2
votes
3answers
380 views

How to prove $r^2=2$ ? (Dedekind's cut)

Let a (Dedekind) cut $r=\{p \in \mathbb{Q} :p^2<2 \text{ or } p<0\}$ and a cut $2^*=\{t\in \mathbb{Q} : t<2\}$. I want to prove $r^2=2^*$. I could show that $r^2 \subset 2^*$ easily, but I ...
2
votes
2answers
2k views

Runge-Kutta algorithm for a given ODE system

consider the system given by: $$x'_{1}=9x_{1}+24x_{2}+5\cos t-\dfrac{1}{3}\sin t$$ $$x'_{2}=-24x_{1}-51x_{2}-9\cos t+\dfrac{1}{3}\sin t$$ with initial values $$x_{1}(0)=\dfrac{4}{3}$$ and ...
1
vote
3answers
396 views

Discrete Math - Counting

Prove that among any 100 integers there are always two whose difference is divisible by 99. How can I prove this?
4
votes
1answer
74 views

Functional Interpretation of Variety?

I am simultaneously taking courses in functional analysis and commutative algebra. In doing so, I found that there is, at least heuristically, some similarity between the notion of an algebraic ...
0
votes
1answer
78 views

Ordered set of integers

$\{x_i\}_{i = 1}^7$ is a set of 7 integers that satisfy $1≤ x_i ≤ 8$. How many such ordered sets of $7$ integers are there, such that $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 - x_1x_2x_3x_4x_5x_6x_7 ...
2
votes
1answer
126 views

Expectations Homework Questions Clarification

This is another one of the questions I didn't get a chance to ask the TA at my school today. I thought I had a pretty good grasp on Expectations but apparently I could still use some clarification. ...
0
votes
1answer
45 views

Help with the proof that $A^*$ has the UMP of the free monoid

In the following proof $A^*$ here is a Kleele closure and $*$ is a product of $a$'s or "concatanation": Proposition 1.9. $A^*$ has the UMP of the free monoid on A. Proof. Given $f:A\to|N|$, ...
0
votes
1answer
214 views

How to find the coefficient of this Fourier sine series?

From $$1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$$ I want to find $a_k.$ My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, ...
1
vote
3answers
161 views

Intuitive understanding of relations and their basic properties

Can anyone explain relations and the four basic properties of them (reflexive, symmetric, antisymmetric, transitive)- or direct me to a source that does. Particularly, are these statements ...
1
vote
0answers
149 views

A weak convergence in Sobolev space.

Let $u \in C^0([0,T], H^{s-1}(\Bbb R^n)). $ Let $\{t_n \} \subset [0,T]$ such that $\lim_{n \to \infty} t_n = t_0$. Let $ u(t_n ) \to u(t_0) $ in the Sobolev space $H^{s-1} ( \Bbb R^n )$ for $s = ...
11
votes
1answer
8k views

Simultaneously Diagonalizable Proof

Two $n$ x $n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a.) Show ...
1
vote
1answer
78 views

mean and distribution problem

A certain job is completed in three steps in series. The means and standard deviations for the steps are (in minutes): Step 1 2 3 Mean 17 13 13 Standard Deviation 2 1 2 Assuming independent steps and ...
0
votes
1answer
161 views

Simple Fourier Series

$$f(x)=\begin{cases}\tfrac{1}{2a},& 0<|x|<a\\ 0,&\text{otherwise} \end{cases}$$ Note that $a<\pi$. $$a_0=\frac{2}{a}\int_0^a \frac{1}{2a}dx=\frac{1}{a}$$ $$a_n=\frac{2}{a}\int_0^a ...
1
vote
1answer
558 views

How to find isomorphism classes of transitive actions

I have not been able to find a proper definition of what an isomorphism class is (in the context of group theory). If one could define it properly for me and give me some help with the following two ...
0
votes
3answers
45 views

Very Simple Relations

For Set A = {1,2,3,4} Is it possible to generate a relation that is reflexive and symmetric, but not transitive? The textbook says, (1,1),(2,2),(3,3),(4,4),(1,2),(2,1)(2,3)(3,2) but isn't this ...
0
votes
0answers
102 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? I'm ...
2
votes
1answer
93 views

How to prove that the number $\pi$ is irrational through differential equations?

How can it be proved that $\pi$ is an irrational number through differential equations?
0
votes
1answer
100 views

Transferability of space properties via continuous functions

Let $f:(X, \tau_x) \to (Y,\tau_y)$ be a continuous and onto function. I need to show that if $X$ is separable (Lindelöf), then $Y$ is respectively separable (Lindelöf).
3
votes
1answer
244 views

Why is free monoid called Free?

Why in the world do they use the word "free" in "free monoid"? It driving me crazy to see where the "freedom" comes from. Here is the Awodey's explanation of it, in terms of "baby lagebra" (sic.) ...
2
votes
1answer
146 views

Mutually independent distribution

Let the mutually independent random variables $X_1, X_2$, and $X_3$ be $N(0,1)$, $N(2,4)$, and $N(-1,1)$, respectively. Compute the probability that exactly two of these three variables are less than ...
1
vote
1answer
252 views

Unique path between any pair of vertices in $G$

I'm having trouble with this question: Suppose there is a unique path between any pair of vertices in $G$. Prove that $G$ is a tree. I know that a path is a trail where all vertices are distinct ...
0
votes
1answer
106 views

One-to-one, Compact, Preimages, Continuous.

Recall that the set $\mathbb{Q}$ of rational numbers is countable; thus we can put them all in a sequence ({ i.e.}~we can enumerate them). Let $(r_n)$ be such a sequence and let $f(x)=\begin{cases} ...
0
votes
1answer
305 views

Orthogonal complement and Legendre polynomial

a.) Find a basis for the orthogonal complement to the following subspaces of $\mathbb{R^4}$. The subspace spanned by $(1,2,-1,3)^T$,$(-2,0,1,-2)^T$,$(-1,2,0,1)^T$. b.) Use the Legendre ...
1
vote
1answer
1k views

Example of a general random variable with finite mean but infinite variance

Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite, ...
2
votes
1answer
91 views

When is a ring considered as a module over a Noetherian subring itself Noetherian?

If $S$ is a ring and $R$ is a Noetherian subring of $S$, and we know $_RS$ (i.e. $S$ viewed as a left $R$-module) is finitely generated (hence Noetherian), is $_SS$ necessarily Noetherian? I can't ...
4
votes
1answer
364 views

Hurwitz's Theorem Proof Question

Hurwitz's Theorem Can anyone explain in the proof of Hurwitz's Theorem on the wikipedia page, the line where it says $ \frac{f_k'(z)}{f_k(z)}$ converges uniformly by Morera's Theorem? I do not see ...
0
votes
1answer
206 views

chi-square distribution: determining the constants c and d

If $X$ follows $\chi^2_5$, determine the constants $c$ and $d$ so that $P(c < X < d) = 0.95$ and $P(X < c) = 0.025$. $\chi^2$ is chi-squared distribution
1
vote
2answers
2k views

Meaning of amortized analysis of an algorithm

From Introduction to Algorithms by Cormen et al: In an amortized analysis, the time required to perform a sequence of data structure operations is averaged over all the operations performed. ...
0
votes
1answer
79 views

Help me out with this related rate problem dealing with a cone?

So a right circular cone is 25m deep with a radius of 9m. It's being filled with water at a rate of $3m^3$ per minute. a) find dh/dt when its height is 10m. What I did first was use a proportion to ...
0
votes
1answer
148 views

Find adjoint operator with integrals

I'm trying to find the adjoint operator of vector spaced defined by $V=\mathbb{P}_1[x]$, and $f:V\rightarrow V$ Now I have an inner product space defined by $\langle p,q ...
1
vote
1answer
89 views

Prove the binary quadratic forms $5x^2+xy+y^2$ and $x^2-xy+5y^2$ are equivalent

I need to show the binary quadratic forms $$5x^2+xy+y^2$$ and $$x^2-xy+5y^2$$ are equivalent. We've only touched on quadric forms, and the only definition I have for "equivalence" is that one can be ...
4
votes
1answer
126 views

Find the Hardy-Littlewood maximal function of $\chi_{[-1,1]}$ on $\Bbb R$

Find the Hardy-Littlewood maximal function $Mf$ of the $\Bbb R^\Bbb R$ function $f=\chi_{[-1,1]}$. How do we find $Mf(x)$ for $|x| > 1$? I see that it should decrease like $1/x$, but I can't find ...
0
votes
1answer
50 views

Prove that set contains at least two co-prime integers

I am asked: Prove that if n+1 distinct numbers are selected from the first 2n positive integers {1,2,3,...,2n-1,2n} then at least two of the n+1 numbers are co-prime where n is a positive ...
1
vote
1answer
177 views

linear equations - under modulo

I'm having difficulties proceeding with this problem: We have the following linear equations: $$\begin{array} 1x + 2y + 2z = 0 \\ 3x – 2y + 2z =1\\ 2x + y + z =3\end{array}$$ ...
3
votes
1answer
80 views

Integration questions on $\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$ and $\int \frac{x^4}{x^4+5x^2+4} \, dx$

I would appreciate any hints on how to solve the following integration problems, they are my homework questions btw: $$\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$$ $$\int ...
2
votes
2answers
395 views

Relation between $\sigma (N)$, $\tau (N)$, and $\varphi (N)$

How to prove $$\sum\limits_{d\mid n} {\sigma (d)\varphi (n/d) = n\tau (n)}$$ and $$\sum\limits_{d\mid n} {\tau (d)\varphi (n/d) = \sigma (n)}$$ where ${\sigma (N)}$ is the divisor function, ${\tau ...
1
vote
1answer
668 views

Fourier Sine Transform of $e^{-ax^2}/x$

I'm trying to do this integral essentially: $$\int^\infty_0 \frac{e^{-ax^2}\sin(kx)}{x}dx$$ which I realized to be $$\frac{1}{2}\operatorname{Re}\left[F\left(\frac{e^{-ax^2}}{x}\right)\right]$$ where ...
6
votes
1answer
189 views

Primes of the form $x^2+5xy+5y^2$

Trying to describe all primes of the form: $$x^2+5xy+5y^2$$ A hint was given with the question to show all primes $p$ for which 5 is a quadratic residue mod $p$. I've been able to show that all ...
1
vote
1answer
55 views

Conditional Distributions

Suppose X is uniform in the interval [0, Y] and Y can either be 1 or 2 with equal probability. How would I calculate P(X>1)? This is what I'm trying to do: $$f_x(x) = \int_a^b \! f_{X,Y}(x,y) \, ...
2
votes
1answer
39 views

Let $q$ be a prime of the form $4k+3$. If $2q+1$ is prime, then $2q+1$ divides $2^q−1$.

This is the theorem I need to show: "Let $q$ be a prime of the form $4k+3$. If $2q+1$ is prime, then $2q+1$ divides $2^q−1$." I need to show this to show that $2^{251} - 1$ is not a Mersenne Prime.
1
vote
4answers
3k views

Equation of a circle that touches a line and both x and y-axes

As shown in the graph below, a circle touches the $x$-axis, the $y$-axis and a line that has equation $y = x/2 +2$. How to find the equation of the circle? Thanks very much!
1
vote
0answers
94 views

Determine the smallest symmetric group for this condition

a) Determine the smallest symmetric group $S_n$ that contains a subgroup isomorphic to H, generated by $x^4=y^3=1$, $xy=y^2x$. b) Find a subgroup of $SL_2(F_5)$ that is isomorphic to that group. My ...
3
votes
1answer
139 views

Fundamental group of $G/H$

Let $G$ be a connected topological group. And $H$ be a discrete subgroup. Theorem : $\pi_1(g/H)=H$ This is the content in the book Algebraic Topology -Greenberg and Harper I want to know the proof. ...
2
votes
2answers
127 views

Marriage modeling [closed]

Let $x(t)$ and $y(t)$ be measures of happiness for the husband and wife, respectively. Negative values indicate unhappiness. Let $x_0$ and $y_0$ be the "natural disposition" of the husband and wife, ...
1
vote
2answers
134 views

Modeling guerilla forces

I can't seem to figure this one out Two guerilla forces, with troop strength $x(t)$ and $y(t)$, are in combat with each other without reinforcement. Suppose the territory is rather large and full of ...
1
vote
1answer
196 views

Multinomial Distributions

I'm a little bit confused about what the meaning of multinomial distributions, at least from what i've gleaned from the wikipedia page on Multinomial Distribution. In essence, a multinomial ...
5
votes
1answer
101 views

$2^{251 }-1$ not Mersenne Prime

I need to show that $2^{251} - 1$ is not a Mersenne prime. Hard because $251$ is prime. If i can show that a prime $p$ is congruent to $3 \bmod 4$, and $q = 2p + 1$ is a prime, then $2^p$ is congruent ...
1
vote
3answers
165 views

Infinite number of primes

This is an axiomatic proof, but I don't know how to start the exercise first one, I have to proof that if $q$ divides $a^p-1$ then $q$ divides $a-1$ or $q=2pt+1, t\in \mathbb{Z}$, and, if $q$ ...
0
votes
1answer
68 views

simplicity of G

I took an exam today. If I remember correctly question was like this: let G be a group. if it has "a" element which has exact two conjugates, then G cant be simple. I answered: let that two ...
3
votes
5answers
173 views

Summation over exponent $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$

Why does $\sum_{i=0}^k 4^i= \frac{4^{k+1}-1}3$, where does that 3 comes from? Ok, from your answers I looked it up on wikipedia Geometric Progression, but to derive the formula it says to multiply by ...

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