Or do you just add your bonus like it was any other number? I'm asking because of  about traps and how easy it is to find them at mid to high level.   My first thought was that a natural 1 on a skill check would still ensure that there was at least a 5% chance on the die to miss the Pendulum of Doom hidden in the ceiling. However, I want to make sure this isn't just some kind of house rule I've been playing with forever. Thank you!
We had a rule check tonight that we couldn't answer. On the Pathfinder wiki I could only find info on natural 20s and 1s in regards to attack rolls and saves in combat.  Do they also apply to skill checks?
Is it ethical to use a recommendation letter from a deceased professor? She was a really nice professor and she wrote me a good recommendation letter that resulted in a lot of good results with applications. Recently, this professor died from brain cancer, which was a fast and shocking death for me.  Tomorrow is the deadline for a grant. What should I do? Is it professional to use letters from the deceased? Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved.   While I'm at it, I might as well add that I'd be most interested in an example from a classical field theory. Furthermore, I am looking for a nontrivial example, e.g. where $H$ doesn't just differ from $E$ by some gradient function that integrates to 0 in the action.   In particular, I am trying to get a better understanding of the physical implications of $H$ being conserved when nontrivial variables are used, i.e. physically, what does $H$ this correspond to. Conversely, can one have a system where $E = H$ in one frame, and is conserved, while in another frame $E'\neq H'$, and $H'$ is conserved while $E'$ is not? Let $n\geq 2$ and let $k$ be "considerably larger" than $n$ (like some large multiple of $n$). Then for each $i$ such that $0&lt;i&lt;n$ and $\gcd(i,n)=1$ let's define $$c_i=\left|\{p_j\;|\; p_j\equiv i \mod n,\;\mbox{where $p_j$ is the $j$-th prime, $1\leq j\leq k$}\}\right|$$ so $c_i$ represents how many of the first $k$ primes are congruent to $i$ modulo $n$.  What can we say about $c_i$s, that is about the distribution of the first $k$ primes modulo $n$?  I thought the distribution will be seemingly random, and that is mostly true - for example $c_i$s are always very close together. But there are observable non-random patterns. For example for $n=3$, for various $k$s I've tried  (up to $10^6$) I always got $c_2&gt;c_1$. If this were just some kind of random discrepancy due to distribution of small primes, it would eventually vanish for large $k$s, which I don't observe. Let $A$ be a $3\times 4$ and $b$ be a $3\times 1$ matrix with integer entries.Suppose that the system $Ax=b$ has a complex solution. Then which of the following are true? (CSIR December 2014)         $Ax=b$ has an integer solution.    $Ax=b$ has a rational solution.   The set of real solutions to $Ax=0$ has a basis consisting of rational solutions.   If $b$ is not equal to zero then $A$ has positive rank.      Is it possible to say that 1 and 2 are true as $Ax=b$ has a complex solution? If not  what we have to understand from the statement "the system $Ax=b$ has a complex solution " ? What about 3 and 4  ? I need to add 30 minutes to values in a Oracle date column.  I do this in my SELECT statement by specifying   to_char(date_and_time + (.000694 * 31)  which works fine most of the time.  But not when the time is on the AM/PM border.  For example, adding 30 minutes to 12:30 [which is  PM] returns 1:00 which is AM. The answer I expect is 13:00.  What's the correct way to do this? I came across this formula in a list:  The number of ways of distributing $n$ identical objects among $r$ groups such that each group can have $0$ or more $(\le n)$ objects  I know that standard way of doing this is to solve the problem of distributing n identical objects and $(r-1)$ partitions among themselves which can be done in $C(n+r-1,r-1)$ ways.   But I am unable to prove to myself why it is not $(r+1)^n$.  Because each of the n objects has r+1 choices, either group1, group2,... group r or none at all. The context is soccer (association football) and the sentence in question is:       The forward received the ball from the midfielder's pass.     I have already ask about this  and received alot of good answers, and I tried to leave it at that.  But then the World Cup 2014 is upon us, and I am starting to have doubts about this football usage of "from", to the point that I have semi-nightmares about it.  The use of "from" in the football example continues to bother me.  I seem to disagree (subconsciously) about how a ball (physical object) could be "from" a pass (action) unless that action creates that object.       Some people were hit by debris from the explosion.   The use of "from" in "debris from the explosion" makes excellent sense to me, because the explosion does create the debris.  But in soccer/football, the ball cannot possibly be "created" by a pass.    Is there any non-football usage of "from" similar to my football example sentence? "They're going gorilla watching" - is there a verb at the end of the sentence? In this sentence:     They're going gorilla watching.   what's the function of the verb watching?   Is it a verb? a noun? ... maybe gorilla watching means "mirador de gorilas", in spanish?  In English, I have not seen a verb at the end of a sentece. How can I find the difference between two angles? Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean? It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our sampling and CI-computation procedures  many times, 95% of thusly computed CIs would contain the population mean. It also seems to be the consensus that this definition does not permit one to conclude from a single 95%CI that there is a 95% chance that the mean falls somewhere within the CI. However, I don't understand how the former doesn't imply the latter insofar as, having imagined many CIs 95% of which contain the population mean, shouldn't our uncertainty (with regards to whether our actually-computed CI contains the population mean or not) force us to use the base-rate of the imagined cases (95%) as our estimate of the probability that our actual case contains the CI?   I've seen posts argue along the lines of "the actually-computed CI either contains the population mean or it doesn't, so its probability is either 1 or 0", but this seems to imply a strange definition of probability that is dependent on unknown states (i.e. a friend flips fair coin, hides the result, and I am disallowed from saying there is a 50% chance that it's heads).  Surely I'm wrong, but I don't see where my logic has gone awry... Why are there gaps in my IDENTITY column values? Is there a way to check that the conditions on your plane ticket are actually what your travel agent said they are? When I´m in a secure part of a web-site I´m developing (entering a specific page or folder using https), the relative links to the normal pages are automatically converted to https as well. Is there a way to tell an &lt;a&gt; tag to always use the http protocol instead of the https protocol (or the other way around)? I know it´s easy using the complete link ( or ), but I would like it to work with relative links (index.html, ../index.html, etc.). how to add xsl attribute Proof that Epicycloids are Algebraic Curves? Epicycloids are most commonly described by the parametric equations,   $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$  $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right).$  Where $R$ is the radius of the fixed circle and $a$ is the radius of the rolling circle.   With $R = ka$ we also have,   $x(t) = a[(k + 1)\cos(t) - \cos((k + 1)t)],$  $y(t) = a[(k + 1)\sin(t) - \sin((k + 1)t)].$  Several books discussing epicycloids mention that if the ratio of the radii of the circles $\left( \frac{R}{a} = k \right)$ is rational, then they are algebraic curves. However, I’ve only been able to find the Cartesian equations for the cardioid, nephroid and ranunculoid. With the cardioid being,   $(ax + x^2 + y^2)^2 = a^2(x^2 + y^2).$   The nephroid,   $(-4a^2 + x^2 + y^2)^3 = 108a^4y^2.$  And the ranunculoid,   $-52521875a^{12} – 1286250a^{10} (x^2 + y^2) – 32025a^8 (x^2 + y^2)^2 + 93312a^7 (x^5 – 10x^3y^2 + 5xy^4) – 812a^6 (x^2 + y^2)^3 – 21a^4 (x^2 + y^2)^4 – 42a^2(x^2 + y^2)^5 + (x^2 + y^2)^6 = 0.$  Clearly this doesn’t cover all epicycloids where $\frac{R}{a}$ is rational.   What is the proof that shows that epicycloids, where the ratio of the radii are rational, are algebraic curves? I am using LaTeX with the article document class. The Bibtex bibliography style that I am using is agsm, part of the Harvard family of bibliography styles.  I have two questions:  (1) If I cite an entry of type @book, then in the reference list I get publisher, address. I would like to re-arrange this order to address: publisher. Which part of agsm.bst should I modify?  (2) I would like to change single quotation marks ' ' to double quotation marks " " for setting off the title of entries of type @article. Which part of agsm.bst should I focus on? A quark star is a hypothetical type of compact exotic star composed of quark matter. These are ultra-dense phases of degenerate matter theorized to form inside particularly massive[citation needed] neutron stars. -Wikipedia   If you add enough mass to a neutron star it forms a black hole but how much mass is needed to form a quark star? When do neutrons and protons start to break down into a mush of quarks and gluons? Is there any uncountably infinite set that does not generate the reals? Does there exist an uncountably infinite set $X \subseteq \mathbb R$ such that $\mathbb R \neq \left&lt;X\right&gt;$? I can't think of any, but I'm also having trouble trying to prove that no such subset exists.  For example: $\mathbb R$ is uncountable and obviously $\mathbb R = \left&lt;\mathbb R\right&gt;$. The Cantor set $C$ is uncountable, and we know that $C - C = [0, 1]$, so then since $\mathbb R = \left&lt;[0, 1]\right&gt;$ we know that $C$ also generates $\mathbb R$. Also the set of irrationals $\mathbb R \setminus \mathbb Q$ is uncountable, but we can generate all the rational numbers by fixing one irrational number $\alpha$ and then saying the any rational number $x$ shall be $(\alpha + x) - \alpha$, since both $\alpha + x$ and $\alpha$ are irrational.  So the examples that quickly come to mind all generate the reals. Is there a simple counterexample? I have seen several times on TV documentaries where the presenter is taken to something like a library archive, and shown a book which they proceed to read an excerpt from.  On a couple of occasions the camera has shown the text they were reading and generally I have thought that I would struggle to read the words because of how the letters were formed (e.g. , or ).  This got me to wondering about how pronunciation and sentence formation may have changed over time as well, and how far back in time could I travel and still be understood while speaking my "modern" English in England (while obviously not talking about modern concepts like mobile phones or space travel)? WAMP Server not working Okay so a continuation from , where you experts intro'd me to , which can basically execute PHP within a Windows XP environment.  So now I've got it installed, but the tray icon forever shows YELLOW, and when I visit any PHP page in my browser, it just shows me the PHP source!  Also, when I visit "" in IE7 it gives me a 404 Not Found, FF3 just shows a blank page.    BTW I've tried "Restart All Services" and restarting my machine, but it still won't work.  Any ideas? Any of you had this problem and solved it? Please help me here, I'm desperate to execute PHP client-side and I'm just reverting to testing on-server for now!