What are vintage magic cards worth?
What's the most efficient way of determine individual card values for my collection?
How to batch watermark JPG files? I have around 15,000 JPG files that need to be watermarked. Is there any way to process these images en masse and add a watermark?  The images may be different sizes, I want the watermark placed in the center of the images. Let $\varphi:G\rightarrow H$ be a group homomorphism from group $G$ to group $H$. Show that, if $H$ is abelian, all subgroups of $G$ that contain $\mathrm{ker} (\varphi)$ are normal in $G$. I marinated chicken five days ago in lemon juice, seasoned salt, and garlic powder, is it still safe to cook in the oven? Could someone provide me with a good explanation of why $0^0=1$?  My train of thought:  $x&gt;0$  $0^x=0^{x-0}=0^x/0^0$, so  $0^0=0^x/0^x=\,?$  Possible answers:   $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined   PS. I've read the  on mathforum.org, but it isn't clear to me. Check if email exist php Associated elements in a ring Please help me to find elements $a,b$ in a ring $R$ such that $a\mid b$ and $b\mid a$, but there does not exist any unit $u$ in $R$ such that $a=ub$. An element of a group has the same order as its inverse If $a$ is a group element, prove that $a$ and $a^{-1}$ have the same order.   I tried doing this by contradiction.    Assume $|a|\neq|a^{-1}|$  Let $a^n=e$ for some $n\in \mathbb{Z}$ and $(a^{-1})^m=e$ for some $m\in \mathbb{Z}$, and we can assume that $m &lt; n$.  Then $e= e*e = (a^n)((a^{-1})^m) = a^{n-m}$. However, $a^{n-m}=e$ implies that $n$ is not the order of $a$, which is a contradiction and $n=m$.  But I realized this doesn't satisfy the condition if $a$ has infinite order. How do I prove that piece? US citizens travelling to Cuba are restricted to a set of "reasons" for travel, of which tourism is not one. I have US-Philippines dual citizenship, and both passports. Would I just buy the ticket using my Filipino passport? What kind of documentation/proof would I need to return back to the US? And as a US citizen travelling with a foreign passport, where would I go for US customs?  Thanks. Supremum of a product measurable function... How to differ sessions in browser-tabs? Show discontinuity of $\frac{xy}{x^2+y^2}$ What Is Energy? Where did it come from? What does “a couple” mean to you, and what does “a few” mean to you? What does “a couple” mean to you, and what does “a few” mean to you?  Is there a proper way to use these words?  It was striking to hear that “a couple” meant two (2) to someone.  My reaction was, “how/why do you make a short word longer by adding an extra syllable to just say ‘two?’”  I think Few: 2-3, Couple 4-6, Several 7+.  What are your thoughts? Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$ Any hint to prove this? I tried with properties of binomial coefficient but I can't get anything. $$\sum_{n=k}^{\infty}{{n}\choose{k}}x^n=\dfrac{x^k}{(1-x)^{k+1}}$$ This question has an inverse:     What would be the most efficient way to translate a combination of $k$-tuple into its positions within the $\left(\!\!\binom{n}{k}\!\!\right)$ combinations? I need this to be fast for combinations of $\left(\!\!\binom{70}{7}\!\!\right)$ order of magnitude - very large, but not exceeding 2 billion (fits into int32 maximum value).  Below is an example of $\left(\!\!\binom{6}{3}\!\!\right)$, where the aim is to quickly translate (a, d, f) tuple to value 9, while in the real problem $k$ ranges between 5 and 8.  $$\begin{array} {cccccc|c} a&amp;b&amp;c&amp;d&amp;e&amp;f&amp;^{combination}/_{sort\_order}&amp; \\\hline x&amp;x&amp;x&amp; &amp; &amp; &amp;1\\ x&amp;x&amp; &amp;x&amp; &amp; &amp;2\\ x&amp;x&amp; &amp; &amp;x&amp; &amp;3\\ x&amp;x&amp; &amp; &amp; &amp;x&amp;4\\ x&amp; &amp;x&amp;x&amp; &amp; &amp;5\\ x&amp; &amp;x&amp; &amp;x&amp; &amp;6\\ x&amp; &amp;x&amp; &amp; &amp;x&amp;7\\ x&amp; &amp; &amp;x&amp;x&amp; &amp;8\\ x&amp; &amp; &amp;x&amp; &amp;x&amp;9\\ x&amp; &amp; &amp; &amp;x&amp;x&amp;10\\ .&amp;.&amp;.&amp;.&amp;.&amp;.&amp;.\\  &amp; &amp; &amp;x&amp;x&amp;x&amp;20\\ \end{array}$$  I know that I could pre-calculate all the combinations and reverse the lookup dictionary. However, such dictionary would not be efficient in terms of memory usage. Therefore I am looking for either calculation-based approach, or a more efficient data structure to perform this mapping. On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ Real-life examples of common distributions Profitable strategy in coin tossing? There is a coin with a probability $p$ of heads, and $1-p$ of tails. Tosses are independent of each other. When you bet an amount of money $x$, you receive $2x$ if it lands heads, and you lose what you bet if it lands tails.  We are going to use the following strategy:  You start betting an initial amount $x_0$. Every time you lose, you bet twice what you bet in the previous throw. You keep doing this, until you win, at which point you retire, or start again, betting $x_0$.  The interesting case of course is when $p &lt; 0.5$.  The question is: How profitable is this strategy? Intuitively I feel that eventually I can always win an arbitrarily large amount of money. But that can't be entirely right ("the house always wins!"). So what am I missing here? What's the expected win in each run?  Also, does this sort of strategy have a name in the literature?  P.S.: Found this later, Collapsing a stream of multiple lines into one line I have a file containing a a large number of lines, each of which contains a bunch of numbers which are separated by spaces. I process this data in a pipe in some way, and then I want to collapse the multiple lines into a single line of all the numbers separated by spaces.  Is there a standard command-line utility I can use to do this? It seems like most line-by-line utilities won't mess with the newlines... Q1: Does a DNS rule for *.domain.com match multi-level domain names? In other words, if I just set *.domain.com in my DNS records, will it match all of:  a.domain.com b.a.domain.com c.b.a.domain.com   as many levels deep as needed, with just that one wildcard DNS specifier?  We have a use case for multi-level DNS and I would like to confirm that we are doing the right thing in setting one wildcard DNS setting for the root domain.  Q2: will this work with round-robin DNS?