How write this Function with many arguments?
Can a function be made to accept a variable amount of inputs?
In other words, why don't we always use Schottky diodes if they are so much better? What diode properties do Schottky diodes have that makes them unfit for certain applications? Is a 1st-level spell cast using a 4th-level slot a 1st-level spell, or a 4th-level spell? As the title says: Is a 1st-level spell cast using a 4th-level slot a 1st-level spell, or a 4th-level spell? Prove that functions map countable sets to countable sets One of my homework problems asks to prove that if $f:X\to Y$ is a function and $A$ is a countable subset of $X$, then $f(A)$ is countable.  I believe I have a valid proof by partitioning the set A into equivalency classes based in image under $f$. Then using that partition as domain for a choice function $C_f$ back to $A$. So now the function $f$ restricted to the domain (range $C_f$) $f|_{range(C_f)}$ maps to $f(A)$ 1-1 and onto. Therefore  $$|f(A)|=|range(C_f)|\leq |A|=\aleph_0$$ $\therefore f(A)$ is a countable set. $\quad \blacksquare$  My problem is that I used the axiom of countable choice $AC_\omega$ in my proof but that isn't a part of the course I am taking. Does anyone have a simpler proof or one that does not involve Choice? skype 4.3 cannot connect to account on ubuntu 14.04 LTS I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. The supremum norm is submultiplicative Is the following proof correct:  Let $X$ be a compact Hausdorff space and $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. We can equip $C(X)$ with the (edit: sorry, semi-)norm $\lVert f \rVert_\infty = \sup_{x\in X} |f(x)|.$ I want to prove that for $f,g \in C(X)$, $\lVert fg \rVert_\infty \leq \lVert f \rVert_\infty \lVert g \rVert_\infty $. Let $z \in X$ be arbitrary then $$ \lvert f(z)g(z) \rvert \leq \lvert   \; \sup_{x} \lvert f(x) \rvert  g(z)  \;\rvert=\sup_{x} \lvert f(x) \rvert  \lvert    g(z)\rvert \leq \sup_{x} \lvert f(x) \rvert  \sup_{y} \lvert g(y) \rvert.  $$  Since $z$ was arbitrary $$ \sup_{z} f(z)g(z)\leq \sup_{x} f(x) \sup_{y} g(y) $$  that is the sought. What is the difference between data mining, statistics, machine learning and AI?  Would it be accurate to say that they are 4 fields attempting to solve very similar problems but with different approaches? What exactly do they have in common and where do they differ? If there is some kind of hierarchy between them, what would it be?  Similar questions have been asked previously but I still don't get it: Vertical spaces between lines in cases environment \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation*}   X_i =    \begin{cases}     \left[\begin{smallmatrix}       0 \\ 1 \\ 2     \end{smallmatrix}\right]     &amp;     \text{if $i$ is even;}     \\     \left[\begin{smallmatrix}       2 \\ 1 \\ 0     \end{smallmatrix}\right]     &amp;     \text{otherwise.}   \end{cases} \end{equation*} \end{document}   This is the LaTeX code that I wrote based on a cases environment. There's no space between the matrices on the right. What's the best way to add some vertical spaces to separate them?    p.s. This is a simplified form of my original writing, so I cannot just accept suggestions like 'use a matrix transpose to transform the column vector into a row vector.' Android, how do I stop a pointer appearing below an EditText Restrict node list in entity reference field based on node field My "Article" nodes have a "domain" field that links to a taxonomy term. My "Article" nodes also have a repeatable "entity reference" field "more like this" that links to other "article" nodes.  How can i restrict the list in the "more like this" field to show only the nodes that have the same "taxonomy term" that the node i am currently editing ?  Thanks a lot ! Do users get notifications when @ was edited in If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of $n$ proper subspaces of $V$ Difference between "try to do" and "try and do" What is the difference between try to do and try and do?  To me (non-native speaker), asking someone try and do this seems a bit rude. It's like saying you can try all you want but this must be done: try and fail is not an option. However, asking someone try to do this means I am asking you to try: success is bonus, failure is OK. evaluate the limit as $x$ approaches $0$ of $(\cos x-1)/x$ without l'Hopital's rule I know with l'Hopital's rule it becomes $-\sin(x)$ which has the limit $0$. However, I have been wondering how to evaluate this limit without l'Hopital's rule. Different Google Searches return different titles for the same page Do MOSFETs have a diode built into them? I have noticed that a MOSFET symbol has a little diode in it (or at least what looks like a diode). Does this mean I do not have to worry about using a diode in a circuit that runs a motor using one? I would have used a diode in order to prevent reverse voltage from turning the motor by hand. How did Rutherford conclude that most of the mass (as well as the positive charge) was concentrated in the nucleus? Geiger and Marsden's experiment led Rutherford to believe that the positive charge and most of the mass of the atom was concentrated in a small region. I understand what led him to conclude the way the positive charge is positioned in the atom. But how did he conclude that most of the mass was in a small region (the nucleus)?  How did the distribution of the mass matter after all? Given that the electric force is greater than the gravitational force by many magnitudes, the force between the positice charge and the electrons was predominantly electric.   So how did Rutherford conclude that most of the mass is in the nucleus? UnboundLocalError on local variable when reassigned after first use The following code works as expected in both Python 2.5 and 3.0:  a, b, c = (1, 2, 3)  print(a, b, c)  def test():     print(a)     print(b)     print(c)    # (A)     #c+=1       # (B) test()   However, when I uncomment line (B), I get an UnboundLocalError: 'c' not assigned at line (A). The values of a and b are printed correctly. This has me completely baffled for two reasons:   Why is there a runtime error thrown at line (A) because of a later statement on line (B)? Why are variables a and b printed as expected, while c raises an error?   The only explanation I can come up with is that a local variable c is created by the assignment c+=1, which takes precedent over the "global" variable c even before the local variable is created. Of course, it doesn't make sense for a variable to "steal" scope before it exists.  Could someone please explain this behavior? I have a manuscript that is under review in a major journal which is a main result out of a few recent years of my research. When applying for faculty positions, I would like to send the draft as a part of my writing sample. But since the material is not yet published, I am not sure if it is a good idea.  What are the foreseeable disadvantages of sending an article in review to the faculty search committee?  My area is neuroscience / biomedical engineering.  Update: The paper got into a high-profile journal, and I got a job at a different place which did not require a writing sample. Find the supremum of the following set Find the supremum of the following set:  $$A:=\left\{{(n-1)\over(2n+3)} : n \in\mathbb{N}\right\}$$  So I have as my answer that $\sup(A) = \frac12$ but need to justify. We were taught to justify in two steps, first show our answer is an upper bound for the set then show it is a least upper bound.   For step $1$: if $a$ is a member of $A$ then $a = (n-1)/(2n+3)$ for some natural number $n$ and $\frac{n-1}{2n+3} \leq \frac{n}{2n} = \frac12$ so $\frac12$ is an upper bound for the set  Step 2 is where I am uncertain. I think I need to show that for any $y&lt;\frac12$ there exists a member of the set between $y$ and $\frac12$ but not sure how to do this. Thanks for all the answers so far! How do I coat popcorn with flavor?