# All Questions

56 views

### Finding the characteristic ODE from a nonlinear PDE

I am studying for a PDE exam on Tuesday, and I am getting pretty confused about one specific type of problem and I am thinking that perhaps I am misinterpreting the correct procedure to follow. The ...
115 views

### On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
423 views

### Quotient of two smooth functions is smooth

Let $f:\mathbb R\to \mathbb R$ be a $C^\infty$-smooth function. Suppose that $f^{(k)}(0)=0$ for $k=0,\dots,n-1$. Prove that the function $g(x)=f(x)/x^n$ extends to a $C^\infty$-smooth function on ...
42 views

### Prove a language is NP-Complete

$A$ is NP-complete. $B$ is P. $A \cap B = \emptyset$ $A \cup B \neq \sum^{*}$ Prove that $A \cup B$ is NP-complete. How can I prove this ? I think if anything can be P-reducible to A then it ...
88 views

### Integrate $f(x)=x^2\ln\left(2\sqrt{\frac{a^2-x^2}{a^2+4x^2}}+\sqrt{\frac{5a^2}{a^2+4x^2}}\right)$

I am trying to find the integral of $$f(x)=x^2\ln\left(2\sqrt{\frac{a^2-x^2}{a^2+4x^2}}+\sqrt{\frac{5a^2}{a^2+4x^2}}\right)$$ And I am having no luck with it. Does anyone have any ideas? Is it even ...
99 views

### Let V denote the Klein 4-group. Show that $\text{Aut} (V)$ is isomorphic to $S_3$

After a week in my Abstract Algebra class, the professor proposed this as a problem. I'm not entirely sure where to begin. $V = \{ e, \tau, \tau_1, \tau_2 \}$, so I'm not sure exactly what is meant ...
105 views

### How can I compute the sum of the primes (with powers) that occur in the factorization of an integer?

For example, $40=2^3\cdot 5$, so the sum $S(40)=2^3+5=13$. Also, $200=2^3\cdot 5^2$, so $S(200)=2^3+5^2=8+25=33$. For a fixed $n$, I'd like to find some properties about $S(n)$. But I could find ...
65 views

### Is this a correct way to prove uniqueness using limits?

I have a question about the following proof: Claim: A sequence in $\mathbb{R}$ can have at most one limit. Proof: Assume a sequence $X = (x_n)$ has two limits. Call them $x$ and $x'$. For ...
19 views

211 views

145 views

### Find cumulative distribution function of a continuous random variable.

$X$ is a random variable with density $f(x)=0.5e^{-|x|}, (-\infty<x<\infty)$. Find c.d.f of $x^2$. I dont quite get the hang of these. I tried for just x and got the following. for $x<0$: ...
47 views

### Adding remainders 1 to different coprime modulos

If i have $a^{\varphi(b)}+b^{\varphi(a)}\equiv 1\pmod b$ and $a^{\varphi(b)}+b^{\varphi(a)}\equiv 1\pmod a$ where $(a,b) = 1$, I was wondering why am I allowed to conclude that ...
39 views

### Predicate Logic formula

I came across this problem and found it quite challenging to solve in predicate logic. Here is the signature of the logic: $$\sigma=\{a,P/1,Q/2\}$$ where $a$ represents 10, $P(x)$ represents "$x$ is ...
69 views

### Should I be able to prove Law of Cosines, Half Angle formula, etc?

This is more of a general question then the title suggests, but the laws in the title are what I'm currently studying. I can read the proofs of both and understand them after a while, but I could ...
A population starts with one amoeba. In each generation, each amoeba divides in two with probability $\frac{1}{2}$, or dies, with probability $\frac{1}{2}$. Let $p_n$ be the probability that the ...