MacBook filesystem FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF after deleting linux partition
MacOS partition showing as "FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFF"
Convergence of the series $\sum a_n$ implies the convergence of $\sum \frac{\sqrt a_n}{n}$, if $a_n>0$ I need help to solve following problem from Rudin's Mathematical analysis book:    Convergence of the series $\sum a_n$  implies the convergence of $\sum \dfrac{\sqrt {a_n}}{n}$, if $a_n&gt;0$  I tried to construct a suitable convergence sequence $b_n$ such that $\sum b_n$ converges and $a_n \leq b_n$ but, I am not able to find such sequence $b_n$ .  Thanks for the help and sugestions. Detect whether Excel workbook is already open Best way to implement keyboard shortcuts in a Windows Forms application? I'm looking for a best way to implement common Windows keyboard shortcuts (for example Ctrl+F, Ctrl+N) in my  application in C#.  The application has a main form which hosts many child forms (one at a time). When a user hits Ctrl+F, I'd like to show a custom search form. The search form would depend on the current open child form in the application.  I was thinking of using something like this in the ChildForm_KeyDown event:     if (e.KeyCode == Keys.F &amp;&amp; Control.ModifierKeys == Keys.Control)         // Show search form   But this doesn't work. The event doesn't even fire when you press a key. What is the solution? Does Windows still rely on MS-DOS? I am just about to start an OS course and as an Apple user I am not very familiar with the underlying details of Windows OS. I was wondering, is MS DOS still used with Windows running on top or is ONLY Windows used now as the OS? I was a little confused because I read somewhere that MS-DOS is used for booting but Windows has all other OS capabilities built into it and thus is used for all other OS operations... My sisters and I all have iPhones under the same account. I also have an iPad under the same account. I am trying to get the iMessage on my iPad compatible with my phone number, not email address.   The only phone number choice to choose from is my sister's iPhone number. How can I choose my phone number for iMessage on my iPad? Detecting superfluous #includes in C/C++? How do I get started with 'painting with light' photography? Tunnels being naturally dark may not be the most intuitive choice for a place to take pictures. But I love the '' work that artists such as '' are putting out and they're clearly seeking out places like tunnels, caves, garages... Dark places in other words, to create their works of art. Which brings two questions to mind:   How do I go about finding places that I can accomplish this sort of 'tunnel photography.' What are some techniques that I can use to 'paint with light' in my photography once I have found some locations? $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite I want to subtract a constant number from my data to get new data file. For example, I want to subtract  10 from my data.   data.txt  Id  a   b   c   d   e 1   1   2   3   4   5 2   2   3   4   5   6 3   3   4   5   6   7 4   4   5   6   7   8 5   5   6   7   8   9 6   6   7   8   9   10 7   7   8   9   10  11 8   8   9   10  11  12 9   9   10  11  12  13 10  10  11  12  13  14 11  11  12  13  14  15 12  12  13  14  15  16 13  13  14  15  16  17 14  14  15  16  17  18   Required output:  out.txt   Id  a   b   c   d   e 1   -9  -8  -7  -6  -5 2   -8  -7  -6  -5  -4 3   -7  -6  -5  -4  -3 4   -6  -5  -4  -3  -2 5   -5  -4  -3  -2  -1 6   -4  -3  -2  -1  0 7   -3  -2  -1  0   1 8   -2  -1  0   1   2 9   -1  0   1   2   3 10  0   1   2   3   4 11  1   2   3   4   5 12  2   3   4   5   6 13  3   4   5   6   7 14  4   5   6   7   8 Blender 2.8 - Transparent background Forgive me for the unclear title. Some time ago, my HTC M8, which is rooted and runs on a  showed something weird. Firstly, in the Google Maps App a mysterious blue dot appeared, that captures the screen completely and makes it impossible to quit or use the application. It looks like this (I could make a screenshot)     When this happens in Maps, the only thing I can do is a hard restart with Power+Volume Up.  Since I have seen this occasionally in the camera app as well, I suspect it is not a bug of Maps itself. I know that the Venom ROM is not standard but maybe this is connected to some setting and anyone has seen this. Btw, the Venom ran smoothly for almost 3 years now.  Question: Has someone a clue what this could be and how I can get rid of it? Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and  $f(1)=1$.For every integer $n$ show  that there must exist $n$ distinct points $\alpha_1,\alpha_2\cdots,\alpha_n$ in that interval so that    $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $   Is it theorem or i can show it with basic differentiation knowledge .Please give me hint . Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.   The  $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative system $R\setminus I$) gives a morphism $$ f_1:\operatorname{Spec}(R_I)\to \operatorname{Spec}(R)=:X $$ The  $\widehat{R}_I$ of $R$ at $I$ gives a morphism $$ f_2:\operatorname{Spec}(\widehat{R}_I)\to X $$ For the special case $I=(a)$, the localization $R_a$ (i.e. the localization of $R$ at the multiplicative system $\{1,a,a^2,\ldots\}$) gives a morphism $$ f_3:\operatorname{Spec}(R_a)\to X $$   My question is:     What is the geometric meaning of all three constructions and how are they related?   This is what I ''know'' already or describes at least the style of answer I would appreciate. As remarked in the comments, $I$ has to be a prime ideal.   $R\to R_I$ is injective iff $R\setminus I$ contains no zero divisors which is the case if $I$ is a prime ideal. The scheme $\operatorname{Spec}(R_I)$ is the intersection of all the neightbourhoods of $I$ in $X$. This is a little contra-intuitive for me since the last statement sounds like ''$f_1$ is injective'' but the $\operatorname{Spec}$-operator should turn 'injective'' and ''surjective'' around somehow (I know this is literally not true but I only want to get a feeling like ''is contained'', ''is bigger'', ''is smaller'', etc.). $\operatorname{Spec}(R_a)$ is somehow the opposite of $\operatorname{Spec}(R_{(a)})$ (= the intersection of all the neightbourhoods of $(a)$ in $X$) because $\operatorname{Spec}(R_a)$ seems to be something like the union of all the open sets of $X$ not containing the point $(a)$. $R\to\widehat{R}_I$ is injective iff $\cap I^n=(0)$ and this holds very often (e.g. for $R$ noetherian and either an integral domain or a local ring). Hence (this is probably false intuition as remarked before), $f_2$ should be something like a ''projection'' (from something ''big'' into something ''small''). But what is geometrically the difference between localization and completion. I don't have a geometric idea of completion at all.   As remarked by Qiaochu Yuan in the comments below, one should not think of $\operatorname{Spec}$ as injective-surjective switching. Greek pdf bookmarks I am puzzled with sections titled with Greek characters. They do appear in the pdf's Content list, though not correctly.  For example, in my thesis--written in English with LyX, based on the KOMA-script--I choose to include a translation of the summary in Greek (=Περίληψη). The "raw" code for the (unnumbered) chapter title reads: \addchap{\textgreek{Per\char208lhyh}}. In the final .pdf file (using pdflatex) the respective entry appears as Per208lhyh. Using the unicode option for the hyperref package does not fix the problem.  Update detail: the document (should) not only contain Chapter or Section titles in Greek, but normal text as well.  I have found some relevant references within tex.stackexchange and elsewhere (e.g., , the manual of the textgreek package, , ). However, I cannot extract a concrete answer.  How can Greek characters appear correctly as bookmarks in the .pdf's Content list?  Note, the information of this Q&amp;A applies to/under LyX as well. Is is possible to restore the MFT of a NTFS partition after a format? Best way to build a Plugin system with Java How would you implement a Plugin-system for your Java application?  Is it possible to have an easy to use (for the developer) system which achieves the following:   Users put their plugins into a subdirectory of the app The Plugin can provide a configuration screen If you use a framework, is the license compatible with commercial developement? Limit of divided integrals Prove that: $$\lim\limits_{m\to \infty} \frac{\int\limits_0^{\pi/2}(\sin(x))^{2m} dx}{\int\limits_0^{\pi/2}(\sin(x))^{2m+1} dx} = 1$$  I had tried to get it from inequalities like $$(\sin(x))^{2m+1} \le (\sin(x))^{2m} \le (\sin(x))^{2m-1}$$ But I always hit a wall in that regard (I integrate the sin functions and from there keep on going but there it ends). How do you determine the winner of a hand? In Texas Hold'em poker, there are some cases where a particular card, often called kicker, acts as a tiebreaker between players to determine who wins the pot, or if the pot has to be shared.  For some combinations such as three or four of a kind, there is no doubt: the kicker is the 5th card of the chosen combination, and determines the winner.  Example     Alice has 7 and 10   Bob has 7 and K   On the board are 7, 7, 2, 3, 4   In that case, both have a three of a kind. But because Bob has a king and Alice only a 10, he wins.  Most of poker websites explaining the rules clearly mention the role of the kicker applying to three of a kind and double pairs. But much fewer say something about colors, straights and full houses, and I found many contradicting answers. I also asked the question to some used online poker players and they also gave me contradicting answers.  I have tried to search for official tournament rules, but most of them only explain what happens with bad behaviors, bad deals, incorrect or confusing betting, showing cards when you shouldn't, acting when it's not your turn, etc. without mentioning subtleties about combinations at all.  To simplify my question, I will take three examples; I think it's better to start with examples before getting to the general answer if one exists. So, what's happening in the following 3 examples? Could you give a more generalized answer?  Example 1 - Flushes     Alice has 2&clubs; and 3&clubs;   Bob has 4&clubs; and 5&clubs;   On the board are 6&clubs;, 8&clubs;, 10&clubs;, 2&hearts;, 5&diams;   Contradicting answer 1: it's a tie, because the highest card included in the flush is the 10, which everybody chooses to include in their 5 showdown cards. Contradicting answer 2: Bob wins, because he has the greatest private card that is part of the flush  Example 2 - Straight     Alice has 6 and K   Bob has 6 and J   On the board are 4, 5, 7, 8, 10   Contradicting answer 1: it's a tie, because the greatest card in the straight is the 8 for both Alice and Bob Contradicting answer 2: Alice wins, because she owns an extra king, compared to the jack of Bob  Example 3 - Full house     Alice has 3 and 7   Bob has 3 and 6   On the board are 3, 3, 2, 2, 5   Contradicting answer 1: it's a tie, because one is supposed to choose only five cards to make a combination, and a full house is already five cards. There couldn't be any kicker, and thus their showdowns are strictly equals. Contradicting answer 2: Alice wins because of her extra 7, compared to the 6 of Bob  Thank you very much for your answers.  Note: I'm unable to post next to you, so I edit my own post; strange not be able to answer to an answer.  Ok, So if I summarize what you are saying :   In the flush case, Bob wins because at some point, their hand differs. Technically, I can continue comparing the highest private card to decide who wins. I had it correct. In the straight case, if the highest card of the straight is public, then it's technically always a tie, no matter what the players had as second private card (asuming that only one of the two cards was part of the straight). In a full house if both players have the same triplet and the same pair, it's always a tie, no matter what the players had as second private card (assuming again that only one of the two was part of the full house).   Please confirm me that what I summarized is correct.  Thank you for your answer. "Total number of votes for 'N' answers" is wrong in "Deleted Recent Answers" Below are the two screenshots. First one shows the number of votes          In above image a post which was deleted shown ) votes. Now in below image which is actual post deleted is having 1 vote. When to use "have" and "have got"