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. marcus

    Meissner sees Penrose-type circles in Planck CMB map

    Meissner et al just posted a paper where they see those circles in the high res. microwave sky of Planck. Who knows if this is real, or what it would mean if it were confirmed? Meissner has a followup paper in preparation with Penrose and others. Either way I think it's pretty interesting...
  2. N

    Why is 4 Colors Sufficient to Color a 2D Map?

    while it has been extensively proven that any 2D map can be colored with at most 4 colors, has any hypothesized why that is (outside the computer programmed brute force method)?
  3. M

    Linear algebra-Basis of a linear map

    Homework Statement Let ##L: R^{2} → R^{2}## be a linear map such that ##L ≠ O## but## L^{2} = L \circ L = O.## Show that there exists a basis {##A##, ##B##} of ##R^{2}## such that: ##L(A) = B## and ##L(B) = O.## The Attempt at a Solution Here's the...
  4. E

    Proving that a matrix is an inverse map

    OK, this is one where I am having trouble starting because I am not sure I am reading the question correctly to begin with. So start with: M_{m×n}(K) denotes the set of all matrices with m rows and n columns with entries in the field K. Let β⊂V and β′⊂W be bases of vector spaces V and W...
  5. T

    Mathematical map - diagram illustrating what you need to know to learn

    I was curious if anyone is aware of anything resembling a mathematical map, or learning tree. By this I mean a diagram that illustrates the sequence in which one must learn particular topics. I ask because I find myself encountering topics (today it was finite element method) and clicking the...
  6. H

    Why Does the Kraus Map Need to be Completely Positive?

    In Quantum Computation we define a map that takes on density matrix to another. It is represented by some kraus matrices. I do not know why it has to be completely positive.
  7. H

    Mapping the Concepts of Mathematics: A Comprehensive Guide

    Hi, Is there a concept map that actually shows all the concepts of math, and how they relate to each other? Kind like this one but more complete. I alreaady searched the web and didn't find anything, that's why I'm here so, Please Help. Thanks.
  8. U

    Find interval for contraction map

    Homework Statement Find an interval [a, b] for which the Contraction Mapping Theorem guarantees convergence to the positive fixed point or verify that there is no such interval. Homework Equations x = g(x) = \frac{14}{13} - \frac{x^{3}}{13} The Attempt at a Solution I know...
  9. B

    SO(2,1) - Haar measure, exponential map

    I wasn't quite sure where to put this, so here goes: I am trying to find out some facts about the group SO(2,1). Specifically; Is the exponential map onto? If so, can the Haar measure be written in terms of the Lebesgue integral over a suitable subset of the Lie algebra? What is that subset...
  10. Barioth

    Mathematica Calculate Pure Function in Map - Mathematica

    Hi! Let's say in mathematica I declare this function t[x_,y_]:= (x'[s]+y'[s])^2 Now I can call it with L=1; t[(#^2)+L &, (#^3)+L &] if I call it this way it will remplace the # with s and evalute the derivative. Now let's say I wana do this for for every L from 1 to 10. so i got...
  11. B

    MHB Calculating Scale on a School Map

    Not sure if this is the right category but i need help. On a map of a school, 3 inches represents 9 feet. How many inches would represent 1 foot 6 inches?
  12. K

    Surjectivity of induced map via hom functor implies injectivity

    Homework Statement Let R be an arbitrary ring, B and B' be left R-modules, and i: B' \to B be an R-module morphism. Show that if the induced map i^*: \operatorname{Hom}_R(B,M) \to \operatorname{Hom}(B',M) is surjective for every R-module M, then i: B' \to B is injective. The...
  13. M

    Drawing a contour map for e^(variable)sine(x-t*variable)

    Homework Statement Draw a contour map for T(x,t)=10e^{-\lambda x}\sin(\omega t-\lambda x) 0\leq\lambda x\leq2\Pi and 0\leq\omega t\leq2\Pi Homework Equations The Attempt at a Solution Because those two variables are within that given range I'm not sure how to do this...
  14. Fernando Revilla

    MHB Lisa's question at Yahoo Answers (Matrix of a linear map)

    Here is the question: Here is a link to the question: Linear Algebra Problem *Help Please*? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  15. G

    A multiplication map as a matrix in Q(sqrt2)

    Homework Statement Consider the field Q(\sqrt{2}), viewed as a vector space of dimension 2 over Q. Let r + s\sqrt{2} \in Q(\sqrt{2}), and defi ne the multiplication map M_{r+s\sqrt{2}}: Q(\sqrt{2}) → Q(\sqrt{2}) by M_{r+s\sqrt{2}}(\alpha)= (r+s\sqrt{2})*\alpha In other words...
  16. beyondlight

    MATLAB How to Create a Grid on a Map in MATLAB?

    I am working on a Bachelors thesis and need to make a grid on a map over a entire floor. Can someone show me how to do this in matlab?
  17. V

    How to determine whether a route overlaps another route in Google Map?

    I've two routes in google map: i) A ---> B ii) C ---> D How can I determine if these routes overlap or not?
  18. D

    Why Must the Image of U Be Open in \(\mathbf{R}^n\)?

    My textbook says that "a chart or coordinate system consists of a subset U of a set M, along with a one-to-one map \phi :U\rightarrow\mathbf{R}^n, such that the image \phi(U) is open in \mathbf{R}^n." What's the motivation for demanding that the image of U under \phi be open?
  19. N

    Karnaugh Map from Boolean Expression

    Hi, I'm trying to figure out a few questions on a practice exam that I'm working on for my Intro to Logic Systems class and could use some help. One of the questions (and the others are similar) says: Determine the minimized realization in the sum-of-produicts form using literals of the...
  20. P

    Why Can Every Element of SO⁺(1,3) Be Expressed as an Exponential?

    Hi! I was wondering why it is possible to write any proper orthochronous Lorentz transformation as an exponential of an element of its Lie-Algebra, i.e., \Lambda = \exp(X), where \Lambda \in SO^{+}(1,3) and X is an element of the Lie Algebra. I know that in case for compact...
  21. ssamsymn

    Is there a map from real numbers to non integers?

    Can you help me to construct a 1-1 mapping from real numbers onto non-integers? thanks
  22. D

    What is an "induced map"? Is it a Quotient Map?

    I have what I hope to be just a simple notation/definition question I can't seem to find an answer to. I'm not going to post my homework question, just a piece of it so I can figure out what the question is actually asking. I have a function i:A --> X I also have a continuous function g: A...
  23. Fernando Revilla

    MHB Surjective and injective linear map

    I quote an unsolved question from MHF posted by user jackGee on February 3rd, 2013. P.S. Of course, I meant in the title and instead of an.
  24. Fernando Revilla

    MHB Linear Map Problem: Proving Equivalence of a & b

    I quote an unsolved problem from MHF (Linear map problem) posted by user jdm900712
  25. J

    Linear Map Problem: Proving a and b Equivalent

    Homework Statement Let V be a vector space over the field F. and T \in L(V, V) be a linear map. Show that the following are equivalent: a) I am T \cap Ker T = {0} b) If T^{2}(v) = 0 -> T(v) = 0, v\in V Homework Equations The Attempt at a Solution Using p -> (q -> r) <->...
  26. B

    Post-Karnaugh map simplification

    So I have to implement a 4 input 1 output circuit. I am given the Karnaugh map (obviously a 4by4) and have to build the circuit. I have already determined the essential prime implicants for my map and three possible permutations of nonessential prime implicants. So let's say I pick a...
  27. K

    On continuous and locally one-to-one map

    Whether a continuous and locally one-to-one map must be a (globally) one-to-one map? If the answer is not. Might you please give a counter-example? Thank in advance.
  28. T

    Definition of the boundary map for chain complexes

    I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined. There's two definitions I've seen floating around. The first is at: http://en.wikipedia.org/wiki/Simplicial_homology The second, at...
  29. Fernando Revilla

    MHB Communitative ring, map R / ( I /\ J) -> R/I x R/J

    Commutative ring, map R / ( I /\ J) -> ( R/I ) x ( R/J ) I quote an unsolved question posted in MHF (November 25th, 2012) by user needhelp2. P.S. Communicative note: Of course I meant in the title, commutative instead of communitative.
  30. J

    Example of a Quotient Map That Is Neither Open Nor Closed

    We are just looking for an example of a quotient map that is not open nor closed. Let π: ℝxℝ -> ℝ be a projection onto the first coordinate. Let A be the subspace of ℝxℝ consisting of all points (x,y) such that x≥0 or y=0 or both. Let q:A -> ℝ be a restriction of π. ( Note: assume that q was...
  31. V

    What type of map can i project onto a sphere to eliminate distortion

    This is not actual "homework." I am building a POV globe and I want to get as accurate of a projection as possible. The images I will upload into the globe will be a simple 2 dimensional image projected onto a 3 dimensional plane. I have researched and it appears that a Mercator type map is the...
  32. C

    How Can Fixed Points Determine Solutions in Differential Equations?

    Homework Statement The Attempt at a Solution set x(t)=1+∫2cos(s(f^2(s)))ds(from 0 to t) then check x(0)=1+∫2cos(s(f^2(s)))ds(from 0 to 0)=1 then the initial condition hold, by FTC, we have dx(t)/dt=2cos(tx^(t)), then solutions can be found as fixed points of the map but for...
  33. N

    Differentiation Map of a Complex Transformation

    Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...
  34. STEMucator

    Is H Isomorphic to Z if H is an Infinite Cyclic Subgroup?

    Homework Statement Suppose H is an infinite cyclic subgroup of Z. Show that H and Z are isomorphic. Homework Equations We know that any infinite cyclic group H isomorphic to Z. H = <a> ≠ <0> |a| = ∞ The Attempt at a Solution Define f : Z → H | f(k) = ak for all k in Z. We...
  35. V

    Linear Algebra : Proving that Every map is an identity operator

    Suppose T belongs to L(V,V) where L(A,W) denotes the set of linear mappings from Vector spaces A to W, is such that every subspace of V with dimension dim V - 1 is invariant under T. Prove that T is a scalar multiple of the identity operator. My attempt : Let U be one of the sub spaces of V...
  36. C

    Please judge my statistics knowledge based on the following mind map

    How bad is my statistics knowledge based on the following mind map? Any concepts which aren't bold are the concepts that I know; the bold ones are the ones I'm currently learning. The mind map in question: http://i.imgur.com/4He3f.png What should I learn next based on my current...
  37. J

    Prove that a retraction is a quotient map

    Homework Statement As in title. Homework Equations Described in my attempt. The Attempt at a Solution Where do I go from here? I need to show that those 2 unioned sets are open in A. I'm not seeing it
  38. S

    Finding rank and nullity of a linear map.

    Homework Statement let a be the vector [2,3,1] in R3 and let T:R3-->R3 be the map given by T(x) =(ax)a State with reasons, the rank and nullity of THomework Equations The Attempt at a Solution Im having trouble understanding this... I know how to do this with a matrix ie row reduce and no. of...
  39. N

    Find the matrix representations of the Differentiation Map in the Basis

    Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...
  40. B

    Efficiency map of the drive train of a car

    Hi there, I'm reading a report about the efficiency of the drive train of an electric car. The author recorded the speed and acceleration of the car over a period of time and created the graph below to illustrate the efficiency. Could anybody tell me what the relationship is between the...
  41. A

    What does it mean for a Ring to be Stabilized by a map

    Homework Statement Let D be a division ring, C its center and let S be a division subring of D which is stabilized by every map x -> dxd-1, d≠0 in D. Show that either S = D or S is a subset of C. 2. The attempt at a solution I haven't actually started working on it yet because I am not...
  42. D

    Continuous Map to Single Point: Clarifying Confusion

    Hi all, I need help with a paragraph of my book that I don't understand. It says: "the map sending all of ℝ^n into a single point of ℝ^m is an example showing that a continuous map need not send open sets into open sets". My confusion arising because I can't figure out how this map can be...
  43. L

    Proving a surjective map iff the map of the inverse image is itself

    In the recommended format :) Homework Statement First we say that f:S→T is a map. If Y ⊆ T and we define f-1(Y) to be the largest subset of S which f maps to Y: f-1(Y) = {x:x ∈ S and f(x) ∈ Y} I must prove that f[f-1(Y)] = Y for every subset Y of T if, and only if, T = f(S). Homework...
  44. V

    Homomorphism as a Structure-preserving Map.

    The definition of a homomorphism is that it must preserve some algebraic structure, so if I transform a vector space using homomorphism between vector spaces (linear map), the result must be a vector space too, correct? Now, if "v" and "w" are two vectors in a vector space V, than "(v + w)"...
  45. T

    Sky Map Animation: 400,000 Galaxies

    A nice animation of sky map containing about 400,000 galaxies. http://www.youtube.com/watch?v=FUAu9ulnwRw
  46. S

    Logistic Map and cobweb diagrams

    Hi all. I am new here and I am having difficulty figuring out what exactly is required of me in this question. If someone could be so kind as to explain. For this part of the project we will consider the evolution of a discrete dynamical system given by a logistic map. We will consider a...
  47. S

    Well-defined map: transvections in symmetric space

    Hi! I'm trying to understand a proof for the fact that the isometry group of a symmetric space is a Lie group. The proof uses a lemma and I don't see how the lemma works. Here is the statement in question: (Let me give you the definition for \tau_v: Let M be a symmetric space and c:\mathbb R...
  48. A

    Understanding Monodromy Map of a Covering Space: Can Anyone Help?

    Hi! Suppose we have a topological space X, a point x\in X and a homomorphism \rho:\pi(X,x) \rightarrow S_n with transitive image. Consider the subgroup H of \pi(X,x) consisting of those homotopy classes [\gamma] such that \rho([\gamma]) fixes the index 1\in \{1,\ldots,n\}. I know that H...
  49. W

    Topology question concerning global continuity of the canonical map.

    Homework Statement If the set \Z of integers is equipped with the relative topology inherited from ℝ, and κ:\Z→\Z_n (where κ is a canonical map and \Z_n is the residue class modulo n) what topology/topologies on \Z_n will render κ globally continuous? Homework Equations The Attempt...
  50. K

    Why are you using a K map to implement pass transistors?

    Hello All, I am a Masters student in Microelectronics and stuck at something very trivial. In implementing Pass transistors using K Map, i am facing some probs. For eg. consider the function bc(bar) now if you draw a k map the left downward 4 blocks will be filled with 1s. I don't understand...
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