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 |
|---|---|---|---|---|
1,255,629 | <p>Show that </p>
<p>$$\sin\left(\frac\pi3(x-2)\right)$$ </p>
<p>is equal to </p>
<p>$$\cos\left(\frac\pi3(x-7/2)\right)$$</p>
<p>I know that $\cos(x + \frac\pi2) = −\sin(x)$ but i'm not sure how i can apply it to this question.</p>
| 5xum | 112,884 | <p><strong>Hint</strong>:</p>
<p>Try to use addition formulas. Let me get you started:</p>
<p><span class="math-container">$$\sin\left(\frac\pi3(x-2)\right) = \sin\left(\frac{\pi x}{3} - \frac{2\pi}{3}\right) = \sin\frac{\pi x}{3}\cos\frac{2\pi}3 - \cos\frac{\pi x}3\sin\frac{2\pi}{3}=\\=-\frac12 \sin\frac{\pi x}{3} - \... |
2,927,374 | <p>In a right triangle, relative to a <span class="math-container">$45^\circ$</span> angle, if we have
<span class="math-container">$$\text{adjacent} = 1$$</span>
<span class="math-container">$$\text{opposite} = 1$$</span></p>
<p>then
<span class="math-container">$$\text{hypotenuse}=\sqrt{o^2+a^2}=\sqrt{1^2+1^2}=\sqr... | NKRsolutions | 504,365 | <p>You have the square root of 1/2. Separate the entire square root into the square root of the numerator and the denominator and second result is identical to your first result!</p>
|
1,030,327 | <p>Im trying to do an exercise from the book Algebraic Curves of Fulton (Exercise $\:6.26^{*}$).</p>
<p>It says:</p>
<p>Let $f:X\rightarrow Y$ be a morphism of affine varieties. Show that $f(X)$ in dense in $Y$ if and only if the homomorphism $\tilde{f}:\Gamma(Y)\rightarrow\Gamma(X)$ is one-to-one.</p>
| Alex Heinis | 178,368 | <p>a) Suppose that <span class="math-container">$f(X)$</span> is dense in <span class="math-container">$Y$</span> and that <span class="math-container">$\tilde{f}(\phi)=0$</span>.
Then <span class="math-container">$\phi \circ f=0\Rightarrow \phi=0\ {\rm on} f(X) \Rightarrow \phi=0$</span> where the last step follows si... |
1,313,056 | <p>Got this matrix: </p>
<p>\begin{bmatrix} 1 & 2 \\ -2 & 5 \end{bmatrix}</p>
<p>I should determine if the matrix is diagonalizable or not.
I found the eigenvalues ( only one) = 3.
My eigenvector is then \begin{bmatrix} 1 \\ 1 \end{bmatrix}
This matrix is not diagonizable (from my teachers notes) but i do... | Miz | 205,000 | <p>Basically you need to find the set $E$ all the eigenvalues associated with the matrix. (In this case $E =\{3\}$). Next for each eigenvalue in $E$ you find the eigenvectors. Let $W$ be the set of all eigenvectors of the matrix. Then if $W$ spans the column space of the matrix we say that the matrix is diagonalizable... |
446,959 | <p>I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points:</p>
<ul>
<li>in the volume of a n-dimensional hypersphere</li>
<li>on the surface of a n-dimensional hypersphere</li>
</ul>
<p>knowing the dimension $n$, the center of the hypersphere $\vec{x}$ and its radiu... | David K | 139,123 | <p>You have an answer for the surface of the sphere, so I'll just address the question about the interior of the <span class="math-container">$n$</span>-ball.</p>
<p>For sufficiently small <span class="math-container">$n,$</span>
the rejection method is a reasonably efficient way to get a
uniformly distributed random p... |
4,004,157 | <p>A firm wants to know how many of its employees have drug problems. Realizing the sensitivity of this issue, the personnel director decides to use a randomized response survey.</p>
<p>Each employee is asked to flip a fair coin,</p>
<p>If head (H), answer the question “Do you carpool to work?”</p>
<p>If tail (T), answ... | Hagen von Eitzen | 39,174 | <p>As the multiplicative group of a finite field <span class="math-container">$F$</span> is cyclic, it follows that <span class="math-container">$b=c^2a$</span> with come <span class="math-container">$c\in F^\times$</span> whenever <span class="math-container">$a,b$</span> are both non-squares. Then wlog. <span class="... |
2,680,297 | <p>I had been given in my complex analysis examination the following problem. </p>
<p><strong>Evaluate the following integral by using contour integration:</strong></p>
<p>$$
\int_0^{\pi}\dfrac{\sin(2\theta)}{5-3\cos(\theta)}d\theta
$$</p>
<p>The answer to which is:
$$
\dfrac{2}{9}(log_e(1024)-6).
$$ </p>
<p>I trie... | learningmaths | 1,021,431 | <p>I just saw this question and found it very interesting! :)</p>
<p>You can use a contour integral but by using the identity <span class="math-container">$\sin 2t = 2\sin t\cos t$</span> first.</p>
<p>So let's try to calculate
<span class="math-container">$$\int_0^{2\pi}\frac{\sin 2t}{5-3\cos t}dt = \int_0^{2\pi}\frac... |
1,237,618 | <p>I really need Your help.</p>
<p>I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||x||^2-x^2}{||x||^3}$</p>
<p>Thank You!</p>
| Michael Grant | 52,878 | <p>Here's a second-derivative proof for <span class="math-container">$\mathbb{R}^n$</span>. The gradient and Hessian do not exist at the origin, but everywhere else they are given by
<span class="math-container">$$\nabla f(x) = \|x\|^{-1} x, \quad \nabla^2 f(x) = \|x\|^{-1} I - \|x\|^{-3} xx^T$$</span>
Strict convexity... |
267,121 | <h2>UPDATE on FINAL RESULT</h2>
<p>Thanks to @SquareOne effort I generated higher-resolution videos with smoothing transitions that can be seen here:</p>
<ul>
<li><p><a href="https://www.linkedin.com/feed/update/urn:li:activity:6926902980323512320/" rel="noreferrer">https://www.linkedin.com/feed/update/urn:li:activity:... | user5601 | 5,601 | <p>Here's a first pass at naively grabbing and animating the main map frames:</p>
<pre><code>april = Table[
dayImgs = Import["https://www.understandingwar.org/backgrounder/russian-offensive-campaign-assessment-april-"<>ToString[n],"Images"];
img = dayImgs[[5]]; (* brittle *)
ymax=Max@ImageValu... |
2,456,961 | <p>I thought of this myself and could not begin to think of a solution without becoming confused:
There are two people, A and B.
A says, "B is telling the truth."
B says, "A is lying."
Are both lying or telling the truth? Or is one lying and the other telling the truth?</p>
| zwim | 399,263 | <p>If you have $n(x,y)$ continuous and $d(x,y)$ continuous and $d$ does not annulate then the fraction $f(x,y)=\dfrac{n(x,y)}{d(x,y)}$ is continuous.</p>
<p>I have shown it here for a simplified version $\dfrac{f(x)}{g(y)}$ but the principle is exactly the same.</p>
<p><a href="https://math.stackexchange.com/question... |
2,338,508 | <p>Given this matrix that stretches to infinity to the right and up:
$$
\begin{matrix}
...&...&...\\
\frac{1}{4}& \frac{1}{8}& \frac{1}{16}&... \\
\frac{1}{2} & \frac{1}{4}& \frac{1}{8}&... \\
1 & \frac{1}{2}& \frac{1}{4}&... \\
... | Masacroso | 173,262 | <blockquote>
<p>There is a theorem that says that a double series is summable if</p>
<p><span class="math-container">$$\sup_{n\in\Bbb N}\sum_{j,k=0}^n|a_{j,k}|<\infty$$</span></p>
</blockquote>
<p>Your method was wrong because the triangular matrix that you derived is not summable as a double series, that is</p>
<p>... |
3,520,538 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a Hausdorff topological space.
Prove that if <span class="math-container">$\{ C_i | i \in I \}$</span> is an infinite collection of compact subsets of <span class="math-container">$X$</span> such that <span class="math-container">$\cap_{i \in I} C_i = ... | HallaSurvivor | 655,547 | <p>One famous, and perhaps surprising, reply:</p>
<p><span class="math-container">$$\sum_{p \text{ prime}} \frac{1}{p}$$</span></p>
<p>It grows like <span class="math-container">$\log \log n$</span>, and shows that there are "lots of primes" heuristically, since they form enough of the harmonic series to still diverg... |
2,744,981 | <blockquote>
<p>Let $X$ be a Banach space. Let $T\in \mathbb{B}(X)$. If $T$ is an
isometry and not invertible, prove that $\sigma(T) = \overline{\mathbb{D}}$.</p>
</blockquote>
<p>I can show that $\sigma(T) \subset \overline{\mathbb{D}}$. Since $T$ is not invertible, then $0 \in \sigma(T)$. Suppose $\sigma(T) \neq... | bitesizebo | 547,202 | <p>So you've gotten to the point that there must exists $\lambda \in \sigma_{ap}(T)$ with $\lvert \lambda \rvert < 1$. Then by definition of the approximate point spectrum there exists a sequence $(x_i) \subset X$ such that $\lVert x_i \rVert = 1$ for all $i$, and $Tx_i - \lambda x_i \to 0$ as $i \to \infty$. But
$$... |
4,694 | <p>While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we <em>normally</em> call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commu... | Jason DeVito | 331 | <p>This example is certainly not "widely used", but I think it's worth thinking about anyway. This answer comes from an MO post by John Goodrick: <a href="https://mathoverflow.net/questions/9402/pedagogical-question-about-linear-algebra">https://mathoverflow.net/questions/9402/pedagogical-question-about-line... |
4,694 | <p>While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we <em>normally</em> call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commu... | Agustí Roig | 664 | <p>What do you "normally" call addition and multiplication?</p>
<p><em>Just</em> those operations with real numbers, or <em>all</em> kind of addition and multiplication "derived" from the well-known operations with real numbers, or that "look like" these operations?</p>
<p>Because, in the first case, you have plenty ... |
3,112,263 | <p>Given that there are three integers <span class="math-container">$a, b,$</span> and <span class="math-container">$c$</span> such that <span class="math-container">$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{6}{7}$</span>, what is the value of a+b+c?</p>
<hr>
<p>Immediately, I see that I should combine the left hand... | Dan Uznanski | 167,895 | <p>Looks like the greedy algorithm wins here, and quite quickly.</p>
<p>Let's assume wlog that <span class="math-container">$a\le b\le c$</span>.</p>
<p>If <span class="math-container">$a=1$</span>, then we're already higher than <span class="math-container">$6/7$</span>, so let's try <span class="math-container">$a=... |
1,703,169 | <p>At a recent maths competition one of the questions was to find for which $x,y,z$ this equation holds true:</p>
<p>$$\sqrt{x}-\sqrt{z+y}=\sqrt{y}-\sqrt{z+x}=\sqrt{z}-\sqrt{x+y}$$</p>
<p>where $x,y,z \in \mathbb{R} \cup \{0\}$. So how am I supposed to approach this problem?</p>
<p>Also sorry for not explaining my p... | user322840 | 322,840 | <p>Take square on both sides?
An obvious solution would be $x = y = z$</p>
|
1,703,169 | <p>At a recent maths competition one of the questions was to find for which $x,y,z$ this equation holds true:</p>
<p>$$\sqrt{x}-\sqrt{z+y}=\sqrt{y}-\sqrt{z+x}=\sqrt{z}-\sqrt{x+y}$$</p>
<p>where $x,y,z \in \mathbb{R} \cup \{0\}$. So how am I supposed to approach this problem?</p>
<p>Also sorry for not explaining my p... | Roman83 | 309,360 | <p>$$\sqrt{x}-\sqrt{z+y}=\sqrt{y}-\sqrt{z+x}=\sqrt{z}-\sqrt{x+y} \Rightarrow$$
$$\sqrt{x}+\sqrt{x+y}=\sqrt{z}+\sqrt{z+y}$$
Let $f(t)=\sqrt t+ \sqrt{t+y},$ where $y -$ fixed. </p>
<p>$f(t) -$ increasing function $\Rightarrow$ if $f(t_1)=f(t_2)$ then $t_1=t_2$. So $x=z$.</p>
<p>Similarly, $x=y$. So $$x=y=z$$</p>
|
3,758 | <p>When implicitly finding the derivative of: </p>
<blockquote>
<p>$xy^3 - xy^3\sin(x) = 1$</p>
</blockquote>
<p>How do you find the implicit derivative of:</p>
<blockquote>
<p>$xy^3\sin(x)$</p>
</blockquote>
<p>Is it using a <em>triple</em> product rule of sorts?</p>
| WWright | 249 | <p>It is possible to derive a 'triple product rule.' In fact, you can generalize it to an arbitrary number of products. You may ask why these rules aren't presented in these general forms. One reason for this is that writing the general formula out is a more complicated to memorize and more intimidating towards student... |
3,758 | <p>When implicitly finding the derivative of: </p>
<blockquote>
<p>$xy^3 - xy^3\sin(x) = 1$</p>
</blockquote>
<p>How do you find the implicit derivative of:</p>
<blockquote>
<p>$xy^3\sin(x)$</p>
</blockquote>
<p>Is it using a <em>triple</em> product rule of sorts?</p>
| Arturo Magidin | 742 | <p>You get a formula for the derivative of a product of $n$ factors from the formula for the product of $2$ factors by doing induction. Intuitively, you do it the same way as you go from $2$ to $3$: $(fgh)' = ((fg)h)' = (fg)'h + (fg)h' = (f'g + fg')h + (fg)h' = f'gh + fg'h + fgh'$. The pattern should now be clear.</p>
... |
2,849,303 | <p>Given two distinct factorizations of a positive integer with the same number of factors (not necessarily prime or all distinct), must the sums of the respective sets of factors also be distinct? This question arises frequently in puzzles of the KenKen or Killer Sudoku type. I have found no obvious counter examples ... | Xerxes | 93,162 | <p>I thought it might be fun to write a <em>Mathematica</em> code to find instances of same-sum factorizations. Using the following code, you can find integers such that <code>num</code> different sets of <code>len</code> different same-length factorizations of an integer less than <code>max</code> have the same sum. I... |
763,073 | <p>I came across the symbol $|v_1 \wedge \dots \wedge v_m|^{-1}$ in a paper - this is the norm of the wedge product of vectors $v_k \in \mathbb{R}^n$ .
I thought it's meaning was self-evident until I tried to compute it. </p>
<p>First of all $|v_1 \wedge \dots \wedge v_m|$ is a <em>number</em> since we are taking th... | Muphrid | 45,296 | <p>In clifford algebra, this could be easily computed. A wedge product of $m$ linearly independent vectors can be rewritten as a geometric product of $m$ orthogonal vectors $u_1 u_2 \ldots u_m$. The product of this $m$-vector with its reverse $u_m u_{m-1} \ldots u_2 u_1$ is guaranteed to be a scalar; call this $|v_1 ... |
421,198 | <p>An object <span class="math-container">$X$</span> of a <a href="https://ncatlab.org/nlab/show/cartesian+closed+category" rel="nofollow noreferrer">cartesian closed category</a> <span class="math-container">$\mathbf C$</span> is <a href="https://ncatlab.org/nlab/show/amazing+right+adjoint" rel="nofollow noreferrer">a... | Maxime Ramzi | 102,343 | <p>This is a partial answer : if <span class="math-container">$C$</span> has finite coproducts, then <span class="math-container">$C$</span> must be posetal. In fact, I only need biproducts of the form <span class="math-container">$X\coprod X$</span>.</p>
<p>Indeed, because <span class="math-container">$C$</span> is ca... |
421,198 | <p>An object <span class="math-container">$X$</span> of a <a href="https://ncatlab.org/nlab/show/cartesian+closed+category" rel="nofollow noreferrer">cartesian closed category</a> <span class="math-container">$\mathbf C$</span> is <a href="https://ncatlab.org/nlab/show/amazing+right+adjoint" rel="nofollow noreferrer">a... | Tim Campion | 2,362 | <p>Building on Maxime's answer -- if <span class="math-container">$C$</span> is cartesian closed and has an initial object <span class="math-container">$0$</span>, and if <span class="math-container">$0$</span> is atomic (or even just tiny), then <span class="math-container">$C$</span> is the terminal category. For <sp... |
3,557,398 | <p>Here are two second-order differential equations.</p>
<p><span class="math-container">$$ y''+9y=\sin(2t) \tag 1 $$</span></p>
<p><span class="math-container">$$ y'' +4y =\sin(2t) \tag 2 $$</span></p>
<p>I am told to use undetermine coefficients method to solve.</p>
<p>For 1), I use <span class="math-container">... | fleablood | 280,126 | <p>Just do it in cases and note the trivial: <span class="math-container">$\min(a,b) \le a$</span> and <span class="math-container">$\min(a,b) \le b$</span></p>
<p>If both <span class="math-container">$x \le z$</span> and <span class="math-container">$y\le z$</span> than <span class="math-container">$\min(x+y, z) \le ... |
84,897 | <pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
</code></pre>
<p>$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$</p>
<p>gives $0$</p>
<p>This cannot be true. What is going on?</p>
<p>If I insert a number into <code>a</code>, it gives a reasonable result:</p>
<pre><code>NIntegrate[2/(Sin[t]^2 + 4), {t, 0,... | Silvia | 17 | <p>Mathematica 10.1 seems to have fixed this bug:</p>
<p><img src="https://i.stack.imgur.com/1Rnvm.png" alt="answer from 10.1"></p>
|
84,897 | <pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
</code></pre>
<p>$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$</p>
<p>gives $0$</p>
<p>This cannot be true. What is going on?</p>
<p>If I insert a number into <code>a</code>, it gives a reasonable result:</p>
<pre><code>NIntegrate[2/(Sin[t]^2 + 4), {t, 0,... | J. M.'s persistent exhaustion | 50 | <p>A bit late, but consider the following function:</p>
<p>$$f_c(a,x)=\frac1{\sqrt{1+a^2}}\left(x+\arctan\left(\frac{\sin\,x\cos\,x}{a^2+a\sqrt{1+a^2}+\sin^2 x}\right)\right)$$</p>
<p>You can verify the following identity in <em>Mathematica</em>:</p>
<pre><code>fc[a_, x_] := (x + ArcTan[(Sin[x] Cos[x])/(a^2 + a Sqrt... |
3,810,856 | <p>I'm having trouble understanding how this form of the principle (<a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow noreferrer">on wiki</a>) results in the form below.</p>
<p>Wiki form:
<a href="https://i.stack.imgur.com/6IXQZ.png" rel="nofollow noreferrer"><img src="https://... | Michael Hardy | 11,667 | <p><span class="math-container">$$
\frac \pi {\sin^2 \frac\pi x} > \frac x {\tan \frac{\pi}{x}}
$$</span>
Let <span class="math-container">$u = \pi/x.$</span> then <span class="math-container">$x>2$</span> means <span class="math-container">$0<u<\pi/2.$</span> Then the inequality becomes
<span class="math-c... |
4,490,117 | <blockquote>
<p>(a) Let <span class="math-container">$f : [a, b] \to \mathbb{R}$</span> be continuous and suppose that <span class="math-container">$f(x) \gt 0$</span> for all <span class="math-container">$x$</span>. Show that there is some <span class="math-container">$L\gt 0$</span> such that <span class="math-contai... | blargoner | 129,912 | <p>I'd recommend <em>Understanding Numbers in Elementary School Mathematics</em> by <a href="https://math.berkeley.edu/%7Ewu/" rel="nofollow noreferrer">Hung-Hsi Wu</a>. Be sure to reference the errata listing on Wu's website.</p>
|
305,373 | <p>I'm doing some self-studying on discrete mathematics and so far its going well however I came upon a question that whilst I can make sense of and I know that set $A$ is a subset of set $B$, I cannot think of how to express it and the book does not feature an answer for this question.</p>
<p>The question is, prove t... | Dominic Michaelis | 62,278 | <p>Ok if $A\subset B$ you have to show that $x\in A \implies x \in B$.
$$x\in A \implies \exists n \in \mathbb{Z}: x=2 n$$
Because of $2 \in \mathbb{Z}^+$ and $n \in \mathbb{Z}^+$ and $(\mathbb{Z}^+, \cdot)$ is a semi-group (you just need to say that when $a\in \mathbb{Z}^+$ and $b\in \mathbb{Z}^+$ than
$$a\cdot b\in... |
2,653,713 | <p>Let $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ be a convex function. If $y<\varphi(x)$ why does there exist a line through $(x,y)$ which lies strictly below the graph of $\varphi$? I ask because this is a step in the proof of Jensen's inequality for conditional expectation.</p>
| g.kov | 122,782 | <p>Consider the point $(x_1,g(x_1))$.
At this point
the distance to the line $y=x$
is $d(x_1)=\exp(x_1)$. </p>
<p><a href="https://i.stack.imgur.com/4rFBp.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/4rFBp.png" alt="enter image description here"></a></p>
<p>As we can see, for every $x$
there ar... |
1,255,215 | <p>How to prove that:</p>
<p>$$\cos{n\theta}=\cos^n{\theta}-
\binom {n} {2}\cos^{n-2} \theta \cdot \sin^2 \theta+
\binom {n} {4}\cos^{n-4} \theta \cdot \sin^{4} \theta -\cdots$$</p>
<p>$$\sin n\theta =
\binom {n} {1}\cos^{n-1} \theta \cdot \sin \theta -
\binom {n} {3}\cos^{n-3} \theta \cdot \sin^3\theta +\cdots$$</p>... | L F | 221,357 | <p>$\textbf{Hint}$: $cos(t)^2+\sin(t)^2=1$, do $x^2+y^2$ and you will get a function $y$ who depends of $x$, then apply chain rule and dont forget the term $dx/dt$</p>
|
1,255,215 | <p>How to prove that:</p>
<p>$$\cos{n\theta}=\cos^n{\theta}-
\binom {n} {2}\cos^{n-2} \theta \cdot \sin^2 \theta+
\binom {n} {4}\cos^{n-4} \theta \cdot \sin^{4} \theta -\cdots$$</p>
<p>$$\sin n\theta =
\binom {n} {1}\cos^{n-1} \theta \cdot \sin \theta -
\binom {n} {3}\cos^{n-3} \theta \cdot \sin^3\theta +\cdots$$</p>... | Community | -1 | <p>Work backwards from the desired endpoint. There's really only one thing to be done with $\tan(t-\frac\pi4)$, and that's applying the addition formula:
$$ \tan(t-\tfrac\pi4) = \frac{\tan(t)-\tan(\tfrac\pi4)}{1+\tan(t)\tan(\tfrac\pi4)} = \frac{\tan(t)-1}{1+\tan(t)} $$
What could you do next with this, to make it look... |
4,632,869 | <p>Suppose <span class="math-container">$\Omega \subset \mathbb{R}$</span> is a bounded domain. Then using
Poincare-Wirtinger, one can prove that for functions <span class="math-container">$f \in W^{1,2}(\Omega)$</span> there exists <span class="math-container">$C>0$</span> such that</p>
<p><span class="math-contain... | notpron | 887,931 | <p>If <span class="math-container">$u$</span> satisfies this second inequality, we should already have <span class="math-container">$u \in W^{1,2}(\Omega)$</span> or <span class="math-container">$u \in C^1(\Omega) \subseteq W^{1,2}(\Omega) $</span>, or else how do you define <span class="math-container">$u'$</span> ins... |
2,862,166 | <p>In a paper I've been studying it says:</p>
<blockquote>
<p>Let $x$ in the cone $\mathbb R_+^n$ of all vectors in $\mathbb R^n$ with nonnnegative components ($n\in\mathbb N$)</p>
</blockquote>
<p>Somebody tell me what does it means, please? $\mathbb R_+^n$ should be $[0,\infty)\times\dots\times [0,\infty)$ ($n$ t... | nicomezi | 316,579 | <p>A subset $A$ of a vector space $X$ on a field $K$ with a notion of positivity is said to be a cone if, for all $x \in A$, $\lambda>0$, we have $\lambda x \in A$.</p>
<p>The infite version of the cone you probably have in mind satisfies this property if the vertex, or apex, is at the origin.</p>
|
2,862,166 | <p>In a paper I've been studying it says:</p>
<blockquote>
<p>Let $x$ in the cone $\mathbb R_+^n$ of all vectors in $\mathbb R^n$ with nonnnegative components ($n\in\mathbb N$)</p>
</blockquote>
<p>Somebody tell me what does it means, please? $\mathbb R_+^n$ should be $[0,\infty)\times\dots\times [0,\infty)$ ($n$ t... | M. Winter | 415,941 | <p>A <em>cone</em> is a set $C\subseteq\Bbb R^n$ with $x\in C\implies\alpha x\in C$ for all $\alpha\ge0$. The name is motivated by the usualy geometric cones, but $\Bbb R^n_+ := [0,\infty)^n$ satisfies this too.</p>
<p>See this picture of different cones in $\Bbb R^3$, two are familiar, and the right most one is $\Bbb... |
166,202 | <p>If we have $k(x,t)= \frac {1}{(4t)^{\frac{n}{2}}} \exp\left(\frac{-|x|^2}{4t}\right)$ is the fundamental solution of heat equation. If we consider $n \ge 3 $, I would like to show that $\int_0^\infty k(x,t) dt$ is the fundamental solution of lLaplace equation.
I would like some hints.
I thought of integrating but ... | Davide Giraudo | 9,849 | <p>We use the substitution $s=\frac t{|x|^2}$ (then $dt=|x|^2ds$) to get
\begin{align}
\int_0^{+\infty}k(x,t)dt&=\int_0^{+\infty}\frac 1{(4t)^{n/2}}\exp\left(-\frac{|x|^2}{4t}\right)dt\\
&=\frac 1{4^{n/2}}\int_0^{+\infty}\frac 1{(s|x|^2)^{n/2}}\exp\left(-\frac 1{4s}\right)|x|^2ds\\
&=\frac 1{4^{n/2}}|x|^{2... |
101,256 | <p>Can you please help me finding an exact description of the set:</p>
<p>$$ E_{R}=\{\cos{z} | z \in \mathbb{C}, |z|>R\} $$</p>
<p>For any $0<R \in \mathbb{R}$.</p>
<p>My feeling is the $E_R = \mathbb{C}$, for any $R$, but I don't know how to show it, if it's true.</p>
| Julián Aguirre | 4,791 | <p>A different approach. Let $w\in\mathbb{C}$. We must find $z\in\mathbb{C}$ such that $|z|>R$ and
$$\cos z=\frac{e^{iz}+e^{-iz}}{2}=w\implies e^{2iz}-2\,w\,e^{iz}+1=0\ .
$$
Solving for $e^{iz}$ we obtain
$$e^{iz}=w\pm\sqrt{w^2-1}\ .$$
One at least of $w+\sqrt{w^2-1}$ or $w-\sqrt{w^2-1}$ is non-zero. Assume $w+\sqr... |
4,023,342 | <p>Blue: <span class="math-container">$x \ln x-x$</span></p>
<p>Brown: <span class="math-container">$\ln \frac{\Gamma(x+1/2)}{\sqrt{\pi}}$</span></p>
<p><a href="https://i.stack.imgur.com/CbVYi.png" rel="noreferrer"><img src="https://i.stack.imgur.com/CbVYi.png" alt="enter image description here" /></a></p>
| 5201314 | 869,482 | <p>Using Sterling’s approximation for the Gamma function, <span class="math-container">$$\Gamma(z)\sim \sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z$$</span>
In this case <span class="math-container">\begin{align*}
\Gamma\left(x+\frac{1}{2}\right)&\sim \sqrt{\frac{2\pi}{x+\frac{1}{2}}}\left(\frac{x+\frac{1}{2}}{e... |
3,239,689 | <p><span class="math-container">$$n\in\mathbb{N}^{*}; S_{n}=\sum_{k=1}^{n}k!(k^2+1)$$</span></p>
<p>I need to find <span class="math-container">$S_n$</span></p>
<p>I started like this: <span class="math-container">$S_{n}=\sum_{k=1}^{n}(k+2)!-3(k+1)!+2k!$</span></p>
<p>How to continue?I tried to give the k values but... | J.G. | 56,861 | <p>Next we use <span class="math-container">$$S_n=\sum_{k=1}^n\left\{[(k+2)!-(k+1)!]-2[(k+1)!-k!]\right\},$$</span>which telescopes to <span class="math-container">$$(n+2)!-2-2((n+1)!-1)=(n+2)!-2\cdot(n+1)!=n\cdot (n+1)!$$</span></p>
|
3,239,689 | <p><span class="math-container">$$n\in\mathbb{N}^{*}; S_{n}=\sum_{k=1}^{n}k!(k^2+1)$$</span></p>
<p>I need to find <span class="math-container">$S_n$</span></p>
<p>I started like this: <span class="math-container">$S_{n}=\sum_{k=1}^{n}(k+2)!-3(k+1)!+2k!$</span></p>
<p>How to continue?I tried to give the k values but... | Hagen von Eitzen | 39,174 | <p>So far you arrived at
<span class="math-container">$$\begin{array}c S_n=&&&&\hphantom{+}3!&+4!+\ldots+n!&+(n+1)!&+(n+2)!\\
&-3(&&\hphantom+2!&+3!&+4!+\ldots +n!&+(n+1)!&&)\\
&+2(&1!&+2!&+3!&+4!+\ldots +n!&&&)\end{array}$$</sp... |
2,996,640 | <p>Maybe trivial for number theorists, but not for me: is the title meaningful to ask for (<span class="math-container">$p$</span> is a prime)? If so, what's the answer? Thanks</p>
| Keen-ameteur | 421,273 | <p>You have to see how the set <span class="math-container">$\{ (x,y): x>y \}$</span> intersects with <span class="math-container">$\{(x,y): x>0, y<1 \}$</span>. The intersection can be written as:</p>
<p><span class="math-container">$D_1\sqcup D_2$</span>, where <span class="math-container">$D_1:= \{ (x,y)... |
1,483,343 | <p>For equation
$$
x_1+x_2+x_3 = 15
$$
Find number of positive integer solutions on conditions:
$$
x_1<6, x_2 > 6
$$
Let: $y_1 = x_1, y_2 = x_2 - 6, y_3 = x_3$
than, to solve the problem, equation $y_1+y_2 +y_3 = 9$ where $y_1 < 6,0<y_2, 0<y_3 $ has to be solved. <strong>Is this correct?</strong></p>
<p... | Community | -1 | <p>There is no infinite descending sequence of natural numbers. That is, there is no sequence of natural numbers
$$ n_0 > n_1 > n_2 > n_3 \ldots .$$
Intuitively, that fact is obvious: $n_0$ is some finite number, and you can't start at some finite number and descend infinitely. More formally, you can show the ... |
150,518 | <blockquote>
<p>Analyze the convergence behavior of the following series: $$\sum_{k=0}^{\infty}\frac{x^{2k}}{2^{2k}}-\frac{x^{2k+1}}{3^{2k+1}}.$$</p>
</blockquote>
<p>I came across this problem as I was preparing for an exam. It is supposed to be an easy one but I am not sure which test to apply.</p>
| ncmathsadist | 4,154 | <p>Break it in two using the - sign. Then examine the pieces using the root or ratio test. More simply, realize that both series are geometric, which converges when the common ration is less than 1 in absolute value.</p>
|
2,309,986 | <p>This question is from the text <em>Geometry and Complexity Theory</em>, from J.M. Landsberg. Before start talking about the question, I think it is good to show the definition of border rank. Consider a tensor $T \in \mathbb{C}^r \otimes \mathbb{C}^r \otimes \mathbb{C}^r$. Then the <em>border rank</em> of $T$ is the... | Zach Teitler | 343,280 | <p>Let $V = \mathbb{C}^r$ have basis $\{x_1,\dotsc,x_r\}$. Consider the tensor
$$
\begin{split}
T &= x_1^2 x_2 + x_3^3 + x_4^3 + \dotsb + x_r^3 \\
&= x_1 \otimes x_1 \otimes x_2 + x_1 \otimes x_2 \otimes x_1 + x_2 \otimes x_1 \otimes x_1 + x_3 \otimes x_3 \otimes x_3 + \dotsb + x_r \otimes x_r \otimes x_r... |
240,620 | <p>I would like to train some recreational probability (Puzzles). </p>
<p>Does any of you know a good collection? Preferably with hints or answers.</p>
<p>I've been studying quite a bit of probability theory, but I don't suspect that will do me any real good.</p>
<p>Thanks in advance,</p>
| Mehness | 226,027 | <p>definitely agree that Mosteller's book is excellent. In case you exhaust that, there's also quite a few problems in the recreational vein in: Challenging Mathematical Problems with Elementary Solutions, Volume I: Combinatorial Analysis and Probability Theory.</p>
<p><a href="http://www.amazon.co.uk/Challenging-Math... |
139,818 | <p>I'm currently working with large datasets of "discrete data". Each of these is a <em>multiset</em> of real numbers of the form $kS + B$, where $k$ is an integer, and $S$ (the <em>"scale</em>") and $B$ (the <em>"bias"</em>) are real values.</p>
<p>These datasets are such that it is often economical to represent the... | Leonid Shifrin | 81 | <h2>General</h2>
<p>A late-to-the party post. To complement other answers, which already show most important ingredients for constructing data types with overloaded system functions, I'd like to show an approach that follows the same ideas a couple steps further. </p>
<p>While <code>UpValues</code> are the tool to us... |
12,091 | <p>It looks like the traffic is really high today, probably because of the introduction of hats. </p>
<p>I was wondering, is there a way to check statistics about the traffic on math.SE, and on the other SE websites?</p>
| Community | -1 | <p>There are publicly visible statistics available on <a href="https://www.quantcast.com/math.stackexchange.com" rel="nofollow">Quantcast</a>. As SE uses the Quantcast tracker, they don't have to guess and this is actual traffic data.</p>
<p>You'll have to wait a bit though, the graphs are only updated later the next ... |
71,670 | <pre><code>TableForm[Table[i/j + 4*Boole[j > i] // N, {i, 3}, {j, 4}], TableHeadings -> {{"Row1", "Row2", "Row3"}, {"Col1", "Col2", "Col3", "Col4"}}]
</code></pre>
<p>Produces the following table:</p>
<p><img src="https://i.stack.imgur.com/Dqnmr.png" alt="enter image description here"></p>
<p>How select the ma... | george2079 | 2,079 | <pre><code> TableForm[ (m = Max[#]; (# /. m :> Style[m, Red] ) ) & /@
Table[i/j + 4*Boole[j > i] // N, {i, 3}, {j, 4}] ,
TableHeadings -> {{"Row1", "Row2", "Row3"}, {"Col1", "Col2", "Col3", "Col4"}}]
</code></pre>
<p><img src="https://i.stack.imgur.com/HwgOl.png" alt="enter image des... |
4,132,907 | <p>I'm trying to figure out the identity above though I'm having difficulties towards figuring it out and would kindly appreciate your support!</p>
<p><span class="math-container">$n\binom{n-1}{r-1} = r\binom{n}{r}$</span></p>
<p>What I have tried:
Given that</p>
<p><span class="math-container">$$\binom{n}{r}=\binom{n-... | David C. Ullrich | 248,223 | <p>Of course this just falls out from the identity <span class="math-container">$\begin{pmatrix}p\\q\end{pmatrix}=\frac{p!}{(p-q)!q!}$</span>. That's no fun (also unenlightening). <strong>Outline</strong> of a "combinatorial" proof:</p>
<p>Say <span class="math-container">$S=\{1,2,\dots,n\}$</span>. Let <span... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | E.Sokol | 921,142 | <p>A more general formula exists, for star polygon too and its smoother variations:</p>
<p><span class="math-container">$$1=\frac{\sqrt{x^2+y^2} \cos \left(\frac{2 \sin ^{-1}\left(k \cos \left(n \tan ^{-1}(x,y)\right)\right)+\pi m}{2 n}\right)}{\cos \left(\frac{2 \sin ^{-1}(k)+\pi m}{2 n}\right)}$$</span></p>
<p>or, ... |
4,237,056 | <p>Can any subgroup of a cyclic group <span class="math-container">$\left<a\right>$</span> be presented as <span class="math-container">$\{x\in \left<a\right>|x^m=e\}$</span>?<br />
Note that m is a factor of n, where n is the order of <span class="math-container">$\left<a\right>$</span>.<br />
I foun... | subrosar | 602,170 | <p>The first partial of the Lagrangian simplifies to <span class="math-container">$2(x_1-1)(1+\lambda),$</span> so either <span class="math-container">$x_1=1$</span> or <span class="math-container">$\lambda=-1.$</span> If <span class="math-container">$x_1=1$</span> then the constraint tells us <span class="math-contain... |
3,910,015 | <p>My impression has been that <span class="math-container">$\mathbb{Z_n}$</span> is the set <span class="math-container">$\{0,1,...,n-1\}$</span> under binary operation addition modulo <span class="math-container">$n$</span>. However I'm also coming across this notion that <span class="math-container">$\mathbb{Z_n}$</... | Duncan Ramage | 405,912 | <p>To put it shortly: yes.</p>
<p>What isn't important is what <span class="math-container">$\mathbb{Z}_n$</span> "is" as a set. Rather, what's important is that all these different formulations of <span class="math-container">$\mathbb{Z}_n$</span> (equivalence classes of integers, select integers with a cert... |
3,910,015 | <p>My impression has been that <span class="math-container">$\mathbb{Z_n}$</span> is the set <span class="math-container">$\{0,1,...,n-1\}$</span> under binary operation addition modulo <span class="math-container">$n$</span>. However I'm also coming across this notion that <span class="math-container">$\mathbb{Z_n}$</... | Community | -1 | <p>To my knowledge, possibly the main plus of the definition of <span class="math-container">$\Bbb Z_n$</span> as the quotient set (group under addition modulo <span class="math-container">$n$</span>, actually) <span class="math-container">$\Bbb Z/\sim$</span>, where <span class="math-container">$x\sim y \stackrel{(def... |
2,870,039 | <p>Just for the sake of simplicity, assume that the centre of rotation is $(0, 0)$. I know that I can find the destination point $(x', y')$ given angle of rotation $a$ and source point $(x, y)$ using
$$\begin{cases}
x' = x \cos a - y \sin a \\
y' = x \sin a + y \cos a \\
\end{cases}$$</p>
<p>However, I want to ask ... | amd | 265,466 | <p>Changing notation a bit, all of the possible rotations of $\mathbf p_0=(x_0,y_0)$ lie on a circle with radius $r = \|\mathbf p_0\| = \sqrt{x_0^2+y_0^2}$. An equation of this circle is $x^2+y^2=x_0^2+y_0^2$. Your problem comes down to finding the points on this circle with $x$-coordinate equal to $x_1$. Plugging $x_1... |
671,437 | <p>A random variable $X \sim N(0,1)$, compute $\Bbb E(X^n)$ .</p>
<p>I manage to do this by characteristic function. Now I try to compute this by moment generating function or do it directly. So I have 2 questions for 2 ways to compute that expectation.</p>
<ol>
<li>Moment generating function, I have $$M_X(t)= e^{t^2... | Gurvan | 127,051 | <p>If you name $u_n$ a term of the sequence, you can check the sign of $u_{n+1}-u_n$, if the sign is always negative (resp. positive), the sequence is decreasing (resp. increasing).</p>
|
671,437 | <p>A random variable $X \sim N(0,1)$, compute $\Bbb E(X^n)$ .</p>
<p>I manage to do this by characteristic function. Now I try to compute this by moment generating function or do it directly. So I have 2 questions for 2 ways to compute that expectation.</p>
<ol>
<li>Moment generating function, I have $$M_X(t)= e^{t^2... | imranfat | 64,546 | <p>The numerator can be written as $\frac{n^2+n}{2}$ (that's the closed formula Daniel is hinting at) and so if you combine the fractions into one, you get a numerator of $-n$ and a denominator of $2(n+2)$ Taking the limit for $n$ to inf gives -0.5</p>
|
359,528 | <p>Let $P$ be the set of permutations all of whose cycles are of even length. Prove that the exponential generating function for $P$ is $\dfrac{1}{\sqrt{1-x^2}}$.</p>
| Community | -1 | <p>A bit of notation to make the answer clearer. I shall write $P_n$ to be the set of all partitions of the set $\{1,\ldots,n\}$ and $\sigma=\{S_1,\ldots,S_k\} \in P_n$ to mean that $\sigma$ is a partition of the set $\{1,\ldots,n\}$ into "parts" $S_1,\ldots,S_k$</p>
<p>Define
$$
a_n= \begin{cases}
(n-1)! &... |
391,572 | <p>I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?</p>
| JohnWO | 73,241 | <p>$e$, the constant, is the limit of $(1 + {1\over n})^n$ as n approaches infinity; As expressed by the series: $$e = 1 + {1\over 1} + {1\over 1\cdot2 } + {1\over 1\cdot2\cdot3} + {1\over 1\cdot2\cdot3\cdot4} + ... + {1\over 1\cdot 2\cdot...\cdot\infty}$$</p>
<p>It is the base of the <a href="http://en.wikipedia.org/... |
391,572 | <p>I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?</p>
| Ron Gordon | 53,268 | <p>The idea of $e$ may be explained through the concept of compound interest. Let's say that you earn an annual interest rate $r$ on a principal $P$ dollars. Let's say that the money is compounded annually. Then after $n$ compounding periods, you have $P (1+r)^n$ dollars.</p>
<p>Now lets say we compound semiannual... |
4,547,151 | <p>Say I have a set of ordered points, like in a polygon, I can name a point in this set <span class="math-container">$P_i$</span>.</p>
<p>Say I have a high dimensional point and I want to denote its x coordinate, I can do <span class="math-container">$P_x$</span> and if I want it's ith coorddinate I can do <span class... | Doug M | 317,176 | <p>I would do this with cylindrical shells.</p>
<p><span class="math-container">$\int_0^1 2\pi x(1-x^4)\ dx$</span></p>
|
679,341 | <p>The 4 bytes used in CDROM sectors is just a usual 32 bit CRC. It uses the polynomial </p>
<pre><code>P(x) = (x^16 + x^15 + x^2 + 1).(x^16 + x^2 + x + 1)
</code></pre>
<p>which expands to </p>
<pre><code>x^32 + x^31 + 2x^18 + 2x^17 + 3x^16 + x^15 + x^4 + x^3 + 2x^2 + x + 1
</code></pre>
<p>The CRC process rever... | gammatester | 61,216 | <p>For a CRC32 algorithm with the polynomial
$$x^{32} + x^{31} + 2x^{18} + 2x^{17} + 3x^{16} + x^{15} + x^4 + x^3 + 2x^2 + x + 1$$
you first omit the $x^{32}$ term and reduce the remaining coefficients mod 2. This gives
$$x^{31} + x^{16} + x^{15} + x^4 + x^3 + x + 1$$
Now simply substitute $x=2\;$ and evaluate, this gi... |
262,601 | <p>What does the following statement mean, that Nature likes to minimize things (like energy) and this equation describes one particular minimization problem? </p>
<p>What is the Variational Principle?</p>
| Siminore | 29,672 | <p>It is, to some extent, just folklore. In particular, Nature likes to find <em>stationary</em> points, which can be maxima or saddle points, not only minima.
Usually we first meet variational principles when studying mechanics (maxima/minima of the action functional), and it is proved that most laws of general physic... |
2,821,514 | <p>so I'm trying to work through this problem: </p>
<p>How many elements does a 2-Sylolw subgroup of $\left(\mathbb Z^\times_{11} \times \mathbb Z^\times_{13} , \cdot \right)$ have (i.e. what is the order) and how many 2-Sylow subgroups there are in $\left(\mathbb Z^\times_{11} \times \mathbb Z^\times_{13} , \cdot \ri... | lhf | 589 | <p>We have $\mathbb{Z}_{11}^\times \times \mathbb{Z}_{13}^\times \cong C_{10} \times C_{12}$, whose $2$-Sylow subgroup is $C_{2} \times C_{4}$.</p>
<p>Therefore, the $2$-Sylow subgroup of $\mathbb{Z}_{11}^\times \times \mathbb{Z}_{13}^\times$ is $\langle g_{11}^5 \rangle \times \langle g_{13}^3 \rangle$, where $g_{p}$... |
1,713,841 | <p>I think $\pi$ is algebraic of degree 3 over $\mathbb{Q}(\pi^3)$. To prove it, I need to show that $\pi \notin \mathbb{Q}(\pi^3)$ (which implies that $x^3-\pi^3$ is the minimum polynomial of $\pi$ over $\mathbb{Q}(\pi^3)$). I have no experience solving this kind of problem for transcendental elements like $\pi$.</p>
| Eric Wofsey | 86,856 | <p>Suppose $\pi\in\mathbb{Q}(\pi^3)$. That means that there are some polynomials $f(x)$ and $g(x)$ with rational coefficients (with $g(x)\neq 0$) such that $\pi=f(\pi^3)/g(\pi^3)$, or $\pi g(\pi^3)-f(\pi^3)=0$. But this is then a polynomial in $\pi$ which vanishes, so since $\pi$ is transcendental, it must be identic... |
4,013,630 | <blockquote>
<p>The population of bacteria triples each day on a petri dish. If it takes 20 days for the population of bacteria to fill the entire dish, how many days will it take bacteria to fill half of the petri dish?</p>
</blockquote>
<ul>
<li><p>My doubt 1: if we don't have a initial population can be solve this p... | Neat Math | 843,178 | <p><span class="math-container">$$a_n-2=4-a_{n-1}-2=(-1)(a_{n-1}-2)$$</span>
<span class="math-container">$\{a_n-2\}$</span> is a geometric sequence so
<span class="math-container">$$a_n-2=(-1)^{n-2}(a_2-2)=5(-1)^{n-1} \implies a_n = 2+5(-1)^{n-1}.$$</span></p>
|
4,623,427 | <p><span class="math-container">$f$</span> is a twice differentiable function from <span class="math-container">$R^n $</span> to <span class="math-container">$R$</span>. I want to show that
<span class="math-container">$$ \| \nabla^2 f(x) \| \le L \implies \| \nabla f(x) - \nabla f(y) \| \le L \| x-y \| $$</span> f... | copper.hat | 27,978 | <p>Here is an outline:</p>
<p>Pick some <span class="math-container">$w$</span> and let <span class="math-container">$\phi(x) = w^T \nabla f(x)$</span>.</p>
<p>Compute <span class="math-container">$\nabla \phi(x)$</span>.</p>
<p>Apply the mean value theorem to <span class="math-container">$\phi$</span> to get
<span cla... |
262,985 | <p>$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.</p>
<p>What should $A$ satisfy to guarantee the equation set have only zero solution?</p>
| Max Alekseyev | 7,076 | <p>The paper "<a href="https://people.ece.cornell.edu/atang/pub/09/allerton09_uniqueness.pdf" rel="nofollow noreferrer">Conditions for a Unique Non-negative Solution to
an Underdetermined System</a>" seems to answer your question.</p>
|
4,174,481 | <p>I'm going through a proof in which at one point, the authors make the following statement :</p>
<blockquote>
<p>Let <span class="math-container">$(X,d)$</span> be a complete metric space and <span class="math-container">$f:X\to X$</span> and <span class="math-container">$f_m:X\to X$</span> Lipschitz with <span class... | Danny Pak-Keung Chan | 374,270 | <p>If there exists <span class="math-container">$c\in(0,1)$</span> such that <span class="math-container">$lip(f_{n})\leq c$</span> for all <span class="math-container">$n$</span>, then it is easy. For, observe that
<span class="math-container">\begin{eqnarray*}
d(x_{n}^{\ast},x^{\ast}) & = & d(f_{n}(x_{n}^{\as... |
2,071,054 | <p>Let $f(x)$ be continuous in $[a,b]$</p>
<p>Let $A$ be the set defined as:
$$A = \{ x \in [a,b] \mid f(x) = f(a) \}$$</p>
<ol>
<li><p>Does $A$ have a maximum? I guess it it is the max value in $[a,b]$ which $f$ sends to $f(a)$ but I don't know how to prove it.</p></li>
<li><p>Would it still have a maximum if $[a,b]... | Teoc | 190,244 | <p>Since you want all the variables to be distinct, we can sum it over all values of i,j,k then subtract the sum that occurs when they are not distinct, and apply PIE(Principle of Inclusion/Exclusion). </p>
<p>Can you take it from there?</p>
|
1,379,878 | <blockquote>
<p>Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.</p>
</blockquote>
<p>My Try:</p>
<p>I proved that $C(M)$ is ... | GAVD | 255,061 | <p>HINT: A square matrix $A$ over a field $F$ commutes with every $F$-linear combination of non-negative powers of $A$.</p>
<p>That is, for every $a_0$, $\dots$ ,$a_n \in F$,</p>
<p>$$A(\sum_{k=0}^n a_kA^k) = \sum_{k=0}^n a_k A^{k+1} = (\sum_{k=0}^n a_k A^k) A.$$ </p>
|
1,685,621 | <p>I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. </p>
<p>So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real vector space), we can say that they are orthogonal iff $\langle \vec x,\vec y\rangle=0$, where $\langle \vec x,\ve... | Balloon | 280,308 | <p>Your reasoning is good, and, as Daniel Fisher said, the assertion "be orthogonal to" depends only on your inner product. </p>
<p>What makes the standard basis
$$\mathcal{C}=(e_1,\dots,e_n) \hspace{0.5cm}\text{ with }\hspace{0.5cm} e_i=(0,\dots,0,\underset{i \text{ rank}}{\underbrace{1}},0,\dots,0)$$
special when ... |
1,685,621 | <p>I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. </p>
<p>So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real vector space), we can say that they are orthogonal iff $\langle \vec x,\vec y\rangle=0$, where $\langle \vec x,\ve... | amd | 265,466 | <p>To expand a bit on Daniel Fischer’s comment, coming at this from a different direction might be fruitful. There are, as you’ve seen, many possible inner products. Each one determines a different notion of length and angle—and so orthogonality—via the formulas with which you’re familiar. There’s nothing inherently co... |
1,183,643 | <p>Given a integer $h$</p>
<blockquote>
<p>What is $N(h)$ the number of full binary trees of height less than $h$?</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/XcNVi.jpg" alt="enter image description here"></p>
<p>For example $N(0)=1,N(1)=2,N(2)=5, N(3)=21$(As pointed by <a href="https://math.stackexc... | Qudit | 210,368 | <p>Let's consider all possible ways of constructing a full binary tree of height at most $h \geq 1$. The root has either $0$ or $2$ children, so one possibility is that the tree is just a single node. On the other hand, the number of full binary trees such that the root has two children, is equal to the square of the... |
5,998 | <p>I have restricted the context of my notebooks to each individual notebook. So variables in each notebook are local and are not seen in another notebook.</p>
<p>But in two of my notebooks I have two functions that I want to plot simultaneously with the command <code>Show</code>. Please note that I want variables of... | user22889 | 22,889 | <p>Globally define as follows
$Assumptions = b >= 0 && c >= 0 && {u11, u13, u14} [Element] Reals
then use globally defined variables as follows
Simplify[expression with global variables]</p>
|
398,535 | <p>I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am curious why the word "recursive" appears in their name. What does the concept of decidability/recognizability have to do... | William | 13,579 | <p>Today in the context of the branch of mathematics called recursion theory or computability theory, the term recursive and computable mean the same thing. The term decidable also mean the same thing but more often used in the context of logical theories or in textbook more focused on computer science. All these terms... |
398,535 | <p>I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am curious why the word "recursive" appears in their name. What does the concept of decidability/recognizability have to do... | Peter Smith | 35,151 | <p>There's a history here. In thumbnail version, in the 1930s various attempts where made to formally characterize the intuitively <strong>computable</strong> numerical functions, and relatedly the <strong>effectively decidable</strong> sets of numbers (i.e. those whose membership can be decided by a computable functio... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Martin Argerami | 475 | <p>I would do <span class="math-container">$$3.9\times7.5=\frac{3.9\times30}4=\frac{39\times3}4=\frac{(40-1)\times3}4=\frac{40\times 3}4-\frac34=30-\frac34.$$</span></p>
|
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Paul_Pedant | 13,841 | <p>Trachtenberg methods: this depends on calculating the answer one digit at a time, starting at the units. Maybe this is what you call "lattice". This was in vogue around 1960 -- maybe it gets rediscovered every generation.</p>
<p>Units can only come from 9 x 5. Write down 5, carry 4.</p>
<p>Tens can come from 3 x 5... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Mark | 346 | <p>Pull out your slide rule. Align the second "1" on the C scale with the "3.9" on the D scale. Move the slide to the "7.5" on the C scale and read "2.92" off the D scale. Observe that there are two decimal places in the problem, and write down the answer as approximately 29.2.</p>
<p>(If you want more precision, g... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Toby Mak | 13,864 | <p>This is similar to Xander Henderson's approach, but the calculations have been made slightly easier by using two FOIL multiplications instead of just one:</p>
<p><span class="math-container">$$3.9 \times (3.8 + 3.7)$$</span>
<span class="math-container">$$=3.9 \times 3.8 + 3.9 \cdot 3.7$$</span>
<span class="math-c... |
946,738 | <p>I ran into a nice question from one book in Discrete Mathematics. I want to someone lean me how solve such a problem, because I prepare for entrance exam. </p>
<blockquote>
<blockquote>
<p>if the time is "Wednesday 4 afternoon", after $47^{74}$ hours, we are in what hours? and what day? </p>
</blockquote>
<... | egreg | 62,967 | <p>If now it's “Wednesday, 4pm”, then $16$ hours ago it was “Wednesday, 0:00”.</p>
<p>Thus the problem is to know what hour it is $47^{74}+16$ hours after “Wed, 0:00”.
This is obviously solved by computing the remainder of $47^{74}+16$ divided by $24$; since $47$ is coprime with $24$ and $\varphi(24)=8$, from Fermat-E... |
2,879,915 | <p>I am attempting to prove $\lim_{k\to\infty}\dfrac{k^2}{k^2+2k+2}=1$. However, I am getting tripped up on the algebra. I believe I want to show that there exists some $N\in\mathbb{N}$ such that when $k\geq N$, $\left|\dfrac{k^2}{k^2+2k+2}-1\right|<\epsilon$ for arbitrary $\epsilon>0$. I set out to find a choice... | Community | -1 | <p>You clearly get something that goes to zero. Try dividing by $k^2$ in both numerator and denominator: $$\left|\frac{-2k-2}{k^2+2k+2}\right|=\left|\frac{\frac{-2}k-\frac 2{k^2}}{1+\frac2k+\frac2{k^2}}\right|\lt\left|\frac2k +\frac2{k^2}\right |\lt\left|\frac2k+\frac2k \right |=\left|\frac4k\right |$$, for $k\gt1$.<... |
1,843,369 | <p>Consider the congruence </p>
<p>$$2x+7y \equiv 5\pmod{12}$$</p>
<p>Here $(2,7,12)=1$. Since $(2,12)=2$, we must have
$$7y \equiv 5\pmod{2}$$</p>
<p>Which clearly gives $y \equiv 1\pmod{2}$, or $y \equiv 1,3,5,7,9,11\pmod{12}$</p>
<p>Why does the previous statement follow?. This is not a problem. It's something ... | 3x89g2 | 90,914 | <p>Suppose $[7y]_2 = [5]_2$, then $[7]_2 [y]_2 = [5]_2$, which implies that $[1]_2 [y]_2 = [1]_2$, that is, $y \equiv 1 (\mod 2)$. This implies that $y$ has to be odd. </p>
|
1,843,369 | <p>Consider the congruence </p>
<p>$$2x+7y \equiv 5\pmod{12}$$</p>
<p>Here $(2,7,12)=1$. Since $(2,12)=2$, we must have
$$7y \equiv 5\pmod{2}$$</p>
<p>Which clearly gives $y \equiv 1\pmod{2}$, or $y \equiv 1,3,5,7,9,11\pmod{12}$</p>
<p>Why does the previous statement follow?. This is not a problem. It's something ... | Arthur | 15,500 | <p>$2x + 7y$ must be odd (since $5\mod 12$ is odd). But $2x$ cannot be odd, and therefore $7y$ must be odd. That means $y$ must be odd. What you've written is saying the same thing, only with modulo $2$ instead of even / odd.</p>
|
2,328,677 | <p>I wish to find the area between the curves:</p>
<p>$y=\sqrt{x}$</p>
<p>and </p>
<p>$y=x^{2}-3x-2$</p>
<p>and between $x=1, x=9$</p>
<p>Now, I think that the first thing to do is to find the intersection between the two curves, and then to find which curve is the upper one before and after the intersection point... | Mundron Schmidt | 448,151 | <p>You ask how to solve $x^2-3x-2=\sqrt{x}$? Compute the square on both sides and you get
\begin{align}
x^2-3x-2=\sqrt{x}&\Leftrightarrow (x^2-3x-2)^2=x\\
&\Leftrightarrow x^4-6x^3+5x^2+12x+4=x\\
&\Leftrightarrow x^4-6x^3+5x^2+11x+4=0.
\end{align}
If you are lucky then you get one of $-4,-2,-1,1,2,4$ is a s... |
220,170 | <p>My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?</p>
<p>For example i would start with :</p>
<ol>
<li... | Liviu Nicolaescu | 20,302 | <p>Here are two examples suggesting the complexity of the world of $3$-manifolds.</p>
<p>The first is the classical result that any $3$-manifold can be obtained by integral surgery on a link in $S^3$. If you believe that knots and links form a complex Universe, than this result should suggest that $3$-manifold... |
3,900,435 | <p>Suppose that <span class="math-container">$\mathbf{Y}\sim N_3(0,\,\sigma^2\mathbf{I}_3)$</span> and that <span class="math-container">$Y_0$</span> is <span class="math-container">$N(0,\,\sigma_0^2)$</span>, independently of the <span class="math-container">$Y_i$</span>'s.
My question is that does <span class="math-c... | perpetuallyconfused | 570,087 | <p>So apparently, <a href="https://sbseminar.wordpress.com/2007/10/30/theme-and-variations-schroeder-bernstein/" rel="nofollow noreferrer">it is possible for this to fail</a>, though I don't profess to understand the counterexamples given in the comments. You are asking if fields satisfy the <a href="https://en.wikiped... |
4,432,047 | <p><span class="math-container">$3\times 2$</span> is <span class="math-container">$3+3$</span> or <span class="math-container">$2+2+2$</span>. We know both are correct as multiplication is commutative for whole numbers. But which one is <strong>mathematically</strong> accurate?</p>
| mweiss | 124,095 | <p>I think this is an interesting and non-trivial question, but essentially impossible to answer. Let me explain why I think this.</p>
<p>The usual (I think?) way to define multiplication in the natural numbers is recursively: we define, for each <span class="math-container">$n \in \mathbb N$</span>, a function <span ... |
80,452 | <p>When can the curvature operator of a Riemannian manifold (M,g) be diagonalized by a basis of the following form</p>
<p>'{${E_i\wedge E_j }$}' where '{${E_i}$}' is an orthonormal basis of the tangent space? If the manifold is three dimensional then it is always possible. But what about higher dimensional cases?</p>
| Anton Petrunin | 1,441 | <p><em>This is not what you asking, but maybe you want to know.</em></p>
<ol>
<li><p>$\mathbb C\mathrm P^2$ does not admit a metric with splitting tensor in your sense.
It has a non-zero Pontryagin number and any Pontryagin number can be expressed as an integral of some function of curvature tensor which is zero for ... |
893,839 | <p>I am dealing with the dual spaces for the first time.</p>
<p>I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.</p>
| Jonas Dahlbæk | 161,825 | <p>Locally convex spaces have a very rich theory for their dual spaces, and much of functional analysis (as the name suggests) is devoted to the study of such spaces. Consider how important the concept of a basis is in the finite dimensional setting and then note that the continuous linear functionals are a natural gen... |
13,616 | <p>Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a short description.</p>
<p>I think that the tags are relevant, but feel free to change them.</p>
<p>Also, have ther... | Vectornaut | 1,096 | <p><em>An Introduction to Families, Deformations and Moduli</em>, by T.E. Venkata Balaji, has a beautiful little appendix that introduces smooth manifolds, complex manifolds, schemes, and complex analytic spaces in a unified way as locally ringed spaces. Although it doesn't say much in general about arbitrary locally r... |
1,014,614 | <p>That Theorem 3.2 says: Every finite orthogonal set of nonzero vectors is linearly independent.</p>
<p>The proof is simple, but it seems to me that the finiteness is redundant, for the argument in the proof applies to an infinite set. Am I right?</p>
<p>The proof runs as follows: If $k > 0$ is an integer, if $a_... | Alfred Chern | 42,820 | <p>Suppose that $f(x)$ is defined on $\mathbb{R}$.</p>
<p>$f(x)$ uniformly continuous </p>
<p>$\Rightarrow\forall\epsilon>0,\exists\delta>0,\forall x_1,x_2\in\mathbb{R}:|x_1-x_2|\leq\delta,|f(x_1)-f(x_2)|<\epsilon$, then fix $\epsilon=1$.</p>
<p>$\forall x\in\mathbb{R},\exists k\in\mathbb{N}^+$, such that $... |
515,915 | <p><strong>Definiton</strong> A $p$-adic integer is a (formal) series
$$\alpha=a_0+a_1p+a_2p^2+\ldots$$
with $0\leq a_i<p$.</p>
<p>The set of $p$-adic integers is denoted by $\mathbb{Z}_p$. If we cut an element $\alpha\in\mathbb{Z}_p$ at its $k$-th term
$$\alpha_k=a_0+a_1p+\ldots+\alpha_{k-1}p^{k-1}$$
$\textbf{we g... | Eric Stucky | 31,888 | <p><strong>Hint:</strong> Any non-identity rotation always fixes a unique point. Which points (if any) are fixed by $RT$ and $TR$?</p>
|
2,842,481 | <p>If the minimum and the maximum values of $y = (x^2-3x+c)/(x^2+3x+c)$ are $7$ and $1/7$ , then the value of $c$ is?</p>
<p>I cross multiplied the equation and tried to find it's discriminant, but I don't think it gets me anywhere. A little hint would be appreciated!</p>
| mengdie1982 | 560,634 | <h1>Solution</h1>
<p>First, we should constrain the value of <span class="math-container">$c$</span> such that <span class="math-container">$x^2+3x+c \neq 0$</span> for all <span class="math-container">$x \in \mathbb{R}.$</span> Otherwise, there necessarily exists at least one infinite discontinuity for <span class="ma... |
40,618 | <p>Let $f_{=}$ be a function from $\mathbb{R}^{2}$ be defined as follows:
(1) if $x = y$ then $f_{=}(x,y) = 1$;
(2) $f_{x,y} = 0$ otherwise.</p>
<p>I would like to have a proof for / a reference to a textbook proof of the following theorem (if it indeed is a theorem):</p>
<p>$f_{=}$ is uncomputable even if one restr... | Thierry Zell | 8,212 | <p><a href="http://people.bath.ac.uk/masdr/" rel="nofollow">Dan Richardson</a> in Bath has extensively studied the problem of recognizing zero under various hypotheses. I would be hard-pressed to give you an account of the details, because there is a lot of subtle and surprising results, but his page has all his papers... |
1,456,262 | <p>Show that the equation to a circle with center at $z_0$ and radius $r$ can be written as $$|z-z_0| = r$$ or as $$z\bar{z} - z\bar{z_0} - \bar{z}z_0 + |z_0|^2 = r^2$$</p>
<p>I let $z_0 = (x_0 + iy_0) = (x_0,y_0)$. Now I have $$(x-x_0)^2 + (y-y_0)^2 = r^2$$</p>
<p>I know $x=\frac{1}{2}(z+\bar{z})$ and $y = \frac{1}{... | Julián Aguirre | 4,791 | <p>Variations of constants is used to find a particular solution of the complete system once you know the general solution of the homogeneous system. Let $U=(1,1)$ be an eigenvector of the eigenvalue $1$ and let $V$ be another vector. Look for a solution of the form
$$
t\,e^t\,U+e^t\,V.
$$
This leads to the equation $(... |
2,873,449 | <p>For a continuous function $f: R \to R $ Define:
$$ \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} f(t)t \space dt $$
Since the function is continuous I can assume that it's also integrable since continuity implies integrability. I assume furthermore that there exists a function $F$ which is an antiderivative of $f$ for w... | Martin Argerami | 22,857 | <p>The limit is indeed $f(0)/2$.
L'Hôpital's Rule applies because the limit of the quotient of the derivatives of numerator and denominator exists. Thus
$$
\lim_{x\to0}\frac1{x^2}\int_0^x f(t)\,t\,dt
=\lim_{x\to0}\frac{x\,f(x)}{2x}=\frac{f(0)}2
$$</p>
|
1,729,220 | <p>Let</p>
<p>$$f(x)=\frac{1}{x^2+3x+2}$$</p>
<p>I must find $$\lim_{n\to\infty}\sum_{i=0}^n \frac{(-1)^i}{i!}f^{(i)}(1)$$</p>
<p>How should I proceed?</p>
| oren revenge | 306,175 | <p>I figured it out, here is the solution:</p>
<p>by decomposing f(x) and applying some derivates you get (note that I have skipped induction)
$$f^{(n)}(1)=\frac{(-1)^nn!}{2^{n+1}}-\frac{(-1)^nn!}{3^{n+1}}$$</p>
<p>after some simple algebraic manipulation the limit becomes
$$\lim_{n->\infty}{\sum^n_{k=0}{\frac{1}... |
221,026 | <p>I need to find the value of <span class="math-container">$z$</span> for a particular value of <span class="math-container">$D_c$</span> (eg. <span class="math-container">$500$</span>), but <span class="math-container">$z$</span> is inside an integral, and I'm not able to use <code>Solve</code> since the integral is ... | SuperCiocia | 35,368 | <p>From your integral I get the following equation (same as yours when you plot it):</p>
<pre><code>eqn[z_] :=
3.2566440560469836` - (
3.5857498598223954` Hypergeometric2F1[1/6, 1/2, 7/
6, -(2.2144005143040824`/(1 + z)^3)])/Sqrt[1 + z]
</code></pre>
<p>Both <code>Solve</code>and <code>NSolve</code> fail.</p>
<p... |
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