Mapping Definition and 364 Threads

Hello folks, I am trying to find a conformal mapping transform function that maps the following region in z-plane into interior of a unit circle in w-plane: |z-i|<\sqrt{2}\text{ ...AND... }|z+i|<\sqrt{2} Many thanks in advance for help & clues. Max.

I have a mapping function: x_{n+1} = \mu (1-x_n) I have some condition that occurs for: \mu (1-x_0) > 1 (1) which is: x_0 < 1- \frac{1}{\mu} but via the map function, there's an initial condition that leads to the above solution: **UNDER CONSTRUCTION, ERROR FOUND**

Homework Statement Show that the function w = e^z maps the shaded rectangle in Fig X one-to-one onto the semi-annulus in Fig y. Fig x is the rectangle -1<x<1 ; 0<y<(x+pi(i)) Fig y is the semi-annulus such that y>0 and -e<r<-1/e Homework Equations ... The Attempt at a...

Hello, My problem is as follows: I want to generate a series of 24 dimensional random numbers to act as the starting population for a genetic algorithm. These numbers need to fully span the space which is limited by a series of nonlinear boundary conditions. The 24 dimensional vector is...

Homework Statement Show that the mapping f carrying each point (x_{1},x_{2},...,x_{n+1}) of E^{n+1}-0 onto the point (\frac{x_{1}}{|x|^{2}},...,\frac{x_{n+1}}{|x|^{2}}) is continuous. [b]2. Continuity theorems I am given. A transformation f:S->T is continuous provided that if p is a limit...

Hey guys, I was wondering if you could help me out with a question I've got, I really don't know where to go or really where to start! Here's the question: Let S be a subspace of a finite dimensional vector space V. Show that there exists a Linear Mapping L: V → V such that the kernel of L is...

Can we think of a linear transformation from R^m-->R^n as mapping scalars to vectors? Let me say what I mean. Say we have some linear transformation L from R^m to R^n which can be represented by a matrix as follows: L=[ a11x1+a12x2+...+a1mx m a21x1+... . . . anmx1+...+ anmxm...

Hey I have a x-ray setup as in the figure, where alpha is the angles between the incoming x-ray beam and the sample. The x-ray are scattered, and measured by a 2D detector in the two outgoing angles. From this i will get a "slice" of the 3D recipricol map. If a want a 3D map, i think i will get...

A theorm I took down in class says: Consider the analytic function f(z). The mapping w=f(z) is conformal at the point z0 if and only if df/dz at z0 is non-zero. However, if df/dz does not exist at that point z0, is that point still a conformal mapping? That would make the function...

Hi all, I used below bash command to plot my data (.dat file) #!/bin/bash ( echo 'set term jpeg' echo 'set style data lines' echo 'set yrange [0:200]' echo 'set xrange[0:200]' echo 'set pm3d map' echo 'set palette defined (-2 "yellow", 0 "green", 2 "red")'for f in "$@" do # echo "Processing...

Homework Statement Using the precise denition of a function and a little logic, show that, for every set Y , there is exactly one function f from \emptyset to Y . When is f injective? Surjective? Let X be a set. Show that there are either no functions from X to \emptyset or exactly one...

I want to show that if the complex variables ζ and z and related via the relation z = (2/ζ) + ζ then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane. Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y? Any help would be...

Hi, I have the following problem that is solved, but I get lost at one step and cannot find how to do it in the notes. I would really appreciate it if someone could tell me where my teacher gets the result from. The problem says: "Find the matrix of linear mapping T:P_3 → P_3 defined by...

hi, I need a conformal mapping that changes the superellipse to an easier shape. if anyone send me any helpful thing (relative article, idea) I will be so pleased.

Homework Statement From Serge Lang's "Linear Algebra, 3rd Edition", pg 51 exercise 9. Prove that the image is equal to a certain set S by proving that the image is contained in S, and also that every element of S in in the image. 9. Let F:R2→R2 be the mapping defined by F(x,y)=(xy,y)...

Attached is my attempt at a proof. Please critque! :shy: Thank you!

Homework Statement http://img684.imageshack.us/img684/779/334sn.jpg The Attempt at a Solution The first part was fairly straightforward, solve for z + 1, and then get w in terms of u + iv, rationalise the denominator, and then we get (x,y) in terms of u and v, which we substitute back...

Prove or disprove that f is a quotient mapping. f:R^3\{(x1,x2,x3):x1=0}--->R^2 defined by (x1,x2,x3)|->(x2/x1,x3/x1)

I have 2D elements distributed in a space of [-4, +4] and want to convert any point in the 2D space to a 1D real-valued number 0~1.0 such that 1st quadrant [+, +] should have higher values (importance) suppose 0.4~1 , 2nd and 3rd quadrant [+, -] and [-, +] should be next 0.2~0.4, and the 4th...

Hi, Can someone here help me understand how to illustrate maps of analytically-continuous paths over algebraic functions onto their normal Riemann surfaces? For example, consider w=\sqrt{(z-5)(z+5)} and it's normal Riemann surfaces which is a double covering of the complex plane onto a single...

Homework Statement find linear fractional transformation from D={z:|Arg z| < \alpha}, \alpha≤\pi to the upper half plane Homework Equations The Attempt at a Solution The problem I am having here what exactly D is.. (visualizing it) D is just z such that |Arg z|≤\pi right? so...

Can't find (or maybe recognize when I see it) anything that discusses this question: A group G is a set of members. We normally assign familiar labels on the members such as a five member group with members labeled as 0, .. , 4. Then, a group operation + is defined as GxG -> G so that a look...

Homework Statement Find the image of the circle |z| = 3 in the complex plane under the mapping a) w = \frac{6}{z} b) w = \frac{6}{z} + 2i The Attempt at a Solution a) w = \frac{6}{3} = 2 So this is a circle in the w-plane of radius 2, centered on the origin? b) w =...

Homework Statement part ii of http://gyazo.com/0754ea00b2a4ea4a4d171906f6bf28bf Answers http://gyazo.com/821f370c502cd20210925f8498d18fa1 Homework Equations I did part i. I had to spot that 1/(x+iy)^2 = 1/(x^2+y^2)^2... (I subbed y = y-1) is this a standard result? Should...

Homework Statement let f(z) be a 1-1 analytic mapping of unit disc |z|<1 onto itself with two fixed points in |z|<1 Show that f(z)=z Homework Equations none The Attempt at a Solution I'm thinking there has to be a theorem or something that I am missing for this.. But I'm not...

Prove that the function F(x) = 4x(1-x) maps [0,1] into itself and it not contraction to prove it is not contraction it is enough to prove that there exist a number in [0,1] such that the first derivative exceed 1 F'(x) = 4(1-x) - 4x = 4 - 8x 4-8x > 1 \Rightarrow \frac{3}{8} > x...

Suppose f is a biholomorphic mapping from Ω to Ω, if f(a) = a and f'(a) = 1 for some a in Ω, can we prove that f(z) = z for all z in Ω?

Describe the image of the strip $\{z: -1 < \text{Im} \ z < 1\}$ under the map $z\mapsto\dfrac{z}{z + i}$ So I know that $-\infty < x < \infty$ and $-1 < y < 1$. Then $$ \frac{x + yi}{x + i(y + 1)} $$ Now if I take the the line y = -1, I have $$ \frac{x-i}{x} $$ Then find out what happens...

Find necessary and sufficient conditions on the real numbers $a$, $b$, $c$, and $d$ such that the fractional linear transformation $$ f(z) = \frac{az + b}{cz + d} $$ maps the upper half plane to itself. I just need some guidance on starting this one since I am not sure on how to begin.

Homework Statement The transformation z=1/2(w + 1/w) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface. (b) Use...

Homework Statement The transformation z=\frac{1}{2}(w + \frac{1}{w}) maps the unit circle in the w-plane into the line −1≤x≤1 in the z-plane. (a) Construct a complex potential in the w-plane which corresponds to a charged metallic cylinder of unit radius having a potential Vo on its surface...

Hi all, Suppose there is a bump at the origin, is there a conformal mapping between the bumped half-space (y>|b-x|, |x|<b && y>0, |x|>b) and the flat upper half space (y>0)? Anyone has a hint? Thanks in advance. Regards, Tony

Homework Statement I am trying to run a model in matlab. D is a 2 by 3 matrix, Knowing that DL=0, which means L is mapped to the null space. Homework Equations How can i find L so that it is a 3 by 3 matrix with all its entries being one times a scalar. The Attempt at a Solution...

When f maps E into a metric space Y: (E is subset of metric space X) Is it eqivalent to say that f is a continuous mapping and that for a subset E of X, to say that for every p element of E, f is continuous at p.? thank you

Homework Statement Let a be a complex number for which Im(a) ≠ 0, and f(z) = (z + conj(a))/(z + a). Prove f(z) maps the real axis onto the circle lwl = 1. 2. The attempt at a solution I wrote out f(z) in an a+bi for and then with the Im(a) ≠ 0 I set the equation as f(a+bi) =...

Homework Statement As in in title. Homework Equations Open mapping: maps open sets to open sets. The Attempt at a Solution Not sure.

Homework Statement The point 1 + i is rotated anticlockwise through \frac{∏}{6} about the origin. Find its image. The Attempt at a Solution The point 1 + i creates an angle of arctan(1/1) = ∏/4 The rotation is by a further angle β = ∏/6. So the new point w in the w-plane from...

Homework Statement Prove whether the below equations are linear or not. (iii) U = P^2 -> V = P^6; (Tp)(t) = (t^2)p(t^2) + p(1). (iv) U=P^2 -> V =P^6;(Tp)(t)=(t^2)p(t^2)+1. Homework Equations None. The Attempt at a Solution I really don't know. Thanks Tom

Homework Statement Let F be the set of all functions f mapping ℝ into ℝ that have derivatives of all orders. Determine whether the given map ∅ is an isomorphism of the first binary structure with the second. {F,+} with {ℝ,+} where ∅(f)=f'(0) Homework Equations None The Attempt at...

Homework Statement Let (X,||\cdot||) be a normed vector space and suppose that Y is a closed vector subspace of X. Show that the map ||x||_1=\inf_{y \in Y}||x-y|| defines a pseudonorm on X. Let (X/Y,||\cdot||_1) denote the normed vector space induced by ||\cdot||_1 and prove that the...

Homework Statement Compute values for the electric field at four different points on the point-line plate. Comment on the validity of your values. Homework Equations E = F/q ΔV = ∫E dot ds The Attempt at a Solution I have attached 3 electric field maps that I did in the lab and...

Homework Statement I'm working on a problem that implements the product of two 2-bit binary numbers (wx, yz) and produces such as the output. However, I am having a bit of confusion in regards to implementing the Karnaugh Map. So this is what I have: wx and yz can be 00, 01, 10, 11, the output...

Homework Statement How might I show that the map (in cylindrical polar coordinates) given by f:(1,\phi,z)\to(\sqrt{1-z^2},\phi,z) does not change the area?Homework Equations The Attempt at a Solution I can see this is like having a sphere in a cylinder and we shine "light" on the cylinder...

Given a measurable function f that is not real- or complex valued, but that maps into some vector space, what are the necessary conditions for it to be integrable? I've looked through over 20 books on integration and measure theory, but they all only deal with integration of real (or...

If you reverse the polarities of the electrodes when you are "electric field mapping", wouldn't it show the same results on paper? Just the path it takes is reversed, but you would never know the difference wether it's going away or towards correct? I am diving into this "Electrical Field...

Homework Statement Let the Complex mapping z → f(z) =(1 + z)/(1 − z) 1.What are the images of i and 1 − i and 2.What are the images of the real and the imaginary axes? The Attempt at a Solution For i we have f(i)=(1+i)/(1-i) since i(depending on the power) can be i,-i,1,-1=>0...

Homework Statement Determine the image of the line segment joining e^(i*2*pi/3) to -e^(-i*2*pi/3) under the mapping f = e^(1/2*Log(z)). Homework Equations The Attempt at a Solution The line joining the two points: {z | -0.5 < x 0.5, y = sqrt(3)/2} f = the principle branch of...

Say we have a complex function f, analytic on some punctured open disk D\{a} where it has a pole at a. Is there some theorem which says something like: f must map D\{a} to a horizontal strip in ℂ of at least width 2π, or something like that?

Homework Statement Consider the open interval (0,1), and let's S be the set of points in the open unit square that is, S={(x,y):0<x,y<1}. Find a function that maps (0,1) into S. but not necessarily onto. The Attempt at a Solution so I can describe any point in my square with x and y...

Homework Statement I was reading my textbook and i encountered this...--->> " For instance if f,g,h are in A(S) and fg = fh then g=h " I understand this part... because we can take the the inverse of f both sides and say g=h. then it says--->> " If gf = f^(-1)g but since f ≠ f^(-1)...