Applets:Two-dimensional Gaussian Random Variables - Revision history

Noah at 20:20, 16 April 2023

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Hwang at 18:06, 21 March 2023

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Noah at 20:14, 1 February 2022

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Noah at 20:14, 1 February 2022

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Noah at 20:13, 1 February 2022

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Javier: Text replacement - "”" to """

2021-05-28T14:47:23Z

Text replacement - "”" to """

@@ Line 12: / Line 12: @@
-The applet uses the framework &nbsp;[https://en.wikipedia.org/wiki/Plotly Plot.ly]
+The applet uses the framework &nbsp;[https://en.wikipedia.org/wiki/Plotly "Plot.ly"]
 ==Theoretical Background==
@@ Line 29: / Line 29: @@
 *$∩$&nbsp; denotes the logical AND operation.
 *$X$&nbsp; and&nbsp; $Y$ denote the two random variables, and&nbsp; $x \in X$&nbsp; and &nbsp; $y \in Y$ indicate realizations thereof.
-*The nomenclature used for this applet thus differs slightly from the description in the [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|Theory section]].}}
+*The nomenclature used for this applet thus differs slightly from the description in the [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|"Theory section"]].}}
@@ Line 58: / Line 58: @@
 ===2D&ndash;PDF for Gaussian random variables===
-For the special case&nbsp; '''Gaussian random variables'''&nbsp; - the name goes back to the scientist&nbsp; [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss Carl Friedrich Gauss]&nbsp; - we can further note:
+For the special case&nbsp; '''Gaussian random variables'''&nbsp; - the name goes back to the scientist&nbsp; [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss "Carl Friedrich Gauss"]&nbsp; - we can further note:
 *The joint PDF of a Gaussian 2D random variable&nbsp; $XY$&nbsp; with means&nbsp; $m_X = 0$&nbsp; and&nbsp; $m_Y = 0$&nbsp; and the correlation coefficient&nbsp; $ρ = ρ_{XY}$&nbsp; is:
 : $$f_{XY}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_X \cdot \sigma_Y \cdot \sqrt{\rm 1-\rho^2}}\ \cdot\ \exp\Bigg[-\frac{\rm 1}{\rm 2 \cdot (1- \it\rho^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_X^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_Y^{\rm 2}}-\rm 2\it\rho\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_Y}\rm ) \rm \Bigg]\hspace{0.8cm}{\rm with}\hspace{0.5cm}-1 \le \rho \le +1.$$
@@ Line 106: / Line 106: @@
 For correlated components&nbsp; $(ρ_{XY} ≠ 0)$&nbsp; the contour lines of the PDF are (almost) always elliptic, so also for the special case&nbsp; $σ_X = σ_Y$.
-<u>Exception:</u>&nbsp; $ρ_{XY}=\pm 1$ &nbsp; &rArr; &nbsp; Diracwand; see&nbsp; [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|Exercise 4.4]]&nbsp; in the book "Stochastic Signal Theory", subtask &nbsp;''(5)'''.
+<u>Exception:</u>&nbsp; $ρ_{XY}=\pm 1$ &nbsp; &rArr; &nbsp; "Dirac-wall"; see&nbsp; [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|"Exercise 4.4"]]&nbsp; in the book "Stochastic Signal Theory", subtask &nbsp;''(5)''.
 [[File:Sto_App_Bild3.png|right|frame|height lines of the two dimensional PDF with correlated quantities]]
 Here, the determining equation of the PDF height lines is:
@@ Line 114: / Line 114: @@
 *The ellipse major axis is dashed in dark blue.
-*The&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Regression_line|regression line]]&nbsp; $K(x)$&nbsp; is drawn in red throughout.
+*The&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Regression_line|"regression line"]]&nbsp; $K(x)$&nbsp; is drawn in red throughout.
@@ Line 121: / Line 121: @@
 *The angle of inclination&nbsp; $α$&nbsp; of the ellipse major axis (dashed straight line) with respect to the&nbsp; $x$&ndash;axis also depends on&nbsp; $σ_X$,&nbsp; $σ_Y$&nbsp; and&nbsp; $ρ_{XY}$&nbsp; :
 :$$\alpha = {1}/{2} \cdot {\rm arctan } \big ( 2 \cdot \rho_{XY} \cdot \frac {\sigma_X \cdot \sigma_Y}{\sigma_X^2 - \sigma_Y^2} \big ).$$
-*The (red) correlation line&nbsp; $y = K(x)$&nbsp; of a Gaussian 2D random variable always lies below the (blue dashed) ellipse major axis.
+*The (red) correlation line&nbsp; $y = K(x)$&nbsp; of a Gaussian 2D-random variable always lies below the (blue dashed) ellipse major axis.
 * $K(x)$&nbsp; can be geometrically constructed from the intersection of the contour lines and their vertical tangents, as indicated in the sketch in green color.
 <br><br>
@@ Line 127: / Line 127: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; '''2D cumulative distribution function'''&nbsp; like the 2D-CDF, is merely a useful extension of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|one-dimensional distribution function]]&nbsp; (PDF):
+$\text{Definition:}$&nbsp; The&nbsp; '''2D cumulative distribution function'''&nbsp; like the 2D-CDF, is merely a useful extension of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|"one-dimensional distribution function"]]&nbsp; (PDF):
 :$$F_{XY}(x,y) = {\rm Pr}\big [(X \le x) \cap (Y \le y) \big ] .$$}}
@@ Line 235: / Line 235: @@
 &nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Parameter input via slider:&nbsp; $\sigma_X$, &nbsp;$\sigma_Y$ and&nbsp; $\rho$.
-&nbsp; &nbsp; '''(B)'''' &nbsp; &nbsp; Selection:&nbsp; Representation of PDF or CDF.
+&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Selection:&nbsp; Representation of PDF or CDF.
 &nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Reset:&nbsp; Setting as at program start.

@@ Line 267: / Line 267: @@
 *The first version was created in 2003 by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] &nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
 *In 2019 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; as part of her bachelor thesis&nbsp; (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
-*Last revision and English version 2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp; Translation using DEEPL.com.
+*Last revision and English version 2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp;

@@ Line 200: / Line 200: @@
 '''(7)'''&nbsp; Starting from&nbsp; $\sigma_X=\sigma_Y=1\ \rho = 0.7$&nbsp; vary the correlation coefficient&nbsp; $\rho$.&nbsp; What is the slope angle &nbsp;$\theta$&nbsp; of the correlation line&nbsp; $K(x)$?}}
-*&nbsp;For&nbsp; $\sigma_X=\sigma_Y$&:nbsp;  &nbsp;$\theta={\rm arctan}\ (\rho)$.&nbsp; The slope increases with increasing&nbsp; $\rho > 0$.&nbsp; In all cases, &nbsp;$\theta < \alpha = 45^\circ$ holds. For&nbsp; $\rho = 0.7$&nbsp; this gives &nbsp;$\theta = 35^\circ$.
+*&nbsp;For&nbsp; $\sigma_X=\sigma_Y$:&nbsp;  &nbsp;$\theta={\rm arctan}\ (\rho)$.&nbsp; The slope increases with increasing&nbsp; $\rho > 0$.&nbsp; In all cases, &nbsp;$\theta < \alpha = 45^\circ$ holds. For&nbsp; $\rho = 0.7$&nbsp; this gives &nbsp;$\theta = 35^\circ$.
@@ Line 231: / Line 231: @@
 ==Applet Manual==
 <br>
-[[File:Anleitung_2D-Gauss.png|left|500px|frame|Screen shot (German version)]]
+[[File:Anleitung_2D-Gauss.png|left|500px|frame|Screen shot from the German version]]
-&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Parametereingabe per Slider:&nbsp; $\sigma_X$, &nbsp;$\sigma_Y$ und&nbsp; $\rho$
+<br><br>
-&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Auswahl:&nbsp; Darstellung von WDF oder VTF
+&nbsp; &nbsp; '''(B)'''' &nbsp; &nbsp; Selection:&nbsp; Representation of PDF or CDF.
-&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Reset:&nbsp; Einstellung wie beim Programmstart
+&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Reset:&nbsp; Setting as at program start.
-&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Höhenlinien darstellen anstelle von "1D-WDF"
+&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Display contour lines instead of one-dimensional PDF.
-&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Darstellungsbereich für "2D-WDF"
+&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Display range for two-dimensional PDF.
-&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Manipulation der 3D-Grafik (Zoom, Drehen, ...)
+&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Manipulation of the three-dimensional graph (zoom, rotate, ...)
-&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Darstellungsbereich für "1D-WDF" bzw. "Höhenlinien"
+&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Display range for&nbsp; "one-dimensional PDF"&nbsp; or&nbsp; "contour lines".
-&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Manipulation der 2D-Grafik ("1D-WDF")
+&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Manipulation of the two-dimensional graphics ("one-dimensional PDF")
-&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Bereich für die Versuchsdurchführung:   Aufgabenauswahl
+&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Area for exercises: Task selection.
-&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Bereich für die Versuchsdurchführung:   Aufgabenstellung
+&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Area for exercises: Task description
-&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Bereich für die Versuchsdurchführung:   Musterlösung einblenden
+&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Area for exercises: Show/hide solution
-&nbsp; &nbsp; '''( L)''' &nbsp; &nbsp; Bereich für die Versuchsdurchführung:   Musterlösung
+&nbsp; &nbsp; '''( L)''' &nbsp; &nbsp; Area for exercises: Output of the sample solution
-Werte&ndash;Ausgabe über Maussteuerung (sowohl bei 2D als auch bei 3D)
+<u>Note:</u> &nbsp; &nbsp;Value output of the graphics&nbsp; $($both 2D and 3D$)$&nbsp; via mouse control.
 <br clear=all>

@@ Line 256: / Line 256: @@
 &nbsp; &nbsp; '''( L)''' &nbsp; &nbsp; Bereich für die Versuchsdurchführung:   Musterlösung
 Werte&ndash;Ausgabe über Maussteuerung (sowohl bei 2D als auch bei 3D)
 <br clear=all>

@@ Line 139: / Line 139: @@
 :$$F_{XY}(-\infty,\ -\infty) = 0,\hspace{0.5cm}F_{XY}(x,\ +\infty)=F_{X}(x ),\hspace{0.5cm}
 F_{XY}(+\infty,\ y)=F_{Y}(y ) ,\hspace{0.5cm}F_{XY}(+\infty,\ +\infty) = 1.$$
-*In the limiting case $($infinitely large&nbsp; $x$&nbsp; and&nbsp; $y)$&nbsp; thus the value&nbsp; $1$ is obtained for the "2D&ndash;CDF". From this we obtain the&nbsp; '''normalization condition'''&nbsp; for the 2D probability density function:
+*In the limiting case $($infinitely large&nbsp; $x$&nbsp; and&nbsp; $y)$&nbsp; thus the value&nbsp; $1$ is obtained for the "2D&ndash;CDF". From this we obtain the&nbsp; '''normalization condition'''&nbsp; for the two-dimensional probability density function:
 :$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}x \,\,{\rm d}y=1 . $$

@@ Line 77: / Line 77: @@
 ===Contour lines for uncorrelated random variables===
-[[File:Sto_App_Bild2.png |frame| Contour lines of 2D/PDF with uncorrelated variables | right]]
+[[File:Sto_App_Bild2.png |frame| Contour lines of 2D-PDF with uncorrelated variables | right]]
 From the conditional equation&nbsp; $f_{XY}(x, y) = {\rm const.}$&nbsp; the contour lines of the PDF can be calculated.

@@ Line 127: / Line 127: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; '''2D cumulative distribution function''''&nbsp; like the 2D-CDF, is merely a useful extension of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|one-dimensional distribution function]]&nbsp; (PDF):
+$\text{Definition:}$&nbsp; The&nbsp; '''2D cumulative distribution function'''&nbsp; like the 2D-CDF, is merely a useful extension of the&nbsp; [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|one-dimensional distribution function]]&nbsp; (PDF):
 :$$F_{XY}(x,y) = {\rm Pr}\big [(X \le x) \cap (Y \le y) \big ] .$$}}
@@ Line 145: / Line 145: @@
 $\text{Conclusion:}$&nbsp; Note the significant difference between one-dimensional and two-dimensional random variables:
 *For one-dimensional random variables, the area under the PDF always yields $1$.
-*For two-dimensional random variables, the PDF volume always equals $1$}}
+*For two-dimensional random variables, the PDF volume always equals $1$.}}
 <br><br>

@@ Line 106: / Line 106: @@
 Bei korrelierten Komponenten&nbsp; $(ρ_{XY} ≠ 0)$&nbsp; sind die Höhenlinien der WDF (fast) immer elliptisch, also auch für den Sonderfall&nbsp; $σ_X = σ_Y$.
-<u>Ausnahme:</u>&nbsp; $ρ_{XY}=\pm 1$ &nbsp; &rArr; &nbsp; Diracwand; siehe&nbsp; [[Aufgaben:Aufgabe_4.4:_Gaußsche_2D-WDF|Aufgabe 4.4]]&nbsp; im Buch "Stochastische Signaltheorie&rdquo;, Teilaufgabe &nbsp;'''(5)'''.
+<u>Ausnahme:</u>&nbsp; $ρ_{XY}=\pm 1$ &nbsp; &rArr; &nbsp; Diracwand; siehe&nbsp; [[Aufgaben:Aufgabe_4.4:_Gaußsche_2D-WDF|Aufgabe 4.4]]&nbsp; im Buch "Stochastische Signaltheorie", Teilaufgabe &nbsp;'''(5)'''.
 [[File:Sto_App_Bild3.png|right|frame|Höhenlinien der 2D-WDF bei korrelierten Größen]]
 Hier lautet die Bestimmungsgleichung der WDF-Höhenlinien:
@@ Line 131: / Line 131: @@
-Es ergeben sich folgende Gemeinsamkeiten und Unterschiede zwischen der "1D-VTF&rdquo; und der" 2D-VTF&rdquo;:
+Es ergeben sich folgende Gemeinsamkeiten und Unterschiede zwischen der "1D-VTF" und der" 2D-VTF":
-*Der Funktionalzusammenhang zwischen "2D&ndash;WDF&rdquo; und "2D&ndash;VTF&rdquo; ist wie im eindimensionalen Fall durch die Integration gegeben, aber nun in zwei Dimensionen. Bei kontinuierlichen Zufallsgrößen gilt:
+*Der Funktionalzusammenhang zwischen "2D&ndash;WDF" und "2D&ndash;VTF" ist wie im eindimensionalen Fall durch die Integration gegeben, aber nun in zwei Dimensionen. Bei kontinuierlichen Zufallsgrößen gilt:
 :$$F_{XY}(x,y)=\int_{-\infty}^{y} \int_{-\infty}^{x} f_{XY}(\xi,\eta) \,\,{\rm d}\xi \,\, {\rm d}\eta   .$$
 *Umgekehrt lässt sich die Wahrscheinlichkeitsdichtefunktion aus der Verteilungsfunktion durch partielle Differentiation nach&nbsp; $x$&nbsp; und&nbsp; $y$&nbsp; angeben:
@@ Line 139: / Line 139: @@
 :$$F_{XY}(-\infty,\ -\infty) = 0,\hspace{0.5cm}F_{XY}(x,\ +\infty)=F_{X}(x ),\hspace{0.5cm}
 F_{XY}(+\infty,\ y)=F_{Y}(y ) ,\hspace{0.5cm}F_{XY}(+\infty,\ +\infty) = 1.$$
-*Im Grenzfall $($unendlich große&nbsp; $x$&nbsp; und&nbsp; $y)$&nbsp; ergibt sich demnach für die "2D-VTF&rdquo; der Wert&nbsp; $1$. Daraus erhält man die&nbsp; '''Normierungsbedingung'''&nbsp; für die 2D-Wahrscheinlichkeitsdichtefunktion:
+*Im Grenzfall $($unendlich große&nbsp; $x$&nbsp; und&nbsp; $y)$&nbsp; ergibt sich demnach für die "2D-VTF" der Wert&nbsp; $1$. Daraus erhält man die&nbsp; '''Normierungsbedingung'''&nbsp; für die 2D-Wahrscheinlichkeitsdichtefunktion:
 :$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}x \,\,{\rm d}y=1  .   $$
@@ Line 184: / Line 184: @@
 {{BlueBox|TEXT=
-'''(5)'''&nbsp; Is the statement true:&nbsp;"Elliptic contour lines exist only for &nbsp;$\rho \ne 0$&rdquo;.&nbsp; Interpret the&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; and&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$&nbsp; for &nbsp;$\sigma_X=1, \ \sigma_Y=0.5$&nbsp; and&nbsp; $\rho = 0$.}}
+'''(5)'''&nbsp; Is the statement true:&nbsp;"Elliptic contour lines exist only for &nbsp;$\rho \ne 0$".&nbsp; Interpret the&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$&nbsp; and&nbsp; $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$&nbsp; for &nbsp;$\sigma_X=1, \ \sigma_Y=0.5$&nbsp; and&nbsp; $\rho = 0$.}}
 *&nbsp;No!&nbsp; Also, for&nbsp; $\ \rho = 0$&nbsp; the contour lines are elliptical&nbsp; (not circular)&nbsp; if &nbsp;$\sigma_X \ne \sigma_Y$.
@@ Line 238: / Line 238: @@
 &nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Reset:&nbsp; Einstellung wie beim Programmstart
-&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Höhenlinien darstellen anstelle von "1D-WDF&rdquo;
+&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Höhenlinien darstellen anstelle von "1D-WDF"
-&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Darstellungsbereich für "2D-WDF&rdquo;
+&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Darstellungsbereich für "2D-WDF"
 &nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Manipulation der 3D-Grafik (Zoom, Drehen, ...)
-&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Darstellungsbereich für "1D-WDF&rdquo; bzw. "Höhenlinien&rdquo;
+&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Darstellungsbereich für "1D-WDF" bzw. "Höhenlinien"
-&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Manipulation der 2D-Grafik ("1D-WDF&rdquo;)
+&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Manipulation der 2D-Grafik ("1D-WDF")
 &nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Bereich für die Versuchsdurchführung:   Aufgabenauswahl
@@ Line 267: / Line 267: @@
 This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich].
 *The first version was created in 2003 by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]] &nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
-*In 2019 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; as part of her bachelor thesis&nbsp; (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5&rdquo;.
+*In 2019 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; as part of her bachelor thesis&nbsp; (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
 *Last revision and English version 2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp; Translation using DEEPL.com.