Why am I sometimes 1 penny short?
Is floating point math broken?
Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ Find the least value of $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$$ I tried AM greater than equal to GM But that's for finding maximum value.  This can probably be solved with calculus but I don't know for some reason I can't find the answer . Which is $80$. What formula should be used to determine "hot" questions? Right now  is fairly broken -- it's a simple descending sort by views. As Joel said , it is &quot;a self-fulfilling prophecy.&quot; But this is not intentional, it's only because we haven't had time to improve it yet! As I sit down to write a better algorithm, I thought you guys might have some insight into what would make a good formula to determine the &quot;hot&quot; questions -- over the last 24 hours, the last week, the last month, and all time. Here are the variables we have to work with:  Votes [aka Score] Views Answers Answer Votes [aka Score] Whether the question has an accepted answer or not Time question was originally asked Time of last activity on question Reputation of asker Reputation(s) of answerers  Joel cited the reddit formula as an example, but I think our site is a little different than just a basic link submission site. Plus, we have more variables and data to work with. I found a  describing a few formulas: Y Combinator Hacker News  (p - 1) / (t + 2)^1.5 p = votes (points) from users t = time since submission in hours  Reddit  t = (time of entry post) - (Dec 8, 2005) x = upvotes - downvotes y = {1 if x &gt; 0, 0 if x = 0, -1 if x &lt; 0) z = {1 if x &lt; 0, otherwise x} log(z) + (y * t)/45000  Thoughts? Ideas? I don't want to overthink this; I just want the &quot;hot&quot; questions to be relevant and useful to us. So the simpler the algorithm the better! Based on my analysis of the above and the comments so far, here's the second version of what I have implemented so far. This might suck. I don't know:  (log(Qviews)*4) + ((Qanswers * Qscore)/5) + sum(Ascores) -------------------------------------------------------- ((QageInHours+1) - ((QageInHours - Qupdated)/2)) ^ 1.5  Note that accepted answers weight not at all in hotness. This is intentional, as I feel accepted answers are a fine social contract, but not a good data point for question or answer quality. This algorithm will heavily favor questions with LOTS of answers, as the sum(Ascores) are now included -- one assumes if there are lots of answers, there will be a lot more voting on the answers, too. Update: Note that this formula is what is used on the  linked from the homepage of each site. It is not the formula used to determine the  list. See . A loss and gain problem This is a very simple but confusing puzzle.  A customer buys goods worth $200$ rupees from a shop. The shopkeeper selling these goods makes zero profit from this purchase.  The lady gives him a $1000$ rupee note.  The shopkeeper has no change, so he goes next door to another shopkeeper to get change for the $1000$ rupee note. He keeps $200$ for himself and returns $800$ to the customer.  Later, the second shopkeeper from next door comes back with the $1000$ rupee note with a stamp on it saying "counterfeit" and takes his money back.  How much loss does the first shopkeeper face? Glossary with entries without reference Is there a command so that the glossary will also display elements without links (an equivalent of \nocite in BibTeX)? My question is simple.  Let $f(x)=x^3\sin(\frac{1}{x}), x\in \mathbb{R}$. Is this function uniformly continuous on $\mathbb{R}$?  I think it is not uniformly continuous when $x$ goes to $\infty$ Smoke shows on render, Fire doesn't Good day! I am having a problem wherein fire does not render. I followed a . Followed everything in his tutorial. However, the fire in my scene does not render at all. I wanted it to look like rocket thrusters. As of now, this is what I currently have for my Smoke Domain material.  I temporarily set the smoke to 0 because it may be blocking the flame. I've provided the screenshots where the smoke is enabled and disabled. (Forgive the quality of the rendered images, I lowered down the samples to quickly render the image.  And for my scene, this is what it looks like.  I am rendering it using GPU compute. If there's anything missing in the  current scene setup, just ask for it and i'll provide the necessary requirements. I tried searching for solutions but nothing worked out for me. Thank you and have a nice day! If $-1&lt;x&lt;1$ show that $\lim\limits_{n \to \infty}{n\,x^n} = 0$. I don't have idea. I only prove that $n\,x^n$ is decreasing. \bordermatrix with brackets [ ] instead of parentheses ( ) I have the following code:  \documentclass{article} \begin{document} $ M = \bordermatrix{~ &amp; x &amp; y \cr               A &amp; 1 &amp; 0 \cr               B &amp; 0 &amp; 1 \cr} $ \end{document}   But this code makes a matrix with () and i would like it to be with [] How do i do this? Spring prototype beans in combination with singleton beans and dependency injection. Is there an approach that is configuration only? Is static HTML the fastest? Do I have to code each case of this Grid full of plots separately? First, my understanding is that the long-term capital gains tax rate is 0% for those whose marginal rate on ordinary income is 10% or 15%, and (ignoring the highest 39.6% bracket) the rate is 15% for everyone else. (See  and .)  My interest is in what happens at the crucial break between the 15% and 25% bracket, which is $73,800 for married/joint returns in 2013 (and presumably will be higher in 2014, though I haven't been able to find out what that is or if it has even been set yet). Let's say a household has $64,000 in wage income, and sells stock for a $10,000 profit.   Does their income for the purposes of establishing their tax bracket go to $74,000? If so, is the entire $10,000 profit subject to the 15% long-term capital gains rate? Is any part of the wage income subject to the higher marginal tax rate?   Since the amount of income subject to the higher marginal rate is, well, marginal, part 3 seems somewhat less important than part 2. The real issue is whether the capital gains rate hits the entire profit, since, in that case, it seems the household should only sell enough stock to generate a $9000 profit, and sell the remainder in a future year. What is a reasonable sample size for correlation analysis for both overall and sub-group analyses? I am running Pearson's correlation on an overall sample of 400 respondents.   When I isolate male and female responses, my sample becomes 220 male responses and 180 female responses.   If I further isolate male and female responses by (say) age groups, some sample sizes become as low as 35 responses (for example, for females over the age of 65).  My question: How good are these sample sizes for correlation analysis? (I am looking at the relationship between income levels and overseas travel.)   (I think this has something to do with margin of error but how does this apply to inferential analysis which is based on probability. I can understand its role in descriptive statistics such as results of a political poll). $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$ I've been doing some more exercises in , in particular the following:  Show that $H_0(X,A) = 0$ iff $A$ meets each path-component of $X$.  "$\Leftarrow$": Let $x_i \in A \cap X_i \neq \emptyset \forall $ path-components $X_i$. Then for any $x \in X_i$ there is a path $\gamma$ from $x_i$ to $x$. They therefore differ by a boundary: $\partial \gamma = x - x_i$ and therefore $\{&#160;x \} = \{ x_i \}$ in $H_0(X,A) $. Therefore $H_0(X,A) = 0$.  "$\Rightarrow$" Now for this direction I'm not so sure and I'd be glad if you could give me a hint. Let $H_0(X,A) = 0$. By definition, the relative homology group is calculated from the following sequence of chain groups:  $$ 0 \rightarrow C_0(A) \rightarrow C_0(X) \xrightarrow{\partial_1} C_0(X) / C_0(A) \xrightarrow{\partial_0} 0$$  Then $H_0(X, A) = H_0(C_0(X) / C_0(A)) = ker \partial_0 / im \partial_0 = (C_0(X) / C_0(A) ) / im \partial_1 = 0$.  There are two cases where this equality holds:  (i) $C_0(X) / C_0(A) = 0$  (ii) $im \partial_1 = C_0(X) / C_0(A)$  In case (i), $A \cap X_i \neq \emptyset$ for all $X_i$.  In case (ii) I'm not sure how to proceed. Can you give me a hint? Many thanks for your help!  Edit Or maybe I could do the second direction like this:  $$0 = H_0(X,A) = \oplus_i H_0(X_i, A) \implies  H_0(X_i, A) = 0 \forall i$$  $$ \implies A \cap X_i \neq \emptyset \forall i$$? How do you combine two jQuery search results? eg:  var $allFoos = $('.foo'),     $allBars = $('.bar')     $allFoosAndBars = $allFoos + $allBars;   Obviously, I just made up that last line, but I hope it makes it sorta clear what I mean. To be clear, the example is greatly simplified, and it could be any arbitrary sets i'm talking about, so $('.foo, .bar') is not what I'm after. How can I further diagnose my central air conditioning compressor fan motor and capacitor? I have a 15 year old Bryant Central Air unit in my house.  Yesterday it stopped working.  Here is what I have checked.   The breaker in the basement is not tripped. The furnace works fine from the thermostat. The blower downstairs by the furnace runs when AC is on and blows room temp air into the house, but the fan on the compressor is not spinning.  The fan on the compressor does spin freely when I manually spin it with a stick.  Checked for fuses in the outside box.  There are no fuses outside, only an on/off pull thingy with two metal plates that you have to turn upside down to turn off. The capacitor is not flat on top, but far less rounded then any of the examples of blown AC dual capacitors I've seen on youtube. I get 220v in on the contactor. I get 220v out on the contactor. I get 24v from the thermostat. When the air is on at the thermostat the contactor solenoid is in, when the air is off it is out.   When the air is on at the thermostat I hear a buzz coming from the outside compressor but I can't tell what the source of the buzz is. If I push the solenoid in when the AC is off, the compressor does not start up and the fan does not spin, I just hear the buzz. If I wait by the compressor outside for a few seconds while the AC is on there's another buzz sound that goes off for about 3 seconds then stops.  From listening to the AC when it was working this is the buzz sound that goes off before the compressor and fan usually kick in. After leaving the non-functioning unit on for a few minutes while I diagnosed I noticed that the metal on top of the compressor fan motor was VERY hot to the touch.  It cooled after I turned the AC off.   I'm thinking the fan motor on the compressor is dead or the capacitor is bad.  Is there a way to check the capacitor and the fan motor while they're still in the unit?  With the power off of course.  Does anyone think it's something else?  Do I have to buy the overpriced ($485 for the manufacturer fan motor and $200 for the dual capacitor) replacement part from the manufacturer or can I use a third party manufacturer?  Any other help would be appreciated.  As much as I'd love to bring an HVAC guy in here I just can't afford it.  Either I go without AC, or I fix it myself.  EDIT:  With the AC on I manually spun up the fan on the compressor and the fan started spinning and the AC unit fired up. SEO (or other) problems with a 404? How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$ Does iOS have a firewall? Does iOS support a built-in firewall like macOS does? I don't see anything about it in Settings. Can't use method return value in write context