Subgaussian norm of a random variable $X$ is defined as: $$ \|X\|_{\psi_2} = \inf\left\{t &gt; 0: \mathbb{E}\left[e^{\frac{X^2}{t^2}}\right] \leq 2\right\}. $$ For $L_2$ norm of a random variable we can write $\|X - \mathbb{E}[X]\|_2 \leq \|X\|_2$ which is called centering inequality. It is easy to prove that a similar inequality holds for subgaussian random variables as well. Namely, $$ \|X - \mathbb{E}[X]\|_{\psi_2} \leq C\|X\|_{\psi_2} $$ for some $C \geq 1$ which does not depend on $X$. I would like to see if this inequality holds for $C = 1$?
Specifically, define the sub-gaussian norm for a r.v. X as  $$ \|X\|_{\psi_2}=\inf\{t&gt;0:e^{X^2/t^2}\leq 2\}. $$  How do we prove that the centering inequality with $C=1$ does not hold in general? i.e. $$ \|X-\mathbb{E}X\|_{\psi_2}\leq C\|X\|_{\psi_2}.  $$ A counter example would work too. Thanks in advance!
Is there an idiom that means "revealing a secret unintentionally"? I am thinking there could be an idiom that states that you or someone revealed a secret unintentionally? Can you think of an idiom like that? I am looking for an idiom I can use to write a blog post about the recent incident where Secretary Pompeo unintentionally revealed the dirty secrets behind the CIA. Names of higher-order derivatives Specific derivatives have specific names. First order is often called tangency/velocity, second order is curvature/acceleration. I've also come across words like Jerk, Yank, Jounce, Jolt, Surge and Lurch for 3rd and 4th order derivatives. Is there a widely agreed list of names for these? How many orders have specific names?  In this case, I'm dealing with NURBs curves, so the "tangency" and "curvature" related words are to be preferred over "velocity" and "acceleration" words. How to round pixel values of a raster in QGIS? I have a raster with float number values for example: 2.544459104537964 and I want to round these up to integers. I have seen other posts on ArcGIS like  but when I use Raster Calculator in QGIS, Int(myraster + 0.5) only makes the first value (which is 0) to 0,5 and the last one (which is 100) to 100,5. The other pixels stay float. Haproxy not logging captured request headers with custom log format Earlier i was not specifying a custom log format, and the captured request headers were getting printed fine. I wanted to log the unique-id-header, but could not find any way of doing so. So i copy pasted the log-format mentioned on , and unique id started getting logged.  Unfortunately, now the captured request headers are not getting logged. The %hrl and %hsl are coming out as empty strings.  Very weird. Here is my config, plzz help :(  listen server   bind *:80   mode http    option  httplog   option  http-server-close   no option logasap   # disable early logging of HTTP requests so that total transfer time is logged   option  forwardfor   capture request  header    X-Forwarded-For      len 500    capture request  header    Host                 len 500    capture request  header    X-Request-UID        len 500    rspadd  X-Haproxy-Backend:\ server    # Generate the X-Haproxy-Unique-ID and log it to make it easy to track requests   log-format %Ci:%Cp\ [id=%ID]\ [%t]\ %f\ %b/%s\ %Tq/%Tw/%Tc/%Tr/%Tt\ %st\ %B\ %cc\ %cs\ %tsc\ %ac/%fc/%bc/%sc/%rc\ %sq/%bq\ {%hrl}\ {%hsl}\ %{+Q}r   unique-id-format %{+X}o\ %Ci:%Cp_%Fi:%Fp_%Ts_%rt:%pid   unique-id-header X-Haproxy-Unique-ID   Here is the log i see:  Sep 13 17:28:57 localhost haproxy[11979]: 10.161.27.218:41592  [id=0AA11BDA:A278_0AA15B71:0050_505217C5_0014:2ECB] [13/Sep/2012:17:28:37.567]  server www-example-com-healthz/- 19998/0/2/1/+20001 200 +326  - - ---- 10/10/1/1/0 0/0 {} {}  "GET /testing/healthz?merchant=www.example.com&amp;source=elb HTTP/1.1"   Note that %hrl and %hsl are empty strings :( How do i capture the request headers ?? Correct use of "is" or "are" Possible Duplicate:          Would I use is or are in the following sentence?      The Pohutakawa Coast, Hauraki Gulf, and Waiheke Island is/are the backdrop for our day.   If I rewrote it so the subject (backdrop) starts the sentence, then it seems to me that as it is singular, the verb should be the singular is. But as it is written above, are seems less clumsy. Is the wave-particle duality a real duality? Why do we use the simple past but not the present or future in the following expressions:  Don't you think it's time we went a little further Don't you think it's time we ate Don't you think it's time we went home Selecting the Real Analysis Textbooks I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational microbiology, prepare for upcoming math research (starting on this Fall) on the analytic number theory, and prepare for the real analysis course I will take on Fall and Putnam competition. I just finished "Calculus with Analytic Geometry" by G. Simmons, "How to Prove It" by Daniel Velleman, and "How to Solve It" by G. Polya. I also read some portions of Apostol's Calculu Vol.I to get a deeper view on the calculus theories. I was originally planned to read Apostol's Calculus Vol.I and Spivak's Calculus first, but I think it would be a better idea to start with real analysis since it covers all ideas in those "advanced calculus" textbooks and much more. My current plan is to start with one "dumbed-down" real analysis textbook and one "comprehensive, detailed, and intermediate" textbook, and advance into Rudin's PMA (required textbook for my real analysis course) starting on Summer and use it in accordance with other real analysis textbooks. Could you help me on selecting one book from each category?   Elementary Real Analysis textbooks:  **Elementary Analysis: The Theory of Calculus (Kenneth Ross)  **Understanding Analysis (Steven Abbott)  **The Way of Analysis (Robert Strichartz)  **Real Mathematical Analysis (Charles Pugh)   Intermediate, detailed Real Analysis textbooks:  **Mathematical Analysis (Tom Apostol)  **Undergraduate Analysis (Serge Lang)  **Introduction to Real Analysis (Bartle, Sherbert) **Elements of Real Analysis (Bartle, Sherbert)  **Mathematical Analysis I (Vladimir Zorich)   Thank you very much for your time, and I look forward to your advice!   Sincerely, PK Can suspended users post comments? Get notifications? What happens to their reputation? Will stationary charge radiate E.M waves if observer is accelerating? I've recently read the Yahoo manifesto . They recommend to put the JavaScript inclusion at the bottom of the HTML code when we can.  But where exactly and when?  Should we put it before closing &lt;/html&gt; or after ? And above all, when should we still put it in the &lt;head&gt; section? When to spell out numbers Command/macro name cannot include numbers and symbols I would like put picture into table, and my idea is define new command as below>   \newcommand{\EJ471}{\includegraphics[scale=0.150]{EJ_471.jpg}}   But compiler reports an error. If I use the command name for example \EJa, everything is all right. I've read similar questions as      But still I don't understand why it does not work. When is the present perfect tense used instead of the past tense? Can Rails Routing Helpers (i.e. mymodel_path(model)) be Used in Models? How to use pagenumbering in the document? In my Master thesis, I would like to have the title page without any page number and an acknowledgments section and the table of contents with roman numbers. The rest of the document should be numbered arabic.  I have tried this, to have roman numbers for the Table of contents and normal numbers for the rest.  \documentclass[12pt,ngerman,openany]{report} \usepackage{titlesec} \usepackage{titletoc} \usepackage{cite} \usepackage[a4paper]{geometry} %\usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[ngerman]{babel} %\usepackage{ngerman}{babel} \usepackage[T1]{fontenc} \usepackage{graphicx} \usepackage{float} \selectlanguage{ngerman} \usepackage{amssymb} \usepackage{amsmath} \usepackage{url} \usepackage{epstopdf} \usepackage{scrhack} \usepackage{listings} \usepackage{caption} \titleformat{\chapter}{\rmfamily\huge}{\textbf\thechapter.}{25pt}{\huge\textbf} \defcaptionname{ngerman}{\lstlistingname}{Code} \defcaptionname{ngerman}{\lstlistlistingname }{Codeverzeichnis}  \lstloadlanguages{R}  \lstset{% general command to set parameter(s)    basicstyle=\sffamily\footnotesize, % print whole listing small    keywordstyle=\sffamily\footnotesize\bfseries, % ubold black keywords    identifierstyle=, % nothing happens    commentstyle=\sffamily\footnotesize\slshape, % green comments    stringstyle=\sffamily\footnotesize, % typewriter type for strings    showstringspaces=false, % no special string spaces    numbers=left,    numberstyle=\sffamily\footnotesize,    stepnumber=1,    numbersep=10pt,    showspaces=false,    showtabs=false,    frame=lines,    morecomment=[l]{\%},    float=htbp,    numberbychapter=true  }   \begin{document} \let\cleardoublepage\clearpage   \pagenumbering{roman}  \tableofcontents  \clearpage  \vspace*{75mm} \chapter{Einleitung} \section{Motivation} bla bla bla \section{Ziel}  bla bla \chapter{Stand der Technik zur Mustererkennung} \section{Die Zeitreihen} bla bla \subsection{Beschreibung} bla bla \subsection{Erklärung}  bla bla  \subsection{Vorhersage} bla bla \subsection{Kontrolle} bla bla \section{Feature Extraktion} bla bla \subsection{Autokorrelation} \subsection{Was ist Autokorrelation?} bla bla   \subsection{Autokorrelation und Zeitreihen} bla bla  \subsubsection{Zeitreihe-Diagramm}  bla bla \subsubsection{Lagged Streudiagramm} bla bla  \subsubsection{Autocorrelation-Function (Korrelogramm)} bla bla \subsection{Spektralanalyse} \subsection{Was ist Spektral-Analyse} bla bla \end{document} Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$.    Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof.  I spend part of the day studying this question   and its answers. Only one answer is clear and give clues to solve the problem. This answer is the first one. However, this answer does not answer the question nor directly, nor the proof follows from this answer.   This answer has some mistakes, he use that $[(k_1-k_2)\alpha]=[k_1\alpha]-[k_2\alpha]$ which is not true. Consider $k_1=3, k_2=1, \alpha=\sqrt{2}$ we have $[(k_1-k_2)\alpha]=2\not= 3=[k_1\alpha]-[k_2\alpha] $. We only can assure that $[k_2\alpha]-[k_1\alpha]-1\leq [(k_2-k_1)\alpha]\leq[k_2\alpha]-[k_1\alpha]$.   Who answered said something interesting about additive subgroups of $\mathbb{R}$, but unfortunately the set $X=\{nx-[nx] : n\in \mathbb{N} \}$ is not a subgroup. Considering the additive subgroup $G=\langle X \rangle$, if we prove the part (a) of the link, we get that indeed $G$ is dense on $\mathbb{R}$ but we can not conclude that $X$ is dense on $[0,1)$.  I think this problem has not been solved.  Thanks! Create a variable in .CSS file for use within that .CSS file Possible Duplicate:          We have some "theme colors" that are reused in our CSS sheet.  Is there a way to set a variable and then reuse it?  E.g.  .css OurColor: Blue  H1 {   color:OurColor; } How can we improve the inconsistent tagging of the questions relating to the recent uproar and associated issues? Question description  There are many questions (~250 in the last five weeks) relating to      sometimes they are about licensing, legal issues, , , , etc.  These questions relate to the same growing polarization between community and Stack&nbsp;Exchange staff. But, it is not easy to search or filter them because the tags are not uniform for those questions, they do not have the same tags.  Could we group all these posts together by some common tag (new or not)? (in addition, maybe we should create less of these questions, they tend to become noise now, water down the core message, and it is ).    Relevance  Overview of current tags (aside from the discussion tag) for the first ten  (           ) questions relating to  (ordered by relevance)                              1   2   3   4   5   6   7   8   9   10     moderators              x           x   x   x     community               x           x   x     etiquette                   x     users                       x     company                     x     moderation                      x     code-of-conduct                     x       x       x     deleted-questions                               x     legal                                                   x   These tags are obviously not the same while it is arguably the same topic (it is even for the same, specific, search term) .  Instead of using multiple searches one might wish to control tracking these question by adding some single relevant tag to watched tags or ignore tags. But, this is difficult when they do not use the same tag. And in the case of ignoring one might ignore too much when the tag is not specific.  Another indicator that these posts need consistent tagging is the fact that there exist highly upvoted answers that function as an archive for these questions (see  and ). But it is only a hack/workaround for the absence of a common tag; they are not an optimal solution.    Why now?  In an older question () it was suggested to create a specific tag. That was rejected at that point. But, now we are several weeks, and many many questions, further. The situation is more escalated than what might have foreseen at that point. The overview becomes a mess and something needs to be done.  See the bump in the image below which indicates that something is 'going on' and it entails a large fraction of the current post. (obtained from ) I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup of order $n$"). This is an exercise from a book, and it is in the section on finite abelian groups, so I know I have to use the fundamental theorem of finite abelian groups. I have come up with a proof, but it seems a bit messy, and I'm not entirely sure if it's correct. It is given below.     Let the order of $G$ be $m$ = $p_1^{\alpha_1} \ldots p_k^{\alpha_k}$. It is known that $G$ is a direct product of $p$-groups, say:      $$G = G_1 \times \ldots \times G_k$$      where each $G_i$ is a $p_i$-group. By the fundamental theorem of finite abelian groups, each $G_i$ is isomorphic to a direct product of cyclic groups of the form      $$\mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}},$$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$. Now if $n$ divides $m$, then we must have      $$n = p_1^{\gamma_1} \ldots p_k^{\gamma_k}$$      for some $\gamma_1, \ldots, \gamma_k$ with $0 \leq \gamma_i \leq \alpha_i$.      Claim: Each $G_i$ has a subgroup of order $p_i^{\gamma_i}$      Proof: As above, we have that       $$ G_i \cong \mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}} $$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$.      Now since $0 \leq \gamma_i \leq \alpha_i$, we can find   $l$ numbers $\delta_1, \ldots , \delta_l$ such that $\gamma_i = \sum_{j=1}^l \delta_j$, and $0 \leq \delta_j \leq \beta_j$. (This choice of numbers is not necessarily unique).      Then $p_i^{\delta_j} | p_i^{\beta_j}$ for each $j = 1, \ldots , l$. Hence, for each factor $\mathbb{Z}_{{p_i}^{\beta_j}}$, there exists a subgroup of order $p_i^{\delta_j}$, namely $\mathbb{Z}_{{p_i}^{\delta_j}}$ (using the fact that the converse of Lagrange's theorem is true for finite cyclic groups). Taking the direct product of each of these subgroups, we get a new subgroup $G_i'$ of $G_i$:      $$G_i' \cong \mathbb{Z}_{{p_i}^{\delta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\delta_l}}$$      The order of this subgroup is $p_i^{\delta_1} \times \ldots \times p_i^{\delta_l} = p_i^{\delta_1 + \ldots + \delta_l} = p_i^{\gamma_i} $. So we have found a subgroup of $G_i$ of order $p_i^{\gamma_i}$, as required.      So each factor $G_i$ in the product $G = G_1 \times \ldots \times G_k$ has a subgroup $G_i'$ of order $p_i^{\gamma_i}$.       Therefore, $G$ has a subgroup   $$G_1' \times G_2' \times \ldots \times G_k'$$       of order $p_1^{\gamma_i}...p_k^{\gamma_k} = n$, which completes the proof.   I have two questions about this: firstly, does this proof seem to work? Secondly, is there a way to make the proof more concise (e.g. a way to prove the statement without using all these indices)?