I think this is not a duplicate of the question John mentions. That question asks specifically why the balloon makes a spiraling movement, which in general is not the case (the answers pay only attention to two variables: the nozzle and air resistance, for which there isn't a fixed centre of drag) while I ask the broader question which variables make the movement chaotic (which it is for sure while its velocity responds in a non-linear way to these variables). The non-spiraling movement can be seen as a sequence of (infinite) little pieces of circular trajectories because there are other variables influencing the motion of the balloon too (like the chaotic air movement around the moving balloon, which causes a non-fixed center of drag).  When a balloon goes flat it makes a chaotic movement through the air. Is it the interaction with the air, the behavior of the part through which the air escapes (in which case the balloon will also trace out a chaotic trajectory on the moon), or what?
When an air-filled balloon is released without its opening tied up, it moves in a circular path rather than a straight line. Why is that?
If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity This is a reworking of a previous question  which was marked as a duplicate. Some nice folks have referred me to solutions to similar problems. I still have a couple of questions, since one of the solutions used the Poisson kernel, which is not the desired solution in the text, because the Poisson kernel has not been introduced yet. The other solutions were for problems that were similar, but whose solutions were only partially applicable, as I understand them.  I'm working on a problem from p. 166 of Lars Ahlfors' Complex Analysis:     If $u$ is harmonic and bounded in $0 &lt; |z| &lt; \rho$, show that the origin is a removable singularity in the sense that $u$ becomes harmonic in $|z|&lt;\rho$ when $u(0)$ is properly defined.   My Solution So Far:   Here's what I've got for a non-Poisson kernel solution: We want to find a harmonic conjugate for $u$ in the punctured disk, and then we can apply big Picard to show that $0$ is a removable singularity. The problem is that the punctured disc is not simply connected.   If we take $\sigma = (u \circ \exp)(z): L \to \mathbb{C}$, where $L$ is the left half-plane, now we have $\sigma$ harmonic, and defined in a simply connected set $L$, so it has a harmonic conjugate $\tau$. Now we wish to take $v = (\tau \circ \log):\{0 &lt; |z| &lt; \rho\}\to \mathbb{C}$, but in order to do so we need $\tau$ to be $2\pi i$-periodic. This is where my proof breaks down.  One of the similar problems which readers referred me to used the following strategy: note that since $\sigma$ is $2\pi i$-periodic, and $\sigma, \tau$ satisfy the Cauchy-Riemann equations, we have that $\nabla\tau(z) = \nabla\tau(z + 2 \pi i)$ for all $z\in L$, or in other words, $\tau(z) - \tau(z+2\pi i)$ is a constant function, say, equal to $\alpha \in \mathbb{R}$. Then, the solution suggests, we can set $$g(z) = u + iv - \frac{c}{2\pi}z,$$  in order to make $g$ $2\pi i$-periodic, and then we can use $(g\circ \log)$ on $\{0 &lt; |z| &lt; \rho\}$. But the reason why this won't work for my purposes is: now we have changed $u$! On the other hand, we can't simply subtract $\frac{c}{2\pi}\Im{(z)}$, and try to leave $u$ untouched, because now $g$ is not analytic. How can we get around this?  Another strategy entirely might be to try to use the mean-value property. Ahlfors discusses that a lot in the preceding section, and his main example is the punctured disc. Since $u$ is harmonic, we know that the average over circles centered at zero is a constant... How can I find the codes related to a function in GAP? When I use "??" in front of the name of function there is no help, so I want to find the code of function among packages. Are there any versions of t-SNE for streaming data? What is a simple example of a limit in the real world? This morning, I read Wikipedia's informal definition of a limit:     Informally, a function f assigns an output $f(x)$ to every input $x$. The   function has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever   $x$ is "close" to $p$. In other words, $f(x)$ becomes closer and closer to $L$   as $x$ moves closer and closer to $p$.   To me that sounds like something that might be better described as a 'target'.  If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function's output when I use inputs other than $p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure).   So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.   What is a simple example of a limit in the real world?  Thank you  -Hal. I'm more of a math guy than a physics guy so bear with me....  In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal dimension of around $1.2$. You can also define a concept called fractal extent (which is the experimental version of hausdorff measure). This extent is a measure of how large a fractal is. It's even been used to more accurately model the distribution of coastline bird nesting.   Even though fractals work as an interpretation of geometrical units, I don't think they would describe things such as $kg^{1/2}$ very well. In addition, I've heard about white Gaussian noise that has square root time units.   What is the physical meaning of fractional units such as the above? (Since it obviously has little geometric meaning) My system is not dual-boot, I run a standard Ubuntu desktop system "on the metal" (I think running it in a VM is the same).  Pressing c while booting does not cause the GRUB menu to appear (which I believe it does for a dual-boot system).  How can I get the GRUB-menu to present itself on a single-boot system? Direction of angular velocity Angular velocity is the rate of angular displacement about an axis. Its direction is determined by right hand rule.  According to right hand rule, if you hold the axis with your right hand and rotate the fingers in the direction of motion of the rotating body then thumb will point the direction of the angular velocity.  The direction of angular velocity is above or below the plane. But what does it mean? I mean in linear velocity the direction of velocity is in the direction of motion of body but what does it mean that body is moving in one direction while the direction of its angular velocity is in another direction? CS0120: An object reference is required for the nonstatic field, method, or property 'foo' What Constitutes Spam? Over the course of the past several months, there have been one or more users who have been spamming about an XML parsing alternative named &quot;VTD-XML&quot;. I have occasionally assisted the moderators by pointing these answers out to them, and they have responded by deleting the answers, and the offending user (which hasn't stopped this person from immediately creating another user account and continuing). Now, there is a user , who claims to be the actual author of VTD-XML. In my mind, several of his answers, like his  to  looks a great deal like the original spam.  has asked me why I consider his answer to be spam. I decided to do so in public, in an attempt to get him to engage with this community, assuming he's not another sock puppet of the original spammer.  My reason is simple: he's been answering with one or two lines of text, followed by the exact same link to the exact same site that the spammer was using:  VTD-XML may well be worth a look as it is 1/3~1/5 the memory usage of DOM   Now, maybe I'm wrong. Maybe this does not constitute spam. Maybe Mr. Zhang can explain to us why we should not treat these posts as spam. Let's see.  Somehow, I missed the fact that in the same question, on July 31 2009:  how about vtd-xml  http://vtd-xml.sf.net   This one was from a user named &quot;Jimmy Zhang&quot;, which has since been deleted. Should I not be suspicious? Additionally, as pointed out by , he's answering a question about Delphi by posting a link to a site that doesn't even mention Delphi! Throughout the nonsense with the sock puppets, I kept asking for some real engagement from the sock puppet. It all just made VTD-XML look bad that instead of truly answering questions, we just kept getting incessant spam against old XML questions. Appeal Mr. Zhang, if you're not just desperately spamming StackOverflow because your product isn't selling, then please answer appropriate questions in a way that will actually help someone use your product to solve their problems. Reference request: convergence for periodic Markov chains Consider an irreducible but possibly periodic Markov chain on a finite state space with transition matrix $P$. We know there exists a unique stationary distribution $\pi$. If the Markov chain were aperiodic, we would have $P^n_{ij} \to \pi(j)$ as $n \to \infty$. This fails if the chain is periodic, but we have convergence of the Cesaro averages: $$\frac{1}{n}\sum_{k=1}^n P^k_{ij} \to \pi(j) \text{ as } n \to \infty.$$ Can anyone point me to a reference that states this fact? Every reference I've seen only considers convergence for aperiodic chains, or "fixes" periodicity by considering a lazy version of the chain. Alternatively, is there a simple way to obtain this result using the result for aperiodic chains? Why are there no byte or short literals in Java? I can create a literal long by appending an L to the value; why can't I create a literal short or byte in some similar way? Why do I need to use an int literal with a cast?  And if the answer is "Because there was no short literal in C", then why are there no short literals in C?  This doesn't actually affect my life in any meaningful way; it's easy enough to write (short) 0 instead of 0S or something. But the inconsistency makes me curious; it's one of those things that bother you when you're up late at night. Someone at some point made a design decision to make it possible to enter literals for some of the primitive types, but not for all of them. Why? Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. 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Given Ohm's law, how can current increase if voltage increases, given fixed resistance? According to Ohm's law, $V=IR$ (voltage equals current times resistance).  So if the voltage increases, then the current increases provided that the resistance remains constant.  I know that Voltage or potential difference means work done per unit positive charge in bringing that charge from one point to another.  So according to Ohm's law, if the work done per unit charge increases then current will increase. How can this be true? Point out my mistakes.