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2:54 AM
4
Q: Proving unboundedness of the natural numbers via the Axiom of Completeness

independentvariableIn the book "Understanding Analysis, second edition" by Stephen Abbot, the unboundedness of the set of natural number $\mathbb{N}$ is proven as the following proof: Assume, for contradiction, that $\mathbb{N}$ is bounded above. By the Axiom of Completeness (AoC), $\mathbb{N}$ should then have a ...

 
3:19 AM
1
Q: A Cauchy-Schwarz Inequality

HasIf $a, b, c, d$ are positive real numbers such that $c^2+d^2=(a^2+b^2)^2$, prove that $$\frac{a^2}{c}+\frac{b^2}{d} \geq1 $$ and equality hold iff $ad=bc$.

 
4:08 AM
2
Q: Conditions for convergence of fixed-point iterations (not necessarily to a unique fixed-point)

Sampath KumarLet $X \in R^n$ be a compact convex set, and $f:X \to X$ be a continuous function. Then, can we say that from all $x_0 \in X$, the fixed-point iterations $x_{k+1}=f(x_k)$ to converge to some fixed-point $\bar{x}(x_0) \in X$? If not, what are the conditions that $f$ must satisfy such that the iter...

 
 
1 hour later…
5:35 AM
1
Q: Show that this structure is an affinely regular polygon

polygonlink1Let $\{x_1,...,x_n\}$ be a set of convex points labeled in a cyclic order. I am trying to show that the following structure is equivalent with an affinely regular polygon: Fix $j$ for some $j\in \{1,...,n\}$. Then we know the following to be true: $\{x_j,x_{j-1}\}\parallel \{x_{j+1},x_{j-2}\}$ mo...

 
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3
Q: Show inequality using the Jensen inequality

gunsenX is a non-negative random variable, with $\mathbb{E}(X) < \infty $. My goal is to show this inequality: $$\sqrt{1+(\mathbb{E}(X))^2} \leq \mathbb{E}(\sqrt{1+X^2})$$ x² is a convex function, so with the Jensen inequality I get that: $$\sqrt{1+(\mathbb{E}(X))^2} \leq \sqrt{1+\mathbb{E}(X^2)} = \sq...

 
 
3 hours later…
8:30 AM
1
Q: How to derive the solution for $F(x)=P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)$?

nashynashI have the following CDF \begin{align*} F(x)=&P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right), \end{align*} where the RHS is found to be \begin{align*} P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)...

 
8:53 AM
@HNQmath.se Removed from HNQ by closure.
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3
Q: 2D bifurcation problem

Free BooksI come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you please help me? Consider the vector field on plane \begin{align} \dot{x} &= x - xy +1\\ \label{sys1...

 
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> I would like some hint about how to solve the last case.
 
 
1 hour later…
10:20 AM
2
Q: Law of sines question

Shooting StarsI came across the following question just now, A triangle $\Delta ABC$ is drawn such that $\angle{ACB} = 30^o$ and side length $AC$ = $9*\sqrt{3}$ If side length $AB = 9$, how many possible triangles can $ABC$ exist as? Here is a diagram for reference: Here is what I did: I used the Law of Sine...

1
Q: Upper bound of product

vibeIf $x \in [a,b]$, then I want to find the optimal upper bound of the product $$ |x - a| |x - b| \leq M $$ It seems obvious that $$ |x - a| |x - b| \leq |b-a|^2 $$ however it seems that the optimal upper bound is in fact $$ |x-a||x-b| \leq \frac{|b-a|^2}{4} $$ Does anyone know how to prove this?

3
Q: Eigenvalues of a random Gaussian matrix

AABBCCI'm studying with the book 'Numerical Linear Algebra' written by L. N. Trefethen, and I wrote code regarding the text below: Here is a numerical example. Let A be a 200 x 200 matrix whose entries are independent samples from the real normal distribution of mean 2 and standard deviation 0.5/$\sqr...

 
11:21 AM
1
Q: Cantor curve is not absolutely continuous

edamondo Definition : A curve $\omega : [0,1]\to X$ is defined absolutely continuous whenever there exists $g\in L^1([0,1])$ such that $d(\omega(t_0),\omega(t_1))\le\int_{t_0}^{t_1}g(s)ds$ for every $t_0<t_1$. I would like to show that, according to the above definition, the graph of the Cantor function...

 
11:51 AM
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@HNQmath.se Removed from HNQ by adding MathJax.
 
12:49 PM
2
Q: Limit with sequence of functions and integrals

GregI need some help with a question. I have to calculate $$\lim_{x \to 3}\frac{x^2}{x - 3}\int_3^x \frac{\sin t}{t}dt.$$ If I'm not wrong, we can write $$\sin(x) = \sum_{n = 0}^{\infty}\frac{x^{2n+1}}{(2n+1)!} \Longrightarrow \frac{\sin(x)}{x}=\sum_{n = 0}^{\infty}\frac{x^{2n}}{(2n+1)!},$$ then $$...

 
 
1 hour later…
1:50 PM
1
Q: Relation between magnitude of diagonal and non-diagonal entries when given information about the inverse

SABOYThink of $\Sigma$ as a covariance matrix, or any positive semidefinite matrix. Let $A(\lambda)$ be a $n \times n$ positive semidefinite matrix with $\lambda > 0$ and the following specifications to its inverse: $(A(\lambda))^{-1}_{jk}=\begin{cases}\Sigma_{jk}+\lambda, & j=k\\ \Sigma_{jk}+\lambda ...

 
2:09 PM
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2:28 PM
3
Q: Why the answer for double integral is coming as zero?

Umesh shankarI am trying to evaluate $$\iint_{R} x+y \:d A$$, where $R$ is the region formed by the vertices $$(0,0),(5,0),\left(\frac{5}{2}, \frac{5}{2}\right) \text { and }\left(\frac{5}{2},-\frac{5}{2}\right)$$. My try: Here is the picture of the region which has two triangular regions. Let the top traing...

 
 
1 hour later…
4:22 PM
1
Q: Existence of jump intensity (random) measure of a Lévy process

0xbadf00dLet $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued càdlàg Lèvy process on $(\Omega,\mathcal A,\operatorname P)$. How can we prove that there is a (unique) transition kernel $\pi$ from $(\Omega,\mathcal A...

 
4:44 PM
2
Q: The integral of the product of a Bessel function and a trigonometric function $\int J_0(x)\sin(ax)\mathrm{d}x$

Jiaxin ZhongI was wondering if the following integral has a closed-form solution? $$I(x) = \int J_0(x)\sin(ax)\mathrm{d}x$$ where $a$ is a constant. I know the answer for the case when $a=1$, see here. I tried the similar method in that link but I was stuck. Integrating by parts yields $$ I(x) =x J_0(x)\sin(...

 
5:20 PM
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-1
Q: System of equations with 3D vectors and constraints

DanGoodrickI have 6 (A…F) noisy 3D normal vectors <x_hat, y_hat, z_hat > and noisy point cloud points <x, y, z> that form a cube and are related by the following vector operations: A cross B = B cross C = C cross D = D cross A A cross E = E cross C = F cross A = C cross F A=-C B=-D E=-F (A cross B) dot E = ...

 
 
1 hour later…
6:51 PM
0
Q: On the existence and uniqueness of the maximum likelihood estimators of a Student-t.

MonoliteTake the density of a generalized student-t, i.e., \begin{align*} p( y_t | \sigma , \mu, \nu ) = \frac {\Gamma (\frac {\nu +1}{2})}{\Gamma (\frac {\nu }{2}){\sqrt {\pi \nu }}{\sigma }\,}\left(1+{\frac {1}{\nu }}\left({\frac {y_t- \mu }{\sigma }}\right)^{2}\right)^{-\frac {\nu +1}{2}} \end{alig...

 
7:04 PM
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> Please let me know how I can improve the question if it isn't clear.
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7:27 PM
3
Q: Let $z$ be a complex number. The number $1$ is written on a board. You perform a series of moves, where in each move you may either

Knocker379Original Problem: Let $z$ be a complex number. The number $1$ is written on a board. You perform a series of moves, where in each move you may either replace the number $w$ written on the board with $z$$w$ or replace the number $w$ with a different complex number $w'$ so that $$\max(\lvert\oper...

 
 
2 hours later…
9:04 PM
4
Q: Applying Zorn's lemma on a set of all sets.

JaswantLet U be the set of all sets. Define a partial ordering on U by inclusion: A≤B iff A ⊆ B for A, B ∈ U. Consider a chain C of U under this partial ordering: C : A1 ≤ A2 ≤ A3 ≤ · · · . Define B = ∪i⩾1A{i}. Clearly, B ∈ U and it is an upper bound of the chain C. Hence, Zorn’s Lemma implies that U ha...

4
Q: Line of circles and star made of circles, are they homeomorphic?

u2718281How can I determine if this two figures are homeomorphic? I'm guessing they're not homeomorphic. I have tried using cut points but from what I understand both figures have the same number of cut points. I can see that in the first picture the circle in the center is connected to the four other ci...

 
9:57 PM
4
Q: How to construct submodules with GAP / MeatAxe?

Bernhard BoehmlerLet $G= \langle g_1, g_2 \rangle$ be a finite group. Let $k$ be a finite field with ${\rm char}(k)=p>0$ such that $p \mid |G|$. Let the $kG$-module $M$ be a MeatAxe-module in GAP. The generators of $M$ are given by the two matrices $m_1$ and $m_2$, respectively, which reflect the actions of $g_1$...

 
 
2 hours later…
11:59 PM
2
Q: Curve of fastest descent between three points.

SAWbladeLet $A, B$, and $C$ be three points in $\mathbb{R}^2$ such that $A = (x_1,y_1), B=(x_2,y_2)$, and $C= (x_3,y_3)$ where $x_1<x_2<x_3$ and $y_1 > y_2 > y_3$. In other words, $B$ is to the upper left of $C$ and $A$ is to the upper left of $B$. What is the curve $y$ of fastest descent that contains t...

 

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