Let $I=[a,b]$, and consider some $f\in BV(I)$ (the space of all functions with bounded total variation on $I$). Let  $$\mathcal Q=\{[a_k,b_k]:k\in\mathbb N,~\operatorname{cl}\left(\bigcup_{k=1}^\infty[a_k,b_k]\right)=I,~(a_k,b_k)\cap(a_j,b_j)=\emptyset~\forall i\neq j \}.$$ In their book, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Gora and Boyarsky claim in a problem that it is possible for  $$\sum_{k=1}^\infty V_{[a_k,b_k]}(f)&lt;V_I(f).$$ I struggle to see how this is true, and believe I have a proof that  $$\sum_{k=1}^\infty V_{[a_k,b_k]}(f)=V_I(f).$$ Now this question is pretty much the same as  one, but I do not think the accepted answer there is actually correct, as the example given shows that we can have  $$V_{[a,c)}(f)+V_{[c,d]}(f)&lt;V_I(f),$$ but the variation of a function over half open intervals is very different from that over closed intervals, so this does not actually address the question. In fact we know that we always have $$V_{[a,c]}(f)+V_{[c,d]}(f)=V_I(f).$$ My supposed proof for the countable case is a standard approximation argument showing inequalities both ways, but it is quite long to type out, and Boyarski and Gora are much wiser than me so my proof is most probably wrong. Thus before I go through the effort of typing it out I'd appreciate some input from the learned members here, because I might be being silly and missing an easy counterexample.  EDIT: On inspecting the fantastic answer by mathcounterexamples.net I realised that I misinterpreted the given counterexample in the linked post, and it is actually shows perfectly what fails in the countable case. Thus I will be flagging this post as a duplicate, as it adds nothing new.
Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$  such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], [0,1/3], [1/3,1/3+1/3^2], \cdots$ ) Can we write  $$ \operatorname{Var}_{[0,1]} f=\sum_{k=1}^\infty \operatorname{Var}_{[a_k,b_k]} f? $$
I've been working on the Android SDK platform, and it is a little unclear how to save an application's state. So given this minor re-tooling of the 'Hello, Android' example:  package com.android.hello;  import android.app.Activity; import android.os.Bundle; import android.widget.TextView;  public class HelloAndroid extends Activity {    private TextView mTextView = null;    /** Called when the activity is first created. */   @Override   public void onCreate(Bundle savedInstanceState) {     super.onCreate(savedInstanceState);      mTextView = new TextView(this);      if (savedInstanceState == null) {        mTextView.setText("Welcome to HelloAndroid!");     } else {        mTextView.setText("Welcome back.");     }      setContentView(mTextView);   } }   I thought it would be enough for the simplest case, but it always responds with the first message, no matter how I navigate away from the app.  I'm sure the solution is as simple as overriding onPause or something like that, but I've been poking away in the documentation for 30 minutes or so and haven't found anything obvious. Start new chapter on same page Adding coordinate system to layout in QGIS composer? What's the difference between $(stuff) and `stuff`? What is the range of the validity of Coulomb's law? What is the smallest and biggest distance in which  is valid?  Please provide a reference to a scientific journal or book. Just saying that this law is valid from this range to that range is not enough. Which experiments did result to these ranges? I needed to create a list of lists in Python, so I typed the following: my_list = [[1] * 4] * 3  The list looked like this: [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]    Then I changed one of the innermost values: my_list[0][0] = 5  Now my list looks like this: [[5, 1, 1, 1], [5, 1, 1, 1], [5, 1, 1, 1]]    which is not what I wanted or expected. Can someone please explain what's going on, and how to get around it? Everybody perhaps knows of Jon Favreau's movie  which is going to be released in 2016. But quite recently I learned (and to my utter surprise) that another Jungle book movie is coming in 2017 and it's called .   Now, this movie has big names too with Christian Bale as Baghera, Benedict Cumberbatch as Sher Khan, Cate Blanchett as Kaa and a young Indian boy in the live action role as Mowgli.   How often does this happen, with two big productions with big cast making a movie on the same subject? I want to ask why this is happening and if there are any really significant differences between the two (apart from the cast, of course). Parsing input with scanf in C How do I access previous promise results in a .then() chain? Why can't Chromium suddenly access any partition except for /home? I just upgraded to Ubuntu 19.10 and to my horror I discovered that the default browser Chromium can no longer access other partitions than /home.  This means that my partitions containing music and videos (/multimedia) for instance are no longer accessible e.g. to upload into Google Drive. This includes USB partitions.   I found another description of this phenomenon here:  where it says  "Also, while we acknowledge that most users will only need to access their home folders, it’s worthy to mention that you no longer can access system files &amp; folders from inside Chromium (You can’t access anything except your home folder). This means that if you are to upload a file or an image for example into the Internet using an uploading website, then this file/image must be in your home folder, as you can not access directories such as /usr, /etc or other directories located in your root partition. In fact, you no longer can access any other partition at all, this includes USB sticks and other booted partitions:"  I wonder why this behaviour been introduced? it violates the whole idea of multiple disks, partitions and symlinks. Btw, this still works for Firefox.  My questions are   Are there any plans to fix this? I expect one reason for this behaviour is about security. If so, is it possible to explain why and how that makes for better security? The files on other partitions are legitimately accessible by the user anyway e.g. through symlinks. But with this change, I'll have to move them to /home before uploading so still accessing them although more slowly and cumbersome.  If this behaviour is now permanent, the question is how should we incorporate multiple disks in our filesystem?  Alternatively, can someone tell me how to find and install the debian package. I haven't been able to locate it?   This is not a duplicate of  because that is about a specific error during installation that I didn't get. My question is about the access chromium has in Ubuntu 19.10.  Anyway, I solved the problem via the following command by using the original debian package from Google:  sudo apt-get install google-chrome-stable   After that I can now access my other partitions again. I know this is hacky in the sense that the package isn't native to Ubuntu so I would really like to get some advice on how to incorporate multiple disks and partitions with this new restricted behaviour of Chromium.  Thank you very much So: I have a Gantt chart created with pgfgantt where in order to fit my pages I have included  x unit=1.4cm,y unit chart=6mm   in its preamble.  This means that milestone shapes are spread out horizontally.  How can I change the vertical and horizontal units of a Gantt chart in a way which doesn't affect the aspect ratios of elements like milestones? Why do we say that the Earth moves around the Sun? In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.  But then in physics we learn that movement is relative, and it depends on the reference point that we choose.  Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?  Was movement considered absolute in physics back then? Finding probability of other child also being a boy A man visits a couple who have two children.One of them, a boy, comes in to the room. Find the probability p that other is also a boy  a) 1/3 b) 2/3 c) 1/2 d) 3/4  Correct Ans: a) 1/3  However I am getting answer as 1/2.My solution is as follows.  The couple has two children, hence the sample space is as follows.  B- Boy G - Girl  BB BG GG ( BG and GB mean the same thing as Girl,Boy), but it is already known that one is boy.So the sample space reduces to  BB BG  So now p is 1/2  Is there some mistake is my approach ? Detecting structured data in Google Webmaster Tools What nutrients can humans absorb in the mouth? For instance, I realise being able to absorb simple sugars in the mouth is pivotal in the rapid action of oral glucose gel. Thus I was wondering what nutrients in general can be absorbed directly within the mouth, and at what speed? I have 14.04.5/16.04 or newer and an AMD GPU. I'm not getting the performance I want, and some programs say they need things like OpenCL.  I've heard about fglrx, but if I try to install it on my machine, it breaks and I have no GUI.  Is there anything I can do to get my AMD card working to its full potential? I just bought a new MTB and am planning to change my previous MTB into a Road Bike.   How do I do it? What kind of parts do I need to change or add or remove?  I am not looking to for a full conversion into specialised road bike. But like the answers are suggesting, I am looking for a bridge between commuter or road bike. My majors concerns to be met are:   Should be fast enough to compete with road racers over short distance Should be light as a road  Should be as close as possible in as cheap as possible. pyPdf for IndirectObject extraction Following this example, I can list all elements into a pdf file  import pyPdf pdf = pyPdf.PdfFileReader(open("pdffile.pdf")) list(pdf.pages) # Process all the objects. print pdf.resolvedObjects   now, I need to extract a non-standard object from the pdf file.   My object is the one named MYOBJECT and it is a string.  The piece printed by the python script that concernes me is:  {'/MYOBJECT': IndirectObject(584, 0)}   The pdf file is this:  558 0 obj &lt;&lt;/Contents 583 0 R/CropBox[0 0 595.22 842]/MediaBox[0 0 595.22 842]/Parent 29 0 R/Resources   &lt;&lt;/ColorSpace &lt;&lt;/CS0 563 0 R&gt;&gt;     /ExtGState &lt;&lt;/GS0 568 0 R&gt;&gt;     /Font&lt;&lt;/TT0 559 0 R/TT1 560 0 R/TT2 561 0 R/TT3 562 0 R&gt;&gt;     /ProcSet[/PDF/Text/ImageC]     /Properties&lt;&lt;/MC0&lt;&lt;/MYOBJECT 584 0 R&gt;&gt;/MC1&lt;&lt;/SubKey 582 0 R&gt;&gt; &gt;&gt;     /XObject&lt;&lt;/Im0 578 0 R&gt;&gt;&gt;&gt;   /Rotate 0/StructParents 0/Type/Page&gt;&gt; endobj ... ... ... 584 0 obj &lt;&lt;/Length 8&gt;&gt;stream  1_22_4_1     ---&gt;&gt;&gt;&gt;  this is the string I need to extract from the object  endstream endobj   How can I follow the 584 value in order to refer to my string (under pyPdf of course)?? How to compute the  formula $\sum  \limits_{r=1}^d r \cdot 2^r$? Why I cannot use universal instantiation rule here? I have the following premise:      $$\forall x (Fx \vee \neg Fx) \rightarrow Ga$$   Why I cannot use UI and get: $(Fa \vee \neg Fa) \rightarrow Ga$ And what is the difference between the first premise and the below premise:     $$\forall x ((Fx \vee \neg Fx) \rightarrow Ga)$$   Thanks in advance.