Why are my Regular Solids Hollow Sometimes?
Can I "fill" an empty object?
Order of an element in a finite cyclic group Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if:  1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive,  2) $\mathcal{C}$ has a zero object,  3) $\mathcal{C}$ has binary products,  4) $\mathcal{C}$ has kernels and cokernels.  A category $\mathcal{C}$ is abelian if it is preabelian and satisfies:  5) every monomorphism is a kernel and every epimorphism is a cokernel.  Define the coimage of a map to be the cokernel of its kernel, and the image to be the kernel of its cokernel. We have the following commutative diagram:    where $\overline{f}$ is the only existing map (because of universality of kernel and cokernel).  I'm having trouble proving the following:     A preabelian category $\mathcal{C}$ is abelian iff $\overline{f}$ is an isomorphism.   The converse is easily shown, I'm having trouble proving $\Rightarrow$... Prove that every terms of a sequence defined by a recurrence relation is a perfect square. The three files are read in this order?  .bash_profile .profile .bashrc   This is not happening when I first open a terminal.   I have trace statements in the files appending to the file init.log. Please take a look at the following. It begins after opening a terminal. I have placed a comment to show where the log picks up after the su command.  stephen@debian:~$ cat init.log reading .bashrc done reading .bashrc stephen@debian:~$ su - stephen Password:  stephen@debian:~$ cat init.log reading .bashrc done reading .bashrc # # after su # reading .bash_profile reading .profile reading .bashrc done reading .bashrc done reading .profile done reading .bash_profile stephen@debian:~$    So the su - login triggers the expected sequence however the initial login reads only the bashrc. This cannot be correct. Can someone explain under what conditions this would occur. I could modify the bashrc and profile files so that the initial read includes all expected files but I would rather get to the root of the problem and fix it there. How to attach my repo to heroku app Is Tolkien prejudiced against the East? iCloud says Password is Wrong Even Though I Know It's Right iCloud says that my password is wrong - even though I know it's right.  Here's whats happened: Yesterday, I reset my iCloud password. Got the reset email, changed password. Tried to log in to the account, didn't accept it. I thought that maybe my finger had slipped typing in the password (twice). I called support, got the reset email.  Now, here's the really, really weird thing: I wanted to make sure I got the password right - so I typed my password into TextEdit, copied it into the reset fields and reset the password. After reseting it, I immediately went to icloud.com and tried to log in using the copied password from TextEdit. The passwords are the same. iCloud won't let me in. Does the order of declaration matter in Java/C#? In C++ I can't use a method if I declare it after the calling method.    Does this order matter in other languages like Java or C#? Why disable Fast Boot on Windows 8 when having dual booting? Why is everybody keep mentioning to disable Fast Boot on Windows 8 if you have installed alongside Ubuntu? Is it something that is recommended only for UEFI machines or is it a suggestion for Legacy BIOS machines as well? Is it because it makes the Windows partition inaccessible from Linux or there is another more serious reason to disable it? is there a way to add hyperlinks in a Illustrator document? I am creating a CV in Illustrator and exporting it to a PDF. I just read an article which stated that it is a good idea that we use hyperlinks in the CV so that the employer can just click and not have to copy the address. I like this idea, but I don't know how to add a hyperlink to text in Illustrator. Is it possible? Finding a set of continuous functions with a certain property 2 I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:  $$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$  I think it might be the set of constant functions but i havent been able to prove it :(   I was thinking that you can use the stone weiestrass theorem considering the set of continuous functions on a closed  interval(non trivial) ,and a subset which contains the set of continuous functions whose integral above diverges in some point in that interval along with  with the set of constant functions. So in order to solve the problem i need only to prove that if two functions do not meet the condition of the problem then their product does not as well .  I hope you can provide some insight and thank you . I need to delete some rows from one large table. The rows to delete shouldn't be in another table, example:  DELETE FROM LargeTable WHERE id IS NOT IN (SELECT DISTINCT foreign_id from EvenLargerTable)  but my server can't handle such blunt query, because there are almost a million records in the LargeTable and few million records in the EvenLargerTable  How can I solve it? Looping through array and removing items, without breaking for loop Hello everyone interested. Here's a seemingly simple problem.  Suppose $G$ is a complete, simple graph with vertices $\{1,2,..,n\}$, $n\geq k\geq 3$.  How many cycles of length $k$ can we have on $G$?  The statement seems simple but I have trouble putting everything together.  So I say fix a vertex and start forming cycles with length k. But I get nowhere... What is the proper approach to this problem since counting is not easy? Is there a relevant theorem that I am unaware of?  Thanks for any help... Why are there so many different ways to measure disk usage? When I sum up the sizes of my files, I get one figure. If I run du, I get another figure. If I run du on all the files on my partition, it doesn't match what df claims is used. Why are there so many different figures for the total size of my files? Can't computers add?  Speaking of adding: when I add the “Used” and “Available” columns of df, I don't get the total figure. And that total figure is smaller than the size of my partition. And if I add up my partition sizes I don't get my disk size! What gives? I want to programmatically create a new column in an MS Access table.  I've tried many permutations of ALTER TABLE MyTable Add  MyField DECIMAL (9,4) NULL; and got:      Syntax Error in Field Definition   I can easily create a number field that goes to a Double type, but I want decimal.  I would very strongly prefer to do this in a single ALTER TABLE statement and not have to create a field and then alter it.   I am using Access 2003. Cross tabulation of two categorical variables: recommended techniques Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign? I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the PCA scores differed in sign. How can it be?  Consider this:  set.seed(999) prcomp(data.frame(1:10,rnorm(10)))$x              PC1        PC2  [1,] -4.508620 -0.2567655  [2,] -3.373772 -1.1369417  [3,] -2.679669  1.0903445  [4,] -1.615837  0.7108631  [5,] -0.548879  0.3093389  [6,]  0.481756  0.1639112  [7,]  1.656178 -0.9952875  [8,]  2.560345 -0.2490548  [9,]  3.508442  0.1874520 [10,]  4.520055  0.1761397  set.seed(999) princomp(data.frame(1:10,rnorm(10)))$scores          Comp.1     Comp.2  [1,]  4.508620  0.2567655  [2,]  3.373772  1.1369417  [3,]  2.679669 -1.0903445  [4,]  1.615837 -0.7108631  [5,]  0.548879 -0.3093389  [6,] -0.481756 -0.1639112  [7,] -1.656178  0.9952875  [8,] -2.560345  0.2490548  [9,] -3.508442 -0.1874520 [10,] -4.520055 -0.1761397   Why do the signs (+/-) differ for the two analyses? If I was then using principal components PC1 and PC2 as predictors in a regression, i.e. lm(y ~ PC1 + PC2), this would completely change my understanding of the effect of the two variables on y depending on which method I used! How could I then say that PC1 has e.g. a positive effect on y and PC2 has e.g. a negative effect on y?    In addition: If the sign of PCA components is meaningless, is this true for factor analysis (FA) as well? Is it acceptable to flip (reverse) the sign of individual PCA/FA component scores (or of loadings, as a column of loading matrix)? How do I change extension of multiple files recursively from the command line? Iterate a list with indexes in Python