2
votes
1answer
112 views

How many homomorphisms $\Psi : S_3 \rightarrow S_3$ exist?

How many homomorphisms $\Psi : S_3 \rightarrow S_3$ exist? Attempt: I found $16$ homomorphisms in total. $S_3 ={(1). (12),(13),(23), (123),(132)}$ There are three normal subgroups in $S_3 = \{(...
1
vote
1answer
59 views

Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
1
vote
1answer
62 views

With $G=S_5$ prove that the only proper normal subgroup is $A_5$

I can find proofs (but I admit I couldn't do them without referencing them) that $A_5$ is a normal subgroup of $S_5$, I also know (and the question says I may use - but without is more than welcome) ...
5
votes
1answer
423 views

Picard group of a smooth projective curve

I have two (related) questions regarding the Picard group: 1) Are there examples of smooth projective curves with large Picard groups (say $Pic(X)\simeq\mathbb{Z}^n)$ for any $n$)? 2) In general, ...
7
votes
2answers
9k views

Plotting Differential Equation Phase Diagrams [closed]

I haven't got Matlab, nor have I found a suitable online tool. Could someone plot the phase diagram for the following, or point me in the right direction? $$\frac{dx}{dt} = y - x, \frac{dy}{dt} = x(4 ...
4
votes
1answer
37 views

Finding a subgroup of the Galois group of $E/(\mathbb{Z}/(p))$ where $E = (\mathbb{Z}/(p))(t)$, $t$ transcendental.

As the title states, the setup is Let $E = (\mathbb{Z}/(p))(t)$, and we are looking at it over $\mathbb{Z}/(p)$ and $t$ is transcendental. Let $G$ be a group of automorphisms of $E$ generated by $\...
0
votes
2answers
42 views

Show that B has infimum

Let $(A,\leq)$ is a lattice. if $B\subseteq A$ and $|B|=3$ then show that there is $\inf(B)$ If A is a lattice we know for every $x,y\in A$ there are $\inf\{x,y\}=x\wedge y $ and $\sup\{x,y\}=x\...
1
vote
1answer
31 views

Gaussian kernels for arbitrary metric spaces

Let $(I,d)$ be an arbitrary (pseudo-)metric space. Define the function $$c(i,i') := \exp\big( - d(i,i')^2 / 2 \big)$$ Is $c$ necessarily nonnegative-definite, hence a kernel function?
1
vote
0answers
37 views

Volume of the set of two bodies

I have been given two bodys: $(x^2+y^2) < R^2(x^2-y^2)$ $x^2+y^2+z^2 < R^2$ Now I am supposed to calculate the volume of the set of the two bodies. I can see that the second body is a ...
1
vote
2answers
82 views

How do I find $\sum_{i=1}^n\frac{1}{x_i}$ where $x_i = x_{i-1} + \frac{1}{x_{i-1}}$

I have a recurrence: $x_n = x_{n-1} + \frac{1}{X_{n-1}}$. I need the value of n for which $x_n$ IS CLOSEST TO $2*x_0$, where $x_0$ is a positive integer. I tried the following: $x_1 = x_0 + \frac{1}{...
1
vote
2answers
71 views

Integration by substitution - $\int (x+1)\sqrt{5x-6} \text{dx}$

$\int (x+1)\sqrt{5x-6} \text{dx}$ My working so far: $u=5x-6$ $du=5dx$ $\int\dfrac{x+1}{5}\sqrt u \ \text{du}$ Substituting x $\int \dfrac{(u+6)/5)+(5/5)}{5}\cdot\sqrt u \ \text{du}$ $\int (u+...
2
votes
1answer
38 views

Simple module over $A$ has finite dimension over $K$

Could someone explain the proof of the following statement to me: Let $A$ be a $K$-algebra. $A$ is finite $\implies$ a simple module over $A$ has finite dimension over $K$ Proof: If $M$ is ...
4
votes
1answer
66 views

$Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective

I am doing this exercise: Let $R = \mathbb{Z}[t]$ and let $K$ be its fraction field. Show that the $R$ module $K/R$ is divisible but not injective. I have done the divisible part, but I am stuck on ...
1
vote
1answer
388 views

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please ...
0
votes
1answer
123 views

Infinite product convergence for cosine

I have trouble proving the following: if $$\sum_k^\infty|z_k|^2 < \infty$$, then $$\prod_k^\infty \cos(z_k)$$ converges. (Note $z_k$ are complex numbers). I think some relevant proof of convergence ...
1
vote
1answer
63 views

Optimization of a convex target function with inequality constraints

I want to solve the following optimization problem: \begin{equation} \begin{split} \text{maximize} &\;\;\; \ln x_1+\ln x_2+\ln x_3+\ln x_4 \\ \text{s.t} &\;\;\; x_4\le4 \\ \text{and}&\;\;\;...
2
votes
1answer
128 views

Difference between upper integral and lower integral

I have the following problem: Let $f: [a,b]\longrightarrow \mathbb{R}$ a bounded function. Prove that $$\bar\int_a^b f- \underline\int_a^b f=\bar\int_a^b w(f,x)dx$$ where $w(f, x)$ is the ...
1
vote
2answers
100 views

Stuck on finding the value of $\sum_{n=0}^\infty {n(n+1) \over 3^n}$

I am trying to find the value of the series $$ \sum_{n=0}^\infty {n(n+1) \over 3^n} $$ Here's what I have done so far: $$ \sum_{n=0}^\infty {n(n+1) \over 3^n}=\sum_{n=0}^\infty {n^2 \over 3^n}+\sum_{...
0
votes
1answer
87 views

Triple intergration find the moment of inertia of a cylinder

What is the Moment of Inertia of a cylinder of Radius R, length l and mass M when rotated about an axis along the length of the cylinder? I've had a lot of examples of calculating the Moment of ...
0
votes
1answer
173 views

It's sufficient to prove collatz conjecture for $3+6k, k \geq 0$?

Thinking about this problem, I saw two interesting properties of Collatz graph. Firstly, if we consider that every even number $e$ can be represented (on a single way) as $e = o 2^n$, where $o$ is an ...
-1
votes
1answer
48 views

integrate $ \int^L_{-L}{\frac{1}{(z^2+x^2)^{\frac{3}{2}}}}dx $ [closed]

How would I perform the following please. $$ \int^L_{-L}{\frac{1}{(z^2+x^2)^{\frac{3}{2}}}}dx\tag1 $$ My first idea was to perform u-substitution $$u=z^2+x^2\tag{2.1}$$ $$\frac{du}{dx}=2x=2\sqrt{z^...
2
votes
1answer
71 views

Integration by substitution - $\int x^2 \sqrt{x-2} \text{dx}$

$$\int x^2\sqrt{x-2} \, dx,u=x-2$$ Using the given substitution $u=x-2$ $\text{du}=\text{dx}$ Attempting to express integral in terms of u... $\int u^2+4x-4\cdot \sqrt{u} \ \text{du}$ This is ...
1
vote
0answers
60 views

Does This Ring have a Name?

Let $M_1=\{0,1,2,4,5,8,9,10,\cdots\}$ be the set of nonnegative integers that can be written as a sum of two perfect squares. Let $M_2=\{\sqrt{m}: m\in M_1\}=\{0,1,\sqrt{2},2,\sqrt{5},\cdots\}$. Let $...
7
votes
1answer
123 views

Compute $\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$

Does the following series have a closed form ? $$\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$$ Motivation : The original exercise is Compute $\int_{1}^{+\infty} \sum_{n=1}...
3
votes
2answers
51 views

$M \otimes_R K$ is uniquely divisible as $R$-module

A module $M$ over a domain $R$ is called uniquely divisible if for every $r \in R \setminus\lbrace 0 \rbrace$ the endomorphism $$\phi_r : M \to M$$ $$m \mapsto r \cdot m$$ is a bijection. Let $K$ be ...
1
vote
0answers
241 views

Finite geometric sequence with a ratio greater than 1

I am trying to solve the recurrence relation: $ t(n)= 3T(n/2)+Cn$ (if $n>2$) $t(2)=C$ (otherwise) I know that there is the Master Theorem but I am trying to use the tree method. This ...
3
votes
1answer
91 views

approximate this fancy looking double integral

$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
0
votes
3answers
221 views

Translating a sentence into a logical expression.

I am having trouble understanding the solution given for a problem in my discrete mathematics text book. Any help would be much appreciated. Question: Let L(x, y) be the statement "x loves y", where ...
0
votes
1answer
2k views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= $$\int_{...
2
votes
6answers
88 views

$\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}$

$$\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}=\lim_{x\rightarrow\infty}1-x+x\sqrt{\frac{2}{x^2}+\frac 2x+1}=\lim_{x\rightarrow\infty}1=1\neq2$$ as Wolfram Alpha state. Where I miss something?
0
votes
1answer
109 views

Set theoretic disjoint union

In this page we read Let $X$ and $Y$ be topological spaces and $Z := X \coprod Y$ be a set-theoretic disjoint union. We wish to define a topology on $Z$ in a most natural way. Definition. ...
0
votes
2answers
277 views

How to prove continuity of the retraction map in proof of Brouwer's fixed point theorem? [duplicate]

I was reading Allen Hatcher's proof of the Brouwer's fixed point theorem using the no retraction theorem. That is, we want to prove that every continuous $f: \mathbb{D}^n \to \mathbb{D}^n$ has a fixed ...
2
votes
0answers
58 views

Condition on degree to ensure that a generic line bundle of that degree is very ample

Let $C$ be any smooth genus $g\geq 1$ curve (exclude $g=1$ as well if you want). What is the smallest possible integer $d$ to make the following statement correct: "A general line bundle $L$ of degree ...
1
vote
3answers
67 views

probability of blue 3 times before red 4 times

hi: interesting probability question here. you have a bag with a red ball and a blue ball in it. The rules of the game are 1) At each trial, a ball is taken out of the bag without looking by an ...
0
votes
0answers
261 views

Frequency distribution for N balls in m urns (distributed equiprobably)

Suppose that I have $N$ balls and m bins, where $N > m$. Each ball is randomly assigned to a bin (with equal probability of being assigned to any bin $i = 1 ...m$). What is the probability of there ...
1
vote
0answers
59 views

Determine all points where 2 variable function is differentiable: when to use definition?

My function is: $$f(x,y) = \frac{x\sin^2 y}{x^2+y^2}\text{ when }(x,y) \ne (0,0)$$ and $0$ when $(x,y) = (0,0)$ When I use the definition (limit as $h$ approaches $0$), the limit for both partiales ...
4
votes
2answers
121 views

does $4097$ divide $2^{4097}-2$?

does $4097$ devide $2^{4097}-2$ ? It was long since I did number theory. $4097 = 17\times241$. I know have that both 17 and 241 does not divide $2^{4097}-2$ (with fermats little theorem) Is there a ...
0
votes
1answer
18 views

Probability of staying in a sub graph

Assume that I have an undirected graph $G$ with $n_1+n_2$ vertices and $e_1+e_2+e_3$ many edges, and a subgraph $G^{\prime}$ of $G$. Assume further, I now that $G^{\prime}$ has $n_1$ vertices and $...
3
votes
2answers
215 views

Variance of a function of independent random variables

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if $X>Y$,...
2
votes
2answers
39 views

Proof of characterization of splitting fields

I'm trying to prove that if $K$ is a finite field extension of $F$ such that $K$ is the splitting field of some collection $C$ of polynomials in $F[x]$, then every irreducible polynomial in $F[x]$ ...
4
votes
1answer
175 views

Derivable doesn't exist in english?

I have a question about terminology. See this is what happens: someone says "this function is derivable", and then another, more experienced Anglo-Saxon mathematician goes on to correct this someone, ...
0
votes
1answer
35 views

Function with 2 outputs

I have written the following code: function [ z,a ] = complx( numb ) z=abs(numb); a=angle(numb); end but I get back just z and not a
2
votes
1answer
43 views

uniform continuity of the function $t\mapsto\langle x^*,f(t)\rangle$

Let $X$ be a Banach space. $f:\mathbb{R}\to X$ a function. If we have $t\mapsto\langle x^*,f(t)\rangle$ uniformly continuous on $\mathbb{R}$ for each $x^*\in D$ where $D$ is a dense subset of $X^*$ (...
1
vote
2answers
51 views

Solve a special differential equation

How can I rigorously solve this differential equation ? $$\frac{dy}{dx}=a y^{-1/6}$$ I know how to solve linear ODEs but I don't know how to do it with this one.
1
vote
1answer
60 views

Lie groups. How to show that the group operations are smooth.

$N:=\{g\in GL(n,R) : g_{ij}=0 \forall j>i , g_{ii}=1 ∀i\}$. For this matrix group, how can we show that it is a Lie group? I am at the beginning of the subject of Lie groups so I can not ...
1
vote
0answers
37 views

Testing for a power law

How can we show wether or not a given probability distribution is a power law distribution? So for example it is know that a normal distribution is not a power law distribution where a student t ...
2
votes
5answers
473 views

Eigenvalues of $AB$ and $BA$ matrices.

Show that if $A,B \in M_{n \times n}(K)$ where $K=\mathbb{C} \vee K=\mathbb{R}$ then matrices $AB$ and $BA$ have same eigenvalues. I do that like this: let $\lambda$ be the eigenvalue of $B$ and $v\...
2
votes
2answers
90 views

Product of Gamma functions I

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{8} \Gamma\left( \frac{k}{8} \right) \end{align} and can it be shown that the product \begin{align} \prod_{k=1}^{16} \...
6
votes
3answers
153 views

Series Question: $\sum_{n=0}^\infty\frac{2n!}{(2n+1)!!}=\pi$

How to prove: $$\sum_{n=0}^\infty\frac{2n!}{(2n+1)!!}=\pi$$ I tried to use property of double factorial $$(2n+1)!!=\frac{(2n+1)!}{2^nn!}$$ then $$\frac{2n!}{(2n+1)!!}=\frac{2^{n+1}(n!)^2}{(2n+1)!}$$ ...
3
votes
1answer
164 views

Epsilon-delta proof for a sum

Consider $\displaystyle \begin{array}{ccccc} f & : & \mathbb R^+ & \to & \mathbb R \\ & & x & \mapsto & \frac{x}{\sqrt{1+x}} \\ \end{array}$ Prove that $\...

15 30 50 per page