We have the matrix: $$M = \left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ I have got three eigenvectors:               (1   0   0)               (0,  1,  0)               (0   0   1)  And this is my task: We are interested in what happens to the system when we repeat the stepwise process several times. To get an idea of the development of the system we compute some powers of $M$. c) Compute $M^2, M^3, M^4$ and $M^5$. What can you say about $M^n$ when $n \to \infty$ in this case? Any suggestions? I do not fully believe my way of approaching this problem is the right way.
We have the matrix: $$\left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ Finding the eigenvalues and eigenvectors are easy. But finding ${\bf M}^n$ when $n \rightarrow \infty$ is the hard part. Any suggestions?
Can we use CAPITAL LETTERS to indicate actions? While designing for the web or the mobile applications, a natural free flowing text can be used for the general content, but when it comes to actions... can we use CAPITAL LETTERS. How to find a Java Memory Leak How do you find a memory leak in Java (using, for example, JHat)? I have tried to load the heap dump up in JHat to take a basic look. However, I do not understand how I am supposed to be able to find the root reference () or whatever it is called. Basically, I can tell that there are several hundred megabytes of hash table entries ([java.util.HashMap$Entry or something like that), but maps are used all over the place... Is there some way to search for large maps, or perhaps find general roots of large object trees?  [Edit] Ok, I've read the answers so far but let's just say I am a cheap bastard (meaning I am more interested in learning how to use JHat than to pay for JProfiler). Also, JHat is always available since it is part of the JDK. Unless of course there is no way with JHat but brute force, but I can't believe that can be the case.  Also, I do not think I will be able to actually modify (adding logging of all map sizes) and run it for long enough for me to notice the leak. I use 'Helvetica' as my font and build my document using Texmaker. But the PDF generated has jagged edges while using this font and when I use a different font (say Times or even Helvetica sans-serif) it builds just fine.     Please suggest if there is anyway I can get over this. The problem with these jagged edges is that it appears very dull and unreadable at 100% resolution.  OK guys, now here is the MWE  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% HEADER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[a4paper,twoside,12pt]{book}  %% Language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} %\usepackage[utf8]{inputenc} \usepackage{graphicx,color,subfigure}  %% Line Spacing %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{setspace} %\singlespacing        %% 1-spacing (default) %\onehalfspacing       %% 1,5-spacing \doublespacing        %% 2-spacing  %% Other Packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[left=1.5in, right=1.5in, top=1in, bottom=1in, includefoot, headheight=13.6pt]{geometry} \usepackage[pagebackref=true,breaklinks=true,letterpaper=true,colorlinks,bookmarks=false]{hyperref} \usepackage{helvet} \usepackage[square,numbers]{natbib} \usepackage{hyphenat} \usepackage{fancyhdr}   \begin{document}  \pagestyle{empty} %No headings for the first pages.  \cleardoublepage %The first chapter should start on an odd page.  \pagestyle{plain} %Now display headings: headings / fancy / ...  \pagebreak  %------------------------------------------------------------------------ \chapter{Divorce}   A major change that has occurred in the Western family is an increased incidence in divorce. Whereas in the past, divorce was a relatively rare occurrence, in recent times it has become quite commonplace. This change is borne out clearly in census figures. For example thirty years ago in Australia, only one marriage in ten ended in divorce; nowadays the figure is more than one in three (Australian Bureau of Statistics, 1996: p.45). A consequence of this change has been a substantial increase in the number of single parent families and the attendant problems that this brings (Kilmartin, 1997).   \end{document} Should I keep my swap file on an SSD drive? I'm considering getting an SSD drive to run as the primary OS partition.  As I understand, this should provide a substantial improvement in performance.     My question is this: Should I leave the swap file on that drive?     The swap partition will be largely random seeks and so should benefit from the speed.  On the other hand, it will be constantly written to which will wear out the drive faster. Adjective for "terrified yet calm and capable" What is an adjective to describe the duality of a first responder (such as an EMT responding to a terrorist attack) who is terrified for the wounded, yet simultaneously calm, detached, and capable enough to give aid? I accidently ran  sudo chmod 755 -R /   instead of  sudo chmod 755 -R ./   I stopped it after few seconds, but now there is some problems such as  sudo: must be setuid root   How can I revert permissions back? Can you Start Skyrim 'silent' without the launcher showing up? I'm trying to make a logo that will be on the header of a website. What I'd like to achieve, and the reason I'm using Blender, is the projected shadows on the background looking perfectly real (as opposed to doing a drop-shadow in Photoshop).  The only thing is, the background has to be transparent because the website might use different background colors / gradients that make it impossible to match with the exported 3D background.  I made this dummy file (not the real logo... duh!):    If I use the file as is, the grey borders of the image will be impossible to match with the website's background color.  Here what I want:    So obviously, I tried masking the plane, which made the shadows disappear. I tried having a plane set to glass, which didn't work either.  And I looked around for a similar question, but didn't find one.   I am not trying to render , or a . I've seen  which has a very difficult solution to this problem, and  that is very similar but in Blender Internal.  is similar too but has never been solved. I'm only a beginner at Compositor, and I'm trying to learn more any chance I get, so any solution in Compositor is welcome if detailed enough. Although if there's a simpler or faster way, it's welcome too!  .blend file: Regression: Transforming Variables When transforming variables, do you have to use all of the same transformation? For example, can I pick and choose differently transformed variables, as in:   Let, $x_1,x_2,x_3$ be age, length of employment, length of residence, and income.  Y = B1*sqrt(x1) + B2*-1/(x2) + B3*log(x3)   Or, must you be consistent with your transforms and use all of the same? As in:   Y = B1*log(x1) + B2*log(x2) + B3*log(x3)    My understanding is that the goal of transformation is to address the problem of normality. Looking at histograms of each variable we can see that they present very different distributions, which would lead me to believe that the transformations required are different on a variable by variable basis.   ## R Code df &lt;- read.spss(file="http://www.bertelsen.ca/R/logistic-regression.sav",                  use.value.labels=T, to.data.frame=T) hist(df[1:7])      Lastly, how valid is it to transform variables using $\log(x_n + 1)$ where $x_n$ has $0$ values? Does this transform need to be consistent across all variables or is it used adhoc even for those variables which do not include $0$'s?  ## R Code  plot(df[1:7]) Why is Lamarck's evolutionary theory considered incorrect? Many sources postulate without much argumentation that Lamarck's evolutionary theory is "obviously incorrect". A classic example provided in such sources is the one with giraffes, in which Lamarck argued that the necks of these animals became long as a result of continually stretching to reach high foliage. This mechanism for adaptive change in response to the environment is considered wrong.  Now the counterargument to Lamarck's ideas comprises modern findings in genetics. So it is stated that organisms pass down traits through predetermined genetic information, not based on environmental adaptations during their lifetime. Furthermore, genetic research shows that living organisms cannot alter their genetic material as needed.   Doesn't the latter contradict the evolutionary theory itself, which states that nature (i.e. environment) shapes species through natural selection in the environmental milieu? If it is not the environmental/natural factor that is responsible for genetic evolution, then what is? If genetic material were to be unchangeable, then how would adaptation to the environment and consequent evolutionary change would ever be possible? The genotype (following this logic) would remain the same. And lastly, maybe giraffes do not develop long necks at such a rapid pace (as Lamarck probably thought) by simply stretching their necks in one generation; however, what, if not neck stretching, is responsible for their consistently longer necks throughout thousands or millions of years? Hadn't the environment gradually shaped their genotype? Similarly, if I run more often, maybe my children will not become more fit, but isn't it just a matter of time (thousands of years) that if all of us consistently run a lot, then our offspring will eventually become more fit? Wouldn't our voluntary efforts eventually enhance our genetics? I do not see any problem with Lamarck's logic at all, except probably the time frame. I'm having trouble with this functional equation, to find all functions $f:\mathbb R \to \mathbb R$ for which the following is valid for all $x,y\in \mathbb R$: $f(x) + f(x+f(y)) = y + f(f(x) + f(f(y)))$ I haven't been able to pinpoint any solution, nor to prove surjectivity or injectivity. So 0 progress. Tried with $x=y=0, x=0, y=0, x=-f(y), y=f(x)$, and even with some assumptions on bijectivity, but nothing helped. I don't know the source but seems like a pretty difficult problem, maybe even impossible to solve. What's the best counter strategy against battlecruisers? How should I prepare my army when attacking someone who has 12+ battlecruisers? (Protoss/Zerg/Terran answers are more than appreciated) Itemize: Identation is different between first and second line How to force the Y axis to only use integers in Matplotlib? Execute vs Read bit. How do directory permissions in Linux work? In my CMS, I noticed that directories need the executable bit (+x) set for the user to open them. Why is the execute permission required to read a directory, and how do directory permissions in Linux work? Why didn't Count Dooku expose Palpatine? Group cohomology is one thing I don't seem to get my head around.  I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.  For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!  Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places? 'sudo su -' vs 'sudo -i' vs 'sudo /bin/bash' - when does it matter which is used, or does it matter at all? When I'm doing something that requires root be typed in dozens of times in a row, I prefer to switch my session to a root session. In the various tutorials and instructions I have used on the Internet, I see sudo su, sudo su -, sudo -i and sudo /bin/bash being used to open a root session, but I'm not clear on the difference between these and when or if that difference matters.  Can someone clear this up for me? What does this depiction of a black hole in the movie Interstellar mean? Is my US visa still valid after change in name? I'm a Malawian national and I got US Visa years back.  Recently, due to some reasons, I had to get my first name partially changed on my passport. I have got an affidavit certificate stating the reason and proof.  Since the visa on my old passport has still not expired, but there is a difference in first name, will I face any problem?