Integrate: $\int_0^\infty \frac{\sin^2(x)}{x^2}dx$
Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$
InDesign: How to restore text-frame position/size to be managed by the master page instead of the actual page? I have some designer that moved slightly a text-frame and in order to be managed again by the master page we need to remove the page and create a new one inserting the content in it.  Is there any different way to do it?  I would like to be able to restore the "power" of positioning/size to the master page because now we need to change the master-page to position the text-frame a little bit to the right and in the case above mentioned will not be updated.  Thanks Prove $(1-a)(1-b)(1-c)(1-d)&gt;1-a-b-c-d$ and $ a,b,c,d$ are real and between 0 and 1.    I can do this with $$(1-a)(1-b)&gt;1-a-b  \\ 1-a-b+ab&gt;1-a-b  \\ ab&gt;0  $$  But with $c$ and $d$, this becomes much more difficult. Can someone nudge me in the right direction? Can you wait to add optional damage dice that apply on a hit (such as Divine Smite) until after seeing if the roll is a critical? Is division of matrices possible? Opinions on closing an older question as a duplicate of a newer question What's everyone's opinion on closing an older question as a duplicate of a newer one, if the newer one is clearly the better resource? I'm looking at a special case right now, where the two were posted by the same person; the  was migrated to SU before a commenter pointed out it was a . Is it considered good practice to close the older as a duplicate of the newer?  More generally, is it ok to do that for separate questions posted by different people, if the older question went unanswered but the newer has a good answer?  (I should probably mention I saw , but I think this is separate) Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s or two consecutive 1s. Note: Problem from "Kenneth Rosen's DM and it's applications" and solution from "Students solution guide for use with ... applications"  Let P(n) be the number of strings not containg two containing two consecutive zeros or two consecutive ones.  I got the answer  P(n)=P(n-1)+2P(n-2)+2P(n-3)+...+2P(0) +2  but then the book says the sequence also satisfies the recurrence relation  P(n)=2P(n-1)+P(n-2). I am not able to understand this part. How do I see all foreign keys to a table or column? In MySQL, how do I get a list of all foreign key constraints pointing to a particular table? a particular column? This is the same thing as , but for MySQL. Which Mac is suitable for Blender 2.8? I am considering purchasing a Mac for my godson who will be using it to run Blender 2.8.  I know very little about this program.  All I can find are requirements for a Mac that will run earlier versions of Blender.  Would someone please recommend a list of requirements for a Mac (or a link to a list of requirements) that I could use to find a model that can run 2.8?  Thank you in advance for all your help! Sincerely, M. Lap Arduino-controlled slot cars - alternative to MOSFETs for speed control? I'm attempting to control the speed of two slot cars in a set using Arduino.    Each car is controlled by a controller with a variable resistor that looks like this:    They're wired in parallel to the same 17V DC power supply:    Simplified:    For Arduino control, I experimented with manipulating a single car's speed using a MOSFET. It worked really well. Using a PWM output on the Arduino, I was able to pulse the MOSFET on and off to throttle the current and change the car's speed. The behavior of the MOSFET seemed to be exactly the same as the variable resistor controller; the resistance, voltage, and current fluctuated the same way on various parts of the circuit.  Enter car B. I added a MOSFET to the setup and mimicked the MOSFET wiring for car A, grounding them both to the Arduino ground. The result looks like this:    Here's a sketch:    Close up, MOSFET A:    MOSFET B:    At first, all seemed well. I sent signals to MOSFET A and successfully changed Car A's speed. I stopped Car A and pulsed MOSFET B, and I was able to control Car B's speed just fine.  Then, I turned Car B to a constant speed, and sent a signal to MOSFET A while Car B was still running. Car A started, but Car B's speed immediately dropped as a result of running Car A.  I hooked a multimeter up across the rails of Car B and watched its voltage while it ran on its own. Then, sure enough, when Car A started running alongside it, Car B's voltage dropped dramatically.  I figured this was because of my wiring configuration, so I left the multimeter on Car B, and made one change. I replaced MOSFET A with the original slot car controller:    I started running Car B again using its MOSFET. The multimeter showed a steady voltage, so I began to squeeze Car A's controller (the variable resistor), and Car A's speed increased. This time, however, unlike using a MOSFET to control Car A, the variable resistor did not affect Car B's voltage at all. Car B remained at a steady speed the whole time.  Any idea why this happens? The MOSFET seemed to function the same as the variable resistor (empirically) with a single car, but had totally different behavior with two cars in parallel. Is it because the MOSFET is not really changing resistance to current, but just switching it on and off, so it opens up another path for current to flow fully every time it closes? Is there an Arduino-controllable alternative to a MOSFET that would fluctuate actual resistance like the variable resistor? Digital potentiometer? Servo hooked up to a potentiometer? ;) I'm really curious what the difference is and what causes the major differences in behavior. Calculating the average area of the states listed in states.txt Need someone to calculate the average area, in square miles, of the states listed in states.txt. States.txt (file): State           Population   SqMi      Region  ------------------------------------------ California           39.37   163.7       West Texas                29.36   268.6      South Florida              21.73    65.7      South New_York             19.33    54.7  NorthEast Pennsylvania         12.78    46.1  NorthEast Illinois             12.59    57.9    Midwest Ohio                 11.69    44.8    Midwest Georgia              10.71    59.4      South North_Carolina       10.60    53.8      South Michigan             9.97    96.7    Midwest  Output should look like this: Average Area: 82.85 square miles  Note my awk program will be ran like this: awk –f &lt;file.awk&gt; states.txt Frattini subgroup $Φ ( G )$ is not trivial Frattini subgroup $Φ ( G )$ of a group G is the intersection of all maximal subgroups of $G$. The question is  Let $G$ be an abelian group.What is the sufficient and necessary condition of $Φ ( G ) ≠｛e｝$？  I think if $G$ has no torsion part, we cannot say $Φ ( G ) ≠｛e｝$,because $\Bbb Z ×\Bbb Z×・・・ ×\Bbb Z$ has a maximal subgroup $p\Bbb Z ×\Bbb Z×・・・×\Bbb Z$, with intersectioning distinct primes, we gain just a trivial group. So, we need torsion part. But I cannot proceed from here.I only know the answer, $G$ has an element of order $p^2$, that  is, $G$ has theree subgroups. I want to deduce this consequence from steady discussion. P.S  According to this answer, in a ring that is a finite direct product of quotient rings of $\mathbf{Z}$, we can say rattini subgroup is the same as the Jacobson radical of the corresponding ring. LEFT JOIN vs. LEFT OUTER JOIN in SQL Server What is the difference between LEFT JOIN and LEFT OUTER JOIN? Show finite complement topology is, in fact, a topology What word would describe someone who doesn't generally leave much of an impression on people? Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$ About a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$  If we'll take a line $f(x)=\begin{cases} ax &amp;, x\in\mathbb R\\ 0 &amp;, x=\infty  \end{cases}$ then we'll have a bijection.   But about this function: $g(x)=\begin{cases} x &amp;, x\in\mathbb R\setminus \{0\}\\ 0 &amp;, x=\infty  \end{cases}$ ? it appears to be a bijection and it's even simpler. I want to write $ k = \{k1,k2,k3,...,kn\} $. How could I do this? I am reading Tom Apostol's Analysis and come across this theorem.   Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon &gt;0$?   I don't doubt the proof in the book but I don't understand the intuition or geometric explanation behind this. Could somebody shed some light on this equation? I just started studying analysis on my own.$\ \ $ How can I force a line break in the table of contents? Convert a string representation of a hex dump to a byte array using Java? I am looking for a way to convert a long string (from a dump), that represents hex values into a byte array.  I couldn't have phrased it better than the person that posted .  But to keep it original, I'll phrase it my own way: suppose I have a string "00A0BF" that I would like interpreted as the  byte[] {0x00,0xA0,0xBf}   what should I do?  I am a Java novice and ended up using BigInteger and watching out for leading hex zeros. But I think it is ugly and I am sure I am missing something simple. Why is $5^{n+1}+2\cdot 3^n+1$ a multiple of $8$ for every natural number $n$? I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is $$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\in\mathbb{N}$$.