## One root of the polynomial f(x) = 5$x^2$ + $x^4$-5$x^3$-$x^2$-10x-2 over $\mathbb C$ is $i$. November 30

One root of the polynomial f(x) = 5$x^2$ + $x^4$-5$x^3$-$x^2$-10x-2 over $\mathbb C$ is $i$.(a) Write $f(x)$ as a product of irreducible polynomials in $\mathbb Q[x]$.(b) Write $f(x)$ as a product of irreducible polynomials in $\mathbb R[x]$.(c) Writ

## Cavalieri's Principle in measure theory November 30    1

The first part of Cavalieri's principle (in measure theory) states if $E$ is measurable, then almost every slice $E_x$ of $E$ is measurable.Here, it uses "almost every", so what is an example where not all slices of E is measurable?Thanks!Couldn

## Are supersets of non-empty measurable sets measurable November 30    1

Challenging conventional wisdom questionLet $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{M}(\mathbb{R}), \lambda)$, where $\mathscr{M}(\mathbb{R})$ is the collection of all $\lambda$-measurable subsets of $\mathbb{R}$If $A \in \Sigma, A \ne \emptyset$ a

## How is math used in computer graphics closed November 30    3

I'm doing a research paper on the mathematics of computer graphics and animation (3D) and I do not know where to start. What mathematical equations and concepts are used for computer graphics and animation (3D)?Like many questions of the form: How mu

## Linear Algebra and Geometry by Kostrikin and Manin: Remark regarding diagrams and graphic representations. November 30    1

On page 5 of this book there is a particular section of the book that I am having trouble trying to understand as to what the authors' are trying to point across. It is concerning linear algebra. I will place in bold the parts I need additional expla

## Calculating normals for a polygon mesh (3D computer graphics) November 30    1

I want to write a program to generate arches, a common architectural form, and export them to a wavefront object format for sharing with various three dimensional graphics editors. To do this, I need to generate normals for inclusion in the wavefront

## How to get angle bewteen two vectors in range -1 to 1 without using arc cosine November 30    2

Given two normalized vectors in 3d space, how can I get a value from $-1$ to $1$ based on their angle without using arc cosine?With use of arc cosine, I think this would give me the correct result. But since arc cosine is a computational expensive fu

## Choosing best representation for hit detection in computer graphics November 30    1

Suppose I have 3D spheres. What is the best representation for the spheres in order to detect whether the end of an arrow hit them? and why?I thought about an isoparametric representation since it's possible to put the values inside in order to test

## 3D triangle computer graphics November 30    3

We are given a 3D triangle with vertices $(0,0,0), (5,0,10), (0,20,0)$. What is the $z$ value of the point in the triangle with $x=3, y=1$? How do we find the $z$ value?There is no such point. All the points of the triangle have $x$ coordinates that

## does any polyhedral partition admit a convex piecewise quadratic surface defined over November 30

Given a polyhedral partition, i learnt that there exist some conditions for the existence of a convex piecewise affine surface over this partition for example the following study. http://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/ftp03/NQ45268.pdf No

## Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective. closed November 30    1

Let $D$ denote the closed unit disc in the plane with boundary the unit circle $S^1$. Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.If the map omitted the point $p$ in t

## Integration using Fourier Transform November 30    1

How to integrate the function $(\sin x)^2/x^2$ using Fourier transform of function $g(x)=1$ if $|x|<1$ else $g(x)=0$ which is $(sin w/w)*2/pi$?!thank you in advance.Parseval/Plancherel's identity/theorem (note that Wikipedia uses a different conventi

## Literature Reference for transformations through vector spaces November 30

I am trying to understand the transformations through vector spaces:Problem 1. Let's say we have orthonormal basis $B=\{v_1, v_2, \ldots, v_n\}$ spanning the vector space $V$ and basis $B_1=\{u_1, u_2, \ldots, u_n\}$ spanning the vector space $U$ and

## Transformation between Ideal and Warped Surface November 30

I work on manufacturing metal panels with holes drilled in them. Suppose I have an ideal 3D surface from CAD. I want to compare it to the actual part using reference points to compare between the two. Ideally yielding some sort of function that when

Let $D\subset \Bbb{R}^2$ be the closed unit disk and $f:D\to\Bbb{R}^2$ a continuous map such that $f(x)=x$ for every $x\in\partial D$. Show that $f(D)\supset D$. This exercise in algebraic topology looks very similar to the following one:Let $f : D \ ## Fast Fourier transform on 8 points November 30 describe the Fast Fourier Transition F(f) for a 1-periodic function f(t) given at eight points t=0,1/8,2/8,3/8,4/8,5/8,6/8,7/8. ## Proof (a b and a not divide b)a not divide (b+c) November 30 Prove$\forall a\in \mathbb Z, \forall b\in \mathbb Z, \forall c\in \mathbb Z, (a | b \land a\nmid c) \rightarrow a\nmid(b + c)$.Maybe a gentle nudge in the right direction ## the probability of two spy in the same group November 30 I was watching a TV show, and the setting is pretty simple:There are 10 players in total, within them there are 2 "spies", players initially form 2 teams of 5 people, and there are 3 rounds of game, where at the end of each round, we can switch ## A Borel subset of a topological space November 30 1 Is every Borel subset of a topological space$(X, \tau)$either open or closed?I'm thinking that if$B \in \mathcal B(X)$, then there are many possibilities:$B$is open for$\tau$, or it is the complement of some open set$B$in$\tau$, and so closed ## Any positive measure subset of$\mathbb R$contains a positive measure Cantor set November 30 1 A question asks to show any positive measure subset of$\mathbb R$contains a positive measure Cantor set. How to start with this? I have been staring on this for a while, but can not come up with any useful idea.What is the correct way to start with You Might Also Like • I've received some code (which I didn't write) and decided at some point to write test cases for the Quatern ... • I have to find the parametric equation of the ... •$f(S \cup T) = f(S) \cup f(T)$f(S) encompasses all x that is in S f(T) encompasses all x that is in Tthus th ... • Prove the following limits by using$\epsilon$and δ1) Show$\lim_{z\to 2}z2 + iz = 4 + i2$.2) Show$\lim_{z ...
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