Why does this inequality always hold true? $$\|x\|_2 \leq\|x\|_1$$ where $\|x\|_2$ is the euclidean norm : $\|x\|_2 = \sqrt{x_1^2 + \dots + x_n^2}$ , and $\|x\|_1$ the $L_1$ norm : $\|x\|_1 = |x_1| + \dots + |x_n|$.
Sorry if this is very basic but here's a question.   Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$.      Why do we have   $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|,  $$   where the left-hand side can be thought of as the $2$-norm $\|\mathbf{v}\|_2$ on $L^2(k^n)$?   $\mathbf{General \; case}$: If this is true for $p=2$-norm, I am guessing that this is true for all $p\geq 1$:  $$ \left( \sum_{i=1}^n |v_i|^p\right)^{1/p}\leq \sum_{i=1}^n |v_i|.  $$     In fact, does this inequality hold when $p$ is a rational number?
I've found several answers on switching between an Adapter and a Battery. But I need to autmoatically switch between two batteries. One of them is 'hotplugable'.  Problem: I've got a Solarcell which charges a ~16000mAh Li-Ion battery (BATa). Once in a while, the Solarcell doesn't manage to charge the battery enough. It's crucial that I have 3.0 to 4.1 Volts on my VSYS but the voltage will drop below this value if the Solarcell wasn't able to do its job (cloudy weather, vulcano eruption [joke], dirty cells, etc).  My Plan: I've come up with the idea to feed the VSYS with a additional, hotplugable backup battery (BATb) which does get connected to VSYS if BATa drops below ~3.1v.  So basically I need a Low-Voltage-Cutoff (using a Z-Diode with a Mosfet?) which, when cutoff happens, opens another Mosfet..?         &ndash; Schematic created using   I'm not sure about the selection of M2 and how to wire it, so it will be conductive if M1 won't forward power anymore.  And I know about another electrical problem: Just using a Z-Diode (D1) with M1 will render M1 to something like a resistor, not a switch. I've read, that I could use something like a Shut Reference (i.e. TI's TL43xx). Again I'm not sure how to chose the correct one and how to wire it. Easy way to concatenate two byte arrays In Ubuntu 18.04, when moving the windows there is a resize effect that occurs when accidentally touching one of the screen edges.  This control was available in CCSM and Ubuntu Tweak in previous versions with Unity.  I'm not finding it in GNOME Tweaks.  I also searched the repository with:  apt search ubuntu.*tweak How can I account for a nonlinear variable in a logistic regression? How can I account for a nonlinear independent variable in a logistic regression?  For example, consider this data set:  b1  b2  b3  b4  b5 1   1   0   0   20 0   1   0   0   20 1   0   0   0   13 1   1   1   0   8 1   1   0   1   5 0   1   0   1   4 1   1   0   1   5 0   0   0   0   8 0   0   1   0   13 1   0   0   1   20 1   0   0   0   29 1   1   0   1   40 1   0   0   0   53 0   1   1   0   68   Suppose I gathered some data like the above (suppose I had 10,000 rows) and I want to use them to predict future sets of similar data with an equation generated by a logistic regression. I get the above data. My independent variables are b1 through b5. By looking at the data, I can clearly see that independent variables b1 through b4 are binary (maybe they are male/female, rent/own, etc).   However I see that b5 is clearly not binary. Upon graphing it and looking at the numbers I determine that the b5 variable has a bathtub shaped U curve with an equation of b5=(x-4)^2+4.   I understand that using the b5 variable as it is would screw up my logistic regression - since any interval independent variables must have a linear relationship with the dependent variable.   Yet, I must account for the variable b5 in my data set while doing a logistic regression for it to be accurate and predict well (right?).  Given the above data, how would I go about making a predictive model with a logistic regression? Any references online or in books would greatly be appreciated as I am having trouble finding a good explanation of the above.  Cheers =)  EDIT:  Sorry if my question is unclear.  Consider this: Suppose we had the following data on many people:  Y: Whether they had a stroke before the age of 50 or not 1. Whether they own a pet or not. 2. Whether they graduated college or not. 3. Whether they are male or female. 4. The number of hours they exercise every week.  We want to train a logistic regression model based on historical data to predict if someone in the future with the same data will be more apt or not to have a stroke before the age of 50.  The problem I am having is with the last variable. A continuous ratio variable like "number of hours exercised each week."  Exercising more each week may have diminishing returns on health and thus having a stroke. For example decreasing an increasing rate. Or maybe (totally made up) exercising a little bit is really good for you, exercising a moderate amount is really bad for you, and exercising a lot is really good for you again. We might have a U shaped curve like (x-4)^2+4 for example when plotting health to exercise.   I would image that having a curved, non-linear independent variable in a logistic regression like the one I described above would cause problems. Am I right? And if so, what kinds of things can you do to still include the variable in the logistic regression. Perhaps a transformation? Is that all? Mark questions to receive notifications and updates? I want to run a command line that opens a new chrome window with another profile. I tested the following commands: Preconditions:  I have two profiles: Profile 1 and Profile 2  Chrome is already running with Profile 1   open -a &quot;Google Chrome.app&quot; --args --profile-directory=&quot;Profile 2&quot; The above command does nothing other than bringing focus to the currently running chrome with Profile 1 I also tried: open -a &quot;Google Chrome.app&quot; --args --profile-directory=&quot;Profile 2&quot; --new-window The effect is the same. I want to run a command line that opens a new chrome window with Profile 2 while there is already a Chrome window running with Profile 1 How to read ext4 partitions in Windows? I have two questions on ext4:   Is there currently a way to mount an ext4 partition in Windows (XP/Vista/7) to get basic read support? If not, are there any plans to support this someday? My final goal is to generate Impulse Response Functions in R.  I have variables that are non stationary when I set k = 5 in a Unit Root test, and they are cointegrated which to my understanding prompts the use of the VECM, from which the Vec2Var argument is used to then generate IRFs. However, my response functions from this methodology do not decay over time and mostly do not revert to the zero line.  Furthermore, I noticed that when I input first differenced variables into the VECM as opposed to level data above, response function do revert to the zero line. Therefore my first question is: 1) is it appropriate to use differenced variables in a VECM model?  Secondly, as an alternative I am considering restricting the number of lags used in the Unit Root tests so that the variables are not all non-stationary; in this case 2) would the use of VECM be void and the VAR with first differences be a more appropriate model? Again, using differenced data is giving me better response functions that revert to zero in the long run. However, 3) is it ok to use the VAR for stationary/non-stationary data in levels where these variables are still cointegrated?  Also, I am using growth rate variables, 4) should I still long transform all variables and use logs in all tests?  Thank you for the help! Explicit solution for linear regression with two predictors I have some samples of data of the form $x,y$ and $z=f(x,y)$. I wish to fit a plane $$z = Ax + By + C$$ to the data with the smallest mean square errors.  I have found an "answer" in section 3 of  document, but it is left in the form of some equations left to be solved. I just about have the ability to solve these equations, but the process gets so messy that the likelihood of me making a mistake is quite high. Surely someone has written out the full solution longhand somewhere (it might be called "closed form") anyway, in the form $$A=\ldots,$$ $$B=\ldots,$$ $$C=\ldots.$$  EDIT: Maybe "closed form" is the wrong phrase. So let me be clear. I want an explicit solution for $A$, $B$, and $C$, and not a solution that ends with "So now if you can solve these equations then you can discover the values of $A$, $B$ and $C$". matlab based program where output comprises of sum of rows of input Anime about a boy and a girl who ride ostrich-like creatures through a valley A method for evaluating sums/discrete functions by assuming they can be made continuous and differentiable? Is there a keyboard command for 'show desktop'? Like Windows, I want to hit Super+D to jump to the desktop or minimize all windows. I don't see command like that in Keyboard Shortcuts. Any idea? Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$? Does there exist a function $f:\mathbb R \to \mathbb R$ such that $\lim_{x \to a} f(x) = L$ for   all $a \in \mathbb R$ but $f(x) \neq L$ for all $x$?   I found such a function in $\mathbb Q \to \mathbb Q$, where $f(\frac{p}{q})= \text{first p+q digits of }\pi$ satisfies the above condition. However, I have not been able to extend it to reals. So, is such a function possible, and if it is, is there any explicit example preferably related to the above function in rationals? When did the 5-foot square/hex become the standard unit of measurement in Dungeons & Dragons? Trouble with batch conversion of .png to .pdf using convert convert /home/bill/TempScan/*.png  myfile.pdf   gives error message:  convert-im6.q16: not authorized `myfile.pdf' @ error/constitute.c/WriteImage/1037.   Any help would be appreciated! When to use "do" in phrases which are not questions? Example: You do need to drink more water.   or   You need to drink more water.   What is the difference? How can I tell which cards or sets are currently legal in various formats? To be short, I want to copy a folder to a location /usr/share/screenlets/.... in Ubuntu 10.04 system. I tried by logging in as root from terminal giving su.   I even changed my user account type to ADMINISTRATOR; Yet, no use. PASTE option in the context menu's list in the folder /usr/share/... is INACTIVE.  How can I copy those files? Best resources to learn bash scripting?