could you help me with this problem, please?   If $U$ is a principal ultrafilter on $I$ such that $\{a\}\in U$. Show that $Ult(\mathfrak{A}_x:x\in I)$ is isomorphic to $\mathfrak{A}_a$ and $[f]=f(a)$ for each $f$ and $j$, the embedding is the identity.
Show that:  If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
How do I concatenate JavaScript files into one file? I'm trying to typeset a table that would have a header in the first row (I'm not looking for a different/better solution to adding a header, but for a fix to this solution. Thanks). I want to achieve this by having a multicolumn cell spanning all the columns of the first row. The problem is that if the text is too long, the column widths are uneven, with the last column accumulating excess width. I would expect all of the columns to be of approximately the same width (although I cannot find a decent source that would say they should), given the fact that all the text in the second row is of approximately the same width.  I find this behaviour quite puzzling and have no idea what the cause could be. I know I could perhaps remedy this by manually setting the column widths, but am curious as to how to fix this without doing that (I do not need or indeed want the columns to be of the same width, but roughly proportional to the length of their content), preferably but not necessarily by using the tabular environment only (some other conflict with the document class I am using).   What is it that I'm doing or assuming wrong? And why?  MWE:  \documentclass{article} \begin{document}  \begin{tabular}{ccccc} \multicolumn{5}{c}{This is a piece of text. The longer it is, the longer the last column.} \\  1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\ \hline \end{tabular}  \end{document}   Resulting table. Notice how the last column of the second row accumulates excess width. I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$ How can I show that the order of an element modulo $n$ divides $\phi(n)$?  I know that if $a$ and $n$ are relatively prime, then the least positive integer $x$ such that $a^x\equiv1\pmod n$ is its order modulo $n$. I also know that, by Euler's theorem, $a^{\phi(n)}\equiv1\pmod n$. Therefore, it must be the case that $x\leq\phi(n)$.  However, all that I am left to do is to show that $kx=\phi(n)$, for some integer $k$. Do you guys have an idea on how to do this? Thanks in advance! Add a way for moderators to cast a normal, non binding close/open vote I think moderators should have the ability to cast a normal, non binding close and open vote like they were a normal user (while of course retaining their ability to cast a binding vote where necessary). This can be used in &quot;grey areas&quot; where a moderator can choose to give his or her opinion, but not make a decision alone; a good example of this is . This also helps in areas not so grey (questions that everyone would agree should be closed) since the OP has the time to see close (1) and possibly edit his or her question. The moderator can then optionally come back after a few minutes and close it for real if it's not improved (or maybe during that time other 4 people voted to close it). This would also fit perfectly in :  Since we’re about to add community moderators on both Stack Overflow and Server Fault, I need to document what it is, exactly, we expect moderators to do. The short answer is, as little as possible! Drawing Day and Night on a Google Map I'm looking to plot day/night on a Google map, for an arbitrary point in time. I'm familiar with generating map tiles; I'm just looking for an algorithm to tell me whether a particular point on the globe is currently in daylight or darkness, or to otherwise plot the curve of the day/night interface onto the map.  I've done some searching, but it's possible I don't even know enough about the problem domain here to know what terms to search for!  Any ideas? Doesn't have to be perfect -- basically, I'm comparing Flickr geolocation data of sunrise and sunset photos (and their "date taken" timestamps) with reality, and this is to help me visualise it. When are infinities okay? I'm a little confused about how to handle infinities in physics. On the one hand, I always here things like infinities are not physically tenable answers. GR can't answer anything about the origin of the universe because of singularities. Throw away the radial solution that gives an infinite potential when solving the Laplace equation in spherical coordinates. But then there are other times that the infinities are okay. For instance, it is usually okay to set the reference point for your potential at infinity. There are certain times in David Tong's quantum field theory lecture notes where he seems to almost to hand wave away infinities. So, philosophically, physically, and mathematically when are infinities okay, and when are they not okay? Atheros AR9462 WiFI very unstable & package loss I'm having severe issues w/ package-loss and a generally bad connection w/in a WiFi-network at my uni. I'm usually encountering a package loss of ~50% and aborting connections which makes working w/ SaaS impossible.  I'm using Ubuntu 13.04 and my network-adapter is an Atheros AR9462.  Applying the nohwcrypt=1-option doesn't change anything.  Also very noticable: The connection works as intended in other networks; the problematic network uses WPA2 w/ MSCHAPv2.  Anyone can help my out here?  $ sudo lshw -class network  *-network UNCLAIMED             description: Ethernet controller        product: NetLink BCM57780 Gigabit Ethernet PCIe        vendor: Broadcom Corporation        physical id: 0        bus info: pci@0000:04:00.0        version: 01        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list        configuration: latency=0        resources: memory:d3500000-d350ffff   *-network        description: Wireless interface        product: AR9462 Wireless Network Adapter        vendor: Atheros Communications Inc.        physical id: 0        bus info: pci@0000:09:00.0        logical name: wlan0        version: 01        serial: e0:06:e6:76:ba:89        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list rom ethernet physical wireless        configuration: broadcast=yes driver=ath9k driverversion=3.8.0-22-generic firmware=N/A ip=131.173.74.130 latency=0 link=no multicast=yes wireless=IEEE 802.11abgn        resources: irq:19 memory:d3400000-d347ffff memory:9fb00000-9fb0ffff I've got an HP Pavilion Touchsmart notebook that came with Windows 8. I've upgraded to 8.1, and now I want to dual-boot an Ubuntu distribution with Windows 10. When dual-booting with Windows 8.1, the Windows Boot Manager always took priority, even after installing GRUB. What do I need to do to get GRUB to be the default bootloader instead of Windows Boot Manager? How do I disable the profiles button in Chrome? Since Chrome version 38, there's a pretty ugly button in the title bar of the Chrome window that allows you to switch profiles quickly.  Since I'm the only one that uses my computer, this is not really useful to me, and I'd rather reclaim the space for more open tabs.  I've looked in the settings but don't see a way to disable it.  How can I get rid of the button? Securing WMS against unauthorized access? In a project the customer wants the images of the WMS to be secured against unauthorized access.   What are the common solutions to realize that?   Are they compatible with the OGC-standards? How do I remove Steam? Anyone know of a way to remove Steam from Ubuntu 12.04? Looked around the net but could not find an straight forward solution.    Thanks. Is there a Google Drive client available? Use $\delta-\epsilon$ to show that $\lim_{n\to\infty} a^{\frac{1}{n}} = 1$? Hope this is a meaningful question, but I'm curious if is possible to show that   $$\lim_{n\to\infty} a^{\frac{1}{n}}=1, \text{where  }a&gt;0$$  using $\delta-\epsilon$ directly or other methods. One method that I am aware of is to use the following:     If $\{s_n\}$ is a nonzero sequence, then $\liminf\bigl|\frac{s_{n+1}}{s_n}\bigr|\le \liminf |s_n|^{\frac{1}{n}}\le \limsup |s_n|^{\frac{1}{n}}\le\limsup\bigl|\frac{s_{n+1}}{s_n}\bigr|$ Automorphism of rational numbers Let $\mathbb Q$ be group of rational numbers under addition.  Then I want to find out the automorphism group of $\mathbb Q$. In a book it is given that Aut($\mathbb Q$) is isomorphic to $\mathbb Q^*$, the  multiplicative group of rational numbers. I stuck in this. How do I install pdfedit? data & synchronization - manually sync mail. calender and contacts $\max(\{n: n+5 | n^5 + 5\})$ What is the maximum value of $n$ if $n + 5$ divides $n^5 + 5$, $n$ being a natural number? I tried to solve by binomial theorem but it failed. Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$ I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. London Heathrow airport - do I need a transit visa?