UNIQUE constraint on a specific pairing
Custom unique constraint, only enforced if one column has a specific value
Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$   My (very limited) attempt so far: after creating a small list of the values $F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21, F_9=34, F_{10}=55$ i can see that yes it does seem to work for instance $F_{6+3}=F_5 F_3 +F_6 F_4 = 10 +24 = 34 = F_9$. However, I really don't know where to begin as showing that this must hold in general terms. Should I be looking to use limits? Or perhaps induction? What is the best way to solve this? Expressing Gain in Decibels? Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal, but that the set of finite sums of products of elements of I and J is an ideal. This ideal is called the product ideal, and is denoted by IJ.   I want to check if my work on the first part makes any sense. I am not sure if I am picking the right example.  If I let R = Z and I = (2) and J = (3) then I has multiples of 2, and J has multiples of 3  6 will divide the product IJ if we add 2 and 3 though we get 5 in IJ, but I am not sure if 5 Divides IJ. A scene I made is not rendering, at all. This grey screen stays the same and nothing changes.  Here are my settings:  [ In the movies, the Death Eaters leave a smoky trail behind them as they fly. As the Order arrive in the Ministry of Magic in Order of the Phoenix, they leave a silvery trail behind them. In the books, there’s no mention of how they’re doing this. And J.K. Rowling said in the books that only Voldemort and Snape could fly like this; first seen in the Battle of the Seven Potters.  Please can somebody explain how the Death Eaters are able to fly like this? I know that its is the possessive and it's is the contraction, and know when to use them.  But why doesn't the possessive have an apostrophe?   "The bear's eating a fish." [contraction] "The bear's coat is brown." [possessive] "It's eating a fish." [contraction] "Its coat is brown." [possessive] "One's eating a fish." [contraction] "One's coat is brown." [possessive]    lists the etymology as "From  +‎ ", and  says that this is actually the original form:     Originally written it's, and still deliberately spelled thus by some writers until early 1800s.   So what happened to the apostrophe?  When did people stop using it, and why did they?  It seems that it's as the possessive is more natural, as most people do this until they're taught that it is wrong (or even after).  Update: Online Etymology Dictionary has been updated to include two potential explanations:     The apostrophe came to be omitted, perhaps because it's already was established as a contraction of it is, or by general habit of omitting apostrophes in personal pronouns (hers, yours, theirs, etc.).   Can anyone back up either of these arguments?  The possessive one's still has the apostrophe, despite these. What is a good complex analysis textbook, barring Ahlfors's? I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.) $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ I came across the following problem that says:  Let $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|&lt;1\}$ such that the range of $f$ is contained in the set $\mathbb{C}\setminus (-\infty,0]$. Then  $f$ is necessarily a constant function. there exists an analytic function $g$ on $\mathbb{D}$ such that $g(x)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$. there exists an analytic function $g$ on $\mathbb{D}$ such that $\operatorname{Re}g(z)\geq 0$ and $g(z)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$. there exists an analytic function $g$ on $\mathbb{D}$ such that $\operatorname{Re}g(z)\leq 0$ and $g(z)$ is a square root of $f(z)$ for each $z\in\mathbb{D}$.   I have to determine which options are correct. Can someone help in the right direction? Thanks in advance for your time. It is very easy to put a tilde over the symbol in math mode. Simply type $\tilde{a}$. But how to put a tilde under the symbol in math mode? Algebraic Property inherited by a "lift"? Suppose $K$ is a field, and $K\subset E$ is an algebraic extension, and $K\subset F$ is any extension.  Suppose that $L$ is some field which contains both $F$ and $E$ as subfields.  I want to show that $FE$ (the smallest subfield of $L$ containing both $F$ and $E$) is algebraic over $F$.  Lang mentions that the result is true because an element remains algebraic under lifting, but I can't prove the claim with this.  I interpret his comment to mean that if $x\in E$ is algebraic over $K$, then $x\in FE$ is algebraic over $F$.  This is certainly true.    But this doesn't seem enough if I start with any element of $FE$. Provide an easily discoverable way to get the full URL to an answer , clicking &quot;link&quot; &quot;share&quot; for anwers yields a short (traceable) URL in the &quot;share a link to this answer&quot; dialog, such as †:  These short URLs show no clue about the linked question whatsoever. Also, when different people use a short URL for the very same post, then the user id suffix makes those be different URLs for a browser. Such links will then not be shown as visited‡. I know how to get URLs that have a proper slug. But as I see  pop up on these sites (where referrals ), I assume many others just don't realize full URLs would be better for use on these sites themselves? Or many don't want to do the little effort to get the full URL? So: can we please have an easy way to get the full URL as well, in a way that is easily found for those who don't know about the different links? (Currently for questions, clicking &quot;link&quot; has the same effect. But then one can also right-click the title, or take the URL from the location bar, to get the full human-readable URL. Once, Jeff .)  † The short links for answers are not (yet) [automatically converted][6] into a clickable question title either, unlike links such as `https://meta.stackexchange.com/questions/74274/privacy-leak-in-permalink#75963`, which renders as https://meta.stackexchange.com/questions/74274/privacy-leak-in-permalink#75963 But I guess that will change soon.  ‡ On Meta the CSS for posts doesn't indicate such difference anyway, which is a different issue. Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps.... Can a 20V 4.51A power supply be used with a computer that requires a 19V 3.42A power supply? Can every even integer be expressed as the difference of two primes? Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof? What does a tilde in angle brackets mean when creating a Java generic class? Verify my proof of the irrationality of $\sqrt{6}$. So uh i've got this problem with a penetrating object and another object that serves as something like a condom if you will    So what I want, is for the penetrating object to not deform while moving into this "rubber cylinder", while the rubber cylinder does deform and reshape itself to the outer form of the penetrating object. That is, if the pen. object is bigger than the entrance hole ofc.  Also, when the object leaves the "rubber cylinder" I want the outer edges to follow (mid object in picture) and then jump back (right object in picture)    Parenting the size and position of the "rubber cylinder"-entrance to the position of the penetrating object would work i guess, but I think it'd create problems when the pen. object enters from another angle (not perpendicular to the entrance). I just read another post where someone mentions Vertex Weight Proximity modifier in a combination with a cast modifier. Don't know if thats gonna work yet. Couldn't find the time to test it. I am trying to build a custom Adventure Island. At this moment I am trying to limit the allowed items on this island. I'm trying to do this by filtering on a certain enchantment which is not item-specific.  For example.   When you want to filter an wooden sword with a custom name, you use this command:  /testfor @p[r=5] {Inventory:[{tag:{display:{Name:"Custom Sword"}}}]} &lt;   When you want to filter a wooden sword with a custom name and specific enchantement, you can use:  /testfor @p[r=5] {Inventory:[{tag:{display:{Name:"Custom Sword"},ench:[{id:16,lvl:1}]}}]} &lt;   But how can I filter on items with a specific enchantment on ANY item? I tried:  /testfor @p[r=5] {Inventory:[{tag:{ench:[{id:16,lvl:1}]}}]} &lt;   But this ain't work. Does anyone know how to solve this issue? What is the limit of Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?