How do fixed lenses "focus" on things at different distances?
How can a lens with a single focal length focus on more than one plane?
I have been trying to get a sense of the different problems in frequentist settings where MCMC is used. I am familiar that MCMC (or Monte Carlo) is used in fitting GLMMs and in maybe Monte Carlo EM algorithms. Are there more frequentist problems where MCMC is used? How do I write a correct micro-benchmark in Java? How do you write (and run) a correct micro-benchmark in Java?  I'm looking for some code samples and comments illustrating various things to think about.  Example: Should the benchmark measure time/iteration or iterations/time, and why?  Related: Is there a way to write binary literals in C#, like prefixing hexadecimal with 0x? 0b doesn't work.  If not, what is an easy way to do it? Some kind of string conversion? For integers $a$ and $b$, $ab=\text{lcm}(a,b)\cdot\text{hcf}(a,b)$ I was reading a text book and came across the following:  Important Results (This comes immediately after LCM:) If 2 [integers] $a$ and $b$ are given, and their $LCM$ and $HCF$ are $L$ and $H$ respectively, then $L \times H = a \times b$  Can some please help me understand why the above result is true? Thanks in advance. Signed to unsigned conversion in C - is it always safe? Suppose I have the following C code.  unsigned int u = 1234; int i = -5678;  unsigned int result = u + i;   What implicit conversions are going on here, and is this code safe for all values of u and i? (Safe, in the sense that even though result in this example will overflow to some huge positive number, I could cast it back to an int and get the real result.) Is there such a thing as a Ubuntu Certified Engineer? How do I know if food left at room temperature is still safe to eat? Can I run a natural gas dryer from a portable propane tank? Is there a way to send a thanks message to Stack Overflow or any of the individual Stack Exchange sites, aside from sending an email? I want to thank Stack Overflow for being an awesome live QA site. It helps people a lot. dotted line with appropriate length for answering a question? Kindly have a look at this code   \documentclass{article} \usepackage{fontspec} \defaultfontfeatures{Ligatures=TeX} \usepackage[small,sf,bf]{titlesec}  \setromanfont{Georgia} \usepackage{amssymb}  \newcommand{\checkbox}{$\square$} \begin{document}  If appropriate \checkbox  . Why appropriate, specify? ........... If inappropriate \checkbox . Why inappropriate, specify? .............. If appropriate \checkbox . Why appropriate, specify? ...........\\ If inappropriate \checkbox . Why inappropriate, specify? ...........  \end{document}   How can I make a dotted like, of appropriate length so that it is depressed, fainter, has consistent length so that it occupies the remaining white space in the  line or has specified length. I've created a Lightning component to view key details for the Account and Contact lookups from a Case, like so:    The problem is, I have been unable to get field content to render when  using the &lt;force:inputField/&gt; or &lt;force:outputField/&gt; components when I place them inside the markup to style them as cards. I've used &lt;ui:outputText/&gt; as a workaround, but that means I don't get any of the built-in behaviors I would get with the &lt;force:*Field/&gt; components -- like Lightning styling, which means I have to manually do it using SLDS.  This displays the contact name as expected:  &lt;aura:component implements="flexipage:availableForRecordHome,force:hasRecordId" access="global" controller="LightningCaseTestController"&gt;   &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}" /&gt;   &lt;aura:attribute name="recordId" type="Id" /&gt;   &lt;aura:attribute name="contact" type="Contact" default="{sobjectType: 'Contact'}"/&gt;    &lt;force:outputField value="{!v.contact.Name}"/&gt;   &lt;force:inputField value="{!v.contact.Name}"/&gt;  &lt;/aura:component&gt;   This does not:  &lt;aura:component implements="flexipage:availableForRecordHome,force:hasRecordId" access="global" controller="LightningCaseTestController"&gt;   &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}" /&gt;   &lt;aura:attribute name="recordId" type="Id" /&gt;   &lt;aura:attribute name="contact" type="Contact" default="{sobjectType: 'Contact'}"/&gt;    &lt;article&gt;     &lt;force:outputField value="{!v.contact.Name}"/&gt;     &lt;force:inputField value="{!v.contact.Name}"/&gt;   &lt;/article&gt;  &lt;/aura:component&gt;   Basically, it appears that as soon as any markup is placed around the &lt;force:*Field/&gt; tags -- even just a &lt;div&gt; -- they no longer render. Examples in Trailhead and other places all seem to use &lt;ui:outputText/&gt; or &lt;ui:input*/&gt;.  This seems like a lot of extra manual work. Am I missing something?  Update: it only fails when the component is included in a Lightning page -- such as the record detail page -- from Lightning App Builder. If I add it to a simple Lightning app, it works:  &lt;aura:application &gt;     &lt;c:LightningCaseTest recordId="SOMECASEID" /&gt; &lt;/aura:application&gt;   So it appears to be a bug? Prolog count the number of times a predicate is true This copy of the Install OS X El Capitan application can't be verified. It may have been corrupted or tampered with during downloading Macbook Pro 13" 2015 with OS X Yosemite on board. I've been trying to install OS X El Capitan, but always get the following message:  "This copy of the Install OS X El Capitan application can't be verified. It may have been corrupted or tampered with during downloading."  By the way this is not a date time issue, date is correct on my computer.    How can I fix this? Who knows the solution, please help. Pass the output of previous command to next as an argument UMVUE of $e^{-\lambda}$ from poisson distribution Let $X_1,\ldots,X_n \sim$ Poisson$(\lambda)$. I wish to find UMVUE of $e^{-\lambda}$. Here $\overline{X}_n$ is complete and sufficient for $\lambda$ (hence for $e^{-\lambda}$??). Define $Y_i=1$ if $X_i=0,$ $Y_i=0$ otherwise. Clearly (I checked!) each $Y_i$ and hence $\overline{Y}_n$ is unbiased for $e^{-\lambda}$. Then I should be able to use the Lehmann Scheffe theorem to deduce the UMVUE. The estimator I get looks like $$\frac{1}{n} E\left\{\#X_i:X_i=0 \mid \sum_{i=1}^n X_i\right\}$$ whereas the correct answer is $$\left(\frac{n-1}{n}\right)^{\sum_{i=1}^n X_i}$$ I'd appreciate if anyone lets me know where I'm going wrong and guides me into the right path. Thank you. This is what we are taught: $$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$  but I don't understand why we take the inverse of the base when we have a negative exponent.  Can anyone explain why? What is the gradient of $f(x, y, z) = \sqrt{x^2+y^2+z^2}$?  I know that $\nabla f = \left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle$ or equivalently $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$.  Eventually, I am going to evaluate this gradient at a point and then dot it with some given unit vector to find a directional derivative. I think maybe what is troubling me is the radical in $f$. Will I have to use the chain rule in order to find the three partial derivatives?  I tried this:  $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-\frac{1}{2}}2x$$ $$= \frac{x}{\sqrt{x^2 + y^2 + z^2}}$$  That result looks ugly to me. However, if I did that partial derivative correctly, then I think I know what I am doing. Solving a cubic polynomial equation. Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for quadratic equation : I am still a student, and when I first saw the quadratic formula I wanted to derive it myself and I did it. This was my logic: the equation $x^2 + bx + c=0$ is obviously supposed to contain 2 roots (real or imaginary). So any 2 points in an axes can be represented as: $n+k$ or $n-k$. so (1) : putting $x = n+k$, (2): putting $x=n-k$ we get two equations; on subtracting $(1) - (2)$ we get an equation in $n$ and $k$ as, $4nk+2kb=0$ which gives $n=-b/2$.  Now substituting this value of $n$ in $(1)$ and $(2)$ and adding them we get $k= (+or-) \sqrt{b^2 -4c}/2 $. My approaches for trying to solve cubic equations: Let $x^3+bx^2+cx+d = 0$ be the equation. My first trials were focused on extracting the roots by various substations. Let me explain some of them. Originally, when I derived the quadratic equation, I wanted to literary &quot;represent the solution&quot; by $n+k$ and $n-k$. most methods that I have thought of for cubic equation similar to that of the quadratic (like assigning 3 unambiguous  variables to uniquely represent 3 roots- $n$ and $k$ represents unique values), by considering 3 unique values, but they end up being ambiguous. Then I was suddenly struck with the idea that I could define certain random operators to &quot;transform&quot; the equation in some way changing the co-coefficients of $x^2, x, x^0$. I have defined the first transformation as: $x^3+bx^2+cx+d=0 \to x^3+ b'x^2 + c'x+d' =0   $ then $b' = b$, $c' =b^2+c$, $d' = bc-d$. Applying this operation changes the modified equation's zeros as follows: $\alpha'=(\alpha+\beta)/2, \beta' = (\beta+\gamma)/2, \gamma' = (\alpha+\gamma)/2 $ if the initial zeros are $\alpha, \beta, \gamma$. My latest assumption is that creating $n$ number of transformations and applying them can help us obtain the roots in some combination. The other operation I have defined is &quot;increment roots by by $\theta$ operation&quot;. in this case, $x^3+ bx^2+ cx+ d \to x^3+b'x^2+ c'x+d=0$. here, $b' = b + 3\theta, c' = 3\theta^2+ 2\theta b+ c, d' = \theta^3 +b\theta^2 + c\theta + d$. This modification transforms the original roots $\alpha, \beta, \gamma$ to $\alpha' = \alpha + \theta, \beta' = \beta + \theta, \gamma' = \gamma + \theta$ Now for my questions: How far am I from solving the cubic polynomial problem myself without any external help? More over, I give up. How to find the roots of a cubic equation? Could my latest idea for solving cubic equations actually be applied (the transformation method)? Is it possible to find the roots with my transformation method? (my second transformation definition was successful in converting the coefficient b or c to zero by transforming the roots linearly) A maximal ideal among those avoiding a multiplicative set is prime Let $S$ be a multiplicatively closed subset of a ring $R$, and let $I$ be an ideal of $R$ which is maximal among ideals disjoint from $S$. Show that $I$ is prime. If $R$ is an integral domain, explain briefly how one may construct a field $F$ together with a ring homomorphism $R\to F$. Deduce that if $R$ is an arbitrary ring, $I$ an ideal of $R$, and $S$ a multiplicatively closed subset disjoint from $I$, then there exists a ring homomorphism $f\colon R\to F$, where $F$ is a field, such that $f(x)=0$ for all $x\in I$ and $f(y)\neq 0$ for all $y\in S$.   [You may assume that if $T$ is a multiplicatively closed subset of a ring, and $0\notin T$, then there exists an ideal which is maximal among ideals disjoint from $T$.]  Here is a question I need to answer. For the first part I can show if $x, y$ are not in $I$ then nor does $xy$. The second part is just the field of fractions.   For the third part, I think I need to find an ideal $J$ containing $I$ such $J$ is prime, so that $R/J$ is an integral domain, and use the second part. To find $J$ prime, I think as suggested by the hint, I should go for a maximal ideal disjoint from $S$ containing $I$, but how can I do that? Headers of \chapters* take the headers of their previous \chapter When using book.cls and using \chapter* after a \chapter, the header used in \chapter* is wrong. An example is pasted below.   Is there any known workaround for that? I have tried quite some tricks without success.   \documentclass[]{book}  \begin{document}  \chapter{First Chapter} aslkjdf  \chapter*{Unnumbered Chapter} The GNU General Public License is a free, copyleft license for software and other kinds of works.  The licenses for most software and other practical works are designed to take away your freedom to share and change the works. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change all versions of a program--to make sure it remains free software for all its users. We, the Free Software Foundation, use the GNU General Public License for most of our software; it applies also to any other work released this way by its authors. You can apply it to your programs, too.  When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for them if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs, and that you know you can do these things.  To protect your rights, we need to prevent others from denying you these rights or asking you to surrender the rights. Therefore, you have certain responsibilities if you distribute copies of the software, or if you modify it: responsibilities to respect the freedom of others.  For example, if you distribute copies of such a program, whether gratis or for a fee, you must pass on to the recipients the same freedoms that you received. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.  Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.  For the developers' and authors' protection, the GPL clearly explains that there is no warranty for this free software. For both users' and authors' sake, the GPL requires that modified versions be marked as changed, so that their problems will not be attributed erroneously to authors of previous versions.  Some devices are designed to deny users access to install or run modified versions of the software inside them, although the manufacturer can do so. This is fundamentally incompatible with the aim of protecting users' freedom to change the software. The systematic pattern of such abuse occurs in the area of products for individuals to use, which is precisely where it is most unacceptable. Therefore, we have designed this version of the GPL to prohibit the practice for those products. If such problems arise substantially in other domains, we stand ready to extend this provision to those domains in future versions of the GPL, as needed to protect the freedom of users.  Finally, every program is threatened constantly by software patents. States should not allow patents to restrict development and use of software on general-purpose computers, but in those that do, we wish to avoid the special danger that patents applied to a free program could make it effectively proprietary. To prevent this, the GPL assures that patents cannot be used to render the program non-free.  The precise terms and conditions for copying, distribution and modification follow.  \end{document}