1
vote
0answers
10 views

Is it possible to factor out monomials in a rational function?

after thinking about the problem for some hours I thought I come here to ask. My problem is that I want to do a coordinate transformation on the following equation $y=\frac{a}{x^2}+\frac{b}{x}+c+dx+...
0
votes
0answers
9 views

Trigonometry from two graphs

My problem is that i have an image or rectangle which has line inside it which is rigid it will not move. I also have a graph with the same line but may be slightly bigger and may have a slightly ...
1
vote
0answers
8 views

Find Probability that latency exceeds 10 ms given sample mean and variance

I am working on a statistics problem for my Engineering Statistics class. The problem goes like this: You are measuring the communications latency between two processors. You take 6 million data ...
0
votes
0answers
5 views

Least Squares Algorithm with Inverse Norm

Given an overdetermined linear system $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m \times 1}$ with $A < 0$ and $b < 0$. What is a good way to numerically determine $$ \min_x \left\lVert ...
1
vote
0answers
15 views

Real and imaginary part of tensors of matrices

Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...
-2
votes
0answers
21 views

what is Expected Mean

Thus the expected mean $\mu$ of the set $\mathcal S$ can be given as \begin{align*} \mathbb E \mu&= \sigma^2+\frac 1r \sum_{i=1}^m\left(\mathbb E\lambda_i-\sigma^2\right)\\ &\geq \sigma^2+\...
1
vote
0answers
16 views

Residue Theorem to solve $\int_{x=-\infty}^\infty \frac{sin(x \, l)}{x \, (x^2+a^2) \, (x^2+b^2)} \, e^{\, \mathrm{j} \,x\,c} \, \mathrm{d}x$

I would like to solve the following integral using residue theorem $\int_{x=-\infty}^\infty \frac{sin(x \, l)}{x \, (x^2+a^2) \, (x^2+b^2)} \, e^{\, \mathrm{j} \,x\,c} \, \mathrm{d}x$ with $a^2 \in\...
1
vote
0answers
7 views

Infinitely many Martingale Equivalent Measures:

Question: Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. Let $Y_t=(Y_1(t),Y_2(t))$ be the solution to the SDE: $dY_1=\beta_1dt+dB1+2dB2+3dB3$ $dY_2=\beta_2dt+dB1+2dB2+2dB3$ Where $\...
0
votes
1answer
11 views

Find locus of points for which $\nabla f$ is parallel to the $y$-axis

If $f(x, y, z)=x^2 + y^2$, what is the locus of points for which $\nabla f$ is parallel to the $y$-axis? I know that $\nabla f=(2x, 2y, 0)$ and I know that the $y$-axis is the set of all points such ...
2
votes
0answers
11 views

Slick proof $\mathcal M^\perp$ is limit-stable (strong factorization systems)

Theorem 1.17 of Emily Riehl's Factorization Systems says that given a class of maps $\mathcal M$, $\mathcal M^\perp$ is closed under limits and dually $^\perp\mathcal M$ is closed under colimits. ...
1
vote
0answers
10 views

Inequality derived from $f:\mathbb{D}\to \mathbb{D}$ moving two points to another two points

Consider $f:\mathbb{D}\to \mathbb{D}$ (where $\mathbb{D}$ is the unit disc), which moves two points $z_1, z_2\in \mathbb{D}$ to $w_1, w_2\in \mathbb{D}$. I want to prove that if such an $f$ exists, ...
4
votes
1answer
17 views

Horizontal Asymptotes

Find all asymptotes of: $$f(x) = \frac{a + be^x}{ae^x+b}$$ The way I've been taught is that the $+a$ and $+b$ in the numerator and denominator respectively do not contribute when x tends to infinity,...
0
votes
0answers
13 views

Integral closure of $R[x]$ in its field of fractions over R

I feel like this might have been discussed before but I couldn't find it so I apologise if this is a very common question. If $S$ is a ring and we have a subring $R$ and an element $x\in S$ integral ...
1
vote
0answers
18 views

Is there any notation to denote the solution of a particular equation?

Let's say that I have an equation: $$x + 2 = 7 + y$$ If I need the solution for $x$, I can solve it and then use $5 + y$. No problem there. But say I have an abstract equation. For now I am ...
0
votes
1answer
20 views

Showing a quotient ring is commutative

Question: Let R be the ring of all continuous function from R to $\mathbb{R}$ under point-wise addition and multiplication. Show that $ I=\left \{ f \in R \mid \left ( 0 \right )f=0\right \}$...
0
votes
1answer
12 views

Calculating the missing two points of rectangle if 2 points and the aspect ratio are known

How can I calculate the missing two points of a rectangle if I know 2 points (top left and top right) and the aspect ratio i.e 16:10. For example: Top left: A(834, 449) and Top right: B(1675, 423)
0
votes
1answer
23 views

if $(f_n)$ is $\mathcal C^1(]a,b[)$ and converge to $f$, then $f$ is Lipschitz.

Let $(f_n)_{n\in\mathbb N}$ a sequence of function $\mathcal C^1(]a,b[)$ such that $(f_n')_{n\in\mathbb N}$ is bounded. We suppose that $f_n$ converge uniformly to $f$. Show that $f$ is Lipschitz. ...
0
votes
0answers
10 views

Intuition on the necessity of the Lipschitz condition and a physical example of an ODE

The Picard-Lindelöf theorem states that the initial value problem $$ y'(x) = F(x,y(x)), \ y(x_0) = y_0$$ will always have a unique solution on some closed interval containing $x_0$ assuming that the ...
0
votes
0answers
28 views

Integrating square root with condition

I am an engineer working on a problem that requires the use of integration to calculate compression force within a segment. I have worked out the formula, I just need help with the integration as I ...
0
votes
2answers
20 views

Roots of polynomial with positive coefficients

My question is very simple. Suppose we have a polynomial defined as follows: $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0 $$ where all of the $a_n$'s are all real and positive. Is there something ...
1
vote
3answers
29 views

Help with simplification of a rational expression (with fractional powers)

Can you please help me see what I don't see yet. Here's a problem from a high school textbook (ISBN 978-5-488-02046-7 p.9, #1.029): $$ \frac{ (a^{1/m}-a^{1/n})^{2} \cdot 4a^{(m+n)/mn} }{ (a^{2/m}-a^{...
0
votes
0answers
5 views

The second differential as a differential on the double tangent bundle

I know what the second differential of $f : \Bbb R^n \to \Bbb R$ means. Nevertheless, when working with abstract manifolds and in the absence of a connection, one cannot come up with a 2-covariant ...
0
votes
1answer
10 views

Rademacher theorem for manifolds

Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which generates the topology of $M$. Let $f:M \to R$ be Lipschitz w.r.t the metric $d$. Is it true that $f$ is differentiable a.e? Note: ...
0
votes
0answers
22 views

Finding an algorithm TSP with additional conditions

Can someone solve this? Basically I want to find an algorithm for Traveling Salesman Problem (TSP) with additional conditions. 1) It should be non-convex. 2)At least pass one time at every given ...
2
votes
0answers
19 views

$I=mI$, when $I$ is not finitely generated.

Let $(R,m)$ be a commutative local ring with unit. Suppose $I$ is an ideal(not finitely generated). If $I=mI$, what can we say about $I$. If $I$ were finitely generated, then Nakayama's lemma would ...
2
votes
1answer
13 views

Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
2
votes
0answers
9 views

Behaviour of a clamped Bspline curve at t=1.0

A bspline curve of order $k$ is given by $$C(t) = \sum_{i=0}^n P_i N_{i,k}(t).$$ where $P_i$ are the control points and $N_{i,k}(t)$ a basis function defined on a knot vector $$T = (t_0,t_1,...t_{n+k})...
1
vote
0answers
14 views

Could we claim that the system has the only one solution there?

Given $$x_1>0,\quad x_2>0,\quad x_3>0,\quad x_4>0,\quad\textrm{and}\quad m>0,$$ could we claim that the system $$\begin{cases} [x_1+x_2]_m = 0 \\ [x_1+x_4]_m = 0 \\ [x_3+x_2]_m = 0 \\ ...
-24
votes
0answers
79 views

Proof that $\pi=1$…

We can think of numbers as sets, so let's look at two numbers: $\pi$ and $1\over \pi$. We know that both are irrational and they both probably contain all possible numerical combinations, which means ...
1
vote
1answer
23 views

A scalene triangle with no right nor obtuse angle

I want to find the area of the perfect triangle, i.e. a triangle with no particularity whatsoever : no side shall be equal to another, no right angle, no obtuse angle. So I gave myself a segment $[...
0
votes
0answers
19 views
1
vote
1answer
13 views

Proving that $\exists f: \mathbb{D}\to\mathbb{D}$, biholomorphic, which maps $z_1$ to $w_1$

Consider a pair of points, $z_1, w_1 \in \mathbb{D}$, where $\mathbb{D}$ is the unit disc centred at the origin. Is it sufficient to argue that $f(z)=z+a$ (for $a\in \mathbb{C}$) is biholomorphic, ...
1
vote
2answers
11 views

Is the orthogonality between Associated Legendre polynomials preserved on an interval [-a,a]

So I am aware of the orthogonality between the Associated Legendre polynomials on the interval $[-1,1]$, that is: \begin{equation} \int_{-1}^{1}P^m_kP^m_ldx\propto\delta_{k,l} \end{equation} where $\...
2
votes
1answer
14 views

Problem with dimension in nonlinear Gauss-Newton algorithm

I have a fundamental reasoning error, which I simply can't solve. Given the Problem to find the best solution for a nonlinear least square Problem, given m functions $f_{1...m}(\bf{x})$ with n ...
0
votes
1answer
66 views

Is there a proof for this or we should accept that?

Why are two independent parameters necessary and enough for determining position of a point with respect to a reference point in a plane? In other words, I want to address a point from another ...
0
votes
2answers
31 views

calculate E[X^n] with moment generating function

Say random variable X has a density function $ f(x)=1 $ when $0<x<1$. So this means $E[X^n]= \int_0^1 x^n.1 dx = \frac{1}{n+1}$. At the same time we can get that the moment generation function ...
1
vote
2answers
39 views

integral solution needed for general powers of $x$

I want to find the solution to the following integral $$\int_0^{\infty} \frac{dx}{x^{\frac{\alpha}{2}}+1}$$ where $\alpha$ can be any value greater than 2 such that $\alpha /2 >1$ but can be any ...
1
vote
0answers
11 views

Raise part of the integrand to positive power and determine the sign of the integral

Given function $h_1(s)\ge0, h_1'(s)\ge0$, $h_2(s)\ge0, h_2'(s)\ge0$ for $s\in[0,\bar s]$ and $g(s)$ is a density function on $[0,\bar s]$. I already shown that $$ \int_0^{\bar s}h_1(s)g(s)ds\ge \...
0
votes
1answer
22 views

Determine all points at which the surfaces share the same tangent line.

Determine all points at which the surfaces $x^2+y^2+z^2=3$ and $x^3+y^3+z^3 =3$ share the same tangent line. I know how to get the same tangent line for the curves, but I'm not sure how to go about ...
0
votes
1answer
19 views

fourier series, prove that the following is true

I'm having a few questions regarding the following problem: Calculate the Fourier series of $f(t)=|t|$ in $[-\pi, \pi)$ and then prove with $$\sum_{k=-n}^n |ck^2| = \frac{1}{2\pi}\int_0^\pi{|f(...
1
vote
1answer
38 views

The graph of $f(x)=ax^2+bx+c$ contains the points $(m,0)$ and $(n,2016^2)$. How many values of $n-m$ are possible?

Let $a,b,c,m$ and $n$ be integers such that $m<n$ an define a quadratic function as $f(x)=ax^2+bx+c$ where $x$ is real. The $f(x)$ has a graph that contains the points $(m,0)$ and $(n,2016^...
1
vote
0answers
11 views

Convention on Cauchy's two line notation for permutations

Let $n\in\mathbb{N}$. A permutation $\sigma\in S_n$ is denoted in Cauchy's two line notation as follow: \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma(1) & \sigma(2) & \cdots & \...
0
votes
1answer
16 views

multiplicative order $ord_{a }(k)$ if $gcd (a, k) > 1$

The question concerns the multiplicative order $ord_{a}(k)$ if $gcd (a, k) > 1$. $2^{0} \pmod 4 = 1$ $2^{1} \pmod 4 = 0$ $2^{2} \pmod 4 = 0$ $2^{3} \pmod 4 = 0$ $2^{4} \pmod 4 = 0$ ... $4^{0}...
-1
votes
2answers
41 views

Let $A=\{1,2,3,…,2^n\}$. Consider the greatest odd factor of each element of A and add them…

Let $A=\{1,2,3,...,2^n\}$. Consider the greatest odd factor (not necessarily prime) of each element of A and add them. What does this sum equal?
0
votes
0answers
25 views

problem solving involving time

This a summary of the question (because it was really long) Everyday A leaves home before 5pm to pick B up from the train station at 5pm and drive home One day B catches an earlier train that ...
0
votes
0answers
11 views

determine the indical equation for the following differential equation

$x^3y''+(\cos2x-1)y'+2xy=0$ One have to make the anstaz that $y= x^m \sum_{j=0}^{\infty}a_j x^j$ and I have been solving problems like this before, but for this one what really confuses me is what to ...
1
vote
1answer
36 views

Riemann Hypothesis and $\sum\limits_{k|n}\left(\frac{\mu(k)}{k}\right)^2$

I know that Riemann Hypothesis is equivalent to the following statement $\sum\limits_{k|n}\frac{\mu(k)}{k}=O(n^{-1/2+\epsilon})$ Is there any relation between Riemann Hypothesis and $\sum\...
0
votes
0answers
24 views

Parametric Equation part A

Hi everyone I am in need of some guidance solving this parametric equation question and was wondering if you guys could give me some pointers and to see if I am doing this correctly. Here I have two ...
0
votes
4answers
51 views

Evaluate $\lim_{x\to 0} \frac{1}{x^3}\int_{0}^{x} \sin^{2}(3t)dt$

$$\lim_{x\to 0} \frac{1}{x^3}\int_{0}^{x} \sin^{2}(3t)dt$$ $$\lim_{x\to 0} \frac{1}{x^3}\int_{0}^{x} \sin^{2}(3t)dt=\lim_{x\to 0} \frac{\int_{0}^{x} \sin^{2}(3t)dt}{{x^3}}$$ I know that the limit ...
1
vote
0answers
17 views

Term for a function that is an involution on its image

Is there a specific term for a function $f:X\to X$ that obeys the law $f^3(x)=f(x)$? It's not necessarily an involution, but it is an involution when its domain is restricted to its image. A simple ...

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