I just had a quick question about a notation I saw.  I know that a $C^2$  function is a function whose second partial derivative exists. But, what does it mean when one writes $C^2(U)$ where $U$ is a subset of $R^n$? Thanks in advance.
The following is an excerpt from Folland's Introduction to Partial Differential Equations:  Let the open set $\Omega\subset{\mathbb R}^n$, and $k$ be a positive integer. $C^k(\Omega)$ will denote the space of functions possessing continuous derivatives up to order $k$ on $\Omega$, and $C^k(\overline{\Omega})$ will denote the space of all $u\in C^k(\Omega)$ such that $\partial^{\alpha}u$ extends continuously to the closure $\overline{\Omega}$ for $0\leq|\alpha|\leq k$.  As I understand, an extension means that there exists $\widetilde{\partial^{\alpha}u}$ which is defined on $\overline{\Omega}$ and $\widetilde{\partial^{\alpha}u}|_{\Omega}=\partial^{\alpha}u$. And "extends continuously" means $\widetilde{\partial^{\alpha}u}$ is continuous with respect to the relative topology on $\overline{\Omega}$.  Here are my questions:        How do people usually do such extension?   When does such extension exist and when does not?   Let $\Omega$ be an open subset of ${\mathbb R}$. Is there any   non-trivial counterexample such that   this kind of extension does not exist?
Prove: The pre-image of an ideal is an ideal. Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$. Strange Recurrence: What is it asymptotic to? So I have the following recurrence relation for the growth rate of an algorithm:  $T(n)$ = time taken by algorithm to solve problem of size n:  $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$  Clearly this does not obey any polynomial law but now my question is what does it obey?  I merely want an asymptotic equivalence.  My guess is subexponential since this is faster than polynomial but clearly slower than any exponential form.  What though is the question.... Duplicate application icons in Ubuntu dock upon launch How to create live autofill search? I am currently trying to create a wordpress search function that shows live results underneath the search bar. There is an example on the  website (screens below). I'm not looking for an autofill like you would find on Google.com which completes words you type, rather I want it to find actual posts on the site.   I tried doing a scrub through Wordpress Answers and other similar resources but have only run into implementing a Google type search which is not what I am looking for. Any help or points in the right direction would be greatly appreciated. How to connect to Salesforce REST API without OAuth? How does Jitterbit do it? Is it possible to restart the unity panel without restarting compiz? I'm being hit (quite often) by that bug where you can't click on some indicators. It's easy to fix by restarting unity, but that usually also means running to the risk of upsetting some window on re-parenting (rare, but I'd rather avoid any such problems).  So, I know there is `unity-panel-service', and I'm guessing this might have something to do with the solution?  Is it possible to restart the unity panel (or the entire unity plugin) without restarting compiz itself? Deriving Newton's Third Law from homogeneity of Space component.set not working for picklist field I have a number of lightning:inputField that i'm setting based on certain criteria, these all work except for picklist field types. The code below shows how i am setting the fields. The country field is the picklist and the 2 alerts have the correct value in them, however the value is not getting set on country. Is there a different method for setting picklist fields?  var address = res.getReturnValue(); var country = component.find('Country__c'); var pcodeFld = component.find('Postcode__c');  if(country != undefined){  alert(country);  alert(address.Country);  country.set('v.value', address.Country); } if(pcodeFld != undefined){  pcodeFld.set('v.value', address.Postcode); }   EDIT: I can set a default value whilst using $A.createComponents using .set("v.value", "United Kingdom");  it just seems that i can't set it after the component has rendered based on some user action.  Reproduceable Example:  I just can't seem to set the value of any picklist fields when using lightning:inputField.  Component  &lt;aura:component implements="flexipage:availableForRecordHome, force:hasRecordId"&gt; &lt;div class="slds-p-bottom_large slds-p-left_large" style="width:500px"&gt;     &lt;lightning:recordEditForm aura:id="recordViewForm"                                   recordId="{!v.recordId}"                                  objectApiName="Contact"&gt;         &lt;lightning:messages /&gt;         &lt;lightning:inputField fieldName="FirstName" /&gt;         &lt;lightning:inputField fieldName="LastName" aura:id="lastname"/&gt;         &lt;lightning:inputField fieldName="Birthdate" /&gt;         &lt;lightning:inputField fieldName="Phone" /&gt;         &lt;!--Picklist--&gt;         &lt;lightning:inputField fieldName="Country__c" aura:id="Country__c"/&gt;         &lt;lightning:button label="Update" variant="brand" onclick="{!c.update}"/&gt;     &lt;/lightning:recordEditForm&gt; &lt;/div&gt;     &lt;/aura:component&gt;   Controller  ({         update: function(component, event, helper) {         //alert(component.find("Country__c"));        component.find("Country__c").set("v.value", "Ireland");     },  }) Expectation of truncated normal Can the preposition 'in' be omitted here? Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$ How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.) Workflow for converting LaTeX into Open Office / MS Word Format I often have to write up reports based on the analysis of some data. I use R to analyse the data and export tables, figures, and text. This is then included into a LaTeX document either using input or Sweave ().  However, when I collaborate with others, I sometimes need to provide a document in Open Office / MS Word format.  Question:  Thus, assume the simplest scenario   I have a LaTeX document with text, tables, and figures I need to export this reliably into Open Office or MS Word format: this includes mathematical formulas, table formatting, and quality figures) I don't need to go back from MS Word to LaTeX   What is a reliable, efficient, and preferably free process?  Initial Thoughts:   Export to HTML,  LaTeX - LyX - ODF Use odfWeave  that deals with some specific issues associated with LaTeX to Open Office conversion    I was hoping that there is an expert out there who has worked out a good system already. If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$ Let , $\Delta = B(0,1)$.  If $f:\Delta \rightarrow \Delta$ is holomorphic, and $f(0)=0$, prove that $|f(z)+f(-z)| \leq 2|z|^2$.  My attempt so far:  $\displaystyle f(z) = \sum_{n=0}^{+\infty} a_n z^n = \sum_{n=0}^{+\infty} a_{2n} z^{2n} + \sum_{n=0}^{+\infty} a_{2n+1} z^{2n+1}$  $\displaystyle f(-z) = \sum_{n=0}^{+\infty} a_n (-z)^n = \sum_{n=0}^{+\infty} a_{2n} z^{2n} - \sum_{n=0}^{+\infty} a_{2n+1} z^{2n+1}$  Then, $\displaystyle f(z)+f(-z) = 2  \sum_{n=0}^{+\infty} a_{2n} (z^2)^n =: 2 g(z^2)$  If $g(\Delta) \subset \Delta$, as $g(0) = 0$, I can invoke Schwarz Lemma and I win. However, I don't know how to prove this. Could anyone help me? Thanks! I was reviewing the  recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a vector space.   Take for example the set of all ordered pairs of integers. Allow for scalar multiplication over the integers and componentwise vector addition as usual. It seems to me that this is a perfectly well-defined vector space.  The integers do not form a , which begs the question: Is there any reason that the "field" over which a vector space is defined must be a mathematical Field? If so, what is wrong with the vector field I attempted to define above? If not, what are the requirements for the scalars? (For instance, do they have to be a  - Abelian or otherwise?) Gravatar links on "Filtered Questions" page are not HTTPS What laptop specs are important for intensive image editing work? How important is the following in purchasing a laptop that will be used for image editing:   processor speed amount of RAM integrated vs. dedicated graphics card HD speed (SSD; or 5400/7200/10000 rpm) In superluminal phase velocities, what is it that is traveling faster than light? I understand that information cannot be transmitted at a velocity greater than speed of light. I think of this in terms of the radio broadcast: the station sends out carrier frequencies $\omega_c$ but the actual content is carried in the modulated wave, the sidebands $\omega_c\pm\omega_m$. The modulation envelop has its group velocity and this is the speed at which information is being transmitted. We also know, for example, that x-rays in glass have a phase velocity which is greater than the speed of light in vacuum.   My question is, what exactly is it that is travelling faster than the speed of light?    EDIT: I know it couldn't be anything physical as its superluminal. My problem is, what is it that has absolutely no information content yet we associate a velocity with it. Every vector space has a basis Prove that every vector space has a basis. I am going to use Zorn's lemma for this also here is a necessary definition regarding totally ordered subsets: one element will be contained in the other.   proof: A basis for $V$ is a maximally linearly independent set. Let $X$ be equal to all linearly independent subsets of $V$. On, $X$ define $A\leq B$ means $A\subset B$. Let $S$ be a totally ordered subset of $X$, let $T = \bigcup_{A_{s}\in S} A_s$ then $A_s\subset T$ for all $s\in S$. Does $T\in S$? i.e. is $T$ a linearly independent set?. Let $$O = \alpha_1 v_1 + \alpha_2 v_2 + \ldots + \alpha_n v_n \ \ \text{where} \ \ v_i\in T = \bigcup_{A_{s}\in S} A_s$$ Then there exists $S_i$ such that for all $i\in A_{s_{i}}$ all $i$ $A_{s_{1}},A_{s_{2}},\ldots, A_{s_{n}}$ (these will have one contained in the other then we would take the larger). So, there exists $A_{s_{k}}$ such that $A_{s_{k}}\supset A_{s_{i}}$ all $i$ just compare two at a time. Therefore, $V_i \in A_{s_{k}}$ all $i$, but $A_{s_{k}}$ is a linearly independent set. So, $\alpha_i = 0$ all $i$. Hence, $T\in X$ so $T$ is an upper bound for $S$. Hence every totally ordered subset of $X$ has an upper bound, hence by Zorn's lemma, there exists a maximal element and this would be the basis.  I am not sure if my proof is right, I apologize if it is a bit messy. If it is right, any suggestions on cleaning it up would be greatly appreciated. System of congruences with variable I am looking over , and I came across a solution containing this:  Problem Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10,\!444$ and $3,\!245$, and LeRoy obtains the sum $S = 13,\!689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 25$ Solution 1 First, we can examine the units digits of the number base 5 and base 6 and eliminate some possibilities. Say that $N \equiv a \pmod{6}$ also that $N \equiv b \pmod{5}$ Substituting these equations into the question and setting the units digits of $2N$ and $S$ equal to each other, it can be seen that $b &lt; 5$ (because otherwise $a$ and $b$ will have different parities), and thus $a=b$. $N \equiv a \pmod{6}$, $N \equiv  a \pmod{5}$, $\implies N=a \pmod{30}$, $0 \le a \le 4$ Therefore, $N$ can be written as $30x+y$ and $2N$ can be written as $60x+2y$ Just keep in mind that $y$ can be one of five choices: $0, 1, 2, 3,$ or $4$, ; Also, we have already found which digits of $y$ will add up into the units digits of $2N$. Now, examine the tens digit, $x$ by using $\mod{25}$ and $\mod{36}$ to find the tens digit (units digits can be disregarded because $y=0,1,2,3,4$ will always work) Then we take $N=30x+y$ $\mod{25}$ and $\mod{36}$ to find the last two digits in the base $5$ and $6$ representation.[N \equiv 30x \pmod{36}][N \equiv 30x \equiv 5x \pmod{25}]Both of those must add up to[2N\equiv60x \pmod{100}] $N\equiv{30x\mod{36}}\\N\equiv{30x}\equiv{5x\mod{25}}$ Both of these must add up to $2N\equiv{60x\mod{100}}$ [...]  I'm not really sure which modular arithmetic rule to apply in the final three congruences, or where to start. How would I go about understand this? I've tried different ways to install iTunes, but to no avail. I tried with Wine, but it didn't work. Is it possible to have iTunes on Ubuntu?