Prove that if f '(x) <= 1/2 for all x in the domain of f, then f has at most one fixed point.
If $f:\mathbb R \to \mathbb R$ satisfies $f'(x) <L$ for some $L<1$, then $f$ has a unique fixed point
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. QGIS calculates main soil type of polygons from another polygon layer I have some survey response data that are inconsistent and I'm not sure what is the best way to handle them.  This is the nature of the inconsistency:    Have you ever had a food poisoning episode that you think might have been caused by the food in a particular restaurant ? a) Yes b) No Did any of these episodes prevent you from eating at that restaurant again?  a) Does not apply, never have had a food poisoning episode b) Yes c) No Following these episodes, did you come back to your normal condition after 24 hours?  a) Does not apply, never have had a food poisoning episode b) Yes c) No   I believe that the intent of the designers of the survey was that anyone answering "No" to 1) should answer "Does not apply" to 2) and 3), but that's not how it worked. A large group of respondents picked "No" for 1) and "No" for at least one of 2) or 3).    What is the best way to deal with this inconsistency?    Where are some good references (books, papers, websites, etc.) to look at discussions and solutions for issues of this kind, and data cleaning in general? Are dark energy and zero-point energy the same thing? According to Quantum Mechanics is it possible that the famous  and  are the same thing that drives the accelerated expansion of the universe or maybe related to each other? Why am I getting no power when I pedal? Fuzzy string search library in Java Could someone provide me with a good explanation of why $0^0=1$?  My train of thought:  $x&gt;0$  $0^x=0^{x-0}=0^x/0^0$, so  $0^0=0^x/0^x=\,?$  Possible answers:   $0^0\cdot0^x=1\cdot0^0$, so $0^0=1$ $0^0=0^x/0^x=0/0$, which is undefined   PS. I've read the  on mathforum.org, but it isn't clear to me. How to use articles with scientific words, such as mass, force, impulse, field and others? How are articles used with scientific quantities, such as mass, force, impulse, field and others? I am writing a journal which has two column. I am using {figure} as well as {figure*} to place the figures in one column as well as stretch across two columns.   My problem is the order of the figures are not correct. e.g. their order of appearance is:                  Figure 1.                 Figure 2.                 Figure 5.                 Figure 3.                 Figure 4.   What should I do to make them appear in order? unable to printout arrayOfInt initialised within the class constructor $\limsup$ and cluster points Let $(x_n)$ be a sequence in $\mathbb R$. We call $y$ a cluster point of $(x_n)$ iff for every neighborhood $N$ of $y$ there are infinitely many $n$ such that $x_n \in N$. Let $C$ denote the set of cluster points of $(x_n)$.   By definition, $\limsup_{n \to \infty} x_n = \inf_n \sup_{k \geq n} x_k$.  Can you show me how to prove $\limsup_n x_n = \sup C$? Thanks. It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of physics to be preserved. But nevertheless the question about the generality of this approach keeps coming to my mind. My question is how one can be sure that this approach has to be always right and fruitful. Isn't it possible, at least from the mathematical point of view that a future theory of physics does not subscribe to this approach? I'm writing a latex document which contains a table. This table is doesn't fit the actual page so it's placed in the next page.  The problem is that vertical spaces are added automatically in the previous page in order to fill the page and I want to avoid that from happening.  I use H in order to add the table figure. Recently, I found this , that allows you to generate a 3D model from a 2D image.   It works very well and you can download an OBJ file. Vertex are colored and when I import it using Blender, I can't see any Vertex color.   Using Cycle (I prefer a internal solution if it exists) according to some answers I've found on Blender Stack Exchange, I patched some nodes : Attribute -> Diffuse BDSF -> Material Output.  Unfortunately, I can't get it works, the model stills black.    When I open the OBJ file, I see that color per vertex are there :  v 43.747554 107.000000 69.000000 0.470588 0.380392 0.309804 v 43.780163 115.000000 64.500000 0.427451 0.337255 0.266667 v 43.782527 107.000000 68.500000 0.470588 0.380392 0.309804 v 43.801902 115.000000 65.000000 0.427451 0.337255 0.266667 v 43.806461 107.000000 69.500000 0.470588 0.380392 0.309804 v 43.821703 115.000000 64.000000 0.427451 0.337255 0.266667 v 43.838352 108.000000 68.500000 0.466667 0.376471 0.305882 v 43.854670 106.000000 69.000000 0.482353 0.392157 0.321569 v 43.856176 108.000000 69.000000 0.466667 0.376471 0.305882 v 43.873876 106.000000 69.500000 0.482353 0.392157 0.321569 v 43.888855 110.000000 68.000000 0.466667 0.376471 0.305882 v 43.891471 109.000000 68.000000 0.466667 0.376471 0.305882 v 43.892733 110.000000 67.500000 0.466667 0.376471 0.305882 v 43.895170 108.000000 68.000000 0.466667 0.376471 0.305882 v 43.900904 109.000000 68.500000 0.466667 0.376471 0.305882 v 43.907584 107.000000 68.000000 0.470588 0.380392 0.309804 v 43.917898 115.000000 65.500000 0.427451 0.337255 0.266667 v 43.925000 111.000000 67.500000 0.454902 0.364706 0.294118   According to some answers here, Blender import more easily the vertex color of a PLY file so I tried to convert to PLY with two different sites :   1. meshconvert .com : Give me a 3.5 Mo file  2. greentoken.de/onlineconv : Give me a 2.3 Mo file  (Of course, if you know some other sites I could try, it interest me)  I guess that color data are lost with greentoken (not sure).   When I import the 3.5 Mo PLY file in blender, I need to be in textured display mode, then I think I see my vertex color value but in black &amp; white instead of RGB, that is weird. Anyway, I can't get it work  EDIT 1 : I opened PLY files and color vertex aren't there... I need to find a better converter   EDIT 2 : I tried www.ofoct.com/3d-model-file-for-3d-printer-converter it doens't take vertex color into account. In fact, in each PLY files I converted with the various sites here, there is a common line :      comment Created by Open Asset Import Library -    (v3.2.0)   (the version varies)  EDIT 3 : In fact it seems that model converted by Green token, keep the vertex color. That is weird because that's the smallest file :   ply format ascii 1.0 comment Created by Open Asset Import Library - http://assimp.sf.net (v3.1.187496374) element vertex 29793 property float x property float y property float z property float nx property float ny property float nz element face 59582 property list uint uint vertex_index end_header 82 103 6.37528 0.40963 0.0331735 0.911648 82 102 6.57043 0.272087 0.235193 0.933088 82.6603 103 6.5 -0.14071 0.043367 0.9891 82 103.935 6.5 0.201161 -0.136681 0.969976 81.5475 103 7 0.825371 0.0489484 0.562465 83 102 6.62244 -0.248588 0.147929 0.957247 83 103 6.56417 -0.411487 -0.0867124 0.907281 81.5819 104 7 0.787841 -0.0338904 0.614946 82 105 6.63886 0.0715144 -0.131383 0.988749 83 104 6.77321 -0.431167 -0.171652 0.885794 83.6781 102 7 -0.470137 0.061235 0.880467 83.6327 103 7 -0.543939 -0.0905203 0.834228 83 105 6.96709 -0.445324 -0.163091 0.880391 84 103 7.23724 -0.520118 -0.141377 0.842313 84 104 7.47775 -0.546517 -0.195833 0.814229 84 102 7.16782 -0.416323 -0.021798 0.908955 83.3508 101 7 -0.314426 0.156435 0.936304 84 101 7.22581 -0.280388 0.0301462 0.959413 84.9009 101 7.5 -0.339848 -0.0219793 0.940223 84.6939 102 7.5 -0.435242 -0.0925323 0.895546   As you can see, nx, ny and nz correspond to RGB values.. Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group. Let $G$ be a group and $c$ be a fixed element of $G$. Define a new operation $*$ on G by:   $$x*y=xdy$$   for all $x$ and $y$ in $G$, where $d$ is the inverse of $c$. Prove that $G$ is a group under the operation $*$.    If you want to check, this problem is in the book “A Course In Group Theory”, by John F. Humphreys (problem $3$, page $28$)   I know what are the axioms that $(G,*)$ must obey in order to be a group and I did show that all the axioms where obeyed except the inverse axiom.  My problem is that, whenever I try to verify what would the inverse be for some element $x$, I get an equation where the $x$ is cancelled in both sides.  Can please someone guide me?  Thank you!  (P.S.: This is NOT a homework assignment, I’m studying Group Theory by this book and got problems solving this question) i m using ubuntu 10.10. please tell me how to type in hindi fonts in open office word. please help I have a Redstone Power loop that connects to a Command Block that is set to kill all entities except for myself (/kill @e[type=!Player]), because I'm building a creative map, but don't want slimes all over the place. Problem is if I want to place an Item Frame than the Command Block kills it.  So, is there anyway to exclude items from getting deleted, or make it so it will only target mobs? What are the relative strong and weak points between tcolorbox and mdframed? I think practically we will not use both tcolorbox and mdframed in a single input file. Correct me if I am wrong. So we have to choose either tcolorbox or mdframed. However, without knowing the strong and weak points of both, we can only choose one by gambling.  Shortly speaking, what are the weak and strong points of both packages? The answers really influence the path of my life in TeX. Specifying latest revision of a particular branch with bower If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive