I know that an ordinal is regular if its equal to is cofinality, but how is that related to cardinal?
In the proof of the lemma     For every limit ordinal $\alpha$, $\textrm{cf } \alpha$ is a regular cardinal    Jech says that     It is easy to see that if $\alpha$ is not a cardinal, then using a mapping of $|\alpha|$ onto $\alpha$ one can construct a cofinal sequence in $\alpha$ of length $\leq |\alpha|$, and therefore $\textrm{cf } \alpha &lt; \alpha$.    I've spent a while trying to construct this cofinal sequence and was wondering if I was on the right path:   From a bijection $f$ between $\kappa = |\alpha|$ and $\alpha$ take $S = \{\beta \in \kappa : \ f(\gamma) &lt; f(\beta)$ for all $\gamma &lt; \beta\}$ then if $\xi$ is the order type of $S$ and $g$ is the isomorphism from $\xi$ to $S$ the function $f\circ g$ is a cofinal sequence in $\alpha$ and $\xi &lt; \alpha$.   Is this on the right track, or is there an easier way of doing it?
Dependency error while installing WINE Every time I try to install "WINE" VIA the Software Center on Ubuntu 12.10, I get this error:  The following packages have unmet dependencies:  wine1.4: PreDepends: dpkg (&gt;= 1.15.7.2~) but 1.16.7ubuntu6 is to be installed          Depends: libc6 (&gt;= 2.14) but 2.15-0ubuntu20 is to be installed          Depends: wine1.4-amd64 (= 1.4.1-0ubuntu1) but 1.4.1-0ubuntu1 is to be installed          Depends: wine1.4-i386 (= 1.4.1-0ubuntu1) but it is not going to be installed   When I try installing it VIA the terminal, this is what I get:  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  wine1.4 : Depends: wine1.4-i386 (= 1.4.1-0ubuntu1) but it is not installable            Recommends: gnome-exe-thumbnailer but it is not going to be installed or                        kde-runtime but it is not going to be installed            Recommends: ttf-droid            Recommends: ttf-mscorefonts-installer but it is not going to be installed            Recommends: ttf-umefont but it is not going to be installed            Recommends: ttf-unfonts-core but it is not going to be installed            Recommends: winbind but it is not going to be installed            Recommends: winetricks but it is not going to be installed E: Unable to correct problems, you have held broken packages.   how do I fix this? Asymptotic behavior of a trigonometric series Consider the following series $$S_N = \sum_{k=1}^N \frac{\sin[k\pi/(N+1)]}{k}.$$ What is the asymptotic behavior of this series for $N \to \infty$?  If we replace the sine function with the factor $1$, we get the standard harmonic series which diverges $\sim \ln N$. However, if we replace the sine function with $(-1)^k$, we get a convergent series with the result $\ln 2$. Therefore, I think that $S_N$, defined above, should diverge, but more slowly than the logarithm. Am I correct? I have looked everywhere for lava or obsidian in 0.7.1. I've looked all the way from top to bottom and all around, in about a 100 block diameter, but I can't find any. Where can I get some? Getting around the FAT32 4GB file size limit Difference between "does have" and "has" Possible Duplicate:                 What is the difference between does have and has? For example, compare she does have a car and she has a car. Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| &gt; 1$.  There has been a similar question on MSE here:  Unfortunately, I cannot comment on other user's questions yet, so I made my own question.  I know that the trick is to use conjugation but I can't follow the logic. Can someone elaborate? Especially on the last part of the top answer. Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$ This is a question from Stromberg related to Steinhaus' Theorem:     If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.   I can't quite see how to modify the Steinhaus proof though. If a function can only be defined implicitly does it have to be multivalued? This question is about cosmology and general relativity. I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.   Does it have infinite mass or is it dishomogeneous?  How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all? How does quantization solve the ultraviolet catastrophe? Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? How to install mysql? Create Java console inside a GUI panel How can I create an instance of the Java console inside of a GUI panel? Twitter Bootstrap uses these break points for responsive design:  Large Display: 1200px Default: 980px Tablet: 768px Mobile: 480px  A client said they are using 1280px for their Large Display breakpoint.  Is there an industry standard for these or are they just made up as we go right now? What are the barriers to understanding pointers and what can be done to overcome them? Why are pointers such a leading factor of confusion for many new, and even old, college level students in C or C++?  Are there any tools or thought processes that helped you understand how pointers work at the variable, function, and beyond level?  What are some good practice things that can be done to bring somebody to the level of, "Ah-hah, I got it," without getting them bogged down in the overall concept? Basically, drill like scenarios. Meaning of Conditional sentences with 'should' and 'happen to' I'm a bit familarized with conditional sentences. But the following sentences with 'should' and 'happen to' are quite confusing too me. They are:   If it should rain, I will stay at home. 2.If it should happen to rain, I'll stay at home   If it happens to rain, I'll stay at home.    What are the meaings of those sentences ? Do they all mean the same ? Thank you a lot in Advance for your answer. Equicontinuity on a compact metric space turns pointwise to uniform convergence I know that   If $\{f_n\}$ is an equicontinuous sequence,  defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly.  I'm having trouble proving this.  I see the same problem  but am having trouble following the proof, particularly with part (3).  Can someone guide me through a proof of this result? How do I get current date/time on the Windows command line in a suitable format for usage in a file/folder name? Update: Now that it's 2016 I'd use PowerShell for this unless there's a really compelling backwards-compatible reason for it, particularly because of the regional settings issue with using date. See @npocmaka's     What's a Windows command line statement(s) I can use to get the current datetime in a format that I can put into a filename?  I want to have a .bat file that zips up a directory into an archive with the current date and time as part of the name, for example, Code_2008-10-14_2257.zip. Is there any easy way I can do this, independent of the regional settings of the machine?  I don't really mind about the date format, ideally it'd be yyyy-mm-dd, but anything simple is fine.  So far I've got this, which on my machine gives me Tue_10_14_2008_230050_91:  rem Get the datetime in a format that can go in a filename. set _my_datetime=%date%_%time% set _my_datetime=%_my_datetime: =_% set _my_datetime=%_my_datetime::=% set _my_datetime=%_my_datetime:/=_% set _my_datetime=%_my_datetime:.=_%  rem Now use the timestamp by in a new ZIP file name. "d:\Program Files\7-Zip\7z.exe" a -r Code_%_my_datetime%.zip Code   I can live with this, but it seems a bit clunky. Ideally it'd be briefer and have the format mentioned earlier.  I'm using Windows Server 2003 and Windows&nbsp;XP Professional. I don't want to install additional utilities to achieve this (although I realise there are some that will do nice date formatting). How does Linux kernel compare to microkernel architectures? I read once that one advantage of a microkernel architecture is that you can stop/start essential services like networking and filesystems, without needing to restart the whole system. But considering that Linux kernel nowadays (was it always the case?) offers the option to use modules to achieve the same effect, what are the (remaining) advantages of a microkernel?