Once someone asked me the question "What is linearity?" in a proficiency exam. I went hot and cold all over. Although, I heard and even used the term linearity many many times, I had not really thought about it until that time. After a hopeless discussion, he said linearity is the satisfaction of the following conditions:  $$f(x+y)=f(x)+f(y)$$ $$f(ax)=af(x)$$  Since then I have no idea about linearity. Because there are:   Linear equations Linear differential equations Linear algebra Linear programming Linear interpolation and so forth...   According to the definition even a straight line $y=mx+n$ cannot be considered linear as long as $n\neq0$. But there are so many linearities.   So, what is this term linearity?
I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear electrical circuits, linear filters, linear electrical elements, linear approximation, linear optimization.  I'm getting crazy trying to understand the application of the "linear concept" to all this aspects (function, equation, mapping, system, operator, transformation, algebra, etc.) and I wish to know the one essence that is to be linear. What something has to be to be linear? If I say something is linear, what do I know for sure about that something (no matter what the something is)?       I heard a definition of linearity is by homogeinity (scaling the input results in a scaled output) and addition (summing the inputs results in summing the outputs). Can I apply this simple definition to all the branches (operator, mapping, system, transformation, algebra, ...) I mentioned ? Do they all behave like a line ?  y = ax + b, for example, is a line but doesn't behave like a line because y is not linear.
Mars needs water and women I read this book in high school too many years ago for me to think about. Martian colonist are raiding earth's oceans for water and because of the conditions on Mars where only male children are conceived, they are also on the look out for young women as potential mates. Our main character while on a raid, bags not only a young woman but her brother as well. Returning to Mars they find that Earth has attacked Mars can't remember much more. Is it appropriate to use short form of "have" ('ve) when it means possession? I feel uncomfortable saying sentences like the following:    "I've a car" instead of "I have a car" "They've a great time" instead of "They have a great time" "He's a pen" instead of "He has a pen" etc   I ask this because I read this sort of thing in a book.  Are they correct? And what is the rule? Can you use such forms in a formal setting? How do you prevent an ENTER key press from submitting a form in a web-based application? How to prove $\vdash\neg P\to (P\to Q)$? I am trying to prove,  $$\vdash\neg P\to (P\to Q)$$  where $P$ and $Q$ are arbitrary but otherwise fixed formulas. The only axioms and rule of inference that I can use are,  $\color{crimson}{\text{Axiom 1.}}\  P\to (Q\to P)$ $\color{crimson}{\text{Axiom 2.}}\  (S\to (P\to Q))\to((S\to P)\to (S\to Q))$ $\color{crimson}{\text{Axiom 3.}}\  (\neg Q\to\neg P)\to(P\to Q)$ $\color{crimson}{\text{Rule of Inference.}}$ .  I know that in Angelo Margaris's book First Order Mathematical Logic there is a proof (page 53) but as I have mentioned , the dot notation seems very confusing and translating them to the nested parentheses notation seems very difficult to me. So, I am trying on my own to prove the result. Can anyone help? "Accountsd wants to use the 'iCloud' keychain" I, too, am suddenly getting this message.  I don't see a Keychain First Aid option in Keychain, and the two Login Keychain Settings were already unlocked.  Any other ideas for what the error message is all about, and what I should do about it? Linux network troubleshooting and debugging From time to time Linux and Unix users faced with various network problems. Many of these problems are presented here and at some others troubleshooting forums, but they are very concrete and contains a lot of additional technical information, and sometimes it's rather difficult to understand the main point and the real reason of buggy system behavior.   By asking this question, my intention is to start a community wiki page which allows generalizing our network troubleshooting and debugging experience. I hope the Linux and Unix users could easier recognize and solve("divide and conquer") their network problems using this page.   The parent of this page should be  . But here we should focus on troubleshooting the network problems from user- and kernel-space.  I suppose, if you:    Share the information about using some great network diagnostic tool with concrete usage examples and examples of network bugs, which they help to catch. Share the link to the great network tutorial connected with this subject Tell about a general method or recipe which allows to tackle some class of network problems Share information about your tool-set for network debugging and troubleshooting   it would perfectly fits for this topic.    I'll begin from sharing the link to varios  and . Also  seem to have actual information about our subject. And for diving into linux networking we definetely need to visit . let ${v_1}, {v_2} $ be orthonormal basis for subspace $W$. Prove that if $z$ is a vector in $W$, then Neutrino Oscillations and Conservation of Momentum I would like to better understand how neutrino oscillations are consistent with conservation of momentum because I'm encountering some conceptual difficulties when thinking about it. I do have a background in standard QM but only rudimentary knowledge of particle physics.   If the velocity expectation value of a neutrino in transit is constant, then it would appear to me that conservation of momentum could be violated when the flavor eigenstate at the location of the neutrino source is different from that at the location of the interaction, since they are associated with different masses.  For this reason I would think that the velocity expectation value changes in transit (for instance, in such a way to keep the momentum expectation value constant as the neutrino oscillates), but then it seems to me that the neutrino is in effect "accelerating" without a "force" acting on it (of course, since the momentum expectation value is presumed constant, there may not be a real problem here, but it still seems strange).   Any comments? Can't see a glass object through another glass object in Eevee I was just testing Eevee and found that you cannot view a glass object through another glass. Is there any workaround? It works fine only If I rotate the glass window or put my camera to the back side of the scene Which is correct: "the below information" or "the information below"? I frequently see statements that refer to something later in the text that use a phrase such as "the below information". Is it more correct instead to say "the information below" (or "the following information")? I am traveling from Chennai to Atlanta. M first flight is to Frankfurt reaches at 8 30 AM. My second flight is scheduled at 10 45 AM.   Will the two hours transit time enough for boarding my second flight.  What should be my do's and dont's?  Update  Its more than enough. I was able to make it without any tension. vim-clang-format: clang-format is not found My Folders Become Hidden System Files And Access Denied? I have an external hard disk , just connected to another system and when i connected to my own computer one of my folders which contain photos become hidden system files AND all folders in it rename to same name !?!? and my access is denied ( Screenshot )    I tried to change ownership but i can't  Any solution !? I'm unable to upgrade Xcode 11GM to 11.1 for some time on my Mid 2014 MacBook Pro running macOS Catalina. I've installed Xcode from Mac AppStore ages ago (IIRC circa 2014), and upgraded step by step since then. Currently, I'm experiencing the following loop:   AppStore reports that I have an update. I open AppStore, open updates, click "Update" on Xcode. Mac Downloads ~2.5GB first. Installs that patch. Continues to get the full package (~7.6GB). It starts to install. After some churning and install, it reports "Cannot download Xcode now, please try again later."   I have more than enough space on the disk, and nothing is modified in Xcode or in the OS (no homebrew or something similar).  Did anyone experience something similar? Are there solutions to this? QuickStart basic layout/template for web development? I'm looking for a quick start or template, or something, to get a web site's basic layout done quickly and looking fairly professional, and then let me do the coding.  I don't want a CMS, as this is for a highly customised reporting application, but I need something like frame header and two columns and frame footer.  I can design the pretty reports for the wide right hand column myself, as the left hand column menu. Unable to mount NTFS external hard drive I have been reading about the new Salesforce async triggers. Currently, I tend to handle these sort of needs with a trigger that passes data to a 'Handler class' that would evaluate if a certain change has occurred. If it has then we pass those Ids to a queueable class for further processing.   What advantage do CDC and async triggers offer over using queueable or batch apex? What is the best way to save values (like strings) for later use? I'm fairly new to LaTeX and I'm struggling with something apparently simple, but for which I can't find a fair solution. The problem (simplified) is as follows (please take care terms used may not have the traditional meaning in LaTeX):  I need to references to files (many) throughout a text. As example, the files:  "C:/Work/this_file1.c", "C:/MoreWork/this_file1.c" and "D:/MoreWork/another_file2.h".  So, now I would like to create an "alias" to each directory and each name file so I can reference them. A "pseudocode" could be:  % Definition \alias \c_dir = C:/Work/ \alias \d_dir = D:/MoreWork/ \alias \file1 = this_file1.c \alias \file2 = another_file2.h  % Reference in text \c_dir\file1    % For referencing C:/Work/this_file1.c \d_dir\file1    % For referencing D:/MoreWork/this_file1.c \c_dir\file2    % For referencing C:/Work/another_file2.h   My first approach was using \newcommand, but that caused numerous problems due to naming limitations. (I also tried other methods like pgfkeys, but it always gets stuck somewhere!)  Can you suggest a method? Why are my SVG imports so small? I'm definitely a newbie with blender still, as I've dipped my toe in a bit being a video editor.  My issue is importing .svg saved from illustrator.  Just simple vectors with not too many paths.  Just a thick text company logo.  I want to bring in and extrude, light, etc.  I've done it before in older versions of blender with no problem.  Am I not doing something right? Can someone tell me how to get an eps saved as an svg from illustrator to size correctly? XML attribute vs XML element