If $\langle T(x), x \rangle = 0$ for all $x$, must $T$ must be zero?
In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$
Prevent specific sites from being overrepresented in the hot questions list Some sites appear far more often in the hot questions list than other sites. I don't have any hard data on that, as that kind of data is just not publicly available, but I'm pretty sure my subjective impression on that is reasonably accurate.  Those sites don't appear more often because they are inherently more interesting, but because they have certain voting and answering patterns that favor them in the hot questions algorithm. I would argue that the goal of the hot questions list is to present questions from a large variety of SE sites to the whole network, and to show possibly interesting questions to users that would not have found them otherwise.   To achieve this goal, the spots in the hot questions list should be distributed among all SE sites. The algorithm should try to get a good sampling of all sites, and it should try to normalize the results for varying voting behaviour. At the moment, sites that intrinsically get more votes and more answers per question seem favored in the algorithm.  So I think one part of the algorithm should try to identify questions that get an unusally high amount of votes and views for that site, and give the smaller sites a chance this way.  I'm not sure on how exactly this weighting should be done, as a complete normalization would likely not work across the huge difference in scale for all the SE sites.  As a specific example that the current algorithm is broken I'll present . I don't think any site should occupy a third of the whole list.   Here's another instance where one site is overrepresented at the top of the list: Directory-tree listing in Python Blending colors in photoshop and illustrator I am working on a print for my wall and I am trying to create something like this:      I have already created the letters in Illustrator and now I am adding the colors in Illustrator and then exporting everything to Photoshop and then using the liquify tool to blend the colors to create a similar effect. I also used the healing patch tool to blend in the hard edges.  Is that the right way of doing it? Or is there another way? While I was reading Enderton's "A mathematical introduction to Logic", I came across the proof of the following sentence: "The set of all finite sequences of members of the countable set A is also countable".   Proof: The set S of all such finite sequences can be characterized by the equation  $$S=\bigcup_{n \in N} A^{n+1}$$ Since A is countable, we have a function f mapping A one-to-one into N. The basic idea is to map S one-to-one into N by assigning to $(a_0,a_1,...,a_m)$ the number $2^{f(a_0)+1}3^{f(a_1)+1}\cdot ... \cdot p_m^{f(a_m)+1}$, where $p_m$ is the $(m+1)$st prime. This suffers from the defect that this assignment might not be well-defined. For conceivably there could be $(a_0,a_1,...,a_m)=(b_0,b_1,...,b_n)$, with $a_i$ and $b_j$ in A but with $m\neq n$. But this is not serious; just assign to each member of S the smallest number obtainable in the above fashion. This gives us a well-defined map; it is easy to see that it is one-to-one.  Note: P is a finite sequence of members of A iff for some positive integer $n$, we have $P=(x_1,...,x_n)$, where each $x_i \in A$.  First of all, I cannot understand why the former assignment might not be well-defined and the latter assignment is well-defined. Secondly, I cannot understand what Enderton means by "just assign to each member of S the smallest number obtainable in the above fashion". By the way, is $(a,b,c,d) = ((a,b),(c,d))$ true? Also, in which cases can I omit/add parentheses in a tuple so as to have an equal tuple? Assign VNC password using script I am using a script to automatically set up a computer. I need to assign a password to the vnc server for the user, which is normally done using the vncserver command. However, it prompts for the user to enter and re-enter their password, neither of which the script is capable of doing. So, how can I set up the VNC password without an interactive prompt?   Thanks. How can I take pictures through a skyscraper, airplane, or train window? Why do I need to type `./` before executing a program in the current directory? While executing a C program, a.out, using the Ubuntu terminal, why do I always need to type ./ before a.out, instead of just writing a.out? Is there solution for this? What gear do I need for a newborn photography session? I am an amateur photographer and I have a newborn session coming up for a family friend.   My current gear is:   Canon T5i (crop sensor) Canon 18-55 IS STM Kit lens 85mm f/1.8 lens   I plan on buying a reflector for the shoot and will be using natural light as my light source.  So my question is: is this enough gear to get some nice photos? Should I get another prime lens besides the 85mm? I don't really use the kit lens.  I also have another 8 month old baby shoot coming up and an informal wedding shoot coming up so it would be nice to use the lenses in all situations. I don't need a telephoto at the moment and will probably not use one any time soon. Are submodules of finitely generated free modules always free? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? Which facility upgrade unlocks maneuver nodes in KSP Which  unlocks  in career mode in KSP?  The  hints that maneuver nodes are now unlocked with a facility upgrade:     Are there any perks for upgrading?      The building in question   will upgrade visually, both from the outside and the inside. More   gameplay mechanics will be also be unlocked as the player progresses   through the different building tiers. For example, allowing EVAs in   space or being able to use maneuver nodes in the map view. I want to clear the inputtext used to search, and clear all search results using a commandbutton. I tried the following on the 'Reset' button I added but it only clears the inputtext and the search results are not clearing.  &lt;apex:pageBlockSectionItem &gt;                     &lt;apex:outputPanel layout="block" Rendered="true"&gt;                     &lt;apex:inputText value="{!ReportSearchString}" /&gt;                     &lt;apex:commandButton value="Search" action="{!searchReports}" reRender="ReportSearchResults, "/&gt; &amp;nbsp;&amp;nbsp;                      &lt;apex:commandButton value="Back" action="{!BackToCreateFeedPage}"/&gt;&amp;nbsp;                     &lt;apex:commandButton value="Reset" onclick="this.form.reset();return false;"/&gt;                     &lt;/apex:outputPanel&gt;                 &lt;/apex:pageBlockSectionItem&gt;                             &lt;/apex:pageBlockSection&gt;                         &lt;apex:pageBlockSection id="ReportSearchResults"&gt;                 &lt;apex:outputText value="{!warningMessage}" rendered="{!!showTable}" /&gt;                 &lt;apex:pageBlockTable value="{!reportList}" var="item" rendered="{!showTable}"&gt;                     &lt;apex:column headerValue="Report Name" &gt;                         &lt;apex:commandLink value="{!item.Name}" action="{!BackToCreateFeedPage}"&gt;                             &lt;apex:param value="{!item.Id}" name="SelectedReport" assignTo="{!selectedReportId}" /&gt;                         &lt;/apex:commandLink&gt;   This is my search function, it works fine. I need to clear the results of this search on clicking 'Reset' button.   public void searchReports()     {         showTable = false;               reportList = new List&lt;Report&gt;();          reportSearchResult = new Map&lt;Id, Report&gt;();          selectedReport = new Report();         if((ReportSearchString != null &amp;&amp; ReportSearchString != '') &amp;&amp; ReportSearchString.length() &gt;= 3)         {             reportSearchResult = new Map&lt;Id, Report&gt;([Select Id, Name From Report Where Name Like :'%'+ReportSearchString+'%' limit 2000]);              warningMessage = 'No Result Found';         }         else         {             warningMessage = 'Search text should have atleast 3 characters';         }     } Ability to update profile from the Stack Exchange Android App Can we please hold the hats? I really like the hats game. But I would love to keep the hats longer then only one winter. I want to leave them in my wardrobe during summer. Last year, I collected some cool hats, and I apparently lost them during summer.  Maybe we could make a system that allows hats to last for about 2 or 3 years, so we don't have to start over each winter. Or even specify per type of hat a number of years they last? Very very rare hats could last for ever.  I asked this, simply because I love hats! :D Word to describe someone who publishes many works? What word describes an author who publishes many works?  "Profuse" comes to mind, but this word has a broader meaning. I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any application for the integral of position with respect to time?  Maybe with respect to something else? If diary Tom Riddle had succeeded, would there have been two Voldemorts? why does fdisk add one additional sector when creating partitions? I'm playing around with fdisk and I tried to create a partition on an empty USB drive. Here's how that went: Command (m for help): n Partition number (1-128, default 1): 1 First sector (34-61187102, default 2048): 2048 Last sector, +/-sectors or +/-size{K,M,G,T,P} (2048-61187102, default 61187102): +30000000           Created a new partition 1 of type 'Linux filesystem' and of size 14,3 GiB.  so just one partition with 30 million sectors. However, when I check how many sectors it contains I get:  I'm checking with a different program and I see again that additional sector:  Sorry for the pettiness :) I'm learning. Why is there no overload of Interlocked.Add that accepts Doubles as parameters? Workflow rule order of execution Logarithm as limit lists this as a limit representation of the natural log:  $$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$  Is there a quick proof of this? Thanks