Not a duplicate since the linked question does not answer this question: A measure of similarity should be maximal for instances which are the same (e.g. similarity between (1,1) and (1,1) should be higher than the similarity between (1, 1) and (1, 2) ). This is not the case. Therefore the dot product is not a measure of similarity. Where am I wrong?  All explanations of the kernel trick I read take for granted that the dot product is a good measure of similarity for instances. I know that the dot product is zero if the vectors are orthogonal.  However, in this image, from my intuitive understanding $x_1$ and $x_3$ are quite similar, while $x_1$ and $x_2$ are very different. Yet, the dot product between $x_1$ and $x_2$ is large, and between $x_1$ and $x_3$ is smaller.  Take for example $x_1 = (1, 1)^T$, $x_2 = (20, 20)^T$, $x_3 = (1, 0.9)^T$. In what sense are $x_1$ and $x_2$ more similiar than $x_1$ and $x_3$? Why does the angle matter so much in practise?
I know how to CALCULATE dot product, but still don't understand MEANING of it.  I mean, If I have to explain vector addition I would say that "Sum of two vectors give you the third vector that has the same effect as other two combined"  But I don't know how to explain dot product. For example we have two vectors: v=(3,1) and w=(2,4). Dot product of these two is 3*2 + 1*4 = 10. What does this result (10) tell us?
I uninstalled my wine, and installed a higher version of wine by compiling source.  However when I run an exe executable (whose path is in $PATH), it doesn't work as it used to:  $  PDFXCview.exe     run-detectors: unable to find an interpreter for /home/tim/program_files/document/pdfxchange_portable/PDFXCview.exe   I am not sure why I was able to run this executable directly before I uninstalled my old wine and installed the new one. How can I make $  PDFXCview.exe work now?  Note:  $ wine /home/tim/program_files/document/pdfxchange_portable/PDFXCview.exe   will work.   $ wine PDFXCview.exe   wine: cannot find L"C:\\windows\\system32\\PDFXCview.exe"   doesn't work. Note that the path of PDFXCview.exe is still in $PATH  Thanks. Intuition for multiple temporal dimensions It’s easy, relatively speaking, to develop an intuition for higher spatial dimensions, usually by induction on familiar lower-dimensional spaces. But I’m having difficulty envisioning a universe with multiple dimensions of time. Even if such a thing may not be real or possible, it seems like a good intellectual exercise. Can anyone offer an illustrative example? My understanding is that heat is essentially atomic vibrations. If there are almost no atoms, how can there be residual heat?  Also, as I understand in space there is no heat transfer via convection and it is mainly through EM radiation. In this case, why is it that if i were to go into space with no suit and just an airtight helmet through which I could breathe, I would still freeze to death very quickly? Does each season of Game of Thrones roughly equal a book? Does each current Game of Thrones season roughly equal one A Song of Ice and Fire book? And do we know which future seasons will cover which books?  I am planning to read the first book after the TV show has finished depicting the events in the first book. I would like to know if after each TV season ends I can expect to be able to read a book covering the events depicted on screen? Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$ Trying to find a story about an unborn telepathic baby who can save a spaceship from doom Merging banking and casual style in moderncv (2) I really like the solution given in  but what to do if I want to keep the footer information of the casual style from page 2 on? That is, with this solution, I can put the header as I want in the first page, but the footer disappears from the rest of the pages and I do not want that.  I want the header of the first page to look like this (with an empty footer):   And the footer from the second page on to look like this: In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality? In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$  Alternatively, we can leverage the notion of a partial surjection, by which I mean a partial function that hits every point of its codomain at least once.  $$Y \succsim X \leftrightarrow \mbox{ there exists a partial surjection } Y \rightarrow X.$$  Now in ZFC, these approaches are equivalent, in the sense that $$X \lesssim Y \iff Y \succsim X.$$  But if we just consider ZF (that is, we drop choice) then the equivalence breaks down. However, we've still got the forward implication, because the converse of an injection is always a partial surjection.  A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."  A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.  Remark. The order symbols can be written \lesssim and \succsim. Why won't QGIS recognize my GPS? Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} $ Evaluate $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$.   $\bf{My\; Try::}$ Although we can solve it by converting into definite Integration.  But I want to solve it without Using Integration.  So $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)} = \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{(n-1)(n-2).......(n-k+1)}{k!\cdot n^{k}\cdot (k+3)}$  Now How can i solve after that, Help required, Thanks How to draw a crisp, opaque hairline in JavaFX 2.2? Cards entering the battlefield simultaneously and "Enter the Battlefield" effects I was playing modern with the owner of a shop.  He played a  which he used to bring in 7 lands: 5 basic mountains and 2  cards.  He then claimed that each  triggered 5 times (killing me).  I am doubting this now.    Here is my reason: The two  cards entered at the same time as the other 5 basic lands.  That means when they were on the battlefield and their ability was in effect, the other 5 basic lands were already there too.  That seems too late to trigger a "Enters the Battlefield" effect.  So here is my question.  Am I right or wrong?  If I am wrong, please show me how the enter the battlefield effect can trigger for something that enters the battlefield at the same time as itself.  If I am right, please show me the rules that prove it.  I now have two shop owners that are convinced that this is a legal play.  If it is not legal, I am going to need clear rules citations to convince them.    Side note:  If I am wrong and it is a legal play, then I have a follow up scenario.  Say I have 2  cards and 5 other human cards in my graveyard.  If I play a spell that lets me return all of them to the battlefield at the same time (say ), would the  cards get 6 +1/+1 counters on them?    To me, if the trick that was played on me with  is legal then my  cards should also get the +1/+1 counters. I am superimposing CGI over video footage, and have run into a problem in making materials for objects that act as 'shadow catchers'.  Essentially I want a shader that is shadeless—like an emitter—but catches shadows and light from other objects. This is so I can reconstruct objects in footage so other objects in the scene interact convincingly with them.  Here's an overcomplicated diagram to illustrate my point:   (In this particular scenario, the problem could be glossed over by using RGB curves, but that is definitely not the case with more complicated scenes)  Isn't this a common situation? How to the real guys do it? Thank you.  (I am welcome suggestions of other ways of doing this.) Problem with the header/ footer width I have recently started using the geometry package to format the margins of my documents. Although, I've noticed that the width of the header/ footer remains the same. How can I correct this?    Here's a MWE:  \documentclass{article}  \usepackage{geometry} \usepackage{fancyhdr}     \pagestyle{fancy} \usepackage{blindtext}  \lhead{} \chead{Header} \rhead{} \lfoot{} \cfoot{Footer} \rfoot{\thepage} \renewcommand{\headrulewidth}{1pt} \renewcommand{\footrulewidth}{1pt}  \begin{document}  \newgeometry{margin=0.5in, bottom=1in, top=1in} \Blinddocument  \end{document} I just ran the following command in a directory with several files, some owned by root, some by a normal user. I got a strange message:  $ sudo chmod -rwx * chmod: example-file: new permissions are ----w----, not ---------   Why the heck does chmod refuse to clear all permissions, if a file had group write rights? What is the thought behind this behaviour?  And is there a way to force removing all permissions in one step, without having to execute sudo chmod g-w * afterwards? What should I do when my neural network doesn't generalize well? How to change background-color on text links on hover but not image links Is leaving airport at a stop before the final destination illegal? There are tricks such as  or  which involve booking a ticket composed of several connecting flights, then leaving the airport at a stop, possibly even "hopping off" a plane (intermediate landing). Rationale: Sometimes these tickets are cheaper than a ticket to one of the stops. So one adds an additional leg and discards it.  Some people say that when using such tricks, one enters a legal gray area. But I wonder:  Could leaving the airport at a stop really be illegal, i.e. violating an airline's or a travel agency's terms?  Also there are suggestions that one should not request an upgrade, because then the trick may be discovered by the airline. Here I wonder:  How can they know that one will throw away the last leg? Effects:  +25 attack damage +15% life steal +40% attack speed Passive  Unique: Your attacks deal 5% of the target's current health in physical damage (60 max vs minions and monsters). Active   Unique: Drains target champion, dealing 15% of the champion's maximum Health in physical damage and healing you by the same amount. Additionally you steal 30% of their movement speed for 4 seconds - 60 second cooldown. (500 range)  Cost: 3200g VS.   Effects:  +70 attack damage +12% life steal Passive: Gain +1 attack damage and +0.2% life steal per kill (maximum +30 attack damage and +6% life steal). Half of the bonuses are lost upon death.  Cost: 3200g What is better for what situations, and when should you build BT instead of BotRK and Vice Versa. Which provides more lifesteal, the only variable being the stats of the weapon? I find myself constantly questioning my decision on which weapon to get and would really love some in depth answers. Would be great if any examples given were given from the ADC side of the house. Is it possible to install Office 365 (2013) on Ubuntu 15.04 Vivid Vervet? I would like to install Office 2015/365 on Ubuntu 15.04. I tried to install Office using Wine, but that did not work. Please note that I do not want virtualisation unless it has a -like behaviour, where application windows are treated like regular Ubuntu application windows by the system.