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
0answers
5 views
Proof that the cartesian plane is an incidence geometry with only vector definition of a line.
I can get up to showing that it is an abstract geometry, but I cannot figure out how to show that for every two points, there is a unique line. The definition of a line in vector form is given
L (AB) ...
2
votes
0answers
27 views
Trigonometry Airplane question. Finding bearing and distance.
A little background(if you don't care for my story, skip straight to the question): I've missed a few lectures from my teacher because I fell ill. Since I have no information to work with other than ...
2
votes
3answers
42 views
How to factor a four term polynomial without grouping?
$$2x^3 + 9x^2 +7x -6$$
This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?
4
votes
1answer
147 views
Probability - Balls and Buckets; variance question
I've been working on this problem for a while and its giving me no end of trouble! The question is this: Suppose we have 2k buckets, numbered 1 through 2k. We throw x black balls and y white balls, at ...
1
vote
1answer
13 views
Inverse fourier transform of a function which is a fundamental solution
Let $f:\mathbb{R}^3 \to \mathbb{R}$ be $f(x)=(1+|x|^2)^{-1}$.
I need to calculate $\mathcal{F}^{-1}(f)$.
I've proven that $f\in L^2(\mathbb{R}^3)$ and I know that the fourier transform is an ...
2
votes
0answers
19 views
Minimum degree of a planar minimal 5-chromatic graph.
Let $G$ be a planar minimal 5-chromatic graph. That is, any of its proper subgraphs has chromatic number at most 4. I need to prove that its minimum degree is at least 5. I want to prove by ...
0
votes
2answers
73 views
How to prove when Möbius transformation determinant ad - bc < 0, the upper half plane does not map to itself? [duplicate]
I proved the part where $$\operatorname{Im} f(z) = \frac {(ad - bc) \operatorname{Im}z} {|cz + d|^2} $$
But now I need to show that when $ ad - bc < 0 $, then the upper half plane does not map ...
0
votes
1answer
120 views
Lebesgue integrability of continuous function in closed interval
I'm trying to show that a continuous function $f$ in $[a,b]$ is Lebesgue integrable, using approximation through step functions.
It is pretty trivial to show using the connection between Riemann ...
0
votes
1answer
35 views
Linear Differentiation
I have to determine whether there is normal linear differentiation equation $a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ on $\mathbb{R}$ such that $u_1, u_2 \in C^2(\mathbb{R})$ defined by $u_1(x) = x, u_2(x) ...
1
vote
1answer
20 views
What's wrong with this Kuhn-Tucker optimization?
The function $u(x,y,z) = xyz$ is to be maximized, under constraints: $ 0 \le x \le 1, y \ge 2, z \ge 0 $ and $ 4 - x - y - z \ge 0 $
Now I'm not quite sure how to translate the x-constraint into ...
2
votes
2answers
36 views
Calculate the length of curve $f(x)=\arcsin(e^x)$, check solution, please.
As in the topic, my task is to calculate the length of $f(x)=\arcsin(e^x)$ between $-1, 0$. My solution: I use the the fact, that the length of $f(x)$ is equal to $\int_{a}^b\sqrt{1+(f'(x))^2}dx$ ...
2
votes
2answers
310 views
Sum of independent Gamma distributions is a Gamma distribution
If $X\sim \mathrm{Gamma}(a_1,b)$ and $Y \sim \mathrm{Gamma}(a_2,b)$, I need to prove $X+Y\sim(a_1+a_2,b)$ if $X$ and $Y$ are independent.
I am trying to apply formula for independence integral and ...
0
votes
0answers
36 views
Question on derivative
I want to differentiate $H(p(t),q(t))=1 $ with respect to $t$, where $H:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} $ is a convex function.
I think that it is:
$\displaystyle \frac{dp}{dt} ...
3
votes
1answer
53 views
Linearization of an implicitly defined function.
Problem:
Given the equation: $xz^{2}+y^{2}z^{5}=19$
Also given: (3,4,1) is a solution to the equation. This point is not the only solution.
1) Find dz/dx and dz/dy (through implicit ...
0
votes
2answers
65 views
Topological manifold example
$\theta(x,x^2)=x$
$\Bbb X =${$(x,x^2)| x$ in $\Bbb R$}
And V is subset of $\Bbb R$
$dim\Bbb X=1$
My instructor said that this is topological manifold.
Why?
Please can you explain me? This ...
1
vote
3answers
52 views
finding nth term
Let
3,8,17,32,57 . . . . .
be a pattern.How do we find the nth number?My brains are completely jammed,I am tired.I do not even recognize the pattern.I calculated a few ways,but all I want is a ...
0
votes
0answers
29 views
Ordinary differential equations with double resonance
I want to know what is the definition of "resonance, double resonance" in
ordinary differential equations with double resonance
Please,
Thank you.
2
votes
0answers
36 views
Linear Difference Equations
Let $g_1, g_2 \in \mathbb{R}^{[0,5]}$ be defined by $g_1(x) = x$ and $g_2(x) = 1-x$ for each $x \in [0,5]$.
Find a second-order homogeneous linear difference equation on $[0,3]$ such that {$g_1, g_2 ...
3
votes
1answer
32 views
Using a definite integral, to create a specific recurrence relation.
Hello i have the integral:
$$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$
I need to show that the integral can be represented by the recurrence relation below;
$$y_n= ...
0
votes
1answer
36 views
Homework involving unitary diagonalization
I was given as an assignment to diagonalize the following matrix:
$\left(\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{array}\right)$
I started by finding ...
3
votes
6answers
53 views
Determine the inverse function of $f(x)=3^{x-1}-2$
Determine the inverse function of $$f(x)=3^{x-1}-2.$$ I'm confused when you solve for the inverse you solve for $x$ instead of $y$
so would it be $x=3^{y-1}-2$?
1
vote
1answer
29 views
Odd and even functions- a direct sum?
Question:
Let V be the vector space of all functions $\Bbb R\to \Bbb R$.
Show that $V=U \oplus W$
for $U=${$f | f(x)=f(-x) \forall x$}$, $W={$f | f(x)=-f(-x) \forall x$}
What I did:
I did prove ...
1
vote
0answers
40 views
$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis.
I don't ...
0
votes
0answers
18 views
Distance between two affine lines using determinant of Gramian matrix.
I've a task to find the distance in $E^4$ between:
$L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$
and
$M = [2,3,1,5] + \text{lin}((2,1,0,2))$
My efforts to find the correct solution:
Let
...
1
vote
2answers
286 views
Sum of two stopping times is a stopping time?
Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions.
Question: Is $\sigma + \tau$ a stopping time?
Attempt at a solution:
I ...
1
vote
2answers
30 views
Question on Contractions
Let $S = \{x \in \mathbb{R}^n ; \|x\| \le 1 \}$ and $f: S \to S$ be a contraction. Determine one can have $f(S) = S$.
I really need some help with this question. In advance I wanted to give all ...
2
votes
1answer
35 views
Proof that the interior of any union of closed sets with empty interior in a compact Hausdorff space is empty
The question is pretty much in the title, I need to show that given $X$ is a compact Hausdorff space and $\left\{ A_n\right\}_{n=1}^\infty$ is a collection of closed subsets of $X$ each with empty ...
3
votes
0answers
66 views
Compute limit of the sequence $x_n$
Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
0
votes
4answers
37 views
rewriting equation in terms of $y$
From Stewart, Precalculus, 5th ed, P98, Q.45
$$x^2+xy+y^2=4$$
how can I re-write this equation in terms of $y$? I want to put this equation into graphing software but don't know to put $y$ on one ...
11
votes
5answers
260 views
Show determinant of matrix is non-zero
I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help ...
1
vote
0answers
21 views
Example on Correspondences
Give an example of correspondences F: X $\rightarrow$ Y, G: Y $\rightarrow \mathbb{R}^s$ such that F and G are closed, but (G o F) is not, if any, where $ \varnothing \neq X \subset\mathbb{R}^m, ...
3
votes
2answers
43 views
On the Hurwitz Zeta Function
In my mathematics course in Uni. (I'm a physics student) my prof. gave us the following exercise: to express the Hurwitz Zeta function $\zeta(2k+1,\frac{1}{4})$ with $k=1,2,3,\dots$ in terms of the ...
2
votes
1answer
30 views
Find the smallest possible integer that satisfies both modular equations
Find the smallest positive integer that satisfies both. x ≡ 4 (mod 9) and x ≡ 7 (mod 8) Explain how you calculated this answer.
I am taking a math for teachers course in university, so I'm worried ...
-4
votes
1answer
32 views
Cost and Marginal cost [closed]
Nissan has determined the following for its Sunny model:
Price function: $p(x)= \frac{1}{5} (45-x)$
Cost function: $C(x)= \frac{1}{4}x^2+3x+67$
Find the following:
Revenue & Profit functions.
...
0
votes
0answers
28 views
Is Multiplication A System?
I don't understand how to identify the properties of a system.
What possible properties could a system have?
Are a certain number of properties required in order to be classified as a system?
The ...
0
votes
2answers
35 views
Shading A Venn Diagram Using A Specific Equation
The expression is: $A\triangle(B\cap C')$. The $\triangle$ refers to $-$ and the $\cap$ refers to an intersection, whilst the $\;{}'$ refers to the prime of $C$.
There are no numbers or items ...
0
votes
1answer
28 views
Solving Modular Equations With Identities
$4+2x≡7 \pmod 8$
Find all possible solutions and note any identities.
Identify how you found the solutions.
Explain what identities are.
1
vote
1answer
146 views
Pre-Calculus Vector Problem.
In this question vector i represents a vector due east and vector j represents a vector 1 km due north.
An aircraft flies (at a constant height) with a speed of $800$ km/h. It flies in a fixed ...
3
votes
1answer
290 views
Fourier Sine Transform
There is a question from my book which I find hard. Here it goes:
Consider
$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}-v_0\frac{\partial u}{\partial x} ...
0
votes
3answers
75 views
plot any irrational number on number line.
I have a basic question that can we plot any irrational number on number line?As I can plot all integers and rational number but how to plot any irrational number on it like $\sqrt2$,$\sqrt3$ etc..
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 ...
2
votes
1answer
26 views
how to dot product two vectors with different planes?
how to dot product two vectors with different planes?
I have vectors $A$,$B$ and $C$, vectors $A$ and $B$ is on $xy$ plane while vector $C$ is on $xz$ plane. I need to find the dot product of $A.C$ ...
2
votes
2answers
83 views
Ratio of areas of similar triangles given SS
What is the ratio $ A_I:A_{II} $ ?
I know that the given angles and common angle prove the triangles are similar. Using proportionality, I found the length of the middle lengthed side of ...
0
votes
0answers
15 views
Finding rate of maximum temperature increase along surface
So I know that the rate of maximum increase of some function (say, $f(x,y)$) is given by the gradient ($\nabla f$), where the direction is the direction of maximum increase of the function, and the ...
1
vote
2answers
45 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} = ...
6
votes
1answer
95 views
+250
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 ...
1
vote
1answer
45 views
Prove that a cylinder have a infinite number of planes of symmetry.
My definition of cylinder is:
A cylinder is the surface formed by parallel lines, where each line contains a point of a curve called guideline. Each lines is called a generatrix of that cylinder.
...
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 ...
0
votes
2answers
181 views
Elementary Row Operations To Find Inverse Matrix
I have to find the inverse matrix of this matrix that represents a relation. My question is, is it possible to use elementary row operations on a one-zero matrix to find the inverse? I've done it ...
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)\,|\, ...
