Are there pokemons on the 2nd generation that only appear on specific continents? Which ones are those and from which regions?
I have heard that there are some Pokémon that are region specific.  Could someone please clarify which Pokémon are region specific and in which regions that they can be found in?
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$ I'm undergraduate student of mathematics.  I need to prove: $$\sum_{k=0}^{n} \binom{2n}{2k}= 2^{2n-1}$$ Can you please help me I would like to put an address in my document as follows:  |                             | |                     Name    | |                     Street #| |                     City    | |                     ZIP     | |                             |   where the | mark the left and right page borders. I would like the items of the address (Name, Street etc.) to be left-flushed, but the box right-flushed so that the longest item touches the right page border.  Is there a simple way to accomplish this? Ubuntu 20.04 Font and UI elements too big on login I'm running Ubuntu 20.04 on an ASUS X99-E-10G WS w/ an NVIDIA 1080Ti and an LG 4k 32&quot; monitor which persistently gets recognized as 27&quot;. When I log in, the text of certain UI elements is ~2x the normal size. After I go to settings/universal access and turn large text on and then off again, everything is the normal size I expect. Screenshot after logging in  After turning &quot;large text&quot; on  After turning &quot;large text&quot; off  How do I get the conditions I have after cycling the &quot;large text&quot; feature to be present immediately after booting? I have the 440 version of NVIDIA's drivers installed. Want to delay startup of program but can't find it in Task Scheduler I'm trying to delay the startup of a specific program that loads at login. As per an answer in , I should be able to do it using Task Scheduler. The problem is, I can't find the program listed anywhere. Is there a particular pane I should be looking in? The program is Razer Synapse so I don't think it would be in the Microsoft or Western Digital folders.    If it's not in Task Scheduler, is there another way I can delay it at startup?  Out of curiosity how many ways are there for a program to be run at Startup in Windows? Prove that:  $$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$  I have tried induction on $m$ but it does not work very well. Any suggestion? Dot Product Intuition I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).      For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$ and vector $\mathbf{b}=(b_1,b_2)$.  What's the physical or geometrical meaning that   $$a_1b_1 + a_2b_2 = |\mathbf{a}||\mathbf{b}|\cos(\theta)\;?$$  Why is multiplying $|\mathbf{b}|$ times $|\mathbf{a}|$ in direction of $\mathbf{b}$ the same as multiplying the first and second components of $\mathbf{a}$ and $\mathbf{b}$ and summing ?       I know this relationship comes out when we use the law of cosines to prove, but even then i cant get a intuition in this relationship.  This image clarifies my doubt:    Thanks Automatically convert quotations in the form of "abc" to become ``abc" I did not know that latex had different symbol for opening and closing quotation marks. I've used it like "abc" every where and just realised that I really should have used ``abc". Is there anyway I can convert all the quotations to become like ``abc"? Doing it manually will take forever as I have a lot of quotations. Sum of real numbers that multiply to 1 I've seen a question in my math book with this explanation above it: "If the product of n positive real numbers is 1, then the sum of these numbers must be more than n". I was wondering if this is correct, and if it is, could someone give an explanation or a proof for it? How do I install a .tar.gz (or .tar.bz2) file? Why use a z test rather than a t test with proportional data? When testing the difference between 2 proportions why do we use a z-test rather than a t-test?    Further, is there a simple way to conduct an omnibus test for significant differences between more than 2 proportions (in the form of percentages). Is there an equivalent to a one-way ANOVA for this?  I imagine that you could use logistic regression (assuming you have the original data for the proportions in the form of 0s and 1s) Are there other options? So first let me state my homework problem:  Let $X$ be a set, let $\{A_k\}$ be a sequence of subsets of $X$, let $B = \bigcup_{n=1}^{+\infty} \bigcap_{k=n}^{+\infty} A_k$, and let $C = \bigcap_{n=1}^{+\infty} \bigcup_{k=n}^{+\infty} A_k$. Show that (a) $\liminf_k\; {\xi_A}_{_k} = \xi_B$, and $(b)$ $\limsup_k \;{\chi_A}_{_k} = \chi_C.$  I know that, in the context I am familiar with, that $$\liminf_{k\rightarrow +\infty}\; X_k = \bigcup_{k=1}^{+\infty} \bigcap_{n=k}^{+\infty} X_n$$ and $$\limsup_{k\rightarrow +\infty}\;X_k = \bigcap_{k=1}^{+\infty} \bigcup_{n=k}^{+\infty} X_n.$$   I also know that the characteristics (indicator) function is defined as $\chi_A(x) = \begin{cases} 1, &amp; x \in A \\ 0, &amp; x \notin A .\end{cases} $  So I wrote out $B$ in some of its `glory': $B= (A_1 \cap A_2 \cap A_3 \cap \cdots) \cup (A_2 \cap A_3 \cap \cdots) \cup (A_3 \cap A_4 \cap \cdots) \cup \cdots$, and as the first argument is the smallest, with increasing size to the right, the last term in the expression for $B$ would be $B$, which would be the largest.  So if I replace the $X_k$'s above with $\chi$'s I still don't see how I can get the correct answer - though it looks pretty clear from the definition of $B$ and that of $\liminf$ being basically the same, except in this case for the $\chi$.  Any direction would be greatly appreciated. By the way, I have checked out  but I was somewhat confused by the topology, the meets/joins, etc.  Thanks much, Nate How to change programmatically exposed filters in views I have a page view with exposed filters. One of the filters is term vocabulary, and that's dropdown list, user can select only one category (term).   I want to set programmatically one term from dropdown list when I came from specific URL. I tried with views_query_alter where $category is tid.    $set_exposed_input  = array();  $set_exposed_input['field_category_tid'] = $category;   if ($set_exposed_input) {     $view-&gt;set_exposed_input($set_exposed_input);  } Durrett Example 1.9 - Pairwise independence does not imply mutual independence? The example in question is from Rick Durrett's "Elementrary Probability for Applications", and the setup is something like this:     Let $A$ be the event "Alice and Betty have the same birthday", $B$ be the event "Betty and Carol have the same birthday", and $C$ be the event "Carol and Alice have the same birthday".   Durrett goes on to demonstrate that each pair is independent, since for example,   $P(A \cap B) = P(A)P(B)$.   However, he concludes that $A, B$, and $C$ are not independent, since   $P(A \cap B \cap C) = \frac{1}{365^2} \neq \frac{1}{365^3} = P(A)P(B)P(C)$.  I understand the reasoning here, and that one can generally show that arbitrary events $X$ and $Y$ are not independent by showing that $P(X\cap Y) \neq P(X)P(Y)$.   I am a little new to probability, though, and don't understand why exactly $P(A \cap B \cap C) = \frac{1}{365^2}$.   My progress so far:   I do see why $P(A) = P(B) =P(C) = \frac{1}{365}$, and thus why $P(A)P(B)P(C) = \frac{1}{365^3}$.  It seems like the sample space $\Omega = \{ (a, b, c) \mid a,b,c \in [365] \}$ -- i.e., all of the possible triples of numbers from 1 to 365, where 1 denotes January 1st, 2 denotes January 2nd, etc. From that, I can conclude $|\Omega| = 365^3$, but I'm not sure where to go from here.   It seems like once a single birthday is chosen, the rest are completely determined if they're all equal to each other - is this a good direction to go in? Field that is a subfield of own of its subfields Let $K$ and $L$ be fields. We have homomorphisms $f: K \to L$ and $g: L \to K$. Are $K$ and $L$ necessarily isomorphic? I'm new to Ubuntu. I tried installing it with Windows 8.1, but failed I lost all my files.  I have successfully formatted my PC and installed Ubuntu, but how do I install codecs to play audio and videos files ? Maximize $\mathrm{tr}(Q^TCQ)$ subject to $Q^TQ=I$ Let $C \in \mathbb{R}^{d \times d}$ be symmetric, and   $$Q = \begin{bmatrix}       \vert &amp; \vert &amp;    &amp; \vert \\       q_1   &amp; q_2   &amp; \dots &amp; q_K  \\       \vert &amp; \vert &amp;    &amp; \vert       \end{bmatrix} \in \mathbb{R}^{d\times K}$$  where $d \geq K$. Using Lagrange multipliers,   $$\begin{array}{ll} \text{maximize} &amp; \mbox{tr} \left( Q^T C Q \right)\\ \text{subject to} &amp; Q^T Q = I\end{array}$$    I am unfamiliar with these kind of constraints with this method, and after reading  I believe the same specific and simple result given is also applicable, and therefore the lagrangian would be:  $$\mathcal{L}(Q,\lambda)=\mathrm{tr}(Q^TCQ)-\left&lt;\lambda,Q^TQ-I\right&gt;$$  where $\lambda\in\mathbb{R}^{K\times K}$, and $\left&lt;\cdot,\cdot\right&gt;$ is the element wise inner product (what kind of makes sense to me since we're actually adding as many constraints as there are elements in these matrices.  In attempting to do that I start taking $\frac{\partial \mathcal{L}}{\partial Q}=O\in\mathbb{R}^{d\times K}$, and compute that LHS element by element; for the $(l,m)$ one:  \begin{equation} 0=\frac{\partial \mathcal{L}}{\partial Q_{lm}}=(CQ+C^TQ)_{lm}-\underbrace{\frac{\partial}{\partial Q_{lm}}\sum_{i,j}\lambda_{i,j}(Q^TQ-I)_{ij}}_{=\lambda_{lm}\frac{\partial (Q^TQ)_{lm}}{\partial Q_{lm}}}=2(CQ)_{lm}-\lambda_{lm}\frac{\partial (q_l^Tq_m)}{\partial q_m(l)} \tag{1}\end{equation}  where in the last step I've used the definition I made at the beginning for $Q$, and $q_m(l)$ denotes the $l$-th component of the column vector $q_m$.  In trying to compute the very last term:  $$\frac{\partial (q_l^Tq_m)}{\partial q_m(l)}=\frac{\partial \left[q_l(1)q_m(1)+ \ldots + q_l(d)q_m(d)\right]}{\partial q_m(l)}= \begin{cases}     q_l(l)\equiv Q_{ll}    &amp; \text{if } l\neq m\\     2q_l(l)\equiv 2Q_{ll}  &amp; \text{if} l=m \end{cases}$$  The whole equality (1) then can be written:  $$0=2(CQ)_{lm}-\lambda_{lm}Q_{ll}(1+\delta_{lm})$$  where $\delta_{lm}$ is the Kronecker delta.  The equation for the other stationary point of the lagrangian, $\frac{\partial \mathcal{L}}{\partial \lambda}=O\in\mathbb{R}^{K\times K}$, for the $(l,m)$ element as well:  $$ 0=\frac{\partial \mathcal L}{\partial \lambda_{lm}}= \frac{\partial }{\partial \lambda_{lm}}\sum_{i,j}\lambda_{i,j}(Q^TQ-I)_{ij}=(Q^TQ-I)_{lm}\tag{2}$$  what obviously leads to $(Q^TQ)_{lm}=\delta_{lm}$.  All this should tell that the columns of $Q$ are eventually the $K$ first eigenvectors of $C$, but I don't know how to continue from here to prove that, supposing I didn't make a mistake. Please I would sincerely appreciate any help.    Edit:  I have rewritten the inner product as a trace of a product of matrices (after seeing ):  $$\left&lt;\lambda,Q^TQ-I\right&gt;=\sum_{i,j}\lambda_{i,j}(Q^TQ-I)_{ij}=\mathrm{tr}(\lambda^TQ^TQ) $$  and have thus managed to do the derivative without losing the matrix format (using formulas from the ):  \begin{align} O=&amp;\frac{\partial \mathcal{L}}{\partial Q}=\frac{\partial}{\partial Q}\mathrm{tr}(Q^TCQ)-\frac{\partial}{\partial Q}\underbrace{\mathrm{tr}(\lambda^T(Q^TQ-I))}_{\mathrm{tr}(\lambda^TQ^TQ)-\mathrm{tr}(\lambda^T)}\\=&amp;(CQ+C^TQ)-(Q(\lambda^T)^T+Q\lambda^T)=2CQ+Q(\lambda+\lambda^T) \end{align}  And this leads to:  $$CQ=Q\underbrace{\left(-\frac{\lambda+\lambda^T}{2}\right)}_{:=\widetilde{\lambda}};\quad CQ=Q$$  If the defined matrix $\widetilde{\lambda}=Q^TCQ$ were diagonal we would already have the result. Two paths that show that $\frac{x-y}{x^2 + y^2}$ has no limit when $(x,y) \rightarrow (0,0)$ I grew up learning that the comma must be placed there, but it seems like an unnecessary interruption in a phrase that isn't ever spoken that way. Cardinality of separable metric spaces Using Regular Expressions to Extract a Value in Java I have several strings in the rough form:  [some text] [some number] [some more text]   I want to extract the text in [some number] using the Java Regex classes.  I know roughly what regular expression I want to use (though all suggestions are welcome). What I'm really interested in are the Java calls to take the regex string and use it on the source data to produce the value of [some number].  EDIT: I should add that I'm only interested in a single [some number] (basically, the first instance). The source strings are short and I'm not going to be looking for multiple occurrences of [some number].