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I saw him crossing the road.   I saw him cross the road.   Which one is correct and why? Three linked question on non-negative definite matrices. 1.a symmetric matrix in $\mathbb{M}_n(\mathbb{R})$ is said to be non-negative definite if $x^Tax≥0$ for all (column) vectors $x\in \mathbb{R}^n$. Which of the following statements are true? (a) If a real symmetric $n\times n$ matrix is non-negative definite, then all of its eigenvalues are non-negative. (b) If a real symmetric $n\times n$ matrix has  all of its eigenvalues are non-negative , then it is non-negative definite. (c) If $ A\in \mathbb{M}_n(\mathbb{R})$, then $AA^T$ is non-negetive definite. 2. only one of the following matrices is non-negetive definite. Find it. (a) $\begin{pmatrix} 5 &amp; -3 \\ -3 &amp; 5 \end{pmatrix}$. (b) $\begin{pmatrix} 1 &amp; -3 \\ -3 &amp; 5 \end{pmatrix}$. (c) $\begin{pmatrix} 1 &amp; 3 \\ 3 &amp; 5 \end{pmatrix}$.      3.let $B$ be the real symmetric non-negative definite $2×2$ matrix such that $B^2=A$ where $A$ where is the non-negetive definite matrix in question $2$.write down the characteristic polynomial of $B$.      My thoughts..  For1.  (a) &amp; (b) are true but not sure about (c). For 2. (a) is the correct option since it has all positive eigen values.they are $2,8$. For 3. Eigen values of $B$ will be $\sqrt{2}$ and $\sqrt{8}$. So the answer is $x^2-3√2x+4=0$.      Can anybody help me to verify the solutions of the above problems. Using iOS 5 rich text editor If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $. I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 &gt; 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality. Adding Items to the List Using JQuery Mobile in Javascript Helvetica font has jagged edges 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} I've made some high resolution pixel art in PS by drawing it down at the pixel level and then increasing the image size with the nearest neighbor setting to keep it crisp. In the web browser, however, I need to display these images as icons with a dynamic size, so I can't just size the images correctly to begin with, and the broswer seems to blur the images when rendering them smaller, while vector images seem to stay more crisp as they shrink.   So I'm wondering if there's a way to convert the pixel art neatly into vector. I'm aware of the trace function in illustrator, but it seems to do a rough job iirc. Photoshop CS6 Resize Rectangle with Rounded Corners This question is about Photoshop CS6. Hopefully the feature was added in this version. Before posting I looked up Google and  and  but didn't find an answer for CS6.  So the question is: Is there a feature within CS6 or a script that allows to resize a vector element with rounded corners without using the Direct Selection Tool (white arrow tool) and not changing the border radius? What would be great would be something like simply using the transform tool or something like a grid resize tool. Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$ How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it? Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry,  on January $29$th, $2010.$  "Let $\mathcal{C}$ be a circle, $A,B$ two distinct points on $\mathcal{C}$ and $M$ be the midpoint of the chord $[AB]$. Take two other chords,$[PQ]$ and $[SR]$, that pass through $M$. Let $C$ (resp. $D$) be the intersection of $[AB]$ with $[PS]$ (resp. $[RQ]$). Prove that $M$ is the midpoint of the chord $[CD]$."    To prove it I've written the following (failed) argument, in the  of this post (translation of mine):    The figure is symmetric with respect to $M$: $\overline{AM}=\overline{MB}$, $\overline{PM}=\overline{MU}$, $\overline{RM}=\overline{MW}$, $\overline{QM}=\overline{MV}$, $\overline{QR}=\overline{VW}$, $\overline{SM}=\overline{MT}$. From $\dfrac{\overline{SC}}{\overline{DT}}=\dfrac{\overline{CM}}{\overline{MD}}=1$ follows that $\overline{CM}=\overline{MD}$.  Here is an extract of the author's reply (translation of mine):   "It is not clear that  $\overline{QM}=\overline{MV}$. Is the point $V$ on the line $(QM)$ defined by this equality or is $V$ defined as the intersection point of the lines $(QM)$ and $(WC)$? Why are both definitions to give the same point?  Let $C^{\prime }$ be the intersection of $(SP)$ and $(VW)$, and $D^{\prime }$ the intersection of $(TU)$ and $(QR)$. (...)  One still has to show that $C^{\prime }=C$ and $D^{\prime }=D$."   I have agreed with these objections.  Until now no proof has been posted. The author considers that the "proof is not quite simple".  Q. What is the theorem this exercise refers to? Or how does one prove it? I understand that in a lot of big-lab fields it is common for the principal investigator to append their name to a paper even if they did not write the paper, design the experiment, or collect data since they spend energy securing funding, and managing the whole lab. What about for small labs?  What are the requirements for a supervisor to be included as an author on a paper, as opposed to just appearing in the acknowledgements? If you are working on your own projects independently of your supervisor, but using funding provided by your supervisor (how does this change when the funding provides resources versus just your salary), are you suppose to add them as authors or just acknowledge the source of funding? I have to give a combinatorial proof of  $$\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}.$$  I find it difficult to solve such problems. I'm not a brilliant person and never will be so I need to have algorithms to solve most problems. Usually I won't just "see" solutions unless I have a thorough enough understanding of a theory. This is also the case here. I know what the symbols mean, but I don't see what I should count and how to obtain the identity.  I think the right-hand side counts $n$-element subsets of a $(2n-1)$-element set with one element chosen to be "special". I think I should find a partition of the set of such subsets into $n$ parts so that the left-hand side is the sum of the cardinalities of the parts. Is there $n$ of anything on the right-hand side? Well, there are $n$ choices for the "special" elements, but only after I've chosen the $n$-element subset. Otherwise I can choose the "special element" in $2n-1$ ways.  This is a homework problem so I probably shouldn't ask for a full solution. But I would like to have some guidelines for approaching such problems, perhaps based on this example. They agreed to stay connected for hours in their provisional support group, looking for answers in their counterparts.   I want to change the bold part to say that they all shared the same problem.  Might end up using one of the above, but I feel like I need some more options to weigh in.    Edit for context: they all share the same rare cognitive impairment What happens if my order exceeds the bid or ask sizes? What will happen if my order exceeds the bid or ask size? For example, if the bid or ask is 100 shares, but I want to buy 100,000 shares, is it possible to do so? Prove that the additive groups of $\Bbb Z$ and $\Bbb Q$ are not isomorphic.    It is hard to create a map that will show homomorphism. D'Alemberts Principle to solve a Classical Mechanics problem I found this question on this very site and was curious on how to solve it using D'Alembert's Principle  .I already know how to do it by balancing the torque about toppling point so please don't post the latter as an answer . My attempt was to displace the upper sphere by $d\theta$ which gives its virtual displacement as  $rd{\theta}cos(\theta)$j(notice we are only interested in the y component becuz the only component of the applied force which is gravity in this case is the y component in the dot product the other components vanish) similarly the cylinder is also displaced $r'd{\phi}cos(\phi)$ where r' is the distance from the point of toppling of the center of mass. $r'cos(\phi)=H/2=r(1+sin(\theta))$ ,$r'sin(\phi)=R=r(1+cos(\theta))$ and $tan(\phi)=H/2R$ the equation coming from D'Alemberts principle is $mgrcos(\theta)d\theta=Mgr'cos(\phi)\phi$ I tried solving this but couldn't arrive at the right answer. Any help would be appreciated! What do I set the 'Opens with...' property to in order to get the system to run .bat files again (when they are double-clicked)?  Somehow my boss convinced his workstation that the handler for .bat files was supposed to be Word. Now, when double-clicking .bat files, they're opened up in Word. MicroSD card is set to Read-only state. How can I write data on it? Logic soundness and completeness I have to do a number of similar type questions, but I am having trouble grasping the general concepts around soundness and completeness. I have read up on the general definitions and think I understand them, but when asked to show something like the following I come up blank. Thanks for any ideas/help.   Show that the following propositional proof system, PCa, is sound but not complete.  PCa  Connectives (and constants) ∨, ∧, →, ↔, ¬, 0, 1  Axioms All tautologies of the form F→F  Rules of inference Hypothetical Syllogism SharePoint People.aspx Page is Visible for All Users The SharePoint People.aspx page is visible for all users by default. Is there any way to restrict this page from certain readers?