2:54 AM
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In 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
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If $a, b, c, d$ 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
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Let $X \in R^n$ 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
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Let $\{x_1,...,x_n\}$ 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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X 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 I 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. @Feeds Answers to this question are eligible for a +50 reputation bounty. nashynash is looking for an answer from a reputable source. 3 I 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... @Feeds Answers to this question are eligible for a +200 reputation bounty. Free Books wants to draw more attention to this question: > I would like some hint about how to solve the last case. 1 hour later… 10:20 AM 2 I 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 If 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} $$Does anyone know how to prove this? 3 I'm studying with the book 'Numerical Linear Algebra' written by L. N. Trefethen, and I wrote code regarding the text below: Here is a numerical example. Let A be a 200 x 200 matrix whose entries are independent samples from the real normal distribution of mean 2 and standard deviation 0.5/\sqr... 10:45 AM 11:21 AM 1 Definition : A curve \omega : [0,1]\to X is defined absolutely continuous whenever there exists g\in L^1([0,1]) such that d(\omega(t_0),\omega(t_1))\le\int_{t_0}^{t_1}g(s)ds for every t_0<t_1. I would like to show that, according to the above definition, the graph of the Cantor function... 11:51 AM @Feeds Answers to this question are eligible for a +50 reputation bounty. edamondo wants to draw more attention to this question. @HNQmath.se Removed from HNQ by adding MathJax. 12:49 PM 2 I need some help with a question. I have to calculate$$\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
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Think 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 @Feeds Answers to this question are eligible for a +200 reputation bounty. SABOY is looking for an answer from a reputable source. 2:28 PM 3 I 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... 2:39 PM 2:54 PM 1 hour later… 4:22 PM 1 Let$(\Omega,\mathcal A,\operatorname P)$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
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I 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 @Feeds Answers to this question are eligible for a +50 reputation bounty. 0xbadf00d is looking for a canonical answer. @Feeds Answers to this question are eligible for a +50 reputation bounty. Jiaxin Zhong wants to draw more attention to this question. -1 I 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 Take 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 @Feeds Answers to this question are eligible for a +50 reputation bounty. DanGoodrick wants to draw more attention to this question: > Please let me know how I can improve the question if it isn't clear. @Feeds Answers to this question are eligible for a +50 reputation bounty. Monolite wants to draw more attention to this question. 7:27 PM 3 Original 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 zw$or replace the number$w$with a different complex number$w'$so that$$\max(\lvert\oper... 2 hours later… 9:04 PM 4 Let 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 How 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 Let$G= \langle g_1, g_2 \rangle$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 Let$A, B$, 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...