Let $\beta&gt;2$ be a real number, is it always possible to construct a real number $x$ such that $\mu(x)=\beta$? where $\mu$ denotes the irrationality measure.    The problem as I see it is to find $(a_n)_n$ an unbounded sequence of positive integers such that $\limsup \ln_{q_n}(q_{n+1})=\beta-1$ where $q_n$ is the numerator of the $n$-th convergent of $[a_0;a_1,a_2,\dots]$. But I was not able to continue.
Is it possible to generate transcendental numbers of arbitrary irrationality measure ? I am thinking of something like an infinite series which depends on a parameter fixing the irrationality measure.   All I have found so far is bounds on the irrationality measure of famous numbers, I have not been able to find something going the other way around.
find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are positive and decreasing.  Edit: This problem is taken verbatim from Exercise 2.7.11 on page 68 of Abbott's Understanding analysis. Which configuration of armor enchantments gives maximum protection? How can I start up an application with a pre-defined window size and position? I'm wondering is there any way to achieve the affect of the Ctrl-Alt-Keypad shortcuts in Unity using terminal commands instead? I want a command that sets a gui window to half the size of the screen, either left or right aligned.  By way of background, I'm writing a script that runs after log in. It uses Zenity to ask whether or not I want to open my development environment (GVim and IPython side-by-side). I have been trying to achieve two equal-sized windows for these programmes by using set lines= columns= in my .gvimrc and c.IPythonWidget.width = and c.IPythonWidget.height = in my ipython_qtconsole_config.py. However, there are problems associated with this approach. Inadvertent use of = instead of == It seems that   if (x=y) { .... }   instead of  if (x==y) { ... }    is a root of many evils.  Why don't all compilers mark it as error instead of a configurable warning?  I'm interested in finding out cases where the construct if (x=y) is useful. How can i show with quite simple methods that a group of order 15 is cyclic? How to find $\operatorname{Gal}(S/\mathbb{Q})$ I have to prove the  by tomorrow and I am really unconfident about what I did:  $$\sum_{n=1}^\infty a_n\text{ converges } \iff \sum_{n=1}^\infty 2^n a_{2^n}\text{ converges}$$  I did the following:  Let $(b_n)$ be a sequence as follow: $b_{2^k+m}:=a_{2^k}$ with $k\in\mathbb N_0$ and $0\leq m&lt;2^k$.  It's $a_{n+1}\leq a_n$ and so $0\leq a_{n+p}\leq a_n$ for all $n,p\in\mathbb N$.  So $\sum\limits_{n=1}^\infty b_n$ converges by the majorizing series $\sum\limits_{n=1}^\infty a_n$. And it's $\sum\limits_{n=0}^\infty b_n=\sum\limits_{n=0}^\infty\sum\limits_{m=0}^{2^n-1}a_{2^{n+1}}=\sum\limits_{n=1}^\infty 2^{n-1}a_{2^n}$ so $\Rightarrow$ is done.  For $\Leftarrow$ consider $c_{2^k+m}:=a_{2^k}$ with $k\in\mathbb N_0$ and $0\leq m&lt;2^k$.  It's $|a_n|\leq c_n$ and $\sum\limits_{n=0}^\infty c_n=\sum\limits_{n=0}^\infty\sum\limits_{m=0}^{2^n-1}a_{2^{n}}=\sum\limits_{n=1}^\infty 2^{n}a_{2^n}$ and so $\sum\limits_{n=1}^\infty a_n$ converges by the majorizing series $\sum\limits_{n=0}^\infty c_n$.  Is this in form and content correct? In AVR datasheets under the Electrical Characteristics section you will typically find a graph like this (this one is from the ATMega328):    I've seen designs that seem to "work" but operate outside the shaded envelope. Specifically, I've seen 3.3V (Arduino) designs that run the clock from an external 16MHz crystal. Clearly, this is out of spec. What are the practical negative consequences of running outside this envelope? Given this HTML:  &lt;div&gt;foo&lt;/div&gt;&lt;div&gt;bar&lt;/div&gt;&lt;div&gt;baz&lt;/div&gt;   How do you make them display inline like this:     foo bar baz   not like this:     foo   bar   baz Weird evaluation order in chained functions Does a run scored on a tag out on a forced runner? The situation:   Runners at third and first, ball is hit to the second basemen, 2nd basemen goes to tag the runner out. Runner stops, runner on third scores and then 2nd basement tags the runner, does not step on base.Does the run count? I can get the element like this $("#txtEmail") but I'm not sure how to get the actual value. Show that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$ Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$  My attempt: Since $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} \leq \lim_{n\rightarrow \infty}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}} = \lim_{n\rightarrow \infty} \sqrt[n]{n}\sqrt[n]{\max\{c_1,c_2,\ldots,c_m\}}=\lim_{n \rightarrow \infty} \max\{\sqrt[n]{c_1^n},\sqrt[n]{c_2^n},\ldots,\sqrt[n]{c_m^n}\}=\lim_{n \rightarrow \infty}\max\{c_1,c_2,\ldots,c_m\}=\max\{c_1,c_2,\ldots,c_m\}$$  it follows that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n}$ is bounded, but I don't think it's monotonically decreasing, at least I can't prove this. Can anybody tell me whether the approach I have chosen is a good one, whether what I have done is correct and how to finish the proof? How to run updmap-sys after installing cm-super? My original question was:   The answer that seems suitable to my cause is to install cm-super. I did that (using package manager [winxp 32 bit, MikTex installation 2.9 recently updated]). But still the command  \usepackage{cm-super}    yields an error.  User H Cavendish suggested to run updmap-sys, but I am not sure about how to do that on WinXP.   2011 November 04 edit: I made it sure that I have installed 'cm-super' - but still get the same error:   I have also tried updating the file name databases - but no avail. If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: $$(r+I)\cdot m=rm.$$ (a) Show that this product is well defined. (b) Show that this product, together with the sum of elements of $M$, turns $M$ into an $R/I$-module.  My answer for (a): Let $r+I,s+I \in R/I$ such that $r+I=s+I$. Then $(r+I)m=(s+I)m$, for all $m \in M$. Then $rm=sm \implies r=s$  (since the inverse of $m$ is in $M$), so the product is well defined.  Is my answer to (a) correct?  How can I use Ann(M) to answer (b)? Is it safe to use the command apt-get autoremove in this particular scenario? Fortran IF statement with numbers/labels rather than another statement Height of an HTML select box (dropdown) Why does `type which` say that `which is hashed`? How to create multiplied identity t-sql?