I'm learning NN and trying to understand the implication that architecture selection has in the model. Am I inferring correctly that:   Wider neural networks can approximate more interactions between input variables Deeper neural networks can model more complex nonlinearities
I haven't seen the question stated precisely in these terms, and this is why I make a new question.  What I am interested in knowing is not the definition of a neural network, but understanding the actual difference with a deep neural network.  For more context: I know what a neural network is and how backpropagation works. I know that a DNN must have multiple hidden layers. However, 10 years ago in class I learned that having several layers or one layer (not counting the input and output layers) was equivalent in terms of the functions a neural network is able to represent (see Cybenko's ), and that having more layers made it more complex to analyse without gain in performance. Obviously, that is not the case anymore.  I suppose, maybe wrongly, that the differences are in terms of training algorithm and properties rather than structure, and therefore I would really appreciate if the answer could underline the reasons that made the move to DNN possible (e.g. mathematical proof or randomly playing with networks?) and desirable (e.g. speed of convergence?)
How do I deal with a compromised server? What happens to the view.frame.size attribute after a CGAffineTransform (Rotate)? Help! My master database is corrupt, I can't even bring the SQL instance online! What are my options to get my server back up?   I do have a backup of master, but the  asks me to start the instance in single-user mode, which I can't do!  (Note: I'm leaving this question unspecified as to SQL version, so as to be a more widely-applicable reference. There are some similar questions on DBA.SE, but none that involve the server unable to start.) Joining two rigged models I currently trying to join two rigged (and animated) models, which both has separate meshes, separate bones and separate animations. The first mesh is the horse while the second mesh is human:   So I am interested to join human model onto horse model to make a rider:   I think the best way would be to make horse as a parent and human as a child during the join process. In that case I would like that all horse animations (there are 3 of them - idle, walk and run) would remain as they are, while I can lose human animations, as I will need different ones for it. I was also thinking to make human root bone as a child of one horse back bones, that when horse moves up and down it would move entire human mesh together. It would be also good to keep the same weights that human bones would not start deforming horse mesh and vice versa.  I tried to use Ctrl+J by selecting various combinations between bones and meshes in object mode, however, I noticed that it messes up scaling and rotation of the child object. So I was wondering how it would be possible to properly join these models to create a rider? Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$ Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$  I've tried subbing in heaps of values but I keep getting things like $f(0)=f(0)$ and other such useless results.  Any help would be hugely appreciated. What do the nomodeset, quiet and splash kernel parameters mean? What do Kernel command line parameters (also also known as boot options) mean?  When you go to Ubuntu (or better, GRUB) options, you have things like nomodeset, quiet and splash. What do these options mean?  I am also interested in other parameters and would be interested to see descriptions of any kernel parameter (like acpi), but these are not essential. In the book I'm reading at the moment () it says that if you don't declare a default constructor for a class, the compiler supplies one for you, which "zeroes out each data member". I've experimented with this, and I'm not seeing any zeroing -out behaviour. I also can't find anything that mentions this on Google. Is this just an error or a quirk of a specific compiler? What is the difference between a symbolic link and a hard link? Recently I was asked this during a job interview. I was honest and said I knew how a symbolic link behaves and how to create one, but do not understand the use of a hard link and how it differs from a symbolic one. Resizing an iframe based on content I am working on an iGoogle-like application. Content from other applications (on other domains) is shown using iframes.   How do I resize the iframes to fit the height of the iframes' content?  I've tried to decipher the javascript Google uses but it's obfuscated, and searching the web has been fruitless so far.  Update: Please note that content is loaded from other domains, so the  applies. I am running QGIS 2.18.14 (QGIS code revision: ff83b9a479) on a windows 7 machine, it comes with GRASS 7.2.2 (Code Revision r71496), but it does not come with r.in.lidar. I have noticed QGIS 2.18.16.1 has the same problem and it shares the same GRASS 7.2.2 (Code Revision r71496).  I know there were some issues with it and later solved as I read , I also certailny used it short ago in 2017, so I wonder which version should I go for before starting the always risky attempt of a new installation. Method to create a live USB disk (with persistence) which actually works? I've tried so many combinations - all I want is a fast Ubuntu Live USB disk with persistence that works. What I've tried so far:   Using unetbootin on my Mac, choosing some "Space used to preserve files across reboots (Ubuntu only)".   Result: Doesn't boot at all.  Using unetbootin on my Mac, choosing no such space.  Result: Boots fine and very fast (doesn't even show the annoying dialog "Try ubuntu/install ubuntu" at startup!), but has no persistence. (So I have to choose my keyboard layout, install programs I need, etc. every time)  Using Startup Disk Creator from my Ubuntu 11.04 installation.  Result: I cannot choose "reserved extra space" at all. It's greyed out. I know I know, there was bug , and many people say that the workaround from  works for them. But it doesn't work with Ubuntu 11.04 anymore (see also ).  Using Startup Disk Creator from my Kubuntu 10.10 installation, gnome version (usb-creator-gtk).   Result: I can select the "reserved extra space" option, but installation doesn't work (fails when installing the bootloader).  Using Startup Disk Creator from my Kubuntu 10.10 installation, kde version (usb-creator-kde).   Result: Actually works: Reserved space can be selected, installation works. But after booting, I get the screen "Try ubuntu/install ubuntu". Clicking on "Try ubuntu" takes me to the desktop - but only after waiting 5 minutes (every time!). It also doesn't seem to keep my keyboard settings.    I know, that instead of creating a startup disk, I could install a full ubuntu system on the USB stick. But as far as I know, this isn't very good, because /tmp and similar directories will then also be on the USB disk, which is a) slow, and b) not very good for the USB disk due to frequent overwriting.  I'm frustrated. So has anybody managed to create a working Ubuntu 11.04 Live USB disk with persistence? tense and time and composite sentence If the eating meal starts at 7:00 AM and finishes at 8:00 AM. the drinking liquor starts at 7:00 AM and finishes at 7:50 AM.  can I say that :   I drank the liquor when I ate the meal. I ate the meal when I drank the liquor.   Is I need to use progressive tense ? "need to do" vs "need do" Consider:  I need to do this. I need do this.  My English grammar knowledge tells me that &quot;need&quot; doesn't have the same status as the modal verbs &quot;may&quot;, &quot;can&quot;, &quot;should&quot; and what not. Hence the second usage where two verbs appear consecutively is incorrect. But yesterday, my native English speaking friends (Americans) told me that &quot;I need compute this.&quot; is a perfectly grammatical sentence, and one is simply omitting the &quot;to&quot;. How can this be? Is it a colloquial usage but grammatically incorrect, or is it grammatically correct? If it is grammatically correct, is it because &quot;need&quot; is a semi-modal verb? EDIT: in particular, is it okay to use &quot;need compute&quot; in a scientific paper? EDIT2: the exact phrase that raised the question was &quot;The advantage of this representation is that we need only compute sums and products&quot; I want to install Windows XP on my desktop computer (only XP), but my CD drive is not working.  I've only found installation tutorials on how to install it on the pen drive rather than from the pen drive.  How can I install Windows XP from a USB flash drive? I have 2 graphics cards on my laptop. One is IGP and another discrete.   I've written a shell script to to turn off the discrete graphics card.  How can I convert it to systemd script to run it at start-up? For the finite-dimensional case, we have a canonical isomorphism between $\mathbf{V}$, a vector space with the usual addition and scalar multiplication, and $(\mathbf{V}^*)^*$, the "dual of the dual of $\mathbf{V}$." This canonical isomorphism means that the isomorphism is always the same, independent of additional choices.   We can define a map $I : \mathbf{V} \to (\mathbf{V}^*)^*$ by $$x \mapsto I(x) \in (\mathbf{V}^*)^* \ \text{ where } \ I(x)(f) = f(x) \ \text{for any } \  f \in \mathbf{V}^*$$  My Question: what can go wrong in the infinite-dimensional case? The notes I am studying remark that if $\mathbf{V}$ is finite-dimensional, then $I$ is an isomorphism, but in the infinite-dimensional case we can go wrong? How? Correct usage of ‘on’, ‘at’ and ‘in’ Previously, someone posted his/her proof of "every neighborhood is open" . I am focusing on the same statement, but it seems that my approach is different from his/hers. Could someone comment on my own proof? Here we go:  Let $X$ be a metric space and $p\in X$. Fix $r&gt;0$. Define a neighborhood around $p$ as $N_r (p)=\{q\in X| d(p, q)&lt;r\}$. Now, take $w = \frac{r}{2}$. Then, $N_w (p)\subset N_r(p)$. In other words, every point of $N_r(p)$ is interior. Therefore, $N_r (p)$ is open. Since $r$ is arbitrary, this proves every neighborhood is open.  Update:  Under my definition of "neighborhood," can I prove in this way instead: For all $m\in N_r(p)$, take $w=\frac{r−d(m,p)}{2}$. Then, $N_w(m)\subset N_r(p)$.   Update2  From the hints given in comments, I solve it:  Assume $m\in N_r(p)$ and $q\in N_w(m)$. Take $w=r-d(m, p)$. Then, $d(p, q)\leq d(p, m) + d(m, q)&lt; d(p, m) + r-d(m,p) =r$.   Thanks to @JoséCarlosSantos and @fleablood ! Why doesn't [Required] work for me? MVC3 Would time freeze if you could travel at the speed of light?