Prove that $\det A=0$ if $I \notin \operatorname{span} \{A,A^2,\dots,A^n\}$
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
Fast way to find period-n points of a tent map? I need to find a period-3 orbit for a tent map defined by $$T(x)=\begin{cases}2x &amp; 0\le x\le0.5 \\[2ex] 2-2x &amp;0.5 &lt; x \le 1 \end{cases}$$  So, what is a fast and efficient way of finding it? I only know how to do it by guessing-and-checking. For example, I picked a random point, $x = \frac{2}{7}$ and found that $$T\left(\frac{2}{7}\right)=\frac{4}{7}\\[4ex]T\left(\frac{4}{7}\right)=\frac{6}{7}\\[4ex]T\left(\frac{6}{7}\right)=\frac{2}{7}$$ which is a period-3 orbit. Is there any better way of finding a period-3 orbit , more specifically, a period-3 point? Is the below brief explanation correct and complete (at least in context)? Can it be improved/simplified without getting technical? I am especially concerned about the last sentence: Is that reasoning absolutely clear to a layman?            The problem can be viewed as drawing the above graph without lifting your hand and without retracing a line. Observe that each dot is connected by three lines. Also observe that you have to draw a line to arrive at a dot, and you have to draw a line to leave that dot. Because each dot is connected by three lines, each must be visited twice. But it is impossible to visit each dot twice (even number of times) and end up with each dot connected with three lines (odd number of times). Comparing values between two tables How to compare values between two different tables?  I have two tables as shown below:   Table 'A' sql_id       OPERATION       OPTIONS  63kf3an2j0pk SELECT STATEMENT 63kf3an2j0pkc TABLE ACCESS FULL  Table 'B' sql_id       OPERATION     OPTIONS 63kf3an2j0pk SELECT STATEMENT 63kf3an2j0pkc INDEX UNIQUE SCAN IDX_THU   I need to compare the values in Table 'A' and Table 'B' for the given sql_id. My question is how to find the changed values.  In this case in Table 'A' the second line is TABLE ACCESS FULL. Table 'B' the the second line is INDEX UNIQUE SCAN IDX_THU. So here the value has changed. How can I find that using query? Something like when matched then "matched" when not changed "value changed".  I'm looking for output something like below:     sql_id           comaparion outcome dmqb1hzrf5uxh   No change in the plan 74anujtt8zw4h   plan changed   Can I do something like this?  select sql_id,case when ( select sql_id,OPERATION,options  from  a where sql_id='dmqb1hzrf5uxh' minus  select sql_id,OPERATION,options from  b where sql_id='dmqb1hzrf5uxh') is null then 'matched'  else 'changed' end as status;   I am comparing the explain plans before and after patch. The table 'A' is before patch result of the sqlid and table 'B' is after patch result. Now I want to compare the plans in both the tables. I want to compare the plan by using a query. Find $ \lim\limits_{ x\to 100 } \dfrac { 10-\sqrt { x } }{ x-100 }$  (without using a calculator and other machines...?) Not able to add 3 JPanels to a main panel I'm looking for a book that starts with the premise that (once the population size had exceeded a certain number) the future development of mankind (revolutions and wars in so and so many years, colonization of other planets in so and so many years, and so on) could be predicted completely by applied statistics. (IIRC, they used slide rules for those exact predictions, but that is certainly owed to the time the story was written). Next, the predictability was used to take a few tiny measures that would change mankind's fate in a favourable way. After a long time, the predictive system fails(?) because a mutation occurs in some person(s).  And now is the part where I'm not sure if it belongs to this or another story: The mutation results in the ability to "beam" by thinking of a place where one has been before... How do I access the terminal without logging in to the desktop? iMac with 8GB RAM -- memory always almost full I have a 27'' iMac (Late 2012, OS X 10.9.2) with 8GB RAM. The usage of memory is always above 7GB, even though I don't think I am running that much stuff. This leads to the occasional hanging of a program and occasional feeling of the computer being lagged.  The heaviest thing I am running is a Parallels virtualization of a defunct Windows laptop (for my Windows-only programs; work-related). Then I mostly surf with Chrome (10-20 tabs, some with youtube videos). Of course I have other things opened, like Mail, Evernote and a few more. But I don't think a 8GB RAM computer should be overwhelmed by that. I could be wrong.  I am willing to provide an export of processes from the Activity Monitor. I tried looking into it, of course, and I am running A LOT of small processes (unknown to me) -- system stuff I presume. I am a long-time PC user and I know my way around process management in PCs, but not on Mac.  What do you think? Do I just need more RAM or it doesn't feel like it?  Regards, What is the best way to call a script from another script? Block of equations with only some being subequations I'm trying to make a block of equations where the first two are standalone equations, and the second two are subequations, ie  X=XXXX (1)  X=XXXX (2)  X=XXXX (3a)  =XXXX (3b)  I'm using the amsmath package, with align and subequation, but I can't figure out how to do this without making a separate block for Eqs 1-2 and one for Eqs 3a-3b. That solution is unsatisfactory, because then there is a big vertical gap between the two blocks, and I want them to appear all as one.  Here's the code I have so far, in case it's unclear.  \begin{align} T(x) &amp;= \frac{m}{2{k_{B}}} \left\langle \mathbf{\tilde{v}}{(\mathbf{x}_{i},t)}^{2} \right\rangle\\  \mathbf{u}(x) &amp;= \left\langle \mathbf{v}_{i}(\mathbf{x}_{i},t) \right\rangle \end{align} \begin{subequations}  \begin{align} P_{xy}(x) &amp;= \frac{1}{A} \left\langle \sum\limits_{i} {m \tilde{v}_{i,x} \tilde{v}_{i,y}} + \frac{1}{2} \sum\limits_{i} {\sum\limits_{j\ne i} {r_{ij,y} F_{ij,x}}} \right\rangle \\ \begin{split}  &amp;=\frac{m}{A} \Bigg\langle \sum\limits_{i}{{{{\tilde{v}}}_{i,x}}{{{\tilde{v}}}_{i,y}}} + \frac{a^3\omega _{pd}^2}{4} \\  &amp;\quad \times \sum\limits_{i}{\sum\limits_{j\ne i}{r_{ij,y}{\left\{ \frac{e^{(-{{r}_{ij}}/\lambda )}}{{{r}_{ij}}}\left( \frac{1}{r_{ij}}+\frac{1}{\lambda} \right) \right\}_{x}}}} \Bigg\rangle \end{split} \end{align} \end{subequations}   Any suggestions would be greatly appreciated! How to get my own IP address and save it to a variable in a shell script? How to inset a face? I am trying to set up a Minecraft rig mouth that will have more controls and will be rounded, but I am having issues trying to figure out what this guy did here. First, he did what looked like a loop cut and slide, but instead of drawing a line it selected all the edges it passed through. He then cut to a new clip where the mouth was already set up like this.    The video is in 3 times speed but if you download a youtube plugin you can slow it down to 0.25 speed of the video and then if you do the spacebar trick you can slow it down so it shows 1 frame at a time and it is really slow.  if you want or need to see the video.  All I am asking is how can I create those 8 basic segments. Groups of order $pq$ without using Sylow theorems If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced. Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced.   Attempt:  $$\gcd(4n-5,60-12n)=(4n-5,-8n+55)=(4n-5,-4n+50)=(4n-5,45)$$  $$n=1: (4-5,45)=1\quad \checkmark\\ n=2: (3,45)=3\quad \times\\ n=3: (7,45)=1\quad \checkmark\\ n=4: (11,45)=1\quad \checkmark\\ n=5: (15,45)=15\quad \times\\ n=6: (19,45)=1\quad \checkmark\\ n=7: (23,45)=1\quad \checkmark\\ n=8: (27,45)=9\quad \times\\ \vdots$$ So the answer is that it can't be reduced for $n=1,3,4,6,7,..$ i.e  $$\bigg\{n\bigg|n\notin \begin{cases}a_1=2\\a_n=a_{n-1}+3\end{cases}\bigg\}$$  I want to verify that my solution is correct At the end of the game, both remaining players have a Baron.   The sum of all cards discarded by both players is equal.  Who wins? I am trying to analyze data based on a sample size of around 50 and would appreciate advice on which statistical test would be most appropriate. Participants, on a single questionnaire, were asked to complete two different sections. Variable 1: rating Likert items on a scale from 1-5 (strongly agree to disagree). Variable 2: rating Likert items of a validated tool on a scale of 1-9, which is averaged to obtain a single number.  My research question is: Do those who 'agree' with the statement in the first section significantly differ in terms of their scores on the variable 2, compared to those who 'disagree'? (ex. Do participants tend to 'agree' with "I like to read books for pleasure." have higher IQs (variable 2) than those who 'disagree'?)  I was advised by a colleague to run a one-way ANOVA with variable 1 (categorical) as IV and variable 2 (continuous) as DV.   Question 1: My independent variable is the three groups: 'agree', 'neutral', and 'disagree.' (I combined strongly agree with agree because of low frequencies and likewise for strongly disagree and disagree.) However, there are many more participants in the 'agree' category than in the 'disagree' category. The distribution is, on average, around 44 agrees, 3 neutrals, and 3 disagrees. Do the (very) unequal sample sizes prevent me from performing ANOVA, or any other tests? Which test would be most appropriate for this analysis?  Question 2: I would like to run analyses on all of the Likert items, which is more than 10 statements. (ex. Seeing how IQs correlate with agreement to many statements including “I like to read books for pleasure,” “I watch a lot of TV,” “I exercise more than 30 minutes each week,” etc.) I was planning to do a Bonferroni correction to adjust for the multiple tests. Would this be correct, and help make my analyses more robust? Note: The continuous data are not normally distributed (there is negative skewness), but I seem to be able to correct the skewness using logarithmic transformation.  Any advice or suggestions for papers to read would be appreciated. Thank you! Why does universal generalization work? (the rule of inference) Is $0 \, \mathrm m = 0 \, \mathrm s = 0 \,\mathrm {kg} = 0$? How do we define $[0 \, \mathrm m]$?  I once was given an assignment where I was asked to deduce and write down some physical quantity. It turned out that this quantity was $0$ in one unit or another, so I decided to drop the physical units because I figured that it didn't matter anyhow.  But I got to thinking:  If $0 \, \mathrm m = 0$, then we could add units of different dimensions (an operation not normally defined) like so: $5 + 0 \,\mathrm m = 5$.   Does $[ \cdot ]$ map $0 \, \mathrm m$ to both $0$ and $L$? Is there an established regime of thinking about physical quantities as an abstract algebra? Perhaps as a vector space where formal sums such as $5 \, \mathrm m + 10 \, \mathrm s$ are allowed? Would it be improper to drop the units of measurement of $0 \,\mathrm m$ like this in an academic paper? How can I rewire half switched receptacles? In my bedroom the bottom half of all receptacles are controlled by one switch. Is it possible to have just one receptacle with top and bottom hot all the time?  The top half is hot all the time the bottom half is controlled by one switch.  Keep in mind there are six other receptacles in this room that are the same way controlled by the same switch. When I took the receptacle out the right side tab had been removed.  The top right screw had a black wire, the bottom right had a red wire, the top and bottom left screws had separate white wires.  I changed the receptacle out as suggested, I capped off the red wire and I capped off the white wire that was attached to the bottom left screw. I attached the other white wire to the top left screw and attached the black wire to the top right screw. I now have none of the receptacles working at all. What did I do wrong? Default constructor with empty brackets