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Characterizing all ring homomorphisms $C[0,1]\to\mathbb{R}$. This is something I've been trying to work out this evening.      Let $R$ be the ring of continuous real-valued functions on $[0,1]$ with pointwise addition and multiplication. For $t\in [0,1]$, the map $\phi_t\colon f\to f(t)$ is a ring homomorphism of $R$ to $\mathbb{R}$. I'm trying to show that every ring homomorphism of $R\to\mathbb{R}$ has this form.   Suppose otherwise, that there is some $\phi\neq\phi_t$, and thus there is some $f_t\in R$ such that $\phi(f_t)\neq \phi_t(f_t)=f_t(t)$. Define $g_t=f_t-\phi(f_t)1\in R$. Here $\phi(f_t)1$ is the constant function sending $[0,1]$ to $\phi(f_t)$. Then $g_t(t)\neq 0$. My first small question is why does $\phi(g_t)=0$? It seems only that $\phi(g_t)=\phi(f_t)-\phi(\phi(f_t)1)$.  I would like to conclude that there are only finitely many $t_i$ such that $g(x)=\sum g_{t_i}^2(x)\neq 0$ for all $x$. Then $g^{-1}=1/g(x)\in R$, but $\phi(g)=0$, contradicting the fact that homomorphisms map units to units. How can we be sure there are only finitely many $g_{t_i}$ such that the sum of their squares is never $0$? Thanks. Perhaps this is trivial: how does one prove that the work done by a homogeneous gravitational field on a system of particles is equal to the work done on a point mass with the total mass of the system, located at the center of mass of the system. Office 2016 Advanced Setup I just started the Office 2016 Setup/Installation on my Windows 10 PC. The installation worked fine and started automatically showing only this dialog:    But there are no additional settings possible! I need to choose which programs I want to install and also add additional tools that are often missing when doing a default installation.  How can I change the Office Installation Settings?   Choose programs to install Add/Remove Office Tools (Maybe) Office installation directory   When I check after the installation in the the windows settings, the Office can only be uninstalled - no modifications available: Convert list of ints to one number? I have a list of integers that I would like to convert to one number like:  numList = [1, 2, 3] num = magic(numList)  print num, type(num) &gt;&gt;&gt; 123, &lt;type 'int'&gt;   What is the best way to implement the magic function?  EDIT  I did find , but it seems like there has to be a better way. How I can create a graph of acceleration and speed - tikz? How can I create a graph of acceleration and speed like the one in the picture. Knowing that they are each other's consequence?    \documentclass{article} \usepackage{tikz}  \begin{document}  \begin{tikzpicture}               \end{tikzpicture} \end{document} How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$ Why Listen to Music, why not Listening Music How can I obfuscate (protect) JavaScript? What are some examples of awkward sounding but grammatically correct sentences? What are some examples of awkward sounding but grammatically correct sentences? Need help in the longtable with the figure Can u please help me in this table. The problem in this table that is not move to the next page in a proper way and kindly mention the size of the figure as well as the place of the figure. \begin{longtable}[c]{@{}|c|c|c|c|c|c|c|c|c|@{}} \caption{Bacteria isolates on different media} \label{tab:my-table}\\ \toprule \textbf{Sr.No} &amp;   \textbf{Reference Code} &amp;   \textbf{Sample Location} &amp;   \textbf{Colony Color} &amp;   \textbf{Colony Morphology} &amp;   \textbf{Pigmentation} &amp;   \textbf{Media} &amp;   \textbf{Predicted Name Of Strain} &amp;   \textbf{Figures} \\* \midrule \endhead % 1 &amp;   EMB(P)1 &amp;   Islamabad &amp;   Green sheen &amp;   Circular &amp;   Green Sheen &amp;   \begin{tabular}[c]{@{}c@{}}Eosin methylene\\    \\ blue agar\end{tabular} &amp;   \textit{\textbf{E.coil}} &amp;    \\* \midrule 2 &amp;   MSA(P)1 &amp;   Islamabad &amp;   Yellow &amp;   Punctiform &amp;   Yellow &amp;   Mannitol salt agar &amp;   \textit{\textbf{\begin{tabular}[c]{@{}c@{}}Staphylococcus\\    \\ Epidermidis\end{tabular}}} &amp;    \\* \midrule 3  &amp; MACC(P)1 &amp; Islamabad &amp; Off white       &amp; Circular  &amp; Off white       &amp; MacConkey agar     &amp; \textit{\textbf{Salmonella}}              &amp;  \\* \midrule 4  &amp; MSA(P)2  &amp; Islamabad &amp; White           &amp; Circular  &amp; White           &amp; Mannitol salt agar &amp; \textit{\textbf{Staphylococcus   aureus}} &amp;  \\* \midrule 5 &amp;   B.A(P)1 &amp;   Islamabad &amp;   White &amp;   Smooth &amp;   White &amp;   \begin{tabular}[c]{@{}c@{}}Blood \\    \\ Agar\end{tabular} &amp;   \textit{\textbf{Pseudomonas   aeruginosa}} &amp;    \\* \midrule 6  &amp; B.A(P)2  &amp; Islamabad &amp; White           &amp; Circular  &amp; White           &amp; Blood agar         &amp; \textit{\textbf{Lactobacilli}}            &amp;  \\* \midrule 7  &amp; MACC(P)2 &amp; Islamabad &amp; Pink            &amp; Circular  &amp; Pink            &amp; MacConkey agar     &amp; \textit{\textbf{Klebsiella}}              &amp;  \\* \midrule 8  &amp; MACC(P)3 &amp; Islamabad &amp; Purple          &amp; Irregular &amp; Purple          &amp; MacConkey agar     &amp; \textit{\textbf{pseudomonas}}             &amp;  \\* \midrule 9  &amp; MSA(P)3  &amp; Islamabad &amp; Crystal White   &amp; spherical &amp; Crystal White   &amp; Mannitol salt agar &amp; \textit{\textbf{Salmonella}}              &amp;  \\* \midrule 10 &amp; B.A(P)3  &amp; Islamabad &amp; White           &amp; Puntiform &amp; White           &amp; Blood agar         &amp; \textit{\textbf{Staphylococcus}}          &amp;  \\* \midrule 11 &amp; MACC(P)4 &amp; Islamabad &amp; Orange to amber &amp; Circular  &amp; Orange to amber &amp; MacConkey agar     &amp; \textit{\textbf{Shigella}}                &amp;  \\* \midrule 12 &amp; MSA(P)4  &amp; Islamabad &amp; Yellow          &amp; Circular  &amp; Yellow          &amp; Mannitol salt agar &amp; \textit{\textbf{S. aureus}}               &amp;  \\* \midrule 13 &amp; MACC(P)5 &amp; Islamabad &amp; Pink to purple  &amp; Circular  &amp; Pink to purple  &amp; MacConkey          &amp; \textit{\textbf{Klebisella}}              &amp;  \\* \bottomrule \end{longtable} I have a 2x2 experimental design. In the experiment, I also collected the participants' professional qualifications (categorical variable- yes /no). I would like to test the effect of controlling this variable on the MSE. I understand that in ANCOVA, the covariate is either an interval or ordinal variable. So should I refer to the test as an ANCOVA if I include professional qualification as an additional variable? Or should it be a blocking variable since &quot;prof. qualification&quot; is categorical. What's the recommended max width for the main container and for the main column, for large screens? Based on this question,  and its comments and answers, I did some research, but I didn't find a complete answer.   For me it is obvious that it is a good decision to limit the width of the main container, but what is the recommended max-width for the main container for large screens, in 2017, when the screens become larger and larger? What variables should be considered in this decision?  Also, let's say a website has multiple columns in the main container, what is the max-width for the main column?  Update: I have now a related question. I recommended to one client to reduce the width of the main container on the homepage, but I can't convince him. It is a website with photos, like pexels.com. It is in this case important to limit the width if the elements are not one more important than other? Unintentional mouse movements, am I paranoid? Model Selection: Logistic Regression Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$. CPU Speed stuck at 0.78 GHz, How do I increase it to its base speed of 2.30 GHz? My 1yr old laptop, HP Pavilion 15 has it's CPU, Intel i5-8300H stuck at 0.78 GHz whenever it's plugged and charging, and the moment I remove the charger plug or switch it off, it increases suddenly and keep fluctuating freely, with good utilisation percentage more than 1-2GHz or more as I experience Thermal Throttling. I tried sneaking into the BIOS to see if I can do anything, but found nothing. I also downloaded Intel Extreme Tuning Utility (XTU) but couldn't control it. My question is how do I increase the speed to say atleast 2 GHz (fixed, not fluctuating) when it's charging, irrespective of getting thermal throttle, and also maybe even control the fan speed if it does get hot? Recursive search in nautilus 3.6 completely breaks my keyboard navigation. I often navigate network volumes, and typing to get directly to a folder and clicking enter was very quick. Now typing removes everything in the view and imposes a 30 second delay which is completely useless.  If it showed whats in the current directory immediately, and then searched it would be fine, but as it is now I just want to turn it off. I couldn't find anything in dconf-settings. Anyone know how to disable it? Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n=2,3,\dots$. Define $$P_n=\left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right)\dots\left(1+\frac{1}{a_n}\right)$$Find $$\lim_{n \to \infty}P_n$$   Trial: $$1+\frac{1}{a_n}=1+\frac{1}{n(a_{n-1}+1)}\;.$$ But I can't simplyfy.Please help. Why are Perl 5's function prototypes bad? The real meaning of time dilation