Mapping Definition and 364 Threads

Considering a vector space ##W = V\oplus V^*## equipped with quadratic form Q such that we have a clifford algebra ##Cl(W, Q)##. How can I map elements of ##V\otimes V^*## into elements of ##Cl(W, Q)##? What about elements of ##V^* \otimes V##, ##V\otimes V## and ##V^* \otimes V^*## into ##Cl(W...

In the context of the mappings of a set S into itself, when S is not number system with a zero, what is the customary definition for "zero mapping"? ( ChatGPT says that its a mapping that maps each element of S to some single element of S , i.e. maps all elements to some constant. )

Recently I've came to some references on mathematical aspects on string theory that deal with the Polyakov euclidean path integral. An example is the book "Quantum Fields and Strings: A Course for Mathematicians. Volume 2", where it is stated roughly that the path integral is $$A =...

Zero does not have an inverse. And y=0 does not have an inverse. Does the wave form y=0 for all x have an inverse?

Given example for what is induced mapping ? In basic level

I came across an article in Scientific American while browsing the news app on my smartphone. The article was originally published with the title "Every Inch of the Seafloor" in Scientific American 327, 2, 40-47 (August 2022) doi:10.1038/scientificamerican0822-40 Summary...

I apologize for the simple question, but it has been bothering me. One can write a relationship between groups, such as for example between Spin##(n)## and SO##(n)## as follows: \begin{equation} 1 \rightarrow \{-1,+1 \} \rightarrow \text{Spin}(n) \rightarrow \text{SO}(n) \rightarrow 1...

In a previous exercise I have shown that for a $$C^{*} algebra \ \mathcal{A}$$ which may or may not have a unit the map $$L_{x} : \mathcal{A} \rightarrow \mathcal{A}, \ L_{x}(y)=xy$$ is bounded. I.e. $$||L_{x}||_{\infty} \leq ||x||_{1}$$, $$x=(a, \lambda) \in \mathcal{\hat{A}} = \mathcal{A}...

Hi, I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one). So, from a formal...

I have two data frames df1 and df2 df1 has two columns 'player_name' and 'player_id'. Similarly df2 has 'player_id' column. From this configuration I want to pass 'player_name' column to df2 by using 'player_id'. For this reason I have tried something like this, df2['player_name'] =...

Hi I am just starting to studying games with C++ by Horton for a few days. It uses SFML for graphics, sound and fonts. I am puzzled with the screen map location. In the program below, you can see the resolution is 1920 X 1080 full screen. But there are something funny when I create a sprite and...

Apologies in advance for my ignorance, I don't really have a reference to consult and Google hasn't been too helpful! In standard probability theory there are a few common useful formulae, e.g. for two events ##S## and ##T## $$P(S\cup T) = P(S) + P(T) - P(S\cap T)$$ $$P(S \cap T) = P(S) \times...

I need a mapping from the unit Hypercube ##C^n:= \left[ 0,1\right] ^n## to a given simplex, namely ##S^n:=\left\{ \vec{x} \in\mathbb{R}^n |0\leq x_1\leq 1, 0\leq x_{k}\leq x_{k-1}\text{ for } k=2,3,\ldots , n\right\}##. Anybody know one? I have other requirements I need satisfy, so if you know...

I understand that the Dual Space is composed of elements that linearly map the elements of the Vector Space onto Real numbers If my preamble shows that I have understood correctly the basic premise, I have one or two questions that I am trying to work through. So: 1: Is there a one to one...

I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...

When I'm going from a smooth manifold to R with V* X V -> R what does the R scalar stand for. Is it some length in the manifold? and Does this have to do with the way V* and V are defined, since one is a contra-variant and one is a co-variant, are they related in the way the Pythagoras formula...

Let us assume a functor ##\mathscr{F}\, : \,\mathcal{G} \times \mathcal{K}\longrightarrow \mathcal{Set}## which is contravariant on ##\mathcal{G}## and covariant on ##\mathcal{K}##. The question whether for any object ##G \in \mathcal{G}## the covariant functor...

How would one tackle this using the definition? (i.e. for some function ff that f(x)=f(y)⟹x=yf(x)=f(y)⟹x=y implies an injection and y=f(x)y=f(x) for all yy in the codomain of ff for a surjection, provided such x∈Dx∈D exist.) One can solve the system of equations for x1x1 and x2x2 and that...

Sorry, I'm not an astronomer. This question relates to the book "S." by Doug Dorst. I understand that the celestial coordinates have a zero-point at the vernal equinox. (0h, 0m, 0s RA, 0⁰, 0", 0' Dec.) I also understand that it's possible to map these coordinates to spherical, or...

Homework Statement If I have an affine camera with a projection relationship governed by: \begin{equation} \begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b \end{equation} where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix...

Hello! (Wave) Let $F$ be a field with infinite elements and $V$ a $F$-linear space of dimension $n$ and $W_1, \dots, W_m$ subspaces of $V$ of dimension $n_i<n, i=1, \dots, m$. We want to show that $V \setminus{(W_1 \cup \cdots \cup W_m)} \neq \varnothing$. Fix a basis $\{ v_1, \dots, v_n\}$...

why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?

Homework Statement Homework Equations As the book says , an affine function of a line is A\rightarrow \mathbb{R} and represent the real number that, multiplied for a basis and starting from an origin of the line gives a certain point of the line, so a origin of the line and a basis is...

Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...

Hi PF! Does anyone know the conformal map that takes a wedge of some interior angle ##\alpha## into a half plane? I'm not talking about the potential flow, just the mapping for the shape. Thanks!

I want a way to map natural numbers to possible values of set A. It would also be helpful if you could tell me how many possible values are for set A(it depends of k). All that is known about set A is: Non of set A elements is subset to another element of A, but all elements of set A are proper...

Hey! :o We have the function $f(x)=x^5-\frac{5}{16}$. I have approximated the root of that function using three steps of Newton's method with initla value $x_0=\frac{1}{2}$ : \begin{align*}x_1&=x_0-\frac{f(x_0)}{f'(x_0)}\approx \frac{7}{5} \\ x_2&=x_1-\frac{f(x_1)}{f'(x_1)} \approx...

Homework Statement This is for a linear algebra class, but it's taught my mathematicians, for mathematicians and not physicists or engineers so we write pseudo-proofs to explain things. In Exercises 37–40, let T be the linear transformation whose standard matrix is given. In Exercises 37 and...

Hi, I'm studying about Conformal Mapping in Complex Analysis and see its applications in Heat transfer, Fluid and Static Eletrocity. But it is said that this subject is very useful in many branches of Physics. Can you tell me about that? Thanks.

Hi, I have an operator given by the expression: L = (d/dx +ia) where a is some constant. Applying this on x, gives a result in the subspace C and R. Can I safely conclude that the operator L can be given as: \begin{equation} L: \mathcal{H} \rightarrow \mathcal{H} \end{equation} where H is...

I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings: a) w=e^(ipi)z b) w = 2z c) w = 2e^(ipi)z d) w = z + 2 + 2i I have managed to complete the question for a square and a rectangle as the points are easy to map as they are...

Self studying here :D... Let X and Y be noncompact, locally compact hausdorff spaces and let f: X--->Y be a map between them; show that this map extends to a continuous map f* : X* ---> Y* iff f is proper, where X* and Y* are the one point compactifications of X and Y. (A continuous map is...

The search for the missing aircraft of flight MH370 has yielded information on the region of the Indian Ocean under survey. http://www.upworthy.com/amp/they-looked-for-missing-flight-mh370-for-3-years-heres-what-they-found-instead...

Consider the mapping ##f: \mathbb{D}^1 \longrightarrow [0,1] \times (0,2 \pi]## where ##\mathbb{D}^1## is the unit disk. This is the familiar polar coordinate system. The area element is ##dx \wedge dy## in ##\mathbb{D}^1## and ##r dr \wedge d\theta## in ##[0,1] \times (0,2 \pi]##*. Now at ##r...

Homework Statement Let V be of finite dimension. Show that the mapping T→Tt is an isomorphism from Hom(V,V) onto Hom(V*,V*). (Here T is any linear operator on V). Homework Equations N/A The Attempt at a Solution Let us denote the mapping T→Tt with F(T). V if of finite dimension, say dim...

Homework Statement Suppose T:V→U is linear and u ∈ U. Prove that u ∈ I am T or that there exists ##\phi## ∈ V* such that TT(##\phi##) = 0 and ##\phi##(u)=1. Homework Equations N/A The Attempt at a Solution Let ##\phi## ∈ Ker Tt, then Tt(##\phi##)(v)=##\phi##(T(v))=0 ∀T(v) ∈ I am T. So...

I am interested in plotting contours (and integrals) over the algebraic function ##w^2-(1-z^2)(1-k^2 z^2)=0## on it's normal Riemann surface, a torus. Anyone here interested in helping me with this? I have the basic idea just the details I'm having problems with. Would be a nice educational...

This is going to be a really silly question, but here it goes. In a ring theory lecture, I was given a definition to a polynomial ##P \in R[X]## evaluated at the element ##\lambda\in R##. I understand the evaluation bit as it is trivial to substitute a lambda into X. At the end of the...

I've seen in books things like "G is mapping of plane into itself", "map of a set into itself" or "map of set/plane onto itself". What exactly to map into/onto itself means? Do this means that when G maps into itself we get G as a result or we can also associate points on G to other points as...

Homework Statement Let γ : I → Rn be a regular smooth curve. Show that the map γ is locally injective, that is for all t0 ∈ I there is some ε > 0 so that γ is injective when restricted to (t0 − ε , t0 + ε ) ∩ I. Homework Equations The Attempt at a Solution [/B] So I know a function (or a...

Hey! :o Let $B=\{b_1, \ldots , b_5\}$ be a basis of the real vector space $V$ and let $\Phi$ be an endomorphis of $V$ with \begin{align*}\Phi (b_1)& =4b_1+2b_2 -2b_4-3b_5 \\ \Phi (b_2)& = -2b_3 +b_5 \\ \Phi (b_3)& = -4b_2+2b_3 -b_5 \\ \Phi (b_4)& =-2b_1 +3b_3+b_4-b_5 \\ \Phi (b_5)& = 3b_2...

Homework Statement Illustrate the mapping of f(z)=z+\frac{1}{z} for a parametric line. The Attempt at a Solution the equation for a parametric line is z(t)=z_0(1-t)+z_1(t) so I plug z(t) in for z in f(z), but I don't get an obvious expression on how to graph it, I tried manipulating it...

Hey! :o Let $F$ be a field and $V,W$ finite-dimensional vector spaces over $F$. Let $f:V\rightarrow W$ a $F$-linear mapping. We have to show that $f$ is injective if and only if for each linearly independent subset $S$ of $V$ the Image $f(S)$ is linearly independent in $W$. I have done the...

Hey! :o Let $\mathbb{K}$ be a field. Find a matrix $M\in \mathbb{K}^{2\times 2}$ such that for the linear mapping $f:\mathbb{K}^2\rightarrow \mathbb{K}^2, x\mapsto Mx$ it holds that $f\neq 0$ and $f^2:=f\circ f=0$. Let $V$ be a $\mathbb{K}$-vector space and $\psi:V\rightarrow V$ be a...

Hi all, I have logistically- regressed 3 different numerical variables ,v1,v2,v3 separately against the same variable w . All variables have the same type of S-curve (meaning, in this case, that probabilities increase as vi ; i=1,2,3 increases ). Is there a way of somehow joining the three...

Homework Statement A piece of linear DNA 1000bp long is completely digested by four enzymes, E, B, P, and S. We are given the sizes of fragments (in bp) produced when each of the enzymes are used in isolation, and when they are used in different combinations: E : 227, 773 B : 150, 450, 400 P ...

Hey! :o I am looking at the following exercise: Let $C$ be an algebraic closure of $F$, let $f\in F[x]$ be irreducible and let $a,b\in C$ be roots of $f$. Applying the theorem: "If $E$ is an algebraic extension of $F$, $C$ is an algebraic closure of $F$, and $i$ is an embedding (that is, a...

I am working on a two-by-two real matrix $M$, with a linear mapping $F$ that returns the sum of $M$ and its transpose. I need to find out the matrix that is associated with the mapping. To the best of my understanding: $$ M + M^T = \begin{bmatrix} r &s\\ t &u \end{bmatrix} + \begin{bmatrix} r...

I am working on a problem that goes like this: Show that $Ker (F) \cap I am (F) = \{0\}$ if $F: W \rightarrow W$ is linear and if $F^4 = F.$ I have the solution but there is one step which I need help: (the delineation is mine) (1) Suppose that there exists $x$, such that $x \in Ker(F) \cap...

I have a mapping $L: \mathbb R^3 \rightarrow \mathbb R^3$ as defined by $L(x, y, z) = (x+z, y+z, x+y).$ How do you prove that the $L$ is an onto mapping? I know for sure that $\forall x, y, z \in \mathbb R$, then $x+z, y+z, x+y \in \mathbb R$ too. Then I need to prove that $Im (L) = \mathbb...