Does the word "therefore" always require a comma if it is in the middle of the sentence?
Should there always be a comma after "therefore","However" etc.?
Let $R$ be a commutative ring with $1$. Suppose that all ideals $I \neq R$ are prime. Prove that $R$ is a field.   Help me some hints.  Thanks a lot. Could anyone give me an explanation of the meaning of "get one's finger in the air" and the usage of it? How do I run Command & Conquer Gold on Windows 7 64-bit (in XP mode)? I downloaded the first Command &amp; Conquer game  and burnt some CDs. I couldn't get the Setup.exe program to run in Windows 7 64-bit, as it must need some old DLLs. So I ran up the Windows XP Mode virtual machine, and was able to install it fine.   However, when I try to run the program, I just get an empty message box with an "OK" button. I've set the compatibility to Windows 95, and checked all of the options in the Compatibility tab, but it's not running. I also downloaded the patch suggested by EA, which is an updated version of the THIPX32.DLL file (copied into the C&amp;C installation folder).  Has anyone managed to get this working? How to vertically align underbraces I am trying to balance underbraces in the equation below, how can I ensure that both underbraces at the same depth?  \begin{equation} y = \underbrace{ \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right]}_\text{growth rate} \times  \underbrace{ C_0}_\text{initial} \end{equation} How to understand the LaTeX warning “there were undefined references”? I dropped my camera while on holiday last summer. Thankfully, my camera wasn't damaged but the kit lens has lost the ability to autofocus and the UV filter and 58mm to 52mm filter step-down (I use it to attach a fisheye filter to the lens) I had on its end are stuck and won't come off.  I've replaced the lens with an 18-200 zoom lens so I don't especially need it. I was wondering if it was worth getting the lens looked at and possibly repaired?  I've heard many places will charge you £50 or so just to look at the lens and given the cost of the kit lens I'm not sure it's worth it. When reviewing flags, either from the queue or from the flagged post, I often want to know what the flagger could have done in addition to, or instead of, flagging for mod attention.  Does he have the rep to downvote?  To comment?  To vote to delete?  Knowing this doesn't necessarily affect how I handle the flag, but if there's a possible teaching moment there I use the flag response to point it out.  But I can't find this out unless I click through to the user's profile (unless I already know this user, of course).  Would it be possible to get the flagger's reputation in a tooltip, like what's already done for comments? Google search results showing up a URL on its search which doesn't exist on the website I have a Lenovo T570 with Xubuntu 18.04 LTS. My current issue is that my cpu speed stepping is kind of going nuts and does what ever it thinks its best.  Sample: when I run my IDE and cpu load increases during build time, the governor changes automatically. I caught the moment during compiling my project with the following call:  martin@martin-ThinkPad-T570:~$ while cat /sys/devices/system/cpu/cpu0/cpufreq/scaling_governor; do sleep 1; done powersave powersave performance performance   cpufreq-info tells me this:  martin@martin-ThinkPad-T570:~$ cpufreq-info  cpufrequtils 008: cpufreq-info (C) Dominik Brodowski 2004-2009 Report errors and bugs to cpufreq@vger.kernel.org, please. analyzing CPU 0:   driver: intel_pstate   CPUs which run at the same hardware frequency: 0   CPUs which need to have their frequency coordinated by software: 0   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.25 GHz. analyzing CPU 1:   driver: intel_pstate   CPUs which run at the same hardware frequency: 1   CPUs which need to have their frequency coordinated by software: 1   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.36 GHz. analyzing CPU 2:   driver: intel_pstate   CPUs which run at the same hardware frequency: 2   CPUs which need to have their frequency coordinated by software: 2   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.26 GHz. analyzing CPU 3:   driver: intel_pstate   CPUs which run at the same hardware frequency: 3   CPUs which need to have their frequency coordinated by software: 3   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.36 GHz.   Please take a look at     current policy: frequency should be within 3.50 GHz and 3.50 GHz.       The governor "performance" may decide which speed to use   Afterwards to go back to powersave I use my little script:  martin@martin-ThinkPad-T570:~$ cat cpu_freq_save.sh  #!/bin/bash sudo cpufreq-set -g powersave -c 0 --min 400MHz sudo cpufreq-set -g powersave -c 1 --min 400MHz sudo cpufreq-set -g powersave -c 2 --min 400MHz sudo cpufreq-set -g powersave -c 3 --min 400MHz   Which resets the min frequency and the governor but as soon as I compile again with my IDE every time the governor and the min frequency changes again.  Any idea how I can disable that automatic change?  PS: If you need more details, just tell me where to find them and I will provide them Prove that for all sets A and B, A ∩ B = ∅ implies ( A ∪ B ) - B = A In the next proof we avail ourselves of the next lemma: For all sets A and B, ( A ∪ B ) - B = A. Proof: Let A and B be arbitrary sets and let x ∈ ( A ∪ B ) - B. x ∈ ( A ∪ B ) - B ⇔ x ∈ ( A ∪ B ) ∧ x ∉ B ⇔ ( x ∈ A ∨ x ∈ B ) ∧ x ∉ B ⇔ ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B) For p ∧ ~ p ≡ F and p ∧ q ⇒ p where p and q are prepositions, ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B ) ⇔ x ∈ A ∨ F By modus tollendo ponens, x ∈ A ∨ F ⇔ x ∈ A So then, x ∈ ( A ∪ B ) - B ⇔ x ∈ A And ( A ∪ B ) - B = A Therefore, the statement for all sets A and B, A ∩ B = ∅  implies ( A ∪ B ) - B = A is true trivially. Is this proof right? If I write the following class:  public class Example {        int j;       int k;        public Example(int j, int k) {            j = j;            k = k;       }        public static void main(String[] args) {            Example exm = new Example(1,2);            System.out.println(exm.j);            System.out.println(exm.k);       }  }   The program compiles, but when I run the program, the main method will print out two 0s. I know that in order to say that I want to initialize the instance variables in the constructor I have to write:  this.j = j; this.k = k;   But if I don't write it, then which variable is evaluated (or considered) in the constructor (on the left and on the write hand side of the expressions)? Is is the argument or the instance variable? Does it make a difference?  Are there other cases where the use of this is obligatory? Defaults to true.  If set to true replaces all instances of known swear words with replacements (comical or * based). If false renders the content as originally presented.  Benefits:    Reduces moderation requirements necessary to ensure that people behind corporate firewalls are not blocked by an errent swear word.   Stops idiots like me complaining about it. Zombies - Why do they need to feed on living animals/humans? Why do some AC adapters and power supplies generate a whining noise, and what can I do about it? If we have 2 batteries one of emf x and the other is of emf y and we connect them in series we get an effective emf of x+y.  But what if we connect them in parallel, how to calculate the emf now? Installing an addon There is this addon that was built called Window Generator 2. It can be . However, when I go to the download link it shows up in a new tab as a bunch of code.  How can I download and use this addon in Blender? I'm trying to model an , but it's not going so well when it comes to the magazine of the weapon.   Next to the magazine is a mesh. I want to cut holes in the mag with that mesh. But the magazine just goes invisible or connects with the mesh when trying to execute the Boolean Modifier. I have tested to invert the normals and some other stuff, but it isn't working. How to run SuperHot at full screen on a Mac? I have SuperHot downloaded through Steam, and am attempting to play it on my 1080p external monitor. It launches , and I don't see a setting to adjust the resolution or switch out of windowed mode.  I'm running a 15" retina Macbook Pro with OS X 10.11.3. Gilbert, MacWilliams, and Sloane defined a solution to "perfect integrity" in , unconditionally secure message authentication, not vulnerable to any cryptanalysis on its own even by an adversary with unbounded computational power, provided you have a key chosen uniformly at random the same length as the message. It sounds counter-intuitive that a MAC could have perfect integrity, any hash function can be broken with enough computational power. How does their solution have "perfect integrity"? I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks          Show that the polynomial $h(x) = (x-1)(x-2)\cdots(x-n) + 1$ is irreducible over $\mathbb Z$ for all $n\ge1$ and $ n\ne4$.      Suppose $h(x) = f(x) g(x)$, then we must have $f(i)g(i) = 1$ for all $i = 1,2,...n$. So both $f(i)$ and $g(i)$ are $1$ or $-1$. In either case, $m(x) = f(x) - g(x)$ has degree smaller than $n$ and have $n$ different roots ($1,2,...,n$). So we must have $m(x) = 0$. Then $h(x) = f(x)^{2}$. So $n$ must be even. Let $n = 2k$. Because $f(x)$ has degree $k$, there are $k$ values from $\{1,2,...,2k\}$ at which $f(x)$ is $1$ and $k$ values at which $f(x)$ is $-1$.  This is where I got stuck. Hope some one can help me solve this. Thanks.