How may one show that $|\sin x-\sin y|\le |x-y|$?
Show that $\left | \sin a - \sin b \right | \leq \left | a-b \right |$
Centre of Lie Algebra $sl_2(\mathbb{F})$ For $L=sl_2(\mathbb{F})$ i.e. matrices with trace zero,  what is the centre i.e. $Z(L)$= {$x\in L : [x,y]=[y,x] \ \forall\ y \in L$}.  I will have to find matrices $A \in L$ such that $AB=BA$ for all $B \in L$. But how to appoach this considering trace is zero here. Is there a way to find self-answers with the advanced search? Preferably, not using the  and including results of all time and not just the recent ones with the new share your knowledge, Q&amp;A-style This question exists as it fills a specific criterion.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will likely be closed per the . More information on the .   What kind of desktop environments or shells are there for Ubuntu users to install?  Please list a desktop environment or shell for each:   a description on why you like or suggest it (features, performance, etc.). a good screenshot, preferably of it running on Ubuntu and showing off some of its features. the minimal requirement for it to be usable, If there's any setting to lower it's requirement (Like kde's low-fat profile); how to enable the said setting some instructions on how to set it up if in the repositories, please provide a software centre link How can I get the list of all fields(API name,Label,type) of any sObject? I'm trying to build a Dynamic list(table) of records by selecting the Sobject and its fields.So I need to get all the fields with its API name,label,type.But I can't get the label and type.(Tried DescribeSObjectResult Class methods to get label,but it didn't help to get the label dynamically)  I tried it with the following code.  System.debug('Selection-----'+selectedObject); Map&lt;String, Schema.SObjectType&gt; gd = Schema.getGlobalDescribe();  Schema.SObjectType ctype = gd.get(selectedObject);  Map&lt;String, Schema.SobjectField&gt; fmap = ctype.getDescribe().fields.getMap();    List&lt;FieldWrap&gt; strList = new List&lt;WrapMap&gt;(); for(String s: fmap.keySet()) {     WrapMap wmp = new WrapMap();     wmp.name = s;     wmp.label = String.valueOf(fmap.get(s));     strList.add(wmp);  }   Where FieldWrap is my wrapper class with name and field of String types. I'm getting the List of API name of fields,but not the Label and type of it.Is there a way to get the label and type of field.If so, please help me. Input password into SSH without typing Does lower resolution result in better signal-to-noise ratio, given the same sensor size? With the release of the Canon EOS R5/R6, there have been many saying that they prefer the 20MP R6 camera over the 45MP R5 camera, due to its lower price but also because of &quot;better low-light performance.&quot; In camera comparison sites, I've also seen the sentiment that bigger pixels (lower resolution) = better low light performance. However, is this a myth? Here is my reasoning for this. Let's say that there are two full-frame sensors, one 20MP sensor and a 80MP sensor. According to many people, the 80MP sensor would be terrible for low-light photography since the pixels are smaller. However, each 2x2 block in the 80MP sensor could be mapped, and in post averaged to a 1x1 pixel, effectively making the image into a 20MP image. This could also be done in-camera as an option. Would this image have just the same amount of usable information as the image from the 20MP sensor? In other terms, since both sensors are 35mm sensors (i.e. equal area), the amount of light falling onto the sensor is the same. So, at the end of the day, there will be no difference between a 20MP image and a 20MP image that has been downsampled from 80MP, right? So their low-light performance would be the same. You even get an advantage from having the option to get 80MP photos if you wanted. Why do people say that a lower-resolution camera has better low-light performance, when it should theoretically be the same as one with higher resolution (all other factors equal)? Note: this question is about sensors with the same size but with different pixel density. Hence, it is different from  asking about different sensor sizes. I am aware that a full-frame camera will have better low-light signal due to its bigger area than an APS-C sensor, but what about the similarly sized sensors? Automatically keep current version of config files when apt-get install Could we send a man safely to the Moon in a rocket without knowledge of general relativity? How can I remove a suggestion from the Chrome address bar? I've got a couple of URL autocomplete suggestions where Chrome suggests the wrong URL consistently.   How can I delete the incorrect entry from the address bar's suggestion list? The page I want shows up too but it's never the first result. Which characters are valid in CSS class names/selectors? Why aren't hash values for "nothing" (empty) defined as all zeros? Any decent hash algorithm will produce basically "random like" values for any given input including the empty input.  For example MD5 produces d41d8cd98f00b204e9800998ecf8427e on an empty input.  I wonder, however, why the MD5 algorithm wasn't defined to have a final step where the output is XORed with d41d8cd98f00b204e9800998ecf8427e. This should not affect the cryptographic strength of the hash algorithm, and it would have the convenient property that a hash of all zeros represented a zero input.  Would it simply just because adding that step would slow the hash down a bit? Why was the Triwizard Cup used as the portkey? In Harry Potter and the Goblet of Fire the Triwizard Cup was used as the portkey to send Harry and Cedric to the graveyard in Little Hangleton to face Voldemort.    Why was the Triwizard Cup chosen?    Couldn't Crouch Jr. have used anything as a portkey; which would have been much easier and without as much risk (after all of their planning Cedric almost ended up getting the cup first). How to forward specific log file outside of /var/log with rsyslog to remote server? How can I forward message from a specific log file like /www/myapp/log/test.log with rsyslog client to remote rsyslog server? This log file is outside of the directory /var/log. In a  a doubt was voiced about my answer, which claims GCM to requires $2^{128}$ for a successful forgery. The doubt was that the square root needs to be taken meaning the security would be $2^{64}$.  So of course I immediately checked the relevant security theorem ():  $$\mathbf{Adv}^{\text{auth}}_{\operatorname{GCM}[\operatorname{Perm}(n),\tau]}(\mathcal A)\leq \frac{0.5(\sigma+q+q'+1)^2}{2^n}+\frac{q'(\ell_A+1)}{2^\tau}$$  with $\sigma$ being the total plaintext size in blocks, $q$ being the total number of encryption queries, $q'$ being the total number of decryption queries, $n$ being the size in bits of the underlying permutation, $\ell_A$ being maximum authenticated input length in blocks and $\tau$ being the tag size in bits.  Now clearly from this we can see that performing $2^{n/2}=2^{64}$ queries yields a strong enough advantage for a break.  Now my question is: When talking about "n-bits of security" or "needs $2^n$ operations to break", do we usually talk about "online" security, ie queries to an oracle can be done and cost $1$ or "offline" security where no oracles are available or do we need to make this decision depending on the situation?  TL;DR: Does GCM offer 64 or 128 bit security? Javascript Remoting - sending sObject as argument I was wondering what is the purpose of piping the downloaded file by curl into sudo -E bash - in the following shell:   curl -sL https://deb.nodesource.com/setup_5.x | sudo -E bash - I add video files to the video editor,  Both audio and video channels appear but only the audio plays.  The video is just a white box.  Anyone know what is wrong.  Added video in WMV.  Also tried MP4 same thing. Where were the cannons in final fight in Wakanda in Infinity War? In Black Panther, Ulysses Klaue describes his arm cannon as a leaf blower in comparison to the cannons/weapons that Wakanda has.  Infinity War comes after the events of Black Panther, so the weapons should exist. During the final fight with the children of Thanos and Space Dogs (as Rocket refers to them), Wakandans could have used these powerful cannons, pointing them at the opened section of barrier. Instead they chose a fist fight.  Even M'Baku says that it would be the end of Wakanda, so the magnitude is huge and risks are very high. Why were these powerful weapons not used?    Edit 1: I watched the movie again and planes are there in the fight. But they were only used to destroy the spaceships that crashed. Recursive feature elimination and one-hot & dummy encoding? When using RFE in linear regression and logistic regression, do we one-hot encode the features (K levels and K dummy features) or dummy-encode the features (K levels and K-1 dummy features leaving one out).   As per a comment by @Matthew Drury in an answer (URL below), one hot encoding is applied for a regularized linear model and for unregularized linear model dummy encoding. My doubt is what type of encoding when using RFE without any L1/L2 penalties.    My understanding is since in RFE some features gets eliminated so if for a categorical variable with say 4 levels we do dummy encoding and have 3 features/levels in model &amp; RFE eliminated 1, we will only have 2 features/levels left and the interpretation of its coefficient would not make sense in absence of the one level which was left out as reference.  Whereas if we have done one-hot encoding and RFE considers 2 features as important and eliminates other 2 then we can very well judge/interpret the coefficients or importance of 2 features RFE keeps.  So question which type of encoding is needed to be done when using RFE with linear and logistic regression? How run a .jar file with a double-click? I have a problem running .jar files by double-clicking them. When I go to the Properties dialogue I can't find Java--only Archive Manager in the list of programs. But java -jar somefile.jar opens the program.