Math.Round of 14.5 rounds to 14 instead of 15, why? I have two decimal values which are 865 and 720. I subtract the two, divide by 10, then round the result. Why is this coming to 14 instead of 15 when the value needing to be rounded is 14.5?  Here's the code:  public decimal CalculateExpoDollars(decimal previousYearActual, decimal goal, decimal currentYearActual) {     if (currentYearActual &lt; goal &amp;&amp; currentYearActual &gt;= previousYearActual)         return 0.00m;      return Math.Round((currentYearActual - previousYearActual) / 10); }
Why does .NET use banker's rounding as default? According to the documentation, the  method uses a round-to-even algorithm which is not common for most applications. So I always end up writing a custom function to do the more natural round-half-up algorithm:  public static decimal RoundHalfUp(this decimal d, int decimals) {     if (decimals &lt; 0)     {         throw new ArgumentException("The decimals must be non-negative",              "decimals");     }      decimal multiplier = (decimal)Math.Pow(10, decimals);     decimal number = d * multiplier;      if (decimal.Truncate(number) &lt; number)     {         number += 0.5m;     }     return decimal.Round(number) / multiplier; }   Does anybody know the reason behind this framework design decision?  Is there any built-in implementation of the round-half-up algorithm into the framework? Or maybe some unmanaged Windows API?  It could be misleading for beginners that simply write decimal.Round(2.5m, 0) expecting 3 as a result but getting 2 instead.
For a submerged object, buoyant force ($F_b$) is defined as:  $$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$   Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to   and found the following explanation, but it seems like non-sequitur to me:      Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object. Assume: $p$ is a prime that satisfies $p \equiv 3 \pmod 4$  Show: $x^{2} \equiv -1 \pmod p$ has no solutions $\forall x \in \mathbb{Z}$.  I know this problem has something to do with Fermat's Little Theorem, that $a^{p-1} \equiv 1\pmod p$. I tried to do a proof by contradiction, assuming the conclusion and showing some contradiction but just ran into a wall. Any help would be greatly appreciated. How can I recover data from a 2TB hard drive? I have a Western Digital 2TB External Hard Drive. I basically use it to keep all my movies and videos. Its file system is NTFS, and currently I'm using 1.4TB. One night I turn it on and copied a 4GB file and there was no problem. Then I tried to copy another 6GB file and Ubuntu throw the following error: "Input/output error". I tried to create a new folder with no luck. I still could read the disk an play a file, so I thought restarting the computer would solve the problem, but it didn't. The next time, Ubuntu didn't mount the disk.   After that I tried in another PC with Window 7. It throwed a message telling me: first, that the disk needs to be formatted, and second, a "Local Disk #: Is not accessible. Data error (cyclic redundancy check)". Here is when I started to worry. I tried to fix the disk with CHKDSK (chkdsk \r). After about 5 hours, it crashed with the following error: "An unspecified error occurred".  My guess to all this is that the disk is dying. :'(  Lastly I did one more thing. I installed GSmartControl on Ubuntu to see the SMART info and the attributes tab appears in red with the following line highlited:  ID# ATTRIBUTE_NAME          FLAG     VALUE WORST THRESH TYPE      UPDATED  WHEN_FAILED RAW_VALUE ... 197 Current_Pending_Sector  0x0032   197   197   000    Old_age   Always       -       977 ...   Just to check, I left the disk on for about 2 minutes and then refreshed the page, and the Current_Pending_Sector RAW_VALUE increased from 977 to 979. My guess is that sectors on the disk are dying. Am I right?  If every time I turn on the disk more sectors are going to die, I need to copy the data fast. I don't know if there is a solution on Windows, and I'm worried about the time that it could take to do it with the slowness of Windows. Basically my question is: How can I recover the data? (Or at least some of it). Is there any program on Ubuntu that can help me with this (If I'm still on time)?. My main concern is the size of the disk (2TB), so I think I would need at least another 2TB disk to copy to.  Any help would be appreciated. I have a question related to multiple regression and interaction, inspired by this CV thread:    When checking for a moderation effect I do center my independent variables and multiply the centered variables in order to calculate my interaction term. Then I run my regression analysis and check for main and interaction effects, which may show the moderation.  If I redo the analysis without centering, apparently the coefficient of determination ($R^2$) does not change but the regression coefficients ($\beta$s) do. That seems clear and logical.  What I do not understand: The p-values of the main effects change substantially with centering, although the interaction does not (which is right). So my interpretation of main effects could change dramatically - just determined by centering or not. (It's is still the same data, in both analyses!)  Can somebody clarify? - Because that would mean that the option to center my variables would be mandatory and everybody should do it in order to get the same results with the same data.    Thanks a lot for distributing to that problem and your comprehensive explanations. Be assured that your help is very much appreciated!  For me, the biggest advantage of centering is to avoid multicollinearity. It is still quite confusing to establish a rule, whether to center or not. My impression is, that most of the resources suggest to center, although there are some "risks" when doing it. Again I want to put out the fact, that 2 researchers dealing with the same material and data might conclude different results, because one does centering and the other does not. I just read some part of a book by Bortz (he was a Professor and kind of a Statistics Star in Germany and Europe), and he does not even mention that technique; just points out to be careful in interpreting main effects of variables when they are involved in interactions.  After all, when you conduct a regression with one IV, one moderator (or second IV) and a DV, would you recommend to center or not? How to create script with auto-complete? I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? For US federal tax, whether a taxpayer is subject to AMT is determined by a complicated way in Instructions for Form 1040, Line 45, which involves an incomprehensible "worksheet to see if you should fill in form 6251".  By simplification and approximation, is there a rule of thumb for an estimate of a salary value of a taxpayer, so that a salary above the value might cause the taxpayer to pay AMT with a big possibility, and a salary below the value might not with a big possibility?  Why do the following two sources seem to differ greatly in estimates of the boundary salary value:    says that  the possible boundary salary value for a single taxpayer  to pay AMT is $53,900.  says that      Taxpayers with small incomes fall within the AMT exemption limits.   That leaves the upper middle class to face the tax, especially those   with incomes between $200,000 and $500,000    Thanks. Finding an unknown linear transformation given that $T(1,1)=(1,0,2)$ and $T(2,3) = (1,-1,4)$ Let $T\colon\mathbb{R}^2\to\mathbb{R}^3$ be the linear transformation such that $T(1,1)=(1,0,2)$ and $T(2,3) = (1,-1,4)$.       Does such a linear transformation exist?  So far I've worked out that it cannot exist, as the first entry for $T(1,1)$ is 1, whereas the first entry for $T(2,3)$ is 1, which can only be obtained through 3-2, or half of 2.  Is there any way of presenting this formally? Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$? I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1  {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum seems to converge to 1.6449.  But I still want to know:  Does this sum converge? Is there a name of this sum (or the series $\frac{1}{k^2} $)? Seven-sided dice from five-sided dice with finite rolls I have a 5-sided dice, and I want to use it to simulate a 7-sided dice. Is there a way to do this with a finite number of rolls? How to obtain the sum of the following series? $\sum_{n=1}^\infty{\frac{n^2}{2^n}}$ How to correctly post a Q&A style question (i.e.self answer) without it failing minimal understanding As I understand it Q&amp;A style questions are  but a self answered question should be judged independently of the answer. Often after I've solved a problem I would like to share the question and answer on stack overflow (and have on a couple of occasions) but how to show &quot;minimal understanding of the problem&quot; is always challenging. For example I have been looking into the following question  How can I plot a straight line in the JMonkey Engine library I'm trying to plot straight lines between vertices that I specify using the JMonkey Engine 3D graphics library. So for example if I was to try to plot between: (2,0,0) (-1,0,1) (0,1,1) (1,1,1) (1,4,0) Then I would get:   This is something I do (and forget how to do) relatively often and the answer is harder to find than you'd imagine ( provides everything needed to figure it out but its spread out over several posts and isn't a complete solution on its own). Therefore I want to provide a Q&amp;A for it but at present this question feels like it would fail the minimal understanding test  Questions must demonstrate a minimal understanding of the problem being solved. Tell us what you've tried to do, why it didn't work, and how it should work  Should I make up some imaginary problem? (Probably the final source code messed up a bit) Or mention in the question that I’m self answering to assuage some of the normal &quot;This is not a do your work for you site&quot; comments. How do I free up more space in /boot? My /boot partition is nearly full and I get a warning every time I reboot my system. I already deleted old kernel packages (linux-headers...), actually I did that to install a newer kernel version that came with the automatic updates.  After installing that new version, the partition is nearly full again. So what else can I delete? Are there some other files associated to the old kernel images?  Here is a list of files that are on my /boot partition:  :~$ ls /boot/ abi-2.6.31-21-generic         lost+found abi-2.6.32-25-generic         memtest86+.bin abi-2.6.38-10-generic         memtest86+_multiboot.bin abi-2.6.38-11-generic         System.map-2.6.31-21-generic abi-2.6.38-12-generic         System.map-2.6.32-25-generic abi-2.6.38-8-generic          System.map-2.6.38-10-generic abi-3.0.0-12-generic          System.map-2.6.38-11-generic abi-3.0.0-13-generic          System.map-2.6.38-12-generic abi-3.0.0-14-generic          System.map-2.6.38-8-generic boot                          System.map-3.0.0-12-generic config-2.6.31-21-generic      System.map-3.0.0-13-generic config-2.6.32-25-generic      System.map-3.0.0-14-generic config-2.6.38-10-generic      vmcoreinfo-2.6.31-21-generic config-2.6.38-11-generic      vmcoreinfo-2.6.32-25-generic config-2.6.38-12-generic      vmcoreinfo-2.6.38-10-generic config-2.6.38-8-generic       vmcoreinfo-2.6.38-11-generic config-3.0.0-12-generic       vmcoreinfo-2.6.38-12-generic config-3.0.0-13-generic       vmcoreinfo-2.6.38-8-generic config-3.0.0-14-generic       vmcoreinfo-3.0.0-12-generic extlinux                      vmcoreinfo-3.0.0-13-generic grub                          vmcoreinfo-3.0.0-14-generic initrd.img-2.6.31-21-generic  vmlinuz-2.6.31-21-generic initrd.img-2.6.32-25-generic  vmlinuz-2.6.32-25-generic initrd.img-2.6.38-10-generic  vmlinuz-2.6.38-10-generic initrd.img-2.6.38-11-generic  vmlinuz-2.6.38-11-generic initrd.img-2.6.38-12-generic  vmlinuz-2.6.38-12-generic initrd.img-2.6.38-8-generic   vmlinuz-2.6.38-8-generic initrd.img-3.0.0-12-generic   vmlinuz-3.0.0-12-generic initrd.img-3.0.0-13-generic   vmlinuz-3.0.0-13-generic initrd.img-3.0.0-14-generic   vmlinuz-3.0.0-14-generic   Currently, I'm using the 3.0.0-14-generic kernel. Roll back from OS X Mavericks ( or any newer OS) to Mountain Lion I installed the developer preview of OS X Mavericks, &amp; too many things broke. I can't stay on it &amp; need to roll back to Mountain Lion. Any ways of doing that? Please help!  I originally installed Mountain Lion &amp; Mavericks through the Mac App Store. Deriving Schrodinger equation from Klein-Gordon QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$ In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, , the position space wavefunction is defined as   $$\psi(x)=\langle 0|\phi(x)|\psi\rangle, \tag{2.82+2.83}$$  where $|\psi\rangle$ is any state in the Fock space. Then he uses the equations (i) $\partial_t^2\phi_0=(\nabla^2-m^2)\phi_0$ (i.e., the Klein-Gordon equation for the free massive scalar field $\phi_0(\textbf{x},t)$) and (ii) $[H,\phi_0]=-i\partial_t\phi_0$ to derive Eq. 2.85, in following 3 steps:  $$i\langle 0|\partial_t\phi_0(\textbf{x},t)|\psi\rangle=\langle 0|\int \frac{d^3\textbf{p}}{(2\pi)^3}\frac{\sqrt{\textbf{p}^2+m^2}}{\sqrt{2\omega_\textbf{p}}}(a_p e^{-ip\cdot x}-a^\dagger_p e^{ip\cdot x})|\psi\rangle\\ =\langle 0|\sqrt{m^2-\nabla^2}\phi_0(\textbf{x},t)|\psi\rangle.\tag{2.85}$$  This equation is used to successfully derive the Schrodinger equation in quantum mechanics for the state $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$.   The first equality follows from differentiation of $\phi_0(x)$ w.r.t $t$. How does the second equality follow from the first? How are the inputs (i) and (ii) are utilized to derive the second equality from the first? Is it true that the operator $$i\partial_t\sim\sqrt{m^2-\nabla^2}~?$$ If yes, what is the use of doing the middle step? How to install and configure Wine? What is the purpose of the bin directory? What are some examples? Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-) Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$? Is there any $2 \times 2$ matrix $A \neq I$ such that $A^3=I$?  In my opinion: no. Thank you very much. Evaluation of Indefinite Integral resulting in Hypergeometric Function I am attempting to derive the result:  $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$  First, I start off with the binomial expansion of the integrand to get:  $$\int\sum_{k=0}^{\infty}\frac{(1/m)_k}{k!}\left(-x^n\right)^kdx  $$  Then, I pull out a $-1$ and interchange summation and integration:  $$\sum_k\int (-1)^k\frac{(1/m)_k}{k!}x^{nk}dx  $$  $$ \sum_k (-1)^k\frac{(1/m)_k}{k!}\frac{x^{nk+1}}{nk+1} $$  $$ x\sum_k \frac{(1/m)_k}{k!}\frac{(-x^n)^k}{nk+1}  $$  So, I'm just right there, but not sure how to express this series in the general form of a hypergeometric series with the two additional Pochhammer symbols.  I even typed this last series into Mathematica, and it returned the Hyper. Function. What am I missing, or how do I transform this last series into the desired form?