I am writing some complicated formulas that involve several layers of fractions.  Example:  \begin{displaymath} \scalebox{2.5}[3]% {$ 1+\frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+\frac{9^2}{2+\cdots}}}}} $} \end{displaymath}   but the visual result, although readable, could be improved with some white space over all the squared odd terms. How would I achieve that? The vertical scaling of \scalebox does not help since I guess that the spacing between the fraction line and the exponent under it is too close to zero.
I have complicated math expression:    Vertical spacing between text and fraction line is too small. I tried to define new \frac command (with amsmath package) with adjustable vertical spaces. As you can see, the result is not satisfying:    \documentclass[a4paper,fleqn,leqno]{article} \usepackage{polyglossia} \usepackage{color,pifont,fontspec} \usepackage[top=2.54cm,bottom=2.54cm,left=2.54cm,right=2.54cm]{geometry} \usepackage{tikz,tikz-3dplot,graphicx} \usepackage{mathtools,breqn,amsmath} \usepackage[fontsize=12]{scrextend} \usepackage{array,multirow} \usepackage{pgfplots} \DeclareMathSizes{12}{14}{14}{10} \newcommand{\myfrac}[3][0pt]{\genfrac{}{}{}{}{\raisebox{#1}{$#2$}}{\raisebox{-#1}{$#3$}}} \begin{document} \begin{dmath*}[breakdepth={0},style={\mathindent=0pt},spread={10pt}] {n=\myfrac[3pt]{\myfrac[3pt]{0,0625N+0,0215}{2}-0,043}{\myfrac[3pt]{\myfrac[3pt]{\scriptstyle 0,0625N+0,0215}{\scriptstyle 2}-0,043}{\myfrac[3pt]{\scriptstyle N}{\scriptstyle 2}-1}+0,0215}} \end{dmath*} \end{document}   How can I enlarge verticall spacing in whole fraction?
How do I show that there exists variational/action principle for a given classical system? We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's principle) and even General Relativity (we get Einstein's equation from the Einstein-Hilbert action). However, how do we explain this very principle, i.e., more mathematically, I want to ask the following:      If I am given a set of generalized positions and velocities, say, $\{q_{i}, \dot{q}_{i}\}$, which describes a classical system with known dynamics (equations of motion), then, how do I rigorously show that there always exists an action functional $A$, where    $$A ~=~ \int L(q_{i}, \dot{q}_{i})dt,$$    such that $\delta A = 0$ gives the correct equations of motion and trajectory of the system?    I presume historically, the motivation came from Optics: i.e., light rays travel along a path where $S = \int_{A}^{B} n ds$ is minimized (or at least stationary). (Here, $ds$ is the differential element along the path). I don't mind some symplectic geometry talk if that is needed at all. Consecutive spoiler blocks (>!) are shown as blockquotes The markdown &gt;! a  &gt;! b  &gt;! c  Will be shown as a blockquote with ! in front of it:   a    b    c  Instead of a spoiler for 3 paragraphs (or 3 spoiler blocks, whatever). Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative test and it seems to suggest that $\Delta u(\vec x) \leq 0$ if $\vec x$ is a min and $\Delta u(\vec x) \geq 0$ if $\vec x$ is a max. However, I'm not convinced of this result. Can somebody point me towards a statement (theorem, etc.) involving the above. I appreciate the help. Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$.  I can't figure out where to start other that I need to prove some biconditional statement. Any help? How OPTIONS are settled can be an eye opener and I hope I can get some answers The only reason why I put it here is to find out whether I am a victim or just ignorant of the way brokers deal with option positions that were exercised. Whether I am a victim or just ignorant depends on the answer that I posted here. The facts are: I had a 2 SHORT QS Option at 75 which was exercised against me on the 23 Dec 00.45 AM and I later exercised my 2 LONG QS Option at 70 on 23 Dec 9.31 AM. The broker charged me Borrow Fee for 24-27 Dec for $1102, which means instead of getting $1000 profit I turn this into a loss of $102 plus comm. My question:  Since both long and short options were exercised on the same day one at 45 mins on 23 Dec and the later at 9.31 am on the same day, how can there be a SHORT POSITION in the stocks since both are net-off in my account ?  The answer I got from my broker and this is the point that I want to clarify.  The response I got from IB was that the initial exercise at 45 mins on 23 Dec was actually on 22 Dec (not 23 as I alleged). They refused to show me how they got 22 Dec.  This is my explanation to my Broker in reply.  Even if the short call and long call was exercised on different dates 22 and 23 Dec respectively, IB can only charge me for the SHORT position for one day and not from 24-27 Dec by ignoring my long call position was not there.  Further Facts:  My Broker ignored my answer and insisted the dates are correct and therefore I was charged to borrow shares that was need to satisfy the short position on exercised. So I am lost as I do not know whether the broker is correct and they can do this by ignoring my long call position after exercised or there is something they had not tell me. That is the reason I came to this site soliciting answers from those who know more than I do or had experienced the same.  I hope I have explained this clearly now and why would I want to give an opinion when I am looking for an answer ? E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable) One-dimensional vector space and it's subspaces I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign), functions and relations.  Now is the "=" taken as undefined? I have seen it been defined in terms of the identity relation.  But in order to talk about functions and relations you need set theory.  However,  seems to be a part of mathematical logic.  Does this mean that (naive) set theory comes before sentential and predicate logic? Is (naive)set-theory at the absolute bottom, where we can define relations and functions and the eqality relation. And then comes sentential logic, and then predicate logic?  I am a little confused because when I took an introductory course, we had a little logic before set-theory. But now I see in another book on introduction to proofs that set-theory is in a chapter before logic. So what is at the bottom/start of mathematics, logic or set theory?, or is it circular at the bottom?  Can this be how it is at the bottom?  naive set-theory $\rightarrow$ sentential logic $\rightarrow $ predicate logic $\rightarrow$ axiomatic set-theory(ZFC) $\rightarrow$ mathematics  (But the problem with this explanation is that it seems that some naive-set theory proofs use logic...)  (The arrows are of course not "logical" arrows.)  simple explanation of the problem:  a book on logic uses at the start: functions, relations, sets, ordered pairs, "="  a book on set theory uses at the start: logical deductions like this: "$B \subseteq A$", means every element in B is in A, so if $C \subseteq B, B \subseteq A$, a proof can be "since every element in C is in B, and every element in B is in A, every element of C is in A: $C \subseteq A$". But this is first order logic? ($(c \rightarrow b \wedge b \rightarrow a)\rightarrow (c\rightarrow a)$).  Hence, both started from each other? Given a subset $A$ of $\mathbb{R}^{n}$, a point $x \in \mathbb{R}^{n}$ is said to be in the boundary of A if and only if for every open rectangle $B\subseteq\mathbb{R}^{n}$ with $x\in B$, $B$ contains both a point of $A$ and a point of $\mathbb{R}^{n}\setminus A$.  My question is from Spivak's Calculus on Manifolds: Construct a set $A \subseteq [0,1]\times [0,1]$ such that $A$ contains at most one point on each horizontal and vertical line but has boundary equal to $[0,1]\times[0,1]$. Can a goalkeeper chip the ball to the head of a teammate from a goal kick to circumvent the backpass restriction? When to use "respectively"? I have been wondering what it means when people use "respectively" in, before, and after sentences. For example:     We are looking for a babysitter to pick up and supervise our kids ages   6 and 3, respectively, on Monday and Friday.   I don't know if this person used it correctly, but if they did what does it mean? If they didn't use it properly what would be the proper usage?  Thanks in advance. hyperlink name with biblatex authoryear (biblatex 1.4b) I'm new here so I'm not sure if this is the right way to express my question:  I'm using the excellent biblatex package with the author-year style and want the links in my citations to include the name of the author as well as the parentheses. The answer presented at  did exactly what I wanted but it seems the solution is broken with the newest biblatex version 1.4b. Can anyone help me by adapting the code to the newest biblatex version? Fixed buffers and interior buffers are not working for a polygon that contains a hole I'm following the online qgis manual available on qgis website. I'm doing the step 21.11.3. ().   To make it simple : I have a raster converted to vector, so now it is a polygon (several polygons). I'm doing an interior buffer exactly how it is described in the manual. I didn't obtain the same result. I can see the interior buffer just in some polygons, not in all of them ->    The interior buffers appear just for those polygons, that don't contain a hole. I use qgis 2.18.3.  This is how i created the interior buffer ->    In green you see the input layer : suitable_terrain, in blue the output layer, the interior buffer.   It is not a problem of a projection. I use WGS84 UTM Zone 34S where the units are in meters. I did also the Check validity and i obtain the same result. Which one is correct: '3 hour journey' or a '3-hour journey'? Which one of the following is correct (bold part)?      It took us quite a long time to get here. It was 3 hour journey.   or     It took us quite a long time to get here. It was a 3-hour journey. Old-style/Antique typesetting in LaTeX/TeX How can I prove that $xy\leq x^2+y^2$? How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ? Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa? How do I change the email address used for Google Play? My email address I use for the Play Store will be invalid soon, so I need to change it. However I don't want to make a new account because I want to keep all my apps. How can I do that? 1) a ten years old boy is sitting on the couch.      2) a ten year old boy is sitting on the couch.      3) a ten-years old boy is sitting on the couch.      4) a ten-year old boy is sitting on the couch.   Which is correct? accounting for LED resistance