Usage of Filters and Ultrafilters I don't know why we need the concept of Filters and Ultrafilters.  they just seem nothing, and I don't know where to use them  can you tell me where do we use those concepts.
Applications of ultrafilters I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?  I'm at the beginning, so I'd prefer some applications that also a beginner could read.
How to find historical stock price for a de-listed or defunct company? I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help! Reference: What is variable scope, which variables are accessible from where and what are "undefined variable" errors? My laptop (running Win10) repeatedly "steals" the bluetooth connection for my headphones while they're connected to my phone. The only way I've been able to prevent the headphones' connection from switching to the laptop is to play some media or make a phone call immediately after reconnecting to my phone, or to unpair the headphones from the laptop.  The headphones are Plantronics Backbeat Go 2, and the specifications from the Plantronics product page give the following information for the Bluetooth entry:      Bluetooth v2.1:   A2DP 1.2, AVRCP, HFP v1.5, HSP v1.1   Thanks in advance! Mobile web refresher, part 2 After a sizeable amount of work we are releasing a new version of the mobile web site, this time focusing on the question and answer page. We have also fixed or implemented a minimal large part of the issues and requests mentioned in the . Changes Question and answer page  SVG icons for upvote, downvote, favorite, comment upvote white background color for legibility general styling issues with notices (they actually exist now) bounty link is now in question menu (next to flag)   Ask question page  general styling and sizing of element use of placeholders where appropriate instead of page text styling of error messages   Pop-ups  Popups are now inlined in the page instead of being sometimes floating and draggable Many overdue styling issues Close button is SVG Flag indicators are readable Buttons   Misc pages  404 page did not exist. Now it does. Login, logout are formatted Pre-ask and pre-answer warnings are now formatted  Bugfixes  made search magnifying glass bigger fixed language in user timeline so it's consistent with the desktop site added login link on top bar made topbar icons ever so slightly smaller  Restating the obvious: send your &lt;3 to  and   for the most excellent design! Feedback , bugs and suggestions while we move to the next iteration? Please mark bugs with What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$?  Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle.  But what is the fundamental group of the half circle? Calculate E[X] from incomplete data? The exercise I'm doing describes the random variable $X$ as the following  | Number of cars |  0  |  1 |  2 |  3 |  4+ |  | % of families  |  15 | 45 | 25 | 13 |  2  |   Then it asks me to evaluate $E[X]$. But if there could be cases with 4+ cars, wouldn't that make it impossible to calculate the expectancy? Related:  (not a dupe, that's a feature request and was closed as a dupe of another one)        (old) post does indeed say that Link-only answers can be flagged as NAA, but there are a lot of posts like  which define Not An Answer is a way that it doesn't apply to link-only answers.  Can we get some up-to-date consensus on this? That link only answers are not allowed is . I am using the document class article. Once I place a part in the document, all subsequent sections belong to the part automatically. How can I reset a section such that it does not belong to a part?  The following sample demonstrates what I want to achieve:  \section{Introduction}  \part{Part 1} \section{Section 1 belonging to part 1} \section{Section 2 belonging to part 1}  \part{Part 2} \section{Section 1 belonging to part 2} \section{Section 2 belonging to part 2}  % should not belong to Part 2 \section{Conclusion} Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$ Calculation of  $\displaystyle \lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$  $\bf{My\; Solution::}$  Let $y=\displaystyle \lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$, Now taking $\log_{e}$ on both side  $\displaystyle \log_{e}(y) = \lim_{n\to \infty}\log_{e}\left(\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}\right)=\lim_{n\to \infty}\frac{1}{n}\cdot \sum_{r=1}^{n}\log\left(1+\frac{r}{n}\right)$  Now Using Reinman Sum of Integral ...  $\displaystyle \log_{e}(y) = \lim_{n\to\infty}\frac{1}{n}\cdot \sum_{r=1}^{n}\log\left(1+\frac{r}{n}\right) = \int_{0}^{1}\ln(1+x)dx = \log_{e}(4)-\log_{e}(e) = \log_{e}\left(\frac{4}{e}\right)$  So $\displaystyle y =\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}} =\frac{4}{e}$  But how can i solve above question using Stirling Approximation,  May I ask for help?  Thank you in advance! So, if we have a large pipe which is constricted in the middle, the fluid in the middle section  will move faster. Why is it so? What dictates that liquid that an X amount of it must pass on the other side regardless of the size of the encasement?  The only reason I could think about is because that the initial pressure which causes the movement somehow is being transferred to a smaller area, but that makes no sense. Why do airplanes experience negligible Coriolis force while bullets experience the Coriolis force in long range shooting, even though the mass of airplane is much bigger than a bullet? How to extend USB beyond the recommended maximum cable length of 5m? I'm planning to use USB port for an application(from a device to a PC) but the distance between the device and the PC will be around 15m. The recommended maximum USB cable length is 5m.  How can I buffer the data without loosing any information, safely? Would a USB hub work? When in a group chat, if many messages are exchanged between other people in the group, when I check my Whatsapp, the messages are grouped according to sender and not chronologically. I know that my phone isn't the problem as I have recently changed my phone, however, I'm still experiencing the same problem.  I have tried uninstalling and re-installing WhatsApp on my phone, and the same problem still occurs. What it's the best way to generate a string of \t's in C#  I am learning C# and experimenting with different ways of saying the same thing.    Tabs(uint t) is a function that returns a string with t amount of \t's  For example Tabs(3) returns "\t\t\t"  Which of these three ways of implementing Tabs(uint numTabs) is best?  Of course that depends on what "best" means.     The LINQ version is only two lines, which is nice.  But are the calls to Repeat and Aggregate unnecessarily time/resource consuming? The StringBuilder version is very clear but is the StringBuilder class somehow slower? The string version is basic, which means it is easy to understand.   Does it not matter at all?  Are they all equal?   These are all questions to help me get a better feel for C#.     private string Tabs(uint numTabs) {     IEnumerable&lt;string&gt; tabs = Enumerable.Repeat("\t", (int) numTabs);     return (numTabs &gt; 0) ? tabs.Aggregate((sum, next) =&gt; sum + next) : "";  }    private string Tabs(uint numTabs) {     StringBuilder sb = new StringBuilder();     for (uint i = 0; i &lt; numTabs; i++)         sb.Append("\t");      return sb.ToString(); }    private string Tabs(uint numTabs) {     string output = "";     for (uint i = 0; i &lt; numTabs; i++)     {         output += '\t';     }     return output;  } For aer.bst, citations with 3 or more authors are listed using et al. I need to modify the bibliography so that citations with 4 or more authors are listed using et al., but that citations with 3 authors have all the names written out. For example, I want "See Adams, Brown, and Clark (2003)" as opposed to "See Adams et al. (2003)." I'm using natbib with LaTex: \usepackage{natbib,har2nat}  In order to do that, I need to modify the function format.lab.names.abbr.  Might someone know how to do this?  FUNCTION {format.lab.names.abbr} { 'name.list :=   name.list num.names$ 'numnames :=   numnames #1 &gt;     { numnames #2 &gt;         { name.list #1 "{vv~}{ll}" format.name$ " et al." * }         { name.list #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =             { name.list #1 "{vv~}{ll}" format.name$ " et al." * }             { name.list #1 "{vv~}{ll}" format.name$ " and " *               name.list #2 "{vv~}{ll}" format.name$ *             }           if$         }       if$       field.used editor.field = {", eds" *} {} if$     }     {         name.list #1 "{vv~}{ll}" format.name$         field.used editor.field = {", ed" *} {} if$     }   if$ }   FUNCTION {format.lab.names.full} { 'name.list :=   #1 'nameptr :=   name.list num.names$ 'numnames :=   numnames 'namesleft :=     { namesleft #0 &gt; }     { name.list nameptr "{vv~}{ll}" format.name$ 'temp :=       nameptr #1 &gt;         { namesleft #1 &gt;             { ", " * temp * }             { temp "others" =                 { " et~al." * }                 { " and " * temp * }               if$             }           if$         }         'temp       if$       nameptr #1 + 'nameptr :=       namesleft #1 - 'namesleft :=     }   while$   numnames #1 &gt; field.used editor.field = and {", eds" *} {} if$   numnames #1 = field.used editor.field = and {", ed" *} {} if$ } I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.   $\dfrac {x}{0}$ is Impossible (  If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only $2. $ has infinite solutions so as $1.$ has none solution, how and why ?) $\dfrac {0}{0}$ is Undefined and has infinite solutions. (How come one be Undefined and yet has infinite solutions ?) $\dfrac {0}{x}$ and $x \ne 0$, it's okay for me, no problem, but if someone else wants to add something about it, feel free to do it. "The following" vs. "Following" I wrote      "Following examples show ..."   and my colleague revised it as      "The following examples show ..."   Why do we need "the" here even though the examples have not been mentioned before? Fullscreen YouTube video crash Too much space between links in post menu