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 |
|---|---|---|---|---|
4,262,888 | <p>My task is to prove that if an atomic measure space is <span class="math-container">$\sigma$</span>-finite, then the set of atoms must be countable.</p>
<p>This is my given definition of an atomic measure space:</p>
<blockquote>
<p>Assume <span class="math-container">$(X,\mathcal{M},\mu)$</span> is a measure space w... | Michael Hardy | 11,667 | <p>Here I will take "countable" to mean finite or countably infinite.</p>
<p>For <span class="math-container">$n\in\{1,2,3,\ldots\},$</span> if the set <span class="math-container">$A_n = \{ x : \mu(\{x\}) \ge 1/n \}$</span> were not countable then the space would not be <span class="math-container">$\sigma$<... |
2,202,724 | <p><strong>Method 1:</strong></p>
<p><img src="https://i.stack.imgur.com/vRVgX.png" alt="Method 1 image hyperlink"></p>
<p><strong>Method 2:</strong></p>
<p><img src="https://i.stack.imgur.com/pwww8.png" alt="Method 2 image hyperlink"></p>
<p>In these two images, you will see that I have integrated $\sin^3 x$ using... | David K | 139,123 | <p>Your teacher was referring to a true fact that is worthwhile for you to know,
although as it turns out, it does <em>not</em> apply to your two calculations.</p>
<p>To understand what your teacher is talking about, we need to remember
that $\int f(x)\,dx$ does not describe a single function, but rather a
family of f... |
2,202,724 | <p><strong>Method 1:</strong></p>
<p><img src="https://i.stack.imgur.com/vRVgX.png" alt="Method 1 image hyperlink"></p>
<p><strong>Method 2:</strong></p>
<p><img src="https://i.stack.imgur.com/pwww8.png" alt="Method 2 image hyperlink"></p>
<p>In these two images, you will see that I have integrated $\sin^3 x$ using... | user428838 | 428,838 | <p>Integration is just opposite of differentiation so whenever we integrate without limits we add a constant C which woud vanish on differentiating it.
You can always write a function f(x) as (f(x) +0) and on integrating it, the integration of 0 is a constant ( differentiation of constant is zero) so we write a consta... |
4,422,824 | <p><strong>Edit: This question involves derivatives, please read my prior work!</strong></p>
<p>This question has me stumped.</p>
<blockquote>
<p>A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration th... | Doug M | 317,176 | <p><span class="math-container">$a = \frac {dv}{dt}$</span> and <span class="math-container">$v = \frac {dx}{dt}$</span></p>
<p>By the chain rule <span class="math-container">$a = \frac {dv}{dx}\frac {dx}{dt} = v\frac {dv}{dx}$</span></p>
<p><span class="math-container">$\int a\ dx = \int v\ dv\\
ax = \frac 12 v^2\\
a ... |
180,296 | <p>I need an algorithm to decide quickly in the worst case if a 20 digit integer is prime or composite.</p>
<p>I do not need the factors.</p>
<p>Is the fastest way still a prime factorization algorithm? Or is there a faster way given the above relaxation?</p>
<p>In any case which algorithm gives the best worst case... | Radu Titiu | 37,056 | <p>If $\overline{a}\cdot \overline{b}= \overline{1}$ then there is $k \in \mathbb{Z}$ such that $ab+kn=1$, therefore $\gcd(a,n)=1$. </p>
<p>Conversely, let $a \in \mathbb{Z}$ such that $\gcd(a,n)=1$.Using Euclid's algorithm one can find $b,k \in \mathbb{Z}$ satisfying $ab+kn=1$, so $\overline{a}\cdot \overline{b}=\ove... |
180,296 | <p>I need an algorithm to decide quickly in the worst case if a 20 digit integer is prime or composite.</p>
<p>I do not need the factors.</p>
<p>Is the fastest way still a prime factorization algorithm? Or is there a faster way given the above relaxation?</p>
<p>In any case which algorithm gives the best worst case... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Over $\,\Bbb Z\,$ (or any <a href="http://en.wikipedia.org/wiki/B%C3%A9zout_domain" rel="nofollow">$\rm\color{#C00}{Bezout}$ domain</a> $\rm\,Z)\,$ we have
$$\rm gcd(a,b) = 1\color{#C00}{\iff} \exists\, j,k\in Z\!:\ j\,a + k\,b = 1\iff \exists\, j\in Z\!:\ j\,a \equiv 1\,\ (mod\ b)$$ </p>
... |
164,002 | <p>When I am reading a mathematical textbook, I tend to skip most of the exercises.
Generally I don't like exercises, particularly artificial ones.
Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..</p>
<p>Sometimes I try to prove a theorem before reading the proof.
Sometimes I try... | Qiaochu Yuan | 232 | <p>Depends on the textbook, I suppose. Some textbooks introduce a lot of material in the exercises that isn't developed in the main text. </p>
|
164,002 | <p>When I am reading a mathematical textbook, I tend to skip most of the exercises.
Generally I don't like exercises, particularly artificial ones.
Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..</p>
<p>Sometimes I try to prove a theorem before reading the proof.
Sometimes I try... | Matt E | 221 | <p>If your goal is to become a research mathematician, then doing exercises is important. Of course, there will be the rare person who can skip exercises with no detriment to their development, but (and I speak from the experience of roughly twenty years of involvement in training for research mathematics) such people... |
164,002 | <p>When I am reading a mathematical textbook, I tend to skip most of the exercises.
Generally I don't like exercises, particularly artificial ones.
Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..</p>
<p>Sometimes I try to prove a theorem before reading the proof.
Sometimes I try... | Amitesh Datta | 10,467 | <p>I think that the most important point in mathematics is to think about the subject for long periods of time. If you think about mathematics, then you will often develop intuition which is very important. Of course, if you think about something for long periods of time, then your memory of the material is better as w... |
3,078,176 | <p>Given an equation in partial derivatives of the form <span class="math-container">$Af_x+Bf_y=\phi(x,y)$</span>, for example <span class="math-container">$$f_x-f_y=(x+y)^2$$</span> How do I know which change of coordinates is appropiate to solve the equation? In this example, the change of coordinates is <span class=... | lightxbulb | 463,794 | <p>Use <span class="math-container">$\cos^2\theta = 1 - \sin^2\theta$</span> and <span class="math-container">$\tan\theta = \frac{\sin\theta}{\cos\theta}=c$</span>. Then <span class="math-container">$\sin\theta = c\sqrt{1-\sin^2\theta}$</span>, then <span class="math-container">$\sin^2\theta = c^2-c^2\sin^2\theta$</spa... |
3,078,176 | <p>Given an equation in partial derivatives of the form <span class="math-container">$Af_x+Bf_y=\phi(x,y)$</span>, for example <span class="math-container">$$f_x-f_y=(x+y)^2$$</span> How do I know which change of coordinates is appropiate to solve the equation? In this example, the change of coordinates is <span class=... | B. Goddard | 362,009 | <p>Draw a right triangle that shows <span class="math-container">$\tan \theta = 7/24.$</span> There are infinitely many, but choosing the one with legs <span class="math-container">$7$</span> and <span class="math-container">$24$</span> is a swell choice. Now use Pythagorean Theorem to find the length of the hypoten... |
1,127,596 | <p>I am trying to show that the value of $\int^\infty_0$$\int^\infty_0$ sin($x^2$+$y^2$) dxdy is $\frac{\pi}{4}$ using Fresnel integrals. I'm having trouble splitting apart the integrand in order to actually be able to use the Fresnel integrals. Any help is appreciated. </p>
<p>Answer:
$\int^\infty_0$ $\int^\infty_0$... | Random Jack | 140,701 | <p>Let's recall the definition of the convergence of an improper multiple integral (of the first kind). Assume that a function $f \colon \mathbb{R}^m \to \mathbb{R}$ is continuous almost everywhere. Consider the following sequence of sets $\{E_n\}_{n = 1}^\infty$:</p>
<ol>
<li>Each $E_n$ is an open Jordan-measurable s... |
3,197,683 | <p>Here is the theorem that I need to prove</p>
<blockquote>
<p>For <span class="math-container">$K = \mathbb{Q}[\sqrt{D}]$</span> we have</p>
<p><span class="math-container">$$\begin{align}O_K = \begin{cases}
\mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\
\mathbb{Z}\left[\frac{1 + \sqrt{D}}{2}\... | lonza leggiera | 632,373 | <p>You can find a proof in many number theory textbooks. In Hardy and Wright's classic <em>Introduction to the Theory of Numbers</em>, for instance, its Theorem 238 on <a href="https://archive.org/details/Hardy_and_Wright_-_Introduction_to_the_Theory_of_Numbers/page/n221" rel="nofollow noreferrer">p.207</a>.</p>
|
1,375,958 | <p>I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or compositions of them, but I still haven't found an adequate one. Any ideas would be greatly appreciated.</p>
| 3SAT | 203,577 | <p>$$f(x)=\sin^2 (x)$$</p>
<p>$$f(0)=0$$</p>
<p>$$f'(x)=\cos(x)2\sin(x)=\sin(2x)$$</p>
<p>$$f'(0)=0$$</p>
|
939,725 | <p>Given that $a_0=2$ and $a_n = \frac{6}{a_{n-1}-1}$, find a closed form for $a_n$.</p>
<p>I tried listing out the first few values of $a_n: 2, 6, 6/5, 30, 6/29$, but no pattern came out. </p>
| Claude Leibovici | 82,404 | <p>I think that Semiclassical proposed a very nice solution rewriting $$a_n = \frac{6}{a_{n-1}-1}$$ $$\dfrac{1}{a_n+2}=\dfrac{1}{2}-\dfrac{3/2}{a_{n-1}+2}$$ So, let us define $$b_n=\dfrac{1}{a_n+2}$$ (with $b_0=\dfrac{1}{4}$); so the recurrence equation is simply $$b_n=\dfrac{1}{4}-\dfrac{3}{2}b_{n-1}$$ from which $$b... |
3,177,343 | <p>I have the following minimization problem in <span class="math-container">$x \in \mathbb{R}^n$</span></p>
<p><span class="math-container">$$\begin{array}{ll} \text{minimize} & \|x\|_2 - c^T x\\ \text{subject to} & Ax = b\end{array}$$</span></p>
<p>where <span class="math-container">$A \in \mathbb{R}^{m \t... | Reinhard Meier | 407,833 | <p>Using the derivatives of the Lagrange function <span class="math-container">$\mathcal{L}(x,\lambda) =\|x\|-c^Tx-\lambda^T(Ax-b),$</span> we get
<span class="math-container">$$
\frac{x}{\|x\|} - c - A^T \lambda = 0
$$</span>
Note that <span class="math-container">$\lambda\in\mathbb{R}^m.$</span> This results in <span... |
123,269 | <p>Consider the following differential equation:</p>
<p>$$\begin{align*}&\rho C_p\left(\frac{\partial T}{\partial t}\right)=k\left[\frac{\partial^2 T}{\partial x^2}\right]+\dot{q}\\
&\text{at }x=0,\;\frac{\partial T}{\partial x}=0\\
&\text{at }x=1,\frac{\partial T}{\partial x}=C_1(T(t,1)-C_2)\\
&\text{... | xzczd | 1,871 | <h1>Short Answer</h1>
<p>Set</p>
<pre><code>Method -> {"MethodOfLines",
"DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> 1}}
</code></pre>
<p>inside <code>NDSolve</code> will resolve the problem. It's not necessary to set <code>"ScaleFactor"</code> to <code>1</code>, it just needs to b... |
189,068 | <p>I am trying to derive a meaningful statistic from a survey where I have asked the person taking the survey to put objects in a certain order. The order the person puts the objects is compared to a correct order and I want to calculate the error.</p>
<p>For example:</p>
<p>Users order: 1, 3, 4, 5, 2</p>
<p>Corre... | Santosh Linkha | 2,199 | <p>At $\sin x + \cos x = {1 \over 2} \sin (2x)$ and $\sin (2x) $ over the interval $[0, {\pi \over 2}]$ looks like this,</p>
<p><img src="https://i.stack.imgur.com/Ga4zO.gif" alt="enter image description here"></p>
<p>The limits of $t$ are the values of $y$ axis, so you should split up the limits as $[0, {\pi \over 4... |
4,188,106 | <p>Let's say we have the following diagram
<span class="math-container">$$\require{AMScd}\begin{CD}
0 @>>> A @>>> B @>>> C @>>> 0\\
{} @V{\alpha}VV @V{\beta}VV @V{\gamma}VV {} \\
0 @>>> A' @>>> B' @>>> C' @>>> 0
\end{CD}$$</span>
where the top and... | Hagen von Eitzen | 39,174 | <p>Two much work.</p>
<hr />
<p>Simply observe that there are pairs of short exact sequences such as
<span class="math-container">$$ 0\to \Bbb Z_2\to\Bbb Z_4\to \Bbb Z_2\to 0$$</span>
and
<span class="math-container">$$ 0\to \Bbb Z_2\to\Bbb Z_2\oplus \Bbb Z_2\to \Bbb Z_2\to 0$$</span>
where the middle terms are not iso... |
3,902,418 | <p>Ok, so I know that the mean value of a function, <span class="math-container">$f(x)$</span>, on the interval <span class="math-container">$[a,b]$</span> is given by (or defined by?)
<span class="math-container">$$\frac{1}{b-a}\int_a^bf(x)~dx$$</span>
but I have <span class="math-container">$2$</span> basic questions... | Ivo Terek | 118,056 | <p>Let <span class="math-container">$f:[a,b] \to \Bbb R$</span> be integrable. Consider the equidistant partition of <span class="math-container">$[a,b]$</span> into <span class="math-container">$n$</span> subintervals: <span class="math-container">$$\mathcal{P}_n: \quad a < a + \frac{b-a}{n} < a + 2\frac{(b-a)}{... |
3,902,418 | <p>Ok, so I know that the mean value of a function, <span class="math-container">$f(x)$</span>, on the interval <span class="math-container">$[a,b]$</span> is given by (or defined by?)
<span class="math-container">$$\frac{1}{b-a}\int_a^bf(x)~dx$$</span>
but I have <span class="math-container">$2$</span> basic questions... | Stefan Lafon | 582,769 | <p>I'd like to complement the answers from @Lee Mosher and @Ivo Terek with a data compression angle. That's the kind of angle that helped me develop an intuition behind the formal concept.</p>
<p>Your function <span class="math-container">$f$</span> typically assumes several values on <span class="math-container">$[a,... |
3,372,832 | <blockquote>
<p><strong>9)</strong> Is
<span class="math-container">$$ \sum_{n=1}^\infty \delta_n \tag{7.10.1} $$</span>
a well-defined distribution? Note, to be a well-defined distribution, its action on any test function should be a finite number. Provide an example of a function <span class="math-container">$f... | Henno Brandsma | 4,280 | <p><span class="math-container">$\gamma[[0,1]] \subseteq D$</span> where <span class="math-container">$D$</span> is some closed disk. </p>
<p>It's easy to see (e.g. by path-connectedness, as you say) that <span class="math-container">$\Bbb C\setminus D$</span> is connected. And as <span class="math-container">$\gamma^... |
3,330,938 | <p>On Wikipedia page about Weierstrass factorization theorem one can find a sentence which mentions a generalized version so that it should work for meromorphic functions. I mean:</p>
<blockquote>
<p>We have sets of zeros and poles of function <span class="math-container">$f$</span>. How could we use that sets to f... | C. Brendel | 529,214 | <p>Given a meromorphic function <span class="math-container">$f$</span> with poles <span class="math-container">$(p_i)_{i\in I}$</span> and zeros <span class="math-container">$(z_i)_{i\in J}$</span> repeated according to multiplicity we have the corresponding weierstrass product for the poles <span class="math-contai... |
3,840,253 | <blockquote>
<p>How to show that <span class="math-container">$\csc x - \csc\left(\frac{\pi}{3} + x \right) + \csc\left(\frac{\pi}{3} - x\right) = 3 \csc 3x$</span>?</p>
</blockquote>
<p>My attempt:<br />
<span class="math-container">\begin{align}
LHS &= \csc x - \csc\left(\frac{\pi}{3} + x\right) + \csc\left(\frac... | player3236 | 435,724 | <p>Using the identities</p>
<p><span class="math-container">$$\sin A \sin B = \frac12 (\cos (A-B) - \cos (A+B))$$</span>
<span class="math-container">$$\sin A \cos B = \frac12 (\sin (A+B) + \sin (A-B))$$</span></p>
<p>we have
<span class="math-container">\begin{align}
&\phantom{=}\sin x \sin\left(\frac{\pi}{3} + x\... |
4,082,588 | <blockquote>
<p><strong>Definition:</strong> <span class="math-container">$\beta X$</span> is the Stone-Čech compactification of <span class="math-container">$X$</span>.</p>
</blockquote>
<blockquote>
<p><strong>Theorem A:</strong> If <span class="math-container">$K$</span> is a compact Hausdorff space and <span class=... | Alessandro Codenotti | 136,041 | <p>A different approach which doesn't give an explicit bijection is to use the characterization of <span class="math-container">$\beta\Bbb N$</span> as the space of ultrafilters on <span class="math-container">$\Bbb N$</span>, which are <span class="math-container">$2^{2^{|\Bbb N|}}$</span>, together with the fact that... |
4,082,588 | <blockquote>
<p><strong>Definition:</strong> <span class="math-container">$\beta X$</span> is the Stone-Čech compactification of <span class="math-container">$X$</span>.</p>
</blockquote>
<blockquote>
<p><strong>Theorem A:</strong> If <span class="math-container">$K$</span> is a compact Hausdorff space and <span class=... | Henno Brandsma | 4,280 | <p>If you want to be more precise:</p>
<p>Let <span class="math-container">$(e_1, \beta \Bbb N)$</span> be the Stone-Čech compactification of <span class="math-container">$\Bbb N$</span>, and <span class="math-container">$(e_2, \beta \Bbb Q)$</span> that of <span class="math-container">$\Bbb Q$</span>.</p>
<p>So indeed... |
3,566,469 | <p>I am confused by a discussion with a colleague. The discussion is about the period of a periodic function.</p>
<p>For example, the periodic function <span class="math-container">$$f(x)=\sin(x), \quad x\in (0,\infty)$$</span> has period <span class="math-container">$2\pi$</span>. If I change the scale and build the ... | mathcounterexamples.net | 187,663 | <p><span class="math-container">$g$</span> is not periodic as the difference between two consecutive roots is unbounded as we consider the roots going to <span class="math-container">$\infty$</span>.</p>
|
3,805,089 | <p>I have directional vectors <span class="math-container">$a, b, c, d$</span> in vector 2 space as seen in the images below. Unfortunately I don't have the sufficient vocabulary to explain this in more mathematical terms. In rough terms I need to check if vector <span class="math-container">$c$</span> and <span class=... | Mithrandir | 793,719 | <p>Find <span class="math-container">$\angle ba$</span>, <span class="math-container">$\angle ca$</span>, and <span class="math-container">$\angle $</span>da. If <span class="math-container">$|\angle ba|$</span> is greater than both <span class="math-container">$|\angle ca|$</span>| and <span class="math-container">$... |
2,972,085 | <p><a href="https://i.stack.imgur.com/pcOfx.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/pcOfx.jpg" alt="enter image description here"></a></p>
<p>My friend show me the diagram above , and ask me </p>
<p>"What is the area of a BLACK circle with radius of 1 of BLUE circle?"</p>
<p>So, I solved it by alge... | Federico | 180,428 | <p>I you perform a circular inversion w.r.t the black circle, the red circle becomes the red tangent line in the picture below, while the blue circle gets reflected to another circle, tangent to the black circle, the black line and the red line. The diameter of this new circle must be <span class="math-container">$1$</... |
964,372 | <p>I have a general question.</p>
<p>If there is a matrix which is inverse and I multiply it by other matrixs which are inverse.
Will the result already be reverse matrix?</p>
<p>My intonation says is correct, but I'm not sure how to prove it.</p>
<p>Any ideas? Thanks.</p>
| symmetricuser | 125,084 | <p>For such an arrangement for $n$ people, there are two cases: $n$ is by itself or $n$ is paired with someone.</p>
<p>For the first case, if you remove $n$, then it's just an arrangement for $n-1$ people, and so, there are $A_{n-1}$ arrangements where $n$ is by itself.</p>
<p>Moving on to the second case, suppose th... |
1,828,042 | <p>This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this.</p>
<p>I am a sophomore math major at a fairly good math department (top 20 in the U.S.), and after taking some upper-level math courses (second courses in abstract algebra and real an... | MathematicsStudent1122 | 238,417 | <p>The answers here are rather idealistic. They seem to be based more on trite cliches rather than concrete reasoning or evidence. </p>
<p>The fact is, academia is competitive and jobs are scarce. Your grades matter. Your performance relative to your peers matters. Being passionate at math or being interested in the ... |
2,886,460 | <blockquote>
<p>Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
$$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.$$</p>
</blockquote>
<p>I have tried combining the first and third terms & first and la... | Anas c | 832,806 | <p><span class="math-container">$\frac{w}{1+w^2 }+\frac{w^3 }{1+w} +\frac{w^2 }{1 +w^4}+\frac{w^4}{1+w^3 }=\frac{w+w^2 +w^3 +w^5}{(1+w)(1+w^2) }+\frac{w^2 +w^5+w^4+w^8}{(1+w^4)(1+w^3) }=\frac{w+w^2 +w^3 +w^5}{w+w^2 +w^3 +1}+\frac{w^2 +1+w^4+w^3 }{1+w^3 +w^4+w^2} =2$</span>
Because <span class="math-container">$w^5 =1\R... |
3,059,857 | <p>We are supposed to use this formula for which I can't find any explaination anywhere and our teacher didn't explain anything so if anyone could help me I would appreciate it. </p>
<p><span class="math-container">$ x = A + k \times 2\pi$</span></p>
<p>and</p>
<p><span class="math-container">$x = \pi - A + k \times... | Vasili | 469,083 | <p>When you have equation <span class="math-container">$\sin x=a$</span>, the solutions are <span class="math-container">$x=\arcsin a +2\pi n$</span> and <span class="math-container">$x=\pi-\arcsin a + 2\pi n$</span>. This is based on the identity <span class="math-container">$\sin(\pi-a)=\sin a$</span>. In your case <... |
2,054,175 | <p>This problem is giving me loads of confusion. I just need someone to walk through it because I have the answer and I can't get to it to save my life. I have been on it for days. Please help.</p>
<p>$$\frac{x + 3}{x - 4}\le 0$$ </p>
| Dr. Sonnhard Graubner | 175,066 | <p>we have only two cases:
a) $$x\geq -3$$ and $$x<4$$
or
b) $$x\le -3$$ and $$x>4$$ and this is impossible.
Thus we have $$-3\le x<4$$</p>
|
1,212,000 | <p>I was trying to solve this square root problem, but I seem not to understand some basics. </p>
<p>Here is the problem.</p>
<p>$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2} - \sqrt[3]{\bigg(1 - \sqrt{2}\bigg)^3}\Bigg)^2$$</p>
<p>The solution is as follows:</p>
<p>$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\b... | Mankind | 207,432 | <p>Nicely put question.</p>
<p>You are right about the absolute value missing somewhere. Indeed, we have:</p>
<p>$$\sqrt{x^2} = |x|.$$</p>
<p>In your case, we have</p>
<p>$$\sqrt{\left(\sqrt{2}-\frac{3}{2}\right)^2}=\left|\sqrt{2}-\frac{3}{2}\right|.$$</p>
<p>But $\sqrt{2}-\frac{3}{2}$ is negative, so the absolute... |
3,340,686 | <p>The <span class="math-container">$7$</span>th floor of a building is <span class="math-container">$23$</span>m above street level and <span class="math-container">$13$</span>th floor is <span class="math-container">$41$</span>m above street level. What is the height (above street level) of the first floor and what i... | NoChance | 15,180 | <p>Yes you are correct. The first floor's hight above street level is (in meters): <span class="math-container">$$a_1=5 $$</span></p>
<p>Also, with <span class="math-container">$d$</span> indicating d is the difference between terms of the arithmetic progression, <span class="math-container">$$d=3$$</span></p>
<p>You... |
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| Bill | 361,593 | <p>We can solve this problem by using our knowledge of limits for composite functions.</p>
<p>Let $u = \sin^2 x.$</p>
<p>$x \to 0 \implies \sin^2 x \to 0 \implies u \to 0$</p>
<p>$\frac{\ln (\cos x)}{\sin^2x} = \frac{\ln (\cos^2 x)}{2\sin^2x} = \frac{\ln (1 - \sin^2 x)}{2\sin^2x} = \frac{\ln (1 - u)}{2u} = 1/2 \ln(1... |
2,834,864 | <p>Is it safe to assume that if $a\equiv b \pmod {35 =5\times7}$</p>
<p>then $a\equiv b\pmod 5$ is also true?</p>
| Billy | 13,942 | <blockquote>
<p>A=hH=kK</p>
</blockquote>
<p>Correct - and, therefore, $H = h^{-1}kK$. But $H$ is a group, so must contain the identity element $e$. So, as $e\in h^{-1}kK$, it's easy to see that $K$ must contain the element $k^{-1}h$. But now $K$ is a group too, so it's closed under inverses, and so it contains $(k^... |
2,482,250 | <p>Consider the generating function $$\frac1{1-2tx+t^2}=\sum_{n=0}^{\infty}y_n(x)t^n$$. I wish to find a second order differential equation of the form $$p(x)y_n''(x)+q(x)y_n'(x)+\lambda_ny_n(x)=0$$ and a recurrence relation satisfied by $y_n(x)$ of the form $$a_ny_{n+1}(x)+b_ny_n(x)+c_ny_{n-1}(x)=xy_n(x)$$. How should... | vidyarthi | 349,094 | <p>After a little search, the generating function is just the one for Chebyshev polynomials of the second kind, $U_n(x)$. Thus from <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials" rel="nofollow noreferrer">Wikipedia</a>, the desired differential equation and recurrence relation are:</p>
<p>Recurrence rel... |
2,482,250 | <p>Consider the generating function $$\frac1{1-2tx+t^2}=\sum_{n=0}^{\infty}y_n(x)t^n$$. I wish to find a second order differential equation of the form $$p(x)y_n''(x)+q(x)y_n'(x)+\lambda_ny_n(x)=0$$ and a recurrence relation satisfied by $y_n(x)$ of the form $$a_ny_{n+1}(x)+b_ny_n(x)+c_ny_{n-1}(x)=xy_n(x)$$. How should... | Chappers | 221,811 | <p>The three-term relation is easy in this case: simply multiply both sides by $1-2xt+t^2$, and then we have
$$ 1 = \sum_{n=0}^{\infty} (1-2xt+t^2)y_n(x)t^n, $$
and equating powers of $t^{n+1}$ gives
$$ y_{n-1}(x) -2xy_n(x)+y_{n+1}(x) = 0 $$
for $n \geq 1$ (this falls apart at the bottom $n$, of course, as one would ex... |
960,010 | <p>Two sides of a triangle are 15cm and 20cm long respectively.
$A)$ How fast is the third side increasing if the angle between the given sidesis 60 degrees and is increasing at the rate of $2^\circ/sec$? $B)$ How fast is the area increasing?</p>
<p>$A)$ I used $c^2=a^2+b^2-2ab\cos(\theta)$ so I got the missing side ... | JimmyK4542 | 155,509 | <p>In part a), $\theta$ is the only variable mentioned as changing. </p>
<p>So $\dfrac{d\theta}{dt} = 2^{\circ}/\text{sec} = \dfrac{\pi}{90}\text{rad/sec}$ and $\dfrac{da}{dt} = \dfrac{db}{dt} = 0 \text{cm/sec}$. </p>
<p>Can you show what you plugged into the formula to get $c \approx 28.72$? I got $c = 5\sqrt{13} \a... |
2,725,019 | <blockquote>
<p>Define $S := \{x ∈ \mathbb Q : x^2 ≤ 2\}$. Prove that $a:=\inf \{S\}$
satisfies $a^2 = 2$.</p>
</blockquote>
<p>Since a is a lower bound for S, we have $a^2\le 2$. if $a^2\neq 2$ then $a^2 < 2$ and we may set $\epsilon:= a^2 − 2 > 0$ and then I am not sure how to show it satisfies $a^2 = 2$.<... | Jimmy R. | 128,037 | <p>Consider $a=-\sqrt{2}$ and show that $a=\inf\{S\}$. This is equivalent to showing that</p>
<ul>
<li>for every $x\in S$, it holds that $x\ge -\sqrt{2}$.</li>
<li>for every $\epsilon>0$, there is $x\in S$ such that $x<-\sqrt{2}+\epsilon$.</li>
</ul>
<p>The first one should be straightforward, for the second ta... |
2,725,019 | <blockquote>
<p>Define $S := \{x ∈ \mathbb Q : x^2 ≤ 2\}$. Prove that $a:=\inf \{S\}$
satisfies $a^2 = 2$.</p>
</blockquote>
<p>Since a is a lower bound for S, we have $a^2\le 2$. if $a^2\neq 2$ then $a^2 < 2$ and we may set $\epsilon:= a^2 − 2 > 0$ and then I am not sure how to show it satisfies $a^2 = 2$.<... | Piquito | 219,998 | <p>HINT.- $x^2\le2\iff-\sqrt2\le x\le \sqrt2$ and $(-\sqrt2)^2=(\sqrt2)^2=2.$</p>
|
2,491,448 | <p>We roll a die ten times. What's the probability of getting all the six different numbers of the dice?</p>
<p>If $A_i$ is the event of getting at least of the numbers from $i=1$ to $6$. What the problem is asking is $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ . So I guess I will get this probability if I ... | N. F. Taussig | 173,070 | <p>If there were no restrictions, there would be six possible outcomes for each of the ten throws, so there are $6^{10}$ possible sequences of throws. From these, we must exclude those in which fewer than six outcomes occur. </p>
<p>There are $\binom{6}{k}$ ways to exclude $k$ of the $6$ outcomes and $(6 - k)^{10}$ p... |
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | Grace Note | 14,141 | <p><a href="http://twitter.com/#!/stackmath" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hfof3.png" alt="Follow us on Twitter!"></a></p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | Ilmari Karonen | 9,602 | <p><a href="http://citeseer.ist.psu.edu/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Mll8O.png" alt="Scientific Literature Digital Library & Search Engine"></a></p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | Ilmari Karonen | 9,602 | <p><a href="http://planetmath.org/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GaFJ9.png" alt="PlanetMath - Math for the people, by the people."></a></p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | JDH | 413 | <p><a href="http://cantorsattic.info" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UAtF2.png" alt="Climb into Cantor's Attic, containing infinities large and small"></a></p>
|
1,670,074 | <p>We know that a necessary and sufficient condition for a path-connected, locally path-connected space to have a universal cover is that it is semi-locally simply connected.</p>
<p>Now since $\mathbb R^2\setminus\{0\}$ is such a space, it must have a universal cover. However I can't see what the universal cover of $\... | Jyrki Lahtonen | 11,619 | <p>Hint: Think of the mapping $z\mapsto e^z$ from $\Bbb{C}$ to $\Bbb{C}\setminus\{0\}$. Its derivative is also $e^z$, which is always non-zero. Therefore the mapping is conformal everywhere, i.e. a local homeomorphism.</p>
|
1,820,036 | <p>I'd be thankful if some could explain to me why the second equality is true...
I just can't figure it out. Maybe it's something really simple I am missing?</p>
<blockquote>
<p>$\displaystyle\lim_{\epsilon\to0}\frac{\det(Id+\epsilon H)-\det(Id)}{\epsilon}=\displaystyle\lim_{\epsilon\to0}\frac{1}{\epsilon}\left[\de... | Nicholas Stull | 28,997 | <p>As I suggested in my comment, we proceed by expanding
$$|Id + \varepsilon H| = |A| = \left| \begin{array}{cccc}
1+\varepsilon h_{11} & \varepsilon h_{12} & \cdots & \varepsilon h_{1n}\\
\varepsilon h_{21} & 1+\varepsilon h_{22} & \cdots & \ \\
\vdots & \ & \ddots & \ &\\
\vare... |
1,820,036 | <p>I'd be thankful if some could explain to me why the second equality is true...
I just can't figure it out. Maybe it's something really simple I am missing?</p>
<blockquote>
<p>$\displaystyle\lim_{\epsilon\to0}\frac{\det(Id+\epsilon H)-\det(Id)}{\epsilon}=\displaystyle\lim_{\epsilon\to0}\frac{1}{\epsilon}\left[\de... | Sheldon Axler | 256,061 | <p>Here is a conceptual proof that avoids expanding a complicated determinant:</p>
<p>The determinant of a linear operator (or of a square matrix) is the product of the eigenvalues, counting multiplicity. The trace of a linear operator (or of a square matrix) is the sum of the eigenvalues, counting multiplicity.</p>
... |
747,561 | <p>I'm having trouble figuring out the limits. What messes me up is that the limit approaches infinity. Usually it approaches a specific number. Is that a trick to solve problems like these? </p>
<p>So for example, use the root test to find convergence/divergence. (n!)^n/(n^n)^7. n=1 and it's to infinity </p>
| JEET TRIVEDI | 115,676 | <p>$$\sum_{n=1}^{\infty}=\dfrac{(n!)^n}{(n^n)^7}$$
so,we let $a_n=\dfrac{(n!)^n}{(n^n)^7}.$</p>
<p>To use the root test,we take
$$L=\lim_{n\rightarrow \infty}\sqrt[n]{\left|a_n\right|}$$
$$L=\lim_{n\rightarrow \infty}\sqrt[n]{\dfrac{(n!)^n}{(n^n)^7}}$$
$$L=\lim_{n\rightarrow \infty}\dfrac{(n!)}{(n)^7}$$
And as n! inc... |
46,236 | <p>Apologies for the uninformative title, this is a relatively specific question so it was hard to title. </p>
<p>I'm solving the following recurrence relation:</p>
<blockquote>
<p>$a_{n} + a_{n-1} - 6a_{n-2} = 0$<br>
With initial conditions $a_{0} = 3$
and $a_{1} = 1$</p>
</blockquote>
<p>And I have it mostly... | Shai Covo | 2,810 | <p>$(2)2^n$ is equal to $2^{n+1}$, not to $4^n$.</p>
|
3,768,086 | <p>Show that <span class="math-container">$(X_n)_n$</span> converges in probability to <span class="math-container">$X$</span> if and only if for every continuous function <span class="math-container">$f$</span> with compact support, <span class="math-container">$f(X_n)$</span> converges in probability to <span class="... | IanFromWashington | 635,153 | <p>Thanks to the comment of @JMoravitz I realized my mistake. I was interpreting turns as the rolls <span class="math-container">$A$</span> AND <span class="math-container">$B$</span>, as in <span class="math-container">$\{A_1,B_1\}, \{A_2,B_2\}, \dots$</span>. In reality the question is merely asking what the probabil... |
1,428,377 | <p>So I was watching the show Numb3rs, and the math genius was teaching, and something he did just stumped me.</p>
<p>He was asking his class (more specifically a student) on which of the three cards is the car. The other two cards have an animal on them. Now, the student picked the middle card to begin with. So the c... | Marconius | 232,988 | <p>This is the classic version of the Monty Hall problem.</p>
<p>Note that there is only one car, so the host is always able to reveal a goat behind one of the other two doors that were unselected. So the fact that he does this reveals no new information if the contestant does not switch.</p>
<p>So by not switching:<... |
371,318 | <p>The original problem was to consider how many ways to make a wiring diagram out of $n$ resistors. When I thought about this I realized that if you can only connect in series and shunt. - Then this is the same as dividing an area with $n-1$ horizontal and vertical lines. When each line only divides one of the current... | OctarineBean | 111,531 | <p>This is a question for surreal numbers. Surreal numbers are a really amazing thing invented by John Conway that include numbers like 0 and 3/4, but also things like "twice the square root of infinity, all plus an infinitesimal". This question depends on the values of the infinite and infinitesimal, but the way it wo... |
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| Glen O | 67,842 | <p>I figured I'd go for a more "proper" proof. Notice that, if we let $a_n = n^n/n!$ we can write that</p>
<p>$$
a_{n+1}-a_n = \frac{(n+1)^{n+1}}{(n+1)!}-\frac{n^n}{n!} = \frac{(n+1)^{n+1}-(n+1)n^n}{(n+1)!}
$$
which can then be written as
$$
\frac{(n+1)^n-n^n}{n!}
$$</p>
<p>Now, using the binomial theorem, the first ... |
237,197 | <p>I'm new here. If there is anything not appropriate pls let me know.</p>
<p>I am currently working on a differential equation with one of which term is a integral of the variable.</p>
<p><span class="math-container">$$
\frac{d^2u(x)}{dx^2}=cosh(G(x))+\frac{1}{C_1}\int_{0}^{1}{u(x)sinh(G(x))dx }+C_2
$$</span>
with th... | Akku14 | 34,287 | <p>You can get an analytical solution for free c1,c2,c3.</p>
<pre><code>G[x_] = 2 Log[(1 + c3 Exp[-x])/(1 - c3 Exp[-x])]
</code></pre>
<p>Since the integral is a number, name it nint and solve for it later. Get anylytical solution for this reduced equation. Do indefinite integration. (I don't show the lengthy intermedi... |
65,631 | <pre><code>Ticker[comp_String] :=
Interpreter["Company"][comp] /. Entity[_, x_] :> x
ticks = Ticker /@ {"Apple", "Google"}
</code></pre>
<blockquote>
<p>{"NASDAQ:AAPL", "NASDAQ:GOOGL"}</p>
</blockquote>
<pre><code>DateListPlot[{
FinancialData[ticks[[1]], "CumulativeFractionalChange", {2010}],
FinancialDa... | Kuba | 5,478 | <p>You may never know how many wrappers there are but those functions have this in common that first argument is what we only care about.</p>
<pre><code>f[x_?NumericQ] := N @ x;
f[x_] := f @ First[x]
</code></pre>
|
794,912 | <p>I am reviewing Calculus III using <a href="http://www.jiblm.org/downloads/dlitem.aspx?id=82&category=jiblmjournal" rel="nofollow">Mahavier, W. Ted's material</a> and get stuck on one question in chapter 1. Here is the problem:</p>
<p>Assume $\vec{u},\vec{v}\in \mathbb{R}^3$. Find a vector $\vec{x}=(x,y,z)$ so t... | guest196883 | 43,798 | <p>Do you remember the definition of the cross product? Given vectors $u = (u_1,u_2, u_3)$ and $v = (v_1,v_2,v_3)$ define $u\times v$ as the unique vector such that</p>
<p>$$(u\times v)\cdot a = \left| \matrix{a_1&a_2&a_3\\u_1&u_2&u_3\\v_1&v_2&v_3}\right|$$</p>
<p>where $a=(a_1,a_2,a_3)$, the ... |
1,672,080 | <p>I have troubles understanding the concepts of quotient topology and product topology (in the infinite case). </p>
<p>I know that we want to give a topology to new spaces built from the old ones, but the thing is that I can't figure out why is the definition for quotient topology natural since we only require that t... | Stahl | 62,500 | <p>Both of these can be understood via <a href="https://en.wikipedia.org/wiki/Universal_property" rel="nofollow">universal properties</a>. Let's look at the product first.</p>
<p>If you want to form the product of two sets $S$ and $T$, what do you do? You form the set $S\times T = \{(s,t)\mid s\in S, t\in T\}$. Howeve... |
796,199 | <p>As far as I know, Brent's method for root finding is said to have superlinear convergence, but I haven't been able to find any more concrete information.</p>
<p>Is its convergence rate known to be at least bounded between some known values?</p>
<p>What is a good bibliographic reference for that?</p>
<p>[EDIT]</p>... | hardmath | 3,111 | <p>Brent proposed his method as combining bisection steps, with guaranteed linear convergence, with <a href="http://en.wikipedia.org/wiki/Inverse_quadratic_interpolation#Behaviour" rel="noreferrer">inverse quadratic interpolation</a>, whose order of convergence is the positive root of:</p>
<p>$$ \mu^3 - \mu^2 - \mu - ... |
2,027,044 | <p>Prove:
$$
(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}, \qquad \forall n \in \mathbb{N}
$$
I have have tried using the triangle inequlity $ |a + b| \le |a| + |b| $, without any success.</p>
| Dominik | 259,493 | <p>I assume that $a$ and $b$ are supposed to be positive? Then this inequaltiy is equivalent to $a + b \le (a^{1/n} + b^{1/n})^n$, which follows immediately from expanding the term on the right-hand side with the binomial theorem.</p>
|
917,276 | <p>If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference?</p>
<p>I will present my answer here. I am hoping to know if I am right or wrong.</p>
<p>Using the method of moment generating functions, we have</p>
<p>\begin{align*}
M_{U-V}(t)&=E\left[e^{t... | Qaswed | 333,427 | <p>In addition to the solution by the OP using the moment generating function, I'll provide a (nearly trivial) solution when <a href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables" rel="noreferrer">the rules about the sum</a> and <a href="https://en.wikipedia.org/wiki/Normal_distribution#Op... |
348,748 | <p>Find the solution for $Ax=0$ for the following $3 \times 3$ matrix:</p>
<p>$$\begin{pmatrix}3 & 2& -3\\ 2& -1&1 \\ 1& 1& 1\end{pmatrix}$$</p>
<p>I found the row reduced form of that matrix, which was </p>
<p>$$\begin{pmatrix}1 & 2/3& -1\\ 0& 1&-9/7 \\ 0& 0& 1\end{pm... | Community | -1 | <p>First note that any linear system of the form $\mathbf{Ax} = \mathbf{0}$ has either one solution (which is $\mathbf{x} = \mathbf{0}$) or infinite solutions. In your case, once you reduce $\mathbf{A}$ to row-echelon form, none of the entries on the leading diagonal are zero. This means your matrix is invertible.</p>
... |
24,055 | <p>Running this code:</p>
<pre><code>Histogram[{RandomVariate[NormalDistribution[1/4,0.12],100],
RandomVariate[NormalDistribution[3/4, 0.12], 100]},
Automatic, "Probability", PlotRange -> {{0, 1}, {0, 1}},
Frame -> True, PlotRangeClipping -> True,
FrameLabel -> {Style["x axis", 15], Style["probability... | Mr.Wizard | 121 | <p>This seems to be related to, or a manifestation of:<br>
<a href="https://mathematica.stackexchange.com/q/18988/121">Poor anti-aliasing in Rotated text with ClearType on</a></p>
<p>On my system Simon's workaround is successful.<br>
Using <code>Style["probability", 15, FontOpacity -> 0.999]</code>:</p>
<p><img sr... |
1,005,291 | <p>I understand that in order to prove this to be one to one, I need to prove $2$ numbers, $a$ and $b$, in the same set are equal. </p>
<p>This is what I did:</p>
<p>$$\sqrt{a} + a + 2 = \sqrt{b} + b + 2$$
$$\sqrt{a} + a = \sqrt{b} + b$$
$$a + a^2 = b + b^2$$</p>
<p>How would I arrive at $a = b$? Is it possible?</p>... | John | 105,625 | <p>$f(x)=\sqrt{x}+x+2$ is strictly increasing on $(0,\infty)$, so it's one-one (suppose not, use the strict monotonicity to draw a contradiction).</p>
<p>Or, from your second step,</p>
<p>$$\sqrt{a} + a = \sqrt{b} + b\iff \sqrt{a} - \sqrt{b}+ a -b=0 \iff(\sqrt{a} - \sqrt{b})(1+\sqrt{a}+\sqrt{b})=0 $$</p>
<p>Since $1... |
1,242,001 | <p>The following is the notation for Fermat's Last Theorem </p>
<p>$\neg\exists_{\{a,b,c,n\},(a,b,c,n)\in(\mathbb{Z}^+)\color{blue}{^4}\land n>2\land abc\neq 0}a^n+b^n=c^n$ </p>
<p>I understand everything in the notation besides the 4 highlighted in blue. Can someone explain to me what this means?</p>
| Kraxxus | 304,794 | <p>If you have a complicated function $g(x)$ involving a lot of products and exponentials ask <a href="https://en.wikipedia.org/wiki/John_Napier" rel="nofollow">John Napier</a> for help. He would tell you to make it even more complicated $f(x)=\log(g(x))$. Then by the chain rule
$f'(x)=\frac{g'(x)}{g(x)}$ or $g'(x)=g... |
654,617 | <p>$v$ being a vector.
I never understood what they mean and haven't found online resources. Just a quick question.</p>
<p>Thought it was absolute and magnitude respectively when regarding vectors. need confirmation</p>
| imranfat | 64,546 | <p>The double bar indicates the magnitude of the vector. In essence algebraically that is still the absolute value, meaning the square root of $x^2+y^2$ (in case of 2D) </p>
|
3,306,747 | <p>Here is my attempt </p>
<p>h = 3k -7 ----(1)</p>
<p>(h-1)^2 + (k -1)^2 = 10/4</p>
<p>(h-1)^2 + (3h - 8)^2 = 10/4</p>
<p>This second one doesn't working.Is my approch wrong?</p>
<p>P.S: Sorry for the typo.Also I assumed the center is C(h,k)</p>
| Ali Shadhar | 432,085 | <p>From <a href="https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Reihenentwicklungen#Potenzen_des_Arkussinus" rel="nofollow noreferrer">here</a>, we have </p>
<p><span class="math-container">$$\frac{\arcsin z}{\sqrt{1-z^2}}=\sum_{n=1}^\infty\frac{(2z)^{2n-1}}{n{2n \choose n}}$$</span></p>
<p>substitute <span... |
277,250 | <p>Let $\mathbb{N}$ be the set of natural numbers and $\beta \mathbb N$ denotes the Stone-Cech compactification of $\mathbb N$. </p>
<p>Is it then true that $\beta \mathbb N\cong \beta \mathbb N \times \beta \mathbb N $ ? </p>
| YCor | 14,094 | <p>The negative answer is equivalent to showing that there are two disjoint subsets $A,B$ of $\mathbf{N}^2$ with non-disjoint closures in $(\beta\mathbf{N})^2$. This be made explicit: take $A=\{(n,m):n=m\}$ and $B=\{(n,m):n>m\}$. Let $\omega$ be a non-principal ultrafilter on $\mathbf{N}$. Let $V$ be a neighborhood ... |
277,250 | <p>Let $\mathbb{N}$ be the set of natural numbers and $\beta \mathbb N$ denotes the Stone-Cech compactification of $\mathbb N$. </p>
<p>Is it then true that $\beta \mathbb N\cong \beta \mathbb N \times \beta \mathbb N $ ? </p>
| M.González | 39,421 | <p>An indirect argument: </p>
<p>Since the Banach space of continuous functions $C(\beta\mathbb{N})$ is isomorphic to $\ell_\infty$, it contains no complemented copies of $c_0$. </p>
<p>Since $C(\beta\mathbb{N}\times\beta\mathbb{N})$ is isomorphic to $C\big(\beta \mathbb{N},C(\beta\mathbb{N})\big)$,
it contains a c... |
2,050,724 | <p>In the proof of Jensen's inequality in a probabilistic setting, the book gives the following demonstration:
<br>
Expand the Taylor series of $f(x)$ around $\mu =\mathbb {E}[X]$.
$$f(x)=f(\mu)+f'(\mu)(x-\mu)+\frac {f''(\epsilon)(x-\mu)^2}{2}$$
For $\epsilon$ between $x$ and $\mu$. Since $f$ is convex, $f''(\epsilon)\... | Sam Blattner | 397,101 | <p>Notice that the last term is "for $\epsilon$ between $x$ and $\mu$." That is the error term, so it isn't an approximation. It is exact.</p>
<p>The error term is given by the mean value theorem. </p>
|
2,050,724 | <p>In the proof of Jensen's inequality in a probabilistic setting, the book gives the following demonstration:
<br>
Expand the Taylor series of $f(x)$ around $\mu =\mathbb {E}[X]$.
$$f(x)=f(\mu)+f'(\mu)(x-\mu)+\frac {f''(\epsilon)(x-\mu)^2}{2}$$
For $\epsilon$ between $x$ and $\mu$. Since $f$ is convex, $f''(\epsilon)\... | Fozz | 341,955 | <p>In general, if we have an <a href="https://en.wikipedia.org/wiki/Analytic_function" rel="nofollow noreferrer">analytic</a> function $f$, we can write out the Taylor series for $f$ around a point $x$ by
$$f(y)=\sum_{i=0}^\infty f^{(i)}(x)\frac{(y-x)^i}{i!}$$
and we'd have equality for all $y$ in some neighborhood of ... |
1,289,626 | <blockquote>
<p>find the Range of $f(x) = |x-6|+x^2-1$</p>
</blockquote>
<p>$$ f(x) = |x-6|+x^2-1 =\left\{
\begin{array}{c}
x^2+x-7,& x>0 .....(b) \\
5,& x=0 .....(a) \\
x^2-x+5,& x<0 ......(c)
\end{array}
\right.
$$</p>
<p>from eq (b) i got $$f(x)= \left(x+\frac12\right)^2-\frac{29}4 \ge-\fra... | Surb | 154,545 | <p>$$f'(x)=\frac{x-6}{|x-6|}+2x$$</p>
<p>$f'(x)=0\iff x=\frac{1}{2}$ and $f'(x)<0$ if $x<\frac{1}{2}$ and $f'(x)>0$ if $x>\frac{1}{2}$, therefore the range is $[f(\frac{1}{2}),+\infty [$.</p>
|
3,371,888 | <p><span class="math-container">$$\left(\!\!{{a+b}\choose k}\!\!\right)= \sum_{j=0}^k \left(\!\!{a\choose j}\!\!\right) \cdot \left(\!\!{b\choose {k-j}}\!\!\right)$$</span></p>
<p>I am quite confused about the case of multichoose. I was able to prove this equation if only "n choose k" form was used as both sides would... | Certainly not a dog | 691,550 | <p>Consider the ways to choose any <span class="math-container">$k$</span> objects from two piles (of size <span class="math-container">$a$</span> and <span class="math-container">$b$</span>).</p>
<p>One way is to simply combine the piles and choose them (the ways to do this is <span class="math-container">$\binom{a+b... |
822,711 | <p>In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of Non-Riemannian Geometry that need not be the case, so the question is how do you actually construct such connections... | PascExchange | 311,814 | <p>A connection is an abstract non-unique $\mathbb{R}$-linear map on a smooth manifold $M$ with tangent bundle $TM$ and smooth vector fields $\Gamma(TM)$ given by
$$\nabla:\begin{cases}\Gamma(TM)\times \Gamma(TM)\to \Gamma(TM)\\
(X,Y)\mapsto \nabla_XY\end{cases}$$
satisfying the following properties</p>
<ol>
<li>$\nab... |
4,298,951 | <p>Let us define a sequence <span class="math-container">$(a_n)$</span> as follows:</p>
<p><span class="math-container">$$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$</span></p>
<p>Prove that the sequence <span class="math-container">$(a_n)$</span> is Cauchy and find the limit.</p>
<hr />
<... | daㅤ | 799,923 | <p>Rewrite <span class="math-container">$a_n$</span> as <span class="math-container">$$a_1=1,\ a_2=2,\ a_{n+2}=\dfrac{3}{4}a_{n+1}+\dfrac{1}{4}a_n \mathrm{\ for\ } n\geqq 1.$$</span>
We can get
<span class="math-container">\begin{align}
&a_{n+2}-a_{n+1}=-\dfrac{1}{4}(a_{n+1}-a_{n}) \cdots (A)\\
&a_{n+2}+\dfrac{... |
355,888 | <p>Consider
$x''-2x'+x= te^t$</p>
<p>Determine the solution with initial values $x(1) = e,$ $x'(1) = 0.$</p>
<p>I know this looks like and probably is a very easy question, but i'm not getting the right answer when i try and solve putting into quadratic form. Could someone please demonstrate or show me a different m... | Ron Gordon | 53,268 | <p>This is actually a tricky problem because the right-hand side is a solution of the left-hand side set to zero (the homogeneous solution). </p>
<p>The homogeneous solution $x^{(H)}$ is </p>
<p>$$x^{(H)}(t) = A e^{t} + B t e^{t}$$</p>
<p>This is because the characteristic equation has $1$ as a double solution, so ... |
1,128,414 | <blockquote>
<p>Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$.</p>
</blockquote>
<p>I read this solution:</p>
<blockquote>
<p>Let $f\in C[0,2]$. Let $\epsilon>0$. Choose $\delta=\epsilon... | Brian Fitzpatrick | 56,960 | <p>Here $T:P_1\to P_1$ is defined by $T(a+b\,x)=6\,(a-b)+(12\,a-11\,b)\,x$ and $\beta$ is the basis $\{p,q\}$ for $P_1$ where $p(x)=3+4\,x$ and $q(x)=2+3\,x$.</p>
<p>Note that
$$
\begin{array}{rcrcr}
T(p) & = & \color{red}{-2}\,p & + & \color{blue}{0}\,q \\
T(q) & = & \color{green}{0}\,p &am... |
1,458,579 | <p>Suppose $f_n$ and $g_n$ be two sequences of functions. Also, $f_n.g_n$ converges to $f.g$ and $g_n$ converges to $g$. Can we prove $f_n$ converges to $f$? How?</p>
| André Nicolas | 6,312 | <p>Remark: Thanks to OP for correcting a serious error I made.</p>
<p>The probability the first collection starts with $1$ is $p$, and therefore the probability the second collection starts with $0$ is $p$, and the probability it starts with $1$ is $1-p$.</p>
<p><strong>Given</strong> the first bit in the second coll... |
197,441 | <p>I have a list,</p>
<pre><code>l1 = {{a, b, 3, c}, {e, f, 5, k}, {n, k, 12, m}, {s, t, 1, y}}
</code></pre>
<p>and want to apply differences on the third parts and keep the parts right of the numerals collected.</p>
<p>My result should be</p>
<pre><code>l2 = {{2, c, k}, {7, k, m}, {-11, m, y}}
</code></pre>
<p>I... | Roman | 26,598 | <p>This is very similar to kglr's first solution but picks the relevant quantities a bit more explicitly:</p>
<pre><code>l2 = BlockMap[{#[[2, 3]] - #[[1, 3]], #[[1, 4]], #[[2, 4]]} &, l1, 2, 1]
</code></pre>
<blockquote>
<p>{{2, c, k}, {7, k, m}, {-11, m, y}}</p>
</blockquote>
<p>With a parameter to change the... |
197,441 | <p>I have a list,</p>
<pre><code>l1 = {{a, b, 3, c}, {e, f, 5, k}, {n, k, 12, m}, {s, t, 1, y}}
</code></pre>
<p>and want to apply differences on the third parts and keep the parts right of the numerals collected.</p>
<p>My result should be</p>
<pre><code>l2 = {{2, c, k}, {7, k, m}, {-11, m, y}}
</code></pre>
<p>I... | Edmund | 19,542 | <p>A solution with <code>MapThread</code> on an offset <code>Partition</code>.</p>
<pre><code>MapThread[Sequence @@ #@#2 &, {{Differences, Identity}, Transpose@#}] & /@
Partition[l1[[All, 3 ;;]], 2, 1]
</code></pre>
<blockquote>
<pre><code>{{2, c, k}, {7, k, m}, {-11, m, y}}
</code></pre>
</blockquote>
<p>... |
1,223,823 | <p>How can one simplify
$$\arctan\left(\frac{1}{\tan \alpha}\right)?$$
$0<α<\dfrac{\pi}{2}.$ Here is what I tried so far,
$$\arctan\left(\dfrac{1}{\tan \alpha}\right)=θ$$ for some θ.
$$\frac{1}{\tan \alpha}=\tan(θ)$$</p>
<p>I didn't know what to do next because there is no significant relationship between ${θ}$... | Christian Blatter | 1,303 | <p>When $\alpha=n\pi$ then $\arctan{1\over\tan\alpha}$ is undefined. Therefore we may assume that
$$-{\pi\over2}<\beta:=\bigl(n+{1\over2}\bigr)\pi-\alpha<{\pi\over2}$$
for a certain $n\in{\mathbb Z}$. I claim that
$$\arctan{1\over\tan\alpha}=\beta\ ,$$
whereby $\alpha=\bigl(n+{1\over2}\bigr)\pi$, i.e., $\tan\alph... |
1,378,633 | <p>It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series.</p>
<p>So this got me thinking about the mathematical proof for such argument.</p>
<p>But even after going through some resources about the Fourier series (whic... | Rousan | 696,544 | <p>Well there are 3 conditions for a Fourier Series of a function to be exist:
1. It has to be periodic.
2. It must be single valued, continuous.it can have finite number of finite discontinuities.
3. It must have only a finite number of Maxima and minima within the period.
4. The Integral over one period of |f(X)| mus... |
1,923,034 | <p>A bagel store sells six different kinds of bagels. Suppose you choose 15 bagels at random. What is the probability that your choice contains at least one bagel of each kind? If one of the bagels is Sesame, what is the probability that your choice contains at least three Sesame bagels?</p>
<p>My approach to the firs... | G Ghost | 641,931 | <p>Your answers are both correct, you just have one tiny little error which I'm guessing is a typo. You said the total number of solutions to the equation <span class="math-container">$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 15, x_i \geq 0$</span> is <span class="math-container">${22 \choose 5}$</span> . But if we solve fo... |
264,572 | <p>I am using <code>Table</code> to plot the time steps of a function. However, since there are a lot of decimal places in the x-axis, they are all cramped up :
<a href="https://i.stack.imgur.com/SCSbK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SCSbK.png" alt="enter image description here" /></a... | Shin Kim | 85,037 | <p>Those are called <a href="https://reference.wolfram.com/language/ref/Ticks.html" rel="nofollow noreferrer"><code>Ticks</code></a>. You could try adding the following option in your <code>Plot</code>:</p>
<pre><code>Ticks -> {Table[{i/10^4, i}, {i, 0, 15}], Automatic}
</code></pre>
|
1,021,753 | <p>Any idea on how to compute the expected value of product of Ito's Integral with two different upper limit?</p>
<p>For example:
$$\mathbb{E}\left[\int_0^r f(t)\,dB(t) \int_0^s f(t)\,dB(t)\right]$$</p>
<p>I only know how to compute when the upper limit r and s are the same...but don't know how when r and s are diffe... | pbierre | 259,171 | <p>If you're familiar with [ Euler axis , Euler angle ] representation of a generalized 3D rotation, it is fairly easy to prove your conjecture.</p>
<p>Axis 1 rotation. Project the Euler axis down onto the x-y plane. Begin by rotating around this axis by the Euler angle.</p>
<p>Axis 2 rotation. Calculate the elevati... |
524,870 | <p>This is adapted from 1.7.7 in Friedman's "Foundations of Modern Analysis":</p>
<blockquote>
<p>Let $\mathscr{B}$ be the $\sigma$-ring generated by the class of open subsets of $X$ [a fixed set], and $\mathscr{D}$ the $\sigma$-ring generated by the class of closed subsets of $X$. Show that $\mathscr{D} = \mathscr{... | Dylan Zhu | 63,526 | <p>you can try to show that for a set $b \in \mathcal B $, b also belongs to $\mathcal D$</p>
|
200,931 | <p>I want to generate a layered drawing of the <a href="https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_graph" rel="noreferrer">Hoffman–Singlelton graph</a>. As an example of what I want, here is a layered drawing of the Petersen graph:</p>
<p><a href="https://i.stack.imgur.com/doEt5.png" rel="noreferrer"><img... | halmir | 590 | <p>Here's the one way by using the custom edge function:</p>
<pre><code>arcRight[{a:{x1_,y1_},___,b:{x2_,y2_}}]/;y1>y2:=arcRight[{b, a}];
arcRight[{a:{x1_,y1_},___,b:{x2_,y2_}}]/;y1<=y2:=BSplineCurve[{a, {x1 + (y2-y1).7, (y1+y2)/2},b}]
iLayeredDrawing[g_, spos_Integer:1, opt___?OptionQ] :=
Module[{s, vlist,... |
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | Mark Bennet | 2,906 | <p>The <a href="http://en.wikipedia.org/wiki/Trigonometric_functions" rel="nofollow">wikipedia article</a> gives some infinite series, which are probably what your calculator uses. The formulae for sine and cosine are the ones to focus on first. They converge very quickly, but you have to realise that the angles are me... |
351,642 | <p>So I'm proving that a group $G$ with order $112=2^4 \cdot 7$ is not simple. And I'm trying to do this in extreme detail :) </p>
<p>So, assume simple and reach contradiction. I've reached the point where I can conclude that $n_7=8$ and $n_2=7$. </p>
<p>I let $P, Q\in \mathrm{Syl}_2(G)$ and now dealing with cases th... | Dean Gurvitz | 283,215 | <p>I wanted to write out Mikko Korhonen's first idea for proof in detail as a separate answer, since it is not trivial at all, and provoked some questions in the comments.</p>
<p>As mentioned in the original question, we assumed $n_2=7$. From Sylow's second theorem, we know that all the 2-Sylow subgroups are conjugate... |
4,579,084 | <p>It was a new contributor's question. I answered, got my -1 again and then deleted. Then I asked myself. Then gave it up again. Actually I was gonna ask a different question NOW. When I pressed ask a question, to my surprise, the question I intended to ask yesterday was in the memory!</p>
<p>I wanted to evaluate the ... | Brian M. Scott | 12,042 | <p>Let <span class="math-container">$\sigma\in\Bbb S_n$</span>, and suppose that <span class="math-container">$\sigma$</span> is <span class="math-container">$i$</span>-orderly and <span class="math-container">$j$</span>-orderly for some <span class="math-container">$i,j\in[m]$</span> such that <span class="math-contai... |
1,873,370 | <p>I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the following way: consider an automorphism of raising each entry to the $q$-th power, (taking $n = 2$ as an example)</p>
<p>... | Qiaochu Yuan | 232 | <p>In general, let $K \to L$ be a field extension. $GL_n(L)/GL_n(K)$ can be interpreted as the set of "$K$-structures" on $L^n$. One of many equivalent ways to describe a $K$-structure is that it is a $K$-subspace $V$ of $L^n$ such that the induced map</p>
<p>$$V \otimes_K L \to L^n$$</p>
<p>is an isomorphism. Conseq... |
1,869,564 | <p>i tried to derive logistic population model, and need to integrate this
$\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t$. here is my solution</p>
<p>$\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t=\int \frac{1}{k-N_t} dN_t=-\int \frac{1}{k-N_t}d{(k-N_t)}=-\ln\mid k-N_t\mid+C_1$. i think i have done something wrong he... | smcc | 354,034 | <p>Any function of the form $f(x)=g(|x|)$ where $g$ is an increasing concave function with $g(0)=0$ and $\lim_{x\to\infty}g(x)=a$ will work. </p>
<p>Letting, $g(x)=a[1-h(x)]$ we need $h$ convex with $h(0)=1$ and $\lim_{x\to\infty}h(x)=0$. </p>
<p>For example, you could take $h(x)=\frac{1}{1+x}$, so that
$$f(x)=a-\fr... |
2,312,968 | <p>If $t=\ln(x)$, $y$ some function of $x$, and $\dfrac{dy}{dx}=e^{-t}\dfrac{dy}{dt}$, why would the second derivative of $y$ with respect to $x$ be:
$$-e^{-t}\frac{dt}{dx}\frac {dy}{dt} + e^{-t}\frac{d^2y}{dt^2}\frac{dt}{dx}?$$</p>
<p>I know this links into the chain rule. I don't have a good intuition for why the fi... | Saketh Malyala | 250,220 | <p>$\sin(2π-k)+c=0$</p>
<p>This is equivalent to $\sin(k)=c$</p>
<p>$\sin(\frac{4π}{3}-k)+c=0$</p>
<p>$-\frac{\sqrt{3}}{2}\cos(k)+\frac{1}{2}\sin(k)+\sin(k)=0$</p>
<p>$-\sqrt{3}\cos(k)=-3\sin(k)$</p>
<p>$\tan(k)=\frac{\sqrt{3}}{3}$</p>
<p>$k=\frac{π}{6}$</p>
<p>$c=\sin(\frac{π}{6})=\frac{1}{2}$</p>
|
2,296,724 | <p>I need to calculate $(A+B)^{-1}$, where $A$ and $B$ are two square, very sparse and very large. $A$ is block diagonal, real symmetric and positive definite, and I have access to $A^{-1}$ (which in this case is also sparse, and block diagonal). $B$ is diagonal and real positive. In my application, I need to calculate... | theo | 718,585 | <p>Unless I am seriously mistaken, the above answer is incorrect. In my opinion, the problem lies with
<span class="math-container">$$
A_i+diag(a_{i,1},\cdots,a_{i,n_i})=P_i\begin{bmatrix}
\lambda_{i,1}+a_{n_i} & 0 & 0 & \cdots & 0 & 0 \\
0 & \ddots & \ddots & \ddot... |
1,187,376 | <p>Let $c(n,k)$ be the unsigned Stirling numbers of the first kind, i.e., the number of
$n$-permutations with exactly $k$ cycles.
Apparently, $$\sum_{k=1}^n c(n,k)2^k = (n+1)!$$</p>
<p>I want to prove the equality. </p>
<p>I am most interested in a combinatorial explanation. </p>
<p>The exponential generating fun... | Qiaochu Yuan | 232 | <p>The more general result is that</p>
<p>$$\sum_{k=1}^n c(n, k) x^k = x(x + 1) \dots (x + n - 1).$$</p>
<p>Your result follows straightforwardly by substituting $x = 2$. This identity has the following cute proof: dividing both sides by $n!$ we get</p>
<p>$${x + n - 1 \choose n} = \frac{1}{n!} \sum_{k=1}^n c(n, k) ... |
1,187,376 | <p>Let $c(n,k)$ be the unsigned Stirling numbers of the first kind, i.e., the number of
$n$-permutations with exactly $k$ cycles.
Apparently, $$\sum_{k=1}^n c(n,k)2^k = (n+1)!$$</p>
<p>I want to prove the equality. </p>
<p>I am most interested in a combinatorial explanation. </p>
<p>The exponential generating fun... | Brian M. Scott | 12,042 | <p>Here is a tedious but extremely elementary combinatorial argument for the more general result.</p>
<p>Let $\pi$ be a permutation of $[n]$ having $k$ cycles. The standard representation of $\pi$ is </p>
<p>$$(a_{11}a_{12}\ldots a_{1m_1})(a_{21}a_{22}\ldots a_{2m_2})\ldots(a_{k1}a_{k2}\ldots a_{km_k})\;,\tag{1}$$ </... |
33,622 | <p>I am looking for differentiable functions $f$ from the unit interval to itself that satisfy the following equation $\forall\:p \in \left( 0,1 \right)$:</p>
<p>$$1-p-f(f(p))-f(p)f'(f(p))=0$$</p>
<p>Is there a way to use <em>Mathematica</em> to solve such equations?<br>
<code>DSolve</code> is of course unable to han... | akater | 1,859 | <p>At least you can eliminate $p$:</p>
<pre><code>continue[{exprs__}, f_] :=
Append[#,
Switch[Head@f
, Rule | RuleDelayed, # /. f &
, _, f]@Last@#]& @ {exprs}
toRHS[term_][lhs_ == rhs_] := lhs - term == rhs - term
multiplyBothSidesBy[x_]@e_Equal := # x & /@ e
privateDRule = f_'[x_] :> d[f@x]/d[x]... |
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