qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,115,566 | <p>Convergence of the series for <span class="math-container">$a \in \mathbb R$</span> <span class="math-container">$$\sum_{n=1}^\infty\sin\left(\pi \sqrt{n^2+a^2} \right)$$</span> </p>
<p>I saw this problem in a calculus book and it gave a hint that says </p>
<p><strong>HINT</strong> First show that <span class="mat... | Sangchul Lee | 9,340 | <p>The first half of the hint is enough to conclude the convergence of the series.</p>
<p>Indeed, notice that <span class="math-container">$a_n := \pi a^2/(\sqrt{n^2+a^2}+n)$</span> decreases monotonically to <span class="math-container">$0$</span> as <span class="math-container">$n\to\infty$</span>, and so, <span cla... |
2,592,999 | <p>Prove that $$\lim_{x \to 2} x^3 = 8$$</p>
<p>My attempt, </p>
<p>Given $\epsilon>0$, $\exists \space \delta>0$ such that if $$|x^3-8|<\epsilon \space \text{if} \space 0<|x-2|<\delta$$ </p>
<p>$$|(x-2)(x^2+2x+4)|<\epsilon$$</p>
<p>I'm stuck here. Hope someone could continue the solution and exp... | Jack D'Aurizio | 44,121 | <p>I am not sure there is a simple way without using Euler's Beta function... Anyway, once
$$ \int_{0}^{1}\frac{x^{2k}}{x^{2/3}(1-x)^{1/3}}\,dx = \frac{\Gamma\left(\frac{2}{3}\right)\Gamma\left(2k+\tfrac{1}{3}\right)}{\Gamma(2k+1)} \tag{A}$$
is proved (for instance by integration by parts and induction on $k$), the ori... |
2,246,522 | <p>I found a form of the lagrange theorem that I don't know,</p>
<p>I didn't find something similar on the internet.</p>
<blockquote>
<p>Suppose that $f:[a,b]\times[c,d] \subset \mathbb{R}^2 \longrightarrow \mathbb{R}$ is a continuous function. Consider the function</p>
<p>\begin{align}
I:[c,d] &\longright... | Chappers | 221,811 | <p>It's actually the same thing in different notation: if we write $f(x,y) = F_x(y)$, then the quotient inside the integral is
$$ \frac{F_x(y+k)-F_x(y)}{k}. $$
By Lagrange (AKA the mean value theorem), there is a $c$ with $y<c<y+k$ so
$$ \frac{F_x(y+k)-F_x(y)}{k} = F_x'(c). $$
Any such $c$ can be written as $(1-\... |
2,352,211 | <blockquote>
<p>Consider $n$ seats in which $k$ distinct men and $m$ distinct women
are going to be seated ($n \ge \max \{m+k,2k\}$). In how many ways is
this possible given that no two men sit next to each other?</p>
</blockquote>
<p>My approach is the following: </p>
<p>First seat the $k$ distinct men in $k!$... | true blue anil | 22,388 | <p>I'll change the notation slightly, <em>m</em> for men and <em>w</em> for women.</p>
<p>Take an <em>unnumbered</em> row of <em>(n-m)</em> seats, which have <em>(n-m+1)</em> "gaps" (including the ends), where seats for men can be put in $\binom{n-m+1}{m}$ ways.</p>
<p>Place the women in $\binom{n-m}{w}$ ways in vaca... |
3,174,153 | <p>Let <span class="math-container">$X_1,X_2,...X_n$</span> be a random sample from <span class="math-container">$f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$</span>.We know by Factorisation theorem that <span class="math-container">$\frac{\sum |X_i|}{n}$</span> is sufficient for <span class="math-container"... | heropup | 118,193 | <p>Yes, it's quite easy. All you have to do to show that a statistic is not sufficient is to show that there has been some loss of information (about the parameter) as compared to what was present in the original sample.</p>
<p>If you know that one statistic is sufficient, then computing its value for a specific samp... |
653,451 | <p>For $X$ a Banach space, let me define the space $C^0([0,T];X)$ to consist of elements $u:[0,T] \to X$ such that
$$\lVert u \rVert_{C^0} := \max_{t \in [0,T]}\lVert u(t) \rVert_X < \infty.$$
So the difference is that I don't care about continuity of $u$ in $t$. This defines a norm. </p>
<p>For completeness, let $... | Thomas Andrews | 7,933 | <p>hint: What is $hgf$? Write it in two different ways.</p>
|
653,451 | <p>For $X$ a Banach space, let me define the space $C^0([0,T];X)$ to consist of elements $u:[0,T] \to X$ such that
$$\lVert u \rVert_{C^0} := \max_{t \in [0,T]}\lVert u(t) \rVert_X < \infty.$$
So the difference is that I don't care about continuity of $u$ in $t$. This defines a norm. </p>
<p>For completeness, let $... | Martin Brandenburg | 1,650 | <p>From the graph we see that $f$ <em>and</em> $h$ are inverse to $g$. Now it is a general fact about categories that inverses of morphisms are unique. The proof is the same as the one for groups.</p>
|
1,690,248 | <p>Hello I am hoping to find some direction on solving this try it yourself problem in my textbook.</p>
<p>Let S be an arbitrary set of symbols and let $\Phi = \{v_0 \equiv t | t \in T^S\} \cup \{\exists v_0 \exists v_1 \neg v_0 \equiv v_1\}$. </p>
<p><em>Note: $T^S$ is the set of all S-terms.</em></p>
<p>Show that ... | Arif Burhan | 313,035 | <p>Calculate the integral wrt. Time using Trapezoidal Rule- high frequency of readings, so this is accurate enough.</p>
<p>Calculate between -T and +T , which is your interval of interest, dividing by 2*T to get the average.</p>
<p>For the next moment discard the earliest reading(s) , adding the same number of fresh ... |
2,956,460 | <p>Consider the "formal definition" here <a href="https://en.wikipedia.org/wiki/Limit_of_a_sequence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Limit_of_a_sequence</a>. I checked some references and this is often precisely the definition in all words and terms used in this article. I claim that this defini... | Christopher | 73,985 | <p>You are right that <span class="math-container">$N$</span> depends on <span class="math-container">$\epsilon$</span> - and missing this fact is the cause of a lot of confusion among people learning analysis for the first time. Sometimes it is made explicit: "for all <span class="math-container">$\epsilon > 0$</sp... |
2,029,279 | <blockquote>
<p>Let <span class="math-container">$T:\Bbb R^2\to \Bbb R^2$</span> be a linear transformation such that <span class="math-container">$T (1,1)=(9,2)$</span> and <span class="math-container">$T(2,-3)=(4,-1)$</span>.</p>
<p>A) Determine if the vectors <span class="math-container">$(1,1)$</span> and <span cla... | mfl | 148,513 | <p><strong>Hint</strong></p>
<p>$\{(1,1),(2,-3)\}$ is a basis if and only if the vectors are linearly independent (note that the vector space is two-dimensional). Are they linearly independent?</p>
<p>To compute $T(x,y)$ use that $$(1,0)=\frac 35(1,1)+\frac 15(2,-3),$$ $$(0,1)=\frac 25(1,1)-\frac 15(2,-3)$$ and $$(x,... |
2,272,831 | <blockquote>
<blockquote>
<p>$$f_{x,y}(x,y) = \begin{cases}\frac{4}{3}(2-x)y, & 0\leq x\leq 1, 0\leq y\leq 1\\
0, & \text{otherwise.}\end{cases}$$
Obtain $P(X\leq 2Y)$. Sketch the region of integration.</p>
</blockquote>
</blockquote>
<p>Is this the correct way to solve it?</p>
<p>$$\int_0^1\int_0... | Servaes | 30,382 | <p><strong>HINT:</strong> What does $X^TAX=0$ mean when $X$ is a standard basis vector? And if $X$ is the sum of two distinct standard basis vectors?</p>
|
2,272,831 | <blockquote>
<blockquote>
<p>$$f_{x,y}(x,y) = \begin{cases}\frac{4}{3}(2-x)y, & 0\leq x\leq 1, 0\leq y\leq 1\\
0, & \text{otherwise.}\end{cases}$$
Obtain $P(X\leq 2Y)$. Sketch the region of integration.</p>
</blockquote>
</blockquote>
<p>Is this the correct way to solve it?</p>
<p>$$\int_0^1\int_0... | πr8 | 302,863 | <p>Suppose that $x^TAx=0$ for all $x$, and define $B=A+A^T$. Then it is clear that:</p>
<ul>
<li>$B$ is symmetric</li>
<li>$x^TBx=0$ for all $x$</li>
</ul>
<p>As a symmetric matrix, $B$ has a complete set of orthonormal eigenvectors $v_i$ with eigenvalues $\lambda_i$, such that $Bv_i = \lambda_i v_i$, and any $x$ can... |
986,875 | <p>A service center consists of two servers, each working at an exponential rate
of two services per hour. If customers arrive at a Poisson rate of three per hour, then,
assuming a system capacity of at most three customers,
What fraction of potential customers enter the system?</p>
<p>I was thinking if I could thi... | Mr.Spot | 155,516 | <p>Time between arrivals is exponential with mean $1/\lambda=1/3$ hour. Service rate $\mu=2$ per hour. Length of a service time is exponential with mean $1/\mu=0.5$ hours. </p>
<p>This is an $M/M/2/3$ queue. 2 servers with $N=3$ max capacity which means a waiting room of size $1.$</p>
<p>You can use the total number ... |
280,292 | <p>Find $a,c$ such that:</p>
<p>$$f(x)= \begin{cases} a\frac{\exp(tgx)}{(1+\exp(tgx))} &\text{for }|x|<\pi/2 \\[2ex] \exp(cx)-2 &\text{for } |x|\ge\pi/2 \end{cases}$$</p>
<p>is continuous. </p>
<p>How do I evaluate the left- and right-hand limits to see if they are equal?</p>
| Hagen von Eitzen | 39,174 | <p>Note that
$$ \lim_{x\to(\frac\pi2)^-}\frac{a\exp\tan x}{1+\exp\tan x}=\lim_{x\to+\infty}\frac{a\exp x}{1+\exp x}=\lim_{x\to+\infty}\frac{a x}{1+ x}=a.$$</p>
|
594,810 | <p>I need multiple ways to solve this question. Thank you! </p>
<p><strong>There are $A$ black balls, and $B$ white balls in an urn. You select balls one by one from this urn randomly without replacement. What is the probability that the $k$-th ball you selected is black? $(1\leq k \leq A+B)$, and $A,B$ are positive i... | M.B. | 2,900 | <p><strong>HINT</strong>: think about it as having aligned $A+B$ persons labeled $1, 2, \ldots, A+B$ in a straight line, and some guy with a bag of $A+B$ balls wherein $A$ are black and $B$ are white. This funny looking guy with a long beard wraps all of the balls into red paper and hands them out, one by one, to each ... |
594,810 | <p>I need multiple ways to solve this question. Thank you! </p>
<p><strong>There are $A$ black balls, and $B$ white balls in an urn. You select balls one by one from this urn randomly without replacement. What is the probability that the $k$-th ball you selected is black? $(1\leq k \leq A+B)$, and $A,B$ are positive i... | Community | -1 | <p><strong>The Hard Way:</strong> You can calculate the probability directly. To motivate my calculation, let's first look at the case $k=3$. The probability
that the 3rd ball is black depends on what happens in the first two draws. Taking all four cases into account we get
\begin{eqnarray*}\mathbb{P}(3^{\rm rd} \mbox... |
4,490,258 | <p>I've recently started learning hyperbolic functions and inverse hyperbolic functions, and I came across this equation involving inverse hyperbolic functions. I tried to solve it numerically (I got x=-0.747), but how would you solve it analytically? I don't know how to type in latex so please forgive me.</p>
<p><span... | Narasimham | 95,860 | <p>HINTS:</p>
<p>Drawing circular and hyperbolic function triangles for inverse functions may be helpful:</p>
<p><a href="https://i.stack.imgur.com/mpTea.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mpTea.png" alt="enter image description here" /></a></p>
<p>Identities</p>
<p><span class="math-con... |
3,243,440 | <p>Find the number k such that:</p>
<p><span class="math-container">$$det\begin{bmatrix}
3a_1 & 2a_1 + a_2 - a_3 & a_3\\\
3b_1 & 2b_1 + b_2 - b_3 & b_3\\\
3c_1 & 2c_1 + c_2 - c_3 & c_3\end{bmatrix}$$</span></p>
<p><span class="math-container">$$ = k \bullet det\begin{bmatrix}
a_1 & a_2 &... | JimmyK4542 | 155,509 | <p><strong>Hint</strong>: The most concise solution is to notice that <span class="math-container">$$\begin{bmatrix}
3a_1 & 2a_1 + a_2 - a_3 & a_3\\\
3b_1 & 2b_1 + b_2 - b_3 & b_3\\\
3c_1 & 2c_1 + c_2 - c_3 & c_3\end{bmatrix} = \begin{bmatrix}
a_1 & a_2 & a_3\\\
b_1 & b_2 & b_3... |
1,903,884 | <p>3 dice are rolled simultaneously. Event A: sum is multiple of 3 and event B: sum is multiple of 5. </p>
<p>Are this events independent? </p>
<p>There are 216 points in the sample space for this problem. Is there any way to do it without writing all points. </p>
<p>If they are independent or dependent..is there an... | robjohn | 13,854 | <p>If $A$ and $B$ are <a href="https://en.wikipedia.org/wiki/Independence_(probability_theory)" rel="nofollow">independent</a>, $P(A\text{ and }B)=P(A)\cdot P(B)$.</p>
<p>We need to convolve the distribution for one die, $\mathcal{P}=\left(\frac16,\frac16,\frac16,\frac16,\frac16,\frac16\right)$, with itself a couple o... |
1,903,884 | <p>3 dice are rolled simultaneously. Event A: sum is multiple of 3 and event B: sum is multiple of 5. </p>
<p>Are this events independent? </p>
<p>There are 216 points in the sample space for this problem. Is there any way to do it without writing all points. </p>
<p>If they are independent or dependent..is there an... | BruceET | 221,800 | <p><strong>Comment</strong> (already answered): A simulation of a million 3-roll experiments in R statistical software (correct to about three places) confirms the exact
probabilities in @robjohn's Answer (+1). Thus according to @GrahamKemp's answer (+1),
the events are not independent.</p>
<pre><code>m = 10^6; n = ... |
877,850 | <p>I'm trying to calculate the expected area of a random triangle with a fixed perimeter of 1. </p>
<p>My initial plan was to create an ellipse where one point on the ellipse is moved around and the triangle that is formed with the foci as the two other vertices (which would have a fixed perimeter) would have all the ... | SuperAbound | 140,590 | <p>The correct identity should be
$$\cot^{-1}{x}=\tan^{-1}\left(\frac{1}{x}\right)$$
The equation becomes
$$\tan^{-1}{x}+\tan^{-1}\left(\frac{1}{x}\right)=\frac{\pi}{2}$$
We will take the tangent then the inverse tangent of the LHS to get
\begin{align}
\tan^{-1}\left(\tan\left(\tan^{-1}{x}+\tan^{-1}\left(\frac{1}{x}\ri... |
1,619,464 | <p>I tried using the universal property of tensor products to show that there are mutually inverse maps from $M \times N$ to $M' \times N$, and use this to show that $M \cong M'$, but I didn't get far. I know that this is true if $N = R$, but I couldn't think of a counterexample.</p>
| Martin Brandenburg | 1,650 | <p>No: We have $M \otimes 0 \cong 0 \cong M' \otimes 0$ for all modules $M,M'$.</p>
<p>Even if $N$ is a very nice $R$-module, say free of finite rank, and non-zero, then $M \otimes N \cong M' \otimes N$ does not imply $M \cong M'$: This is because there are non-isomorphic modules $M,M'$ with $M^2 \cong M'^2$, i.e. $M ... |
200,063 | <p>I am looking to evaluate</p>
<p>$$\int_0^1 x \sinh (x) \ \mathrm{dx}$$</p>
| crackpotHouseplant | 41,981 | <p>Use integration by parts!
General format:</p>
<p>$$\int f(x) g'(x)dx = f(x)g(x)-\int f'(x)g(x)dx$$</p>
<p>Remember! If you have bounds on the integral, they must be applied to the $f(x)g(x)$ part, too (this proceeds in the normal way).</p>
<p>So, if I recall correctly, the integral of $\sinh(x)$ is $\cosh(x)$...<... |
1,622,041 | <p>Let $\rho: G \to GL(V)$ be a finite dimensional irreducible representation of a group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and let $R$ be a commutative ring with unity on which $G$ acts. Suppose also that $\mathbb{F} \subseteq R$. We can form the extension of scalars
$$M=R \oti... | dbluesk | 217,671 | <p>I think it might be false. Take for instance $R=\mathbb{F}[t]$ with trivial $G$-action, and $N=tM$.</p>
|
1,622,041 | <p>Let $\rho: G \to GL(V)$ be a finite dimensional irreducible representation of a group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and let $R$ be a commutative ring with unity on which $G$ acts. Suppose also that $\mathbb{F} \subseteq R$. We can form the extension of scalars
$$M=R \oti... | Eric Wofsey | 86,856 | <p>If $I\subseteq R$ is any $G$-invariant ideal, then $I\otimes V$ is a $G$-invariant submodule of $R\otimes V$. So if $R$ has any nontrivial $G$-invariant ideals (e.g., if $G$ acts trivially on $R$ and $R$ is not a field), then $R\otimes V$ will have nontrivial $G$-invariant submodules.</p>
|
2,615,427 | <p>I saw the following question in my linear algbra book, and found it rather strange:</p>
<p>"Let $V$ be a real vector space of dimension $n$. Let $L,K \colon V \rightarrow \mathbb{R}$ be linear transformations, so that $\ker(L) \subset \ker(K)$.
Prove that $K=\lambda L$ for a $\lambda \in \mathbb{R}$, using the foll... | dxiv | 291,201 | <p>Hint: $u^2-v^2=(u-v)(u+v)\,$, and if $u,v \ge 0$ then $u+v \ge 0$, so $u-v$ and $u^2-v^2$ have the same sign. Use that twice, once for $u=a, v=b$, then for $u=\sqrt{a}, b=\sqrt{b}\,$.</p>
<p><hr>
[ <em>EDIT</em> ] For example, with $u=a, v=b$ where $a,b \ge 0\,$, it follows from the above that:</p>
<p>... |
1,416,661 | <p>Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, measuring the resulting angle (divided by 2π) as a number from 0 to 1.</p>
<p>I am bit confused, when i look into the... | Ashutosh Gupta | 215,160 | <p>1) Remember, that for a continuous random variable, $Pr(X=x) = 0 $. To derive the PDF write the CDF as : </p>
<p>$$ Pr[X \leq x] = \int_{0}^{x} \frac{1}{2\pi}dx' = \frac{x}{2\pi} $$</p>
<p>Hence PDF is:
$$ f(x) = \frac{1}{2\pi}$$</p>
<p>2) It is inappropriate to think that there are exactly 360 points between 0 a... |
3,859,737 | <p>I saw this problem on a math board / insta:</p>
<p><span class="math-container">$$\lim_{x\rightarrow3}\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt{2}}$$</span></p>
<p>My first step would be take a derivative of the numerator and denominator to see if the limit exists or not, since just plugging in gets me 0/0 which is undef... | egreg | 62,967 | <p>Uhm, you get <span class="math-container">$2^{-1/2}$</span> in the denominator, not <span class="math-container">$2^{1/2}=\sqrt{2}$</span>.</p>
<p>Let's try in a different way: first
<span class="math-container">$$
\lim_{x\to3}\frac{\sqrt{3x}-3}{x-3}=
\lim_{x\to3}\frac{3x-9}{x-3}\frac{1}{\sqrt{3x}+3}=\frac{3}{6}=\fr... |
2,484,027 | <p>I've been trying to solve this permutation problem. I know that it's been posted on this site before, but my question is about the specific approach I take to solving it.</p>
<p>Here's what I thought I'd do : I could first figure out the total number of permutation where AT LEAST $2$ girls are together, and then su... | hmakholm left over Monica | 14,366 | <p>You're overcounting the number of ways "at least two" girls can sit together, because for example</p>
<pre><code> Boy1 Boy2 Girl1 Girl2 Girl3 Boy3 Boy4 Boy5
</code></pre>
<p>is produced twice: Once considering <code>Girl1 Girl2</code> as the unit, and once considering <code>Girl2 Girl3</code> as the unit. This mea... |
2,484,027 | <p>I've been trying to solve this permutation problem. I know that it's been posted on this site before, but my question is about the specific approach I take to solving it.</p>
<p>Here's what I thought I'd do : I could first figure out the total number of permutation where AT LEAST $2$ girls are together, and then su... | nonuser | 463,553 | <p>Neglect for the moment the personality and just watch the gender.</p>
<p>First arrange girls in line so that there is an empty place between them. Then we put 2 boys between 1. and 2. and between 2. and 3. girl.
To every ''good'' arrangement of gender we assign 6-couple so that between two girls we ''delete'' one ... |
2,484,027 | <p>I've been trying to solve this permutation problem. I know that it's been posted on this site before, but my question is about the specific approach I take to solving it.</p>
<p>Here's what I thought I'd do : I could first figure out the total number of permutation where AT LEAST $2$ girls are together, and then su... | Allawonder | 145,126 | <p>First arrange the boys in a row (this can be done in exactly <span class="math-container">$5!$</span> ways):</p>
<p><span class="math-container">$$B_1,B_2,B_3,B_4,B_5.$$</span></p>
<p>Then you have six available spaces to place each of the three girls so as to satisfy the given condition. You may do this in <span ... |
2,924,831 | <p>Suppose you have <span class="math-container">$100$</span> coins whose probabilities of obtaining the outcome "head" are <span class="math-container">$p_1,\ldots,\,p_{100}$</span>. These probabilities are not necessarily equal each other. Consider the following random experiment divided into two rounds.</p>
<ul>
<l... | btilly | 6,708 | <p>There is no efficient theoretical way to do this, but it is a straightforward dynamic programming problem for a computer. Here is sample code for it in Python.</p>
<pre><code>#! /usr/bin/env python3
def transition_distribution (prob_vector):
transitions = {(0,0): 1.0}
for p_i in prob_vector:
new_tr... |
2,060,418 | <p>We have</p>
<p><span class="math-container">$$ \begin{cases} \dot{x} = - y - y^3 \\ \dot{y} = x \end{cases} $$</span></p>
<p>where <span class="math-container">$x,y \in \mathbb{R}$</span>. Show that the critical point for the linear system is a <span class="math-container">$\mathbf{center}$</span>. Prove that the ty... | Binh Ly | 398,931 | <p>The differential equation is the same as y"+(y+y^3)=0, which describes free vibration of an oscillator with a nonlinear spring. The system is conserve in energy. So its orbit is a closed curve.</p>
|
2,991,811 | <p>I think it converges to <span class="math-container">$\frac{\pi}{2}$</span>, but I am not sure how to prove it. I've tried using induction on <span class="math-container">$n$</span> but have had no luck. Any help would be appreciated thanks.</p>
| Angina Seng | 436,618 | <p>Use <span class="math-container">$\sin x=x+O(x^3)$</span> as <span class="math-container">$x\to0$</span>. Then
<span class="math-container">$$\sin\frac\pi{2^{n+1}}=\frac{\pi}{2^{n+1}}+O(2^{-3n})$$</span>
and
<span class="math-container">$$2^n\sin\frac\pi{2^{n+1}}=\frac{2^n\pi}{2^{n+1}}+O(2^{-2n})$$</span>
etc.</p>
|
2,878,073 | <p><strong>Problem:</strong> $\{a_n\}_{n\in \mathbb N}, \quad a_{n+1}=\sqrt{2+a_n}, \quad \forall n\geq 1, \quad a_1=\sqrt{2}$</p>
<p><strong>Solution:</strong></p>
<p>We assume $a_n$ converges. Then is $\lim_{n\to\infty}a_n=\lim_{n\to\infty}a_{n+1}$</p>
<p>So we get</p>
<p>$$$a_n=\sqrt{2+a_n} \Rightarrow a_n^2-a_n... | user | 505,767 | <p>We need to refer to monotonic sequences theorem and show by induction that $a_n$ is strictly increasing and bounded above, then we can claim that the limit exists and therefore set</p>
<p>$$L=\sqrt{2+L}\implies L=2$$</p>
|
116,417 | <p>To be honest, I don't really know, whether or not the following is a research level
question:</p>
<p>Let $M$ be a smooth manifold, $C^\infty(M)$ the smooth function ring on $M$ and
suppose $R\subset C^\infty(M)$ is a subring. What are conditions, such that
$R$ is the smooth function ring of a smooth manifold ?</p>... | Nevermind | 21,302 | <p>Seems like I found something useful by myself:</p>
<p>In case someone else is interested, here a short article of Peter Michor on that topic </p>
<p><a href="https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0CE4QFjAC&url=http%3A%2F%2Fwww.mat.univie.ac.at%2F~michor%2F... |
1,540,756 | <p>I need help trying to solve this question, been cracking my head for the whole week and my professor said he used an online solver but in exams we have to solve by hand!</p>
<p>Given these 8 equations, we are supposed to solve for $i_0, i_1, \dots, i_7$:
$$\begin{array}{rl}
i_1+i_2 &= 12 \\
i_2+i_5+i_6 &= 0... | Bryan | 292,388 | <p>yeah you are right, it should be -2, I guess the book have some misprints. But it is too tedious for a 5 marks question in exam,
don't know if there is a faster way to do it. I did an online solver and it takes 28 steps! Aint no time for that!</p>
|
511,921 | <p>Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of <strong>similar</strong>: "We say that operators $S,T \in \mathcal{L}(V)$ are <strong>similar</strong> if there exists an isomorphism (in this context of linear maps, a bijecti... | Callus - Reinstate Monica | 94,624 | <p>Sure, there are a couple different ways to think about this. One pretty general way is like this. Say you have two objects $X$ and $Y$ and an isomorphism between them. That isomorphism translates anything you do on one object to the other. So if you turn, or stretch $X$, what happens to $Y$? </p>
<p>Think for exam... |
453,502 | <p>Let $I$ be a generalized rectangle in $\Bbb R^n$ </p>
<p>Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$</p>
<p>Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$. </p>
<hr>
<p>My idea is that</p>
<p>For $(\impliedby)$</p... | Michael Hardy | 11,667 | <p>This may be only a minor variation on an earlier answer, but maybe it adds something.</p>
<p>Suppose there's some point $x_0$ where $f(x_0)>0$. Let $\varepsilon=f(x_0)/2$. Then by continuity, there is some $\delta>0$ such that for $x$ in the open interval with endpoints $x_0\pm\delta$, the distance between ... |
1,419,105 | <p>I had a course in the construction of numbers last semester. I understand the potencial of most of the proofs, for example: I guess I can answer decently why commutativity is important.</p>
<p>But when it comes to the proof of uniqueness of $0,1$, I have no idea why that is important. For $\Bbb{N}$, I guess it's im... | Asaf Karagila | 622 | <p>You seem to think about the order, and ask about arithmetic. </p>
<p>What would $1'+0$ be? What would $3+0'$ be? If you have two chains, you need to handle "cross chain addition" or multiplication. Of course that is doable, but then adding two additive neutral element would be $0=0+0'=0'$. So the chain is unique. <... |
1,077,119 | <p>I am starting to learn about tensor products of abelian groups.</p>
<p>Why is the tensor product defined for <strong>abelian</strong> groups? In which part of the construction the commutativity of the groups is needed?</p>
| Ronnie Brown | 28,586 | <p>@Laters: Just to add to the answer of laters, the following should explain the idea of the nonabelian tensor product. More details are in the Brown-Loday paper linked in that answer. </p>
<p>Let $M,N$ be normal subgroups of the group $P$. Consider the commutator map </p>
<p>$$c=[\, ,\, ]: M \times N \to P, (m.n) \... |
2,351,629 | <p>First of all I don't understand why we need Banach's theorem, as a result I can't make it intuitive for me to understand how it works but I tried to solve an example.</p>
<p><a href="https://en.wikipedia.org/wiki/Banach_fixed-point_theorem#Statement" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Banach_fi... | hardmath | 3,111 | <p>Many authors use $\mathbf{J_n}$ to represent the $n\times n$ matrix of all ones entries. Since the size $n\times n$ is unchanged throughout this problem, we will omit the subscript:</p>
<p>$$ \mathbf J = \begin{pmatrix} 1 & 1 & \cdots & 1 \\
1 & 1 & \cdots & 1 \\
\vdots & \vdots & \... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | Celia Escalante | 98,846 | <p>If you master pre-algebra, then you can figure out almost any other branch of mathematics using the appropriate study material. Geometric formulas will be second nature to you. Trigonometry and Calculus are not required to graduate from every high school. If you are strong in Algebra, then your college placements sc... |
186,555 | <p>I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks ... | galib20 | 372,443 | <p>Algebra:
1. Algebra for Beginners.
2. Elementary Algebra.
3. Higher Algebra.
All three by by Hall and Knight.</p>
|
61,697 | <p>The hamiltonian flow box theorem, as stated in Abraham and Marsden's Foundations of Mechanics, says that:</p>
<p>Given an hamiltonian system $(M,\omega,h)$ with $dh(x_0)\neq 0$ for some $x_0$ in $M$, there is a symplectic chart $(U,\phi)$ on $M$ centered at $x_0$ such that $\phi_{\ast}h(x)=h(x_0)+\omega_0(\phi_{\as... | Michael Bächtold | 745 | <p>My guess would be that you can find such a proof in the literature, since the Moser trick is such a powerful tool, though I don't know where. </p>
<p>Instead let me sketch a proof of the fact that any two Hamiltonian systems $(M_i,\omega_i,h_i)$ are locally isomorphic around non deg. points $x_i\in M_i, i=0,1$ usin... |
326,501 | <p>Arrange the numbers $1,2,...,9$ in such an order that no four of them appear (adjacently or otherwise) in ascending or descending order.
Show that there is no arrangement of the numbers $1,2,...,10$ with this property.</p>
<p>Now suppose $n>1$, and find the maximum $k$ such that the numbers $1,2,...,k$ can be a... | Ashot | 51,491 | <p>For the first question consider this sequence $5\ 3\ 7\ 1\ 9\ 4\ 2\ 8\ 6$. None $4$ of them are increasing or decreasing.</p>
|
374,194 | <p>Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be <strong>recurrent</strong> iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, there exists an increasing sequence $(n_k)$ of natural numbers with $n_k \rightarrow \infty$ such that $... | Brian M. Scott | 12,042 | <p><span class="math-container">$\newcommand{\cl}{\operatorname{cl}}\newcommand{\orb}{\operatorname{orb}}\newcommand{\corb}{\overline{\operatorname{orb}}}$</span>For <span class="math-container">$x\in X$</span> let <span class="math-container">$\orb(x)=\{T^n(x):n\in\Bbb N\}$</span>, and let <span class="math-container"... |
2,265,819 | <p>I need to parameterize the intersection of $$4x^2 + y^2 + z^2 = 9\tag{1}$$ and $$z=x^2+y^2\tag{2}$$. </p>
<p>First, I'll solve (2) for $y^2$ and substitute the result into (1):</p>
<p>$$3x^2+z+z^2 = 9 \tag{3}$$</p>
<p>Next, I'll make the substitution $u=\sqrt{3}x$, such that we can complete the square in (3) by a... | rtybase | 22,583 | <p><strong>As you stated $f(z)$ can't be constant</strong>, otherwise, from $f(z) \in \mathbb{R} \Leftrightarrow z \in \mathbb{R}$, $f(z)$ must be a real constant which happens to be real constant for non real $z$ too.</p>
<p>Now, by contradiction, let's assume
$$f'(z)=0, \forall z \in \mathbb{R} \tag{1}$$
which also... |
35,539 | <p>I express multilinear functions in the following format. Is there any ready command to convert them to multilinear functions easily?</p>
<p><strong>Input</strong></p>
<pre><code>posTerms = {2, 4, 9, 13, 19};
negTerms = {6, 11, 26};
IntegerString[posTerms, 2]
IntegerString[negTerms, 2]
{"10", "100", "1001", "1101"... | ybeltukov | 4,678 | <p>How about</p>
<pre><code>FromCoefficientRules[#, Subscript[x, #] & /@ Reverse@Range@Length@#[[1, 1]]] &@
Thread[PadLeft@IntegerDigits[posTerms, 2] -> 1] -
FromCoefficientRules[#, Subscript[x, #] & /@ Reverse@Range@Length@#[[1, 1]]] &@
Thread[PadLeft@IntegerDigits[negTerms, 2] -> 1]
</... |
655,064 | <p>Let $G$ be a group and $a\in G$. Define the centralizer of $a$ to be </p>
<p>$\hspace{150pt} C(a)=\{g\in G : ga=ag\}$.</p>
<p>That is, $C(a)$ consists of all the elements that commute with $a$. Show that $C(a)$ is a subgroup of $G$. </p>
<hr>
<p>Clearly $C(a)$ is nonempty. Since $G$ is a group, $\exists e\in G$ ... | aPaulT | 77,929 | <p>Note that in your proof you have taken one of your elements to be $a$, the element we've taken the centralizer of. To show $C(a)$ is a subgroup, you would have to show that $gh^{-1}\in C(a)$ for <em>any</em> choice of $g$ and $h$ in $C(a)$, neither of which need be $a$ itself.</p>
<p>(Also you have not stated in yo... |
164,871 | <p>i want to choose optimal decision from following problem
Imagine having been bitten by an exotic, poisonous snake. Suppose the ER
physician estimates that the probability you will die is $1/3$ unless you receive
effective treatment immediately. At the moment, she can offer you a choice of
experimental antivenins fro... | Jack D'Aurizio | 44,121 | <p>The number of $k\in[0,p-1]$ such that $k$ and $k+1$ are both quadratic residues is equal to:
$$ \frac{1}{4}\sum_{k=0}^{p-1}\left(1+\left(\frac{k}{p}\right)\right)\left(1+\left(\frac{k+1}{p}\right)\right)+\frac{3+\left(\frac{-1}{p}\right)}{4}, $$
where the extra term is relative to the only $k=-1$ and $k=0$, in order... |
656,724 | <p>In Principia Mathematica's Introduction, there is a definition for "incomplete" symbol:</p>
<blockquote>
<p>By an "incomplete" symbol we mean a symbol which is not supposed to
have any meaning in isolation, but is only defined in certain
contexts. -- Chapter III, Principia Mathematica, 1st edition, page 69.<... | symplectomorphic | 23,611 | <p>Russell is developing his <a href="http://en.wikipedia.org/wiki/Theory_of_descriptions" rel="nofollow">theory of descriptions</a> here. Roughly, he takes an "incomplete symbol" to be one that does not refer -- one that does not have a denotation in the way that proper names do. So, for example, the expression "the a... |
656,724 | <p>In Principia Mathematica's Introduction, there is a definition for "incomplete" symbol:</p>
<blockquote>
<p>By an "incomplete" symbol we mean a symbol which is not supposed to
have any meaning in isolation, but is only defined in certain
contexts. -- Chapter III, Principia Mathematica, 1st edition, page 69.<... | Mauro ALLEGRANZA | 108,274 | <p>Following your own answer, recall that [page 66] :</p>
<blockquote>
<p>By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts.</p>
</blockquote>
<p>The issue is similar to that of <em>terms</em> [i.e.names] without denotation.</p... |
1,085,668 | <blockquote>
<p>Let $X$ and $Y$ be i.i.d. $\operatorname{Geom}(p)$, and $N = X + Y$. Find the joint PMF of $X, Y, N$.</p>
</blockquote>
<p>I have generally difficulties with such problems, as I get easily confused. Below I detailed my (most probability incorrect) approach. Besides the correct approach, I also would... | pragyy | 985,284 | <p>We have,
<span class="math-container">$$
P_{X,Y,N}(x,y,n) =
\begin{cases}
P(X=x).P(Y=y) & if x+y=n \\
0 & otherwise
\end{cases}
$$</span>
So,
<span class="math-container">\begin{align*}
P_{X,Y,N}(x,y,n) = P(X=x, Y=y, N=n)
= P(X=x).P(Y=y)
= pq^x.pq^y
... |
3,288,534 | <p><span class="math-container">$7^{-1} \bmod 120 = 103$</span></p>
<p>I would like to know how <span class="math-container">$7^{-1} \bmod 120$</span> results in <span class="math-container">$103$</span>.</p>
| P Vanchinathan | 28,915 | <p>Imagine a clock that has a large enough dial so that one can write numbers 0 to 119 (totally 120 numbers).</p>
<p>Now start at 0. Move clockwise jumping in steps of seven, so the first step will land in 7, 2nd step will land in 14, .... 10th step at 70, ... 15th step at 105, 18th step at 6 .....etc</p>
<p>Now you... |
1,958,152 | <blockquote>
<p>I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$. </p>
</blockquote>
<p>I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!</p>
| MPW | 113,214 | <p>$\require{cancel}$ It looks like you are thinking of doing this
$$\frac{\cancel{\log}4}{\cancel{\log}b}=\frac4b$$</p>
<p>But don't make the mistake of thinking of "$\log 4$" as "$(\log)\times(4)$". It isn't multiplication (which is what it would need to be if you were to cancel), but it's applying a function. In ot... |
1,835,295 | <blockquote>
<p>What is $\gcd(12345,54321)$?</p>
</blockquote>
<p>I noticed that after trying $\gcd(12,21),\gcd(123,321),$ and $\gcd(1234,4321)$ that they are all less then or equal to $3$. That leads me to question if there is an easy way to calculate such greatest common divisors.</p>
| Michael Hardy | 11,667 | <p>When $54321$ is divided by $12345$, the quotient is $4$ and the remainder is $4941$:
$$
54321 = (4\times12345) + 4941.
$$
Therefore (as Euclid taught us),
$$
\gcd(12345,54321) = \gcd(12345,4941).
$$
When $12345$ is divided by $4941$, the quotient is $2$ and the remainder is $2463$:
$$
12345 = (2\times4941) + 2463.
$... |
1,835,295 | <blockquote>
<p>What is $\gcd(12345,54321)$?</p>
</blockquote>
<p>I noticed that after trying $\gcd(12,21),\gcd(123,321),$ and $\gcd(1234,4321)$ that they are all less then or equal to $3$. That leads me to question if there is an easy way to calculate such greatest common divisors.</p>
| Erick Wong | 30,402 | <p>In order to generalize this cleanly to values greater than $9$ digits, we should probably think of $54321$ as $\sum_{k=1}^n k\cdot 10^{k-1}$ for $n=5$, so when $n=11$ we get $120987654321$ instead of, say, $110987654321$.</p>
<p>Similarly for $12345$ can be generalized to $\sum_{k=1}^{n} (n-k+1) \cdot 10^{k-1}$. W... |
272,752 | <p><strong>Bug introduced in 12.2.0 and persisting through 13.1</strong></p>
<hr />
<p>I have selected the following minimal example from the documentation. I am running v12.2.0 Win7-x64.</p>
<pre><code>GeoRegionValuePlot[
EntityClass["Country", "SouthAmerica"] -> "MerchantShips"]
</co... | Bob Hanlon | 9,362 | <p>Using <a href="https://reference.wolfram.com/language/ref/GeoScaleBar.html" rel="nofollow noreferrer"><code>GeoScaleBar</code></a> with <a href="https://reference.wolfram.com/language/ref/GeoRegionValuePlot.html" rel="nofollow noreferrer"><code>GeoRegionValuePlot</code></a> works with v12.0 or v12.1.1 but appears to... |
203,627 | <p>I initially asked this question on <a href="https://math.stackexchange.com/q/1219052/39599">MSE</a> but I haven't had any luck.</p>
<hr>
<p>The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous... | Ben McKay | 13,268 | <p>The result is not stated in Grauert's paper. On the other hand, Grauert proves that every real analytic manifold <span class="math-container">$M$</span> sits as a real analytic totally real submanifold, and analytic deformation retraction, in a Stein manifold <span class="math-container">$M_{\mathbb{C}}$</span>. So ... |
1,134,323 | <p>Say you pick a number $x$, like $\frac 43$. Its inverse is of course $\frac 34$. $x$ is a distance of $\frac 13$ away from 1, and its inverse is a distance of $\frac 14$ away from 1. Is there any number $x$ that is a distance $d$ away from 1, whose inverse $\frac 1x$ is also a distance $d$ away from 1? I came up... | marty cohen | 13,079 | <p>From
$|x-1|
= \left|1-\frac{1}{x}\right|.
$,
squaring both sides,
$x^2-2x+1
=1/x^2-2/x+1
$
or
$x^2-1/x^2-2x+2/x
=0
$
or
$2(x-1/x)
=x^2-1/x^2
=(x-1/x)(x+1/x)
$.</p>
<p>If
$x=1/x$,
then $x = \pm 1$.
These both satisfy
$|x-1|
= \left|1-\frac{1}{x}\right|
$.</p>
<p>If
$x\ne 1/x$,
then $x +1/x=2$.
THis implies that
$... |
1,948,163 | <p>I don't know how to differentiate this function. $$u=\left(\frac{1}{t}-\frac{1}{\sqrt{t}}\right)^{2}$$</p>
<p>Should I use the quotient rule or just the power chain rule?</p>
| Mark Viola | 218,419 | <p>Note that we can expand the term in parentheses to find</p>
<p>$$\begin{align}\frac{du}{dt}&=\frac{d}{dx}\left(\frac1{t^2}-\frac{2}{t^{3/2}}+\frac{1}{t}\right)\\\\
&=-\frac{2}{t^3}+\frac{3}{t^{5/2}}-\frac{1}{t^2}
\end{align}$$</p>
<p>Alternatively, we can apply the chain rule to find</p>
<p>$$\begin{align... |
725,746 | <p>I am not sure I understand the $N - \epsilon$ method for proving the equality of a limit.</p>
<p>I have a past mid-semester exam question that has:
$$\lim \limits_{x \to 1} (x^2 - 4x) = -3$$</p>
<p>Now it seems I want to take the $-3$ over $\rightarrow$ $|x^2 - 4x + 3| \lt \epsilon$ $$\text{ }\text{ }\text{ }\t... | Ishfaaq | 109,161 | <p>$N$ is generally used to represent a natural number and hence is used in proving limits of sequences. I would prefer to use $\delta$ here since this is the limit of a function. You are required to prove the fact that $\lim_{x \to a}f(x) = l$ The way to approach this is as follows. </p>
<p>You begin with $ |f(x) - l... |
3,600,065 | <p><strong>Problem</strong></p>
<p>Solve the system:</p>
<p><span class="math-container">$$
\begin{align}
Y+Z*\cos\left ( \frac{\pi}{4}\left ( 1982-t_0 \right ) \right )&=1.5 \\
Y+Z*\cos\left ( \frac{\pi}{4}\left ( 1984-t_0 \right ) \right )&=1 \\
Y+Z*\cos\left ( \frac{\pi}{4}\left ( 1985-t_0 \right ) \right ... | mathlove | 78,967 | <p>Since <span class="math-container">$$\frac{1982}{4}\pi=247\times 2\pi+\frac 32\pi$$</span>
<span class="math-container">$$\frac{1984}{4}\pi =248\times 2\pi$$</span>
<span class="math-container">$$\frac{1985}{4}\pi=248\times 2\pi+\frac{\pi}{4}$$</span></p>
<p>the system is equivalent to
<span class="math-container">... |
4,243,005 | <p>I was having trouble with this integral</p>
<blockquote>
<p>Prove
<span class="math-container">$$\int_{-\infty}^{\infty} e^{2x}x^2 e^{-e^{x}}dx=\gamma^2 -2\gamma+\zeta(2)$$</span></p>
</blockquote>
<p>Where <span class="math-container">$\gamma$</span> is the euler mascheroni constant. Let <span class="math-container... | Z Ahmed | 671,540 | <p>One typo: it should be <span class="math-container">$e^{-x}$</span> in your last three equations.
Yes go ahead, your integral is <span class="math-container">$I=\Gamma"(2)$</span>, which can be evaluated as
<span class="math-container">$\psi(z)\Gamma(z)=\Gamma'(z)$</span> d.w.r.t. <span class="math-container">$z$</s... |
3,525,017 | <p>a) Show that the generating function by length for binary strings where every block of 0s has length at least 2, each block of ones has length at least 3 is:</p>
<p><span class="math-container">$$\frac{(1-x+x^3)(1-x+x^2)}{1-2x+x^2-x^5}$$</span></p>
<p>b) Give a recurrence relation and enough initial conditions to ... | robjohn | 13,854 | <p><strong>Generating Function</strong></p>
<p>We can piece together the generating function as follows
<span class="math-container">$$
\overbrace{\vphantom{\left(\frac{x^2}1\right)}\ \ \frac1{1-x}\ \ }^{\substack{\text{any number}\\\text{of ones}}}\overbrace{\vphantom{\left(\frac{x^2}1\right)}\frac1{1-\underbrace{\qu... |
3,434,656 | <p>We are given a <span class="math-container">$3 \times 3$</span> real matrix <span class="math-container">$A$</span>, and we know it has three eigenvalues. One eigenvalue is <span class="math-container">$\lambda_1=-1$</span> with corresponding eigenvector <span class="math-container">$v_1=\left[\begin{matrix}
0 \... | Alexander Geldhof | 560,477 | <p><span class="math-container">$$A {v}_1 = (1 + i) {v}_1 \Rightarrow \overline{A} \overline{{v}_1} = \overline{(1 + i)} \overline{{v}_1} \iff A \overline{{v}_1} = (1 - i) \overline{{v}_1}.$$</span></p>
|
95,341 | <p>I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?).</p>
<p>Is it possible to extend the proof also to polynomials
$P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$ ?</p>
<p>Thanks!</p>
| Neil Strickland | 10,366 | <p>$SL(n,\mathbb{C})^N$ is a smooth and irreducible variety, and thus a manifold. The zero set $Z$ of a nonzero polynomial is a subvariety with $\dim_{\mathbb{C}}(Z)\leq Nn^2-1$ and so $\dim_{\mathbb{R}}(Z)\leq 2Nn^2-2$. If $a$ and $b$ are points in $Z^c$ then we can choose a smooth path $\mathbb{R}\to SL(n,\mathbb{C... |
1,838,596 | <p>I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$
where $0<r<R$.</p>
<p>I know I need to compute the metric tensor and the Gramian determinant etc, but in order to so, I need a regular global parametrization of $T^2$, I guess? How... | Hrhm | 332,390 | <p>Here's an easy derivation of the surface area of a torus (without using metric tensor or the Gramian determinant). </p>
<p>All we have to do is compute the following integral:</p>
<p>$$2\pi\int_{0}^{2\pi}\left(R+r\cos\theta\right)\cdot rd\theta$$</p>
<p>Which can be explained through this diagram:
<a href="https:... |
27,759 | <p>Suppose $N$ is an RSA modulus (ie, it's the product of two distinct primes), 256 bits long. What is the best method to factor it?</p>
<p>Trial division is out of the question, Pollard's Rho is probably out as well (without significant parallelization). I doubt there are any online tools or math libraries that can... | InterestedGuest | 3,731 | <p>One thing that jumps out at me is that this matrix has two eigenvalues. Find them: det
$
(\begin{bmatrix}
2-\lambda & 3\\
3 & 2-\lambda
\end{bmatrix})$=
$(2-\lambda)^{2}-9$=$-5-4\lambda+\lambda^{2}$. Factoring this you get $(\lambda+1)(\lambda-5)=0$, so your eigenvalues are -1 and 5. Now, find bases for the... |
4,039,655 | <p>I'm currently having trouble evaluating the following sum to get a formula in terms of <span class="math-container">$k$</span>:
<span class="math-container">$$\sum_{i=0}^{k-1} 2^i\cdot 4(k-i-1)$$</span></p>
<p>I know that <span class="math-container">$$\sum_{i=0}^n 2^i = 2^{n+1}-1$$</span>
but since my <span class="... | CHAMSI | 758,100 | <p>Let <span class="math-container">$ k\in\mathbb{N}^{*} $</span>, we have :<span class="math-container">\begin{aligned}\sum_{i=0}^{k-1}{\left(k-1-i\right)2^{i}}&=\sum_{i=0}^{k-1}{\left(k-1-i\right)\left(2^{i+1}-2^{i}\right)}\\ &=\sum_{i=0}^{k-1}{\left(k-1-i\right)2^{i+1}}-\sum_{i=0}^{k-1}{\left(k-1-i\right)2^{... |
2,488,218 | <p>I encountered this problem while practicing for a mathematics competition. </p>
<blockquote>
<p>A cube has a diagonal length of 10. What is the surface area of the cube? <strong>No Calculators Allowed.</strong></p>
</blockquote>
<p>(Emphasis mine)</p>
<p>I'm not even sure where to start with this, so I scribble... | Qi Zhu | 470,938 | <p>Well. Let $a$ be the side-length of a cube. Then the diagonal of a face is $a \sqrt 2$. Now, again, using Pythagoras, the diagonal of the cube is
$$ d= \sqrt{\left(a \sqrt2 \right)^2 + a^2} = a \sqrt3.$$</p>
|
196,155 | <p>I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqr... | Stan Tendijck | 526,717 | <p>One way of approaching this problem is by viewing it as a zero of an equation. Let me explain. Let's say you want to compute <span class="math-container">$\sqrt{x_0}$</span> where <span class="math-container">$x_0$</span> is a zero of some quadratic polynomial of the form <span class="math-container">$x^2-bx+1$</spa... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | John Gowers | 26,267 | <p>Two silly, brute-force, examples, which (I hope) give two fairly extensible ways of constructing an answer to this problem. $n$ refers throughout to the girl's number.</p>
<p><strong>The unsolved problem in mathematics approach:</strong> (E.g., @alex's answer)</p>
<hr>
<p>Is there a <a href="https://en.wikipedia... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Glen O | 67,842 | <p>"If two less than your number is the second derivative of a function at a turning point, is that point a local minimum?"</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Peter Olson | 7,994 | <p>Pick any bijective mapping from $\{1,2,3\}$ to $\{\text{Yes},\text{No},\text{I don't know}\}$ and then it is easy to contrive a question.</p>
<p>Here's an example question using this method.</p>
<blockquote>
<p>Let $f =\{(1,\text{Yes}),(2,\text{No}),(3,\text{I don't know})\}$ and let $x$ be the number that you a... |
910,414 | <p>My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that "two parallel lines intersect." If you type projective space into Wikipedia, or look it up in a textbook, a lot of t... | OuttaSpaceTime | 851,242 | <p>Have a look at this picture</p>
<p><a href="https://i.stack.imgur.com/xiSuE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xiSuE.png" alt="enter image description here" /></a></p>
<p>taken from <a href="https://en.wikipedia.org/wiki/Vanishing_point" rel="nofollow noreferrer">wikipedia</a></p>
<p>... |
763,381 | <p>There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the ... | Pedro | 23,350 | <p>Suppose $r=r_0$ is a root. Then $r_1=r_0^2+1$ if distinct, is another, and $r_2=r_1^2+1$, if distinct, is another, and so on. Hence, either two situations arise. Which are...?</p>
|
624,672 | <blockquote>
<p>Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove $H\rtimes_{\phi}K\simeq H\rtimes_{\psi}H$?</p>
</blockquote>
<p>My idea is to use the following</p>
<p>Theor... | Moishe Kohan | 84,907 | <p>I think, this is false. Take $H=Z^5$, $K$ cyclic of order 5 generated by $k$, $\phi(k)$ an element $c$ of order 5 and $\psi(k)=c^2$. I think, the corresponding semidirect products are not isomorphic since $c$ is not conjugate to $c^2$ in the group $GL(5,Z)$. </p>
|
624,672 | <blockquote>
<p>Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove $H\rtimes_{\phi}K\simeq H\rtimes_{\psi}H$?</p>
</blockquote>
<p>My idea is to use the following</p>
<p>Theor... | Andreas Caranti | 58,401 | <p>If $$\zeta \varphi(a)\zeta^{-1} = \psi(a)^t$$ I believe the map
$$
(h, a^i) \mapsto (h^{\zeta}, a^{it} )
$$
provides the required isomorphism from the semidirect product with respect to $\varphi$ to the one with respect to $\psi$. Here I am writing morphisms as exponents. </p>
<p>(Here I am only addressing the fir... |
4,584,609 | <p>Which is bigger</p>
<p><span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\sin x}{1+x^2}dx$$</span> or <span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\cos x}{1+x^2}dx~?$$</span></p>
<p>I let <span class="math-container">$x=\frac{\pi}{2}-t$</span> in the second integral, and I obtain this
<span ... | Martin R | 42,969 | <p><span class="math-container">$g(x) = 1/(1+x)^2$</span> is strictly decreasing on <span class="math-container">$[0, \pi/2]$</span>, therefore is
<span class="math-container">$$
\int_0^{\pi/2} (\cos(x)-\sin(x))g(x) \, dx = \int_0^{\pi/4} (\cos(x)-\sin(x))g(x) \, dx + \int_{\pi/4}^{\pi/2} (\cos(x)-\sin(x))g(x) \, dx... |
455,642 | <p>I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the $16$ elements a unique number id. Can you suggest me where to start ? </p>
| SomeOne | 87,286 | <p>Any one-to-one mapping $\mathbb{N}^2 \to \mathbb{N}$ will work.</p>
<p>This one is easy <a href="https://math.stackexchange.com/a/444454/26306"> $\mathbb{N}^2 \to \mathbb{N}$ bijection</a></p>
<p>Or this one is easier :-) If you have index $(i,j)$, your mapping will be $f(i,j)=2^i3^j$</p>
|
455,642 | <p>I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the $16$ elements a unique number id. Can you suggest me where to start ? </p>
| jkn | 37,377 | <p>You can simply "string out" the elements of the matrix. That is, you label the $(1,1)$ entry with $1$, the $(1,2)$ entry with $2$, $\dots$, the $(2,1)$ entry with $5$, $\dots$, the $(4,4)$ entry with $16$.</p>
<p>You can construct such bijection $f:\mathbb{N}\to\mathbb{N}^2$ using the <a href="https://en.wikipedia.... |
41,707 | <p>Is there a slick way to define a partial computable function $f$ so that $f(e) \in W_{e}$ whenever $W_{e} \neq \emptyset$? (Here $W_{e}$ denotes the $e^{\text{th}}$ c.e. set.) My only solution is to start by defining $g(e) = \mu s [W_{e,s} \neq \emptyset]$, where $W_{e,s}$ denotes the $s^{\text{th}}$ finite approxim... | ItsNotObvious | 9,450 | <p>Matrices are heavily used in mathematical finance in various ways. One specific example is a correlation matrix where an entry (i,j) specifies the degree to which price movements in instrument i and instrument j are correlated over a specified time period. A huge number of computer cycles is spent daily on computing... |
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | Fimpellizzeri | 173,410 | <p>Substitute $x$ for the appropriate values in your expression $$p(x)=ax^5+bx^4+cx^3+dx^2+ex+f$$and you will have a system of linear equations in $6$ variables ($a,b,c,d,e,f)$ that can be dealt with your favorite method, including linear algebra tech.</p>
<p>For instance, $p(2)=1$ yields $32a+16b+8c+4d+2e+f=1$.</p>
... |
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | lhf | 589 | <p>The <a href="http://en.wikipedia.org/wiki/Tarski-Seidenberg_theorem">Tarski–Seidenberg theorem</a> says that the set of semialgebraic sets is closed under projection. It's a pure real-algebraic statement that was originally proved with logic.</p>
<p>Jacobson says this in chapter 5 of his <em>Basic Algebra I</em>:</... |
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | Doug Spoonwood | 11,300 | <p>So automated theorem provers consist of an application of mathematical logic. Consequently, the solution of the <a href="http://en.wikipedia.org/wiki/Robbins_algebra">Robbins conjecture</a> by William McCune using EQP qualifies as solving an open problem outside of mathematical logic by using mathematical logic.</p... |
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | Dan Piponi | 80 | <p>The <a href="http://en.wikipedia.org/wiki/Compactness_theorem" rel="nofollow noreferrer">compactness theorem</a> gives many ways to generate theorems of the form "if all finite $X$s have property $P$, then so do all infinite $X$s".</p>
<p>A random example from Bell and Slomson <em>Models and Ultraproducts</em>:</p>... |
283,119 | <p>Let $X$ be a Riemannian $n$-manifold with tubular end $\mathbb R^+\times Y$, where $Y$ is a closed $n-1$-manifold. Suppose $L:L^{p,w}_2(X)\to L^{p,w}(X)$ is the Laplacian operator which is translation invariant on the cylindrical end; here $L^{p,w}_i$ is a weighted $L^{p}_i$ space, i.e. $w$ is a function on $X$ suc... | Olivier | 2,284 | <blockquote>
<p>What I don't understand is why the exactly the same terms should appear in both sums.</p>
</blockquote>
<p>The Galois action on CM points is described in adelic terms via the fundamental theorem of complex multiplication (or Shimura's explicit reciprocity law) so identifying the terms appearing in th... |
672,265 | <p>Find the normal form of the matrix $A$: $$\begin{bmatrix}2 & 0 & 0 & 0\\1 & 2 & 0 & 0\\0 & 0 & 1 & 1\\0 & 0 & 0 & 1\end{bmatrix}$$</p>
<p>It looks like A's jordan decomposition should be: </p>
<p>$$\begin{bmatrix}2 & 1 & 0 & 0\\0 & 2 & 0 & 0\\... | Martin Sleziak | 8,297 | <p>The matrix $P=\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}$ is the <a href="http://en.wikipedia.org/wiki/Permutation_matrix" rel="nofollow">permutation matrix</a> corresponding to interchanging 1 and 2. It is c... |
339,289 | <blockquote>
<p>Prove that if $f$ is defined for $x\ge 0$ by $f(x)=\sqrt x$, then $f$
is continuous at every point of its domain.</p>
</blockquote>
<p>Definition of a continuous function is:</p>
<p>Let $A\subseteq\mathbb{R}$ and let $f:A\to\mathbb{R}$. Denote $c\in A$.</p>
<p>Then $f(x)$ is continuous at $c$ iff... | AXH | 29,068 | <p>The answer above is not an answer. Try to show that the square root function is continuous at the point $c=0$ for the "choices" of $\delta$ given above and see the bizarre world that you end up in. More frustratingly, the people giving the answers make bigger mistakes or have bigger confusions about continuity than ... |
1,571,478 | <p>Let $X, Y$ have the joint pdf</p>
<p>$f(x, y) = 2, \quad 0 < x < y < 1 \quad 0, \quad$ otherwise</p>
<p>Find $P(0 < X < 1/2$ | $y = 3/4)$</p>
<p>The solutions say</p>
<p>$\int_0^{1/2} f_{X|Y}(x | y = 3/4)dx$</p>
<p>I know that $f_{X|Y}(x | Y = y) = 2$, but how do I find $f_{X|Y}(x | y = 3/4)$?</p... | David G. Stork | 210,401 | <p>Consider the figure, which shows the region in which $f(x,y) = 2$ and the particular value $y = 3/4$. What is the chance that a point on the red line ($y = 3/4$) has value $x<1/2$?
<a href="https://i.stack.imgur.com/zbk9t.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zbk9t.png" alt="enter im... |
1,428,319 | <p>If we have a non-zero matrix $A\in\mathrm{Mat}_n(\mathbb{R})$ that is non-invertible. How do we prove that $A$ is both a left and right zero divisor?</p>
| Oussama Boussif | 258,472 | <p>$a$ doesn't divide $3$ so:</p>
<p>$$
a\equiv\pm 1[3]\\
a^2 \equiv 1[3]\\
a^2+23 \equiv 24\equiv 0[3]
$$</p>
<p>So $3$ divides $a^2+23$ and since you correctly showed that $8|a^2+23$ and knowing that $\gcd(8,3)=1$ So :</p>
<p>$$
24|a^2+23
$$</p>
<p>I used the property that:</p>
<p>$$
a|b\quad \mathrm{and}\quad b... |
533,812 | <p>A field is quadratically closed if each of its elements is a square.</p>
<p>The field <span class="math-container">$\mathbb{F}_2$</span> with two elements is obviously quadratically closed.</p>
<p>However, testing some more finite fields with this property, I didn't find any more. Hence my question is:</p>
<blockquo... | Keenan Kidwell | 628 | <p>Consider the squaring map from the multiplicative group of a finite field $F$ to itself. The kernel is $\{\pm1 \}$, i.e., it is trivial if and only if the characteristic of $F$ is $2$. Since this map is surjective if and only if it is injective, every element of $F$ is a square if and only if the characteristic of $... |
3,134,854 | <p>In a 2D environment, I have a circle with velocity <em>v</em>, a stationary point (infinite mass), and I am trying to calculate the velocity of the circle after a perfectly elastic collision with the point. </p>
<p>This is what I've came up with: </p>
<p><span class="math-container">$p$</span> is the position of t... | David K | 139,123 | <p>The arrow you have drawn for "new velocity" is actually the direction of the <em>impulse</em> that the point can give to the circle.
That is, it is the direction in which the point can "push" the circle.
The point does not have any ability to cancel out parts of the circle's incoming velocity that are not parallel t... |
72,794 | <p><img src="https://i.stack.imgur.com/jabWi.png" alt="enter image description here"></p>
<p>I am looking for something like this:</p>
<pre><code>SetOptions[EvaluationNotebook[], "PaperSize" -> "A3"]
</code></pre>
<p>The options are listed, but where do I find the arguments? Say for A3 paper, do I use <code>A3</c... | Mike Honeychurch | 77 | <p>According to <a href="http://www.prepressure.com/library/paper-size/din-a3" rel="nofollow">this</a> A3 is {842, 1190} printers points, so to set that and a variety of print related options programmatically:</p>
<pre><code>SetOptions[EvaluationNotebook[],
RulerUnits -> "Points",
PageHeaders -> {{
Ce... |
355,801 | <p>The Fubini's Theorem states that for any two $\sigma$-finite measure spaces $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\upsilon)$, there exists a unique measure $(\mu \otimes \upsilon)(A\times B)=\mu A \cdot \upsilon B$, $\forall A\in\mathcal{S},B\in\mathcal{T}$.
Further more, for any measurable function $f:S\times T... | Lord_Farin | 43,351 | <p>We have, just expanding the definitions:</p>
<p>$$\begin{align}
\Bbb E(\xi^p) &= \int \xi^p \, \mathrm dP = \int \left(p \int_0^\infty 1_{\{t < \xi\}}t^{p-1} \,\mathrm dt\right)\,\mathrm dP
\end{align}$$</p>
<p>Can you see what Fubini's Theorem has to do with this?</p>
|
2,296,968 | <p>So far I have this: Let P(n) be the statement "$n2^n \lt n!$". $k_{0}=6$. $(6)2^6 = 384 <720=6!$. $P(k_{0})$ is true. Let $n \geq 6$ and assume P(n) to be true. By the induction hypothesis, $(n+1)2^{n+1}=(n+1)(2)2^n ...$ Somehow this gets to be $n!(n+1)(2)2^n$. I am clearly missing a few steps in my proof. </p>
| DeepSea | 101,504 | <p>The key step is the last step: $(n+1)! = (n+1)n! > (n+1)(n\cdot 2^n)\ge(n+1)(2\cdot 2^n) = (n+1)2^{n+1}$. I hope this helps.</p>
|
97,579 | <p>Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?</p>
| Will Sawin | 18,060 | <p>Probably not unless $A$ and $B$ are positive-definite, since if $B$ is very close to $-A$ then $B^{-1}+A^{-1}$ is very small and so its inverse is very large. In fact, depending on the norm, they probably need to be close only on one shared or almost-shared eigenvector.</p>
<p>For spectral norm of positive-definite... |
3,463 | <p>I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to <em>itself</em>?" Call these maps endomaps. Permutations of elements of the set are a special case of endomaps: they are bijective ... | Qiaochu Yuan | 232 | <p>The generalization of cycle decomposition to endomaps is quite interesting; rather than just cycles, endomaps break up into cycles in which each vertex is the root of a tree. Counting endomaps (of which there are $n^n$) is therefore relevant to counting trees, and this is the basis of a beautiful proof due to Joyal... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.