Mixing
The cone $mathbb R_+^n$ of all vectors in $mathbb R^n$
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
In a paper I've been studying it says:
Let $x$ in the cone $mathbb R_+^n$ of all vectors in $mathbb R^n$ with nonnnegative components ($ninmathbb N$)
Somebody tell me what does it means, please? $mathbb R_+^n$ should be $[0,infty)timesdotstimes [0,infty)$ ($n$ times), but I don't understand why the cone $mathbb R_+^n$. Maybe the cone $mathbb R_+^n$ is different from $[0,infty)timesdotstimes [0,infty)$?
notation
asked Jul 25 at 8:15
Mark
3,56051746
add a comment |Â
up vote
1
down vote
favorite
In a paper I've been studying it says:
Let $x$ in the cone $mathbb R_+^n$ of all vectors in $mathbb R^n$ with nonnnegative components ($ninmathbb N$)
Somebody tell me what does it means, please? $mathbb R_+^n$ should be $[0,infty)timesdotstimes [0,infty)$ ($n$ times), but I don't understand why the cone $mathbb R_+^n$. Maybe the cone $mathbb R_+^n$ is different from $[0,infty)timesdotstimes [0,infty)$?
notation
asked Jul 25 at 8:15
Mark
3,56051746
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In a paper I've been studying it says:
Let $x$ in the cone $mathbb R_+^n$ of all vectors in $mathbb R^n$ with nonnnegative components ($ninmathbb N$)
Somebody tell me what does it means, please? $mathbb R_+^n$ should be $[0,infty)timesdotstimes [0,infty)$ ($n$ times), but I don't understand why the cone $mathbb R_+^n$. Maybe the cone $mathbb R_+^n$ is different from $[0,infty)timesdotstimes [0,infty)$?
notation
asked Jul 25 at 8:15
Mark
3,56051746
In a paper I've been studying it says:
Let $x$ in the cone $mathbb R_+^n$ of all vectors in $mathbb R^n$ with nonnnegative components ($ninmathbb N$)
Somebody tell me what does it means, please? $mathbb R_+^n$ should be $[0,infty)timesdotstimes [0,infty)$ ($n$ times), but I don't understand why the cone $mathbb R_+^n$. Maybe the cone $mathbb R_+^n$ is different from $[0,infty)timesdotstimes [0,infty)$?
notation
asked Jul 25 at 8:15
Mark
3,56051746
asked Jul 25 at 8:15
Mark
3,56051746
asked Jul 25 at 8:15
Mark
3,56051746
asked Jul 25 at 8:15
Mark
3,56051746
3,56051746
add a comment |Â
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
4
down vote
accepted
A cone is a set $CsubseteqBbb R^n$ with $xin Cimpliesalpha xin C$ for all $alphage0$. The name is motivated by the usualy geometric cones, but $Bbb R^n_+ := [0,infty)^n$ satisfies this too.
See this picture of different cones in $Bbb R^3$, two are familiar, and the right most one is $Bbb R^n_+$.
answered Jul 25 at 8:29
M. Winter
17.6k62764
add a comment |Â
up vote
1
down vote
A cone $C$ is a set in a vector space with two propertes: $x+y in C$ whenever $x,y in C$ and $tx in C$ whenever $xin C$ and $t geq 0$. Since $[0,infty ) times [0,infty ) times ...[0,infty ) $ has these properties it is a cone.
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
add a comment |Â
up vote
1
down vote
A subset $A$ of a vector space $X$ on a field $K$ with a notion of positivity is said to be a cone if, for all $x in A$, $lambda>0$, we have $lambda x in A$.
The infite version of the cone you probably have in mind satisfies this property if the vertex, or apex, is at the origin.
answered Jul 25 at 8:23
nicomezi
3,4121819
add a comment |Â
up vote
1
down vote
Yes, we have $mathbb R_+^n= [0,infty)timesdotstimes [0,infty)$.
If $C ne emptyset$ is a subset of $mathbb R^n$, then $C$ is called a cone if $x in C$ and $t ge 0$ imply that $tx in C$.
Hence $mathbb R_+^n$ is a cone in $mathbb R^n$
answered Jul 25 at 8:24
Fred
37.2k1237
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
A cone is a set $CsubseteqBbb R^n$ with $xin Cimpliesalpha xin C$ for all $alphage0$. The name is motivated by the usualy geometric cones, but $Bbb R^n_+ := [0,infty)^n$ satisfies this too.
See this picture of different cones in $Bbb R^3$, two are familiar, and the right most one is $Bbb R^n_+$.
answered Jul 25 at 8:29
M. Winter
17.6k62764
add a comment |Â
up vote
4
down vote
accepted
A cone is a set $CsubseteqBbb R^n$ with $xin Cimpliesalpha xin C$ for all $alphage0$. The name is motivated by the usualy geometric cones, but $Bbb R^n_+ := [0,infty)^n$ satisfies this too.
See this picture of different cones in $Bbb R^3$, two are familiar, and the right most one is $Bbb R^n_+$.
answered Jul 25 at 8:29
M. Winter
17.6k62764
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
A cone is a set $CsubseteqBbb R^n$ with $xin Cimpliesalpha xin C$ for all $alphage0$. The name is motivated by the usualy geometric cones, but $Bbb R^n_+ := [0,infty)^n$ satisfies this too.
See this picture of different cones in $Bbb R^3$, two are familiar, and the right most one is $Bbb R^n_+$.
answered Jul 25 at 8:29
M. Winter
17.6k62764
A cone is a set $CsubseteqBbb R^n$ with $xin Cimpliesalpha xin C$ for all $alphage0$. The name is motivated by the usualy geometric cones, but $Bbb R^n_+ := [0,infty)^n$ satisfies this too.
See this picture of different cones in $Bbb R^3$, two are familiar, and the right most one is $Bbb R^n_+$.
answered Jul 25 at 8:29
M. Winter
17.6k62764
answered Jul 25 at 8:29
M. Winter
17.6k62764
answered Jul 25 at 8:29
M. Winter
17.6k62764
answered Jul 25 at 8:29
M. Winter
17.6k62764
17.6k62764
add a comment |Â
add a comment |Â
up vote
1
down vote
A cone $C$ is a set in a vector space with two propertes: $x+y in C$ whenever $x,y in C$ and $tx in C$ whenever $xin C$ and $t geq 0$. Since $[0,infty ) times [0,infty ) times ...[0,infty ) $ has these properties it is a cone.
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
add a comment |Â
up vote
1
down vote
A cone $C$ is a set in a vector space with two propertes: $x+y in C$ whenever $x,y in C$ and $tx in C$ whenever $xin C$ and $t geq 0$. Since $[0,infty ) times [0,infty ) times ...[0,infty ) $ has these properties it is a cone.
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
add a comment |Â
up vote
1
down vote
up vote
1
down vote
A cone $C$ is a set in a vector space with two propertes: $x+y in C$ whenever $x,y in C$ and $tx in C$ whenever $xin C$ and $t geq 0$. Since $[0,infty ) times [0,infty ) times ...[0,infty ) $ has these properties it is a cone.
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
A cone $C$ is a set in a vector space with two propertes: $x+y in C$ whenever $x,y in C$ and $tx in C$ whenever $xin C$ and $t geq 0$. Since $[0,infty ) times [0,infty ) times ...[0,infty ) $ has these properties it is a cone.
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
answered Jul 25 at 8:23
Kavi Rama Murthy
20.1k2829
20.1k2829
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
add a comment |Â
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
Are you sure about $x+y in C$ ? This seems to belong to the definition of a convex cone instead, not necessarily a cone.
– nicomezi
Jul 25 at 8:24
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@Kavi: $ x+y in C$ is not required !
– Fred
Jul 25 at 8:25
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
@nicomezi Some authors add it to the definition because all they gonna do is discuss convex cones. However, I think the preferred definition of only cone should not contain it.
– M. Winter
Jul 25 at 8:26
add a comment |Â
up vote
1
down vote
A subset $A$ of a vector space $X$ on a field $K$ with a notion of positivity is said to be a cone if, for all $x in A$, $lambda>0$, we have $lambda x in A$.
The infite version of the cone you probably have in mind satisfies this property if the vertex, or apex, is at the origin.
answered Jul 25 at 8:23
nicomezi
3,4121819
add a comment |Â
up vote
1
down vote
A subset $A$ of a vector space $X$ on a field $K$ with a notion of positivity is said to be a cone if, for all $x in A$, $lambda>0$, we have $lambda x in A$.
The infite version of the cone you probably have in mind satisfies this property if the vertex, or apex, is at the origin.
answered Jul 25 at 8:23
nicomezi
3,4121819
add a comment |Â
up vote
1
down vote
up vote
1
down vote
A subset $A$ of a vector space $X$ on a field $K$ with a notion of positivity is said to be a cone if, for all $x in A$, $lambda>0$, we have $lambda x in A$.
The infite version of the cone you probably have in mind satisfies this property if the vertex, or apex, is at the origin.
answered Jul 25 at 8:23
nicomezi
3,4121819
A subset $A$ of a vector space $X$ on a field $K$ with a notion of positivity is said to be a cone if, for all $x in A$, $lambda>0$, we have $lambda x in A$.
The infite version of the cone you probably have in mind satisfies this property if the vertex, or apex, is at the origin.
answered Jul 25 at 8:23
nicomezi
3,4121819
answered Jul 25 at 8:23
nicomezi
3,4121819
answered Jul 25 at 8:23
nicomezi
3,4121819
answered Jul 25 at 8:23
nicomezi
3,4121819
3,4121819
add a comment |Â
add a comment |Â
up vote
1
down vote
Yes, we have $mathbb R_+^n= [0,infty)timesdotstimes [0,infty)$.
If $C ne emptyset$ is a subset of $mathbb R^n$, then $C$ is called a cone if $x in C$ and $t ge 0$ imply that $tx in C$.
Hence $mathbb R_+^n$ is a cone in $mathbb R^n$
answered Jul 25 at 8:24
Fred
37.2k1237
add a comment |Â
up vote
1
down vote
Yes, we have $mathbb R_+^n= [0,infty)timesdotstimes [0,infty)$.
If $C ne emptyset$ is a subset of $mathbb R^n$, then $C$ is called a cone if $x in C$ and $t ge 0$ imply that $tx in C$.
Hence $mathbb R_+^n$ is a cone in $mathbb R^n$
answered Jul 25 at 8:24
Fred
37.2k1237
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Yes, we have $mathbb R_+^n= [0,infty)timesdotstimes [0,infty)$.
If $C ne emptyset$ is a subset of $mathbb R^n$, then $C$ is called a cone if $x in C$ and $t ge 0$ imply that $tx in C$.
Hence $mathbb R_+^n$ is a cone in $mathbb R^n$
answered Jul 25 at 8:24
Fred
37.2k1237
Yes, we have $mathbb R_+^n= [0,infty)timesdotstimes [0,infty)$.
If $C ne emptyset$ is a subset of $mathbb R^n$, then $C$ is called a cone if $x in C$ and $t ge 0$ imply that $tx in C$.
Hence $mathbb R_+^n$ is a cone in $mathbb R^n$
answered Jul 25 at 8:24
Fred
37.2k1237
answered Jul 25 at 8:24
Fred
37.2k1237
answered Jul 25 at 8:24
Fred
37.2k1237
answered Jul 25 at 8:24
Fred
37.2k1237
37.2k1237
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2862166%2fthe-cone-mathbb-r-n-of-all-vectors-in-mathbb-rn%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password