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 |
|---|---|---|---|---|
881,520 | <p>Take half a square with side length $1$.
The resulting right-angled triangle ABC
has two angles of $45^\circ$.
By Pythagoras’ theorem, the hypotenuse
AC has length $\sqrt{2}$. Applying the definitions on the previous page gives the values in the table below. that $\sin 30^\circ= \frac{1}{2}$</p>
<p>Sorry I cannot ... | 5xum | 112,884 | <p>The right angled triangle you describe has angles of $45$ degrees, not $30$ degrees. What you proved is $\sin 45=\frac{1}{\sqrt2}$, which is correct.</p>
|
881,520 | <p>Take half a square with side length $1$.
The resulting right-angled triangle ABC
has two angles of $45^\circ$.
By Pythagoras’ theorem, the hypotenuse
AC has length $\sqrt{2}$. Applying the definitions on the previous page gives the values in the table below. that $\sin 30^\circ= \frac{1}{2}$</p>
<p>Sorry I cannot ... | Fermat | 83,272 | <p>First draw a equilateral triangle and find the sine or cosine of the angle of $60^\circ$. Now for the angle of $30$ degrees apply the identity
$$\cos x=\sin (90^\circ -x)$$</p>
|
881,520 | <p>Take half a square with side length $1$.
The resulting right-angled triangle ABC
has two angles of $45^\circ$.
By Pythagoras’ theorem, the hypotenuse
AC has length $\sqrt{2}$. Applying the definitions on the previous page gives the values in the table below. that $\sin 30^\circ= \frac{1}{2}$</p>
<p>Sorry I cannot ... | John Joy | 140,156 | <p><img src="https://i.stack.imgur.com/zO5p0.png" alt="enter image description here"></p>
<p>The trigonometric function values for all the special angles are derived from the first three regular polygons (equilateral triangle, square, and regular pentagon). The trigonometric function values for the first two shapes ma... |
1,610,700 | <blockquote>
<p>$$\int \frac{x-3}{\sqrt{1-x^2}} \mathrm dx$$</p>
</blockquote>
<p>I know that $\int \frac{1}{1-x^2}\mathrm dx=\arcsin(\frac{x}{1})$ but how can I continue from here? </p>
| Error 404 | 206,726 | <p>Write $\int \frac {x-3}{\sqrt {1-x^2}} = \int \frac {x}{\sqrt {1-x^2}}-3\int \frac 1{\sqrt {1-x^2}}$.</p>
<p>Use the substitution $1-x^2=u$ in the first integral of the RHS. Also you know the second integral of the RHS as per your post.</p>
|
2,753,548 | <p>Let $F$ be a field with $7^5$ elements.
$$X=\{a^7-b^7 \mid a,b \in F\}$$
I have no idea how to solve.
Please help me.</p>
| Antoine Giard | 554,642 | <p><strong>Hint:</strong> Prove that \begin{align*}
\phi \ \colon \ &F \to F\\
&x \mapsto x^7,
\end{align*}
is an isomorphism.</p>
|
2,974,747 | <p><strong>Q</strong>:Solve the equation <span class="math-container">$x^4+x^3-9x^2+11x-4=0$</span> which has multiple roots.<br><strong>My approach</strong>:Let <span class="math-container">$f(x)=x^4+x^3-9x^2+11x-4=0$</span>.And i knew that if the equation have multiple roots then there must exist H.C.F(Highest Common... | Community | -1 | <p>There is no need to apply the Euclidean algorithm to solve this problem. Note that the sum of coefficients of <span class="math-container">$$f(x)=x^4+x^3−9x^2+11x−4$$</span> is <span class="math-container">$0$</span>, so <span class="math-container">$f(1)=0$</span>. Next, <span class="math-container">$$f'(x)=4x^3+... |
2,974,747 | <p><strong>Q</strong>:Solve the equation <span class="math-container">$x^4+x^3-9x^2+11x-4=0$</span> which has multiple roots.<br><strong>My approach</strong>:Let <span class="math-container">$f(x)=x^4+x^3-9x^2+11x-4=0$</span>.And i knew that if the equation have multiple roots then there must exist H.C.F(Highest Common... | Will Jagy | 10,400 | <p>I quite like the Euclidean algorithm. The Extended part can be done as a continued fraction, same steps as for integers.</p>
<p><span class="math-container">$$ \left( x^{4} + x^{3} - 9 x^{2} + 11 x - 4 \right) $$</span></p>
<p><span class="math-container">$$ \left( 4 x^{3} + 3 x^{2} - 18 x + 11 \right)... |
3,969,598 | <p>Let <span class="math-container">$f: \mathbb{R} \rightarrow \mathbb{R}$</span> a monotonically increasing function and <span class="math-container">$A \subset \mathbb{R}$</span> where <span class="math-container">$A \neq \emptyset$</span> and boundend.</p>
<p>i) If f is continuous function, prove that <span class="... | Henry | 6,460 | <p><span class="math-container">$$s = r - \sqrt{r^2 - y^2}$$</span> can be rearranged to
<span class="math-container">$$\sqrt{r^2 - y^2} = r - s$$</span> and squaring both sides (possibly introducing a spurious root)
<span class="math-container">$${r^2 - y^2} = r ^2 -2rs +s^2$$</span> and simplifying
<span class="math-... |
271,105 | <p><strong>tl;dr</strong> What are some good workflows for developing and running data processing pipelines with Mathematica?</p>
<hr />
<p>I sometimes develop data processing pipelines with Mathematica. I load some data, transform it, and derive some summary results. I tend to experiment quite a bit when doing this, c... | Lukas Lang | 36,508 | <p><em>Disclaimer: I have faced similar issues as well, but I have not "field-tested" the solution I present below, so I can't speak about its issues and limitations.</em></p>
<p>The idea is the following: We use <a href="https://reference.wolfram.com/language/ref/AutoGeneratedPackage.html" rel="noreferrer"><... |
2,114,636 | <p>Let $V$ is a finite dimensional vector space over a field $\mathbb F.$ Let $\rm dim (V)=n>0$ and $\mathcal {B}=\{v_1,\ldots,v_n\}$ be a basis of $V.$ Now we know dimension of $V \otimes_\mathbb {F}V$ is $n^2$ as $V \otimes_\mathbb {F}V \cong \mathbb {F^{n^2}}.$ Now since the set $\mathcal {A}=\{v_i\otimes v_j:1 ... | JWL | 161,058 | <p>Whatever your definition of tensor is, it should be true that any bilinear function
$$
V\times V\longrightarrow \mathbb{F}
$$
must factor uniquely through
$$
V\times V\longrightarrow V\otimes_\mathbb{F} V\longrightarrow \mathbb{F}
$$
Now take any linear functional $\ell :V\to\mathbb{F}$ such that $\ell(v_i)\neq0 $ a... |
2,114,636 | <p>Let $V$ is a finite dimensional vector space over a field $\mathbb F.$ Let $\rm dim (V)=n>0$ and $\mathcal {B}=\{v_1,\ldots,v_n\}$ be a basis of $V.$ Now we know dimension of $V \otimes_\mathbb {F}V$ is $n^2$ as $V \otimes_\mathbb {F}V \cong \mathbb {F^{n^2}}.$ Now since the set $\mathcal {A}=\{v_i\otimes v_j:1 ... | Community | -1 | <p>If $(v_i,v_j)$ are non-zero there is a bilinear application $\beta : V \times V \to k$ with $\beta(v_i,v_j) = 1$ and by the universal property of tensor product we get a map $b : V \otimes V \to k $ with $b(v_i \otimes v_j) = 1$ so it cannot be zero. </p>
|
1,290,363 | <p>So I already proved Closure and Associativity, now I'm trying to find the identity element of this operation defined as:
$$
a * b = a + b - ab
$$</p>
<p>But my identity element gets cancelled...</p>
<p>(The set defined in this exercise is the real numbers.)</p>
<p><img src="https://i.stack.imgur.com/ZchjC.jpg" al... | Matt Samuel | 187,867 | <p>As stated in the other answers, the identity element is $0$. If the goal was to prove or disprove that this is a group, you checked the axioms in an unfortunate order, because inverses don't exist. In particular, $1$ does not have an inverse, because $a\ast 1=1$ for all $a$.</p>
|
3,093,660 | <p>This is an introducory task from an exam. </p>
<p><strong>If</strong> <span class="math-container">$z = -2(\cos{5} - i\sin{5})$</span>, <strong>then what are:</strong></p>
<p><span class="math-container">$Re(z), Im(z), arg(z)$</span> and <span class="math-container">$ |z|$</span>?</p>
<p>First of all, how is it p... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$z=a+bi$</span>, with <span class="math-container">$a,b\in\mathbb R$</span>, then <span class="math-container">$\lvert z\rvert=\sqrt{a^2+b^2}$</span>; in particular, <span class="math-container">$\lvert z\rvert\geqslant0$</span> for any <span class="math-container">$z\in\mathbb C$</sp... |
575,597 | <p>I don't understand how to solve $3^{1/4} \cdot 9^{-5/8}$. Help please?</p>
<p>I have tried many different things, but they're not working. Once I plug the problem into a math equation solver, the answer $1/3$ appears, but I don't understand how they got that. </p>
| Newb | 98,587 | <p>We have your equivalence relation <strong>R</strong> as $a \sim b$ if $3|(a^2 - b^2)$.</p>
<p>To find the class of elements equivalent to $0$, we need to set one of the elements to $0$ (just one because of reflexivity, symmetry, and transitivity, as this is an equivalence relation): suppose $a \sim 0$, i.e. $3|(a^2... |
575,597 | <p>I don't understand how to solve $3^{1/4} \cdot 9^{-5/8}$. Help please?</p>
<p>I have tried many different things, but they're not working. Once I plug the problem into a math equation solver, the answer $1/3$ appears, but I don't understand how they got that. </p>
| Cory Crowley | 397,663 | <p>The set has the following equivalence relations.
<a href="https://i.stack.imgur.com/xm0Vi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xm0Vi.png" alt="[0] equivalence relation on set"></a></p>
|
3,692,877 | <p>Consider the following expression in three variables, <span class="math-container">$0 \leq p,s \leq 1$</span> and <span class="math-container">$n >0$</span></p>
<p><span class="math-container">$$S_{n, p, s} = \sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$$</span></p>
<p>If <span class="math-contai... | SchrodingersCat | 278,967 | <p>Just an attempt in trying to simplify the sum (without any idea to reach a closed form):</p>
<p><span class="math-container">$$S_{n, p, s} = \sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$$</span>
<span class="math-container">$$ = \sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} \sum_{m=0}^\infty (-1)^m \fra... |
3,191,345 | <p>Evaluate the following definite integral :
<span class="math-container">$$\int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \qquad \qquad \qquad (1)$$</span> </p>
<p><span class="math-container">\begin{align}
& = \int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \ \ I \ used \ u=1-\sin x \ and \ dx= \cfrac{-du}{cosx... | fGDu94 | 658,818 | <p>Symbolab is wrong here, <span class="math-container">$\cos(x) \geq 0$</span> on the interval and <span class="math-container">$\sqrt{1-sin(x)} \geq 0$</span> on the interval, with regions where both functions are strictly positive.</p>
|
3,191,345 | <p>Evaluate the following definite integral :
<span class="math-container">$$\int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \qquad \qquad \qquad (1)$$</span> </p>
<p><span class="math-container">\begin{align}
& = \int_0^{\pi/2}\cfrac{\cos x}{\sqrt{1-\sin x}} \ \ I \ used \ u=1-\sin x \ and \ dx= \cfrac{-du}{cosx... | Dr. Sonnhard Graubner | 175,066 | <p>I would substitute <span class="math-container">$$t=\sqrt{1-\sin(x)}$$</span> then we get <span class="math-container">$$dt=\frac{\cos(x)}{2\sqrt{1-\sin(x)}}dx$$</span></p>
|
672,736 | <p>Let $A = \begin{bmatrix}1&2&1\\0&1&0\\1&3&1\end{bmatrix}$. Find the eigenvalues of $A$.</p>
<p>I think I got a pretty steady ground on how I approached this, I just have some difficulty getting the right answer.</p>
<p>What I have done so far:</p>
<p>$P(\lambda) = det(A - \lambda I)$</p>
... | Eleven-Eleven | 61,030 | <p>Your multiplication is wrong...since the middle row has two zeros, you only have to evaluate using the middle cofactor...
$$\det{(A)}=(1-\lambda)((1-\lambda)^2-1)=(1-\lambda)(1-2\lambda+\lambda^2-1)=(1-\lambda)(\lambda^2-2\lambda)$$
$$=\lambda(1-\lambda)(\lambda-2)$$</p>
|
3,243,733 | <p><strong>Use induction to show that the Fibonacci numbers satisfy F(n) <span class="math-container">$\ge$</span> <span class="math-container">$(2 ^ {(n-1) / 2})$</span> for all n <span class="math-container">$\ge$</span> 3</strong></p>
<p>My work thus far:</p>
<blockquote>
<p>Base Case: F(3) <span class="math-co... | YiFan | 496,634 | <p>You seem to be misunderstanding how a proof by induction works. Say you have a proposition <span class="math-container">$P(n)$</span> to be verified for all natural numbers <span class="math-container">$n=0,1,2,\dots$</span>. The <strong>base case</strong> is to show that <span class="math-container">$P(0)$</span> i... |
1,700,689 | <p>Let $A, B$, and $C$ be sets. If $A\backslash B$ is a subset of $C$, then $A\backslash C$ is a subset of $B$. Is this a direct proof where I let $x$ be an element of $A$ and then work from there? I can't seem to figure out all of the cases. Thanks for help in advance.</p>
| martini | 15,379 | <p>As you want to prove $A \setminus C \subseteq B$, start with an element $x \in A \setminus C$, that is $a \in A$, $a \not\in C$. Then either $x \in B$, or $x \not\in B$. If $x \not\in B$, then $x \in A \setminus B$, hence $x \in C$, which is not possible. If $x \in B$, we are done.</p>
<p>Hence $A \setminus C \subs... |
1,700,689 | <p>Let $A, B$, and $C$ be sets. If $A\backslash B$ is a subset of $C$, then $A\backslash C$ is a subset of $B$. Is this a direct proof where I let $x$ be an element of $A$ and then work from there? I can't seem to figure out all of the cases. Thanks for help in advance.</p>
| johnnycrab | 171,304 | <p>Let $x \in A \setminus C$, then $x \in A$, but $x \notin C$. If $x \notin B$, then $x \in C$ (because $A \setminus B \subset C$), which is a contradiction. Thus $x \in B$ and $A \setminus C \subset B$.</p>
|
272,846 | <p>Suppose I have a List of numbers:</p>
<pre><code>num = Range[5]
</code></pre>
<p>I want to combine the second and the third element into a sublist to get the result as {1,{2,3},4,5}.<br />
I tried using this:</p>
<pre><code>MapAt[List, num, {{2}, {3}}]
</code></pre>
<p>which is not giving me the desired result. What... | Artes | 184 | <p>I guess this is the simplest approach:</p>
<pre><code>num /. {a_, b_, c_, d___} :> {a, {b, c}, d}
</code></pre>
<blockquote>
<pre><code> {1, {2, 3}, 4, 5}
</code></pre>
</blockquote>
<p>At the last position we have <code>BlankNullSequence</code> to make it work.</p>
|
3,014,453 | <p>If there is a number somewhere between 0 and 100 and you have to find it with the least attempts possible. Every attempt consists of you checking if the number is smaller (or bigger) than a number in the said interval (0 to 100). My guess would be you start with the half way point.</p>
<p>Is it smaller than 50?
yes... | badjohn | 332,763 | <p>The other answers have confirmed that this is the best method but did not mention how you can see that you cannot get much better. With <span class="math-container">$7$</span> tests and only <span class="math-container">$2$</span> possibilities each, there are <span class="math-container">$2 ^ 7 = 128$</span> possi... |
95,598 | <p>I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants.</p>
<p>I am trying to find the probability density, so I need to find the product of $\psi$ with it's complex conjugate. The problem is, im not sure what is it's complex conjugate, I know the complex conj... | MrOperator | 21,887 | <p>The complex conjugation factors through sums and products. So you can take the complex conjugate of the factor with A and B separately. The constant A and B form know problem, this goes according to the usual rules. This leaves something of the form $e^{(a+bi)}$. Now note that $e^{(a+bi)}= e^a(\cos(b)+i \sin(b))$ Ta... |
2,890,625 | <p>Suppose $f(x)$ is differentiable on $[0,1]$, and $f(0)=0$, $f(x)\ne 0,\forall x\in(0,1)$ , Prove for every $n,m\in\mathbb{N^+}$, there exists $\xi=\xi_{n,m}\in(0,1)$ such that
$$n\cdot\frac{f'(\xi)}{f(\xi)}=m\cdot\frac{f'(1-\xi)}{f(1-\xi)}$$</p>
| xbh | 514,490 | <p>MVT method:</p>
<p>Consider an auxiliary function
$$
F(x) =f(x)^n f(1-x)^m, \quad x \in [0,1],
$$
then Rolle's theorem achieves the goal. </p>
|
4,313,593 | <p>I am asked to determine the derivative of the function
<span class="math-container">\begin{align*}
f(\textbf{x}) = \left\|\textbf{A}\textbf{x}-\textbf{y}\right\|_{2} ^{2}+\alpha \textbf{x}^{\mathsf{T}}\textbf{M}\textbf{x}
\end{align*}</span>
with <span class="math-container">$\textbf{A} \in \mathbb{R}^{m, n}, \ ... | Will Jagy | 10,400 | <p>ADDED next morning: took forever, I finally confirmed that my <span class="math-container">$A$</span> satisfies <span class="math-container">$31A^4 - 22 A^2 + 8A - 1$</span> which reduces, eventually
<span class="math-container">$$ 31 A^3 - 31 A^2 + 9A - 1 = 0$$</span>
Meanwhile, the dependence is <span class="... |
4,313,593 | <p>I am asked to determine the derivative of the function
<span class="math-container">\begin{align*}
f(\textbf{x}) = \left\|\textbf{A}\textbf{x}-\textbf{y}\right\|_{2} ^{2}+\alpha \textbf{x}^{\mathsf{T}}\textbf{M}\textbf{x}
\end{align*}</span>
with <span class="math-container">$\textbf{A} \in \mathbb{R}^{m, n}, \ ... | Claude Leibovici | 82,404 | <p>Just in case you want the exact formulae</p>
<p><span class="math-container">$$c_1=\frac{1}{3} \left(1+2 \cosh \left(\frac{1}{3} \cosh
^{-1}\left(\frac{29}{2}\right)\right)\right)$$</span></p>
<p>and</p>
<p><span class="math-container">$$d_1=\frac 13 \Bigg[1+\frac{4}{\sqrt{31}}\cosh \left(\frac{1}{3} \cosh ^{-1}\... |
1,107,317 | <p>I've got this hypergeometric series</p>
<p>$_2F_1 \left[ \begin{array}{ll}
a &-n \\
-a-n+1 &
\end{array} ; 1\right]$</p>
<p>where $a,n>0$ and $a,n\in \mathbb{N}$</p>
<p>The problem is that $-a-n+1$ is negative in this case. So when I try to use Gauss's identity</p>
<p>$_2F_1 \left[ \begin{array}{ll}... | balping | 208,441 | <p>Here I post the full solution</p>
<p>The problem: we are looking for the closed form of this sum:</p>
<p>$\sum\limits_{i=0}^n \binom{a+i-1}{i} \binom{a-i+n-1}{n-i}$</p>
<p>The first term of the sum for $i=0$ is $\binom{a+n-1}{n}$</p>
<p>The ratio two consecutive terms:</p>
<p>$\dfrac{t_{i+1}}{t_i} = \dfrac{P(i)... |
1,523,287 | <p>We choose a random number from 1 to 10. We ask someone to find what number it is by asking a yes or no question. Calculate the expected value if the person ask if it is $x$ number till you got it right?</p>
<p>I know the answer is around 5 but i can't find how to get there.</p>
<p>I tried $\frac{1}{10}(1)+\frac{1}... | Bernard | 202,857 | <p>We have the obvious *Bézout's identity: $\;3\cdot 5-2\cdot 7 =1$. The solution of the system of congruences
$$\begin{cases}n\equiv \color{cyan}1\mod5\\n\equiv \color{red}3\mod 7\end{cases}\quad \text{is}\enspace n\equiv \color{red}3\cdot3\cdot 5-\color{cyan}1\cdot2\cdot 7\equiv 31\mod 35 $$</p>
|
1,344,161 | <p>Suppose $k\geq 2$ is an integer. I want to show $$\frac{1+k+k(k-2)}{1+\frac{k-1}{k}+\frac{(-1-\sqrt{k-1} )^2}{k(k-2)}}$$ is not an integer. It is equal to $$\frac{(k-2) k (k^2-k+1)}{2 (k^2-2 k+\sqrt{k-1}+1)}.$$</p>
<p>If I can show this then I will be able to finish my proof of the <a href="https://en.wikipedia.org... | Jyrki Lahtonen | 11,619 | <p>Hint: For that number to be rational it is necessary that the square root is rational, which happens only when $k=n^2+1$ for some integer $n$. Show that then the denominator (of the latter formula) is divisible $n$ but the numerator <strike>is not</strike> leaves remainder $-1$ when divided by $n$. Therefore the fac... |
3,436,219 | <p><img src="https://i.stack.imgur.com/VplT3.jpg" alt="enter image description here"></p>
<p>I could use gaussian elimination if I make some assumptions or does any one have another suggestion?</p>
| Glowing0v3rlord | 725,413 | <p>Say the first number is <em>x</em>, the second is <em>y</em>, and the answer is <em>A</em>.</p>
<p>I believe the formula is <em>A</em> = <em>x</em>-(2+<em>y</em>).</p>
<p>Another that works is <em>A</em> = <em>x</em>-<em>y</em>-2</p>
<p>According to this logic:</p>
<p>a) 9 ● 2=5</p>
<p>b) ● = -2-</p>
<p>I hope... |
687,352 | <p>How many experiments should we conduct so that we could state that with more than $0.9$ probability the event occurs at least once. The probability that the event occurs is $0.7$. </p>
<p>I have tried the following:</p>
<p>Let's say the number of experiments is equal to $n$.
The opposite of 'occurs at least once'... | NovaDenizen | 109,816 | <p>First, for $0 < x < \frac{\pi}{2}$, $\sin x \ge \dfrac{2x}{\pi}$. Likewise for any interval $[k\pi, (k + \frac12)\pi]$ (with integer $k$), $|\sin{x}| \ge \dfrac{2(x-k\pi)}{\pi}$.</p>
<p>For each interval $[k\pi, (k + \frac12)\pi]$, $\dfrac{|\sin x|}{x} \ge \dfrac{2(x - k\pi)}{\pi x} \gt \dfrac{2(x-k\pi)}{(k+... |
2,448,349 | <p>Would the cubic equation that is the best approximation for $e^x$ just be the first 4 terms of its Taylor Series expansion?
$$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}$$
I guess it would depend on the bounds you are inspecting though? I'm not well-versed in dealing with infinities. Is there a cubic equation that is th... | zwim | 399,263 | <p>It depends what is it you try to minimize ?</p>
<p>Taylor expansion is the one that optimize the fitness of the polynomial for $||\cdot||_{\infty}$ to the exponential in a neighbourhood of $0$.</p>
<p>But you can try to minimize other things, by having other norms and/or trying to minimize locally or globally.</p>... |
3,047,241 | <blockquote>
<p>Let <span class="math-container">$X_1, X_2, \cdots, X_n$</span> be i.i.d. <span class="math-container">$\sim \text{Bernoulli}(p)$</span>. Then <span class="math-container">$\bar{x}$</span> is an unbiased estimator of <span class="math-container">$p$</span>.</p>
</blockquote>
<p>How should I approach ... | Vishaal Sudarsan | 414,699 | <p>Use the fact that <span class="math-container">$X_i$</span> are Identical. And the linearity property of Expectation.</p>
<p><span class="math-container">$E(\bar{X})=E(X_1)=p$</span></p>
|
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | Joseph O'Rourke | 6,094 | <p>If you don't mind specializing Stokes theorem to Green's theorem, then one of the most practical applications is computation of the area of a region by integrating around its contour.
I am old enough to have used a <a href="http://en.wikipedia.org/wiki/Planimeter" rel="nofollow noreferrer">planimeter</a>, a delightf... |
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | orbifold | 19,007 | <p>A nice application in fluid mechanics is <a href="http://en.wikipedia.org/wiki/Kelvin%27s_circulation_theorem" rel="nofollow">Kelvin's circulation theorem</a>. You could also discuss how it fails to hold, if there are obstacles in the fluid flow. In the same spirit stokes theorem is applied in the canonical formalis... |
2,746,222 | <p>Problem:</p>
<p>A boats speed is <strong>1,70 m/s</strong> in still water.<br />
It must cross a river with a width of <strong>260 m</strong>.<br />
The boats starting point is the <strong>origin on the xy-axsis</strong> (on the shore).<br />
It has to dock <strong>110 m</strong> to the right(in the positive x-direc... | CiaPan | 152,299 | <p>Sailing on a still 'river', the boat would arrive to the opposide side at the point $260\,m$ across the river <em>and</em> $260\,m$ along the river, due to the angle of $45^\circ$.</p>
<p>What is the length of this route? What time would it take the boat to sail it?</p>
<p>You know the boat actually ends its trave... |
2,704,394 | <p>Here is the formal statement:</p>
<blockquote>
<p>Let $\lambda_1, \lambda_2, \lambda_3$ be distinct eigenvalues of $n\times n$ matrix $A$. Let $S=\{v_1, v_2, v_3\}$, where $Av_i = \lambda_i v_i$ for $1\leq i\leq 3$. Prove $S$ is linearly independent. </p>
</blockquote>
<p>Many resources online state the general ... | copper.hat | 27,978 | <p>Suppose $v=\sum_k \alpha_k v_k = 0$.</p>
<p>$(A-\lambda_1 I)v = \sum_{k>1} \alpha_k (\lambda_k-\lambda_1)v_k = 0$.</p>
<p>$(A-\lambda_2 I)(A-\lambda_1 I)v = \sum_{k>2} \alpha_k (\lambda_k-\lambda_2)
(\lambda_k-\lambda_1)v_k = 0$.
$$\vdots$$
$\prod_{i=1}^{n-1} (A-\lambda_i I) v = \alpha_n \prod_{i=1}^{n-1}(\... |
3,999,699 | <p>I want to show that <span class="math-container">$a_{n} = \sqrt{n}$</span> is not a bounded sequence.</p>
<p>Definition: We say that a sequence is bounded if it is bounded above and below. A sequence <span class="math-container">$a_{n}$</span> is bounded above if there exists <span class="math-container">$C$</span> ... | Joe | 623,665 | <p>Ted Shifrin has already explained the problem with your derivation, but if you're looking for a 'safer' way of finding the derivative then consider this: if <span class="math-container">$\DeclareMathOperator{\arcsec}{arcsec} y = \arcsec x$</span>, then using the identity <span class="math-container">$\arcsec x = \ar... |
1,171,492 | <p>I was trying to obtain the square root of $-5-12i$ by the formula for square root (given below) and also by De Moivre's theorem and verify that both give the same result. But the two results are somehow not matching for this complex number. I am writing my solution below in two cases for each method:</p>
<p>Case - ... | Toby Mak | 285,313 | <p>There is a way without using $\arctan$.</p>
<p>Let the square root be $a+bi$. Then, $a^2+2abi-b^2 = -5-12i$, so $a^2-b^2 = -5 \text{(eq. 1)}$ and $2abi = -12i \ \text{(eq. 2)}; ab = -6 \ \text{(eq. 3)}$.</p>
<p>From equation $3$, $a = -\frac{6}{b}$. Substituting into equation $1$, $(-\frac{6}{b})^2-b^2+5=0$, and s... |
1,317,143 | <blockquote>
<p><em>Notation</em>: $\log:=\log_{10}$</p>
</blockquote>
<p>$\log x+\log_x 10$</p>
<p>$=\log x+ \frac{1}{\log x}$ </p>
<p>$=\log(x \cdot \frac{1}{x})$ </p>
<p>$=\log 1$ </p>
<p>$=0$ </p>
<p>Is the process correct? I doubt this is wrong. Please help.
Thanks.</p>
| user26486 | 107,671 | <p>I assume $\log_{10} x>0$ (iff $x>1$), because without constraints arbitrary small values can be gained.</p>
<p>$$\log_{10}x+\log_x 10=\log_{10}x+\frac{1}{\log_{10} x}\ge 2$$ </p>
<p>with equality iff $x=10$, because $a+\frac{1}{a}\ge 2$ for $a>0$ with equality iff $a=1$. </p>
<p>To prove it, $$a+\fra... |
1,201,002 | <p>I´m trying to find a vector $\vec{c} = $ , which is orthogonal to vector $\vec{a}$ and $\vec{b}$:</p>
<p>As far I understood, I have to show that:</p>
<p>$$\langle a,c\rangle=0 $$
$$\langle b,c\rangle=0 $$ </p>
<p>So if I would like to determine an orthogonal vector regarding: \begin{bmatrix}-1\\1\end{bmatrix}
... | Nick Lim | 500,921 | <p>I discovered a "quick" <span class="math-container">$(O(d^2))$</span> algorithm to generate <span class="math-container">$d-1$</span> mutually orthogonal vectors that are perpendicular to <span class="math-container">$\vec{x}$</span> where <span class="math-container">$d$</span> is the size of <span class="math-cont... |
2,634,701 | <p>Let $ f: {{\mathbb{R^n}} \rightarrow {{\mathbb{R}} }}$ be continuous and let $a$ and $b$ be points in $ {{\mathbb{R} }} $
Let the function $g: {\mathbb{R}} \rightarrow {\mathbb{R}}$ be defined as:
$$ g(t) = f(ta+(1-t)b) $$
Show that $g$ is continuous .</p>
<p>If I define a function $ h(t)=ta+(1-t)b$, then I have ... | José Carlos Santos | 446,262 | <p>If $t_1.t_2\in\mathbb R$, then\begin{align}\bigl\|h(t_2)-h(t_1)\bigr\|&=\bigl\|t_2a+(1-t_2)b-t_1a-(1-t_1)b\bigr\|\\&=\bigl\|(t_2-t_1)a-(t_2-t_1)b\bigr\|\\&=|t_2-t_1|.\|a-b\|.\end{align}If $a=b$, $h$ is the null function and therefore ir is continuous. Otherwise, if $\varepsilon>0$ then take $\delta=\f... |
1,727,339 | <p>What am I doing wrong?</p>
<p>I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix:</p>
<p>$A=\begin{bmatrix}
2 & 2 & 0 & -1 \\
0 & 0 & 0 & 1 \\
1 & 5 & 2 & -1 \\
0 & -4 & 0 & 4 \\
\end... | Isaac mathsgod | 675,881 | <ol>
<li><p>Look at the image space (column-wise) of <span class="math-container">$S = ( A - 2I )$</span>, which yields two vectors <span class="math-container">$S(v), v = (0,0,2,0) , (-1,1,-1,2)$</span> [SPECIFICALLY IN THIS ORDER]</p></li>
<li><p>Now find <span class="math-container">$u$</span> such that <span class... |
148,807 | <p>I'm not sure if these types of questions are accepted here or not (I'm very sorry if it's not), but it would be great if anyone could explain me this.</p>
<blockquote>
<p><strong>Question:</strong>
Using his bike, Daniel can complete a paper route in 20 minutes. Francisco, who walks the route, can complete it i... | André Nicolas | 6,312 | <p>Suppose that $N$ newspapers have to be delivered. Since Daniel can do the job in $20$ minutes, he distributes $\frac{N}{20}$ newspapers per minute.</p>
<p>Similarly, Francisco delivers $\frac{N}{30}$ newspapers per minute.</p>
<p>So if they work together as described, they deliver a total of $\frac{N}{20}+\frac{N}... |
31,767 | <p>I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight. </p>
<p>Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to construct a bijective mapping $f: T_k^n \to \{1, 2, \ldots, \binom{n}{k}\}$ such that computing each $f(x_1^n)$... | Anweshi | 2,938 | <p>Harald says:</p>
<blockquote>
<p>My basic question is how to go about this.</p>
</blockquote>
<p>Answer:</p>
<ol>
<li><p>If you want to have an intensive discussion with someone over this, through the internet: For communication that may happen burst-by-burst, leading to something definite later, google wave is... |
1,117,592 | <blockquote>
<p>Let <span class="math-container">$k$</span> be a finite field and <span class="math-container">$V$</span> a finite-dimensional vector space over <span class="math-container">$k$</span>.</p>
<p>Let <span class="math-container">$d$</span> be the dimension of <span class="math-container">$V$</span> and <sp... | quid | 85,306 | <p>Note that it suffices to do this for a vector-space of dimension $2$. You can then simply "blow up" each line to a hyperplane in a larger space. </p>
<p>For dimension two, check that $q+1$ is in fact equal to the number of hyperplanes, that is lines in this case. </p>
|
3,397,548 | <p>For a sequence <span class="math-container">$\{x_n\}_{n=1}^{\infty}$</span>, define <span class="math-container">$$\Delta x_n:=x_{n+1}-x_n,~\Delta^2 x_n:=\Delta x_{n+1}-\Delta x_n,~(n=1,2,\ldots)$$</span> which are named <strong>1-order</strong> and <strong>2-order difference</strong>, respectively. </p>
<p>The pro... | hskimse | 691,294 | <p>I wonder if this prove is correct.</p>
<p>Assume <span class="math-container">$\Delta x_n$</span> does not converge to 0, than there will be infinitely many <span class="math-container">$n$</span> with <span class="math-container">$\Delta x_n>c$</span>(or <span class="math-container">$\Delta x_n<c$</span>). a... |
3,033,943 | <blockquote>
<p><span class="math-container">$\textbf{Problem}$</span> Let <span class="math-container">$\Omega$</span> be an open, bounded and connected subset of <span class="math-container">$\mathbb{R}^n$</span>. Suppose that <span class="math-container">$\partial \Omega$</span> is <span class="math-container">$C^... | user9077 | 9,077 | <p>Assume <span class="math-container">$T$</span> is continuous at <span class="math-container">$0$</span>. It means given <span class="math-container">$\epsilon >0$</span> there exist <span class="math-container">$\delta>0$</span> such that for all <span class="math-container">$|z|<\delta$</span> one has <spa... |
487,123 | <p>How to evaluate the following limit?
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$</p>
<p>For this problem I have two methods. But I'd like to know if there are better methods.</p>
<p><strong>My solution 1:</strong></p>
<p>Using Stolz-Cesaro Theorem, we have
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}=\lim_... | wendy.krieger | 78,024 | <p>You can write this as a kind of 'added fraction', or fraction of continued numerator. Such fractions were used, for example, by Fibonacci in <em>Liber Aceri</em></p>
<p>Thus $1 \frac {a+}A \frac{b+}B \dots = 1 \frac{a+\frac{b+ \dots}B}A$ For example, one might regard decimals, as a series of added tenths, as $1m... |
487,123 | <p>How to evaluate the following limit?
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$</p>
<p>For this problem I have two methods. But I'd like to know if there are better methods.</p>
<p><strong>My solution 1:</strong></p>
<p>Using Stolz-Cesaro Theorem, we have
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}=\lim_... | Barry Cipra | 86,747 | <p>Let </p>
<p>$$b_n={1!+2!+\cdots(n-1)!\over n!}$$ </p>
<p>It suffices to show that $\lim_{n\rightarrow\infty}b_n=0$.</p>
<p>Note that $0\lt b_n\lt1$ for all $n\gt1$. (There are fewer than $n$ terms in the numerator, none larger than $(n-1)!$.) This implies</p>
<p>$$0\lt b_n={1\over n}\left({1!+\cdots+(n-2)!+(n... |
543,712 | <p>I am stuck on the following problem that says:</p>
<blockquote>
<p>Let <span class="math-container">$p,q$</span> be 2 complex numbers with <span class="math-container">$|p|<|q|$</span>. Let <span class="math-container">$$f(z)=\sum\{3p^n-5q^n\}z^n$$</span> Then the radius of convergence of <span class="math-conta... | Kirk Fogg | 83,162 | <p>Permutations are used when one is concerned with order. For example, if you wanted to choose how many ways are there to arrange five people in a line, the answer will be different. In your case, a person first in line is the same as a person fourth in line, i.e., order does not matter. </p>
<p>So combinations are u... |
543,712 | <p>I am stuck on the following problem that says:</p>
<blockquote>
<p>Let <span class="math-container">$p,q$</span> be 2 complex numbers with <span class="math-container">$|p|<|q|$</span>. Let <span class="math-container">$$f(z)=\sum\{3p^n-5q^n\}z^n$$</span> Then the radius of convergence of <span class="math-conta... | Ross Millikan | 1,827 | <p>It is a combination problem because you don't care the order the people are selected. A committee of ABCDE is the same as one of EBADC. Note that $C(13,5)=\frac {13!}{5!(13-5)!}$ so your $3!$ in the denominator should be $5!$</p>
|
892,114 | <p>i have three number
1 2 3 which will always be in this order {123}, i want to find out number of cases can be made,
like {1},{2},{23},{13},{12},{123}{3},{}. but each number has two states like "a" "b", i.e, each one will become different entity,like 2a,2b,3a,3b,1a,
with only exception i.e. 1 will have only one stat... | Vikram | 11,309 | <p>Same as others but with some colors</p>
<p>$(x-2)^2=\color{red}{(x-2)}\color{blue}{(x-2)}$</p>
<p>$\color{red}{(x-2)}\color{blue}{(x-2)}=\color{red}x\color{blue}{(x-2)}\color{red}{-2}\color{blue}{(x-2)}$</p>
<p>$\hspace{65 pt}=\underbrace{\color{red}x\times \color{blue}x}\hspace{5 pt}+\underbrace{\color{red}x \ti... |
331,962 | <p>We have an first order ODE : </p>
<p>Equation1 : $y' + y = x$ ?
We can view the left-hand side as an operator acting on $y$. </p>
<p>In that case $L=(d/dx + 1)$ </p>
<p>$L(y_1) = x$<br>
$L(y_2)=x$<br>
$L(y_1+y_2)=x$<br>
So, clearly $L(y_1+y_2) = x \neq L(y_1)+L(y_2) = 2x$ </p>
<p>So why is $y'+y=x$ ... | Ross Millikan | 1,827 | <p>The equation is considered a linear differentail equation because the operator $1+\frac {d}{dx}$ is linear. In this light $1+\frac {dy}{dx}y=f(x)$ is linear regardless of $x$. This allows us to say that if we find any solution to the homogeneous part, the equation without the $f(x)$, we can add it to a solution of... |
1,474,867 | <p>I was trying to prove </p>
<p>$$\left|\int_{0}^{a}{\frac{1-\cos{x}}{x^2}}dx-\frac{\pi}{2}\right|\leq \frac{3}{a}$$ or $\leq \frac{2}{a}$. My work: I would like to use Fubini's theorem to prove it. </p>
<p>I notice that $\frac{1}{x^2}=\int^{\infty}_{0}{ue^{-xu}}du$. </p>
<p>Then, I got $\int_{0}^{a}{\frac{1-\cos{x... | tired | 101,233 | <p>To circumvent possible divergence issues at the origin, </p>
<p>write $\int_0^afdx=\int_0^{\infty}fdx-\int_a^{\infty}fdx$</p>
<p>because the first integral is just $\frac{\pi}{2}$ as @Julian Rosen pointed out, we have to inspect</p>
<p>$$
J(a)=\int_a^{\infty}\frac{1-\cos(x)}{x^2}=2\int_a^{\infty}\frac{\sin^2(x/2)... |
1,108,832 | <p>Q: A team of $11$ is to be chosen out of $15$ cricketers of whom $5$ are bowlers and $2$ others are wicket keepers. In how many ways can this be done so that the team contains at least $4$ bowlers and at least $1$ wicket keeper?</p>
| barak manos | 131,263 | <p>The number of ways to choose exactly $4$ bowlers and exactly $1$ keeper:</p>
<p>$$\binom{5}{4}\cdot\binom{2}{1}\cdot\binom{15-5-2}{11-4-1}=280$$</p>
<p>The number of ways to choose exactly $5$ bowlers and exactly $1$ keeper:</p>
<p>$$\binom{5}{5}\cdot\binom{2}{1}\cdot\binom{15-5-2}{11-5-1}=112$$</p>
<p>The numbe... |
439,302 | <p>@HansEngler Left the following response to <a href="https://math.stackexchange.com/questions/260656/cant-argue-with-success-looking-for-bad-math-that-gets-away-with-it">this question</a> regarding "bad math" that works,</p>
<blockquote>
<p>Here's another classical freshman calculus example: </p>
<p><strong>F... | Michael Hardy | 11,667 | <p>I relied on something similar to this in a published paper. There's an identity that, in the concrete instance where the number of independent variables is $3$, says
\begin{align}
& \phantom{{}=} \frac{\partial^3}{\partial x_1\,\partial x_2\,\partial x_3} e^y \\[10pt]
& = e^y\left(\frac{\partial^3 y}{\parti... |
439,302 | <p>@HansEngler Left the following response to <a href="https://math.stackexchange.com/questions/260656/cant-argue-with-success-looking-for-bad-math-that-gets-away-with-it">this question</a> regarding "bad math" that works,</p>
<blockquote>
<p>Here's another classical freshman calculus example: </p>
<p><strong>F... | Fabian | 458,126 | <p>I beg to differ with the suggested differential to the x to the power of x. The result would be best expressed using ln x or the log to base e of x. Ie </p>
<p>d/dx(x ^x) = x ^x (ln x + 1)</p>
<p>When using log(believed to be in base 10), the result will be expressed as</p>
<p>d/dx(x ^x) = x^x ln10(log x + log e)... |
2,756,798 | <p>Consider the sequence space $l^2:=\{(x_n)_n\mid \sum^\infty_{n=0}x_n<\infty\}$ together with the norm
$$
||(x_n)_n||=(\sum^\infty_{n=0}|x_n|^2)^{1/2}
$$
How can I show that the triangle inequality holds for $||\cdot||$?</p>
| Clement C. | 75,808 | <p>Let $x,y\in\ell^2(\mathbb{N})$. Then we have
$$\begin{align}
\lVert x+y\rVert^2 &= \sum_{n=0}^\infty \lvert x_n+y_n\rvert^2 \leq \sum_{n=0}^\infty (\lvert x_n\rvert+\lvert y_n\rvert)^2 \tag{Triangle}\\
&= \lVert x\rVert^2+\lVert y\rVert^2 + 2\sum_{n=0}^\infty \lvert x_n\rvert\lvert y_n\rvert\\
&\leq \lVe... |
742,216 | <p>$a$, $b$, $c$, $d$ are rational numbers and all $> 0$.</p>
<p>$\max \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\} \geq \dfrac{a+c}{b+d}\geq \min
\left\{\dfrac{a}{b} , \dfrac{c}{d}\right\}$</p>
<p>Hope someone can help me with this one. How would you go on proving the validity? Thanks in advance.</p>
| lab bhattacharjee | 33,337 | <p>$$\frac{a+c}{b+d}-\frac ab=\frac{b(a+c)-a(b+d)}{(b+d)b}=\frac{bc-ad}{b(b+d)}$$</p>
<p>Similarly, $$\frac{a+c}{b+d}-\frac cd=\cdots=\frac{ad-bc}{(b+d)d}$$</p>
<p>Observe that the signs of the terms are opposite as $a,b,c,d>0$</p>
|
759,087 | <p>I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this? </p>
<p>Looking for something like
$$\sup_{a_{i,j} \in A}|a_{i,j}|$$
but the notation $a_{i,j} \in A$ in reality doesn't make much sense in my opi... | user1551 | 1,551 | <p>Although not really <strong>a</strong> notation but a combination of notations, the quantity in question is $\|\operatorname{vec}(A)\|_\infty$.</p>
|
1,754,931 | <p>If a sequence has a pattern where +2 is the pattern at the start, but 1 is added each time, like the sequence below, is there a formula to find the 125th number in this sequence? It would also need to work with patterns similar to this. For example if the pattern started as +4, and 5 was added each time.</p>
<block... | Jeevan Devaranjan | 220,567 | <p>Let $a_1 = 2$. From the way you defined the sequence you can see that $a_n - a_{n-1} = n$. We can use this to find
\begin{align}
a_n &= a_{n-1} + n\\
&= a_{n-2} + (n-1) + n\\
&= a_{n-3} + (n-2) + (n-1) + n\\
&\vdots \\
&= a_1 + 2 + \cdots + (n - 2) + (n-1) + n
\end{align}
which is just the sum of... |
670,813 | <p>Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence
$$ 0 \to K(X,Y) \to K(X) \to K(Y) \to 0.$$
Then I have to verify that $K(X) \simeq K(X,Y) \oplus K(Y)$. How can I do it? I thin... | Thomas Rot | 5,882 | <p>For the exactness at the left, you can use the long exact sequence in $K$ theory. Then every map involving (suspensions) of $i$ will split (why). Then the long exact sequence breaks down in short exact sequences.</p>
|
95,819 | <p>I think I have solved a problem in <em>Topology</em> by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight what is troubling me.</p>
<p>We prove by contradiction: Assume $X$ is not Hausdorff. Then there exist points... | Clive Newstead | 19,542 | <p>The definition of Hausdorff is that for all distinct pairs $x,y \in X$ there exist disjoint open $U \ni x$ and $V \ni y$. Hence the negative is that there <em>exist</em> $x,y \in X$ each of whose open neighbourhoods $U \ni x$, $V \ni y$ have nonempty intersection. You seem to have said that this is the case for <em>... |
95,819 | <p>I think I have solved a problem in <em>Topology</em> by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight what is troubling me.</p>
<p>We prove by contradiction: Assume $X$ is not Hausdorff. Then there exist points... | Zarrax | 3,035 | <p>You made the statement "Since $X$ is not Hausdorff, for any neighbourhoods $U$ and $V$ about $x,y$ respectively the intersection of $U$ and $V$ is not trivial." This actually has to only be true for a single $(x,y)$, not all $x$ and $y$.</p>
<p>It's probably best to not use proof by contradiction... If $\Delta$ is ... |
3,288,010 | <p>The following snippet is from Adamek, Rosicky:Algebra and local presentability,how algebraic are.</p>
<p>It is unclear to me the end of Example 5.1:</p>
<p>Since <span class="math-container">$e$</span> is the coequalizer of <span class="math-container">$\bar{u}_1,\bar{u}_2$</span> in <span class="math-container">$... | Julian Mejia | 452,658 | <p>From <span class="math-container">$x^2\equiv 1\pmod{2}$</span>. You got that <span class="math-container">$x$</span> has to be odd.
You have two options:</p>
<p><span class="math-container">$x,y$</span> odd. Then <span class="math-container">$x^2+2y^2\equiv 1+2=3\pmod{8}$</span>.</p>
<p><span class="math-containe... |
7,575 | <p>How could I display text that flashed red for a half second or so and then reverted to black? (Or was put in bold and reverted to normal, etc.)</p>
| ragfield | 15 | <p>Use Dynamic and Refresh:</p>
<pre><code>Dynamic[Refresh[
Style["text",
FontColor -> If[Mod[Round[AbsoluteTime[]], 2] == 0, Red, Black]],
UpdateInterval -> 1]]
</code></pre>
|
7,575 | <p>How could I display text that flashed red for a half second or so and then reverted to black? (Or was put in bold and reverted to normal, etc.)</p>
| Rojo | 109 | <pre><code>Style["TESTESTEST", FontColor -> Dynamic[If[Clock[] 2 - 1. > 0, Red, Black]]]
</code></pre>
<p>If you want it to flash only once</p>
<pre><code>Style["TESTESTEST", FontColor -> Dynamic[If[Clock[{-1, 1, 1}, 1, 1] > 0, Black, Red]]]
</code></pre>
|
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | André Nicolas | 256 | <p>One disadvantage of allowing "memory aids" is that there will then be little or no credit given for the appropriate formula. This makes it more difficult for a student with only a modest level of understanding to get a C, or even to pass.</p>
|
2,926,270 | <p>The base step is pretty obvious: <span class="math-container">$1 \geq \frac{2}{3}$</span>.</p>
<p>Then we assume that <span class="math-container">$P(k)$</span> is true for some <span class="math-container">$k \in \mathbb{Z}^{+}$</span> and try to prove <span class="math-container">$P(k+1)$</span>. So I have</p>
<... | trancelocation | 467,003 | <p>If you can drop "by induction" there is another way to show the inequality.</p>
<p>At least, it shows how "others" may invent such inequalities:</p>
<p><span class="math-container">$$\frac{2}{3}n\sqrt{n} \leq \sum_{i=1}^n \sqrt{i} \Longleftrightarrow \color{blue}{\frac{2}{3} \leq} \frac{1}{n\sqrt{n}}\sum_{i=1}^n ... |
2,926,270 | <p>The base step is pretty obvious: <span class="math-container">$1 \geq \frac{2}{3}$</span>.</p>
<p>Then we assume that <span class="math-container">$P(k)$</span> is true for some <span class="math-container">$k \in \mathbb{Z}^{+}$</span> and try to prove <span class="math-container">$P(k+1)$</span>. So I have</p>
<... | Peter Szilas | 408,605 | <p>Need to show: </p>
<p><span class="math-container">$(2/3)k√k+\sqrt{k+1} \ge$</span></p>
<p><span class="math-container">$ (2/3)(k+1)\sqrt{k+1}$</span>, or</p>
<p><span class="math-container">$\sqrt{k+1} \ge (2/3)[(k+1)^{3/2}-k^{3/2}]$</span>.</p>
<p><span class="math-container">$\displaystyle {\int_{k}^{k+1}} x^... |
541,644 | <p>I want to know why $p \leftrightarrow q$ is equivalent to $(p \wedge q) \vee (\neg p \wedge \neg q)$? Without using the truth table.</p>
<p>Thanks all</p>
| mathematics2x2life | 79,043 | <p>Just think about the statement. </p>
<p>$p \leftrightarrow q$: This says that $p$ occurs only if $q$ occurs and that $q$ happens only if $p$ does. Meaning, that either they both happen or nothing happens at all.</p>
<p>But look at my last sentence. They BOTH happen OR NEITHER happens. They both happen is $p \wedge... |
2,654,538 | <p>If $2\tan^2x - 5\sec x = 1$ has exactly $7$ distinct solutions for $x\in[0,\frac{n\pi}{2}]$, $n\in N$, then the greatest value of $n$ is?</p>
<p>My attempt:</p>
<p>Solving the above quadratic equation, we get $\cos x = \frac{1}{3}$</p>
<p>The general solution of the equation is given by $\cos x = 2n\pi \pm \cos^{... | user | 505,767 | <p>I agree with your answer, indeed note that</p>
<p>$$2\tan^2x - 5\sec x = 1\iff2(1-\cos^2x)-5\cos x=\cos^2x\\\iff3\cos^2x+5\cos x-2=0$$</p>
<p>and</p>
<p>$$3t^2+5t-2=0\implies t=\frac{-5\pm\sqrt{25+24}}{6}\implies t=\frac{-5+\sqrt{47}}{6}= \frac13$$</p>
<p>thus we have 2 solution on the interval $[0,2\pi]$ and no... |
2,299,768 | <p><em>It may have been already done, but I have found the answer nowhere...</em></p>
<hr>
<p><strong>Context.</strong></p>
<p>We already know by Stirling's formula that </p>
<p>$$n!\sim \sqrt{2\pi n}\left(\frac ne\right)^n.$$</p>
<p>We can deduce from this that</p>
<p>$$\log(n!)\sim n\log n.$$</p>
<p><strong>Th... | Angina Seng | 436,618 | <p>The natural approach is to consider
$$\log P_n=\sum_{k=2}^n\log\log k$$
and apply the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula" rel="nofollow noreferrer">Euler-Maclaurin summation method</a>.</p>
|
2,299,768 | <p><em>It may have been already done, but I have found the answer nowhere...</em></p>
<hr>
<p><strong>Context.</strong></p>
<p>We already know by Stirling's formula that </p>
<p>$$n!\sim \sqrt{2\pi n}\left(\frac ne\right)^n.$$</p>
<p>We can deduce from this that</p>
<p>$$\log(n!)\sim n\log n.$$</p>
<p><strong>Th... | Jack D'Aurizio | 44,121 | <p>The natural approach is to consider
$$ \log P_n = \sum_{k=2}^{n}\log\log k $$
then notice that $\log\log k$ is approximately constant on short intervals.<br>
By applying <a href="https://en.wikipedia.org/wiki/Abel%27s_summation_formula" rel="nofollow noreferrer">Abel's summation formula</a> we get:</p>
<p>$$\begin{... |
1,115,117 | <p>Consider the initial value problem
$$y'=ty(4-y)/(1+t)$$ $$y(0)=y_{0}>0$$</p>
<p>(a)Determine how the solution behaves as $t$ tends to infinity.</p>
<p>(b)If $y_{0}=2$,find the time $T$ at which the solution first reaches the value of 3.99</p>
<p>(c)Find the range of initial values for which the solution lies... | voldemort | 118,052 | <p>Half angle formula: We will manipulate the RHS-1.</p>
<p>RHS= $2\cos^2(\theta/2)-1+i2\cos(\theta/2)\sin(\theta/2)=\cos(\theta)+i\sin(\theta)=z$</p>
|
1,115,117 | <p>Consider the initial value problem
$$y'=ty(4-y)/(1+t)$$ $$y(0)=y_{0}>0$$</p>
<p>(a)Determine how the solution behaves as $t$ tends to infinity.</p>
<p>(b)If $y_{0}=2$,find the time $T$ at which the solution first reaches the value of 3.99</p>
<p>(c)Find the range of initial values for which the solution lies... | Autolatry | 25,097 | <p>Two relations may help</p>
<p>\begin{eqnarray}
\sin \theta \cos \phi &=& \frac{\sin(\theta + \phi)+\sin(\theta-\phi)}{2} \\
\cos^{2} \frac{\theta}{2} &=& \frac{1+\cos \theta}{2}
\end{eqnarray}</p>
|
121,924 | <p>Why is <span class="math-container">$\mathrm{Hom}_{\mathbb{Z}}\left(\prod_{n \geq 2}\mathbb{Z}_{n},\mathbb{Q}\right)$</span> nonzero?</p>
<p>Context: This is problem <span class="math-container">$2.25 (iii)$</span> of page <span class="math-container">$69$</span> Rotman's Introduction to Homological Algebra:</p>
<... | Mariano Suárez-Álvarez | 274 | <p>Let $G=\prod_{n\geq2}\mathbb Z_n$ and let $t(G)$ be the torsion subgroup, which is properly contained in $G$ (the element $(1,1,1,\dots)$ is not in $t(G)$, for example) Then $G/t(G)$ is a torsion-free abelian group, which therefore embeds into its localization $(G/t(G))\otimes_{\mathbb Z}\mathbb Q$, which is a non-z... |
1,130,142 | <p><img src="https://i.stack.imgur.com/NXr1V.png" alt="enter image description here"></p>
<p>This is how I solved this problem but I have some reservations regarding my answer.</p>
<p>1st house = x ; 2nd house = 3x ; 3rd house = [3x + x] - 2610</p>
<p>12(x) + 12(3x) + 12(4x - 2610) = 186,390</p>
<p>96x = 155,070</p... | coolcheetah | 195,418 | <p>Let the rent for each house be A, B and C for the House 1, 2 and 3 respectively.
Therefore,
3A = B
C = A + B - 2610</p>
<p>It is also given that only 6 months' rent is collected from the tenant of House 1.
Therefore,
186390 = 12[A + B - 2610] + 36A + 6A
186390 = 90A - 31320
A = 2419;</p>
<p>B=7257</p>
<p>Therefor... |
1,753,620 | <p>How do I find the matrix exponential $e^{tA}$ with </p>
<p>$$A = \left(\begin{matrix} 2 & 8 \\ 0 & 2\end{matrix}\right)$$</p>
<p>The eigenvalue is 2 with multiplicity 2, but it yields only 1 eigenvector {${1, 0}$}, so the matrix isn't diagonalizable. I'm confused what to do. One option is to convert it int... | Edward Pickman Derby | 305,007 | <p>a) Note: $|f_n(x)| = \Bigg| \frac{\sin(nx)}{1+n^3}\Bigg| \leq \Bigg|\frac{1} {1+n^3} \Bigg|$ for all $n \in \mathbb{N}$ and for all $x \in \mathbb{R}$. Observe also that $\sum_{n=1}^{\infty} \frac{1} {1+n^3}\leq \infty$. </p>
<p>So by the Weierstrass M-Test, $\sum_{n=1}^{\infty} \frac{\sin(nx)}{1+n^3}$ converges u... |
2,426,361 | <p>What would be the best mathematical tool/concept to measure how far a matrix is from being singular? Could it be the condition number?</p>
| user1551 | 1,551 | <p>Given a matrix norm induced by a vector norm of your choice, the distance of an invertible matrix $A$ to its nearest singular matrix, i.e. $\min\{\|A-B\|:\ B \text{ is singular}\}$, is known to be $\|A^{-1}\|^{-1}=\|A\|/\kappa(A)$.</p>
<p>Note that this is a concept different from (but closely related to) the cond... |
3,443,137 | <p>Find the radius of the circle tangent to <span class="math-container">$3$</span> other circles <span class="math-container">$O_1$</span>, <span class="math-container">$O_2$</span> and <span class="math-container">$O_3$</span> have radius of <span class="math-container">$a$</span>, <span class="math-container">$b$</s... | achille hui | 59,379 | <p>I recently need to compute something like this and I finally use
<a href="https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant" rel="nofollow noreferrer">Cayley-Menger determinant</a> to find the radius.</p>
<p>Let <span class="math-container">$ABCD$</span> be an tetrahedron. Let <span class="math-contai... |
728,495 | <p>I'm taking Discrete Math this semester. While I understand the mechanics of proofs, I find that I must refine my understanding of how to work them. To that end, I'm working through some extra problems on spring break. Please read over this proof I did from an exercise from the book. I apologize in advance for po... | MarnixKlooster ReinstateMonica | 11,994 | <p>Here is an alternative proof which goes back to the definitions and solves the problem using the rules of logic:
\begin{align}
& A \cap B \subseteq C \;\land\; A^c \cap B \subseteq C \\
\equiv & \qquad \text{"definition of $\;\subseteq\;$, twice"} \\
& \langle \forall x :: x \in A \cap B \Rightarrow x \i... |
3,501,879 | <p>I have been stuck at this problem for some time now. I'd really apprechiate your help. Thanks.</p>
<p><span class="math-container">$$2\sin^2(x)+6\cos^2(\frac x4)=5-2k$$</span></p>
| Claude Leibovici | 82,404 | <p><em>Too long for a comment but I cannot resist when I see an equation !</em></p>
<p>As @lhf answered, we are looking for the maximum of
<span class="math-container">$$f(x)=2 \sin ^2(x)+6 \cos ^2\left(\frac{x}{4}\right)$$</span> so for the zero's of
<span class="math-container">$$f'(x)=2 \sin (2 x)-\frac{3}{2} \sin ... |
3,862,408 | <p>This is the second example of 1. in <a href="http://www-personal.umich.edu/%7Ebhattb/teaching/mat679w17/lectures.pdf" rel="nofollow noreferrer">Ex. 2.0.3 </a> of Bhatt's notes in perfectoid space.</p>
<p>We define <span class="math-container">$R^{perf}:= \varprojlim ( \cdots R \xrightarrow{\phi} R)$</span> where <sp... | Arturo Magidin | 742 | <p>If <span class="math-container">$g\in N_G(H)$</span>, <span class="math-container">$h\in H$</span>, and <span class="math-container">$x\in X^H$</span>, then you know that <span class="math-container">$g^{-1}hg(x) = x$</span> (since <span class="math-container">$g^{-1}hg\in H$</span>). That means that <span class="ma... |
408,717 | <p>Let $n\in \mathbb N$ and $A_1,A_2,..,A_n$ be arbitrary sets. Now define $X=[x_{ij}]_{n \times n}$ where
$$x_{ij}=
\begin{cases}
1 , & \text{$A_i$$\subsetneq$}A_j \\
0 , & \text{otherwise} \\
\end{cases}.$$
How do you prove $X^n=0$?</p>
<p>Thanks in advance.</p>
| Julien | 38,053 | <p><strong>Hint:</strong> by iteration of the formula $(XY)_{ij}=\sum_{k=1}^nx_{ik}y_{kj}$ for the coefficients of a matrix product, we have
$$
(X^n)_{ij}=\sum_{1\leq i_1,\ldots,i_{n-1}\leq n}x_{ii_1}x_{i_1i_2}\cdots x_{i_{n-1}j}.
$$</p>
<blockquote class="spoiler">
<p>For a term of this sum to be nonzero, we need ... |
19,815 | <p>Problem:</p>
<blockquote>
<p>Prove that if gcd( a, b ) = 1, then gcd( a - b, a + b ) is either 1 or 2.</p>
</blockquote>
<p>From Bezout's Theorem, I see that am + bn = 1, and a, b are relative primes. However, I could not find a way to link this idea to a - b and a + b. I realized that in order to have gcd( a, b ) =... | Arturo Magidin | 742 | <p>The gcd of $x$ and $y$ divides any linear combination of $x$ and $y$. And any number that divides $r$ and $s$ divides the gcd of $r$ and $s$.</p>
<p>If you add $a+b$ and $a-b$, you get <code><blank></code>, so $\mathrm{gcd}(a+b,a-b)$ divides <code><blank></code>.</p>
<p>If you subtract $a-b$ from $a+b... |
1,757,260 | <p>A little box contains $40$ smarties: $16$ yellow, $14$ red and $10$ orange.</p>
<p>You draw $3$ smarties at random (without replacement) from the box.</p>
<p>What is the probability (in percentage) that you get $2$ smarties of one color and another smarties of a different color?</p>
<p>Round your answer to the ne... | M47145 | 188,658 | <p>Your options are "Exactly two yellow smarties, exactly two red smarties, or exactly two orange smarties."</p>
<p>If $P$ represents your final probability, you need to add up the following probabilities:</p>
<p>$P($exactly two yellows$) \, + \, P($exactly two reds$) \, + \,P($exactly two orange$)$</p>
<p>The proba... |
1,375,085 | <p>It is the first time I met such a question:</p>
<blockquote>
<p>Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?</p>
</blockquote>
<p>Intuitively I think $f(n)$ would gradually become larger as $n$ gets larger, but I find it hard to produce an argument.
Is there any trick to use for... | Jonathan Aronson | 83,164 | <p>Try taking the logs. Log is a monotonic transformation.</p>
|
2,699,621 | <p>To show $1 + \frac12 x - \frac18 x^2 < \sqrt{1+x}$ is it enough to tell that the taylor series expansion of $\sqrt{1+x}$ around $0$ has more positive terms?</p>
| user284331 | 284,331 | <p>Let $\varphi(x)=\sqrt{1+x}-1\dfrac{1}{2}x+\dfrac{1}{8}x^{2}$, $x\geq 0$, then for $x>0$,
\begin{align*}
\varphi(x)&=\varphi(x)-\varphi(0)\\
&=\varphi'(\xi)x\\
&=\left(\dfrac{1}{2}(1+\xi)^{-1/2}-\dfrac{1}{2}+\dfrac{1}{4}\xi\right)x.
\end{align*}
Now let $\eta(x)=\dfrac{1}{2}(1+x)^{-1/2}-\dfrac{1}{2}+\d... |
2,669,277 | <p>In his textbook <em>Calculus</em>, Spivak presents integration by parts as follows: </p>
<p>If $f'$ and $g'$ are continuous then
\begin{align*}
\int fg'&=fg-\int f'g\\
\int f(x)g'(x)\,dx&=f(x)g(x)-\int f'(x)g(x)\,dx\\
\int_a^b f(x)g'(x)\,dx&=f(x)g(x)\bigg|_a^b-\int_a^b f'(x)g(x)\,dx\\
\end{align*}
I und... | ryang | 21,813 | <blockquote>
<p>why isn't it enough to have <span class="math-container">$f'$</span> and <span class="math-container">$g'$</span> be integrable functions?
<span class="math-container">$$\int_a^b f(x)g'(x)\,dx =f(x)g(x)\bigg|_a^b-\int_a^b f'(x)g(x)\,dx$$</span></p>
</blockquote>
<p>Yes, <span class="math-container">$f′$... |
363,391 | <p>In <a href="https://math.stackexchange.com/q/2602271/682690">this MathSE question</a>,
classification of finite simple groups with Abelian Sylow 2-subgroups,
credit is rightly given to John Walter. But in the introduction to his paper, Walter explicitly states that "It seems to be a very difficult problem to sh... | Derek Holt | 35,840 | <p>The remark of Walter in his paper is referring specifically to the groups of Type (3) in his classification, that is, simple groups <span class="math-container">$S$</span> such that, for each involution <span class="math-container">$\tau \in S$</span>, we have <span class="math-container">$C_S(\tau) \cong \langle \t... |
825,318 | <p>Can someone please help me with these True and False questions? I've tried them myself, but I'm not very good at discrete math... Thank you in advance!</p>
<ol>
<li><p>Any set $A$ and $B$ with $B\subseteq A$ and $f: B \to A$ be $1$-$1$ and onto, then $B = A$</p>
<p>False?</p></li>
<li><p>Let $A$ and $B$ be nonempt... | heropup | 118,193 | <p>The variance of the <strong>sum</strong> of the two measurements $X_1$ and $X_2$ is equal to the sum of the variances of each individual measurement; i.e., $${\rm Var}[X_1 + X_2] = {\rm Var}[X_1] + {\rm Var}[X_2].$$ Thus, the variance of the <strong>mean</strong> of the measurements is $${\rm Var}[\bar X] = {\rm Va... |
50,227 | <p>The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in $(x,y,z)$ form, and want to map them onto the plane described by the normal of the triangle, such that I end up with three points in $(x,y)$ form.</p>
<p>My guess would be it'd assign an arbitrary up vector and then doing some... | Ralph Dratman | 120,442 | <p>Suppose the three vertices of the 3D triangle are given by three coordinate triples <span class="math-container">$a, b, c$</span>. For example, <span class="math-container">$b = \{x_b,y_b,z_b\}$</span>. In Mathematica 10, a 2D triangle congruent to this 3D triangle is</p>
<pre><code>SSSTriangle[Norm[b-a], Norm[c-b],... |
50,227 | <p>The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in $(x,y,z)$ form, and want to map them onto the plane described by the normal of the triangle, such that I end up with three points in $(x,y)$ form.</p>
<p>My guess would be it'd assign an arbitrary up vector and then doing some... | Radim Cernej | 161,576 | <p>If you only care about the shape of the triangle, not about its orientation, then it is reasonably simple -just calculate the distances between the points. I believe that is what Ralph Dratman is doing above.</p>
<p>Let us have the three 3D points</p>
<pre><code>x1, y1, z1
x2, y2, z2
x3, y3, z3
</code></pre>
<p>Let ... |
232,930 | <p>Let $f(n)$ denote the number of integer solutions of the equation $$3x^2+2xy+3y^2=n $$</p>
<p>How can one evaluate the limit $$\lim_{n\rightarrow\infty}\frac{f(1)+...f(n)}{n}$$</p>
<p>Thanks</p>
| EuYu | 9,246 | <p>The ellipse $Ax^2 + Bxy + Cy^2 = n$ with discriminant $\Delta=-B^2 + 4AC > 0$ has an area of $$\rm{Area} = \frac{2\pi n}{\sqrt{\Delta}}$$
In this case we have $A=3$, $B=2$ and $C=3$ for an area of $\frac{\pi n}{2\sqrt{2}}$. It is rather well known that the number of lattice points inside an ellipse is given by
$$... |
1,814,823 | <p>In this question , multiple concepts of graphical transformations are involved. I am facing problems in applying all of them in a single question.</p>
| Roman83 | 309,360 | <p>$$|a|=\begin{cases}a, a\ge 0\\ -a, a<0\end{cases}$$</p>
<p>If $x\ge0$ then $$y=\frac x{1+x}=\frac {1+x-1}{1+x}=1-\frac 1{1+x} -$$ hyperbola</p>
<p>If $x<0$ then $$y=\frac x{1-x}=-\frac {1-x-1}{1-x}=-1+\frac 1{1-x} -$$ hyperbola</p>
<p><a href="https://i.stack.imgur.com/WGXxn.png" rel="nofollow noreferrer"><... |
3,363,944 | <p>A group consisting of <span class="math-container">$3$</span> men and <span class="math-container">$6$</span> women attends a prizegiving ceremony. If <span class="math-container">$ 5$</span> prizes are awarded at random to members of the group, find the probability that exactly <span class="math-container">$3 $</sp... | drhab | 75,923 | <p>b) There are <span class="math-container">$5$</span> <em>independent</em> events in the form of prizes that are awarded that can succeed each (i.e. the prize is awarded to a woman) with (the same) probability <span class="math-container">$\frac69$</span>, or fail (i.e. the prize is not awarded to a woman). </p>
<p>... |
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