Multiplicatively separable function: Difference between revisions

Revision as of 23:35, 10 April 2012

Definition

For a function of two variables

Suppose $G$ is a function of two variables $x$ and $y$ . We say that $G$ is multiplicatively separable if there exist functions $f, g$ of one variable such that:

$G (x, y) = f (x) g (y)$

on the entire domain of $G$ .

Note that the concept of multiplicatively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally multiplicatively separable need not remain multiplicatively separable.

For a function of many variables

Suppose $G$ is a function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . We say that $G$ is completely multiplicatively separable if there exist functions $g_{1}, g_{2}, \dots, g_{n}$ , each a function of one variable, such that:

$G (x_{1}, x_{2}, \dots, x_{n}) = g_{1} (x_{1}) g_{2} (x_{2}) \dots g_{n} (x_{n})$

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially multiplicatively separable: if we express the set ${1, 2, \dots, n}$ as a union of two disjoint subsets $A, B$ , $G$ is multiplicatively separable with respect to the partition if there exist functions $g_{A}, g_{B}$ such that:

$G (x_{1}, x_{2}, \dots, x_{n}) = g_{A} (only the variables x_{i}, i \in A) g_{B} (only the variables x_{i}, i \in B)$

@@ Line 3: / Line 3: @@
 ===For a function of two variables===
-Suppose <math>F</math> is a function of two variables <matH>x</math> and <math>y</math>. We say that <math>F</matH> is '''multiplicatively separable''' if there exist functions <math>f,g</math> of one variable such that:
+Suppose <math>G</math> is a function of two variables <matH>x</math> and <math>y</math>. We say that <math>G</matH> is '''multiplicatively separable''' if there exist functions <math>f,g</math> of one variable such that:
-<math>F(x,y) = f(x)g(y)</math>
+<math>G(x,y) = f(x)g(y)</math>
-on the entire domain of <math>F</math>.
+on the entire domain of <math>G</math>.
 Note that the concept of multiplicatively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally multiplicatively separable need not remain multiplicatively separable.
@@ Line 13: / Line 13: @@
 ===For a function of many variables===
-Suppose <math>F</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. We say that <math>F</math> is '''completely multiplicatively separable''' if there exist functions <math>f_1,f_2,\dots,f_n</math>, each a function of one variable, such that:
+Suppose <math>G</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. We say that <math>G</math> is '''completely multiplicatively separable''' if there exist functions <math>g_1,g_2,\dots,g_n</math>, each a function of one variable, such that:
-<math>F(x_1,x_2,\dots,x_n) = f_1(x_1)f_2(x_2) \dots f_n(x_n)</math>
+<math>G(x_1,x_2,\dots,x_n) = g_1(x_1)g_2(x_2) \dots g_n(x_n)</math>
 (note that the subscripts here are ''not'' to be confused with subscripts used for partial derivatives).
-There is a weaker notion of ''partially multiplicatively separable'': if we express the set <math>\{ 1,2,\dots,n\}</math> as a union of two disjoint subsets <math>A,B</math>, <math>F</matH> is multiplicatively separable with respect to the partition if there exist functions <math>f_A,f_B</math> such that:
+There is a weaker notion of ''partially multiplicatively separable'': if we express the set <math>\{ 1,2,\dots,n\}</math> as a union of two disjoint subsets <math>A,B</math>, <math>G</matH> is multiplicatively separable with respect to the partition if there exist functions <math>g_A,g_B</math> such that:
-<math>F(x_1,x_2,\dots,x_n) = f_A(\mbox{only the variables } x_i, i \in A)f_B(\mbox{only the variables } x_i, i \in B)</math>
+<math>G(x_1,x_2,\dots,x_n) = g_A(\mbox{only the variables } x_i, i \in A)g_B(\mbox{only the variables } x_i, i \in B)</math>
+<center>{{#widget:YouTube|id=9pXmMkHG248}}</center>