Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...
4
votes
4answers
119 views
Does every ordinal have cardinality no greater than $\aleph_\mathbb{0}$?
My notes say that the ordinals $\omega + 1, \omega + 2, ... , 2 \omega, ... , 3 \omega, ... \omega^2, ... $ are all countable, and hence have cardinality equal to $\omega = \aleph_\mathbb{0}$. So I ...
0
votes
0answers
27 views
distribution function and density function
A lion is standing $30$ meters from one end of a $100$-meter road. The lion will attack any zebra that appears on the road. Suppose that a zebra appears on the road, and suppose that the position at ...
3
votes
1answer
36 views
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it.
Thanks for ...
1
vote
1answer
23 views
How do I work out what percent of my customers will be girls and what percent will be boys?
I know that 33.3333% of all girls questions would buy my product and that 80% of all boys questioned would buy it.
What i don't know is how to work out is statistically what percentage of our ...
1
vote
3answers
42 views
Poisson Distribution - sum of RVs
Question:
$X$ balls are thrown to $n$ bins (each ball has an equal chance to get to each bin). Let $X_1,\dots, X_n$ be the amount of balls in each cell.
a. Show that if $X \sim ...
4
votes
1answer
59 views
Does there exist $g$ s.t $g'=f$?
I have the following homework question:
Let G be the bounded open set shown in gray in this picture, whose
boundary consists of eight line segments. The endpoints of those
segments are, as ...
1
vote
1answer
25 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
2
votes
1answer
53 views
Two problems about Structure Theorem for finitely generated modules over PIDs
1) Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either
$$\begin{pmatrix}
0&-1&0&0\\
...
1
vote
2answers
28 views
Find the closed solution of $s_{n} = 3s_{n-1} + 2^{n-2} - 1$
Find the closed solution of $s_{n} = 3s_{n-1} + 2^{n-2} - 1$ if $s_1 = 0, s_2 = 0, s_3 = 1$
I have attempted to use $p_n = c2^{n-2} - d$ [where $h_n = A(3)^n$, but to no avail] - i ended up with ...
1
vote
1answer
22 views
no. of real roots of exponential equation in three questions
How Can i calculate no. of real roots of exponential equation in three questions
(1) $2^x = 1+x^2$
(2) $2^x+3^x+4^x = x^2$
(3) $3^x+4^x+5^x = 1+x^2$
My Try::
(1) Let $f(x) = 1+x^2-2^x$
now ...
6
votes
0answers
54 views
How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$
How can I calculate $\displaystyle \int \frac{\sec x\tan x}{3x+5}\,\mathrm dx$
My Try:: $\displaystyle \int \frac{1}{3x+5}\left(\sec x\tan x \right)\,\mathrm dx$
Now Using Integration by Parts::
We ...
0
votes
0answers
31 views
Number Plate Problem
I'm having trouble with a question that seems to perplex:
A number plate contains three letters followed by three numbers. A number plate is selected at random.
Calculate the probability that the ...
1
vote
1answer
18 views
Marble Possibility P(At least one yellow)
There are $2$ black and $3$ yellow marbles in a bag. $2$ marbles are drawn randomly without replacement. What is the possibility that at least $1$ yellow marble is selected.
2
votes
3answers
37 views
Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$
I was asked the following (homework) question:
For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\,
z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$
whose sum ...
1
vote
1answer
19 views
solving for one variable in terms of others
A question from Steward's Precalculus textbook 5th, Pg 55,
the original formula is $$h=\frac{1}{2}gt^2+V_0t$$
the question asks to write the formula in terms of $t$, the answer is ...
-2
votes
2answers
46 views
Find area of triangle ABC
BD Perpendicular AC , AB =BC=a
Find the area of triangle ABC
I have tried Googling , I used formula 1/2 (base X Height) . Used Pythagorean theorem. Anyone can suggest me solution.
-3
votes
1answer
49 views
Probability of purple party voters in a samaple prediction of two cities A and B
City A has 1,000,000 people; City B has 4,000,000 people. Suppose the goal is to try to predict the percent of Purple Party voters in a sample. Other things being equal, a simple random sample of 1% ...
1
vote
0answers
27 views
Proving a strict inequality (Application of Hölder's Inequality)
Rudin 6.15 asks one to show that, for $f$ a real, continuously differentiable function on $[a,b]$, $f(a)=f(b)=0$, and $\int_a^b f^2(x)dx=1$, $\int_a^b xf(x)f'(x)dx=-\frac{1}{2}$. This is a simple ...
1
vote
1answer
52 views
Is the following differentiating under the integral sign correct?
Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial ...
4
votes
4answers
74 views
Prove $\ln{(\frac {x}{y})} = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.
Prove $\ln (\frac{x}{y}) = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.
I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am ...
1
vote
2answers
27 views
Matrix involving values of polynomials
I've been doing this problem but im stuck.
Be $f_1 f_2 f_3 \in \mathbb{R}_2$[$x $]. Proove that {$f_1$,$ f_2$,$ f_3$} form a base of $\mathbb{R}_2$[$x $] as $\mathbb{R}$ vector space, if and only ...
2
votes
2answers
24 views
Find the average temperature between $t=0$ and $t=24$ when $T(t) = 49+8t-(1/2)t^2$ degrees.
What was the average temperature during that period?
My initial thought was to take the derivative of the problem, plug in 24 for $t$ and solve. I was wrong. This is what I have
$T'=8-t=8-24=-16$ ...
1
vote
2answers
48 views
Help solving recurrence relation, $a_n = 3a_{n-1} + 4a_{n-2} - 12a_{n-3}$
This is in my homework, and I am not sure how to go about this, I've read the book but I can't seem to grasp what to do. Help?
$$a_n = 3a_{n-1} + 4a_{n-2} - 12a_{n-3}$$
where $a_0 = 2$, $a_1 = -1$, ...
3
votes
3answers
68 views
Does $n n^{1/n} =O(n)$?
I was asked does $n n^{1/n} =O(n)$ ?
I can see that the left hand side is always bigger than $n$ but how would you prove the equality is false?
2
votes
2answers
36 views
Topological extension property
Let $X$ and $Y$ be topological spaces. We say that the extension property holds if, whenver $S$ is a closed subset of $X$ and $f:S\rightarrow Y$ is continuous, $f$ can be extended to a continuous ...
1
vote
1answer
28 views
We are to evaluate the problem at the given limit using pi and redicals in our answer as needed.
The Problem:
$$
\int\!\sin^5(4x)\,dx
$$
The formula that I used from the integration tables is:
$$
\int\!\sin^n(u)\,du
$$
My final answer is
$$
...
-6
votes
0answers
40 views
Direct products and Semidirect products [closed]
If $G$ is a group of order $35$
a) Explain why $G$ is a direct product $\Bbb Z_7 \times\Bbb Z_5$, or is one of the possible semidirect products $\Bbb Z_7 \times \Bbb Z_5$.
b) Determine all possible ...
-4
votes
0answers
39 views
Group automorphisms and Sylow subgroups [closed]
Consider the automorphism group G = Aut(Z25,+) isomorphic to (U25,*)
a) What is the order of G? Is G cyclic? What are the orders of its Sylow subgroups?
b) What are the isomorphism types of the ...
-5
votes
0answers
36 views
Group direct products [closed]
The direct product $\Bbb Z_{45}\times\Bbb Z_{98}$ is cyclic and isomorphic to $\Bbb Z_{4410}$ because $\gcd(45,98) = 1$; furthermore the element $1 = \left([1]_{45},[1]_{98}\right)$ is a cyclic ...
-4
votes
0answers
45 views
Dihedral groups [closed]
Determine the center $Z(G)$ for the dihedral group $G = D_n$ for $n$ greater or equal to $3$. The answer will depend on whether $n$ is even or odd.
Please be precise, thanks in advance
-2
votes
0answers
45 views
Abstract Algebra automorphisms and isomorphisms [closed]
Consider the automorphism group $\mathrm{Aut}(Z_{16},+)$ isomorphic to $(U_{16},\times)$.
a) By examining the cyclic subgroups in $U_{16}$ show that $\mathrm{Aut}(Z_{16},+)$ is isomorphic to ...
5
votes
3answers
60 views
Solve the recursion, $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$
Bring the following recursion relation to an explicit expression:
$$a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$$
$a_{0} = 0$, $a_1 = 1$, $a_2 = 2$
All the examples I have seen were with maximum 2 steps back ...
4
votes
1answer
27 views
Characterization for compact sets in $\mathbb{R} $ with the topology generated by rays of the form $\left(-\infty,a\right) $
I'm trying to find a sufficient and necessary condition for a subset to be compact in $\mathbb{R} $ when the topology is generated by the basis $\left\{ \left(-\infty,a\right)\,|\, ...
1
vote
1answer
42 views
functions and infinite intersections
let $f$ be a function.
If it is given that $$ f\left[\bigcap_{a\in A}F_{a}\right]\subseteq\bigcap_{a\in A}f[F_{a}]$$
then, if it is further given that $f$ is one to one, prove that the $\subseteq$ ...
2
votes
0answers
57 views
“Let A be a set. We are able to quotient all possible well-orders over the set A.” What does this mean?
"Let A be a set. We are able to quotient all possible well-orders over the set A."
This was the first line in the set-up of some exercises I have to do (which ask specific questions depending on ...
0
votes
0answers
44 views
Use mathematical induction to prove the equality, Thank you
prove that for arbitrary sets $\{A_1 , A_2 , \dots , A_n , n\geq 2;\}$ we have :$$A_1 \cup A_2 \cup \dots \cup A_n=(A_1-A_2) \cup (A_2-A_3) \cup \dots \cup (A_{n-1}-A_n) \cup (A_n-A_1) \cup (A_1 \cap ...
1
vote
3answers
35 views
Eigenvector Proof $(I+A)^{-1}$.
Show that the eigenvectors of the $n \times n$ matrix A are also eigenvectors of the matrix $$M = (I+A)^{-1} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues.
My Work:
...
-2
votes
3answers
52 views
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set.
(Can we use ...
6
votes
7answers
127 views
Complete the square for $f(x) = 2x^2 + 4x - 6$
I'm studying for a math test. This is the question:
$f(x) = 2x^2 + 4x - 6$. complete the square.
This is how much I get out of the question:
$$2x^2 + 4x - 6$$
$$2(x^2 + 2x - 3)$$
$$2(x^2 + 2x + ...
3
votes
0answers
49 views
Eigenbasis of a Hilbert space: isomorphism
Let $K$ be a matrix containing the dot product between points in a Hilbert space $\mathcal{H}$ (assume that it is finite-dimensional). Then, we could form a basis using the eigenvectors of a normal ...
0
votes
1answer
30 views
Prove $\int_2^\infty{\frac{\ln(t)}{t^{3/2}}},\mathrm{d}t$ converges
Show, using a comparison test, that $\displaystyle \int_2^\infty{\frac{\log{t}}{t^{\frac32}}}\mathrm{d}t$ converges.
All the answers I've tried shows it diverges, taking $\log{t} \le t^{1/2}$ and ...
0
votes
1answer
40 views
distance travelled after nth bounce
A ball is thrown vertically to a height of $625$ meters from ground. Each time it hits the ground it bounces $\frac{2}{5}$ of the height it fell in the previous stage. How much will the ball travel ...
0
votes
1answer
33 views
Current carrying wires
A current carrying wire takes the form of a plane circular loop of radius a. If the current in the loop is I, find the magnetic firld strength B at a point on the axis of the circle, distance b from ...
2
votes
1answer
79 views
Gravitational fields
Could anyone help me with the following question?
Consider a right circular cone of constant mass density sigma, height h, and semi-vertical angle alpha. By dissecting the cone into discs, show that ...
3
votes
0answers
28 views
How to prove this Stirling related equation
Here is what I need to prove, but have no idea were to start. I know there is some connection with the Stirling theorem.
$$
\sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right )^{d}
$$
I ...
0
votes
1answer
77 views
If sent the same message m to Alice and Bob, how someone who follow the channel can find m ?
Alice has public key (n,ea) and Bob has public key (n,eb) with gcd(ea,eb)=1. If sent the same message m to Alice and Bob, how someone who follow the channel can find m ?
1
vote
0answers
29 views
Describing domain of integration of triple integral
I'm struggling to visualize the following problem:
This question concerns the integral $\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ ...
0
votes
1answer
38 views
Simple Math Equation find sum of 4 numbers and if greater then number X reduce all 4 numbers respectively
Im not the greatest at Math but i have the following problem:
impressions = 791.
watched 100 = 500
watched 75 = 383
watched 50 = 600
watched 25 = 700
The sum of all watched fields is 2183.
...
0
votes
0answers
22 views
The sum of variable whith inverse Gauassian distribution
Let $ X_1,X_2,...,X_n$be a sample from inverse Gaussian pdf whith parameter $\mu$ and $\lambda$ .I want to show that
$\overline{X}$ has an inverse Gaussian distribution with parameter $\mu$ and ...
0
votes
1answer
71 views
probability density functions
Suppose $Y$ is a random variable pdf $f(y)=ky , y=3/n,6/n,9/n...,3n/n$
Find the value of the constant $k$ and write down $Y$'s cdf.
Find simple general expressions for $EY, \text{Var} \,Y, P(Y=3/2)$ ...
