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 |
|---|---|---|---|---|
332,927 | <p>there are two bowls with black olives in one and green in the other. A boy takes 20 green olives and puts in the black olive bowl, mixes the black olive bowl, takes 20 olives and puts it in the green olive bowl. The question is -</p>
<p>Are there more green olives in the black olive bowl or black olive in the green... | joriki | 6,622 | <p>There are just as many green olives in the black olive bowl as there are black olives in the green olive bowl. The number of olives in each bowl hasn't changed; hence there has merely been an exchange of some black olives for an equal number of green olives.</p>
|
194 | <p>In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier relationship with math. But in the present situation, how can we help the students who come to our classes, which the... | Sue VanHattum | 60 | <p>When I posted this question over a year ago, I meant to post my own answer after giving others a bit of time to post. I apparently forgot.</p>
<p>My students have had some success in decreasing their anxiety with books like <em><a href="http://www.betterworldbooks.com/mind-over-math-put-yourself-on-the-road-to-succ... |
1,723,942 | <p>The four color theorem declares that any map in the plane (and, more generally, spheres and so on) can be colored with four colors so that no two adjacent regions have the same colors. </p>
<p>However, it's not clear what constitutes a map, or a region in a map. Is this actually a theorem in graph theory, something... | Plutoro | 108,709 | <p>Here is a formal statement. Let $G=(V,E)$ be a finite planar graph with vertices $V$ and edges $E$. It is possible to find a collection $P=\{S_1,S_2,S_3,S_4\}$ of dijoint subsets of $V$ such that $V=S_1\cup S_2\cup S_3\cup S_4$ and for every $a,b\in S_i$, there is no edge in $E$ joining $a$ and $b$.</p>
<p>The map ... |
1,281,967 | <p>This is a dumb question I know.</p>
<p>If I have matrix equation $Ax = b$ where $A$ is a square matrix and $x,b$ are vectors, and I know $A$ and $b$, I am solving for $x$.</p>
<p>But multiplication is not commutative in matrix math. Would it be correct to state that I can solve for $A^{-1}Ax = A^{-1}b \implies x =... | OKPALA MMADUABUCHI | 98,218 | <p>If $A$ is invertible, the answer is yes. Otherwise, it doesn't make sense to write that. Also in that case you may not have any solution. </p>
|
669,582 | <p>Let $X,τ$ be a topological space. Prove that a subset $A$ of $X$ is dense if and only if every open subset of $ X$ contain some point of $A$</p>
<p>this is what I got</p>
<p>Let $X,τ$ be a topological space</p>
<p>Part 1: Assume that a subset $A$ of $X$ is dense, show that every open subset of $X$ contain some po... | Hagen von Eitzen | 39,174 | <p>The complement of the closed set $\operatorname{Cl}A$ is open and disjoint to $A$. By assumption the only open set disjoint to $A$ is the empty set.</p>
|
1,447,852 | <p>Compute this sum:</p>
<p><span class="math-container">$$\sum_{k=0}^{n} k \binom{n}{k}.$$</span></p>
<p>I tried but I got stuck.</p>
| Calvin Khor | 80,734 | <p><strong>Hint.</strong> Let $X\sim \text{Bin}(n,p)$. Then it is known that
$$\Bbb EX = np$$
But of course, $\Bbb EX = \sum_{k=0}^n k \Bbb P(X=k) = \sum_{k=0}^n k \binom{n}{k}p^k(1-p)^k$. So</p>
<p>$$np = \sum_{k=0}^n k \binom{n}{k}p^k(1-p)^k$$</p>
<p>Try finishing from here.</p>
|
4,036,761 | <blockquote>
<p>Let G be the additive group of all polynomials in <span class="math-container">$x$</span> with integer coefficients.
Show that G is isomorphic to the group <span class="math-container">$\mathbb{Q}$</span>* of all positive rationals (under
multiplication).</p>
</blockquote>
<p>This question is from my ab... | lhf | 589 | <p><em>Hint:</em> The fundamental theorem of arithmetic implies that
<span class="math-container">$$
\mathbb{Q}^\times_+ \cong \bigoplus_p \mathbb Z
$$</span></p>
|
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Duncan Ramage | 405,912 | <p>$44^2 = 1936 < 2017 < 2025 = 45^2$.</p>
<p>Really, I don't think there's much to this one except for "try squaring small integers until you find the right ones".</p>
|
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Robert Soupe | 149,436 | <p>Try graphing $\sqrt x$ for $x \geq 0$. You should see a fairly smooth curve that goes upwards, which means that if $a < b$ and they're both positive, then $\sqrt a < \sqrt b$.</p>
<p>From this, it's clear that the integers you want are $\lfloor \sqrt{2017} \rfloor$ and $\lceil \sqrt{2017} \rceil$. A calculato... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Will Jagy | 10,400 | <p>I can only imagine this was intended to be about
$$ (10a + 5)^2 = 100 a (a+1) + 25, $$
$$ 15^2 = 225, $$
$$ 25^2 = 625, $$
$$ 35^2 = 1225, $$
$$ 45^2 = 2025. $$
Then
$$ 44^2 = 2025 - 2 \cdot 45 + 1 = 2025 - 90 + 1 < 2017. $$</p>
<p>EXAMPLE: factor $10001 = 10^4 + 1$</p>
<p>$$ 105^2 = 11025 $$
$$ 105^2 ... |
1,994,922 | <p>Given $B_1, B_2,\ldots$ are independent and bounded variables with $E(B_i) = 0$ for all $i=1,2,\ldots$. Define $S_n = B_1+ B_2+\ldots + B_n$ with variance $s_n^2\rightarrow \infty$. Prove that $\frac{S_n}{s_n}$ has a central limit.</p>
<p><strong>My attempt:</strong> Due to the given condition, without i.i.d proper... | Remy | 325,426 | <p>Since each <span class="math-container">$X_k$</span> is bounded then for any <span class="math-container">$\epsilon>0$</span> there exists <span class="math-container">$N\in\mathbb N$</span> such that</p>
<p><span class="math-container">$$\mathbf{1}\left(|X_k|>\epsilon s_n\right)=0$$</span></p>
<p>for all <spa... |
1,905,863 | <p>I'm on the section of my book about separable equations, and it asks me to solve this:</p>
<p>$$\frac{dy}{dx} = \frac{ay+b}{cy+d}$$</p>
<p>So I must separate it into something like: $f(y)\frac{dy}{dx} + g(x) = constant$</p>
<p>*note that there are no $g(x)$</p>
<p>but I don't think it's possible. Is there someth... | Ian Miller | 278,461 | <p>You can always rearrange it as follows:</p>
<p>$$\frac{dy}{dx} = \frac{ay+b}{cy+d}$$</p>
<p>$$\frac{cy+d}{ay+b}\frac{dy}{dx} = 1$$</p>
<p>It is clearly separated and you can integrate with respect to $x$ to get:</p>
<p>$$\int\frac{cy+d}{ay+b}\frac{dy}{dx}dx = \int dx$$</p>
<p>$$\int\frac{cy+d}{ay+b}dy = x$$</p>... |
327,750 | <p>$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$</p>
<p>The results is obvious enough, but how to prove this</p>
| Brian M. Scott | 12,042 | <p>For $n\in\Bbb Z^+$ let $B_n=A_n\setminus\bigcup_{k=1}^{n-1}A_k=A_n\cap\bigcap_{k=1}^{n-1}A_k^c$; you want to show that $$\bigcup_{n\ge 1}A_n=\bigcup_{n\ge 1}B_n\;.$$</p>
<p>Clearly it suffices to show that </p>
<p>$$\bigcup_{k\ge 1}A_k\subseteq\bigcup_{k\ge 1}B_k\;.$$</p>
<p>For $x\in\bigcup_{k\ge 1}A_k$ let $n(x... |
1,074,534 | <p>How can I get started on this proof? I was thinking originally:</p>
<p>Let $ n $ be odd. (Proving by contradiction) then I dont know.</p>
| ajotatxe | 132,456 | <p>If $n$ is odd, let $p$ be its smallest prime divisor, and $p^r$ the greatest power of $p$ that divides $n$. Then, the number
$$\frac{2^rn}{p^r}$$
has the same number of divisors, it is smaller than $n$ and it is even.</p>
|
187,618 | <p>I am trying to solve the following problem.</p>
<p>The time $T$ required to repair a machine is an exponentially distributed random variable with mean 10 hours.</p>
<p>a) What is the probability that a repair takes at least 15 hours given that its duration
exceeds 12 hours?
b) What is the probability that the comb... | Dilip Sarwate | 15,941 | <p>While the numerical answer you get for part (a) is correct, I think that your work
indicates some misinterpretation of the concepts. $T$ is the time required to <em>complete</em>
a repair, and its <em>complementary cumulative distribution function</em> is $\exp(-t/10)$,
that is,
$$P\{T > t\} = 1 - F_T(t) = e^{-t... |
522,289 | <p>It is an exercise on the lecture that i am unable to prove.</p>
<p>Given that $gcd(a,b)=1$, prove that $gcd(a+b,a^2-ab+b^2)=1$ or $3$, also when will it equal $1$?</p>
| Zafer Sernikli | 98,237 | <p>Let $d|a+b \quad(1)$ and $d|a^2-ab+b^2 (2)$</p>
<p>$ d|a^2-ab+b^2 \quad (2)\implies d|(a+b)^2 - 3ab \qquad \qquad(3)$ </p>
<p>And as $d|a+b$ then, $(3) \land (1) \implies d|-3ab \implies d|3ab \implies d|3 \vee d|a \vee d|b$ </p>
<p>On the other hand, as $d|a+b$, if $d|a \implies d|b$, and vice verse. And as $gc... |
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | Daniel Loughran | 5,101 | <p>It seems like you want to discover analytic number theory. There is a lot of it. A good comprehensive modern book I would recommend is </p>
<p>Iwaniec, Kowalski - Analytic number theory.</p>
<p>Example areas with applications of harmonic analysis include the circle method and modular forms.</p>
|
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | Brendan Murphy | 36,862 | <p>The fourth volume of Stein & Shakarchi's series on analysis has a nice account of the application of harmonic analysis to lattice point problems (e.g. Gauss' circle problem and the Dirichlet divisor problem).</p>
<p>Martin Huxley's "Area, Lattice Points, and Exponential Sums" is a more thorough (but very readab... |
528,456 | <p>I have a question regarding L'hospital's rule. </p>
<p>Why can I apply L'hospital's rule to $$\lim_{x\to 0}\frac{\sin 2x}{ x}$$ and not to $$\lim_{x\to 0} \frac{\sin x}{x}~~?$$</p>
| hmakholm left over Monica | 14,366 | <p>You <em>can</em> apply L'Hospital's rule just fine to $\lim_{x\to 0}\frac{\sin x}{x}$. It just doesn't tell you anything you didn't already know, because it concludes that the limit is
$$ \frac{\sin'(0)}{1} = \sin'(0) = \lim_{h\to 0}\frac{\sin(0+h)-\sin(0)}{h} = \lim_{h\to 0}\frac{\sin h - 0}{h} = \lim_{h\to 0}\frac... |
2,871,729 | <p>Given any $\alpha > 0$, I need to show that for $ x \in [0,\infty)$
\begin{equation}
\lim_{x\to 0} x^{\alpha}e^{|\log x|^{1/2}}=0
\end{equation}</p>
<p>I have tried using L'Hospital's rule. But I am not able to arrive at answer. </p>
<p>Thank you in advance.</p>
| Ahmad Bazzi | 310,385 | <p>Let $$f(x) = x^{\alpha} e^{\sqrt{\vert \log x \vert}}$$
Consider
$$\log f(x) = \alpha \log x + \sqrt{\vert \log x \vert} = \log x (\alpha + \frac{ \sqrt{\vert \log x \vert}}{\log x} )=\log x (\alpha - \frac{ \sqrt{\vert \log x \vert}}{\sqrt{\vert \log x \vert}\sqrt{\vert \log x \vert}} )$$
which is
$$\log f(x) = \l... |
2,062,398 | <p>May you tell me if my translation to symbolic logic is correct? </p>
<p>Thank you so much! Here is the problem:</p>
<p>To check that a given integer $n > 1$ is a prime, prove that it is enough to show that $n$ is not divisible by any prime $p$ with $p \le \sqrt{n}$.</p>
<p>$$\forall p \in P ~\forall n \in N ~(... | C. Falcon | 285,416 | <p>First of all, let us find a nonzero annihilator polynomial of $\sqrt{2}+\sqrt[3]{5}.$ Let $p=x^2-2$ and $q=x^3-5$, $p$ is a nonzero annihilator of $\sqrt{2}$ and $q$ is a nonzero annihilator polynomial of $\sqrt[3]{5}$. Hence, the <a href="https://en.wikipedia.org/wiki/Resultant" rel="nofollow noreferrer">resultant<... |
2,064,284 | <blockquote>
<p>Prove that the sequence $\{y_n\}$ where $y_{n+2}=\frac{y_{n+1} +2 y_{n}}{3}$ $n\geq 1$, $0<y_1<y_2$, is convergent by using subsequencial criteria, <strong>by showing $\{y_{2n}\}$ and $\{y_{2n-1}\}$ converges to the same limit. Find the limit also</strong>.</p>
</blockquote>
<p>I can solve it b... | Mark | 310,244 | <p>Yes. By definition, the minimal polynomial $m_A$ divides any other polynomial that annihilates $A$. So, if some polynomial $f$ of equal degree annihilates $A$, then $m_A\mid f$, so $m_A = cf$ where $c$ is some constant.</p>
|
3,385,420 | <p>The question is from <em>Cambridge Admission Test 1983</em></p>
<blockquote>
<p>A room contains m men and w women. They leave one by one at random until only people of the same sex remain. show by a carefully explained inductive argument, or otherwise, that the expected number of people remaining is <span class="... | Floris Claassens | 638,208 | <p>If <span class="math-container">$m+w=1$</span> we have that <span class="math-container">$\frac{m}{w+1}+\frac{w}{m+1}=1$</span> which is the expected number of people remaining. </p>
<p>Now let <span class="math-container">$N$</span> be arbitrarily given and suppose the expected number of people remaining if <span ... |
2,431,548 | <p>Okay, so, my teacher gave us this worksheet of "harder/unusual probability questions", and Q.5 is real tough. I'm studying at GCSE level, so it'd be appreciated if all you stellar mathematicians explained it in a way that a 15 year old would understand. Thanks!</p>
<p>So, John has an empty box.
He puts some red cou... | green frog | 351,828 | <p>So the ratio of the number of red counters to blue is 1:4. That is, our first pull for a red has a probability of 1/5. </p>
<p>Let $R$ be the number of red counters. Then the probability to pull out a red after our first is $(R - 1)/(5R - 1).$ So we have $(1/5)((R - 1)/(5R - 1)) = 6/155.$ Does that make sense? I th... |
2,431,548 | <p>Okay, so, my teacher gave us this worksheet of "harder/unusual probability questions", and Q.5 is real tough. I'm studying at GCSE level, so it'd be appreciated if all you stellar mathematicians explained it in a way that a 15 year old would understand. Thanks!</p>
<p>So, John has an empty box.
He puts some red cou... | john doe | 476,378 | <p>Take number of red counters to be $x$, number of blue counters to be $4x$. Then required probability is $^{x}C_2/^{5x}C_2 = 6/155.$
$\qquad^nC_r = n!/((n-r)!r!)\quad$<br>
Solving, we get $x(x-1)/(5x(5x-1))=6/155$, which gives $x=25.$</p>
|
3,364,016 | <p>Can the following expression be written as the factorial of <span class="math-container">$m$</span>?</p>
<p><span class="math-container">$m(m-1)(m-2) \dots {m-(n-1)}$</span></p>
| Ross Millikan | 1,827 | <p>No, but you can write <span class="math-container">$$m(m-1)(m-2) \dots {m-(n-1)}=\frac {m!}{(m-n)!}$$</span>
Note which factors are missing from <span class="math-container">$m!$</span> in your original expression</p>
|
853,774 | <blockquote>
<p>If $(G,*)$ is a group and $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is abelian for all $a,b \in G$.</p>
</blockquote>
<p>I know that I have to show $G$ is commutative, ie $a * b = b * a$</p>
<p>I have done this by first using $a^{-1}$ on the left, then $b^{-1}$ on the right, and I end up with and expres... | Asinomás | 33,907 | <p>$abab=a^2b^2\implies a^{-1}abab=a^{-1}a^2b^2\implies bab=ab^2\implies bab b^{-1}= ab^2b^{-1}\implies ba=ab$</p>
|
1,547,122 | <p>If $\lim_{x \to x0} f(x) = L$, then $\lim_{x \to x0} \lvert f(x)\rvert = \lvert L \rvert$.
<br>I know this is true, because $\lvert f(x) \rvert - \lvert L \rvert <= \lvert f(x) - L \rvert < \epsilon$ but why is it bigger than minus Epsilon?</p>
| Ethan Alwaise | 221,420 | <p>It follows from the fact that
$$\vert \vert a \vert - \vert b \vert \vert \leq \vert a - b \vert$$
for all $a,b \in \mathbb{R}$. So if $\vert x - x_0 \vert < \delta$, then
$$\vert \vert f(x) - \vert L \vert \vert \leq \vert f(x) - L \vert < \epsilon,$$
i.e. $\lim_{x \to x_0}\vert f(x) \vert = \vert L \vert$.</... |
1,547,122 | <p>If $\lim_{x \to x0} f(x) = L$, then $\lim_{x \to x0} \lvert f(x)\rvert = \lvert L \rvert$.
<br>I know this is true, because $\lvert f(x) \rvert - \lvert L \rvert <= \lvert f(x) - L \rvert < \epsilon$ but why is it bigger than minus Epsilon?</p>
| Empiricist | 189,188 | <p>Just use the other side of the triangle inequality</p>
<p>$$\epsilon > \vert f(x) - L \vert = \vert L - f(x) \vert \geq \vert L \vert - \vert f(x) \vert \implies -\epsilon < \vert f(x) \vert - \vert L \vert$$</p>
|
1,648,587 | <blockquote>
<p><strong>Problem.</strong> Consider two arcs <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> embedded in <span class="math-container">$D^2\times I$</span> as shown in the figure. The loop <span class="math-container">$\gamma$</span> is obviously nullhomotopic ... | ChesterX | 54,151 | <p>You can split the space $Y=D^2\times I \setminus \alpha\cup\beta$ in the following way :
<a href="https://i.stack.imgur.com/aiirI.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/aiirI.jpg" alt="Hatcher 1.2.10"></a></p>
<p>Here $X=A\cap B$. Carefully label all the "missing lines" in $A$ and $B$, w... |
181,110 | <p>On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true? </p>
| davidlowryduda | 9,754 | <p>Every now and then it's nice to <a href="https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts">nuke a mosquito</a>.</p>
<p>Let's assume that the path connecting two points $(a,y(a))$ and $(b,y(b))$ can be expressed as a function, and the curve $C(x)$ is given by $C(x) = (x,y(x))$. T... |
181,110 | <p>On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true? </p>
| user02138 | 2,720 | <p>Let $\gamma(s)$ be a continuous curve in the plane with end-points $\gamma(0) = a$ and $\gamma(1) = b$. Using the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation#Examples" rel="nofollow">Euler-Lagrange</a> equations, the only stationary solution is $\gamma(s) = bs + (1 - s)a$, which is a line c... |
88,788 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/51292/relation-of-this-antisymmetric-matrix-r-beginpmatrix-01-10-endpmatr">Relation of this antisymmetric matrix $r = \begin{pmatrix} 0&amp;1\\ -1&amp;0 \end{pmatrix}$ to $i$</a> </p>
</blockquote>
... | Patrick Da Silva | 10,704 | <p>Here's a hint : If you consider a complex number $z$, it can be written as $a+bi$ with $a,b \in \mathbb R$. (I haven't chosen the letters $a$ and $b$ for no reason.) </p>
<p>If you want a proof I don't mind showing. Just ask.</p>
<p>Hope that helps,</p>
|
4,304 | <p>I am trying to understand <a href="http://en.wikipedia.org/wiki/All-pairs_testing" rel="nofollow noreferrer"><strong>pairwise testing</strong></a>.</p>
<p>How many combinations of tests would be there for example, if</p>
<blockquote>
<p><code>a</code> can take values from 1 to m</p>
<p><code>b</code> can take values... | Bill Dubuque | 242 | <p>If each parameter had $10$ choices you'd be testing $300$ vs $1000$ combinations, namely hold $\rm a$ constant and vary $\rm b,c$ through $10\cdot 10 = 100$ values. Similarly hold, $\rm b$ constant; then $\rm c$. As the number of variables $\rm k$ increases you get better savings, roughly $\rm (k N)^2$ vs. $\rm N^k$... |
2,005,798 | <p>I have the following equality:
$$
\lim_{k \to \infty}\int_{0}^{k}
x^{n}\left(1 - {x \over k}\right)^{k}\,\mathrm{d}x = n!
$$</p>
<p>What I think is that after taking the limit inside the integral ( maybe with the help of Fatou's Lemma, I don't know how should I do that yet ), then get</p>
<p>$$
\int_{0}^... | Théophile | 26,091 | <p>The first $n-1$ flips in some sense don't matter at all, because the last flip will make the total number of heads even or odd with equal probability. So the probability is $0.5$.</p>
|
1,686,568 | <p>I am learning about tensor products of modules, but there is a question which makes me very confused about it! </p>
<p>If $E$ is a right $R$-module and $F$ is a left $R$-module, then suppose we have a balanced map (or bilinear map) $E\times F\to E\otimes F$. If some element $x\otimes y \in E\otimes F$ is $0$, then ... | dfsfljn | 320,073 | <p>By the universal property of tensor product, an elementary tensor <span class="math-container">$x\otimes y$</span> equals zero if and only if for every <span class="math-container">$R$</span>-bilinear map <span class="math-container">$B:E\times F\to M$</span>, <span class="math-container">$B(x,y)=0$</span>. While th... |
1,686,568 | <p>I am learning about tensor products of modules, but there is a question which makes me very confused about it! </p>
<p>If $E$ is a right $R$-module and $F$ is a left $R$-module, then suppose we have a balanced map (or bilinear map) $E\times F\to E\otimes F$. If some element $x\otimes y \in E\otimes F$ is $0$, then ... | paul garrett | 12,291 | <p>To add to other good answers and comments: perhaps a one slightly abstracted version of the question can be construed as asking about the <em>exactness</em> of the tensor product functor(s) <span class="math-container">$M\to M\otimes N$</span>. This is not an exact functor (in many interesting categories), and its f... |
446,272 | <p>let
$$\left(\dfrac{x}{1-x^2}+\dfrac{3x^3}{1-x^6}+\dfrac{5x^5}{1-x^{10}}+\dfrac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$$</p>
<p>How find the $a_{2^n}=?$</p>
<p>my idea:let
$$\dfrac{nx^n}{1-x^{2n}}=nx^n(1+x^{2n}+x^{4n}+\cdots+x^{2kn}+\cdots)=n\sum_{i=0}^{\infty}x^{(2k+1)n}$$
Thank you everyone</... | Ethan Splaver | 50,290 | <p>Let $\chi_2(n)$ be the Dirichlet character modulo $2$</p>
<p>Define $\sigma'(n)=\sum_{d\mid n}d\chi_2(d)$</p>
<p>$$\sum_{n=1}^\infty\sigma'(n)x^n=\sum_{n=1}^\infty \frac{\chi_2(n)nx^n}{1-x^{2n}}=\sum_{n \text{ odd}}\frac{nx^n}{1-x^{2n}}$$</p>
<p>$$(\sum_{n \text{ odd}}\frac{nx^n}{1-x^{2n}})^2=\sum_{n=2}^\infty\su... |
933,487 | <p>How do I find it?</p>
<p>I know that $\mathcal{L}(e^t \cos t) =\frac{s-1}{(s-1)^2+1^2}$ but what is it when multiplied by $t$, as written in the title?</p>
| Mosk | 175,514 | <p>$$\text{Another approach: } \mathcal{L}(e^ttcost)=F(s-1)$$
$$\mathcal{L}(tcost)=-\frac{d}{ds}(\frac{s}{s^2+1})=\frac{s^2-1}{(s^2+1)^2}$$
$$\text{so the final answer is:}$$
$$\ F(s-1)=\frac{(s-1)^2-1}{[(s-1)^2+1]^2}$$</p>
|
3,457,277 | <p>why Pi is transcendental number if <span class="math-container">$\pi$</span> also have algebraic equation like below which have root at <span class="math-container">$x =\pi/3$</span> as <span class="math-container">$n$</span> tends to infinity.
<span class="math-container">$$\Biggl(\biggl(\Bigl(\bigl((x^2-2^{(n2... | Noah Schweber | 28,111 | <p>All you've shown is that <span class="math-container">$\pi$</span> can be <strong>approximated by</strong> algebraic numbers. But that's not the same as <strong>being</strong> an algebraic number! When you write </p>
<blockquote>
<p><span class="math-container">$\pi$</span> can also be represented as a root of al... |
299,471 | <p>I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product identifying the wedge (or smash product) , what just remains is the 0.cell and the 2-cell...This is very theoretical and I don... | Steve Costenoble | 58,888 | <p>I like to think about it this way: Start with a circle and cross with an interval rather than another circle, to get a cylinder. You're next going to identify the two ends of the cylinder and a path connecting them. If you can't picture that immediately, start by pinching off each end of the cylinder, to get a spher... |
1,849,797 | <p>Complex numbers make it easier to find real solutions of real polynomial equations. Algebraic topology makes it easier to prove theorems of (very) elementary topology (e.g. the invariance of domain theorem).</p>
<p>In that sense, what are theorems purely about rational numbers whose proofs are greatly helped by the... | Mikhail Katz | 72,694 | <p>Real numbers together with all the other completions of the rationals known as the $p$-adics are very useful indeed in finding <em>rational</em> solutions of quadratic forms. A general principle known as <a href="https://en.wikipedia.org/wiki/Hasse_principle" rel="nofollow">Hasse principle</a> asserts that a quadrat... |
109,754 | <p>Please help me get started on this problem:</p>
<blockquote>
<p>Let <span class="math-container">$V = R^3$</span>, and define <span class="math-container">$f_1, f_2, f_3 ∈ V^*$</span> as follows:<br />
<span class="math-container">$f_1(x,y,z) = x - 2y$</span><br />
<span class="math-container">$f_2(x,y,z) = x + y + ... | David Mitra | 18,986 | <p>First, you should verify that the $f_i$ are elements of $V^*$; that is, that they are functions on $\Bbb R^3$ that satisfy:</p>
<p>$\ \ \ $1) $f({\bf x}+{\bf y})=f({\bf x})+f({\bf y})$ for all ${\bf x},{\bf y}\in\Bbb R^3$</p>
<p>and</p>
<p>$\ \ \ $2) $f(c{\bf x})=cf({\bf x})$ for all $c\in\Bbb R$ and all ${\bf x... |
1,046,687 | <p>I have this math problem: "determine whether the series converges absolutely, converges conditionally, or diverges."</p>
<p>I can use any method I'd like. This is the series:$$\sum_{n=1}^{\infty}(-1)^n\frac{1}{n\sqrt{n+10}}$$</p>
<p>I though about using a comparison test. But I'm not sure what series I can compare... | Learnmore | 294,365 | <p>Converges absolutely</p>
<p>$|u_n|=\frac{1}{n\sqrt{(n+10)}}<\frac{1}{n}.\frac{1}{n^{\frac{1}{2}}}=\frac{1}{n^{\frac{3}{2}}}$</p>
|
1,046,687 | <p>I have this math problem: "determine whether the series converges absolutely, converges conditionally, or diverges."</p>
<p>I can use any method I'd like. This is the series:$$\sum_{n=1}^{\infty}(-1)^n\frac{1}{n\sqrt{n+10}}$$</p>
<p>I though about using a comparison test. But I'm not sure what series I can compare... | GorTeX | 196,508 | <p>We can use Leibniz theorem: if <code>a_n</code> is a monotone sequence that converges to 0, then the series $\sum_{n=1}^{\infty}(-1)^n a_n$ converges. Let´s have $a_n=\dfrac{1}{n\sqrt{n+10}}$.
So we have to prove that $lim$ an=0 and that it is monotone. If you can show that, you have that it converges </p>
|
916,794 | <p>I have to find the value of $m$ such that:</p>
<p>$\displaystyle\int_0^m \dfrac{dx}{3x+1}=1.$</p>
<p>I'm not sure how to integrate when dx is in the numerator. What do I do?</p>
<p>edit: I believe there was a typo in the question. Solved now, thank you!</p>
| Jack D'Aurizio | 44,121 | <p>Since:
$$\int_{0}^{m}\frac{dx}{3x+1}=\frac{1}{3}\log(3m+1)$$
we have:
$$ m = \frac{e^3-1}{3}.$$</p>
|
2,337,524 | <p>We have $p(x)$ a degree $m$ polynomial and $q(x)$ a degree $k$ polynomial. We also know that $p(x) = q(x)$ has at least $n+1$ solutions. And, $n\geq m\land n\geq k$.</p>
<p>Now, I tried graphing a little to see if I see a pattern </p>
<p>I tried making $$y= x^{2} $$
and
$$y = -x^{2}+5 $$
There were two points of... | hamam_Abdallah | 369,188 | <p><strong>just a hint</strong></p>
<p>If the equation $R (x)=P (x)-Q (x)=0$ has at least $n+1$ solutions, it means that degree of $R (x)\ge n+1$.</p>
<p>but degree of $R (x)\le \max(m,k) $
thus
$m\ge n+1$</p>
<p>or $k\ge n+1$</p>
|
1,850,418 | <p>An argument has two parts, the set of all premises, and the conclusion drawn from said premise. Now since there's only 1 conclusion, it would be weird to choose a name for the 'second' part of the argument. However, what is the first part called? I used to think that this was actually called the premise, however tha... | lemontree | 344,246 | <p>I'd say the answer is already in the title of your question: You can simply call it the <strong>set of premises</strong>, or the <strong>premise set</strong>.<br>
This indicates that it is (possibly multiple, but might in principle be the empty set or a singleton set) a set of propositions serving altogether as the ... |
2,916,306 | <p>Let $f(x,y) = x\ln(x) + y\ln(y)$ be defined on space </p>
<p>$S = \{(x,y) \in \mathbb{R}^2| x> 0, y > 0, x + y = 1\}$.</p>
<p>My question is, how do I take the partial derivative for this function, given that the parameters are coupled through $x+y = 1$.</p>
<p>A first idea would be to do it ignoring the co... | Nosrati | 108,128 | <p>You can't use $y=1−x$, these variables are independent. Although we choose our points from
$$S = \{(x,y) \in \mathbb{R}^2| x> 0, y > 0, x + y = 1\}$$
but it doesn't mean these variables are dependent. Then the substitute $y = 1-x$ makes no sense in this <strong>two variables function</strong>.</p>
|
3,848,179 | <blockquote>
<p>The velocity <span class="math-container">$v$</span> of a freefalling skydiver is modeled by the differential equation</p>
<p><span class="math-container">$$ m\frac{dv}{dt} = mg - kv^2,$$</span></p>
<p>where <span class="math-container">$m$</span> is the mass of the skydiver, <span class="math-container... | Lion Heart | 809,481 | <p><span class="math-container">$\sqrt{2^2 - x^2}=2\sqrt{1-(\frac{x}{2})^2}=2\sqrt{1-sin^2t}=2cost$</span> and say <span class="math-container">$x=2sint, t=arcsin\frac{x}{2}, \implies dx=2costdt$</span></p>
|
462,569 | <blockquote>
<p>Consider the polynomial ring <span class="math-container">$F\left[x\right]$</span> over a field <span class="math-container">$F$</span>. Let <span class="math-container">$d$</span> and <span class="math-container">$n$</span> be two nonnegative integers.</p>
<p>Prove：<span class="math-container">$x^d-1 \... | Prism | 49,227 | <p>Suppose $x^{d}-1\mid x^{n}-1$. By division algorithm, we can write $n=qd+r$ for some $q,r\in\mathbb{N}_0$ with $0\le r<d$. Now, observe that
$$x^{d}-1 \mid (x^{d}-1)(x^{n-d}+x^{n-2d}+\cdots + x^{n-qd}+1)$$
Expanding the above, and cancelling many terms, we get that
$$x^{d}-1 \mid x^{n}+x^{d}-x^{n-qd}-1=x^{n}-1+x^... |
118,070 | <p>I have the following code:</p>
<pre><code>a = 6.08717*10^6;
b = a/3;
c = a*1.5;
d = a^2;
matrix={{a, b, c, d},{b, c, d, a},{c, d, a, b},{d, a, b, c}};
matrix // EngineeringForm
</code></pre>
<p>Normally, I use this result by copying (<code>Copy As ► MathML</code>) and pasting into Microsoft Word.</p>
<p>However, ... | Alexey Popkov | 280 | <h2>Original answer</h2>
<p>The reason for getting extra quotes is that these extra quotes are explicitly present in the box form of the expression generated by such functions as <code>EngineeringForm</code>, <code>NumberForm</code> etc.:</p>
<pre><code>ToBoxes@EngineeringForm[6.08717*10^6]
</code></pre>
<blockquot... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | Evan Kearney | 152,838 | <p>I have the blessing of not being quite sure what type of mathematical insight and clarity each day will bring. Some days, despite little sleep, i may be physically tired, but it allows me to stay at the desk scribbling away and I make some progress (even if it is not directly research related - just coming to terms ... |
809,499 | <p>The no. of real solution of the equation $\sin x+2\sin 2x-\sin 3x = 3,$ where $x\in (0,\pi)$.</p>
<p>$\bf{My\; Try::}$ Given $\left(\sin x-\sin 3x\right)+2\sin 2x = 3$</p>
<p>$\Rightarrow -2\cos 2x\cdot \sin x+2\sin 2x = 3\Rightarrow -2\cos 2x\cdot \sin x+4\sin x\cdot \cos x = 3$</p>
<p>$\Rightarrow 2\sin x\cdot ... | juantheron | 14,311 | <p><strong>Using Inequality</strong></p>
<p>Starting from $\bf{L.H.S\;\;}$</p>
<p>$$\sin x+2\sin 2x-\sin 3x = -2\sin x\cos 2x+2\sin 2x\leq \left|2\sin x\cos 2x+2\sin 2x\right|$$</p>
<p>$$\leq 2|\sin 2x+\cos 2x|\leq 2\sqrt{2}\left|\sin \left(2x+\frac{\pi}{4}\right)\right|\leq 2\sqrt{2}<3\;(\bf{R.H.S})$$</p>
<p>So... |
174,149 | <p>How many seven - digit even numbers greater than $4,000,000$ can be formed using the digits $0,2,3,3,4,4,5$?</p>
<p>I have solved the question using different cases: when $4$ is at the first place and when $5$ is at the first place, then using constraints on last digit.</p>
<p>But is there a smarter way ?</p>
| Giles Gardam | 32,305 | <p>I think considering the two different cases you mentioned separately is best. To avoid further case division, I'd proceed like this:</p>
<ul>
<li><p>Case with leading digit 4: since an even digit has to go in the rightmost position, there are $5 \choose 3$ ways to choose the positions of the 3 odd numbers amongst t... |
176,691 | <p>Let $A'$ denotes the complement of A with respect to $ \mathbb{R}$ and $A,B,T$ are subsets of $\mathbb{R}$. I am trying to prove $A' \cap (A' \cup B') \cap T= A' \cap T$, but I got some problems along the way.</p>
<p>$A' \cap (A' \cup B') \cap T= (A' \cap A') \cup (A' \cap B') \cap T= A' \cup (A \cup B) \cap T =(A'... | Dilip Sarwate | 15,941 | <p>$A' \cap (A' \cup B') \cap T= ((A' \cap A') \cup (A' \cap B')) \cap T= (A' \cup (A' \cap B')) \cap T$. But, $A' \cap B' \subset A'$ and so $A' \cup (A' \cap B') = A'$ giving
$(A' \cup (A' \cap B')) \cap T = A' \cap T$ without dragging DeMorgan's Laws into it.</p>
|
2,673,835 | <p>I need to prove that the sequence
$$
f_n = \sum_{i=0}^n\prod_{j=0}^i \left(z+j\right)^{-1} = \frac{1}{z}+\frac{1}{z(z+1)}+\cdots + \frac{1}{z(z+1)(z+2)\cdots (z+n)}$$
converge uniformly to a function in every compact subset of $\mathbb{C}\setminus\{0,-1,-2,-3,\dotsc\}$. The problem has other questions, but this is ... | G Cab | 317,234 | <p>For the negative <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials" rel="nofollow noreferrer">Falling Factorials</a>
we have the partial fraction expansion:
$$
\eqalign{
& \left( {x - 1} \right)^{\,\underline {\, - n\,} } = {1 \over {x^{\,\overline {\,n\,} } }} = \cr
& = \left[ {n... |
3,264,693 | <p>For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space... | xrfxlp | 678,937 | <p>Let <span class="math-container">$z_o$</span> and <span class="math-container">$z_d$</span> be two complex number, lets call the first one operator and second one operand, and let <span class="math-container">$z'_d$</span> be the transformed version( state after it being operated by <span class="math-container">$z_o... |
3,264,693 | <p>For context, I have been relearning a lot of math through the lovely website Brilliant.org. One of their sections covers complex numbers and tries to intuitively introduce Euler's Formula and complex exponentiation by pulling features from polar coordinates, trigonometry, real number exponentiation, and vector space... | fedja | 12,992 | <p>Think of ot this way. <span class="math-container">$2iz=2(iz)$</span>, so you have two steps here: <span class="math-container">$z\mapsto iz$</span>, which is a pure rotation and <span class="math-container">$(iz)\mapsto 2(iz)$</span>, which is a pure stretching. Of course, you can also write <span class="math-conta... |
1,177,605 | <p>Since the problem uses $y^2=x$, I first assumed that the element must be horizontal (parallel to the $x$-axis). However, the bounded region has all $y$ values greater than $0$, so I could also use a vertical element. This problem has me stumped; I know how to set up the integral but for the shell method I need to fi... | Daniel Valenzuela | 156,302 | <p>Consider $SO(R^3)$ which acts on $R^3$ by rotations, but restricts to an action of $S^2$. For every point $x\in S^2$ we have a unique orthogonal plane $V$, hence $SO(V)\subset SO(R^3)$ will fix $x$. It is easy to see that in fact $Stab(x)=SO(V) \cong SO(2)$. Hence we have a fiber bundle
$$
SO(3) \to S^2
$$</p>
<p>... |
3,786,553 | <p>I was trying to follow the logic in a similar question (<a href="https://math.stackexchange.com/questions/643352/probability-number-comes-up-before-another">Probability number comes up before another</a>), but I can't seem to get it to work out.</p>
<p>Some craps games have a Repeater bet. You can bet on rolling ac... | Botond | 281,471 | <p>I'll assume that the integral is a Riemann-Stieltjes integral. We can use integration by parts, which states that if either <span class="math-container">$\int_a^b f(x) \mathrm{d}g(x)$</span> or <span class="math-container">$\int_a^b g(x) \mathrm{d}f(x)$</span> exists then the other one exists as well and
<span class... |
876,896 | <p>What are the poles of a polynomial? Are they the same as the roots?</p>
| 5xum | 112,884 | <p>This is the shortest answer possible for your question:</p>
<p>There are no poles of a polynomial.</p>
|
388,523 | <p>I have this question:</p>
<p>Evaluate $\int r . dS$ over the surface of a sphere, radius a, centred at the origin. </p>
<p>I'm not really sure what '$r$' is supposed to be? I would guess a position vector? If so, I would have $r . dS$ as $(asin\theta cos\phi, a sin\theta sin\phi, acos\theta) . (a^2sin\theta d\thet... | user642796 | 8,348 | <p>The <a href="http://en.wikipedia.org/wiki/Fast-growing_hierarchy" rel="noreferrer">fast-growing hierarchy</a> (or Wainer hierarchy) has the property that a recursive function is $\sf{PA}$-provably total iff it is dominated by $f_\alpha$ for some $\alpha < \varepsilon_0$. (Similarly with the <a href="http://en.w... |
4,646,773 | <p>I started with an integral <span class="math-container">$ \int_{0}^{2\pi} \sqrt{2[\sin^2(t) + 16\cos^2(t) - 4\sin(t)\cos(t)]} \,dt $</span></p>
<p>And I simplified it to <span class="math-container">$ \int_{0}^{2\pi} \sqrt{17 + 15\cos(2t) - 4\sin(2t)} \, dt$</span></p>
<p>My question: I know this can be simplified w... | mathlove | 78,967 | <p>It is given by
<span class="math-container">$$a_n=\begin{cases}(n+1)\cdot 2^{n-\left\lceil\sqrt{n-1}\right\rceil^2+\left\lceil\sqrt{n-1}\right\rceil-2}&\text{if $\left\lceil\sqrt{n-1}\right\rceil\leqslant \left\lfloor\dfrac{1+\sqrt{4n-3}}{2}\right\rfloor$}
\\\\(n+1)\cdot 2^{\left\lfloor\sqrt{n-1}\right\rfloor^2+... |
3,106,696 | <p>I am confused on the notation used when writing down the solution of x and y in quadratic equations.
For example in <span class="math-container">$x^2+2x-15=0$</span>, do I write :</p>
<p><span class="math-container">$x=-5$</span> AND <span class="math-container">$x=3$</span></p>
<p>or is it</p>
<p><span class=... | fleablood | 280,126 | <p>Just do it.</p>
<p><span class="math-container">$f(n+1) = (n+1)^2 + (n+1) + 1 = n^2 +2n + 1 + n + 1 + 1 = n^2 + n + 1 + 2n + 2 = f(n) + 2n + 2$</span>.</p>
<p>So <span class="math-container">$\gcd(f(n+1), f(n)) = \gcd(f(n), f(n) + 2n + 2) = \gcd(f(n), 2n + 2)$</span>.</p>
<p><span class="math-container">$f(n) = n... |
2,943,892 | <p>I have heard it said that completeness is a not a property of topological spaces, only a property of metric spaces (or topological groups), because Cauchy sequences require a metric to define them, and different metrics yield different sets of Cauchy sequence, even if the metrics induce the same topology. But why w... | Paul Frost | 349,785 | <p>You are interested in metrizable spaces <span class="math-container">$X$</span> and ask how the set <span class="math-container">$\mathfrak{C}(X)$</span> of sequences <span class="math-container">$(x_n)$</span> in <span class="math-container">$X$</span> which are Cauchy with respect to <strong>all</strong> metrics o... |
1,581,456 | <p>Given functions $g,h: A \rightarrow B$ and a set C that contains at least two elements, with $f \circ g = f \circ h$ for all $f:B \rightarrow C$. Prove that $g = h$. </p>
<p>My logic is to take C = B and h(x) =x for all x in particular and the result follows immediately. But, I don't see the use of the condition on... | bilaterus | 287,278 | <p>First we observe that:</p>
<p>$$\sum_{k=1}^{2n-1} {k\over {1-z^k}} - \sum_{k=1}^{n-1} {2k\over {1-z^{2k}}} = \sum_{k=1}^{n-1} {2k+1 \over {1-z^{2k+1}}}
$$</p>
<p>Now, by partial fractions,</p>
<p>$${2k \over {1-z^{2k}}} = {k\over {1-z^k}} + {k\over {1+z^k}}
$$
Hence: </p>
<p>$$\sum_{k=1}^{2n-1} {k\over {1-z^k}}... |
211,175 | <p>In Gradshteyn and Ryzhik, (specifically starting with the section 3.13) there are several results involving integrals of polynomials inside square root. These are given in terms of combinations of elliptic integrals. See for instance: <a href="https://i.stack.imgur.com/6Cqyb.jpg" rel="nofollow noreferrer"><img src=... | TheDoctor | 2,263 | <p>You may find DLMF <a href="https://dlmf.nist.gov/19" rel="nofollow noreferrer">Chapter 19</a> on Elliptic Integrals, authored by B. C. Carlson interesting and helpful. Carlson introduced <em>symmetric</em> standard integrals that simplify many aspects of theory, applications, and numerical computation of Elliptic In... |
184,772 | <p>Is there any way to write a code that has a <strong>function</strong> include <strong>Block[ ]</strong> and <strong>Do[ ]</strong> loop instead of my code?</p>
<p>Here is my code:</p>
<pre><code>(* m = Maximum members of "list" *)
list = {{12, 9, 10, 5}, {3, 7, 18, 6}, {1, 2, 3, 3},
{4, 5, 6, 2}, {1, 13, 1, 1}}... | kglr | 125 | <pre><code>ClearAll[f]
f[l_List] := Block[{m={}, i=1}, Do[AppendTo[m, Max[l[[All, i++]]]], {Length @ l[[1]]}]; m]
f[list]
</code></pre>
<blockquote>
<p>{12, 13, 18, 6}</p>
</blockquote>
|
3,079,929 | <p>Find all positive triples of positive integers a, b, c so that <span class="math-container">$\frac {a+1}{b}$</span> , <span class="math-container">$\frac {b+1}{c}$</span>, <span class="math-container">$\frac {c+1}{a}$</span> are also integers. </p>
<p>WLOG, let a<span class="math-container">$\leqq b\leqq c$</span>... | Ross Millikan | 1,827 | <p>Hint: given <span class="math-container">$a \le b$</span> and <span class="math-container">$\frac {a+1}b$</span> is an integer, you must have <span class="math-container">$b=a+1$</span> or <span class="math-container">$a=b=1$</span>.</p>
|
3,079,929 | <p>Find all positive triples of positive integers a, b, c so that <span class="math-container">$\frac {a+1}{b}$</span> , <span class="math-container">$\frac {b+1}{c}$</span>, <span class="math-container">$\frac {c+1}{a}$</span> are also integers. </p>
<p>WLOG, let a<span class="math-container">$\leqq b\leqq c$</span>... | fleablood | 280,126 | <p>If <span class="math-container">$a \le b$</span> but <span class="math-container">$b|a+1$</span> then <span class="math-container">$b \le a+1$</span> so either <span class="math-container">$a = b$</span> or <span class="math-container">$b=a+1$</span>. </p>
<p>If <span class="math-container">$a = b$</span> then <sp... |
3,245,428 | <p>Is it true that every tame knot has at least an alternating diagram?</p>
<p>If yes, is it true that we can always obtain an alternating diagram by a finite number of Reidemeister moves from a diagram of a knot? </p>
<p>If yes, how can we do it?</p>
<p>I am reading GTM Introduction to Knot Theory and find they sor... | auscrypt | 675,509 | <p>The former is true and is essentially a "checkerboard colouring" argument -- colour all the regions black and white, no two of the same colour sharing a border ( (ab)use Jordan curve theorem if you want to prove this ). Then as you travel towards a crossing, if there's black on your right as you approach, go under; ... |
3,166,947 | <p>I want to transform the following
<span class="math-container">$$\prod_{k=0}^{n} (1+x^{2^{k}})$$</span>
to the canonical form <span class="math-container">$\sum_{k=0}^{n} c_{k}x^{k}$</span></p>
<p>This is what I got so far
<span class="math-container">\begin{align*}
\prod_{k=0}^{n} (1+x^{2^{k}})= \dfrac{x^{2^{n}}-... | Community | -1 | <p>Looks good as</p>
<p><span class="math-container">\begin{align}
\prod_{k=0}^{n-1}\left(1+x^{2^{k}}\right)&=\prod_{k=0}^{n-1}\frac{1-x^{2^{k+1}}}{1-x^{2^{k}}}\\ &=\frac{1}{1-x}\prod_{k=0}^{n-1}\left(1-x^{2^{k+1}}\right)\prod_{k=0}^{n-2}\left(1-x^{2^{k+1}}\right)^{-1}\\ &={\frac{1-x^{2^{n}}}{1-x}}.
\end{a... |
3,166,947 | <p>I want to transform the following
<span class="math-container">$$\prod_{k=0}^{n} (1+x^{2^{k}})$$</span>
to the canonical form <span class="math-container">$\sum_{k=0}^{n} c_{k}x^{k}$</span></p>
<p>This is what I got so far
<span class="math-container">\begin{align*}
\prod_{k=0}^{n} (1+x^{2^{k}})= \dfrac{x^{2^{n}}-... | trancelocation | 467,003 | <p>You may also approach this from a combinatorical point fo view:</p>
<ul>
<li>Expanding would give a sum of <span class="math-container">$2^{n+1}$</span> summands.</li>
<li>Each summand is a product of powers of <span class="math-container">$x$</span> where the exponent corresponds to a uniquely determined sequence ... |
3,617,600 | <p>I am trying to understand the proof of the First and Second Variation of Arclength formulas for Riemannian Manifolds. I want some verifaction that the following covariant derivaties commute. I find it intuitive but I want to also have a formal proof.</p>
<p>Some notation: Let <span class="math-container">$\gamma(t,... | HK Lee | 37,116 | <p>Recall : <span class="math-container">$$(1) \ Xf = \frac{d}{dt}\bigg|_{t=0}\ (f\circ c)(t)$$</span> where <span class="math-container">$c$</span> is a curve with <span class="math-container">$
c'(0)=X$</span>.</p>
<p><span class="math-container">$$ (2) \ [X,Y] f = X(Yf)-Y(Xf) $$</span> </p>
<p>Hence if <span clas... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | paul garrett | 12,291 | <p>"Indefinite integral" and "anti-derivative(s)" are the same thing, and are the same as "primitive(s)".</p>
<p>(Integrals with one or more limits "infinity" are "improper".)</p>
<p>Added: and, of course, usage varies. That is, it is possible to find examples of incompatible uses. And, quite seriously, $F(b)=\int_a^... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | Hagen von Eitzen | 39,174 | <p>An anti-derivative of a function $f$ is a function $F$ such that $F'=f$.</p>
<p>The indefinte integral $\int f(x)\,\mathrm dx$ of $f$ (that is, a function $F$ such that $\int_a^bf(x)\,\mathrm dx=F(b)-F(a)$ for all $a<b$) is an antiderivative if $f$ is continuous, but need not be an antiderivative in the general ... |
586,112 | <p>Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$</p>
<p>I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothing\}\}$ which is not i... | Community | -1 | <p>(<a href="http://math.mit.edu/suppnotes/suppnotes01-01a/01ft.pdf" rel="nofollow noreferrer">http://math.mit.edu/suppnotes/suppnotes01-01a/01ft.pdf</a>) p 1 sur 7 </p>
<p>Antiderivative is an indefinite integral. </p>
<p><img src="https://i.stack.imgur.com/JlzdN.png" alt="enter image description here"></p>
|
1,981,948 | <p>Is there a relationship between group order and element order? </p>
<p>I know that there is a relationship between group order and subgroup order, which is that $[G:H] = \frac{|G|}{|H|}$ where $H$ is the subgroup of $G$ and $[G:H]$ is the index of $H$ in $G$. But is there a relationship between group order and th... | Stefan4024 | 67,746 | <p>Note that by definition the order of an element is the order of the group generated by it, i.e we have that $|a| = |\langle a \rangle|$. Now obviously $\langle a \rangle \le G$, so we can easily correlate an order of an element with the order of the corresponding subgroup.</p>
<p>This proves that if $a \in G$, then... |
2,878,412 | <p>I've been working on a problem that involves discovering valid methods of expressing natural numbers as Roman Numerals, and I came across a few oddities in the numbering system.</p>
<p>For example, the number 5 could be most succinctly expressed as $\texttt{V}$, but as per the rules I've seen online, could also be ... | mau | 89 | <p>Please keep in mind that strict rules for Roman numerals were established in a recent time. Besides having different signs for the digits, subtractive system was not fixed. In an inscription found in Forum Popilii and dated between 172 BC and 158 BC, 84 is written as XXCIIII (quoted from Georges Ifrah, The Universal... |
1,360,835 | <p>Reading "A First Look at Rigorous Probability Theory", and in the definition of outer measure of a set A, we take the infimum over the measure of covering sets for A from the semi-algebra (e.g., intervals in [0,1] ).</p>
<p>Is this set over which we are taking the infimum well-defined? For a given real
number x, ho... | mweiss | 124,095 | <blockquote>
<p>I have a function f(x) and I want to prove that x∗>y where x∗ is the number that satisfies f′(x∗)=0 and y is just an arbitrary constant. So what I did is that I assume x∗>y and show second derivative f"(x∗)|x∗>y<0, then I show that f′(y)>0, representing that x∗>y. Thus the first assumption is true ... |
4,821 | <p>A quick bit of motivation: recently a question I answered quite a while ago ( <a href="https://math.stackexchange.com/questions/22437/combining-two-3d-rotations/178957">Combining Two 3D Rotations</a> ) picked up another (IMHO rather poor) answer. While it was downvoted by someone else and I strongly concur with the... | BlueTrin | 36,894 | <p>I think that, as long, as you are downcasting indepedently of your participartion, you should be doing it: i.e. would you have downvoted the reply even if you did not post an answer. As long as your reasons for downvoting have nothing to do with yourself having posted a reply I do not see a problem there.</p>
<p>Of... |
966,835 | <p>I want to find the asymptotic complexity of the function:</p>
<p>$$g(n)=n^6-9n^5 \log^2 n-16-5n^3$$</p>
<p>That's what I have tried:</p>
<p>$$n^6-9n^5 \log^2 n-16-5n^3 \geq n^6-9n^5 \sqrt{n}-16n^5 \sqrt{n}-5 n^5 \sqrt{n}=n^6-30n^5 \sqrt{n}=n^6-30n^{\frac{11}{2}} \geq c_1n^6 \Rightarrow (1-c_1)n^6 \geq 30n^{\frac{... | Community | -1 | <p>For a log-polynomial expression $\sum a_kn^{i_k}\log^{j_k}(n)$, take the highest power of $n$ and in case of equality take the highest power of the $\log$.</p>
<p>In your case, the dominant term is $n^6$, and
$$\frac{g(n)}{n^6}=1-\frac{9\log^2 n}n-\frac{16}{n^5}-\frac5{n^3}.$$
The limit is clearly $1$.</p>
<p>Also... |
141,115 | <p>I know only some basics about mathematica. However I need to write down the following sum: </p>
<p>$\sum_{\{m_k\}_N}\prod_{k=1}^N\frac{1}{m_k}[T_k(Z(\tau))]^{m_k}$. </p>
<p>Where $\{m_k\}_N$ denotes partitions of $N$ i.e. $\sum_{k=1}^Nkm_k=N$. The argument in brackets [..] is the Hecke Operator, for now not that i... | matheorem | 4,742 | <p>the number of coordinates in VertexTextureCoordinates should be the same as the number of vertex in polygon.</p>
<p>As an example</p>
<pre><code>Graphics[{Texture[pic], EdgeForm[Black],
Polygon[{{0, 0}, {1, 0}, {2, 2}, {0.5, 2.5}},
VertexTextureCoordinates -> {{0, 0}, {2, 0}, {2, 2}, {0, 2}}]}]
</code></pr... |
141,115 | <p>I know only some basics about mathematica. However I need to write down the following sum: </p>
<p>$\sum_{\{m_k\}_N}\prod_{k=1}^N\frac{1}{m_k}[T_k(Z(\tau))]^{m_k}$. </p>
<p>Where $\{m_k\}_N$ denotes partitions of $N$ i.e. $\sum_{k=1}^Nkm_k=N$. The argument in brackets [..] is the Hecke Operator, for now not that i... | andre314 | 5,467 | <p><em>Note first : it is recommended to read this answer after the two other ones (from Szabolcs and matheorem)</em></p>
<p>Here is a tool intended to explore how <code>VertexTextureCoordinates</code> works.</p>
<p>The texture is the image : </p>
<p><a href="https://i.stack.imgur.com/6EFBg.jpg" rel="noreferrer"><i... |
273,127 | <p>Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\lim_{r \rightarrow \infty} V(r) = 0$. Now consider a function $f: X \rightarrow \mathbb R$, defined by
$$
f(V)(\mathbf{r... | vibe | 111,071 | <p>I don't have a solution for the general question, but consider a special case where we pick $\phi$ to be a simple dipole,
$$
\phi = {\cos{\theta} \over r^2} = \phi_{10} {Y_{10} \over r^2}
$$
where $\phi_{10} = {1 \over 2} \sqrt{{3 \over \pi}}$ is a constant which isn't really important. This choice of $\phi$ is harm... |
717,980 | <p>In a certain 2-player game, the winner is determined by rolling a single 6-sided die in turn, until a
6 is shown, at which point the game ends immediately. Now, suppose that k dice are now rolled simultaneously by each player on his turn, and the
first player to obtain a total of k (or more) 6’s, accumulated over all... | Steve Kass | 60,500 | <p><strong>Note:</strong> The initial answer here was incorrect. Thanks to Markus, who solved the problem in a different way and got different results from my first ones, I found the mistake and rewrote the answer. (If there's any reason for me to repost the original, wrong answer somewhere, let me know.)</p>
<hr>
<p... |
717,980 | <p>In a certain 2-player game, the winner is determined by rolling a single 6-sided die in turn, until a
6 is shown, at which point the game ends immediately. Now, suppose that k dice are now rolled simultaneously by each player on his turn, and the
first player to obtain a total of k (or more) 6’s, accumulated over all... | epi163sqrt | 132,007 | <p><em>Note:</em> This question seemed to be a nice challenge for me. Here I provide an <em>explicit formula</em>, which is admittedly not very attractive, because it is rather complicated. In fact, I really appreciated the elegant approach of <a href="https://math.stackexchange.com/users/60500/steve-kass">Steve</a> an... |
2,343,993 | <blockquote>
<p>Find the limit -$$\left(\frac{n}{n+5}\right)^n$$</p>
</blockquote>
<p>I set it up all the way to $\dfrac{\left(\dfrac{n+5}{n}\right)}{-\dfrac{1}{n^2}}$ but now I am stuck and do not know what to do.</p>
| Michael Rozenberg | 190,319 | <p>Just $$\left(1-\frac{5}{n+5}\right)^{-\frac{n+5}{5}\cdot\frac{-5n}{n+5}}\rightarrow e^{-5}$$</p>
|
84,076 | <p>I think computation of the Euler characteristic of a real variety is not a problem in theory.</p>
<p>There are some nice papers like <em><a href="http://blms.oxfordjournals.org/content/22/6/547.abstract" rel="nofollow">J.W. Bruce, Euler characteristics of real varieties</a></em>.</p>
<p>But suppose we have, say, a... | Liviu Nicolaescu | 20,302 | <p>This is tricky even in the simplest case. Suppose we are given a real polynomial in one real variable. The Euler characteristic of its zero set is equal to the number of real roots (not counted with multiplicity). </p>
|
3,283,724 | <p>Let A, B ⊆ R, and let f : A → B be a bijective function. Show that if <span class="math-container">$f$</span> is strictly increasing on A, then <span class="math-container">$f^{-1}$</span> is strictly increasing on B.</p>
<p>How would I write this proof? I think by contradiction but I don't know where to start.</p... | Kavi Rama Murthy | 142,385 | <p>Suppose <span class="math-container">$x<y$</span>. We have to show that <span class="math-container">$f^{-1}(x) <f^{-1}(y)$</span>. Let us prove this by contradiction.</p>
<p>Suppose <span class="math-container">$f^{-1}(x) \geq f^{-1}(y)$</span>. Since <span class="math-container">$f$</span> is increasing thi... |
4,220,518 | <p>A point is moving along the curve <span class="math-container">$y = x^2$</span> with unit speed. What is the magnitude of its acceleration at the point <span class="math-container">$(1/2, 1/4)$</span>?</p>
<p>My approach : I use the parametric equation <span class="math-container">$s(t) = (ct, c^2t^2)$</span>, then ... | mathcounterexamples.net | 187,663 | <p>What is wrong is that using the parametric coordinates</p>
<p><span class="math-container">$$s(t)=(ct,c^2t^2)$$</span></p>
<p>the point is not moving with unit speed. This can be easily seen by computing the norm of the speed, which is not constant. You need to normalize the speed <strong>for all <span class="math-... |
280,500 | <p>I would like to pose a question about the range of validity of the expansion of Binomial Theorems. </p>
<p>I know that for non-positive integer, rational $n$
$$
\left(1+x\right)^{n}=1+nx+\frac{n\left(n-1\right)}{2!}x^{2}+\dots,
$$
where the range of validity is $\left|x\right|<1$.</p>
<p>My question is that if ... | Robert Israel | 8,508 | <p>Yes, it really is that straightforward.</p>
|
1,823,187 | <blockquote>
<p>There are $n \gt 0$ different cells and $n+2$ different balls. Each cell cannot be empty. How many ways can we put those balls into those cells?</p>
</blockquote>
<p>My solution:</p>
<p>Let's start with putting one different ball to each cell.
for the first cell there are $n+2$ options to choose a b... | Christian Blatter | 1,303 | <p>Consider at $x=k\in{\mathbb N}_{\geq2}$ a trapezoidal spike of height $k$ with base $\left[k-{2\over k^3},\ k+{2\over k^3}\right]$ and top $\left[k-{1\over k^3},\ k+{1\over k^3}\right]$. Let $f: \>[1,\infty)\to{\mathbb R}$ be the function obtained by "concatenation" of these spikes. The area of the spike at $k... |
1,338,999 | <p><img src="https://i.stack.imgur.com/hXfn2.png" alt="sat question"></p>
<p>My friend selected option B, I did C. We're confused. Can someone please explain this for my friend?</p>
| James Pak | 187,056 | <p>\begin{align}
\left[\frac{(m!)^2}{(m-k)!(m+k)!}\right]^{m}
& = \left[\frac{m!}{(m-k)!}\frac{m!}{(m+k)!}\right]^{m}
\\& = \left[\frac{m\times(m-1)\times\cdots\times(m-k+1)}{(m+k)\times(m+k-1)\times\cdots\times(m+1)}\right]^{m}
\\& = \left[\left(1+\frac{k}{m}\right) \left(1+\frac{k}{m-1}\right)\cdots\lef... |
537,021 | <p>Say I divide a number by 6, will a number modulus by 6 always be between 0-5? If so, will a number modulus any number (N) , the result should be between $0$ and $ N - 1$?</p>
| Henry | 6,460 | <p>It depends whether you regard modulus as a function or as an equivalence relation. </p>
<p>It is often useful to identify numbers which are equivalent to "$-1 \mod p$".</p>
<p>It also depends on what you mean by "number": what would you say "$11.4 \mod 6$" was?</p>
<p>If you say "$61 \equiv 79 \mod 6$" then I wou... |
2,183,390 | <p>So, I need to solve a hard problem, which reduces to this: </p>
<blockquote>
<p>Prove that $3^{\frac{1}{3}} \notin \mathbb{Q}[13^{\frac{1}{3}}]$.</p>
</blockquote>
<p>The only thing that comes into my mind is to suppose the opposite, <em>i.e.</em>, $3^{\frac{1}{3}} \in \mathbb{Q}[13^{\frac{1}{3}}]$, and then to ... | Thomas Andrews | 7,933 | <p>The vector space approach is to consider $\mathbb Q[\sqrt[3]{13}]$ as a vector space over $\mathbb Q$ with the basis $1,\sqrt[3]{13},\sqrt[3]{13}^2$. Then multiplication by $a+b\sqrt[3]{13}+c\sqrt[3]{13}^2$ can be represented by the matrix:</p>
<p>$$\begin{pmatrix}a&13c&13b\\
b&a&13c\\
c&b&a... |
1,742,982 | <p>I was trying to solve the equation using factorial as shown below but now I'm stuck at this level and need help.</p>
<p>$$C(n,3) = 2*C(n,2)$$</p>
<p>$$\frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$$</p>
<p>$$3! (n - 3)! = (n - 2)!$$</p>
| user5713492 | 316,404 | <p>Definitely don't need calculus nor a graphing calculator for this problem. We are trying to find
$$-x^4-2x^3+15x^2=y_{max}$$
Rewrite as
$$x^4+2x^3-15x^2+y_{max}=0$$
Now if we knew where the maximum occurred, say at $x=a$, then $a$ would be a root of this equation. Also it must be a double root of else the graph of t... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | Artem | 29,547 | <p>So
$$
n+1\choose m
$$
means the number of ways to choose $m$ elements out of $n+1$. Now fix one element out of $n+1$. This element can be among these $m+1$, to pick the rest we need to pick $n\choose m-1$, or this element is not among these $m$, and we should pick then $n\choose m$ elements. Since this is exactly th... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | Zachary Hamm | 11,091 | <p>I struggled with this problem too a bit and I feel like the answers here don't explain all the steps here.</p>
<p>For my solution, there are three key insights: that ${m! = m(m-1)!}$ and that ${(m + 1)! = m!(m+1)}$, and that if the proposition is true, then ${m!(n - m + 1)!}$ must be valid as the common denominator... |
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