Proof that $n^3-n$ is a multiple of $3$.
Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
Java toString() using reflection? Structure of groups of order $pq$, where $p,q$ are distinct primes. I don't know about the Sylow Theorems.  But I have been wondering about a proof of the fact that a group or order $pq$ where $p$ and $q$ are distinct primes must be cyclic. I can't quite work out the details, but here is the general idea. I would like help with filling in details. I assume that it is already known that $G$ has subgroup(s) of order $p$ and subgroup(s) of order $q$.  If $G$ is a group of order $pq$ ($p\neq q$), then I know that $G$ has a subgroup $H$ of order $p$ and a subgroup $K$ of order $q$. Then $H\simeq \mathbb{Z}_p$ and $K\simeq \mathbb{Z}_q$.   But then $H\oplus K \simeq \mathbb{Z}_{pq}$, so I would think that $H\oplus K \simeq G$. I guess one could do an internal direct product instead of an external direct product, but I don't know that $H$ and $K$ are normal subgroups.  I am asking for help completing this argument.  Edit: I see from the comments below that I might need to assume that the smaller prime does not divide the larger prime minus $1$. Or maybe it is enough to assume that the primes are greater than or equal to $3$ (Still distinct). I have an Asus Sabertooth Z170 motherboard, with a Samsung 950 Pro m.2 NVMe drive on it. I've heard in the past that installing Windows 7 on an NVMe drive was a significant amount of work. Its now August 2016, is it still difficult? I've seen updates from Microsoft saying they've added support for NVMe in Windows 7.  Can I just download Windows 7 from Microsoft () and put it on a USB drive and install?  Is it harder than that? $\delta$ differential notation Various textbooks that I am currently consulting (including Spacecraft Dynamics and Control An Introduction - Anton H.J. De Ruiter | Christopher J. Damaren | James R. Forbes Section 1.4, page 32) use $\delta$, not $d$ or $\partial$ to express an infinitesimal quantity. In the context of the reference text above, the symbol is used as part of a general definition for the derivative of a vector. Specifically, given   $$\mathbf{\vec{r}}=\mathcal{\vec{F^{T}_{1}}}\mathbf{\vec{r_{1}}}$$   where   $$\mathcal{\vec{F^{T}_{1}}}$$   is a vectrix   $$[\mathbf{\vec{x_1}} \phantom{s} \mathbf{\vec{y_1}} \phantom{s} \mathbf{\vec{z_1}}].$$  The time derivative of the vector is defined as   $$\dot{\mathbf{\vec{r}}}\triangleq \lim_{\delta t\to 0}\frac{\delta\mathbf{\vec{r}}}{\delta t}.$$  In this context, what is the difference between $\frac{\delta\mathbf{\vec{r}}}{\delta t}$ and $\frac{d \mathbf{\vec{r}}}{d t}$ ?  Note about thermodynamic use of $\delta$: My understanding is that $\delta$ is used in thermodynamic equations to express path dependence of a scalar quantity such as heat or work. What does it mean in the context of an abstract physical vector? Regular Expression Uppercase Replacement in C# I have the following C# which simply replaces parts of the input string that look like EQUIP:19d005 into URLs, like this:  input = Regex.Replace(input, @"(EQUIP:)(\S+)", @"&lt;a title=""View equipment item $2"" href=""/EquipmentDisplay.asp?eqnum=$2""&gt;$1$2&lt;/a&gt;", RegexOptions.IgnoreCase);   The HTML ends up looking like this.  &lt;a title="View equipment item 19d005" href="/EquipmentDisplay.asp?eqnum=19d005"&gt;EQUIP:19d005&lt;/a&gt;   The only trouble is that the destination page expects the eqnum querystring to be all UPPERCASE so it returns the correct equipment when eqnum=19D005 but fails if it receives eqnum=19d005.  I guess I can modify and correct EquipmentDisplay.asp's errant requirement of uppercase values however, if possible I'd like to make the C# code comply with the existing classic ASP page by uppercasing the $2 in the Regex.Replace statement above.  Ideally, I'd like the HTML returned to look like this:  &lt;a title="View equipment item 19d005" href="/EquipmentDisplay.asp?eqnum=19D005"&gt;EQUIP:19d005&lt;/a&gt;   Notice although the original string was EQUIP:19d005 (lowercase), only the eqnum= value is uppercased.  Can it be done and if so, what's the tidiest way to do it? I want to generate some formatted output of data retrieved from an MS-Access database and stored in a DataTable object/variable, myDataTable. However, some of the fields in myDataTable cotain dbNull data. So, the following VB.net code snippet will give errors if the value of any of the fields lastname, intials, or sID is dbNull.     dim myDataTable as DataTable    dim tmpStr as String    dim sID as Integer = 1     ...    myDataTable = myTableAdapter.GetData() ' Reads the data from MS-Access table    ...     For Each myItem As DataRow In myDataTable.Rows      tmpStr = nameItem("lastname") + " " + nameItem("initials")      If myItem("sID")=sID Then         ' Do something     End If      ' print tmpStr     Next   So, how do i get the above code to work when the fields may contain dbNull without having to check each time if the data is dbNull as in ? Using a web service is often an excellent architectural approach. And, with the advent of WCF in .Net, it's getting even better.  But, in my experience, some people seem to think that web services should always be used in the data access layer for calls to the database. I don't think that web services are the universal solution.  I am thinking of smaller intranet applications with a few dozen users. The web app and its web service are deployed to one web server, not a web farm. There isn't going to be another web app in the future that can use this particular web service. It seems to me that the cost of calling the web service unnecessarily increases the burden on the web server. There is a performance hit to inter-process calls. Maintaining and debugging the code for the web app and the web service is more complicated. So is deployment. I just don't see the advantages of using a web service here.  One could test this by creating two versions of the web app, with and without the web service, and do stress testing, but I haven't done it.  Do you have an opinion on using web services for small-scale web app's? Any other occasions when web services are not a good architectural choice? When I go to , my tablet is listed, but not my phone. I had signed up just as soon as ADM was available, and at that time both devices were listed. Now, for the last several days, I can only see my tablet. (For what it's worth, both devices appear in the "My Devices" section of the Google Play Store.)  I have done the following while troubleshooting:   Ensured that the Android Device Manager settings in Google Settings are both on. I have even toggled them off and back on again Ensured that Location Services are on Ensured that all sync services for my account are toggled on and work Have connected to the Play Store several different times Ensured Google Play Services is at the latest version Made sure I'm using the right account. (In fact, I removed the other secondary accounts from the phone) Power-cycled the phone several different times   I could understand if ADM couldn't find the phone due to it being off, out of range, poor signal, etc., but ADM would at least display the phone and tell me it couldn't find it.  So, any suggestions to get my phone to show up on Android Device Manager?  I would like this to be working before something happens to the phone. ( and all that.)  Although it shouldn't make any difference:   Tablet: Nexus 7 (2012) with Android 4.3 Phone: Verizon Galaxy Nexus with Android 4.2.2   Addendum:  There's apparently some (relatively new) advice on the :   Open  Google Settings from your device's apps menu. Touch Android Device Manager. Uncheck Allow remote factory reset. Go to your device's main Settings menu, then touch Apps > All > Google Play services. Touch Clear Data. Note that this action doesn't remove personal data. Go back to Google Settings and select Allow remote factory reset. Restart your device.   I have followed these steps, but it doesn't appear to have worked. How to understand a volatility based ETF like VXX The ETF VXX is based on the volatility index VIX. If volatility goes up, VIX goes up and VXX goes up. Similar thing for the downside. Now when I buy an ETF like SPY, I can conceptually understand that I am buying a small portion of each of the underlying companies. So if an underlying company stock makes a move, the ETF SPY makes a move in  the same direction.  But when I buy VXX, what exactly is the underlying? It is VIX. I am unable to grasp the concept of 'volatility' as the underlying. I can visualise only stocks and commodities as the underlying because in my understanding, the unerlying needs to be something physical (let us say stocks are physical commodities for this example).  Can someone explain the concept of 'volatility' as the underlying?  Is the word 'underlying' even appropriate in this case or is the word 'underlying' only to be used for derivative products? What exactly is a hat?  I have seen a lot of people arguing the significance of hats and the limited edition hats. I know the hats are sort of linked to the Winterbash stuff, but what exactly is a hat, what does it do, how do I get one?  Is there a full list of hats? I'm using terminal-ide  which is basically a terminal for android with all sorts of utilities. One is able to create bash scripts for example. Is there a way to create an icon on the home screen that will launch one of these scripts? "(" or "\left(" parentheses? What is the difference between ( and \left( in LaTeX? Sometimes, when the content is small, it does not seem to matter which pair I use.  What is the best practice when it comes to which parentheses use? I made a series of outdoor shots with Nikon D5000, with the 18-55mm 1:3.5-5.6G VR kit lens.  On some shots, I see in the EXIF info f/11, and in another shot, f/9.    How can that be, that the f-number is greater than 5.6, when that's listed as the maximum in the lens spec? I want to know if this is grammatically correct " How long would the conference be for?". My PC has 8 Gb of ram. Is there any way to make ubuntu use most of it? I mean rarely drop caches and keep once opened programms in ram longer, preload apps on boot, etc. What's the 'correct' way of registering/installing an Assembly to the GAC? Is fractional scaling ( ) available in 18.04 LTS yet? I am waiting until this is available before I install. What is the probability that $x_1+x_2+...+x_n \le n$? I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$ Setting up LVM Snapshot as a backup/restore point in ubuntu Setting up LVM Snapshot as a backup/restore point in ubuntu. Why? Well, as I see it, with all the great usefulness of Linux, its biggest drawback is with its mixed up way of installing apps. All the dlls go to one directory. All the configuration files to another. And the binaries are also put in a bunch. Yes, Windows works the same way more or less, but with the nature of system/server software installed on Linux, being that most of the software installed is only about 95% mature at any given time, messing up your system twice a month is very probable.. Of course, we can backup the system before and after each installation, but that takes a million years. System backups (as oppose to just data/files backup) requires you to turn off you computer and use some form of bootable live CD and a backup media. And, if the drive is more than 15% full, it might take a long time to do the backup, even with modern hardware. So, how can this be done?