Why can't I double my sample to decrease the variance of the estimator in the OLS?
What are the consequences of "copying" a data set for OLS?
Problem:  Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow  rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the Hilbert Schmidt Norm.  Please read my solution and tell me whether it is correct. If not, let me know where the mistake is.  Proof of the implication $\Leftarrow $:   If $rank(A)=0$, then in this case $A=0$. It follows that $\left \| A \right \|=\left \| A \right \|_{HS}=0$ If $rank(A)=1$, than: $ A=\begin{pmatrix} A_{1}\\  \alpha _{2}A_{1}\\  ...\\  \alpha _{n}A_{1} \end{pmatrix}$ where $A_{1}$ is the first row of $A$   and in this case: $\left \| A \right \|=\left \| A \right \|_{HS}=\left \| A_{1} \right \|\sqrt{1+\alpha _{1}^{2}+...+\alpha _{n}^{2}}$  Proof of the implication $\Rightarrow $   We know that $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|$  On the other hand: $ A=\begin{pmatrix} A_{1}\\  A_{2}\\  ... \\ A_{n}  \end{pmatrix}$. So, $$\left \| Ax \right \|=\left \| \begin{pmatrix} \left \langle A_{1},x \right \rangle\\  \left \langle A_{2},x \right \rangle\\  ...\\   \left \langle A_{n},x \right \rangle\end{pmatrix} \right \|=\sqrt{\left \langle A_{1},x \right \rangle^{2}+\left \langle A_{2},x \right \rangle^{2}+...+\left \langle A_{n},x \right \rangle^{2}} $$  Where $ \left \langle A_{i},x \right \rangle$ is the inner product of $A_{i}$ and $x$  Using the Cauchy-Schwarz inequality: $\left \langle A_{i},x \right \rangle^{2}\leq \left \| A_{i} \right \|^{2}\left \| x \right \|^{2}$, we get: $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|\leq max_{\left \| x \right \|=1}\left \| x \right \|\left \| A \right \|_{HS}=\left \| A \right \|_{HS}$. In order to have the equality: $\left \| A \right \|=\left \| A \right \|_{HS}$, we should have: $\left \langle A_{i},x \right \rangle^{2}=\left \| A_{i} \right \|^{2}\left \| x \right \|^{2} $ which occurs only if $A_{i}=\lambda _{i} x$. So, the rows of A are dependent, which implies that $rank(A)=1$. Note that the case $rank(A)=0$ happens when $A=0$  Please let me know if my solution makes sense. How to solve this equation $x^{2}=2^{x}$? How to solve this equation $$x^{2}=2^{x}$$   where $x \in \mathbb{R}$.  Por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?(*)  (Translation: By trying different values I've found that $2$ is a solution, but I couldn't find any method to this though. Any suggestions? ) Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$ Who can use magic scrolls? I'm confused about who can use scrolls in 5e. In the DMG, p139 under Magic Item Categories: Scrolls, it states that:     Any creature that can understand a written language can read the   arcane script on a scroll and attempt to activate it.   This implies that anyone can try to cast a spell using a scroll. However, on p.200 in the Spell Scroll entry it states that:     If the spell is on your class's spell list, you can use an action to read the scroll and cast its spell without having to provide any of the spell's components. Otherwise, the scroll is unintelligible.   Then there is the bit about a DC 10 + scroll spell level ability check if the spell is higher than a spell slot that you can cast.     If the spell is on your class's spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + spell's level. On a failed check, the spell disappears from the scroll with no other effect.    So, which is it? Can anyone who can read a language use a scroll (like a fighter) if they make an arcana check, or just a PC with that specific spell in their spell list (only spell casters)? Is there a tool to get the lines in one file that are not in another? I inherited a project using SQL Server 2005.  Recently there were some performance problems and I started looking at the indexes, and found  _dta_index_survey_25_135059617__K1_K10_K19:     id ASC, sent_date ASC, group_id ASC   and  _dta_index_survey_21_364632442__K18_K1_K2_K9:     group_id ASC, id ASC, campaign_id ASC, sent_date ASC   which appears to be generated by some kind of automatic performance tuning tool.  I think SQL Server comes with something like that but I'm not familiar with it.  Anyway, id is the primary key, and what I'm trying to understand here is how it could possibly be beneficial to include the primary key within a secondary index... because you have to have the leftmost pieces of information to use an index and if you know id then you've got the row anyway.  Can someone explain why a tool would have generated these indexes and if there's any point in keeping them?  Just because the indexes are getting used doesn't mean they're a good design.  SQL Server is using the wrong indexes lately, and I'm trying to determine why. More specifically, queries against this table that ran in milliseconds a month ago are now taking multiple seconds.  The table is huge (4M rows), and growing steadily, but at the same pace it's been going at for years.  Almost every query joins across group_id or campaign_id.  Often both.  The application almost never specifically selects or sorts by the PK, but I could see how ID could be useful when comparing or merging lists of IDs within the database engine.  ... But id is also an identity column, and an index holds lists of row ids, so it still seems redundant to have the ID in the key when it's also in the index "buckets".  And the additional fields could still be covered by the index without being part of the key.  I discovered that (id, sent_date, group_id) hardly ever gets read.  Meanwhile (group_id, id, campaign_id, sent_date) is one of the hottest indexes.  Next I found that an index which I thought was pretty reasonable (campaign_id, sent_date, group_id, id) also has a index_columns_include of every other column in the table, and occupies more space than the primary clustered index on id!  (but it's also pretty hot...)  Surely I can replace it with a simple (campaign_id, sent_date, group_id)? OK guys I have this problem:  For $x,y,p,q&gt;0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$  It says I should use Jensen's inequality, but I can't figure out how to apply it in this case. Any ideas about the solution? Calculation of $\int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx$, where $n\in \mathbb{N}$ Compute the definite integral      $$ \int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx $$      where $n\in \mathbb{N}$.   My Attempt:  Using $\cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get  $$ \begin{align} \int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx&amp;=\int_{0}^{\pi/2} \left(\frac{e^{ix}+e^{-ix}}{2}\right)^n\left(\frac{e^{inx}+e^{-inx}}{2}\right)\,dx\\ &amp;= \frac{1}{2^n}\mathrm{Re}\left\{\int_{0}^{\pi/2} \left(e^{ix}+e^{-ix}\right)^n\cdot e^{inx}\,dx\right\}\\ &amp;=\frac{1}{2^n}\mathrm{Re}\left\{\int_{0}^{\pi/2}\left(e^{2ix}+1\right)^n\,dx\right\} \end{align} $$  Letting $z=e^{4ix}$ gives us  $$ \begin{align} e^{4ix}dx &amp;= \frac{1}{4i}dz\\ dx &amp;= \frac{dz}{4iz} \end{align} $$  So the integral becomes  $$\frac{1}{4\cdot 2^n}\mathrm{Re}\left\{\int_{C}\left(\sqrt{z}+1\right)\cdot \frac{dz}{iz}\right\}$$  How can I complete the solution from here? For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$ Suppose $f$ is a Riemann integrable function on $[0,1]$. Prove that $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$   This is what I am thinking: Fix $n$. Then by Jensen's Inequality we have $$0\leq\left(\int_0^1x^nf(x)dx\right)^2 \leq \left(\int_0^1x^{2n}dx\right)\left(\int_0^1f^2(x)dx\right)=\left(\frac{1}{2n+1}\right)\left(\int_0^1f^2(x)dx\right).$$Thus, if $n\to\infty$ then $$0\leq \lim_{n\to \infty}\left(\int_0^1x^nf(x)dx\right)^2 \leq 0$$ and hence we get what we want. How correct (or incorrect) is this? Received W-2 but was never paid I just now received a W-2 from a company that I worked for 3 days as a contractor. I had to leave country for 2 months due to a family emergency. Since the employment duration was very short, I was not paid. I was told that they don't want to charge the client because 3 days is not enough contribution.  Since I had to leave country the same day, I did not argue with them and told it's okay if they feel right.  Now I have a W-2 from them saying they have paid me about $1500. I contacted them but I have not received any reply yet.  I just want to be prepared. What can I do here? Is it possible to reference an excel cell in a comment? I have an Excel balance sheet with a total at the bottom. On that cell I have a comment that says that the reason the total is X is because of what happens in J2. What happens if I insert another row before that, now my comment is wrong because all the rows jumped up a value.  Is there any way to reference a cell dynamically in an Excel commment? What's wrong with these two cron jobs? I have the following cron jobs defined.  55  8   *   *   3   /usr/bin/php /home/mark/dev/processes/customClient/events.php &gt; /home/mark/dev/processes/customClient/events-`date +%Y-%m-%d --date='last Wednesday'`-`date +%Y-%m-%d`.csv 0   9   *   *   3   /usr/bin/echo 'The csv for last week, trying my hand at automatiging this' | /usr/bin/mutt &lt;emailaddress&gt; -s 'Events from `date +%Y-%m-%d --date='last Wednesday'`-`date +%Y-%m-%d`' -a '/home/mark/dev/processes/customClient/events-`date +%Y-%m-%d --date='last Wednesday'`-`date +%Y-%m-%d`.csv'   It seems to work properly if I run it the above command directly from the command line. But when I checked the running of the script this morning I got an e-mail stating (I'm paraphrasing because I accidentally deleted them) that the back ticks weren't closed properly. Recently, Firefox has changed the default folder for newly created bookmarks (using Ctrl+D) from the Bookmarks Menu to "Other Bookmarks".  Is there some way to change that back? Or some alternate method instead of Ctrl+D?  EDIT  A native way to do this (without an add-on) would be best. Is there a word for the sadness over "What might have been"? Trouble Viewing Whole HDR Image as cycles world When I load an HDR image into an environment texture using Cycles Render, only a small fraction of the image appears. I have a feeling it's an easy fix, but I can not find what needs to be changed. Unexpected Results From Clicking The Unanswered Button Question of remainder on dividing by 7 Question :  What is the remainder when   $$ 10^{10} + 10^{10^2} +10^{10^3} + \ldots + 10^{10^{100}} $$  is divided by $7$? Why can't a normal user `chown` a file? Console flashes on launching, then closes (ms-windows) I installed blender on a windows 7 64 bit system.  I opened it one time then went to the tutorials.  in the meantime I had a power failure and when I tried to start blender again I get a dos window that closes itself then nothing.  I reinstalled Blender and get the same thing.  I am going to try to uninstall then reinstall but in the mean time, any other suggestions are welcome. I read it as a paperback book somewhere in the mid-90s, I think. The protagonist lived in an apartment building. He's out on the balcony when a birdlike (but humanoid) alien swooped down and attacked him. The presence of the aliens was commonly known and was a known danger. He kills it with his watering can and is rewarded by either the aliens or the humans for it with money. Other than that, all I remember was that the chapters were started with an internet-like news report which I think chronicled things getting worse and more violent. One of the updates involved the bombing of a bridge with the casualties raised by having two slow-moving trucks moving side by side obstructing traffic and ensuring a full bridge.  The internet system was pretty ubiquitous as I recall. I think that he looked up his watering can shortly after the incident with the aliens, but decided to keep this one, even if it leaks a bit. I want to say that it was by Piers Anthony, but none of the book titles look familiar. I think the cover I had showed the man being attacked by the bird-man, which looked kind of like the illustrations on the cover of The Stand.