1
vote
1answer
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
2
votes
2answers
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 ...
0
votes
2answers
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 ...
1
vote
3answers
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 ...
0
votes
2answers
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 ...
3
votes
1answer
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 ...
6
votes
4answers
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 ...
0
votes
1answer
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?
0
votes
1answer
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 ...
2
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
3
votes
2answers
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 ...
0
votes
1answer
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 ...
2
votes
0answers
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 ...
0
votes
1answer
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 ...
6
votes
1answer
311 views

Green's function in 2D

How does one compute the Green's function of the laplacian on $ \mathbb{R}^2 $? Can someone point me to a reference? In particular, the fourier transform: $$ \int_0^\infty \int_0^\infty \frac{e^{i m ...
4
votes
2answers
189 views

Finding derivatives of Trigonometric Functions

I'm having a very difficult time understanding how find the derivative of trig functions. Here's one that I've been trying to crack for the half hour: $$y = \frac{11}{\sin x} + \frac{1}{\cot x}$$ I ...
0
votes
1answer
77 views

convex conjugate $f^*$ is proper if both $f$ and $f^{**}$ are

If $f$ and $f^{**}$ on $\mathbb R^d$ are proper functions where $f^*$ stands for the convex conjugate of $f$ why does that follow that $f^*$ is proper, too? Thanks a lot...
0
votes
1answer
145 views

exponential random variable question

let X be an exponential random variable with parameter λ=4 and Y be an exponential random variable with parameter λ=5. X and Y are independent. Find the probability that 3X<2Y. I'm thinking of ...
5
votes
5answers
1k views

Show that open segment $(a,b)$, close segment $[a,b]$ have the same cardinality as $\mathbb{R}$

a) Show that any open segment $(a,b)$ with $a<b$ has the same cardinality as $\mathbb{R}$. b) Show that any closed segment $[a,b]$ with $a<b$ has the same cardinality as $\mathbb{R}$. ...
2
votes
1answer
167 views

Equation of birationally equivalent curve?

I'm reading through Shafarevich volume 1 and am a bit confused about some of his examples. So I get how if we use the line $y=tx$ then we can show certain curves are birational to a line. One of his ...
3
votes
2answers
439 views

When is the number 11111…1 a prime number?

For which $n$ is the sum: $$\sum_{k=0}^{n}10^k$$ a prime number? Are they finite?
6
votes
1answer
108 views

What theorems/examples will make me really understand representation theory?

Okay, so I've been through some basic results on representation theory. I've gone over the proof of Burnside's $pq$ theorem using characters. I've also read though the basics of Lie groups and ...
1
vote
1answer
360 views

finding $\int_0^\infty \dfrac{dx}{1+x^4}$ through complex analysis

I am trying to find $\int_0^\infty \dfrac{dx}{1+x^4}$ by setting it equal to $\dfrac{1}{2}\oint_C \dfrac{dz}{1+z^4}$ and solving that. By a computer program I've calculated it to be $\approx 1.11072$; ...
1
vote
2answers
284 views

Inverse of natural log

I have a problem: Let $f(x)=\ln(x)$ solve each of the following equations for $x$. the question is in three parts $(f(x))^{-1}=5$ $f^{-1}(x)=5$ $f(x^{-1})=5$ My understanding is that $\ln(x)$ is ...
3
votes
1answer
133 views

Is there any deformation retraction between a point and a circle?

A point is a retraction of every topological space but how it isn't a Deformation Retraction of s1(circle)?
4
votes
1answer
325 views

On the definition of ideal (is it a subring?)

My book defines an ideal to be a subring of R such that $xr \in I$ and $rx \in I$, whenever $r \in R$ and $x \in I$. However, by definition any subring of a ring with unity must contain unity. So it ...
0
votes
2answers
239 views

Graph Theory Question (Bipartite graph/Cartesian)

Prove that G and H are bipartite if any only if G x H is bipartite. Can anyone give me an idea of how to start this proof?
4
votes
2answers
73 views

Pathological function needed

Give a differentiable function that has a positive derivative at $0$, yet is not increasing on any open neighbourhood of $0$. I believe that the required function needs to have a derived ...
1
vote
1answer
238 views

Fourier transform of a function is square integrable

Is there a result stating that if a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable and decays at infinity, then its Fourier transform is also square integrable?
1
vote
1answer
184 views

How to find interior, closure, limit points, and isolated points of subsets of $\mathbb{R}$?

(a) $\{m+nπ:m,n\in \mathbb{Z}\}$ (b) $\{m+nπ:m,n \textrm{ are positive integers}\}$ I know the interior of a and b are both empty and the answer said the closure is $\mathbb{R}$,i got troubled in ...
-1
votes
2answers
805 views

Show that $V = \{\mathrm{id}, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $\operatorname{Sym}(4)$…

Show that $V = \{\mathrm{id}, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $\operatorname{Sym}(4)$, and that $G/V$ is isomorphic to $\operatorname{Sym}(3)$. (The group $V$ is the Klein ...
1
vote
0answers
59 views

Compute the covariance of $R_2 R_1^T$ where $R_2$ and $R_1$ are rotation matrices with Gaussian uncertainty

I have estimates of two 3x3 rotation matrices $R_1$, $R_2$ expressed in terms of their expected values $R_{1\mu}$ and $R_{2\mu}$ and covariances $\Sigma_1$, $\Sigma_2$. The latter are expressed in the ...
1
vote
0answers
191 views

Signed Measure Integral

Let $\mu$ a signed measure, and $f$ a integrable function with respect to $|\mu|$, and if $\nu$ is defined for every mensurable set by $$\nu(E) = \int_E f \, d\mu$$ then $\nu \ll \mu$. I need some ...
3
votes
2answers
182 views

Does the limit as $(x,y) \to (1,2)$ of $3x^3-x^2 y^2$ exist?

The title says it all: does the following limit exist? $$\lim_{\large(x, y) \to (1, 2)} \; 3x^3 - x^2y^2$$ It approaches $\,-1\,$ with direct substitution, but if you approach the point with the ...
1
vote
2answers
482 views

Rouche's Theorem problem for a general polynomial

Prove that for any $a\in \mathbb{C}$ and $n\geq 2$, the polynomial $az^n+z+1$ has at least one root in the disk $|z| < 2$. My instinct is to use Rouche's Theorem for this problem, however I have ...
1
vote
3answers
679 views

Complex number inequality.

If z and w are distinct complex numbers such that $|z| =|w| = r$, prove that $|\frac{1}{2}(z + w)| < r$.
5
votes
4answers
318 views

If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?

If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded? ($a$ and $b$ being finite numbers). I tried proving and disproving it. Couldn't find an example for a ...
0
votes
2answers
39 views

If two infinites series obey an inequality from $k = N,\ldots,\infty$ does this imply that the inequality is obeyed for $k=1,\ldots, \infty$?

If $$ \frac{1}{k^{\log(k)}} < \frac{1}{k^2} $$ for $100 < k$, does this automatically imply that: $$ \sum_{k=1}^\infty \frac{1}{k^{\log(k)}} < \sum_{k=1}^\infty \frac{1}{k^2}, $$ or only ...
2
votes
0answers
207 views

Desired Z axis and Yaw to ZXY Euler Angles?

I'm trying to calculate a desired pair of pitch and roll Euler angles (the XY in ZXY format) given a desired z-axis of the rotated frame (expressed in the world frame) and a specified yaw angle ...
0
votes
1answer
26 views

Finding integrator

I am currently studying Riemann Stieltjes integrals and I am stuck on understanding the following: $\int_a^bfd\alpha=f(c_1)+5f(c_2)+12f(c_3)+1/2\int_c^df(x)dx$ implies that $\alpha$ has jump ...
1
vote
1answer
125 views

inverse of function

Thanks for the help! Here is the solution.. i had a problem: $$f(x)=\frac{(\sqrt x+8)}{(5-\sqrt x)}$$ i had to find the inverse, so lets begin... 1) first i write in terms of $y$ ...
0
votes
1answer
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.
1
vote
0answers
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 ...
1
vote
0answers
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 ...
0
votes
1answer
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 ...
2
votes
1answer
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 $[0,1]$.
2
votes
3answers
277 views

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 ...
1
vote
1answer
204 views

Calculate Statistics (Check if the answers are correct)

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; ...

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