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 |
|---|---|---|---|---|
4,551,407 | <p>Here's the question:<br />
If we have m loaves of bread and want to divide them between n people equally what is the minimum number of cuts we should make?<br />
example:<br />
3 loaves of bread and 15 people the answer is 12 cuts.<br />
6 loaves of bread and 10 people the answer is 8 cuts.</p>
<p>for example 1, I f... | Átila Correia | 953,679 | <p><strong>HINT</strong></p>
<p>You can rearrange the LHS in order to get:</p>
<p><span class="math-container">\begin{align*}
x^{2}y'' + 4xy' + 2y & = (x^{2}y'' + 2xy') + (2xy' + 2y)\\\\
& = (x^{2}y')' + (2xy)'
\end{align*}</span></p>
|
808,389 | <p>I roll a dice $3$ times. What is the probability that only $2$ of the sides show up, or put equivalently, what is the probability that 4 of the sides don't show up at all?</p>
<p>More generally lets say I have a $20$ numbered balls in a bag. I pull one out, write down its number and put it back, I then pull another... | Locke | 153,184 | <p>The probability of $2$ of them showing up in one is $\frac13$. Because you roll it $3$ times you get $\left(\frac13\right)^3=\frac1{27}$. For the ball one the probability of not getting $14$ on one turn is $\frac6{20}=\frac3{10}$.<br>
$\left(\frac3{10}\right)^8=\frac{6561}{100000000}=6.561\cdot 10^{-5}$ which is a v... |
1,366,372 | <p>In this <a href="https://math.stackexchange.com/questions/1365989/testing-pab-using-2-dice">question</a>
: </p>
<p>$$ P_r(a\cap b)=P_r(a,b)=P_r(a)P_r(b)$$</p>
<p>However in this <a href="https://stats.stackexchange.com/questions/156852/what-do-did-you-do-to-remember-bayes-rule/156866#156866">question</a>: </p>
<p... | Ted | 15,012 | <p>The second example is the general case of Bayes' rule. In the first example, the two events are independent so $p(b|a) = p(b)$. </p>
|
1,366,372 | <p>In this <a href="https://math.stackexchange.com/questions/1365989/testing-pab-using-2-dice">question</a>
: </p>
<p>$$ P_r(a\cap b)=P_r(a,b)=P_r(a)P_r(b)$$</p>
<p>However in this <a href="https://stats.stackexchange.com/questions/156852/what-do-did-you-do-to-remember-bayes-rule/156866#156866">question</a>: </p>
<p... | wythagoras | 236,048 | <p>$P(A \cup B) = P(A)P(B)$ only holds if $A$ and $B$ are independent.
In that case they have no effect on each other. Then it shouldn't be a suprise that $$P(B | A) = \frac{P(A\cup B)}{P(A)} = \frac{P(A)P(B)}{P(A)}=P(B)$$</p>
|
139,487 | <p>I have a not-so-complicated piecewise cubic function, shown below as the yellow curve on the right. It's derivative is on the left; the blue lines are references. Please see the code below where I call it <strong>myF</strong> (and its derivative myf).</p>
<p>To my surprise, the integration with a parameter $u$ and ... | Michael E2 | 4,999 | <p>An action in <a href="http://reference.wolfram.com/language/ref/WhenEvent.html" rel="nofollow noreferrer"><code>WhenEvent</code></a> should be a <a href="http://reference.wolfram.com/language/ref/Rule.html" rel="nofollow noreferrer"><code>Rule</code></a> or a keyword string (see docs), or a list of these. Instead of... |
238,627 | <p>Kindly consider it soft question as I am a software engineer.and I know only software but I have a doubt in my mind that there would be something like null in mathmatics as well. </p>
<p>If Mathematics <code>NULL</code> IS Equivalent to <code>ZERO</code>?</p>
| Three.OneFour | 48,324 | <p>In my mind there is no need for a concept like <code>NULL</code> in mathematics if you think of <code>NULL</code> as in <code>NULL</code>-pointers.</p>
<p><code>NULL</code> in this sense is a technical necessity because you cannot un-define a variable: Once a variable has been assigned a value, a certain bit of mem... |
238,627 | <p>Kindly consider it soft question as I am a software engineer.and I know only software but I have a doubt in my mind that there would be something like null in mathmatics as well. </p>
<p>If Mathematics <code>NULL</code> IS Equivalent to <code>ZERO</code>?</p>
| Manish Shrivastava | 49,656 | <p>Thats what I wanted to get </p>
<h1>Ref : Wikipedia</h1>
<p>In mathematics, the word null (from German null, which is from Latin nullus, both meaning "zero", or "none")[1] means of or related to having zero members in a set or a value of zero. Sometimes the symbol ∅ is used to distinguish "null" from 0.</p>
<p>In... |
4,617,031 | <p>How would I order <span class="math-container">$x = \sqrt{3}-1, y = \sqrt{5}-\sqrt{2}, z = 1+\sqrt{2} \ $</span> without approximating the irrational numbers? In fact, I would be interested in knowing a general way to solve such questions if there is one.</p>
<p>What I tried to so far, because they are all positive ... | Brevan Ellefsen | 269,764 | <p>Clearly <span class="math-container">$0 < x, y < 1$</span> and simple calculation shows <span class="math-container">$y = \sqrt{7-2\sqrt{10}}$</span> so the claim reduces to showing <span class="math-container">$y^2 - x^2 = 3 - 2\sqrt{10} + 2\sqrt{3} > 0$</span>, i.e. that <span class="math-container">$\sqr... |
3,130,195 | <p>I'm interested of finding a closed formula for the fundamental matrix to the system
<span class="math-container">$$\eqalign{
& y'(t) = a(t)z(t) \cr
& z'(t) = \delta a(t)y(t) \cr} $$</span>
<span class="math-container">$$(y(0),z(0)) = ({y_0},{z_0})$$</span>
where <span class="math-container">$\delta$</s... | Intelligenti pauca | 255,730 | <p>A <a href="https://math.stackexchange.com/questions/665837/prove-that-the-foot-of-the-perpendicular-from-the-focus-to-any-tangent-of-a-para/3060866#3060866">nice property of the parabola</a> states that: the perpendicular from the focus to any tangent intersects it, and the tangent through the vertex, at the same po... |
4,551,497 | <p>How do I find the Taylor series of <span class="math-container">$\cos^{20}{(x)}$</span> for <span class="math-container">$x_0 = 0$</span>, knowing the Taylor series of <span class="math-container">$\cos{(x)}$</span>?</p>
| Claude Leibovici | 82,404 | <p>If you are looking for a limited number of terms, just compose
<span class="math-container">$$\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+\frac{x^8}{40320
}-\frac{x^{10}}{3628800}+O\left(x^{12}\right)$$</span></p>
<p>Now, square a few times
<span class="math-container">$$\cos^2(x)=1-x^2+\frac{x^4}{3}-\... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Community | -1 | <p>You're right. The random, anonymous person you met online is not competent. This is basic mathematical literacy, as taught in every freshman chemistry and physics class.</p>
|
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | JTP - Apologise to Monica | 64 | <p>When a tutoring student asks me about rounding, I tell them that absent specific instructions from a teacher, common sense should apply.</p>
<p>For a conversion, 22 miles isn’t 22.0000 miles, there’s the assumption it’s been rounded. You can’t convert and find yourself with 6 digits of accuracy beyond the decimal. A... |
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | Acccumulation | 8,989 | <p>The product of two numbers should be given with as many significant digits as the least precise of the numbers multiplied (see <a href="https://www.nku.edu/%7Eintsci/sci110/worksheets/rules_for_significant_figures.html" rel="noreferrer">https://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html... |
747,519 | <p><img src="https://i.stack.imgur.com/jYzfz.png" alt="enter image description here"></p>
<p>I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did:</p>
<p>Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ with $a,b \in I$, $a \... | Andres Mejia | 297,998 | <p><strong>Hint</strong>: Extreme value theorem will yield nice results</p>
<p>More to the point: Consider $[f(x_{min}),f(x_{max})]$</p>
<p>A comment on your attempt:</p>
<p>All you've shown is that there is a closed interval contained in $f(I)$ so it doesn't really get at the heart of the problem. </p>
<p>Your sec... |
613,105 | <p>I was observing some nice examples of equalities containing the numbers $1,2,3$ like $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$ and $\log 1+\log 2+ \log 3=\log (1+2+3)$. I found out this only happens because $1+2+3=1*2*3=6$.<br> I wanted to find other examples in small numbers, but I failed. How can we find all of the s... | Community | -1 | <p>Here's a start of a full solution: The right side grows <em>way</em> faster than the left side, so it's unlikely that there are very many solutions. More formally, suppose that $a, b, c \ge 2$, and that $c$ is at least as large as $a, b$. Then we have</p>
<p>$$abc \ge 4c > c + c + c \ge c + b + a$$</p>
<p>so it... |
1,815,662 | <blockquote>
<p>Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but it has two different irreducible factors in $\mathbb{R}[X]$.</p>
</blockquote>
<p>I've tried to use the cyclotomic polynomial as:
$$X^5-1=(X-1)(X^4+X^3+X^2+X+1)$$</p>
<p>So I have that my polynomial is
$$\frac{X^5-1}{X-1}$$ and now... | Behrouz Maleki | 343,616 | <p>let $$P(x)=x^4+x^3+x^2+x^1+1$$
We know if $x=\frac{a}{b}$ is root of $P(x)$ then $b|1\,$ , $\,a|1$. In the other words $a=\pm 1 $ and $b=\pm 1 $ but $P(1)=5$ and $P(-1)=1$, thus we let
$$P(x)=(x^2+ax+b)(x^2+cx+d)$$
as a result
\begin{align}
& bd=1 \\
& ad+bc=1 \\
& b+d+ac=1 \\
& a+c=1 \\
\... |
3,167,571 | <p>Let consider a square <span class="math-container">$10\times 10$</span> and write in the every unit square the numbers from <span class="math-container">$1$</span> to <span class="math-container">$100$</span> such that every two consecutive numbers are in squares which has a... | marty cohen | 13,079 | <p>Letting
<span class="math-container">$y_1 = x_1$</span>
and
<span class="math-container">$y_i
=x_i-x_{i-1}$</span>
for <span class="math-container">$i>1$</span>,
this is
<span class="math-container">$\sum_{i=1}^n p_iy_i
=1
$</span>
with <span class="math-container">$n=5, y_i \ge 0$</span>.</p>
<p>As Maria Mazu... |
3,167,571 | <p>Let consider a square <span class="math-container">$10\times 10$</span> and write in the every unit square the numbers from <span class="math-container">$1$</span> to <span class="math-container">$100$</span> such that every two consecutive numbers are in squares which has a... | David M. | 398,989 | <p>This answer addresses the case when
<span class="math-container">$$
\frac{5}{x_5}\geq\max_{i=1,\dots,4}\big\{\frac{i}{x_i}\big\}.
$$</span>
In this case, the solution using the AM-GM inequality fails to satisfy the ordering constraint. There are still other cases to be considered, for example, <span class="math-cont... |
3,340,093 | <p>Is the following statement true?</p>
<blockquote>
<p>Two real numbers a and b are equal iff for every ε > 0, |a − b| < ε.</p>
</blockquote>
<p>I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.</p>
| Allawonder | 145,126 | <p>Yes, it is true because the condition is satisfied for <em>every</em> <span class="math-container">$\epsilon>0,$</span> no matter how little. When we say a <em>number</em> is less than such an <span class="math-container">$\epsilon,$</span> we simply mean that the number vanishes, or is <span class="math-containe... |
1,757,556 | <p>Solve the recursion $p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$</p>
<p>$p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$</p>
<p>$= p + (1-2p)p_{n-1}$ I can see that this step simply rearranges the expression, but what's the point of it? What are we trying to accomplish here? Is it to combine the $p_{n-1}$ terms?</p>
<p... | zyx | 14,120 | <p>It's a 2-state Markov chain where with probability $p$ one switches to the other state and with probability $q=(1-p)$ stays in the same place. The binomial expansion of $(pX + qY)^n$ gives the distribution of the number of switch (X) and stay (Y) steps. Therefore the sum of the probabilities of even/odd number of X... |
3,509,912 | <blockquote>
<p>Given <span class="math-container">$$A = \begin{pmatrix} 0&& 1&& 0&& 0 \\ 0&& 0&& 2&& 0 \\ 0&& 0&& 0&& 3\end{pmatrix}$$</span> and <span class="math-container">$$B = \begin{pmatrix} 0&& 0&& 0 \\ 1&& 0&am... | mvw | 86,776 | <p><a href="https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#Definition" rel="nofollow noreferrer">pseudoinverse</a></p>
<p>Only showing 1. and 2. using an Octave session, 3. and 4. hold for real valued symmetric matrices.</p>
<pre><code>>> A = [0,1,0,0;0,0,2,0;0,0,0,3]
A =
0 1 0 0
0 0... |
3,509,912 | <blockquote>
<p>Given <span class="math-container">$$A = \begin{pmatrix} 0&& 1&& 0&& 0 \\ 0&& 0&& 2&& 0 \\ 0&& 0&& 0&& 3\end{pmatrix}$$</span> and <span class="math-container">$$B = \begin{pmatrix} 0&& 0&& 0 \\ 1&& 0&am... | Rodrigo de Azevedo | 339,790 | <p>Note that the SVD of matrix <span class="math-container">$\rm A$</span> is</p>
<p><span class="math-container">$$\rm A = \mathrm I_3 \underbrace{\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 3 & 0 \end{bmatrix}}_{=: \Sigma} \, \underbrace{\begin{bmatrix} 0 & 1 &... |
1,319,767 | <p>If we know that $\frac{2^n}{n!}>0$ for every $n\in \mathbb{N}$ and $$\frac{2^n}{n!}=\frac{2}{1}\frac{2}{2}...\frac{2}{n}$$ how to bound this sequence above?</p>
| Crostul | 160,300 | <p>For $n \ge 4$ you have
$$\frac{2}{1} \frac{2}{2} \frac{2}{3} \cdots \frac{2}{n} \le 2 \cdot 1 \cdot 1 \cdot \dots 1 \cdot \frac{2}{n} = \frac{4}{n}$$</p>
|
1,319,767 | <p>If we know that $\frac{2^n}{n!}>0$ for every $n\in \mathbb{N}$ and $$\frac{2^n}{n!}=\frac{2}{1}\frac{2}{2}...\frac{2}{n}$$ how to bound this sequence above?</p>
| Jack D'Aurizio | 44,121 | <p>You cannot know that $\lim_{n\to +\infty}\frac{2^n}{n!}\color{red}{>}0$ since the limit is <strong>exactly</strong> zero. </p>
<p>That follows from the trivial inequality:
$$ \forall n\geq 3,\qquad \frac{2^n}{n!}\leq \frac{2^n}{2\cdot 3^{n-2}}=\frac{9}{2}\cdot\left(\frac{2}{3}\right)^n\xrightarrow[n\to +\infty]{... |
300,900 | <p>I am learning random matrix theory. Unfortunately I do not like combinatorics, and have never really been good at it. But I found that random matrix theory has heavily relied on combinatorics, particularly in finding the limiting spectral distribution, at least for symmetric matrices, where the moment method is the ... | Jochen Glueck | 102,946 | <p><strong>Setting.</strong> Throughout, let $E$ be a complex Banach space and denote the space of bounded linear operators on $E$ by $\mathcal{L}(E)$. Let $X \subseteq E$ be a closed subspace and let $\mathcal{T} = (T(t))_{t \ge 0}$ be a $C_0$-semigroup on $E$ with generator $A: E \supseteq D(A) \to E$.</p>
<p>As alr... |
3,415,266 | <p>I cannot find how or why this,</p>
<p><span class="math-container">$$5\sin(3x)−1 = 3$$</span></p>
<p>Has one of two solutions being this,</p>
<p><span class="math-container">$$42.29 + n \times 120^\circ$$</span></p>
<p>I am lost on how to get this solution. I have found the other one, so I will not mention it.</... | Allawonder | 145,126 | <p>Rewrite your equation as <span class="math-container">$$\sin 3x=\frac45,$$</span> and find one angle whose sine is <span class="math-container">$4/5.$</span> Call such an angle <span class="math-container">$\alpha$</span> (I can't be bothered to work this out explicitly here, since it's tangential to my main point).... |
3,999,652 | <p>Let triangle <span class="math-container">$ABC$</span> is an equilateral triangle. Triangle <span class="math-container">$DEF$</span> is also an equilateral triangle and it is inscribed in triangle <span class="math-container">$ABC \left(D\in BC,E\in AC,F\in AB\right)$</span>. Find <span class="math-container">$\cos... | Quanto | 686,284 | <p>You may calculate <span class="math-container">$\cos\alpha $</span> directly. Per the sine rule for <span class="math-container">$\triangle CDE$</span>
<span class="math-container">$$\frac{\sin \alpha }{CD}= \frac{\sin (120-\alpha) }{AB-CD}= \frac{\sin 60 }{DE}
$$</span>
Eliminate <span class="math-container">$CD$</... |
358,184 | <p>We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality)
<span class="math-container">$$
P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[X]) \geq c, 0<\theta < 1
$$</span></p>
<p>However, It would be great to know if t... | Clement C. | 37,266 | <p>Not sure how interesting it is, given that computing <span class="math-container">$\mathbb{E}[|X-\mathbb{E}[X]|]$</span> may be unwiedly, but Iosif Pinelis' argument can be adapted to give the following statement, which does not require existence of a finite second moment nor a lower bound on the support.</p>
<p>Sup... |
4,226,030 | <blockquote>
<p>I want to solve
<span class="math-container">$$C\cos(\sqrt\lambda \theta) + D\sin(\sqrt\lambda \theta) = C\cos(\sqrt\lambda (\theta + 2m\pi)) + D\sin(\sqrt\lambda (\theta + 2m\pi))$$</span>
The solution must be valid for all <span class="math-container">$\theta$</span> in <span class="math-container">$\... | Barry Cipra | 86,747 | <p>If <span class="math-container">$C=D=0$</span>, then <em>every</em> <span class="math-container">$\theta\in\mathbb{R}$</span> is a solution, since the equation is just <span class="math-container">$0=0$</span>.</p>
<p>If <span class="math-container">$D=iC\not=0$</span>, we can divide by <span class="math-container">... |
79,270 | <p>For integer $n\ge 3$, consider the graph on the set of all even vertices of the $n$-dimensional hypercube $\{0,1\}^n$ in which two vertices are adjacent whenever they differ in exactly two coordinates. This is an $(n(n-1)/2)$-regular graph on $2^{n-1}$ vertices. Is there any standard name / notation for this graph? ... | Gordon Royle | 1,492 | <p>This graph is known as the half-cube.</p>
<p>I don't know about the other question.</p>
|
1,255,803 | <p>My understanding is that the thesis is essentially a <em>definition</em> of the term "computable" to mean something that is computable on a Turing Machine.</p>
<p>Is this really all there is to it? If so, what makes this definition so important? What makes this definition so significant as to warrant having it's ow... | Andreas Blass | 48,510 | <p>The answer from user73985 explains the content of the Church-Turing thesis, but I'd like to add a few words about its value; why do we want it. </p>
<p>The first benefit that we get from this thesis is that it lets us connect formal mathematical theorems to real-world issues of computability. For example, the theor... |
3,644,870 | <p><strong>Give an example or argue that it is impossible.</strong></p>
<p>I argue that it is impossible because, if <span class="math-container">$(x_n)_n$</span> is a sequence which converges to 0, then <span class="math-container">$(x_n)_n$</span> must be bounded above or below by 0. As <span class="math-container">... | cqfd | 588,038 | <p>I would argue as follows: since <span class="math-container">$K $</span> is compact, every sequence in <span class="math-container">$K $</span> has a convergent subsequence with the limit in <span class="math-container">$K$</span>. Now, as <span class="math-container">$(x_n)$</span> converges to <span class="math-c... |
25,778 | <p>I am going to teach a Calculus 1 course next semester, and I have 15 weeks for the course material. The class meets MWF for 50 minutes each. I have taught this class before using the same syllabus, but my colleague shared concerns that my pacing is too fast:</p>
<p>Week 1: Review of Functions</p>
<p>Week 2: Limits a... | fedja | 19,946 | <p>Just do what works for you and your students. If they get a good grasp of the material (which you should check regularly) and do not look terribly overworked, I see nothing wrong with going fast. However, if you see that a noticeable portion of the class is falling behind, slow down and allow them more time to diges... |
1,186,516 | <p>Please the highlighted part in the image below. I don't understand why w(c2) must be larger than s(c1, c2) considering s(c1, c2) is counting the position where c1 + c2 = 0, c1 != 0 and c2 != 0 while w(c2) is only counting position where w(c2) != 0.</p>
<p>Should s(c1, c2) be larger than w(c2)?</p>
<p>Thanks for he... | Alp Uzman | 169,085 | <p>Observe that each summand has a factor of $\left(\dfrac{|h|}{H}\right)^2$. First take that factor outside. Then as $\dfrac{|h|}{H}\leq1$, erasing its powers that appear as factors in the sum will increase the value of the expression. Finally adding two more terms will increase the value once more. Observe that all t... |
1,488,752 | <p><s>I would like to know if the following question has an intelligent solution:</p>
<p>Determine the largest bet that cannot be made using chips of $7$ and $9$ dollars.</p>
<p>After not being able to solve it I found a solution online which writes out all combinations of $7$ and $9$ up to $90$ and then notes that w... | Christian Blatter | 1,303 | <p>It helps to imagine $a<b$. As ${\rm gcd}(a,b)=1$ the $a$ numbers
$$r_j:=j\>b\quad(0\leq j\leq a-1)$$
represent the $a$ different remainders modulo $a$. At the same time $r_j$ is the smallest representable number having that remainder modulo $a$: You need at least $j$ summands $b$ to produce that remainder. It ... |
3,757,222 | <p>Let <span class="math-container">$n_{1}, n_{2}, ... n_{k} $</span> be a sequence of k consecutive odd integers. If <span class="math-container">$n_{1} + n_{2} + n_{3} = p^3$</span> and <span class="math-container">$n_{k} + n_{k-1} + n_{k-2} + n_{k-3} + n_{k-4} = q^4$</span> where both p and q are prime, what is k?<... | Keith Backman | 29,783 | <p>It is a little appreciated fact that every power <span class="math-container">$k\ge 2$</span> of any positive integer <span class="math-container">$n$</span> can be expressed as the sum of exactly <span class="math-container">$n$</span> consecutive odd numbers, viz: <span class="math-container">$$n^k=\sum_{i=\frac{... |
584,171 | <blockquote>
<p>Show that every graph can be embedded in $\mathbb{R}^3$ with all
edges straight. </p>
</blockquote>
<p>(Hint: Embed the vertices inductively, where should you
not put the new vertex?)</p>
<p>I've also received a tip about using the curve ${(t, t^2 , t^3 : t \in \mathbb{R} )}$ but I'm not sure ho... | achille hui | 59,379 | <p>Let $\vec{p} : \mathbb{R} \to \mathbb{R}^3$ be the function
$\vec{p}(t) = (t, t^2, t^3)$ and $\mathscr{C}$ be the curve
$\Big\{\;\vec{p}(t) : t \in \mathbb{R}\;\Big\}$.</p>
<p>For any four distinct $t_1, t_2, t_3, t_4$ in $\mathbb{R}$, the volume of the tetrahedron $\mathscr{T}$ formed by
$\vec{p}(t_i) \in \mathsc... |
500,973 | <p>I am presented with the question:</p>
<blockquote>
<p>The photoresist thickness in semiconductor manufacturing has a mean of
10 micrometers and a standard deviation of 1 micrometer. Assume that
the thickness is normally distributed and that the thicknesses of
different wafers are independent.</p>
<p>(a... | QED | 91,884 | <p>You have certain observations $X_1,\cdots,X_n$ which follow $N(10,1)$ distribution, where $X_i$ denote the thickness of the $i^{th}$ wafer. Then the the average thickness of the $n$ wafers $\bar{X}_n=\sum_{i=1}^nX_i/n$ follows $N(10,1/n)$ distribution. Then the probability that
$$P(\bar{X}_n<11~or~\bar{X}_n>9... |
315,386 | <p>I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in this format -
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Ellipse_Properties_of_Directrix_a... | Rick Goldstein | 62,318 | <p>It's easiest to start with the equation for the ellipse in rectangular coordinates:</p>
<p>$$(x/a)^2 + (y/b)^2 = 1$$</p>
<p>Then substitute $x = r(\theta)\cos\theta$ and $y = r(\theta)\sin\theta$ and solve for $r(\theta)$.</p>
<p>That will give you the equation you found on Wikipedia.</p>
|
315,386 | <p>I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants <span class="math-container">$a$</span> and <span class="math-container">$b$</span> in this format -
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Ellipse_Properties_of_Directrix_a... | Sharat V Chandrasekhar | 400,967 | <p>You're making the common mistake of using the polar coordinate instead of the eccentric anomaly which is the parameter in the ellipse coordinates.</p>
|
120,992 | <p>An algorithm book <a href="http://rads.stackoverflow.com/amzn/click/1849967202" rel="nofollow">Algorithm Design Manual</a> has given an description:</p>
<blockquote>
<p>Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neigh... | Henry | 6,460 | <p>If you have exactly 15 columns (avenues) of vertices in 200 rows (streets) then basic properties of multiplication give $15 \times 200 = 3000$ vertices.</p>
<p>As for edges, there are 14 edges in each row and 199 edges in each column so there are $14 \times 200 + 199 \times 15 = 5785$ edges. $6000$ was only an a... |
120,992 | <p>An algorithm book <a href="http://rads.stackoverflow.com/amzn/click/1849967202" rel="nofollow">Algorithm Design Manual</a> has given an description:</p>
<blockquote>
<p>Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neigh... | BARUCH | 1,147,075 | <p><a href="https://i.stack.imgur.com/ONMuU.jpg" rel="nofollow noreferrer">look here</a></p>
<p>Avenue : form North ----> South
Streets : From East ----> West</p>
|
2,724,686 | <blockquote>
<p>Set $B = \{1,2,3,4,5\}$, $S$ - equivalence relation. It is given that for all $x,y \in B$ if $(x,y)\in S$ and if $x+y$ is an even number then $x = y$. In such case is it true that:</p>
<ol>
<li>the number of elements in each equivalence class of $S$ is at most $2$</li>
<li>any relation $S$ wo... | 57Jimmy | 356,190 | <p>Your are right about the fact that you cannot have $(x,y)$ with both even or both odd and different, but there is a flaw in the second part of your argument. If you add to the identity relation also a single couple of pairs $(x,y)$ and $(y,x)$ with $x$ odd and $y$ even, everything is still fine: why should transitiv... |
1,639,232 | <p>A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$?</p>
<p>Thank you very much.</p>
| Clement C. | 75,808 | <p>The $Q$ is a parameter, and $q$ is a variable ranging from $0$ to $Q$: basically, you have $Q+1$ parameters $\textrm{ceps}_0,\dots, \textrm{ceps}_Q$; or, in programming terms, you have an array $\textrm{ceps}[0\dots Q]$.</p>
<p>Similarly, the LPC coefficients are a list of $p$ values $a_1,\dots, a_p$ (i.e., $a_q$ f... |
4,562,451 | <p>I had this maths question:</p>
<blockquote>
<p>Given that <span class="math-container">$$8\sqrt{p} = q\sqrt{80}$$</span> where <span class="math-container">$p$</span> is prime, find the value of <span class="math-container">$p$</span> and the value of <span class="math-container">$q$</span></p>
</blockquote>
<p>I di... | Mike | 544,150 | <p><strong>If <span class="math-container">$q$</span> is restricted to the integers:</strong> From the equation <span class="math-container">$4p = 5q^2$</span>, it follows that <span class="math-container">$5|5q^2=4p$</span>, or in particular, <span class="math-container">$5$</span> has to divide <span class="math-cont... |
3,567,563 | <p>We have the polynomial <span class="math-container">$ f= X^4+X^3+X^2+X+2$</span> with <span class="math-container">$f\in \Bbb C[X] $</span>, it asks to determine the quotient of the division of the polynomial <span class="math-container">$f$</span> by the polynomial <span class="math-container">$g$</span>, <span cla... | user744868 | 744,868 | <p>By remainder theorem,
<span class="math-container">$$1 + i(1 + \sqrt{2}) = f(\cos \alpha - i \sin \alpha) = f(e^{-i\alpha}).$$</span>
Therefore
<span class="math-container">\begin{align*}
&(e^{-i\alpha})^4 + (e^{-i\alpha})^3 + (e^{-i\alpha})^2 + e^{-i\alpha} + 2 = 1 + i(1 + \sqrt{2}) \\
\iff \, &(e^{-i\alpha... |
1,369,076 | <p>Are there any good "analysis through problems" type books? I've tried reading analysis books but I literally get bored to death, and, until I manage to concoct a way of transforming a normal textbook into a problem book (maybe by trying to prove all the theorems myself, but that probably requires more math maturity ... | Community | -1 | <p>P. M. Fitzpatrick - Advanced Calculus. <a href="http://rads.stackoverflow.com/amzn/click/B008VRZWTS" rel="nofollow">Here</a> is the reviews. </p>
<p>Fitzpatrick's Advanced Calculus is enough to cover Calculus and Real Analysis, and it includes also many exercises as well as it's a rigorous text and very readable fo... |
1,724,419 | <p>I can create a large collection of normalized real valued $n$-dimensional vectors from some random process which I hypothesis should be equidistributed on the unit sphere. I would like to test this hypothesis.</p>
<ul>
<li>What is a good way numerically to test if vectors are equidistributed on the unit sphere? I ... | Alehud | 649,614 | <p>From <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4196685/" rel="nofollow noreferrer">this paper by Cai, Fan, Jiang</a>: you can calculate angles <span class="math-container">$\theta_{ij}$</span> between pairs of vectors <span class="math-container">$\vec{v}_i$</span>, <span class="math-container">$\vec{v}_... |
3,229,789 | <p>For proving <span class="math-container">$$\frac {16}{\cos (4x)+7} =\frac{1}{\sin^4x +\cos^2x} +\frac{1}{\sin^2x +\cos^4x} $$</span></p>
<p>I tried to use that:</p>
<p><span class="math-container">\begin{align}
\sin^4 x +\cos^4 x&=\sin^4 x +2\sin^2x\cos^2 x+\cos^4 x - 2\sin^2x\cos^2 x\\
&=(\sin^2x+\cos^2 x... | CY Aries | 268,334 | <p><span class="math-container">$\displaystyle \sin^4x+\cos^2x=\frac{(1-\cos2x)^2}{4}+\frac{1+\cos 2x}{2}=\frac{3+\cos^22x}{4}=\frac{3+\frac{1+\cos 4x}{2}}{4}=\frac{7+\cos4x}{8}$</span></p>
<p><span class="math-container">$\displaystyle \sin^2x+\cos^4x=\sin^4\left(\frac \pi2-x\right)+\cos^2\left(\frac \pi2-x\right)=\f... |
3,229,789 | <p>For proving <span class="math-container">$$\frac {16}{\cos (4x)+7} =\frac{1}{\sin^4x +\cos^2x} +\frac{1}{\sin^2x +\cos^4x} $$</span></p>
<p>I tried to use that:</p>
<p><span class="math-container">\begin{align}
\sin^4 x +\cos^4 x&=\sin^4 x +2\sin^2x\cos^2 x+\cos^4 x - 2\sin^2x\cos^2 x\\
&=(\sin^2x+\cos^2 x... | Robert Z | 299,698 | <p>Note that
<span class="math-container">$$\begin{align}\sin^4x +\cos^2x &=
\sin^2x(1-\cos^2x) +\cos^2x\\
&=\sin^2x+\cos^2x(1-\sin^2x)=\sin^2x +\cos^4x.
\end{align}$$</span>
Hence, according to your work,
<span class="math-container">$$\begin{align}
2(\sin^4x +\cos^2x)&=(\sin^4x +\cos^2x) +(\sin^2x +\cos... |
1,746,776 | <p>I am wondering how I could solve the integral </p>
<p>$$\iiint \frac{1-e^{-(x^2+y^2+z^2)}}{[x^2+y^2+z^2]^{2}}$$</p>
<p>over $\mathbb{R}^{3}$</p>
<p>I thought maybe I could break it up into three single integrals and multiply or something. I think it is not supposed to be difficult to solve. How should it be appro... | DeepSea | 101,504 | <p>Hint: Use <strong>spherical coordinates</strong>, and note that each copy of $\mathbb{R} = \left(-\infty,\infty\right)\Rightarrow I = \displaystyle \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \dfrac{1-e^{-\left(x^2+y^2+z^2\right)}}{(x^2+y^2+z^2)^2}dxdydz= 8\displaystyle \lim_{r \to \infty... |
125,661 | <p>When typing the name of a built-in function like <code>Integrate</code>, the button (<em>ℹ︎</em>) appears next to that name in the autocomplete:</p>
<p><img src="https://i.stack.imgur.com/ISWj1.png" alt="enter image description here"></p>
<p>But I don't get that (<em>ℹ︎</em>) button for my package functions, even ... | Jeff Henning | 80,135 | <p>Very curious if there's been any progress on this. It's now 2021, and I've built my own paclet application with documentation using Workbench 3 (Eclipse add-in 10.1.822) with MM 12.1. Workbench is creating the index file but I get the same behavior as above for my own functions.</p>
<p><a href="https://i.stack.img... |
3,772,534 | <p>Tangents to a circumference of center O, drawn by an outer point C, touch the circle at points A and B. Let S be any point on the circle. The lines SA, SB and SC cut the diameter perpendicular to OS at points A ', B' and C ', respectively. Prove that C 'is the midpoint of A'B'.</p>
<p><a href="https://i.stack.imgur.... | J.G. | 56,861 | <p>For <span class="math-container">$0\le j\le 3$</span> define <span class="math-container">$S_j(z):=\sum_{n\ge0}\frac{z^{4n+j}}{(4n+j)!}$</span> so, for <span class="math-container">$0\le k\le3$</span>,<span class="math-container">$$\sum_ji^{jk}S_j=\sum_{jn}\frac{(zi^k)^{4n+j}}{(4n+j)!}=e^{i^kz}.$$</span>In other wor... |
6,831 | <p>I would like for the autocomplete feature to search through contexts, for example if I have a symbol named A`B`C`MyFunction, when I type A` and press "cmd + shift + k" it will complete it.</p>
<p><em>Edit</em></p>
<p>To be clear, I don't want to have to type the path because it's usually very long, and I don't wan... | Community | -1 | <p>One option is to put the context on the path:</p>
<pre><code>$ContextPath = AppendTo[$ContextPath, "A`B`C`"]
</code></pre>
|
2,522,342 | <p>So far I have only got 9 from just guess and check. I am thinking of using Vieta's Formula, but I am struggling over the algebra. Can someone give me the first few steps?</p>
| TomGrubb | 223,701 | <p>One way to go about this is to determine when $2^n-1$ and $2^n+1$ are individually prime. For the first case, try to show that $n$ must be a prime number. For the second case, try to show that $n$ must be a power of $2$. What can you conclude?</p>
|
1,111,041 | <p>Given: $y=\log(1+x)$</p>
<p>Show that $y≈x$ if $x$ gets small (less than 1).</p>
<p>I don't think we're supposed to use Taylor series (because they were never formally introduced in class), but I do think we have to differentiate and show that the derivative of $\log(1+x)$ is approximately equal to $\log(1+x)$ on ... | Anurag A | 68,092 | <p>Use linear approximation around $x=0$. What it means is that in the neighborhood of $x=0$ you are using a tangent line to approximate the actual function.</p>
<p>The tangent line at $x=0$ is given by $y-\ln (1)=f^{'}(0)(x-0)$.</p>
|
27 | <p>Burt Totaro has a result that for a certain class of algebraic stacks, having affine diagonal is equivalent to the stabilizers at closed points begin affine. Is there an example of this equivalence failing in general?</p>
| Anton Geraschenko | 1 | <p>Let $X$ be an algebraic stack. Given a point $f:T\to X$, we define $Stab(f)=X\times_{X\times X}T$, where the map $X\to X\times X$ is the diagonal and $T\to X\times X$ is $(f,f)$. We say that the stabilizer if affine if $Stab(f)\to T$ is an affine morphism.</p>
<p>Since affine morphisms are stable under base extensi... |
808,144 | <p>Here is a fun looking one some may enjoy. </p>
<p>Show that:</p>
<p>$$\int_{0}^{1}\log\left(\frac{x^{2}+2x\cos(a)+1}{x^{2}-2x\cos(a)+1}\right)\cdot \frac{1}{x}dx=\frac{\pi^{2}}{2}-\pi a$$</p>
| Norbert | 19,538 | <p>Denote
$$
I(r)
=\int_{0}^{1}\log\left(\frac{x^{2}+2x r +1}{x^{2}-2x r+1}\right)\cdot \frac{1}{x}dx
$$
then
$$
\begin{align}
\frac{dI}{dr}
&=\int_0^1 \frac{4 \left(x^2+1\right)}{\left(2-4 r^2\right) x^2+x^4+1} dx\\
&=\int_0^1 \left(\frac{2}{x^2+2rx+1}+\frac{2}{x^2-2 r x+1} \right)dx\\
&=\frac{2 \tan ^{-1... |
808,144 | <p>Here is a fun looking one some may enjoy. </p>
<p>Show that:</p>
<p>$$\int_{0}^{1}\log\left(\frac{x^{2}+2x\cos(a)+1}{x^{2}-2x\cos(a)+1}\right)\cdot \frac{1}{x}dx=\frac{\pi^{2}}{2}-\pi a$$</p>
| Omran Kouba | 140,450 | <p>Starting from
$$
{\rm Log}(1+x e^{ia})=\sum_{n=1}^\infty\frac{(-1)^{n-1}e^{ina}}{n}x^n
$$
we see that
$$
\int_0^1{\rm Log}(1+x e^{ia})\cdot \frac{1}{x}dx=\sum_{n=1}^\infty\frac{(-1)^{n-1}e^{ina}}{n^2}
$$
Taking real parts we get
$$
\int_0^1 \log|1+x e^{ia}|\cdot \frac{1}{x}dx=\sum_{n=1}^\infty\frac{(-1)^{n-1}\cos(na... |
28,104 | <p>It occured to me that the Sieve of Eratosthenes eventually generates the same prime numbers, independently of the ones chosen at the beginning. (We generally start with 2, but we could chose any finite set of integers >= 2, and it would still end up generating the same primes, after a "recalibrating" phase).</p>
<p... | Jon Awbrey | 1,636 | <p>There is a branch of literature that goes under the name of Generalized Primes. I remember spending some time with a book by Knopfmacher?</p>
|
117,432 | <p>All varieties are over $\mathbb{C}$. Notions related to weights etc. refer to mixed Hodge structures (say rational, but I would be grateful if the experts would point out any differences in the real setting).</p>
<p>I am trying to get some intuition for the geometric meaning of/when to expect the weight filtration ... | Donu Arapura | 4,144 | <p>Here is a simple example to keep in mind. Let $\bar X$ be a smooth
projective curve, and let $X=\bar X-\lbrace p_1, \ldots, p_n \rbrace$.
The rational mixed Hodge structure on $H^1(X)$ is given by an extension
$$0\to H^1(\bar X)\to H^1(X)\to \mathbb{Q}(-1)^{n-1}\to 0$$
This splits if and only if for each $i,j$ th... |
669,207 | <p>For any function $f$ and any $x∈Dom(f)$, if for any neighbourhood $S$ of $x$,</p>
<p>$\qquad f(t)=0$ for some $t∈S$ </p>
<p>$\qquad f(u)=1$ for some $u∈S$ </p>
<p>then $ f$ is discontinuous at $x$.</p>
<p>Why is this true? I find it very hard to understand it :(.</p>
| janmarqz | 74,166 | <p>Note that $f^{-1}(2)=\{\sqrt{2},-\sqrt{2}\}$, and in general for $x>0$ we have
$$f^{-1}(x)=\{\sqrt{x},-\sqrt{x}\}.$$</p>
|
347,214 | <p>We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog starts running toward the woman from (0,0) they are both running at constant speed, the dogs path is curved and we w... | Ishan Banerjee | 52,488 | <p>Hint : Do not try to find the equation of curve.It will be complicated and unnecessary. Try to take relative velocities with respect to the dog and the qoman and find the time taken for the dog to reach the woman. As it's moving with constant speed, you can then find the distance.</p>
<p>Ok, might as well post a c... |
347,214 | <p>We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog starts running toward the woman from (0,0) they are both running at constant speed, the dogs path is curved and we w... | Paul Safier | 98,088 | <p>Sounds like this is a variant of the prototypical pursuit curve. In this case you're asked for the length of the pursuit curve. Researching pursuit curves, you can find that the working equation (formed by stipulating that the dog is always running towards the woman's position) is:</p>
<p>dy/dx = (y - w t)/(x - 50)... |
1,465,167 | <p>Given 2 points $p_1=(x_1^1, x_2^1, ..., x_n^1)$ and $p_2=(x_1^2, x_2^2, ..., x_n^2)$ in $n$-dimensional Euclidean space, how would you define the straight-line from $p_1$ to $p_2$ with these 2 points being the endpoints of the line.</p>
<p>There are a few of things I've been completely unable to figure out. One is ... | CIJ | 159,421 | <p>Use the binomial theorem:</p>
<p><strong>Case I.</strong> $x\geqslant1.$ For each $n,$ let $x_n:=\sqrt[n]{x}-1.$ Then we have $x=(1+x_n)^n\geqslant nx_n$ and hence $0\leqslant x_n\leqslant\dfrac{x}{n}$ and so $\lim\limits_{n\to\infty}x_n=\lim\limits_{n\to\infty}(\sqrt[n]{x}-1)=0$ which means that $\lim\limits_{n\to... |
1,465,167 | <p>Given 2 points $p_1=(x_1^1, x_2^1, ..., x_n^1)$ and $p_2=(x_1^2, x_2^2, ..., x_n^2)$ in $n$-dimensional Euclidean space, how would you define the straight-line from $p_1$ to $p_2$ with these 2 points being the endpoints of the line.</p>
<p>There are a few of things I've been completely unable to figure out. One is ... | LIR | 608,434 | <p>There is a theorem that states that
If <span class="math-container">$x_n>0$</span> and <span class="math-container">$\lim_\limits{n \to \infty } \frac{x_{n+1}}{x_n} = L$</span> then <span class="math-container">$\lim_\limits{n \to \infty} \sqrt[n]{x_n}
= L$</span>. </p>
<p>For <span class="math-container">$x_n... |
1,388,565 | <p>Given for example $\omega_1$ coin tosses (i.e. a mapping from the elements of $\omega_1$ to $\{H,T\}$ with independent probabilities half), what is the probability that there is an infinite <del>subsequence</del> subinterval [<em>corrected following comments</em>] consisting only of heads?</p>
<p>Is this question e... | Brent | 219,983 | <p>In reference to the countable sequence of tosses, the probability of any individual subsequence might be zero, but that is not the same as the probability of there <em>existing</em> an infinite subsequence of heads. The only way to <em>not</em> have an infinite subsequence of heads is if, after some finite point, y... |
1,607,190 | <p>Prove by induction that $8^{n} − 1$ for any positive integer $n$ is divisible by $7$. </p>
<p>Hint: It is easy to represent divisibility by $7$ in the following way: $8^{n} − 1 = 7 \cdot k$ where k is a positive integer.</p>
<p>This question confused me because I think the hint isn't true. If $n = 1$ and $k = 2$ f... | JnxF | 53,301 | <p>Proving $7 | 8^n - 1$ is the same as proving $8^n - 1 \equiv 0 \pmod 7$.</p>
<p><strong>Prove</strong>:</p>
<blockquote>
<p>If $n \geq 1$, we see that $8 \equiv 1 \pmod 7$, so using module $7$: $$8^n - 1 \equiv
0 \iff 1^n - 1 \equiv 0$$ which is obviously true as $1^n$ is
always $1$ for $n\geq 1$. $\qquad ... |
4,116,252 | <p>I'm trying to prove (or disprove) the following:</p>
<p><span class="math-container">$$ \sum_{i=1}^{N} \sum_{j=1}^{N} c_i c_j K_{ij} \geq 0$$</span>
where <span class="math-container">$c \in \mathbb{R}^N$</span>, and <span class="math-container">$K_{ij}$</span> is referring to a <a href="https://en.wikipedia.org/w... | Anand | 782,298 | <p>Fix <span class="math-container">$x_i\in\mathbb{R}^n$</span>, <span class="math-container">$i = 1, 2, \ldots, N$</span>. We will assume without loss of generality that no <span class="math-container">$x_i$</span> is identically <span class="math-container">$0$</span>. Define <span class="math-container">$N\times N$<... |
4,157,472 | <p>Let <span class="math-container">$V$</span> be a vector space, and <span class="math-container">$T:V→V$</span> a linear transformation such that:</p>
<p><span class="math-container">$T(2v_1 + 3v_2) = -5v_1 - 4v_2$</span> and <span class="math-container">$T(3v_1 + 5v_2) = 3v_1 -2v_2$</span></p>
<p>Then:</p>
<p>T(v<su... | Coriolanus | 439,201 | <p>Let <span class="math-container">$v_i$</span> = <span class="math-container">$\text{sign }u_i$</span>. Now Let <span class="math-container">$\bar{v} = -\alpha v$</span>. So</p>
<p><span class="math-container">$$\langle u, \bar{v} \rangle = \langle u, -\alpha v \rangle = -\alpha\langle u, v \rangle = -\alpha \|u \|_... |
2,502,255 | <p>We say that if $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $g(a)$ then $(f\circ g)(x)$ is continuous at $a$. </p>
<p>The converse: If $(f\circ g)(x)$ is continuous at $a$ then $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $g(a)$ is not necessarily true.</p>
<p>For example consider $g(x) = x+1... | Siong Thye Goh | 306,553 | <p>Let $f$ be a constant function. </p>
<p>$$\forall x \in \mathbb{R}, f(x) = 1$$</p>
<p>Hence $f(g(x))=1$ which is continuous.</p>
|
3,254,290 | <p>Consider two random variables <span class="math-container">$X,Y$</span>. <span class="math-container">$Y$</span> has support <span class="math-container">$\mathcal{Y}\equiv \mathcal{Y}_1\cup \mathcal{Y}_2$</span>. <span class="math-container">$X$</span> has support <span class="math-container">$\mathcal{X}$</span>. ... | Explorer | 630,833 | <p>You can view <span class="math-container">$E\{f(X,Y)|Y=y\}$</span> as a special case of conditioning on a set. <span class="math-container">$E\{f(X,Y)|Y=y\}=E\{f(X,Y)|Y\in\{y\}\}$</span>. In general,
<span class="math-container">\begin{align}
E\{f(X,Y)|Y\in\mathcal{Y}_1\} &= \sum_{x\in \mathcal{X}, y\in \mathca... |
2,730,407 | <p>How many different numbers must be selected from the first 25 positive integers to be certain that at least one of them will be twice the other ?</p>
| Fimpellizzeri | 173,410 | <p>Generalizing saulspatz' answer:
partition $\{1,\dots, n\}$ into groups of the form $G_a=\{a\cdot2^k\}$, where $a$ is odd.
There are $\lceil n/2 \rceil$ such groups, one for each odd number on $\{1,\dots, n\}$.</p>
<p>On each group $G_a$, we can alternate between picking a number $($starting from $a)$ and not pickin... |
96,437 | <p>In Mathematica 9.0, the documentation for the Curl function states that in n-dimensions "the resulting curl is an array with depth n-k-1 of dimensions". Accordingly, if a 2-dimensional array is feeded in the Curl function in 3-D space, it returns a scalar value. </p>
<p>However, it does not agree with the definitio... | Hosein Rahnama | 34,873 | <p>Recently, I communicated with wolfram support about this issue and they send me back an illustrative notebook which compares the definition that you got from <a href="https://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Curl_of_a_tensor_field" rel="nofollow noreferrer">wikipedia</a> with that of Mat... |
96,437 | <p>In Mathematica 9.0, the documentation for the Curl function states that in n-dimensions "the resulting curl is an array with depth n-k-1 of dimensions". Accordingly, if a 2-dimensional array is feeded in the Curl function in 3-D space, it returns a scalar value. </p>
<p>However, it does not agree with the definitio... | OA Fakinlede | 65,652 | <p><a href="https://1drv.ms/b/s!AgbbD-KyVrKGheA2lvdi5-qTRRdtmQ" rel="nofollow noreferrer">The curl of a tensor field</a></p>
<p>Alas, the example for the curl of a tensor field in Mathematica 12 is not what is defined in the literature for continuum mechanics. The scalar result they have is the negative of half the tr... |
656,423 | <p>This is a really simple problem but I am unsure if I have proved it properly.</p>
<p>By contradiction:</p>
<p>Suppose that $x \geq 1$ and $x< \sqrt{x}$. Then $x\cdot x \geq x \cdot 1$ and $x^2 < x$ (squaring both sides), which is a contradiction.</p>
| Dave L. Renfro | 13,130 | <p>Assume $x \geq 1.$ Then $x - 1 \geq 0$ and $x > 0,$ and hence</p>
<p>$$ x(x-1) \; = \; x^2 - x \; \geq \; 0$$</p>
<p>since the product of two non-nonegative expressions is non-negative.</p>
<p>Factoring $x^2 - x$ as a difference of squares gives</p>
<p>$$ \left(x - \sqrt{x} \right) \left( x + \sqrt{x} \right)... |
1,071,866 | <p>I am confused by the statement of Sylow's Fourth Theorem:<br>
Let $G$ be a finite group, $p$ a prime. The Sylow $p$-subgroups of $G$ form a single conjugacy class of subgroups.<br>
In particular, I do not understand what it means for the subgroups to form a single conjugacy class?
Thanks!</p>
| Hayden | 27,496 | <p>The Theorem is saying that given two Sylow $p$-subgroups $H$ and $H'$ of the group $G$, then there exists an element $g\in G$ such that $H=gH'g^{-1}$. Moreover, given any $g\in G$ and any Sylow $p$-subgroup $H$, then $gHg^{-1}$ is another Sylow $p$-subgroup.</p>
<p>That is, every pair of Sylow $p$-subgroups are c... |
98,298 | <p>This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it:</p>
<p>For which $p$ is $3$ a cube root in $\mathbb{Q}_p$?</p>
<p>The case $p=3$ for which $X^3-3$ is not separable modulo $p$ can easily be ruled out by checking that $3$ is not a cube modulo $9$. Is there an app... | Henry | 6,460 | <p>It may depend on what you mean by uniformity, but if you mean random and independent with a uniform distribution then you can probably move on to the next issue. </p>
<p>There is a problem that powers of $N$ are not powers of 256 (unless this is an infinite stream) so with your suggestion you cannot be sure that al... |
98,298 | <p>This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it:</p>
<p>For which $p$ is $3$ a cube root in $\mathbb{Q}_p$?</p>
<p>The case $p=3$ for which $X^3-3$ is not separable modulo $p$ can easily be ruled out by checking that $3$ is not a cube modulo $9$. Is there an app... | fedja | 12,992 | <p>If you have a decent random number generator available, then just choose the least $K=2^k>N$ and do the following at each step: generate a random number from $0$ to $K-1$. If it is greater than $N-1$, accept it and write down its bits just halting the stream for this step. If not, discard it, accept one number fr... |
4,146,858 | <blockquote>
<p>Q) For every twice differentiable function <span class="math-container">$f:\mathbb{R}\longrightarrow [-2,2] $</span> with <span class="math-container">$[f(0)]^2+[f'(0)]^2=85$</span> , which of the following statement(s) is(are) TRUE?</p>
</blockquote>
<blockquote>
<p>(A) There exists <span class="math-c... | jjagmath | 571,433 | <p>Why to have a rule for the derivative of <span class="math-container">$A^3$</span> if we can write it as <span class="math-container">$A\cdot (A\cdot A)$</span> and apply two times the product rule?</p>
|
1,585,772 | <p>I am finding this problem confusing :</p>
<blockquote>
<p>If,for all $x$,$f(x)=f(2a)^x$ and $f(x+2)=27f(x)$,then find $a$.</p>
</blockquote>
<p>When $x=1$ I have that $f(1)=f(2a)$ using the first identity.</p>
<p>Then when $x=2a$ I have by the second identity that $f(2a+2)=27f(2a)$,after that I simple stare at ... | user665248 | 665,248 | <p>You can't apply f^-1 to both sides since the function isn't 1:1, by your method, then for x=2, a would be +1/sqrt2 or -1/sqrt2. If you try x=3 you also get a different answer. So your method doesn't work at all; horrendous logic.</p>
|
3,310,193 | <p>I want to show:
<span class="math-container">$$\mu(E)=0 \rightarrow \int_E f d\mu=0$$</span> </p>
<p><span class="math-container">$f:X \rightarrow [0, \infty] $</span> is measurable and <span class="math-container">$E \in \mathcal{A} $</span></p>
<p>Consider a step function <span class="math-container">$s=\sum_i ... | Aloizio Macedo | 59,234 | <p>No. For instance, the map
<span class="math-container">\begin{align*}
F: \mathbb{R}^2 &\to \mathbb{R}^2 \\
(x,y) &\mapsto \frac{1}{4\pi}\left(\cos(2\pi x), \sin(2\pi x)\right)
\end{align*}</span>
takes the unit ball to a circle, which is not contractible. That the map is a contraction can be seen by the der... |
4,498,296 | <p>Is there any subtle way to compute the following integral?</p>
<p><span class="math-container">$$\int \frac{\sqrt{u^2+1}}{u^2-1}~ \mathrm{d}u$$</span></p>
<p>The solution i had in mind was substituting <span class="math-container">$u=\tan (\theta)$</span>,then after a few calculations the integral became <span class... | MathFail | 978,020 | <p>Use reciprocal substitution: <span class="math-container">$u=\frac{1}t$</span>,</p>
<p><span class="math-container">$$I=\frac{1}{2}\int \frac{\sqrt{t^2+1}}{t^2(t^2-1)}d(t^2+1)$$</span></p>
<p>Next, let <span class="math-container">$x=\sqrt{1+t^2}$</span></p>
<p><span class="math-container">$$I=\int \frac{x^2}{(x^2-1... |
1,566,215 | <p>Can someone explain to me the difference between joint probability distribution and conditional probability distribution?</p>
| Brian Tung | 224,454 | <p>Broadly speaking, joint probability is the probability of two things* happening together: e.g., the probability that I wash my car, <em>and</em> it rains. Conditional probability is the probability of one thing happening, given that the other thing happens: e.g., the probability that, given that I wash my car, it r... |
746,750 | <p>This is a consequence of the exponential rule, but how do I actually prove it to be true?</p>
| Spencer | 71,045 | <p>Every definition of $\exp(x)$ leads to the property, $\exp(a+b)= \exp(a)\exp(b)$. Furthermore every definition also tells us that $\exp$ is defined on the entire real line and takes real values when given a real number as an argument.</p>
<p>So we will consider the exponential function evaluated at $x$ and notice t... |
746,750 | <p>This is a consequence of the exponential rule, but how do I actually prove it to be true?</p>
| Haha | 94,689 | <p>We have that $e>0$. Then for every $n\in \Bbb N$ we have that $e^n>0$.Because $\sqrt[m] .$ is increasing then $\sqrt [m] {e^n}>0$. Thus $$e^q>0$$ for every $q\in \Bbb Q$. Now $\Bbb Q$ is dense in $\Bbb R$ thus $e^x>0$ for every $x\in \Bbb R$.</p>
|
1,179,195 | <p>Good day everyone. </p>
<p>I need to know automata theory. Can you advice me the best way to study math?
What themes will I need to know to understand automata theory. What a sequence of study? What level will I need to study intermediate themes? Maybe can you say something yet, what can help me quickly learn autom... | Ross Millikan | 1,827 | <p>If you have two fixed points $a,b$, you have $\phi(a)=a, \phi(b)=b$, so $|\phi(a)-\phi(b)|=|a-b|$, violating the requirement of a contraction that $|\phi(a)-\phi(b)| \lt |a-b|$</p>
|
2,475,617 | <p>EDIT: I first designated $x$, $y$ as irrational numbers. I mean rational.</p>
<p>I have this, In the question it says: For every $x$, $y$ being rational,there exists $z$ being rational so that: $x<z$ or $z<y$ Now, I have this: $\forall(x,y) \in\Bbb Q^2, \exists z\in\Bbb Q/(x<z)∨(z<y)$ Does this the sig... | pseudocydonia | 381,572 | <p>The second statement you have written down is not a well-formed statement in first-order logic. </p>
<p>One way to see this is to look at any logic textbook where they inductively define what statements are admissible, and see that it's impossible to create something that looks like what you've written down.</p>
<... |
55,965 | <p>I'm a games programmer with an interest in the following areas:</p>
<ul>
<li>Calculus</li>
<li>Matrices</li>
<li>Graph theory</li>
<li>Probability theory</li>
<li>Combinatorics</li>
<li>Statistics</li>
<li>More linguistic related fields of logic such as natural language processing, generative grammars</li>
</ul>
<... | Geoff Robinson | 13,147 | <p>There is a nice generalization of this fact due to John Thompson, known as the "Thompson transfer Lemma". It goes as follows: let $G$ be a finite group which has a subgroup $M$ such that $[G:M] = 2d$ for some odd integer $d$, and suppose that $G$ has no factor group of order $2$. Then every element of order $2$ in $... |
3,019,506 | <p>I am stuck on this problem during my review for my stats test. </p>
<p>I know I have to use the convolution formula, and I understand that:</p>
<p><span class="math-container">$f_{U_1}(U_1) = 1$</span> for <span class="math-container">$0≤U_1≤1$</span> </p>
<p><span class="math-container">$f_{U_2}(U_2) = 1$</span... | Pere | 354,985 | <p>I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.</p>
<p>Taking in account that the OP hasn't even stated that all right angles must turn... |
2,246,137 | <p>Let
$$\parallel\overrightarrow{a}\parallel =6\text{ and}\parallel\overrightarrow{b}\parallel =3$$</p>
<p>$$2\overrightarrow{a}+(k-3)\overrightarrow{b}\text{ and } k\overrightarrow{a}-\overrightarrow{b}\text{ are parallel}$$</p>
<p>Find all the value(s) of k.</p>
<p>How to get the value(s) of k?</p>
<p>I tried th... | Jean Marie | 305,862 | <p>These vectors are collinear (I wouldn't use the term "parallel") if and only the determinant of their coordinates with respect to basis $\{a,b\}$ is zero:</p>
<p>$$\det \begin{pmatrix}2 & (k-3) \\ k & -1\end{pmatrix}=0 \ \iff \ -2-k(k-3)=0$$</p>
<p>which is an easy-to-solve quadratic equation.</p>
<p>Rema... |
93,099 | <p>Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?</p>
| Joseph O'Rourke | 6,094 | <p>Let $A$ be the expected area. Then:
$$\lim_{n \rightarrow \infty} \frac{n}{\ln n} (1 - A) = \frac{8}{3} \;.$$
This can be found in many places, e.g., <a href="http://mathworld.wolfram.com/SquarePointPicking.html" rel="noreferrer">this MathWorld article</a>.</p>
<p>[<em>Updated with comparisons between the above fo... |
1,998,391 | <p>To define a <strong>real</strong> exponential function
$$f(x)=a^x=e^{x \,\mathrm{lg} a}$$</p>
<p>It is strictly necessary that $a>0$.</p>
<p>But is the same true if the exponent is a <strong>natural</strong> number?</p>
<hr>
<p>In function series and moreover in power series I see things like
$$\sum_{n\geq0} ... | dxiv | 291,201 | <blockquote>
<p>On the one hand this passage should be valid because it is just the definition of logaritm</p>
</blockquote>
<p>No, it is <em>not</em> valid in general, since $x^n = e^{n \ln x}$ is <em>only</em> valid for $x \gt 0$.</p>
<p>One could technically use $x = |x| \cdot \operatorname{sgn}(x)$ to write it ... |
947,626 | <p>What are the conditions under which the center of a group will have a cyclic subgroup? (with proof, of course)</p>
| Belgi | 21,335 | <p>Hint:every group have a cyclic subgroup of order 1</p>
|
478,713 | <p>I have this logic statement:</p>
<pre><code> (A and x) or (B and y) or (not (A and B) and z)
</code></pre>
<p>The problem is that accessing A and B are rather expensive. Therefore I'd like to access them only once each. I can do this with an if-then-else construct:</p>
<pre><code>if A then
if x then
tru... | Brian M. Scott | 12,042 | <p>The $x$-intercepts are the points where $f(x)=0$, i.e., where the graph touches the $x$-axis. You have $f(3)=(3-3)^2=0$, so you have an $x$-intercept at $x=3$.</p>
|
511,150 | <p>I want to check if I understand proof by induction, so I want to proof the following:</p>
<p>$a^n<b^n$ for $a,b \in \mathbb{R}$, $0<a<b$, $n \in \mathbb{N}$ and $n>0$</p>
<p>Here's my attempt:</p>
<blockquote>
<p><strong>Base Case</strong></p>
<p>If $a=1$ and $b=2$, then $1^n < 2^n$ for any ... | Cameron Buie | 28,900 | <p>As it stands, your proposition is still incorrect. Consider for example $a=-2,b=-1$. Then $a<b,$ but $b^2<a^2$. Now, if we make the further assumption that $a,b$ are both <em>positive</em>, then it works out fine.</p>
<p>Your base case is actually immediate, since $a^1=a<b=b^1$. (We must allow $a,b$ to be ... |
108,594 | <p>I would like to know if one can weaken conditions of Proposition 2.8 in
<a href="http://www.jmilne.org/math/xnotes/CA.pdf" rel="nofollow">http://www.jmilne.org/math/xnotes/CA.pdf</a></p>
<p>The proposition says that if an ideal $a$ in a ring $A$ is contained in the union of ideals $p_1,...,p_r$ with $p_2,...,p_r$ ... | Hans Schoutens | 22,873 | <p>Some variants of prime avoidance (as this property is usually called) are in Eisenbud's "Commutative Algebra with a View..." on p. 114, including an example where it fails: for instance the ideal $(x,y)$ in $\mathbb Z/2\mathbb Z[x,y]/(x,y)^2$ is the union of three (smaller) non-prime ideals.</p>
|
1,176,098 | <p>Here are some of my ideas:</p>
<p><strong>1. Addition Formula:</strong> <span class="math-container">$\sin{x}$</span> and <span class="math-container">$\cos{x}$</span> are the unique functions satisfying:</p>
<ul>
<li><p><span class="math-container">$\sin(x + y) = \sin x \cos y + \cos x \sin y $</span></p>
</li>
<li... | Jade Vanadium | 813,439 | <p>Here's a way to formalize the geometric intuition. We can define <span class="math-container">$\sin,\cos$</span> as the projections of the helix <span class="math-container">$\text{cis}(x)=:(\cos(x),\sin(x))$</span>, where <span class="math-container">$\text{cis}:\mathbb{R}\to\mathbb{R}^2$</span> is the unique <span... |
1,025,671 | <p>When we have something in this form</p>
<p>$$\sqrt{x + a} = \sqrt{y + b},$$</p>
<p>a common technique to solve is to square both side so that:</p>
<p>$$(\sqrt{x + a})^2 = (\sqrt{y + b})^2 \implies x + a = y + b.$$</p>
<p>I'm an engineer and not a mathematician. As I understand it engineers do lots of things that... | Michael Albanese | 39,599 | <p>Travelling down the alternative road, we see that</p>
<p>\begin{align*}
\sqrt{x+a} &= \sqrt{y+b} & \\
\sqrt{x+a}\times\sqrt{y+b} &= \sqrt{y+b}\times\sqrt{y+b} &(\text{multiplying both sides by}\ \sqrt{y+b})\\
\sqrt{x+a}\times\sqrt{y+b} &= y + b &(\text{simplifying})\\
\sqrt{x+a}\times\sqrt{x... |
4,151,381 | <p>I am a nube just getting into mathematics and set theory.</p>
<p>I am learning about how we can produce the list of ordinal numbers by purely using the null set, with 0 standing for Ø, 1 standing for {Ø}, 2 standing for {Ø, {Ø}} and so forth. What I am confused about is the operation at play here to produce the larg... | Thomas Andrews | 7,933 | <p>An idea, too long for a comment.</p>
<p>For any <span class="math-container">$y\in X,$</span> <span class="math-container">$h^{-1}(y)$</span> is closed, and hence compact subset of <span class="math-container">$X.$</span></p>
<p>If <span class="math-container">$y\in \Omega(g),$</span> if you want <span class="math-c... |
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