2
votes
1answer
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 ...
1
vote
1answer
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. ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
0
votes
2answers
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 ...
0
votes
1answer
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 ...
0
votes
2answers
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 ...
1
vote
1answer
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 ...
2
votes
2answers
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 ...
1
vote
1answer
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 ...
0
votes
1answer
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 ...
1
vote
0answers
62 views

Regions formed by polar coordinates in double integration.

I need to sketch the region of integration of the following double integral in the $xy$ plane: $$\int_0^{\pi/2}\int_0^{1/\cos\theta} f(r,\theta) \ dr \ d\theta,$$ where $f(r,\theta)= ...
1
vote
2answers
78 views

Solving for distance and time. Please help, I am stuck, and need to know how to do this please.

A ship is 4 degrees off course. If the ship is traveling at 10 miles per hour, how far off course will it be after 6 hours
0
votes
1answer
246 views

Natural Deduction please help!

I am sorry for posting this here, but this is my last resort. I have been fighting with these natural deduction problems for the last two weeks. I take an online college logic course and it makes it ...
0
votes
3answers
129 views

Find the max value of a function at a given interval

Trying to determine local max for a function at interval $[-4, 6]$. $$f(x)= x^3 -3x^2-24x + 7$$ Is the proper next step to take the derivative of $f(x)$ and find the roots, set roots = to zero?
1
vote
1answer
85 views

That submodule generated by one element leads to submodule being finitely generated

In Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, in the prove of Proposition 1.4, the auther seems to use the following fact. Let $R$ be a Noetherian ring, $M$ is a ...
0
votes
2answers
37 views

Are homomorphisms a finite group?

I have a question from my homework that says: Find all the homomorphisms φ:ℂ->ℂ such that for any $x$ in ℝ: φ($x$) = $x$ I don't even know how to begin. The field of complex numbers is infinite. ...
4
votes
1answer
128 views

Is the direct limit of Noetherian rings necessarily Noetherian?

Is the direct limit of Noetherian rings necessarily Noetherian? And if it is, how to prove; if it is not,what is a counterexample. (I was thinking this question: if $A_{m}$ are Noetherian for $m\in ...
0
votes
2answers
56 views

limit of a finite sequence (if I read it correctly)

this is the limit to evaluate: $$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } \root n \of {{a_1}^n + {a_2}^n + ...{a_k}^n} = \max \{ {a_1}...{a_k}\} \cr & {a_1}...{a_k} \ge 0 ...
0
votes
1answer
153 views

Proving a vector equality in a triangle without using Thales' theorem.

Problem Let $\text{ABC}$ be a triangle, and $\text{M}$ and $\text{N}$ are points where: $\vec{\text{AM}}=\frac{1}{3}\vec{\text{AB}}$ and $\vec{\text{AN}}=\frac{1}{2}\vec{\text{AB}}$ and ...
0
votes
1answer
83 views

Use the definition of derivative to find $f'(1)$ for $f(x) = \frac{x}{\sqrt{x^2+1}}$

This is analysis.. So I am using the definition that $$ f'(x)=\lim_{x\to c} \frac{f(x)-f(c)}{x-c} $$ So far I have, $$ \lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - ...
4
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
2answers
621 views

Show that a positive operator on a complex Hilbert space is self-adjoint

Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A : \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in ...
2
votes
1answer
52 views

Show that a random variable is not dominated

Let $((0,1], \mathcal{B}_{(0,1]}, \lambda)$ be a probability space, and define $$ X_n = n 1_{(0,1/n]} $$ This is an example where $\lim E(X_n) \neq E( \lim X_n)$, and the dominated convergence theorem ...
0
votes
1answer
52 views

Abstract Algebra (Ring Homomorphisms and Ideals)

Show that the equation $y^2=4$ has at least $4$ solutions in the ring $\mathbb{Z}/5[x]/\langle x^2+1\rangle$. What do you conclude? My main question about this is what $\mathbb{Z}/5[x]/\langle ...
1
vote
1answer
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
106 views

Differential equation-issue

I need some help with this task: Determine the constant $k$ such that the function $y$ given by the following differential equation: $$\frac {dy}{dx}-y=xe^{kx}$$ satisfies $\lim_{x\to \infty} ...
0
votes
1answer
46 views

Show that $\mathrm{Ann}(M) = (8) \dots$, possible typo?

The question I was given is stated as follows: Let $M = \{1+a_1x+\dots a_7x^7 : a_i \in \{0,1 \} \} \subset \mathbb{Z}/2\mathbb{Z}[x]$. With group law $a(x)*b(x) = a(x)b(x) \mod x^8$, M becomes ...
0
votes
1answer
71 views

Orthogonal projection on span $x$

If $x = (2,1)$ and $y = (1,-1)$ how can I find orthogonal projection on the span of $x$? And projection on the span of $x$ along the span of $y$? What I have done is, for the first question, I did ...
2
votes
2answers
40 views

Determine a set H of all vectors (x, y, z) ϵ R³ that are L.C. of vectors U, V and W

Let me put the enunciation first: Let be the vectors $(U, V, W) \in \mathbb{R}^3: U = (1, 1, 5), V = (2, 1, 4), W = (-3,-1,-7)$, then determine the set $H$ of all vectors $(x, y, z) \in ...
1
vote
0answers
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 ...
2
votes
1answer
78 views

Norms Induce by Inner Products (Complex Case)

I have just proved that if $||\cdot||$ satisfy $||u+v||^{2}+||u-v||^{2}=2||u||^{2}+2||v||^{2}$, then there exists an inner product such that $||u||^{2}= \langle u,u \rangle$ is given by $\langle u,v ...
2
votes
1answer
97 views

compute Pr($X_1>X_2$)

I have two iid random variables $X_1,X_2$ with cdf $F$. I want to compute Pr($X_1>X_2$) and express it with $F$. Is it correct to use the following? Pr($X_1>X_2$)=Pr($X_1>x|X_2=x$)=$1-F(x)$ ...
3
votes
0answers
96 views

Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
1
vote
2answers
406 views

lim n tends to infinity, $\frac{\sqrt{n}\sqrt{n+1} }{(n+1)}$

I am interested in the limit $\frac{\sqrt{n}\sqrt{n+1} }{(n+1)}$ as n grows without bound. There is already one question on this site asking about this limit. However this is not duplicate, since my ...
2
votes
2answers
2k views

How many ways to seat 9 couple around a round table

You are a host/hostess at your local Applebee’s. You are seating a group consisting of 9 couples at a round table. A)In how many different ways can you do this, provided that each couple will sit ...
0
votes
1answer
68 views

Exponential derivative using L'Hopital

Obviously $f(x) = e^x$ but we have yet to learn about that officially in this class. The hint that was given was:In the definition of the derivative, set $t = 1/x$ to convert to a limit as $t → ...
0
votes
2answers
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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 $y$?
2
votes
2answers
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 ...
2
votes
1answer
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 $f*\phi_k\to cf$ ...
0
votes
2answers
43 views

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 ...
1
vote
0answers
441 views

What are some good ideas in teaching combinations and permutations

I am a student teacher trying to brainstorm some effective lesson plans for combinations and permutations for a high school statistics course. My master teacher has decided that he will introduce ...

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