For posts looking for feedback or verification of a proposed solution.

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0
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1answer
27 views

Radius and Interval of Convergence of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{2n-1}$

This is my first time finding the radius and interval of convergence of a series, so please bear with me. I would like to find the radius and interval of convergence of ...
0
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0answers
26 views

Isomorphism of vector spaces

Let $S$ be the space of all $3\times k$ matrices,$T$ be the space of all column vectors consists of seven components.If $S$ is isomorphic to a subspace of $T$ then what are possible values of $k$? I ...
2
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2answers
27 views

Absolute/Conditional Convergence or Divergence of $\sum\limits_{n=1}^{\infty}\frac{(-1)^{n}e^{1/n}}{n^3}$

I'd just like to verify that my solution is right. This is for the series $$\sum\limits_{n=1}^{\infty}\frac{(-1)^{n}e^{1/n}}{n^3}\text{.}$$ Note ...
4
votes
1answer
43 views

Examining the convergence of $\int_{1}^{\infty}\frac{1}{x^2+x}\text{ d}x$

I'd like to have my solution verified for this one. I'd like to show that $$\int\limits_{1}^{\infty}\dfrac{1}{x^2+x}\text{ d}x$$ is convergent. Notice, by partial fraction decomposition, that ...
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0answers
33 views

Explaining the divergence of $\displaystyle\int\limits_{-\infty}^{0}\dfrac{1}{3-4x}\text{ d}x$

This is mostly to see if my solution is correct. I would like to show that $$\int\limits_{-\infty}^{0}\dfrac{1}{3-4x}\text{ d}x$$ is a divergent integral. Notice $$\begin{align} ...
4
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2answers
63 views

$\displaystyle\int\limits_{\sqrt{2}}^{2}\dfrac{1}{t^3\sqrt{t^2-1}}$

This is my very first trigonometric substitution integral, so please bear with me here. I would mostly like verification that my methods are sound. ...
2
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0answers
66 views

measure theory exercise (verification)

Hi I found the following exercise in the Dudley's book and I'd like to see if my answer is correct; the last part is what I'm not entirely sure, since I'm not completely familiar with this kind of ...
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1answer
46 views

Can the depicted function be a solution of an ODE with locally Lipschitz autonomous vector field?

Problem: Can x(t) depicted be a solution of a scalar differential equation x(dot)=f with locally Lipschitz autonomous
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0answers
21 views

Counting similar pairs

I was given a simple programming assignment: Your task is to quickly find the number of pairs of sentences that are at the word-level edit distance at most 1. Two sentences S1 and S2 they are ...
0
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1answer
17 views

T/F Limit question.

True or False? a) If $f(x) → 0$ as $x → a^+$,(from the right) and $g(x) \ge 1$ for all $x$ in $\Bbb R$, then $g(x)/f(x) → ∞$ as $x → a^+$. True: take $f(x) = \sin x$ and $g(x) = x^2$ as $x → pi/2$ ...
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1answer
29 views

True or False, limit, functions questions. Does limit exist?

True or False Let a be a real number, and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. a) For each natural n, the function ...
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0answers
30 views

Nullity and rank of a linear transformation

Let $T:V\to W$ be a linear transformation, where $$T=d^2/{dx}^2,\\V=\{f(x): f \text{ polynomial of degree}\leq n\},\\W=\{f(x): f \text{ polynomial of degree}\leq n-2\}.$$ What is the nullity of ...
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0answers
26 views

Ideals of a field

I had the following - apparently straightforward - question on one of my past assignments: Show that a field has no other ideals except $\{0\}$ and the field itself. This was the proof I gave: ...
0
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1answer
23 views

Eigen values of a positive semidefinite matrix and its transpose

$A\in M_n(\mathbb{C})$ is positive semi-definite so there there exists unitary matrix $U$ such that $A=U^*DU$ where $D$ is the real diagonal matrix consisting of eigen values $(\ge 0)$ of $A$, now I ...
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2answers
8 views

Find the solutions of an equation with arctan?

I have to show that $1$ and $\frac{-1}{\sqrt{3}}$ are (maybe not) solutions of the following equation: $arctan(x)+arctan(x\sqrt{3})= \frac{7\pi}{12}$. How can I do that ? Thank you in advance
2
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2answers
36 views

A measure is sigma-finite if, and only if, there exists a integrable function w such that its image is contained in (0,1)

I have to prove the following proposition: Consider a measure space $(S,\Sigma,\mu)$. Prove that $\mu$ is $\sigma$-finite if, and only if, there exists $w\in\mathcal{L}^1(S,\Sigma,\mu)$ such that ...
2
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2answers
22 views

Write out the following set by listing its elements between braces.

Write out the following set by listing its elements between braces: $\begin{align} \{X \subseteq \mathbb{N}: |X| \leq 1 \} &=\{\emptyset,\{1\},\{2\},\{3\},\ldots\}\end{align} $ Is my answer ...
0
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1answer
36 views

Wrong solution set in textbook, quadratic equation

So we have an the equation $\frac{2}{3}t^2+\frac{4}{3}t=\frac 15$, when you finish solving the equation you get $t = \frac{-10 + \sqrt{130}}{10} $ and $\frac{-10 - \sqrt{130}}{10}$. The text book ...
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0answers
32 views

Prove that function is differentiable at $0$ if and only if $a>3/2$

Let $a>0$. I have to show that function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ $$f(x,y):=\frac{x^{2a}+y^{2a}}{x^2+y^2}$$ when $(x,y)\ne(0,0)$ and $f(0,0):=(0,0)$ is differentiable at $(0,0)$ if ...
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1answer
29 views

Conflict between geometric intuition and computed answer

Evaluate the integral $\int_{C} z ds$ where C is the intersection of $x^{2}+y^{2}=4$ and $z=0$ (oriented clockwise as viewed from above). My interpretation of this problem yields the following ...
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0answers
21 views

Negative integrals spherical coordinates

Need help interpreting a result prompted by the following question. Solve the triple integral of $\sqrt{x^{2} + y^{2} +z^{2}}$ on the region Q where Q is bound by $z=-\sqrt{9-x^2-y^2}$ and the x-y ...
1
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1answer
32 views

A simple problem of the equation of a plane.

Two planes given $$x-y+z=5 , \hspace{0.5cm}x+y+z=3 $$ Their intersection is a line $l$.Find the equation of a plane such that the line $l$ is perpendicular to that required plane and this plane ...
0
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1answer
32 views

Bernoulli distribution solving for n

So we have this missile protection system that has $n$ radar sets that are all independent. Each have a probability of $0.9$ of detecting a missile. How large must $n$ be if we want the probability ...
0
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1answer
14 views

Conditional entropy of repetition code over BSC

Consider the channel that takes in a bit, repeats it $k$ times, then sends the result over a binary symmetric channel with transition probability $p$. For example, if $0$ was sent over the channel ...
1
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1answer
31 views

Statement about solutions for diff. equations.

Let $f_1,f_2:\mathbb{R}^2\to\mathbb{R}$ be $C^{\infty}$ functions such that $f_1(x,y)\leq f_2(x,y)\; \forall(x,y)\in\mathbb{R}^2$. Suppose that $\psi_1:I_1\to \mathbb{R}$ and $\psi_2:I_2\to ...
2
votes
1answer
26 views

Concise description of Lebesgue measure in $\mathbb{R}^{n}$

I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$. The outer Lebesgue measure $E \subset \mathbb{R}^{n}$: $$\lambda^{*}(E) = ...
0
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0answers
26 views

Verifying the Solutions of an Equation

I wanted to verify the solutions of the equation $y^2+xz=0$. Solution Let $x=r$ and $y=s$ (I choose these as the free parameters). Then, $z=-\frac{s^2}{r}$, provided that $r\neq 0$. The solution ...
0
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1answer
31 views

Which of the following is not uniformly continuous?

Which of the following is not uniformly continuous? 1.$f_1(x)=|x|$ 2.$f_2(x)=\frac{1}{1+x^2}$ 3.$f_3(x)=\sin x^2$ 4.$f_4(x)=\ln(1+x^2)$ 5.$f_5(x)=e^{-x}$ My solution:$f_1(x)=|x|$ is lipschitz so ...
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2answers
66 views

Why does $\lambda (e^x - 1) = x$ have two solutions for $\lambda > 1$?

Apparently $\lambda (e^x - 1) = x$ has two solutions for $\lambda > 1$. My textbook is kind of handwaving and saying that this is true without explaining. Can somebody prove this for me or show me ...
0
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2answers
71 views

Multilinear Algebra, finding $z \wedge z.$

Define $A^k(V)$ to be the set of all alternating multilinear functions from $V^k \to \mathbb{R}$. Consider the space $A^2(\mathbb{R}^4)$, does there exists $z\in A^2(\mathbb{R}^4)$ such that $z ...
1
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1answer
60 views

Conditional probability plane problem

I was presented with this problem and am not sure where to take it. A plane is missing and is presume to have equal probability of going down in any of 3 regions. If a plane is actually down in ...
2
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1answer
65 views

Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$

Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$. My Solution: For all $n$ that is an element of Natural number there is ...
1
vote
1answer
17 views

Differential equation system changing the eigen-values

The following is a problem from a past exam, that I couldn't answer (or at least I didn't knew how to answer). Consider $\Omega:=\begin{pmatrix} \alpha & 1 \\ 0 & \alpha \end{pmatrix}$, ...
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0answers
26 views

Number of distinguishable arrangements of the word INDOOROOPILLY with three different conditions

I have the following three questions on a past final exam, I wanted to ask if I have done everything correctly. Thank you! How many distinguishable arrangements are there for the letters of the ...
0
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1answer
18 views

Proving convergence iff projection converges in product topology

This question is regarding the same problem. I wish to present my proposed solution and get feedback on my argument, and as such, I claim that it is not a duplicate. (In particular, the other asker ...
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0answers
25 views

How to show proper set inclusion/exclusion? Please don't give me the solution.

I found this problem from an online source. I've just got two question 1) I think there is a typo in the solution, it should be $(x_n) \in \ell_1$ right? 2) I am guessing $c_0 \subsetneq ...
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votes
1answer
45 views

If $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

can someone please correct me if i made some mistakes here on ma solution. i used Lynx hea Problem D 1.) Give the definition of $S\subseteq T$ for general sets $S$ and $T$. Solution The set $S$ is ...
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0answers
33 views

Partial Derivative Chain Rule verify solution

for a function $g(u,v) = f(x(u,v),y(u,v))$ find $\frac{\partial^{2} g}{\partial u \, \partial v}$ To begin with, we know by the chain rule that the first order partial derivative is: $$ \frac{ ...
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vote
0answers
37 views

Prove that if $X \subset [a,b]$ isn't a measure-zero set, then there exists $\varepsilon >0$

Please, check my solution to this problem: "Prove that if $X \subset [a,b]$ isn't a measure-zero set, then there exists $\varepsilon >0$ such that, for every partition $P$ of $[a,b]$, the sum of ...
1
vote
1answer
17 views

Figuring out the variables rows and colums for matrices

Let P be a 2 × 3 matrix, Q a m × 5 matrix and R a p × q matrix. Find the values of m, p and q such that the operation Q - PR is possible. So I figured that p = 3 Is m=2 and q=5? Just need to make ...
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0answers
50 views

Introductory Topology True/False check - Topology without tears - Exercises 1.1

I have just started to learn Topology, using specifically the book mentioned in the title. I have placed that information in the title with SEO in mind, if this is not acceptable practice in this ...
2
votes
2answers
56 views

Integral of $x^2 \cos(a x)\; \mathrm{d}x$

I am trying to solve the following problem: $\int x^2 \cos(a x)\; \mathrm{d}x$ I thought this would be simple and I am pretty sure this is the answer: $I ...
1
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1answer
34 views

Number of distinct colourings for the regular pentagon

This is an answer check for the number of distinct colouring's for the regular pentagon given only four colour choices. I have the rotational group action ...
1
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0answers
20 views

Constrained optimization minima and maxima and non-degeneracy answer check

Find the critical points of $$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\l{\lambda}$$ $$\f=\1\2+\2\3+\3\1$$ subject to constraint $\1+\2+\3=1$ First I will construct the Lagrangian: $$L ...
5
votes
3answers
113 views

Use the binomial theorem to show that for any positive integer $n$, $\displaystyle\sum_{i=0}^{n} {n \choose i} = 2^n$.

Can somebody check to see if this is good enough just to show? It's very simple but the question doesn't say prove or anything like that. So the binomial theorem states that ...
2
votes
1answer
41 views

Example with almost every convergence where the dominated convergence theorem fails

So I ran into this exercise, and I want someone to check the accuracy of my answer, because I feel pretty sure that I make some mistakes which I can't see. Let $f_n(x) : \mathbb{R} \to \mathbb{R}, ...
0
votes
1answer
25 views

Laplace transform quick answer check :) using second shift theorem

I want to get $L((t-4)^2u(t-4))$ I say this is a second shift with $g(t)=(t^2-4t)$ and my friend says "NO you are wrong, you are dumb!!!!!! $g(t)$ is MOST CERTAINLY equal to $t^2$" Mine gives me ...
1
vote
1answer
22 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
1
vote
1answer
43 views

Laplace Transform assistance

Find the inverse laplace transform of: $\frac{25}{(s-1)^2(s^2+4)}$ $\frac{25}{(s-1)^2(s^2+4)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{s^2 + 4}$ $$25=A(s^2+4)(s-1)+B(s^2+4)+C(s-1)^2$$ ...
2
votes
1answer
53 views

How many ways are there to color the $H$-shaped tree with $3$ colors such that each color is used exactly twice?

How many ways are there to color this graph with the following constraints? We have three colors: blue, red, green, and we require that the number of nodes of color green is 2, and blue 2, and red ...