# All Questions

101 views

### PDE method of characteristics solving $e^{t^2}u_t+tu_x=0$ with $u(x,0)=x+2$

The equation is given as: $$e^{t^2}u_t+tu_x=0$$ with $u(x,0)=x+2$ I've got $x=\frac{1}{2}e^{t^2}+x_0$ but I'm not sure where to go from there
2k views

### Finding the Solutions of the two systems by using the inverse.

I am having a difficult time understanding where I went wrong with the following: $$\begin{matrix}4x-y = 1 \\ 2x+3y = 3 \end{matrix}$$ $$\begin{matrix}4x-y = -3 \\ 2x+3y = 3 \end{matrix}$$ I found ...
92 views

### Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...
56 views

### Probability with time

The time to fly between New York City and Atlanta is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight takes more than 140 ...
924 views

### Probability using Percentages

In a management trainee program, 80 percent of the trainees are female, 20 percent male. 90 percent of the females attended college, 78 percent of the males attended college. A management trainee is ...
550 views

### $(L^\infty)^*$ is not isomorphic to $L^1$

I am working on this problem in Rudin: Let $L^\infty=L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$. Show that there is a bounded linear functional $\lambda\neq 0$ on $L^\infty$ that is ...
254 views

### Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$

Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$ My thoughts: By the theorem: Suppose $a_n\ge0$ for all $n$, and let ...
117 views

### Binomial or permutation probability?

A bowl contains 20 white balls, 10 red balls, and 10 blue balls. Assuming replacement, what is the probability you draw three red balls in a row?
97 views

### Explicitly Proving a parametrization for $x^2 + y^2 - z^2 = a$ for $a < 0$ is a Diffeomorphism.

Problem: I'd like to parametrize the manifold given by $\{(x,y,z)\in{\mathbb R}^{3}\,|\, x^2 + y^2 - z^2 = a\}$ for $a < 0$. The two mappings we'd use are $f(x,y) = (x,y,\sqrt{x^2 + y^2 - a})$ and ...
691 views

### Finding the limit by using roots, quotient and sum laws.

$$lim_{x\to-0.2^+} \sqrt{\frac{x+11}{x+2}}$$ I have tried working this problem out and the limit I got was 2.44. My first attempt was just plugging in $-0.2$ and solving for the limit. I was told to ...
52 views

### Lower bound and monotonocally decreasing functions

I may be missing something quite obvious, but why does it follow that if $$(d/dx)(f(x)+{g(x)\over x^2 })<0$$ then for some $a>0$, we can say $$f(0)>{g(a)\over a^2 }$$? Sorry about my ...
45 views

### Predictions for recurrence relations

Given the recurrence relation $a_n = 2a_{n-1} + a_{n-2}$ $a_0 = 1$ and $a_1=1$ Is it true that $a_n < 6a_{n-2}$ for all $n\ge4$ I'm not really sure how to go about solving this problem. I've ...
301 views

### Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$,$\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ converge, their product converges too

Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ both converge, $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ also converge then Show if ...
71 views

### Understanding the proof that if two loops in $S^1$ are equivalent then their degrees are equal

I am trying to understand the proof of the following: Theorem: For loops $\alpha$, $\beta$ in $S^1$ with base point $1=(1,0)$, $[\alpha]=[\beta]$ if and only if ...
93 views

### Invariant Subspaces and Differential Equations

Given I'm given a marginally stable system, $\dot{x}(t)=Ax(t)$, where$A=\begin{bmatrix} -1 & -10 & -10\cr 1 & 0 & 0\cr 0 & 1 & 0 \end{bmatrix}$, and $x(0)=x_o$.The eigenvalues ...
86 views

### Finding Eigenvectors

Let $A$ be a $2\times2$ matrix: $$\begin{bmatrix} 1 & 1\\ 1 & 0\\ \end{bmatrix}.$$ I found the eigen values $$\lambda_1=1-\sqrt{5} \\\text{and} \\ \lambda_2=1+\sqrt{5}.$$ But for some reason ...
311 views

35 views

### Need help with Volume question.

I have to find the Area of the Vertical cross section A and the Volume. I have no idea how to do this problem we never learned this in class. Need all the help I can get. Thank you.
59 views

### Are finitely additive measures 'topological'?

The category of measurable spaces are topological over $Set$ in that they support initial & final structures similarly to that topological spaces. A measurable space is a set supporting ...
263 views

### Is the composition of trace inverse and convex matrix product convex?

Is the trace of the inverse of the matrix product $B^TB$, i.e. $\mathrm{trace}((B^TB)^{-1})$, convex where $B\in M_{n,m}$. I know that $S\longrightarrow \mathrm{trace}(S^{-1})$ is a convex function ...
101 views

### Equiprobable model for Pearson's goodness-of-fit method.

There is a very long introduction to this problem. I can provide this if needed but for now I will stick with the actual question. "A question of interest to the researchers was whether there were ...
72 views

### How to use Rolle's Theorem to prove the following?:

For each $\lambda$, the function $f(x)=x^3-\frac 32x^2+\lambda$ does not have two distinct zeros in $[0,1]$.
### Complex Logarithms: Detailed explanation for why $\operatorname{Log} z^2$ is not equal to $2\operatorname{Log}z$
Why is $\operatorname{Log} z^2$ not equal to $2\operatorname{Log} z$ where $z$ is a complex number. $\operatorname{Log} z$ here refers to just the principal Log. Detailed explanation would be ...
Calculate the statistics below using the following data on a sample of the variable $X$: Data ($X$ sample) = $\{9, -1, 7, 0, -2, 5, 4, 9, 5 \}$ Using the sample data, calculate: Mean; Median; ...