what is a good book (about number theory)
Books on Number Theory for Layman
I have a dataset for which I have multiple sets of binary labels. For each set of labels, I train a classifier, evaluating it by cross-validation. I want to reduce dimensionality using principal component analysis (PCA). My question is:  Is it possible to do the PCA once for the whole dataset and then use the new dataset of lower dimensionality for cross-validation as described above? Or do I need to do a separate PCA for every training set (which would mean doing a separate PCA for every classifier and for every cross-validation fold)?  On one hand, the PCA does not make any use of the labels. On the other hand, it does use the test data to do the transformation, so I am afraid it could bias the results.  I should mention that in addition to saving me some work, doing the PCA once on the whole dataset would allow me to visualize the dataset for all label sets at once. If I have a different PCA for each label set, I would need to visualize each label set separately. how to pass objects between intents How can I programmatically delete a line from a Word document using c#? Is the Echo of a Echo Knight actually a creature? I've always been certain that the echo created by an Echo Knight fighter using their 3rd-level feature Manifest Echo is not defined in game terms as a creature, since the description of the echo never explicitly tells it is a creature. But the presence of people believing the echo to be a creature in lieu of the fact that it is immune to all conditions and might make saving throws (game characteristics typically associated with creatures only) brought this doubt to my mind.  from Jeremy Crawford implies it is just an image and not a creature, further confirming that the feature should tell you, but  implies that the echo is an object whilst the feature doesn't explicitly say so. (Included for reference and insight, since tweets are not eligible as official ruling) Is the echo a creature? Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact? I have a string that contains both double-quotes and backslashes that I want to set to a variable in Python. However, whenever I try to set it, the quotes or slashes are either removed or escaped. Here's an example:  &gt;&gt;&gt; foo = 'baz "\"' &gt;&gt;&gt; foo 'baz ""'   So instead of baz "\" like I want I'm getting baz "". If I then try to escape the backslash, it doesn't help either:  &gt;&gt;&gt; foo = 'baz "\\"' &gt;&gt;&gt; foo 'baz "\\"'   Which now matches what I put in but wasn't what I originally wanted. How do you get around this problem? How to call a space in which every G$_\kappa$-set is open Recall that in a topological space $X$, we say that a subset $A\subset X$ is a G$_\delta$-set if $A$ is the intersection of countably many open sets of $X$. Also, we say that $X$ is a  if G$_\delta$-sets are open.  Now, call as G$_\kappa$-set the intersection of $\kappa$ many open sets, where $\kappa\geq\aleph_0$ is a cardinal. I would like to know if there is some standard name for those spaces in which G$_\kappa$-sets are open, for a fixed cardinal $\kappa\geq \aleph_0.$  Motivation: this kind of space arose in my PhD thesis, but I do not know any references in which they appear. I'm temporary calling them as $P_\kappa$-spaces. Capitalization for a bullet list How to draw a fish? I would like to draw an image like the following.     This is what I've done so far (I used tikz, but any other package is welcome):  \documentclass{article} \usepackage{tikz}  \begin{document}     \begin{tikzpicture}         \draw[fill=blue] (.5,0) rectangle (4,2);         \draw[fill=red] (1,1) to[bend left=50] (3,1) to[bend left=50] (1,1);         \draw[fill=red] (3,1) -- (3.5,1.5) -- (3.3,1) -- (3.5,.5) -- cycle;         \draw[fill=white] (1.6,1.1) circle (.15cm);          \draw[fill=blue] (1.55,1.1) circle (.05cm);              \end{tikzpicture} \end{document}     Edit: April 2nd, Time for explanation  In Italy (but I also think in other countries), a practical joke done on April 1st is called "pesce d'Aprile" (= April's fish, that is an ) and is often related to fishes.  The usual one is to stick a drawing of a fish on someone's back without he/she realizes it.  Some years ago a colleague of mine stuck some "pesci d'Aprile" with written below "sono disponibile" (= I'm easy) on the rear of all the cars in our company car parking. Some other colleagues didn't realize it, and they had gone around with that drawing for days.  Another funny example happened in Milan . Someone put on the windshield of a lot of cars a false fine for illegal parking. On the fine sheet was written "per il pagamento rivolgersi agli uffici comunali di via Gadio, 2" (= for payment, please contact the municipal offices in via Gadio, 2), the address of the Civic Aquarium.  Hence, this question was simply a way to put    P.S. = Since I earned a lot of reputation points with this trick question, I decided to give them back to the Community awarding a bounty to the two answers posted on April, 1st.  Moreover, I accepted Robert's one since he guessed without spoiling, and I thank the other two answerers.  To the downvoters: you're probably technically right, but:   there's already too much sadness in the world, let's laugh now and then! humor is part of the LaTeX philosophy, Prof. Knuth himself docet if only serious questions are accepted, why did they create a fun tag? "This question is not available, it may have been deleted" when submitting a question on the Android app after losing internet connection Can't view deleted questions via Android app Why does Li-Po battery voltage drop under load? I have very low background in electronics.  I have a Li-Po battery connected to an electric motor. The question is:   Why does battery voltage drop under load? Why is each cell of a Li-Po battery limited to 3.0 V (discharging)? Why is each cell of a Li-Po battery limited to 4.2 V (charging)? Triggered send status queued, need to get an alert Is it possible to bring built-in lightning components into aura component? I have a form that needs to let the user select 3 out of 5 products, something similar to Order > Add Product (Modal). I was wondering if there was a way to bring that piece into my component (markup) and add some sort of filter that only shows the products I want to show.  What I found so far:  &lt;lightning:listView objectApiName="Products2" listName="productsView" showActionBar="true" enableInlineEdit="true" showRowLevelActions="true"/&gt;   But Product2 is not supported for that component.  also these from browser inspector:  forceMultiAddUsingLVM forceListViewManager   These are data-aura-class attributes when rendered.  I'm looking for directions here. Thanks in advance. expected true to respond to true? Duplicate flag lists "you have already raised this flag" after flagging as "Unclear what you're asking" Is it possible to turn off the Two-Factor Authentication for an Apple ID? Is it possible to turn off the Two-Factor Authentication for an Apple ID?  I have more than one device linked to this Apple ID and I know the password and security questions. String manipulation with & or + in VB.NET I have now enough reputation points on Stack Overflow to edit every post directly. But sometimes (especially if this is about a field where I'm not savvy, or when I try to edit a question title without having understood the question itself) I might be not totally sure about whether my edit really is an improvement.  For now, I have these options:   Do the edit anyway, hope that someone else will catch it if it is wrong Don't do the edit, and hope that someone else will see this post and edit it in the right way Let a comment and hope that someone will see it (either after editing or instead of it). Flag the post for moderator attention (but will the moderators know it better?) Search the right chat room to find someone who knows more about the topic.   All these are not really optimal, I think.  I would like to have a check box     [_] Submit this edit for approval   which, when checked, would submit the edit into the usual edit approval queue. Then I would get at least one or two other pairs of eyes looking over my edit, and maybe even improving it.  (I don't really care about the +2 reputation for the edit - I think it should be without any reputation difference here.)  There's an  for nearly giving everyone the ability to suggest edits that the post owner can approve, but I'm simply interested in being able to insert them into the general edit queue where anyone can approve them.  Actually, I could do this right now by having a low-reputation sock puppet (or a totally anonymous edit), but I think this is not the intended use for sock puppets, is it? Also, I would like to be able to get feedback for my edit. Importing only certain rows between spreadsheet documents Document 1 has many rows of data.  How do I import into document 2 only those rows that have a certain value in column C?