A question on a sequence in a Banach algebra If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If $$\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0,\quad \text{for every $\lambda\in \mathbb{D}(0, 1/3)$},$$ where the $\mathbb{D}(0, 1/3)~$ denotes an open disc centered at $0$ and radius $1/3$.   Then can we conclude that $\boldsymbol{ u_{n}=0}$, for every $\boldsymbol{n}$?
An easy question on complex Let $\{u_{k}\}_{k=1}^{\infty}$ be a complex number sequence. If $\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0$, for each $\lambda\in \mathbb{D}(0, 1/3)$(where the $\mathbb{D}(0, 1/3)~$denotes an open disc centered at $0$ and radius $1/3$). Then can we conclude $u_{n}=0$, for $n=1, 2, 3...$ ?
OrderBy and OrderByDescending are stable? Which punctuation is correct in this case (colon, semicolon, or period)? Why are real symmetric matrices diagonalizable? In statistical learning, implicitly or explicitly, one always assumes that the training set $\mathcal{D} = \{ \bf {X}, \bf{y} \}$ is composed of $N$ input/response tuples $({\bf{X}}_i,y_i)$ that are independently drawn from the same joint distribution $\mathbb{P}({\bf{X}},y)$ with  $$ p({\bf{X}},y) = p( y \vert {\bf{X}}) p({\bf{X}}) $$  and $p( y \vert {\bf{X}})$ the relationship we are trying to capture through a particular learning algorithm. Mathematically, this i.i.d. assumption writes:  \begin{gather} ({\bf{X}}_i,y_i) \sim \mathbb{P}({\bf{X}},y), \forall i=1,...,N \\ ({\bf{X}}_i,y_i) \text{ independent of } ({\bf{X}}_j,y_j), \forall i \ne j \in \{1,...,N\} \end{gather}  I think we can all agree that this assumption is rarely satisfied in practice, see this related  and the wise comments of @Glen_b and @Luca.  My question is therefore:      Where exactly does the i.i.d. assumption becomes critical in practice?    [Context]   I'm asking this because I can think of many situations where such a stringent assumption is not needed to train a certain model (e.g. linear regression methods), or at least one can work around the i.i.d. assumption and obtain robust results. Actually the results will usually stay the same, it is rather the inferences that one can draw that will change (e.g. heteroskedasticity and autocorrelation consistent HAC estimators in linear regression: the idea is to re-use the good old OLS regression weights but to adapt the finite-sample behaviour of the OLS estimator to account for the violation of the Gauss-Markov assumptions).   My guess is therefore that the i.i.d. assumption is required not to be able to train a particular learning algorithm, but rather to guarantee that techniques such as cross-validation can indeed be used to infer a reliable measure of the model's capability of generalising well, which is the only thing we are interested in at the end of the day in statistical learning because it shows that we can indeed learn from the data. Intuitively, I can indeed understand that using cross-validation on dependent data could be optimistically biased (as illustrated/explained in ).   For me i.i.d. has thus nothing to do with training a particular model but everything to do with that model's generalisability. This seems to agree with a paper I found by Huan Xu et al, see "Robustness and Generalizability for Markovian Samples" .   Would you agree with that?  [Example]   If this can help the discussion, consider the problem of using the LASSO algorithm to perform a smart selection amongst $P$ features given $N$ training samples $({\bf{X}}_i,y_i)$ with $\forall i=1,...,N$ $$ {\bf{X}}_i=[X_{i1},...,X_{iP}] $$ We can further assume that:   The inputs ${\bf{X}}_i$ are dependent hence leading to a violation of the i.i.d. assumption (e.g. for each feature $j=1,..,P$ we observe a $N$ point time series, hence introducing temporal auto-correlation) The conditional responses $y_i \vert {\bf{X}}_i$ are independent. We have $P \gg N$.    In what way(s) does the violation of the i.i.d. assumption can pose problem in that case assuming we plan to determine the LASSO penalisation coefficient $\lambda$ using a cross-validation approach (on the full data set) + use a nested cross-validation to get a feel for the generalisation error of this learning strategy (we can leave the discussion concerning the inherent pros/cons of the LASSO aside, except if it is useful). Probability of the union of $3$ events? I need some clarification for why the probability of the union of three events is equal to the right side in the following:  $$P(E\cup F\cup G)=P(E)+P(F)+P(G)-P(E\cap F)-P(E\cap G)-P(F\cap G)+P(E\cap F\cap G)$$  What I don't understand is, why is the last term(intersection of all) added back just once, when it was subtracted three times as it appears from a Venn Diagram?  Here on page 3, this is explained but not in enough details that I can understand it: How do you assert that a certain exception is thrown in JUnit 4 tests? How can I use JUnit4 idiomatically to test that some code throws an exception?  While I can certainly do something like this:  @Test public void testFooThrowsIndexOutOfBoundsException() {   boolean thrown = false;    try {     foo.doStuff();   } catch (IndexOutOfBoundsException e) {     thrown = true;   }    assertTrue(thrown); }   I recall that there is an annotation or an Assert.xyz or something that is far less kludgy and far more in-the-spirit of JUnit for these sorts of situations. Going back twice required to return to search results in the tags view This looks a little bit like a variation on  but specific to the tags view.  Reproduction (app version 1.0.71):   Select any site (say Meta) from the sidebar Switch to the tags view using the top bar Choose a tag, any tag ( is nice)   optionaly refine your search with more terms  Go to any question found, go back   you end on the unfiltered tags view  Go back again   lucky you, your search is here While using the mobile site on my Android phone, I saw a comment that I wanted to flag but there was no flag button. After searching, I saw  that says flagging comments is not supported on mobile site.  Could the ability to flag comments be added to the mobile site? How to make cmd correctly display arabic script? Does more reflective aluminum foil make a room cooler compared to less reflective foil? Aluminum foil is said to be not absorbing light at all. It reflects light. So, does it mean that a more shiny aluminum foil will reflect more light and thus make the room more cooler as compared to less shiny foil?  If yes, then where does unreflected light go in case of less shiny aluminum foil?  Does less shiny and more shiny actually make a difference? Why?  In order to keep whole room cool will it help if I stick the foil to the outside of wooden door (exposed to the sunlight) too? What "exactly" happens when someone turns to the Dark Side? Typsetting the Ponny Today, once again, the  came to my attention and I have decided to try to  using XeLaTeX. The result is far from perfect, worse even than Chrome's rendering.  Is there a way I can improve the result? Is XeLaTeX and LuaLaTeX incapable of handling the Ponny?  :  \documentclass[]{article} \usepackage{lmodern} \usepackage{amssymb,amsmath} \usepackage{ifxetex,ifluatex} \usepackage{fontspec} \defaultfontfeatures{Mapping=tex-text,Scale=MatchLowercase} \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{\usepackage{microtype}}{} \usepackage[normalem]{ulem} \setmainfont{Doulos SIL}  \title{html - RegEx match open tags except XHTML self-contained tags - Stack Overflow} \date{} \author{StackOverflow}  \begin{document} \maketitle  Rege̿̔̉x-based HTML parsers are the cancer that is killing StackOverflow \emph{it is too late it is too late we cannot be saved} the trangession of a chi͡ld ensures regex will consume all living tissue (except for HTML which it cannot, as previously prophesied) \emph{dear lord help us how can anyone survive this scourge} using regex to parse HTML has doomed humanity to an eternity of dread torture and security holes \emph{using rege}x as a tool to process HTML establishes a brea\emph{ch between this world} and the dread realm of c͒ͪo͛ͫrrupt entities (like SGML entities, but \emph{more corrupt) a mere glimp}se of the world of reg​\textbf{ex parsers for HTML will ins}​tantly transport a p\emph{rogrammer's consciousness i}nto a w\emph{orl}d of ceaseless screaming, he comes\sout{, the pestilent sl}ithy regex-infection wil​\textbf{l devour your HT}​ML parser, application and existence for all time like Visual Basic only worse \emph{he comes he com}es \emph{do not fi}​ght h\textbf{e com̡e̶s, ̕h̵i}​s un̨ho͞ly radiańcé de\emph{stro҉ying all enli̍̈́̂̈́ghtenment, HTML tags \textbf{lea͠ki̧n͘g fr̶ǫm ̡yo​͟ur eye͢s̸ ̛l̕ik͏e liq}​uid p}ain, the song of re̸gular exp​re\sout{ssion parsing} will exti\emph{​nguish the voices of mor​\textbf{tal man from the sp}​here I can see it can you see ̲͚̖͔̙î̩́t̲͎̩̱͔́̋̀ it is beautiful t​}he f\texttt{inal snuf}fing o\emph{f the lie​\textbf{s of Man ALL IS LOŚ͖̩͇̗̪̏̈́T A}}\textbf{LL I​S L}OST th\emph{e pon̷y he come}s he c̶̮om\sout{es he co}\textbf{\sout{me}s t\emph{he} ich​}or permeat\emph{es al}l MY FAC\emph{E MY FACE ᵒh god n\textbf{o NO NOO̼}}\textbf{O​O N}Θ stop t\emph{he an​*̶͑̾̾​̅ͫ͏̙̤g͇̫͛͆̾ͫ̑͆l͖͉̗̩̳̟̍ͫͥͨ}e̠̅s\texttt{ ͎a̧͈͖r̽̾̈́͒͑e} n\textbf{​ot rè̑ͧ̌aͨl̘̝̙̃ͤ͂̾̆ ZA̡͊͠͝LGΌ ISͮ̂҉̯͈͕̹̘̱ T}O͇̹̺ͅƝ̴ȳ̳ TH̘\textbf{Ë͖́̉ ͠P̯͍̭O̚​N̐Y̡ H̸̡̪̯ͨ͊̽̅̾̎Ȩ̬̩̾͛ͪ̈́̀́͘ ̶̧̨̱̹̭̯ͧ̾ͬC̷̙̲̝͖ͭ̏ͥͮ͟Oͮ͏̮̪̝͍M̲̖͊̒ͪͩͬ̚̚͜Ȇ̴̟̟͙̞ͩ͌͝}S̨̥̫͎̭ͯ̿̔̀ͅ  \end{document}   UPDATE  Using the Doulos SIL font gets me something closer to what Firefox renders but still far from perfect. Much less than, what does that mean? What exactly does $\ll$ mean?  I am familiar that this symbol means much less than.   ...but what exactly does "much less than" mean? (Or the corollary, $\gg$)  On , the example they use is that $1\ll 9999999999$ But my thought on that is that $10^{10^{10^{11}}}\ll 10^{10^{10^{11}}}+9999999999$, based on the same logic. But I am confused because the numbers are comparitively.   Logically, I am really kinda spent dealing with this issue of mine. Thanks for any input. Cannot set LD_LIBRARY_PATH in /etc/profile.d How to prevent my screens from turning off, when the screen is locked? I would like to have my screen fully powered at all times, but it goes to standby once I lock my PC. The distance between touching objects What is the distance between, say, a cup of coffee and the table it rests on? What is the distance between two touching hands? Reward for Passing Review tests? Just wanted to see what the community thought of adding a reward to passing the review tests? Maybe a 1 point reward?  I know seeing the tests in the review area has helped me be extra careful when reviewing, but I thought perhaps a little boost in incentive would be cool. Tags on badge question list I'm trying to get the Weed-Eater hat on Stack Overflow. To do so, I'm looking at the list of questions . The problem is, no tags show up there, and so it's rather difficult to see which ones I have a shot at. Any chance of showing the tags on this page?  I realize that getting this before Winter Bash is over would be difficult, but it would be good to keep such things in mind for the future. How to use an alpha map in Blender Render? Message appearing when a user was removed.     It seems I lost 1765 reputation points on Stack Overflow yesterday. The reason given in my history is "removed", with the description of "User was removed ()". I don't understand what that means. What caused this?