Systematic increase in runtime
Mathematica9: NDSolve slows down after repeated calls
What is a good book to learn number theory? What would be a good book to learn basic number theory? If possible a book which also has a collection of practice problems? Thanks. How do I install HP P1102W printer driver (exercise from Tao's analysis book) Proof of a lemma relating to power set of X I'm stuck at one exercise from chapter of sets from Terence Tao's analysis book. I need to proof the lemma:  Lemma: Let $X$ be a set. Then the set $\{Y : Y \:\text{is a subset of}\: X\}$ is a set.  Note: The set $\{Y : Y \:\text{is a subset of}\: X\}$ is known as the power set of $X$, defined as  $2^X $  I can understand why the lemma is true, but I have no clue how to prove it. Furthermore, the author gave a hint that I found very confusing, it follows:  Hint: start with the set $\{0, 1\}^X$ and apply the replacement axiom, replacing each function $f$ with the object $f^{-1}(\{1\})$. It seems that with some profiles the Gravatar image is broken. I am not sure if this is an issue with SE or Gravatar, or both, so posting here.  As an example,  shows an error on , which shows a HTTP 500 on Gravatar.  Profile image on SO:        Image on Gravatar:        Switching to the MSE profile of the user, the image is working (but has a different hash).  Is this a bug on the SE side? Can anything be done to make Gravatar fix this?    Update 2016-10-04:  It happened to me again two times today. The last user profile was . Calling the service two or three times fixes the issue. Thanking users who answered my question Quite often a number of users will give similar answers to one of my questions. Sometimes someone will give an answer that is interesting, but does not really help with my question.  In all cases, I feel a need to thank the people who answered my question. It isn't always appropriate for me to upvote every answer, and I can only accept one answer. Giving a "thanks" comment on every answer appears to be frowned upon.  So, what's the etiquette? How do I work recursively through a directory tree and execute a specific command on each file, and output the path, filename, extension, filesize and some other specific text to a single file in bash. How can I use colours to separate texture node treatment? Wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$? What is correct wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$?  How are these wave equations used in the positive $x$-direction and negative $x$-direction? What exactly is a bad backlink that is Disavow worthy? I have been reading Webmasters SE and many other blogs on this topic but I could not understand how do we truly evaluate a bad link that we must disavow in GWT. Everybody seem to have a different opinion on this.  A little background  I am a WordPress developer and 3 years ago I gave away a WordPress theme to my blog visitors which has a dofollow link back to my business website. Like this.  WordPress theme designed by &lt;a href="http://domain.com"&gt;MyBusinessName&lt;/a&gt;   From last 2 years I was not paying any attention on organic traffic to my business website because of other projects.  That free WordPress theme helped us getting ~10K backlinks from variety of domains and websites. All most 99% backlinks are from non-related websites and in many different languages.  Now  Last week it came to my attention that in October 2013, organic traffic from Google started decreasing and in just 2 months it became as low as 25% of October. Before October it was gradually increasing ~5-10% per month.  So I researched and draw to the conclusion that it could be because of Penguin Update. Since  was introduced at the same time.  Now I need information on what a bad/toxic link is.  I was about to disavow almost all links but after reading many topics on Webmasters SE I think I need more information on how to identify a bad link.   Does credit link in footer consider as spam because it build thousands of backlinks? Do foreign language website backlinks consider as spam? Should I remove all backlinks from theme users because almost all are from non-related content domains? Although some links from non-related content domains have nice DA/PA. Should I remove those too? I understand foreign language anchor text backlink could be a spam but what if it is from related foreign language content.   The question is how exactly one can find a link that is bad and must be removed. Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4):     If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in G$, show that $G$ is abelian.   I managed to prove it, but I'm not very happy with my result (I think there's a neater way to prove this). Anyway, I'm just looking to see if there's a different approach to this.   My approach:  Let $j=i+1, k=i+2$ for some $i\in \mathbb{Z}$.  Then we have that $(ab)^i = a^ib^i$, $(ab)^j = a^jb^j$ and $(ab)^k = a^kb^k$.  If $(ab)^k = a^kb^k$, then $a^jb^jab =a^jab^jb$.   We cancel on the left and right and we have $b^ja = ab^j$, that is $b^iba = ab^j$.  Multiply both sides by $a^i$ on the left and we get $a^ib^iba = a^jb^j$, so $(ab)^iba = (ab)^j$. But that is $(ab)^iba = (ab)^iab$.   Cancelling on the left yields $ab=ba$, which holds for all $a,b \in G$, and therefore, $G$ is abelian.  Thanks! Complex numbers equation: $z^4 = -16$ As the title states, I'm having quite a lot of trouble with the integral:  $$\int_{-\ln 2}^{\ln 2}  \frac{(\sin x+x)^{1/3}}{e^x}dx$$  The problem is that my standard (admittedly sparse) repertoire of tricks seems to have no effect at all. I can't think of any clever substitution that would work, and it looks like a nasty integral all-round.   I was hoping to apply some symmetry arguments (especially considering the limits of integration), but the exponential function is neither even nor odd, so I don't see how this would work.   I'm afraid I have absolutely no idea how to approach this. Why is the Petersen graph no Cayley graph? I'm suppose to prove the following...  Let $A$ be a nonempty subset of $\mathbb{R}$. If $\alpha = \sup A $ is finite show that for each $ \epsilon &gt; 0 $ there is an $a \in A $ such that $\alpha - \epsilon &lt; a\leq \alpha $.  This is what I have so far...  Proof: We claim that $\alpha = \sup A $   Then by definition of a supremum $\alpha $ is the least upper bound of set A  Thus $\forall a \in A$,  $a \leq \alpha$   Suppose $\epsilon &gt; 0$   Then $\alpha - \epsilon &lt; \alpha $.   Thus $\alpha - \epsilon &lt; a\leq \alpha$.   I feel like I'm missing some step between my last and second to last step so what am I forgetting? Also is the rest of my proof right? Sending a message to nil in Objective-C As a Java developer who is reading Apple's Objective-C 2.0 documentation: I wonder what "sending a message to nil" means - let alone how it is actually useful.  Taking an excerpt from the documentation:     There are several patterns in Cocoa   that take advantage of this fact. The   value returned from a message to nil   may also be valid:         If the method returns an object, any pointer type, any integer scalar   of size less than or equal to   sizeof(void*), a float, a double, a   long double, or a long long, then a   message sent to nil returns 0.   If the method returns a struct, as defined by the Mac OS X ABI Function   Call Guide to be returned in   registers, then a message sent to nil   returns 0.0 for every field in the   data structure. Other struct data   types will not be filled with zeros.   If the method returns anything other than the aforementioned value   types the return value of a message   sent to nil is undefined.      Has Java rendered my brain incapable of grokking the explanation above?  Or is there something that I am missing that would make this as clear as glass?  I do get the idea of messages/receivers in Objective-C, I am simply confused about a receiver that happens to be nil. How can a credit card company make any money off me? I have a no-fee card and pay my balance on time My understanding of the Stern–Gerlach experiment is that neutral (0 total charge) particles are sent through a non-homogeneous magnetic field, with the expectation that the field will push that particle's path higher or lower on a detector because of the collective spin of that particle. While the detector can detect particles in a two-dimensional surface, the results of the experiment are that particles appear in only two localized areas directly above the path of the particle stream, or directly below the path of the particle stream - spin up or spin down. The conclusions from these measurements are that the particle, when measured, will always have about the same magnitude.  Why does this not follow from classical mechanical theory related to magnetism? If you shot a magnet through a similar apparatus, I would expect the magnet to be rotated to align with the magnetic field in some way which, at high enough field strengths relative to the mass of the magnet, would cause us to measure basically the same magnitude as if the magnet entered the apparatus pre-aligned with the field.  How is my explanation incorrect? What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.  I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$  Can someone please explain this to me, possibly with a link to a general topic? How to play blu ray .mkv files on ubuntu 20.04 LTS? How to play blu ray .mkv files on Linux? e.g:1080p.BluRay.DD5.1.x264-NTb.mkv Why is it that when you add normally distributed random variables the variance gets larger but in the Central Limit Theorem it gets smaller? When you add two independent normal distributions the resulting distributions' variance is the sum of the variances i.e. it gets larger.  However, the Central Limit Theorem states that  when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed.   and also says that as you add more variables the variance gets smaller. Why does the variance increase in one case but decrease in the other? Is it because in the CLT case the random variables come from the same underlying distribution?