What is Map: Definition and 441 Discussions

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. R

    Show that a distance preserving map T:X->X is onto

    Homework Statement I'm trying to show that a distance preserving map is 1:1 and onto. The 1:1 part was easy, but I'm stuck on proving it's onto... Homework Equations X is compact T(X)\subseteqX THere's a hint saying to consider a point y in X\T(X) and consider the minimum distance...
  2. J

    MATLAB How can I overlay a GPS track on a city map using GIS software?

    I have a GPS track I'd like to plop on top of a map. The region of the track is very small, no bigger than the area of a city. I also have a bunch of shape files to build up a map of the city with all the buildings, roads, etc. in separate shp files. Although the shape files actually contain...
  3. M

    Why Can't There Be a Continuous Antipode-Preserving Map from S2 to S1?

    I don't understand the proof of this theorem: There is no continuous antipode-preserving map g: S2→S1. The proof is like this: Suppose g: S2→S1 is continuous and antipode-preserving. Take S1 to be the equator of S2. Then the restriction of g to S1 is a continuous antipode-preserving map h of...
  4. I

    How to compute the exponential map

    I need help calculating the exponential map of a general vector. Definition of the exponential map For a Lie group G with Lie algebra \mathfrak{g}, and a vector X \in \mathfrak{g} \equiv T_eG, let \hat{X} be the corresponding left-invariant vector field. Then let \gamma_X(t) be the maximal...
  5. E

    Differential of map from surface to surface

    Homework Statement Does anyone know the process for finding the differential of of f:S→S' where S,S' are surfaces. My textbook explains how to do this when f is a vector valued function but in the problem that I am working on I have something like f(x,y)=(g(x),h(x),j(y)) rather than something...
  6. J

    Homotopy between identity and antipodal map

    Homework Statement Prove that the identity map \mathrm{id}_{S^{2k+1}} and the antipodal map -\mathrm{id}_{S^{2k+1}} are smoothly homotopic. Homework Equations N/A The Attempt at a Solution My attempt: Fix k \in \mathbb{Z}_{\geq 0} and let \{e_i\}_{i=1}^{2k+2} be the standard basis for...
  7. P

    Map energy eigenstates to cartesian unit vectors - Harmonic Osillator

    Homework Statement Evaluate the matrix elements x_{nn'} = \left<n\left|x\right|n'\right> and p_{nn'} = \left<n\left|p\right|n'\right> and map the energy eigenstates \left|n\right> to Cartesian unit vectors. Homework Equations x = \sqrt{\frac{\hbar}{2m...
  8. S

    Inverse Maps: What Makes a Map Reversible?

    consider we have a map. what condition should have our map that it has inverse?
  9. Greg Bernhardt

    Wind Map US: Near Live Wind Trails Visualized

    Top down visual of near live wind trails in the US. Really neat! http://hint.fm/wind/ Now we just need it overlayed on google maps!
  10. F

    Mathematica (Mathematica) Poincarè map of restricted three body problem

    Hi everybody, sorry for the inconvenience. I try to plot the poincarè map of the restricted three body problem. I find in this forum the follow script that do this for the Lorenz system: mysol = NDSolve[{x'[t] == -3 (x[t] - y[t]), y'[t] == -x[t] z[t] + 26.5 x[t] - y[t], z'[t] ==...
  11. C

    Prove map σ:y→xyx⁻¹ is bijective

    1. Let G be any group and x∈G. Let σ be the map σ:y→xyx⁻¹. Prove that this map is bijective. It seems to be written strangely, since it never really says anywhere that y is in G, but I guess that must be an assumption.2. bijective=injective+surjective. in order to prove injective, we need to...
  12. H

    Showing differentiation is a linear map

    Homework Statement The Attempt at a Solution For part ii) I wrote it out as a matrix, getting \begin{array}{ccccccc} 0 & 0 & 0 & 0 & ... & 0 \\ 0 & 0 & 2 & 0 & ... & 0 \\ 0 & 0 & 0 & 6 & ... & 0 \\ . & . & . & . & . & . \\ 0 & 0 & 0 & 0 & ... & N(N-2) \end{array} So...
  13. L

    MATLAB Create Logistic Map M-File in MATLAB

    I want to make a m-file that show the behavior of the logistic map for di erent values of r using the bifurcation diagram. This is what i currently have, but i don't know why it is wrong. Can anyone help me? c=0; hold on while c < 4 y=0.5; for i=1:100; y =...
  14. D

    MATLAB Altering Matlab Code for a Logistic Map Cobweb Plot

    I found some Matlab code that works. However, I am not sure how to alter it for my needs. How can I make the code work for this:##N_{t+1} = \frac{(1+r)N_t}{1+rN_t}##What needs to be changed? %%% MAKES A COBWEB PLOT FOR A LOGISTIC MAP % compute trajectory a=3.0; % parameter x0=0.2...
  15. A

    Inner product-preserving map that isn't unitary?

    Suppose you've got a linear map U between two Hilbert spaces H1 and H2. If U preserves the inner product - that is, (Ux,Uy)_2 = (x,y)_1 for all x and y in H1 - is it necessarily unitary? Or are there inner product-preserving linear mappings that aren't one-to-one or onto?
  16. Rasalhague

    What {in,sur}jectivity of composite map implies for components

    I'm looking at Munkres: Topology Problems 1.2.4(c), 1.2.4(e), and 1.2.5(a). Problem 1.2.4(c) asks, "If g\circ f is injective, what can you say about the injectivity of f and g?" Problem 1.2.4(e) asks, "If g\circ f is surjective, what can you say about the surjectivity of f and g?" I concluded...
  17. Matterwave

    Understanding the Inverse of a Fiber Bundle Projection Map

    Hey guys, I've often seen in the definition of a Fiber bundle a projection map \pi: E\rightarrow B where E is the fiber bundle and B is the base manifold. This projection is used to project each individual fiber to its base point on the base manifold. I then see a lot of references to...
  18. TheStatutoryApe

    Google France Sued by Bottin Cartographes for Providing Free Map Services

    Google France Sued by Bottin Cartographes for Providing Free Map Services... So this is an interesting story... Unfortunately all the articles I have seen seem to be based on the same AFP article and the AFP article is rather lacking in information. Add to that my lack of french and I am...
  19. S

    A geometric property of a map from points to sets?

    I'm interested in the proper way to give a mathematical definition of a certain geometric property exhibited by certain maps from points to sets. Consider mappings from a n-dimensional space of real numbers P into subsets of an m-dimensional space S of real numbers. For a practical...
  20. N

    What does this notation mean? Linear map A = [A^\mu \nu]_\mu \nu

    It's used in a certain proof that I'm reading. A is a linear map from a vectorspace V onto itself. They say they can rewrite the vector space as \mathcal V = \bigoplus_\mu \mathbb C^{m_\mu} \otimes \mathcal V^\mu and I understand this, but they then claim one can (always, as any linear map)...
  21. Femme_physics

    Boolean Algebra VS Karnaugh map

    We were taught both methods to minimize gates. I frankly just want to pick one method all the time and become an expert in it, rather then try them both. So, according to your experience, which method do I better pick?
  22. O

    Find the minimum SOP function (Karnaugh map)

    first i hope is the correct forum to ask this kind of question. so i asked to find the minimum SOP function from the Karnaugh map (given in the picture). so i started to solve it as you can see in the picture. and this what i got:fsop=AB+BC+ACD'X' till here i think is ok. now this is...
  23. E

    Quick order preserving map question

    Homework Statement Let X and Y be ordered sets in the order topology. I want to show that a function f:X→Y is injective. We are given that f is surjective and preserves order. Homework Equations Definition of an order preserving map: If x≤y implies f(x)≤f(y) The Attempt at a...
  24. B

    Does there Exist a Continuous Map ?

    Hi, All: I saw this question somewhere else: we are given any two topological spaces (X,T), (X',T'), and we want to see if there is always at least one continuous map between the two. The idea to say yes is this: we only need to find f so that f-1(U)=V , for every U in T', and some V in T. So...
  25. M

    Is there a C^infty map that is one to one from R^n to R

    Is there a C^infty map that is one to one from R^n to R? Thanks.
  26. C

    What is the metaphysical dichotomy of objects and processes?

    Throughout most of the discussions I have had about science, philosophy, physics, math, and life in general over the past 35 years (since beginning my first undergraduate class in philosophy) there is one element of every discussion that returns. It has to do with the statement "the map is not...
  27. rhody

    Medical Ted Video: Allan Jones: A map of the brain

    http://www.ted.com/talks/allan_jones_a_map_of_the_brain.html?utm_source=newsletter_weekly_2011-11-11&utm_campaign=newsletter_weekly&utm_medium=email" http://www.ted.com/speakers/allan_jones.html" http://human.brain-map.org/explorer.html" Lots of explore and research here, cheers Paul...
  28. N

    Ordering Line Segments to form a 2D Polygon after slicing a 3D Tri Map

    I have a 3D shape described by a triangulation map i.e. a map between the vertices to the faces of the shape which are all triangles. I then sliced the shape by a plane and computed the intersections of the plane and the triangle faces. Each triangle face that intersects the plane, will have...
  29. D

    Anyone know of a good, detailed San Andreas Fault map?

    I live less than a mile from the SAFZ near Frazier Park, and would like to identify related surface features like escarpments, tuff outcrops, etc. After much Googling, I have found no maps that would help me locate the identified fault line locations within even 1000ft. None. Has anyone here...
  30. I

    Does anyone know the name of this type of map

    I have been exploring an algebraic structure with a map (_)* such that (x)*** = (x)* but in general it is not an involution. Also, the set of elements e such that e** = e do not form a substructure because they are not closed to addition. Has anyone seen such maps before, or know/can...
  31. M

    Chemistry: Road map question.

    Chemistry: "Road map" question. Borax can be converted to pure boron through the series of chemical reactions shown below: Pure boron is isolated in the final step of this reaction series. The starting material Y, is a non-polar gas with terminal chlorine atoms. Compound Y (2 mol) is heated...
  32. E

    How Can We Map Two Omega Tuples to One Bijectively?

    Homework Statement Find a bijective map : χωxχω\rightarrowχω Homework Equations An omega tuple is a function x:N\rightarrowχ, where χ is a set. χω is the set of all omega tuples of elements of χ. A bijective function is both injective and surjective. The Attempt at a...
  33. D

    Karnaugh map - Any reason to flip it, ever?

    Why, hello there. I'm doing Karnaugh maps. I'm using them to device gates to express the numbers 0, 1, 2, 3, 4 in a seven segmented digital display. Our teacher has provided us with predrawn Karnaugh maps, where we simply fill in the 1's and 0's. However, he's decided to invert the...
  34. D

    Showing that a map from factor group to another set bijective

    Let G have a transitive left action on a set X and set H = G_x to be the stabilizer of any point x. Show that the map defined by f: G/H \rightarrow X where f(gH) = gx is well defined, one to one, and onto. i think i know how to show well defined. letting g1 H = g2 H, if i multiply on the...
  35. jinksys

    Verify that any square matrix is a linear operator when considered as a linear map.

    Homework Statement Verify that any square matrix is a linear operator when considered as a linear transformation. Homework Equations The Attempt at a Solution If a square matrix A\inℂ^{n,n} is a linear operator on the vector space C^{n}, where n ≥ 1, then the square matrix A is...
  36. K

    Mathematica Can Mathematica handle complex values in iterative mapping?

    I have no programming experience and trying to get mathematica to do what I need it to do is frustrating. I have the following functions that I need to iterate. For notational purposes, k[t+1] is the value of K in the next period. w is a parameter. k[t+1] = -x[t] - y[t] + w x[t+1] =...
  37. S

    NASA release new world salinty from satellite map

    Don't know whether this belongs in chemistry or Earth science but there has been discussion about salinity here. http://www.bbc.co.uk/news/science-environment-15033532
  38. Y

    Relation between a map and its lifting

    I have the following question: Let $\mathbb{D}$ denote the unit disk. Let $f:X_1 \longrightarrow X_2$ be a continuous mapping between Riemann Surfaces. Let $ \pi_1 : \mathbb{D} \longrightarrow X_1$ , and $ \pi_2 : \mathbb{D} \longrightarrow X_2$ be the universal covering spaces of $X_1$ and...
  39. W

    Finding the Degree of a Map on S^3 and its Homotopy with the Identity

    Hi: More on Prelims: We have a map f: S^3 -->S^3 ; S^3 is the 3-sphere , given by: (x1,x2,x3,x4)-->(-x2,-x3,-x4,-x1). We're asked to find its degree, and to determine if f is homotopic to the identity. I computed that f^4 ( i.e., fofofof ) is the identity, and we have that...
  40. B

    Is the Mod2-Reduction Map Onto?

    Hi, Algebraists: The modN reduction map r(N) from a matrix group (any group in which the elements are matrices over Z-integers) over the integers, in which r is defined by r(N) : (a_ij)-->(a_ij mod N) is not always commutative, e.g.: r(6) :Gl(2,Z) --Gl(2,Z/N) is not...
  41. G

    Draw Map Based on Speed and G Forces

    Hi All, I am trying to draw a map, the raw data I have is Speed and G_Force (Lat and Acce). I have attached a XLS of the RAW data I have. Its from a race track, (I had to trim the file to 100kb, so it might or might not loop). I don't have a GPS, but I have seen software draw maps based on...
  42. N

    [Complex Analysis] Finding a conformal map

    Homework Statement I have to find a conformal map from \Omega = \{ z \in \mathbb C | -1 < \textrm{Re}(z) < 1 \} to the unit disk D(0,1) Homework Equations an analytical function f is conformal in each point where the derivative is non-vanishing specifically, we can think of linear...
  43. N

    [Complex Analysis] prove non-existence of conformal map

    Homework Statement "Show that there is no conformal map from D(0,1) to \mathbb C" and D(0,1) means the (open) unit disk Homework Equations Conformal maps preserve angles The Attempt at a Solution I don't have a clue. I thought the clou might be that D(0,1) has a boundary, and C...
  44. wolram

    Gravity Map of Earth: Have You Seen It?

    Has anyone ever drawn a gravity map of the Earth? i mean one that looks at mountain ranges, Vallie's, oceans, and shows the gravity (topology), or the shape of the deformation from a perfect sphere. edit, may be this should go in the Earth forum but i think it is more, gp.
  45. D

    Conformal map of unit disk to itself

    Homework Statement This problem is an already solved one in Marsden and Hoffman's Basic Complex Analysis, but I can't seem to work out the last step. Here's the problem: Suppose a,b,c,d are real and ad-bc>0. Then show that T(z) = \frac{az+b}{cz+d} leaves the upper half plane invariant. Show...
  46. W

    Cartesian products and the definition of a map

    Hello, I was wondering if there were alternative definitions to a "function" ( alternative to the standard f is a subset of A X B if f : A -> B ). I was introduced to the "general" definition of a cartesian product ( with respect to an indexing set H ) , it is weird to me because the general...
  47. narrator

    Spacetime, like an isometric weather map?

    Novice question :blushing: I've been reading Brian Greene's "The Hidden Reality". It occurred to me that spacetime across the 2D analog of the U could be much like an isometric weather map. Could it be that the U isn't expanding, but that our region, like a High on a weather map is rushing...
  48. S

    Proving the parametrization of a Torus imbedded in R3 is a Quotient map

    Homework Statement Let b > a > 0. Consider the map F : [0, 1] X [0, 1] -> R3 defined by F(s, t) = ((b+a cos(2PIt)) cos(2PIs), (b+a cos(2PIt)) sin(2PIs), a sin(2PIt)). This is the parametrization of a Torus. Show F is a quotient map onto it's image. Homework Equations Proving that any subset...
  49. B

    Map complex line to complex circle

    Homework Statement Find the Linear Fractional Transformation that maps the line Re\left(z\right) = \frac{1}{2} to the circle |w-4i| = 4. Homework Equations For a transform L\left(z\right), T\left(z\right)=\frac{z-z_{1}}{z-z_{3}}\frac{z_{2}-z_{3}}{z_{2}-z_{1}}...
  50. B

    Rainbow Spectrum Topology Map to Heightmap Grayscale.

    Hello, I am working with python's Image Library (PIL), Sympy, and Matlab. I have a topographical map of the earth, ( see 3d warehouse from google ). I am wondering if with an rgb matrix from a jpeg heightmap is traditionally the value of black and white ignored, because it seems that the...
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