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 ...
1
vote
1answer
13 views
Neumann problem, stuck on a boundary condition.
I am stuck on a problem that I am trying for exam practice and I would very much appreciate a hint to help me out, here is the section where I am stuck:
A solution is sought to the Neumann problem ...
-1
votes
0answers
15 views
Off topic Question [migrated]
I am fed up of using MS paint for creating diagrams and posting images on our board , do you have any application which will me to create diagrams easily . which must be portable or is there any ...
1
vote
1answer
38 views
Using Semi-circle find side of triangle
The figure below above shown a bicycle path. If semicircular portion $ABC$ is $100$ $\pi$ and $CD$ is $100$$ft$ then what is $AD$?
I have tried to find the diamenter of the circle and the ...
1
vote
2answers
75 views
inverse of laplace transform
How to compute this inverse Laplace transform ?
$$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$
Thanks.
2
votes
2answers
30 views
Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad ...
1
vote
0answers
28 views
Using a matrix to organise values into groups
Let's say I have a matrix of size 6 x 6.
Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
3
votes
3answers
35 views
Finding side of rectangle using given information
Really simple question but I am stuck. The following information is given:
$$BD=8,\quad AB = 6,\quad ED =5,\quad EF = EC$$
and we want to find $AF$.
If we have three $90^\circ$, what does that ...
2
votes
2answers
70 views
Show that $[l_1 \cdot l_2 \cdot l_3 ] = [l_1 + l_2 + l_3] \in H_1(X)$ The first Homology group of X
Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) =
l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$).
Show that ...
1
vote
0answers
52 views
Relationship between curls, gradients, and divergences; and the Isomorphism Theorem
I am trying to develop a geometric intuition for the relationship between the curl, the gradient, and the divergence based on the Isomorphism Theorem, where the Isomorphism Theorem says that "If ...
0
votes
0answers
23 views
find area of Triangle ABF
In the figure given below , rectangle CDEF with perimeter 32 has the maximum area. find area of Triangle ABF
So , i had following try.
P = 2.W+2.H where ...
3
votes
5answers
102 views
Polynomials - The sum of two roots
If the sum of two roots of
$$x^4 + 2x^3 - 8x^2 - 18x - 9 = 0$$ is
$0$, find
the roots of the equation
3
votes
2answers
30 views
Matrix multiplication related to complex numbers?
Evaluate and simplify the product
$\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
13
votes
3answers
357 views
Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$
Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid?
I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
4
votes
1answer
68 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
0
votes
0answers
21 views
What does this mean: Symmetry of the KDV generated by a vector field
What is
a symmetry of the KDV
$$\frac{\partial u}{\partial t}=6u\frac{\partial u}{\partial x}-\frac{\partial^3 u}{\partial x^3}$$ generated by $$V=A(t,x,u)\frac{\partial }{\partial ...
-1
votes
0answers
50 views
Product topology question
Consider $\mathbb R$ with the usual topology and let
$S^1=\{(x,y)\in \mathbb R^2: x^2+y^2 = 1\} $.
a) $( \mathbb R^2, \Gamma_{p})$, where $\Gamma_p$ is the product topology, coincides with ...
4
votes
1answer
31 views
How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle $?
Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle $. Prove that a valid inner product on $V$ is defined with the formula $[x, y] = ...
6
votes
1answer
107 views
Decompose $P$ into the direct sum of irreducible representations.
Note: I need help with part (c).
Consider the representation $P: S_3 \rightarrow GL_3$ where $P_{\sigma}$ is the permutation matrix associated to $\sigma$.
a) Determine the character $\chi_P : S_3 ...
3
votes
2answers
62 views
Exercise about prime ideals
Let $A$ be a ring. Prove that the following conditions are equivalent:
$i)$ All ideals $I \subsetneq A$ are prime.
$ii)$ The set of all ideals of $A$ is totally ordered by
inclusion and all ideals ...
0
votes
2answers
37 views
$\sum \frac{ln(n)}{\sqrt{n^5}}$ test for convergence
Let $\sum a_{n}=\sum \frac{ln(n)}{\sqrt{n^5}}$. To find if the serie is convergence or not, I had some difficult on finding the proper serie to test the given one.
After some work around, I found ...
6
votes
1answer
74 views
If $0<a<1, 0<b<1$, $a+b=1$, then prove that $a^{2b}+ b^{2a} \le 1$
I have been really struggling with this problem ... please help!
Let a,b be real numbers. If $0<a<1, 0<b<1, a+b=1$, then prove that $a^{2b} + b^{2a} \le 1$
What I have thought so far:
...
4
votes
1answer
69 views
The Baer-McCoy (a.k.a. prime) radical of $A$
Let $B$ a ring and let $A$ a subring of $B$.
Show that $A\cap \mathrm{Nil}_{*}(B)\subset \mathrm{Nil}_{*}(A)$.
If $A$ is contained in the center of $B$, show that $A\cap ...
0
votes
2answers
32 views
Trigonometrical Question
the question is solve the following equation in the interval
$$0<\theta\leq 360$$
$$\tan(\theta) = \tan(\theta)(2+3\sin(\theta))$$
I got 199.5 and 340.5 as my answers like so:
$\tan(\theta) = ...
2
votes
0answers
45 views
Conformal mapping from exterior of closed unit disk onto exterior of horizontal interval.
This is a problem from Bak-Newman's "Complex Analysis", #4 from Chapter 14 "The Riemann Mapping Theorem".
The question is this:
Verify directly that $F(z) = z + \frac{1}{z}$ is the unique conformal ...
0
votes
2answers
43 views
generalizing De Morgan's Laws [duplicate]
Show: $$ B - \bigcup_{a \in A}F_{a} = \bigcap_{a \in A} (B - F_{a}) $$
and show $$ B - \bigcap_{a \in A}F_{a} = \bigcup_{a \in A} (B - F_{a}) $$
I struggle with proofs. This is what I have for the ...
0
votes
5answers
58 views
Prove that if $G$ is abelian, then $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$ [duplicate]
Let $G$ be an abelian group. Prove that $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$, where
$e$ is the neutral element of $G$.
I need some help to approach this question.
2
votes
0answers
38 views
Structure Theorem for finitely generated modules over PIDs [duplicate]
Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either
$$\begin{pmatrix}
0&-1&0&0\\
...
0
votes
0answers
17 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
1
vote
3answers
33 views
quadratic equation precalculus
from Stewart, Precalculus, 5th, p56, Q. 79
Find all real solutions of the equation
$$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{x^2-4}$$
my solution
...
4
votes
3answers
145 views
Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest ...
0
votes
1answer
160 views
I want help with $4\times 4$ symmetric matrix
I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
1
vote
0answers
51 views
count the number of connected induced subgraphs in a graph with bounded degree
Let $G=(V,E)$ be a graph where the maximum degree of a vertex is 4. I've been asked to find an efficient algorithm for counting how many connected induced subgraphs are in $G$.
What I have so far is a ...
1
vote
1answer
47 views
distribution function of time T
an ambulance station is located 30 miles from one end of a 100-mile road. the station services accidents along the entire road. suppose that an accident occurs. suppose that Suppose accidents occur ...
2
votes
1answer
34 views
Question about limits with variable on exponent
So I have to find the following limit $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{1/n}.$$I said that this is ...
1
vote
3answers
36 views
Solving Equation Using Algebraic Method
How to solve these equations using an algebraic method?
I need to show my working, don't you do something in reverse, like 7 multiplies by something. I haven't done it in class.
...
0
votes
1answer
18 views
please prove the following proof related to F distribution.
Suppose $S_1^2$ and $S_2^2$ are two independent unbiased estimate of the common population variance $\sigma^2$ from two random sample of sizes $n_1$ and $n_2$ respectively.
Then show that
...
1
vote
2answers
49 views
Spherical Trigonometry: Spherical triangle
ABC is an equilateral spherical triangle in which small displacements are made, in the sides and angles, of such a nature that the triangle remains equilateral. Prove that
$$
\frac{da}{dA} = ...
4
votes
4answers
121 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
55 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.
