Figuring out an ideal balance to carry on credit cards
Should I really pay off my entire credit card balance each month or should I maintain some balance?
ALTER TABLE [dbo].[REMINDER_EMAIL]  WITH CHECK ADD  CONSTRAINT    [FK__REMINDER_EMAIL__FACILITY_ID] FOREIGN KEY ( [FACILITY_ID] )   REFERENCES [dbo].[FACILITY] ( [FACILITY_ID] )  ALTER TABLE [dbo].[REMINDER_EMAIL] CHECK CONSTRAINT [FK__REMINDER_EMAIL__FACILITY_ID] Is there a string math evaluator in .NET? If I have a string with a valid math expression such as:  String s = "1 + 2 * 7";   Is there a built in library/function in .NET that will parse and evaluate that expression for me and return the result? In this case 15. I have a very long equation that must be separated into two lines, and it has several pairs of nesting \left \right delimiters. Unfortunately, it seems that they need to be on the same line in order for them to work.  Also I'm using the align environment because I need the aligning functionality. Is there a way to have the size of delimiters automatically adjusted over multiple lines? Find $|(\mathbb{Z} \oplus \mathbb{Z}) / \langle(3,6)\rangle|$ and determine if its cyclic Intuition for Smooth Manifolds Consider the graphs of the functions $f_1(x) = |x|$, and $f_2(x) = x$ under the subspace topology of $\mathbb{R}^2$.   Both of these graphs are smooth manifolds, just pick coordinate charts to be $(x, f_i(x)) \leftrightarrow x$.   Moreover, they are diffeomorphic via the map $(x, f_1(x)) \rightarrow (x, f_2(x))$.   This seems to clash with my intuition. For example, the graph of $f_1$ has a corner, so it "shouldn't" be smooth, much less diffeomorphic to $f_2$, which is just a straight line.   Can someone explain what's going on here?   In light of these examples, how should I visualize smooth manifolds and diffeomorphisms? How can I send the stdout of one process to multiple processes using (preferably unnamed) pipes in Unix (or Windows)? I'd like to redirect the stdout of process proc1 to two processes proc2 and proc3:           proc2 -&gt; stdout        /  proc1        \           proc3 -&gt; stdout   I tried   proc1 | (proc2 &amp; proc3)   but it doesn't seem to work, i.e.   echo 123 | (tr 1 a &amp; tr 1 b)   writes   b23   to stdout instead of    a23  b23 How to calculate $\lim \limits_{n \to \infty}(1 +\frac{a}{n})^n$? The $n$ complex $n$th roots of a complex number $z$ Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$.   My proof: Let $z=re^{i\theta}\in\mathbb{C}$ and $a\in\mathbb{C}$ s.t. $a^{n}=z$. Let $a=\rho e^{i\varphi}$. Then, $a^{n}=(\rho e^{i\varphi})^{n}$. Setting $\rho^{n}e^{i\varphi n}=re^{i\theta}$ and rewriting in polar form, we get $$\rho^{n}(\cos(n\varphi)+i\sin(n\varphi))=r(\cos(θ)+i\sin(\theta)).$$ Thus $\rho^{n}=r$ $\rightarrow$ $\rho=r^{1/n}$ and $n\varphi=\theta\rightarrow\varphi=\theta/n$. Thus the roots of $a\in\mathbb{C}$ have the form $$r^{1/n}(\cos(\tfrac{\theta}{n})+i\sin(\tfrac{\theta}{n}))=r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))$$ for some $j\in\mathbb{Z}$ due to the periodicity of trig functions. Then, $$r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))=r^{1/n}e^{i\left(2\pi j+\tfrac{\theta}{n}\right)}=r^{1/n}e^{i(\theta+2\pi k)/n}$$ for $0\leq k\leq n-1 ,j=nk\in\mathbb{Z} $.  I feel like there could be a more concise way to put this. Discrete mathematics - Find all integer solutions of the equation $a^2+ b^2 + c^2=a^2 b^2$. Find all integer solutions of the equation $a^2+ b^2 + c^2=a^2 b^2$.  This is one of the questions we presented in one session to contest preparation PUTNAM. It turns out that I can't get from the problem. Could someone just give me a hint? (Please, don't give me the answer. Simply, an argument that can help me advance in the problem or theorem might suffice.) When does the 'Lost City' become Atlantis in Stargate In Stargate SG-1 they are searching for the Lost City in season 8, however in the Stargate Atlantis pilot it is referred to as Atlantis. When is the name found? Or the connection between the Lost City and Atlantis made? pdflatex, \includegraphics, and multi-page PDF files I have a PDF file with multiple figures; each figure is a separate page in the PDF file.  I am using pdflatex, and I would like to use \includegraphics or equivalent to include some of the figures.  Of course I could use an external tool to split the PDF file into multiple one-page PDF files. Then I could use \includegraphics as usual. However, I would have a much simpler workflow if I could avoid this extra step. Hence the question:  How can I include, e.g., page 3 of foo.pdf as a figure?  I am aware that there is a Latex package called pdfpages, but it seems to be designed for something else: with pdfpages, each page in the input file will produce a full page in the final document, not just a box. Uniform convergence on two separate intervals Test the sequence of functions for uniform convergence       $$f_n(x)=\frac{nx}{1+n^2x^2}$$      $a) \text{on} \ [0,1]$      $b) \text{on} \ [1,{\infty})$   a)  If we differentiate this with respect to x we get the following :  $$(\frac{nx}{1+n^2x^2})'=\frac{n(1-n^2x^2)}{(1+n^2x^2)^2}$$  This is equal to $0$ for $x=^{+}_{-}\frac{1}{n}$  From here we can either check these two values in the original function and take the larger one as the supremum or we can find the second derivative and check whether it's negative. In either case we find that for $x=\frac{1}{n}$ we get the value $f_n(x)=\frac{1}{2}$ and for $x=-\frac{1}{n}$ we get $f_n(x)=-\frac{1}{2}$. Note that $x=-\frac{1}{n}$ doesn't even belong in our interval. Now we can finish the problem.  $$\lim_{n\to \infty}{\frac{nx}{1+n^2x^2}}=0$$  So it is piecewise converging to $f(x)=0$  $$\lim_{n\to \infty}\sup{\left \lvert{f_n(x)-f(x)}\right \rvert}=\lim_{n\to \infty}{\frac{n\frac{1}{n}}{1+n^2 \frac{1}{n^2}}}=\frac{1}{2}$$  So $f_n(x)$ does not convergence uniformly on $[0,1]$ to $f(x)=0$  b)  Now we need to test the on the interval $[1,\infty)$.From the first derivative we notice that if $x\in [1,\infty)$ then the numerator seems to be negative for all $x&gt;1$. So $f_n(x)$ is monotonically decreasing. That means that the supremum will be at the first point of the interval or $x=1$. Taking the limit :  $$\lim_{n\to \infty}{\frac{n}{1+n^2}}=0$$  We can see that the function convergences uniformly on this interval.  Is there something wrong with either of these solutions?  I'm having doubts mostly about the second interval since $x=\frac{1}{n}$ is a supremum but only for $n=1$. However I believe that we need to have the supremum that's satisfied for all $n \in N$. Pointing even the tiniest mistake in my solutions would be very helpful. Thank you. How do I make a Theorem n followed by a Theorem n' ? Is SHA-256 secure as a CTR block cipher? Generate a 256-bit random nonce. XOR it with a 256-bit reusable symmetric key. This is x.  We represent numbers in simple binary instead of a counting function. 0 in dec = [256 zeros] in binary, 1 = [255 zeros]1, 23092348 = [241 zeros]1011000000101110001111100, etc.  For each nth block needed for the length of the plaintext, XOR x with n and hash it with SHA-256 to form b[n].  Let p[n] be the nth 256-bit block in the plaintext and c[n] be the nth block in the ciphertext. Then c[n] := b[n] XOR p[n].  Send the unencrypted nonce along with the ciphertext.  To decrypt c[n], XOR it with b[n] to retrieve p[n].  Is this secure? If so, then why do we need AES-256?  If this is not secure, then is there any mode of operation for which SHA-256 is secure as a block cipher? Specifically, it would need to be used in cipher feedback, output feedback, or any other mode that only uses the cipher in the encryption direction. I don't understand why su - is preferred over su to login as root. Will ESM also become available for Ubuntu 14.04? ESM is available for 12.04, which enables Advantage customers to keep receiving kernel updates and updates for many user space packages. In our case it's very important to know if we'll be able tot use Ubuntu 14.04 safely after April '19, because it is not possible to upgrade our application and migrate it to say, 16.04. I heard the following dialog in a British English movie:     Words go round quick. All the little tongues go clack, clack, clack.   What does this mean, and why was the word "quick" used instead of "quickly"? Is it possible to write some C or C++ code and compile to binaries, then use those binaries with php? Is it also possible to write a php library using C and C++ ?  If so, please tell how can I do so? I am a US citizen and have recently visted the UK  with simply a passport which allowed me to vist for 6months I have heard rumors that I am obligated to stay in the US for a Equal amount of time that I have visited the UK before I am able to return, but I am struggling tour no any proof of this obligation .. Am I missing somthing or is it simply not true How to return multiple objects from a Java method? I want to return two objects from a Java method and was wondering what could be a good way of doing so?  The possible ways I can think of are: return a HashMap (since the two Objects are related) or return an ArrayList of Object objects.  To be more precise, the two objects I want to return are (a) List of objects and (b) comma separated names of the same.  I want to return these two Objects from one method because I dont want to iterate through the list of objects to get the comma separated names (which I can do in the same loop in this method).  Somehow, returning a HashMap does not look a very elegant way of doing so.