How is this possible, technically? What is the logic behind this?
Below is a visual proof (!) that $32.5 = 31.5$. How could that be?    (As noted in a comment and answer, this is known as .)
Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$ If $a$ and $b$ are positive real numbers such that $a&lt;b$ and if $$a_1=\frac{a+b}{2}, b_1=\sqrt{(a_1b)},..., a_n=\frac{a_{n-1}+b_{n-1}}{2},b_n=\sqrt{a_nb_{n-1}},$$ then show that   $$\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}.$$ I tried to calculate explicitly the first few terms $a_2,b_2$ etc but the terms got too complicated really quickly and I couldn't spot any pattern. Problems in Creating InfoWindow on each individual circle with mouseover using Google Map API $(function(){   $('.slider').mobilyslider({     content: '.sliderContent',     children: 'div',     transition: 'horizontal',     animationSpeed: 500,     autoplay: true,     autoplaySpeed: 9000,     pauseOnHover: true,     bullets: true,     arrows: true,     arrowsHide: true,     prev: 'prev',     next: 'next',     animationStart: function(){},     animationComplete: function(){}   });    });   I am integrating the above JavaScript file in a Drupal theme, but it is giving me the "$ is not function" error. I googled it, and changed the code bro the below, but it doesn't work either.  (function($) {   $('.slider').mobilyslider({     content: '.sliderContent',     children: 'div',     transition: 'horizontal',     animationSpeed: 500,     autoplay: true,     autoplaySpeed: 9000,     pauseOnHover: true,     bullets: true,     arrows: true,     arrowsHide: true,     prev: 'prev',     next: 'next',     animationStart: function(){},     animationComplete: function(){}   });    })(jQuery);   The above code gives me following error:     $('.slider').mobilyslider is not a function Examples in which the light maximizes the optical path length I posted a similar  about geodesics on Math.SE. Many sources ( for instance) claim that the light could maximize the optical path length in some cases. But I don't think it's actually true since between two points I can always imagine a path with arbitrary length. Therefore, or the light minimizes the optical length or it's stationary (neither maximum nor minimum). My question is: is the trajectory in red in the figure below a local minimum or a stationary curve?   If I'm wrong, please provide some examples which the light propagates in trajectory of maximum length. Why is "If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct" true? In Chiswell and Hodges Mathematical Logic the authors define a sequent as such      "A sequent is an expression   (Γ ⊢ ψ) (or Γ ⊢ ψ when there is no ambiguity)   where ψ is a statement (the conclusion of the sequent) and Γ is a set of statements (the assumptions of the sequent) ...There is a proof whose conclusion is ψ and whose undischarged assumptions are all in the set Γ.".   They then go on to provide the axiom as follows:     Sequent Rule (Axiom Rule) If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct.   Upon further reading about what exactly it means to be a sequent  I understand that for $\psi$ to be satisfied or "correct", every element of   $\Gamma$ must be true as their all linked by AND conjunctions.    What confuses me is that suppose that there is an element in $\Gamma$ which is false then the whole antecedent is false according to the wikipedia definition by conjunction and then even though $\psi$ exists and is true in $\Gamma$ ultimately $\psi$ would be false. Could someone please explain to me and help clarify how the axiom and the definition can both be true. Thank you for all of your help. What happens if my ESTA expires on the day of arrival? How do I apply a gradient across multiple objects in Illustrator? I have two separate objects that I have applied gradient fill to.  The gradient fill seems to only apply to the objects as individuals. I tried grouping the shapes, but that doesn't seem to change the gradient.    How do I get one gradient to cover all my shapes? Punctuation with units I remember reading somewhere that if a unit is abbreviated as one character, there must not be a space between the number and the unit (e.g., 5m, 26K). If the unit is abbreviated as two or more Characters, there must be a space between the number and the unit (e.g., "10 km", "USD 5").   Can you please help me find the source again? Is this recommendation correct? I want to do a survey of textbooks in set theory. Amazon returns 3582 books for the keywords "set theory". A small somewhat random selection with number of references in Google scholar is the following.   Two questions:    What textbooks in set-theory are considered standard?  What is a good place to aggregate standard references (of textbooks)?    (I'm only interested in text-books, which as a format already provide a kind of standardization)  Textbooks on set theory (author: title (year) #Google Scholar citations )   Thomas J. Jech: Set Theory 3rd Edition (1978) #2441  Kenneth Kunen: Set Theory (1980) #1881  P. R. Halmos: Naive Set Theory (1974) #1079 AA Fraenkel, Y Bar-Hillel, A Levy: Foundations of set theory (1973) #696 K Kuratowski, A Mostowski, M Mączyński: Set theory (1976) #609 Suppes: Axiomatic set theory  (1972) #589 Quine: Set theory and its logic (1969) #442 A Levy: Basic set theory (1979) #442 Herbert B. Enderton: Elements of Set Theory (1977) #349 Karel Hrbacek, Thomas J. Jech: Introduction to set theory (1999) #284 RR Stoll: Set theory and logic (1979) #281  N. Bourbaki: theory of sets (1970) #211 K Devlin: The joy of sets: fundamentals of contemporary set theory (1994) #182 Ciesielski: Set theory for the working mathematician (1997) #156 YN Moschovakis: Notes on set theory (1994) #138 FW Lawvere: Sets for mathematics (2003) #118 G Takeuti, WM Zaring: Introduction to axiomatic set theory (1982) #106 Kaplansky: Set Theory and Metric Spaces (2001) #103 Potter: Set Theory and Its Philosophy: A Critical Introduction (2004) #91 PT Johnstone: Notes on logic and set theory (1987) #87 Judith Roitman: Introduction to Modern Set Theory (1990) #47 Derek Goldrei: Set Theory: For Guided Independent Study (1996) #31 Basic set theory: A Shen, NK Vereshchagin (2002) #14 Ralf Schindler: Set Theory (2014) #13 Foreman, Kanamori: Handbook of Set Theory (2009) #38   Note:  Please feel free to add textbooks to the list, as long as they are not just on a smaller subset of set theory. (Some of the dates might not be the first published edition.) Whenever my cheese gets too old and has a bit of blue/white mold on the side, I'm not sure whether I should throw it completely out or not. Is it safe to eat if I cut off the edge? Should I cut a bit off of all the sides just to be safe?  I'm particularly interested in the answer for cheddar cheese, but answers about other cheese are welcome. Ubuntu 16.04 VIM without python support! Although Ubuntu 16.04 release notes mention that:  The default VIM package has been built against python3 instead of python2.  Yet after I installed vim with the following command:  sudo apt-get install vim   And checked:  vim --version   It showed:  -python -python3   Is it a bug? How do I get vim with Python support? Which one of this two sentences is correct?   I need you to help me do this job I need you to help me to do this job   Can you also provide me some explanation? So I have seen the similar question and answers on here for $x^4 +1$, but I am having trouble extending anything there to this polynomial... I understand it is fairly trivial with Galois theory, but my class has just barely covered Field Extensions, so suffice it to say we have no Galois theory to play with.  I managed to prove it for the primes such that $p \equiv 1, 7 \pmod 8$, by noting that $2$ is a square modulo those primes and thus $x^4 - 2x^2 +1 = (x^2 -1 + 2qx)(x^2 - 1 - 2qx)$ for those $\mathbb{Z}_p$... however, trying to get a similar result for $3 \pmod 8$ and $5 \pmod 8$ has been stumping me for a long time, I am having a hard time making $q^2 = -1$ and $q^2 = -2$ give me something factorable...  I guess the worst part of all of this is that I don't think this solution is even particularly enlightening, in terms of abstract algebra. It's really just some number theory trickery. I don't think my course has prepared me theoretically for this problem, does anyone have an elementary approach to it? What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)? How do I pass the forest stage on Candy Box 2? I cannot pass the forest stage on Candy Box 2. The dogs and forest mosquitos kill me fast.  My stats and equipment are:   220 health Polished silver sword (7 damage, also have the trolls bludgeon; 12-16 damage) Leather boots and gloves (no stat bonus) Basic and advanced spells from the witch across the bridge   On another save I also increased my health to 1000 and still could not pass the forest stage, any advice?  Thanks. Error to calculate date diff with DateTime class Branch of the n-th root of a holomorphic function is holomorphic It is true that given a domain $D$ and a function $f$ that is holomorphic and not vanishing in $D$, then a branch $\varphi$ (with $\varphi$ continuous) of $f^{1/n}$ in $D$ is holomorphic in $D$ ().   My question is, does the statement remain true if we remove the non-vanishing condition? I haven't been able to find a counterexample or a proof on that more general case. Remove HTML Tags from an NSString on the iPhone Why is my StackExchange OpenID so hard to find? Pluralization of names If I were to use the sentence "There are lots of John Smiths" in the world, would that be the correct use for saying that there are a lot of people named John Smith in the world?  I don't think there should be an apostrophe as that would imply ownership of something.  If my first example is correct, then what would you do if the name referenced already ended with an 's'?