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The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of ...
If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t. Function defined on finite sets which change the order of their elements such as permutations can be composed on the same set, this being composition of permutations.
In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa). For example, in the depicted pentagon lattice N 5, the element x is distributive, [2] but not dual distributive, since x ∧ (y ∨ z) = x ∧ 1 = x ≠ z = 0 ∨ z = (x ∧ y) ∨ (x ∧ z).
The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. [1] It is a synthesis of the work of many authors in the information theory , digital communications , signal processing , statistics , and artificial intelligence communities.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects,,, the canonical map [,]: + (+)is an isomorphism, and for all objects , the canonical map is an isomorphism (where 0 denotes the initial object).
The Dirac comb of period 2 π, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped Dirac delta function . It represents a discrete probability distribution concentrated at 2 π n — a degenerate distribution — but the notation treats it as if it were a continuous distribution.