For a continuous function $f: \mathbb R^+ \rightarrow \mathbb R$ if: $$f\left(\sqrt{xy}\right) \leq 1/2(f(x)+f(y))$$ for all $x,y&gt;0$ then $f(e^t)$ is convex for all $t \in \mathbb R$. What I've tried I am going after showing the inequality: $$f\left (\exp (\lambda t_1 + (1-\lambda) t_2)\right) \leq \lambda f(e^{t_1}) + (1-\lambda) f(e^{t_2})$$ for all $t_1,t_2 \in \mathbb R$ and $\lambda \in [0,1]$. My plan was to rewrite the first inequality as: $$f\left(\exp 1/2\log x + 1/2 \log y\right) \leq 1/2(f(\exp \log x)+f(\exp \log y))$$ and claim that since $\log$ is a bijection from $\mathbb R^+$ to $\mathbb R$ then inequality holds for $\lambda = 1/2$. I next tried to show using the continuity hypothesis on $f$ that this holds for any $\lambda \in [0,1]$ but I am struggling with that argument and think I might be missing a simpler way.
Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that $f$ is convex. Thanks.      Edit: I'm sorry for all the confusion. $f$ is assumed to be continuous on an interval $(a,b)$.
How to use paging in a WFS query? I am running a GetFeature request against a WFS server which does not support to download all data at once. Can I use ogc:PropertyIsGreaterThanOrEqualTo to split the dataset into chunks at download them step by step? I wonder how I can construct the actual URL to include the paging filter. Here is what I tried:  http://example.com/wfs.aspx?request=GetFeature&amp;service=WFS&amp;version=1.1.0 \ &amp;typeName=example:example&amp;maxFeatures=50000 \ &amp;FILTER=&lt;ogc:Filter xmlns:ogc="http://www.opengis.net/ogc"&gt; \ &lt;ogc:PropertyIsGreaterThanOrEqualTo&gt;&lt;ogc:PropertyName&gt;OBJECTID&lt;/ogc:PropertyName&gt; \ &lt;ogc:Literal&gt;50000&lt;/ogc:Literal&gt;&lt;/ogc:PropertyIsGreaterThanOrEqualTo&gt;&lt;/ogc:Filter&gt;   This is not a valid URI though... That's why uri-encoded the filter part:  http://example.com/wfs.aspx?request=GetFeature&amp;service=WFS&amp;version=1.1.0 \ &amp;typeName=example:example&amp;maxFeatures=50000 \     &amp;FILTER=%3Cogc%3AFilter%20xmlns%3Aogc%3D%22http%3A%2F%2Fwww.opengis.net \ %2Fogc%22%3E%3Cogc%3APropertyIsGreaterThanOrEqualTo%3E%3Cogc%3APropertyName%3EOBJECTID \ %3C%2Fogc%3APropertyName%3E%3Cogc%3ALiteral%3E50000%3C%2Fogc%3ALiteral \ %3E%3C%2Fogc%3APropertyIsGreaterThanOrEqualTo%3E%3C%2Fogc%3AFilter%3E   This actually works. Is there a "nicer" way to do this? Joining non-spatial CSV file to spatial data (shapefile) using QGIS? I have a CSV file which has postcode data in a field, but it is non-spatial e.g. no Easting and Northing.   I have a shapefile with postcode in a field and some additional data.  I want to join the CSV with the shapefile based on attributes (postcode fields). I know I can do this in ArcMap, but how can I achieve the same thing in QGIS? How to check if list item already exist when adding it with how to check Title( Country) if exists in the list. if exists it is necessary to do the update if not it is necessary to add it ?  function AddListItem() {      var listTitle = "Test";     //Get the current client context       context = SP.ClientContext.get_current();     var airportList = context.get_web().get_lists().getByTitle(listTitle);          //Create a new record       var listItemCreationInformation = new SP.ListItemCreationInformation();     var listItem = airportList.addItem(listItemCreationInformation);      //Set the values      var country= $("#Country").val();     var client= $("#Client").val();     // var nb1 = $("#nb1").val();      // var nb2 = $("#nb2").val();       // var nb3 = $("#nb3").val();      // var nb4 = $("#nb4").val();       // // var nb5 = $("#nb5").val();        listItem.set_item('Country', Country);     listItem.set_item('Client', Client);     listItem.update();     context.load(listItem);     context.executeQueryAsync(AddListItemSucceeded, AddListItemFailed); } function AddListItemSucceeded() { } function AddListItemFailed(sender, args) {     alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace()); } "as much as you and I" vs. "as much as you and me" Increasing extent of Raster? What are the rules for animal mating? Perhaps this is trivial: how does one prove that the work done by a homogeneous gravitational field on a system of particles is equal to the work done on a point mass with the total mass of the system, located at the center of mass of the system. Mount ext4 partition with >4KiB block size I'm attempting to mount a partition from a NAS and the block size is 65536 bytes, this is above my distribution's precompiled kernel memory paging size of 4KiB, so I cannot mount it normally.  Which kernel compilation parameters would I need to change to get a memory paging size large enough?  Would it be possible under the x86-64 architecture?  EDIT: There doesn't seem to be an option for it in make xconfig unless I'm just missing it.  From page_types.h under arch\x86\include\asm  /* PAGE_SHIFT determines the page size */ #define PAGE_SHIFT  12 #define PAGE_SIZE   (_AC(1,UL) &lt;&lt; PAGE_SHIFT) #define PAGE_MASK   (~(PAGE_SIZE-1))`   It doesn't look to be configurable.  I could change this number directly to 16 and it would probably make pretty explosions and destroy Tokyo.  I shall probably try it later this evening. How can I separate a single object file into multiple objects based on closed geometry? Let's say you have a single obj with a number of separate closed 3D geometries in it. For example's sake, imagine you have an obj called neighborhoodfirehydrants.obj that includes the model for all of the fire hydrants on a street, which themselves are closed, but as I said above, are all within the same obj file.  Now let's say you wanted to be able to process each of those fire hydrants one at a time, separately. To the user, it's very obvious to see visually that these faces can be subgrouped into separate "objects" but in the Blender world, it sees them all as one.  I'm looking for a way (script would be ideal) to iterate through a blender obj neighborhoodfirehydrants.obj and find each of the "objects" (fire hydrants) that make it up. I've been looking into using the looping tools, but that's not making sense in this context. How long can cooked food be safely stored at room/warm temperature? Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  My work: So I think I have to do a proof by induction and I just wanted some help editing my proof.  My attempt:  Let $P(n)=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  Then $$P(1)=1^2=\frac{1(1+1)(2+1)}{6}$$ $$1=\frac{6}{6}.$$ So $P(1)$ is true.  Next suppose that $P(k)=1^2+2^2+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}$ for $k \in \mathbb{N}$. Then adding $(k+1)^2$ to both sides of $P(k)$ we obtain the following: $$1^2+2^2+\cdots+k^2+(k+1)^2=\frac{k(k+1)(2k+1)}{6}+(k+1)^2$$ $$=\frac{2k^3+3k^2+k+6(k^2+2k+1)}{6}$$ $$=\frac{2k^3+9k^2+13k+6}{6}$$ $$=\frac{(k^2+3k+2)(2k+3)}{6}$$ $$=\frac{(k+1)(k+2)(2k+3)}{6}$$ $$=\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$$ $$=P(k+1).$$ Thus $P(k)$ is true for $k \in \mathbb{N}$. Hence by mathematical induction, $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ is true for $n \in \mathbb{N}$. Which of these two phrases has more negative undertone?      "The interview I had was not very good"    or      "The interview I had was not good" Redirect all requests to subdirectory except a few ip addresses Some packages, have a Provides: header. For example, the package postfix on Debian Wheezy has Provides: mail-transport-agent.   The package mail-transport-agent which doesn't exist physically is considered as a "virtual" package.   How can I know, on a Debian based system, if a given virtual package is "installed" (or "provided")? Can I list every "provided" virtual package?  Hint: to list every existing virtual package, installed or not, do: aptitude search "~v" Tag info not updated regularly on profile? From my , I learned that all tag related info on profile will be updated daily at UTC 03:00.  However, something is wrong here. It's now UTC 05:00+, but my tag number, say 93 (c++) is not updated against yesterday. I am pretty sure about this and I did earn some more points throughout the day. Sorry for not be able to capture the screenshot yesterday, but I did check this number 10 hours ago (which belongs to yesterday, also 93).  Is there a bug happening here? Or updating time different?    UPDATE: As of 24 Jan, the tags are still not updated with the proper values.  UPDATE2: Fixed around 25 Jan, UTC 03:00, at least for me. Reduce minimum in call volume I find that the minimum in-call volume on my HTC Desire is still far to loud on some calls.  Since I got the phone, I have flashed several different custom ROMs, radio updates, etc, and have had the issue throughout.  Do you know of any way (setting, hack, app, whatever) to reduce the minimum volume?  My device is rooted and I am currently running one of the CyanogenMod 7 (Gingerbread) nightly builds. How can I prove the immigrant officers that I have significant ties in my home country?  Are there any documents that I need? In Data Extension one field 'HTML' have Data that I have shown below, Now I have to Find the 'Href', and and at the last of URL append some text like '&amp;MC='. It may have Multiple 'Href'. Please help me on this.   &lt;table width="540" style="margin: 0px; padding: 0px;" bgcolor="#ffffff" border="0" cellspacing="0" cellpadding="0"&gt;  &lt;tbody&gt;   &lt;tr&gt;    &lt;td valign="top"&gt;    &lt;h5 style="margin: 0px; padding: 0px 0px 2px; line-height: normal; font-family: &amp;quot;Segoe UI&amp;quot;, &amp;quot;Lucida Grande&amp;quot;, Verdana, Arial, Helvetica, sans-serif; font-size: 16px; font-weight: normal;"&gt;    &lt;a style="text-decoration: none;" href="https://test.4.com/testRegistration/testLobbyServlet?target=reg20.jsp&amp;eventid=897605&amp;sessionid=1&amp;key=2F3D1EB963D9DB7738BF2877DFFDBA01&amp;partnerref=Social&amp;sourcepage=register&amp;loc=zOTlocz&amp;prod=zWAz&amp;tech=zsecz&amp;prog=zOTprogz&amp;type=zOTtypez&amp;media=zWCz&amp;country=zUSz" alias="Success with Enterprise Mobility: Making email and Office 365 secure on mobile devices"&gt;    Success with Enterprise Mobility: Making email and Office 365 secure on mobile devices     &lt;/a&gt;    &lt;/h5&gt;    &lt;p style="margin: 0px; padding: 0px; line-height: 20px; font-family: &amp;quot;Segoe UI&amp;quot;, &amp;quot;Lucida Grande&amp;quot;, Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-weight: normal;"&gt;    &lt;span style="color: rgb(103, 103, 103);"&gt;    &lt;em&gt;December 9, 2014, 10:30 A.M.&amp;ndash;11:30 A.M. Pacific Time&lt;/em&gt;&lt;br&gt;    An enterprise mobility strategy involves identity, management, and productivity. In this session featuring Brad Anderson, you will learn about the integration and the Office mobile apps while enabling secure mobile productivity.    &lt;/span&gt;    &lt;/p&gt;    &lt;/td&gt;   &lt;/tr&gt;   &lt;tr&gt;    &lt;td valign="top"&gt;&amp;nbsp;    &lt;/td&gt;   &lt;/tr&gt;   &lt;tr&gt;    &lt;td valign="top"&gt;    &lt;h5 style="margin: 0px; padding: 0px 0px 2px; line-height: normal; font-family: &amp;quot;Segoe UI&amp;quot;, &amp;quot;Lucida Grande&amp;quot;, Verdana, Arial, Helvetica, sans-serif; font-size: 16px; font-weight: normal;"&gt;    &lt;a style="text-decoration: none;" href="http://test.TTr.com/?Wt.mc_id=IG15W2NWTN&amp;loc=zOTlocz&amp;prod=zWAz&amp;prod=zWSz&amp;tech=zCLz&amp;prog=zOTprogz&amp;type=zEVz&amp;media=zOTmediaz&amp;country=zUSz" alias="Test: Everything about enterprise technology under one roof"&gt;    Test: Everything about enterprise technology under one roof    &lt;/a&gt;    &lt;/h5&gt;    &lt;p style="margin: 0px; padding: 0px; line-height: 20px; font-family: &amp;quot;Segoe UI&amp;quot;, &amp;quot;Lucida Grande&amp;quot;, Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-weight: normal;"&gt;    &lt;span style="color: rgb(103, 103, 103);"&gt;    &lt;em&gt;May 4&amp;ndash;8, 2015, Chicago, IL&lt;/em&gt;&lt;br&gt;    If you&amp;rsquo;re serious about technology, you don&amp;rsquo;t want to miss Test. This event will be the intersection of everything in enterprise technology. Get ahead in your job and your career.    &lt;/span&gt;    &lt;/p&gt;    &lt;/td&gt;   &lt;/tr&gt;  &lt;/tbody&gt; &lt;/table&gt; Is there a 2D generalization of the coefficient of restitution? The  characterizes a collision in one dimension by relating the initial and final speeds of the particles involved,  $$C_R = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$$   In a 2D collision, the velocity can be split into components parallel and perpendicular to the "plane of collision" (the plane tangent to the two objects' surfaces at their contact point), and the equation above applies to the perpendicular components of the velocities only. One could write  $$\begin{pmatrix}v_{2f\shortparallel} - v_{1f\shortparallel} \\ v_{2f\perp} - v_{1f\perp}\end{pmatrix} = \begin{pmatrix}1 &amp; 0 \\ 0 &amp; -C_R\end{pmatrix}\begin{pmatrix}v_{2i\shortparallel} - v_{1i\shortparallel} \\ v_{2i\perp} - v_{1i\perp}\end{pmatrix}$$   My question is, is it useful (i.e. does it produce a more accurate description of realistic 2D collisions) to generalize that matrix? Perhaps by allowing the top left element to be unequal to 1, or allowing nonzero off-diagonal elements? Or is there some nonlinear relation that works better?  As usual, if you know of any relevant published research, references would be much appreciated. When the ball is being bounced by the goalkeeper from his hands and a player steals it without touching the goalkeeper (while the goalkeeper is not touching the ball) and scores, is it legal or not?   shows what I'm talking about.