Even Digit Series (2,4,6,8,20,....) I have a series of numbers whose Nth term is a number whose all of the digits are even. The series is of course divergent , but i am interested in finding a formula to find the Nth term of this series.The beginning terms of the series is as follows:- 2,4,6,8,20,22,24,26,28,40,....  We can see that each of the numbers in the above series consists of digits that are all even numbers. Can anyone please help me in finding the Nth term of this series.
Generating ordered combination of numbers I can form numbers with only 0,2,4,6,8.  The sequence is as follows  0,2,4,6,8,20,22,24,26,28,40,42,.....  How to generate an ordered sequence of numbers from combinations of 0,2,4,6,8.
How can I replace Windows with Ubuntu? My laptop is filled with viruses and Windows XP is just becoming impossible to work with.   I've been interested in Ubuntu for a while, so, would I be able to use something like Debian to clear my HDD and OS and then install Ubuntu and start fresh? I have an ASUS N550JK laptop (BIOS version 204) running Ubuntu 14.04 (3.13.0-43-generic kernel).  My issue is that the battery indicator ALWAYS shows that it's using battery power - even when plugged in and fully charged. So even though I have selected to show the battery status in the menu bar only when charging or in use (in power settings), it is ALWAYS displayed. When it reaches a full charge, the indicator icon will either show just the percentage (usually 99% or 100%), or sometimes a very big number for time (like 22:14) along with the percentage. If I go to the power settings menu, it always says "Using battery power", with something like 40 hours remaining. This is all while plugged in and fully charged.  Are there any fixes less drastic than trying to update BIOS, or is a BIOS update necessary to fix this? The latest BIOS version is 208 according to the ASUS website, but the (very brief) descriptions of the updates don't mention anything about battery issues. If at all possible I'd like to avoid updating BIOS because of the risk involved.  I've already tried letting the battery drain all the way and then charging to full, with no luck.  Thanks in advance! Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that   $$          f: \mathfrak{g} \rightarrow T_eG \\              X \mapsto X(e) $$ is a linear isomorphism.  The only thing that I have left to prove is surjectivity.Letting $v\in T_eG$, we set $X\in \mathcal{X}(G)$ such that $\forall x\in G,\,X(x)=L_{x_{*,e}}(v)$ (I still have to show that such an $X$ is indeed a vector field on $G$). We get that  $$ f(X)=X(e)=L_{e_{*,e}}(v)=v $$  where the last equality holds because  $$       L_{e_{*,e}}(v)f = v(f \circ L_e) = v(f \circ id_G) = v(f) $$ for any $f\in C^\infty(G)$. (I am thinking of the tangent vectors as derivations).  Now all we have left to do is prove that such an $X$ is left-invariant, and it is therefore an element of $\mathfrak{g}$, but I do not know how. I tried to prove that  $$            \forall f\in C^\infty(G),\, L_{{x}_*,y}(X(y))f=X(L_x(y))f $$ but with no success. Evaluate $$\int \frac{2-3x}{2+3x}\sqrt\frac{1+x}{1-x}d x$$ What substitution should I use ? $\sqrt\frac{1+x}{1-x}$ suggests $x=cos2\theta $ but its not useful in $\frac{2-3x}{2+3x}$ What is the difference between an indeterminate and variable? From Wikipedia-     In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.   So, how it is different from variable which is also used as a placeholder? What is the use of the indeterminate? Complete residue system proof I was given a proof of this in class, but it does not make sense to me, can someone provide a better proof, or explain the one I have provided, thanks.  Suppose $\gcd(a,n) = 1$, where $a$ and $n$ are natural numbers. Prove that $0$, $a$, $2a$, $3a,\ldots, (n-1)a$, is complete residue system modulo $n$.  Proof given in class:  \begin{align*} ka &amp;\equiv ra \pmod{n}\\ &amp;\Rightarrow  k \equiv r \pmod{n}\\ &amp;\Rightarrow  \text{$k-r$ divides $n$}\\ &amp;\Rightarrow \text{$k=r$ because $0\leq k\lt n$, $0\leq r\lt n$} \end{align*} patching SQL Server with Availability Group and Secondary Replica Should I use #define, enum or const? In a C++ project I'm working on, I have a flag kind of value which can have four values. Those four flags can be combined. Flags describe the records in database and can be:   new record deleted record modified record existing record   Now, for each record I wish to keep this attribute, so I could use an enum:  enum { xNew, xDeleted, xModified, xExisting }   However, in other places in code, I need to select which records are to be visible to the user, so I'd like to be able to pass that as a single parameter, like:   showRecords(xNew | xDeleted);   So, it seems I have three possible appoaches:  #define X_NEW      0x01 #define X_DELETED  0x02 #define X_MODIFIED 0x04 #define X_EXISTING 0x08   or  typedef enum { xNew = 1, xDeleted, xModified = 4, xExisting = 8 } RecordType;   or   namespace RecordType {     static const uint8 xNew = 1;     static const uint8 xDeleted = 2;     static const uint8 xModified = 4;     static const uint8 xExisting = 8; }   Space requirements are important (byte vs int) but not crucial. With defines I lose type safety, and with enum I lose some space (integers) and probably have to cast when I want to do a bitwise operation. With const I think I also lose type safety since a random uint8 could get in by mistake.  Is there some other cleaner way?  If not, what would you use and why?  P.S. The rest of the code is rather clean modern C++ without #defines, and I have used namespaces and templates in few spaces, so those aren't out of question either. Standard graded algebra In a Feistel cipher structure, what exactly does the "round function $F$" do? In a Feistel cipher structure, what exactly does the "round function $F$" do to the right half of the data, before taking XOR of the output with the left half of the data? I was wondering how to create a theorem environment similar to the following example:   Specifically, the environment should resemble the box that contains "Claim -- The function g is linear".  This comes from a , who has his style files uploaded  I have tried to mimic what I believed to be the relevant code (as there's a lot of stuff there that isn't related to this), to no avail. I have attached my attempt below, but it may be better to ignore it.  \usepackage{amsmath} \usepackage{amsthm} \usepackage{mdframed} \usepackage{thmtools} \usepackage{tikz} \usepackage{xcolor} \usepackage[framemethod=TikZ]{mdframed}     \mdfdefinestyle{mdgreenbox}{%         linewidth=0.5pt,         skipabove=12pt,         frametitleaboveskip=5pt,         frametitlebelowskip=0pt,         skipbelow=2pt,         frametitlefont=\bfseries,         innertopmargin=4pt,         innerbottommargin=8pt,         nobreak=true,         backgroundcolor=SpringGreen!15,         rightline=false,         leftline=false,         topline=false,         bottomline=false,         linecolor=green,     }     \declaretheoremstyle[         headfont=\bfseries\color{OliveGreen},         mdframed={style=mdgreenbox},         headpunct={\\[3pt]},         postheadspace={0pt},     ]{thmgreenbox} \declaretheorem[style=thmgreenbox]{thrm1}   This returns an error, one for xcolor not recognizing the colors, and the following.  l.17        \mdfdefinestyle                       {mdgreenbox}{% The package mdframed has already been loaded with options:   [] There has now been an attempt to load it with options   [framemethod=TikZ] Adding the global options:   ,framemethod=TikZ to your \documentclass declaration may fix this. Try typing  &lt;return&gt;  to proceed.  Package thmtools Info: Key `mdframed' (with value `style=mdgreenbox') (thmtools)             is not a known style key. (thmtools)             Will pass this to every \declaretheorem (thmtools)             that uses `style=thmgreenbox' on input line 39. Package thmtools Info: Automatically pulling in `thmdef-mdframed' on input line 40. (/usr/local/texlive/2017/texmf-dist/tex/latex/thmtools/thmdef-mdframed.sty Package: thmdef-mdframed 2014/04/21 v66 ) \c@thrm1=\count282 (/compile/output.aux) \openout1 = `output.aux'.   I am looking for a way to mimic the environment more than to fix my methods.  Thank you in advance! Explaining to laypeople why bootstrapping works I recently used bootstrapping to estimate confidence intervals for a project. Someone who doesn't know much about statistics recently asked me to explain why bootstrapping works, i.e., why is it that resampling the same sample over and over gives good results. I realized that although I'd spent a lot of time understanding how to use it, I don't really understand why bootstrapping works.  Specifically: if we are resampling from our sample, how is it that we are learning something about the population rather than only about the sample? There seems to be a leap there which is somewhat counter-intuitive.  I have found a few answers to this question here which I half-understand. Particularly . I am a "consumer" of statistics, not a statistician, and I work with people who know much less about statistics than I do. So, can someone explain, with a minimum of references to theorems, etc., the basic reasoning behind the bootstrap? That is, if you had to explain it to your neighbor, what would you say? I'd like to know if there is any alternative to Deep Freeze for Linux that still under development, because I've read some things on the internet and looks like the same company which developed Deep Freeze had a Linux version of it, but the project was discontinued. Using \@-ified punctuation while using \frenchspacing After reading  about spacing and punctuation, I was left wondering about an issue that didn't seem to be addressed: using \frenchspacing together with the modified punctuation \@., \@?, etc.  As far as I can tell visually, using \frenchspacing together with these commands works correctly (i.e., it seems like nothing breaks, and the extra space is suppressed as normal), but are there problems that can arise from doing so?   Moreover, when using \frenchspacing, "should" I still use \@. and its friends where it would ordinarily be necessary, in case someone wants to remove \frenchspacing later? Would accommodating (hypothetical) future editors in this way be "standard", "above and beyond the call of duty", or "crazy"? Install suspension forks on non-suspension bike Is a music band a singular or a collective entity, grammatically speaking? Duplicate:    What is correct to say?  Korn* is a great band  OR  Korn* are a great band.  (* You can replace your favourite band's name here) Of course everybody there is no doubt about the following sentences:  The Beatles are a great band Led Zeppelin is a great band  But with certain kinds of names, the situation becomes difficult.  Metallica are a great band  OR  Metallica is a great band  Is there any consensus on this matter? How does RNA transcription determine which half of the DNA to use? I feel that I might have a complete misunderstanding here. If DNA has two strands, how does the machinery of RNA transcription determine which one to transcribe? QGIS installation error Why couldn't Voldemort view the prophecy without Harry's help? After drinking an energy drink, there are little yellow and white bars above the health bar. What are those? What do they represent?