Let $S$ be open subset of $R^2$ and let $f$ be twice differentiable real valued function on $S$. Suppose that there exists $(a,b) \in S$ such that $f(x,y) \leq f(a,b)$ on $S$, then show that $(\Delta f)  (a,b) \leq 0$ where $\Delta f = f_{xx} + f_{yy}$  I have proved a similar result for functions define on $R$. If $g$ is a function defined on an open interval $I$ such that $a\in I$ and $g(x) \leq g(a) $ on $I$, then $g''(a) \leq 0$ I proved the above using the fact that $g''(a) =\lim_{h\to 0} \frac{g(a+h) + g(a-h) - 2g(a)}{h^2}$ and by observing that $g(a+h) + g(a-h) - 2g(a) \leq 0$ Can I use this somehow to arrive at the required result?
Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative test and it seems to suggest that $\Delta u(\vec x) \leq 0$ if $\vec x$ is a min and $\Delta u(\vec x) \geq 0$ if $\vec x$ is a max. However, I'm not convinced of this result. Can somebody point me towards a statement (theorem, etc.) involving the above. I appreciate the help.
I'm looking for a Drupal Commerce Shipping in-cart estimate What if MSE used a different rep model for privileges? How can I permanently enable cheating? Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$   Just to begin,  $(1-x)(1+2x) = -2x^2 + x + 1$  $(1-x)(1+2x)(1-3x) = 6x^3 - 5x^2 - 2x + 1$   But expanding on like this take too long.  In the end the terms will be like:  $(1 - 15x)(a_0 + a_1x^1 + a_2x^2 + ....)$  Then just looking at the $x^2$, it will be:  $(a_2x^2 - 15a_1x^2) = x^2(a_2 - 15a_1)$  But it it still a very hard problem, any hints? When we find the electric field between the plates of a parallel plate capacitor we assume that the electric field from both plates is $${\bf E}=\frac{\sigma}{2\epsilon_0}\hat{n.}$$ The factor of two in the denominator comes from the fact that there is a surface charge density on both sides of the (very thin) plates. This result can be obtained easily for each plate. Therefore when we put them together the net field between the plates is $${\bf E}=\frac{\sigma}{\epsilon_0}\hat{n}$$ and zero everywhere else. Here, $\sigma$ is the surface charge density on a single side of the plate, or $Q/2A$, since half the charge will be on each side.   But in a real capacitor the plates are conducting, and the surface charge density will change on each plate when the other plate is brought closer to it. That is, in the limit that the two plates get brought closer together, all of the charge of each plate must be on a single side. If we let $d$ denote the distance between the plates, then we must have $$\lim_{d \rightarrow 0}{\bf E}=\frac{2\sigma}{\epsilon_0}\hat{n}$$ which disagrees with the above equation. Where is the mistake in this reasoning?  Or more likely, do our textbook authors commonly assume that we are in this limit, and that this is why the conductor behaves like a perfectly thin charged sheet? I'm using OSX 10.9. I'm trying to set up Internet Sharing to connect my Android to my Mac directly over Wi-Fi. Every tutorial I've come across (including ) mentions checking a box for "AirPort" in a "To computers using" list. However, the option is simply not available on my machine:    I have a few related questions:   Why isn't the AirPort option available? Is there anything I can do to make it available? If there isn't, how can I work around it to finish setting up Internet Sharing so I can connect my Android and my Mac directly over Wi-Fi? Fermat: Last two digits of $7^{355}$ I am doing this problem mentioned above and I know the answer because I know Euler's Theorem that $$a^{\varphi{(m)}}\equiv{1}\pmod{m}.$$  I used 100 as my modulus and got that the last two digits of $7^{355}$ are $43$.  My question is this.  What prime should I pick to solve this problem using Fermat's theorem $$a^{p-1}\equiv{1}\pmod{p}$$  As a precursor problem, the author (Dudley; Elementary Number Theory) asked to find the last digit of the number above; so using a modulus $5$, $7^4\equiv{1}\pmod{5}$ by Fermat.  Since the answer is $7^{355}\equiv{3}\pmod{5}$, the last digit is either $3$ or $8$.  Since $7$ is odd, the answer will clearly be odd, so $3$ is it.  The book is working linearly, building from the previous chapters and Euler's Theorem isn't mentioned for another two chapters (I remembered Euler from a problem solving class in college).    So again, what prime would be best?  How should I procede?    EDIT: I've seen a bunch of problems on the site and none that I looked at did it using Fermat.  If there is one, that could be posted and I would be happy.  EDIT 2:  it seems that the problem of finding of finding the last two digits using is rather difficult because of the implication that we need something mod 100.  Is there a way around this , like adding a step or complexifying, in order to finish the problem with Fermat? Double and triple substitution in bash and zsh Follow-up to the background part in .  In bash I can use ${!FOO} for double substitution, in zsh ${(P)FOO}. In both, the old-school (hack-y) eval \$$FOO works.  So, the smartest and most logical thing for me would be ${${FOO}}, ${${${FOO}}}… for double/triple/n substitution. Why doesn’t this work as expected?  Second: What does the \ do in the eval statement? I reckon it’s an escape, making something like eval \$$$FOO impossible. How to do a triple/n substitution with that that works in every shell? Electric Potential produced by Hollow Sphere Let there be a hollow sphere (Throughout the question we will ignore the thickness of the sphere) which is positively uniformly charged,q of radius,  R. Suppose there to be a test positive charge, q' at point P which is r distance away from the hollow sphere(r>R, P is outside of hollow sphere).  Now I have an doubt on evaluating Electric Potential,V at point P.  Both source charge,q and test charge,q' are +ve, so source charge will repel the test charge which will make the charge, q' to go to infinity.I have to apply external force, F' to displace the charge, q' from infinity to point P against the equal but repulsive force, F.In short, F' is  displacing the q' from infinity to P.  So External Force, F' will do +ve work and Repulsive Force, F will do -ve work. This way the Electric Field,E produced by hollow sphere is Opposite to Displacement of test charge(i.e.,θ=180).  Now we know that E=k(q/(r^2))  [Please Notice that infinity is the lower limit of integration]  Also V=-∫r∞(E dr cosθ)=-∫r∞(E dr cos 180)=∫r∞(E dr)  At last I ended up to V=- k(q/r). I know that V=k(q/r).  I just want to know why I am having this shitty negative sign in my equation. I am trying to hide Authoring information, Revision information, Publishing options tabs in create/edit node for any content type. I tried different modules with no luck. Please give me suggestions how to achieve this. Thanks a lot I have a problem with the endfloat package and unicode characters.  When I compile the following, everything works as expected:  \documentclass{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{graphicx}  \begin{document}  \begin{figure} \caption{Ö} \end{figure}  \end{document}   However, when I also load the endfloat package (\usepackage{endfloat}), I get the following error message:  ! Package inputenc Error: Unicode char \u8:ÖÖ not set up for use with LaTeX.   This could be a bug, but my suspicion is that I'm making a mistake somewhere.  I'm hardly the first person to use special characters with the endfloat package.  However, the above example is so simple, I don't see where I could have made a mistake.  Any advice is appreciated.  Edit: The file is of course encoded using utf-8. What causes javac to issue the "uses unchecked or unsafe operations" warning \overbrace and \underbrace with square bracket Can I submit/ArXiv without my advisor? I am being blocked from publishing my work I joined a research group of a faculty member in 201x, and worked with her for three years. During that time, I collaborated with a large hospital to perform research in the domain of medical informatics. I spoke about my research at two conferences; one was a talk, and the other was a poster presentation. My advisor asked me to write up the research in a paper around the second year, which I did. Around the third year, I decided to switch research groups, because of the relationship between me and my advisor going sour, and not having funding from her (I used to work part-time to fund my PhD research).  Once this happened, she withdrew her support from me publishing my paper and ordered that I give up my work to a fellow grad student. I told her, that my manuscript is in its Nth revision, and that I would like to take it to completion as first author. She was dismissive and didn’t offer any comments on my work. I told her, that, should I pass on my work, I would like to discuss authorship issues in advance, which she wasn’t ready to discuss. She ganged up my new advisor against me, and together they forced me to give up my code, results, paper drafts. I had to then leave the university without a degree.  Fast forward couple of years, I receive an email from a student in her lab (the student whom I had to hand over my work to), saying that they are publishing a paper, and have decided I am the fourth author. I read the paper and found out that it uses my experimental setup and design, data collection, and one of the ideas I had presented (the idea wasn’t the main crux of my paper, but I believe it was novel).  I want to   Publish my original draft in ArXiv/BioArXiv, as first author. Request my advisor to cite my work in their publication.    Given how she behaved with me, I know that despite what I am asking for sounds fair, she is likely to object.  I am asking the community: Can I go ahead and ArXiv it without her permission? I could add her as an author, but not take her permission. Also, if you have any other ideas on how to deal with this situation please let me know.  (For the record, this was at a top-10 school in the US in an engineering department). Can I get a \widebar without using the mathabx package? One can use \bar to put a bar over a letter in math mode, but often the bar isn't wide enough. An alternative is \overline, but this isn't an accent, so it doesn't take into account the skew of the letter. In the following example, the bar extends too far to the left:    Now the mathabx package offers a \widebar accent, but this has several drawbacks:   mathabx changes a lot of other things, The \widebar sits slightly higher than \bar and \overline, In previewers (checked with kpdf and acroread), the \widebar appears fuzzy, which \bar and \overline don't.   So my question is: How can I put a wide bar over a letter in math mode without the above drawbacks? Add an Image Upload button to the Mobile Site With the release of iOS 6, the most commonly used mobile browser now supports standard generic image upload buttons. This is actually really handy.  I can take screenshots on my phone or iPad, and use them to answer questions about apps made to run on those devices (or ) to answer questions. Until the release of iOS 6 however, the only way to use these images in my answer was to either sync them over to my computer, and then answer from a desktop, or use a specialized app to upload the images, such as PicUp or DropBox, and then manually write out the markdown code.  After iOS 6, the Image Upload button works just fine, if I switch to viewing the full site from my iDevice, rather than the mobile view. I can upload images directly, and the markdown is handled for me. It's awesome, except for the fact that the non-mobile site is decidedly non-awesome on the small screen of my phone.  Which brings me to my feature request: Add the Image Upload button (and other markdown shortcuts, as deemed worthwhile) to the ask/answer editors on the mobile version of SE sites. 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. For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous. This is from Apostol's Mathematical Analysis.  Let $f$ be a function defined in $[0, 1]$ such that for each real number $y$ either there is no value of $x \in [0, 1]$ such that $f(x) = y$ or there are exactly two such values of $x$. Prove that a) $f$ must be discontinuous on $[0, 1]$. b) $f$ has an infinite number of discontinuities in $[0, 1]$.  Long back I had tried to solve the problem. I reasoned for a) as follows: Suppose that $f$ is continuous and let $m, M$ be its minimum and maximum values on $[0, 1]$. Clearly if $m = M$ then $f$ is constant and clearly for every $x \in [0, 1]$ we have $f(x) = m = M$ which contradicts given property of $f$. So we must have $m &lt; M$. Clearly there are now four distinct points $a, b, c, d$ in $[0, 1]$ such that $f(a) = f(b) = m, f(c) = f(d) = M$. By checking possible linear orderings of $a, b, c, d$ I could find a find a value of $y \in (m, M)$ for which there exist more than 2 values of $x$ such that $f(x) = y$. However this approach looks very clumsy and I suspect much simpler and easier proof is possible. Part b) seemed bit complex and my only guess was that probably $f$ was discontinuous on each subinterval of $[0, 1]$ but I could not prove this claim (may be the claim is wrong!). Perhaps there is a simpler solution to this part of the problem. Apostol's also asks to provide an example of such a function. Any hints on the overall problem will be appreciated.  Update: After looking at Marc's answer, I think I have found a way to show that there are infinitely many discontinuities. Earlier the solution was a part of the question itself but it made the question look quite lengthy hence I moved it to a separate answer. Some months ago I posted  about exporting Sentinel 1 scenes from a collection and was successful with help from .   I now have a new region of interest, which sits on the overlap between 2 scene footprints (sequential images in descending orbit). When I make a mosaic of the 2 scenes I get a small no data gap and lack of alignment between the 2 images:  // Create a geometry representing an export region. var roi = ee.Geometry.Rectangle([-62.67, -8.31, -63.25, -8.85]); var zoomPoint = ee.Geometry.Point([-62.35, -8.06])  // Load the Sentinel-1 ImageCollection. var sentinel1 = ee.ImageCollection('COPERNICUS/S1_GRD') .filterBounds(roi);   // Filter by metadata properties. var IW_H = sentinel1  // Filter to get images with VV and VH dual polarization. .filter(ee.Filter.listContains('transmitterReceiverPolarisation', 'VV')) .filter(ee.Filter.listContains('transmitterReceiverPolarisation', 'VH'))  // Filter to get images collected in interferometric wide swath mode. .filter(ee.Filter.eq('instrumentMode', 'IW'))  // Filter IWs to get High res. .filter(ee.Filter.eq('resolution', 'H'))  // Filter IW-Highs to get 10m res .filter(ee.Filter.eq('resolution_meters', 10));  // Filter to get images from different look angles var col_desc = IW_H.filter(ee.Filter.eq('orbitProperties_pass', 'DESCENDING'));  // Spatially mosaic the images in the collection and display var col = col_desc.mosaic(); var colVV = col.select(0) Map.addLayer(colVV, {min:-17, max:-3}, "VV"); Map.centerObject(zoomPoint, 20);   What am I missing?  Are these scenes not acquired sequentially? How to cite own unpublished work in text and reference? I would like to cite one of my own unpublished work (under review). Is it okay to cite as following?  In text According to author et al. (year of submission)................. In reference authors list, year of submission, title, name of the Journal (status)   How about the results?   How will the reviewer access those results cited here?   Is it okay to provide them as supplements?   If I provide couple of figures as supplements, whether it will violate the copyright of the under reviewed paper?