Munive Hernandez Morales 2022a - Revision history

LRMH at 05:39, 31 December 2022

2022-12-31T05:39:12Z

Gstinoco: Gstinoco moved page Review 481725695542 to Munive Hernandez Morales 2022a

2022-12-13T04:51:22Z

Gstinoco moved page Review 481725695542 to Munive Hernandez Morales 2022a

LRMH at 01:33, 13 December 2022

2022-12-13T01:33:57Z

LRMH at 01:31, 13 December 2022

2022-12-13T01:31:12Z

LRMH at 01:21, 13 December 2022

2022-12-13T01:21:09Z

LRMH: /* 3.5 Pair correlation function g */

2022-12-13T01:13:49Z

‎3.5 Pair correlation function g

LRMH: /* 3.4 K function */

2022-12-13T00:55:44Z

‎3.4 K function

LRMH: /* 3.4 K function */

2022-12-13T00:24:41Z

‎3.4 K function

LRMH at 00:18, 13 December 2022

2022-12-13T00:18:15Z

LRMH at 00:09, 13 December 2022

2022-12-13T00:09:57Z

@@ Line 417: / Line 417: @@
 | style="text-align: center;" | <math> g_{\Delta r}(r) = \frac{\lambda \left(K(r + \Delta r) - K(r) \right)}{\lambda \; \mu_L\left(B_1(\boldsymbol{0}) \right)\left( \left(r + \Delta r\right)^d - r^d  \right)} </math>
 |-
-| style="text-align: center;" | <math>  = \frac{K(r + \Delta r) - K(r)}{\mu_L\left( B_1(\boldsymbol{0}) \right)\left(\sum_{k=0}^d \binom{d}{k} r^{d - k} \Delta r^k - r^d \right)}.
+| style="text-align: center;" | <math>  = \frac{K(r + \Delta r) - K(r)}{\mu_L\left( B_1(\boldsymbol{0}) \right)\left(\displaystyle\sum_{k=0}^d \dbinom{d}{k} r^{d - k} \Delta r^k - r^d \right)}.
 </math>
 |}

@@ Line 386: / Line 386: @@
 for any compact set <math display="inline">C \subset \mathcal{S} \times \mathcal{S}</math>, where <math display="inline">\nu </math> is a suitable measure on <math display="inline">\left(\mathcal{S}, \mathcal{B}\left(\mathcal{S} \right)\right)</math> (e.g., if <math display="inline">\mathcal{S} = \mathbb{R}^d</math>, so <math display="inline">\nu = \mu _L</math>), and <math display="inline">\nu _{[2]}(A_1 \times A_2) = \mathbb{E}\left(Y(A_1) \; Y(A_2) \right)- \mathbb{E}\left(Y\left(A_1 \cap A_2 \right)\right)</math>, with <math display="inline">A_1, A_2 \subset \mathcal{S}</math>, is the second factorial moment measure of <math display="inline">Y</math>.
-If <math display="inline">Y</math> is stationary and isotropic, it follows from ([[#eq-3|3]]) that <span id='citeF-16'></span>[[#cite-16|[16]]], <span id='citeF-20'></span>[[#cite-20|[20]]],
+If <math display="inline">Y</math> is stationary and isotropic, it follows from ([[#eq-2|2]]) that <span id='citeF-16'></span>[[#cite-16|[16]]], <span id='citeF-20'></span>[[#cite-20|[20]]],
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 423: / Line 423: @@
 |}
-All binomial expansion components in the denominator of the second line in ([[#lb-3.5|3.5]]) lose significance except for <math display="inline">d \; r^{d - 1} \Delta r</math>, so
+All binomial expansion components in the denominator of the second line in ([[#eq-3|3]]) lose significance except for <math display="inline">d \; r^{d - 1} \Delta r</math>, so
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 420: / Line 420: @@
 </math>
 |}
 |}

@@ Line 409: / Line 409: @@
 If <math display="inline">Y</math> is stationary and isotropic, the expected number of locations in the ring is <math display="inline">\lambda \; K (r + \Delta r) - \lambda \; K(r)</math>. Dividing it by the expected value of points assuming a Poisson process, we obtain
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-

@@ Line 414: / Line 414: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> g_r(r)  = \frac{\left(K(r + r) - K(r) \right)}{\lambda \; \mu_L\left(B_1(\boldsymbol{0}) \right)\left( \left(r + r\right)^d - r^d  \right)}
+| style="text-align: center;" | <math> g_{\Delta r}(r) = \frac{\lambda \left(K(r + \Delta r) - K(r) \right)}{\lambda \; \mu_L\left(B_1(\boldsymbol{0}) \right)\left( \left(r + \Delta r\right)^d - r^d  \right)} </math>
-= \frac{K(r + r) - K(r)}{\mu_L\left( B_1(\boldsymbol{0}) \right)\left(\sum_k=0^d \binom{d}{k} r^{d - k} \Delta r^k - r^d \right)}.
+|-
 </math>
 |}

← Older revision	Revision as of 04:51, 13 December 2022
(No difference)

@@ Line 316: / Line 316: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>K(r) = \frac{1}{\lambda } \mathbb{E} \left(Y\left(\boldsymbol{Y} \cap B_r(\boldsymbol{s}) - \{ \boldsymbol{s} \} \right)\; \mid \; \boldsymbol{s} \in \boldsymbol{Y} \right),</math>
+| style="text-align: center;" | <math>K(r) = \frac{1}{\lambda } \mathbb{E} \left(Y\left(\boldsymbol{Y} \cap \left\lbrace B_r(\boldsymbol{s}) - \{ \boldsymbol{s} \} \right) \right\rbrace \; \mid \; \boldsymbol{s} \in \boldsymbol{Y} \right),</math>
 |}
 |}

@@ Line 352: / Line 352: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>K(r) = \mu _L\left(B_1(\boldsymbol{0}) \right)\; r^d.</math>
+| style="text-align: center;" | <math>K(r) = \mu_L\left(B_1(\boldsymbol{0}) \right)\; r^d.</math>
 |}
 |}

@@ Line 166: / Line 166: @@
 Note that
-<span id="eq-2"></span>
+<span id="eq-1"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 174: / Line 174: @@
 | style="text-align: center;" | <math>\mathrm{d}(\boldsymbol{s}, \boldsymbol{Y}) \leq r \Leftrightarrow Y\left(B_r(\boldsymbol{s}) \right)> 0,  </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
 |}
 where <math display="inline">B_r(\boldsymbol{s})</math> is the neighborhood of radius <math display="inline">r</math> centered on <math display="inline">\boldsymbol{s}</math>.
-In other words, as shown in Figure [[#img-4|4]], the empty-space distance satisfies the logical equivalence of the biconditional ([[#eq-2|2]]), <math display="inline">\mathrm{d}(\boldsymbol{s}, \boldsymbol{Y}) > r \Leftrightarrow Y\left(B_r(\boldsymbol{s}) \right)= 0</math>.
+In other words, as shown in Figure [[#img-4|4]], the empty-space distance satisfies the logical equivalence of the biconditional ([[#eq-1|1]]), <math display="inline">\mathrm{d}(\boldsymbol{s}, \boldsymbol{Y}) > r \Leftrightarrow Y\left(B_r(\boldsymbol{s}) \right)= 0</math>.
 Moreover, because <math display="inline">\left\lbrace  Y\left(B_r(\boldsymbol{s}) \right)> 0 \right\rbrace </math> is measurable, the event <math display="inline">\left\lbrace \mathrm{d}(\boldsymbol{s}, \boldsymbol{Y}) \leq r \right\rbrace </math> is measurable, implying that the contact distance is a well-defined random element.

@@ Line 332: / Line 332: @@
 If <math display="inline">\mathcal{S} = \mathbb{R}^d</math> and the point process <math display="inline">Y</math> is assumed to be stationary, then hold <math display="inline">\lambda _2 \left(\boldsymbol{s}, \boldsymbol{u} \right)= \lambda _2 \left(\boldsymbol{s} - \boldsymbol{u} \right)</math>. Also, if <math display="inline">Y</math> is isotropic, hence <math display="inline">\lambda _2(\boldsymbol{s} - \boldsymbol{u}) = \lambda _2(r)</math>, where <math display="inline">r = \lVert \boldsymbol{s} - \boldsymbol{u} \rVert </math>. These conditions implies that <span id='citeF-15'></span>[[#cite-15|[15]]], <span id='citeF-16'></span>[[#cite-16|[16]]],
-<span id="eq-3"></span>
+<span id="eq-2"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"
 |-
@@ Line 340: / Line 340: @@
 | style="text-align: center;" | <math>\lambda \; K(r) = \frac{d \; \mu _L\left(B_1(\boldsymbol{0}) \right)}{\lambda } \int _0^r \lambda _2 (z) \; z^{d-1} \; \mathrm{d}z.   </math>
 |}
-| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
 |}