I would like to know whether Morita equivalence is a useful tool when dealing with right and left ideals of a ring. If so, could someone illustrate it on the example of $2\times 2$ matrices?  Thanks
I have been thinking a bit about Morita equivalence   and I would like to know whether it also applies to subrings such as right or left ideals.  And, if so, how specifically? Could you illustrate if with the example of 2x2 matrices with zero entries in the bottom row? Since Morita equivalence concerns non-commutative rings, I am assuming it should apply, but I am just guessing.  Thanks in advance.
How many elements of order $k$ are in $S_n$? Our Epson dot matrix printer is not printing on the following DOS command.:  Type text.txt&gt;prn   But it does print perfectly on:  open TEXT.txt&gt;File&gt;Print    What is the issue here? Is the usage of text embedded in an image a bad practice, for very controlled uses? What are your thoughts on writing your text in images?  I'm not saying throughout the whole website... But in certain key locations like image carousels. And I'm not talking about the special cases where you have font types that browsers can't render. I'm talking about text that uses standard fonts and is not too big (3/4 bullets).  Should I embed the text in the images that compose the carousel? Or should I apply the text to the images as "real text" HTML?  What are the advantages/ disadvantages of using text in an image? Proof verification: column rank = row rank $\newcommand{\span}{\operatorname{span}}\newcommand{\rank}{\operatorname{rank}}$I would like to prove that for every martix, column rank = row rank Let $A\in M_{m\times n}$ be some matrix. fix $R_A$ = vector space of the rows of $A$ , $C_A$ = vector space of the columns of $A$  $$\rank (R_A) = dim \ (\span \{R_1, R_2, \ldots, R_m\})$$ $$\rank (C_A) = dim \ (\span \{C_1, C_2, \ldots, C_n\})$$  let $x$ be some vector. $$x\in \operatorname{Null}(A) \Leftrightarrow \forall i  \ \ (1 \leq i \leq m) : \langle x,R_i\rangle = 0$$ (to be clear - I'm referring to the inner product of $x$ with each row of $A$) Using the rank nullity theorem, $\dim \operatorname{Null}(A) =  n - \rank(R_A)$ as $n$ =  number of columns. what I would like to do is to say: $$\dim \operatorname{Null}(A) =  n - \rank(R_A)$$ $$\dim \operatorname{Null}(A) = n - \rank(C_A)$$ therefore,  $\dim \operatorname{Null}(A) = n - \rank(R_A)= n - \rank(C_A)  \Longrightarrow \rank(R_A) = \rank(C_A) $ it that false? is using the rank- nullity  theorem this way is cheating? or just doesn't prove what needs to be proven formally? Can PS4 Minecraft join with PC Minecraft? Does getting disconnected from an SSH session kill your programs? I am making a character for a campaign that is starting soon. The character is going to start out as a cleric (War Domain) and will multiclass into artificer (Battle Smith).   I have looked at the multiclassing rules, but I am having trouble making heads or tails of it for a multi-class involving a 2/3 caster (the Artificer) and a full caster (the Cleric).  That leads me to my question: what would a level 15 War Domain cleric/level 5 Battle Smith artificer look like? How many spells per level? How many cantrips? Does the artificer's ability to replace cantrips still apply? I have a WordPress site which is 90% based on images.  I've installed Wp-Super-Cache, now it seems to load a little bit faster.   I've made a speed test on Pingdom Tools:   Page size 4.7MB Load time 5.09s Requests 153 Perf. grade 74/100   And also the Time Spent per Content Type:   Image    88.23%     So the problem is obviously the images.   My plan is to try as much as I can to optimize the images in terms of size, and then to add CDN support.  Is this enough in terms of improving the loading of images? underline omitting the descenders I'm very well aware that it is considered best practice not to use underlining other than on typewriters and in handwriting. But sometimes it is needed nonetheless.  In my opinion the typographically least offending way to underline in print is to leave out the descenders. This is also how I learned to do it in handwriting. Very few fonts offer a pre-made underlined variant, but some text-processors (e.g. OS X text-engine) do -- depending on the font used -- an acceptable job at emulating this effect:    How would I tweak any of the existing ways/packages for underlining to do the same?  As a bonus, I'd like to be able to tweak the behaviour (extent of the omitted part, apply to which letters) per font and text style, preferably working with XeLaTeX. How can I change the margins for only part of the text? Sometimes the margins need to be changed for a particular page, paragraph, or other section of text.  For example, if I'm writing a letter and want the left margin to be almost at the right side of the page for the four lines of my address, or if I am combining prose with poetry and want the poems to be indented relative to the rest of the work.  What is the best way to change both left and right margins on the fly? How to prompt for user input and read command-line arguments What's the deal with a leading underscore in PHP class methods? While looking over various PHP libraries I've noticed that a lot of people choose to prefix some class methods with a single underscore, such as  public function _foo()   ...instead of...  public function foo()   I realize that ultimately this comes down to personal preference, but I was wondering if anyone had some insight into where this habit comes from.  My thought is that it's probably being carried over from PHP 4, before class methods could be marked as protected or private, as a way of implying "do not call this method from outside the class".  However, it also occurred to me that maybe it originates somewhere (a language) I'm not familiar with or that there may be good reasoning behind it that I would benefit from knowing.  Any thoughts, insights and/or opinions would be appreciated. Characterization convex function. Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that for all $a,b\in [0,1]$ with $a&lt;b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is a convex function? Thank you for your help. Teenage girl in life-sustaining valley on poisoned Earth visited by another survivor I'm trying to identify a book I read in school, though I can't recall at what grade level; let's say it had to have been published before I graduated high school in 2000. Despite being required reading, it didn't feel much like a classic.  The story took place on Earth, after some catastrophe had caused the air to become poisonous. However, one teenage girl was able to survive because her house was in a valley which was somehow protected from the poisonous winds and was able to sustain its own ecosystem. One day, a man wearing a protective full-body suit and gas mask appears in the valley, and both are rather surprised to find someone else alive. She feeds and shelters the man and shows him how she had come to survive in the valley. Rather abruptly…      the man attempts to rape her, but she overpowers him somehow. She steals his protective suit and starts to leave the valley. He tells her that he had seen birds in a certain direction away from the valley, indicating there might be life there as well, and she leaves.   I believe there was an author's note either at the beginning or end of the book stating that the author had actually died while writing the manuscript, and that their friends or family had finished the book by going off the author's original notes, which I suppose may explain the abruptness of the ending.  In case it wasn't clear from the above, I didn't really like the book. But I'm curious about it regardless. Core Data - Storing Images (iPhone) If $x,y,z\gt 0$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ If $x,y,z\gt 0$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$   $\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\ge \frac{(x+y+z)^2}{2(x+y+z)} = \frac{x+y+z}{2}\ge 3\frac{\sqrt[3]{xyz}}{2} = \frac{3}{2}$$  and equality holds when $$x=y=z=1$$  My question is how can we solve it without the above lemma, like using Jensen's Inequality or other inequality.  Please explain me.  Thanks Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$ $$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$  Does it converge or diverge?   Can we have a rigorous proof that is not probabilistic?   For reference, this question is supposedly a mix of real analysis and calculus. QM without complex numbers I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier? Workflow for converting LaTeX into Open Office / MS Word Format Password Reset for Users on a Multisite Subsite