How to add confidence regions around a graph, from a table
pgfplots legend style for plots with error bands
I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power.  Here's the theorem (with proof) and two applications:    (Baire) A non-empty complete metric space $X$ is not a countable union of nowhere dense sets.  Proof: Let $X = \bigcup U_i$ where $\mathring{\overline{U_i}} = \varnothing$. We construct a Cauchy sequence as follows: Let $x_1$ be any point in $(\overline{U_1})^c$. We can find such a point because $(\overline{U_1})^c \subset X$ and $X$ contains at least one non-empty open set (if nothing else, itself) but $\mathring{\overline{U_1}} = \varnothing$ which is the same as saying that $\overline{U_1}$ does not contain any open sets hence the open set contained in $X$ is contained in $\overline{U_1}^c$. Hence we can pick $x_1$ and $\varepsilon_1 &gt; 0$ such that $B(x_1, \varepsilon_1) \subset (\overline{U_1})^c \subset U_1^c$.   Next we make a similar observation about $U_2$ so that we can find $x_2$ and $\varepsilon_2 &gt; 0$ such that $B(x_2, \varepsilon_2) \subset \overline{U_2}^c \cap B(x_1, \frac{\varepsilon_1}{2})$. We repeat this process to get a sequence of balls such that $B_{k+1} \subset B_k$ and a sequence $(x_k)$ that is Cauchy. By completeness of $X$, $\lim x_k =: x$ is in $X$. But $x$ is in $B_k$ for every $k$ hence not in any of the $U_i$ and hence not in $\bigcup U_i = X$. Contradiction. $\Box$    Here is one application (taken from ):  Claim: $[0,1]$ contains uncountably many elements.  Proof: Assume that it contains countably many. Then $[0,1] = \bigcup_{x \in (0,1)} \{x\}$ and since $\{x\}$ are nowhere dense sets, $X$ is a countable union of nowhere dense sets. But $[0,1]$ is complete, so we have a contradiction. Hence $X$ has to be uncountable.     And here is another one (taken from ):  Claim: The linear space of all polynomials in one variable is not a Banach space in any norm.  Proof: "The subspace of polynomials of degree $\leq n$ is closed in any norm because it is finite-dimensional. Hence the space of all polynomials can be written as countable union of closed nowhere dense sets. If there were a complete norm this would contradict the Baire Category Theorem." I'm currently an M. Phil student in Math and I failed a course last semester because I was not able to hand in a home work assignment (which was worth 40% of my grade) on time (I was 30 minutes late).   Apart from that I'd say I'm a fairly decent student, I got an upper second class honours bachelors degree, I have a B+ and A's for all my other courses in my current programme and I'm almost finished with my first paper to be published.   I've however been deeply worried about this failure as it would reflect poorly on my transcript if I were to apply to a Ph.D or another Master's programme.   Will my failing grade affect my chances of getting into a competitive programme or greatly lower my chances of getting a scholarship? Why are continuous functions the "right" morphisms between topological spaces? I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does this happen anymore now that we have computers? I imagine not. But it seems totally possible that this could have happened back in the Enlightenment.   Feel free to interpret this how you wish! What is the "En passant" rule in chess? How useful is that rule and how many times we can use that in one game?  Under what conditions can the en passant rule be used?  How often does it come up and is it a good idea to use it? Can a PC be a lich? The heading says it all, can a PC be a lich? If so, how? The DMG hints that the Book of Vile Darkness contains the secrets of how; there is nothing official otherwise. I just picked up a copy of Windows 7 Pro OEM for install on my computer.  Guy at the store said the only real difference between OEM and standard retail box was tech support.  However, the outside of the box has this big warning that says "The software must be installed with an OPK"?  What is the OPK?  Is it really necessary?  The MS site about it seemed to say it was optional.  EDIT:  BTW - I'm just an individual tinkerer, and not a System Builder Company Employee.  I should have made that clear.  I've used sysprep in a professional setting in the past, so I'm somewhat familiar with its concepts. Is there a way to capture the viewport in Blender the same way as it is on 3DS Max? I know that by default I can render the viewport and everithing that is in the viewport to be the same as in the render. But, when comparing 3DS Max to Blender, I have certain limitations. In Blender I can render the view-port with vertices/edges/faces to be shown, to be selected, but not the reflections, shadows and such.  Is there a way to render the viewport in Blender as it is in 3DS Max?  Here's how I can do it in 3DS Max:  Viewport look    Render result of the viewport    Here is how I can do it in Blender:  Viewport look:      Render result of the viewport Revision history suggests that I edited the post body, whereas I didn't I was doing some re-tagging and I noticed that the automatically generated revision comment says that I edited the post body where I only edited the tags. "is" vs "are" when followed by a number I read the following sentence on YouTube and initially thought it was grammatically incorrect:     Here are 10 minutes of the movie Black Hawk Down!   I thought "are" should have been "is" but then I realized minutes is plural so the plural form, are, is used correctly. I also thought how most people would naturally say "Here is ten dollars in change" but is the use of the singular form, is, correct since dollars is plural?   Am I misunderstanding a rule? Function $f(z)$ is holomorphic at least within the unit disk.   If for all arguments $|z|&lt;1$ we have $|f(z)| \le 1$, and there exists a zero of order $n$ at $z=z_0$, then show that $|f(0)| \le |z_0|^n$.  How should I tackle this problem? Can it be generalized if for $|z|&lt;R$ we have $|f(z)| \le M$ ?    So far I only understand that there exists a holomorphic $g(z)$ with $g(z_0) \ne 0$ such that $f(z) = (z-z_0)^n g(z)$,then $|f(z)| = |z-z_0|^n |g(z)| \le 1$ and moreover $|f(0)| = |z_0|^n |g(0)|$, but that doesn't seem useful since apparently I know too little about $g(z)$. In particular why should $|g(0)| \le 1$? Bought a new battery, do I need to calibrate it? My original battery is bloated and is dying quickly after a full charge so I now bought a replacement. Do I need to calibrate this new battery using a calibration app? How to extract Time from Date/Time field within formula Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$ I need to prove that $\sup \{-x \mid x \in A\} = -\inf\{x \mid x \in A\}$ and am having trouble moving the $-x$ out of the $\sup$ to $\inf$. Another thing is that I don't quite know how to prove $b = \inf$ (see below). Any help would be appreciated! Here my attempt:  Let $a = \sup\{-x \mid x \in A\}$  Then by definition $a \geq -x$ for $x \in A \implies -a\leq x$ for $x \in A$  Thus $-a$ is a lower bound of $A$. Assume $b$ is a lower bound of $A$  Then $b \leq -a \implies  a \geq -b$ and $a = \sup \{−x \mid x\in A\} = -\inf(A)$ Is a hexagonal house feasible? Recently I have been considering the economics and feasability of building a house versus buying a used one. In my design I settled on a hexagon as the logical shape for the frame, which I am considering for masonry construction. This would be a big house, each side being at least 30 feet wide (footprint 2340 sq ft). The advantages of a hexagon are obvious: simple roof shape, no valleys, space efficient, heat efficient (this would be in New England and subject to cold winters), good wind resistance.  What I can't understand is why there seems to be no-one building hexagonal houses? The only notable building I could find at all was the Supreme Court.  Possible explanations would be that the shape is expensive for small houses due to the 6 corners, but for a large house like mine the room shapes work out fine, in fact they are better than a box house. Maybe its just tradition? People don't like it because it looks like church? Builders have some objection to 60-degree angles?  What is the reason there are no hexagonal houses? so i have a 30A disconnect available for use at a customer site for running 2 5HP motors ( at 480V)  the only question i have though is if the line is sized for a 10HP motor would both the 5HP motors need additional fuse protection?   Or could i get away without having addtional fuses protecting the motors ? Landscape printing from HTML I have a HTML report, which needs to be printed landscape because of the many columns. It there a way to do this, without the user having to change the document settings?  And what are the options amongst browsers. How do I delete a Git branch locally and remotely? Can I take a sex-toy on plane through UAE? I will be traveling from Australia to Austria, transferring in Abu Dhabi. Is it alright for me to bring a vibrator in the check-in luggage, or are there laws against that?  It is approximately 17cm long. Are there any precautions to take? How to find the id of the last added feature using pyQGIS? Is it possible to programmatically (in python) access the attribute info of the last added feature in QGIS 2.0 before edits are saved? I'm failing to find a way to get the ID and subsequently other attributes of my feature of interest (the one most recently added).  idx = layer.fieldNameIndex('myField') myList = [] lastEdited = ID THAT I'D LIKE TO RETRIEVE for feature in layer.getFeatures():     if feature.id() == lastEdited:         print feature.attributes()[idx]   From this I'd like to get 'myField' of my most recent addition.   UPDATE: Following lead of this , I was able to achieve my goals via the Python console, but am failing to get it to work for my plugin.  def updateTableWidget(self, fid):     idx = layer.fieldNameIndex('myField')     myList = []     for feature in layer.getFeatures():         if feature.id() == fid:             myList.append(feature.attributes()[idx])   Via the python console, I was able to connect to my layer like so:  layer.featureAdded.connect(updateTableWidget)   After adding a QgsMessageLog to my function, the attributes were correctly identified. I now have a beginner python issue, however. Calling the function in my plugin as shown below gives me TypeError: updateTableWidget() takes exactly 2 arguments (1 given):  self.updateTableWidget()   The function was called from feature form python file.