Stone–von Neumann theorem - The Heisenberg group, Stone–von Neumann theorem - Relation to the Fourier transform, Stone–von Neumann theorem - Representations of finite Heisenberg groups, Stone–von Neumann theorem - Generalizations Read more here: » Stone–von Neumann theorem: Encyclopedia II - Stone–von Neumann theorem - Relation to the Fourier transform |
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