Additively separable function - Revision history

Vipul: /* For a function of many variables */

2012-04-10T23:33:43Z

For a function of many variables

← Older revision		Revision as of 23:33, 10 April 2012
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	(note that the subscripts here are ''not'' to be confused with subscripts used for partial derivatives).		(note that the subscripts here are ''not'' to be confused with subscripts used for partial derivatives).

	There is a weaker notion of ''partially additively 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 additively 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 additively 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 additively separable with respect to the partition if there exist functions <math>f_A,f_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>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>

Vipul: /* Partial derivatives */

2012-04-10T23:33:03Z

Partial derivatives

← Older revision		Revision as of 23:33, 10 April 2012
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	\| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> \|\| <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.\|\| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.		\| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> \|\| <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.\|\| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.
	\|-		\|-
	\| partially additively separable function <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> \|\| Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. \|\| Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.		\| partially additively separable function <math>F(x_1,x_2,\dots,x_n)</math> equals <math>f_A(\mbox{only the variables } x_i, i \in A) + f_B(\mbox{only the variables } x_i, i \in B)</math> \|\| Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. \|\| Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.
	\|}		\|}

Vipul: /* Partial derivatives */

2012-04-10T23:32:21Z

Partial derivatives

← Older revision		Revision as of 23:32, 10 April 2012
Line 22:		Line 22:

	<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>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>

			<center>{{#widget:YouTube\|id=9pXmMkHG248}}</center>

	==Partial derivatives==		==Partial derivatives==
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	! Version type !! Statement about first-order partial derivatives !! Statement about [[second-order mixed partial derivative]]s		! Version type !! Statement about first-order partial derivatives !! Statement about [[second-order mixed partial derivative]]s
	\|-		\|-
	\| additively separable function <math>F</math> of two variables <math>x,y</math>, both pieces are differentiable functions, written as <math>F(x,y) = f(x) + g(y)</math> \|\| <math>F_x(x,y) = f'(x)</math> (independent of <math>y</math>)<br><math>F_y(x,y) = g'(y)</math> (independent of <math>x</math>) \|\| <math>F_{xy}(x,y) = 0</math><br>F_{yx}(x,y) = 0</math>		\| additively separable function <math>F</math> of two variables <math>x,y</math>, both pieces are differentiable functions, written as <math>F(x,y) = f(x) + g(y)</math> \|\| <math>F_x(x,y) = f'(x)</math> (independent of <math>y</math>)<br><math>F_y(x,y) = g'(y)</math> (independent of <math>x</math>) \|\| <math>F_{xy}(x,y) = 0</math><br><math>F_{yx}(x,y) = 0</math>
	\|-		\|-
	\| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> \|\| <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.\|\| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.		\| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> \|\| <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.\|\| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.

Vipul at 23:30, 10 April 2012

2012-04-10T23:30:21Z

← Older revision		Revision as of 23:30, 10 April 2012
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	There is a weaker notion of ''partially additively 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 additively 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 additively 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 additively separable with respect to the partition if there exist functions <math>f_A,f_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>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>

	==Partial derivatives==		==Partial derivatives==
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	\| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> \|\| <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.\|\| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.		\| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> \|\| <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.\|\| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.
	\|-		\|-
	\| partially additively separable function <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> \|\| Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. \|\| Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.		\| partially additively separable function <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> \|\| Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. \|\| Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.
	\|}		\|}

Vipul: Created page with "==Definition== ===For a function of two variables=== Suppose $F$ is a function of two variables $x$ and $y$ . We say that $F$ is '..."

2012-04-10T23:26:31Z

Created page with "==Definition== ===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 '..."

New page

==Definition==

===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 '''additively separable''' if there exist functions <math>f,g</math> of one variable such that:

<math>F(x,y) = f(x) + g(y)</math>

on the entire domain of <math>F</math>.

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

===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 additively separable''' if there exist functions <math>f_1,f_2,\dots,f_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>

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

There is a weaker notion of ''partially additively 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 additively separable with respect to the partition if there exist functions <math>f_A,f_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>

==Partial derivatives==

Additively separable functions are the exceptions to the general rule that [[value of partial derivative depends on all inputs]].

{| class="sortable" border="1"
! Version type !! Statement about first-order partial derivatives !! Statement about [[second-order mixed partial derivative]]s
|-
| additively separable function <math>F</math> of two variables <math>x,y</math>, both pieces are differentiable functions, written as <math>F(x,y) = f(x) + g(y)</math> || <math>F_x(x,y) = f'(x)</math> (independent of <math>y</math>)<br><math>F_y(x,y) = g'(y)</math> (independent of <math>x</math>) || <math>F_{xy}(x,y) = 0</math><br>F_{yx}(x,y) = 0</math>
|-
| completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> || <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.|| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.
|-
| partially additively separable function <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> || Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. || Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.
|}

Additively separable function - Revision history

Vipul: /* For a function of many variables */

Vipul: /* Partial derivatives */

Vipul: /* Partial derivatives */

Vipul at 23:30, 10 April 2012

Vipul: Created page with "==Definition== ===For a function of two variables=== Suppose F is a function of two variables x and y. We say that F is '..."

Vipul: Created page with "==Definition== ===For a function of two variables=== Suppose $F$ is a function of two variables $x$ and $y$ . We say that $F$ is '..."