prove the equation by generating functions Im studying a book about stochastic processes and it is mentioned in the book that the equation below is true and we can obtain it by generating functions can anyone prove it. Thanks a lot. $$\sum_{m=0}^{\infty}\binom{2m}{m}p^m = (1 - 4p)^{-1/2}$$
Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$ How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.)
How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$ Let $n$ and $r$ be positive integers with $n \ge r$. Prove that   Still a beginner here. Need to learn formatting.  I am guessing by induction? Not sure what or how to go forward with this.  Need help with the proof. Is it possible to know who voted my question/answer up or down? If not, then is it possible to include this feature? By this feature we can come to know if you are being voted up by question owner or persons giving other answers to same question or just the visitors of that question thread.  Like...  If it is a group of people giving other answers, then your answer is most correct If it is readers of that question, then your answer is more popular rewrite access to collection to avoid "double" finding Home folders accessible to other users and should not be, how to fix? Why are my home folders accessible by other users on the same system by default?  I understand that I can change the permission of the home folders but want to know why this happened?  I have done nothing to make the folders "shareable" and yet I can easily navigate to any users home folder using Nemo 2.4.5 and see all their contents.  This is quite a privacy and security flaw and I'm not sure why it has defaulted this way on my system.  Under Users and Groups (items checked =) (both are administrator accounts)  User 1 = adm, cdrom, dip, lpadmin, plugdev, sudo  User 2 = User 2, sudo  Also of note is why User 2 has User 2 clicked but User 1 does not have User 1 clicked.  Both accounts appear to behave exactly the same.  Thank you for any tips in this matter. How do I identify which Linux distro is running? We have some new hardware in our office which runs its own customized Linux OS.   How do I go about figuring which distro it's based on? How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?  Edit:  This answer came to mind.  Is it correct?  This answer just came to mind.  By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$.  Define the function $f: \mathbb{N} \to \{0,1\}$ by $f(n) \ne f_n(n)$.  Then $f \notin\{f_n\}_{n \in \mathbb{N}}$. I have been experiencing an issue that once I have a material created and looking the way I want it, I cannot seem to export it correctly with the textures mixed.  I don't know the correct words to use, but in blender I have the material created using the nodes editor, and I have placed two textures inside of there (one for the rock texture and one for moss, my object is a rock). When I export this model and import to Unity (the game engine I am using), I find a materials folder but the material inside of it does not have the textures assigned, it's just a plain white material?  How can I export my material from blender in a way that it will allow me to import it WITH TEXTURES into Unity? I want to create a   How can I create a formfield with N fields, each 5 mm wide with separations like in the picture?    At the moment I use a workaround with a for loop, but this renders the code hard to maintain. Finally I need a function \formline{N}{cellwidth}{label}{value} to create the complete formfield and fill it with an optional value:   The following code is available on .   \documentclass[a4paper,11pt]{article} \usepackage[utf8]{inputenc} \usepackage{tikz} \usetikzlibrary{shapes}  \thispagestyle{empty} % no pagenumber. The form should be a stand alone macro... later.  \definecolor{SEPAOrange}{RGB}{254,213,161} \definecolor{SEPADOrange}{RGB}{253,185,19} \definecolor{SEPABlindcolor}{RGB}{255,0,0} \begin{document} \begin{tikzpicture}[x=1 mm, y=-1mm] \pgfmathsetmacro{\yh}{4.2333} % y heigth step defined as 1/6 inch =&gt; 1/6 * 25.4 mm = 4.2333 mm \pgfmathsetmacro{\xs}{9} % x start (own definition) \pgfmathsetmacro{\xe}{141.5} % x end (own definition) \pgfmathsetmacro{\widefield}{4.9859} % def: 134.62 mm / 27 \pgfmathsetmacro{\narrowfiels}{3.9594} % def: 134.62 mm/ 34  \filldraw[draw=black,color=SEPAOrange] (7.62, 4.5*\yh) rectangle (149.86-7.62,105.83-1); %orange background  \filldraw[draw=black,color=white] (\xs, 5*\yh) rectangle (\xe, 6.5*\yh -1.5); %Recepient 27 Char \foreach \x in {1,...,26} {  \draw[color=SEPAOrange, line width=0.3mm] (7.62 + \x*\widefield, 5*\yh ) -- (7.62 + \x*\widefield, 6.5*\yh);  \draw[color=SEPAOrange, line width=0.7mm] (7.62 + \x*\widefield, 5.8*\yh ) -- (7.62 + \x*\widefield, 6.5*\yh); }  \filldraw[draw=black,color=white] (\xs, 7*\yh) rectangle (\xe, 8.5*\yh -1.5); %IBAN 34 Char \foreach \x in {1,...,33}  \draw[color=SEPAOrange, line width=0.3mm] (7.62 + \x*4, 7*\yh ) -- (7.62 + \x*4, 8.5*\yh);  \filldraw[draw=black,color=white] (\xs, 9*\yh) rectangle (\xe, 10.5*\yh -1.5); %BIC \filldraw[draw=black,color=white] (\xe-12*5, 11*\yh) rectangle (\xe, 12.5*\yh -1.5); %Value 12 Char  \filldraw[draw=black,color=white] (\xs, 13*\yh) rectangle (\xe, 14.5*\yh -1.5); %Subject1 \filldraw[draw=black,color=white] (\xs, 15*\yh) rectangle (\xe, 16.5*\yh -1.5); %Subject2 \filldraw[draw=black,color=white] (\xs, 17*\yh) rectangle (\xe, 18.5*\yh -1.5); %Subject3  \filldraw[draw=black,color=white] (\xs, 19*\yh) rectangle (\xs+22*5, 20.5*\yh -1.5); % IBAN 22 Char  \filldraw[draw=black,color=white] (149.86-17.59, 19* \yh) rectangle (\xe, 20.5*\yh -1.5); % counter \filldraw[draw=black,color=white] (\xs, 21* \yh) rectangle (42.52, 24.5*\yh); % date \filldraw[draw=black,color=white] (42.52+5, 21*\yh) rectangle (149.86-17.59, 24.5*\yh); % signature  \draw[color=SEPADOrange] (0, 4.5 *\yh) --(149.86, 4.5*\yh); upper dark orange line \draw[color=SEPADOrange] (0, 20.5 *\yh) --(149.86, 20.5*\yh); lower dark orange line \draw[draw=black,color=black, line width=0.3mm] (0,0) rectangle (149.86,105.83); % black border %\draw[align=left] at (\yh,\yh) {SEPA-Überweisung};  \end{tikzpicture} \end{document} Footnotes layout in multicols environment I am using the package multicol to typeset a one-column (spanning the entire text width) heading and a two-column main text block. In order to do so, I begin the multicols environment along with the start of the text block. The one-column heading spans a small fraction of the text height in the front page and I need to set a footnote there, before starting the two-column environment. I'd like to have that footnote typeset under the first column in the two-column running text. How could I possibly do this?   Moreover, I'd like to have the footnotes typeset in a single column under one of the text columns -- I don't even mind having them as two-column footnotes, but what I do not want is a footnotes layout spanning the entire text width. I have tried with the command \twocolumnfootnotes from the class memoir, with the packages ledmac, dblfnote, and even with ftnright (the latter complains about its being used in a single-column document, due to that one-column heading). None has proved useful in this setting. As a less than MWE, just to convey a more concrete depiction of my scenario, let us consider:  \documentclass{memoir}  \usepackage{multicol}  \begin{document}  A single-column heading\footnote{Here I have my footnote.}  \begin{multicols}{2} Here I am in the two-column environment, and I want my footnotes being set in accordance with the document's ``columnwidth''.\footnote{A footnote in the two-column environment.} \end{multicols}   \end{document} The formal solution of the time-dependent Schrödinger equation Consider the time-dependent  (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi . $$ Usually, one likes to write that it has a formal solution of the form $$ \tag 2 \Psi (t) ~=~ \exp\left[-i \int \limits_{0}^{t} \hat{ H}(t^{\prime}) ~\mathrm dt^{\prime}\right]\Psi (0). $$ However, this form for the solution of $(1)$ is actually built by the method of successive approximations which actually returns a solution of the form $$ \tag 3 \Psi (t) ~=~ \hat{\mathrm T} \exp\left[-i \int \limits_{0}^{t} \hat{H}(t^{\prime})~\mathrm  dt^{\prime}\right]\Psi (0), \qquad t&gt;0, $$ where $\hat{\mathrm T}$ is the time-ordering operator.  It seems that $(3)$ doesn't coincide with $(2)$, but formally $(2)$ seems to be perfectly fine: it satisfies $(1)$ and the initial conditions. So where is the mistake? Two numbers are chosen randomly and independently from the unit interval $(0,1)$.  What is the probability that they differ by more than $\frac{1}{2}$? Hint: Think of the $2$ numbers chosen as coordinates in the Cartesian plane. Draw a square with vertices at $(0,0)$,$(1,0)$,$(1,1)$, and $(0,1)$. Allowable choices for the $2$ points lie below the line $y=x-\frac{1}{2}$ or the above $y=x+\frac{1}{2}$. Find the combined area of these regions.  I need help on this. I know what $y=x-\frac{1}{2}$ and $y=x+\frac{1}{2}$ looks like but I am not understanding the problem as a whole What hardware does Ubuntu Touch support? With the announcement of the Ubuntu Phone OS I'd like to know what phone (and tablet) models are supported at this time.  Note from foss &amp; Oli: We are making this the master question for all future "Will this work on &lt;insert random tablet/phone/device here&gt;?!" questions I recently reviewed a paper (general applied mathematics context). During the review, I realized that the fairly general ideas introduced in the paper can potentially be applied to a specific problem in a similar context. The authors did not mention this application in their paper, so I suggested to add a short comment about it in my review.  I later started thinking about the application in more detail and tried out some things, which actually seem to work quite well for this application. Now, the paper is still in the review process and might take some time until it gets published. What is the proper way how to proceed here in order to use the ideas and results which I in my exclusive position as a reviewer saw? How can I refer to the paper that is still under review? How can I best discuss the ideas with my co-workers (and potential co-authors) without disclosing that I reviewed the paper?  The paper under review is not available online on any preprint servers such as arXiv etc Beryllium Vacuum Sphere Boat/Aircraft Is it possible to make a solid rigid evacuated "balloon" out of Beryllium or other elements or alloys?  The critical buckling pressure at which an evacuated sphere is given as $$ P_1=\frac{2E\cdot\left(h/R\right)^2}{\sqrt{3\left(1-\nu^2\right)}} $$ where $E$ is , $h$ the shell's thickness, $R$ the radius to the midsurface shell, and $\nu$ is .  Of course this is theoretical and experimental and literature values for submersibles hover around 1/4 of the theoretical pressure.  Beryllium has the following properties:   Density=1.85 g/cm$^3$ Young's Modulus=287*10$^9$ Pa Poisson Ratio=0.032   Calculating the upward force, buoyancy at 101.3 kPa atmospheric pressure and Radius of 10m gives 4.24*10$^8$ N. The weight of this sphere with shell thickness 1mm has 2.32*10$^7$ N downward force of Beryllium material, so it will theoretically float at ground level.   Calculating the critical buckling stress of this beryllium sphere is 331.57 kPa, which is 3.27 times the atmospheric pressure of 101.3 kPa, so theoretically it won't buckle. But in experimentation done in earlier experiments as stated before, due to construction/material non-ideal situations, the critical buckling formula predicts about 4 times stronger than actual, so it would need about 400 kPa, not 331 kPa as Beryllium sphere with this specification would hold. Increasing the strength capability also increases weight and therefore lowers it's ability to float, but Beryllium is the closest element with this situation I have calculated that could work if ideal (I have only been using periodic table elements and household materials)  Is it possible with an alloy of elements? Using the equations for buckling and buoyancy, the following is needed. What alloys/materials come to your mind?  Material considerations:   Material with larger Poisson ratio Material with larger Young's Modulus Less dense material Construction (is it possible)   Material considerations affect on lift   Thickness can be larger, increases strength by the squared. But also increases weight. If a material with lower density with similar structural properties as Beryllium is used, it would help. Radius can be lower for same reason, but also lowers buoyancy (volume)   My other designs (don't worry about these just yet, just if you are curious is a vacuum sphere with a pressurized "outer tube" with spokes in tension (carbon fiber nanotubes maybe) to keep the 2 spheres in position. Another is a spherical truss out of carbon fiber nanotubes with a light membrane around. The truss prevents the outer membrane from collapsing due to the atmospheric pressure  The "point" of this if you ask, as to why not use a traditional balloon is it can be used as a boat by filling the sphere with air, and then used as a aircraft at will by evacuating the sphere. An alternative mode of transportation for sea vessels etc. I know references are syntactic sugar, so code is easier to read and write. But what is the difference between a pointer variable and a reference variable? Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show that $f(z)=0$. What is the current URL to create a new Google Document please? Since Google last updated their Docs the old URL does not work. How to prove that minimum of two exponential random variables is another exponential random variable? How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y) What does outputting to /dev/null accomplish in bash scripts? delegate keyword vs. lambda notation Once it is compiled, is there a difference between:  delegate { x = 0; }   and  () =&gt; { x = 0 }   ?