My Math Forum  

Go Back   My Math Forum > College Math Forum > Topology

Topology Topology Math Forum


Thanks Tree1Thanks
Reply
 
LinkBack Thread Tools Display Modes
May 13th, 2015, 02:35 PM   #21
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

Quote:
Originally Posted by v8archie View Post
Well, my guess might be wrong (see here), but yours is definitely not correct either.

What's a 'viewable space' then?

In portuguese "viewable" means "visualizavel", when a space had more than 3 dimensions, can't be viewable, understand now?

about the link than you give, a got a prove, with think solve the question.

We can write U1+U2+⋯ as the sum of two subspaces, for example U1+(U2+U3+⋯), and then:
dim(U1+U2+⋯)=dim(U1)+dim(U2+U3+⋯)−dim(U1∩( U2+U3+⋯)).
So then applying the same argument to the term dim(U2+U3+⋯) recursively you can then eventually arrive at an equation consisting of the sum of the dimensions of each subspace minus a bunch of nasty terms involving intersections similar to dim(U1 ∩(U2+U3+⋯))..
In our example, this term uderline, being the dimension of minor, in case, U1,
then, 3 dimensions, the other term will be 3n dimensions, so, the sum will be 3(n-1). You can get in adress linear algebra - The dimension of the sum of subspaces $(U_1,\ldots,U_n)$ - Mathematics Stack Exchange.

seems the same results, than I show in my post.

Lucio

Can You viewable (n-1) spheres above your head?

Last edited by lemgruber; May 13th, 2015 at 02:38 PM.
lemgruber is offline  
 
May 14th, 2015, 01:23 PM   #22
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

(xo,yo,zo)= (0,0,0) be the center of sphere with rn it's ray of n-esima sphere:

Change the equation of sphere,

(x1 - x0)² + (y1 - y0)² + (z1 - z0)²= r²1 ∈ R³1
(x2 - x0)² + (y2 - y0)² + (z2 - z0)²= r²2 ∈ R³2
. . . . .
. . . . .
. . . . .
(xn - x0)² + (yn - y0)² + (zn - z0)²= r²n ∈ R³n

to coordinates who made a sphere in subspace vector, we got:

(ρ1 sin φ cos θ)² + (ρ1 sin φ sin θ)² + (ρ1 cos φ)²= r²1 ∈ R³1
(ρ2 sin φ cos θ)² + (ρ2 sin φ sin θ)² + (ρ2 cos φ)²= r²2 ∈ R³2
. . . .
. . . .
. . . .
(ρn sin φ cos θ)² + (ρn sin φ sin θ)² + (ρn cos φ)²= r²n ∈ R³n

where R³1 ⊂ R³2 ⊂.........⊂ R³n where R³n are subspaces vectors and n ∈ N ( natural set numbers)

ρ1 ≠ ρ2 ≠.............≠ ρn where n ∈ N ( natural set numbers)

Dimension of Sum of subspaces vector:
Proposition-
Be V a vector space of dimension finite, if U1 and U2 are vectors subspace of V, then, Dim(U1 + U2)= Dim(U1) + Dim(U2) - Dim(U1 ∩ U2).

in case Un, where n ∈ N ( natural set numbers), we got:

dim(U1+U2+...Un)=dim(U1)+dim(U2+U3+....+Un) - dim(U1 ∩(U2+U3+....+Un)).
see:http://math.stackexchange.com/questi...ldots-u-n?rq=1

In case of subspaces of spheres, we got:

Dim (R³1 + R³2 +.....+ R³n) = Dim (R³1) + Dim(R³2 + .... +R³n) - Dim( R³1 ∩ (R³2 + ... +R³n) = 3 + 3(n - 1) - 3 = 3(n - 1)

Dim (R³1 + R³2 + .... + R³n) = 3(n - 1) Dimensions

this sum make a space vector compound of n -1 spheres concentric, that can be drawn on a graph, a graph 3(n - 1) dimensional. This, at least to me, a new thing.

Apologize delay to make right form, but now seems ready.

First I understand this with faith, after I could demonstrate mathematically.
First I receive the idea, and I like thanks to being who make this possible.

Lucio

Last edited by lemgruber; May 14th, 2015 at 02:13 PM.
lemgruber is offline  
May 14th, 2015, 05:55 PM   #23
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

In time:

ρ1=r1 ; ρ2=r2 ;....; ρn=rn

and

ρ1<ρ2<......<ρn
lemgruber is offline  
May 15th, 2015, 09:52 AM   #24
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

The utility for this, are inumerous, although the geometry of n-dimensional space, was much developed, the utilization of figures geometrics depicts on space n dimensional, let the understand about the geometry of fenomenous much more easy and fast.


Lucio
lemgruber is offline  
May 15th, 2015, 11:21 AM   #25
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
The utility is 0 until someone can understand you.
CRGreathouse is offline  
May 15th, 2015, 11:46 AM   #26
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

In economy, for example, the utility function operates in n dimensional space, but we can't
draw the geometric figures, start of four dimensions, with this tool, became possible viewable the geometric figures and understand in another way, more intuitive what be said in language not geometric. The utility function don't have a spheric equation, but can be transformed in equations of sphere or became a subspace vector trough another transformation like a example in this post.

Please, if someone understand, what I try to say, help me. Say something!


Lucio

Last edited by lemgruber; May 15th, 2015 at 11:57 AM.
lemgruber is offline  
May 15th, 2015, 12:06 PM   #27
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
Originally Posted by lemgruber View Post
In economy, for example, the utility function operates in n dimensional space, but we can't
draw the geometric figures
I don't understand why this would be a problem. I mean, there are ways to visualize things in more than three dimensions (see, for example, literally any complex analysis course), but it's not a limitation to fail to draw something.
CRGreathouse is offline  
May 15th, 2015, 12:18 PM   #28
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

I agree Greathouse, but when we draw geometry in graphic, the understand gain in clarity, again, I don't say what not exist ways to interpret things without draw figures, but it's much better when we do. At least I think so.


Lucio
lemgruber is offline  
May 19th, 2015, 02:42 PM   #29
Senior Member
 
Joined: Dec 2014
From: Brazil

Posts: 203
Thanks: 1

Quote:
Originally Posted by lemgruber View Post
In time:

ρ1=r1 ; ρ2=r2 ;....; ρn=rn

and

ρ1<ρ2<......<ρn

Just possible if, ρ1≠r1 ; ρ2≠r2 ;....; ρn≠rn
lemgruber is offline  
Reply

  My Math Forum > College Math Forum > Topology

Tags
dimensions, space, viewable, visual, whith



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Visual Differentiation help ProJO Calculus 4 February 10th, 2011 04:59 PM
visual interactive math blog jamin1001 New Users 0 October 23rd, 2007 01:20 AM
flatland visual pun ^e^ Algebra 1 February 14th, 2007 10:20 AM





Copyright © 2019 My Math Forum. All rights reserved.