0
votes
2answers
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 ...
2
votes
0answers
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 ...
4
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
2answers
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 ...
3
votes
1answer
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 ...
3
votes
1answer
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 ...
1
vote
1answer
19 views

How do I get an answer of $14$ using simpsons rule for $\frac{152e}{180n^4}<.0001$

I must have the algebra wrong somewhere but here is the original equation: $$\frac{152e}{180n^4}<.0001$$ If I then multiply like this: $$152e<.0001(180)n^4$$ Which then gives: $$152e < ...
3
votes
1answer
77 views

Semi-direct product question

I have a semi-direct product that I feel must be nonabelian, but my thought process is telling me it is abelian. I have $G\cong Z_7\rtimes Z_4$ and I have indeed found nontrivial homomorphisms ...
0
votes
1answer
45 views

fundamental complex integral( in Conway's book)

I am reading Conway's Functions of One Complex Variable. I have some trouble in doing some exercise. Find $\int_\gamma(z^2-1)^{-1}dz$, where $\gamma$ is a path. $\gamma(t)=1+e^{it}$ for $0\leq t\leq ...
5
votes
2answers
74 views

How prove $A=B=C$?

in $\Delta ABC$, such $$\sin{A}+\cos{B}+\tan{C}=\dfrac{3\sqrt{3}+1}{2}$$ prove that $$A=B=C=\dfrac{\pi}{3}$$ My try: use $$\sin{x}+\sin{y}=2\sin{\dfrac{x+y}{2}}\cos{\dfrac{x-y}{2}}$$ ...
0
votes
1answer
135 views

What's the probability that nine people were born in the same two months (but not the same month)?

Find the probability that nine people were born in the same two months (but not all in the same month). No clue how to approach this. I was thinking well you have to choose 8 out of the 9 people and ...
0
votes
1answer
53 views

Congruence question with divisibility

I have this question and I have proved that a/d is congruent to b/d mod(m/d) However, I don't know how to go forward to prove a/k is congruent to b/k mod(m/d) Can anyone help me out? THX
1
vote
2answers
112 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
1
vote
0answers
58 views

hypothesis test and execution? did I do this right?

I am trying to figure this out and I'm not sure if I am doing it right. a) did I select the correct type of hypothesis testing??? b) am I using the numbers in the right place (I have SD for 3.5 for ...
1
vote
2answers
121 views

Let $F$ be a field. Could the ring $F[x]$ be a field? [duplicate]

$F$ is a field, so by definition, $F$ is a commutative ring with unity in which every non-zero element is a unit. Then, $F[x]$ is a set of polynomials in which the coefficients come from $F$, so all ...
3
votes
0answers
50 views

Find the natural numbers $n$ for which $\varphi(n)$ is not divisible by $4$.

This one is a toughie... I thought I had it, but then my proof didn't work out... Let $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}\dots p_r^{\alpha_r}$ be the canonical form of $n$. Therefore: ...
0
votes
2answers
40 views

Deduce an unspecified polynomial?

I'm having a little trouble with problems that have an unspecified polynomial, for example $p(x)$, and having to get properties of them. A problem I ran across had something along the lines of $p(x) ...
3
votes
1answer
73 views

Real projective $n$ space

We define $\sim$ on $\mathbf{R}^n - \{0\}$ by $x \sim y$ if $x = \lambda y$ for some $\lambda \in \mathbf{R}$. We define projective $n$ space by $X = (\mathbf{R}^n - \{0\})/{\sim}$. I am having ...
2
votes
1answer
125 views

Questions on “painless conjugate gradient”: take gradient of a quadratic form

I am reading this paper: http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf I have difficulties on the derivation of equation (6) on page 4. It is to take gradient of a quadratic ...
0
votes
1answer
29 views

Determining a sequence generation variable

Let $$ \begin{align} r_1 &= c + d + \frac{d}{2} + \frac{d}{4} + \dots + \frac{d}{2^n}\\ r_2 &= c + d + \frac{d}{2} + \frac{d}{4} + \dots + \frac{d}{2^n} + \frac{d}{2^{n+1}} \\ r_3 &= c + ...
1
vote
1answer
211 views

Error analysis of exponential function

By definition: $$ e^x = \lim_{n \rightarrow \infty} ( 1 + \frac{x}{n} ) ^ n$$ I am interesting in calculating the error $$\left | e^x - \left( 1 + \frac{x}{n} \right) ^ n \right|$$ for some fixed $n ...
0
votes
1answer
64 views

Let $T: V→V$ be a linear operator on a vector space $V$. Assume that $T^n = 0v$ for some $n\ge1$. Prove that the linear map $Iv + T$ is an isomorphism

Does $T^n$ refer to $T(T(T(\cdots T(v))))$ - $n$ total compositions of $T$? If so, I'm not exactly sure what information this gives us on the map $T$ aside from the obvious that it slowly converges ...
-1
votes
1answer
161 views

Relation between an integer represented by a binary quadratic form and a certain Dirichlet character defined by Jacobi symbol

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $f$. It's easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). If ...
2
votes
3answers
142 views

If a polynomial does not have roots, does that imply it is irreducible?

One of my homework problems wants me explain why $x^2 +2$ is irreducible in $\mathbb{Z}_5$. The possible roots of $x^2+2$ are $x=\pm \sqrt{-2} \notin \mathbb{Z}_5$. Is it enough to say that since ...
1
vote
2answers
61 views

Perfect square given by ${r}^{2}+u\,r+v$, where r is variable and u,v are constants

I am looking for solution for a problem of finding a perfect square given by $$ {r}^{2}+u\,r+v $$ where $u > 0$ and $v > 0$ are integer constants and expected $r >= 0$. The closest I was ...
1
vote
1answer
28 views

the set of points that are annhilated by a subset $N\subset X'$

Let $X$ be a normed vector space over $\mathbb k$ ($\mathbb k=\mathbb R$ or $\mathbb C$). Let's consider $X'$ the set of continuous linear functionals $f:X\to \mathbb k$ called the dual of $X$. We ...
21
votes
2answers
1k views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
0
votes
1answer
84 views

Let $V=C^n$ be the $n$-dimensional space of $n$-vectors over the field $\Bbb C$.

$R/C$ represent real/complex respectively. a) Explain why we can regard $V$ as a vector space $V_r$ over $R$. b) Determine the dimension $d$ of $V_r$. c) Find an isomorphism of $V_r$ with ...
0
votes
1answer
323 views

Prove the dominant strategy of Game Theory

A row $r$ of the payoff matrix is said to dominate a row $s$ if $a_{rj}\geq a_{sj}$ for all $j$ = 1,2,......,$n$. Similarly, a column $r$ of the payoff matrix is said to dominate a column $s$ if ...
7
votes
1answer
305 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
2
votes
0answers
50 views

Example of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$ but $|X|,|Y| \leq |\mathbb{R}|$

I am interested in knowing examples of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$. By allowing $|X|,|Y|$ to be large we can provide a trivial counterexample, as in the one ...
4
votes
1answer
145 views

Probability convergence in distribution

$Y_1, Y_2,..., Y_n$ are i.i.d and uniformly distributed on the set $\{1, 2,..., n\}$. Define $X_n = \min\{k: Y_k = Y_j\; \text {for some}\; j < k\}$, and prove that $\frac {X_n}{\sqrt n}$ converges ...
0
votes
2answers
1k views

Rotation Matrix inverse using Gauss-Jordan elimination

I'd like to calculate the inverse of a rotation matrix, let take the simplest case which is a $2$ x $2$ rotation matrix: $R =\begin{bmatrix} \cos \theta & -\sin \theta \\[0.3em] \sin \theta & ...
1
vote
1answer
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 $x<0$: ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
2
votes
2answers
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 ...
1
vote
1answer
165 views

Survival Probability of a Population

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 ...
2
votes
0answers
84 views

Skorohod space measurable function

Consider the space of RCLL (Cadlag) functions on the domain $[0,1]$ and endowed with the Skorohod topology. Let us consider the set $S := \{x: \omega_x (\delta) \leq \epsilon\}$, where $\omega_x ...
1
vote
2answers
68 views

If a sequence starts from 0 or -1, how do you refer each item?

If a sequence starts numbering from $0$ or even $-1$, how do you refer the first/second item? I would like to know if $(-1)$-th, $0$-th, $1$-st, $2$-nd acceptable?
2
votes
1answer
152 views

A different approach to finding the area of a triangle

Is it possible to find the area of a triangle with just the circumradius and inradius?
3
votes
3answers
188 views

sequences with cosines are they always divergent?

I have to show if this sequence is converging or diverging : $$a_n=\cos(n/2)$$ I know that $\lim_\infty n/2= \infty $ and I also know that the cosines function is alternating between $[-1,1]$. So by a ...
2
votes
1answer
601 views

Proving if the limit of $f(x)$ approaches zero, then the limit of $1/|f(x)|$ approaches infinity.

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood of ...
1
vote
1answer
43 views

How to find the range of this function?

I need to find the range of the function $g:(0,1)\times (0,1) \to \mathbb{R}^2$ given by \begin{align} g_1(x,y)&= \frac{x}{x+y}\\ g_2(x,y)&= x+y \end{align} I can see that it is ...
1
vote
1answer
147 views

Induction over negative number

Suppose I were given the task of proving that for all negative integers $3n^{2} \equiv 3n \pmod{6}$. The original intent was to use negative induction. But, I was wondering if another, perhaps simpler ...
4
votes
0answers
105 views

Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
1
vote
3answers
313 views

Examples of complex functions with infinitely many complex zeros

What are some examples of complex functions with infinitely many complex zeros? There are no particular restrictions on the functions I am just curious and having a hard time finding examples. Also ...
0
votes
3answers
153 views

Question about direct factors

this is a problem from hungerford: "A normal subgroup $H$ of a group $G$ is said to be a direct factor if there exists a (normal) subgroup $K$ of $G$ such that $G = H \times K$. if $H$ is a direct ...

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