what is the technical term for such pictures?  Those with the special characteristics black&amp;white, with a single color?
There is an option to filter only one color (R,G,B,Y) in my camera. What I want to know is can this effect be achieved in photoshop? And what is this called? I tried googling monochrome and came up with nothing. My camera calls it part color.
Value of $x=\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of second digit $1$ of $x$ What is the value of the series   $\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}}$   and can one find closed forms for the sums   $\sum\limits_{n=1}^N {\frac{1}{10^n-1}}$?   I found that $\sum_{n=1}^N {\frac{1}{10^n-1}} $  =  $\sum_{n=1}^N (\lim_{m\to\infty  }{\frac{10^{-n}-10^{-nm}}{1-10^{-n}}})$  =  $0.111\ldots+0.0101\ldots+\ldots$ but I can't go farther.     The original question is: "Where does the second '1' appear?" How can one use multi threading in PHP applications Is there a realistic way of implementing a multi-threaded model in PHP whether truly, or just simulating it. Some time back it was suggested that you could force the operating system to load another instance of the PHP executable and handle other simultaneous processes.  The problem with this is that when the PHP code finished executing the PHP instance remains in memory because there is no way to kill it from within PHP.  So if you are simulating several threads you can imagine whats going to happen.  So I am still looking for a way multi-threading can be done or simulated effectively from within PHP. Any ideas? When is it appropriate to end a sentence in a preposition? How did Asuna and Kirito survive? In the last episode of Aincrad arc of Sword Art Online, how did Asuna survive even though her avatar disintegrated after being attacked by Kayaba? Also Kirito was stabbed by Kayaba as well. How did he survive?  Was it in the novel on how they really survived? override at runtime __setattr__ QR factorization of complex matrix If you have two complex numbers $a,b$ how can you find the QR factorization of   $ M = \begin{bmatrix}        aI_n\\          bI_n \end{bmatrix} $,  I can't seem to be able to do it. I tried an earlier trick, but I now know it's not necessary true for complex numbers. Is there anyway to do this?  I think if I use a Givens algorithm I should be able to reduce it to a QR factorization. Nested sequence of non-empty compact subsets - intersection differs from empty set I have the following proposition:  "If $(S_n)$ is a sequence of compact subsets of a metric space such that  $S_1 \supseteq S_2 \supseteq ... \supseteq S_n$  Then $\bigcap^\infty_{i=1} S_i$ $\neq$ $\emptyset$"  My lecture notes provide the following proof, whose last step I don't understand:    Proof: Assume by way of contradiction that $\bigcap^\infty_{i=1} S_i$ = $\emptyset$. Let $V_n := (S_n)^c$ for each $n$. The class $\mathbb{C} :=\{V_n: n \in \mathbb{N} \}$ is an open covering of $S_1$. Since $S_1$ is compact, $\exists I$ finite, $I \subset \mathbb{N}$, s.t $\{V_i: i \in I \}$ covers $S_1$. Let $n:=max(I)$. The class $\{V_1, ..., V_n \}$ also covers $S_1$ and we get the following contradiction   $\emptyset = \bigcap^n_{i=1} S_i = S_n$    So, I don't get the part from "Let $n:=max(I)$..." onwards, especially the contradiction. I have tried applying DeMorgan's Law and the only way the last equalities makes sense is if the union of $\{V_1, ..., V_n \}$ equals the whole metric space, a property I haven't been able to understand from the steps of the proof. This is a corollary to the finite intersection property, so it must have a connection but again I can't quite get it.  Thanks in advance Prove Uncountable set minus a countable set is uncountable Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable How to set a font for math mode? I would like to have something like Segoe UI of something similar. Thank you  \documentclass{article} \usepackage{fontspec} \usepackage{firamath-otf} \begin{document} \begin{equation} x+y=45 \end{equation}  \end{document}   Error: Finding the number of non-negative integral solutions of $x + y + z = 10$ I have been playing for almost 2 days, and today it is asking me to sign up again.  I have all my game saved in my google account, but I wonder if I just log in from a trainer account will I keep my progress? How Do I Choose Between a Hash Table and a Trie (Prefix Tree)? So if I have to choose between a hash table or a prefix tree what are the discriminating factors that would lead me to choose one over the other. From my own naive point of view it seems as though using a trie has some extra overhead since it isn't stored as an array but that in terms of run time (assuming the longest key is the longest english word) it can be essentially O(1) (in relation to the upper bound). Maybe the longest english word is 50 characters?  Hash tables are instant look up once you get the index. Hashing the key to get the index however seems like it could easily take near 50 steps.  Can someone provide me a more experienced perspective on this? Thanks! For equal cost, why would you select the MacBook Air over the MacBook Pro? I can see that the 13" version of MacBook Pro and MacBook Air is same $1199, but the Pro is more feature-loaded than Macbook Air. Why would anyone buy MacBook Air? What's the catch in this?    I was thinking of buying a MacBook for development purposes, and worried if there's any problem in the Pro version, because it comes with more features (like more storage).  Edit:    Thanks for the answer guys, I bought a MacBook Pro a week ago and already loving it. Why won't any faces show up in the UV Image Editor? Show the Euclidean metric and maximum metric are strongly equivalent. I need to show that the Euclidean metric and maximum metric (or square metric??) are strongly equivalent. I have no idea how to start this proof. Any help?  $d_1, d_2$ are called strongly equivalent if there exist positive constants $K, M$ such that for all $x, y\in X$: $Md_1(x,y)\leq d_2(x,y)\leq Kd_1(x,y)$ Why is polyploidy lethal for some organisms while for others is not? How do you type a single backslash, code-formatted, in a comment? Is there a way to type something in a comment that renders like this?     You'll need to escape it with a backslash - that is, the \ character.   I've tried various combinations, but a single backslash formatted as code seems to be impossible. Can this be fixed, or can I be enlightened please?  And to clarify: I expect this:  You'll need to escape it with a backslash - that is, the `\\` character.   to work properly, but it doesn't. Instead, one backslash escapes the ` and I'm left with a mess. This question has been asked before, at least twice by people who were trying to cheat on the PROMYS Europe 2020 Application, and once in a  because the OP showed no effort. I haven't been able to solve it, so I'm going to to ask it again.  PROMYS applications have closed, and I hope the effort shown below is enough to keep this question from being closed.     The Mathematical Forest is grown in a two-dimensional plane, where   trees can only grow on points with integer coordinates. To start with,   there are no trees at all. The foresters plant the first tree at   $(0,0)$. Each year, they carry out tree planting according to the   following rule. If there is a tree on the point (𝑚,𝑛) but there are   no trees on the points $(m+1,n)$ and $(m,n+1)$, then they can choose   to remove the tree on $(m,n)$ and plant new trees on the points   $(m,n+1)$ and $(m+1,n)$. For an integer $k\geq1$, the $k$th diagonal   consists of all points $(m,n)$ with $m+n=k−1$. Is it possible for the   foresters to arrange their planting so that eventually there are no   trees on the first $2$ diagonals? What about the first $3$ diagonals?    $4$ diagonals? Can you generalize?   It's easy to get to a position with no trees on the first two diagonals.  (It only takes $4$ plantings.)  I can prove it's impossible to reach a position with no trees on the first $4$ diagonals, and I believe it's impossible to reach a position with no trees on the first $3$ diagonals, but I can't prove it.  For the $k=4$ case, I used a pagoda function, as in a peg solitaire problem.  For $k=1,2,3,\dots$ define the potential of a tree on the $k$th diagonal as $\frac1{2^{k-1}}$, and the potential of the forest as the sum of the potentials of all the trees in it.  When a tree of potential $\frac1{2^{k-1}}$ is removed, it is replaced by $2$ trees of potential $\frac1{2^k}$ so the potential of the forest never changes.  Initially, the potential of the forest is $1$.  Since there are $k$ trees on diagonal $k$, the potential of all points beyond the fourth diagonal is $$\sum_{k=5}^\infty\frac k{2^{k-1}}=\frac34&lt;1,$$ so it is impossible that there are no trees on the first $4$ diagonals.  The potential of the fourth diagonal is $\frac12$, so this argument doesn't show that it's impossible to have no trees on the first $3$ diagonals.  I've done a lot of experimenting, though, and I believe that the statement is true.  (I even wrote a little computer game so I could experiment quickly.)  The best I've been able to do is to get down to one man on the third diagonal.   The black circles are trees eligible for removal, and the gray circles are other trees.  You can see that there is a "traffic jam" in front of the lone tree on diagonal $3$, and there seems to be no way to clear it.  I've been trying to come up with an argument that reflects this traffic jam, but I haven't been at all successful. We know that after $n$ removals there are $n+1$ trees, and I've been trying to prove somehow that they can't all be sufficiently far from the origin to allow all trees to get past diagonal $3$, but I haven't come close.   In the diagram above, if the tree at $(1,1)$ were moved to $(0,2)$, and the tree at $(1,2)$ were moved to $(2,0)$, then it would be possible to remove the tree at $(0,2)$, emptying the second diagonal.  In the fictitious position, we still have one tree on diagonal $2$ and two on diagonal $3$.   So, I think an argument along the lines I was trying cannot just be concerned with the distance from the origin; it must somehow take into account how the trees interfere with one another.  I haven't been able to do this.    I'd be grateful for solutions, hints, or counterexamples.  I wasn't really sure what tags to attach.  Please add any that seem appropriate. how to count dates entries monthwise from a resultset Books or resources for learning unified view of classical statistics? I did a science PhD doing applied bayesian statistics, and I liked it so much that I decided to shift to statistics.  For the last 2-3 years have been working at a firm that has more of a frequentist focus.  It seems to me that the bayesian approach (at least what I learned) has an underlying theory that was laid out and I could understand.  Now I find this - more classical - field fascinating but I am looking for some resources to get a unified view of the matter.  For example, I am learning and applying anovas, F-tests and chi-squares, thinking about degrees of freedom, etc but part of me is looking for deeper understanding that would make it more than just methods.  Does anyone have any suggestions on books or more resources that provide such a unified, deeper focus?