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. M

    Discovering the Image of a Linear Map: Methods and Examples

    Hi all! Does anyone know a general method for determining the image of a lin map? I´m aware of the definition if im, but how could I determine it. Maybe it would be useful to show this on some examples :)
  2. K

    Degree of a Map: Finding Antipodal Points for Odd Degree Functions

    Homework Statement Two questions: 1) Show that deg(f(g(x)) = deg(f)*deg(g) 2) f: Sn -> Sn deg(f) is odd then show there exists a pair of antipodal points that are mapped to antipodal points The Attempt at a Solution 1) I have tried the method of just counting preimages, but i don't...
  3. glondor

    How Does Botany Impact Our Daily Meals?

    http://www.stumbleupon.com/toolbar/#url=http%2525253A//www.eigenfactor.org/map/maps.htm
  4. J

    Draw a contour map of the function showing several level curves.

    Homework Statement Draw a contour map of the function showing several level curves. f(x,y) = x^3 - y Homework Equations f(x, y) = x^3 - y The Attempt at a Solution I think I should be finding the domain and range, but other than that I am not sure what else I need to do.
  5. S

    Proving Nonempty Fibers of a Map Partition the Domain

    Homework Statement Prove that the nonempty fibers of a map form a partition of the domain. The Attempt at a Solution Ok so we have some map phi: S -->T And we want to show that its pre-image phi-1(t) = {s in S | phi(s)=t} forms a partition of the domain. Im really confused here. I assume...
  6. S

    How to Set Up a Karnaugh Map with Three Inputs and Binary Outputs?

    Derive the circuit that implements the state table http://silvercurvemedia.com/alex/flyers/state%20table.jpg The Attempt at a Solution I know you can get your equations for the circuit from either the table itself or through a Karnaugh map, but I prefer using a karnaugh map. How exactly would...
  7. S

    Conformal map to convert circle to a line

    I need a conformal mapping that would map an ellipse or a circle to a line. I need this for the http://physics.indiana.edu/~berger/p506_fall2008/p506ps6.pdf" . As far as I can understand, z^2 + 1/z^2 makes the geometry similar to that of a plane on the horizontal axis with a circle centered...
  8. J

    Understanding the Degree of a Continuous Map g:Circle --> Circle

    Hi, I am having some problems understanding the degree of a continuous map g:circle --> circle I have found a definition in Munkres (pg 367) that I can't really understand (I'm an engineering student with little algebraic topology) and one in Lawson (pg 181), Topology:A Geometric approach...
  9. J

    Proof that a specific map is an injective immersion

    Homework Statement Consider f: R^{m+1} - {0} -> R^{(m+1)(m+2)/2}, (x^{0},...,x^{m}) -> (x^{i} x^{j}) i<j in lexicografical order a) prove that f is an immersion b) prove that f(a) = f(b) if and only if b=±a, so that f restricted to Sm factors through an injective map g from Pm. c) show g...
  10. M

    Suppose T is a linear map and dim(Im(T))=k

    Please, help me! Suppose T is a linear map and dim(Im(T))=k. Prove that T has at most k+1 distinct eigenvalues. Thank you in advance!
  11. J

    Prove "dim U \geq dim V - dim W" Linear Map Question

    Homework Statement Suppose that V and W are finite dimensional and that U is a subspace of V. Prove that there exists T \in L(V,W) such that null T = U if and only if dim U \geq dim V - dim W. Homework Equations thm: If T \in L(V,W), then range T is a subspace...
  12. C

    NASA Gamma ray map of the galaxy completed by NASA

    Very cool image. Does anyone know, is anything in the information we got from this so far at all surprising? Is it likely we will learn anything about gamma ray bursts from this or is more information
  13. F

    Contour Map Help: Make & Label -50V to +50V Lines

    Homework Statement 50V | /\ | / \ | / \ |/ \ 0 |---1------2---3---- x(cm) | \ / | \ / | \/ -50V| best i could do, sorry don't know how to open picture up... its not to scale obviously but i...
  14. W

    Degree of a Map: Directly Proving k is Degree of f

    Consider f: S1 -> S1: (cos 2 pi x, sin 2 pi x) -> (cos 2 k pi x, sin 2 k pi x). How to show directly from the homological definition (without using Hurewicz etc) that degree(f) = k?
  15. wolram

    Can sound waves reveal the history of dark energy in galaxies?

    http://space.newscientist.com/channel/astronomy/cosmology/dn14546-biggest-3d-galaxy-map-to-probe-dark-energys-history.html In this cosmic cacophony, one particular note was louder than the rest, and it survives to this day as a characteristic wavelength in the clustering of galaxies...
  16. quasar987

    The (i,j)-minor of a matrix: multilinear map?

    Recall that for an nxn matrix A, the (i,i)-minor of A is defined as M_{ij}(A)=detA(i|j), where A(j|i) stands for the matrix (n-1)x(n-1) obtained from A by removing the ith line and jth column. Also note that we can view det as a map from R^n x ... x R^n to R taking n vectors from R^n, staking...
  17. F

    Need to map from cartesian R3 to a plane in R2

    Homework Statement This is actually a programming assignment, however it's very math involved. Given a set of points in R3 (x,y,z coordinates plus a weighted value) that are known to be coplanar, I need to draw an appropriately rotated, scaled, and colored plane intersecting the data. We...
  18. MathematicalPhysicist

    Proof of Quotient Map Q: A -> R for Exam Review

    Im reviewing material for the exam and came across this question: Let pi_1:RxR->R be the projection on the first coordinate. Let A be the subspace of RxR consisitng of all points (x,y) s.t either x>=0 or (inclusive or) y=0. let q:A->R be obtained by resticting pi_1. show that q is quotient...
  19. P

    Understanding the Power Map in Tensor Analysis: Bishop and Goldberg Page 6

    I'm a bit confused as to how the text Tensor Analysis on Manifolds, by Bishop and Goldberg on page 6. The authors define the term power set as follows _________________________________________ If A is a set, we denote by PA the collection of all subsets of A, PA = {C| C is a subset of A}...
  20. P

    Four Color Map Theorem: Proving the Impossibility?

    [SOLVED] The four color map?? so i guess most of you know about the four color map theorm. i read about it in a book a couple of days ago and had some idle brain time today while driving a tractor. i scribbled out some maps on the dirt on the windows and always seems to enclose one region...
  21. P

    Linear Map f:R^2 to R^3 with Given Inputs (1,2) and (2,1)

    Homework Statement Find the linear map f:R^2 \rightarrow R^3, with f(1,2) = (2,1,0) and f(2,1)=(0,1,2) Homework Equations The Attempt at a Solution I actually don't understand this task. PLease help! Thank you...
  22. S

    Representing map F in another basis

    Homework Statement Let V = P(3)(R) be the vector space of all polynomials P : R −> R with degree less than 3. We consider the mapping F : V −> V defined for all P belonging to V by F(P(x)) = P(−x) for all x 2 R. I have to represent the linear operator F as a matrix in the basis {1 + x, 2x, x2...
  23. quasar987

    Proving Injectivity of the Map T: L^p(E) --> (L^q(E))* for 1<p<2 and q>=2

    [SOLVED] Show map is injective Homework Statement Going crazy over this. Let 1<p<2 and q>=2 be its conjugate exponent. I want to show that the map T: L^p(E) --> (L^q(E))*: x-->T(x) where <T(x),y> = \int_Ex(t)y(t)dt is injective. This amount to showing that if \int_Ex(t)y(t)dt=0 for all...
  24. M

    Proving Injectivity and the Identity Map for Finite Dimensional Linear Maps

    The question is to prove for finite dimensional T: V to W, T is injective iff there exists an S: W to V such that ST is the identity map on V. I can't quite make the connection between injectivity and the identity map. any suggestions? thanks in advance.
  25. F

    Dimensions of the logistic map state space and quantum chaos.

    Apologies if this is the wrong forum, but I have a pair of thematically connected questions that I can't really fit anywhere else. Please move if this is better suited to the quantum physics forums. My first question being: The Poincare-Bendixon theorem states that chaos can only occur for...
  26. quasar987

    Linear Map Form: E x R to R [SOLVED]

    [SOLVED] form of a linear map Homework Statement Say E is a linear space (not necessarily of finite dimension), and R is the real numbers. Say we have a (contiuous) linear form T from E x R to R. Can we say T is of such and such a form? Particularily, can we say that T=g1+g2 where g1:E-->R...
  27. S

    Homoclinic orbit in the Logistic Map

    My question is the following: " For the logistic map x_{k+1} = rx_k(1-x_k) the band-merging point, where the period-1 orbit undergoes its first homoclinic bifurcation, is at r=3.678573510. Draw a trajectory to the map that illustrates the homoclinic orbit. " The period-1 orbit is at the...
  28. D

    How to map intervals of Real line

    Can someone explain how to create a function that will map an interval of the real line onto some other interval? Is there a general method? Can you demonstrate? (30 140) to (200, 260)? Thanks, Diffy.
  29. marcus

    Map of QG (Perimeter view, January 2008 update)

    At Perimeter Institute, Bianca Dittrich recently gave a survey introduction to (non-string) QG with Lee Smolin and Leonard Susskind asking questions among other. The talk video is online PIRSA 07120030 and the title was Introduction to Quantum Gravity I think of this as a kind of QG map from...
  30. C

    I'm trying to prove that a linear map is injective

    hello, I've been reading some proofs and in keep finding this same argument tyo prove that a linear map is injective viz, we suppose that t(a,c) = 0 and then we deduce that a,c = 0,0. is it the case that the only way a linear map could be non injective is if it took two elements to zero? i.e. t...
  31. Ivan Seeking

    News Can a New Democracies Organization Level the Playing Field in Globalization?

    I am a believer that globalization is not only unavoidable, but also that in the long term it is good for everyone. However, we see tremendous inequities between developed and developing nations in the labor and environmental protection laws, safety laws, enforcement of these laws, and oversight...
  32. marcus

    Map of nonstring QG (as of November 2007)

    QG has several approaches developing rapidly and it's not easy to maintain perspective. I'll update a rough outline map made earlier. We can use it to help know what papers to expect during the next couple of months and what developments to be prepared for. A. Three main sectors of...
  33. J

    What constitutes a quotient map?

    This is not directly a homework problem, so I opted not to place this question there. From what I have read/gathered from the internet/my textbook, a quotient mapping is any surjective, continuous mapping from a space X to a space comprised of the equivalence classes of all x in X from a...
  34. D

    Decide whether each map is an isomorphism

    Homework Statement Decide whether each map is an isomorphism (if it is an isomorphism then prove it and if it isn’t then state a condition that it fails to satisfy). Homework Equations f : M2×2 ---- P^3 given by: a b c d --- c + (d + c)x + (b + a)x^2 + ax^3 The Attempt at...
  35. D

    Diffrence between a transform a map

    Is there a diffrence between a map and a transform or are they the same thing? My math book uses the term map but i studyed transforms in lin alg and they seem like the same thing. please help me get this straight in my head.
  36. M

    What Is an Example of a Linear Map on R^4 Meeting Specific Dimensional Criteria?

    Homework Statement Give a specific example of an operator T on R^4 such that, 1. dim(nullT) = dim(rangeT) and 2. dim(the intersection of nullT and rangeT) = 1 The attempt at a solution I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2. I also...
  37. N

    All eigenvalues zero => zero map

    I want to prove that if all the eigenvalues of a linear transformation T : V --> V are zero, then T = 0. I think this is obvious but I'm having difficulty putting it into words. If all the eigenvalues of T are zero, then there exists a basis B for V in which [T]_B is the zero matrix. Thus...
  38. cepheid

    Can Karnaugh Maps Simplify Digital Logic Expressions Without Static Hazards?

    Homework Statement Draw the schematic circuit diagram that implements the following expression using as few basic gates as possible (AND, OR, NOT, XOR, NAND, NOR). The prime denotes the complement: f = w^\prime z^\prime + w^\prime xy + wx^\prime z + wxyz The Attempt at a Solution From the...
  39. P

    Proving the Existence of a Map: f, h, and g

    I have 2 maps f and h such f :\, (\mathcal{X}, \mathbb{E}) \rightarrow (\mathcal{Y}, \mathbb{K}) h :\, (\mathcal{X}, \mathbb{E}) \rightarrow (\mathcal{Z}, \mathbb{G}) where \mathbb{K} and \mathbb{G} are \sigma-algebras on the spaces Y and Z respectively, and \mathbb{E} =...
  40. E

    How Do You Return to the Oak Tree After Finding the Treasure?

    Homework Statement While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at 30degrees west of north, then 1.00 km 40.0 degrees north of east where you find a treasure. To return to the oak tree, in what direction would...
  41. DaveC426913

    How can I determine the contour lines for temperature readings on a map?

    I want to map the temperature gradient in my part of town. I live within spitting distance of the lake and want to demonstrate the lake effect. Ideally, I would make the geographical location of reading-taking as a constant (a grid of points across the area) and record the temperature value...
  42. P

    Is Quotient Map Closed? Proof and Explanation | Math Homework

    Homework Statement THe quotient map f is open but is it also closed? The Attempt at a Solution I think it is. Consider f: X->Y FOr every open set V in Y there exists by definition an open set f^-1(V) in X. There is a one to one correspondence between open sets in X and open sets in Y by...
  43. U

    What is the kernel of such a linear map

    Homework Statement This is a problem related to linear map over vector spaces of functions and finding kernels. Let V be the vector space of functions which have derivatives of all orders, and let D:V->V be the derivative. Problem1: What is the kernal of D? Problem2: Let L=D-I,where I...
  44. D

    Can a Proper Map Preserve Integral of Smooth n-Forms with Compact Support?

    Homework Statement Let f:R^n-->R^n be a C^oo proper map. Suppose there is a real number r such that f(x)=x for all x in R^n with |x|> r. Show that for every compactly supported smooth n-form w on R^n integral of f*w = integral of w. Here, integral is defined on R^n. Homework Equations...
  45. B

    Computing Relative Homology Groups from a Given Map

    Hi, I'm working on some homology problems but I need help figuring out the induced map from a given map, say f:X\rightarrow Y. For example, compute H_* (\mathbb{R}, \mathbb{R}^n - p) where p \in \mathbb{R}^n. So for n=1, we have the long exact sequence 0 \rightarrow...
  46. T

    Where Can I Find a Map of Our Galaxy?

    Hey all, I'm writing a piece of music for a competition in England and it requires a map of our galaxy. I've written off to NASA but I suspect I am not going to get any joy from them. In short, does anyone know where I can get a map of our galaxy from? It needs to show the position of the...
  47. L

    Is an Explicit Formula for Beilinson's Regulator Map Feasible?

    X is a smooth quasi-projective variety over Q. Beilinson's regulator map is a map from the motivic cohomology H to the Deligne cohomology H_D. Originally the motivic cohomology was defined by Beilinson as an eigenspace of an Adams operation on an algebraic K-group. Bloch (or Levine or someone...
  48. F

    C/C++ Fixing the Size of a Map in Shared Memory

    Hi, I'm wondering how to give a map a fixed size like you can give an array a fixed size by just "byte data[SIZE*DATASIZE]".. or if it's possible anybody know.. struct Table { map<int, int> table; // }; struct Table *flightChart; The reason I have to give this a fixed size, is...
  49. R

    How to Map a USB Printer to LPT1/2/3 in Citrix Metaframe XP 1.0?

    Hello, I am a new sysop for a company who uses Citrix Metaframe XP 1.0 and ICA remote connections. As far as I can tell, this version of Citrix does not support USB Printers when it auto-configures client printers for application use. However if the same printer is plugged into a LPT...
  50. A

    What is the relationship between holomorphic maps and elliptic functions?

    grrr, so annoyed, can't see the wood from the trees on this problem! I'm trying to get a holomorphic map from C/(Z+iZ) -> C/(Z+iZ) where C=complex numbers and Z=integers. Does this function have to be doubly periodic? Are doubly periodic functions the same as elliptic functions? Are all...
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