Suppose $A$ is a square matrix (over $\mathbb{R}$ or $\mathbb{C}$, take your pick) such that $(I-A)^{-1}$ exists. Then is it necessarily true that  $$I + A + A^2 + \dots + A^n + \dots = (I-A)^{-1}$$ ?  There is a well known theorem which claims that if $||A|| \lt 1$ then $(I-A)^{-1}$ exists and the above is true.   What I am asking is something like the converse of that theorem. My hunch is that this is false, but I am unable to come up with a counter-example.
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm &lt; $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.   Thinking in terms of a converse, if $T$ is any bounded linear operator defined on $X$, then does the existence of a bounded inverse $S=(I-T)^{-1}$ imply that $S$ can be represented as $S=\sum_{n=0}^\infty T^n$?
show all my unanswered questions from SE Any way how to get    my all un-answered questions  from across all SE sites? Ubuntu on a USB stick - boot in both BIOS and UEFI modes How is the electostatic field propagated by the vacuum? How does a charge feel the presence of another charge when there is NOTHING between them? Is the word "vacuum" equal to "nothing"? Imagine two charges of opposite sign, far from one another in vacuum, and moving with constant velocity - s.t. none radiates energy, none emits photons. At some time though, the course of their movement brings them closer. They begin to accelerate toward one another. How do they feel the presence of one another? During their constant movement no photons are emitted, no "information" spreads in the universe about their presence. How do I find out which program and process ID accesses a given IP address in Windows? Is it possible to catch, which program is accessing a specific IP address?   I found my computer is flooding and wish to check addresses one by one.  May be it is possible to set up some audit for this? Using AMPScript, I want to pass from a URL Attribute lookup variables and use these variables in my LookupOrderRows function.   The challenge is that not all lookup variables have to be passed by the URL e.g I might lookup just 'office' or I could also lookup both 'office' and 'score.   I've tried several approaches of Concat the string etc - but can't see to make the lookup work.  Has anyone had any success performing similar?   IF not empty( Requestparameter("office") ) then  Set @field1_name = "OFFICE" Set @field1_value = Requestparameter("office")  EndIf  IF not empty( Requestparameter("specialism") ) then  Set @field2_name = "SPECIALISM" Set @field2_value = Requestparameter("specialism")  EndIf  IF not empty( Requestparameter("score") ) then  Set @field3_name = "SCORE" Set @field3_value = Requestparameter("score")  EndIf   set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc",@Field1_Name,@Field1_Value,@Field2_Name,@Field2_Value,@Field3_Name,@Field3_Value)   My current solution is a lot If then statements and combinations i.e. :  /* OFFICE ONLY*/ IF NOT EMPTY(@input_office) AND EMPTY(@input_organisation) AND EMPTY(@input_specialism)  AND EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office) ENDIF  /* OFFICE + ORG*/ IF NOT EMPTY(@input_office) AND NOT EMPTY(@input_organisation) AND EMPTY(@input_specialism)  AND EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"ORGANISATION_NAME",@input_organisation) ENDIF /* OFFICE + ORG + SPECIALISM */ IF NOT EMPTY(@input_office) AND NOT EMPTY(@input_organisation) AND NOT EMPTY(@input_specialism)  AND EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"ORGANISATION_NAME",@input_organisation,"SPECIALISM",@input_specialism) ENDIF /* OFFICE + ORG + SCORE */ IF NOT EMPTY(@input_office) AND NOT EMPTY(@input_organisation) AND EMPTY(@input_specialism)  AND NOT EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"ORGANISATION_NAME",@input_organisation,"Engagement Score",@input_score) ENDIF /* OFFICE + ORG + SPECIALISM + SCORE  */ IF NOT EMPTY(@input_office) AND NOT EMPTY(@input_organisation) AND NOT EMPTY(@input_specialism)  AND EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"ORGANISATION_NAME",@input_organisation,"SPECIALISM",@input_specialism,"Engagement Score",@input_score) ENDIF  IF NOT EMPTY(@input_office) AND EMPTY(@input_organisation) AND NOT EMPTY(@input_specialism)  AND EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"SPECIALISM",@input_specialism) ENDIF  IF NOT EMPTY(@input_office) AND EMPTY(@input_organisation) AND NOT EMPTY(@input_specialism)  AND NOT EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"SPECIALISM",@input_specialism,"Engagement Score",@input_score) ENDIF  IF NOT EMPTY(@input_office) AND EMPTY(@input_organisation) AND EMPTY(@input_specialism)  AND NOT EMPTY(@input_score) then     set @rows = LookupOrderedRows(@LeadsDE,0,"[Engagement Activity Date] desc","OFFICE",@input_office,"Engagement Score",@input_score) ENDIF Does it matter what I choose for serialVersionUID when extending Serializable classes in Java? Which tool for fixing wobbly crank? Any idea what kind of tool should I but to fix the loose left crank on my bike? I found this  but not sure if it's gonna suit. Or is it enough to buy only nut driver? Ruby block and unparenthesized arguments I extracted simple example:  require 'pp' x = 1..3 pp x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end pp x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} }  pp(x.map do |i| {:value =&gt; i, :double =&gt; (i*2)} end) pp(x.map { |i| {:value =&gt; i, :double =&gt; (i*2)} })   I am wondering why first pp produces:  [1, 2, 3]   While all the oders are giving:  [{:value=&gt;1, :double=&gt;2}, {:value=&gt;2, :double=&gt;4}, {:value=&gt;3, :double=&gt;6}]   I assume it has something to do with operator precedence. Where can I find good explanation? How to get Intel Dual Band Wireless-AC 3168 work on Ubuntu 16.04? Please help me to prove the following result:  I recall that the holomorph $Hol(\Gamma)$ of a group $\Gamma$ is the semidirect product $Hol(\Gamma)=\Gamma\ltimes_{\phi} Aut(\Gamma)$ where $\phi: Aut(\Gamma)\to  Aut(\Gamma)$ is the identity map.  Let $G=Hol(C_8)$. In addition let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $8$, $L$ a splitting field of $f(x)$ over $\mathbb{Q}$ and suppose that $Gal(L/\mathbb{Q})\cong G$     Show that there is a subgroup of $S_8$ isomorphic to $G$   Thanks in advance. PHP regex delimiter, what's the point? Beamer - place text of footnote in footline I must place footnotes and footcites at the bottom of the frames, like shown in the picture:   The line above the footnotes must always be there, and it must have a minimum distance to the bottom of lets say 2.5em.  So I see two possible solutions:   Add the footnote via \setbeamertemplate to the footline. Change the format of the footnoteline and let it always appear, even if no footnote was used.   For the last hours, I tried to accomplish point 1, but I had absolutely no success. Can you please help me? Otherwise I have to use *whisper* Powerpoint. ;)    EDIT: Do you know the phenomenon that only after you ask someone, you can get the right ideas? ;)  Here is a MWE of my current solution:  \documentclass[xcolor={dvipsnames}]{beamer}  \usepackage[absolute, overlay]{textpos} \usepackage{environ}   \usetheme{default} \setbeamertemplate{navigation symbols}{}%remove navigation symbols   \newcommand{\makemyrule}{%     \setlength{\unitlength}{14pt}%      minimum distance to bottom     \ifdim\footnoteheight&gt;\unitlength \setlength{\unitlength}{\footnoteheight} \fi%     check which distance is bigger     \begin{textblock*}{\linewidth}(0pt,\dimexpr\textheight-\unitlength)%    place with use of textpos         \textcolor{blue}{\rule{\paperwidth}{.3ex}}     \end{textblock*} }   \let\oldfootnote\footnote%      redefine the footnote command \renewcommand{\footnote}[1]{\Addheight{#1}\oldfootnote{#1}}  \newlength{\footnoteheight} \newsavebox{\mytext} \newcommand{\Addheight}[1]%     measure height of the text         {%                 \savebox{\mytext}{\small\strut\parbox{\textwidth}%                 {\hspace*{2em}\thefootnote#1}\strut}%       approximated by eye, not perfect!                 \addtolength{\footnoteheight}{\ht\mytext}%                 \addtolength{\footnoteheight}{\dp\mytext}%         }     \NewEnviron{myframe}[1]{%     \begin{frame}{#1}%     \setlength{\footnoteheight}{0pt}%     \BODY%     \makemyrule%     \end{frame}%     }    \renewcommand\footnoterule{}  \begin{document} \begin{myframe}{} \frametitle{Beamer example} using\footnote{Y LUSPIN, et al} footnotes\footnote{P. GHOSEZ, et al and a lot more information to create a linebreak} excessively \footnote{You do not need to read this, this is only blindtext to create even more and more text. Blindtext will go on forever, nobody can stop the allmighty blindtext. I think I am a blindtext author.} \end{myframe}{} \begin{myframe} \frametitle{Beamer example} using\footnote{Y LUSPIN, et al} footnotes\footnote{P. GHOSEZ, et al} \end{myframe}{} \begin{myframe} \frametitle{Another example} without footnotes\\ \end{myframe} \end{document}   What this does: At the beginning of each myframe-environment, the length \footnoteheight is reseted to zero. When a \footnote is placed, the height of the text is added to this length. At the end of each frame, via \makemyrule the bottom rule is created, which is above the footnote texts (my solution is not perfect) or, if there are no footnotes, at least 14pt above the bottom.  My wishes (not real problems, but I want it to work better) are   The height of the footnotetext is only approximated. I would like a solution where you can measure the height of the formatted footnote. The frame environment needs to be renamed to myframe. I would like a solution which works without changing the document. But it all failed, because I could not execute \makemyrule before \end{frame} automatically. To make your opinion known on MSE you need both up and downvotes. Given at the moment you need 125 rep to downvote on this site, the voting patterns will be skewed positive because not everyone is active on the site.   This is to be expected for regular Stack Exchange sites as you are expected to be involved in the community before you are deemed knowledgeable enough to be part of the community voice. However, MSE is different because you are voting for the entirety of the Stack Exchange network and you shouldn't need to be active on MSE itself to have a voice on changes to the entire network.  This is especially important for questions like  because it will effect every network to do with coding at all. This was brought up in .  I would suggest reducing the rep required to downvote on MSE down to 100 (or 101) rep. This is the amount you get for the site association bonus. This is given once you achieve 200 rep on any network site and shows that you are active in at least one community. I think this would be the perfect use of the association bonus in showing that you are a member of the Stack Exchange community. Is it unethical to cite a paper or book that you have never looked at? Suppose I see that Book X is cited by several papers with reference to some particular Fact Y. I wish to cite Fact Y. As far as I know, Book X is the only reference for Fact Y.   I have never looked at Book X. Nonetheless, I cite it like this:  'Fact Y' (see Book X)  My question is: Is this ethical, given that I've never looked at Book X?  More context for my question. I am NOT asking whether this is good scientific practice. I am NOT asking how a good scientist should actually go about citing Fact Y. I am NOT asking if this could undermine your credibility as a scientist.  Instead, I am asking this question because it seems to be a somewhat-common practice. And moreover, as far as I know, no scientist has ever been issued even an official rebuke (much less fired) for engaging in such practice. This would suggest to me that this practice is not considered to be unethical.   Of course, it is not binary as to whether a practice is unethical. But I would simply like to know whether the academic community in general considers this to be even a mildly unethical practice. Or if it is perhaps very slightly unethical, but not a big deal. Or if it is not in the least bit unethical. 403 error after upgrading to apache2.4 After upgrading from Ubuntu 13.04 server, the www pages won't display themselves.     Error 403 Forbidden You don't have permission to access / on this server.   I've changed access control saves from apache2.2 to apache2.4, but that didn't work. What else could I do? Graph for relationship between two ordinal variables What is an appropriate graph to illustrate the relationship between two ordinal variables?  A few options I can think of:   Scatter plot with added random jitter to stop points hiding each other. Apparently a standard graphic - Minitab calls this an "individual values plot". In my opinion it may be misleading as it visually encourages a kind of linear interpolation between ordinal levels, as if the data were from an interval scale. Scatter plot adapted so that size (area) of point represents frequency of that combination of levels, rather than drawing one point for each sampling unit. I have occasionally seen such plots in practice. They can be hard to read, but the points lie on a regularly-spaced lattice which somewhat overcomes the criticism of the jittered scatter plot that it visually "intervalises" the data. Particularly if one of the variables is treated as dependent, a box plot grouped by the levels of the independent variable. Likely to look terrible if the number of levels of the dependent variable is not sufficiently high (very "flat" with missing whiskers or even worse collapsed quartiles which makes visual identification of median impossible), but at least draws attention to median and quartiles which are relevant descriptive statistics for an ordinal variable. Table of values or blank grid of cells with heat map to indicate frequency. Visually different but conceptually similar to the scatter plot with point area showing frequency.   Are there other ideas, or thoughts on which plots are preferable? Are there any fields of research in which certain ordinal-vs-ordinal plots are regarded as standard? (I seem to recall frequency heatmap being widespread in genomics but suspect that is more often for nominal-vs-nominal.) Suggestions for a good standard reference would also be very welcome, I am guessing something from Agresti.  If anyone wants to illustrate with a plot, R code for bogus sample data follows.  "How important is exercise to you?" 1 = not at all important, 2 = somewhat unimportant, 3 = neither important nor unimportant, 4 = somewhat important, 5 = very important.  "How regularly do you take a run of 10 minutes or longer?" 1 = never, 2 = less than once per fortnight, 3 = once every one or two weeks, 4 = two or three times per week, 5 = four or more times per week.  If it  would be natural to treat "often" as a dependent variable and "importance" as an independent variable, if a plot distinguishes between the two.  importance &lt;- rep(1:5, times = c(30, 42, 75, 93, 60)) often &lt;- c(rep(1:5, times = c(15, 07, 04, 03, 01)), #n=30, importance 1            rep(1:5, times = c(10, 14, 12, 03, 03)), #n=42, importance 2            rep(1:5, times = c(12, 23, 20, 13, 07)), #n=75, importance 3            rep(1:5, times = c(16, 14, 20, 30, 13)), #n=93, importance 4            rep(1:5, times = c(12, 06, 11, 17, 14))) #n=60, importance 5 running.df &lt;- data.frame(importance, often) cor.test(often, importance, method = "kendall") #positive concordance plot(running.df) #currently useless   A related question for continuous variables I found helpful, maybe a useful starting point: Attributive nouns vs. of-genitive If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges? If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges?  I want to use this fact in another problem but don't immediately see how to prove it. The Time That 2 Masses Will Collide Due To Newtonian Gravity How do I resize an encrypted LVM to install another copy of Ubuntu? I have an old PowerBook which has been happily running Ubuntu Server for a few months. Now I want to repartition the hard drive and install Ubuntu Client, while keeping my existing installation intact.  When I first installed Ubuntu, I chose to erase the entire hard drive and use it as an encrypted LVM. Now, the installer seems to recognize the partitions on the drive, but it never prompts me for the password to the LVM.  How do I make room for another installation alongside my current one?