Compute $\langle 5 \rangle$ in integers November 29    2

Compute $\langle 5 \rangle$ in integers. I thought the answer would have been $$\{5^n| \text{ for n in integers }\}$$ However my teacher has marked it $$\{5*n|\text{ for n in integers }\}$$ What have I done wrong?The integers do not form a group

How to see at a glance the solution to the exercise Show that $\langle (1,2,3… n),(1,2,3… m)\rangle$ contains a 3 cycle, if $1 n m$ November 29    1

I tried the commutator of the generators and it worked, but I had no real justification for making that computation. Is there a perspective from which trying the commutator is the "obvious" thing to think about?I was inspired to try the commutat

Let $\langle a \rangle$ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism. November 29    4

I'm stuck on this proof. I need to prove:Let $\langle a \rangle$ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism.And this is the work progress I have made so far:We need to show 3 thi

Approximation of integrable function by continuous in norm of $L_2$ November 29

Let $\alpha$ be a fixed increasing function on $[a,b]$. For $u\in \mathcal{R}(\alpha)$, define $$\lVert u \rVert_2=\left(\int \limits_{a}^{b}|u|^2d\alpha\right)^{1/2}.$$ Suppose $f\in \mathcal{R}(\alpha)$ and $\epsilon>0$. Prove that there exists a c

Solid Ellipsoid Volume Evaluation November 29

Evaluate ∭H((x^2/a^2)+(y^2/b^2)+(z^2/c^2))dV, where H is the solid ellipsoid: ((x^2/a^2)+(y^2/b^2)+(z^2/c^2)) <= 1, where a,b&c are positive numbers.We've been using the Jacobian determinant to evaluate transformations, however I am not sure how to

Volume of a general ellipsoid November 29    2

Can somebody please show me at least one derivation of the volume of a general ellipsoid? I've been trying to derive by considering it a surface of revolution. The answer I keep getting is $4\pi(abc^3)/3$ but I know it's supposed $4\pi(abc)/3$. Could

Let G be a cyclic group of order m and we let the number s relatively prime to m. Prove that then the a ^ s = b ^ s follows a = b. November 29

Another homework problem from my Group Theory class. I need a proof for it.

Count Planar Graphs November 29

I have an assignment due tomorrow, i am stuck with these two questions. Can someone give me a hint ? I don't know where to start..5. Planar Graphs. Let G = (V, E) be a planar graph.(a) A graph is k-regular if every vertex degree is exactly k. How man

How many elements of order d are there in Z_10 X Z_10 where d is a divisor of 10 November 29

I completed exercise 6 page 166 in Dummit and Foote, Abstract Algebra. It reads: "Let G be a finite abelian group of type (n_1, n_2, ..., n_s). Prove G contains an element of order m if and only if m divides n_1."This exercies made me wonder if

Prove that the function $f:2,4\rightarrow\mathbb{R}$ is integrable. November 29

Define $$f(x)=\begin{cases}x\qquad\text{if 2\leq x\leq 3}\\ 2\qquad\text{if 3<x\leq4}\end{cases}$$ Prove that the function $f:[2,4]\rightarrow\mathbb{R}$ is integrable. Let $f=f_1+f_2$ where $f_1=x$ with $x\in[2,3]$ and $f_2=2$ with $x\in(3,4]$.

Volume of Ellipsoid with a hole November 29

How would I go about computing the volume of an ellipsoid with spherical caps removed and a cylindrical hole through it? I'm thinking about finding dV of a cross section (the ellipsoid has a circular horizontal cross section). Would this be the best

What is the correct (i.e., precise) term for a vector of the form $(a_1,…,a_n)$ November 29    1

What is the correct (i.e., precise) term for a vector of the form $(a_1,...,a_n)$?I ask this because my definition of vector is that it is an element of some vector space, this means that whenever we say vector it may be non-obvious that I'm talking

what is the handwritten notation for a random vector November 29    1

Not sure if this is a maths question but didn't know where to turn. I'm learning probability theory on my own using a textbook. It uses capital letters with subscripted number to denote random variables (eg, X1,...,Xn) and bold subscripted capital le

What is the meaning of $dA$ in double integrals November 29    1

What is the meaning of $dA$ in $\iint_E\dots dA$, where $E$ is a region in the $xy$ plane?In some integrals we use $dA=dx\,dy$, but in others $dA=\hat{k}\,dx\,dy$. (Here $\hat {k}$ is the unit vector in the $z$ direction.)Why this difference? Also, i

What is the component projection of vector a onto vector b, with the notation (a, b) November 29    2

I stumbled across this notation while reading the article "A handwritten character recognition system using directional element feature and asymmetric Mahalanobis distance" (http://www.researchgate.net/publication/3192962_A_handwritten_character

Is there the shortest notation defined for a vector obtained by projecting $\vec{A}$ onto $\vec{B}$?Is there the shortest notation defined for the complementary vector of a vector obtained by projecting $\vec{A}$ onto $\vec{B}$? Is it ok if I used $\ what is the right notation for this operation on two vectors November 29 1 I am trying to find the right notation for this operation on two vectors of size N and M. I do not believe it is the dot product, because there is no sum for the multiplication of each element. Instead, the result is a vector that is of size N * M:v1 Notation for the sum of the scores of items of an intersection of two sets November 29 1 I have two sets of items. Every item has a score. I'm finding the sum of the scores of the items of the intersection of both sets.What would the best way to notate this be?An example:A = {a, b, c, d} B = {c, d, e} The scores are: a -> 9 b -> 8 c -&g Is$\epsilon _{X_i}(I_n) $an other notation for the indicator function November 29 1 My book uses both of the functions$\epsilon _{X_i}(I_n)$and$1_{X_i\in I_n}$once with an equality sign, otherwise just the first one. Is this different notation for the same function?Thanks in advance!Now I found the answer myselft, he use it to d Finding an ideal such that$\mathbb{Z}x/I \cong \mathbb{Z}i$. November 29 2 On this released exam, it asks at 2g (slightly modified wording):Give a brief example or show there does not exist an ideal$I$,$I \subseteq \mathbb{Z}[x]$such that$\mathbb{Z}[x]/I$is isomorphic to$\mathbb{Z}[i]$the Gaussian integers. I have a You Might Also Like • Consider the set$\mathbb{N}$of all natural numbers; we can assign each natural number a point on a single ... • Let$G$and$H$be Lie groups with associated Lie algebras$\mathfrak{g}:=\text{Lie}(G)$and$\mathfrak{h}:= ...
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