3.3V Logic to 12V Logic (Uni-directional and Bi-directional)
Design Review: 3.3V Logic to 12V Logic (Uni-directional and Bi-directional)
Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$ Encouraging users to create an account (and keep it) Encouraging people to explain downvotes Note: If you're looking for a simple explanation as to why comments aren't required on downvotes, see .   I used to get "upset" (though that is too strong a term) when I got downvoted without comment. If my answer isn't good enough then I'd like to know why. Not only does it improve the answer for the OP but it improves my knowledge too.  Where the down-vote has been explained I've found it useful &amp; it has improved my answer, or forced me to delete the answer if it was totally wrong.  So is there any way we can encourage people to leave a comment? Perhaps they don't lose rep if they explain their down-vote?  I must admit though that I haven't always explained my down-vote either so you could call me a hypocrite. I've also grown a thicker skin over the months of using SO (it seems to have come with the higher rep score ;-)), so I'm less bothered about this now. Does the Sun produce audible sound? When I add a mirror modifier it sometimes doesn't mirror correctly, giving me an overlapping mesh. How can I fix this? What is the best way to increase my reputation and privileges? I'm not sure if this is a stupid question, but here goes anyway... I have only recently started learning PHP and Javascript and I'm reasonably proficient. One of the reasons I'm so proficient is because I LOVE Stack Overflow and find most of the solutions to my dilemmas here.   But I've found that I never get the chance to ask a question because it has normally been asked many times before and when I go through the unanswered questions to try and help someone else, they are generally WAY over my head.  So I'm not sure how to get more privileges. I can't even vote up answers that I like or that helped me. I can't comment to thank someone for their solution. It is most tragic and demoralising... How do I change the situation? In my preamble, you'll find:  \renewcommand{\baselinestretch}{1} \renewcommand{\familydefault}{\sfdefault} \usepackage[sfmath]{kpfonts} \usepackage[mathrmOrig]{sfmath} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{animate} \usepackage{anysize} \usepackage{arcs} \usepackage{array} \usepackage[scale=0]{background} \usepackage{blindtext} \usepackage{calc} \usepackage{cancel} \usepackage{chngcntr} \usepackage{CJK} \usepackage{CJKspace} \usepackage{CJKpunct} \usepackage{colortbl} \usepackage{enumitem} \usepackage{epstopdf} \usepackage{etoolbox} \usepackage{fancybox} \usepackage{fancyhdr} \usepackage{float} \usepackage{framed} \usepackage{graphicx} \usepackage{hyphenat} \usepackage{ifthen} \usepackage{latexsym} \usepackage{lmodern} \usepackage{multicol} \usepackage{multirow} \usepackage{mwe} \usepackage{pgffor} \usepackage{rotating} \usepackage{tabularx} \usepackage{textcomp} \usepackage{tgheros,tgtermes,tgcursor} \usepackage{tikz} \usepackage{times} \usepackage{tipa} \usepackage{varwidth} \usepackage{wasysym} \usepackage{wboiboites} \usepackage[abs]{overpic} \usepackage[breaklinks,colorlinks=true,linkcolor=red,citecolor=red, urlcolor=blue]{hyperref} \usepackage[framemethod=TikZ]{mdframed} \usepackage[breakable, theorems, skins]{tcolorbox,empheq} \usepackage[default]{gillius} \usepackage[document]{ragged2e} \usepackage[latin1]{inputenc} \usepackage[spanish]{babel} \usepackage[T1]{fontenc} \usepackage[total={18.59cm,26.5cm},left=1.5cm,top=2.5cm]{geometry} \usetikzlibrary{matrix,arrows, positioning,shadows,shadings,backgrounds,calc, shapes, tikzmark} \usetikzlibrary{shadows.blur} \usetikzlibrary{shapes.symbols} \usetikzlibrary{patterns,fadings} \tcbuselibrary{skins,breakable,listings,theorems}    I'm trying to draw an arrow for X axis and Y axis.   \begin{tikzpicture} \draw[yellow!10, fill=blue!5](-1, -1) rectangle (7, 5); \draw[dotted] (-0.75,-0.75) grid (6.75,4.75); \node[below] at (-0.25,0){$O$}; \draw[-&gt;,black, thick] (-0.75,0)--(6.75,0); \end{tikzpicture}   But I get this error:    Additionally, if I remove -> from the options of \draw the tikz picture compiles normally. Should "move-to-chat" be provided on demand (as a button perhaps)? There are various methods to find installed applications on the system but to my knowledge none returns exactly the same list as control panel item Add Remove Programs. Some of the ways I tried are:   WMI via Win32_Product and Win32Reg_AddRemovePrograms Registry, both x32 and x64 branches   For instance, I have 217 entries in AR and   gwmi  Win32_Product   returns 261 items.  gp HKLM:\Software\Microsoft\Windows\CurrentVersion\Uninstall\*, HKLM:\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Uninstall\* | ? {![string]::IsNullOrWhiteSpace($_.DisplayName) } | select DisplayName   returns 119 items.   (gwmi Win32Reg_AddRemovePrograms) + (gwmi Win32Reg_AddRemovePrograms64) | sort DisplayName -Unique    returns 107 items.  I tried to nail it down with more complex filter that included InstallLocation, InstallSource, InstallDate etc without success. I remember reading a while back about arithmetics with only a single operator that did all the work of the familiar $+ - \times /$ operators. If there is such a thing, could someone point me to it?  As background: I have a side project where I'm playing with neural nets and genetic search algorithms. Right now it's the good old kind, with matrix multiplication and addition. I thought it would be interesting to implement a sort of neural network that used a combined or "all in one" operator.  I've googled like heck and can't pick up the trail of where I thought I saw that before.  Is there study in that area, or am I mis-remembering, or worse, just making it up? I'd appreciate any pointers. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? Rescuing/Compiling old TeX file starting with “\input amstex” Changing sfdc header and footer without creating new VF pages I need to change the header and footer of standard salesforce pages (sfdc pages) with a custom CSS that I have. Now I am not sure if this is a customer portal (it is probably not). I searched and found out it can be done by changing the Self-Service as given in the following link:   However the problem is when I go to Name | Setup | Customize | Self-Service, I do not see the Self-Service Portal - Settings anywhere (as given in the step 1 of the link). Instead there are two options, 1) Public Solutions 2) Web-to-Case.  Basically what I want is to replace the Blue header and footer of the salesforce standard pages with the CSS given to me. The changes will be in actual org.  Any guesses on how to go ahead with this? I've been trying to change the language of my LaTeX document by replacing \usepackage[swedish]{babel} with \usepackage[english]{babel} but I only get this message:  ! Package babel Error: You haven't loaded the option swedish yet.   I've read in an other question at this forum that compiling enough would solve the problem, but that doesn't work for me. Does anyone know what I should do? QGIS Atlas generation for multiple map extents? What pins on a nRF51822 do I connect to a STM32 programmer? I want to be able to program a  with  but I'm not sure what pins on the actual chip I should use, and all the guides seemed to be using a different chip Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, then $\omega_\partial=\omega(v_{\text{out}},v_1,\dots,v_{k-1})$ orients $\partial M$, where the $v_1,\dots,v_{k-1}$ are basis vectors for a tangent space on a point of the boundary, and $v_\text{out}$ is an outward pointing vector (see here: , which follows the exposition of Hubbard &amp; Hubbard).  In practice I find this very difficult to interpret. How can one find this $v_\text{out}$ in practice?   For instance, take $M$ to be defined by $z=x^2+y^2$ for $x\ge0$ (technically a manifold with boundary, in fact a half-paraboloid). It's a fact that $\partial M$ is defined by the parabola $z=y^2$. If $M$ is oriented by $dx\wedge dy$, how does this induce an orientation on the boundary (in fact we should obtain a 1-form that induces an orientation on the boundary)?   Note: I find the whole discussion of boundary orientation to be rather clumsy in its exposition. In the case of surfaces inducing orientations on boundary curves, I was originally taught to imagine that the normal vector that induces the surface orientation is me walking on the surface. This normal should walk along the boundary such that the surface lies to its left. OK - but how does this translate to the $v_{\text{out}}$ interpretation, so that we can generalize to other dimensions easily?)  Edit: An ideal answer would merge the intuitive approach with the rigorous approach, if possible. I would grateful to anybody who could help order my (at present confused) understanding! Better way of saying "those who are curious"? I am writing a note in which I want to ask (curious) readers to refer another paper. The phrase curious minded comes to mind, but it seems somewhat awkward.  Is there a better way to convey similar import? Why can't I seem to grasp interfaces? Could someone please demystify interfaces for me or point me to some good examples? I keep seeing interfaces popup here and there, but I haven't ever really been exposed to good explanations of interfaces or when to use them.  I am talking about interfaces in a context of interfaces vs. abstract classes. I am given two charged particles of same charge at a distance of $r$. They initially apply force $F$.  Now an infinite dielectric (of dielectric constant $4$) of width $\frac{r}{2}$ is introduced between the particles. What will be the new force?   I find this problem confusing because I have only been told about forces when its either fully dielectric or fully vaccum given by Coulomb's law. How do we get forces when only partial space is dielectric?