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 |
|---|---|---|---|---|
1,939,937 | <p>$(172195)(572167)=985242x6565$</p>
<p>Obviously the answer is 9 if you have a calculator, but how can you find x without redoing the multiplication?</p>
<p>The book says to use congruences, but I don't see how that is very helpful. </p>
| user133281 | 133,281 | <p><strong>Hint:</strong> any natural number $n$ is congruent to the alternating sum of its digits modulo $11$.</p>
<p>Example: $172195 \equiv 5 - 9 + 1 - 2 + 7 - 1 \mod 11$.</p>
<p>This happens because $10^k \equiv \pm 1 \mod 11$, where the sign depends on whether $k$ is odd or even.</p>
|
2,405,767 | <p>This is Velleman's exercise 3.6.8.b (<strong>And of course not a duplicate of</strong> <a href="https://math.stackexchange.com/questions/253446/uniqueness-proof-for-forall-a-in-mathcalpu-existsb-in-mathcalpu-f">Uniqueness proof for $\forall A\in\mathcal{P}(U)\ \exists!B\in\mathcal{P}(U)\ \forall C\in\mathcal{P}(U)\ ... | Community | -1 | <p><strong>Hint</strong>:</p>
<p>This is a quadratic equation in $r$, waiting for you to solve it.</p>
|
19,962 | <p><a href="http://en.wikipedia.org/wiki/Covariance_matrix" rel="nofollow">http://en.wikipedia.org/wiki/Covariance_matrix</a></p>
<pre><code>Cov(Xi,Xj) = E((Xi-Mi)(Xj-Mj))
</code></pre>
<p>Is the above equivalent to:</p>
<pre><code>(Xi-Mi)(Xj-Mj)
</code></pre>
<p>I don't understand why the expectancy of (Xi-Mi)(Xj-... | Ross Millikan | 1,827 | <p>The covariance matrix is $Cov(X_i,X_j)=E[(X_i-\mu_i)(X_j-\mu_j)]$ Each $\mu_i=E[X_i]$. the expected value of variable $i$. If your variables are independent, the off-axis covariance terms are zero because $E[(X_i-\mu_i)(X_j-\mu_j)]=E[(X_i-\mu_i)]E[(X_j-\mu_j)]$ and both terms on the right are zero. The diagonal t... |
2,156,331 | <p>Consider the discrete topology $\tau$ on $X:= \{ a,b,c, d,e \}$. Find subbasis for $\tau$ which does not contain any singleton sets.</p>
<p>The definition of subbasis is as follows: </p>
<blockquote>
<p><strong>Definition:</strong> A <em>subbasis</em> $S$ for a topology on $X$ is a collection of subsets of $X$ w... | bof | 111,012 | <p>The collection consisting of all <span class="math-container">$27$</span> non-singleton sets is a subbase: each of them is an open set (since the topology is discrete), and every singleton set can be expressed as the intersection of two non-singleton sets.</p>
<p>More economically, the five <span class="math-contai... |
297,036 | <p>If $f'(x) = \sin{\dfrac{\pi e^x}{2}}$ and $f(0)= 1$, then what will be $f(2)$?</p>
<p>This is what I tried to find the antiderivative of $f'(x)$ with u-substitution, </p>
<p>$$
\begin{align}
u &=\frac{\pi e^x}{2} \\
\frac{2}{\pi}du &=e^x dx
\end{align}
$$</p>
<p>I don't know what to do next.</p>
| Tony | 534,934 | <p>To get result from Mean Value Theorem we have to make sure that the sine function in the last step is $\ge 0$. The argument of sine function lies between $-\pi/2$ to $\pi/2$. From there we can get $x$ lies between $-\infty$ to $0$. If $x$ lies between $-\infty$ to $0$ then min value of sine function is $0$ and max v... |
908,196 | <blockquote>
<p>Solve $x^2-1=2$</p>
</blockquote>
<p>I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.</p>
| SameOldNick | 57,095 | <p>You could also graph this to find what the 2 possible values for $x$ are. To do this, you can rewrite it using the following method:</p>
<p>$$x^2 - 1 = 2$$</p>
<p>Subtract 2 from both sides:</p>
<p>$$x^2 - 1 - 2 = 2 - 2$$</p>
<p>To make it equal to $0$:
$$x^2 - 3 = 0$$</p>
<p>Then graph it (I used <a href="http... |
646,032 | <p>I'm wondering where the notation for the quotient of a ring by an ideal comes from. I.e., why do we write $R/I$ to denote a ring structure on the set $\{r+I: r\in I\}$, wouldn't $R+I$ be more natural?</p>
| mdp | 25,159 | <p>When $A,B\subseteq R$, with $R$ a ring, it is common to write</p>
<p>$$A+B=\{a+b:a\in A,b\in B\}$$</p>
<p>This is particularly useful when $A$ and $B$ are ideals, in which case $A+B$ is also an ideal. So $R+I$ already has an interpretation (although it would just be $R$).</p>
<p>On the other hand, if $R$ is finit... |
2,414,472 | <blockquote>
<p>Let $(a_n)_{n\geq2}$ be a sequence defined as
$$
a_2=1,\qquad a_{n+1}=\frac{n^2-1}{n^2}a_n.
$$
Show that
$$
a_n=\frac{n}{2(n-1)},\quad\forall n\geq2
$$
and determine $\lim_{n\rightarrow+\infty}a_n$.</p>
</blockquote>
<p>I cannot show that $a_n$ is $\frac{1}{2}\frac{n}{n-1}$. Some helps? </p>
... | Jonatan B. Bastos | 476,786 | <p>Notice that $n^2-1=(n-1)(n+1)$. Write the recursion formula as
$$
\frac{a_{n+1}}{n+1}=\frac{a_n}{n}\cdot\frac{n-1}{n}.
$$
From here we have that
$$
\frac{a_{n+1}}{n+1}=\big(\frac{a_{n-1}}{n-1}\cdot\frac{n-2}{n-1}\big)\cdot\frac{n-1}{n}=\frac{a_{n-1}}{n-1}\cdot\frac{n-2}{n}.
$$
Applying this repeatedly we arrive at
$... |
499,652 | <p>I saw this a lot in physics textbook but today I am curious about it and want to know if anyone can show me a formal mathematical proof of this statement? Thanks!</p>
| Stefan Smith | 55,689 | <p>This is the most basic way:</p>
<p>$$\lim_{\alpha \to 0} \frac{\tan \alpha}{\alpha} = \lim_{\alpha \to 0} \frac{\sin \alpha}{(\cos \alpha)\alpha} = \lim_{\alpha \to 0} \frac{1}{\cos\alpha}\lim_{\alpha \to 0} \frac{\sin \alpha}{\alpha} = \frac{1}{\cos(0)}\lim_{\alpha \to 0} \frac{\sin \alpha}{\alpha}=\lim_{\alpha \t... |
2,761,151 | <p>In the formula below, where does the $\frac{4}{3}$ come from and what happened to the $3$? How did they get the far right answer? Taken from Stewart Early Transcendentals Calculus textbook.</p>
<p>$$\sum^\infty_{n=1} 2^{2n}3^{1-n}=\sum^\infty_{n=1}(2^2)^{n}3^{-(n-1)}=\sum^\infty_{n=1}\frac{4^n}{3^{n-1}}=\sum_{n=1}^... | Szeto | 512,032 | <p>$$\frac{4^n}{3^{n-1}}=\frac{4^1\cdot 4^{n-1}}{3^{n-1}}=4\cdot\frac {4^{n-1}}{3^{n-1}}=4\cdot\left(\frac43\right)^{n-1}$$</p>
|
298,913 | <p>Suppose you have $n$ triangles whose corners are random points on a sphere $S$
in $\mathbb{R}^3$.
Viewing the triangles as built from rigid bars as edges,
two triangles are <em>linked</em> if they cannot be separated without two
edges passing through one another.
A triangle that is not topologically linked with any... | Joseph O'Rourke | 6,094 | <p>Here is @fedja's clever example: The magenta triangle is topologically <em>loose</em>
but metrically "stuck":
<hr />
<a href="https://i.stack.imgur.com/wVP02.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/wVP02.jpg" alt="fedja"></a>
<br />
&n... |
902,592 | <p>Consider,</p>
<p>$$ \displaystyle x\frac{\partial u}{\partial x}+\frac{\partial
u}{\partial t} = 0 $$</p>
<p>with initial values $ t = 0 : \ u(x, 0) = f(x) $ and calculate the
solution $ u(x,t) $ of the above Cauchy problem using the method of
characteristics.</p>
<p>And here is the solution, I will point out ... | Robert Israel | 8,508 | <p>You're solving the initial value problem:
$$ \dfrac{\partial}{\partial \sigma} x(\sigma,n) = x(\sigma,n),\ x(0,n) = n $$
The variable $n$ is not important here: this is just the ordinary differential equation
$$ \dfrac{d}{d\sigma} x = x, \ x(0) = n$$</p>
<p>You do remember how to solve constant-coefficient first-... |
1,450,497 | <p>Consider the class of topological spaces $\langle X,\mathcal T\rangle$ such that the following are equivalent for $A\subseteq X$:</p>
<ul>
<li>$A$ is a $G_\delta$ set with respect to $\mathcal T$</li>
<li>$A\in\mathcal T$ or $X\smallsetminus A\in\mathcal T$</li>
</ul>
<p>Open sets, of course, are always $G_\delta$... | Caleb Stanford | 68,107 | <p>For <em>general</em> integers $x, y$, if $x | y$ and $y | x$ then you can't quite conclude $x = y$. But the following holds:
$$
\forall x, y \in \mathbb{Z} \;:\;
(x \mid y \text{ and } y \mid x) \implies |x| = |y|. \tag{1}
$$</p>
<p>In particular, if $x \ge 0$ and $y \ge 0$, then $x | y$ and $y | x$ implies $x = y$... |
3,581,390 | <p>The problem is as follows:</p>
<p>Mike was born on <span class="math-container">$\textrm{October 1st, 2012,}$</span> and Jack on <span class="math-container">$\textrm{December 1st, 2013}$</span>. Find the date when the triple the age of Jack is the double of Mike's age.</p>
<p>The alternatives given in my book are... | PrincessEev | 597,568 | <p>So first, do some math (or ask Google or Wolfram or something) to find how many days after Mike's birthday Jack's is. The answer is <span class="math-container">$426$</span> days. Therefore, if <span class="math-container">$J$</span> is Jack's age, then <span class="math-container">$M$</span> (Mike's age) satisfies ... |
2,101,756 | <p>From the power series definition of the polylogarithm and from the integral representation of the Gamma function it is easy to show that:
\begin{equation}
Li_{s}(z) := \sum\limits_{k=1}^\infty k^{-s} z^k = \frac{z}{\Gamma(s)} \int\limits_0^\infty \frac{\theta^{s-1}}{e^\theta-z} d \theta
\end{equation}
The identity ... | egreg | 62,967 | <p>Set $u=\log_3x$ and $v=\log_3y$. The system becomes
$$
\begin{cases}
u+v=5\\
uv=6
\end{cases}
$$
and the solutions are quite easy to find.</p>
<p>It's quite unclear how you arrive to the title equation.</p>
|
3,115,347 | <p>Let <span class="math-container">$f:(0,\infty) \to \mathbb R$</span> be a differentiable function and <span class="math-container">$F$</span> on of its primitives. Prove that if <span class="math-container">$f$</span> is bounded and <span class="math-container">$\lim_{x \to \infty}F(x)=0$</span>, then <span class="m... | Ingix | 393,096 | <p>I'm trying to address your metholodigal doubts, GNUSupporter 8964民主女神 地下教會 already gave a (very brief) answer to your mathematical questions.</p>
<p>Probably it's an urban legend, but there was a dispute between two statisticians: If you throw 2 "normal", 6-sided fair dice and add the values, will you get (in the l... |
3,713,395 | <p>I would like the definition of a modular form with complex multiplication and if possible a reference.
Thank you ! </p>
| reuns | 276,986 | <p>The MO post is a mess. From what I understand,</p>
<p>Let <span class="math-container">$f$</span> be a weight <span class="math-container">$k$</span> newform for <span class="math-container">$\Gamma_1(N)$</span>
<span class="math-container">$$f(z) =\sum_n a_n(f) e^{2i\pi nz}, \qquad L(s,f)=\sum_n a_n(f)n^{-s}=\prod... |
3,941,106 | <p>Let <span class="math-container">$K\subseteq\mathbb R$</span> be compact and <span class="math-container">$h:K\to\mathbb R$</span> be continuous and <span class="math-container">$\varepsilon>0$</span>. By the Stone-Weierstrass theorem, there is a polynomial <span class="math-container">$p:K\to\mathbb R$</span> wi... | DanielWainfleet | 254,665 | <p>Assuming <span class="math-container">$K$</span> is not empty and <span class="math-container">$h\ne 0.$</span></p>
<p>Let <span class="math-container">$e=\min (\|h\|/2, \varepsilon /2).$</span></p>
<p>For <span class="math-container">$x\in K$</span> let <span class="math-container">$\bar h(x)=h(x)$</span> if <span ... |
100,739 | <p>Let $a\in (1,e)\cup(e,\infty).$ I'd like to show that the equation $a^x=x^a$ has exactly two positive solutions, and one is larger and one smaller than $e.$ Is it even possible to show? I think I've tried everything.</p>
| Dirk | 3,148 | <p>The equation is equivalent to $\frac{\log(a)}{a} = \frac{\log(x)}{x}$. Now look at the value of the left hand side and the graph of the right hand side...</p>
|
684,076 | <blockquote>
<p>In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of <a href="http://en.wikipedia.org/wiki/Applied_mathematics" rel="nofo... | DanielV | 97,045 | <p>All of them. It's possible to verify if a proof is valid in polynomial time, and it's possible to check if the last step of a proof is the theorem under consideration in polynomial time, so just encode "valid proof of length N" as a satisfiability problem.</p>
<blockquote>
<p>N = 1 <BR>
DO<BR>
IF (there is a... |
2,129,086 | <p>I know that the total number of choosing without constraint is </p>
<p>$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13·12}{2} =78$</p>
<p>Then with x1 ≥ 1, x2 ≥ 2, and x3 ≥ 3. </p>
<p>the textbook has the following solution </p>
<p>$\binom{3+5−1}{5}=\binom{7}{5}=21$ I can't figure out where is the 5 coming from?</... | amWhy | 9,003 | <p>For the given equation: $$x_1+x_2+x_3 = 11, \;\text{ with } x_1, x_2, x_3 \;\text{ non-negative },$$ your solution is correct.</p>
<p>$$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13!}{2!11!} = \frac{13·12}{2} =78$$</p>
<p><strong>Your final answer is correct</strong>, (<strong>Now corrected</strong>: (but you fail... |
2,799,439 | <blockquote>
<p>Prove that if $p$ is a prime in $\Bbb Z$ that can be written in the form $a^2+b^2$ then $a+bi$ is irreducible in $\Bbb Z[i]$ .</p>
</blockquote>
<p>Let $a+bi=(c+di)(e+fi)\implies a-bi=(c-di)(e-fi)\implies a^2+b^2=(c^2+d^2)(e^2+f^2)\implies p|(c^2+d^2)(e^2+f^2)\implies p|c^2+d^2 $ or $p|e^2+f^2$
since... | David R. | 158,279 | <p>For what it's worth, I would rewind all the way back to the problem statement and go from there. Just my two cents.</p>
<p>If $p$ is a prime in $\textbf{Z}$ of the form $a^2 + b^2$ (with both $a, b \in \textbf{Z}$), it follows that $p$ is not prime in $\textbf{Z}[i]$ since $(a - bi)(a + bi) = p$. But you already kn... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| hmakholm left over Monica | 14,366 | <p>The thing is you're not <em>supposed to</em> "wrap your mind around" higher-dimensional shapes.</p>
<p>Instead, what happens is that we take a formalism that is made to describe 3-dimensional shapes (which we can understand more or less intuitively), and then we just see what happens when we replace all of the "$3$... |
3,251,754 | <p>Let <span class="math-container">$M$</span> be the set of all <span class="math-container">$m\times n$</span> matrices over real numbers.Which of the following statements is/are true??</p>
<ol>
<li>There exists <span class="math-container">$A\in M_{2\times 5}(\mathbb R)$</span> such that the dimension of the nulls... | drhab | 75,923 | <p>Let <span class="math-container">$(\Omega,\mathcal A,P)$</span> denote a probability space and let <span class="math-container">$C\in\mathcal A$</span> with <span class="math-container">$P(C)>0$</span></p>
<p>Then the function <span class="math-container">$P(-\mid C):\mathcal A\to\mathbb R$</span> is a <em>prob... |
2,559,564 | <blockquote>
<p>A nonempty subfamily $\mathcal{F}$ of $Z(X)$ is called $z$-filter on $X$ provided that</p>
<ol>
<li>$ \emptyset \not \in \mathcal{F}$ </li>
<li>If $z_{1} , z_{2} \in \mathcal{F}$ , then $z_{1} \cap z_{2} \in \mathcal{F}$ </li>
<li>If $ z \in \mathcal{F} , z^{*} \in Z(X) , z^{*} \sup... | Henno Brandsma | 4,280 | <p>$1 \Rightarrow 2$ is clear by the definition of a prime filter in $Z(X)$ as $X \in Z(X)$ ($0$-function) and $X \in \mathcal{F}$ for any $z$-filter $\mathcal{F}$ (axiom 3).</p>
<p>$3\Rightarrow 1$ is also clear: suppose $z_1 \cup z_2 \in \mathcal{F}$ for some $z_1,z_2 \in Z(X)$. We apply 3. and get $z \in \mathcal{... |
149,769 | <p>Using the <a href="http://szhorvat.net/pelican/latex-typesetting-in-mathematica.html" rel="nofollow noreferrer">MaTeX</a> package from our colleague <a href="https://mathematica.stackexchange.com/users/12/szabolcs">Szabolcs</a> I had a certain problem.
I would like to highlight the result for bold, but I did not get... | Jason B. | 9,490 | <p>Use <code>\boldsymbol{\pi}</code>, which renders as $\boldsymbol{\pi}$, instead of <code>\mathbf{\pi}</code>, which renders as $\mathbf{\pi}$</p>
<p>(don't have LaTeX on my system, so I can't test with MaTeX)</p>
<pre><code>Needs["MaTeX`"]
MaTeX["\\int_0^r 2\\pi r\\,dr = 2\\pi \\int_0^r r\\,dr = \
2\\pi\\bigg\vert... |
2,860,196 | <p>$X$ is a Kahler manifold. Then is it true that the class of Kahler form $[\omega]$ lies in $H^2(X,\mathbb Z)$?</p>
<p>In fact I am not sure I understand $H^2(X,\mathbb Z)$ correctly. Why can we talk about $H^2_{dR}(X,\mathbb Z)$? Because I don't think "forms with integer coefficients" is well-defined.</p>
<hr>
<p... | Bernard | 202,857 | <p>Note this operation is commutative, so $A$ has an inverse under this operation if and only if there exists $x$ such that $\;a@x=ax+x+a=0$.</p>
<p>Can you determine for which $a$ this equation has a solution?</p>
|
2,860,196 | <p>$X$ is a Kahler manifold. Then is it true that the class of Kahler form $[\omega]$ lies in $H^2(X,\mathbb Z)$?</p>
<p>In fact I am not sure I understand $H^2(X,\mathbb Z)$ correctly. Why can we talk about $H^2_{dR}(X,\mathbb Z)$? Because I don't think "forms with integer coefficients" is well-defined.</p>
<hr>
<p... | Cornman | 439,383 | <blockquote>
<p>a) Show that 0 is an identity for the operation.</p>
</blockquote>
<p>(I write $\circ$ instead of @)</p>
<p>We have $a\circ b:=ab+b+a$. Let $a\in\mathbb{R}$ be arbitrary, then:</p>
<p>$0\circ a=0a+a+0=a$</p>
<p>$a\circ 0=a0+0+a=a$</p>
<blockquote>
<p>b) Show that some real numbers have inverses... |
201,122 | <p>A little bit of <em>motivation</em> (the question starts below the line): I am studying a proper, generically finite map of varieties $X \to Y$, with $X$ and $Y$ smooth. Since the map is proper, we can use the Stein factorization $X \to \hat{X} \to Y$. Since the composition is generically finite, $X \to \hat{X}$ is ... | Sasha | 4,428 | <p>$\hat{X}$ can be as bad as you want. For example, take your favorite non-Gorenstein variety $\hat{X}$ in $\mathbb{A}^N$. By Noether Lemma there is a finite morphism $\hat{X} \to \mathbb{A}^n =: Y$. Take $X$ to be a resolution of singularities of $\hat{X}$. Then $X \to Y$ is a quasifinite morphism between smooth vari... |
23,314 | <p>I have been waiting for <em>Mathematica</em> to give me something for the following integral, errors welcome, but it has been "running" for almost 30 minutes now. </p>
<pre><code> h[s_] := If[1 < s, 1, 0]
Integrate[
Abs[1/(b1^2 + b2^2)
2 E^(-b1 s) ((a1 b1 + a2 b2) E^(
... | gpap | 1,079 | <p>This integral diverges; you can see that by looking at the behaviour of the integrand for real values of the parameters.</p>
<p>Compile the integrand:</p>
<pre><code>h[s_] := UnitStep[s - 1];
g = Compile[{{a1, _Real}, {a2, _Real}, {b1, _Real}, {b2, _Real}, {s,_Real}},
Abs[1/(b1^2 + b2^2) 2 E^(-b1 s) ((a1 b1... |
1,407,131 | <p>I need to prove the following integral is convergent and find an upper bound
$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$</p>
<p>I've tried integrating $\frac{1}{1+x^2+y^2} \lt \frac{1}{1+x^2+y^4}$ but it doesn't converge</p>
| zhw. | 228,045 | <p>Let $y=\sqrt t$ to see the integral is</p>
<p>$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{2\sqrt t(1+x^2+t^2)} dt\ dx.$$</p>
<p>Go to polar coordinates $x=r\cos \theta, t = r\sin \theta$ and we have</p>
<p>$$\int_0^{\pi/2} \frac{1}{\sqrt {\sin \theta}}\ d\theta \int_0^\infty \frac{\sqrt r}{2(1+r^2)}\ dr.$$</p... |
1,148,720 | <blockquote>
<p>Toni and her friends are building triangular pyramids with golf balls.
Write a formula for the number of golf balls in a pyramid with n
layers, if a $1$-layer pyramid contains 1 ball, a 2-layer pyramid contains 4
balls, a 3-layer one contains 10 balls, and so on.</p>
</blockquote>
<p>What is th... | JMoravitz | 179,297 | <p>If it is a square pyramid, the length of the side of each level will increase by one each time you go down. Thus the number of balls on each level is $k^2$. Therefore the total number of balls with $n$ levels is $\sum\limits_{k=1}^n k^2$</p>
<p>In simplifying this it becomes the <a href="http://en.wikipedia.org/w... |
2,981,444 | <p><span class="math-container">$AB$</span> is a chord of a circle and the tangents at <span class="math-container">$A$</span>, <span class="math-container">$B$</span> meet at <span class="math-container">$C$</span>. If <span class="math-container">$P$</span> is any point on the circle and <span class="math-container">... | Michael Rozenberg | 190,319 | <p>Consider two cases.</p>
<ol>
<li><span class="math-container">$P$</span> is located on the circle such that <span class="math-container">$P$</span> and <span class="math-container">$C$</span> are placed in two different sides respect to <span class="math-container">$AB$</span>.</li>
</ol>
<p>Since, <span class="ma... |
2,943,329 | <p>There seem to be six essentially different types of cubic polynomials with real coefficients, giving rise to 1, 2 or 3 real roots in different ways. </p>
<p>Consider <span class="math-container">$f(z) = z^3 + a_2z^2 + a_1z + a_0$</span> and let <span class="math-container">$(a_2,a_1,a_0)$</span> be <span class="mat... | orion | 137,195 | <p>A strange but efficient way that uses the geometric series formula 3 times:</p>
<p><span class="math-container">$$(1+x+x^2+x^3)=\frac{1-x^4}{1-x}$$</span>
Your polynomial is then:
<span class="math-container">$$p(x)=(1+x+x^2+x^3)^2-x^3=\frac{(1-x^4)^2-x^3(1-x)^2}{(1-x)^2}$$</span>
<span class="math-container">$$=\f... |
1,936,045 | <p>Simple question. I was reading through a proof of Mordell's Theorem <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ford.pdf" rel="nofollow">here</a>, and in the statement of the Descent Theorem (page 4), the notation $A/mA$ is used, where $m \geq 2$ is an integer and $A$ is an abelian group. I ... | Brian M. Scott | 12,042 | <p>Yes, your interpretation of it is correct. The notation is quite common when dealing with an Abelian group written additively; the most familiar example is probably $m\Bbb Z$, the subgroup of $\Bbb Z$ consisting of the multiples of $m$.</p>
|
1,936,045 | <p>Simple question. I was reading through a proof of Mordell's Theorem <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Ford.pdf" rel="nofollow">here</a>, and in the statement of the Descent Theorem (page 4), the notation $A/mA$ is used, where $m \geq 2$ is an integer and $A$ is an abelian group. I ... | Dietrich Burde | 83,966 | <p>The notation $A/nA$ is indeed quite common, and arises very often in the context of abelian groups. You gave an example in number theory, because the group of an elliptic curve over $\mathbb{Q}$ is finitely-generated abelian; another context is
in homological algebra. For example, for every abelian group $A$ we have... |
43,095 | <p>The continuous max flow problem is posed as follows : </p>
<p>sup $\int_\Omega p_s(x)dx$</p>
<p>subject to : </p>
<p>$|p(x)| \le C(x); \forall x \in \Omega $</p>
<p>$p_s(x) \le C_s(x); \forall x \in \Omega $</p>
<p>$p_t(x) \le C_t(x); \forall x \in \Omega $</p>
<p>$\nabla \cdot p(x) - p_s(x) + p_t(x) = 0; ... | Jim Belk | 1,726 | <p>I'm assuming $\Omega$ is an $n$-dimensional manifold with a volume form (such as a region in $\mathbb{R}^n$).</p>
<p>First of all, we may assume that $C_s$ and $C_t$ have disjoint supports, for otherwise we can simplify the problem by replacing $C_s$ by $\max(C_s-C_t,0)$ and $C_t$ by $\max(C_t-C_s,0)$, and adding $... |
663,363 | <p>I do not know if this is an ill-posed question but ...
is $\delta(t)e^{-\gamma t}$ equal to $\delta(t)$?</p>
<p>Thanks,
biologist</p>
| Cameron Buie | 28,900 | <p>Consider the space $V$ of all functions $\Bbb R\to\Bbb R$ as a vector space over $\Bbb R.$ The set $P$ of all polynomial functions is an infinite-dimensional subspace, but (for example) consider the set $F$ of functions $f:\Bbb R\to\Bbb R$ such that $\{x\in\Bbb R:f(x)\ne 0\}$ is finite. Then $F$ is again infinite-di... |
3,736,706 | <p>Let <span class="math-container">$M$</span> be an <span class="math-container">$A$</span>-module and let <span class="math-container">$\mathfrak{a}$</span> and <span class="math-container">$\mathfrak{b}$</span> be coprime ideals of A.</p>
<p>I must show that <span class="math-container">$M/ \mathfrak{a}M \oplus M/ \... | egreg | 62,967 | <p>There is an obvious homomorphism <span class="math-container">$\varphi\colon M\to M/\mathfrak{a}M\oplus M/\mathfrak{b}M$</span>, namely <span class="math-container">$\varphi(x)=(x+\mathfrak{a}M,x+\mathfrak{b}M)$</span>.</p>
<p>The kernel is obviously <span class="math-container">$\mathfrak{a}M\cap\mathfrak{b}M$</spa... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Michael Hardy | 205 | <p>Quoting the question: "the study of one of the great intellectual and creative achievements of humankind"</p>
<p>The conventional high school course except for statistics and perhaps some aspects of geometry exists only for the purpose of preparing students for calculus. 99% of the students who take calculus will n... |
4,281,028 | <p>I have the following problem, which was asked in a <a href="https://math.stackexchange.com/questions/584375/cumulative-distribution-function-word-problem">similar question</a> but it doesn't help me.</p>
<p><strong>A dart is equally likely to land at any point inside a circular target of unit radius.
Let <span class... | drhab | 75,923 | <p><span class="math-container">$$P(R\leq r)=\frac{\text{area of disc with radius }r}{\text{area of disc with radius }1}=\frac{\pi r^2}{\pi}=r^2$$</span></p>
<p>This is based on the fact that we are dealing with a unifom distribution on a disc with radius <span class="math-container">$1$</span>.</p>
<p>Then for every s... |
2,060,694 | <p>Problem:</p>
<blockquote>
<p>I have 610 friends. Each one of them will invite me to his birthday party, and I will accept every invitation. What is the probability that I will be attending at least one birthday party on every day of the year?</p>
</blockquote>
<p>My attempt has been to try counting how many ways... | duanduan | 390,113 | <p>Here I gave a simulation solution which is based on Monte Carlo sampling. You can set up the number of sampling experiments, but the result is alway 0 since the probability of this kind of event is pretty low. </p>
<p>The C++ code is as below:</p>
<pre><code>#include <iostream>
#include <map>
#include ... |
2,326,564 | <p>Is it true that iff CardA = Card A then A is a set of distinct terms? </p>
<p>[This questions is actually from a confusion on what a set versus multiset is]</p>
| Bram28 | 256,001 | <p>$$A+A'B=(A+A')(A+B)=1(A+B)=A+B$$</p>
<p>The first step is Distribution: $A+BC=(A+B)(A+C)$</p>
|
1,257,193 | <p>Let $f:[0,\infty)\to\mathbb{R}$ continuous and $\lim\limits_{x\to\infty}f(x)=a$. Claim: $\lim\limits_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=a$.</p>
<p>My try: It is $\int_0^xf(t)dt=F(x)-F(0)$ (because of the fundamentaltheorem of calculus) and $\lim\limits_{x\to\infty}F(x)=ax+b$, because $\lim\limits_{x\to\infty}f(x... | Timbuc | 118,527 | <p>You're close: as $\;f\;$ is continuous it has a primitive $\;F\;$ , so</p>
<p>$$\frac1x\int_0^xf(t)dt=\frac{F(x)-F(0)}{x-0}\stackrel{MVT}=F'(c_x)=f(c_x)\;,\;\;c_x\in(0,x)$$</p>
<p>and now just take the limit $\;x\to\infty\;$ ... <strong>and be careful</strong> (for example, for different $\;x$'s one could possibly... |
1,257,193 | <p>Let $f:[0,\infty)\to\mathbb{R}$ continuous and $\lim\limits_{x\to\infty}f(x)=a$. Claim: $\lim\limits_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=a$.</p>
<p>My try: It is $\int_0^xf(t)dt=F(x)-F(0)$ (because of the fundamentaltheorem of calculus) and $\lim\limits_{x\to\infty}F(x)=ax+b$, because $\lim\limits_{x\to\infty}f(x... | Community | -1 | <p>Use L'Hôpital's rule and the fundamental theorem of calculus. There are two cases.</p>
<p><strong>Case 1</strong>: $\int_0^\infty f(t)dt = \pm \infty$. In this case, by L'Hôpital's rule and the FTC, $$\lim_{x \to \infty} \frac{1}{x} \int_0^x f(t)dt = \lim_{x \to \infty}\frac{\frac{d}{dx}\int_0^x f(t)dt}{\frac{d}{dx... |
25,158 | <p>I'm trying to derive the LTE for CN applied to the linear heat equation;
$u_t = u_{xx}$.</p>
<p>The problem is that I end up with terms of the form $\frac{{\Delta t}^k}{{\Delta x}^2}$
when using a two dimensional Taylor expansion around $(x,t)$ for the term:</p>
<p>${\Delta x}^2 {\delta^2_x} = (u_{i+1}^{n+1} - 2 ... | Community | -1 | <p>I don't know what is the $k$ in your $\frac{\Delta t^k}{\Delta x^2}$. And I cannot tell what's wrong in your result since you didn't provide enough details. </p>
<p>For the LTE (Local Truncation Error) of the C-N (<a href="http://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method" rel="nofollow">Crank-Nicolson sch... |
639,449 | <p>I've seen on Wikipedia that for a complex matrix $X$, $\det(e^X)=e^{\operatorname{tr}(X)}$.</p>
<p>It is clearly true for a diagonal matrix. What about other matrices ?</p>
<p>The series-based definition of exp is useless here.</p>
| Jakub Konieczny | 10,674 | <p>If $X$ is upper triangular, then this is clear.</p>
<p>If the claim holds for a matrix $Y$, then it holds for any $X$ similar to $Y$.</p>
<p>By Jordan decomposition, each $X$ is similar to an upper triangular matrix $Y$ (of a special form, but never mind).</p>
<p>Thus, the claim holds for all $X$.</p>
|
258,205 | <p>I want to know if $\displaystyle{\int_{0}^{+\infty}\frac{e^{-x} - e^{-2x}}{x}dx}$ is finite, or in the other words, if the function $\displaystyle{\frac{e^{-x} - e^{-2x}}{x}}$ is integrable in the neighborhood of zero.</p>
| Kerry | 7,887 | <p>It suffice to expand the function locally:$$e^{-x}-e^{-2x}=(1-x+x^{2}/2)+..-(1-2x+4x^{2}/2)-...=x-3x^{3}/2+...$$ where $...$ are terms of power at least cubic. It is not difficult to see the above expression divided by $x$ should be locally integrable around 0. </p>
|
2,937,990 | <p>I need to prove or disprove that in any Boolean algebra: if <span class="math-container">$a+ab=b$</span> then <span class="math-container">$a=b=1$</span> or <span class="math-container">$a=b=0$</span>.</p>
<p>I build the following truth table:
<span class="math-container">$$
\begin{array}{|c|c|c|}
\hline
a & b ... | Robert Z | 299,698 | <p>Hint. Note that
<span class="math-container">$$d_n=\sum_{k=0}^n\frac{1}{\sqrt{(1+k)(n+1-k)}}\geq \sum_{k=0}^n\frac{2}{(1+k)+(n+1-k)}=\frac{2(n+1)}{n+2},$$</span>
and therefore <span class="math-container">$c_n=(-1)^n d_n$</span> does not tend to zero as <span class="math-container">$n$</span> goes to infinity.</p>
|
418,026 | <p>I am asked to find the power series of the function $f(x)=\arctan(\frac{x}{\sqrt{2}})$. I first found the derivative of this function which is: $f'(x)=\frac{\sqrt{2}}{2+x^{2}}$. Then I found the power series of $f'(x)$ which is: $\sum_{n=0}^{\infty }\frac{1}{\sqrt{2}}(-1)^{n}\left ( \frac{1}{2} \right )^{n}x^{2n}$. ... | Michael Hardy | 11,667 | <p>We know $\displaystyle u_i^T G u_j = \begin{cases} 1 & \text{if }i=j, \\ 0 & \text{if }i\ne j. \end{cases}$</p>
<p>How to prove that
$$
u^T G\left(\sum_{j=1}^n u_j u_j^T\right) = u^T,
$$
so that
$$
G\left(\sum_{j=1}^n u_j u_j^T\right) = I\text{ ?}
$$</p>
<p>It is enough to show this for $u=u_i$, $i=1,\ldot... |
4,473,632 | <p>Let <span class="math-container">$f:\mathbb R^+\to\mathbb R$</span> be a continuous function satisfied <span class="math-container">$f(a)+f(b)\ge f(2\sqrt{ab})$</span> for all <span class="math-container">$a,b>0$</span> , is <span class="math-container">$f$</span> differentiable?</p>
<p>Morever, if for all <span ... | FShrike | 815,585 | <p>Explicit demonstration: Let <span class="math-container">$x\gt0$</span> be such that <span class="math-container">$f(2x)\neq0$</span> and <span class="math-container">$f(x)\neq0$</span>. Define <span class="math-container">$y$</span> such that <span class="math-container">$2x=e^y$</span>, which is possible by surjec... |
2,653,829 | <blockquote>
<p>How can I show $(x^2+1, y^2+1)$ is not maximal in $\mathbb R[x,y]$?</p>
</blockquote>
<p>I know I can mod out the ideal one piece at a time and show $\mathbb C[x]/(x^2+1)$ is not a field since $(x^2+1)$ is not maximal in $\mathbb C[x]$, <strong>but is there another way of showing this?</strong></p>
| hunter | 108,129 | <p>It's not even prime: $(x+y)(x-y) = x^2 - y^2 = (x^2+1)-(y^2+1)$ belongs to this ideal, but neither $(x+y)$ nor $(x-y)$ does (the latter by considering degrees).</p>
|
2,636,931 | <p>Consider the ellipse given by:</p>
<p>$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0.
$$</p>
<p>What is the equation of an ellipse which has major and minor axis equal to $p$ times the major and minor axis length of the above ellipse.</p>
<p>My attempt is as follows:
We can remove rotation, increase axis length and then r... | Ng Chung Tak | 299,599 | <p>Referring to the standard results <a href="https://math.stackexchange.com/questions/1839510/how-to-get-the-correct-angle-of-the-ellipse-after-approximation/1840050#1840050"><strong>here</strong></a>, the centre is given by </p>
<p>$$(h,k)=
\left(
\frac{2CD-BE}{B^2-4AC}, \frac{2AE-BD}{B^2-4AC}
\right)$$</p>
<p>an... |
3,144,197 | <p>The title may be confusing but i want to show that if <span class="math-container">$f:\mathbb R^n \rightarrow \mathbb R^m$</span> <span class="math-container">$f \in C^1$</span> with <span class="math-container">$m<n$</span> then f is not injective and If <span class="math-container">$m>n$</span> f is not surj... | Noah Schweber | 28,111 | <p>The statement you want is (without further hypotheses on <span class="math-container">$f$</span>) <strong>false</strong> - this was observed by Cantor, and is similar to how there is a bijection between <span class="math-container">$\mathbb{N}$</span> and <span class="math-container">$\mathbb{N}\times\mathbb{N}$</sp... |
3,144,197 | <p>The title may be confusing but i want to show that if <span class="math-container">$f:\mathbb R^n \rightarrow \mathbb R^m$</span> <span class="math-container">$f \in C^1$</span> with <span class="math-container">$m<n$</span> then f is not injective and If <span class="math-container">$m>n$</span> f is not surj... | avs | 353,141 | <p>If <span class="math-container">$f$</span> is required to be <span class="math-container">${\cal C}^{1}$</span>, then you assume <span class="math-container">$f(0) = 0$</span> (where each zero denotes the zero vector of the appropriate dimension) and use the Implicit Function Theorem, which says that <span class="ma... |
1,673,452 | <p>Let $\{a_j\}_{j=1}^N$ be a finite set of positive real numbers. Suppose </p>
<p>$$\sum_{j=1}^{N} a_j = A,$$ prove</p>
<p>$$\sum_{j=1}^{N} \frac{1}{a_j} \geq \frac{N^2}{A}.$$ </p>
<p>Hints on how to proceed?</p>
| zhw. | 228,045 | <p>Let $g(x) = 1/x, x > 0.$ Then $g$ is convex on $(0,\infty),$ hence by Jensen,</p>
<p>$$ g(\frac{1}{N}\sum_{j=1}^{N}a_n) \le \frac{1}{N}\sum_{j=1}^{N}g(a_n).$$</p>
<p>The inequality falls right out. </p>
|
25,782 | <p>Hello I'm having trouble showing the following:</p>
<p>Let $u$ be a positive measure. If $\int_E f\, du= \int_E g\, du$ for all measurable $E$ then $f=g$ a.e.</p>
<p>I was trying to argue by contradiction: if $f\neq g$ a.e. then there must exist some set $E=\{x: f(x)\neq g(x)\}$ such that $u(E) \gt 0$. Then let $E... | Eric Naslund | 6,075 | <p><strong>Hint</strong> We have that $f$ and $g$ map into $\mathbb{R}$ from some unknown measure space, say $(X,\mathcal{M})$. Let $h(x)=f(x)-g(x)$. Then $E^+\subset h^{-1}(0,\infty)$ and $E^- \subset h^{-1}(-\infty,0)$. Then recall that the sum of two measurable functions is measurable (addition is continuous), an... |
2,590,068 | <p>$$\epsilon^\epsilon=?$$
Where $\epsilon^2=0$, $\epsilon\notin\mathbb R$.
There is a formula for exponentiation of dual numbers, namely:
$$(a+b\epsilon)^{c+d\epsilon}=a^c+\epsilon(bca^{c-1}+da^c\ln a)$$
However, this formula breaks down in multiple places for $\epsilon^\epsilon$, yielding many undefined expressions l... | Mike Battaglia | 52,694 | <p>The function <span class="math-container">$x^x$</span> is undefined at <span class="math-container">$x = \epsilon$</span> in the dual numbers.</p>
<p>To see this, we note that it's equal to <span class="math-container">$\exp(x \log x) = 1 + x \log x + \frac{x^2 \log^2 x}{2!} + ...$</span>.</p>
<p>If we put <span cla... |
3,489,086 | <p>I was given this integral:</p>
<p><span class="math-container">$$\int^{\infty}_{0}\frac{\arctan(x)}{x}dx$$</span></p>
<p>As the title says, I have to find out whether it is convergent or not.
So far, I have tried integrating by parts and substituting <span class="math-container">${\arctan(x)}$</span>, and neither ... | Hanul Jeon | 53,976 | <p>If <span class="math-container">$\operatorname{atg}$</span> means the inverse tangent function, then your integral does not converge.</p>
<p>One can see that <span class="math-container">$\arctan x\ge \pi/4$</span> if <span class="math-container">$x\ge 1$</span>, so <span class="math-container">$$\frac{\arctan x}{x... |
3,853,351 | <p>Given an n-dimensional ellipsoid in <span class="math-container">$\mathbb{R}^n$</span>, is any orthogonal projection of it to a subspace also an ellipsoid? Here, an ellipsoid is defined as</p>
<p><span class="math-container">$$\Delta_{A, c}=\{x\in \Bbb R^n\,:\, x^TAx\le c\}$$</span></p>
<p>where <span class="math-co... | spiridon_the_sun_rotator | 810,640 | <p>There are already a good answers presented, but I want to add the also one may think in a following way :</p>
<p>The orthogonal projection defines some subspace <span class="math-container">$\langle e_1, e_2 \ldots e_n \rangle$</span>, and we perform an orthogonal transformation <span class="math-container">$R^{T}$<... |
2,476,194 | <p>I am trying to prove that $I =(x^2+1,y-1)$ is a maximal ideal in $\mathbb{Q}[x,y]$, but I am having a hard time understanding what this ideal even looks like. I know that I can prove it's a maximal ideal by proving $\mathbb{Q}[x,y]/I$ is a field, but I'm also having a hard time understanding what this quotient looks... | A. Thomas Yerger | 112,357 | <p>Perhaps the intuition you would like comes from algebraic geometry? There is a correspondence called the Nullstellensatz which gives a connection between prime ideals and algebraic sets in affine space over the given algebraically closed field. This is one way to visually 'see' what an ideal looks like.</p>
<p>It i... |
1,905,898 | <p>The graph of quadratic function drawn on the interval $-1\leq x\leq 5$.</p>
<p><a href="https://i.stack.imgur.com/0eOzE.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0eOzE.jpg" alt="enter image description here"></a></p>
<p>i.If the quadratic function is , $y=(x-2)^2-k$, find the value of $k$.... | Community | -1 | <p>If $\gamma < \alpha$, then since $\alpha$ is the <strong>least</strong> upper bound of $L$, this means that $\gamma$ is <strong>not</strong> an upper bound of $L$, which means there is some $c \in L$ with $\gamma < c$. But $c \in L$, so $c$ is a lower bound of $B$, meaning every element of $B$ is at least as l... |
1,905,898 | <p>The graph of quadratic function drawn on the interval $-1\leq x\leq 5$.</p>
<p><a href="https://i.stack.imgur.com/0eOzE.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0eOzE.jpg" alt="enter image description here"></a></p>
<p>i.If the quadratic function is , $y=(x-2)^2-k$, find the value of $k$.... | lEm | 319,071 | <p>By definition of supremum, if $\gamma < \alpha$, then $\gamma$ is not an upper bound of $L$. Since every element of $B$ is an upper bound of $L$, so $\gamma$ must not be in $B$.</p>
<p>Since $\alpha$ is an upper bound of $L$, if $\beta \in L$, then we have a contradiction that $\beta \leq \alpha $. So $\beta$ is... |
1,905,898 | <p>The graph of quadratic function drawn on the interval $-1\leq x\leq 5$.</p>
<p><a href="https://i.stack.imgur.com/0eOzE.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0eOzE.jpg" alt="enter image description here"></a></p>
<p>i.If the quadratic function is , $y=(x-2)^2-k$, find the value of $k$.... | Sathasivam K | 355,833 | <p>L is the set of all lower bound of B and $\alpha$ is the supreme of L.if an element $\gamma$ <$\alpha$ ,it mean that $\gamma$ is belong to L and not belong to B.</p>
<p>Also,since,L is thethe set of lower bound of B,which means x$\in B$ is the upper bound of L.but $\alpha $ is greatest lower bound ,so x must be ... |
2,929,203 | <p>Suppose we define the relation <span class="math-container">$∼$</span> by <span class="math-container">$v∼w$</span> (where <span class="math-container">$v$</span> and <span class="math-container">$w$</span> are arbitrary elements in <span class="math-container">$R^n$</span>) if there exists a matrix <span class="mat... | M. Winter | 415,941 | <p><strong>Hint:</strong> There are two equivalence classes: <span class="math-container">$\{0\}$</span> and <span class="math-container">$\Bbb R^n\setminus \{0\}$</span>.</p>
|
3,582,585 | <p>Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the
die again and if any other number comes, toss a coin. Find the conditional probability of the event <strong>‘the coin shows a tail’</strong>, given that <em>‘at least one throw of die shows a 3’</em>.</p>
<p>I don't know how to deal w... | fleablood | 280,126 | <p>To get the variables directly take logarithms. (Doesn't matter which base).</p>
<p><span class="math-container">$\log 3^{15a} = \log 5^{5b} = \log 15^{3c}$</span> so </p>
<p><span class="math-container">$15a \log 3 = 5b \log 5 = 3c \log 15$</span>.</p>
<p>So flip a coin and choose which one we should express the ... |
384,700 | <p>This question addresses a hierarchy of linear recurrences
which arise from an attempt to generalize the Nekrasov-Okounkov
formula to the Young-Fibonacci setting.
A related posting</p>
<p><a href="https://mathoverflow.net/questions/384591/extensions-of-the-nekrasov-okounkov-formula">extensions of the Nekrasov-Okounko... | Salvatore Siciliano | 14,653 | <p>Let <span class="math-container">$L$</span> be a finite-dimensional <span class="math-container">$p$</span>-nilpotent restricted Lie algebra over a field of characteristic <span class="math-container">$p>0$</span> and consider its restricted enveloping algebra <span class="math-container">$u(L)$</span>. Then the ... |
3,514,547 | <p>The problem is as follows:</p>
<p>The figure from below shows vectors <span class="math-container">$\vec{A}$</span> and <span class="math-container">$\vec{B}$</span>. It is known that <span class="math-container">$A=B=3$</span>. Find <span class="math-container">$\vec{E}=(\vec{A}+\vec{B})\times(\vec{A}-\vec{B})$</s... | user247327 | 247,327 | <p>The cross product is associative and anti-commutative. (A+ B)X(A- B)= AXA- AXB+ BXA- BXB= -AXB- AXB= -2AXB. Yes, since both A and B lie in the xy-plane -AXB is in the negative z direction. And since <span class="math-container">$|AXB|= |A||B| sin(\theta)$</span>, here |AXB|= 2(3)(3) sin(150)= 9. (A+ B)X(A- B)= -9... |
844,832 | <p>How to find the derivative of this function $$ 7\sinh(\ln t)?$$</p>
<p>I don't know from where to start, so i looked at it in wolfram alpha and it was saying that the $$ 7((-1 + t^2) / 2t) $$ I did not get that. How did they jump from $$ 7\sinh(\ln t) $$ to this step? Is there an equation for it that I am missing?<... | Jean-Claude Arbaut | 43,608 | <p>Hint:</p>
<p>$$\sinh (\ln t)=\frac{e^{\ln t}-e^{-\ln t}}{2}=\frac{t-\frac{1}{t}}{2}=\frac{t^2-1}{2t}$$</p>
|
489,562 | <p>I am teaching a "proof techniques" class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to predicate logic and simple proofs using the rules of first order logic. After that we prove simple math statements via dir... | Willemien | 88,985 | <p>maybe start with some riddles from <a href="http://en.wikipedia.org/wiki/Charles_Lutwidge_Dodgson" rel="nofollow">http://en.wikipedia.org/wiki/Charles_Lutwidge_Dodgson</a> or <a href="http://en.wikipedia.org/wiki/Raymond_Smullyan" rel="nofollow">http://en.wikipedia.org/wiki/Raymond_Smullyan</a> both have made books ... |
3,269,112 | <p>Theorem:</p>
<p><span class="math-container">$ x \lt y + \epsilon$</span> for all <span class="math-container">$\epsilon \gt 0$</span> if and only if <span class="math-container">$x \leq y$</span></p>
<p>Suppose to the contrary that <span class="math-container">$x \lt y + \epsilon$</span> but <span class="math-con... | Arturo Magidin | 742 | <p>I really dislike this way of phrasing this particular problem (or rather, the typical more general statement of which this problem is a special case), <em>precisely</em> because it just leads to confusions because it contradicts (or seems to contradict) the initial hypothesis.</p>
<p>The general statement they are ... |
3,458,962 | <p>If I am given this function: <span class="math-container">$$f(x) = \sum_{i = 1}^{\infty} \frac{1}{i^x}$$</span>
Is there a way to rewrite:
<span class="math-container">$$g(x) = \sum_{ j = 1}^{\infty} \sum_{i = j}^{\infty} \frac{1}{(i \cdot j)^x}$$</span>
In terms of f(x). By simple arthimatic I know that <span class... | marty cohen | 13,079 | <p>For
<span class="math-container">$g(x)
= \sum_{ j = i}^{\infty} \sum_{i = 0}^{\infty} \dfrac{1}{(i \cdot j)^x}
$</span>
and sums like it,
anything independent
of the innermost index of summation
can be pulled out.</p>
<p>However,
you have a mistake in
the way you have written this:
An outer summation
can not depen... |
3,458,962 | <p>If I am given this function: <span class="math-container">$$f(x) = \sum_{i = 1}^{\infty} \frac{1}{i^x}$$</span>
Is there a way to rewrite:
<span class="math-container">$$g(x) = \sum_{ j = 1}^{\infty} \sum_{i = j}^{\infty} \frac{1}{(i \cdot j)^x}$$</span>
In terms of f(x). By simple arthimatic I know that <span class... | Donald Splutterwit | 404,247 | <p>Let
<span class="math-container">\begin{eqnarray*}
f_1(x) &=& \sum_{i \geq 1} \frac{1}{i^x} \\
f_2(x) &=& \sum_{i >j \geq 1} \frac{1}{(ij) ^x} \\
f_3(x) &=& \sum_{i>j>k \geq 1} \frac{1}{(ijk)^x}. \\
\end{eqnarray*}</span>
In your question you have calculated
<span class="math-con... |
2,335,831 | <p>I am trying to implement an Extended Kalman Filter (EKF) and it is becoming harder than I thought.</p>
<p>I have one question. I noticed that the covariance matrix which should get updated over each iteration is not symmetric. I am debugging through MATLAB.
I know that P should be symmetric and stay symmetric.</p>... | thb | 30,109 | <p>Unitarily diagonalize $P=V\Lambda V^{*}$, where the adjoint $V^{*}$ is just $V^T$ if your matrix is real. Having done this, <em>do not store $V^{*}$ separately.</em></p>
<p>The matrix $\Lambda$ is to have the eigenvalues of $P$ along its main diagonal and is to be null elsewhere. The columns of $V$ are to be mutual... |
666,217 | <p>If $a^2+b^2 \le 2$ then show that $a+b \le2$</p>
<p>I tried to transform the first inequality to $(a+b)^2\le 2+2ab$ then $\frac{a+b}{2} \le \sqrt{1+ab}$ and I thought about applying $AM-GM$ here but without result</p>
| N. S. | 9,176 | <p><strong>Hint</strong> Use Cauchy-Schwarz.</p>
<p><strong>Second solution</strong></p>
<p>$$(a+b)^2=a^2+b^2+2ab \leq 2+2ab$$</p>
<p>You got that far, you are almost there:
By AM-GM</p>
<p>$$\sqrt{a^2b^2} \leq \frac{a^2+b^2}{2}$$</p>
<p>which implies $$2ab \leq a^2+b^2 \leq 2$$</p>
|
2,713,873 | <p>We know that if a real valued function $f$ is continuous over an interval $[a,b]$ then the following integral $$\int_a^bf(x)dx$$ represents the area between horizontally the line $y=0$ and the curve of $f$, vertically between the lines $x=a$ and $x=b$. So what represent the following $$\int_{[a,b]\times [c,d]}g(x... | Pritt Balagopal | 433,002 | <p>Two-variable integrations refer to the volume enclosed by the surface, the $xy$-plane and the planes perpendicular to $xy$-plane passing through $x=a$ to $x=b$, and $y=c$ to $y=d$, in the case of:</p>
<p>$$\int_{y=c}^{y=d}\int_{x=a}^{x=b}{g(x,y)dxdy}$$</p>
<p>Going by analogy, the three-variable integrations refer... |
4,310,003 | <p>Suppose you have a non empty set <span class="math-container">$X$</span>, and suppose that for every function <span class="math-container">$f : X \rightarrow X$</span>, if <span class="math-container">$f$</span> is surjective, then it is also injective. Does it necessarily follow that <span class="math-container">$... | Darshan P. | 992,496 | <p><span class="math-container">$$\int_{-\infty}^{\infty}\frac {e^x}{1 + e^{4x}}dx$$</span>
Substitute: <span class="math-container">$ e^x = \sqrt{\tan\theta} \implies e^xdx = \frac {\sec^2{\theta}d\theta}{2\sqrt{\tan\theta}}$</span></p>
<p><span class="math-container">$$\begin{align*}
\frac12\int_0^{\frac\pi2} \sin^{-... |
151,430 | <p>Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore ho... | Mark Grant | 8,103 | <p>This is an answer to your more general question about how to define $$f_\ast:H^\ast(X;\mathbb{Z})\to H^{\ast+\dim(Y)-\dim(X)}(Y;\mathbb{Z})$$
when $f: X\to Y$ is a proper oriented map.</p>
<p>For $\ell$ sufficiently large, there is an embedding $g: X\hookrightarrow \mathbb{R}^\ell$ which is unique up to isotopy. Th... |
3,578,740 | <p>Good day everybody,</p>
<p>I would like to ask a question about undecidability.
May I ask You, if we have some problem that is undecidable but true, for example if RH would be found out to be undecidable it would mean that it is true, does that mean that such undecidable problem is true for no reason at all, or is ... | Esa Pulkkinen | 741,661 | <p>The relationship between infinity and undecidability can for example be seen in the Limit lemma and related Modulus lemma. In particular, with infinite sequences of total computable functions and their limits it is possible to express undecidable problems, and it's possible to determine that the modulus of convergen... |
364,800 | <p>Let <span class="math-container">$V$</span> be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve <span class="math-container">$U$</span> such that there is a non-constant holomorphic map <span class="math-container">$U\to V$</span... | Ariyan Javanpeykar | 4,333 | <p>Just turning my comments into an answer:</p>
<p>Following the OP, let <span class="math-container">$V$</span> be a smooth projective connected curve with negative Euler characteristic (i.e., genus at least two) over <span class="math-container">$\mathbb{C}$</span>. Then <span class="math-container">$V$</span> is hy... |
3,400,766 | <p>I know that by considering projection <span class="math-container">$q : \mathbb{R}^2 \to \mathbb{R}$</span>, <span class="math-container">$(x, y) \to x$</span>, and the closed subset </p>
<p><span class="math-container">$$G = \left\{(x, y) : y \ge \frac 1 x, x > 0\right\}$$</span></p>
<p>will prove that <span c... | Berci | 41,488 | <p>The term 'projection' for the natural quotient map <span class="math-container">$X\to X/M$</span> is rather illustrative.</p>
<p>However, the projection <span class="math-container">$q:\Bbb R^2\to \Bbb R$</span> can also be viewed as a quotient map, namely take <span class="math-container">$X=\Bbb R^2 $</span> (as ... |
172,080 | <p>Here is a fun integral I am trying to evaluate:</p>
<p>$$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$</p>
<p>I thought about integrating by parts $2n$ times and then using the binomial theorem for $\sin(x)$, that is, using $\dfrac{e^{ix}-e^{-ix}}{2i}$ form in the binomial se... | Sasha | 11,069 | <p>Using
$$
\sin^{2n+1}(x) = \sum_{k=0}^n \frac{(-1)^k }{4^n} \binom{2n+1}{n+k+1} \sin\left((2k+1)x\right)
$$
We get
$$ \begin{eqnarray}
\int_0^\infty \frac{\sin^{2n+1}(x)}{x}\mathrm{d} x &=& \sum_{k=0}^n \frac{(-1)^k }{4^n} \binom{2n+1}{n+k+1}\int_0^\infty \frac{\sin\left((2k+1)x\right)}{x}\mathrm{d} x\\ &... |
40,836 | <p>I have a very big notebook with a command repeated a lot of times but with different parameters each time and I want to run all these commands at the same time without having to run separately all of them. Is there a way?</p>
<p><strong>EDIT: more info</strong></p>
<p>Between different calls of the 'command' there... | Red | 7,847 | <p>I get it using this code:</p>
<pre><code>nb = EvaluationNotebook[];
NotebookEvaluate[nb, InsertResults -> True, EvaluationElements -> {"Tags" -> {"evaluate"}}];
</code></pre>
<p>and tagging with the tag <code>evaluate</code> only the cells that I want to evaluate on first run.</p>
|
40,836 | <p>I have a very big notebook with a command repeated a lot of times but with different parameters each time and I want to run all these commands at the same time without having to run separately all of them. Is there a way?</p>
<p><strong>EDIT: more info</strong></p>
<p>Between different calls of the 'command' there... | ciao | 11,467 | <p>As requested:</p>
<p>Use tags on the cells you want (same tag for all), then <code>NotebookLocate</code> and<code>SelectionEvaluate</code> together.</p>
|
2,962,880 | <p>So I am looking at the integral
<span class="math-container">$$ \int_0^{2\pi} \frac{1}{a + \cos\theta} \mathrm{d} \theta$$</span> for <span class="math-container">$a > 1$</span>. Evaluating the integral with complex analysis gives us <span class="math-container">$\frac{2 \pi}{\sqrt{a^2 - 1}} $</span> but I have f... | Szeto | 512,032 | <p>The theory about substitution actually states that when you want to use the substitution <span class="math-container">$t=g(\theta)$</span>:</p>
<blockquote>
<p>Suppose <span class="math-container">$f(\theta)=f^*(t)$</span> and <span class="math-container">$g’(\theta)=g^*(t)$</span>, then <span class="math-contain... |
2,962,880 | <p>So I am looking at the integral
<span class="math-container">$$ \int_0^{2\pi} \frac{1}{a + \cos\theta} \mathrm{d} \theta$$</span> for <span class="math-container">$a > 1$</span>. Evaluating the integral with complex analysis gives us <span class="math-container">$\frac{2 \pi}{\sqrt{a^2 - 1}} $</span> but I have f... | sirous | 346,566 | <p>If you plot the function you will see a maximum <span class="math-container">$\frac{1}{a-1}$</span> at <span class="math-container">$\pi$</span>, so we can consider the bound as <span class="math-container">$[\pi, 0]$</span> and double the resulted value:</p>
<p><span class="math-container">$$2\times[\frac {2}{\sqr... |
1,061,077 | <p>SO, I drop a piece of bread and jam repeatedly. It lands either jam face-up or jam face-down and I know that jam side down probability is <span class="math-container">$P(Down)=p$</span></p>
<p>I continue to drop the bread until it falls jam side up for the first time. What is the expression for the expected number o... | Barry Cipra | 86,747 | <p>Let $X$ be the random variable for the number of drops until the bread lands jam-side up. You found that $P(X=n)=p^{n-1}(1-p)$. The expected value is therefore</p>
<p>$$E(X)=1(1-p)+2p(1-p)+3p^2(1-p)+4p^3(1-p)+\cdots$$</p>
<p>Now the problem mentions the binomial expansion of $(1-p)^{-2}$, so I assume you know th... |
3,189,173 | <p>What will be the remainder when <span class="math-container">$2^{87} -1$</span> is divided by <span class="math-container">$89$</span>?</p>
<p>I tried it solving by Euler's remainder theorem by separating terms:</p>
<p><span class="math-container">$$ \frac {2^{87}}{89} - \frac{1}{89}$$</span></p>
<p><span class="... | saulspatz | 235,128 | <p><span class="math-container">$2^{88}\equiv1\pmod{89}\implies45\cdot2^{88}\equiv45\pmod{89}\implies90\cdot2^{87}\equiv45\pmod{89}$</span> Continue from here.</p>
|
2,477,676 | <p>I'm supposed to prove that for any Random Variable X, </p>
<p>$E[X^4] \ge \frac 14 P(X^2\ge \frac 12)$</p>
<p>I tried substituting the definitions of expected value and of the probability into the inequality, but that gets me no where. </p>
<p>Any tips on where to go with this proof? Would a moment generating fun... | Daniel Ordoñez | 478,506 | <p>Hint:
Chebyshev’s Inequality
$P[\arrowvert x \arrowvert \geq a]\leq \frac{E[x^2]}{a^2}$</p>
|
3,073,832 | <p>I need to understand the meaning of this mathematical concept: "undecided/undecidable". </p>
<p>I know what it means in the English dictionary. But, I don't know what it means mathematically.</p>
<p>If You answer this question with possible mathematical examples, it will be very helpful to understand this issue.<... | Acccumulation | 476,070 | <p>Suppose you have some logical system <span class="math-container">$S$</span> equipped with some initial axioms and some process for generating statements from previous statements. Suppose you have a set <span class="math-container">$T$</span> of statements such that each statement is either an axiom in <span class="... |
1,717,802 | <p>So I assume I rewrite the equation like this:</p>
<p>$\frac{dy}{dx}=x^2e^{-4x}-4y \Rightarrow \frac{dy}{dx}+4y=x^2e^{-4x}$</p>
<p>I then solve the homogenous form of the equation by writing its characteristic equation:</p>
<p>$r+4=0$, which means $r=-4$</p>
<p>This means my general solution is: $y=Ae^{-4x}$</p>
... | Nikolaos Skout | 308,929 | <p>Hint. Assume that you have a solution of the form $A(x)e^{-4x}$ for the non-homogenuous problem $y'+4y=x^2e^{-4x}$ and substitute to this equation to get $A(x)$. The general solution of the ODE is $Ae^{-4x}+A(x)e^{-4x}.$</p>
<p>Edit. (methodology) A standard procedure goes like that: suppose you have found the gene... |
1,717,802 | <p>So I assume I rewrite the equation like this:</p>
<p>$\frac{dy}{dx}=x^2e^{-4x}-4y \Rightarrow \frac{dy}{dx}+4y=x^2e^{-4x}$</p>
<p>I then solve the homogenous form of the equation by writing its characteristic equation:</p>
<p>$r+4=0$, which means $r=-4$</p>
<p>This means my general solution is: $y=Ae^{-4x}$</p>
... | Sentient | 193,446 | <p>$\frac{dy}{dx} = x^2e^{-4x} - 4y$ is a first-order ordinary differential equation as it fits the form $\frac{dy}{dx} + p(x)y = q(x)$.</p>
<hr>
<ol>
<li><p>Calculate the integrating factor.</p>
<p>$u(x) = e^{\int p(x) dx} = e^{\int 4 dx} = e^{4x}$</p></li>
<li><p>Evaluate using this <a href="http://mathworld.wolfr... |
2,115,532 | <blockquote>
<p>Let $\mu$ be a $\sigma$-finite measure on $(A,\mathcal{A})$. Then
there are finite measures $(\mu_n)_{n \in \mathbb{N}}$ on
$(X,\mathcal{A})$ such that $$\mu = \sum_{n \in \mathbb{N}}\mu_n$$</p>
</blockquote>
<p>So if $\mu$ is $\sigma$-finite, we have that $$X = \bigcup_{n \in \mathbb{N}}X_n$$ fo... | Jack D'Aurizio | 44,121 | <p>The residue theorem is an overkill here. We have:
$$ e^{e^{i\theta}} = 1+\sum_{n\geq 1}\frac{e^{in\theta}}{n!} $$
hence the integral is clearly $\color{red}{2\pi}$ since
$$\forall n\in\mathbb{N}^*,\qquad \int_{0}^{2\pi}e^{ni\theta}\,d\theta = 0.$$</p>
|
184,564 | <p>If $\frac{a}{c} > \frac{b}{d}$, then the mediant of these two fractions is defined as $\frac{a+b}{c+d}$ and can be shown to lie striclty between the two fractions. </p>
<p>My question is can you prove the following property of mediants: if $|\frac{a}{c} - x| > |x - \frac{b}{d}|$ then $|b/d - mediant| < |me... | nbubis | 28,743 | <p>The theory behind simulating fluids is called CFD - computational fluid dynamics. This is a wide field, with very high demands on computing power, and numerous methods available, depending on the exact nature of the problem at hand.</p>
<p>One book to start with, <a href="http://books.google.co.il/books?id=58zeqKhV... |
49,238 | <p>How to prove that if $f$ is continuous then
$$
F(x) =\int\limits_{-\infty}^\infty f(y)\frac{1}{x\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy
$$
is also a continuous function? I tried to make it through the definition taking $x_n\to x$ but then I can use neither Lebesgue monotone convergence theorem nor dominated converg... | Andrew | 11,265 | <p>The function $F(x)=u(0,x)\ $ is the Poisson potential (and solution) for the Cauchy problem $u_x-u_{ss}/2=0$, $u(s,0)=f(s)$. If $f$ is bounded and continuous, then $F$ is bounded and contuinuous fo $x\ge0$ and is $C^\infty$ for $x>0$. The proof can be found in most books on parabolic equations. For example, in <a... |
49,238 | <p>How to prove that if $f$ is continuous then
$$
F(x) =\int\limits_{-\infty}^\infty f(y)\frac{1}{x\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy
$$
is also a continuous function? I tried to make it through the definition taking $x_n\to x$ but then I can use neither Lebesgue monotone convergence theorem nor dominated converg... | Eric Naslund | 6,075 | <p>First lets do a change of variables to get rid of the x in the exponent. Let $y=xu$. Then we have</p>
<p>$$F(x)=\int\limits _{-\infty}^{\infty}f(y)\frac{1}{x\sqrt{2\pi}}\mathrm{e}^{-y^{2}/2x^{2}}\, dy=\int\limits _{-\infty}^{\infty}f(xu)\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-u^{2}/2} du.$$</p>
<p>Next, what is the defi... |
49,238 | <p>How to prove that if $f$ is continuous then
$$
F(x) =\int\limits_{-\infty}^\infty f(y)\frac{1}{x\sqrt{2\pi}}\mathrm e^{-y^2/2x^2}\,dy
$$
is also a continuous function? I tried to make it through the definition taking $x_n\to x$ but then I can use neither Lebesgue monotone convergence theorem nor dominated converg... | Shai Covo | 2,810 | <p>Suppose that $f$ is a continuous bounded function. For convenience, replace $x^2$ with $t\,( > 0)$.
If $\{X_t: t \geq 0\}$ is a standard Brownian motion, then
$$
{\rm E}f(X_t) = \int_{ - \infty }^\infty {f(y)\frac{1}{{\sqrt {2\pi t} }}e^{ - y^2 /(2t)} dy} .
$$
Now let $(t_n)$ be a sequence of positive numbers... |
2,056,309 | <p>$\textbf{Question}$: Let $f$ be absolutely continuous on the interval $[\epsilon, 1]$ for $0<\epsilon<1$. Does the continuity of $f$ at 0 imply that $f$ is absolutely continuous on $[0,1]$? What if f is also of bounded variation on $[0,1]$?</p>
<p>$\textbf{Attempt}$:</p>
<p>My thoughts are that $f$ is NOT ab... | user293794 | 293,794 | <p>Here's another counter example: </p>
<p>Let $f(x)=x^2\sin(1/x^2)$ on $(0,1]$ and $f(0)=0$. You can check that this function is continuous on $[0,1]$. On every $[\epsilon, 1]$, the derivative of $f$ is bounded so $f$ is Lipschitz on such intervals and therefore is also absolutely continuous on $[\epsilon,1]$. But $... |
939,509 | <p>Is there a proper name for a shape defined by the volume between two concentric spheres? My understanding is that, formally, a "sphere" is strictly a 2D surface and there's a formal term for volume contained by that surface -- which I forget.</p>
<p>Is there a term that describes the volume between two concentric s... | Dan Uznanski | 167,895 | <p>It's called a "<a href="http://en.wikipedia.org/wiki/Spherical_shell" rel="noreferrer">spherical shell</a>".</p>
|
914,936 | <p>Does anyone know where I can find the posthumously published (I think) chapter 8 of Gauss's Disquisitiones Arithmaticae?</p>
| Frunobulax | 93,252 | <p>According to the <a href="http://de.wikipedia.org/wiki/Disquisitiones_Arithmeticae" rel="nofollow">German edition</a> of Wikipedia you can find the eighth chapter in <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN23599524X&IDDOC=137221" rel="nofollow">here</a>.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.