I'm trying to understand the classification of PDEs into the categories elliptic, hyperbolic, and parabolic. Frustratingly, most of the discussions I've found are "definition by examples.''   I think I more or less understand this classification in the case of quasi-linear second-order PDE, which is what's described on Wikipedia. I want to understand what makes a non-linear second-order PDE elliptic, parabolic, hyperbolic. MathWorld says something vague about this: "A non-linear PDE is elliptic at a solution u if its linearization is elliptic at u" &nbsp; (). What is meant by "linearization" here?   To be concrete, I want to be able to answer a question like the following: where (in the $x-y$ plane) is the non-linear second-order PDE   $$ a \partial_x^2 u +  b \partial_x \partial_y u + c \partial_y^2 u +  d (\partial_x u)^2 + e \partial_y u = 0$$  elliptic, hyperbolic, or parabolic (here all coefficients depend on $x,y,$ and $u$)? This doesn't really fit the form of any discussions I've found, so I'm not sure how to make sense of it.   EDIT: I am aware of the question at , but I believe that the answer there really only addresses linear PDE.   Several people have marked this question as already being answered at that link, but I don't see how. I would be much obliged if you could sketch a procedure for classifying the qualitative behavior for a non-linear example operator of the form I describe.
what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.  So far, I have not found any precise definition in literature.
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$, $\forall x\in G$. Prove that $H$ is a normal subgroup of $G$.    I have tried to using the definition but failed. Can someone help me please. Prove $f(S \cup T) = f(S) \cup f(T)$ Logistic Regression - Adding interactions makes Independent variable statistically insignificant My name is Abhi &amp; I am trying to better understand logistic regression by solving a few practice problem. I am using R and RStudio as the development environment  Problem Statement  Given the age, sex and class(first,second,third) for each passenger can you predict if he survived or died when the titanic sank   Attempt1  Simple Logistic Regression. Use age,sex and passenger class directly. The formula (in R)  Survived~Pclass+Sex+Age  This gives fairly decent results - accuracy of 79% and all of independent variables are statistically significant   Attempt2  Add interactions between age,sex and passenger class. Accuracy has improved to 80% but age is no longer significant. Also none of the new terms(age-sex,age-class,sex-class,age-sex-class) are statistically significant. The formula (in R) Survived~Pclass*Sex*Age   Can some one explain why this is happening? I can accept that the new terms may not be significant but why is age no longer significant?   Any help would be much appreciated How to understand degrees of freedom? How can I use asymmetric encryption, such as RSA, to encrypt an arbitrary length of plaintext? I'm writing a linguistics paper that uses a large amount of linguistic examples from a wide variety of languages, using the gb4e package. I want to be able to label each example with information about the language it comes from, as I've seen in some texts, sort of like this:   Language (family, citation, etc.)      (1) example text          glosses        'translation'    gb4e is great at formatting the examples themselves, but I can't seem to get the label placed where I want it, that is, placed immediately before the example and without an indent. Here's some example code:  \documentclass{article} \usepackage{gb4e} \usepackage{lipsum}  \begin{document} \lipsum[2]  Hungarian (Finn-Ugric, \emph{reference})   \begin{exe}   \ex \gll    J\'anos h\'aza\\               John house.his\\       \glt    `John's house'   \end{exe}  \lipsum[3] \end{document}   If the label text is outside the exe environment, it's treated like a regular paragraph, but if I try to put it inside, it gives me an error. How can I make it look the way I want? I've looked at other packages that do numbered examples, but they don't seem to have solutions either, and they also don't have the same functionality as gb4e that I need.  Any help would be greatly appreciated. Why didn't Kenobi start training Luke earlier? When Anakin was only 9 years old, everyone in the Jedi Council was arguing that he was already too old for Jedi training.  When it comes to Luke, it was a pre-planned agenda to make him a powerful Jedi. So, why did Kenobi wait for Luke to be much older? He could have started Luke's training by himself. $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$ Question is to prove :  $f : S^1 \to \mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$  I guess it would be helpful to use intermediate value theorem  Assuming $f(x)\neq f(-x)$ then given any $p\in (f(-x),f(x))$ (assuming $f(-x)&lt;f(x)$) there exists $y\in (-x, x)$ such that $f(y)=p$  I am not very sure of how to use this..   It would be helpful if someone can give some hint which would help me to solve this..  Thank you. How can I split multiple joined words? I have an array of 1000 or so entries, with examples below:  wickedweather liquidweather driveourtrucks gocompact slimprojector   I would like to be able to split these into their respective words, as:  wicked weather liquid weather drive our trucks go compact slim projector   I was hoping a regular expression my do the trick.  But, since there is no boundary to stop on, nor is there any sort of capitalization that I could possibly key on, I am thinking, that some sort of reference to a dictionary might be necessary?    I suppose it could be done by hand, but why - when it can be done with code! =)  But this has stumped me.  Any ideas? Looking for a Simple Argument for "Integral Curve Starting at A Singular Point is Constant" Let $U$ be an open subset of $\mathbf R^n$ and $V:U\to\mathbf R^n$ be a differentiable vector field on $U$. Let $\mathbf p\in U$ be a singular point of $V$, that is, $V(\mathbf p)=\mathbf 0$.     Then the only integral curve which starts at $\mathbf p$ is the constant curve.   I know that one could simply use the theorem of "uniqueness of integral curves" and in fact adapt the same proof for this particular case.  But is anybody aware of a simple argument for this? If I roll 2d8 and 1d6 for damage, how many "damage rolls" is that? 1, 2, or 3? Can any one help? I purchased a solar powered PIR security light, with the intention of putting it inside my kitchen window. The PIR would be inside the window, but this does not work.  When I go outside to activate the light, it does not work. When I go inside to run my hand across the sensor, it works.  Will a PIR unit penetrate glass, or am I barking up the wrong tree? Understanding relation between Quotient Space and $S^1$ There is this example at the Wikipedia article on Quotient spaces (QS):     Consider the set $X = \mathbb{R}$ of all real numbers with the ordinary topology, and write $x \sim y$ if and only if $x−y$ is an integer. Then the quotient space $X/\sim$ is homeomorphic to the unit circle $S^1$ via the homeomorphism which sends the equivalence class of $x$ to $\exp(2πix)$.   I understand relations, equivalence relation and equivalence class but quotient space is still too abstract for me. This seems like a simple enough example to begin with.  I understand (sort of) the definition but I can't visualize. And by this example and others, there is a lot of visualizing going on here! torus, circles etc. Changing color and fonts of itemize How can I change the font and color of a particular command in a LaTeX document?  For example I want all the text that follows the \item tag to be gray and the font should be different from the rest of the document , which i can specify ... To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$ Proof that there are infinitely many prime numbers starting with a given digit string To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they start with 314159265358979323. I think using a prime number theorem will be helpful . But I am still not getting a point of how to start the proof. If a lightning-radio-group has two options, and a user selects one, I want to programmatically have the UI changed back to the other one.  Link to a . (code copied below)  import { LightningElement, track } from 'lwc';  export default class RadioGroupRequired extends LightningElement {     @track value = 'option1';      get options() {         return [             { label: 'Option one', value: 'option1' },             { label: 'Option two', value: 'option2' },         ];     }      handleOnChange(evt) {         this.value = 'option1';         // how to make the UI select Option one?     } }   And the html:  &lt;template&gt;     &lt;lightning-radio-group name="radioGroup"                            label="Radio Group"                            options={options}                            value={value}                            required                            type="button"                            onchange={handleOnChange}&gt;&lt;/lightning-radio-group&gt;                             value={value} &lt;/template&gt;     I think that there is no rerendering because the bind variable, {value} doesn't actually change. However, the selected state of the lightning-radio-group does, and I want to be able to change it back. How can that be achieved?   I did not notice any methods or attributes that would seem to do the trick in the . Catch Regex too complicated when splitting a string in salesforce Installing PHP 5.3 in Ubuntu 14.04 I have a block of text that I would like to automatically break in blocks of equal length. Something like what \num does with numbers.  What I need is a definition for what I called \makeBlocks. The MWE below should output: aaaaa bbbbb ccccc ddddd followed by abc def ghi  \documentclass{article}  \newcommand{\makeBlocks}[1]{???}  \begin{document} \makeBlocks{5}{aaaaabbbbbcccccddddd}  followed by \makeBlocks{3}{abcdefghi} \end{document}