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. P

    Planck CMB map - what to expect?

    If I'm correct Planck will publish its all-sky CMB map (foreground deducted) this year. What's to be expected from this? Is another revolution coming?
  2. S

    Understanding Tangent Map Derivation in S.S. Chern's Ebook

    Hi, I am trying to understand the concept of tangent map and following the ebook of S S Chern. I am a bit confused about the derivation of the tangent map acting on the basis I tried for sometime to type out the equation but it appears I am having problems with the display and not sure what is...
  3. S

    Eigenvalues of a linear map over a finite field

    Homework Statement Let F be a finite field of characteristic p. As such, it is a finite dimensional vector space over Z_p. (a) Prove that the Frobenius morphism T : F -> F, T(a) = a^p is a linear map over Z_p. (b) Prove that the geometric multiplicity of 1 as an eigenvalue of T is 1. (c) Let F...
  4. A

    If f(x) = 0 for every bounded linear map f, is x = 0?

    Suppose you're looking at a complex vector space X, and you know that, for some x in X, you have f(x) = 0 for every linear map on X. Can you conclude that x = 0? If so, how? This seems easy, but I can't think of it for some reason. (EDIT: Assume it holds for every CONTINUOUS (i.e...
  5. D

    Proper continuous map is closed

    Homework Statement Prove a proper continuous function from R to R is closed. Homework Equations proper functions have compact images corresponding to compact preimages, continuous functions have open images corresponding to open preimages, in R compact sets are closed and bounded...
  6. M

    Diagonalizable map from f to f'

    Homework Statement Hi, i need to show if the map D: Vn maps Vn for f(x) maps to f '(x) is diagonalizable. I know how to do this with matrices i am given, but i don't know how to write D as a matrix. Homework Equations The Attempt at a Solution I'd really appreciate it if someone...
  7. F

    Quotient Map Theorem: Topology Induced by f

    Here is theorem 9.2 from Stephen Willard's General Topology: If X and Y are topological spaces and f:X\to Y is continuous and either open or closed, then the topology \tau on Y is the quotient topology induced by f. So f has to be onto doesn't it? Otherwise there will be multiple...
  8. F

    Defining an Integral for a Map x → g(x)

    Homework Statement Could someone define the notion of an integral for a map, x → g(x), x element of R2 or xn+1=g(xn thanks
  9. marcus

    Bianchi Haggard volume spectrum paper puts UC Berkeley on Lqg map

    I was glad to see this paper for several reasons. The volume operator in Loop Gravity is the locus of some interesting unresolved questions. The kind that requires and attracts creative mathematicians IMHO. This first paper from Gene Bianchi and Hal Haggard is just a 4-page letter I guess for...
  10. H

    Linear map from n-dim space to p-dim space

    I've been thinking about the following question: if x\in R^n and y=Cx\in R^p where matrix C describes the linear map from n dimensional reals to p dimensional reals. If we only have access to y and want to recover the information about x, which components in x are needed? I kind of figured out...
  11. P

    Is the Canonical Map Z to Zsubscript5 1-1 and Onto?

    Determine if the canonical map Z to Zsubscript5 is 1-1 and onto. Prove your answer Im not sure how to prove it but I am almost positive that its onto and not 1-1. I believe it onto because Z contains all the integers and Zsubscript5 contain the equivalence classes [0] [1] [2] [3] [4]. I...
  12. M

    Is this statement about the rank of a linear map true or false?

    Is this statement true or false if false a counterexample is needed if true then an explanation If T : U \rightarrow V is a linear map, then Rank(T) \leq (dim(U) + dim(V ))/2
  13. L

    No simple map between classical and quantum

    In Sean Carroll's GR book I found the following statement: there is no simple map between classical and quantum theories, - there are classical theories with no quantum counterpart - classical theories with multiple quantum versions - quantum theories without any classical analogue Could...
  14. M

    Determining dont-care values in a Karnaugh Map

    Ok I'm not sure if this question belongs here, but I am learning this in a CS class and the people at math.stack wouldn't know about this stuff, so here it goes. I'm having a hard time understanding how to find the don't-care values in a Kmap. What does it even mean? If I have a boolean...
  15. T

    Find the actual length of a wall on a map with scale 1:25000

    Homework Statement The scale on a map is 1:25000. The length of a wall on the map is 3.2 mm. Find the actual length in metres.Homework Equations The Attempt at a Solution
  16. J

    Is It Possible to Map Complex Numbers to Real Numbers Using a Unique Function?

    Can a function be defined such that for a complex argument z = x + iy, the function will uniquely map z onto the real number line? I have a hunch that this would not be possible, but if such a function existed, it could be used to define a unique ordering of the complex numbers without the need...
  17. R

    Proving Linear Map f is a Tensor of Type (1,1)

    Homework Statement Let V and W be vector spaces and let f:V\rightarrow W be a linear map. Show that f is a tensor of type (1,1) Can someone please show how to do this , I have no idea how to do it. Homework Equations The Attempt at a Solution
  18. C

    [Matlab] Correlation function for henon map

    Hi! I need to write MATLAB script, which will be plotting corretation function for two-dimmensial system. Henon map is not my system, but is very popular, so solusion for henon map can be very helpful for me. Have anybody know code for have plot like this...
  19. X

    Need example of a continuous function map cauchy sequence to non-cauchy sequence

    Homework Statement I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. Homework Equations If a function f is continuous in metric space (X, d), then...
  20. E

    Confusion on the definition of a quotient map

    Let X and Y be topological spaces; let p:X -> Y be a surjective map. The map p is said to be a quotient map provided a subset U of Y is open in Y if and only if p^-1(U) is open in X. Let X be the subspace [0,1] U [2,3] of R, and let Y be the subspace [0,2] of R. The map p:X -> Y defined by...
  21. radou

    Solving the Evaluation Map Problem for Metric Spaces X and Y

    Homework Statement This problem is a bit of a digression (at least it seems so) from the problems about imbeddings I'm dealing with currently (and I yet have a few more to complete). Let X and Y be spaces. Define e : X x C(X, Y) --> Y with e(x, f) = f(x). e is called the evaluation map...
  22. O

    Map the Sky: Charting Stars & Planets Pencil & Paper Style

    I was wondering how someone would go about charting stars on their own, pencil and paper style. Particularly, I was curious how astronomers charted the positions of stars and paths of planets hundreds of years ago, and was hoping to replicate this. I understand it's probably very involved, but...
  23. S

    Interesting feature on a weather map

    I was checking out a developing blizzard forecast for the northeast US today when I saw this interesting fog feature over the South Dakota, Nebraska, Oklahoma area. http://www.weather.com/maps/maptype/currentweatherusnational/uscurrentweather_large.html EDIT: The image is changing. It was the...
  24. T

    Derivative of the exponential map for matrices

    Homework Statement exp^\prime(0)B=B for all n by n matrices B. Homework Equations exp(A)= \sum_{k=0}^\infty A^k/k! The Attempt at a Solution Obviously I want to calculate the limit of some series, but I don't know what series to calculate. I wanted to try \lim_{h \to...
  25. H

    Convert grahpic map X/Y pixels from/to Latitude/longitude

    Hi all. I am doing a programming project where I have a map which is 879x436 pixels. At all endpoints I have the geographical coordinates in latitude and longitude corresponding to 0x0, 0x436, 879x0 and 879x436 pixels endpoints X/Y in all corners. My biggest problem is how to calculate the...
  26. E

    Why does a map from simply connected space to U(1) factors through R?

    I'm having trouble seeing why the following is true: let M be a simply connected manifold and s a smooth map from M to U(1). Then why does it follow that s = e^(iu) for some smooth function u from M to R? Thanks!
  27. radou

    Closed continuous surjective map and Hausdorff space

    Homework Statement Here's a nice one. I hope it's correct. Let p : X --> Y be a closed, continuous and surjective map such that p^-1({y}) is compact for every y in Y. If X is Hausdorff, so is Y. The Attempt at a Solution Let y1 and y2 in Y. p^-1({y1}) are then p^-1({y2}) disjoint...
  28. radou

    Closed continuous surjective map and normal spaces

    Homework Statement Let p : X --> Y be a closed, continuous and surjective map. Show that if X is normal, so is Y. The Attempt at a Solution I used the following lemma: X is normal iff given a closed set A and open set U containing A, there is an open set V containing A and whose...
  29. A

    Is there a 3D map of the naked-eye visible nightsky objects?

    I often look up at the sky and wonder how far away this or that star/object is. I know that most discernible stars are in our neighborhood, but I have a hard time figuring out just how close they are. Is there a map/graphic that would map all the objects we can see with a naked eye in our...
  30. M

    Is the Change of Basis Matrix in My Book Wrong?

    Homework Statement I have posted this problem on another website (mathhelpforum) but have received no replies. I don't know whether this is because no one knows what I am talking about or if it's just that no one can find a fault with my reasoning. Please please please could you post a reply...
  31. S

    New Dark Matter Map Solving Galactic Puzzle?

    http://blogs.nationalgeographic.com/blogs/news/breakingorbit/2010/11/new-dark-matter-map-hubble.html?source=link_tw20101112hubble" i found this article interesting , so shared here
  32. M

    Karnaugh Map for 4-bit Multiplication: How to Determine Values for Clusters?

    Make a circuit to multiply two 2-bit numbers AB and CD together to produce a 4-bit output. (a) construct a truth table to represent all states of the inputs and the corresponding outputs. (b) make a Karnaugh map for each output bit. Write down the Boolean algebraic expressions that describe...
  33. E

    Exponential Map of R3: A Closer Look

    I don not know whether I was right or not, please give me a hint. (R3,+) can be considered a Lie group. and its TG in 0 is still R3. suppose X as a infinitesimal generater, it can give a left-invariant vector field and also an one-parameter subgroup. but i think, this one-parameter...
  34. DaveC426913

    Is There a Map Projection Where Distances Remain Constant at All Latitudes?

    I want a projection of Earth where distances are undistorted. i.e. 10 degrees of latitude at the equator is exactly the same map distance as 10 degrees of latitude at the Arctic Circle. As a disqualified example, the Mercator Projection has map distance increasing with increasing latitude...
  35. B

    Homomophim and cannoical map assignement due in 13 hours

    urgent homomophim and cannoical map assignement due in 13 hours due now
  36. M

    Is V Isomorphic to R^2 Under the Given Mapping?

    Homework Statement Let V={a cosx + b sinx | a,b \in R} (a) Show that V is a subspace of the R-vector space of all maps from R to R. (b) Show that V is isomorphic to R^2, under the map f: V\rightarrowR^2 a cosx + b sinx \rightleftharpoons [ a over b ] (this is...
  37. M

    Show that a linear map is linearly independent

    Homework Statement Let f:V\rightarrow V be a linear map and let v\inV be such that f^n(v)\neq0 and f^(n+1)(v)=0. Show that v,f(v),...,f^(n-1)(v) are linearly independent. The Attempt at a Solution I'm really stuck with this one. I know the definition of linear independence and...
  38. A

    Understanding Singular Linear Maps: R^m -> R^n

    Homework Statement the question here said is L, linear transformation/mapping is singular? i'm still googling the definition singular linear map, can anyone give me the definition please T_T p/s; i thought it L maybe the matrix representation, but the question L : R^m -> R^n...
  39. H

    Smooth covering map and smooth embedding

    Now F:S^2->R^4 is a map of the following form: F(x,y)=(x^2-y^2,xy,xz,yz) now using the smooth covering map p:S^2->RP^2, p is the composition of inclusion map i:S^2->R^3 and the quotient map q:R^3\{0}->RP^2. show that F descends to a smooth embedding of RP^2 into R^4. Is the problem asked to...
  40. H

    Smooth manifold and constant map

    Suppose M and N are smooth manifold with M connected, and F:M->N is a smooth map and its pushforward is zero map for each p in M. Show that F is a constant map. I just remember from topology, the only continuous functions from connected space to {0,1} are constant functions. With this be...
  41. T

    What is the Image of a Plane Under a Linear Transformation?

    Homework Statement Let T: \mathbb{R}^3 \to \mathbb{R}^3 be the linear map represented by the matrix \begin{pmatrix} 4 & -1 & 0 \\ 6& 3 & -2\\ 12& 6 & -4\end{pmatrix} What is the image under T of the plane 2x - 5y + 2z = -5? Homework Equations None The Attempt at a Solution I...
  42. M

    Treasure Map Vector Help: Solving Directions and Distance for Buried Treasure

    Vector Help! Homework Statement The treasure map in the figure gives the following directions to the buried treasure: "Start at the old oak tree, walk due north for 490 paces, then due east for 150 paces. Dig." But when you arrive, you find an angry dragon just north of the tree. To avoid the...
  43. D

    Determining whether the map is an isomorphism

    Homework Statement Let F be the set of all functions f mapping R into R that have derivatives of all orders. Determine whether p is an isomorphism of the first binary structure with the second. 1. <F, +> with <R, +> where p(f) = f'(0) 2. <F, +> with <F, +> where p(f)(x) = \int^{x}_{0}...
  44. P

    Why the map get this direction as we see today?

    In all map, we always see the north on top, why's that? when was this way first used? Anybody know? Thanks
  45. 1

    How to Determine the Matrix of a Linear Map with a Non-Standard Basis?

    Homework Statement T(2,1)---> (5,2) and T(1,2)--->(7,10) is a linear map on R^2. Determine the matrix T with respect to the basis B= {(3,3),(1,-1)} Homework Equations The Attempt at a Solution matrix = 5 7 2 10 ?
  46. Fredrik

    The terms function and map .

    The terms "function" and "map". I have noticed that the term "map" is used more often than "function" when a map/function is defined using the "mapsto" arrow, as in "the map x\mapsto x^2 ". It has occurred to me that when a function is defined this way, it's usually not clear what the codomain...
  47. D

    Uncovering the Magic of Karnaugh Maps: A Closer Look at How They Work

    Why do Karnaugh map work? I don't understand how they work. If I follow the rules I get a minimized expression easily enough...it just seems like magic.
  48. X

    Inverse map of a one to one function

    I'm trying to show an inverse map composed with its noninverse results in the identity in terms of the set map f:X-->Y between topological spaces when f is one to one function. If I define the inverse map of a set as the disjoint union of the inverse map of each point in the set in Y, then...
  49. O

    Can Odd Degree Maps from S^3 to RP^3 Be Constructed Smoothly?

    It's easy to construct maps of even degree from the three-sphere to real projective three-space. Do there exist maps of odd degree?
  50. W

    How to construct a map from S^2 to RP^2 with covering time being unity?

    it is easy to construct a map from S^2 to S^2, with covering time being unity but how to do the similar task on the projected manifold RP^2=S^2/Z_2? i tried to use the stereographical trick the points on the lower half semisphere are projected onto the plane the problem is that the...
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