Let $C$ be the Banach space of complex continuous functions on $[0,1]$, with the supremum norm. Let $B$ be the closed unit ball of $C$. Show that there exist continuous linear functionals $\phi$ on $C$ for which $\phi(B)$ is an open subset of the complex plane; in particular $|\phi|$ attains no maximum on $B$.  I had tried it several times but I am not able to generate such a linear  continuous functional on the given space. Any type of help will be appreciated. Thanks in advance
Let $C$ be the Banach space of all complex continuous functions on $[0,1]$ endowed with the supremum norm. Let $B$ be the closed unit ball of $C$. Does there exist a $\Lambda\in C^*$ for which $\Lambda (B)$ is open? Note that $C$ is a locally convex space and $B$ is convex, balanced and closed set, by the separation theorem of convex sets,we know there exists $\Lambda\in C^*$ such that $|\Lambda f|\leq 1$ for all $f\in B$ and $|\Lambda f|&gt;1$ for $f\in B^c$. But how about this when we consider some concrete space above?
How do I run a game for a larger group? I am currently running a Dungeons and Dragons 4 campaign for a group of 9 players.  Whittling the group down to a more manageable size is not really an option since they are all family.  When we do combat it takes forever even if I have one player help me and allow them to go 2 at a time.  When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs.  I've tried encounters with many foes, encounters with only a few tough foes, dialog only encounters, social puzzle encounters, everything I can think of.  What else can I try to get this group more engaged? Travel alone on accompanied visa $f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$ How do I install ndiswrapper support for an unsupported RTL8723DE wireless module? Are the WotC archives of d20 Character Optimization forum forever lost? I was looking for the Wizard Handbook but all the links were broken.  Somewhere on the handbooks of other classes (on other forums) I saw a      This was archived by WotC, here's the link   But the link leads to the welcome page of WotC (since it's broken).  Is there a way to get all that info back? Do you know where to get it from? The Wayback Machine did not archive the forums. Eeeek! Code blocks are misbehaving! So code blocks in questions and answers on the android app (inline included) seem to be scared of me putting my finger in them. When I poke them, they only move over two columns!  (For clarity, these are the code blocks that overlap the TextArea view because they've been eating too much)  The intended behaviour here is that when I press my finger into a code block, the touch event should register a drag; at current, I try to scroll inside a code block and I end up switching to question or answer view instead, over and over.  EDIT:  Something tells me this is status-bydesign because the text is deliberately cut off (probably inviting you to click on the code block and expand it; UX Decision?): Difference between confidence intervals and prediction intervals For a prediction interval in linear regression you still use $\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ to generate the interval. You also use this to generate a confidence interval of $E[Y|x_0]$. What's the difference between the two? Are there any rules for adjudicating a ranged attack fired at a target engaged in melee combat with an ally? While running a session recently, one of the players chose to "assist" his companion who was in melee with a goblin, by firing an arrow at the pair of them. I couldn't find any particular rules for this scenario so just had him roll at a disadvantage, counting a miss as a hit on his companion (damage to be ascertained by a second roll).   This seemed to work okay, and became a running joke for the night as the arrow bounced off the companions back, but are there specific rules for this scenario? Python mechanize javascript \begin{eqnarray*} M&amp;=&amp; \max{\left \{a,b,\frac{a+b}{2},\right}\\ &amp;&amp; a+b,a+b/2\Bigg\} \end{eqnarray*} How to find the number of automorphisms on $\mathbb Z_2\times \mathbb Z_4$ Well, I was trying find out the number of group homomorphisms from $U(20)$ onto $U(15)$ which I reduce to the problem of finding the number of automorphisms for $\mathbb Z_2\times \mathbb Z_4$. But don't know how shall I proceed.   Any help will be appreciated. What is strain? Why does DNA coil upon itself? I have been reading Lehninger's Principles of Biochemistry to understand DNA supercoiling (pg-). Supercoiling is the twisting of a DNA upon itself and this happens when the two helically wound strands of DNA are either underwound or overwound. The book says that this act of underwinding or overwinding generates a structural strain  which is then accommodated by coiling of the DNA upon itself (supercoiling). Here the word strain has been used in the way stress of mechanical physics would be used. I wonder if it means stress. Besides I would like to know how underwinding or overwinding cause the DNA to coil upon itself. Excerpts from the book:  DNA supercoiling is generally a manifestation of structural strain. Most DNA are negatively supercoiled (which means that the supercoiling is left handed). In almost every instance, the strain is a result of underwinding of DNA double helix. Supercoiling is not a random process; the path of the supercoiling is largely prescribed by the torsional strain imparted to the DNA....  Torsional strain, again is it talking about ? P.S. The description of the question has been changed recently. Compute $ \int_{0}^{ \infty }\frac{\ln xdx}{x^{2}+ax+b}$ Is every complex (smooth) manifold a scheme? The question in the title doesn't quite make sense. I was always wondering if the scheme is the generalization of manifold. The precise statement should be like following:  If $X$ is a complex (smooth)manifold, is there a scheme $Y$ over $\mathbb{C}$ such that the corresponding analytic space $Y^{an}$ isomorphic to $X$?  I guess if $X$ is projective, then this is Chow's theorem. And I doubt the question has an affirmative answer if $X$ is non-compact (I guess that $X: y-e^x=0$ in $\mathbb{C}^2$ might be an example, but I do not know how to show this).  Edit 1:  There are also example (as @hunter pointed out) of a complex torus but not an abelian variety shows that a compact manifold might not be a variety, however, is it possible to be a scheme ?(according to Hartshorne, the category of varieties is a subcategory of quasi-projective, integral, separated scheme. But could there be some non-quasiprojective scheme($Y$ has to be integral, separated in our case according to GAGA, I guess) )?  Edit 2: Another question I think of is: is (Artin) Stack a generalization of manifold? I really want to know if manifold could be think of some subcategory of some category of with "algebraic objects". Can an observed event in fact be of zero probability? Can an observed event in fact be of zero probability?  Of course, I know that there exist non-empty events of zero probability. Mu question is the reverse: given that we have observed an event (and we have no other information about it, just the fact that it has been observed), is it possible that the event has in fact zero probability? Or does an observation necessarily mean that the probability is strictly positive?    Example context for the question:  Suppose $x_i$ (countably many) are i.i.d on $[0,1]$, but we do not know the distribution they come from. It may be uniform on $[0,1]$, may be discrete, may be any legitimate distribution (discrete or continuous). Just imagine we have some sort of a machine that shows us one by one a list of randomly drawn numbers between $0$ and $1$. We are comparing the observed numbers one by one to some special number chosen beforehand, for example $\frac12$. Now, given that at some iteration we have observed that special number at least once, does that mean that $\frac12$ has some positive probability under that (unknown) distribution? And if the probability can be zero, can we nevertheless say that we will necessarily observe $\frac12$ again later, if we continue the experiment ad infimum?  Also, disregard the "real world limitations" such as an inability to produce truly uniformly distibuted numbers, or rounding errors or any such thing. Function with limits only at irrational points This is the starred example 5.1.8 in Krantz' Real Analysis and Foundations.     Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x\rightarrow c} f(x)$ exists when $c$ is irrational but does not exist when $c$ is rational.   I'm aware of the Dirichlet and Thomae functions but this seems slightly different. I'm not even sure a solution exists. Thanks for any help! Tufte advocates to "minimize the ink-to-data ratio" for charts. I am trying to style a multi data series bar chart, that can have a grid and axes as shown below (chart at the left is from the Economist and on the right my attempt).      Here is the code:  \documentclass[justified]{tufte-book} \usepackage{pgfplots,lipsum} \pgfplotsset{compat=newest} \begin{document} \section{Bar Charts}  \lipsum[1-2]\vfill   \begin{figure*}[htbp]   \fbox{  \begin{tikzpicture}   \centering   \begin{axis}[         ybar,           title={Cumulative Progress of Works},         height=8cm,         width=15.5cm,         bar width=0.4cm,         ymajorgrids, tick align=inside,         enlarge  y limits={value=.2,auto}, % see the upper          ymin=0,         ymax=100,         axis x line=bottom,          axis y line=right,         %enlarge x limits =e,         legend style={at={(0.5,-0.2)},         anchor=north,legend columns=-1},         ylabel={Percentage (\%)},         symbolic x coords={            Sep-11,Oct-11,Nov-11,Dec-11,            Jan-12,Feb-12,            Mar-12,           Apr-12},        xtick=data,        nodes near coords,        every node near coord/.append style={         anchor=mid west,         rotate=70       }     ]     \addplot coordinates {       (Sep-11,75.4064)       (Oct-11, 72.7961)        (Nov-11,94.4597)       (Dec-11,66.6786)        (Jan-12,67.5600)        (Feb-12,88.2339)       (Mar-12,78.6138)        (Apr-12,58.9129) };    \addplot coordinates {       (Sep-11,75.4064)       (Oct-11, 89.7961)        (Nov-11,94.4597)       (Dec-11,76.6786)        (Jan-12,77.5600)        (Feb-12,78.2339)       (Mar-12,88.6138)        (Apr-12,78.9129) };    \addplot coordinates {       (Sep-11,75.4064)       (Oct-11, 89.7961)        (Nov-11,94.4597)       (Dec-11,76.6786)        (Jan-12,77.5600)        (Feb-12,78.2339)       (Mar-12,88.6138)        (Apr-12,78.9129) };      \legend{First Fix,Second Fix,Third Fix}   \end{axis}   \end{tikzpicture} }  \caption{\protect\raggedright Cumulative progress for all MEP works. Notice the slower rate of production during the last three months.} \end{figure*}  \lipsum[1-3] \end{document}   I have moved the y-axis to the right, but cannot get rid of the axis line. The plot is also not fitting within the plot width, despite using \enlarge.   Edit  And thanks to Jake the ugly duckling has been transformed to the following, with minor variations to the code as suggested by Jake. I have only added a stronger color for one of the bars in order to highlight it: Gauss's law in a uniform charge distribution extending infinitely in all directions Let us assume the universe filled with positive charge. About a particular point, all the positive charged particles will be symmetrical. Now consider a sphere of radius $r &lt; \infty$ and apply Gauss law. According to Gauss law the electric flux through the sphere will be some finite value as it contains some positive charge. But by symmetry the electric field through the sphere and hence electric flux will be zero. This is a contradiction. Preserve bash history in multiple terminal windows I consistently have more than one terminal open. Anywhere from two to ten, doing various bits and bobs. Now let's say I restart and open up another set of terminals. Some remember certain things, some forget.  I want a history that:   Remembers everything from every terminal Is instantly accessible from every terminal (eg if I ls in one, switch to another already-running terminal and then press up, ls shows up) Doesn't forget command if there are spaces at the front of the command.   Anything I can do to make bash work more like that? How to export / dump a MySql table into a text file including the field names (aka headers or column names)