Proving a tricky divisibility by 8
Why is $5^{n+1}+2\cdot 3^n+1$ a multiple of $8$ for every natural number $n$?
check the availability of the WCF Web Service I want to check the availability of the WCF web service i.c service is up or down through  the C# code. How to achieve that? The "Reputation &amp; Moderation" page on any site (/help/site-moderators) still refers to the company behind Stack Exchange as Stack Exchange and not Stack Overflow.     Moderators act as a liaison between the community and Stack Exchange the company.   Seeing as how the name change was recent, I'm not surprised this hasn't been caught yet, and I'm sure there are other instances of this around somewhere. Macros for code annotations Finding the index of an item in a list If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation behind this fact. Perhaps to do with commutators?    Proof: $a \cdot b = (b \cdot b)  \cdot (a \cdot b) \cdot (a \cdot a) = b \cdot (b \cdot a) \cdot (b\cdot a) \cdot a = b \cdot a$. 403 error after upgrading to apache2.4 I have a trained logistic regression model that I am applying to a testing data set. The dependent variable is binary (boolean). For each sample in the testing data set, I apply the logistic regression model to generates a % probability that the dependent variable will be true. Then I record whether the acutal value was true or false. I'm trying to calculate an $R^2$ or Adjusted $R^2$ figure as in a linear regression model.  This gives me a record for each sample in the testing set like:  prob_value_is_true         acutal_value    .34                          0    .45                          1    .11                          0    .84                          0     ....                        ....             I am wondering how to test the accuracy of the model. My first attempt was to use a contingency table and say "if prob_value_is_true > 0.80, guess that the actual value is true" and then measure the ratio of correct to incorrect classifications. But I don't like that, because it feels more like I'm just evaluating the 0.80 as a boundary, not the accuracy of the model as a whole and at all prob_value_is_true values.  Then I tried to just look at each prob_value_is_true discrete value, as an example, looking at all samples where prob_value_is_true=0.34 and measuring the % of those samples where the acutal value is true (in this case, perfect accuracy would be if the % of samples that was true = 34%). I might create a model accuracy score by summing the difference at each discrete value of prob_value_is_true. But sample sizes are a huge concern here, especially for the extremes (nearing 0% or 100%), such that the averages of the acutal values are not accurate, so using them to measure the model accuracy doesn't seem right.  I even tried creating huge ranges to ensure sufficient sample sizes (0-.25, .25-.50, .50-.75, .75-1.0), but how to measure "goodness" of that % of actual value stumps me. Say all samples where prob_value_is_true is between 0.25 and 0.50 have an average acutal_value of 0.45. Is that good since its in the range? Bad since its not near 37.5% (the center of the range)?  So I'm stuck at what seems like should be an easy question, and hoping someone can point me to a resource or method to calculate an accuracy stastic for a logistic regression model. C# 3.0 auto-properties — useful or not? Note: This was posted when I was starting out C#. With 2014 knowledge, I can truly say that auto-properties are among the best things that ever happened to the C# language.  I am used to create my properties in C# using a private and a public field:  private string title; public string Title {     get { return title;  }     set { title = value;  } }   Now, with  3.0, we got auto-properties:  public string Title { get; set; }   I know this is more a philosophical/subjective questions, but is there any reason to use these auto-properties except from saving five lines of code for each field? My personal gripe is that those properties are hiding stuff from me, and I am not a big fan of black magic.  In fact, the hidden private field does not even show up in the debugger, which is OK given the fact that the get/set functions do nothing. But when I want to actually implement some getter/setter logic, I have to use the private/public pair anyway.  I see the benefit that I save a lot of code (one vs six lines) without losing the ability to change the getter/setter logic later, but then again I can already do that by simply declaring a public field "Public string Title" without the need of the { get; set; } block, thus even saving more code.  So, what am I missing here? Why would anyone actually want to use auto-properties? I left my cast iron pot on the gas stove, on high flame, and then forgot about it for 3 hours. It was empty and I was drying it before I oiled it to store it. After I discovered it and let it cool, I oiled it with a paper towel and all this rusty stuff came off and continues to come off. Is my pan ruined? What should I do? Title does not appear when LaTeX code is compiled Permalinks: custom post type -> custom taxonomy -> post I am having trouble working with WordPress rewrite rules, and could use some help.  I have a custom post type called _shows_.  All shows have a single custom taxonomy category _show-category_. A _show_ will never have more than one _show-category_.  I would like my urls to route in this manner:  www.mysite.com/shows/  =&gt;  archive-shows.php  www.mysite.com/shows/%category%/ =&gt; taxonomy-show-category.php  www.mysite.com/shows/%category%/%postname%/ =&gt; single-shows.php   So as a real world example, lets say we have a _show-category_ "Foo" and a _show_ post titled "Bar" that has "Foo" as it's _show-category_. I would expect my WordPress app to look like this:  www.mysite.com/shows/foo/ =&gt; shows all posts under the foo category www.mysite.com/shows/foo/bar =&gt; shows the indivual post   I try to avoid plugins when possible, but am open to any solution. "Do you know where's Linda?" vs "Do you know where Linda is?" Making sense of principal component analysis, eigenvectors & eigenvalues Save site as template using client object model what's different between each and collect method in Ruby Let f : A → B and g : B → C be bijections. Prove that g◦f : A → C is a bijection  Can someone show me the steps I should take to solve this problem? Hide Command Window of .BAT file that Executes Another .EXE File how does the circuit of a basic Variable speed electric drill work? i am trying to 'automate' the control of an electric drill motor that im taking from a garden variety Variable Speed drill (from Harbor Freight)  it has a universal AC motor which I basically undertand and a DPDT switch to reverse direction and a triac citcuit to vary speed    Question 1 . i see 4 wires going to the 'field/Stator coil(s)'. Are there 2 seperate field/stator coils here? is this to provide low/high power.  I think in theory , you only need 1 coil with 2 wires and that coil is wired in series with the brushes... is that right?   how does this circuit work with 2 coils.  Question 2 . how does this variable speed circuit work.  is the wiring diagram correct (it doesnt show a variable resistor and it doesnt show what drives the control input of the traic? why are there two switches in the thing? does switch 2 close at high speed and thus shunt around the triac?  please see the attached 4 diagrams Add/Show comments focuses text box by default If I open a post using the mobile theme, which has many comments, there is a link at the footer "add/show n more comments". If this link is clicked, the comments are loaded and displayed.   At the moment the text area is automatically focused which is not a great experience (in my opinion), as you have to close the editor and scroll back to the top if you just wanted to read the comments, without adding a new one.  Would it be possible to either   not focus the text area by default and to load only the comments add a new link for adding a comment True Definition of the Real Numbers I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really?  I mean what is the definition of a real number?  If nothing else, anyone know of a resource where I could find out myself?  Thanks!