Inconsistencies when calculating the remainder where 220 % 25 gives a remainder of 20 while 420 % 25 gives a remainder of 19?
Is floating point math broken?
Let $\mathbb R$ denote the set of all real numbers. Find all function $f: R\to \ R$ such that   $$f\left(x^2+f(y)\right)=y+(f(x))^2$$   It is the problem. I tried to it by putting many at the place of $x$ and $y$ but I can't proceed. Please somebody help me. C# Unsafe/Fixed Code Can I connect copper wire to aluminum wire? Why does Ubuntu 12.10 not detect my Win7 partition? In a few large projects i have been working on lately it seems to become increasingly important to choose one or the other (XML or Annotation). As projects grow, consistency is very important for maintainability.   My questions are: what are the advantages of XML-based configuration over Annotation-based configuration and what are the advantages of Annotation-based configuration over XML-based configuration? Rubbing eggplants in salt Publish and/or subscribe to Platform Events between separate Orgs? Who was the piano player that Starbuck met in the bar on Galactica? Towards the end of season four, Starbuck meets a piano player in the makeshift bar on Galactica.  Now, it appears that this guy is some kind of ghost/angel like Caprica-6 and the ghost Baltar - but I have an idea that he could also be the original copy of the seventh model (Daniel). It's mentioned right in the episode before that Cavil poisoned all of the bodies that were being prepared to accept a copy of Daniel - but it's ambiguous about the fate of the original cylon with that personality. Daniel was also mentioned as an 'artist' and 'sensitive' person.  So - could it have been Daniel that helped Starbuck realise what the music meant? Global setting of spacing between items in itemize environment for beamer Prevent Serialization of Base Class Handling of special LaTeX characters in text Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$ I have seen this identity on Wolfram mathworld and in a comment to another similar trigonometric proof:    I can't seem to find a proof to this identity online anywhere. What method is used to prove this? What is the difference between '/' and '//' when used for division? Is there a benefit to using one over the other? In Python 2, they both seem to return the same results:  &gt;&gt;&gt; 6/3 2 &gt;&gt;&gt; 6//3 2 Handling large numbers in C++? What is the best way to handle large numeric inputs in C++ (for example 10^100)?   For algorithms I usually switch over to ruby and I sometimes use strings.  Any other good methods? I bought an Acer laptop and installed Ubuntu 14.04LTS but WiFi drivers are not available for it. So I am trying to install a windows driver using wifidocs/driver/ndiswrapper.  On doing lspci -vvnn, it gives:  Network controller ［0280: Qualcomm Atheros Device ［168c:0042］ (rev 30)   Then I also disabled the atheros drivers. Using lspci and lspci -n, I can say PCI id is 168c:0042. But Acer website has three atheros drivers. I could not figure out which one to download but downloaded one on a windows PC and pasted the zip file on my desktop then unzipped it from the terminal using unzip &lt;filename&gt;.zip. The page says you have to go to control panel system hardware and all that on the windows system. I did not do it. Windows computer not being mine, I am afraid I should not install any additional driver on it. After unzipping it has one .inx file and one .bin file but no .sys file. I don't know whether I am completely right till here or have I missed something? And if I am right then how to proceed from here? I also made a new directory and copied the unzipped .inx and .bin files in it. Then in that directory I ran the ndiswrapper -i &lt;filename&gt;.inf but it says ndiswrapper not installed and when I do sudo apt-get install ndiswrapper-common, I get the reply that ndiswrapper-common is already the latest version. What to do? here is my dmesg | grep ath10k result  [   7.882758] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [    8.190016] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [    8.334587] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/board-2.bin failed with error -2 [   10.140677] ath10k_pci 0000:03:00.0: qca9377 hw1.0 (0x05020000, 0x003820ff sub 105b:e09a) fw WLAN.TF.1.0-00267-1 fwapi 5 bdapi 1 htt-ver 3.1 wmi-op 4 htt-op 3 cal otp max-sta 32 raw 0 hwcrypto 1 features ignore-otp [   10.140682] ath10k_pci 0000:03:00.0: debug 0 debugfs 0 tracing 0 dfs 0 testmode 0 Here is the output of lspci -nnk | grep -iA2 net  02:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 15)     Subsystem: Acer Incorporated [ALI] Device [1025:098a]     Kernel driver in use: r8169 03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:0042] (rev 30)     Subsystem: Foxconn International, Inc. Device [105b:e09a]     Kernel driver in use: ath10k_pci I am looking for a way to replace a string in a file with a string that contains a slash by using sed.  connect="192.168.100.61/foo" srcText="foo.bar=XPLACEHOLDERX" echo $srcText | sed "s/XPLACEHOLDERX/$connect"   The result is:  sed: -e Expression #1, Character 32: Unknown option for `s' Can I disable GNOME legacy tray in 17.04? Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. Inequality about infinite cardinals The following is a question I'm interested in solving:    The idea is to solve it without using AC. As @Asaf Karagila pointed out in the comment section:  "Even if $\kappa$ is an infinite cardinal, you can't assume that $\kappa \geq \omega$ without using AC".  I am still trying to think this through: My reasoning being, that as $\kappa &gt; n \forall n \in \omega$, then $\kappa \geq \sup\{n:n\in \omega\}=\omega$ as $\omega$ is a limit ordinal.  Where does my reasoning fail? And also, a detailed solution would be very appreciated. I've recently updated to Ubuntu 15.04 and after installing the latest nvidia drivers (346.59) which worked on ubuntu 14.10 seems to have caused some trouble on boot.  All I get is the following when booting:   A purple screen for about a second and then  [0.514409] ACPI PPC Probe failed. Starting version 219  A black screen with a message of my monitor saying : No Signal detected.   I am using a GTX 970.