$n$ &amp; $m :=$ any value in $\{0,1,2\ldots\}$  $\Omega$ &amp; $\beta :=$ any value in $\{1,2,3\ldots\}$  [1] If there is a function $F_\beta$ such that for some value $\Omega$ and some function $T_{n,m}$ that $\text{if } F_1(\sum_{i=0}^{\Omega}T_{i,\Omega-i}) = T_{1,0}+T_{0,1}  \text{ and }  \\F_2(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)}) = T_{2,0}+2T_{1,1}+T_{0,2}\text{ and }  \\F_3(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)})=T_{3,0}+3T_{2,1}+3T_{1,2}+T_{0,3}\text{ and }\\ \ldots\\\text{then }F_\beta(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)})=\sum_{i=0}^\beta\binom{\beta}{i}T_{i,(\beta-i)}$  [2] if $\phi_i$ represents a variation of some function for every value i. $F_\beta(\sum_{i=0}^{k}\phi_i)=\sum_{i=0}^{k}F_\beta(\phi_i)$ AND $F_1(T_{n,m})=T_{n+1,m}+T_{n,m+1}$  $$:.F_\beta(T_{n,m})=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$$  example of [1]  If $T_{n,m}:=a^n*e^m$ and $F_\beta(T_{1,0}+T_{0,1})=F_\beta(\sum_{i=0}^{1}T_{i,(i-1)}):=(a+e)^{\beta}$  $F_1(\sum_{i=0}^{1}T_{i,(i-1)})\\=(a^{1}+e^{1})^{1}=a^{1}e^{0}+a^{0}e^{1}=T_{1,0}+T_{0,1}$  $F_2(\sum_{i=0}^{1}T_{i,(i-1)})\\=(a^{1}+e^{1})^{2}=a^{2}e^{0}+2a^{1}e^{1}+a^{0}e^{2}=T_{1,0}+2T_{1,1}+T_{0,2}$  $F_3(\sum_{i=0}^{1}T_{i,(i-1)})=(a^{1}+e^{1})^{3}\\=a^{3}e^{0}+3a^{2}e^{1}+3a^{1}e^{2}+a^{0}e^{3}=T_{3,0}+3T_{2,1}+3T_{1,2}+T_{0,3}\\\ldots$  $:.F_\beta(\sum_{i=0}^{1}T_{i,(i-1)})=(a+e)^{\beta} \\=\sum_{i=0}^\beta\binom{\beta}{i}a^{i}*e^{\beta-i}=\sum_{i=0}^\beta\binom{\beta}{i}T_{i,(\beta-i)}$  example of [2]  If $T_{n,m}=u^{(n)}v^{(m)}$ and $F_\beta(G):=\frac{d^\beta}{(dx)^\beta}(G)$  $F_1(T_{n,m})=\frac{d}{dx}(u^{(n)}v^{(m)}) \\=u^{(n+1)}v^{(m)}+u^{(n)}v^{(m+1)}=T_{(n+1),m}+T_{n,(m+1)}$  $:.F_\beta(T_{n,m})=\frac{d^\beta}{(dx)^\beta}(u^{(n)}v^{(m)}) \\=\sum_{i=0}^\beta\binom{\beta}{i}u^{(n+i)}v^{(m+\beta-i)}=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$  What are these distinct properties called? for when $F_\beta(\sum_{i=0}^{\Omega}T_{i,(\Omega-i)})=\sum_{i=0}^\beta\binom{\beta}{i}T_{i,(\beta-i)}$ or $F_\beta(T_{n,m})=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$.  Edit: I found that $F_\beta(T_{n,m})=\sum_{i=0}^\beta\binom{\beta}{i}T_{(n+i),(m+\beta-i)}$ is shown in for n=0, m=0.      Abstract binomial theorem. Assume that we have:         A vector space (or even a module) $V$.   A family of "general terms" $(T_{i,j}\in V)_{i,j\in\mathbb N}$.   A linear operatior $F: V\to V$, such that $F(T_{i,j}) = T_{(i+1),j} + T_{i,(j+1)}$.         Then $F^n(T_{0,0}) = \sum_{i=0}^n \binom{n}{i} T_{i,(n-i)}$.
Possible Duplicate:          The  describes the expansion of the $n$th power of the sum $(a+b)$:  $$(a+b)^n = \sum_{k = 0}^n {n\choose k}a^kb^{n-k}$$  In calculus, there is a generalization of the product rule called , which describes the expansion of the $n$th derivative of the product of two functions $f$ and $g$:  $$(fg)^{(n)} = \sum_{k = 0}^n {n\choose k}f^{(k)}g^{(n-k)}$$  It's not difficult to prove using the product rule and induction, what I'm really trying understand here is why the formula is true. Is there a way to prove it using the binomial formula? How are these two formulas related? Is there a combinatorial proof for Leibniz's rule like there is for the ?   I understand this question is a bit vague, but I hope you'll agree that these two formulas look so similar it just feels like there should be some relation between the two. Of course this is not necessarily true, but I'm hoping it's not just a crazy coincidence.
Java Equivalent to Python Dictionaries I am trying to test the impact of a sale initiative on store revenue. I have created 10 matched pairs of stores and then randomly distributed one store in each pair into a test or control group.  I then ran the sale for 21 days within the test group (keeping control constant) and measured relative lift in revenue compared to baseline.  My questions  1.) Should I be using a paired t-test based on the previous store matching  2.) Should I be measuring across each day of the test such as:          Control Test 1/1/2017    4   3 1/2/2017    3   4 1/3/2017    1   3 1/4/2017    2   3 1/5/2017    2   4 1/6/2017    1   7   Or measuring across my paired markets such as:                 Test Control Market Pair 1   10  5 Market Pair 2   8   3 Market Pair 3   5   1 Is it possible to combine processing power of 2 computers? Here are a few questions, I wish you could enlighten me.   Is it possible to combine processing power of 2 computers? How do I do it? I tried and got this   $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi en!)=\lim_{n\to\infty}n\sin\left(2\pi n!\sum_{k=0}^n\frac{1}{k!}\right)=\lim_{n\to\infty}n\sin(2\pi m)=\lim_{n\to\infty}n\cdot0=0$$  Is it correct? Upper bound for norm of Hilbert space operator It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $\|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|:=M(T)$, where $M(T)$ is called the  of $T$. It seems that for any bounded operator on $H$, we have $||T||\leq 2M$. Why is that so? Understanding QGIS buffer tool units Service not avaliable Unity3d error in Kubuntu I'm trying to use  in Kubuntu, but I'm getting this error:  Service not avaliable, please try again later.   I've found :  Certain systems may experience "Service unavailable" errors when trying to log in. WORKAROUND: Launch Unity with LD_PRELOAD=/usr/lib/libresolv.so.2 /path/to/Unity (check the path to your local libresolv)   I've checked the path of libresolv.so.2 in my computer. I found it under /lib32/libresolv.so.2 and under /libx32/libresolv.so.2. (Don't know what the difference is).  When I run:  LD_PRELOAD=/lib32/libresolv.so.2 /opt/Unity/Editor/Unity   I get:  ERROR: ld.so: object '/lib32/libresolv.so.2' from LD_PRELOAD cannot be preloaded (wrong ELF class: ELFCLASS32): ignored.   and when I run:   LD_PRELOAD=/libx32/libresolv.so.2 /opt/Unity/Editor/Unity   I get:   ERROR: ld.so: object '/libx32/libresolv.so.2' from LD_PRELOAD cannot be preloaded (cannot open shared object file): ignored.   With both commands I also get this:  /opt/Unity/Editor/Unity: /opt/Unity/Editor/Data/Tools/libgpg-error.so.0: no version information available (required by /lib/x86_64-linux-gnu/libgcrypt.so.20)   Any suggestion? I have a problem with arscalssica style in combination with classicthesis.  The text under the chapter title starts always at the same position, independently on how long is the chapter title, which produce this ugly behaviour:   The code I used to generete this output is really basic:  \documentclass[10pt,a4paper,twoside,openright,titlepage,fleqn,headinclude,footinclude,BCOR5mm,numbers=noenddot,cleardoublepage=empty,tablecaptionabove]{scrreprt} \usepackage{subfig} \usepackage[eulerchapternumbers,subfig,beramono,eulermath,pdfspacing]{classicthesis}  \usepackage{arsclassica}  \usepackage{blindtext}  \begin{document}     \chapter{\blindtext}     \blindtext \end{document}   The compilation raises the following warnings:  C:\Program Files (x86)\MiKTeX 2.9\tex\latex\koma-script\scrreprt.cls: Class scrreprt Warning: You've used obsolete option `tablecaptionabove'. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\classicthesis\classicthesis.sty: Class scrreprt Warning: Usage of package `titlesec' together(scrreprt)              with a KOMA-Script class is not recommended. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\titlesec\titlesec.sty: Package titlesec Warning: Non standard sectioning command detected(titlesec)                Using default spacing and no format. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\titlesec\titlesec.sty: Package titlesec Warning: Non standard sectioning command detected(titlesec)                Using default spacing and no format. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\titlesec\titlesec.sty: Package titlesec Warning: Non standard sectioning command detected(titlesec)                Using default spacing and no format. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\titlesec\titlesec.sty: Package titlesec Warning: Non standard sectioning command detected(titlesec)                Using default spacing and no format. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\titlesec\titlesec.sty: Package titlesec Warning: Non standard sectioning command detected(titlesec)                Using default spacing and no format. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\classicthesis\classicthesis.sty:373: Package scrreprt Warning: Activating an ugly workaround for a missing(scrreprt)                feature of package `titlesec` on input line 373. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\arsclassica\arsclassica.sty:192: Package hyperref Warning: Option `hyperfootnotes' has already been used,(hyperref)                setting the option has no effect on input line 192. C:\Program Files (x86)\MiKTeX 2.9\tex\latex\arsclassica\arsclassica.sty:192: Package hyperref Warning: Option `pdfpagelabels' has already been used,(hyperref)                setting the option has no effect on input line 192   The behaviour is ok if I do not use arsclassica but only classicthesis: Holomorphic function has at most countably zeros Launch apps full screen Can you terminate more than one common on the furnace C terminal? How does Natural Selection shape Genetic Variation? Background  Importance of the additive genetic variance  As stated , the fundamental theorem of Natural Selection (NS) by Fisher says:     The rate of increase in the mean fitness of any organism at any time ascribable to NS acting through changes in gene frequencies is exactly equal to its genetic variance in fitness at that time.   NS reduces the additive genetic variance  On the other hand, NS reduces additive genetic variance (discussion about the origin of this knowledge can be found ). The genetic variance of a population for multiple traits is best described by the G-matrix ( is a post on the subject).  What is a G-matrix  A G-matrix is a matrix where additive genetic variance of trait $i$ can be found at position $m_{ii}$. In other words, the diagonal contains the additive genetic variance for all traits. The other positions $m_{ij}$, where $i≠j$ contains the additive genetic covariance between the traits $i$ and $j$.  Question  How can one model how the G-matrix changes over time because of selection (assuming no mutation)? Can anyone help me identify a story about kids with silver blood? Is there a tool to get the lines in one file that are not in another? Is there any tool that can get lines which file A contains, but file B doesn't? I could make a little simple script with, e.g, perl, but if something like that already exists, I'll save my time from now on. find command: -or option doesn't work I want to execute a command for an each file that matches a pattern. But the command that seems to be right, doesn't work for me, I have no idea why.  $ find . -type f -name '*.c' -or -name '*.h' -or -name '*.cpp' -exec ls {} \; $ ls script.sh  test.c  test.h The following MWE produces a PDF whose TOC has two entries: “Foo” and “Bar” with correct page numbers. So far, so good. However, upon clicking on the link to “Bar”, I am taken to the page for “Foo” instead.  \documentclass{report} \usepackage{hyperref} \newcommand*\backmatter{\setcounter{chapter}{0}}  \begin{document} \tableofcontents \chapter{Foo} \backmatter \chapter{Bar} \end{document}   (Produced with pdflatex, run twice.)  This MWE is whittled down from a (much!) more complex document but the above does seem to reproduce the error reliably.  Re-setting the chapter count seems to confuse hyperref, even though the page references are still accurate. Notice that in my real document, \backmatter does other things, including setting the numbering style to alphabetical so there’s no chance of confusion even though both chapters are ostensibly labelled as “Chapter 1”.  How do I fix this? Is this a bug? Are there workarounds? Before we launched our website, we had set up a "coming soon" page and google picked up the title and description from its contents. So the description in the search results said:     Coming soon! Visit example.org for updates.   It's been a few weeks since we launched our website. We've even created a sitemap and submitted it to Google. In Google Search Console, the pages have been crawled and all the pages are appearing as expected on Google, EXCEPT the homepage which is still not updated! The title and description of the homepage in google search results still says "coming soon.".***  I checked on bing too, but it appears to be fine there.  All other pages seem to be indexed fine on Google. I don't even have any crawl errors in my reports. So what seems to be the problem? I've already waited for 2 weeks. Below is my code to create the component dynamically - The component is "ui:inputselect"  $A.createComponent(             RecoveryLineItem.Attributes[i].name,{                 "class" : "slds-input ",                 "aura:id" : 'ExampleId',                 "label" : RecoveryLineItem.Attributes[i].label,                 "multiple" : RecoveryLineItem.Attributes[i].multiple             }, ....   ..  console.log(component.find('ExampleId').get("v.value"))    Or console.log(component.find('ExampleId'))  is Giving as 'Undefined"  How to access the aura:id of the dynamically created components? Non-EU spouse of UK citizen Schengen visa requirements I'm a British citizen and my wife is in the UK on a spouse visa. She holds a BRP card. We're planning to visit Germany and some other Schengen countries, and I've got a question around visa requirements/supporting documents.  I've read in few places (i.e ) that the visa requirements are more relaxed if the applicant is a spouse of a UK citizen, and is travelling with the UK citizen.  How relaxed are the requirements? Does my wife need to submit supporting documents for accommodation, finances, flight booking, etc.? Or is it simply submitting the application form alongside the passports and BRP? (i.e standard documentation for Schengen visa application section from ) Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory?