The problem is as follows:  Let $I\subseteq J$ be ideals in a Noetherian ring. Show that if $I_{p}=J_{p}$ for every associated prime $p$ of $I$,then $I=J$.  It seems reasonable to  consider $J/I\subseteq R/I$ but I couldn't go on.  Let M be an $R-\mathrm{module}$. A prime ideal $p$ is an associated prime of $M$ if there exists nonzero $x\in M$ such that $p=\mathrm{ann}_{R}(x)$.
Let $I$ and $J$ be ideals of a Noetherian ring $A$. If $JA_P\subseteq IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subseteq I$.   I'm reading Matsumura's Commutative Ring Theory book on my own. This is exercise $6.4$ in that book, and I fail to prove it. Could you help me with this? Thanks in advance.
How can I withdraw my paper from a fake conference? How to use both binary and continuous variables together in clustering? I need to use binary variables (values 0 &amp; 1) in k-means. But k-means only works with continuous variables. I know some people still use these binary variables in k-means ignoring the fact that k-means is only designed for continuous variables. This is unacceptable to me.  Questions:     So what is the statistically / mathematically correct way of using binary variables in k-means / hierarchical clustering? How to implement the solution in SAS / R? Same mesh with different colors Using Euler's Totient Function, how do I find all values n such that $\phi(n)=12$? How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works? Is there an equivalent to Google Sketchup? mathtools: \dddot cause the term to rise Compression API on the iPhone Why is the CDF of a sample uniformly distributed I read  that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution.  I have verified this using qualitative simulations in Python, and I was easily able to verify the relationship.  import matplotlib.pyplot as plt import scipy.stats  xs = scipy.stats.norm.rvs(5, 2, 10000)  fig, axes = plt.subplots(1, 2, figsize=(9, 3)) axes[0].hist(xs, bins=50) axes[0].set_title("Samples") axes[1].hist(     scipy.stats.norm.cdf(xs, 5, 2),     bins=50 ) axes[1].set_title("CDF(samples)")   Resulting in the following plot:    I am unable to grasp why this happens. I assume it has to do with the definition of the CDF and it's relationship to the PDF, but I am missing something...  I would appreciate it if someone could point me to some reading on the subject or help me get some intuition on the subject.  EDIT: The CDF looks like this: Why are there no animals in "The Matrix"? I am trying to determine the size in bytes of the contents in a VARBINARY(MAX) field in SQL Server 2005, using SQL. As I doubt there is native support for this, could it be done using CLR integration? Any ideas would be greatly appreciated. I added two planes in Cycles with Image Texture in png with some transparency. I covered one by the other but transparency doesn't work, I mean it is impossible to see the one which is under despite transparency in the one which is above. What could be the problem? Change edit approval threshold to (# approve − # reject ≥ N) Currently, the threshold for approving or rejecting a suggested edit is , whichever is reached first.  This means that, when  or just  edits get approved, it's often with a vote count of something like 1 reject / 2 approve (or 2 reject / 3 approve on SO).  Typically, these edits manage to get one or two very quick approve votes from, um, , after which it's very hard for the slower and more careful reviewers to accumulate enough votes to catch up before the edit is approved.  Conversely, when a valid edit gets wrongly rejected, this very rarely happens unanimously either.  More often, you'll see something like 3 votes to reject / 2 to approve, but that single last vote, no matter how well or poorly considered, is enough to tip the scales.  So I'd like to propose a simple change to the approval / rejection threshold:        Suggested edits should be approved when S = (# of approve votes &minus; # or reject votes) &ge; N, where N = 3 on SO and N = 2 on other sites.   Similarly, suggested edits should be rejected when S &le; &minus;N.      No, this change is not going to magically fix all the problems with suggested edit review.  But I believe that it would be a simple and conservative improvement over the current behavior:   For obviously good or bad edits that get approved / rejected unanimously, this suggestion would not change the current behavior in any way. For edits that get one mistaken or careless reject / approve vote (as happens pretty often), an extra approve / reject vote would be required to balance it.  This would slightly slow the review process for such edits, but not excessively so.  Further, I would argue that these are precisely the kind of edits that deserve that little bit of extra attention to make sure the eventual decision will be correct. Edits that are truly controversial, and for which the reviewer community is nearly exactly 50&ndash;50 split between voting to approve and reject, might stay in the review queue significantly longer (maybe twice as long on average) than under the current system.  However, this is a trade-off for the increased likelihood that the eventual decision will actually reflect the majority opinion among reviewers, rather than mere random chance.  In any case, even if reviewers are exactly 50&ndash;50 split, the  under the proposed scheme is only N&sup2;, i.e. 4 votes for other sites and 9 votes for SO.  Even a slight bias in the voting will further reduce this.  Besides, the longer the suggestion remains in the queue, the more likely it gets that someone will short-circuit the process by casting a binding vote (mod / post owner) or by choosing to improve the edit.     Ps. If it is felt that there's an issue with , a simple further chance to address that could be to make each reject vote count as, say, &minus;2 rather than just &minus;1 approval votes (and change the rejection threshold to S &le; &minus;2 &times; N to compensate).  This still wouldn't help with bad edits getting unanimously approved (which isn't really something that can be fixed, except maybe by slugster's cattle prod), but it would mean that each reject vote would now effectively cancel out two approve votes, titlting the balance for disputed edit suggestions in favor of rejection and making it easier for more careful reviewers to counter the "robo-approvers". "a ten years old boy" or "a ten year old boy" Solve $\int_0^{+\infty} \Big(\frac{\sin t}{t}\Big)^2 \ dt$ Are $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ different? I have seen the claim that $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ are not diffeomorphic, but I have only ever seen the proof that they are not isomorphic as vector bundles (which is a cute application of the hairy ball theorem). How would I prove that they are indeed not diffeomorphic (or better, not homeomorphic)? Is there an easy method for installing binary builds of glibc? Time and time again I see questions such as these:       And these are the types of solutions we're typically pushing:        Is this really the best we can do? Aren't there binary builds of GLIBC that we can simply unzip into a directory such as /opt/myglibc, and set the $LD_LIBRARY_PATH or whatever and run whatever application we want, without issues?   An application, such as the newer builds of Chrome (28+) that appear to require GLIBC 2.14?  NOTE: This thread titled:  on tecmint.com is what ultimately got me thinking about this.  References Importing a Single Symbol From a Different Font How can I mount my partition with executive permissions using the launcher icons? "Effect" vs. "Affect" QGLWidget appears black in Windows 7