# All Questions

60 views

### Does $H_0^1(\Omega)$ embed into $H_0^1(R^d)$?

Given a domain $\Omega$ in $\mathbb{R}^d$ and a function $f\in H_0^1(\Omega)$, the closure of the test functions on $\Omega$, does the extension of f by 0 to all of $\mathbb{R}^d$ necessarily lie in ...
117 views

### Abstract Algebra- Ideals

Given an ideal $I\subset R$, define $J=\{\Phi(a):a\in I\}\subset S$. Prove that $J$ is an ideal, provided $\Phi$ maps onto $S$. Give an example to demonstrate that the latter hypothesis is necessary. ...
218 views

### Solving non-negative least squares by analogy with least squares (Matlab)

There is a least-squares problem Ax = b. It can be solved using backslash in Matlab (x = A \ b). Let's assume that I have the ...
174 views

### Lyapunov Stability

Let $\dot{x}=v(x)$ with $v(x)=Ax+O(\left \| x \right \|^2)$, $v\in C^k(U)$, $U\subset \mathbb{R}^n$, $n\geq 2$, is true or not that if the origin is a singular point Lyapunov stable for ...
303 views

### Use product rule and mathematical induction to show that $f^n$ is differentiable on $I$

Suppose that $f$ is differentiable on $I$. Use the product rule and mathematical induction to show that $f^n$ (the function f is raised to the nth power) is differentiable on $I$ for every positive ...
164 views

### Transition probability in Continuous Time Markov Chain (CTMC)

I know that for a CTMC, the transition matrix $P(t)=e^{tQ}$, where $Q$ is the infinitesimal generator matrix of the irriducible CTMC. My question is how do I deal with situations or problems that ask ...
714 views

### Why is the following integral improper? $\int_0^\frac {\pi}{2} sec(x) dx$

I'm asked to explain why the following integral is improper and determine whether the integral is convergent or divergent. I really am not sure how to do these problems and I am unsure on where to ...
107 views

### Finding Probabilities from moment generating functions

If $$M_X(t) = (1 - p + pe^t)^5,$$ find $P(X \le 1.23).$ I seem not to understand the connection between cdf and mgf. Can I find the $E(X)$ and then use the formula for $E(X)$ to ...
22 views

### For what other values of $a$ is $H_a$ a subset of $V$?

This is NOT homework, but review for a test. This is part b of a 2-part question, where part a was to show that $H$ was a subspace of $V$. I have done that part successfully and need help with the ...
66 views

### Differentiation of $2\arccos (\sqrt{(a-x)(a-b)}$

Okay so the question is: Show that the function $$2\arccos\bigg(\sqrt{\frac{a-x}{a-b}}\bigg)$$ is equal to $$\frac{1}{(\sqrt{(a-x)(x-b)}}$$ I started by changing the $\arccos$ into inverse ...
55 views

### If $|V(G)|=n$ and $e(G)>\frac{n}{4}\{1+\sqrt{4n-3}\}$ then $G$ contains 4-cycle

This question is linked to my former question Special properties of subgraphs I want to practice this technique a little bit more and want to show that if $|V(G)|=n$ and ...
62 views

56 views

124 views

### Combinatorics intersecting sets question

Let $A_1 , . . . , A_m$ and $B_1 , . . . , B_m$ be subsets of $[n]$ such that $| A_i ∩ B_i |$ is odd for all $i$ and $| A_i ∩ B_j |$ is even for all $i \neq j$ . Show that $m ≤ n$. I've tried using ...
64 views

### Fields of polynomials . Proving that a belongs to k as a root

if $f(x)\in k[x]$, where $k$ is a field, then $a\in k$ is a root of $f(x)$ iff $x-a$ divides $f(x)$ in $k[x]$. My result ... If $a$ is a root of $f(x)=q(x)(x-q)$ and if we let $f(x)=q(x)(x-a)$,then ...
621 views

190 views

### Diameter of a subset of a metric space

Let $(\Bbb R,d)$ be the metric space with the metric function $$d(x,y)=\frac{|x-y|}{1 + |x-y|}\;.$$ Calculate $\operatorname{diam}(0,\infty)$. I am thinking the answer is $1$ because ...
139 views

### Series convergence test with geometric series

This is my first question on the math stackexchange-website. This is an assignment question, but I've tried to detail my thought process as granularly as possible to show I'm not just being lazy. My ...
27 views

### Let $X=\{a,b,c\}$ and $\mathcal{T}=\{X,\emptyset,\{a\},\{b\},\{a,b\}\}$. Let $A=\{a,b\}$ Find each of the following

Let $X=\{a,b,c\}$ and $\mathcal{T}=\{X,\emptyset,\{a\},\{b\},\{a,b\}\}$. Let $A=\{a,b\}$ Find each of the following sets: $Int_{A}(\{a\})$ $Int_{X}(\{a\})$ $Int_{A}(\{c\})$ Can anyone explain how ...
106 views

35 views

### Solving for $y$ from multiple $y$ terms

If you were given a problem like, $y^3+7y=16x^2-3x+2$, where there are multiple terms with $y$ of different powers in them, how would you solve for $y$? Also, are there many situations where you ...
78 views

85 views

### Continuous function - how prove?

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f(x_1,x_2) = (\cos(x_1),x_2^2)$. Is it true that this function is continuous? What about $f(x_1,x_2) = (e^{x_1},x_2)$ ? I suppose that these ...
349 views

### How to find the points at which a piecewise defined function is continuous?

Define $$f(x) = \begin{cases} 11 & 0 \leq x \leq 1\\ x & 1< x \leq 2 \end{cases}$$ At what points is the function $f:[0,2]\to \mathbb{R}$ continuous? I am pretty sure that the ...
47 views

### What is $\lim\limits_{i\to 0} \dfrac{2^n}{\frac{(n+1)\sin((n+1)\theta)}{\sin\theta } - \frac{(n-1)\sin((n-1)\theta)}{\sin\theta }}$?

What is $$\lim\limits_{i\to 0} \dfrac{2^n}{\frac{(n+1)\sin((n+1)\theta)}{\sin\theta } - \frac{(n-1)\sin((n-1)\theta)}{\sin\theta }}$$ where $$\theta=\frac{i\pi}{n}$$ The second page of this document ...
95 views

### If $L_2L_1$ is accepted by a DFA, is $L_1$ too?

Given that $L_2, L_2L_1$ are accepted by a DFA, is $L_1$ accepted by a DFA too? What is the general approach to such question? What if instead of $\cdot$ we are given that $L_2 \cup L_1$ is ...
75 views

### Solving third order quadratic nonlinear DE

I would like to solve the following nonlinear DE using a numerical integration algorithm (like runge-kutta) $a_0\dddot{x}^2+a_1\dddot{x}+m\ddot{x}+c\dot{x}+kx=0$ I cannot isolate $\dddot{x}$, how ...
36 views

### Finding a relationship between x and y of a DE

I have the differential equation: $$\frac{dy}{dx}=\frac{-5y-xy}{-4x-xy}$$ How do I go about finding a relationship between $x$ and $y$?
88 views

### Properties of a pseudo-metric on a measure space

Given a measure space $\left(X,\mathcal{F},\mu\right)$ and two $\mathcal{F}-\mbox{measurable}$ functions $f,g:\left(X,\mathcal{F}\right)\to\mathbb{R}$ we define the ...
208 views

### Convolution and uniform continuity

If $f\in L^{\infty}(\mathbb{R}^n)$ and $f$ is continuous at $x$, then $$\lim_{k\to\infty}(f*\phi_k)(x)=cf(x)$$ If $f\in L^{\infty}(\mathbb{R}^n)$ and is uniformly continuous, then $f*\phi_k\to cf$ ...
### Prove that $X$ is a finite set
Base case: $7 \in X$ Recursive case: If $x \in X$, either $\dfrac{x}{2} \in X$ (if $x$ is even) or $3 \times x + 1 \in X$ (if $x$ is odd) Prove that $X$ is a finite set by explicitly listing all of ...