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. M

    MAP for multivariate gaussian?

    The standard multivariate gaussian is given by: http://upload.wikimedia.org/math/1/c/d/1cd250fc27ef7b7a9da469416333d07f.png taken from: http://en.wikipedia.org/wiki/Multivariate_normal_distribution The parameters can be estimated using...
  2. P

    Finding basis for kernal of linear map

    Homework Statement Let A = 1 3 2 2 1 1 0 -2 0 1 1 2 Viewing A as a linear map from M_(4x1) to M_(3x1) find a basis for the kernal of A and verify directly that these basis vectors are indeed linearly independant. The Attempt at a...
  3. W

    Identity map and Inverse Image

    Hello everyone, I would like to ask what's the purpose of identity map? Recently I came across something that apparently use this to find the inverse image of a function F(x) in the form of F(x) = ( f(x) , x ) . Thanks. Wayne
  4. B

    Shape operator and The Gauss tangent map

    Let M be a surface in R3 oriented by a unit normal vector field U=g1U1+g2U2+g3U3 Then, the Gauss Map G: M to E, of M sends each point p of M to the point (g1(p),g2(p),g3(p)) of the unit sphere E. Show that the shape operator of M is (minus) the tangent map of its Gauss map: If S and G are...
  5. B

    Tangent map, gauss map, and shape operator

    Can anyone help me with this problem?? Let M be a surface in R^3 oriented by a unit normal vector field U=g1U1+g2U2+g3U3 Then the Gauss map G:M\rightarrow\Sigma of M sends each point p of M to the point (g1(p),g2(p),g3(p)) of the unit sphere \Sigma. Show that the shape operator of M is...
  6. lisab

    News Nice Map Article - A Chuckle-worthy Story

    This gave me a chuckle. http://www.economist.com/world/europe/displayStory.cfm?story_id=16003661&source=most_commented Some folks have had http://en.wikipedia.org/wiki/File:Jesusland_map.svg" , too.
  7. CFDFEAGURU

    C/C++ Question on using <map> in C++

    Hello all, I am using 2008 C++ Express edition on a Windows XP machine and I have the following question regarding use of a map. How would you use a map to do the following: The user enters a pipe size, say 1/8" NPS, now there are three possible schedules for that pipe size. They are...
  8. B

    Apparent Contradiction: Every Map from a Contractible Space to any X is trivi

    Hi, everyone: I am confused about the result that every map from a contractible space X into any topological space Y is contractible. I think the caveat here is that the homotopy between any f:X-->Y and c:X-->Y with c(X)={pt.} is that the homotopy is free, i.e., the...
  9. J

    Diagonalisation of a linear map

    For the theorem: " If v1,...,vr are eigenvectors of a linear map T going from vector space V to V, with respect to distinct eigenvalues λ1,...,λr, then they are linearly independent eigenvectors". Are the λ-eigenspaces all dimension 1. for each λ1,...,λr.? Is the dimension of V, r? ie...
  10. H

    Complex analysis/holomorphic/conformal map

    D is be a bounded domain in the complex plane. Suppose f : D -->D is a holomorphic automorphism (conformal bijection). Now define f_n(z) = f(f(f(f ..(z) (composed n times ). Trying (and failing) to show: (i) the sequence {f_n} has a subsequence that converges either to a constant or to an...
  11. J

    How do I map a celestial object's declination ?

    to my local meridian at various times of the year. An object's declination will be fixed - relatively speaking - but its altitude on my local horizon varies from season to season because of Earth's movement, doesn't it? Also, if I'm at a lat. 47.3 N shouldn't celestial objects w/dec. of -{90 -...
  12. A

    Can We Map Reflected IR Laser to Object Color?

    Is it be possible to produce from the Intensity return of reflective IR laser the true color of the object causing the reflection of the laser? My though is that surfaces of the same material, but of different colors would cause slight changes in wave length of the laser light. These changes...
  13. R

    Contraction map of geometric mean

    I have the following mapping (generalized geometric mean): y(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N where p(j|i) is a normalized conditional probability. my question is - is this a contraction mapping? in other words, does the following equation have a unique...
  14. C

    Dot product of vector and symmetric linear map?

    Homework Statement My book states as follows: --- If u and v have the coordinate vectors X and Y respectively in a given orthonormal basis, and the symmetric, linear map \Gamma has the matrix A in the same basis, then \Gamma(u) and \Gamma(v) have the coordinates AX and AY, respectively. This...
  15. O

    Solving for a fixed point for a sine map

    Homework Statement Consider the sine map x{sub t+1} = f(x{sub t}) where f(x) = r*sin(x*pi). For r > 1/pi there are two fixed points, one at the origin that is unstable, and one elsewhere on the curve. The non-origin fixed point starts out, as you turn r just slightly above 1/pi, as stable...
  16. K

    Linear Map w/ Matrix: Solve for a + d

    Homework Statement Consider the map L from the space of 2x2-matrices to R given by: L([a b]) = a+ d ([c d]) For clarity, that's L(2x2 matrix) = a + d The Attempt at a Solution Im confused how any function of a matrix could possibly equal addition of two scalars, and thus have no...
  17. P

    How Do Karnaugh Maps Simplify Boolean Algebra?

    help me about karnaugh maps
  18. P

    Can 74ls ICs Help Simplify Karnaugh Maps?

    help me please
  19. A

    What is a phase map and how is it used in structured light 3D scanners?

    What is a "phase map"? I am doing some reading on Structured Light 3D Scanners using digital fringe projection, where a projector shines light (e.g. sinusoidal patterns) onto an object, a camera takes some pictures, and some software uses them to extract a 3D model of the object. The papers...
  20. H

    Proving the Henon Map is the Most General Quadratic Map with Constant Jacobian

    Homework Statement Show that the most general two-dimensional quadratic map with a constant Jacobian is the Henon map: xn+1=yn+1-ax2n yn+1=bxn, where a,b are positive constants. [/b] Homework Equations From the general quadratic map, xn+1=f1+a1xn+b1yn+c1x2n+d1xnyn+e1y2n...
  21. K

    Comp Sci How to navigate the Wumpus Map?

    I'm doing the movement part of the wumpus game. I'm sure many of you are familiar with it, but if not; basically the player starts out in room 1 and s allowed to move into any adjacent room, then from that room into any adjacent room and so on . The map files that are organized as such: 1...
  22. H

    Field of modulo p equiv classes, how injective linear map -> surjectivity

    Field of modulo p equiv classes, how injective linear map --> surjectivity Homework Statement Let Fp be the field of modulo p equivalence classes on Z. Recall that |Fp| = p. Let L: Fpn-->Fpn be a linear map. Prove that L is injective if and only if L is surjective. Homework Equations...
  23. H

    Matrix A for Linear Map T: R3→R3

    Homework Statement Determine the matrix A for the linear map T: R3→R3 which is defined by that the vector u first is mapped on v×u, where v=(-9,2,9) and then reflected in the plane x=z (positively oriented ON-system). Also determine the determinant for A. Homework Equations The...
  24. W

    Topology question - is this function an open map? sin(1/x)

    Homework Statement This problem is from Schaum's Outline, chapter 7 #38 i believe. Let f: (0, inf) -> [-1,1] be given as f(x) = sin(1/x), where R is given the usual euclidean metric topology and (0,inf) and [-1,1] are given the relative subspace topology. Show that f is not an open map...
  25. N

    Solve 0=u''+u*e^x | Poincare Return Map

    How do I write u''+u*e^x = 0 as a planar system?
  26. B

    Linear Map = Function of degree P-1

    If p is prime, prove that for every function f: Fp -> Fp there exists a polynomial Q (depending on f) of degree at most p-1 such that f(x) = Q(x) for each x in Fp.
  27. B

    Shhowing a map is well defined and bijective

    Homework Statement The group G acts transitively from the left on the set X. Let G_x be the little group of the element x \epsilon X. Show that the map i:G/G_x, i(gG_x)=gx is well defined and bijective. Homework Equations transitive action:for any two x, y in X there exists a g in G such...
  28. S

    IBEX, New Map of the Heliosphere

    Here is the press conference. https://www.youtube.com/watch?v=<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/mTnwjd8CF1c&color1=0xb1b1b1&color2=0xcfcfcf&hl=en&feature=player_embedded&fs=1"></param><param name="allowFullScreen" value="true"></param><param...
  29. W

    What is the command for dumping memory segment information in Unix/Linux?

    Here is a small segment that was dumped after an error in a program has occurred - that does a lot of memory allocation.b7ee6000-b7ee7000 rw-p 00157000 08:02 72280 /lib/libc-2.9.so b7ee7000-b7eea000 rw-p b7ee7000 00:00 0 b7eea000-b7ef7000...
  30. P

    Prove a map of a space onto itself is bijective

    Hi, Say F:A->A where A is a metric space and F is onto. I think it should be true that this implies that F is also one to one. Is there a way to formally prove this? Thanks.
  31. S

    Is the Transition Map Smooth in the Intersecting Set?

    Smooth transition map (easy!?) Homework Statement Check the transition map http://img132.imageshack.us/img132/4341/18142532.png is smooth in the set for which their images intersect The Attempt at a Solution I have thought of two ways to show this. (1) Show that Φ is a composition...
  32. F

    Prove that the bth projection map is continuous and open.

    I am trying to prove that the bth projection map Pb:\PiXa --> Xb is both continuous and open. I have already done the problem but I would like to check it. 1) Continuity: Consider an open set Ub in Xb, then Pb-1(Ub) is an element of the base for the Tychonoff topology on \PiXa. Thus, Pb is...
  33. J

    Determine whether the given map is an isomorphism

    Hello, I just cracked open this abstract algebra book, and saw a problem I have no idea how to solve. Instruction: Determine whether the given map \phi is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not? (Note: F is the set of all functions...
  34. T

    Exponential map of a lie group

    I am trying to read through this paper on the standard model. The ideas seem straightforward enough, but as always, I'm tripping over the "physicist's math" it uses. I was wondering if I can get some clarification or general guidance...
  35. B

    MATLAB Matlab Map Projection Plot: How to Plot Supernovae Coordinates on a Hammer Plot

    Hi there, I am trying to plot the coordinates of Supernovae onto what I think is known as a hammer plot i.e a 2D plot representing the surface of a sphere. I have no idea how to do this, and have been searching the internet to no avail. Can anyone offer any advice ? I only have a basic...
  36. S

    Basis for the image of a surjective linear map.

    Homework Statement Let V and W be vector spaces over F, and let T: V -> W be a surjective (onto) linear map. Suppose that {v1, ..., v_m, u1, ... , u_n} is a basis for V such that ker(T) = span({u1, ... , u_n}). Show that {T(v1), ... , T(v_m)} is a basis for W. Homework Equations Basic...
  37. L

    Differential of a map (mathematical analysis)

    Im not sure if this is the right section to post this question.. Calculate the differential of the map f:R^3 -> R^2 , (x ,y ,z)->(xy3 + x2z , x3y2z) at (1 ,2 ,3) in the direction (1 ,-1 , 4) I know how to get the differential of the map (finding the jacobian matrix) but the only part i am...
  38. J

    How Do You Prove Vector Spaces Are Isomorphic?

    Homework Statement For each of the following pairs of vectors, define an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a)P3 and R4 b)P5 and M(2,3) Homework Equations None The Attempt at a Solution a)I know that P3 and R4...
  39. J

    Degree of multiplication map of a topological group

    Hi. I'm trying to find the degree of the map of f(g,h)=g.h (i.e. multiplication in g) for fixed g. It is a map G-->G (if we fix g). We can assign a degree to this map for any topological group for which the last non-zero homology group is Z and proceed like we do for the degree of a map...
  40. L

    Is the Given Map Continuous and Bijective for Cantor Sets?

    Homework Statement Consider the map phi : C -> I which maps each point of the middle third Cantor set C, considered as a subset of real numbers between 0 and 1 written in base 3 and containing only digits 0 and 2, to the set of real numbers I=[0,1] written in base 2, according to the rule...
  41. B

    How can contour map data be used to determine volume?

    if you are given a contour map how could you calculate the volume of the hill.
  42. G

    Jordan Can. Form of Frobenius map

    Hello all, I am trying to solve this exercise here: Let \phi denote the Frobenius map x |-> x^p on the finite field F_{p^n}. Determine the Jordan canonical form (over a field containing all the eigenvalues) for \phi considered as an F_p-linear transformation of the n-dimensional F_p-vector...
  43. K

    Finding the map between two images

    Homework Statement Let D* be the parallelogram with vertices at (-1,3), (0,0), (2,-1), and (1,2), and let D be the rectangle D = [0,1] X [0,1]. Find a T such that D is the image set of D8 under T. Homework Equations Not much to say other than T(D*) = D The Attempt at a Solution...
  44. O

    Map vector A onto line l would that mean the projection of A

    If i were to say: Map vector A onto line l would that mean the projection of A onto l or the rotation of A onto l?
  45. J

    Group theory : inverse of a map.

    Homework Statement Let H, K be subgroups of a finite group G. Consider the map, f : H \times K \rightarrow HK : (h,k)\rightarrow hk. Describe f^{-1}(hk) in terms of h, k and the elements of H\cap K. Homework Equations HK = \{hk : h \in H, k \in K \} f^{-1}(hk)=\{ (h',k') : f(h',k')=hk...
  46. A

    Writing A Map As A Composition of Three Linear/Bilinear Maps

    Homework Statement Consider the map F: C0([a,b],Rn) --> R, F(\phi)=\int\phi(t)\phi(t)dt (This is the integral from a to b). Write F as a composition of three maps, each of which is linear or bilinear. Then use the chain rule and generalized product rule to compute DF(\phi). Homework...
  47. marcus

    Map of Bagger-Lambert papers-links, interviews

    Map of Bagger-Lambert papers--links, interviews http://sciencewatch.com/dr/erf/maps/08decerfBaggETRFM/#156486489 This is an interesting development in String/M, very recent, most papers just appeared in the past year or two. The map shows the most highly cited papers and their approx. degree...
  48. F

    Map Well-Defined: Proving Injectivity of a Map

    This is a general question... What is the difference between showing that a map is well-defined and that it is injective? To prove both can't you show that, given a map x, and elements a,b if x(a)=x(b) we want to show a=b.
  49. M

    Is Every Projection Map a Quotient Map in Topology?

    Is a projection a quotient map? I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that U is open/closed in Y iff (p)-1(U) is open/closed in X. And a projection is a map f:X-->X/~ defined by f(x)=[x] where [x] is the equivalent class (for a...
  50. N

    Automorphisms and Order of a Map

    Homework Statement Let G={(a,b)/ a,b\inZ} be a group with addition defined by (a,b)+(c,d)=(a+c,b+d). a) Show that the map\phi:G\rightarrowG defined by \phi((a,b))=(-b,a) is an automorphism of G. b) Determine the order of \phi. c) determine all (a,b)\inG with \phi((a,b))=(b,a)...
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