If ($x_\alpha)_{\alpha \in A}$ is a collection of numbers $x_{\alpha}\in [0, \infty)$ such that $\Sigma_{\alpha \in A}x_{\alpha}\lt \infty$, show that $x_{\alpha}= 0$ for all but at most countably many $\alpha \in A$, even if $A$ itself is uncountable.  This is exercise 0.0.1 in Terrence Tao's "An Introduction to Measure Theory", haha.   Can somebody give me some insight here? Haven't been in the analysis mindset for awhile. I think proceeding by contradiction might be a good strategy.
Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha&lt;\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).  Proof: Let $\sum_{\alpha\in A}x_\alpha=M&lt;\infty$. Consider $S_n=\{\alpha\in A \mid x_\alpha&gt;1/n\}$.  Then $M\geq\sum_{\alpha\in S_n}x_\alpha&gt;\sum_{\alpha\in S_n}1/n=\frac{N}{n}$, where $N\in\mathbb{N}\cup\{\infty\}$ is the number of elements in $S_n$.  Thus $S_n$ has at most $Mn$ elements.   Hence $\{\alpha\in A \mid x_\alpha&gt;0\}=\bigcup_{n\in\mathbb{N}}S_n$ is countable as the countable union of finite sets. $\square$  First, is my proof correct? Second, are there more concise/elegant proofs?
Let $\alpha(t)$ be a regular curve. Suppose there is a point $a$ in $R^3$ such that $\alpha(t)-a$ is orthogonal to $T(t)$ for all t. Prove that $\alpha(t)$ lies on a sphere.  So, I let $\alpha(t)=(x(t), y(t),z(t))$ and $a=(a,b,c)$.  Since, $\alpha(t)-a$ is orthogonal to $T(t)$,  $&lt;\alpha(t)-a,T(t)&gt;=0$.  But I don't know how to solve this.. What is the use for non-inverting input resistance in a negative feedback opamp? Here in the Figure R5=50K. This resistor value is not in the equations to obtain output voltage, and in many tutorials non-inverting input is directly grounded without a resistor. What might be the reason to use it here? Connection pooling in PHP Are 7 hours enough time to leave the Brussels airport and go to the city center? What does the comma operator do? Catch Regex too complicated when splitting a string in salesforce I have created a function wherein I split a string from the csv filecsvAsString.split('\n') . It is working when the string is not so big. But when I try to split a very large string, an error will occur      regex too complicated   what I want to do is to catch if an error occur during the split process. How will I do it ? I've tried to use try and catch but the problem still occur. Is there any other possible solution? How old is SpongeBob? I don't mean how old is the series, I mean what is SpongeBob's age if he were a real person. I ask because his age seems to be ambiguous in the show.   He has a full-time job He has his own house He is always referred to as a kid (especially in the movie) He acts childish and immature In one episode, he was pretending to be an adult/grown-up \documentclass[11pt]{article}  \begin{document}   The above equations on application of Laplace Transform become:  $$sL_1\hat{i}_{L1}(s) = D\cdot \hat{v}_{C1}(s) + D\cdot \hat{v}_{C2}(s) - D\prime \cdot \hat{v}_g(s) + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)$$ $$sL_2\hat{i}_{L2}(s) = -D\prime \cdot \hat{v}_{C1}(s) + D\cdot \hat{v}+{C2}(s) + D\prime \cdot \hat{v}_g(s) + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)$$ $$sC_1\hat{v}_{C1}(s) = -D\cdot \hat{i}_{L1}(s)+ D\prime \cdot \hat{i}_{L2}(s) + (-I_{L1} -I_{L2} - I_1)\cdot \hat{d}(s) -D\prime \cdot \hat{i}_l(s)$$ $$sC_1\hat{v}_{C1}(s) = D\prime \cdot \hat{i}_{L1}(s)- D\cdot \hat{i}_{L2}(s) + (-I_{L1} -I_{L2} - I_1)\cdot \hat{d}(s) -D\prime \cdot \hat{i}_l(s)$$ $$sL_l\hat{i}_l(s) = D\prime \cdot \hat{c}_{C1}(s) + D\prime \cot \hat{v}_{C2}(s) - D\prime \cdot \hat{v}_g(s) +(V_g - V_D - V_{C1} - V_{C2})\cdot \hat{d}(s) - R_l \cdot \hat{i}_l(s)$$  \end{document}   Error:   181 Underfull \hbox (badness 10000) in paragraph at lines 181--181 181 Overfull \hbox (22.63293pt too wide) detected at line 181 Schur's theorem and numbers Can you give a proof for bounds of  $S(r)$? Please suggest me articles to have better idea of Schur's theorem (Ramsey theory). combinatorics circular arrangement problem If $n$ distinct things are arranged in a circle, then what are the number of ways selecting three of these things so that no two of them are next to each other? unicode-math broken A question about the idea of Laplace transform In Laplace integral transform equation one multiplies the function $f(t)$ by $e^{-st}$. I read in many tutorials that $e^{-st}$ decays much faster than any other function so the integral diverges.  But what I don't understand what makes one to think to multiply a function with  $e^{-st}$ to transform it in $s$ domain? What is the motivation behind it? Why would you suddenly come up with an idea as such: "Oh there is a function $f(t)$ in time domain what can I do to transform it to complex freq. ($s$) domain? Hmm let me multiply it with $e^{-st}$ and integrate it from zero to infinity" What would have motivated this idea of multiplying and integrating for transformation? Find the limit $ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}} $ Find the limit $$ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}}. $$  I know that the limit is of indeterminate form type $\frac{\infty}{\infty}$, but it seems using L'Hopital's rule directly does not help here.  Do I somehow use the fact that $\lim_{x \to \infty} (1+\frac{1}{x})^{x}=e$? How to find solutions of linear Diophantine ax + by = c? For example, you're doing a research for a car to purchase. You get a list of cars with engine number, city where a car was assembled, and a color. What's missing is make, model and price. proof that on a compact manifold a vector field is complete I would like to know if my proof of the following statement is correct     If $M$ is a compact manifold, then every vector field $X$ over $M$ is complete.   Proof:  I take $p\in M$ and $(\gamma_p,I_p)$ a maximal integral curve through $p$. $I_p$ is open and non empty so if $I_p$ is closed, it is equal to $\mathbb{R}$.  So take $s\in \text{Adh}(I_p)$. There exists $t_i\rightarrow s, t_i\in I_p$. Define $q$ as $\text{lim}(\gamma_p(t_i))$ and take a maximal integral curve passing by $q$ as $(\gamma_q,I_q)$. Then since the vector fields are $C^\infty$, $\gamma_q(I_q)\cap\gamma_p(I_p)$ is nonempty (intuitively, this is clear but don't really know how to show it).  On the intersection, $\gamma_p$ and $\gamma_q$ coincide. So we can extend the maximal curve $\gamma_p$ to include $\gamma_q$. This implies that $q\in\gamma_p(I_p)$ and is actually equal to $\gamma_p(s)$  I think the main idea is there even though it probably lacks a bit of rigour. terminal won't launch after install python 3.5 After I installed Python 3.5 and ran the commands below, my terminal and many other programs won't launch any more.  cd /usr/bin sudo rm python3 sudo ln -s python3.5 python3    How can I restore the default Python version again to fix these problems? Approximation of real number to the power of irrational Say I have two numbers, $c\in\Bbb R^+\,,q\in\Bbb{R\setminus Q}$.  Is there a way to approximate the value of $c^q$?  This question bugging me because of how simple it sound in compare to how complicated it is. First I want to understand what a power of irrational number is, when talking about $\Bbb R^+$(I use this domain for sake of simplicity) we have very well defined actions for power a rational number, we can even use some theorems to get $c^{a/b}=\sqrt[b]{c^a}$, but when I am dealing with irrational number I can not do this.  I know that we have some definition for this because the solution of the equation $2^x=3$ is irrational.  So what exactly does it mean to take something to a irrational power and is there a way to approximate a numeric value for this? T-SQL stored procedure that accepts multiple Id values Is there a graceful way to handle passing a list of ids as a parameter to a stored procedure?   For instance, I want departments 1, 2, 5, 7, 20 returned by my stored procedure. In the past, I have passed in a comma delimited list of ids, like the below code, but feel really dirty doing it.   SQL Server 2005 is my only applicable limitation I think.  create procedure getDepartments   @DepartmentIds varchar(max) as   declare @Sql varchar(max)        select @Sql = 'select [Name] from Department where DepartmentId in (' + @DepartmentIds + ')'   exec(@Sql) As a disclaimer, I must admit this may benefit me (I haven't checked), but probably not as significantly as most others. This issue is also something that has prevented me from answering questions by users with 1 rep in the past, especially if they are poorly fashioned. The Problem New users with a programming question find SO from various methods, such as google. They ask a question with haste, hoping for a quick answer -- which they usually receive. Many of these users have no interest in Stack Overflow itself, but are simply interested in getting the solution they require. As a result of this, they'll check the site regularly for the first day or two until a sufficient answer has been posted. Once they get what they need, they bail; without the required rep or understanding that they should up-vote and select a response as the answer. The Result A great many questions on SO are abandoned by users with no intention of ever coming back. The answer they require has been provided, but for one or more reasons they have neglected their duty of contributing back by way of awarding reputation. The Reach (Since Jeff loves Datum) Some queries on the data dump examine the reach of this 'problem'. The query I used is based on extensions of the following base query; SELECT p1.Id, p1.Score, p1.OwnerUserId, p2.OwnerUserId, p2.Score FROM posts p1  INNER JOIN posts p2 ON (p1.Id=p2.ParentId) LEFT JOIN users ON (p1.OwnerUserId=users.Id) WHERE p2.Score &gt;= 2  AND p2.Score = ( SELECT MAX(p3.Score) FROM posts p3 WHERE p3.ParentId=p1.Id ) AND users.Reputation &lt; 15 AND p1.PostTypeId = 1 AND p1.AcceptedAnswerId = 0  Result: 3,288 Questions by users with under 15 rep which have answers with 2+ up votes that have no accepted answer. Adding: AND (users.LastAccessDate-p1.LastActivityDate) &lt; 604800 // 1 week AND p1.CreationDate &lt; 1249084800 // 1249084800 is 1 week before latest dump data  to the where clause further reveals that 877 questions that have answers with 2+ up votes have no accepted answer, and the owner of the question has not returned since the week the question was asked. Modifying there where clause, and removing the left join: AND p1.OwnerUserId = 0    Reveals a staggering 3,675 questions with answers of 2+ votes by deleted users.  These are questions that were probably answered but the users who did the work can never be recognized. Building off the previous and using an even more restrictive WHERE clause WHERE p2.Score &gt;=5  Still 1,178 posts that can never have an accepted answer. How many posts are there with over 5 up votes that don't provide the correct answer? Modifying the where clause for Reputation to AND users.Reputation = 1  Still reveals 388 questions by users with 1 rep have been abandoned. Solution Similar to how bounty questions work, if a user has abandons the site (not active since x time after the question was asked) with a question that has answers with 2+ up votes, force the answer with the most up votes to be the selected answer. This will reduce the amount of questions that are unanswered, and also give users more of an incentive to answer questions by new users without extra hesitation.  Update Added more statistics for deleted users.  Please feel free to add/remove rules to make this a truly practical feature.