first lessons

1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture1.aux

@@ -4,4 +4,6 @@
 \@writefile{toc}{\contentsline {chapter}{\numberline {1}Lecture 1 - 09-03-2020}{1}\protected@file@percent }
 \@writefile{lof}{\addvspace {10\p@ }}
 \@writefile{lot}{\addvspace {10\p@ }}
-\@writefile{toc}{\contentsline {section}{\numberline {1.1}Introduction}{1}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\numberline {1.1}Introduction of the course}{1}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\numberline {1.2}Examples}{1}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Spam filtering}{4}\protected@file@percent }

1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture1.log

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.13)  13 APR 2020 12:41
+This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.13)  15 APR 2020 21:00
 entering extended mode
 **./lecture1.tex
 (lecture1.tex
@@ -232,16 +232,6 @@ Package: titlesec 2019/10/16 v2.13 Sectioning titles
 )
 ("C:\Program Files\MiKTeX 2.9\tex/latex/base\fontenc.sty"
 Package: fontenc 2020/02/11 v2.0o Standard LaTeX package
-)
-("C:\Program Files\MiKTeX 2.9\tex/latex/fncychap\fncychap.sty"
-Package: fncychap 2007/07/30 v1.34 LaTeX package (Revised chapters)
-\RW=\skip55
-\mylen=\skip56
-\myhi=\skip57
-\px=\skip58
-\py=\skip59
-\pyy=\skip60
-\pxx=\skip61
 )))
 ("C:\Program Files\MiKTeX 2.9\tex/latex/l3backend\l3backend-pdfmode.def"
 File: l3backend-pdfmode.def 2020-03-12 L3 backend support: PDF mode
@@ -318,29 +308,68 @@ G,.JBIG2,.JB2,.eps]
 (grfext)             \AppendGraphicsExtensions on input line 504.
 )
 Chapter 1.
-LaTeX Font Info:    Trying to load font information for OT1+pzc on input line 3
-.
-("C:\Program Files\MiKTeX 2.9\tex/latex/psnfss\ot1pzc.fd"
-File: ot1pzc.fd 2020/03/25 font definitions for OT1/pzc.
-)
-LaTeX Font Info:    Font shape `OT1/pzc/m/n' in size <76> not available
-(Font)              Font shape `OT1/pzc/m/it' tried instead on input line 3.
 
-Overfull \hbox (10.0pt too wide) in paragraph at lines 3--3
+Underfull \hbox (badness 10000) in paragraph at lines 8--14
-[][][][][][][][][]  
- []
-
-LaTeX Font Info:    Trying to load font information for T1+cmss on input line 3
-.
-("C:\Program Files\MiKTeX 2.9\tex/latex/base\t1cmss.fd"
-File: t1cmss.fd 2019/12/16 v2.5j Standard LaTeX font definitions
-)
-Underfull \hbox (badness 10000) in paragraph at lines 8--95
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 8--95
+Underfull \hbox (badness 10000) in paragraph at lines 8--14
+
+ []
+
+LaTeX Font Info:    Trying to load font information for U+msa on input line 15.
+
+("C:\Program Files\MiKTeX 2.9\tex/latex/amsfonts\umsa.fd"
+File: umsa.fd 2013/01/14 v3.01 AMS symbols A
+)
+LaTeX Font Info:    Trying to load font information for U+msb on input line 15.
+
+
+("C:\Program Files\MiKTeX 2.9\tex/latex/amsfonts\umsb.fd"
+File: umsb.fd 2013/01/14 v3.01 AMS symbols B
+)
+Underfull \hbox (badness 10000) in paragraph at lines 22--23
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 32--53
 
  []
 
@@ -348,55 +377,85 @@ Underfull \hbox (badness 10000) in paragraph at lines 8--95
 
 
 {C:/Users/AndreDany/AppData/Local/MiKTeX/2.9/pdftex/config/pdftex.map}]
-LaTeX Font Info:    Trying to load font information for U+msa on input line 96.
+Underfull \hbox (badness 10000) in paragraph at lines 61--89
-
-
-("C:\Program Files\MiKTeX 2.9\tex/latex/amsfonts\umsa.fd"
-File: umsa.fd 2013/01/14 v3.01 AMS symbols A
-)
-LaTeX Font Info:    Trying to load font information for U+msb on input line 96.
-
-
-("C:\Program Files\MiKTeX 2.9\tex/latex/amsfonts\umsb.fd"
-File: umsb.fd 2013/01/14 v3.01 AMS symbols B
-)
-Underfull \hbox (badness 10000) in paragraph at lines 98--138
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 98--138
+Underfull \hbox (badness 10000) in paragraph at lines 61--89
 
  []
 
 
-Underfull \hbox (badness 10000) in paragraph at lines 98--138
+Underfull \hbox (badness 10000) in paragraph at lines 61--89
 
  []
 
-[2] [3] (lecture1.aux) ) 
+
+Underfull \hbox (badness 10000) in paragraph at lines 61--89
+
+ []
+
+[2]
+Underfull \hbox (badness 10000) in paragraph at lines 93--104
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 93--104
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 93--104
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 107--123
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 107--123
+
+ []
+
+
+Underfull \hbox (badness 10000) in paragraph at lines 107--123
+
+ []
+
+[3]
+Underfull \hbox (badness 10000) in paragraph at lines 132--135
+
+ []
+
+[4] (lecture1.aux) ) 
 Here is how much of TeX's memory you used:
- 5192 strings out of 480934
+ 5099 strings out of 480934
- 70233 string characters out of 2909670
+ 68776 string characters out of 2909670
- 340605 words of memory out of 3000000
+ 333073 words of memory out of 3000000
- 20940 multiletter control sequences out of 15000+200000
+ 20853 multiletter control sequences out of 15000+200000
- 542533 words of font info for 50 fonts, out of 3000000 for 9000
+ 546309 words of font info for 55 fonts, out of 3000000 for 9000
  1141 hyphenation exceptions out of 8191
- 42i,9n,50p,332b,220s stack positions out of 5000i,500n,10000p,200000b,50000s
+ 42i,9n,50p,332b,152s stack positions out of 5000i,500n,10000p,200000b,50000s
- <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk/
+ <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk/ljfo
-ljfour/jknappen/ec/dpi600\tcrm1200.pk> <C:\Users\AndreDany\AppData\Local\MiKTeX
+ur/jknappen/ec/dpi600\ecti1200.pk> <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9
-\2.9\fonts/pk/ljfour/jknappen/ec/dpi600\ecrm1200.pk> <C:\Users\AndreDany\AppDat
+\fonts/pk/ljfour/jknappen/ec/dpi600\ecbx1440.pk> <C:\Users\AndreDany\AppData\Lo
-a\Local\MiKTeX\2.9\fonts/pk/ljfour/jknappen/ec/dpi600\ecbx1728.pk> <C:\Users\An
+cal\MiKTeX\2.9\fonts/pk/ljfour/jknappen/ec/dpi600\tcrm1200.pk> <C:\Users\AndreD
-dreDany\AppData\Local\MiKTeX\2.9\fonts/pk/ljfour/jknappen/ec/dpi600\ecsx1728.pk
+any\AppData\Local\MiKTeX\2.9\fonts/pk/ljfour/jknappen/ec/dpi600\ecbx1200.pk> <C
->{C:/Program Files/MiKTeX 2.9/fonts/enc/dvips/base/8r.enc}<C:/Program Files/MiK
+:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk/ljfour/jknappen/ec/dpi600\e
-TeX 2.9/fonts/type1/public/amsfonts/cm/cmex10.pfb><C:/Program Files/MiKTeX 2.9/
+crm1200.pk> <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk/ljfour/jknapp
-fonts/type1/public/amsfonts/cm/cmmi12.pfb><C:/Program Files/MiKTeX 2.9/fonts/ty
+en/ec/dpi600\ecbx1728.pk> <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk
-pe1/public/amsfonts/cm/cmr12.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/publi
+/ljfour/jknappen/ec/dpi600\ecbx2488.pk><C:/Program Files/MiKTeX 2.9/fonts/type1
-c/amsfonts/cm/cmsy10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/urw/zapfchan/
+/public/amsfonts/cm/cmex10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/
-uzcmi8a.pfb>
+amsfonts/cm/cmmi12.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts
-Output written on lecture1.pdf (3 pages, 76413 bytes).
+/cm/cmr12.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmsy1
+0.pfb>
+Output written on lecture1.pdf (4 pages, 100195 bytes).
 PDF statistics:
- 143 PDF objects out of 1000 (max. 8388607)
+ 205 PDF objects out of 1000 (max. 8388607)
  0 named destinations out of 1000 (max. 500000)
  1 words of extra memory for PDF output out of 10000 (max. 10000000)
 

1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture1.tex

@@ -2,90 +2,99 @@
 \begin{document}
 \chapter{Lecture 1 - 09-03-2020}
 
-\section{Introduction}
+\section{Introduction of the course}
-This is time for all good men to come to the aid of their party!
+
 
-MACHINE LEARNING
 In this course we look at the principle behind design of Machine learning.
-Not just coding but have an idea of algorithm that can work with the data.
+Not just coding but have an idea of algorithm that can work with the data.\\\\
-We have to fix a mathematical framework: some statistic and mathematics.
+We have to fix a mathematical framework: some statistic and mathematics.\\
-Work on ML on a higher level
+\textbf{Work on ML on a higher level}\\
 ML is data inference: make prediction about the future using data about the
-past
+past\\
-Clustering —> grouping according to similarity
+\begin{itemize}
-Planning —> (robot to learn to interact in a certain environment)
+\item Clustering $\rightarrow$  grouping according to similarity
-Classification —> (assign meaning to data) example: Spam filtering
+\item Planning $\rightarrow$ (robot to learn to interact in a certain environment)
+\item Classification $\rightarrow$ (assign meaning to data) example: Spam filtering\\
 I want to predict the outcome of this individual or i want to predict whether a
 person click or not in a certain advertisement.
-Examples
+\end{itemize}
-Classify data into categories:
+\section{Examples}
-Medical diagnosis: data are medical records and • categories are diseases
+Classify data into categories:\\
-• Document analysis: data are texts and categories are topics
+\begin{itemize}
-• Image analysts: data are digital images and for categories name of objects
+\item Medical diagnosis: data are medical records and categories are diseases
+\item Document analysis: data are texts and categories are topics
+\item Image analysts: data are digital images and for categories name of objects
 in the image (but could be different).
-• Spam filtering: data are emails, categories are spam vs non spam.
+\item Spam filtering: data are emails, categories are spam vs non spam.
-• Advertising prediction: data are features of web site visitors and categories
+\item Advertising prediction: data are features of web site visitors and categories
 could be click/non click on banners.
-Classification : Different from clustering since we do not have semantically
+\end{itemize}
-classification (spam or not spam) —> like meaning of the image.
+Classification : \textbf{ Different from clustering }since we do not have semantically
+classification (spam or not spam) $\rightarrow$ like meaning of the image.\\
 I have a semantic label.
-Clustering: i want to group data with similarity function.
+\\\\
-Planning: Learning what to do next
+Clustering: i want to group data with similarity function. \\\\
-Clustering: Learn similarity function
+Planning: Learning what to do next \\\\
-Classification: Learn semantic labels meaning of data
+Clustering: Learn similarity function \\\\
-Planning: Learn actions given state
+Classification: Learn semantic labels meaning of data\\\\
+Planning: Learn actions given state\\\\
 In classification is an easier than planning task since I’m able to make
-prediction telling what is the semantic label that goes with data points.
+prediction telling what is the semantic label that goes with data points.\\
-If i can do classification i can clustering.
+If i can do classification i can clustering.\\
 If you do planning you probably classify (since you understanding meaning in
-your position) and then you can also do clustering probably.
+your position) and then you can also do clustering probably.\\
-We will focus on classification because many tasks are about classification.
+We will focus on classification because many tasks are about classification.\\\\
-Classify data in categories we can image a set of categories.
+Classify data in categories we can image a set of categories.\\
-For instance the tasks:
+For instance the tasks:\\
-‘predict income of a person’
+‘predict income of a person’\\
-‘Predict tomorrow price for a stock’
+‘Predict tomorrow price for a stock’\\
-The label is a number and not an abstract thing.
+The label is a number and not an abstract thing.\\\\
 We can distinguish two cases:
-The label set —> set of possible categories for each data • point. For each of
+\begin{itemize}
+\item The label set $\rightarrow$ set of possible categories for each data point. For each of
 this could be finite set of abstract symbols (case of document classification,
 medical diagnosis). So the task is classification.
-• Real number (no bound on how many of them). My prediction will be a real
+\item Real number (no bound on how many of them). My prediction will be a real
 number and is not a category. In this case we talk about a task of
 regression.
+\end{itemize}
 Classification: task we want to give a label predefined point in abstract
 categories (like YES or NO)
+\\
 Regression: task we want to give label to data points but this label are
-numbers.
+numbers.\\\\
 When we say prediction task: used both for classification and regression
-tasks.
+tasks.\\
-Supervised learning: Label attached to data (classification, regression)
+Supervised learning: Label attached to data (classification, regression)\\
-Unsupervised learning: No labels attached to data (clustering)
+Unsupervised learning: No labels attached to data (clustering)\\
 In unsupervised the mathematical modelling and way algorithm are score and
 can learn from mistakes is a little bit harder. Problem of clustering is harder to
-model mathematically.
+model mathematically.\\
 You can cast planning as supervised learning: i can show the robot which is
 the right action to do in that state. But that depends on planning task is
-formalised.
+formalised.\\
 Planning is higher level of learning since include task of supervised and
-unsupervised learning.
+unsupervised learning.\\\\
-Why is this important ?
+Why is this important ?\\
-Algorithm has to know how to given the label.
+Algorithm has to know how to given the label.\\
 In ML we want to teach the algorithm to perform prediction correctly. Initially
 algorithm will make mistakes in classifying data. We want to tell algorithm that
 classification was wrong and just want to perform a score. Like giving a grade
 to the algorithm to understand if it did bad or really bad.
-So we have mistakes!
+So we have mistakes!\\\\
-Algorithm predicts and something makes a mistake —> we can correct it.
+Algorithm predicts and something makes a mistake $\rightarrow$ we can correct it.\\
 Then algorithm can be more precisely.
-We have to define this mistake.
+We have to define this mistake.\\
-Mistakes in case of classification:
+Mistakes in case of classification:\\
-If category is the wrong one (in the simple case). We • have a binary signal
+\begin{itemize}
+\item If category is the wrong one (in the simple case). We have a binary signal
 where we know that category is wrong.
-How to communicate it?
+\end{itemize}
+How to communicate it?\\
 We can use the loss function: we can tell the algorithm whether is wrong or
-not.
+not.\\\\
-Loss function: measure discrepancy between ‘true’ label and predicted
+\bred{Loss function}: measure discrepancy between ‘true’ label and predicted
-label.
+label.\\
 So we may assume that every datapoint has a true label.
 If we have a set of topic this is the true topic that document is talking about.
 It is typical in supervised learning.
@@ -97,7 +106,7 @@ How good the algorithm did?
 
 were $y $ is true label and $\hat{y}$ is predicted label
 \\\\
-We want to build a spam filter were $0$ is not spam and $1$ is spam and that 
+We want to build a spam filter were $0$ is not spam and $1$ is spam and that's a
 Classification task: 
 \\\\
 $
@@ -107,33 +116,30 @@ $
 \end{cases}
 $
 \\\\
-The loss function is the “interface” between algorithm and data.
+\textbf{The loss function is the “interface” between algorithm and data.}\\
-So algorithm know about the data through the loss function.
+So algorithm know about the data through the loss function.\\
 If we give a useless loss function the algorithm will not perform good: is
 important to have a good loss function.
-Spam filtering
+\subsection{Spam filtering}
-We have two main mistakes:
+$Y = \{ spam, no \, spam\}$
+\\
+Binary classification $|Y| = 2$ \\
+We have two main mistake:
+\begin{itemize}
+\item False positive: $y = $non spam, $\hat{y} = $ spam 
+\item False negative: $y = $spam, $\hat{y} = $ no spam
+\end{itemize}
 It is the same mistake? No if i have important email and you classify as spam
-that’s bad and if you show me a spam than it’s ok.
+that’s bad and if you show me a spam than it’s ok.\\
-So we have to assign a different weight.
+So we have to assign a different weight.\\
+$$
+\ell\left(y,\hat{y}\right) = \begin{cases}
+2 \quad \textit{if FP}\\
+1 \quad \textit{if FN}\\
+0 \quad otherwise
+\end{cases}
+$$
+\bred{We have to take more attention on positive mistake}\\
 Even in binary classification, mistakes are not equal.
-e Iotf.TFprIuos.uos
-True came
-razee
-Cussler aircN TASK spam ACG FIRM
-ftp.y GO
-IF F Y n is soon
-IF FEY 0 Nor spam
-ZERO CNE Cass
-n n
-Span No Seamy Binary Classification
-I 2
-FALSE PEENE Mistake Y NON SPAM J Spam
-FN Mistake i f SPAM y NO spam
-2 IF Fp Meter Airenita
-f Y F on positive
-y ye en MISTAKE
-0 otherwise
-0 otherwise
 
 \end{document}

1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture2.aux

@@ -6,8 +6,8 @@
 \@writefile{lot}{\addvspace {10\p@ }}
 \@writefile{toc}{\contentsline {section}{\numberline {1.1}Argomento}{1}\protected@file@percent }
 \@writefile{toc}{\contentsline {section}{\numberline {1.2}Loss}{1}\protected@file@percent }
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Absolute Loss}{2}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Absolute Loss}{1}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Square Loss}{2}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.3}Example of information of square loss}{2}\protected@file@percent }
-\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.4}labels and losses}{4}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.4}labels and losses}{3}\protected@file@percent }
 \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.5}Example TF(idf) documents encoding}{5}\protected@file@percent }

1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture2.log

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.13)  13 APR 2020 12:31
+This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 2.9.7300 64-bit) (preloaded format=pdflatex 2020.4.13)  15 APR 2020 21:01
 entering extended mode
 **./lecture2.tex
 (lecture2.tex
@@ -238,14 +238,7 @@ File: l3backend-pdfmode.def 2020-03-12 L3 backend support: PDF mode
 \l__kernel_color_stack_int=\count193
 \l__pdf_internal_box=\box48
 )
-(lecture2.aux
+(lecture2.aux)
-
-Package babel Warning: Unknown language `ngerman'. Very likely you
-(babel)                requested it in a previous run. Expect some
-(babel)                wrong results in this run, which should vanish
-(babel)                in the next one. Reported on input line 21.
-
-)
 \openout1 = `lecture2.aux'.
 
 LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 3.
@@ -340,15 +333,15 @@ Underfull \hbox (badness 10000) in paragraph at lines 17--30
 
  []
 
-[1
 
-
-{C:/Users/AndreDany/AppData/Local/MiKTeX/2.9/pdftex/config/pdftex.map}]
 Underfull \hbox (badness 10000) in paragraph at lines 32--35
 
  []
 
+[1
 
+
+{C:/Users/AndreDany/AppData/Local/MiKTeX/2.9/pdftex/config/pdftex.map}]
 Underfull \hbox (badness 10000) in paragraph at lines 49--52
 
  []
@@ -383,12 +376,12 @@ Underfull \hbox (badness 10000) in paragraph at lines 89--109
 
  []
 
-[3]
+
 Underfull \hbox (badness 10000) in paragraph at lines 110--115
 
  []
 
-
+[3]
 Underfull \hbox (badness 10000) in paragraph at lines 118--156
 
  []
@@ -423,23 +416,23 @@ Underfull \hbox (badness 10000) in paragraph at lines 164--171
 
  []
 
-[5]
+
 Underfull \hbox (badness 10000) in paragraph at lines 172--182
 
  []
 
-
+[5]
 Underfull \hbox (badness 10000) in paragraph at lines 189--199
 
  []
 
 [6] (lecture2.aux) ) 
 Here is how much of TeX's memory you used:
- 5108 strings out of 480934
+ 5099 strings out of 480934
- 68939 string characters out of 2909670
+ 68776 string characters out of 2909670
- 334197 words of memory out of 3000000
+ 333073 words of memory out of 3000000
- 20861 multiletter control sequences out of 15000+200000
+ 20853 multiletter control sequences out of 15000+200000
- 547081 words of font info for 56 fonts, out of 3000000 for 9000
+ 546309 words of font info for 55 fonts, out of 3000000 for 9000
  1141 hyphenation exceptions out of 8191
  42i,10n,50p,316b,142s stack positions out of 5000i,500n,10000p,200000b,50000s
  <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk/ljfo
@@ -460,9 +453,9 @@ es/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmsy10.pfb><C:/Program Files/MiKTe
 X 2.9/fonts/type1/public/amsfonts/cm/cmsy6.pfb><C:/Program Files/MiKTeX 2.9/fon
 ts/type1/public/amsfonts/cm/cmsy8.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/
 public/amsfonts/symbols/msbm10.pfb>
-Output written on lecture2.pdf (6 pages, 167391 bytes).
+Output written on lecture2.pdf (6 pages, 165099 bytes).
 PDF statistics:
- 281 PDF objects out of 1000 (max. 8388607)
+ 276 PDF objects out of 1000 (max. 8388607)
  0 named destinations out of 1000 (max. 500000)
  1 words of extra memory for PDF output out of 10000 (max. 10000000)