General Advice on choosing Mathematics as career path My Background :    I am Bachelors of Arts graduate, 27, I work as a Clerk currently. I have interest in Mathematics so I started studying it. I have studied "How to Prove It ". After getting basics I am doing Real Analysis and starting onto Abstract Algebra. I have covered Calculus 1,2,3 already from Adrian Banner'S Calculus Lifesaver videos.      I have been given advice that since I am 27, so it's too late to graduate in Mathematics and do P.H.d because that will can take 7-8 years which means that i will be around 35-36 years of age. I have been told that i should pursue mathematics as interest only (hobby).   I wish to know advice and opinions of experienced people before this question getting closed. If it gets closed, I have email I.d in my profile description, you can send me responses there. But I want to reach out to mathematicians before it gets closed.  Thanks
How old is too old for a PhD? Are there any age limits (formal, informal, or guidelines) that schools use when deciding to accept someone into a PhD program? I'm most curious about the upper age limits. For example, will most schools accept someone in their 40's? How about their 50's?
How do I enable hardware virtualization technology (VT-x) for use in Virtualbox? I was using my MacBook Pro (with 16GB of RAM) as usual and out of blue I've this popup that:  Your system has run out of application memory. To avoid problems with your computer, quit any applications you are not using.   I've obviously closed few which I could, but it didn't help. After checking on memory, it seems kernel task ate 7GB and 22.36GB swap memory was used of total 23GB (which obviously was the case). However I've still 20GB of space free on my SDD.  Activity Monitor didn't help much whilst my OS X was heading to the destruction.    My top stats before my kernel freeze: Processes: 344 total, 2 running, 5 stuck, 337 sleeping, 2580 threads                                                                                                                         19:23:56 Load Avg: 1.56, 1.62, 2.09  CPU usage: 3.51% user, 8.47% sys, 88.1% idle   SharedLibs: 46M resident, 0B data, 6572K linkedit. MemRegions: 757970 total, 2139M resident, 56M private, 907M shared. PhysMem: 9410M used (6198M wired), 556M unused. VM: 1155G vsize, 1311M framework vsize, 112872658(320) swapins, 122668389(0) swapouts. Networks: packets: 299419263/363G in, 142126838/14G out. Disks: 58970173/1079G read, 20012389/1120G written.  At the end my OS X frozen and I had to do hard reset, repair my SDD in recovery mode and fixing afterwards (recovering lost work, fixing application conflicts, checking my lost+found folder, Chrome/Terminal tabs gone, headache, etc.). My question is, how do I check on high memory usage of kernel task or how to properly deal with that kind of situation? I've tried to take Sample with Activity Monitor, but it's greyed out.  My MacBook Pro details: 2.3GHz Intel Core i7 (late 2013) with 16GB RAM. OS X: 10.9.5 I have a vocabulary like this:   Parent 1  Child 1 Child 2 Child 3   Parent 2  Child 1 Child 2    Now I would like to use a view to create a list of a Parent term with its Child terms. I try to do this with Views UI, but only show the term parent... No the child term's. It is very confusing for me.  Can anyone help with this case? Thanks :D While there be any issue applying for a new passport before stamping? Removing internal geometry So in the image below there are two circles, how can I remove the internal vertices between them without messing up the outside shape. I'm working on a gun and this is just a simple way to illustrate the issue I'm dealing with. If I where to go in and delete the internal faces it would leave holes on the outside of the object, is there a fast way to get around this? Every infinite subset of a countable set is countable. Here is the proof I tried to weave while trying to prove this theorem:     Theorem. Every infinite subset of a countable set is countable.      Proof. Let $A$ be a countable set and $E\subset A$ be infinite. Then $A\thicksim\mathbb{N}$. This implies that there is a sequence $\left\{x_{n}\right\}_{n\in\mathbb{N}}$ where $x_{n}\in A$. Construct a sequence $\left\{n_{k}\right\}_{k\in\mathbb{N}}$ where $x_{n_{k}}\in A$, $n_{1}$ is the smallest positive integer such that $x_{n_{1}}\in A$, and $n_{k+1}&gt;n_{k}$. Then      $$ E=\bigcup_{k=1}^{\infty}x_{n_{k}}, $$      and $E$ is countable, because $E\thicksim\mathbb{N}$. $\blacksquare$   Is it convincing? I have accounts in Stack Overflow, Programmers, Unix &amp; Linux and Area 51. I don't want to continue my Area 51 account, so I planned to delete my Area 51 account. I have performed the following steps to delete my account, but I couldn't!  Case 1     If your account has votes or posts do the following steps:         Edit the "About Me" section of your profile to say “please delete me”.   Contact the SE support to request deletion.      I edited my "About Me" section to "please delete me". When I contacted SE Support to delete my account, I couldn't make a request. I am getting the following error:  This operation is incomplete. please do it in main site.   For deleting Area 51 account, when I do it main site, in link for deletion column I gave the Area 51 link. It says that:  it is not valid link!   Case 2     If the account has not interacted with the site yet (votes or posts), click on the delete button on your user page next to the edit link.   I am not eligible for this case!  Then how to delete my Area 51 account? Any help is appreciated! Workflow for UV orientation How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 Aligning numbers by decimal points in table columns What damage can person/s do if they have your passport number and visa control number? I read  as well as  but it doesn't tell of what damages can be done. I have seen several sites on which people share/show the visa without the numbers and sometimes without name as well.   The only reason which many people perceive is identity theft but am unable to see how is that possible. I have to use public facilities such as printing, scanning etc. where it is easy for the operator to make copies without my consent. Also as shared, both these details are widely circulated when you stay in a hotel or anywhere else (when abroad) where identity information is needed.   Also particular to visas, as I'm given to understand, for short-stay visas (less than 3 months) which most tourist visas operate under, the window of doing anything nefarious activities (such as terrorism) would be extremely small. If I were a terrorist and I needed those bits to make work, then I would probably go for somebody who has a long-term stay as those credentials would look valid and I could jump from place to place under the radar under a fraudulent identity. Also as a terrorist, I would take quite a long view as any such attack takes years of planning, as shown in the investigation of various terrorist attacks in India as well as the U.S. Also, although I don't know (am speculating here) don't most airports might/should be checking on status quo of a passport to ensure issues such as showing 2 entries of a passport without exit (in some database where this information is collected and collated.)   Am I missing something in the logic here ? $X$ is a topological space with no special properties.  If a space $X$ has no isolated points, then show the same for any dense subset of   $X$.  Thanks a lot. Quadratic substitution question: applying substitution $p=x+\frac1x$ to $2x^4+x^3-6x^2+x+2=0$ By using the substitution $p=x+\frac{1}{x}$, show that the equation $$2x^4+x^3-6x^2+x+2=0$$ reduces to $2p^2+p-10=0$.  I can't think of anything that produces a useful result, I tried writing p as $p=\frac{x^2+1}{x}$ and finding areas to substitute but have come with no progress. Could someone offer a slight hint on how to proceed?  Thanks I am a mathematics student with a hobby interest in physics. This means that I've taken graduate courses in quantum dynamics and general relativity without the bulk of undergraduate physics courses and sheer volume of education into the physical tools and mindset that the other students who took the course had, like Noether's theorem, Lagrangian and Hamiltonian mechanics, statistical methods, and so on.  The courses themselves went well enough. My mathematical experience more or less made up for a lacking physical understanding. However, I still haven't found an elementary explanation of gauge invariance (if there is such a thing). I am aware of some examples, like how the magnetic potential is unique only up to a (time-)constant gradient. I also came across it in linearised general relativity, where there are several different perturbations to the spacetime metric that give the same observable dynamics.  However, to really understand what's going on, I like to have simpler examples. Unfortunately, I haven't been able to find any. I guess, since "gauge invariance" is such a frightening phrase, no one use that word when writing to a high school student.  So, my (very simple) question is: In many high school physics calculations, you measure or calculate time, distance, potential energy, temperature, and other quantities. These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge invariance in the same sense as the graduate examples above? Or are these two different concepts? Write $\frac 11 + \frac12 + ...\frac1{(p-1)} =$ $\frac ab$.Such that $(a,b)=1$. Show that $p^2 \mid a$ $\text{if}$ $p\geq 5$.  I was trying to apply something with prime modulus, but I am unsure how exactly to go about it. Position of adverbs I am bit confused when it comes to the positioning of adverbs determinations in a sentence.   I was told that you can place them almost everywhere like in:     To be a teacher not only means to teach a language...   &nbsp;     To be a teacher means not only to teach a language   Placing adverbs at the beginning or the end of the sentence is not a big problem, as far as I know, but what about the rest.   Are there rules for all this or can you place them anywhere including normal adverbs like "passionately" etc.? This tiny little script deletes all small features in all layers. This works fine but the size of .shp files are not reduced. Please tell me why...  canvas = iface.mapCanvas() expr = QgsExpression("$length&lt;100") for l in canvas.layers():     l.startEditing()     it = l.getFeatures(QgsFeatureRequest(expr))     for ft in it:         l.deleteFeature(ft.id())     l.commitChanges() Chrome S3 Cloudfront: No 'Access-Control-Allow-Origin' header on initial XHR request I'm taking Calculus II next semester so I don't have all the tools from it to back me up but something came up in a discussion this evening that asked me to prove that the following sequence converges. As I'm new to formal math I'm hoping someone can verify my proof, especially step $6$. I must prove that $$\lim_{n\to\infty}P_n=b$$ where $$P_{n+1} = P_n + \frac{b-P_n}{2}$$ $$P_0 = a$$ and $a,b \in \mathbb{R}$ such that $a&gt;b$ My proof goes as follows:   We must prove that $$\lim_{n\to\infty}P_n=b$$  That is, $\forall\epsilon&gt;0$ there exists a $\delta$ such that there exists an $n$ satisfying $$n&gt;\delta \iff b-P_n&lt;\epsilon$$  First, let us prove that $P_n$ is bounded by $b$ by contradiction. Let us choose an $i$ such that $P_i &lt; b$ and $P_{i+1}&gt;b$. Then we have $P_{i+1} = P_i + \Delta P$ where $\Delta P &gt; b-P_i$. but this is impossible because by the definition of the recurrence $\Delta P = \frac{b-P_i}{2} &lt; b-P_i$. Therefore, $P_i$ is bounded from above by $b$.  Additionally, it is trivial to show that $P_{i+1} &gt; P_i$.  With this in mind, choose an arbitrary $\epsilon$. We need to show that there exists an $n$ such that $P_n &gt; b-\epsilon$. Assume the contrary, that $P_n$ is bounded by $\epsilon$.  Using a similar argument to what which was used before, assume $P_n$ is bounded by $\epsilon$ and then there exists a $P_i$ such that $P_{i+1}\not &gt; P_i$. But this results in a contradiction with (4)  Therefore, $P_n$ is not bounded by $\epsilon$ and we have proved what we want to prove. What did Dumbledore say his scar on his left knee was? In Philosopher's Stone, we hear Dumbledore replying to McGonagall's query as to whether he could do something about the scar baby Harry has on his forehead:     “Even if I could, I wouldn’t. Scars can come in handy. I have one   myself above my left knee that is a perfect map of the London Underground." - Chapter 1, The Boy Who Lived.   Now, I have no idea what that last sentence means.  It could be because English is not my native tongue. Is it some kind of a joke, or does it really mean something? Is there a backquote on the iPhone keyboard? Is there a backquote (backtick `) on the iPhone keyboard, and if so, where is it?  When I'm writing a post on a site in the SE family (especially SO), I want to use the backquote for inline code blocks. Under the single quote key, I've found two curly/angled quotes, but those (’ and ‘) don't seem to do the trick...