Journey planning
Travelling by bus
Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$. If you know what a number mod two different primes is (3 and 2) in this case, how can you tell what the mod is of the two products? How to write below bullet I have to write something under the symbol \bullet in Latex. Is it possible? I'm using the class beamer. Here it is an example of a frame:  \begin{frame} \begin{itemize} \item&lt;1-&gt; Products Market\mbox{}\\ \footnotesize{where $p_{i,t}=u_{i,t}$, with $u_{i,t}\sim u[0,2]$} \vspace{0.5cm} \item&lt;2-&gt; Credit Market \mbox{} \vspace{0.2cm} \\ \footnotesize{where \hspace{0.1cm} $\bullet_{bank} \xrightarrow{loan} \bullet$} \vspace{0.5cm} \item&lt;3-&gt; Interbank Market\mbox{} \vspace{0.08cm} \\ \footnotesize{where \hspace{0.1cm} $\bullet \xrightarrow{loan} \bullet$} \end{itemize} \end{frame}   In particular I want to write "bank" under the bullet, while now it is only in subscript.  Thanks Verifying that the ideal $(x^3-y^2)$ is prime How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime?    I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel is exactly $I$. But the difficulty is that how to prove the kernel is contained in $I$.  Could you help me? Thanks a lot! What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? I do not understand the difference between $\rightarrow$ and $\Rightarrow$. Sometimes I see implication truth tables labeled with the former, sometimes with the latter.  Aren't they synonyms of logical implication or is there any difference? When is a gerund supposed to be preceded by a possessive adjective/determiner? Which annotation should I use: @IdClass or @EmbeddedId The JPA (Java Persistence API) specification has 2 different ways to specify entity composite keys: @IdClass and @EmbeddedId.   I'm using both annotations on my mapped entities, but it turns out to be a big mess to people who aren't very familiar with JPA.   I want to adopt only one way to specify composite keys. Which one is really the best? Why? Relation between rank and number of distinct eigenvalues of a matrix Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues.   I can realize that the rank  will correspond to the number of non-zero eigenvalues (counted up to multiplicity) and the nullity will correspond to the 0 eigenvalue (counted up to multiplicity), but I cannot design an analytical proof of this.  Thanks for any help . Is a string literal in c++ created in static memory and destroyed only when the program exits? I am unable to figure out simple solution to add "Bookman Old Style"  as a main font in my latex document. I have used below line but does not give me required font.  \usepackage{bookman}   Could not find any Simple correct answer for this. Pl. suggest how to do it.  I am using windows, pdflatex.  thank you How to use "who" vs. "that" I often get confused when trying to use who vs that.  Some examples that often confuse me:  That     The person that went to the store.      The people that went shopping.      The persons that went shopping.      The group that went shopping.   Who     The person who went to the store.      The people who go shopping.   Please explain when to use either for plural subjects and singular subjects.  Animate and inanimate objects as well. Are user accounts on data.stackexchange.com and Stack Exchange two different entities? I tried to search for similar questions, as I always do, but was not able to find one. I just created a new  on Data.Stack Exchange with my existing Google OpenID and it seems to be not associated with my regular Stack Exchange account. Is this by design? and if so, why? What are the pros and cons of the leading Java HTML parsers? Find $\lim\limits_{n \to \infty} \frac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}}$. $$\lim_{n \to \infty} \dfrac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}}$$   $$\lim_{n \to \infty} \dfrac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}} =\lim_{n\to \infty} \dfrac1{n}\sum^{n}_{k = 1} \sqrt{\dfrac k n} $$  While searching this question I found, .   Like in the accepted answer I first compared my series to LRAM,    $$\int_a^b f(x)\,dx=\lim_{n\to\infty}\frac 1n\sum_{i=1}^n f\left(a+\frac{b-a}n i \right)$$  I got $a = 0$, $b = 1$ and $f(x) =\sqrt{x}$ so,  $$\int_0^1 \sqrt{x}\ dx = \dfrac2 3$$ should be the answer.  Is there any simpler method to do this sum ? I have not learnt this method to do infinite sums so I can't use it. Wrong ordering in generated table in jpa Case expressions in Access How can I position my div at the bottom of its container? Given the following HTML:   &lt;div id="container"&gt;   &lt;!-- Other elements here --&gt;   &lt;div id="copyright"&gt;     Copyright Foo web designs   &lt;/div&gt; &lt;/div&gt;    I would like #copyright to stick to the bottom of #container. Can I achieve this without using absolute positioning? What is a good complex analysis textbook, barring Ahlfors's? I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.) Why does $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ instead of $1$? On Wikipedia, it says that $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ :     It [e] is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, ... ()   When I evaluate $(1+\frac{1}{n})^n$ for $n = 10^8$, I get approximately $2.718281798347$ which indeed is pretty close to $e$.  But when I try to "solve" the limit using the laws of limits, I get  $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^n$$  because of the power law  $$ = \left(\lim_{n\to\infty}(1)+\lim_{n\to\infty}\left(\frac{1}{n}\right)\right)^n$$  because of the addition law  $$=\left(1+0\right)^n = 1$$  but that would mean that $e=1$, which is obviously not true.  What am I missing / doing wrong?  Thanks in advance. 1.I will study hard until I get a high score on this exam. 2.I will study hard until I've gotten a high score on this exam.  I think these sentences are grammatically correct. but if I add a specific time, I think these sentences have to be only this one:  3.I will study hard until I get a high score on this exam on July 25th.  And how about these?  a. I'd like to watch all of the holocaust movies present in the world until I've become an expert on it. b. I'd like to watch all of the holocaust movies present in the world until I become an expert on it.  Since there's no a word representing a specific time, both of them are okay, right? And what I'd like to know is the difference between &quot;have gotten&quot; and &quot;get&quot;. What I know about them is we can use &quot;future perfect&quot; to refer to unspecific future event that might happen before some accident. but &quot;get&quot; refers to future event that could be either a specific scheduled event or unspecific future event. that's why if I add a specific time mentioning, I have to use &quot;get&quot;.