Can you please cite a book that treats deep learning from the mathematical perspective as I tried to search, but most of the books are either technical or too superficial in terms of the mathematics behind?
I am looking for a book that goes through the mathematical aspects of neural networks, from simple forward passage of multilayer perceptron in matrix form or differentiation of activation functions, to back propagation in CNN or RNN (to mention some of the topics).  Do you know any book that goes in depth into this theory? I've had a look at a couple (such as Pattern Recognition and Machine Learning by Bishop or Deep Learning by Goodfellow, Bengio and Courville) but still have not found a rigorous one (with exercises would be a plus). Do you have any suggestions?
Problem with a sequence with multiple integrals Given an infinite set of cardinality $X$, is there a chain in $\mathcal P(X)$ of the same cardinalty of $\mathcal P(X)$? $P(\mathbb N)$ = power set of $\mathbb N$. $A \subset P(\mathbb N)$ is a chain if $a,b \in A \implies$ either $a \subseteq b$ or $ b \subseteq a$ That is, we have something like this: $$\cdots a \subseteq b \subseteq c \subseteq\cdots$$ where $a,b,c \in A$ are distinct. We can show easy enough that there is an uncountable chain - this is done by noting $\mathbb N\sim\mathbb Q$ then using Dedekind cuts in $\mathbb Q$ to define $\mathbb R$ we see that a family of (nearly arbitrary) cuts satisfy the condition. For instance the family $L_r=\{q \in \mathbb Q : q &lt; r \}$ for $r &gt; 0$ gives us the sets we need and obviously we can pick others. I tried doing this for $ \mathbb R$ and don't seem to be getting anywhere.  To be more specific, does there exist a chain in $P(\mathbb R)$ with cardinality $2^ \mathbb R$? Given a set of cardinality $X$ (necessarily non-finite), is there a chain in $P(X)$ of the same cardinalty of $P(X)$? If you have an $m \times n$ matrix and the rows are linearly independent, are the columns linearly independent too?  I know that it's the case for $n \times n$ matrices. I've yet tried to think about it, and I think the answer is yes, for example the matrix: $\begin{bmatrix} 1 &amp; 2 &amp; 3 \\ a &amp; 2a &amp; 3c \end{bmatrix}$ has linearly dependent rows and columns, but I'm not sure that it works for every $m \times n$ matrix. "Overclocking" an AVR In AVR datasheets under the Electrical Characteristics section you will typically find a graph like this (this one is from the ATMega328):    I've seen designs that seem to "work" but operate outside the shaded envelope. Specifically, I've seen 3.3V (Arduino) designs that run the clock from an external 16MHz crystal. Clearly, this is out of spec. What are the practical negative consequences of running outside this envelope? What are bitwise shift (bit-shift) operators and how do they work? I have a rather long footnote which breaks a listing in two parts, unnecessarily.  Since the listing is more important then the footnote I'd rather lake to have the footnote split.   Can I customize (one or all) footnotes to have a maximal length before they are continued on the next page? Or can I manually split a footnote in two to get the effect? How to connect a GUI button to function? efficient and accurate approximation of error function I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z  e^{-t^2} \,\mathrm dt$$ I want to draw a perfect rectangle. How can I do that with QGIS v.3+?  .  Is there a similar solution for the latest QGIS version? How did Lucius Malfoy know Sirius's Animagus form? In Order of the Phoenix, Snape eggs on Sirius at Grimmauld Place:  ‘Tell me, how is Lucius Malfoy these days? I expect he’s delighted his lapdog’s working at Hogwarts, isn’t he?’ [Sirius] ‘Speaking of dogs,’ said Snape softly, ‘did you know that Lucius Malfoy recognised you last time you risked a little jaunt outside? Clever idea, Black, getting yourself seen on a safe station platform ... gave you a cast-iron excuse not to leave your hidey-hole in future, didn’t it?’ Order of the Phoenix - Chapter twenty-four, Occlumency - Page 460 - Bloomsbury  How did Lucius Malfoy know that Sirius was a dog Animagus? How did he know it was Sirius in the train station? Some facts to consider: According to the 's timeline, the characters who are considered to be of the Marauders era (born 1959) were only five years younger than Lucius Malfoy (born 1954), so Sirius and Snape, and the other Marauders-era students, and Lucius Malfoy, would have been at Hogwarts together for two years. Prisoner of Azkaban demonstrates it took Sirius, James, and Peter four to five years to master the Animagus transformation, which would have been at least two years after Lucius Malfoy would have left Hogwarts (Although we don't know when Lucius became a school governor.). A canon-based answer would be great; barring one, a speculative answer in the spirit of canon is welcome. We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$.   I'm now trying to say something about the possible cardinalities of subfields of $F$. I can see that there is a subfield of cardinality $p^s$ for each $s$ that divides $r$, given by the fixed field of the group generated by $\phi^s$, where $\phi$ is the Frobenius automorphism.  Now suppose $K$ is a subfield of $F$. Then (since both are additive groups), Lagrange gives us that $|K|$ divides $|F|$, so $|K| = p^t$ for some $1 \leq t \leq r $ (alternatively, $K$ contains $\mathbb{F}_p$ and so is a vector space over $\mathbb{F}_p$ and is thus isomorphic to $\mathbb{F}_p^t$, where $t = [K:\mathbb{F}_p]$). By considering the multiplicative group of units of $K$ and $F$ respectively, we get that $ p^t - 1$ divides $p^r -1$. I want to make the leap to $t|r$, but I'm failing to see why this needs to be true. Any help would be appreciated. Can I gift games from my library? I understand that once a game is in the library it cannot be removed, only hidden.  My question is, can I gift an uninstalled copy of a game to a friend?  I understand that this may be impossible due to legal issues, and that once a game is in the library it stays. If I have a Java source file (*.java) or a class file (*.class), how can I convert it to a .exe file?  I also need an installer for my program. Password hash function for Excel VBA Is there a rational number between any two irrationals? Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 &lt; i_2$.  Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$?  How would you construct such a rational number?  [I posted this only so that the useful answers at  could be merged here before that question was deleted.] Add a property to a JavaScript object using a variable as the name? The Matching Problem/Derangements - n letters to n people There are n letters addressed to n eople at different addresses. The n addresses are typed on n envelopes.  A disgruntled secretary shuffles the letters and puts them in the envelopes in random order, one letter per envelope.  Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.] What is the probability approximately, for large n ?    My attempt at a solution:  P(all correct) = $\frac{1}{n!}$  P(n-1 correct) = ${n \choose n-1} \frac{1}{n!} = \frac{1}{1!(n-1)!}$  ${n \choose n-1}$ ways of selecting n-1 correct letters.  P(n-2 correct) = ${n \choose n-2} \frac{1}{n!} = \frac{1}{2!(n-2)!}$  ...  P(n-n correct) = ${n \choose n-n} \frac{1}{n!} = \frac{1}{n!(n-n)!} = \frac{1}{n!}$  This all looks ok to me, but I tested this for a case of 4 people, (I think the theoretical value should be $\frac{9}{24}$ but obviously that is not what I got. Can someone correct my logic please? Also, with regards to the hint about inclusion/exclusion - where would that come in?  I read about derangements on Wikipedia but it did not make total sense to me (the derivation).  I would appreciate any advice whatsoever including references to other material. What does "Package  has no installation candidate" mean? How do I make the Messages that are sent to me stop going to my Mac? How to add a WiX custom action that happens only on uninstall (via MSI)? I would like to modify an MSI installer (created through ) to delete an entire directory on uninstall.  I understand the RemoveFile and RemoveFolder options in WiX, but these are not robust enough to recursively delete an entire folder that has content created after the installation.  I noticed the similar Stack Overflow question , but I was wondering if this could be done more simply using a call to a batch script to delete the folder.  This is my first time using WiX, and I'm still getting the hang of . What would be a basic example of a custom action that will run a batch script on uninstall?