Map Definition and 441 Threads

A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although most commonly used to depict geography, maps may represent any space, real or fictional, without regard to context or scale, such as in brain mapping, DNA mapping, or computer network topology mapping. The space being mapped may be two dimensional, such as the surface of the earth, three dimensional, such as the interior of the earth, or even more abstract spaces of any dimension, such as arise in modeling phenomena having many independent variables.
Although the earliest maps known are of the heavens, geographic maps of territory have a very long tradition and exist from ancient times. The word "map" comes from the medieval Latin Mappa mundi, wherein mappa meant napkin or cloth and mundi the world. Thus, "map" became a shortened term referring to a two-dimensional representation of the surface of the world.

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  1. Ben2

    How Do Right-Side Homotopies Affect Homotopy Equivalence?

    "[A] map f: X-->Y is called a \mathbf{homotopy~equivalence} if there is a map g: Y-->X such that fg\cong\mathbb{1} and gf\cong\mathbb{1}," where "cong" means "is homotopic." "The spaces X and Y are said to be \mathbf{homotopy~equivalent}..." Additional definitions are in Hatcher, "Algebraic...
  2. F

    B Map of M87 Galaxy & Universe: Find & Explore Here!

    Does anyone know where I can find the map animation of the galaxy / nearby galaxy cluster that illustrate the geometry between Earth and M87 at the 29th minute in the Netflix show ? Alternatively if that's not a public source, can anyone recommend a publicly available "map of the universe", I'd...
  3. S

    I A = norm-preserving linear map (+other conditions) => A = lin isometry

    I'm studying "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Stephen Newman. Theorem 4.4.4 in that book: The proof of part 2 is given like this: Seems a bit incomplete. I'd like to know if my approach is correct: $$\langle A(v+tw),A(v+tw)\rangle=\langle...
  4. S

    Python Animating 2D Heat Map with Varying Time Intervals

    Consider the following heat map: from scipy import special import numpy as np import matplotlib.pyplot as plt u0=200 r0x=25 r0y=25 rmax=2.5 alpha=2 t=0.575 y, x = np.meshgrid(np.linspace(0, 50, 100), np.linspace(0, 50, 100)) r=np.sqrt((x-r0x)**2+(y-r0y)**2)...
  5. BWV

    Is this map of metabolic pathways comprehensive or are there missing components?

    can get some idea of the complexity of biological systems from this graphic https://faculty.cc.gatech.edu/~turk/bio_sim/articles/metabolic_pathways.png
  6. G

    I The best ways to display a map of non-Euclidean 3-D space?

    You might have seen such a 3-D map of the stars before: And I was wondering about a 3-D map if space was non-Euclidean, what would be the best ways to display it. To keep it simple, if you're considering elliptic space, you can consider the entire "global" map. Just like we have world maps of...
  7. N

    Engineering Help with my Karnaugh map circuit: it does not work

    Hi! Good night ;) I have made the simplification of my circuit from the truth table using maps of. Everything looks good, however the simulation does not work in many cases according to the truth table. I notice that my boolean equation taken from the maps looks very big, is there the error...
  8. H

    I Gaia vs DESI: Comparing Largest Star Maps Ever

    I've read that ESA's Gaia space observatory has created the "richest map of our galaxy (and beyond)" https://www.esa.int/Science_Exploration/Space_Science/Gaia/Gaia_creates_richest_star_map_of_our_Galaxy_and_beyond Meanwhile, it's been widely reported tat the "The Dark Energy Spectroscopic...
  9. M

    Dynamical Systems: how to find equation for Poincare map?

    Hi, I was attempting a question on the dynamical systems topic of Poincare maps, and was struggling to understand a certain part of it. Knowledge from prior parts of the questions: There was a system which we converted to polar coordinates to get: (## a ## is an arbitrary real constant)...
  10. dlgoff

    How Are COVID-19 Cases and Deaths Trending Globally?

    I found this site with a map that shows COVID-19 cases and deaths. Scroll down to the bottom to see the US map. If you click on TOTAL at the top of that map, you can see the how many deaths...
  11. DaveC426913

    How can I effectively display different types of worker coverage on a map?

    OK, I'm stuck on a problem implementing this map. (It'll be built in HTML with JavaScript and CSS and is interactive but that's just context - this is really about map-colouring.) The function of the map is to help users see at-a-glance what regions/counties of the province are serviced by...
  12. M

    MHB Understanding Random Variable Mapping and Probability Functions

    Hey! :giggle: What does it mean to give the mapping for a random variable? Do we have to give the outcome space and the probability function? Does it hold that $X: ( \Omega, P)\mapsto \mathbb{R}$ ? :unsure:
  13. Euge

    MHB Projection Map $X \times Y$: Closure Property

    Here is this week's POTW: ----- Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed. -----
  14. S

    A New Dark Energy Map Challenges Einstein's Theory

    A few days ago a new dark energy map was published, which some claim "strays from Einstein's theory of relativity". To quote from the BBC News website: "Dr Niall Jeffrey, of École Normale Supérieure, in Paris, who pieced the map together, said that the result posed a "real problem" for...
  15. M

    Engineering 2D MAP Estimation with a Uniform Prior

    Hi, I was attempting the following question, but got confused on this part: Question: Two radar tracking stations provide independent measurements ##x_1## and ##x_2## of the landing site, ##\mathbf{x} = (x, y) ##, of a returning space probe. Both have Gaussian sensor models, ##p(x_i|X_i ) =...
  16. N

    How to Solve a Trigonometry Word Problem Involving Distances

    The sketch: First of all find the length of PQ on i.e $$4 \ inches - 1 \ mile$$ $$1.2 \ inches - x \ mile$$ $$x = \frac{1.2}{4} = 0.3 \ mile = 1584 \ ft$$ Now, I do not understand where shall I draw the horizontal, and the connection between the lengths of the contours, so I'll be grateful if...
  17. TimeSkip

    Nuclear meltdown in 2015 on antineutrino map?

    I was browsing a antineutrino map and was wondering if there was a meltdown in 2015 when it was taken. I never heard about it in the news; but, here are some pictures from the paper: https://www.nature.com/articles/srep13945 Thoughts?
  18. anorlunda

    Could This New Map Projection Revolutionize Our Understanding of the Globe?

    A recent thread Why Grid North doesn't agree with True North on maps had a lot of discussion on map projections. Here is a new contender for best way to project the globe in 2D. I am interested in their claim. "distances across oceans or across poles are both accurate and easy to measure."...
  19. Mr Davis 97

    Dimension of set of all linear maps that map three elements to zero

  20. D

    Java Java Tree Map always gets initialised but shouldn't, why is that?

    here is the portion of code [Code tags added by the Mentors] public void addPosition(String name, String skillNames) throws ApplicationException { Position ps=new Position (name); //creating object position if(listaPosizioni.get(name)==null) { //checking if position already present...
  21. Q

    A Is The Seiberg-Witten Map Unique?

    From my understanding the Seiberg-Witten map is a way to convert a non-commutative field theory into a commutative field theory. For example for the commutative relation between positions [x,y]=i*n, the common SW map I see in the literature for non commutative quantum mechanics is x->...
  22. M

    MHB Relations between map and matrix

    Hey! 😊 Let $1\leq m,n\in \mathbb{N}$ and let $\mathbb{K}$ be a field. For $a\in M_m(\mathbb{K})$ we consider the map $\mu_a$ that is defined by \begin{equation*}\mu_a: \mathbb{K}^{m\times n}\rightarrow \mathbb{K}^{m\times n}, \ c\mapsto ac\end{equation*} I have show that $\mu_a$ is a linear...
  23. bland

    I Why isn't the Goldstone boson on the Standard Model map?

    They seem to have something to do with Z or W particles, maybe, and they have something to do with mediating weak iso spin, I think, and they may be related to the Higgs field. But they don't have a box dedicated to them on the Standard Model of particle physics. Can someone tell me why this is...
  24. K

    B USGS Moon Map w/ Legend: Help Finding & Enlarging

    I went to the usgs site and found the pdf, but for some reason I could not enlarge the file to read it. Any body got a link to the map with a legend? TIA.
  25. BillTre

    What New Discoveries Await in NASA's Detailed Geological Map of Mars?

    NASA and the US Geological Survey have produced a new geological map of Mars. Better than before. Very detailed and interesting (to me anyway). Lots of download options. Here is one of my favorite Martian regions:
  26. M

    MHB Kernel of Linear Map: Show $\ker \phi$ Equation

    Hey! :o Let $1\leq n,m\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $(b_1, \ldots , b_n)$ a basis of $V$. Let $W:=\mathbb{R}^m$ and let $\phi:V\rightarrow W$ be a linear map. Show that $$\ker \phi =\left \{\sum_{i=1}^n\lambda_ib_i\mid \begin{pmatrix}\lambda_1\\ \vdots \\ \lambda_n\end{pmatrix}\in...
  27. V

    A The map from a complex torus to the projective algebraic curve

    I am following the proof to show that the complex torus is the same as the projective algebraic curve. First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##...
  28. J

    Comp Sci Modeling the queues at a college restaurant

    So far, I have implemented the code below, using Java Priority Queues and Maps. I tried to identify each customer by the time they came in (ranging from 360 and onwards) and their grades. However, this is my first time using priority queues and maps, and I am not very sure if I'm doing things...
  29. Decimal

    A Path between fixed points in a logistic map

    Hello, I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, ##f(x) = 4\lambda x(1-x)##. Let me then compare 1,2 and 4 iterations of this map on fixed points. I assume that ##\lambda## is large enough such that two period doublings...
  30. D

    Find distance using map measurements

    Summary:: That is the problem down there. The scale on a map is 1 inch (in.) ? 80 miles (mi). If two cities are 2 & 3/4in. apart on the map, what is the actual distance between the cities? So what I did was convert the scales 80/1 mi per in. and multiplied it to 2 & 3/4in which I got an answer...
  31. M

    MHB Understanding Symmetry Maps and Rotations in the Plane

    Hey! :o Let $F\subseteq \mathbb{R}^2$. A map $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ is called symmetry map of $F$, if $\pi(F)=F$. A symmetry map of $F$ is a map where the figures $F$ and $\pi (F)$ are congruent. Let $\pi_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$, $x\mapsto \begin{pmatrix}0...
  32. Fochina

    Proof: Relationship between a linear map and the associated matrix

    Hi! I don't understand how to demonstrate the following exercise. Let ##F: R^{n} \rightarrow R^{n}## be a linear map which is invertible. Show that if ##A## is the matrix associated with ##F##, then ##A^{-1}## is the matrix associated with the inverse of ##F##.
  33. M

    MHB Give a linear map that satisfies given properties

    Hey! :o Let $v_1:\begin{pmatrix}1 \\ 1\\ 1\end{pmatrix}, \ \ v_2:\begin{pmatrix}1 \\ 0\\ 1\end{pmatrix}\in \mathbb{R}^3$. Let $w=\begin{pmatrix}1 \\ 0 \\2\end{pmatrix}\in \mathbb{R}^3$. If possible, give a linear map $\phi:\mathbb{R}^3\rightarrow \mathbb{R}^2$ such that $\phi...
  34. DennisN

    Metabolic pathways - Interactive map

    Are you like me and think biology can be messy and difficult? Well, you're right. :nb) Here is a very cool, very large interactive map of metabolic pathways, which a friend of mine sent to me today: http://metabolic-pathway.com/fullMap.html And if you want to be extremely careful with your...
  35. lomidrevo

    B See the first precise pulsar surface map

    Link to article: https://www.nasa.gov/feature/goddard/2019/nasa-s-nicer-delivers-best-ever-pulsar-measurements-1st-surface-map Link to accompanying video:
  36. Cheesycheese213

    Best type of map (projection) for measuring distances?

    Hi! Sorry if this isn't the best question, but I've been trying to do a project where I thought it would be interesting to talk about coastlines. I downloaded a GIS (QGIS) to try and see if I could roughly measure all the continents for myself, but I haven't really done this before, so I wasn't...
  37. Monoxdifly

    MHB [ASK] Minimum Dimension of a Map

    The farthest distance of two places in an area is 200 km. If someone wants to make a map of that area on a 1 m × 1 m paper, the possible scale to make it is ... a. 1 : 210 b. 1 : 2.100 c. 1 : 21.000 d. 1 : 210.000 Can you help? The 200 and 210 makes me think that the distance on map won't be an...
  38. DaveC426913

    What do the tick marks on closed contours indicate?

    I'm trying to make a physical 3D map of this landscape. These isolated curves (red) have no altitude on them, so it's up to me to interpret. How do I know if these are hills or dips? Can I gather from the rising landscape (marked in magenta) that these are probably hills? Can I be sure?
  39. J

    Meteorology - USA Lower 48 National Winds Pattern Map Website

    For those interested but may not have found before, the below site provides a continuously updated animated map of wind pattern throughout the continental US PS I have a serious allergic reaction to a species of tree that grows NW of my location and during their pollination period I use this...
  40. K

    Velocity Map Imaging: Understanding 3D Reconstruction

    Hello! I am a bit confused about the image reconstruction for velocity map imaging. As far as I understand, at the interaction point, ions are produced in a Newton sphere which gets projected on a 2D screen (such that all the particles with the save velocity get mapped on the same point). What...
  41. jedishrfu

    B Is the Milky Way Warped? New 3D Map Reveals Surprising Findings

    https://www.inverse.com/article/58175-study-shows-the-milky-way-is-warped
  42. E

    Discontinuities in a Poincare map for a double pendulum

    I'm generating poincare sections of a double pendulum, and they mostly look okay, but some of them have weird discontinuities that seem wrong. The condition for these sections is the standard ##\theta_1 = 0## and ##\dot{\theta}_1 > 0##. Looking at one of the maps, we see that most of the...
  43. E

    I Derivative of the Ad map on a Lie group

    Hi, let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##. In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a...
  44. RichardWattUK

    I "Map" point between coordinate systems

    I've taken on a new job recently where I'm having to maintain an existing application that generates a points profile to drive a CNC machine and part of it projects points from an axial plane (which represents the machine's working axis) onto another plane which (I think) acts as as a...
  45. jedishrfu

    Interactive Map of Changing Climate City by City

    In this Science Alert article, an interactive map reference is provided to show the effects of climate change on your city in 2080: https://www.sciencealert.com/in-the-year-2080-your-city-will-feel-like-it-s-500-miles-away and the map: https://fitzlab.shinyapps.io/cityapp/
  46. Kitty123

    Map a one dimensional random walk to a two-state paramagnet

    1. The question asks us to map a one dimensional random walk to a two state paramagnet and then write an expression for the number of journeys of N steps which end up at r=Rdelta. Then we are asked to find an expression for the probability that N steps will end up at r. 2...
  47. B

    Healpix: How to convert from data to map?

    Hi. At first,I am sorry that my English is so bad. I've just installed Healpix on cxx(c++), and I'm starting to leraning. then I made txt file containing binary data(RING pixel and corresponding to data like temperture),and I want to make Healpix RING map of the data. but I don't know which...
  48. F

    MATLAB Limiting Map Boundaries in Matlab for Scientists

    Hello everyone. I have generated a world map by plotting coastlat and coastlon. I have done it because I want to plot some data that I have over said map. The problem is that my data goes only from 70N to 40 S and from 45E to 125W, thanks to that, I have a figure which looks like in the adjoint...
  49. joker1319

    Automotive "Engine fuel consumption map" without the torque possible?

    Hello, I am currently facing a problem. I'm on a project and trying to create an "engine fuel consumption map (engine torque / engine speed)" among other things. Unfortunately, the engine torque is not measured in my measurements. More specifically, the "actual engine - percent torque (OBD PID...
  50. J

    MHB Show the Linear Map is Diagonalizable

    Problem: Suppose $V$ is a complex vector space of dimension $n$, where $n > 0$, and suppose that $T$ is a linear map from $V$ to $V$. Suppose that if $\lambda$ is any eigenvalue of $T$, then $ker(T−\lambda I)^2 = ker(T−\lambda I)$. Prove that $T$ is diagonalizable. Here's what I think I need...
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