I try to get the basic understanding behing SVM algorithm, however I have a problem with basic mathematics.  I follow the lecture .     Suppose the two classes can be separated by a hyperplane:$(w \cdot x) + b = 0$   Acoording to , hyperplane is defined as $n(r-r_0)=0$, does it mean that $b=-w \cdot r_0$?  I tried to consider 2-dimensional case, when $w \cdot x +b =0$ is a line, but it's completely doesn't make sense, $b=-w \cdot x$, where $b$ should be a constant, how can I generalize it to a 2 dimensional case.  In addition, why $w$ is actually orthogonal to the plus and minus plane?
How does a  work, and what differentiates it from other linear classifiers, such as the , , or ? *  (* I'm thinking in terms of the underlying motivations for the algorithm, optimisation strategies, generalisation capabilities, and run-time complexity)
Prove $p_n(x) \rightarrow \sqrt{x}$ uniformly as $n \rightarrow \infty$ Define $(p_n)_n$ recursively by $p_0(x) = 0$ and $p_{n+1}(x) = p_n(x) + \frac{1}{2}(x - p_n(x)^2)$.  Prove that $p_n(x) \rightarrow \sqrt{x}$ uniformly for $0 \leq x \leq 1$ as $n \rightarrow \infty$.  I'm not sure where to go with this.  I can show inductively that $\forall n,\  0 \leq p_n(x) \leq 1$ for $0 \leq x \leq 1$, and I think that $p_n(x)$ is increasing with $n$, but I have no idea whether this helps me.  I know that I need to show that $\forall \epsilon&gt;0: \exists N \in \mathbb{N}: \forall n \geq N: \forall x\in [0,1]: \big|p_n(x) - \sqrt{x}\big| &lt; \epsilon$ Edit:  I just had a thought.  I'm given a version of Dini's theorem that reads  Let $X$ be a compact metric space, $(f_n)_n$ a sequence of real-valued continuous functions on $X$, $f$ a continuous function on $X$ such that: (i) $f_n \rightarrow f$ pointwise on $X$, and (ii) $f_n(x) \geq f_{n+1}(x) $ for all $x \in X, n \in \mathbb{N}$. Then: $f_n \rightarrow f$ uniformly on $X$.  Obviously, (ii) is not true for my sequence, but what if I took $f_n(x) = -p_n(x)$ and $f(x) = - \sqrt{x}$?  Then I could prove (ii) to be true, I know that $[0,1]$ is compact, and $f_n(x)$ is real-valued.  The only part I would need to prove is (i), which is where I get confused.  Correct me if I'm wrong, but after that I could say that since $(-p_n)_n \rightarrow -\sqrt{x}$, then $(p_n)_n \rightarrow \sqrt{x}$.  Any advice? Is an apostrophe with a decade (e.g. 1920’s) generally considered “incorrect”? I typically don’t use an apostrophe with plurals in any situation, but I always assumed that the use of an apostrophe in constructions like acronyms:     Forty BA’s were given out to students this year.   or numbers:     Though the greatest period instability occurred during the 1950’s when. . . .   was more or less ok. I would personally use BAs and 1950s here, but I've never thought the apostrophe in this case was necessarily “incorrect”. What say ye? Moment of inertia of a cylinder about its base I've tried to find the moment of inertia of a cylinder rotating about an axis parallel to its base (i.e about the 'End diameter') as one can see  . But when I checked my results with different references ,I've found that it's incorrect!.I need a help to figure out where I did it wrong. since $$I=\int\limits x^2.dm$$ $$dm = \rho.dv$$ where $$dv= r.d\theta . dr .dh$$ &amp; $$\rho=\frac{M}{\pi R^2 H}$$ $M$:cylinder total mass. $H$:total height of cylinder. $R$:the radius of the cylinder. The distance of each infinitesimal element from the axis of rotation would be : $$\sqrt{r^2+h^2}$$ Therfore, $$I= \frac{2M}{ R^2 H}\int\limits_{0}^{R} \int\limits_{0}^{H}\, r(r^2+h^2)dh\,dr$$ and this gives a result of : $$\frac{M~H^2}{3}+\frac{M~R^2}{2}$$ Which is obviously wrong since the second term should be multiplied by a factor of $1/2$. Look at the profile page of , who has just "lost" 100k reputation and all their bronze badges, because of this broken design.  Shouldn't the primary usercard be kept intact no matter how other elements grow?  P.S. So many design bugs these days, did SE do a large CSS refactory?  P.P.S  that squeezes the right pane ( to reproduce) How to test for minimum amount of items in inventory and execute a command if true I am using the AmazingKit plugin on my Minecraft server to make inventory kits for players. In a command block I have the following command:  /tellraw @p ["",{"text":"Iron","clickEvent":{"action":"run_command","value":"_____"}},{"text":" / "},{"text":"Gold","clickEvent":{"action":"run_command","value":"_____"}}]   Which prints the following result: Iron / Gold.  When a player clicks on Iron, what command should I write where the empty space is _____ in order to check if the player has at least (>=) 20 iron ingots in his inventory and if he/she does, remove 20 iron ingots and execute /kit Basic. If the player has less than 20 iron ingots in inventory, the command won't be executed. Instead, Insufficient amount of iron ingots will be printed out.   Same applies when a player clicks on "Gold".  I'm on 1.14 version A while ago, I had a discussion with a colleague about how to insert values in STL . I preferred map[key] = value; because it feels natural and is clear to read whereas he preferred map.insert(std::make_pair(key, value)). I just asked him and neither of us can remember the reason why insert is better, but I am sure it was not just a style preference rather there was a technical reason such as efficiency. The  simply says: &quot;Strictly speaking, this member function is unnecessary: it exists only for convenience.&quot; Can anybody tell me that reason, or am I just dreaming that there is one? I'm setting up a Cronjob that will backup a MySQL database I have in my server, but I don't want it to keep overwriting the same file over and over again. Instead, I want to have an array of backups to choose from, done automatically. For example:  ## Cronjob, run May 21st, 2011: mysqldump -u username -ppasword database &gt; /path/to/file/21-03-2011.sql  ## SAME Conjob, run May 28th, 2011: mysqldump -u username -ppasword database &gt; /path/to/file/28-03-2011.sql   And so on.  Is there any way that I can use the system date and/or time as some kind of variable in my Cronjob? If not, what are your suggestions to accomplish the same? I really can't understand what Leonard Susskind means when he says in the video  that information is indestructible.  Is that information that is lost, through the increase of entropy really recoverable?   He himself said that entropy is hidden information. Then, although the information hidden has measurable effects, I think information lost in an irreversible process cannot be retrieved. However, Susskind's claim is quite the opposite. How does one understand the loss of information by an entropy increasing process, and its connection to the statement “information is indestructible”.  Black hole physics can be used in answers, but, as he proposes a general law of physics, I would prefer an answer not involving black holes. I found an asnwered question concerning this topic already here:  Copying \captionsetup[subfigure]{justification=centering} doesn't help me thought for I am using different packages.  I would like my Captions and Subcaptions to be justified as the text below. Just under my pictures. Any suggestions?  Thanks a lot  \documentclass{article} \usepackage{subfig} \usepackage[labelfont=bf, justification=justified]{caption} \usepackage[demo]{graphicx}% Remove demo option in real document  \begin{document}     \begin{figure} %   \captionsetup[subfigure]{justification=centering}     \centering       \subfloat[Subcaption one \newline including a linebreak ]   {     \includegraphics[width=0.2\textwidth]{image1}     }     \hspace{1cm}     \subfloat[Subcaption two \newline including a linebreak]     {     \includegraphics[width=0.2\textwidth]{image2}     }        \caption{Figure caption \newline with additional text right here} \end{figure}  \begin{center} This is correctly justified Text \\ including a longer linebreak than blabla. \end{center} \end{document} Is a set without limit points necessarily closed? According to the definition on Rudin' Principles of Mathematicial Analysis, closed set is defined as:      $E$ is closed if every limit point of $E$ is a point of $E$.   Then I have a question: if a set has no limit point, is it necessarily closed? I think this idea doesn't contradict the definition of closed sets. Am I right?  And another question is #10 in Chap.2 of Rudin's:      Let $X$ be an infinite set. For $p\in X$ and $q\in X$, define    $d(p,q)=0$ when $p=q$. Otherwises,$d(p,q)=1$.    I have known that every set in the metric space is closed. To prove it, we can show that the complement of any set is open, which is quite trivial. My question is: how can we prove that every set is closed just by checking according to the definition of closed set, i.e. how can we show that every limit point of any arbitrary set $E$ in the metric space is a point of $E$?   Thanks in advanced! Can't use String#trim as a callback for Array#map What are the requirements to use iMac screen as a second monitor - Target Display Mode? I have a Surface Pro 4 and an iMac (late 2019). I would like to connect my Surface Pro to the iMac to use the iMac screen as a second monitor.  Is that possible?  What are the requirements to use an iMac as a second screen? Would depositing large cash amounts cause problems? I saw  recently where a man withdrew $600,000 USD in cash from his bank.  I'm sure many folks here would advise against this move for obvious (and perhaps not-so-obvious) reasons.   My question: suppose you did this and then, some months later, changed your mind and wanted to re-deposit with a bank.  I assume the gent cannot simply walk into a bank with $600,000 USD and deposit it, right?  Wouldn't money-laundering alarms be raised?  Wouldn't the IRS like to know where the money came from? Space faring human race encounters one highly advanced spaceship.  The one ship tells the human space fleet to go away and leave them alone. We don't listen and our ships are destroyed.  Human spy is sent to the alien world. Aliens look virtually identical to humans except they have no cartilage in their nose and the 'females' are only capable of mating every seven years. The human spy tells this story and is a brilliant tactician. He enters into a game tournament where they play a very complicated version of chess.  Human spy interacts with a higher ranking military officer who has a human female wife.  Human spy gets caught and when he is ultimately returned to Earth he advises humanity to unconditionally surrender to the alien race.  Humans eventually win because the alien males take human wives and their alien DNA is diluted in the vast human gene pool.  Looking for the title and author if anyone can recall them.  06/15 - update  I found the anthology. 'Second Game' was in 'The Arbor House Treasury of Great Science Fiction Short Novels' which came out in 1980 and 1981.  I did read the original story from 1958 which was actually a nominee for the Hugo Award for Novelette in 1959. In Revenge of the Sith, we learn that Anakin's final seduction to the Dark Side occurs because of his obsession with power and especially his obsession with the power to cheat death to save Padme.       PALPATINE: Only through me can you achieve a power greater than any Jedi. Learn to know the dark side of the Force, Anakin, and you will be able to save your wife from certain death.   (Note, save your wife from certain death, not resurrect)  This obsession is motivated by Anakin's repeated visions of Padme's death during childbirth.   Also, we have     PALPATINE: To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret.   Yet when Padme confronts Anakin on Mustafar, and then Luke and Leia are born, they are full term or close to it.   So how did Anakin think this would help?   Did he expect to figure out this power with Sidious in a matter of days or weeks? Deriving Heaviside-Feynman formula for the electric field of an arbitrarily moving charge from Lienard-Wiechert potential Substitute for onions and garlic I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm:  given $x\in[1,2)$, do the following:  L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$.  L2: [Test for end] If $x=1$, stop.  L3: [Compare] If $x-z&lt;1$, go to L5.  L4: [Reduce values] Set $x\leftarrow x-z$, $z\leftarrow x/2^k$, $y\leftarrow y+\log_b(2^k/(2^k-1))$, and go to L2.  L5: [Shift] Set $z\leftarrow z/2$, $k\leftarrow k+1$, and go to L2.  (The algorithm needs an auxiliary table which stores $\log_b2$, $\log_b(4/3)$, and so on, to as many values as the precision of the computer.  Then Knuth concludes: this exercise is to explain why the above algorithm will terminate and why it computes an approximation of $y=\log_b x$.  Well, I see that it works, but I am not able to explain why... Enumerated description list How can a list be created, whose items are both    enumerated (a,b,c,d,...)  and allow for setting a description via \item[texthere] ?   Im am using MikTex on Windows. Passport validity requirement to enter USA for travelling