What fallacy is there??
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
\bar vs. \overline - when to use what, semantically? Following the question "" which is simply about the lengths of both, I'd like to know when to use \bar and when to use \overline from a semantically viewpoint.  I always used \bar, because up until now I only had single letter mean values to type. But now I came to a problem where I needed a wider "bar".  Then I just tried  \documentclass{article}  \begin{document}   \[ \bar{a} \overline{a} \]   \[ \bar{abc} \overline{abc} \] \end{document}   where one can see that they also differ in position (\overline is lower), and also in thickness (\overline is thicker).  So, the question here is:  What does \bar represent semantically? And when to choose it over \overline and vice versa? Spivak Chapter 2, problems 27 (and 28) To be honest, I have no idea how to even start this problem. I'm sorry I don't have any work to show, but I'm just at a blank. Help?  Chapter 2: Problem 27:  University B, once boasted $17$ tenured professors of mathematics. Tradition prescribed that at their weekly luncheon meeting, faithfully attended by all $17$, any members who had discovered an error in their published work should make an announcement of this fact, and promptly resign. Such an announcement had never actually been made, because no professor was aware of any errors in her or his work. This is not to say that no errors existed, however. In fact, over the years, in the work of every member of the department at least one error had been found, by some other member of the department. This error had been mentioned to all other members of the department, but the actual author of the error had been kept ignorant of the fact, to forestall any resignations. One fateful year, the department was augmented by a visitor from another university, one Prof. X, who had come with hopes of being offered a permanent position at the end of the academic year. Naturally, he was apprised, by various members of the department, of the published errors which had been discovered. When the hoped-for appointment failed to materialize, Prof. X obtained his revenge at the last luncheon of the year. &quot;I have enjoyed my visit here very much&quot;, he said, &quot;but I feel that there is one thing that I have to tell you. At least one of you has published an incorrect result, which has been discovered by others in the department.&quot; What happened the next year?&quot;   Chapter 2: Problem 28:  After figuring out, or looking up, the answer to Problem 27, consider the following: Each member of the department already knew that Prof.X asserted, so how could his saying it change anything? Could not list layers for this store error when creating new ImageMosaic Parser for C# Which parsers are available for parsing C# code?  I'm looking for a C# parser that can be used in C# and give me access to line and file informations about each artefact of the analysed code. I try to search for lines that start with "1" using   ls -1 | grep ^1*   but it returns lines that do not start with 1. What I am missing here? I'm wondering about the period character . used as a shortcut for the local computer name.  You can use it when logging into Windows 7, for example, to specify that the user account you're entering belongs to the local computer rather than to a domain:  .\MyLocalUserAccount   And you can use it within a Windows Management Instrumentation (WMI) query to indicate that the target is the local computer rather than a remote computer.  You can also use it when working with named pipes to for similar purposes.  However, you cannot use it as the UNC network name for the computer, like this:  \\.\MyShare   Does this character have an official name (e.g. "LocalHostToken") and when and where can it be used for this purpose? It is currently said that MD5 is partially unsafe. Taking this into consideration, I'd like to know which mechanism to use for password protection.  This question,   suggests that hashing multiple times may be a good idea, whereas  suggests using salt.  I'm using PHP. I want a safe and fast password encryption system. Hashing a password a million times may be safer, but also slower. How to achieve a good balance between speed and safety? Also, I'd prefer the result to have a constant number of characters.   The hashing mechanism must be available in PHP It must be safe It can use salt (in this case, are all salts equally good? Is there any way to generate good salts?)   Also, should I store two fields in the database (one using MD5 and another one using SHA, for example)? Would it make it safer or unsafer?  In case I wasn't clear enough, I want to know which hashing function(s) to use and how to pick a good salt in order to have a safe and fast password protection mechanism.  Related questions that don't quite cover my question: Why do polarized sunglasses remove (most) reflections on car windows?  I know that both lenses of polarized sunglasses are polarized in the same direction and I can see that reflections in my own windshield and the windows of other cars appear, disappear, or change depending on how I tilt my head, but why?  Why is the light in those reflections polarized?  edit: And why is most of the light in the various reflections on the interior of my own windshield and various windows/windshields at what seem like a variety of angles all polarized in a sufficiently narrow set of ways as to make polarized sunglasses work? For example, this regex  (.*)&lt;FooBar&gt;   will match:  abcde&lt;FooBar&gt;   But how do I get it to match across multiple lines?  abcde fghij&lt;FooBar&gt; Why is spectrum obtained by sunlight, said to be continuous? My teacher spoke about atomic spectra today, and he explained that, unlike the spectrum obtained by analyzing the sunlight, the spectra of atoms are not continuous.     I have a question about this - even the sunlight has the radiations emitted by the atoms of the elements which compose the sun, still spectra is continuous, which is in opposition to the statement that atomic spectra is discontinuous. So, spectrum obtained by sunlight is continuous even though it atomic spectra. In order to account for spectrum obtained by the sunlight to be continuous and atomic spectra to be discontinuous, can we confirm that sun consists of all those elements (sodium, helium, neon, mercury, etc) which emit the colors of frequency belonging to visible region? Quick question to anyone who has used this LaTeX template:    Any ideas on how to change the left margin to get more words onto one page? Google map API does not plot on different continents How do I prove that a graph is a topological manifold I'm having problems with that:  Prove that M is a topological manifold.  $f:U \to \mathbb R^k$, U $\subset \mathbb R^{n}$ open, f continuous  M = $\{\left(x,y\right)\in \mathbb R^{n+k} \mid x \in U, y=f(x) \}$  Can anyone help me please? How can I get the list of available databases on a SQL Server instance? I'm planning to make a list of them in a combo box in VB.NET. Totally sweet horizontal rules in LaTeX What is the name of the problem where lines appear in a photo where there should just be a smooth gradient? Is it better in C++ to pass by value or pass by constant reference? Is it better in C++ to pass by value or pass by constant reference?  I am wondering which is better practice.  I realize that pass by constant reference should provide for better performance in the program because you are not making a copy of the variable. Use the new helpcenter for the question/answer ban message Why doesn't Normal[] work on GeometricTransformation? Sort order issue I'm having an issue sorting data in ascending order.   Right now the sort order is: 1.0.0  10.0.0 10.0.1 11.0.0 2.0.0 2.0.1 3.0.0 3.0.1 3.1.0  What I want is: 1.0.0 2.0.0 2.0.1 3.0.0 3.0.1 3.1.0 10.0.0 10.0.1 11.0.0  How can I achieve this? The datatype can't be number or currency because of the 2 decimals, so I used text-field.