Prove that in an disordered set of 50 numbers, you can always remove 42 numbers in a way that leaves an ordered set of 8 numbers.
What are some important applications of the Erdős-Szekeres Theorem?
NetGear A6100 AC600 wireless adapter not working in Ubuntu 20.04. Worked fine in 18.04. Not same problem as previously Firstly, I know very well there are answers to this already. But, in my case, it is installed and shows in the terminal as such Bus 001 Device 007: ID 0846:9052 NetGear, Inc. A6100 AC600 DB Wireless Adapter [Realtek RTL8811AU]  It says it is installed in  as well.  I have tried downloading the driver for this adapter from several guides online, linking to GitHub, but no luck there either...  Any help is greatly appreciated! Thanks in advance!  EDIT: The answer from sudo dkms status command reads:  rtl8812au, 4.3.8.12175.20140902+dfsg, 5.4.0-31-generic, x86_64: installed  The other command  modinfo 8812au | grep 9052   spits out the following:  alias:          usb:v0846p9052d*dc*dsc*dp*ic*isc*ip*in* Removing all \n\r characters from a node XSLT? wondered if you could help me please? I have node in xml that is as followed  $LOG: 08880xbpnd $ fhdsafidsfsd df sd fsd f sd fsd  I was wondering is there anyway to make all the text go on to one line so that it then can be passsed through to a javascript function? so it would turn out like this   $LOG: 08880xbpnd $fhdsafidsfsddfsdfsdfsdfsd Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ I haven't been able to do this exercise:     Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse   image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \subseteq B$.   Since $X \subseteq B$, all the elements in $X$ belong to the codomain of $f$.  Since $f$ is surjective, it means that all elements in the codomain $B$ have some preimage in $A$. Given that $X \subseteq B$, all elements in $X$ must also have a preimage in $A$.  Have $\triangle = f^{-1}(X)$, $\triangle$ is now a set containing the preimages of the elements in $X$. Because of this, $\triangle \subseteq A$.  If we evaluate $f(\triangle)$, we...... nope, I don't know what I'm doing now.    What do you think? Is it worth self-funding a PhD to attend a top 10 university? What should I opt for?   Option A) very well paid PhD, around €3000 per month at a respectable university or;  Option B) a non-paid PhD, where I have to pay €2000 per year at one of the top 10 universities of the world?  EDIT: Actually they are both UK universities, Option B  is ranked between the top 10 in the world in engineering and tech, and I do get a scholarship that pays 18000 pounds which covers the difference between international student tuition and EU tuition. In Option A, I will be doing research in a good university and will be working alongside giants in the industry. I would like to see an explicit example of a non-trivial elementary extension of the structure $(\mathbb{N}, +, \cdot, 0, 1)$ where $\mathbb{N}$ includes zero. Most of all I am interested in countable ones. UITextField - capture return button event Can I programmatically open the devtools from a Google Chrome extension? Who killed Ra's Al Ghul first, Batman or Arrow? I am trying to write a script that will delete all comments and everything in between inside C files in my current directory. I've been using sed, and this is what I have so far:   sed -i '/ * [^()] */d' *.c  This works when the comments are on the same line as an asterisk or backslash.   However it doesn't work when there is a commented line without a slash or asterisk.   I know sed goes line by line, I just don't know how to tell it to keep deleting until it sees a */. "EOL while scanning single-quoted string"? (backslash in string) import os xp1 = "\Documents and Settings\" xp2 = os.getenv("USERNAME") print xp1+xp2   Gives me error   File "1.py", line 2  xp1 = "\Documents and Settings\"                                ^ SyntaxError: EOL while scannning single-quoted string   Can you help me please, do you see the problem? On the uniqueness of the Riemann-Christoffel tensor According to Section 6.2, Gravitation and Cosmology by Weinberg, the Riemann-Christoffel tensor is the only tensor that can be constructed out of the second (or lower) order derivatives of the metric tensor and is linear in the second order derivatives. The reasoning behind the same goes like this:   In a class of frames where $\Gamma^{\lambda}_{\mu\nu} =0$, the transformation rule for $\dfrac{\partial \Gamma^{\lambda}_{\mu\nu}}{\partial x^{\kappa}}$ involves an inhomogeneous term which is symmetric in $\mu$, $\nu$, and $\kappa$. Thus, if one is to construct a tensor which is a linear combination of the first order derivatives of the Christoffel symbol then the only way to do so is by eliminating the inhomogeneous part of the transformation and this could be done only by making the combination explicitly antisymmetric in $\mu$ and $\kappa$. Since in these frames, $R^{\lambda}_{\mu\nu\rho}$ $ = \dfrac{\partial \Gamma^{\lambda}_{\mu\nu}}{\partial x^{\rho}} -  \dfrac{\partial \Gamma^{\lambda}_{\mu\rho}}{\partial x^{\kappa}} $, $R^{\lambda}_{\mu\nu\rho}$ is the only tensor that can be formulated using the second (or lower) order derivatives of the metric tensor and is linear in the second order derivatives.  I think the presented argument can only suffice to prove that $R^{\lambda}_{\mu\nu\rho}$ is the only tensor that can be formulated from the first derivatives of the Christoffel symbol and is linear in them. I can't figure out why this suffices to assert that $R^{\lambda}_{\mu\nu\rho}$  is the only tensor that can be constructed out of the second (or lower) order derivatives of the metric tensor and is linear in the second order derivatives.  Edit: As the metric is covariantly constant, $\dfrac{\partial g_{{\mu}{\nu}}}{\partial x^{\rho}} = \Gamma^{\kappa}_{{\mu}{\rho}}g_{{\kappa}{\nu}}+\Gamma^{\kappa}_{{\nu}{\rho}}g_{{\kappa}{\mu}}$.  Therefore,   $\dfrac{\partial^2 g_{{\mu}{\nu}}}{\partial x^{\xi}\partial x^{\rho}} = \bigg(\dfrac{\partial \Gamma^{\sigma}_{{\mu}{\rho}}}{\partial x^{\xi}} + \Gamma^{\sigma_1}_{{\mu}{\rho}}\Gamma^{\sigma}_{{\sigma_1}{\xi}}\bigg)g_{{\sigma}{\nu}}+\bigg(\dfrac{\partial \Gamma^{\sigma}_{{\nu}{\rho}}}{\partial x^{\xi}} + \Gamma^{\sigma_1}_{{\nu}{\rho}}\Gamma^{\sigma}_{{\sigma_1}{\xi}}\bigg)g_{{\sigma}{\mu}} + \bigg(\Gamma^{\sigma}_{{\mu}{\rho}}\Gamma^{\sigma_1}_{{\nu}{\xi}}+\Gamma^{\sigma}_{{\nu}{\rho}}\Gamma^{\sigma_1}_{{\mu}{\xi}}\bigg)g_{{\sigma}{\sigma_1}} $  Now, if I want to show that the Riemann-Christoffel tensor is the only (non-trivial and independent) tensor that can be formulated out of the linear combinations of the second order derivatives of the metric tensor then I can equivalently show that the above expansion can be expressed linearly in terms of the Riemann-Christoffel tensor. But I am stuck over how to do it. Also, I am a little bit unclear as to what a linear combination means in this context. The coefficient of the second order term can be only a scalar constant or can it be the metric or can it be the metric that is getting summed over any of the indices as well? Do you really answer "How do you do?" with "How do you do?" Most common method for deciding when to stop training a neural net on a batch I have created my own neural net which is using batch gradient descent. In other words, it trains on batches of examples all at once.  My issue is trying to figure out when to stop the training of the batch. I'll try to make things as understandable as possible since there are so many options, but please comment if you need clarification.  Each batch allows me to compute a list of output errors that correspond to the error the computer has on each example. Each error is a vector with elements equal to the number of output neurons.  Let each error vector be notated by e1, e2, e3, ... so the list of output errors is [e1, e2, ...]. Let an error vector in general be notated by e.  Now there are 2 things we can do to a vector or a list:   Average: average all of the elements in a vector or list. This will be notated by the function a(). For example to find the average error of an example we would do a(e) Check: make sure that each element in the vector or list is below a certain value. If any of the elements are too high, we continue to train on the entire batch. (This function can also be applied to a single value.) This will be notated by the function c(). For example, if we wanted to make sure the average error in e was low enough, we would do c(a(e)) Sum: add up all of the elements in the vector or list. This will be notated by the function s(). For example, s(e) would be totaling the error found in error vector e.   Here are the methods I have determined for deciding when to stop gradient descent:   Iterative: just run the training a certain number of times Average of averages: Find the average error of each example, then find the average of those and make sure it's below a certain amount. This is given by the formula c(a([a(e1), a(e2), ...])) Specific error: Check to make sure the error in each output neuron is below a certain amount. This is given by the formula c(e1), c(e2), ... Specific averages: Check to make sure the average error in each example is below a certain amount. This is given by the formula c([a(e1), a(e2), ...]) Sum of sums: Total up all the error in the entire batch and then make sure it's low enough. This is given by the formula c(s([s(e1), s(e2), ...]) Sum of averages: Find the average error of each example, then add up all those and make sure it's below a certain amount. This is given by the formula c(s([a(e1), a(e2), ...])) Average of sums: Compute the average total error in each example, and then make sure it's below a certain amount. This is given by the formula c(a([s(e1), s(e2), ...])) Specific sums: Check to make sure the total of the error in each example is below a certain amount. This is given by the formula c([s(e1), s(e2), ...])   So my question is this:  Which one of these options is most commonly used when training neural nets? Why? (Is there another option that I've missed?)  (Edit: I have found a very general version of this question , where the answers seem to say there is no specifically good way. If that is still true, please say so in your answer. Note that this is not a duplicate, as I explain the specific methods I would like to use.)  Edit: as A.D pointed out, summing is pretty useless because it depends on how many values are in each list/array/batch. And if you know how many examples there are, you can just use averaging. So, the methods involving "summing" have been cut. Must an uncountable subset of R have uncountably many accumulation points? If temperature makes particles vibrate faster, and movement is limited by the speed of light, then I would assume that temperature must be limited as well. Why is there no limit? Upon trying to upgrade from 10.10 to 11.04 all seemed to go well until the restart. This error message is what comes up:  Kernel Panic - not syncing: VFS: Unable to mount root fs on unknown-block(0,0)   How do we fix that? I have researched and found out long time no see is a direct translate from the Chinese phrase, 好久不见. As much as we use it almost everyday in our lives, is it actually grammatically correct? Since it is a direct translate, is it grammatically approved? register_globals error in php I was stuck up with the error   directive 'register_globals' is no longer available in PHP in unknown on line 0   when tried to check the php version using "php -v" after enabling register_globals in php.ini file. I am not getting any php version info by doing so. Instead it throws the above mentioned error.After turning off this option, php info works quite well. It is very essential for me to have register_globals to be turned on.How can I have this corrected.  my php.ini is as follows:   ; Default Value: None ; Development Value: "GP" ; Production Value: "GP" ; http://php.net/request-order request_order = "GP"  ; Whether or not to register the EGPCS variables as global variables.  You may ; want to turn this off if you don't want to clutter your scripts' global scope ; with user data. ; You should do your best to write your scripts so that they do not require ; register_globals to be on;  Using form variables as globals can easily lead ; to possible security problems, if the code is not very well thought of. ;  register_globals = On  ; Determines whether the deprecated long $HTTP_*_VARS type predefined variables ; are registered by PHP or not. As they are deprecated, we obviously don't ; recommend you use them. They are on by default for compatibility reasons but ; they are not recommended on production servers. ; Default Value: On ; Development Value: Off ; Production Value: Off ;  register_long_arrays = Off  ; This directive determines whether PHP registers $argv &amp; $argc each time it ; runs. $argv contains an array of all the arguments passed to PHP when a script ; is invoked. $argc contains an integer representing the number of arguments ; that were passed when the script was invoked. These arrays are extremely ; useful when running scripts from the command line. When this directive is ; enabled, registering these variables consumes CPU cycles and memory each time ; a script is executed. For performance reasons, this feature should be disabled ; on production servers. ; Note: This directive is hardcoded to On for the CLI SAPI ; Default Value: On ; Development Value: Off ; Production Value: Off ; register_argc_argv = Off  ; When enabled, the SERVER and ENV variables are created when they're first ; used (Just In Time) instead of when the script starts. If these variables ; are not used within a script, having this directive on will result in a ; performance gain. The PHP directives register_globals, register_long_arrays, ; and register_argc_argv must be disabled for this directive to have any affect. ; auto_globals_jit = On Testing for linear dependence among the columns of a matrix I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be positive definite.)  My hypothesis is that at least one security is linearly dependent on other securities. Is there a function in R that sequentially tests each column a matrix for linear dependence?  For example, one approach would be to build up a correlation matrix one security at a time and calculate the determinant at each step. When the determinant = 0 then stop as you have identified the security who is a linear combination of other securities.  Any other techniques to identify linear dependence in such a matrix are appreciated. How to prove that there are finitely many $n$ such that $f(1+1/n) = \frac{1}{n+1}$? Let $f : \mathbb{C} \to \mathbb{C}$ be analytical. How does one prove that there are finitely many positive $n$ such that $f(1+1/n) = \frac{1}{n+1}$ ? A hint is given: think of the expression $\frac{z-1}{z}$.  I think this is a pretty hard homework assignment. Here's what I tried so far: Let $g(z) = \frac{z-1}{z} $ . Then $g(1 + 1/n) = \frac{1 + 1/n - 1}{ 1 + 1/n} = \frac{1/n}{1+1/n} = \frac{1}{n+1} $. So the given hint is probably indeed useful, but I'm not sure yet how.   Another thing I tried to use to obtain the solution was the given fact that $f$ is analytical. Let $f = g$. Then $f(z) = f(x+iy) = \frac{ x + iy - 1}{x+iy} = \frac{x+iy-1}{x+iy} \cdot \frac{x-iy}{x-iy} = \frac{x^2 - 2 i x y + y^2 + iy}{x^2 + y^2} = 1 + i \frac{y(1-2x)}{x^2 + y^2} $ . Now I have written $f$ in the form $ f = u + i v$, with $u,v$ being real-valued functions. So now we can use the Cauchy-Riemann Integral formulas: $ \frac{d}{dx} 1 = 0 = \frac{d}{dy} \frac{ y(1-2x) }{x^2+y^2} \implies \frac{y(1-2x)}{x^2+y^2} = c $. The other formula yields a similar result.   But now what? I still have no idea how to prove that there are finitely many positive $n$ such that the aforementioned equality holds. Should I use any of the results I already obtained? If so, how? If not, do you know how to prove this?