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 ...
0
votes
1answer
12 views
Eigenvalues and eigenvectors of AB and BA, proof.
A is an n x k matrix and B is an k x n matrix.
If $v_1, ..., v_l$ are linearly independent eigenvectors of $BA$ corresponding to a single nonzero eigenvalue $k$, then $Av_1, ..., Av_l$ are linearly ...
0
votes
0answers
7 views
Tell me the ideal selling price to get back a specified number
If I am buying something at xxx,
what is the price to sell it if I want a profit of $2.50 after minus-ing 0.63% (broker fee) from the selling price?
I need to make this into an excel formula, but ...
0
votes
0answers
10 views
Linear Transformation invertible or not?
$ T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$
It is defined such that the inner product of $\langle T(v), v \rangle = 0$ for all $v$ in $\mathbb{R}^2$.
Attempt:
If $T$ has eigenvalues (i.e., it's ...
0
votes
0answers
14 views
Prove the Recursion Theorem
Let $g$ be a function on a subset $A\times\Bbb N$ into $A$, and $a \in A$
Prove there is a unique sequence, $f$ of elements of $A$ such that
a) $f_{0} = a$
b) $f_{n+1} = g(f_{n}, n)$ for all $n ...
0
votes
0answers
8 views
Please show me that with which formula, I can calculate pooled variance for unequal population variance?
When equal population variances, I can calculate pooled variance (as like part-b)
But when unequal population variances, how to calculate pooled variance ( as like part-e)(also I underlined it ...
0
votes
1answer
22 views
The polynomial subspace
Let $A$ be a set of 6 polynomials in $\mathbb{R}_5[x]$ over $\mathbb{R}$ field,
assume $sp(A) = \mathbb{R}_5[x]$ which of the following is true?
1. It might be that $A$ holds exactly 4 polynomials ...
0
votes
1answer
19 views
Prove that the degrees lie in a range
Let $G=(V,E)$ be a graph with $|V|=n$ and $|E|=m$ prove that
$$
\min_{u\in V} \{d(u)\}\leq 2\frac{m}{n}\leq \max_{v\in V} \{ d(v)\}
$$
now my first intuition is to assume that $\min\limits_{u\in V} ...
2
votes
1answer
23 views
Kinematics stone thrown upwards past a point, show the following.
I know I should be able to do this, but I have tried for 3 hours and can't do it. I know its simple but it's driving me mad.
A particle is projected vertically upwards with speed $ u_{0}$ and passes ...
1
vote
2answers
39 views
How to solve the irrational system of equations?
Solve the system of equations $$\begin{cases} \sqrt{x+2y+3}+\sqrt{9 x+10y+11}=10,&\\[10pt] \sqrt{12 x+13y+14}+\sqrt{28 x+29y+30}=20. \end{cases} $$
0
votes
1answer
15 views
Recovering random variable from its moments
The problem is: can you recover a distribution of random variable if you know all its moments?
My first guess was to use moment-generating function (MGF). It is known that if two random variables ...
2
votes
1answer
45 views
Finding global max./min.
my task is to figure out the critical points of $f(x,y)=e^y(x^4-x^2+y)$, $\ $$\mathbb{R}^2 \rightarrow \mathbb{R}$, and show which of them is a maximum or minimum. As far as I got, I've shown that the ...
0
votes
1answer
23 views
longest path two nodes in common
Let $G$ be a connected graph. Prove that two longest paths in $G$ have atleast one node in common. Note that two longest paths do not neccessarily have the same length.
I began by defining two paths ...
2
votes
1answer
27 views
Does a closed walk necessarily contain a cycle?
[HOMEWORK]
I asked my professor and he said that a counter example would be two nodes, by which the pathw ould go from one node and back. this would be a closed path but does not contain a cycle. But ...
3
votes
1answer
36 views
Several graph theory proofs
Let $G=(V,E)$ be a graph with $|V|=n$. Prove that $G$ is connected if $d(v)\geq \frac{n-1}{2}$ for all $v\in V$.
Let $G=(V,E)$ be a graph with $|V|=n$ and $|E|=m$. Prove that $\min\limits_{u\in V} ...
2
votes
1answer
35 views
Finding eccentricity of an ellipse from latus rectum
The latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is the same as latus rectum of a parabola $y^2=4cx$ . Find eccentricity of the ellipse .
1
vote
2answers
26 views
Given a spanning set, what is the span of the 'transpose' of the set?
Given $$sp\left \{
\begin{pmatrix}
a_1\\
a_2\\
a_3
\end{pmatrix}
,\begin{pmatrix}
b_1\\
b_2\\
b_3
\end{pmatrix}
,\begin{pmatrix}
c_1\\
c_2\\
c_3
\end{pmatrix}
\right \} = \mathbb{R}^3$$
What ...
4
votes
1answer
36 views
trigonometric identity related to $ \sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^{2})}{\sqrt{n}} $
As homework I was given the following series to check for convergence:
$ \displaystyle \sum_{n=1}^{\infty}\dfrac{\sin(n)\sin(n^{2})}{\sqrt{n}} $
and the tip was "use the appropriate identity".
...
1
vote
1answer
29 views
Applying Möbius' theorem
Let $\displaystyle T(Z)=\frac{z+1}{z+i}$. Find a straight line in $\mathbb{C}$ that is mapped by $T$ to a circle.
True or False: Every holomorphic function $f:\mathbb{C}\to\mathbb{C}$ maps any ...
3
votes
2answers
34 views
Given a triangle with points in $\mathbb{R}^3$, find the coordinates of a point perpendicular to a side
Consider the triangle ABC in $\mathbb{R}^3$ formed by the point $A(3,2,1)$, $B(4,4,2)$, $C(6,1,0)$.
Find the coordinates of the point $D$ on $BC$ such that $AD$ is perpendicular to $BC$.
I believe ...
0
votes
0answers
15 views
A version of coupon collector problem
Suppose each new coupon is, independent of the past, a type $i$ coupon with probability $p_i$. A total of $n$ coupons is to be collected. Let $A_i$ be event that there is at least one type $i$ in ...
2
votes
3answers
50 views
Algebra Equation with equals on left hand side
$6 = 4 - 2x =$
Show the answer with the mechanics of working out.
5
votes
1answer
67 views
Calculate the limit of two interrelated sequences?
I'm given two sequences:
$$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$
as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle ...
3
votes
2answers
71 views
How to evaluate $\lim_{x\to 0} (1+2x)^{1/x}$
Good night guys! I'm having some trouble with this:
$$\lim_{x\to 0} (1+2x)^{1/x}$$
I know that $\lim_{x\to\infty} (1 + 1/x)^x = e$
but I don't know if i should take $h=1/(2x)$ or $h=1/x$
Can ...
2
votes
1answer
32 views
Same linear transformation, different basis.
Let $\beta=\{(1,0,0),(0,1,0),(0,0,1\}$be a basis of $\mathbb{R^3}$ and $g: \mathbb{R^3} \to \mathbb {R^3}$ a linear transformation, which matrix is:
$$G=\begin{bmatrix}1 & 0 &-1 \\ 6 ...
3
votes
2answers
29 views
Finding the sum of a Taylor expansion
I want to find the following sum:
$$
\sum\limits_{k=0}^\infty (-1)^k \frac{(\ln{4})^k}{k!}
$$
I decided to substitute $x = \ln{4}$:
$$
\sum\limits_{k=0}^\infty (-1)^k \frac{x^k}{k!}
$$
The first ...
2
votes
1answer
60 views
Prove that a given meromorphic function is rational
I'm doing some exercises in complex analysis, and I've reached one I simply can't figure out on my own, which is why I'm hoping for some help.
The exercise:
We assume that $h:\Bbb C\to \Bbb C \cup ...
0
votes
2answers
52 views
complex analysis- Liouville's theorem
1).Let $f: \mathbb{C}\to \mathbb{C}$ be a holomorphic function. Prove that if $f(0) = f(1) =0 $ and $f:\lvert f'(z) \rvert\leq 1 $ for all $z\in \mathbb{C}$, then $f(z)=0$ for all $z\in \mathbb{C}$.
...
0
votes
0answers
52 views
to find dimension of $(\ker f)^\perp$
Let $V$ be an $n$-dimensional inner product space.Let $f:V\to R$ be a linear form. Find the dimension of $(\ker f)^\perp$
1
vote
2answers
49 views
How to solve this system of equation.
$x^2-yz=a^2$
$y^2-zx=b^2$
$z^2-xy=c^2$
How to solve this equation for $x,y,z$. Use elementary methods to solve (elimination, substitution etc.).
Given answer is:$x=\pm\dfrac{a^4-b^2c^2}{\sqrt ...
1
vote
1answer
43 views
set of natural numbers
Prove each set of natural numbers is the set of all smaller natural numbers.
IE $ n =$ { $m \in N | m < n$ }
Hint: use induction to prove that all elements of a natural number are natural ...
-2
votes
0answers
45 views
Generating function for integers
What is the generating function for the numbers of partitions of an
integer in which each part is either even or divisible by 3?
5
votes
3answers
88 views
How to solve this simultaneous equation of $3$ variables.
I've stuck in this equation system.No clue how to start ?
$x+y+z=a+b+c\,\cdots(1)$
$ax+by+cz=a^2+b^2+c^2\,\cdots(2)$
$ax^2+by^2+cz^2=a^3+b^3+c^3\,\cdots(3)$
Find the value of x,y,z is in the form ...
1
vote
2answers
28 views
Factorization in Gaussian integers
Let $p$ be a natural number, suppose $p$ prime. Show that the following conditions are equivalent:
1) the polynomial $x^2+1\in\mathbb{Z}_p$ has roots in $\mathbb{Z}_p$
2) $p$ is reducible in the ...
0
votes
3answers
49 views
Similar triangles question
If I have a right triangle with sides $a$.$b$, and $c$ with $a$ being the hypotenuse and another right triangle with sides $d$, $e$, and $f$ with $d$ being the hypotenuse and $d$ has a length $x$ ...
0
votes
1answer
40 views
Cardinality of the set of surjective functions on $\mathbb{N}$? [duplicate]
I know that the set of all surjective mappings of ℕ onto ℕ (lets name this set as F) should have cardinality |ℝ|.
How to strictly prove that?
From the fact that cardinality of every possible ...
1
vote
1answer
35 views
natural numbers and successors
I hope this question is not a repeat.
Given there is no $k \in N $ such that $ n < k < n+1 $ for $n \in N$
Prove for all $ m, n \in N $ if $ m < n $ then $ m+1 \leq n$
Conclude $ m < ...
1
vote
4answers
54 views
The successor of a set
Definition : The successor of a set $x$ is the set $S(x) = x \bigcup \{x\}$
Prove that $x \subseteq S(x)$ and there is no $z$ such that $ x \subset z \subset S(x)$
I really battle with proofs :(
...
9
votes
3answers
176 views
Accumulation points of accumulation points of accumulation points
Let $A'$ denote the set of accumulation points of $A$.
Find a subset $A$ of $\Bbb R^2$ such that $A, A', A'', A'''$ are all distinct.
I can find a set $A$ such that $A$ and $A'$ are distinct, but not ...
0
votes
3answers
63 views
Is the following set empty?
$$
sp\left \{
\begin{pmatrix}
1 \\
-1 \\
1 \\
-1
\end{pmatrix} , \begin{pmatrix}
4\\
-2 \\
4 \\
-2
\end{pmatrix} , \begin{pmatrix}
1\\
1\\
1\\
1
\end{pmatrix}
\right \} \bigcap \left \{ ...
3
votes
1answer
58 views
Prove that $A_1,B_1,C_1$ is collinear
Let $ABC$ be a triangle inscribled inside circle $(O)$ . M is a point inside the triangle $ABC$ ($M \notin BC,CA,AB$)
$AM,BM,CM$ meets $(O)$ again at $A',B',C'$ respectively. Midperpendicular of ...
1
vote
1answer
41 views
Prove that if $n$ is sufficently large, there is a prime gap $G(p_k, p_{k+1})$ with $p_k \leq n$ and $p_{k+1} - p_k > \frac{1}{2}\ln(n)$
Let $p_k ( k \geq 1)$ be an enumeration of all the positive primes with $p_1 = 2$ and $p_k < p_{k+1}$ for all $k \geq 1$. Prove that if $n$ is sufficiently large, then there is a prime gap ...
0
votes
0answers
38 views
Bayesian computational problem with R
I can't understand the iter (this is me) to solve this kind of exercise involving the approximation of the posterior density.
Consider this exercise taken from Bayesian computation with R:
We ...
0
votes
1answer
47 views
complex analysis-roots of constant polynomial function
True or False: There exists a value for $w\in \mathbb{C}$ such that the equation $z^{100}+z +1 = w$ has no solution with $z\in\mathbb{C}$.
My answer
$False$-By fundamental theorem of algebra,Every ...
1
vote
3answers
47 views
Which one is the correct series expansion?
Is
$$p^{n+1} = p^0+p^1+ \dots + p^n$$
or
$$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$
I am confused.
please explain the correct one.
1
vote
1answer
31 views
determine whether an integral is positive
Given a standardized normal variable $X\sim N\left(0,1\right)$, and constants $ \kappa \in \left[0,1\right)$ and $\tau \in \mathbb{R}$, I want to sign the following expression:
\begin{equation}
...
2
votes
1answer
27 views
Lagrange's Theorem for further elementary consequences
Question: Let $G$ be a finite group, and let $H$ and $K$ be subgroup of $G$. Prove: suppose $H$ and $G$ are not equal, and both have order the same prime number $p$, Then $H\cap K=\{e\}$.
This is my ...
0
votes
1answer
42 views
Binary operations of addition and multiplication.
Prove that the set $\mathbb{R}$ of all real numbers is an abelian group under the binary operation of addition (+) and a semi-group under multiplication ($\cdot$).
Prove that $\mathbb{R}^*$, the ...
0
votes
1answer
37 views
How do I solve this solution-mixing problem?
A chemist has a 55% acid solution and a 40% acid solution. How many liters of each should be
mixed in order to produce 100 liters of a 46% acid solution?
0
votes
1answer
16 views
Basic question on the transformation of Exponential distribution.
Why central moments coincide for random variables
$V\sim E(a,h)$ and $Y\sim E(h)$
where a=location parameter
h= scale parameter.
-4
votes
0answers
43 views
How find this must conclude who arrived first
Alpine skiing is the Winter Olympic Games for a class of double plate skiing competitions, including rotary, rotary, large rotation and downhill four game. The game is common several players with two ...
