Drorbn - User contributions [en]

Notes for AKT-140305/0:41:57

2018-07-30T21:40:56Z

Leo algknt:

'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W, V\; \mathrm{and}\; W</math> '''are vector spaces'''. '''([[User:Leo algknt|Leo algknt]] and Jesse had a discussion.)
'''

Let <math>\iota : \ker f \rightarrow V</math> be the inclusion of <math>\ker f</math> into <math>V</math>, then <math>V^*/\ker \iota^* \cong (\ker f)^*</math>.

We show that <math>\ker \iota^* = \mathrm{im} f^*</math>.

<center>
<math>
\begin{align}
\ker \iota^* &= \{\phi \in V^* \;\;|\;\; \iota^*(\phi) = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi\circ\iota = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = f^*(\alpha), \; \alpha \in W^* \}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = \alpha\circ f, \; \alpha \in W^* \}\\
&= \{f^*(\alpha) \in V^* \;\;|\;\; (\alpha\circ f)|_{\ker f} = 0, \; \alpha \in W^* \}\\
&= \mathrm{im} f^*.

\end{align}
</math>
</center>

Notes for AKT-140305/0:41:57

2018-07-30T21:37:06Z

Leo algknt:

'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''([[User:Leo algknt|Leo algknt]] and Jesse had a discussion.)
'''

Let <math>\iota : \ker f \rightarrow V</math> be the inclusion of <math>\ker f</math> into <math>V</math>, then <math>V^*/\ker \iota^* \cong (\ker f)^*</math>.

We show that <math>\ker \iota^* = \mathrm{im} f^*</math>.

<center>
<math>
\begin{align}
\ker \iota^* &= \{\phi \in V^* \;\;|\;\; \iota^*(\phi) = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi\circ\iota = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = f^*(\alpha), \; \alpha \in W^* \}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = \alpha\circ f, \; \alpha \in W^* \}\\
&= \{f^*(\alpha) \in V^* \;\;|\;\; (\alpha\circ f)|_{\ker f} = 0, \; \alpha \in W^* \}\\
&= \mathrm{im} f^*.

\end{align}
</math>
</center>

Notes for AKT-140305/0:41:57

2018-07-30T21:35:14Z

Leo algknt: Created page with "'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''(~~~ and Jesse had a discussion.) ''' Let <math>\..."

'''Showing that''' <math>(\ker f)^* = V^*/\mathrm{im}f^*</math> '''for a linear map''' <math>f : V \rightarrow W</math>. '''([[User:Leo algknt|Leo algknt]] ([[User talk:Leo algknt|talk]]) and Jesse had a discussion.)
'''

Let <math>\iota : \ker f \rightarrow V</math> be the inclusion of <math>\ker f</math> into <math>V</math>, then <math>V^*/\ker \iota^* \cong (\ker f)^*</math>.

We show that <math>\ker \iota^* = \mathrm{im} f^*</math>.

<center>
<math>
\begin{align}
\ker \iota^* &= \{\phi \in V^* \;\;|\;\; \iota^*(\phi) = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi\circ\iota = 0\}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = f^*(\alpha), \; \alpha \in W^* \}\\
&= \{\phi \in V^* \;\;|\;\; \phi|_{\ker f} = 0, \; \phi = \alpha\circ f, \; \alpha \in W^* \}\\
&= \{f^*(\alpha) \in V^* \;\;|\;\; (\alpha\circ f)|_{\ker f} = 0, \; \alpha \in W^* \}\\
&= \mathrm{im} f^*.

\end{align}
</math>
</center>

Notes for AKT-140228/0:41:45

2018-07-26T22:02:44Z

Leo algknt:

'''Showing ''''''<math>(A^g)^h = A^{(gh)}</math> '''

<center><math>\begin{align}
A^{(gh)} &= (gh)^{-1}A(gh) + (gh)^{-1}\mathrm{d}(gh) \\
&= (gh)^{-1}A(gh) + (gh)^{-1}\Big((\mathrm{d}g)h + g(\mathrm{d}h)\Big)\\
&= (gh)^{-1}A(gh) + (gh)^{-1}(\mathrm{d}g)h + (gh)^{-1}g(\mathrm{d}h)\\
&= h^{-1}(g^{-1}Ag)h + h^{-1}(g^{-1}\mathrm{d}g)h + h^{-1}\mathrm{d}h\\
&= h^{-1}\Big(g^{-1}Ag + g^{-1}\mathrm{d}g\Big)h + h^{-1}\mathrm{d}h\\
&= (A^g)^h.

\end{align}
</math></center>

The equality shows that the action is a group action.

Notes for AKT-140228/0:41:45

2018-07-26T21:38:46Z

Leo algknt:

'''<math>(A^g)^h = A^{(gh)}</math> ?'''

<center><math>\begin{align}
A^{(gh)} &= (gh)^{-1}A(gh) + (gh)^{-1}\mathrm{d}(gh) \\
&= (gh)^{-1}A(gh) + (gh)^{-1}\Big((\mathrm{d}g)h + g(\mathrm{d}h)\Big)\\
&= (gh)^{-1}A(gh) + (gh)^{-1}(\mathrm{d}g)h + (gh)^{-1}g(\mathrm{d}h)\\
&= h^{-1}(g^{-1}Ag)h + h^{-1}(g^{-1}\mathrm{d}g)h + h^{-1}\mathrm{d}h\\
&= h^{-1}\Big(g^{-1}Ag + g^{-1}\mathrm{d}g\Big)h + h^{-1}\mathrm{d}h\\
&= (A^g)^h.

\end{align}
</math></center>

The equality shows that the action is a group action.

Notes for AKT-140303/0:35:03

2018-07-18T19:25:41Z

Leo algknt:

'''Value of <math>W_{\mathfrak{g},R}</math> on <math>IHX</math>'''.

Computation for <math>I</math>

<center><math>\begin{align}
W_{\mathfrak{g},R}(I) & = f_{ecd}f_{abe^{\prime}}t^{ee^{\prime}} \\
& = \langle[X_e, X_c], X_d \rangle \langle[X_a, X_b], X_e^{\prime} \rangle t^{ee^{\prime}} = f_{ec}^st_{sd} f_{ab}^kt_{ke^{\prime}}t^{ee^{\prime}}\\
& = f_{ec}^st_{sd} f_{ab}^k \delta_{k}^{e} \\
& = f_{ec}^sf_{ab}^et_{sd}
\end{align}</math></center>

Here <math>s</math> is a dummy variable and could be replace by <math>e^{\prime}</math>

Notes for AKT-140228/0:41:45

2018-07-18T19:11:44Z

Leo algknt:

'''<math>(A^g)^h = A^{(gh)}</math> ?'''

<center><math>\begin{align}
(A^g)^h &= h^{-1}(g^{-1}Ag + g^{-1}\mathrm{d}g)h + h^{-1}\mathrm{d}h \\
&= (gh)^{-1}A(gh) + (gh)^{-1}\mathrm{d}(gh) + h^{-1}\mathrm{d}h\\
&= A^{(gh)} + h^{-1}\mathrm{d}h
\end{align}
</math></center>

Notes for AKT-140228/0:41:45

2018-07-18T19:07:09Z

Leo algknt: Created page with "'''<math>(A^g)^h = A^{(gh)}</math> ?''' <center><math>\begin{align} (A^g)^h &= h^{-1}(g^{-1}Ag + g^{-1}\mathrm{d}g)h + h^{-1}\mathrm{d}h \\ &= \end{align} </math></center>"

'''<math>(A^g)^h = A^{(gh)}</math> ?'''

<center><math>\begin{align}
(A^g)^h &= h^{-1}(g^{-1}Ag + g^{-1}\mathrm{d}g)h + h^{-1}\mathrm{d}h \\
&=
\end{align}
</math></center>

Notes for AKT-140207/0:42:32

2018-07-18T18:51:45Z

Leo algknt:

'''Showing <math>\Psi(A) = A\wedge \mathrm{d}A</math> is invariant under <math>A\mapsto A + \mathrm{d}f</math>'''

<center><math>\begin{align}
\Psi(A + \mathrm{d}f) &= (A + \mathrm{d}f)\wedge \mathrm{d}(A + \mathrm{d}f)\\
&= (A + \mathrm{d}A)\wedge \mathrm{d}A,\;\;\;\;\; \mathrm{since \; d\circ d = 0}\\
&= A\wedge \mathrm{d}A.
\end{align}</math> </center>

Notes for AKT-140207/0:42:32

2018-07-18T18:51:31Z

Leo algknt:

Notes for AKT-140207/0:42:32

2018-07-17T23:47:49Z

Leo algknt:

Notes for AKT-140207/0:42:32

2018-07-17T23:45:08Z

Leo algknt: Created page with "'''Showing <math>\Psi(A) = A\wedge \mathrm{d}A</math> is invariant under <math>A\mapsto A + \mathrm{d}f</math>''' <center><math>\begin{align} \Psi(A + \mathrm{d}f) &= (A + \ma..."

Notes for AKT-140303/0:35:03

2018-07-12T17:36:39Z

Leo algknt:

'''Value of <math>W_{\mathfrak{g},R}</math> on <math>IHX</math>'''.

Computation for <math>I</math>

<center><math>\begin{align}
W_{\mathfrak{g},R}(I) & = f_{ecd}f_{abe^{\prime}}t^{ee^{\prime}} \\
& = \langle[X_e, X_c], X_d \rangle \langle[X_a, X_b], X_e^{\prime} \rangle = f_{ec}^st_{sd} f_{ab}^kt_{ke^{\prime}}t^{ee^{\prime}}\\
& = f_{ec}^st_{sd} f_{ab}^k \delta_{k}^{e} \\
& = f_{ec}^sf_{ab}^et_{sd}
\end{align}</math></center>

Here <math>s</math> is a dummy variable and could be replace by <math>e^{\prime}</math>

Notes for AKT-140129/0:27:34

2018-07-12T16:23:40Z

Leo algknt:

[[File:fr_swa.jpg]]

Notes for AKT-140303/0:35:03

2018-07-12T16:22:14Z

Leo algknt:

'''Value of <math>W_{\mathfrak{g},R}</math> on <math>IHX</math>'''.

Computation for <math>I</math>

<center><math>\begin{align}
W_{\mathfrak{g},R}(I) & = f_{ecd}f_{abe^{\prime}}t^{ee^{\prime}} \\
& = \langle[X_e, X_c], X_d \rangle \langle[X_a, X_b], X_e^{\prime} \rangle = f_{ec}^st_{sd} f_{ab}^kt_{ke^{\prime}}t^{ee^{\prime}}\\
& = f_{ec}^st_{sd} f_{ab}^k \delta_{k}^{e} \\
& = f_{ec}^sf_{ab}^et_{sd}

\end{align}</math></center>

Notes for AKT-140303/0:35:03

2018-07-12T16:20:21Z

Leo algknt:

'''Value of <math>W_{\mathfrak{g},R}</math> on <math>\mathcal{IHX}</math>'''.

Computation for <math>\mathcal{I}</math>

<center><math>\begin{align}
W_{\mathfrak{g},R}(\mathcal{I}) & = f_{ecd}f_{abe^{\prime}}t^{ee^{\prime}} \\
& = \langle[X_e, X_c], X_d \rangle \langle[X_a, X_b], X_e^{\prime} \rangle = f_{ec}^st_{sd} f_{ab}^kt_{ke^{\prime}}t^{ee^{\prime}}\\
& = f_{ec}^st_{sd} f_{ab}^k \delta_{k}^{e} \\
& = f_{ec}^sf_{ab}^et_{sd}

\end{align}</math></center>

Notes for AKT-140303/0:35:03

2018-07-12T16:20:00Z

Leo algknt: Created page with "'''Value of <math>W_{\mathfrak{g},R}</math> on <math>\mathcal{IHX}</math>'''. Compution for <math>\mathcal{I}</math> <center><math>\begin{align} W_{\mathfrak{g},R}(\mathca..."

'''Value of <math>W_{\mathfrak{g},R}</math> on <math>\mathcal{IHX}</math>'''.

Compution for <math>\mathcal{I}</math>

<center><math>\begin{align}
W_{\mathfrak{g},R}(\mathcal{I}) & = f_{ecd}f_{abe^{\prime}}t^{ee^{\prime}} \\
& = \langle[X_e, X_c], X_d \rangle \langle[X_a, X_b], X_e^{\prime} \rangle = f_{ec}^st_{sd} f_{ab}^kt_{ke^{\prime}}t^{ee^{\prime}}\\
& = f_{ec}^st_{sd} f_{ab}^k \delta_{k}^{e} \\
& = f_{ec}^sf_{ab}^et_{sd}

\end{align}</math></center>

Notes for AKT-140212/0:19:31

2018-07-12T15:26:08Z

Leo algknt:

'''Question about modulo rotations''':

Can [[User:Leo algknt|Leo algknt]] say since we are dealing with points and therefore rotations do not matter?

Rotations do matter since we are looking at direction (view) map, the reason for not taking modulo rotations.

Notes for AKT-140129/0:27:34

2018-07-11T21:38:56Z

Leo algknt: Blanked the page

Notes for AKT-140129/0:27:34

2018-07-11T21:37:53Z

Leo algknt:

[[File:fr_swa.jpg|center|0.0000000000002x0.00000000001px]]

Notes for AKT-140129/0:27:34

2018-07-11T21:25:26Z

Leo algknt:

[[File:fr_swa.jpg|10px]]

File:Fr swa.jpg

2018-07-11T21:24:25Z

Leo algknt: Linking number; framing and swaddling maps

Linking number; framing and swaddling maps

Notes for AKT-140129/0:27:34

2018-07-11T21:16:25Z

Leo algknt:

[[File:Proof.jpg|10px]]

File:Proof.jpg

2018-07-11T21:14:49Z

Leo algknt: Leo algknt uploaded a new version of "File:Proof.jpg"

File:Proof.jpg

2018-07-11T21:13:43Z

Leo algknt: Leo algknt uploaded a new version of "File:Proof.jpg"

Notes for AKT-140129/0:27:34

2018-07-11T21:07:03Z

Leo algknt:

[[File:Proof.jpg|thumb|10px]]

Notes for AKT-140212/0:19:31

2018-07-11T19:57:01Z

Leo algknt:

'''Question about modulo rotations''':

Can [[User:Leo algknt|Leo algknt]] say since we are dealing with points and therefore rotations do not matter?

Notes for AKT-140212/0:19:31

2018-07-11T19:55:49Z

Leo algknt:

'''Question about modulo rotations''':

Can --[[User:Leo algknt|Leo algknt]] ([[User talk:Leo algknt|talk]]) 15:55, 11 July 2018 (EDT) say since we are dealing with points and therefore rotations do not matter?

Notes for AKT-140224/0:27:53

2018-07-04T06:27:24Z

Leo algknt: Created page with "'''Lie algebra of dimensions 1 and 2''' 1. '''one-dimensional Lie algebras''' are unique up to isomorphism. For if <math>\mathfrak{g} = \langle x \rangle </math> is a one d..."

'''Lie algebra of dimensions 1 and 2'''

1. '''one-dimensional Lie algebras''' are unique up to isomorphism. For if <math>\mathfrak{g} = \langle x \rangle </math> is a one dimensional Lie algebra, then since the bracket is antisymmetric, we have <math>[x, x] = 0</math>. Thus the bracket is zero and <math>\mathfrak{g}</math> is unique up to isomorphism.

2. <math>\mathfrak{g} = \mathbb{F}^2 = \{ax + by \;|\; a, b \in \mathbb{F}\}</math>. <math>\mathfrak{g}</math> is a two-dimensional Lie algebra. There are only two of such up to isomorphism, that is, the one with the bracket equal to zero and the other with bracket <math>[x, y] = x</math>.

Notes for AKT-140214/0:08:40

2018-06-28T04:42:07Z

Leo algknt:

The set of differential <math>k</math>-forms on a manifold <math>M</math> (example <math>\mathbb{R}^3</math>) is a vector space <math>\Omega^k(M)</math> and when <math>k=0</math> then <math>\Omega^0(M)</math> is the set of smooth functions. Thus smooth functions are 0-forms. Now <math>k</math>-forms are integrated on <math>k</math>-manifolds. For example, a 1-form <math> f(x,y) \mathrm{d}x + g(x,y) \mathrm{d}y</math> can be integrated on a curve <math>\gamma</math>. Also differential forms can be differentiated using the operator d called the exterior operator where <math>d</math> acts on a <math>k</math>-form to produce a <math>k+1</math>-form and that <math>\mathrm{d}\circ \mathrm{d} =0</math>.

Now

1. if <math>f \in \Omega^0(\mathbb{R}^3)</math>, then <math>\mathrm{d}f = \sum_i^3 \frac{\partial{f}}{\partial{x_i}}\mathrm{d}x_i</math> is a 1-form so that <math>\mathrm{d}f \in \Omega^1(M)</math>. Thus <math>d: \mathrm{d} \Omega^0(\mathbb{R}^3) \rightarrow \Omega^1(\mathbb{R}^3)</math> is the gradient operator <math>\mathrm{grad}</math>.

2. If we have a 1-form <math>v = v_x\mathrm{d}x + v_y\mathrm{d}y + v_z\mathrm{d}z</math>, then <math>\mathrm{d}v = \left( \frac{\partial{v_z}}{\partial{y}}- \frac{\partial{v_y}}{\partial{z}}\right)\mathrm{d}y\wedge \mathrm{d}z + \left( \frac{\partial{v_x}}{\partial{z}}- \frac{\partial{v_z}}{\partial{x}}\right)\mathrm{d}x\wedge \mathrm{d}z + \left( \frac{\partial{v_x}}{\partial{y}} - \frac{\partial{v_y}}{\partial{x}}\right)\mathrm{d}x\wedge \mathrm{d}y</math> which is a two form. In this case we have <math>d: \mathrm{d} \Omega^1(\mathbb{R}^3) \rightarrow \Omega^2(\mathbb{R}^3)</math> is the <math>\mathrm{curl}</math> operator.

3. If we have 2-form <math>\omega = (\omega_x, \omega_y, \omega_z)</math> then again get a 3-form <math>\mathrm{d}\omega = \left( \frac{\partial{\omega_x}}{\partial{x}} + \frac{\partial{\omega_y}}{\partial{y}} + \frac{\partial{\omega_z}}{\partial{z}} \right)\mathrm{d}x\wedge\mathrm{d}y\wedge \mathrm{d}z</math>. If we think of <math>\mathrm{d}x\wedge\mathrm{d}y\wedge \mathrm{d}z</math> as a function <math>f</math>, then again we get <math>d: \mathrm{d} \Omega^2(\mathbb{R}^3) \rightarrow \Omega^3(\mathbb{R}^3)</math> is the divergence operator <math>\mathrm{div}</math>.

Notes for AKT-140214/0:08:40

2018-06-28T04:38:45Z

Leo algknt: Created page with "The set of differential <math>k</math>-forms on a manifold <math>M</math> (example <math>\mathbb{R}^3</math>) is a vector space <math>\Omega^k(M)</math> and when <math>k=0</ma..."

Notes for AKT-140212/0:19:31

2018-06-20T22:41:31Z

Leo algknt:

'''Question about modulo rotations''':

Can I say since we are dealing with points and therefore rotations do not matter?

Notes for AKT-140212/0:19:31

2018-06-20T22:41:10Z

Leo algknt:

'''Question about modulo rotation''':

Can I say since we are dealing with points and therefore rotations do not matter?

Notes for AKT-140212/0:19:31

2018-06-20T22:40:45Z

Leo algknt: Created page with "'''Question about mod rotation''': Can I say since we are dealing with points and therefore rotations do not matter?"

'''Question about mod rotation''':

Can I say since we are dealing with points and therefore rotations do not matter?

Notes for AKT-140205/0:31:49

2018-06-20T17:49:07Z

Leo algknt:

In the definition of <math>C_A(M)</math>, the disjoint union is taken over partition of <math>A</math>; I would like to know why they are all of length <math>k</math> (<math>k</math> '''is not fixed, it varies'''). Also do the <math>A_\alpha</math>'s represent clusters? ('''Yes they represent the clusters''')

Notes for AKT-140205/0:18:09

2018-06-20T17:45:46Z

Leo algknt:

'''Configuration space'''

<math>C_A^O(M) = \{\mathrm{injections}\; f: A \rightarrow M\}</math>, where <math>A</math> and <math>M</math> are finite set and a topological space respectively, is another way to define the configuration space of <math>M</math>. In this definition, the points on <math>M</math> are labelled by the elements of <math>A</math>. See [[http://drorbn.net/?title=Notes_for_AKT-140115/0:30:33]]

Notes for AKT-140205/0:18:09

2018-06-20T17:40:59Z

Leo algknt:

Notes for AKT-140205/0:18:09

2018-06-20T17:40:42Z

Leo algknt: Created page with "'''Configuration space''' <math>C_A^O(M) = {\mathrm{injections}\; f: A \rightarrow M}</math>, where <math>A</math> and <math>M</math> are finite set and a topological space ..."

'''Configuration space'''

<math>C_A^O(M) = {\mathrm{injections}\; f: A \rightarrow M}</math>, where <math>A</math> and <math>M</math> are finite set and a topological space respectively, is another way to define the configuration space of <math>M</math>. In this definition, the points on <math>M</math> are labelled by the elements of <math>A</math>.

Notes for AKT-140115/0:30:33

2018-06-20T06:22:42Z

Leo algknt:

'''Configuration space''' Given a topological space <math>X</math>, the <math>n</math>th ordered configuration space of <math>X</math> denoted by <math>\mathrm{Conf}_n(X)</math> is the set of <math>n</math>-tuples of pairwise distinct points in <math>X</math>, that is <math>\mathrm{Conf}_n(X):= \prod^n X \setminus \{(x_1, \ldots, x_n) : x_i = x_j \;\mathrm{for} \;i\ne j\}</math>.

In physics, parameters are used to define the configuration of a system and the vector space defined by these parameters is the configuration space of the system. It is used to describe the state of a whole system as a single point in a higher-dimensional space.

'''Examples of Configuration space'''

1. The configuration space of a particle in <math>\mathbb{R}^3</math> is <math>\mathbb{R}^3</math>. For <math>n</math> particles in <math>\mathbb{R}^3</math>, it is <math>\mathbb{R}^{3n}</math>

2. For a rigid body in <math>\mathbb{R}^3</math>, the configuration space is <math>\mathbb{R}^3 \times SO(3)</math>. Generally, it is <math>\mathbb{R}^n \times SO(n)</math>, where <math>SO(n)</math> is the special orthogonal group.

3. The torus with its diagonal removed, <math>S^1 \times \mathbb{R}</math>, is the configuration space of two points on <math>S^1</math>. This is <math>C_2(S^1)</math>

'''Reference:''' [https://en.wikipedia.org/wiki/Configuration_space_(mathematics)]

Notes for AKT-140115/0:30:33

2018-06-20T06:18:47Z

Leo algknt:

'''Configuration space''' Given a topological space <math>X</math>, the <math>n</math>th ordered configuration space of <math>X</math> denoted by <math>\mathrm{Conf}_n(X)</math> is the set of <math>n</math>-tuples of pairwise distinct points in <math>X</math>, that is <math>\mathrm{Conf}_n(X):= \prod^n X \setminus \{(x_1, \ldots, x_n) : x_i = x_j \;\mathrm{for} \;i\ne j\}</math>.

In physics, parameters are used to define the configuration of a system and the vector space defined by these parameters is the configuration space of the system. It is used to describe the state of a whole system as a single point in a higher-dimensional space.

'''Examples of Configuration space'''

1. The configuration space of a particle in <math>\mathbb{R}^3</math> is <math>\mathbb{R}^3</math>. For <math>n</math> particles in <math>\mathbb{R}^3</math>, it is <math>\mathbb{R}^{3n}</math>

2. For a rigid body in <math>\mathbb{R}^3</math>, the configuration space is <math>\mathbb{R}^3 \times SO(3)</math>. Generally, it is <math>\mathbb{R}^n \times SO(n)</math>, where <math>SO(n)</math> is the special orthogonal group.

3. The torus with its diagonal removed is the configuration space of two points on <math>S^1</math>.

'''Reference:''' [https://en.wikipedia.org/wiki/Configuration_space_(mathematics)]

Notes for AKT-140115/0:30:33

2018-06-20T06:16:31Z

Leo algknt:

'''Configuration space''' Given a topological space <math>X</math>, the <math>n</math>th ordered configuration space of <math>X</math> denoted by <math>\mathrm{Conf}_n(X)</math> is the set of <math>n</math>-tuples of pairwise distinct points in <math>X</math>, that is <math>\mathrm{Conf}_n(X):= \prod^n X \setminus \{(x_1, \ldots, x_n) : x_i = x_j \;\mathrm{for} \;i\ne j\}</math>.

In physics, parameters are used to define the configuration of a system and the vector space defined by these parameters is the configuration space of the system. It is used to describe the state of a whole system as a single point in a higher-dimensional space.

'''Examples of Configuration space'''

1. The configuration space of a particle in <math>\mathbb{R}^3</math> is <math>\mathbb{R}^3</math>. For <math>n</math> particles in <math>\mathbb{R}^3</math>, it is <math>\mathbb{R}^{3n}</math>

2. For a rigid body in <math>\mathbb{R}^3</math>, the configuration space is <math>\mathbb{R}^3 \times SO(3)</math>. Generally, it is <math>\mathbb{R}^n \times SO(n)</math>, where <math>SO(n)</math> is the special orthogonal group.

'''Reference:''' [https://en.wikipedia.org/wiki/Configuration_space_(mathematics)]

Notes for AKT-140115/0:30:33

2018-06-20T06:15:08Z

Leo algknt: Created page with "'''Configuration space''' Given a topological space <math>X</math>, the <math>n</math>th ordered configuration space of <math>X</math> denoted by <math>\mathrm{Conf}_n(X)</ma..."

Notes for AKT-140124/0:23:30

2018-06-18T19:03:11Z

Leo algknt:

Let <math>\Lambda</math> be a symmetric, positive definite, non-singular square matrix. Then we have the following:

<math> \langle x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)\rangle = \langle x,\Lambda x \rangle - \langle x, y \rangle - \langle \Lambda^{-1}y, \Lambda x \rangle + \langle \Lambda^{-1}y,y \rangle </math>.

We have <math> \langle \Lambda^{-1}y, \Lambda x \rangle = \langle x,y\rangle </math> and <math> \langle \Lambda^{-1}y,y \rangle = \langle y,\Lambda^{-1}y \rangle</math> since <math>\Lambda</math> is symmetric.

From the above, we see that <math>-\frac12 \langle x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y) \rangle + \frac12 \langle y,\Lambda^{-1}y \rangle = -\frac12 \langle x,\Lambda x \rangle + \langle x, y \rangle</math>

Notes for AKT-140129/0:27:34

2018-06-14T07:52:06Z

Leo algknt: Blanked the page

Notes for AKT-140129/0:27:34

2018-06-14T07:44:08Z

Leo algknt: Created page with "File:Proof.jpg"

[[File:Proof.jpg]]

File:Proof.jpg

2018-06-14T07:40:56Z

Leo algknt:

Notes for AKT-140205/0:31:49

2018-06-14T00:07:45Z

Leo algknt: Created page with "In the definition of <math>C_A(M)</math>, the disjoint union is taken over partition of <math>A</math>; I would like to know why they are all of length <math>k</math>. Also do..."

In the definition of <math>C_A(M)</math>, the disjoint union is taken over partition of <math>A</math>; I would like to know why they are all of length <math>k</math>. Also do the <math>A_\alpha</math>'s represent clusters?

Notes for AKT-140210/0:34:37

2018-06-09T06:02:51Z

Leo algknt: Created page with "Could we have used a different graph instead of the trivalent graph and a different set of relations to get what we want?"

Could we have used a different graph instead of the trivalent graph and a different set of relations to get what we want?

Notes for AKT-140124/0:38:21

2018-06-07T23:47:42Z

Leo algknt: Created page with "It is not clear why <math>Z^{-1}</math> times the integral equals <math>\varphi_1\Lambda^{-1}\varphi_2</math> and why it is the same as the formula obtained previously"

It is not clear why <math>Z^{-1}</math> times the integral equals <math>\varphi_1\Lambda^{-1}\varphi_2</math> and why it is the same as the formula obtained previously

Notes for AKT-140124/0:23:30

2018-06-06T05:23:01Z

Leo algknt: Created page with "Let <math>\Lambda</math> be a symmetric, positive definite, non-singular square matrix. Then we have the following: <math> <x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)> = ..."

Let <math>\Lambda</math> be a symmetric, positive definite, non-singular square matrix. Then we have the following:

<math> <x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)> = <x,\Lambda x> - <x, y> -<\Lambda^{-1}y, \Lambda x> + <\Lambda^{-1}y,y> </math>.

We have <math><\Lambda^{-1}y, \Lambda x> = <x,y> </math> and <math><\Lambda^{-1}y,y> = <y,\Lambda^{-1}y></math> since <math>\Lambda</math> is symmetric.

From the above, we see that <math>-\frac12 <x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)> + \frac12<y,\Lambda^{-1}y> = -\frac12<x,\Lambda x> + <x, y></math>