Let $\{a_n\}$ be a monotone decreasing sequence of real numbers, with limit $0$.   Prove that, for each $0&lt;\delta&lt;2$, the power series  $\sum_{n=0}^{\infty} a_n z^n$  converges uniformly on each $\Omega_\delta = \{z : |z|\leq 1 , |z-1|\geq \delta\}$  I have no idea on how to deal with the points in the boundary. Thanks in advance!
Let $b_n$ be a sequence which decreases monotonically to zero, prove $\sum b_nz^n$ converges for $|z|\leq 1$ and $z\neq 1$.  So I was able to prove it converges on the open unit disk in the complex plane, but except for a few special points on the boundary, such as $-1$ and $i$, where I was able to use the alternating series test, I can't seem to prove it converges in general on the boundary minus $z=1$.  I feel I have to somehow use the fact that all the numbers on the unit circle except $z=1$ rotate around and around as you raise them to higher and higher powers, thus you get the real and imaginary parts of your series to alternate between sets of positive and negative terms, but in a very irregular way for most initial angles.  I tried to simplify the problem by breaking the series into its real and imaginary parts, and then trying to prove they each converge separately, but I still can't prove either converges:  $$\sum_{n=0}^{\infty}b_ne^{n\theta i} = \sum_{n=0}^{\infty}b_n\cos(n\theta) +i\sum_{n=0}^{\infty}b_n\sin(n\theta).$$
Help with Axis Placement when extruding I am just learning Blender. I am going through the .  The first part was good. I was able to follow it. Coming from SketchUp, it is taking some adjustment. But I am getting it.  I am just starting the second video. At the beginning, the instructor selects the top of the mug to extrude it. When he presses the E key and Enter, the red, green blue axis moves to the center of those vertices. That does not happen when I press the E key and Enter.   This is only 1:30 into the video.  How can I get the axis in the middle of this circle of vertices?  The next step is to scale. I need the reference point in the middle of this circle of vertices. I cannot left-click accurately enough to position the axis/reference point.  I am using 2.69. The video uses 2.64. I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).  Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic rules you laid down in the first place on your own.  Now using these laid-down rules and a set of other rules for a subject called logic which was established similarly, you define certain quantities and name them using the undefined entities and then go on to prove certain statements called theorems.  Now what is a proof exactly? Suppose in an exam, I am asked to prove Pythagoras' theorem. Then I prove it using only one certain system of axioms and logic. It isn't proved in all the axiom-systems in which it could possibly hold true, and what stops me from making another set of axioms that have Pythagoras' theorem as an axiom, and then just state in my system/exam "this is an axiom, hence can't be proven".  EDIT : How is the term "wrong" defined in mathematics then ? You can say that proving fermat's last theorem using the number-theory axioms was a difficult task but then it can be taken as an axiom in another set of axioms .  Is mathematics as rigorous and as thought-through as it is believed and expected to be ? It seems to me that there many loopholes in problems as well as the subject in-itself, but there is a false backbone of rigour that seems true until you start questioning the very fundamentals . Double dual mappings $\newcommand{\Hom}{\operatorname{Hom}}$Let $U,V$ be vector spaces. Denote the dual space of $U$ with $U^\intercal$ and the dual mapping of a linear map $\Phi$ be $\Phi^\intercal$. Define the double dual embedding on $U$ to be $\iota_U:U \to U^{\intercal\intercal}$. ($u \in U, f \in U^\intercal$) $$ \iota_U(u)(f) = f(u) $$     If $\Phi \in \Hom(V,U)$, then $\Phi^{\intercal\intercal} \circ \iota_V = \iota_U \circ \Phi:V \to U^{\intercal\intercal}$   Proof: Let $v \in V, f \in U^\intercal$. Then $$ \iota_U \circ \Phi(v)(f) = f(\Phi(v)) $$ but how do I show that $$ \Phi^{\intercal\intercal} \circ \iota_V(v)(f) = f(\Phi(v)) $$ ? Story ID: Boy who is stuck at 5 years old with robot nanny Here's what I remember:  The boy is 5 years old, and has been for some time, because there's a bit in which he recalls turning 5 (celebrating his 5th birthday) many different times  There's a robot nanny in the apartment with him that he learned to disable so that he can hang out with the girl, who he isn't supposed to know. (In my mind the nanny bears somewhat of a resemblance to Muthr from the Wondla series... however I definitely read this story well before reading Wondla)  There's a neighbor girl who teaches him to read (I think she's like 9), and when they first meet she asks how old he is (5 is the last age at which growth/development can be successfully (and safely?) suspended) and he lies, saying that he's 6, but doesn't know why he does.  When a child whose aging process has been stopped learns how to read, they begin aging normally again, so the parents don't want them to learn to read.  The children who have stopped aging are kept apart from other children who grow normally-- they have separate playgrounds and the like  The injection/serum used to halt the aging process was developed by a scientist, who when he first shared his findings had a 5 year old son and injected him, then told the onlookers/potential buyers/other parents to come back in 20(?) years. When they did, their children were adults and no longer needed them or were ungrateful layabouts, and the scientist's son was still 5 years old, cute, needy, etc.  The main character has a discussion about how he wants to turn 6 with his parents and they agree but only because he's already learned to read and it's irreversible.  It's kinda dark and emphasizes the need that parents have to feel wanted. Please help! I remember it so vividly and I'd love to read it again! Can the Next/Prev Post links be ordered by menu order or by a meta key? I have a series of posts that are ordered by a meta_key value. They could also be arranged by menu order, if necessary.  The next/prev post links (generated by next_post_link, previous_post_link, or posts_nav_link all navigate by chronology. While I understand this default behaviour, I don't understand how to change it. I found that it maps through to adjacent_post_link in link-template.php, but then it starts to seem fairly hard-coded. Is it recommended to re-write this from scratch to replace it, or is there a better solution. Can the ship's captain really perform your wedding when in international waters? On , Mark Mayo posted a comment that any marriage has to be on dry land or close to it.  But I'm sure all of us have heard of ships' captains marrying couples at sea.  From the movie Casablanca:     Say, why don't we get married in Marseilles?   ...   Well, let's see. What about the engineer? Why can't he marry us on the train?   ...   Well, why not? The captain on a ship can. It doesn't seem fair that...    So is this fiction after all? If not how does it fit with Mark's comment? What special rules and such should we know?  If I'm travelling at sea with a significant other and wish to tie the knot, can we do so with any captain on any ship on any sea? Must it be international waters? For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold? The rank of Jacobian matrix at a point of affine variety is independent of choice of generators Is it acceptable to edit (correct) grammar or spelling? Some individuals may be sensitive to edits of their posts on the basis of spelling or grammar. Some editors may be concerned about what to put in as a reason/explanation for the edit. New editors, in general, are likely uncertain of what the accepted "etiquette" is, on these sites, for such corrections. Search among posts you've voted for Related posts:    These posts are different from mine in that they merely ask for the &quot;votes&quot; tab back or ask if it is at all possible to get such information, while I propose how to implement this as a search feature. Background I recently found myself facing a problem very similar to one that I had solved before. I remembered that I found the solution on SO and I remembered upvoting the post containing the solution. I wasn't able to find that particular post by just searching with relevant keywords. It would have been nice if I could search only among the questions I have voted on. I searched on meta and found that there had been a tab for looking at what you've voted on, but that it was removed due to lack of interest. This is understandable, since just listing the hundreds/thousands of posts you've voted on isn't particularly useful. Proposal What I want is to be able to search for posts I've voted on similarly to the way I'm able to search on my posts within a specific tag. For instance: user:310121 [iphone] apple  This will return all my posts tagged iPhone, where I mention Apple. This is brilliant. This is just how I want search to work. Now what I want to do is to extend this to include votes. The exact syntax isn't relevant, but I had to make one up to show you what I mean: To show all posts you've voted on: user:310121 {votes}  To show all posts you've upvoted: user:310121 {upvotes}  To show all posts you've downvoted: user:310121 {downvotes}  You could probably omit the the user part, since you're only allowed to see your own votes. (Unless it's desirable to be able to look at other peoples votes. Maybe you should be allowed to do this if you have enough rep? Well, that's another discussion.) I guess that even if you're allowed to see other peoples votes it would make sense to display your own votes if the user part is omitted. So in my case I could have done the following search and found the post I was looking for: {upvotes} keyword  Lastly I just want to suggest that clicking on your vote count will work just like clicking on a tag. If I click on a tag in my profile the following search will be done: user:310121 [tag]  So if I click on my upvote count the following search should be made: user:310121 {upvotes}  The same goes for downvotes: user:310121 {downvotes}  This would be a very useful feature and it wouldn't be intrusive in any way. The users that don't care for this feature might never know that it's there. And just for the record the favorite system doesn't replace this feature. Does being refused entry to Singapore mean I am now blacklisted? My friend who is an Indonesian citizen tried to enter Singapore this week and was refused entry at the Singapore Harbour Front Ferry Terminal by immigration.  This was his first trip overseas and is not used to international travel, so due to his inexperience and that he was excited to go to Singapore, he possibly made some mistakes on trying to be granted access into Singapore.   He was told he was to enter a Singapore address for his stay or a friend or relative he will stay with. He was only staying for one day and was going to catch the ferry back at night, so he had no plan to stay in Singapore and as such had no address to put. They then asked why he was traveling alone and that he should travel with somebody for his first trip. He told them he always travels alone in Indonesia and finds it easier to see and do things alone. He was then asked why he was entering Singapore and the reason he gave was to purchase some electronics which he cannot buy in Indonesia. They asked for how much money he had and they then advised that he did not have enough money to purchase the items he was after. They refused his entry and he was then escorted to a ferry and returned to Indonesia. His passport was not stamped and he was not fingerprinted.   He was given a Notice of Refusal of Entry which had the reason as "Being ineligible for the issue of a pass under current immigration policies" form on his departure. Also included on this form is his passport number etc. He was not advised of him being banned or a time frame that he cannot enter Singapore.  My questions are: Has he been banned or restricted to enter Singapore for a period of time? When entering countries including Singapore, does he need to tick the box stating he has been refused entry into another country? I am hoping to get my friend to Australia at the end of the year for a 10 day holiday and will this hinder his chance of getting a visa and entering Australia? The Earth is spinning, so why don't we jump and land on a different location? I know there are similar questions on StackExchange but I think it is different and detailed.  The earth is spinning 465 meters/second so why don't we jump and land on a different location?  I googled about this question and I got some answers:   When we jump, we (and the atmosphere itself) also spin along with the earth so we don't land at a different location. (But why we are also spinning along with earth ?) The earth is so big and we are very small relative to earth so the tiny jump won't make any difference.   But according to answer 1, We are also spinning with the earth. But the question is, "why are we also spinning along with earth" ??  Regarding the answer 2, it won't make any difference but logically it doesn't make sense or I just don't understand it.  Here is the followup question,   "When we jump why don't we thrown out of earth due to centrifugal force of the spinning earth ??"  Please provide some detailed answer to these questions. With logic if possible.   This phenomenon is not possible I know, but even if it exists, then traveling will be so easy, we just have to hang on space (with help of helicopter etc.. ) and we can travel the whole world in 24 hours :D I get the following error:     Strict warning: Only variables should be passed by reference in include() (line 18 of /home/sites/dev/theparce/sites/all/themes/parce/block--block--3.tpl.php).   This is the block code which is causing that error.  if ($user_gallery) {   print render(node_show($user_gallery));  // Line 18   print drupal_render ($user_gallery_edit);  } else {   print drupal_render($user_gallery_new); }   Why do I get that error, even if I get all printed as expected? How do I beat the Reaper on Rannoch? I'm having trouble with the last of the Rannoch missions, where Shep wakes up and takes on a Reaper.   Shepard has a targeting laser which will call in an orbital strike with the combined might of the Quarian Migrant Fleet, but there's two major problems standing between the commander and victory.   The Targeting Laser takes a while to power up. The Reaper is trying to kill you.   I can consistently avoid the Reaper's attacks for the first volley, but each time you hit it, the Reaper moves forwards, giving you less time to dodge future assaults. Is there some key to avoiding the Reaper's attacks? Strafing doesn't seem to be enough. Get the exact time for a remote server This is a follow up to   I am using a SHA-256 HMAC function on a single-word input: sha256hmac(privatekey,word) = output. The private key length is 128 bits, it does not change. The "word" is literally one short word.  Assuming an attacker knows "word" and "output" for multiple words (e.g., dog = "dsfwrw3r3r323211", cat = "3erwerwetwewer"):   Is it significantly easier for an attacker to identify the private key by brute force with the above word/output pairs, as compared to a situation where they do not have those pairs? Is this brute force vulnerability (with word/output pair knowledge) an actual security concern? Escaping regex string Approximation of real number to the power of irrational Say I have two numbers, $c\in\Bbb R^+\,,q\in\Bbb{R\setminus Q}$.  Is there a way to approximate the value of $c^q$?  This question bugging me because of how simple it sound in compare to how complicated it is. First I want to understand what a power of irrational number is, when talking about $\Bbb R^+$(I use this domain for sake of simplicity) we have very well defined actions for power a rational number, we can even use some theorems to get $c^{a/b}=\sqrt[b]{c^a}$, but when I am dealing with irrational number I can not do this.  I know that we have some definition for this because the solution of the equation $2^x=3$ is irrational.  So what exactly does it mean to take something to a irrational power and is there a way to approximate a numeric value for this? I have an org with hundreds of apex classes, all of which have api versions ranging between 30-37. Is there an easy way to bring them all to the latest api version at once without resorting to manual updates? IP address of host of Wi-Fi ad-hoc network? I often create a wireless ad-hoc network on my laptop running Windows 7. Now I'd like to connect to it using standard HTTP from my . I've tried 192.168.1.1 and that doesn't work. What should the address be so I can connect?