I know two definitions of uniform integrability:   (Rudin - Real and Complex Analysis). Let $(X,\mathfrak{M},\mu)$ be a positive measure space. A set $\Phi\subset L^1(\mu)$ is called uniformly integrable if to each $\epsilon&gt;0$ corresponds a $\delta&gt;0$ such that $$\left|\int_E fd\mu\right|&lt;\epsilon\tag{1}$$ whenever $f\in\Phi$ and $\mu(E)&lt;\delta$. (Royden &amp; Fitzpatrick - Real Analysis) Same thing except that $(1)$ is replaced by $$\int_E|f|d\mu&lt;\epsilon.$$   Are the two equivalent? Clearly 2. implies 1., but I am not sure whether or not 1. implies 2.   The second one a lot more convenient to use in proofs, but sometime I don't know if "uniformly integrable" only assumes the first one.
Let $(X,\mathcal{M},\mu)$ be a measure space and $\{f\}$ be a sequence of functions on $X$, each of which is integrable over $X$. Show that $\{f_n\}$ is uniformly integrable if and only if for each $\varepsilon \gt 0$, there is a $\delta \gt 0$ such that for any natural number $n$ and measurable subset $E$ of $X$, if $\mu(E) \lt \delta$, $$ \left|\int_E f_n~d\mu\right| \lt \epsilon.$$     I think one direction $(\Rightarrow)$ is clear. Since $f_n$ being uniformly integrable imply that for every $\varepsilon\gt 0,~\exists \delta \gt 0$ such that for any $E\in \mathcal{M},~\mu(E)\lt \delta$, $\int_E |f_n|~d\mu$ for every natural $n$. But then   $$ \left|\int_E f_n~d\mu\right|\le \int_E|f_n|~d\mu \lt \varepsilon.$$    Any suggestions for the other direction?   Edit:  Following Davide's suggestions, I have   $$ \begin{align*} \int_E |f_n|~d\mu &amp; \le \int_{E\cap [f_n \ge 0]} f_n~d\mu + \int_{E\cap [f_n \lt 0]} (-f_n)~d\mu\\ &amp; = \left| \int_{E\cap [f_n \ge 0]} f_n~d\mu \right| + \left| \int_{E\cap [f_n \lt 0]}f_n~d\mu \right| \\ &amp; \lt \varepsilon + \varepsilon = 2\varepsilon. \end{align*} $$
How do I install an application from a .dmg disk image? I've downloaded Firefox and it's now sitting on my desktop as Firefox.dmg. When I double click on Firefox.dmg it opens up a window with Firefox in it so I can use it - does this mean the application has been installed?    Community Wiki'd the question because I think it's a worthwhile question to have on the site for pointing people to, it's not exactly intuitive to do and not a question I think I deserve any rep from. How do I compare strings in Java? Android backup/restore: how to backup an internal database? Consider a text file with the following entries:  aaa bbb ccc ddd eee fff ggg hhh iii   Given a pattern (e.g. fff), I would like to grep the file above to get in the output:  all_lines except (pattern_matching_lines  U (B lines_before) U (A lines_after))   For example, if B = 2 and A = 1, the output with pattern = fff should be:  aaa bbb ccc hhh iii   How can I do this with grep or other command line tools?    Note, when I try:  grep -v 'fff'  -A1 -B2 file.txt   I don't get what I want. I instead get:  aaa bbb ccc ddd eee fff -- -- fff ggg hhh iii How can I get suffix matching in a search in SharePoint 2013? I'm new to SharePoint 2013 and I'm quite puzzled, that it seems to be impossible to search for the middle part of a string.  Simple example:  I've added a lot of documents (PDFs, ...) from our Windows Server to the SharePoint Library and I want to add metadata. I try to filter the view of the document library, to only get those documents which I currently want to edit.  The names of the documents are very systematic, composed of naming conventions with abbreviations like:  Superthing2000-b_Suppl_REF_13580_note_1108_04en which means e. g.: Document about Product "Superthing2000", variant "b", Supplement to the manual, ID "13580", from 08/2011 in english language.  Now, I'd like for example to filter out all documents containing ID 13580 or "en" for english, but no chance, as suffix matching is not possible, even if I explicitly search for e. g. *en*.  Is there a way to define a search with kind of a "contains" operator which also finds files when the search string is only a part in the middle of the file name? Even the Windows Explorer search feature is able to do that!! These may be separate questions, but the following MWE:    \usepackage{pdfpages} ...   \addcontentsline{lot}{subsection}{\ref{tab:UserRoles} \textbf{User Roles}}   \includepdf[pages={-}, pagecommand={\label{tab:UserRoles}}, offset=0in -1in, addtolist={9, table, {The User Roles table}, tab:UserRoles}]{UserRolesTable}   ...   has several unexpected behaviors.   I get the following warning:  Package pdfpages Warning: There's something wrong with the entries of `addtolist'. Not all entries were processed. Check `addtolist' [in the MWE]."    Probably due to the \label not being bound to the inserted pdf (in a table environment), my inserted PDF is numbered using the section, and not the expected "Table X" value.   The manually-inserted LoT line (which also uses the section number rather than the incremented table number) is not formatted like the other lines in the LoT - the number lines up, but the table caption itself is not aligned with the others in the list.   The source PDF has no errors and displays correctly in the output PDF.  PDFLaTeX -> BibTex    MWE:  \documentclass[11pt,letterpaper,twoside,final]{article} \usepackage{pdfpages} \begin{document} \listoftables \begin{center} \textbf{Table \ref{tab:UserRoles} - The User Roles Table} \addcontentsline{lot}{subsection}{\ref{tab:UserRoles}  \textbf{User Roles}} % manual entry in ToC  \includepdf[pages={-}, pagecommand={\label{tab:UserRoles}}, offset=0in -1in, addtolist={9, table, {The User Roles table}, tab:UserRoles, lot}]{UserRolesTable} \end{center} \end{document} One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.  Of course you have to change some things in the inductive step, when you want to use it on real numbers.  I guess that using induction on real numbers isn't really possible, since $[r,r+\epsilon]$ with $\epsilon > 0$, $r \in \mathbb R$ is never empty.  Can you either give a good reason, why it isn't possible or provide an example where it is used? I would like to see my review audit history Confession: I have been suspended a couple times from reviewing activities.  Partial Redemption: I have become better and better at passing audits.   I'd like to see my recent and distant history with audits.  Even better, I'd like to see my history with the various types of audits.    I'm not sure this has much value beyond self-reflection and self-analysis. I'd like to see if there are patterns to when I was more likely to fail (do I get sloppy after 10 or 20?), or if there are certain types of reviews at which I struggle more (e.g., suggested edit vs close question).  I suspect that if I reviewed my history and revisited audits I failed, I could create rules and rhythms for myself to become a better reviewer. Convert binary to ASCII and vice versa Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue. Are amounts of money singular or plural? Are the two sentences correct or is one of them incorrect?     1) Five billion dollars is earmarked for the project.   2) Five billion dollars has been earmarked for the project.   I know dollars are countable. So I think both 'is' and 'has been' are correct here.   Also, do they differ in meaning? Proper submodules of Noetherian modules are intersections of indecomposable modules If $M$ is Noetherian, any submodule $N \neq M$ can be written as a finite intersection of intersection-indecomposable ones.   I just don't know how to begin with the proof of this proposition.   Very kind of you for reading or answering. What is a deterministic method for evaluating a sensible buffer pool size? I'm trying to come up with a sane way to understand if the max server memory (mb) setting is appropriate (either should be lower, or higher, or stay the way it is).  I am aware that max server memory (mb) should always be low enough to leave room for the operating system itself, etc.    The environment I'm looking at has several hundred servers; I need a reliable formula I can use to determine if the present size of the buffer pool is appropriate since RAM is costed per GB allocated to each server.  The entire environment is virtualized, and "physical" RAM allocated to a VM can easily be changed up or down.  I have a particular SQL Server Instance I'm looking at now with a PLE of 1,100,052 seconds, which equates to 12.7 days (the amount of time the server has been up).  The server has a max server memory setting of 2560MB (2.5GB), of which only 1380MB (1.3GB) is actually committed.    I've read several items including one by Jonathan Keheyias () and another by Paul Randal (), and several others.  Jonathan advocates monitoring for a PLE below 300 per 4GB of buffer pool as being too low.  For the SQL Server Instance above, 300 * (2.5 / 4) = 187 results in a really really low target PLE below 300.  This instance has 290GB of SQL Server data (not including log files), and is only used for integration testing.  Assuming the last 12 days is representative of typical usage for this server, I'd say the max server memory (mb) setting could be lowered.  At the other end of the scale, I have another integration-test server with a PLE of 294, that has a max server memory (mb) setting of only 1GB.  This server has only 224MB of SQL Server data not including logs, and is running some BizFlow databases.  This server might benefit from a higher max server memory (mb) setting.  I'm thinking a good starting place for targets that might be assigned too much memory could include looking at:  SELECT      RamMB = physical_memory_in_bytes / 1048576     , BufferPoolCommittedMB = bpool_committed * 8192E0 / 1048576     , BufferPoolCommitTargetMB = bpool_commit_target * 8192E0 / 1048576     , PercentOfDesiredSizeMB = CONVERT(INT,(CONVERT(DECIMAL(18,2),bpool_committed)                              / bpool_commit_target) * 100) FROM sys.dm_os_sys_info;   If BufferPoolCommitTargetMB / BufferPoolCommittedMB is greater than 1, the server is not using the entire buffer pool.  If the machine in question also has a PLE of greater than "x" then it might be a good candidate for a decrease in max server memory (mb).  Since the Buffer Manager:Lazy writes/sec performance counter tracks the number of times SQLOS has written pages out to disk between checkpoints due to memory pressure, this might be another good thing to look at.  DECLARE @WaitTime DATETIME; SET @WaitTime = '00:00:15'; DECLARE @NumSeconds INT; SET @NumSeconds = DATEDIFF(SECOND, 0, @WaitTime); DECLARE @LazyWrites1 BIGINT; DECLARE @LazyWrites2 BIGINT;  SELECT @LazyWrites1 = cntr_value  FROM sys.dm_os_performance_counters dopc WHERE (         dopc.counter_name LIKE 'Lazy writes/sec%' COLLATE SQL_Latin1_General_CP1_CI_AS     )     AND dopc.object_name = 'MSSQL$' + CONVERT(VARCHAR(255),                SERVERPROPERTY('InstanceName')) + ':Buffer Manager';  WAITFOR DELAY @WaitTime;  SELECT @LazyWrites2 = cntr_value  FROM sys.dm_os_performance_counters dopc WHERE (         dopc.counter_name LIKE 'Lazy writes/sec%' COLLATE SQL_Latin1_General_CP1_CI_AS     )     AND dopc.object_name = 'MSSQL$' + CONVERT(VARCHAR(255),                SERVERPROPERTY('InstanceName')) + ':Buffer Manager';  SELECT LazyWritesPerSecond = (@LazyWrites2 - @LazyWrites1) / @NumSeconds;   The above code assumes the server is under load during the 15 seconds it takes to run, otherwise it will report 0; which might be a misleading false-negative.  Should I also look at PAGELATCHIO_* wait stats or some other wait type as an indicator of memory pressure, or lack thereof?   My question is, how can I reliably determine a "good" target value for PLE and max server memory (mb)? Uneven movement on camera following a path. I render a short movie. I set the camera to the circle curve to make it fly in circle. The problem is after render, the camera is a little shaky. Can anyone help me? Thanks first. Constructing $g$ holomorphic differentials on $\omega^3=(z-1) \cdots (z-8)$ Let us have a surface $X$ which satisfy the equation $\omega^3 = (z-1)(z-2)\cdots (z-8)$ It's stated in the task, that there are exactly $g$ holomorphic differentials on the curve. How can I find $g$ and construct them?  The genus $g$ of this curve (I tried to use Riemann-Hurwitz theorem)  The map $X(z,\omega) \to CP^1(\omega)$ has degree 3. and we have 8 branching points with 3 leaves at each. $\chi(Domain) = DegreeOfMap * \chi(Target) - \sum_{points}(BranchingLeaves - 1) $ $2-2g = 3 *(2) - 8*2 $ hence $ 12 = 2g$ and  $g = 6$ I know that in case of hyperelliptic curve there are $g-2$ (If i'm not mistaken) holomorphic differetials and they are of sort $\frac{z^k dz}{w}$. But the form of the curve equation was heavily used. UPD: there is a branhing point at $\infty$, because if we go around all points $ 1 \cdots 8$ at once, we get that cubic root of the right side is multiplied by $\exp{4 \pi /3}$ $2-2g = 3 *(2) - 9*2 $ hence $ 12 = 2g$ and  $g = 7$ (I hope that this is true) Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$. Given any set $X$, let $P(X)$ be the set of all subsets of $X$, and let $\{0,1\}^X$ be the set of all functions $X \rightarrow \{0,1\}$. Construct a bijection (and its inverse) between P(X) and $\{0,1\}^X$.    Let us define $f: P(X) \rightarrow \{0,1\}^X$ by $f(\alpha) = g_{\alpha}$ where $\alpha \in P(X)$ and $g_{\alpha} : X \rightarrow \{0,1\}$ is defined by  $$g_{\alpha}(x) =\begin{cases}        1 &amp; x \in \alpha \\       0 &amp; x \in X-\alpha     \end{cases}$$  We need to check if this function is well-defined. Let $\alpha, \beta \in P(X)$. If $\alpha=\beta$, then $\alpha$ and $\beta$ correspond to the same set, and so correspond to the same function since both $g_{\alpha}$ and $g_{\beta}$ send that set to 1 and the ($X-$that set) to 0.   If we have a function $g_{\alpha} \in \{0,1\}$, then $\alpha$ is a set in $X$ that is sent to 1. But since $P(X)$ is the set of all sets of $X$, we must have $\alpha \in P(X)$. So the function is surjective.  Finally, if $g_{\alpha}=g_{\beta}$ then both $g_{\alpha}$ and $g_{\beta}$ send the set $\alpha$ or $\beta$ to 1. So $\alpha$ and $\beta$ must be same set. In other words, the function is injective; and hence bijective.   The inverse function is $f^{-1}: \{0,1\}^X \rightarrow P(X)$ where $f^{-1}(g_{\alpha}) = \alpha$.     Another way to prove this is to first construct the inverse function, show that it is well defined, and then show that $ff^{-1}$ and $f^{-1}f$ are the identity functions, right?  So if we let $g_{\alpha}=g_{\beta}$, then $\alpha=\beta$, because otherwise $g_{\alpha} (\beta)=0$ and $g_{\beta} (\beta)=1$; a contradiction.   Now we have $f \circ f^{-1} (g_{\alpha}) = f(\alpha) = g_{\alpha}$ and $f^{-1} \circ f(\alpha) = f^{-1}(g_{\alpha}) = \alpha$ , and so $f$ and $f^{-1}$ must be bijective. My request is simple: Many times, there will be a question where some people feel like it should be closed. At the same time, I'll feel like it's a perfectly legit question, and it should therefore NOT be closed. Right now, I have to wait till it gets enough close votes (which it always does for some reason, as soon as 1 or 2 people vote to close, e/o else decides to vote as well) and then vote to re-open. How about, as soon as there's one vote to close, there should be an option to counter the close vote, something like "vote not to close" or whatever. This would then bring the "Vote to close" number down by one. Only when the vote to close outnumbers the vote not to close by 5, that's when a question is indeed closed. Any thoughts? Preventing robots from crawling specific part of a page As a webmaster in charge of a tiny site that has a forum, I regularly receive complains from users that both the internal search engine and that external searches (like when using Google) are totally polluted by my users' signatures (they're using long signatures and that's part of the forum's experience because signatures makes a lot of sense in my forum).  So basically I'm seeing two options as of now:   Rendering the signature as a picture and when a user click on the "signature picture" it gets taken to a page that contains the real signature (with the links in the signature etc.) and that page is set as being non-crawlable by search engine spiders).  This would consume some bandwidth and need some work (because I'd need an HTML renderer producing the picture etc.) but obviously it would solve the issue (there are tiny gotchas in that the signature wouldn't respect the font/color scheme of the users but my users are very creative with their signatures anyway, using custom fonts/colors/size etc. so it's not that much of an issue). Marking every part of the webpage that contains a signature as being non-crawlable.   However I'm not sure about the later: is this something that can be done?  Can you just mark specific parts of a webpage as being non-crawlable? I am a little bit confused about the Area Under Curve (AUC) of ROC and the overall accuracy.  Will the AUC be proportional to the overall accuracy? In other words, when we have a larger overall accuracy will we definitely a get larger AUC? Or are they by definition positively correlated?  If they are positively correlated, why do we bother reporting both of them in some publications?  In real case, I performed some classification task and got the results as follows: classifier A got an accuracy 85% and AUC of 0.98 and classifier B got an accuracy of 93% and AUC of 0.92. Question is, which classifier is better? Is it possible to get results similar to these (or do my results indicate a bug in my implementation)? How to tell whether two graphs are isomorphic?