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 |
|---|---|---|---|---|
250,434 | <p>I'm stuck on a couple of practice problems relating to PIDs, they are paraphrased below:</p>
<p>Given a PID $R$ with $a$ and $b$ in $R$ and gcd$(a,b)=1$ I need to show that:</p>
<p>1) There are elements $s$ and $t$ of $R$ such that $sa+tb=1$</p>
<p>2) The $R$-module $R/(a) \bigoplus R/(b)$ is isomorphic to the $R... | Dilip Sarwate | 15,941 | <p>Let <span class="math-container">$Z = X+Y$</span>. Then, if <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> have values in <span class="math-container">$[0,1]$</span>, it follows that <span class="math-container">$0 \leq Z \leq 2$</span>.
Thus, it is obvious that the cumulative pr... |
257 | <p>In my graduate courses, I often have my students write term papers on original mathematical topics. I explain the process in <a href="https://mathoverflow.net/a/133643/1946">this answer over at MathOverflow</a>. It works fairly well for me. </p>
<p>But I'd be very interested in hearing about other experiences or ap... | Community | -1 | <p>This was the term-paper assignment for the graduate class I took on differential geometry:</p>
<p>Take a paper in differential geometry published in the past 10 years. Explain what it draws on, what it means, and why it is important.</p>
|
2,304,714 | <p>In our introductory course on groups, we defined cosets and quotient groups in the following way.</p>
<blockquote>
<p>Let $N\trianglelefteq G$ be a normal subgroup of a group $(G,\,\cdot\,)$. Then the quotient group $G/N$ contains all the cosets of $N$ with respect to the elements of $G$, i.e. $G/N=\{Ng:g\in G\}$... | FWE | 170,600 | <p>Say I is an ideal in a ring R.</p>
<p>Then a definition of a <strong>quotient ring</strong> with multiplicative subsets</p>
<p>R/I := {Ir : r in R}</p>
<p>would not make much sense, since this would be equal to just {I} (since I is an ideal).</p>
|
1,767,751 | <p>I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius.
I considered $\triangle API$ ($P$ is orthocentre and $I$ is incentre). I could find $AP=2R\cos A$, $AI=4R\sin\frac{B}{2}\sin\frac{C}{2} $ and $\angle PAI=\angle ... | claude | 652,088 | <p>Use these facts
1. The Euler circle is tangent to the inscribed circle
2. The distance between the circumcenter and the incenter using the Euler formula.
3. The formula for the power of a point with respect to a circle
4. The properties of the Euler line
5. The fact that the reflection of the orthocenter with respec... |
4,343,792 | <p>Suppose that <span class="math-container">$$</span> and <span class="math-container">$ + 2$</span> are both prime numbers.</p>
<p><strong>a) Is the integer between <span class="math-container">$$</span> and <span class="math-container">$ + 2$</span> odd or even? Explain your answer.</strong></p>
<p>All prime numbers... | Hagen von Eitzen | 39,174 | <p>(a) If <span class="math-container">$p=2$</span> then <span class="math-container">$p+2=4$</span> is not prime, i.e. we can be sure that the two primes <span class="math-container">$p$</span> and <span class="math-container">$p+2$</span> are <strong>odd</strong>.</p>
<p>(b) If <span class="math-container">$p+1=n^2$<... |
175,791 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/173370/compute-the-trigonometric-integrals">Compute the trigonometric integrals</a> </p>
</blockquote>
<p>For $n \in \mathbb N$, $$ \int_{-\pi}^{\pi} \frac{1 - \cos (n+1) x}{1- \cos x} dx = (n+1) 2 \pi$$<... | AFK | 8,882 | <p>Consider
$$
0 \to I \to R \to R/I \to 0
$$
Apply the right exact functor $-\otimes_R M$, and you get
$$
I\otimes_R M \to R\otimes_R M \to (R/I)\otimes_R M \to 0
$$
But $R\otimes_R M$ is canonically identified with $M$ by $a\otimes m \mapsto am$. Then $I\otimes_R M \to R\otimes_R M = M$ is $a\otimes m \mapsto ... |
4,315,858 | <p>When tackling the <a href="https://math.stackexchange.com/a/4313831/732917">question</a>, I found that for any <span class="math-container">$a>1$</span>,</p>
<p><span class="math-container">$$
I_1(a)=\int_{0}^{\pi} \frac{d x}{a-\cos x}=\frac{\pi}{\sqrt{a^{2}-1}}.
$$</span>
Then I started to think whether there is... | Quanto | 686,284 | <p>Express the integral as
<span class="math-container">\begin{align}
\int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}}
=\sum_{k=0}^{[\frac{n-1}2]} \frac{\binom{n-1}{2k}a^{n-2k-1}}{(a^2-1)^{n-1/2}}\int_0^{\pi}\cos^{2k}x \ dx
= \frac{\pi P(a)}{\left(a^{2}-1\right)^{n-{1}/{2}}}
\end{align}</span>
which leads to
<span class="math... |
3,214,136 | <p>I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are <span class="math-container">$\sqrt{5}$</span>, <span class="math-container">$\sqrt{10}$</span>, <span class="math-container">$\sqrt{13}$</span>. After some algebra hell trying to work with Heron's formula, I plugged the qu... | Hyperion | 636,641 | <p>For another interesting approach, consider the Law of Cosines, <span class="math-container">$a^2 = b^2 + c^2 - 2bc\cos(\alpha)$</span>. If we let <span class="math-container">$a = \sqrt{13}$</span>, <span class="math-container">$b = \sqrt{5}$</span>, and <span class="math-container">$c = \sqrt{10}$</span>, then we f... |
594,785 | <p>$f:\mathbb N\to\mathbb N$
such that
$f(x) = 2x$.</p>
<p>$f:\mathbb Z\to\mathbb Z$
such that
$f(x) = 2x$</p>
<p>How are these two different?</p>
<p>And also $h:\mathbb R\to\mathbb R$ where $h(x) = \sqrt x$</p>
<p>$f:\mathbb N\to\mathbb N$ where $f(x) = \sqrt n$</p>
| Carlos Eugenio Thompson Pinzón | 99,344 | <p>In the first case, for any $x$ such as $x\in\mathbb N$ and $x\in\mathbb Z$ then $f_{\mathbb N}(x)=f_{\mathbb Z}(x)$. Given that all natural numbers are integers, then there is no difference for the (sub)domain of natural numbers. However, the second function is also defined for negative integers which are not natu... |
1,713,713 | <p>To my knowledge, the exponential function is the unique function satisfying</p>
<p>$f'=f$ and
$f(0)=1$</p>
<p>however, unless I've made a mistake, we have</p>
<p>$$\frac{\partial}{\partial x} (ax)^x = x (ax)^{x-1} a = ax (ax)^{x-1} = (ax)^x$$</p>
<p>and </p>
<p>$$(a0)^0 = 0^0 =1$$</p>
<p>so I feel like I must ... | Calvin Khor | 80,734 | <p>You differentiated $x^x$ wrong. In fact,</p>
<p>$$ (x^x)' = (e^{x \log x})' \overset{\text{chain rule}}{=} [x \log x]' e^{x\log x} = (\log x +1 )x^x$$</p>
<p>The rule $[x^n]' = n x^{n-1}$ only applies when $n$ is a <em>fixed</em> constant.</p>
|
1,845,517 | <p>The number of primes less than $2^{2^{100}}$ is $(a)101$ $(b)100$ $(c) 2^{100}$ $(c)2^{101}$.</p>
<p>How can I solve this ? Please help.</p>
<p>Thank you.</p>
| BigbearZzz | 231,327 | <p>From the <a href="https://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>, the prime numbers are asymptotically distributed according to
$$
\pi(n) \approx \frac n{\ln n}\ ,
$$
where $\pi(n)$ is the number of primes less than or equal to $n$. This shows that for $n=2^{2^{100}}$ we'd have
$$
\pi... |
1,591,706 | <p>Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$?</p>
<p>It looks like it might be possible to express this in terms of gamma functions and use Stirling's approximation.</p>
<p>It is clear that $\frac{an(an-1)...(an-n+1)}{n!}=\frac{\Gamma(an+1)}{\Gamma(a... | Gary | 83,800 | <p>You can use the <a href="http://dlmf.nist.gov/5.5.E3" rel="nofollow noreferrer">reflection formula</a> and the <a href="http://dlmf.nist.gov/5.11.E7" rel="nofollow noreferrer">known asymptotics</a> for the gamma function to obtain
<span class="math-container">\begin{align*}
\frac{{\Gamma (an + 1)}}{{\Gamma (an - n +... |
2,250,900 | <p>As the title states, I'm trying to determine whether the series $$\sum_{n=1}^\infty \sin(n+1)-\sin(n+2)$$ converges or diverges. My intuition is saying diverging since the sines are oscillating. How will I go about 'formally' proving this? </p>
<p>I searched online for this and I got an 'explanation' saying $s_n=\s... | Aperson123 | 430,504 | <p>First: your sum diverges. To see this, write it out and note that it is a telescoping series- all but two of the terms cancel. Thats how you get your $s_n$. Then note that the value of the nth partial sum is actually a linear combination of $\sin(n)$, whose limit does not exist, therefore the limit of $s_n$ does not... |
1,123,694 | <p>Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$?</p>
<p>The first one seems very obvious, but is there any way to show it very formally? </p>
| copper.hat | 27,978 | <p>We have $a z^{2n} = -b |z|^{2n}$. We can assume $a \neq 0$.</p>
<p>If $z \neq 0$, then ${z^{2n} \over |z|^{2n}} = -{b \over a}$. Then left hand side has unit length, then right hand side $|{ b \over a}| \neq 1$.</p>
|
3,864,442 | <p>I have a propositional formula:</p>
<p><span class="math-container">$$(\neg p \lor \neg q \lor r)$$</span></p>
<p>Can i rewrite it in this way?:</p>
<p><span class="math-container">$$(\neg p \lor \neg q \lor r) = (\neg p \lor (q \land \neg r))= (\neg p \lor q) \land (\neg p \lor \neg r))$$</span></p>
| supinf | 168,859 | <p>Yes, you are correct and the answer is <strong>no</strong>.</p>
<p>Such a possible norm is
<span class="math-container">$$
|(x,y)|:=
\left\| \begin{pmatrix}
x + y
\\
4(x-y)
\end{pmatrix}\right\|_2.
$$</span>
Then we have <span class="math-container">$$|(0,1)|=\sqrt{17}> 2\cdot 2 = 2|(1,1)|.$$</span></p>
<p>The r... |
3,245,517 | <p>As by <a href="https://arxiv.org/abs/0803.3787" rel="nofollow noreferrer">Landau's proof</a>
<span class="math-container">$$\sum_{n=1}^{\infty} \mu(n)/n = 0$$</span>
Therefore for any <span class="math-container">$N \in \mathbb{N}$</span>,
<span class="math-container">$$ \sum_{n=1}^{N} \mu(n)/n = -\sum_{n=N+1}^{\in... | QuantumSpace | 661,543 | <p>This is always true if the series converges. It doesn't even matter if the terms are positive, the convergence is absolute, etc. Indeed, we have
<span class="math-container">$$0 = \sum_{n=1}^\infty\mu(n)/n = \sum_{n=1}^N \mu(n)/n + \sum_{n=N+1}^\infty \mu(n)/n$$</span></p>
<p>Thus, the result follows by substractin... |
1,869,768 | <p>I need to do the following limit without using L'Hopital and I have not been able, please help</p>
<blockquote>
<p>$$\lim\limits_{x \to 3} \left(\frac{x-1}{2x-4}\right)^{\frac{1}{x-3}}$$</p>
</blockquote>
| COOLGUY | 352,180 | <p>Hint......</p>
<p>Just use the formula, </p>
<p>$\lim_{x \rightarrow a}f(x)^{g(x)}$</p>
<p>$= e^{\lim_{x \rightarrow a}g(x)(f(x)-1)}$</p>
|
448,581 | <p>Let $\{X_n\}_{n \geq 1}$ be an independent sequence of random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Fix $n \geq 1$. I want to prove that $X_1, \ldots, X_n$ is independent of $\limsup X_n$.</p>
<p>This result is incredibly obvious, at least intuitively: the limsup does not depend on the first $n$ element... | Did | 6,179 | <p>This is a consequence of <a href="http://en.wikipedia.org/wiki/Dynkin_system" rel="nofollow">Dynkin's $\pi$-$\lambda$ theorem</a>. </p>
<p>To show that $\mathcal F_n=\sigma(X_k;k\leqslant n)$ and $\mathcal G_n=\sigma(X_k;k\geqslant n+1)$ are independent sigma-algebras, consider the class $\mathcal C_n$ of the event... |
3,256,509 | <p>I need to show that <span class="math-container">$ \sum_{k=0}^n (-1)^{k} {{m+1}\choose k }{{m+n-k}\choose m }= 0 $</span> if <span class="math-container">$n>0$</span>. Here <span class="math-container">$m$</span> is a non negative integer.</p>
<p>I am thinking induction, but do I apply it on <span class="math-c... | Marko Riedel | 44,883 | <p>Starting from</p>
<p><span class="math-container">$$\sum_{k=0}^n (-1)^k {m+1\choose k} {m+n-k\choose m}
= \sum_{k=0}^n (-1)^k {m+1\choose k} {m+n-k\choose n-k}$$</span></p>
<p>we find</p>
<p><span class="math-container">$$[z^n] (1+z)^{m+n}
\sum_{k=0}^n (-1)^k {m+1\choose k} z^k (1+z)^{-k}.$$</span></p>
<p>Now we... |
2,301,450 | <p>Can anyone give a brief proof or a reference of a proof for the following property of <a href="https://en.wikipedia.org/wiki/Hilbert_space" rel="nofollow noreferrer">Hilbert spaces</a>?</p>
<blockquote>
<p>If <span class="math-container">$H$</span> is a Hilbert space and <span class="math-container">$M$</span> is... | Oskar Henriksson | 95,044 | <p>First of all, note that $H/M$ is a <strong>normed linear space</strong>, with the norm $$\|[x]\|_{H/M}=\inf_{y\in M}\|x-y\|_H,$$
where $\|\cdot\|_H$ is the norm induced by the scalar product on $H$. To show that $H/M$ is a Hilbert space, we need to show two things: that $H/M$ is a <strong>complete</strong> metric sp... |
370,570 | <p>In the wikipedia page (<a href="http://en.wikipedia.org/wiki/Birthday_problem" rel="nofollow">http://en.wikipedia.org/wiki/Birthday_problem</a>) on birthday paradox the following statement has been said : "the probability that, in a set of $n$ "randomly chosen" people, some pair of them will have the same birthday. ... | Sammy Black | 6,509 | <p>As in the comment from Alex, the problem assumes a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28discrete%29" rel="nofollow">discrete <em>uniform</em> distribution</a>. The word uniform means that each possible outcome is equally likely.</p>
<p>To help clarify the distinction with the word <em>rand... |
1,469,695 | <p><a href="https://i.stack.imgur.com/fLWrM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fLWrM.png" alt="enter image description here"></a></p>
<p>I have the following graph, and i'm trying to determine the maximum. When I use conventional methods, I end up with the ordered pair (2.35, 1.28) roug... | Community | -1 | <p><strong>First Proof</strong></p>
<blockquote>
<p><strong>Claim.</strong>
$kd\mid d$.</p>
<p><strong><em>Proof</em></strong>
$d\mid a \land k\mid r \implies kd\mid ar$</p>
<p>$d\mid b \land k\mid s \implies kd\mid bs$</p>
<p>$\therefore kd\mid ar+bs\implies kd\mid d\implies ??$</p>
</blockquot... |
209,293 | <p>I'm trying to solve a set of 5 nonlinear equations using NSolve:</p>
<pre><code>exp1 := y*E^(x - z) == 18 a*x + b*y
exp2 := E^(x - z) == 8 a*y + b (x + z)
exp3 := -y*E^(x - z) == 72 a*z + b*y
exp4 := 9 x^2 + 4 y^2 + 36 z^2 == 36
exp5 := x*y + y*z == 1
NSolve[{exp1, exp2, exp3, exp4, exp5}, {x, y, z, a, b}, Reals]
<... | Akku14 | 34,287 | <p>Introduce a sixth variable for the exponential term</p>
<pre><code> eq6 = E^(x - z) == exz
{eq1, eq2, eq3, eq4, eq5} ={exp1, exp2, exp3, exp4, exp5} /. E^(x - z) -> exz
</code></pre>
<p>And tell <code>NSolve</code> all variables are Reals</p>
<pre><code> (nsol = NSolve[And @@ {eq1, eq2, eq3, eq4, eq5... |
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| Dror Bar-Natan | 94 | <p>Another nice solution to a similar question is at <a href="http://katlas.math.toronto.edu/drorbn/index.php?title=0708-1300/the_unit_sphere_in_a_Hilbert_space_is_contractible" rel="noreferrer">http://katlas.math.toronto.edu/drorbn/index.php?title=0708-1300/the_unit_sphere_in_a_Hilbert_space_is_contractible</a>:</p>
... |
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| S. Carnahan | 121 | <p><a href="http://en.wikipedia.org/wiki/Contractibility_of_unit_sphere_in_Hilbert_space" rel="nofollow">3 proofs on Wikipedia</a> - basically the same arguments as above. The Hilbert space part is superfluous.</p>
|
198 | <p>Here I mean the version with all but finitely many components zero.</p>
| Ilya Nikokoshev | 65 | <p>Kind of late to the party, but the (weak) contractibility follows from $\pi_i(S^\infty) = 0$ for $i>0$.</p>
|
26,107 | <p>Is there a <em>Mathematica</em>-to-$\LaTeX$ converter, so that I can place <em>Mathematica</em> code, including things like <code>A // MatrixForm</code>, just as they are in the console straight into a $\LaTeX$ document?</p>
| cyclochaotic | 7,692 | <p>This works great for transferring expressions in Traditional Form.</p>
<p>Select an expression, Right-Click, Copy As, LaTeX. Paste it where you want.</p>
<p>You can even highlight part of a traditional form output and grab the LaTex. This also works for MathML in Mathematica 8.</p>
|
171,038 | <p>After some computations I end up with the following expression:</p>
<pre><code>Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]]
</code></pre>
<p>where $z$ is actually complex.</p>
<p>For reasons I don't understand, <em>Mathematica</em> won't simplify this to $1$.</p>
<p>According to the <a href="http://referen... | Carl Woll | 45,431 | <p><em>(Michael already gave a good answer, but I'll leave mine here as it includes a few extra observations)</em></p>
<p>This is a tricky question. First, here's a counterexample:</p>
<pre><code>expr = Sqrt[Exp[-z/(4*Sqrt[2])]] Sqrt[Exp[z/(4*Sqrt[2])]];
FullSimplify[expr /. z->4 Sqrt[2] Pi I]
</code></pre>
<blo... |
2,463,756 | <p>I'm trying to prove that $f(x)=x^2-6x-40$ is injective for $f: [3, \infty) \rightarrow [-49, \infty)$. Note that I cannot use calculus.</p>
<p>I tried letting $f(a)=f(b)$ and I arrived at $a^2-6a=b^2-6b$. Then I tried to find a solution for $a$ in terms of $b$ and a solution for $b$ in terms of $a$ and I got $a=\fr... | Peter Szilas | 408,605 | <p>Complete the square:</p>
<p>$y=(x-3)^2 -49.$</p>
<p>This is a normal parabola with vertex at</p>
<p>$(3,-49)$, which is the minimum.</p>
<p>$3 \le x,$ $x \in \mathbb{R}.$</p>
<p>Let $z: = x-3$, then</p>
<p>$Y=z^2 -49$, $0 \le z$, $z \in \mathbb{R}.$</p>
<p>This function $Y(z)$ is strictly monotonically incr... |
3,209,773 | <p>Is the following result true? Or Is there any known result of fractions like this?</p>
<p>Let <span class="math-container">$n$</span> be fixed.</p>
<blockquote>
<p>There are infinitely many integer solutions for <span class="math-container">$$\sum_{i=1}^n \frac{1}{x_i} = 0,$$</span> where <span class="math-conta... | saulspatz | 235,128 | <p>If <span class="math-container">$k$</span> is fixed, and the <span class="math-container">$x_i$</span> distinct, then there are only finitely many possible values of the negative part, so if there are infinitely many solutions, there must be some positive number <span class="math-container">$x$</span> with infinitel... |
978,927 | <p>How would one prove the equality of the sum of squares of diagonals and twice the sum of squares of the two sides:</p>
<p>$$\left|\mathbf{p} + \mathbf{q}\right|^2 + \left|\mathbf{p} - \mathbf{q}\right|^2 = 2\left|\mathbf{p}\right|^2 + 2\left|\mathbf{q}\right|^2 $$</p>
<p>where $\mathbf{p}$ and $\mathbf{q}$ are vec... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> The rank of $A - \lambda I$ is $n$ minus (the number of Jordan blocks of $A$ that are associated with $\lambda$).</p>
|
1,120,737 | <p>Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$</p>
<p>After a bit of algebra I get that</p>
<p>$$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$</p>
<p>So now how do I justify that it's divisible by $3$? Can I show that both expressions are divisible by $3$ ... | Community | -1 | <p>If $3|(a+b)$ then $3|((a+b)^3-3a^2b-3ab^2)$ so $3|a^3+b^3$.</p>
|
3,472,345 | <p>Problem: calculate <span class="math-container">$\int _0^{2\pi }e^{\cos \left(x\right)}\cos \left(\sin \left(x\right)\right)dx$</span></p>
<p>This is a problem which post in <a href="https://math.stackexchange.com/questions/409171/how-to-evaluate-int-02-pie-cos-theta-cos-sin-theta-d-theta?noredirect=1&lq=1">How... | Lai | 732,917 | <p><strong>Feynman’s Integration Technique without Contour Integration</strong></p>
<p>First of all, we rewrite the integral as
<span class="math-container">\begin{aligned}
I & =\int_0^{2 \pi} e^{\cos x} \cos (\sin x) d x =\operatorname{Re} \int_0^{2 \pi} e^{\cos x+i \sin x} d x = \operatorname{Re} \int_0^{2 \pi} e... |
4,349,993 | <p>Suppose that <span class="math-container">$(X_i, \tau_{X_i})$</span> are path-connected topological spaces for all <span class="math-container">$i \in I$</span>. I know that the product <span class="math-container">$\Pi_{i \in I}X_i$</span> with its product topology is path-connected. But is the converse true ? If <... | Kilkik | 870,351 | <p>As suggested by @Alessandro, axiom of choice gives us the possibility to take an element in every space. Then let <span class="math-container">$x,y \in X_\alpha$</span>, we have that <span class="math-container">$X=(x_1,\cdots,x_\alpha,\cdots)$</span> and <span class="math-container">$Y=(y_1,\cdots, y_\alpha, \cdots... |
835,536 | <p>Can you help me with this limit? What do I have to do? I'm lost.</p>
<p>$$\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$$</p>
<p>The solution given is $\dfrac{1}{2}$.</p>
| Claude Leibovici | 82,404 | <p>Paul gave you the nice way for the solution of your problem.</p>
<p>For your curiosity, I shall not enter into much details but I shall just mention that $$\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}$$ has a closed form which has an asymptotic expansion given by $$\frac{1}{2 n}-\frac{3}{8 n^2}+\frac{7}{48
n^3}+O\left(\left... |
2,197,561 | <p>If $f(x)$ is an irreducible cubic, then $\operatorname{Gal}(f(x))\cong S_3$ or $A_3$. But what about the converse? That is, if $\operatorname{Gal}(K/F)\cong S_3$, is it necessarily true that $K$ is the splitting field of some irreducible cubic in $F[x]$?</p>
| Tom Gannon | 362,020 | <p>Yes. To see this, note that if $K/F$ is a Galois extension, then by the Fundamental Theorem of Galois Theory there is an intermediate subfield $L$ with $[L : F] = 3$, corresponding to the fixed field of $\langle
(1,2) \rangle$. By the <a href="https://en.wikipedia.org/wiki/Primitive_element_theorem" rel="noreferrer... |
40,063 | <p>I'm trying to answer the following question:</p>
<p>"A person collects coupons one at a time, at jump times of a Poisson process $(N_t)_{t\geq 0}$ of rate $\lambda$. There are m types of coupons, and each time a coupon of type j is obtained with probability $p_j$, independently of the previously collected coupons a... | Luboš Motl | 10,599 | <p>Dear Ben, the coupon collector is just running $m$ Poisson processes at the same moment. The coupons of the $j$th type are being collected at rate $p_j\lambda$ and they're totally independent of the other types of the coupons. It's not hard to see why. Imagine you have an infinitesimal period of time ${\rm d}t$. Wha... |
1,610,201 | <p>Will someone please help me with the following problem?</p>
<blockquote>
<p>Calculate the volume bounded between $z=x^2+y^2$ and $z=2x+3y+1$. </p>
</blockquote>
<p>As far as I understand, I need to switch to cylindrical coordinates:
$(h,\theta, r)$. </p>
<p>The problem is, that I can't understand how to find th... | Will Jagy | 10,400 | <p>You have reduced it to the case where $C,D$ are diagonal matrices, and $C$ has $n$ different eigenvalues. That means that the minimal polynomial of $C$ is the same as the characteristic polynomial. That is, no polynomial of degree smaller than $n$ annihilates $C.$ Or,
$$ I,C,C^2, C^3, \ldots, C^{n-1} $$ are linearly... |
22,794 | <p>I have 2d list which is upper triangular. I would like to interpolate it, but I cannot unless I set the interpolation order to 1. I tried making the list rectangular by filling the bottom half with <code>Null</code>. This does allow me to interpolate with higher order. However, the interpolation function ends too ea... | Jens | 245 | <p>You're getting numerical problems with polynomial interpolation because of the effectively discontinuous jump in function values at the diagonal, if you insist on padding the missing triangle with zeros.</p>
<p>To get a good interpolation, you should simply pad the matrix in such a way as to avoid a numerical disco... |
1,947,915 | <p>Can I ask what is the Laplace transform of $\sqrt[3]{t}$ using the Gamma function?
This was my initial answer. Note that there is a theorem, $\Gamma(\frac{1}{3})=3\Gamma(\frac{4}{3})$. $$\int^{\infty}_{0}\exp^{-\beta} \beta^{-\frac{2}{3}}=3\int^{\infty}_{0}\exp^{-\beta} \beta^{\frac{4}{3}}$$ And was able to get the ... | lazyborg | 373,788 | <p>Laplace of $t^n$ is $[\Gamma (n+1)]/[s^{(n+1)}]$</p>
<p>So if we put $n=1/2$, we get $\Gamma (1/2) ÷ (s^{1/2})$ and $\Gamma(1/2) = \sqrt{\pi}$.</p>
<p>Hence the answer is $\sqrt{\pi/s}$. Hope this will help calculating the Laplace of $t^n$.</p>
|
1,947,915 | <p>Can I ask what is the Laplace transform of $\sqrt[3]{t}$ using the Gamma function?
This was my initial answer. Note that there is a theorem, $\Gamma(\frac{1}{3})=3\Gamma(\frac{4}{3})$. $$\int^{\infty}_{0}\exp^{-\beta} \beta^{-\frac{2}{3}}=3\int^{\infty}_{0}\exp^{-\beta} \beta^{\frac{4}{3}}$$ And was able to get the ... | Jan Eerland | 226,665 | <p>$$\mathcal{L}_t\left[t^{\frac{1}{\text{n}}}\right]_{\text{s}}=\int_0^\infty t^{\frac{1}{\text{n}}}\cdot e^{-\text{s}t}\space\text{d}t=\text{s}^{-\frac{1+\text{n}}{\text{n}}}\Gamma\left\{1+\frac{1}{\text{n}}\right\}$$</p>
<p>When $\Re\left[\text{s}\right]>0\space\wedge\space\Re\left[\frac{1}{\text{n}}\right]>-... |
457,790 | <p>$\Big\{ \lim\limits_{x \to a} f(x-a) = L\Big\} \Longleftrightarrow \Big\{ \lim\limits_{h \to 0} f(h) = L\Big\}$</p>
<p>True or false? </p>
<hr>
<p><strong>Edit:</strong> Many thanks to the people who provided answers for this question.</p>
<p>I was confused about the domain of $f(x-a)$. If $D$ is the domain of $... | 23rd | 46,120 | <p>Let
$$f(x)=(x-\alpha)(x-\beta)=x^2+2ax+3a=(x+a)^2-a^2+3a.$$ We may assume that $\alpha\le \beta$. Then $\alpha\le -a\le \beta$. Since $f$ is strictly decreasing on $(-\infty,-a]$ and strictly increasing on $[-a,+\infty)$,
$$-1<\alpha\le\beta<1\iff f(-1)>0, f(-a)\le 0\text{ and } f(1)>0.$$
Note that $f(... |
717,882 | <p>The $x^{2/2}$ can be represented by these ways:
$$\begin{align}
x^{2\over2}=\sqrt{x^2} = |x|\\
\end{align}
$$
And<br>
$$\begin{align}
x^{2\over2}=x^{1} = x\\
\end{align}
$$
Which one is correct?
And what is the domain of $x^{2 \over 2}$?</p>
| Yiorgos S. Smyrlis | 57,021 | <p>If $n$ is even, then the function
$$
x^{\frac{m}{n}},
$$
is defined ONLY for $x>0$.</p>
<p>Otherwise, it might not be equal to a real numbers.</p>
<p>In general the identity
$$
x^\frac{km}{kn}=x^{\frac{m}{n}},
$$
holds ONLY for $x>0$. If $x<0$, then it still holds if $k,n$ are odd numbers.</p>
|
1,522,359 | <blockquote>
<p>Let $W_0, W_1, W_2, \dots$ be random variables on a probability space
$(\Omega, \mathscr{F}, \mathbb{P})$ where</p>
<p>$$\sum_{k=0}^{\infty}P(|W_k|>k) <\infty$$</p>
<p>Prove that $$\limsup \frac{|W_k|}{k} \le 1 \ \text{a.s.} $$</p>
</blockquote>
<hr>
<p>I initially thought the conc... | Daniel Fischer | 83,702 | <p>Let's try to get things straight. We have two conditions,</p>
<p>\begin{align}
P\left(\left\{\omega \in \Omega : \limsup_{k\to\infty} \frac{\lvert W_k(\omega)\rvert}{k} \leqslant 1\right\}\right) &= 1,\text{ and} \tag{$\ast$}\\
P\left(\limsup_{k\to\infty} \left\{\omega\in \Omega : \frac{\lvert W_k(\omega)\rvert... |
17,669 | <p>I just inherited two slide rules from my grandfather-in-law, one wood with smooth action despite nearly a century without
use.<sup>1</sup> (I used a K+E slide rule myself as an undergraduate in the 1970's.) It struck me that the conversion of multiplication/division to addition/subtraction via the
logarithmically ru... | Chris Cunningham | 11 | <p>I have my college algebra students create slide rules each semester.</p>
<p><strong>Key pedagogical question:</strong></p>
<ul>
<li>What is the point of having an algebra student construct a slide rule?</li>
</ul>
<p><strong>Possible answers:</strong></p>
<ul>
<li><p>The goal is to convince the students that thi... |
113,362 | <blockquote>
<p>Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?</p>
</blockquote>
<p>Suppose, for example, that two mathematical concepts, say, doodad and doohickey, originally defined far, far away from the reaches of category theory, are later di... | Brett Frankel | 22,405 | <p>In category theory, cartesian squares, products, kernels, and limits are all maps <em>into</em> a given diagram, while cocartesian squares, coproducts, cokernels and colimits are all maps <em>from</em> a given diagram. Homology and cohomology follow this rule as well (when taking resolutions for covariant functors).... |
1,304,346 | <p>I am trying to solve the following differential equation:</p>
<p>$$x'=\frac{x+2t}{x-t}$$</p>
<p>with initial value condition: $x(1)=2$</p>
<p>This is what I have so far:</p>
<p><strong>Substitution:</strong> $u=\frac{x}{t}$</p>
<p>$$\implies u't+u=\frac{2t+tu}{-t+tu}$$</p>
<p><strong>Separation of variables:</... | Dr. Sonnhard Graubner | 175,066 | <p>it must be $$\frac{u-1}{-u^2+2u+2}du=\frac{dt}{t}$$
we get
$$-\frac{1}{2}\ln|-u^2+2u+2|=\ln|t|+C$$</p>
|
1,304,346 | <p>I am trying to solve the following differential equation:</p>
<p>$$x'=\frac{x+2t}{x-t}$$</p>
<p>with initial value condition: $x(1)=2$</p>
<p>This is what I have so far:</p>
<p><strong>Substitution:</strong> $u=\frac{x}{t}$</p>
<p>$$\implies u't+u=\frac{2t+tu}{-t+tu}$$</p>
<p><strong>Separation of variables:</... | abel | 9,252 | <p>here is another way to do this. we have $$\frac{dx}{dt} = \frac{x+2t}{x-t}. $$ we can rewrite this as an exact differential in the form $$0=(t-x) dx + (x+2t) dt = f_x\, dx + f_t \, dt $$ we have $$f_{xt} = f_{tx} =1 $$</p>
<p>we can integrate $$ f_x = t-x \to f = tx-\frac12x^2 + C(t) \to f_t = x + C'=x+2t\to C= t... |
92,105 | <p>Trying to solve</p>
<blockquote>
<p>$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.</p>
</blockquote>
<p>I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$</p>
<p>Would appreciate your help!</p>
| savick01 | 18,493 | <p>If the integral converges than you have: $\lim_{M\to\infty} \int_a^M f(x) dx = c\ $ for some $c$. Assuming that f does not converge to 0 we have:
$\forall M>0\ \exists p>M: |f(p)|>\varepsilon\ \ $ for some $\varepsilon$.</p>
<p>Without loss of generality we have $\forall M>0\ \exists p>M: f(p)>\... |
2,182,994 | <p>Consider $\text{SL}(2,\mathbb{R})$ with the left-invariant metric obtained by translating the standard Frobenius product at $T_I\text{SL}(2,\mathbb{R})$. (i.e $g_I(A,B)=\operatorname{tr}(A^TB)$ for $A,B \in T_I\text{SL}(2,\mathbb{R})$).</p>
<p>One can show that the geodesics starting at $I$ are of the form of</p>
... | HK Lee | 37,116 | <p>${\rm Tr}\ x=0$ implies that $$ x^2 +({\rm det}\ x) I=0 $$</p>
<p>where $$ x:= \left(
\begin{array}{cc}
a & b \\
c & -a \\
... |
3,119,573 | <blockquote>
<p>Let <span class="math-container">$R$</span> be the equivalence relation on the real numbers given by
<span class="math-container">$$R = \{(x, y) \in \Bbb R^2: (x−y)(x+y) = 0 \} $$</span>
What are the equivalence classes of <span class="math-container">$R$</span>?</p>
</blockquote>
<p>So I wrote t... | Jossie Calderon | 346,651 | <p>1) Make sure that your relation is indeed an equivalence relation, i.e. it is reflexive, symmetric and transitive. Indeed, it is (I just checked, but as an exercise, you should, too.)</p>
<p>2) Now find the ordered pairs , i.e. elements of R. For example, when x = y, or x = -y, R does indeed contain the elements (x... |
3,874,049 | <p>After writing an integral as a limit of a Riemann sum, how do we actually calculate the integral? It seems that generally, we're in some form that isn't simplified. For example, take</p>
<p><span class="math-container">$$\int_0^3e^xdx=e^x|_0^3=e^3-1.$$</span></p>
<p>But this is also <span class="math-container">$$\i... | RRL | 148,510 | <p>Take a uniform partition of <span class="math-container">$[0,3]$</span> and consider the upper sum</p>
<p><span class="math-container">$$\frac{3}{n} \sum_{k=1}^n e^{3k/n} = \frac{3}{n}\frac{e^{3/n} - e^{3(n+1)/n}}{1 - e^{3/n}} = e^{3/n} \frac{3}{n}\frac{1 - e^{3}}{1 - e^{3/n}}\\ = e^{3/n}\frac{3}{n(e^{3/n}- 1)}(e... |
296,737 | <p>Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ ... | Mariano Suárez-Álvarez | 274 | <p>A simple one:</p>
<p>Let $\def\CC{\mathbb C}A=\CC[X,Y]/(X^2-Y^3)$ and let consider the ideal $J=(X,Y)$ of $A$, which is maximal. One can easily see that if $d:A\to A$ is a derivation, then $d(J)\subseteq J$.</p>
<p>On the other hand, let $B=\CC[T]$ and consider the derivation $\delta:f\in B\mapsto f'\in B$ . Then ... |
1,997,050 | <p>Because strange conjectures have the tendency to enter my mind, I've become convinced that the only powers of $10$ that take the form $\frac{k(k+1)}{2}$ for $k\in \mathbb{Z}^{+}$ (i.e. is triangular) are $10^{0}$ and $10^{1}$. However, I'm having a difficult time making progress towards how to prove or disprove this... | Momo | 384,029 | <p>$k(k+1)=2^{n+1}\cdot 5^n$</p>
<p>In $k(k+1)$ one factor is odd and one is even.</p>
<p>So necessarily one of them is $2^{n+1}$ and the other one is $5^n$.</p>
<p>If $n\ge2$:</p>
<p>$5^n>4^n=2^{2n}>2^{n+1}+1$</p>
<p>So $2^{n+1}$ is much smaller than $5^n$ for $n\ge2$, thus they cannot be consecutive number... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | lhf | 532 | <p>See the book <em>The fundamental theorem of algebra</em> by Fine and Rosenberger.</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | David Jordan | 1,040 | <p>I came across this article when I was pondering teaching the FTA to my multi-variable calculus class. In the end, I didn't have time to include it. It is nice in that it relies only on Green's theorem which we get through in the first semester. On the other hand, it is clear that they are largely being clever at ... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Steven Sam | 321 | <p>In Hatcher's book <a href="http://pi.math.cornell.edu/%7Ehatcher/AT/ATpage.html" rel="nofollow noreferrer">http://pi.math.cornell.edu/~hatcher/AT/ATpage.html</a> Theorem 1.8 he deduces the fundamental theorem as a corollary of the fact that the fundamental group of the circle is isomorphic to the integers.</p>
|
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Kevin H. Lin | 83 | <p><a href="https://lists.lehigh.edu/pipermail/algtop-l/2009q4/000645.html">Here</a> is a nice proof that was posted on the ALGTOP list recently. See also the ensuing discussion.</p>
<blockquote>
<p>An algebraic extension of $\mathbb{C}$ is a unital division algebra over $\mathbb{C}$, say
of dimension $n$, so indu... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | timur | 824 | <p>This proof uses the open mapping theorem, which I found in Narasimhan's book.</p>
<p>Theorem: Let $f\in C^\omega(\mathbb{C})$, and suppose that $|f(z)|\to\infty$ as $|z|\to\infty$.
Then $f(\mathbb{C})=\mathbb{C}$.</p>
<p><em>Proof</em>:
Obviously $f$ is not constant, and so the open mapping theorem implies that $f... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | marsupilam | 12,386 | <p>Maybe I should have posted this as a comment to Gian Maria Dall'Ara very nice proof, because is a mere variation.</p>
<p>He uses the lemma : Any open proper map to a locally compact space surjects the connected components it reaches. Now any non-constant polynomial corestricted to its regular value locus is as in t... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Trevor J Richards | 35,158 | <p>Let $p$ be a complex polynomial of degree $n$. </p>
<ul>
<li><p>By analytic continuation arguments applied to any branch of $p^{-1}$, it can be shown that any level curve $\Lambda=\{z:|p(z)|=\epsilon\}$ of $p$ which does not contain a zero of $p'$ is a Jordan curve.</p></li>
<li><p>Since $p'$ has at most $n-1$ zer... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | Moe Hirsch | 16,671 | <p>Re Gauss's first proof, Smale pointed out the following (discussed by Weber in the collected papers of Gauss): In order to show that the zero sets of the real and imaginary parts of a complex polynomial intersect, Gauss states (paraphrasing): "If a [polynomial] curve C in the plane enters a region, it must leave ... |
10,535 | <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p>
<p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p>
<ol>
<li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li>
<li><p>Courant and Robbins, What is... | sobasu | 76,709 | <p>Gauss's first proof contained an enormous gap, since he presumed facts equivalent to the <a href="https://en.wikipedia.org/wiki/Jordan_curve_theorem" rel="noreferrer">Jordan curve theorem</a> to be true. Jordan curve theorem was proven a century later.</p>
<p>There is a modification on Gauss's first proof that uses... |
133,406 | <p>Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$. </p>
<p>Is there a natural mixed Hodge structure on its equivariant cohomology?</p>
<p>Is it pure if $X$ is smooth projective? </p>
<p>What if we ask the analogous question for $l$-adic equivar... | Dan Petersen | 1,310 | <p>The answer to the first question is yes. Although the words "equivariant cohomology" don't appear there, this goes back to Deligne's Hodge III, since he defines the mixed Hodge structure on a simplicial variety and you can define the Borel construction simplicially in the usual way (via the resolution with $X$, $G \... |
1,053,738 | <p>I've been asked to prove that:
$$
\sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)}
$$
When $0<x<2\pi$.</p>
<p>I know there are many similar posts on this site, but using $\cos(kx)$ instead, that's why I created this post, I can't get to the $1/2\cot$ in this one.
Thanks!</p... | Bombyx mori | 32,240 | <p>Here is an elementary way to see it. We know that
$$
\sin[x]\sin[y]=\frac{1}{2}(\cos[x-y]-\cos[x+y])
$$
Therefore the sum $A=\sin[x]+\sin[2x]+\sin[3x]$ be be treated by
$$
2\sin[x/2]A=\cos[x-x/2]-\cos[x+x/2]+\cos[2x-x/2]-\cos[2x+x/2]+\cos[3x-x/2]-\cos[3x+x/2]
$$
After cancel out intermediate terms we get
$$
A=\frac{... |
2,251,501 | <p>How can I calculate the number of perfect cubes among the first $4000$ positive integers?</p>
<p>Is there any trick to solving such questions?</p>
| Mr. Brooks | 162,538 | <blockquote>
<p>Is there any trick to find such questions?</p>
</blockquote>
<p>Google, I would suppose. I think you meant to say "to find the answers to such questions." And yes, there is, though to me it seems too elementary to call it a "trick." To the find how many perfect $k$-th powers there are among the first... |
2,384 | <p>I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, web-sites, etc. with a interesting questions?</p>
<p>I know of the ArtofProblemSolving.com. I am currently using the... | Community | -1 | <p>Try Yakov Perelman, "Mathematics can be Fun".</p>
|
2,494,160 | <p>I've been having trouble using the definition of a limit to prove limits, and at the moment I am trying to prove that
$$\lim_{n\to\infty} \frac{n^x}{n!}=0$$</p>
<p>for all $x$ which are elements of natural numbers.
I'm able to start the usual setup, namely let $0<\epsilon$ and attempt to obtain $\left\lvert\dfr... | Alex Ortiz | 305,215 | <p>If $x \le 0$, then there is really nothing to show. Otherwise, if $x > 0$, let $m$ be a positive integer greater than $x$.</p>
<p>Then,
\begin{align*}
\frac{n^x}{n!} \le \frac{n^m}{n!} &= \frac{n^m}{n\cdot(n-1)\dotsb(n-m+1)}\cdot\frac{1}{(n-m)!}\\
&= \frac{n^m}{n^m\big[1\cdot\big(1-\frac{1}{n}\big)\dots... |
2,494,160 | <p>I've been having trouble using the definition of a limit to prove limits, and at the moment I am trying to prove that
$$\lim_{n\to\infty} \frac{n^x}{n!}=0$$</p>
<p>for all $x$ which are elements of natural numbers.
I'm able to start the usual setup, namely let $0<\epsilon$ and attempt to obtain $\left\lvert\dfr... | egreg | 62,967 | <p>Let $\varepsilon>0$. Considering $k=\lceil x\rceil$ (the least integer greater than or equal to $x$), we have
$$
\frac{n^x}{n!}\le\frac{n^k}{n!}
$$
Let's prove that, from $n>2k$, we have
$$
\frac{n^k}{n!}<\frac{2^{k+1}}{n}
$$
Indeed
$$
n!=
\underbrace{n(n-1)\dotsm(n-k)}_{\text{$k+1$ factors $>n/2$}}\,(n-... |
3,033,601 | <p>I am interested in the following question:</p>
<blockquote>
<p>Does there exist a continuous function <span class="math-container">$f:S^2\to S^2$</span> such that, for any <span class="math-container">$p\in S^2$</span>, <span class="math-container">$|f^{-1}(\{p\})|=2$</span>?</p>
</blockquote>
<p>I suspect the a... | SmileyCraft | 439,467 | <p>If I understand correctly, Florentin MB's answer will be complete if we can prove that <span class="math-container">$f$</span> is locally injective. Feel free to tell me if I'm wrong here, but I think we can at least show that <span class="math-container">$f$</span> is locally injective if <span class="math-containe... |
3,664,795 | <p>I have two binomial expression: <span class="math-container">$S1= \sum_{k=0}^{\frac{n}{2}}{{n}\choose{2k}} $</span> and <span class="math-container">$ S2 =\sum_{k=0}^{\frac{n-1}{2}}{{n}\choose{2k + 1}} $</span>. I have to prove those two expression are equal, and then find their shared value.</p>
<p>So far I'm tryi... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
3,695,971 | <p>The way I understand it currently, saying "<em>only if <span class="math-container">$P$</span>, then <span class="math-container">$Q$</span></em>" is like saying that "<em>only if <span class="math-container">$P$</span> happens, <span class="math-container">$Q$</span> happens.</em>" To me, it seems to say the same t... | PrincessEev | 597,568 | <p>"Only if <span class="math-container">$P$</span>, then <span class="math-container">$Q$</span>" basically means "<span class="math-container">$Q$</span> implies <span class="math-container">$P$</span>" from a logical standpoint. "If and only if <span class="math-container">$P$</span>, then <span class="math-containe... |
1,960,862 | <blockquote>
<p>If $a,b,c>0$, Then prove that $$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\geq a+b+c$$</p>
</blockquote>
<p>$\bf{My\; Try::}$ Using Cauchy- Schwarz Inequality</p>
<p>$$\frac{a^4}{abc}+\frac{b^4}{abc}+\frac{c^4}{abc}\geq \frac{a^2+b^2+c^2}{3abc}$$</p>
<p>Now How can i solve after that , Help re... | Faraad Armwood | 317,914 | <p>The arc-length parameter $g(s) = s^{-1}(t)$ i.e;</p>
<p>$$g'(s) = \frac{1}{s'(g(s))} \ \ \ \ \ \ s(t)= \int_{a}^t \|\gamma'(u)\| \ du$$</p>
<p>The arc-length parametrization is $\gamma(g(s))$ and so;</p>
<p>$$\|(\gamma\circ g)'(s))\| = |\gamma'(g(s)) \cdot g'(s)\| = \left\|\gamma'(g(s)) \cdot \frac{1}{s'(g(s))}\r... |
1,196,633 | <p>I have the following recursive sequences:</p>
<p>$x_n = x_{n-1} + 2y_{n-1} , x_1 = 1$</p>
<p>$y_n = y_{n-1} - x_{n-1}, y_1 = -1$</p>
<p>where $ x_n,y_n \in \mathbb{Z}$ </p>
<p>I have to show that for any $n$ neither $x_n$ or $y_n$ are equal to 0.</p>
| DeepSea | 101,504 | <p><strong>hint</strong></p>
<p>$x_n - x_{n-1} = 2y_{n-1}, x_{n-1} - x_{n-2} = 2y_{n-2} \to x_n - 2x_{n-1} + x_{n-2} = 2(y_{n-1} - y_{n-2}) = -2x_{n-2} \to x_n - 2x_{n-1} + 3x_{n-2} = 0 \to x^2 - 2x + 3 = 0$. Can you continue?</p>
|
655,185 | <blockquote>
<p>Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$</p>
</blockquote>
<p>I try. But I think that I am wrong. $$G=C_{i_1}\oplus C_{i_2}\oplus\ldots C_{i_k},$$
as $C_{i}$ is a cyclic group
If $\chi$ is 1-dimen... | Mikko Korhonen | 17,384 | <p>Apply the orthogonality relations.</p>
<p>Alternatively, consider an irreducible representation $\mathscr{X}$ affording $\chi$. Let $A = \sum_{g \in G} \mathscr{X}(g)$. Prove that either $A = 0$ or $\mathscr{X}$ is the trivial representation.</p>
|
1,838 | <p>I understand what the problem with Gimbal Lock is, such that at the North Pole, all directions are south, there's no concept of east and west. But what I don't understand is why this is such an issue for navigation systems? Surely if you find you're in Gimbal Lock, you can simply move a small amount in any direction... | ridecar2 | 999 | <p>One problem that I have come across is when roller coasters are being designed. If you have the pitch at +/-90 degrees (pointing straight up/down) then using normal Euler-based orientation you can't easily specify an angle of banking as you have no reference to 'up'. To solve this Quaternions are often used.</p>
|
3,934,824 | <p>There are three containers, one container has diamond in it, other containers are empty. Each
container has label as clue whatever is inside the containers. Those labels are:</p>
<hr />
<ul>
<li><strong>Container#1:</strong> The diamond is not present</li>
<li><strong>Container #2:</strong> The diamond is not presen... | k170 | 161,538 | <p>Since the diamond can only be in one out of the two boxes, we can proceed with just one predicate
<span class="math-container">$$p=\mbox{Diamond is in box 1}$$</span>
<strong>Scenario 1</strong>
<span class="math-container">$$\neg p\rightarrow \neg p\land p\equiv\mbox{false}$$</span>
<strong>Scenario 2</strong>
<spa... |
3,992,896 | <p>I had this question, and I was wondering how to do it. The question was what is the probability of not drawing a pair out of a deck of cards, and I wasn't sure how to do it could someone help me out</p>
| Community | -1 | <p>Going back to the definition of (discrete) <a href="https://en.wikipedia.org/wiki/Joint_probability_distribution#Discrete_case" rel="nofollow noreferrer">joint probability distribution</a>, you want to find the quantity:
<span class="math-container">$$
P(X=x,Y=y)
$$</span>
Note that the event <span class="math-conta... |
1,712,769 | <p>I found this question in text book. I am looking at the solution but I can't understand it. Can anyone enlighten me. Thanks!</p>
<p>Let function $f : X → Y$ , with $|X| = m$ and $|Y|= n$.</p>
<ol>
<li>How many functions $f$ are possible ?</li>
</ol>
<p>Solution:$n^m$</p>
<ol start="2">
<li>How many one-to-one fu... | thanasissdr | 124,031 | <p>Question $\#1$.</p>
<p>Someone might think that way. Let's say we have a function
$$f: \{ x_1, x_2, \ldots, x_m\} \to \{y_1, y_2, \ldots, y_n\}.$$ We can correspond the variable $x_1$ to $n$ distinct $y$ values (namely $x_1\mapsto y_1$ or $x_1 \mapsto y_2$ or ... or $x_1 \mapsto y_n$), $x_2$ to $n$ distinct $y$ v... |
59,429 | <p>I am trying to learn more about the Berlin Airlift transport problem. Two links I could find are here:</p>
<p><a href="http://drmohdzamani.com/notes/file/Simplex%20Method.pdf" rel="nofollow noreferrer">http://drmohdzamani.com/notes/file/Simplex%20Method.pdf</a></p>
<p><a href="http://www.cabrillo.edu/%7Emladdon/math... | Jiri Kriz | 12,741 | <p>The problem has already been analyzed by @joriki. You can get a numerical solution by trying an online Simplex solver. Just google for it. I tried the first one that I found: <a href="http://www.zweigmedia.com/RealWorld/simplex.html" rel="nofollow noreferrer">Simplex Method Tool</a> and got for the maximal capacity ... |
2,998,735 | <p>I am trying to solved this inequality for <span class="math-container">$k$</span>.</p>
<p><span class="math-container">$x^{2k}<\varepsilon\cdot k^k$</span></p>
<p>Here <span class="math-container">$k\in\mathbb{N}$</span> and <span class="math-container">$x,\varepsilon$</span> are fixed such that <span class="ma... | Ross Millikan | 1,827 | <p>Because the log function (to a base greater than <span class="math-container">$1$</span>) is monotonically increasing, you can take the log of both sides. You still won't get a clean answer for <span class="math-container">$k$</span> given <span class="math-container">$x,\varepsilon$</span> If we make it an equali... |
593,418 | <p>Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$</p>
<hr>
<p>By computing some partial sums, the answers are 0. It seems an inductive argument is possible.</p>
| Albert Steppi | 7,323 | <p>Note that
$$\sum_{k=1}^{n}{\frac{1}{k{n \choose k}}} = \frac{1}{n}\sum_{k=0}^{n-1}{\frac{1}{{n-1 \choose k}}} = \frac{1}{n}\sum_{k=0}^{n-1}\frac{\Gamma\left(k+1\right)\Gamma\left(n-k\right)}{\Gamma\left(n\right)} =$$
$$\sum_{k=0}^{n-1}\beta\left(k+1,n-k\right)$$</p>
<p>where $\Gamma$ is the <a href="http://en.wiki... |
299,976 | <p>I am not sure how to tackle this problem and yes it is a homework problem.
Here is what I have. I know that $\text{ord}(a) = h$ and h is even so $h = 2 \alpha$ for some $\alpha \in \mathbb{Z}$. I also know Euler's criterion which states that $$a^{\frac{p-1}{2}} = -1 \pmod{p}$$ if there are no integers $x$ which sat... | DonAntonio | 31,254 | <p>The problem must be to show $\,a^{h/2}=-1\pmod p\,$ , but this follows almost immediately from:</p>
<p>$$ord_p(a)=h\Longleftrightarrow a^h=1\pmod p\,\,\,\wedge\,\,\,a^m\neq 1\,\,\,\forall\,m<h\Longrightarrow $$</p>
<p>$$a^{h/2}\,\,\,\text{is a square root of 1 different from 1 itself}\Longrightarrow a^{h/2}=-1$... |
299,976 | <p>I am not sure how to tackle this problem and yes it is a homework problem.
Here is what I have. I know that $\text{ord}(a) = h$ and h is even so $h = 2 \alpha$ for some $\alpha \in \mathbb{Z}$. I also know Euler's criterion which states that $$a^{\frac{p-1}{2}} = -1 \pmod{p}$$ if there are no integers $x$ which sat... | Math Gems | 75,092 | <p><strong>Hint</strong> $\rm\ mod\ p\!:\ 0 \equiv a^h\!-1\equiv (a^{h/2}\!-1)(a^{h/2}\!+1)\Rightarrow a^{h/2}\!+1 \equiv 0,\:$ by cancelling $\rm\,a^{h/2}\!-1$ $(\not\equiv 0,\,$ else the order of $\rm\,a\,$ is $\rm\le h/2).\:$ Elements $\ne 0$ are invertible (so cancellable) because $\rm\,\Bbb Z/p\,$ is a field.</p>
... |
244,207 | <p>Prove that $a^2b + b^2c + c^2a \ge ab + bc+ ac$ for positive real numbers $a,b,c$ such that $a+b+c=3$.</p>
| Dominik | 50,527 | <p>You can use <a href="http://en.wikipedia.org/wiki/Lagrange_multipliesr" rel="nofollow">Lagrange multipliers</a>. Define $f(a,b,c)=a^2b+b^2c+c^2a-ab-bc-ca$. Since the triangle $a+b+c=3$ with $a,b,c \ge 0$ is compact, f has a global minimum on this domain. If this minimum lies on the boundary, WLOG at $a=0$, it's easy... |
2,379,873 | <p>In a text I was reading; this was included in a section on set operations...</p>
<p>(-∞, 0) ∪ (0, ∞)</p>
<p>Is that a valid set operation if the typical "{}" are missing?</p>
<p>If it is valid; I assume this would also be valid...</p>
<p>[1, 3] = {x: 1 <= x <= 3}</p>
| Fred | 380,717 | <p>We have $(-∞, 0) ∪ (0, ∞)= \mathbb R \setminus \{0\}=\{x \in \mathbb R: x \ne 0\}$.</p>
|
1,526,442 | <p>I have problem with proving following equation:</p>
<p>$$
\binom{n}{0}0^2+\binom{n}{1}1^2+\binom{n}{2}2^2+...\binom{n}{n}n^2=n(1+n) \cdot 2^{2n-2}
$$</p>
<p>Thanks for any help!</p>
| Brian M. Scott | 12,042 | <p>HINT: Use the identity $\binom{n}kk=\binom{n-1}{k-1}n$ a couple of times. I’ll get you started:</p>
<p>$$\begin{align*}
\sum_k\binom{n}kk^2&=n\sum_k\binom{n-1}{k-1}k\\
&=n\sum_k\binom{n-1}k(k+1)\\
&=n\sum_k\binom{n-1}kk+n\sum_k\binom{n-1}k
\end{align*}$$</p>
<p>Now repeat the process to simplify the fi... |
2,833,085 | <p>This is basic, I know, but it's been a long time since I've done equations. I'm watching a tutorial video on circuits. </p>
<p>Let's say I have this equation:</p>
<p><a href="https://i.stack.imgur.com/jhU7S.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jhU7S.jpg" alt="enter image description h... | Bernard | 202,857 | <ul>
<li>Solve the associated homogeneous recurrence relation $\; a_n=2a_{n-1}$, which is just a geometric series $a_n=K2^n$ for some constant $K$.</li>
<li>Find a particular solution of the complete equation $\;a_n=2a_{n−1} + 3 · 2^{n}$. As $2^n$ is a solution of the homogeneous equation, you can seek for a particula... |
1,334,557 | <p>Suppose that an entire function $f$ has uncountably many zeros. Is it true that $f=0$?</p>
<p>I have no idea how to proceed with this. Perhaps some theorem that I am not aware of. I have done an undergraduate course in complex analysis.</p>
| Nitin Uniyal | 246,221 | <p>If there are uncountably many zeroes of $f(z)$ and suppose $a$ (say) be their limit point then $f(a+\frac 1n)=0$ where $n\in \Bbb N$. By corollary of Identity theorem $f(z)$ is identically zero on $\Bbb C$.</p>
|
465,555 | <p>Let $f , g : X \rightarrow Y$ be continuous where $Y$ is Hausdorff. Prove that $A = \{x : f(x) = g(x)\}$ is closed in $X$.
I have done the followings.</p>
<p>$f(X)$ and $g(X)$ are two subspaces of $Y$.</p>
<p>As Y is Hausdorff, $f(X), g(X)$ and $f(X) \times g(X)$ are also.</p>
<p>$L = \{(f(X),g(X)) : f(X) = g(X)\... | Brian M. Scott | 12,042 | <p>Your $L$ is empty if $f[X]\ne g[X]$, and it contains the single pair $\langle f[X],g[X]\rangle$ if $f[X]=g[X]$. This is a pair of subsets of $Y\times Y$ and is definitely not what you want. I'll get you started on a correct version of this approach.</p>
<p>Let $Y_f=f[X]$ and $Y_g=g[X]$, and define a function</p>
<... |
1,628,839 | <p>For what values of $p$ does the following integral converge:</p>
<p>$\sum_{n=2}^{\infty} \frac{1}{n(\ln\ n)^p}.$</p>
<p>Ans. (Integral Test) $\int\limits_{n=2}^{n=\infty}\frac{1}{n(\ln n)^p} = \frac{1}{(-p+1)(ln\ n)^{p-1}}$</p>
<p>I know that $p \neq 1$, but I do not understand why the answer is $p > 1$ </p>
| K. Jiang | 302,781 | <p>To begin, let us find the antiderivative by integration by parts. If we allow $u = x - 1$ and $dv = \sin\left(4x\right)dx,$ it follows that $du = dx$ and $v = -\frac{1}{4}\cos\left(4x\right).$ We continue as follows:
$$\int\left(x - 1\right)\sin\left(4x\right)dx$$
$$= -\frac{1}{4}\left(x - 1\right)\cos\left(4x\right... |
2,781,867 | <p>$$
a+\frac{b}{2}+\frac{c}{3}=7 \left(1+\frac{1}{2}+\frac{1}{3} \right)
$$
Find the number of positive integral solution.</p>
| achille hui | 59,379 | <p>There are $2n+1$ terms in the sum, you just need to pair up the terms
symmetrically from both ends, take average and compare with the term in the middle.
$$\begin{align}\sum_{k=n+1}^{3n+1} \frac{1}{k}
&= \sum_{k=-n}^n\frac{1}{2n+1+k}
= \frac12\sum_{k=-n}^n\left(\frac{1}{2n+1+k} + \frac{1}{2n+1-k}\right)\\
&... |
4,625,085 | <p>Let <span class="math-container">$D$</span> be a digraph as follows:
<a href="https://i.stack.imgur.com/sN32Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sN32Z.png" alt="enter image description here" /></a></p>
<p>I want to compute a longest simple path of it.</p>
<p>For an acyclic digraph, <a... | Claude Leibovici | 82,404 | <p><em>Too long for a comment</em></p>
<p>Using <span class="math-container">$x=10^k$</span> and computing
<span class="math-container">$$\left(
\begin{array}{cc}
k & f(x)-f(x-1) \\
1 & 1.80609 \\
2 & 2.25907 \\
3 & 2.48912 \\
4 & 2.62061 \\
5 & 2.70309 \\
6 & 2.75896 \\
\end{array}
\... |
3,626,839 | <p>I have a parking lot with dimensions <span class="math-container">$20 m \times 30 m$</span> which is illuminated by lights placed in different positions and heights as shown below:</p>
<p><a href="https://i.stack.imgur.com/27CaQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/27CaQ.png" alt="Park... | Axel Kemper | 58,610 | <p>Rather than numbering the squares <span class="math-container">$0..599$</span>, introduce two-dimensional indices <span class="math-container">$m=0..19;n=0..29$</span> which correspond to the lower left corners of the squares (in meters).</p>
<p>The center <span class="math-container">$c_{mn}$</span> of square in <... |
3,502,280 | <blockquote>
<p><strong>Problem:</strong> Let <span class="math-container">$x,y,z$</span> are positive integers such that <span class="math-container">$x+y+z=200$</span>. Find maximal value and minimal value of <span class="math-container">$M = x! + y! + z!$</span></p>
</blockquote>
<p>Could you give me some suggest... | Ross Millikan | 1,827 | <p>Factorials grow very fast, so for a maximal value you want one of the numbers to be as large as possible. For a minimal value you want the largest to be as small as possible.</p>
|
252,810 | <p>I have generated a real antisymmetric matrix of order 6 as follows.</p>
<pre><code>k0 = {{0, 1, 0, 0, 0, 0}, {-1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0,
0, -1, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, -1, 0}}
</code></pre>
<p>Any way to generate such a matrix of order 36 X 36 without writing each terms as abo... | creidhne | 41,569 | <p>Use <a href="http://reference.wolfram.com/language/ref/SparseArray.html" rel="nofollow noreferrer"><code>SparseArray</code></a> and <a href="http://reference.wolfram.com/language/ref/Band.html" rel="nofollow noreferrer"><code>Band</code></a> to reproduce your example 6-by-6 matrix:</p>
<pre><code>m = SparseArray[
... |
2,934,529 | <p>I am trying to compute
<span class="math-container">$$\large \lim_{x\to \frac32^-}\frac{x^2+1}{3x-2x^2}$$</span> </p>
<p>I know the numerator will equal one fine and how it becomes <span class="math-container">$x^2+1 * 1/(3x-2x^2)$</span> because <span class="math-container">$a/b = a * 1/b$</span>, but then I'm not... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Note that</p>
<p><span class="math-container">$$3x-2x^2=2x\left(\frac32-x\right)$$</span></p>
|
791,157 | <p>I am stuck on the following question.</p>
<blockquote>
<blockquote>
<p>Suppose $f = u+iv$ is entire and there exists $M > 0$ such that $|u(z)| \leq M$ for all $z\in C$. Show that $f$ is constant.</p>
</blockquote>
</blockquote>
<p>I would figure we would have to use Liouville's Theorem to show this is t... | marty cohen | 13,079 | <p>You integrated the expression
instead of differentiating.</p>
<p>From
$F(x)= \int_{0}^{x}f=x.e^{-x}+ax+b$,
differentiating we get
$f(x) = e^{-x}-x e^{-x}+a
=(1-x)e^{-x}+a$.
Setting $x=1$,
$1=f(1)
= a
$.
Therefore $f(x) = (1-x)e^{-x}+1$.</p>
<p>Setting $x=0$ in the definition of $F$,
$F(0)
= \int_{0}^{0}f=0.e^{-0}... |
61,438 | <p><strong>Bug introduced in 10.0 and fixed in 11.2</strong>
<br> Problem is due to a <code>BezierCurve</code> bug.</p>
<hr>
<p>I don't understand why the following does not work:</p>
<pre><code>Export["graph.pdf", Graph[{1 <-> 1, 1 <-> 2}, EdgeShapeFunction -> "Line",
EdgeStyle -> {Black}... | Szabolcs | 12 | <p>On my system (OS X 10.10.4) it doesn't even display correctly on-screen. This means that rasterization doesn't help.</p>
<p>I can confirm the problem in 10.0.2, 10.1.0 and 10.2.0. The problem doesn't exist in 9.0.1.</p>
<p><img src="https://i.stack.imgur.com/zNoNk.png" alt="enter image description here"></p>
<p... |
238,686 | <p>I want to solve this equation for <span class="math-container">$z$</span>:</p>
<p><span class="math-container">$-\frac{2\pi^2}{\beta^2}z^2+\frac{2i\pi^2u}{\beta^2}z^3+z^4=0$</span></p>
<p><span class="math-container">$\beta$</span> is a positive real constant, <span class="math-container">$u$</span> is a real variab... | Andreas | 69,887 | <p>Execute</p>
<pre><code>s1 = Solve[-2*(Pi^2/\[Beta]^2)*z^2 + 2*I*((Pi^2*u)/\[Beta]^2)*z^3 +
z^4 == 0, z]; s2 =
Solve[I*\[Eta] - 2*(Pi^2/\[Beta]^2)*z^2 +
2*I*((Pi^2*u)/\[Beta]^2)*z^3 + z^4 == 0, z];
</code></pre>
<p>and then compare</p>
<pre><code>s1 /. {\[Beta] -> 0.7, u -> \[Pi]}
</code></pre>
<p>with</p>
... |
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