How do I specify a function without a defined argument?
Notation for functions vs. numbers
Are the WotC archives of d20 Character Optimization forum forever lost? I was looking for the Wizard Handbook but all the links were broken.  Somewhere on the handbooks of other classes (on other forums) I saw a      This was archived by WotC, here's the link   But the link leads to the welcome page of WotC (since it's broken).  Is there a way to get all that info back? Do you know where to get it from? The Wayback Machine did not archive the forums. How can I convert a hex string to a byte array? Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$ I'm learning calculus, specifically derivatives and applications of MVT, and need help with the following exercice:      Show that, for all $n &gt; 1$ $$\frac{1}{n + 1} &lt; \log(1 + \frac1n) &lt; \frac1n.$$    I tried to follow the below steps in order to prove the RHS inequality:   Proving that $f &lt; g$ on $I$ from $a$ to $b$:  Step $1$. Prove that $f' &lt; g'$ on $\operatorname{Int}(I)$.  Step $2$. Show that $f(a) \leq g(a)$ or that $f(a^+) \leq g(a^+)$  Following the above steps, let $f(x) = \log(1 + \frac1x)$ and $g(x) = \frac1x$, for all $x &gt; 1$. One has  $$f'(x) =  -\frac{1}{x^2 + x} \quad \text{and} \quad g'(x) = -\frac{1}{x^2}.$$  We note that, for every $x &gt; 1$, $f'(x) &gt; g'(x)$. Moreover, $f(1^+) = \log(2) &lt; 1 = g(1^+).$   My problem is that I got the wrong inequality sign in Step $1$.    Looking at the solution in my textbook, the author suggests using the MVT but I don't know how to apply it in this case. I’m going to Germany for a semester exchange and I have a long-stay visa from Germany. Can I travel to the other Schengen states during this period with this visa or do I have to apply for a separate short stay Schengen visa?  My visa is German national visa type D. It is for a duration of 4 months, i.e., longer than 90 days. It says on the visa that the long stay visa is valid in Deutschland, which is different from my earlier visa, which I had when I visited Germany earlier; that one was a short stay visa and said it was valid in Schengen Staten.  Perhaps my current visa means that I can stay in Germany for a period of longer that 90 days can visit the other Schengen states if I don’t exceed the 90 day period. Density function for RV The density function for a random variable X is given in terms of a constant $c$.  Find the value of $c$.  What is the corresponding distribution function?  Sketch both the density and the distribution functions. Finally, find the probabilities.  5.1 $f(x)=0$ for $x&lt;0$ and $f(x)=\frac{c}{(x+1)^4}$ for $0&gt;x$   $P(X&gt;4)$  $P(X&lt;2)$  $P(1\le\ X&lt;3)$   I get $c=3$ by taking the integral, and then I am a little confused as to why  $F(x)=1-\frac{1}{(x+1)^3}$  Is that because in solving for $c$, I ended up with $1=\frac{c}{3(x+1)^3}$, so substituting c in I get $1=\frac{1}{(x+1)^3}$?  And, then  $P(X&gt;4)=1-P(X\le\ 4)$ $=1-(1-{1}{(1+4)^3})$ $=1/125$  $P(X&lt;2)=1-{1}{(1+2)^3}$ $=26/27$  I guess I am a little confused about what $F(x)$ and $f(x)$ actually MEAN? And thus that leads to my confusion about what the differences between $&gt;$ and $&lt;$ and greater than or equal to are, etc etc in terms of the equations. The ones above I solved by comparing to the book but I still don't understand what they mean really.  So as a result, I don't really get how to solve $P(1\le\ X&lt;3)$.  5.2 $f(x)=ce^x$ for $x&lt;0$ and $f(x)=ce^{-x}$ for $0&gt;x$  5.3 $f(x)=0$ for $x&lt;-1$ or $x&gt;2$ and $f(x)=cx^2$ for $-1&lt;x&lt;2$ In definition of vector multiplication, the direction of the resulting vector is given by the right hand rule. However I don't know any mathematical requirement to pick right hand instead of left for this purpose. Is there any? Primary key or Unique index? At work we have a big database with unique indexes instead of primary keys and all works fine.  I'm designing new database for a new project and I have a dilemma:  In DB theory, primary key is fundamental element, that's OK, but in REAL projects what are advantages and disadvantages of both?  What do you use in projects?  EDIT: ...and what about primary keys and replication on MS SQL server? Residuals correlated positively with response variable strongly in linear regression Footnote in tabular environment $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$ Average length of the longest segment This post is related to a previous SE post  concerning average length of a smallest segment.  A rope of 1m is divided into three pieces by two random points. Find the average length of the largest segment. My answer is 11/18.  Here is how I do it:  Here we have two independent random variables $X,Y$, both uniform on $[0,1]$. Let  $A=\min (X,Y), B=\max (X,Y)$ and $C=\max (A, 1-B, B-A)$. First we want to find the probability density function $f_C(a)$ of $C$. Let $F_C(a)$ be the cumulative distribution function. Then $$ F_C(a) = P(C\le a)=P(A\le a, 1-B\le a, B-A\le a).$$  By rewriting this probability as area in the unit square, I get $$F_C(a)=\left\{\begin{array}{ll} (3a-1)^2 &amp; \frac{1}{3}\le a\le \frac{1}{2}\\ 1-3(1-a)^2 &amp; \frac{1}{2}\le a\le 1\end{array}\right.$$ from which it follows that  $$f_C(a)=\left\{\begin{array}{ll} 6(3a-1) &amp; \frac{1}{3}\le a\le \frac{1}{2}\\ 6(1-a) &amp; \frac{1}{2}\le a\le 1\end{array}\right.$$ Therefore the expected value of $C$ is $$\int_{1/3} ^{1/2}6a(3a-1) da+\int_{1/2} ^{1}6a(1-a) da= \frac{11}{18}.$$  My questions are:  (A) Is there a "clever" way to figure out this number 11/18?  (B) What is the answer if the rope is divided into $n&gt;3$ segments? Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ For the $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0,$ my text book asks the following questions.  (b) Why is $f$ uniformly continuous on any bounded subset of $\mathbb{R}$?   (c) Is $f$ uniformly continuous on $\mathbb{R}$??  The graph for the function is this.    For the question (b), if I take subset between $[0.2,0.6]$ or the subset where the slope is steep, I don't think the function is uniformly continuous because I think for a given $\epsilon&gt;0$, there is no unique $\delta &gt;0$ for the bounded subset. Therefore, it also cannot be uniformly continuous on $\mathbb{R}.$ However, the questions sounds like the function is uniformly continuous and the book says that it is uniformly continuous. The answer on the book says something but I need more explanation. Thanks. $SU(3)$ irreducible representations with tensor method My son is 5 years old and sexually active I am a 20 year old female student and I have a 5 year old son who is really sexually active. For instance, he kisses girls by force, he forces them to lick his penis, he "fingers" them, he speaks about sex openly with everyone except me, his father and my family.  There is a whole lot more that he does and the most frustrating part is that when teachers ask him where he saw those things, he says he learned it from me.  We do not even live together.  Currently I am in Cape Town doing my degree and he is in KZN with my mother.  Since he was 8 months old, he went to live with his father because I had to finish high school.  We never lived together. We have tried talking to him and it hasn't helped.  We've punished him but still he keeps on getting worse. What should I do?  I am really angry, embarrassed and confused. If I am not mistaken, this is his third year acting like this. how do we know the BBP formula for $\pi$ is valid? I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we can verify for every $\pi$ digit discovered till now with a "standard formula", how do we know that the formula is correct after those digits?  Also, what's the implication of these kind of BBP formulas for other "divine"/fundamental constants? can we learn anything else about the value of such constants and our universe through these? Any insight into the "supreme fascist" mind? Why was my "shopping list" question closed? My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Removing Katoolin from Ubuntu 14.04 Filter "Hot Network Questions" by excluding sites Using more than one set of Locators in a Manipulate Get OS-level system information