1
vote
2answers
940 views

Show every subgroup of D4 can be regarded as an isotropy group for a suitable action of D4

Show every subgroup of D4 can be regarded as an isotropy group for a suitable action of D4 I know that D4={1,R,R2,R3,D1,D2,M1,M2} and the subgroups are {1,R,R2,R3} {1,D1} {1,D2} {1,M1} {1,M2} ...
1
vote
2answers
102 views

Interscholastic Mathematics League Senior B #12

Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$. So here is what I have done so far. I got two of the roots to be zero and 4 since ...
4
votes
3answers
870 views

Another Question in Hatcher

First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again. The problem is 1.3.15. It reads ...
1
vote
1answer
174 views

What does unique “minimal” partition mean (Context: Partitioning of Vertex-Sets)?

I am studying R. Diestel's Book Graph Theory and I encountered a formulation which I don't quiet understand. Mr. Diestel speaks in this proof on page 180 (Google Books Link) in the second last line of ...
1
vote
1answer
217 views

This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG ...
1
vote
1answer
198 views

Pi approximation

If $d(a,b)=$ largest $n$ such that $a$ and $b$ agree on all digits upto $n$. Eg. $d(\pi,3.14)=3$, $d(0.1234667,0.1234669)=7$. What is the asymptotics of $d(\pi/4,1-1/3+1/5-1/7+\cdots(\pm)1/m)$ as ...
2
votes
1answer
118 views

Why can't $\mathbb{Z}/(p^k)$ for $k>1$ be the direct sum of two submodules?

If you mod out $\mathbb{Z}$ be a nontrivial prime power $p^k$, $k>1$, then why can't $\mathbb{Z}/(p^k)=\mathbb{Z}/(n)\oplus\mathbb{Z}/(m)$ for some such submodules? If that where the case, then ...
3
votes
2answers
358 views

Real Analysis: Continuity Proof

This problem has me stumped. I'm not sure how to proceed. Let $A = (0,\infty)$ and let $k: A \to \mathbb{R}$ be defined as follows: $$ k(x) = \begin{cases} 0 & \text{for } x \in ...
3
votes
1answer
92 views

Prove inequality using optional sampling

I proved the inequality below using Wald's identity and some tricky but easy manipulation, but I cannot do it using the suggestion from the source: "Hint: optional sampling!" Here is the problem: ...
-1
votes
1answer
71 views

Interscholastic Mathematic League Senior B Division [closed]

The number 2011 has the property that one of its digits is the sum of its other digits, i.e., 0+1+1=2. Compute the sum of the two largest integers less than 2011 with this property.
3
votes
3answers
496 views

What Does the Associative Property Mean Intuitively Across All Notational Schemes?

You can find descriptions of associativity as intuitively meaning that the order of operations performed does not matter, e. g. such as that of Wikipedia. However, if you write what associativity ...
0
votes
2answers
1k views

Simpler mathematic formula to find latitude coordinate mapping to lines “equally sized” on mercator projection?

I'm implementing a map visualization atop a mercator projected map (e.g google maps) where each circle appears to be the same size on the map: . At the equator, a circle plotted with a one degree ...
1
vote
2answers
120 views

Interscholastic Mathematic League Senior B Division #10

In traingle ABC, Angle A=45 degrees, Angle B is 60 degrees, and AC= radical 15. D is also a point on AB so that AB is perpendicular to CD. The circle with diameter AB intersects CD at point E. Compute ...
1
vote
1answer
78 views

Interscholastic Mathematic League Senior B Division #11

The roots of the equation 3x^3-38x^2+cx-192=0 form a geometric progression. Compute c.
3
votes
2answers
133 views

Infinitely many $n$ such that $p(n)$ is odd/even?

We denote by $p(n)$ the number of partitions of $n$. There are infinitely many integers $m$ such that $p(m)$ is even, and infinitely many integers $n$ such that $p(n)$ is odd. It might be proved ...
0
votes
1answer
103 views

Interscholastic Mathematics League Senior B Division #2

Points P,Q,R, and S are chosen on the sides of parallelogram ABCD, so that P is on line AB, Q is on line BC, R is on line CD, S is on line DA, and AP=BQ=CR=DS=1/3 AB. Compute the ratio of the area of ...
5
votes
2answers
256 views

Is $\mathbb R$ terminal among Archimedean fields?

I was wondering why metrics and norms are always defined to be real, rather than generalized to some other fields (or whatever). The best guess I have so far is: Because every Archimedean ordered ...
0
votes
1answer
86 views

Interscholastic Mathematic League Senior B Division #1

Let n be a positive integer less than 1000. If n^3 has 10 factors, compute the largest value of n.
4
votes
1answer
192 views

Limits of Functions

I'm self studying real analysis and currently reading about the limits of functions. Naturally everything in the chapter is about determining if a limit exists at a single point. But what about ...
7
votes
5answers
668 views

How many arrangements of $\{a,2b,3c,4d, 5e\}$ have no identical consecutive letters?

How many arrangements of $\{a,2b,3c,4d, 5e\}$ have no identical consecutive letters? I find it very tough... Could anyone have some good ways?
1
vote
3answers
108 views

How do I solve this matrix equation?

How do I solve this matrix equation and what is the answer: $$\begin{bmatrix} -122.366667 \\ 37.61666667 \end{bmatrix} = \begin{bmatrix} 0.000046 & 0.000032 & -122.413307 \\ ...
1
vote
0answers
354 views

how to find correlation between 2 arrays of 1's and 0's?

For my case, I have 2 arrays or sets of data, 100 elements, and the values are only 0 and 1. What test or procedure would measure the correlation or independence of the 2 sets? To give an example of ...
11
votes
1answer
389 views

How to “explain” Szemerédi's Regularity Lemma so that classmates may understand its value?

I am a student, preparing myself for a talk in which I want to present and prove Szemerédi's Regularity Lemma. I understand the proof and I am able to reproduce it - that is no problem. But I am ...
3
votes
3answers
800 views

functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere

I am supposed to show that if $f$ and $g$ are continuous, real-valued functions on $\mathbb{R}$, then if $f=g\;\;$, $\lambda$-a.e., then $f=g$ everywhere. So I have been reading and I think that this ...
7
votes
1answer
357 views

Is this a well known determinant identity? Are there any generalizations?

Let $A$ be a $3\times3$ matrix and for any $i,j\subseteq\{1,2,3\}$, let $A^{i,j}$ denote the $2\times2$ matrix resulting from removing row $i$ and column $j$ from $A$. Then: ...
2
votes
1answer
202 views

Interpretation of a question: “group of all p-power roots of unity”

I have a homework problem I'm trying to do, but I'm not sure what it's asking. The problem is as follows: Recall that $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the group of all roots of unity in ...
3
votes
3answers
284 views

Apostol Section 13.25 #13 - Conic Sections

Question: Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. (The next ...
0
votes
0answers
63 views

How can I convert lines intersecting a plane into a focused image?

I am writing a particle transport code. I would like to be able to obtain an image of my geometry when transporting photons given the following information: The photons are incident on a plane. For ...
0
votes
1answer
225 views

how to find the application of ordinary differential equation

This question may be out of the scope of this website, if so let me know and i will delete it. so I am right now taking an ODE class, which i find to be mildly interesting, though it seems to be more ...
5
votes
4answers
332 views

Definition of injective function

From wikipedia I obtain the following definition of an injective function : Let $f$ be a function whose domain is a set $A$. The function $f$ is injective if for all $a$ and $b$ in $A$, if $f(a) = ...
5
votes
2answers
349 views

How to prove a combinatorics identity

How to prove $$n^n=\sum\limits_{\substack{k,i_1,\ldots, i_k\ge1,\\ i_1+\cdots+i_k=n}}\frac{n!}{i_1!\cdots i_k!}i_1^{i_1-1}\cdots i_k^{i_k-1}=\sum\limits_{k=0}^n {n\choose ...
3
votes
1answer
172 views

Confused about $f: \mathbb{R}^{m \times n} \to \mathbb{R}$

I'm currently watching a lecture on Machine Learning at Stanford university. The lecturer defines $f$ to be a mapping from $\mathbb{R}^{m \times n}$ to $\mathbb{R}$. I understand this to be a ...
2
votes
1answer
2k views

Discrete Math - Bézout Coefficients

I'm taking a discrete math course, and were on Bézout Coefficients right now. I kind of understand the algorithm, the generalization. However the example in the book is throwing me off. The steps in ...
13
votes
1answer
2k views

Enumerating all subgroups of the symmetric group

Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group ...
5
votes
1answer
1k views

Distribution of Ratio of Exponential and Gamma random variable

A recent question asked about the distribution of the ratio of two random variables, and the answer accepted there was a reference to Wikipedia which (in simplified and restated form) claims that if ...
4
votes
3answers
447 views

What would be the radius of convergence of $\sum\limits_{n=0}^{\infty} z^{3^n}$?

I know how to find the radius of convergence of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, but how does this apply to the power series $\sum\limits_{n=0}^{\infty} z^{3^n}$? Would the ...
2
votes
1answer
70 views

Function from a sphere

This is a very basic question but i have difficulties understanding the following: If I have a function which has as its domain a 1-Sphere $S^1$ then how do I have to imagine such a function? Will ...
1
vote
1answer
110 views

best way to retrieve similarity $L$ matrix

given the similarity relation between $A$ and $B$ matrices; $$ A = L^{-1} B L $$ if $A$ and $B$ are given, what is the best way to compute the similarity transformation matrix $L$?
9
votes
3answers
633 views

Who introduced the notation $x^2$?

In the book 'Problem Solving and Number Theory' I read The law of quadratic reciprocity was discovered for the first time, in a complex form, by L. Euler who published it in his paper ...
7
votes
0answers
227 views

Representation for Vandermonde's permanent

Permanent of a matrix A = $\|a_{i,j}\|_{i,j=1}^{n}$ is defined as $$ \mathrm{Perm}(A) = \sum\limits_{\sigma \in S_{n}} a_{1,\sigma_{1}},\ldots,a_{n,\sigma_{n}} $$ Is there some representation for ...
0
votes
1answer
76 views

Proving $\mathrm{ann}_{1}(x)=A_{1} \partial^2+ A_{1}(x \partial -1)$

I don't understand the proof of $$\mathrm{ann}_1(x) = A_1\partial^2 + A_1(x\partial - 1),$$ where $\mathrm{ann}_r(S)=\{r \in R :rm=0, \forall m \in S\}$. $A_1$ is a left module. The ring is ...
2
votes
1answer
174 views

Borel $\sigma$-algebra on non- second countable topological space

Is it possible to define a Borel $\sigma$-algebra on a topological space which is not second countable, i.e. one which does not have a countable base? I am trying to learn measure theory and my ...
2
votes
1answer
2k views

Uniform Convergence of $nx(1-x^2)^n$ on $[a,1]$

Hello I am trying to understand why the sequence of functions $f_n=nx(1-x^2)^n$ does not converge uniformly to 0 on the interval $[0,1]$ but does on the interval $[a,1]$ where $a\in (0,1)$. I know ...
2
votes
2answers
165 views

If the condition of differentiability holds for the rationals then the function is differentiable?

$f:\mathbb{R}\rightarrow\mathbb{R}$ continuous, $a\in\mathbb{R}$. Suppose that there exists $L\in\mathbb{R}$ such that for every $\varepsilon>0$ there exists $r(\varepsilon)>0$ such that ...
1
vote
4answers
285 views

Figuring out probabilities with Hidden Markov Models

I'm really new to Math so sorry in advance if this question does not make sense. Also I cross posted this on stats.stackexchange.com also. Background: I'm trying to learn about hidden Markov models ...
7
votes
3answers
300 views

A property of some sequences of natural numbers (and their binary representation)

Let's consider a sequence of natural numbers $a_n$, represented in binary, with the following properties: $\forall n \in \mathbb{N}$ the number $a_n$ is represented with $n$ binary digits $\forall ...
7
votes
4answers
176 views

$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction

I have to show $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction. This is where I am stuck: $$\left( \frac{n+2}{2} \right)^{n+1} \geq \dots \geq =2 \left( \frac{n+1}{2} \right)^{n+1} = ...
1
vote
1answer
230 views

How does this proof show $h_*$ (homomorphism induced by $h\colon (X,a)\to (Y,b)$) is an isomorphism?

Claim: if $h\colon(X,a)\to(Y,b)$ is a homeomorphism of $X$ with $Y$, then $h_*\colon \pi_1(X,a)\to \pi_1(Y,b)$ is an isomorphism. where $\pi_1$ refers to the fundamental group and $h_*$ is the ...
2
votes
2answers
475 views

Independence of sigma algebra

I am trying to establish whether the following is true (my intuition tells me it is), more importantly if it is true, I need to establish a proof. If $X_1, X_2$ and $X_3$ are pairwise independent ...
1
vote
3answers
103 views

For $k\geq 2$ and $m_1,\ldots,m_k \in \mathbb{N}$ with $\gcd(m_i,m_j) = 1$ for $i\neq j$, show that $f(x)$ is a ring homomorphism

Let $k\ge 2$ and $m_{1},…,m_{k} \in \mathbb{N}$ with $\gcd(m_{i},m_{j}) = 1$ for all $i\ne j$. Show that $f(x) = (x,…,x)$ defines a ring homomorphism $f: \mathbb{Z}/m\mathbb{Z} \rightarrow ...

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