Let $f_1(s,\tau)=\delta(e^{(\tau+s)} \sinh \tau )$.  This should be equal to $0$ everywhere unless $\tau=0$, but I think there should be some constant multiplying the delta, i.e. it should be equivalent to $f_2(s,\tau)=c(\tau,s)\delta(\tau)$ where $c(\tau,s)$ is a constant.  How can I manipulate $f_1$ to resemble $f_2$?  Thanks.
I'm trying to show that  $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$  Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by using a dummy function $g(x)$ and carrying out:  $$\int_{-\infty}^{\infty}dx\,\delta\big(f(x)\big)g(x)$$  Then making the coordinate substitution $u$ = $f(x)$ and integrating over $u$. This seems to be on the right track, but I'm unsure where the absolute value comes in in the denominator, and also why it becomes a sum.  $$\int_{-\infty}^{\infty}\frac{du}{\frac{df}{dx}}\delta(u)g\big(f^{-1}(u)\big) = \frac{g\big(f^{-1}(0)\big)}{\frac{df}{dx}\big(f^{-1}(0)\big)}$$  Can any one shed some light? Wikipedia just states the formula and doesn't actually show where it comes from.
What are the highest DC and AC voltages or currents that can be considered as safe? My trackpad is broken so I can tap-to-click but i cannot drag and drop. This is really a problem when it comes to doing certain things.   Does anyone know how to program a sticky key to turn a click into a click-select that can then be dragged about?   I can't resize my browser or open window and I can't drag applications into the folder when installing etc.  free solutions only please.  A mid 2009 macbook-pro 13". 2.53 Ghz Intel Core 2 Due.   Running os lion (grudgingly). Why do we need in RSA the modulus to be product of 2 primes? I'm a newbie in WordPress. I'm trying to understand how it generates the XHTML pages. I guess it's the function of do_action( 'hook-name' ) , but I'm stuck with the understanding of the instruction isset( $wp_filter['all] ) when reading the function do_action because I don't understand what is the meaning of the word all.  Is it a PHP feature or a WordPress one? I want to have a page-filling and -breaking longtable, so I decided to try xltabular. Unfortunately xltabular seems to screw up the booktabs rules in the normal tabular environments, see:  \documentclass{article} \usepackage{booktabs, xltabular} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tabular} \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}     Not loading xltabular resolves the issue.  Did I miss some package option or some incompatibilities? Is there a better package than xltabular which doesn't break booktabs?  (I am using TeXLive 2019 and compile with lualatex)    EDIT (by Skillmon)  I reduced the bug to the following MWE (which doesn't make much sense on its own), it looks like booktabs tries to be smart detecting something, but fails badly.  \documentclass{article} \usepackage{longtable} \let\xltabular\relax \usepackage{booktabs}  \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}   It seems to suffice that longtable is loaded and \xltabular is defined (even if it's just \relax). How to use "who" vs. "that" What exactly does ARM sell to vendors? Assumptions:   Computer architecture: Describes how the different modules of a processor interact with each other. A computer architecture is defined using vhdl files Computer Organization: Describes the physical layout of the processor modules on silicon. A computer organization is defined using a set of photo masks (and manufacturing process eg chemical, that goes at each step) Computer Organization, therefore, requires that the fab process be taken into account. ARM is not in the fabrication business, therefore it does not sell photo masks.     My question(s):   What exactly is ARM selling to a vendor (eg: freescale)? For a SoC (System On Chip), (eg: ), which part is ARM and which is Freescale? Who did the integration? Sum of two closed sets in $\mathbb R$ is closed? Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it?  (By sum of sets $X+Y$ I mean the set of all sums $x+y$ where $x$ is in $X$ and $y$ is in $Y$)  Thanks! The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions? How to compare the output of \ref to a number? I want to compare the output of \ref to a number in LaTeX. I thought it would be easy but after spending lots of hours surfing the net I still don't know the answer. I found similar questions on this site but I would prefer a solution that does not require another package to be included, so that I can see clearly how the \ref command (and some others) works. I tried a lot of things, some of which I share below, therefore my question is rather long, I'm sorry about that. First, I tried the following codes (the resulting error messages are shown after a % sign):  \begin{equation} \label{eq:myeq}   a=b   \end{equation}  \ifnum\ref{eq:myeq}=1 true\else false\fi % error: "Missing number, treated as zero."  \expandafter\ifnum\ref{eq:myeq}=1 true\else false\fi % error: "Missing number, treated as zero."  \edef\blabla{\ref{eq:myeq}} % error: Incomplete \iffalse; all text was ignored after line 42. \ifnum\blabla=1 true\else false\fi \expandafter\ifnum\blabla=1 true\else false\fi   Of course, \ref{eq:myeq} gave the expected result, namely 1.  Then, I found the definition of \ref using the latexdef ref command:       \ref:   macro:#1->\expandafter \@setref \csname r@#1\endcsname \@firstoftwo {#1}   Further examining \@setref and \firstoftwo, I concluded that the equation number (together with the page number) is stored in a control sequence called \r@eq:myeq. More precisely, the content of \r@eq:myeq is {1}{1} where the first number is the equation number and the second number is the page number. (The output of \expandafter{\csname r@eq:myeq\endcsname} is 11.) If I understand well, in order to obtain the equation number we have to expand \csname r@eq:myeq\endcsname by only two levels to get {1}{1} (the first level of expansion yields \r@eq:myeq), and then we choose the first argument by \@firstoftwo. Since there is no simple way to expand a control sequence by only one level, it is done in a rather circuitous (say tricky) way by \ref. This I could also reproduce by the following code:  \def\sth#1#2{\expandafter #2#1} \makeatletter \expandafter\sth\csname r@eq:myeq\endcsname\@firstoftwo \quad \ifnum1=\expandafter\sth\csname r@eq:myeq\endcsname\@firstoftwo true\else false\fi \makeatother   which resulted in 1   true. So far so good but there are still a few questions. First of all, if the above code works then why does the code with \ifnum1=\ref{eq:myeq}... fail (see above)? Seemingly, the two "algorithms" are doing the same. Second, the above code works only if the auxiliary file (with aux extension) contains the label definition. If it doesn't (the reference is undefined) then LaTeX gives the following error message: Missing number, treated as zero. How can I write a code that works even if the reference is undefined? Thank everybody who read all my above struggle and I would appreciate if someone shed light on the points that were not clear to me and who corrected me where I was wrong and confirmed where I was correct. Unfortunately the whole command syntax of (La)TeX is still very strange to me. I would like to cook frozen chicken breasts in a pressure cooker for shredding and making chicken mole.  I do not want to add anything except water, but I'm having a hard time finding specific guidelines on the time to cook frozen meats with no added ingredients or steps.   In my case, I have 3 pounds of frozen chicken breasts to cook.  What are the general rules to follow for frozen meats and, in my case, how long and at what heat should I cook this for? Which Stack Exchange website for machine learning and computational algorithms? I'm currently working on many machine learning and computational algorithms, such as Singular Value Decomposition, Support Vector Machines, and others.  I'd like to ask questions about these topics, but I don't know which Stack Exchange website to use to post my questions.   Should I use ? Should I use ? Should I use ? Should I use ?   How do I tell?  What criteria do I use? Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.  My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this?  $(0)$.  $(f(X))$, where $f(X)$ is an irreducible polynomial.  $(p)$, where $p$ is a prime number.  $(p, f(X))$, where $p$ is a prime number and $f(X)$ is an irreducible polynomial modulo $p$. Print composer custom grid label problem In answer to  Klas Karlsson puts forward a solution to the problem of formatting print composer grid labels for projected maps.  I'm very weak on scripting, but in QGIS 2.14 I've tried to enter the text as shown in Klas's illustration, which generates error messages. In my case I'm working with a coordinate system that has a six-digit Easting and a six-digit Northing and I'm trying to get a result similar to that shown in Klas's picture (but with a six-digit Northing). Can anyone help me with this? Selecting features with NULL attributes in QGIS How do I identify which features have NULL attributes in the attribute table of a shapefile?  When i try [field]= 'NULL' it selects the features with value = 0 (see picture) I want to automatically split all lines in a QGIS layer at their vertices (). I've done a Google search and found , which is quite close but the workflow is complicated. Because I'll have to train my co-workers (who are unfamiliar with GIS) to use this tool, I want something as simple as a plugin or something that doesn't require GRASS. Since it's been a long time from 2010 - the time of , I wonder if there is any better way to achieve the same result? Both DHCP and static IP addresses simultaneously on one interface We can use multiple IP addresses for a single interface in Linux (eg. eth0:0, eth0:1). But can we use DHCP and static IP at the same time? eth0:1 should have a static IP address and eth0:0 should be configured using DHCP.  How can we accomplish both? Show that there is no solution to the congruence $x^2\equiv3\bmod5$.   I'm not sure if the fact that the only numbers divisible by $5$ are those ending in $0$ or $5$ will help&hellip; What is postselection in quantum computing? A quantum computer can efficiently solve problems lying in the complexity class . I have seen a claim the one can (potentially, because we don't know whether BQP is a proper subset or equal to PP) increase the efficiency of a quantum computer by applying postselection and that the class of efficiently solvable problems becomes now postBQP = .  What does postselection mean here? 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?