Binary Search Tree Induction
Proving that a Binary Tree of $n$ nodes has a height of at least $\log(n)$.
What should I do when Ubuntu freezes? All operating systems freeze sometimes, and Ubuntu is no exception. What should I do to regain control when...   just one program stops responding? nothing at all responds to mouse clicks or key presses? the mouse stops moving entirely? I have an    In what order should I try various solutions before deciding to pull the power plug?  What should I do when starting up Ubuntu fails?  Is there a diagnostic procedure I can follow? I have downloaded the offline (consumer, i.e. not msi package) installer for Google Chrome. But when I run it, it tries to contact a Google server to check for a newer version. If it finds it, it will download that first, and then install that version. It doesn't use the version that is already there, as part of the EXE file. For me, that defeats the point of having an offline installer. I keep a repository of all the free and purchased software locally, so that I can install them quickly whenever I need to. But as more and more people get access to broadband Internet connections, many developers and companies are pushing for offline installers with new releases. Many software titles, like Google Chrome, are almost exclusively distributed with online installers.  File name: ChromeStandaloneSetup.exe File version: 1.3.21.115 Size: 36.6 MB (38 416 368 bytes) SHA-1: 2393BAB7F408DC8D30F039FCA5D6B45600A0C4E7   How do I install Chrome as is, without automatically checking for and getting a newer version?  Will  work with Chrome installer? Is there a switch to disable version check when running the installer?  Update:  The "duplicate" question linked to below points out File Hippo as a possible repository where you one can get previous versions of Chrome. They do have a healthy listing of pretty much every single Chrome version ever released on Windows, both beta, dev and stable versions. Unfortunately they don't host the installer files, they only redirect to Google servers for downloads, which only spits out the installers for the latest versions of Chrome. It's not just File Hippo, I also tried downloading an older version of Chrome from CNET Downloads, and they too just redirect you to Google servers for download.  For whatever the reason (security related probably), Google doesn't like anyone to install an older version of Chrome on any system. Apparently not even your common source for downloading an older version of a software, like the CNET Downloads, not even they are allowed to host the Chrome installer files on their own servers. The redirects to Google is probably dictated by Google.     Google is a dictator!   They don't even allow full standalone offline installers to be installed as is, without checking for newer version. I tried running the installer from cmd with some of the command line switches used for Chromium. I only ended up with errors.    The business oriented msi installer (GoogleChromeStandaloneEnterprise.msi) might have some additional command line options you can use, but I wouldn't know if there is one that disables the check for a newer version. I have yet to find a complete list of switches you can use. The only one I've found so far is /update but that only does the opposite thing. So I'll give up here and go back to the trusty Firefox. TZInfo::DataSourceNotFound error starting Rails v4.1.0 server on Windows Jquery, changing background image with timer There are two schools of thought on how to best extend, enhance, and reuse code in an object-oriented system:   Inheritance: extend the functionality of a class by creating a subclass. Override superclass members in the subclasses to provide new functionality. Make methods abstract/virtual to force subclasses to "fill-in-the-blanks" when the superclass wants a particular interface but is agnostic about its implementation. Aggregation: create new functionality by taking other classes and combining them into a new class. Attach an common interface to this new class for interoperability with other code.   What are the benefits, costs, and consequences of each? Are there other alternatives?  I see this debate come up on a regular basis, but I don't think it's been asked on  Stack Overflow yet (though there is some related discussion). There's also a surprising lack of good Google results for it. Why do networks need a common ground cable? Using subcaption package but subfigures end up stacked rather than side by side Under Wild Shape it says "assume the shape of a beast" you have seen before. Does this mean they can only transform into creatures that have the beast type, such as bear, wolf, crocodile, etc.? Or can they turn into things with other types, like dragon, monstrosity, humanoid, fiend, etc.? Organic yet accurate modeling with the golden spiral Foreword  First off, let me say that I am profoundly amazed at the orderliness in nature. If you haven't heard about the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, ...), you should check it out.  This mathematical concept of the Fibonacci numbers can be shaped into a set of squares, where the lengths of the sides of the squares follow the Fibonacci sequence:     This is sometimes called a golden ratio. It can be formed into a (golden) spiral. (a.k.a. logarithmic spiral):      Application  I am looking for ways on how to utilize this concept in Blender. Primarily in organic modeling. The golden spiral is found all throughout nature. From flower seed heads, sea shells, and ram horns to hurricanes and spiral galaxies.  My question is: how can I model based off of one or multiple golden spirals? Specifically, I'm interested in modeling a sunflower head like the one in the following image. I'm mostly interested in the point of origin of the individual seeds (not worried about the shape). Does Blender have any features related to this? Scenario: I have a custom button on a standard layout, which will call an apex class to do some validation on a record. If not validated, it needs to throw out some errors on that page. What are the best UX options to achieve this?  I am not too sure how to achieve this. Within a trigger, you can use the addError method to add the custom errors.  Can my Apex class reference the fields of a standard object? The ideal approach would be to do the following:  Contact.Email.addError('Please enter the right format here');   How can I achieve this via an apex class that the custom button will call?  What I have thought: Having a VF Page popup on the screen so that user can see all the errors, but that's not the best approach as UX is a big deal for this application  Thanks. Is it worth buying a 5D Mark I or an EOS 1Ds II today? What can cause a question to be bumped? What can cause a question to be "bumped" to the first page in the Active tab of a Stack Exchange site? (Changing the "Last Active" date.)  . I need to externally limit a process/session to a certain number of cores.  Are there any other possibilities than CPU affinity (I don't like the need to specify the actual cores) and cgroups (hard to integrate into our project)? Consider the  function     $$ r(x) = \mathbb{E}(Y \mid X = x) $$   This has been called the regression function in a textbook I'm using. I'm trying to figure out the relationship between this function and the classical linear regression model.  So, I know that it is a theorem* that we may write  $$ Y = r(X) + \epsilon $$  for some random variable $\epsilon$ s.t. $\mathbb{E}(\epsilon) = 0$.  Now suppose that we have     $$ Y = \beta_0 + \beta_1 X + \epsilon $$   This is the classical 1-dimensional regression function (assuming the $\beta_0$ and $\beta_1$ minimize the residual sum of squares).  Question: Is it then a mathematical theorem that if $Y$ is defined as above, that   $$ r(X) = \mathbb{E}(Y \mid X) = (\beta_0 + \beta_1 X)? $$  And is this why the function $\mathbb{E}(Y \mid X)$ is called the "regression function"?  EDIT: The theorem that I am making use of is as follows (from All of Statistics pg. 89):     Regression models are sometimes written as      $$ Y = r(X) + \epsilon $$      where $\mathbb{E}(\epsilon) = 0$. We can always rewrite a regression model this way. To see this, define $\epsilon = Y - r(X)$ and hence $Y = Y + r(X) - r(X) = r(X) + \epsilon$. Moreover, $\mathbb{E}(\epsilon) = \mathbb{E}\mathbb{E}(\epsilon \mid X) = \mathbb{E}(\mathbb{E}(Y - r(X)) \mid X) = \mathbb{E}(\mathbb{E} ( Y \mid X) - r(X)) = \mathbb{E}(r(X) - r(X)) = 0$. Pixelated image in ps format! What are the differences between a pointer variable and a reference variable in C++? What is this yellow and white mass inside my red cabbage? In iPhone, you can easily change the default ringtone (Settings → Sounds → Ringtone) and set a custom ringtone for individual people (Contacts → [some person] → Edit → Ringtone).   But can you set a custom tone for a group of contacts? (E.g. "family" or "workmates".)  I think that would be handy—and quite basic functionality, so to speak—but I couldn't find a way to do this through the iPhone UI. Is it possible at all (without jailbreaking)? Minecraft Vanilla 1.9 crash with problematic frame ig8icd64.dll I go to the Minecraft menu and click on singleplayer to test out the 1.9 update. I made a new world (no mods), it said downloading terrain and then it crashes every time (I've tried about 20 times now).  Has anyone else experienced similar issues and know what the fix may be?  Here is the launcher log:  Completely ignored arguments: [--nativeLauncherVersion, 301] [00:35:12] [Client thread/INFO]: Setting user: Teddy_Cromwell [00:35:12] [Client thread/INFO]: (Session ID is &lt;censored&gt;) [00:35:14] [Client thread/INFO]: LWJGL Version: 2.9.4 [00:35:14] [Client thread/WARN]: Removed selected resource pack resources (no region and battle music).zip because it's no longer compatible [00:35:14] [Client thread/INFO]: Reloading ResourceManager: Default [00:35:15] [Sound Library Loader/INFO]: Starting up SoundSystem... [00:35:16] [Thread-5/INFO]: Initializing LWJGL OpenAL [00:35:16] [Thread-5/INFO]: (The LWJGL binding of OpenAL.  For more information, see http://www.lwjgl.org) [00:35:16] [Thread-5/INFO]: OpenAL initialized. [00:35:16] [Sound Library Loader/INFO]: Sound engine started [00:35:18] [Client thread/INFO]: Created: 1024x512 textures-atlas [00:35:26] [Server thread/INFO]: Starting integrated minecraft server version 1.9 [00:35:26] [Server thread/INFO]: Generating keypair [00:35:27] [Server thread/INFO]: Preparing start region for level 0 [00:35:28] [Server thread/INFO]: Preparing spawn area: 5% [00:35:29] [Server thread/INFO]: Preparing spawn area: 8% [00:35:30] [Server thread/INFO]: Preparing spawn area: 15% [00:35:31] [Server thread/INFO]: Preparing spawn area: 21% [00:35:32] [Server thread/INFO]: Preparing spawn area: 27% [00:35:33] [Server thread/INFO]: Preparing spawn area: 34% [00:35:34] [Server thread/INFO]: Preparing spawn area: 42% [00:35:35] [Server thread/INFO]: Preparing spawn area: 48% [00:35:36] [Server thread/INFO]: Preparing spawn area: 56% [00:35:37] [Server thread/INFO]: Preparing spawn area: 66% [00:35:38] [Server thread/INFO]: Preparing spawn area: 74% [00:35:39] [Server thread/INFO]: Preparing spawn area: 81% [00:35:40] [Server thread/INFO]: Preparing spawn area: 91% [00:35:41] [Server thread/INFO]: Preparing spawn area: 98% [00:35:42] [Server thread/INFO]: Teddy_Cromwell[local:E:6d7525a7] logged in with entity id 301 at (244.5, 69.0, 244.5) [00:35:42] [Server thread/INFO]: Teddy_Cromwell joined the game # # A fatal error has been detected by the Java Runtime Environment: # #  EXCEPTION_ACCESS_VIOLATION (0xc0000005) at pc=0x00007fffcf6e6b37, pid=15456, tid=12424 # # JRE version: Java(TM) SE Runtime Environment (8.0_25-b18) (build 1.8.0_25-b18) # Java VM: Java HotSpot(TM) 64-Bit Server VM (25.25-b02 mixed mode windows-amd64 compressed oops) # Problematic frame: # C  [ig8icd64.dll+0x16b37] # # Failed to write core dump. Minidumps are not enabled by default on client versions of Windows # # An error report file with more information is saved as: # C:\Users\Connor\AppData\Roaming\.minecraft\hs_err_pid15456.log # # If you would like to submit a bug report, please visit: #   http://bugreport.sun.com/bugreport/crash.jsp # The crash happened outside the Java Virtual Machine in native code. # See problematic frame for where to report the bug. # AL lib: (EE) alc_cleanup: 1 device not closed Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and will likely be removed in a future release Powering a WS2812B strip