Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show that $f,g$ are homotopic.    I haven't been able to construct any rigorous proof so far, but I do feel it is geometrically easy. My intuition is, for any $x\in X$, we just need to connect $f(x)$ and $g(x)$ in the following way: suppose $f(x)=(a,h_1),a\in\Bbb R^n, h_1\in \Bbb R$ and $g(x)=(b,h_2)$ similarly, if $h_2\ne -1$, then for $b$ obviously there exists a unique "longitude" passing through it, and for $h_1$ there is also a unique "latitude" through it (of course, here we better interpret $S^n$ as "spanned" from $S^{n-1}$ with the additional $(n+1)$-th dimension being the "height"). Then denoting by $p$ the point where the lattitude and longitude meet, we can simply find a path $\gamma$ on $S^n$ which first connects $f(x)$ to $p$ latitudinally, then $p$ to $g(x)$ longitudinally. Moreover, to determine the explicit homotopy $H(x,t)$, we can split the time domain into $[0,1/2],[1/2,1]$, the first part to be spent on the latitudinal trip and the second the longitudinal one. This entire idea seems quite workable for me.   When I'm trying to formalise this idea into a rigorous proof, it becomes quite messy to deal with the case  $h_2=-1$, where we can't guarantee a unique longitude through $g(x)$. Of coure one can always find a longtitude through $g(x)$, but without uniqueness, it would be very troublesome to define a general rule to find one longtitude for each $g(x)$ in a consistent pattern, or in other words, if we allow loose choice of longitude for any special $g(x)$, the resulting "homotopy" might well fall apart due to discontinuity.   So my question is, how to neatly apply the "no antipodal pair" condition to get around this flaw and make the entire idea workable again?
At the moment I'm taking an algebraic topology course. In our current exercise we have to prove the following: Consider any topological space $X$ and any two maps $f, g : X \to S^n$, such that $f(x) \neq -g(x)$. It follows that $f$ is homotopic to $g$.  My idea is to use the mapping cylinders $M_f, M_g$ to construct a homotopy via $$M := M_f \amalg M_g/ (x, 0)_f \sim (x,0)_g$$ But this doesn't seem to work. In addition I don't know how $f(x) \neq -g(x)$ is going to help me (I can see how this could help if $n=1$, but I have no clue how this would work for $n&gt;1$).  I would appreciate any hints towards a solution.  Thanks in advance.
What is the purpose of .bashrc and how does it work? Is there a way to get the count of rows in all tables in a MySQL database without running a SELECT count() on each table? This fire image:    As you can see, it has a black background, which I can't delete (because it would probably destroy the smooth-looking fire effect). I want to place some fire in a photo.  However, I remember it was possible to take such images and do something in photoshop so that when you place them over a photo, the black is gone and the fire is perfectly blended there. But I can't remember what feature or option was it. Any ideas?  Photoshop CS5. Benchmarking hosting providers IO with Bonnie Is it always better to use the whole dataset to train the final model? A common technique after training, validating and testing the Machine Learning model of preference is to use the complete dataset, including the testing subset, to train a final model to deploy it on, e.g. a product.      My question is: Is it always for the best to do so? What if the   performance actually deteriorates?   For example, let us assume a case where the model scores around 65% in classifying the testing subset. This could mean that either the model is trained insufficiently OR that the testing subset consists of outliers. In the latter case, training the final model with them would decrease its performance and you find out only after deploying it.   Re-phrasing my initial question:     If you had a one-time demonstration of a model, such as deploying it   on embedded electronics on-board an expensive rocket experiment, would you trust a   model that has been re-trained with the test subset in the final step without being   re-tested on its new performance? Does Exchange support disposable e-mail addresses? Sometimes it is useful to have e-mail address like foo-randomnumber@example.org, where randomnumber identifies the conversation between a robot such as a mailing list server and a user. Is this possible with Microsoft Exchange? Is it possible to change your single-player skin? In Minecraft, is it possible to customize your skin in single-player mode (not just in multiplayer)? How? Old Budrys(?) end-of-world/alien invasion of NYC story This one, from the late 1950s or early 60s (I think), features the last girl in NYC, who drives her jeep up the stairs of the main library (with the lions). She feels bad about breaking the lock, but what's a girl to do, if she wants something to read? In summertime, she drives topless and, the author writes, her bosom dances enchantingly on the steps. She lives in the boathouse on the lake in Central Park. Mysterious insect-like aliens are eating the buildings in the City.... I didn't recognize a title in Budrys or Bester's isfdb bibliogs, so I may be wrong about the author. But not about the story details: it's a memorable one. Thanks in advance for helping ID this one! Why do return flight costs vary depending on the country of origin? Quickly calculate date differences *.xhtml url-pattern causes java.lang.NullPointerException at com.sun.faces.context.flash.ELFlash.loggingGetPhaseMapForReading Researching English/Christian families in India during first half of 19th century? How do I contact other users? I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in , it is mentioned that this is not true, and that there exist obstructions to this decomposition. Unfortunately, the level of complexity in the answers goes a bit over my head.  This type of decomposition is also addressed in .  With this loss of faith in my intuition, I have three questions:   Is it true that Lie groups do not necessarily decompose into the semidirect product of a connected group and a discrete group? If this decomposition doesn't always work, then is there a particularly nice argument as to why it doesn't? If this decomposition doesn't always work, then is there a nice example I can use to convince myself that my intuition is wrong?   Edit: As much as I love Qiaochu Yuan's answer, which is thorough and (as usual) a role model among Stack Exchange answers, this is not what I'm looking for. I'm aware of the general extension problem, and that not every short exact sequence splits, but my intuition has led me to think that each Lie group can be decomposed this way. I will certainly play around with his outline for a construction, but I would much prefer a more concrete answer. Once again, thank you!  Edit: As of yet, playing with the construction suggested by Qiaochu hasn't yielded anything interesting. This question is still open to answers! I often need to select a number of rows from each group in a result set.  For example, I might want to list the 'n' highest or lowest recent order values per customer.  In more complex cases, the number of rows to list might vary per group (defined by an attribute of the grouping/parent record). This part is definitely optional/for extra credit and not intended to dissuade people from answering.  What are the main options for solving these types of problems in SQL Server 2005 and later? What are the main advantages and disadvantages of each method?  AdventureWorks examples (for clarity, optional)   List the five most recent recent transaction dates and IDs from the TransactionHistory table, for each product that starts with a letter from M to R inclusive. Same again, but with n history lines per product, where n is five times the DaysToManufacture Product attribute. Same, for the special case where exactly one history line per product is required (the single most recent entry by TransactionDate, tie-break on TransactionID. I think everyone has known - we need to install katoolin to use Kali Linux Tools on Ubuntu. Before did anything I had looked through the internet and also found out these solutions: ,   So here are my steps:  apt-get install git git clone https://github.com/LionSec/katoolin.git &amp;&amp; cp katoolin/katoolin.py /usr/bin/katoolin chmod +x  /usr/bin/katoolin sudo katoolin   Then, to add Kali Linux repositories and update them, I selected option 1 from the katoolin Menu  Then, to view the available categories and install all available tools, I selected option 2 from the main menu of katoolin and then pressed 0.   But after that I got this:    What do you want to do ?>    1) Add Kali repositories & Update  2) View Categories 3) Install classicmenu indicator 4) Install Kali menu 5) Help   kat > 2  ************************** All Categories **************************  1) Information Gathering                      8) Exploitation Tools 2) Vulnerability Analysis                     9) Forensics Tools 3) Wireless Attacks                           10) Stress Testing 4) Web Applications                           11) Password Attacks 5) Sniffing & Spoofing                        12) Reverse Engineering 6) Maintaining Access                         13) Hardware Hacking 7) Reporting Tools                            14) Extra  0) All   Select a category or press (0) to install all Kali linux tools .  Furthermore, I got an error in Software Updater - it didn't let me do anything. Ubuntu just hung up. I solved this problem by deleting kali linux repositories in katoolin and then in Software Updater.  Does anyone know how to install Kali Linux tools in Ubuntu 16.04 in a proper way? Iterating over array of objects javascript - odd behaviour? 2D graphics - why use spritesheets? I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. So it's not clear to me whether my choice affects performance.  I suspect that there are some very good reasons most 2d game developers seem to be using them, I just don't understand why. Tried to install both Windows and Ubuntu, now neither will boot Hardware:  I've got an SSD (c),an internal HDD (d), an external HDD (e) and a USB flash drive (f).   How this all went down:   I downloaded Ubuntu and used a program to install it on my (F) drive. I reset my computer, ran the (F) drive for boot, and installed Ubuntu. I reset my computer, and it forcibly automatically booted into Windows 10 instead of Ubuntu. I went into Windows 10, and used a program called Easy BCD, by NeoSmart. I followed their guide, but after step 6, it wasn't the same for me. Their guide said that upon rebooting, I should see a grub screen that lets me choose between Windows/Linux and a few other options, however my computer would only boot into Windows 10.  I decided to follow the rest of the guide regardless, however by the end it would let me boot 'windows' or 'linux', but the linux option would always break. It went to a screen with lots of commands, that I thought -may- be a barebones Linux or something, but it definitely wasn't Ubuntu.  I decided to reinstall Ubuntu using the USB flash drive. Upon resetting, no matter what hard drive or format I select for booting (besides the F drive), it says An operating system wasn't found. Try disconnecting any drives that don't contain an operating system. I can boot the F drive and run Ubuntu, but it's a one time version that doesn't save any of my data. When I went into the installation, it still says that I already have Ubuntu installed on the 4TB HDD and that I have Windows 10 installed on the SSD, however I cannot seem to boot into either of them.   TLDR: Windows 10/Ubuntu are both installed, Windows on the internal SSD and Ubuntu on the external HDD. Ubuntu's installation program on the (F) drive recognizes them, however, when I try to boot anything it says An operating system wasn't found.   When I press F11 and force it to boot from the SSD it says:  error: no such device: 83b6c64d-9a2c0467b-a02c-6374ee6ac3dc.  error: unknown filesystem. Entering rescue mode. . .  grub rescue&gt; _   and it will allow me to type some commands. The following code should find the appropriate project tag and remove it from the XmlDocument, however when I test it, it says:  The node to be removed is not a child of this node.  Does anyone know the proper way to do this?  public void DeleteProject (string projectName) {     string ccConfigPath = ConfigurationManager.AppSettings["ConfigPath"];      XmlDocument configDoc = new XmlDocument();      configDoc.Load(ccConfigPath);      XmlNodeList projectNodes = configDoc.GetElementsByTagName("project");      for (int i = 0; i &lt; projectNodes.Count; i++)     {         if (projectNodes[i].Attributes["name"] != null)         {             if (projectName == projectNodes[i].Attributes["name"].InnerText)             {                                                                 configDoc.RemoveChild(projectNodes[i]);                 configDoc.Save(ccConfigPath);             }         }     } }   UPDATE  Fixed. I did two things:  XmlNode project = configDoc.SelectSingleNode("//project[@name='" + projectName + "']");   Replaced the For loop with an XPath query, which wasn't for fixing it, just because it was a better approach.  The actual fix was:  project.ParentNode.RemoveChild(project);   Thanks Pat and Chuck for this suggestion.