JavaFX returning null to a User that is listed
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Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$ In other words, why don't we always use Schottky diodes if they are so much better? What diode properties do Schottky diodes have that makes them unfit for certain applications? link new object to scene with python in 2.8 How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$ Given the integral $(1)$     $$\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3\tag1$$   This integral seems to be quite tough, I can't even know where to start. I have try substitution $u=1+\cos^3 x$ and $u=\cos^3 x$, can seem to remove the x variables.  We have the following identities   $$\sin^5a={5\over 8}\sin a-{5\over 16}\sin 3a+{1\over 16}\sin 5a\tag2$$  $$\cos^3 a={3\over 4}\cos a+{1\over 4}\cos 3a\tag3$$  We may write $(1)$ as  $$\int_{0}^{\pi}{\sin x(1-\cos^2 x)^2\over 1+\cos x(1-\sin ^2 x)}\mathrm dx\tag4$$  $u=1-\sin^2 x\implies \mathrm du=-2\sin x\cos x\mathrm dx$ then $(4)$ becomes  $${1\over 2}\int_{0}^{1}{(1-u)^2\over \sqrt{u}+u^2}\mathrm du\tag5$$  $u=v^2\implies du=2vdv$ then $(5)$ becomes  $$\int_{0}^{1}{(1-v^2)^2\over v+v^4}\cdot{v\mathrm dv}\tag6$$ Simplify to  $$\int_{0}^{1}{(1-v)(1-v^2)\over 1-v+v^2}\mathrm dv\tag7$$  I guess using partial decomposition of fraction and integrate term by term, seem lengthy process  How else can we prove $(1)$? What happens if you use the 32-bit int 0x80 Linux ABI in 64-bit code? I have been facing hard time understanding meaning of "random sample" as well as "iid random variable". I tried to find out the meaning from several sources, but just got more and more confused. I am posting here what I tried and got to know:  Degroot's Probability &amp; Statistics says:     Random Samples / i.i.d. / Sample Size : Consider a given probability distribution on the real line that can be represented by either a p.f. or a p.d.f. $f$. It is said that $n$ random variables $X_1 , . . . , X_n$ form a random sample from this distribution if these random variables are independent and the marginal p.f. or p.d.f. of each of them is $f$. Such random variables are also said to be independent and identically distributed, abbreviated i.i.d. We refer to the number n of random variables as the sample size.   But one of the other statistics book I have says:       In a Random Sampling, we guarantee that every individual unit in the population gets an equal chance(probability) of being selected.   So, I have a feeling that i.i.d.s are elements that construct random sample, and the procedure to have random sample is random sampling. Am I right?   P.S.: I am very confused about this topic, so I will appreciate elaborate reply. Thanks. java.lang.StringIndexOutOfBoundsException: String index out of range How can I highlight text before a character with listing? Why does a query run slower in a Stored Procedure than in the Query window? Don't show pending suggested edits to me after I run out of votes What license to use while putting papers on the Arxiv? I am new to this and not sure what to make of all the options available. I am interested in putting a machine-learning-related preprint, which will potentially be submitted to places like NIPS/ICLR/ICML in the future.   arXiv.org perpetual, non-exclusive license to distribute this article (Minimal rights required by arXiv.org)  Creative Commons Attribution license (CC BY 4.0)  Creative Commons Attribution-ShareAlike license (CC BY-SA 4.0)  Creative Commons Attribution-Noncommercial-ShareAlike license (CC BY-NC-SA 4.0)  Creative Commons Public Domain Declaration (CC0 1.0) Material will not assign to selected vertices The  mentions that it's possible to post one more question after a six-months waiting period and if the question is good enough, the question ban can be lifted:     If you're unable to improve your existing questions, you'll get the chance to ask one new one 6 months after your last question. If that question is positively received, you may be able to continue asking questions; if not, then the ban will be reinstated.   What is positively received in this context? Is it enough if the question has a zero score and is not closed? Or does it have to have a positive score?  How long does the system wait after posting this new question to evaluate this criteria and possibly lift (or reinstate) the question ban? My book says  \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation}  I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made any sense to me.  For example,   Based on how my ignorant brain is viewing this, I don't see why the definition could not be.  \begin{equation} (a,b)=\{\{a\},\{b\},\{a,b\}\} \end{equation}  aka the power set. What is the significance of the {a} in that definition? Please keep things simple if possible. Normally definitions have a valid and clear reason for being defined that way.    Clarification  First, I understand what an ordered pair is. I just don't see how the set notation says that.   Second,  \begin{equation} (a,b) = \{\{a\},\{a,b\}\}=\{\{a,b\},\{a\}\} \end{equation} Sets don't preserve order, but ordered pairs do. How does the third part of the equality apply to the definition?  Third, another issue with the notation that I have starts with the Product Property of Sets  \begin{equation} \text{Let $X$ and $Y$ be sets} :\ X=\{a,b,c\}\text{ and }Y=\{a,d,e\}. \end{equation} \begin{equation} \text{Then }X \times Y = \{(a,a),(a,d),(a,e),(b,a),(b,d),(b,e),\dots,(c,e)\} \end{equation} If we look at the first ordered pair and our given definition we have  \begin{equation} (a,a)=\{\{a\},\{a,a\}\} \end{equation}   How can this be so, you can't have duplicates in sets? I guess what I am looking for in an answer, is not a proof or a definition of ordered pairs, but rather something like, "This notation says what it says because...". Except for the second to last point I get the terminology, I just don't get the connection between the two different uses of notation. Is there any other size tires that will fit on a 27 x 1 1/4 rim? Can't reach wi-fi signal on Ubuntu, but can do it on other OS/devices (ath9k) How do I get the CPU temperature? How do I get the CPU temperature? Consider the symmetric group $S_5$. I would like to find how many elements of $S_5$ are of order 5, and how many are of order 6. I would also like to determine what the maximum order of an element in this group would be.  Here is what I have so far: elements of order 5 are the 5-cycles. Elements of order 6 (since a 6-cycle is impossible for 5 elements) must have at least an even cycle and a cycle of length divisible by 3, and 2 + 3 = 5, so elements of order 5 are a combination of 2-cycles and 3-cycles.  How can I find the total count of such elements of order 5 and order 6, and the maximum order? I'm not entirely sure where to go from here. Creating Indexes on DB with Hibernate @Index Annotation Can I see my active diplomatic deals in Civ 6? When making deals with AI I'm having a hard time assessing if I'm already trading for the resource or not. Is there a way I can see my currently active deals or deal history?  If not, is there a way I can see which resources I am currently importing?