Aufgaben:Exercise 2.5Z: Compression Factor vs. Residual Redundancy - Revision history

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	*A smaller residual redundancy  $r(N)$  for a given  $N$  therefore says nothing about whether LZW coding is useful for the source at hand.		*A smaller residual redundancy  $r(N)$  for a given  $N$  therefore says nothing about whether LZW coding is useful for the source at hand.
	*For this, the compression factor  $K(N)$  must be considered.   In general, the following relationship between  $r(N)$  and  $K(N)$ applies:		*For this, the compression factor  $K(N)$  must be considered.   In general, the following relationship between  $r(N)$  and  $K(N)$ applies:
	:$$r(N) = 1 - \frac{H}{K(N)}\hspace{0.05cm},\hspace{0.5cm} K(N) = H \cdot (1- r(N))		:$$r(N) = 1 - \frac{H}{K(N)}\hspace{0.05cm},\hspace{0.5cm} K(N) = H \cdot (1- r(N))\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

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	[[Category:Information Theory: Exercises\|^2.2 Lempel-Ziv-Welch Compression^]]		[[Category:Information Theory: Exercises\|^2.2 Lempel-Ziv-Welch Compression^]]
			[[de:Zusatzaufgaben:2.05_LZW-Komprimierung]]

Guenter at 12:13, 10 August 2021

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	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Komprimierung nach Lempel~~, ~~Ziv und Welch~~		{{quiz-Header\|Buchseite=Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch
	}}		}}

Guenter at 11:29, 10 August 2021

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Guenter at 11:22, 10 August 2021

2021-08-10T11:22:15Z

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	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/~~Komprimierung_nach_Lempel~~,~~_Ziv_und_Welch~~#~~Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t~~\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Quantitative_statements_on_asymptotic_optimality\|Quantitative statements on asymptotic optimality]].

Guenter at 11:21, 10 August 2021

2021-08-10T11:21:24Z

← Older revision		Revision as of 13:21, 10 August 2021
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	*In particular, reference is made to the pages		*In particular, reference is made to the pages
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/~~Komprimierung_nach_Lempel~~,~~_Ziv_und_Welch~~#~~Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung~~\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].

Guenter at 11:20, 10 August 2021

2021-08-10T11:20:16Z

← Older revision		Revision as of 13:20, 10 August 2021
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	*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].		*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].
	*In particular, reference is made to the pages		*In particular, reference is made to the pages
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#~~Remainding_redundancy_as_a_measure_for_the_efficiency_of_coding_methods~~\|Remainding redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].

Guenter at 11:19, 10 August 2021

2021-08-10T11:19:12Z

Noah at 20:04, 6 August 2021

2021-08-06T20:04:34Z

← Older revision		Revision as of 13:22, 10 August 2021
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	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/~~Komprimierung_nach_Lempel~~,~~_Ziv_und_Welch~~#~~Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t~~\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Quantitative_statements_on_asymptotic_optimality\|Quantitative statements on asymptotic optimality]].

← Older revision		Revision as of 13:20, 10 August 2021
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	*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].		*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].
	*In particular, reference is made to the pages		*In particular, reference is made to the pages
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#~~Remainding_redundancy_as_a_measure_for_the_efficiency_of_coding_methods~~\|Remainding redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].

← Older revision		Revision as of 17:55, 16 March 2026
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	[[File:P_ID2449__Inf_Z_2_5_neu.png\|right\|frame\|Data length $L(N)$ after LZW coding for $\rm BQ1$ and $\rm BQ2$]]		[[File:P_ID2449__Inf_Z_2_5_neu.png\|right\|frame\|Data length $L(N)$ after LZW coding for $\rm BQ1$ and $\rm BQ2$]]
	As in  [[Aufgaben:2.5_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]  we consider data compression using the  [[Information_Theory/~~Komprimierung_nach_Lempel~~,~~_Ziv_und_Welch~~#The_Lempel-Ziv-Welch_algorithm\|Lempel–Ziv–Welch algorithm]]. Here holds:		As in  [[Aufgaben:2.5_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]  we consider data compression using the  [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#The_Lempel-Ziv-Welch_algorithm\|Lempel–Ziv–Welch algorithm]]. Here holds:
	* Let the input sequence have the length  $N$.		* Let the input sequence have the length  $N$.
	* The length of the LZW coder output is  $L$.		* The length of the LZW coder output is  $L$.
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	*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].		*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].
	*In particular, reference is made to the pages		*In particular, reference is made to the pages
	:: [[Information_Theory/~~Komprimierung_nach_Lempel~~,~~_Ziv_und_Welch~~#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Efficiency_of_Lempel-Ziv_coding\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Quantitative_statements_on_asymptotic_optimality\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch#Quantitative_statements_on_asymptotic_optimality\|Quantitative statements on asymptotic optimality]].
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	[[Category:Information Theory: Exercises\|^2.2 Lempel-Ziv-Welch Compression^]]		[[Category:Information Theory: Exercises\|^2.2 Lempel-Ziv-Welch Compression^]]
	[[de:~~Zusatzaufgaben~~:2.~~05_LZW-Komprimierung~~]]		[[de:Aufgaben:Aufgabe 2.5Z: Komprimierungsfaktor vs. Restredundanz]]

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	{What conclusions can be drawn from the comparison of  $N = 10000$  ~~And~~  $N = 50000$?		{What conclusions can be drawn from the comparison of  $N = 10000$  and  $N = 50000$?
	\|type="[]"}		\|type="[]"}
	+ For both sources  $K(N = 50000)$  is smaller than  $K(N = 10000)$.		+ For both sources  $K(N = 50000)$  is smaller than  $K(N = 10000)$.
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	:$${\rm BQ1:}\hspace{0.3cm} K \hspace{0.2cm} = \hspace{0.2cm} \frac{6800}{10000}\hspace{0.15cm}\underline{= 0.680}\hspace{0.05cm},$$		:$${\rm BQ1:}\hspace{0.3cm} K \hspace{0.2cm} = \hspace{0.2cm} \frac{6800}{10000}\hspace{0.15cm}\underline{= 0.680}\hspace{0.05cm},$$
	:$$ {\rm BQ2:}\hspace{0.3cm} K \hspace{0.2cm} = \hspace{0.2cm} \frac{12330}{10000}\hspace{0.15cm}\underline{= 1.233}\hspace{0.05cm}. $$		:$$ {\rm BQ2:}\hspace{0.3cm} K \hspace{0.2cm} = \hspace{0.2cm} \frac{12330}{10000}\hspace{0.15cm}\underline{= 1.233}\hspace{0.05cm}. $$
	*Of course, the LZW coding only makes sense with the redundant binary source  $\rm BQ1$ .  Here the amount of data can be reduced by  $32\%$ .		*Of course, the LZW coding only makes sense with the redundant binary source  $\rm BQ1$ .  Here the amount of data can be reduced by  $1-0.68=32\%$ .
	*With the redundancy-free binary source  $\rm BQ2$ , on the other hand, the LZW coding leads to a  $23.3\%$  larger data quantity.		*With the redundancy-free binary source  $\rm BQ2$ , on the other hand, the LZW coding leads to a  $1.233-1=23.3\%$  larger data quantity.


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	* For both sources the compression factor  $K(N)$  is smaller for  $N = 50000$  than for  $N = 10000$.		* For both sources the compression factor  $K(N)$  is smaller for  $N = 50000$  than for  $N = 10000$.
	* The same applies to the residual redundancy:   $r(N = 50000) < r(N = 10000)$.		* The same applies to the residual redundancy:   $r(N = 50000) < r(N = 10000)$.
	* Both with regard to  $K(N)$  and  $r(N)$  "more favourable" values result with larger  $N$ , even if, as with the redundancy-free binary source  $\rm BQ2$ , the application of Lempel–Ziv actually leads to a deterioration.		* Both with regard to  $K(N)$  and  $r(N)$  "more favourable"  values result with larger  $N$ , even if, as with the redundancy-free binary source  $\rm BQ2$,  <br>the application of Lempel–Ziv actually leads to a deterioration.

	{{ML-Fuß}}		{{ML-Fuß}}

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	[[File:P_ID2449__Inf_Z_2_5_neu.png\|right\|frame\|Data length $L(N)$ after LZW coding for $\rm BQ1$ and $\rm BQ2$]]		[[File:P_ID2449__Inf_Z_2_5_neu.png\|right\|frame\|Data length $L(N)$ after LZW coding for $\rm BQ1$ and $\rm BQ2$]]
	As in  [[Aufgaben:2.5_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]  we consider data compression using the  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#The_Lempel-Ziv-Welch_algorithm\|Lempel–Ziv–Welch~~–~~algorithm]]. Here holds:		As in  [[Aufgaben:2.5_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]  we consider data compression using the  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#The_Lempel-Ziv-Welch_algorithm\|Lempel–Ziv–Welch algorithm]]. Here holds:
	* Let the input sequence have length  $N$.		* Let the input sequence have the length  $N$.
	* The length of the LZW coder output is  $L$.		* The length of the LZW coder output is  $L$.


	The graph shows for two different binary sources  $\rm BQ1$  and  $\rm BQ2$  the relationship between the sequence lengths  $N$  and  $L$, represented by the function  $L(N)$.		The graph shows for two different binary sources  $\rm BQ1$  and  $\rm BQ2$  the relationship between the sequence lengths  $N$  and  $L$,  represented by the function  $L(N)$.

	The two ~~message~~ sources have the same statistical properties as in  [[Aufgaben:Aufgabe_2.5:_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]:		The two sources have the same statistical properties as in  [[Aufgaben:Aufgabe_2.5:_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]:
	* $\rm BQ1$  is redundant due to unequal symbol probabilities  $(p_{\rm A} = 0.89$  and  $p_{\rm B} = 0.11)$ .  There are no ~~ties~~ between the individual symbols.  The entropy is  $H = 0.5$ bit/source symbol.		* $\rm BQ1$  is redundant due to unequal symbol probabilities  $(p_{\rm A} = 0.89$  and  $p_{\rm B} = 0.11)$ .  There are no dependencies between the individual symbols.  The entropy is  $H = 0.5$ bit/source symbol.
	* $\rm BQ2$  is redundancy-free and thus has entropy  $H = 1$ bit/source symbol.		* $\rm BQ2$  is redundancy-free and thus has the entropy  $H = 1$ bit/source symbol.


	Furthermore, you need two definitions for the solution of this task:		Furthermore, you need two definitions for the solution of this task:
	*Let the ~~<i>~~compression factor~~</i>~~ be by definition		*Let the  "compression factor"  be by definition
	:$$K(N) = \frac{L(N)}{N}\hspace{0.05cm}.$$		:$$K(N) = \frac{L(N)}{N}\hspace{0.05cm}.$$
	* The relative redundancy of the LZW code sequence  (called  ~~<i>~~residual redundancy~~</i>~~  in the following)  is		* The relative redundancy of the LZW code sequence  (called  "residual redundancy"  in the following)  is
	:$$r(N) = \frac{L(N) - N \cdot H}{L(N)}= 1 - \frac{ N \cdot H}{L(N)}\hspace{0.05cm}.$$		:$$r(N) = \frac{L(N) - N \cdot H}{L(N)}= 1 - \frac{ N \cdot H}{L(N)}\hspace{0.05cm}.$$

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	Hints:		Hints:
	*The ~~task~~ belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].		*The exercise belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].
	*In particular, reference is made to the pages		*In particular, reference is made to the pages
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#~~Remaining_redundancy_as_a_measure_for_the_efficiency_of_coding_methods~~\|~~Residual~~ redundancy as a measure of the efficiency of coding methods]],		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Remainding_redundancy_as_a_measure_for_the_efficiency_of_coding_methods\|Remainding redundancy as a measure of the efficiency of coding methods]],
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung\|Efficiency of Lempel-Ziv coding]] and		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Effizienz_der_Lempel.E2.80.93Ziv.E2.80.93Codierung\|Efficiency of Lempel-Ziv coding]] and
	:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].		:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Quantitative_Aussagen_zur_asymptotischen_Optimalit.C3.A4t\|Quantitative statements on asymptotic optimality]].

← Older revision		Revision as of 22:04, 6 August 2021
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	[[File:P_ID2449__Inf_Z_2_5_neu.png\|right\|frame\|Data length $L(N)$ after LZW coding for $\rm BQ1$ and $\rm BQ2$]]		[[File:P_ID2449__Inf_Z_2_5_neu.png\|right\|frame\|Data length $L(N)$ after LZW coding for $\rm BQ1$ and $\rm BQ2$]]
	As in  [[Aufgaben:2.5_Restredundanz_bei_LZW-Codierung\|~~exercise~~ 2.5]]  we consider data compression using the  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#The_Lempel-Ziv-Welch_algorithm\|Lempel–Ziv–Welch–algorithm]]. Here holds:		As in  [[Aufgaben:2.5_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]  we consider data compression using the  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#The_Lempel-Ziv-Welch_algorithm\|Lempel–Ziv–Welch–algorithm]]. Here holds:
	* Let the input sequence have length  $N$.		* Let the input sequence have length  $N$.
	* The length of the LZW coder output is  $L$.		* The length of the LZW coder output is  $L$.
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	The graph shows for two different binary sources  $\rm BQ1$  and  $\rm BQ2$  the relationship between the sequence lengths  $N$  and  $L$, represented by the function  $L(N)$.		The graph shows for two different binary sources  $\rm BQ1$  and  $\rm BQ2$  the relationship between the sequence lengths  $N$  and  $L$, represented by the function  $L(N)$.

	The two message sources have the same statistical properties as in  [[Aufgaben:Aufgabe_2.5:_Restredundanz_bei_LZW-Codierung\|~~exercise~~ 2.5]]:		The two message sources have the same statistical properties as in  [[Aufgaben:Aufgabe_2.5:_Restredundanz_bei_LZW-Codierung\|Exercise 2.5]]:
	* $\rm BQ1$  is redundant due to unequal symbol probabilities  $(p_{\rm A} = 0.89$  and  $p_{\rm B} = 0.11)$ .  There are no ties between the individual symbols.  The entropy is  $H = 0.5$ bit/source symbol.		* $\rm BQ1$  is redundant due to unequal symbol probabilities  $(p_{\rm A} = 0.89$  and  $p_{\rm B} = 0.11)$ .  There are no ties between the individual symbols.  The entropy is  $H = 0.5$ bit/source symbol.
	* $\rm BQ2$  is redundancy-free and thus has entropy  $H = 1$ bit/source symbol.		* $\rm BQ2$  is redundancy-free and thus has entropy  $H = 1$ bit/source symbol.
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	''Hints:''		Hints:
	*The task belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].		*The task belongs to the chapter  [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch\|Compression according to Lempel, Ziv and Welch]].
	*In particular, reference is made to the pages		*In particular, reference is made to the pages