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Is $x_i(t+1)=|S_i(t)|^-1sum_jin S_i(t) x_jquad i=1,dots,n$ a linear system?
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I have $x_i(t)inmathbb [0,1]$ for $i=1,dots, n$, with $n,tinmathbb N$. Let us consider the set
$$S_i(t):=left1leq jleq n Bigl$$
I'm trying to understand if the following system
$$x_i(t+1)=|S_i(t)|^-1sum_jin S_i(t) x_jquad i=1,dots,n$$
(where $|cdot|$ for a finite set denotes the number of elements) is linear or not in $x_1,dots,x_n$.
What do you think?
linear-algebra
asked Jul 27 at 19:02
Mark
3,56051746
add a comment |Â
up vote
1
down vote
favorite
I have $x_i(t)inmathbb [0,1]$ for $i=1,dots, n$, with $n,tinmathbb N$. Let us consider the set
$$S_i(t):=left1leq jleq n Bigl$$
I'm trying to understand if the following system
$$x_i(t+1)=|S_i(t)|^-1sum_jin S_i(t) x_jquad i=1,dots,n$$
(where $|cdot|$ for a finite set denotes the number of elements) is linear or not in $x_1,dots,x_n$.
What do you think?
linear-algebra
asked Jul 27 at 19:02
Mark
3,56051746
Is your $t$ a parameter, where does it come from? Also, what do you mean with $x_i(t)$?
– zzuussee
Jul 27 at 19:31
3
Unless I'm misunderstanding your definition of linear, I don't see how this could possibly be linear, given that $|S_i(t)|$ isn't even continuous in $x_i$. Suppose you have $x_1=0$ and $x_2=1/4$. Then $|S_1(t)| = 2$. Now translate $x_2rightarrow x_2+1$. Then $|S_1(t)|=1$.
– Alex R.
Jul 27 at 19:37
@AlexR. Thank you, but I don't understand why your example would prove the nonlinearity of the system...
– Mark
Jul 28 at 9:48
@zzuussee, I'm sorry I've updated the post
– Mark
Jul 28 at 9:48
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have $x_i(t)inmathbb [0,1]$ for $i=1,dots, n$, with $n,tinmathbb N$. Let us consider the set
$$S_i(t):=left1leq jleq n Bigl$$
I'm trying to understand if the following system
$$x_i(t+1)=|S_i(t)|^-1sum_jin S_i(t) x_jquad i=1,dots,n$$
(where $|cdot|$ for a finite set denotes the number of elements) is linear or not in $x_1,dots,x_n$.
What do you think?
linear-algebra
asked Jul 27 at 19:02
Mark
3,56051746
I have $x_i(t)inmathbb [0,1]$ for $i=1,dots, n$, with $n,tinmathbb N$. Let us consider the set
$$S_i(t):=left1leq jleq n Bigl$$
I'm trying to understand if the following system
$$x_i(t+1)=|S_i(t)|^-1sum_jin S_i(t) x_jquad i=1,dots,n$$
(where $|cdot|$ for a finite set denotes the number of elements) is linear or not in $x_1,dots,x_n$.
What do you think?
linear-algebra
asked Jul 27 at 19:02
Mark
3,56051746
edited Jul 28 at 9:45
asked Jul 27 at 19:02
Mark
3,56051746
asked Jul 27 at 19:02
Mark
3,56051746
asked Jul 27 at 19:02
Mark
3,56051746
3,56051746
Is your $t$ a parameter, where does it come from? Also, what do you mean with $x_i(t)$?
– zzuussee
Jul 27 at 19:31
3
Unless I'm misunderstanding your definition of linear, I don't see how this could possibly be linear, given that $|S_i(t)|$ isn't even continuous in $x_i$. Suppose you have $x_1=0$ and $x_2=1/4$. Then $|S_1(t)| = 2$. Now translate $x_2rightarrow x_2+1$. Then $|S_1(t)|=1$.
– Alex R.
Jul 27 at 19:37
@AlexR. Thank you, but I don't understand why your example would prove the nonlinearity of the system...
– Mark
Jul 28 at 9:48
@zzuussee, I'm sorry I've updated the post
– Mark
Jul 28 at 9:48
add a comment |Â
Is your $t$ a parameter, where does it come from? Also, what do you mean with $x_i(t)$?
– zzuussee
Jul 27 at 19:31
3
Unless I'm misunderstanding your definition of linear, I don't see how this could possibly be linear, given that $|S_i(t)|$ isn't even continuous in $x_i$. Suppose you have $x_1=0$ and $x_2=1/4$. Then $|S_1(t)| = 2$. Now translate $x_2rightarrow x_2+1$. Then $|S_1(t)|=1$.
– Alex R.
Jul 27 at 19:37
@AlexR. Thank you, but I don't understand why your example would prove the nonlinearity of the system...
– Mark
Jul 28 at 9:48
@zzuussee, I'm sorry I've updated the post
– Mark
Jul 28 at 9:48
Is your $t$ a parameter, where does it come from? Also, what do you mean with $x_i(t)$?
– zzuussee
Jul 27 at 19:31
Is your $t$ a parameter, where does it come from? Also, what do you mean with $x_i(t)$?
– zzuussee
Jul 27 at 19:31
3
3
Unless I'm misunderstanding your definition of linear, I don't see how this could possibly be linear, given that $|S_i(t)|$ isn't even continuous in $x_i$. Suppose you have $x_1=0$ and $x_2=1/4$. Then $|S_1(t)| = 2$. Now translate $x_2rightarrow x_2+1$. Then $|S_1(t)|=1$.
– Alex R.
Jul 27 at 19:37
Unless I'm misunderstanding your definition of linear, I don't see how this could possibly be linear, given that $|S_i(t)|$ isn't even continuous in $x_i$. Suppose you have $x_1=0$ and $x_2=1/4$. Then $|S_1(t)| = 2$. Now translate $x_2rightarrow x_2+1$. Then $|S_1(t)|=1$.
– Alex R.
Jul 27 at 19:37
@AlexR. Thank you, but I don't understand why your example would prove the nonlinearity of the system...
– Mark
Jul 28 at 9:48
@AlexR. Thank you, but I don't understand why your example would prove the nonlinearity of the system...
– Mark
Jul 28 at 9:48
@zzuussee, I'm sorry I've updated the post
– Mark
Jul 28 at 9:48
@zzuussee, I'm sorry I've updated the post
– Mark
Jul 28 at 9:48
add a comment |Â
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Is your $t$ a parameter, where does it come from? Also, what do you mean with $x_i(t)$?
– zzuussee
Jul 27 at 19:31
3
Unless I'm misunderstanding your definition of linear, I don't see how this could possibly be linear, given that $|S_i(t)|$ isn't even continuous in $x_i$. Suppose you have $x_1=0$ and $x_2=1/4$. Then $|S_1(t)| = 2$. Now translate $x_2rightarrow x_2+1$. Then $|S_1(t)|=1$.
– Alex R.
Jul 27 at 19:37
@AlexR. Thank you, but I don't understand why your example would prove the nonlinearity of the system...
– Mark
Jul 28 at 9:48
@zzuussee, I'm sorry I've updated the post
– Mark
Jul 28 at 9:48