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 |
|---|---|---|---|---|
2,509,095 | <p>Is there a very simple test to check if a line <em>segment</em> in $3D$ space cuts a plane?
It is assumed we have the coordinates of the endpoints of the line segment, so $p_1,p_2$ and that we have the equation of the plane: $z = d$ (so for simplicity we're assuming it's a plane orthogonal to the z-axis).</p>
| amd | 265,466 | <p>Represent the plane by the equation $ax+by+cz+d=0$ and plug the coordinates of the end points of the line segment into the left-hand side. If the resulting values have opposite signs, then the segment intersects the plane. If you get zero for either endpoint, then that point of course lies <em>on</em> the plane. </... |
438,336 | <p>This a two part question:</p>
<p>$1$: If three cards are selected at random without replacement. What is the probability that all three are Kings? In a deck of $52$ cards.</p>
<p>$2$: Can you please explain to me in lay man terms what is the difference between with and without replacement.</p>
<p>Thanks guys!</p>... | André Nicolas | 6,312 | <p><strong>Without Replacement:</strong> You shuffle the deck thoroughly, take out <strong>three</strong> cards. For this particular problem, the question is "What is the probability these cards are all Kings."</p>
<p><strong>With Replacement:</strong> Shuffle the deck, pick out <strong>one</strong> card, record what ... |
918,788 | <p>How to do this integral</p>
<p>$$\int_{-\infty}^{\infty}{\rm e}^{-x^{2}}\cos\left(\,kx\,\right)\,{\rm d}x$$</p>
<p>for any $k > 0$ ?.</p>
<p>I tried to use gamma function, but sometimes the series doesn't converge.</p>
| Mateus Sampaio | 101,351 | <p>Write
$$\cos kx = \frac{e^{ikx}+e^{-ikx}}{2}.$$
Using the identity that for $a>0$ and $b\in\mathbb{R}$,
$$\int_{-\infty}^{\infty}e^{-ax^2+ibx}dx=\sqrt{\frac{\pi}{a}}e^{-\frac{b^2}{4a}},$$that can be obtained completing squares, we find for $a=1$ and $b=k$ that
$$\int_{-\infty}^{\infty}e^{-x^2}\cos kx dx=\int_{-\i... |
240,700 | <p>How can I prove that every maximal ideal of $B= \mathbb{Z} [(1+\sqrt{5})/2] $ is a principal?</p>
<p>I know if I show that B has division with remainder, that means it is a Euclidean domain. It follows that B is PID, and then every maximal ideal is principal ideal in PID. </p>
<p>However, I haven't been able to sh... | kahen | 1,269 | <p>There are many different ways of proving that $\mathbb R^\infty$ is not a Banach space under <em>any</em> norm. The cleanest is probably to note that is has a countable basis, $(e_i)$, where $e_i(k) = \delta_{ik}$ (q.v. <a href="https://en.wikipedia.org/wiki/Kronecker_delta" rel="noreferrer">Kronecker delta</a>), bu... |
3,511,118 | <p>I can't see how <span class="math-container">$$e^\left(2i\pi\right) = 1$$</span>
will result in:
<span class="math-container">$$e^\left(i\pi\right) +1 = 0$$</span>
thanks</p>
| José Carlos Santos | 446,262 | <p>My guess is that it's a typo and that they want you to compute <span class="math-container">$\displaystyle\frac{\mathrm d}{\mathrm dx}\int_3^xf(t)^2\,\mathrm dt$</span>.</p>
<p>Otherwise, you can say that <span class="math-container">$\displaystyle\int_3^xf(t)^2\,\mathrm dx=f(t)^2(x-3)$</span> and that therefore<sp... |
3,405,914 | <blockquote>
<p>It's known that <span class="math-container">$\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n = e^x$</span>.</p>
<p>Using the above statement, prove <span class="math-container">$\lim_{n \to \infty} \left(\frac{3n-2}{3n+1}\right)^{2n} = \frac{1}{e^2}$</span>.</p>
</blockquote>
<h2>My attempt... | lhf | 589 | <p>Write
<span class="math-container">$$
\frac{3n-2}{3n+1} = 1-\frac{3}{3n+1}
$$</span>
Recall that
<span class="math-container">$$
\left(1-\frac{3}{3n+1}\right)^{3n+1} \to e^{-3}
$$</span>
Then
<span class="math-container">$$
\begin{align}
\left(1-\frac{3}{3n+1}\right)^{3n+1}
= &\left(1-\frac{3}{3n+1}\right)^{3n} ... |
3,405,914 | <blockquote>
<p>It's known that <span class="math-container">$\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n = e^x$</span>.</p>
<p>Using the above statement, prove <span class="math-container">$\lim_{n \to \infty} \left(\frac{3n-2}{3n+1}\right)^{2n} = \frac{1}{e^2}$</span>.</p>
</blockquote>
<h2>My attempt... | Danny Pak-Keung Chan | 374,270 | <p>Observe that <span class="math-container">\begin{eqnarray*}
(\frac{3n-2}{3n+1})^{2n} & = & \left(\frac{3n+1-3}{3n+1}\right)^{2n}\\
& = & \left\{ \left(1+\frac{(-3)}{3n+1}\right)^{-1}\left(1+\frac{(-3)}{3n+1}\right)^{3n+1}\right\} ^{\frac{2}{3}}.
\end{eqnarray*}</span>
We assume the fact without proo... |
3,405,914 | <blockquote>
<p>It's known that <span class="math-container">$\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n = e^x$</span>.</p>
<p>Using the above statement, prove <span class="math-container">$\lim_{n \to \infty} \left(\frac{3n-2}{3n+1}\right)^{2n} = \frac{1}{e^2}$</span>.</p>
</blockquote>
<h2>My attempt... | user | 505,767 | <p>We can use that</p>
<p><span class="math-container">$$\left(\frac{3n-2}{3n+1}\right)^{2n}=\left(1-\frac{3}{3n+1}\right)^{2n}=\left[\left(1-\frac{3}{3n+1}\right)^{-\frac{3n+1}3}\right]^{-\frac{6n}{3n+1}}\to e^{-2}$$</span></p>
|
2,882,985 | <p>Let $f_n(x)=\frac{1}{n}\boldsymbol 1_{[0,n]}(x)$. This sequence is bounded in $L^1(\mathbb R)$ since $\|f_n\|_{L^1}=1$. But why is there no subsequence that convergent weakly ? I know that if such subsequence exist (still denote $f_n$), then $\|f_n\|_{L^1}=1$ Let denote $f$ it's limit. Then, since $f_n\to 0$ pointwi... | Lorenzo Quarisa | 402,046 | <p>As you noticed, if a subsequence $\left\{f_n\right\}$ of $\frac{1}{n}\mathbf{1}_{[0,n]}$ converges, it must converge to $0$. But for a sequence $\left\{f_n\right\}\subset L^1(\mathbb{R})$ we have $f_n\to 0$ weakly in $L^1(\mathbb{R})$ if and only if
$$\int_{-\infty}^{+\infty} f_n\varphi \to 0,\qquad \forall \varph... |
2,050,760 | <p>The question:</p>
<blockquote>
<p>Find a recurrence for the number of n length ternary strings that contain "00", "11", or "22".</p>
</blockquote>
<p>My answer:</p>
<p>$3(a_{n-2}) + 3(a_{n-1} - 1)$</p>
<p>Proof:</p>
<p>Cases:</p>
<p>______________00 (a_(n-2))</p>
<p>______________11 (a_(n-2))</p>
<p>__... | Robert Israel | 8,508 | <p>The point is that
$$ \pmatrix{1 & a\cr 0 & 1\cr} \pmatrix{1 & b\cr 0 & 1\cr} = \pmatrix{1 & a+b\cr 0 & 1\cr}$$</p>
<p>The mapping takes $\pmatrix{1 & c\cr 0 & 1\cr}$ to $c$.</p>
|
3,572,842 | <p><strong>Context:</strong> 1st year BSc Mathematics, Vectors and Mechanics module, constant circular motion.</p>
<p>This may be trivial, but can someone tell me what's wrong with the following reasoning?</p>
<p><span class="math-container">$$\underline{e_r}=\underline{i}\cos\theta+\underline{j}\sin\theta=(1,\theta)... | poetasis | 546,655 | <p>I belive the leg-sums are the same "set" as leg-differences except that the difference-set also contains <span class="math-container">$P^0=1$</span>.</p>
<p>What you have is the set of prime power products where
<span class="math-container">$\quad p_n\equiv\pm 1 \pmod 8$</span>.</p>
<p>The value <span clas... |
4,228,826 | <p>Consider the inequality
<span class="math-container">$$
1-\frac{x}{2}-\frac{x^2}{2} \le \sqrt{1-x} < 1-\frac{x}{2}
$$</span>
for <span class="math-container">$0 < x < 1$</span>. The upper bound can be read off the Taylor expansion for <span class="math-container">$\sqrt{1-x}$</span> around <span class="math... | orangeskid | 168,051 | <p>We have a function
<span class="math-container">$$f(x) =\sqrt{1-x} = 1 - \frac{x}{2} - \frac{x^2}{8} - \frac{x^3}{16} - \frac{5 x^4}{128 }- \frac{7 x^5}{256}- \cdots = \\=1 - a_1 x - a_2 x^2 -a_3 x^3 \cdots $$</span>
The series converges for every <span class="math-container">$x\in [0,1]$</span>, so we have
<span cl... |
193,001 | <p>This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$</p>
| Juan S | 2,219 | <p>This telescopes, using the fact that $\text{arctan}(u)-\text{arctan}(v) = \text{arctan}(\frac{u-v}{1+uv})$</p>
<p>Specifically take $u=n+1$ and $v=n$. Then $$\text{arctan}\left(\frac1{n^2+n+1}\right) = \text{arctan}(n+1)-\text{arctan}(n)$$</p>
<p>This gives $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right)... |
193,001 | <p>This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$</p>
| lab bhattacharjee | 33,337 | <p>By observation,
$\tan^{-1}\frac{1}{n^2+n+1}=\cot^{-1}(n^2+n+1)=\cot^{-1}\frac{n(n+1)+1}{n+1-n}$
$=\cot^{-1}(n)-\cot^{-1}(n+1)$</p>
<p>$\sum_{n=1}^{m}\tan^{-1}\left({\frac{1}{{n^2+n+1}}}\right)$
$=\frac{\pi}{4}-\cot^{-1}(m+1)$ using <a href="http://mathworld.wolfram.com/TelescopingSum.html" rel="noreferrer">this</a... |
1,600,307 | <p>Let $n$ be an integer greater than 1, $\alpha$ be a real number, and consider the quadratic form $Q_{\alpha}$ given by: </p>
<p>for every $(x_1, ... , x_n) \in R^n$, </p>
<p>$$Q_{\alpha}(x_1,...,x_n)= \sum_{i=1}^n x_i^2 - \alpha(\sum_{i=1}^n x_i)^2$$</p>
<p>Find all the eigenvalues of $Q_{\alpha}$ in terms of $\a... | zyx | 14,120 | <p>For constant $S = \sum x_i$, the sum of squares is minimum when all $x_i$ are equal, at which point the form equals the rank 1 quadratic form $n(S/n)^2 - \alpha S^2 = S^2 (\frac{1}{n} - \alpha)$. </p>
<p>Hence definiteness requires $\alpha \leq \frac{1}{n}$. Subtracting the rank 1 form from $Q$ leaves the varianc... |
1,366,023 | <p>Here's a problem I was just working on:</p>
<blockquote>
<p>Let $f$ have an essential singularity at $0$. Show that there is a sequence of points $z_n \to 0$ such that $z_n^n f(z_n)$ tends to infinity.</p>
</blockquote>
<p>I know already that there exists a sequence $z_n \to 0$ such that $f(z_n)$ tends to any c... | amirbd89 | 49,744 | <p>Yes. It is possible, but I think it will be easier to rebuild it from scratch than to extract such a sequence.</p>
<p>You can define $z_n$ in the following way:</p>
<p>For each $n$, there exists $z_n \in B(0,1/n)$ such that $|g_n(z_n)| > n$ (since, as you said yourself, there is a sequence that tends to zero an... |
255,374 | <p>Does there exist any noncomputable set $A$ and probabilistic Turing machine $M$ such that $\forall n\in A$ $M(n)$ halts and outputs $1$ with probability at least $2/3$, and $\forall n\in\mathbb{N}\setminus A$ $M(n)$ halts and outputs $0$ with probability at least $2/3$? What if you only require that $M(n)$ is correc... | Greg Kuperberg | 1,450 | <p>Every such decision problem is computable, even in the harder version of the problem, assuming that the transition probabilities are, say, fixed rational numbers. A deterministic algorithm can calculate the probability distribution on the set of states of this stochastic TM after each $t$ time steps, and then step... |
1,220,923 | <p>Find the value of the integral
$$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx.$$
I tried the substitution $x=t^5$ to obtain
$$\int_0^\infty \frac{5t^6}{1+t^{10}}dt.$$
Now we can factor the denominator to polynomials of degree two (because we can easily find all roots of polynomial occured in the denominator of the form... | Olivier Oloa | 118,798 | <p>One may recall the <a href="http://en.wikipedia.org/wiki/Beta_function">Euler beta function</a>
$$
B(a,b) =\int _0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},
$$ with a remarkable case
$$
\int _0^1 x^{a-1}(1-x)^{-a}dx=\Gamma(a)\Gamma(1-a)=\frac{\pi}{\sin(\pi a)}.
$$</p>
<p>Then, by the change of ... |
2,962,203 | <p>I got stuck at : <span class="math-container">$a^2/b^2 = 12+2 \sqrt 35$</span></p>
<p>I understand that <span class="math-container">$12$</span> is rational and now I need to prove that <span class="math-container">$\sqrt{35}$</span> is irrational.</p>
<p>so I defined <span class="math-container">$∀c,d∈R$</span> ... | lhf | 589 | <p>You're on the right track.</p>
<p>Consider the powers of <span class="math-container">$5$</span> that divide both sides of <span class="math-container">$c^2=35d^2$</span>. You have an even number for the RHS but an odd number for the LHS.</p>
<p>Indeed, if <span class="math-container">$5^m$</span> is the largest p... |
2,962,203 | <p>I got stuck at : <span class="math-container">$a^2/b^2 = 12+2 \sqrt 35$</span></p>
<p>I understand that <span class="math-container">$12$</span> is rational and now I need to prove that <span class="math-container">$\sqrt{35}$</span> is irrational.</p>
<p>so I defined <span class="math-container">$∀c,d∈R$</span> ... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ $</span> Subtract <span class="math-container">$12$</span> then square it again to show that it's a root of a polynomial with integer coef's that's monic (lead coef <span class="math-container">$= 1).\,$</span> Now apply the <a href="https://math.stackexchange.com... |
2,962,203 | <p>I got stuck at : <span class="math-container">$a^2/b^2 = 12+2 \sqrt 35$</span></p>
<p>I understand that <span class="math-container">$12$</span> is rational and now I need to prove that <span class="math-container">$\sqrt{35}$</span> is irrational.</p>
<p>so I defined <span class="math-container">$∀c,d∈R$</span> ... | Rishabh Khandelwal | 606,037 | <p>Assuming √5 + √7 is rational,
√5 + √7 = a ∕ b
Squaring both sides,
5 + 7 + 2√35 = a²/b²
⇒2√35 = a²/b² - 12
⇒2√35 = (a²-12b²)/b²
⇒√35 = (a²-12b²)/2b²</p>
<p>According to our assumption,
(a²-12b²)/2b² should have been rational
but actually it is irrational.
Hence our assumption is wrong.
Therefore, √5 + √7 is irratio... |
42,787 | <p>I am using <code>ListPlot</code> to display from 5 to 12 lines of busy data. The individual time series in my data are not easy to distinguish visually, as may be evident below, because the colors are not sufficiently different.</p>
<p><img src="https://i.stack.imgur.com/PiMMh.png" alt="enter image description here... | Sjoerd C. de Vries | 57 | <p>The key is to use one of the <em>indexed</em> color sets. You can find them in the color schemes palette or generate them using <code>ColorData[c]</code>. The j-th color in scheme i can be obtained using <code>ColorData[i][j]</code>.</p>
<p>To generate a set of colors for use in <code>PlotStyle</code> you can use</... |
779,095 | <p>Let
$$f(x,y)=\left\{ \begin{matrix} \frac{x^2y}{x^4+y^2} & (x,y)\neq(0,0) \\0 & (x,y)=(0,0)\end{matrix}\right.$$</p>
<p>It is easy to prove that the $f$ is not continuous at $(0,0)$ (doing the limit along the curve $y=x^2$).</p>
<p>I want to know whether it is possible to define the partial derivatives of... | Wlod AA | 490,755 | <p>Each finite pos (partially ordered set) has to have at least one minimal element.</p>
<p>Consider 3-set <span class="math-container">$\ X:=\{a\ b\ c\}.\ $</span> First, let <span class="math-container">$a$</span> be the <strong>unique</strong> minimal element. There are exactly three different partial orders in <spa... |
2,642,144 | <p>How would I prove or disprove the following statement?
$ \forall a \in \mathbb{Z} \forall b \in \mathbb{N}$ , if $a < b$ then $a^2 < b^2$</p>
| Cameron Buie | 28,900 | <p>To disprove it, one simply provides a counterexample, as the other answers have addressed.</p>
<p>If I were to attempt to prove it, here's how I might begin--and how one might discover that it is false, and find a key to generating counterexamples.</p>
<blockquote>
<p>Since <span class="math-container">$x<y$</spa... |
8,312 | <p>Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that one differs from the other by a multiple of a square, but I don't know which is which. On a more philosophical note: wh... | Qiaochu Yuan | 232 | <p>The two are the same if the roots of <span class="math-container">$f$</span> form an integral basis of the ring of integers of <span class="math-container">$\mathbb{Q}[x]/f(x)$</span> (e.g. if <span class="math-container">$f$</span> is a cyclotomic polynomial) because then, well, they're defined by the same determin... |
8,312 | <p>Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that one differs from the other by a multiple of a square, but I don't know which is which. On a more philosophical note: wh... | Community | -1 | <p>In response to Qiaochu,</p>
<p>$Disc(f)/Disc(\mathcal{O}_K)$ is the square of the index of $\mathbb{Z}[ \alpha _1, \ldots , \alpha _n ]$ in $\mathcal{O}_K$. The index itself is the determinant of the change of basis matrix from $(\alpha _1, \ldots , \alpha _n )$ to an integral basis for $\mathcal{O}_K$. This matrix... |
8,312 | <p>Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that one differs from the other by a multiple of a square, but I don't know which is which. On a more philosophical note: wh... | Jiangwei Xue | 9,430 | <p>I think there was some confusion about the splitting field and the field $\mathbb{Q}[x]/(f(x))$, which is isomorphic to the field generated by one root of $f(x)$. (We always assume that $f(x)$ is monic irreducible.)</p>
<p>Let $\alpha$ be a root of $f(x)$, and $L=\mathbb{Q}(\alpha)$ be the field generated by $\alph... |
1,716,656 | <p>I am having trouble solving this problem</p>
<blockquote>
<p>Julie bought a house with a 100,000 mortgage for 30 years being repaid with payments at the end of each month at an interest rate of 8% compounded monthly. If Julie pays an extra 100 each month, what is the outstanding balance at the end of 10 years im... | browngreen | 321,445 | <p>The probability of rolling a 1 and 3 is 1/18. Same for the probability of 2&4, 3&5, and 4&6.</p>
<p>So the overall probability of the dice being two apart equals 4/18 = 2/9.</p>
|
1,411,305 | <p>I have been trying to solve the following problem:</p>
<blockquote>
<p>What is the probability that among 3 random digits, there appear
exactly 2 different ones?</p>
</blockquote>
<p>The formula for no repititions is:</p>
<pre><code>(n*(n-1)...(n-r+1))/n^r
</code></pre>
<p>So, for the first digit there are 1... | robjohn | 13,854 | <p>There are $10$ choices for the digit that appears twice and then $9$ choices for the digit that appears once. There are $\binom{3}{1}=3$ ways to arrange the digits. This gives $10\cdot9\cdot3=270$ choices out of $1000$. That is, a $0.27$ probability.</p>
|
2,243,542 | <p>In a previous question here <a href="https://math.stackexchange.com/q/2240195/369757">Can we define the Cantor Set in this way?</a></p>
<p>we defined a family of sets $ \left\{ C_0,C_1,C_2,C_3,\dots \right\}$</p>
<p>We can call this set $S_1$ , where the values of these elements is</p>
<p>$C_0 = \left\{ 0.0 \ri... | Noah Schweber | 28,111 | <p>First, note that this isn't really a question about the Cantor set, just about the combinatorics of infinite sets in general.</p>
<p>E.g. we can recast it in terms of sets of natural numbers: let $A_n=\{1, 2, 3, ..., n\}$, and think about $\bigcup A_n$. Or, even easier, we could just take $B_n=\{0\}$ (that's not a ... |
3,430,812 | <p>Consider the set of integers, <span class="math-container">$\Bbb{Z}$</span>. Now consider the sequence of sets which we get as we divide each of the integers by <span class="math-container">$2, 3, 4, \ldots$</span>.</p>
<p>Obviously, as we increase the divisor, the elements of the resulting sets will get closer and... | Ali Ashja' | 437,913 | <p><b>1)</b> Let <span class="math-container">$S_n = \{ \frac{z}{n} \ | \ z \in \mathbb{Z} \}$</span> and <span class="math-container">$p_i$</span> be <span class="math-container">$i$</span>-th prime integer.</p>
<p><b>2)</b> It has no limit! Because since <span class="math-container">$(n,n+1)=1$</span> we have <span ... |
3,430,812 | <p>Consider the set of integers, <span class="math-container">$\Bbb{Z}$</span>. Now consider the sequence of sets which we get as we divide each of the integers by <span class="math-container">$2, 3, 4, \ldots$</span>.</p>
<p>Obviously, as we increase the divisor, the elements of the resulting sets will get closer and... | Paul Sinclair | 258,282 | <p>The three answers thus far assume by limits of the sets you mean the common value of the set-theoretic <span class="math-container">$\liminf$</span> and <span class="math-container">$\limsup$</span> (where convergence means they agree). This is a highly reasonable assumption, given that you did not specify a meaning... |
316,055 | <p>I have no idea of how to solve the following: </p>
<p>$$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{3x}$$</p>
<p>I know about the notable special limit $$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{x}=1$$, and I know that I have to do some algebraic manipulation and change what I have above to the notable ... | muzzlator | 60,855 | <p>A handy thing to know is that $e^x = 1 + x + O(x^2) \approx 1 + x $ for $x$ near $0$.</p>
<p>Using this, you will see immediately that the limit is $\frac{1}{3}$.</p>
<p>Being a bit more rigorous, you may notice that $$\lim_{x\rightarrow 0} \frac{e^x - 1}{3x} = \frac{1}{3} \lim_{x\rightarrow 0} \frac{e^x - e^0}{x}... |
316,055 | <p>I have no idea of how to solve the following: </p>
<p>$$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{3x}$$</p>
<p>I know about the notable special limit $$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{x}=1$$, and I know that I have to do some algebraic manipulation and change what I have above to the notable ... | Joe | 24,942 | <p>Using the fact that
$$\lim_{x\to 0} \frac{e^x-1}{x} = 1$$</p>
<p>We can simply rewrite your limit as </p>
<p>$$\frac13\lim_{x\to 0} \frac{e^x-1}{x}$$</p>
<p>Which is $\frac13 \cdot1 = \frac13$.</p>
<p><strong>Alternative</strong>: note that this is of form $\frac00$, and both functions are continuous in the nei... |
3,165,781 | <p>Ok, I am a bit confused.</p>
<p>Does first countability mean that for every element <span class="math-container">$x$</span> in the space, there is a collection of open sets and each of those open sets are countable containing <span class="math-container">$x$</span> and for every open neighborhood of <span class="ma... | Henno Brandsma | 4,280 | <p>It's more the second one, put formally: for each <span class="math-container">$x \in X$</span> there is an at most countable collection <span class="math-container">$U_n(x), n \in \mathbb{N}$</span> of open neighbourhoods of <span class="math-container">$x$</span> such that for each open set <span class="math-contai... |
202,034 | <p>Is finding the largest prime factor of a number computationally easier than factoring the number into powers of primes? </p>
| binn | 39,264 | <p>no, see</p>
<p><a href="https://mathoverflow.net/questions/104043/saying-things-rapidly-about-integer-factorisations">https://mathoverflow.net/questions/104043/saying-things-rapidly-about-integer-factorisations</a></p>
|
177,774 | <blockquote>
<p>Find the derivative of $x^x$ at $x=1$ by definition (i.e. using the limit of the incremental ratio).</p>
</blockquote>
<p>The only trick I know is $x^x = e^{x \ln x}$ but it doesn't work.</p>
| user758556 | 31,023 | <p>The trick you mentioned $\frac{d}{dx}[x^{x}] = \frac{d}{dx} e^{x \ln{x}}$ still works. :)</p>
<p>Apply the chain rule:
$e^{x \ln{x}}\frac{d}{dx}[x \ln{x}]$</p>
<p>And then the product rule:
$e^{x \ln{x}}(\ln{x}+x\frac{1}{x})$</p>
<p>Simplify:
$x^x(1+\ln{x})$</p>
<p>Edit: You wanted the value of the derivative ev... |
177,774 | <blockquote>
<p>Find the derivative of $x^x$ at $x=1$ by definition (i.e. using the limit of the incremental ratio).</p>
</blockquote>
<p>The only trick I know is $x^x = e^{x \ln x}$ but it doesn't work.</p>
| Vincenzo Tibullo | 6,266 | <p>Using the definition:
$$
\begin{align}
f'(1)&=\lim_{x\rightarrow1}\frac{x^x-1}{x-1}\\
&=\lim_{x\rightarrow1}\frac{e^{x\log{x}}-1}{x-1}\\
&=\lim_{y\rightarrow0}\frac{e^{(1+y)\log(1+y)}-1}{y}\\
&=\lim_{y\rightarrow0}\frac{e^{(1+y)\log(1+y)}-1}{(1+y)\log(1+y)}\frac{(1+y)\log(1+y)}{y}\\
&=\lim_{t\rig... |
347,385 | <p>Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$</p>
<p>and how to find the smallest constant $C$ which satisfies
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq C\int_0^1[f'(x)]^2\text{d}x$$</p>
| Georges Henry | 77,984 | <p>write $g=f'$ and observe that $f(0)=-\int_0^{1/2}(1-2t)g(t)dt$ from $\int_0^{1/2}f(x)dx=0.$ Therefore $$(\int_0^{1}f(x)dx)^2=(\int_0^1g(t)\min(t,1-t)dt)^2\leq \int_0^{1}g(t)^2dt\times \int_0^1(\min(t,1-t)^2dt$$ from Schwarz. </p>
|
2,153,743 | <p>I have the relation $R = \{(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)\}$ on the set $\{1,2,3,4\}$. I have to find [1] and [4] but I don't really understand what that means. </p>
<p>I get that $[a]$ is the set of all elements of $A$ related (by $R$) to $a$ so $[a]=\{x\in A : x$ $R$ $a\}$ right? But I don't get the signific... | Vlad Z | 287,055 | <p>Your R relationship tells you which two elements are "equal" to each other, for instance 1=1, 1=2, 2=1, and so on...</p>
<p>[1] is a set of all the elements that are equal to 1 (that is the first element of all the pairs in R that have the second element 1), so [1] = {1, 2}.
Also [4]={4}</p>
|
1,812,956 | <blockquote>
<p>Find the equation of the normal to the curve with equation $4x^2+xy^2-3y^3=56$ at the point $(-5,2)$.</p>
</blockquote>
<p>I know that the normal to a curve is $$-\frac{1}{f'(x)}$$
And when I differentiate the curve implicitly I get $$-\frac{8x-y^2}{6y^2}$$</p>
<p>Substituting that into the equation... | Annalise | 332,290 | <p>The derivative is $8x+y^2+2yx\dfrac{dy}{dx}-9y^2\dfrac{dy}{dx}=0 \Rightarrow \dfrac{dy}{dx}=\dfrac{-y^2-8x}{2yx-9y^2}$</p>
<p>Substituting the point $(-5,2)$ we have $\dfrac{dy}{dx}=(-4+40)/(-20-36)=-\frac{9}{14}$</p>
<p>Since we want the normal line, we take the inverse reciprocal of the slope: $\frac{14}{9}$.</p... |
3,387,138 | <p>First Definition.
A modular form of level n and dimension -k is an analytic function <span class="math-container">$F$</span> of <span class="math-container">$\omega_1 $</span> and <span class="math-container">$\omega_2$</span> satisfying the following properties :</p>
<ol>
<li><span class="math-container">$F(\om... | hunter | 108,129 | <p>The recipe to convert:</p>
<p>Given <span class="math-container">$f$</span>, set <span class="math-container">$F(\omega_1, \omega_2) = f(\omega_1/\omega_2)$</span>.</p>
<p>Given <span class="math-container">$F$</span>, set <span class="math-container">$f(\tau) = F(\tau, 1)$</span>. </p>
<p>(You'll have to check t... |
152,880 | <p>I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.</p>
<blockquote>
<p>I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of... | Martin Brandenburg | 1,650 | <p>The number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_p$ equals</p>
<p>$$\frac{1}{n} \cdot \sum_{d|n} p^d \mu\left(\frac{n}{d}\right)$$</p>
<p>where $\mu$ is the Möbius function. This follows rather easily from the Möbius inversion formula. You can find details <a href="http://people.virginia.... |
311,677 | <p>The problem from the book. </p>
<blockquote>
<p>$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ </p>
</blockquote>
<p>I understand the solution till this part. </p>
<p>$\ln \vert 6 - y \vert = x + C$ </p>
<p>The solution in the book is $6 - Ce^{-x}$ </p>
<p>My issue this that this solution, from the book, doesn't s... | ryang | 21,813 | <ol>
<li>Here's a rigorous solution:
<span class="math-container">$$\begin{align}
&\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y \\
\frac1{6-y}\dfrac{\mathrm{d}y}{\mathrm{d}x} &= 1 \ \ \ \ \ \ \ \ \text{or}\ \ \ \ \ \ \ \ y=\bbox[pink]{6} \\
\int\frac{\mathrm{d}y}{6-y} &= \int1{\mathrm{d}x} \\
-\ln \lvert 6-y\rv... |
3,148,076 | <p>CONTEXT: Challenge question set by uni lecturer for discrete mathematics course</p>
<p>Question: Prove the following statement is true using proof by contradiction: </p>
<p>For all positive integers <span class="math-container">$x$</span>, if <span class="math-container">$x$</span>, <span class="math-container">$x... | NazimJ | 533,809 | <p>Assume not, so <span class="math-container">$\exists x$</span> positive integer s.t. <span class="math-container">$x, x+2, x+4$</span> prime but <span class="math-container">$x\neq3$</span>.</p>
<p>But we can look at the numbers modulo 3, and realize that there are 3 possiblilities:</p>
<p>1) <span class="math-con... |
1,436,867 | <p>I don´t know an example wich $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ is not sufficient for the existence of a fixed point .
can anybody help me? please</p>
| Brian M. Scott | 12,042 | <p>HINT: Take your space to be $\Bbb R$, and try $A(x)=x-f(x)$, where $f$ is positive and increasing and satisfies </p>
<p>$$\frac{f(y)-f(x)}{y-x}<1$$</p>
<p>whenever $x<y$. You can get such an $f$ by tinkering with the arctangent function.</p>
|
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | Kaynex | 296,320 | <p>You are allowed to invert both sides, given you invert the entire side, like such:
$$2 + x = \frac1y \rightarrow \frac{1}{2 + x} = y$$
A common mistake is to invert only one term. Note that "inverting" happens because we can multiply both sides of the equation by the product of both sides. Take for example:
$$\frac{... |
2,038,189 | <p>(Note: I didn't learn how to solve equations the conventional way; instead I was just taught to "move numbers from side to side", inverting the sign or the operation accordingly. I am learning the conventional way though because I think it makes the process of solving equations clearer. That being said, I apologize ... | q.Then | 222,237 | <p>Well...
$$\begin{equation}\begin{aligned}5 &= \frac{2}{x} &&\text{From question}\\5x&=2 &&\text{Multiply by x on each side}\\ x&=\frac{2}{5} &&\text{Divide by 5 on each side}
\end{aligned}\end{equation}$$
Or
$$\begin{equation}\begin{aligned}\frac{2}{5} &= x &&\text{Fro... |
229,966 | <p>I want to put a title to the plotlegends I am using. I get a solution <a href="https://mathematica.stackexchange.com/questions/201353/title-for-plotlegends">here</a> which says to use <code>PlotLegends -> SwatchLegend[{0, 3.3, 6.7, 10, 13, 17, 20}, LegendLabel -> "mu"]</code>. But I also want to p... | Tim Laska | 61,809 | <p>You are probably looking for <a href="https://reference.wolfram.com/language/ref/PointLegend.html" rel="noreferrer"><code>PointLegend</code></a>, but you should provide more details so others can reproduce your results.</p>
<pre><code>mu = {0, 3.3, 6.7, 10, 13, 17, 20};
pl = PointLegend[mu, LegendLabel -> "m... |
3,630,421 | <p>If <span class="math-container">$x+y = 5$</span>, <span class="math-container">$xy = 1$</span> and <span class="math-container">$x > y$</span>, then <span class="math-container">$\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}= ?$</span> The answer key gives for the asnwer <span class="math-container">$\frac{\sqrt{... | user170231 | 170,231 | <p>If you're amenable to a multivariate approach, you can place the base of the pyramid in the <span class="math-container">$x,y$</span> plane so that the vertices of the base are located at <span class="math-container">$\left(\pm\frac\ell{\sqrt2},0,0\right)$</span> and <span class="math-container">$\left(0,\pm\frac\el... |
1,363,860 | <p>This problem is for my own exploration, not for class. The problem goes as follows:</p>
<blockquote>
<p>There are <span class="math-container">$n$</span> pairs of people with restraining orders against one another. However, all <span class="math-container">$2n$</span> people are friends with the other <span class="m... | Batominovski | 72,152 | <p>I shall prove that this task is not possible if there are uncountably many persons. Let $J$ be an (uncountable) index set, and for $j \in J$, $x_j$ and $y_j$ are two people with a restraining order against one another. Suppose that it is possible to put these people on $\mathbb{R}^d$ for some $d\in\mathbb{N}$ so t... |
19,495 | <p>I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynman path integral", which, as far as I understand, means "integrating" a functional (action... | Theo Johnson-Freyd | 78 | <p>There is a relatively large literature on path integrals. The best book that I know of is Johnson and Lapidus, <em>The Feynman Integral and Feynman's Operational Calculus</em>, 2000. See also the books and papers by Cecile DeWitt-Morette.</p>
|
4,487,654 | <blockquote>
<p>Demonstrate recursively that</p>
<p><span class="math-container">$$\prod_{k = 0}^\infty (1 + x^{2^k}) = \frac{1}{1-x}$$</span></p>
</blockquote>
<p><strong>My work:</strong></p>
<p>Define</p>
<p><span class="math-container">$$a_n = \prod_{k = 0}^n (1 + x^{2^k}) = (1 + x^{2^n})a_{n - 1} \iff a_n - (1 + x... | Joshua Tilley | 389,601 | <p>Let <span class="math-container">$x_{mn}$</span> be the <span class="math-container">$m^{th}$</span> root of <span class="math-container">$\sin\pi x=x/2n$</span> counting from zero, so <span class="math-container">$m$</span> ranges from <span class="math-container">$0,...,2n-1$</span> as noted. Then <span class="mat... |
576,379 | <p>I know how to show that $f(x)=x^2$ is uniformly continuous, but I am confused when it is $x^2 +x$</p>
| John Hughes | 114,036 | <p>Quick wise-guy answer: every continuous function on a compact set is uniformly continuous. Heine-Borel tells you that $[0, 1]$ is compact. So your function is unif. continuous on $[0, 1]$, hence on the subset $(0, 1)$. </p>
<p>Alternative: In general, doing an epsilon-delta proof for a differentiable function $f$ ... |
1,874,634 | <blockquote>
<p>Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional.</p>
</blockquote>
<p>My question is now, why has the group to be abelian? As far as I know, we want the representation $\rho(g)$ to be a $Hom_G(V,V)$, where $V$ is the representation... | dasWesen | 306,898 | <p>I had the same question, for maybe longer than a year, but because of a stupid mistake in understanding Schur's. Here it is, in case ( hoping ;) ) that someone else might do the same mistake:</p>
<p>Schur's lemma says something about >any< linear $\psi$ so that </p>
<p>(*) $ \;\;\;\; \psi \rho (g) = \rho (g) \p... |
1,491,484 | <p>Let $a,b,x \in Z^+$. Prove that $\operatorname{lcm}(ax,bx) = \operatorname{lcm}(a,b)\cdot x$.</p>
<p>Here are my thoughts: </p>
<p>Let $d = \operatorname{lcm}(ax, bx)$. By definition $ax|d$ and $bx|d$. Now it can be seen that $a|d$ and $b|d$. So, let e = lcm(a,b). e is merely the lcm(ax, bx) (which equals d) multi... | Oiler | 270,500 | <p>A nice fact that you can use is that for any $a,b\in \mathbb{Z}$, $$\text{lcm}(a,b) = \frac{ab}{\gcd(a,b)}.$$ So then you have that $$\text{lcm}(ax,bx)=\frac{x^2ab}{\gcd(ax,bx)}= \frac{x^2ab}{x \cdot \gcd(a,b)} = x \cdot \left( \frac{ab}{\gcd(a,b)} \right) = x \cdot \text{lcm}(a,b).$$</p>
|
207,778 | <p>I want to save expressions as well as their names in a file.</p>
<pre><code> func[i_] := i;
Do[func[i] >>> out.m,{i,1,3}];
</code></pre>
<p>The output is </p>
<pre><code> cat out.m
1
2
3
</code></pre>
<p>However the desired output is</p>
<pre><code> cat out.m
func[1] = 1;
func[2... | Fraccalo | 40,354 | <pre><code>list = ToString[#] <> "=" <> ToString[ReleaseHold@#] <> ";" &@
HoldForm@func[#] & /@ Range[3]
(# >>> out.m) & /@ list
</code></pre>
|
1,660,289 | <p>I want to find the line that passes through $(3,1,-2)$ and intersects at a right angle the line $x=-1+t, y=-2+t, z=-1+t$. </p>
<p>The line that passes through $(3,1,-2)$ is of the form $l(t)=(3,1,-2)+ \lambda u, \lambda \in \mathbb{R}$ where $u$ is a parallel vector to the line. </p>
<p>There will be a $\lambda \i... | Julián Aguirre | 4,791 | <p>This is an answer to your last demand:</p>
<pre><code>Do[
q = Prime[n] Times@@@Rest[Subsets[Table[Prime[k], {k, 2, n - 1}]]];
twin = Intersection[Select[q - 4, PrimeQ] + 2, Select[q - 2, PrimeQ]];
Print["n = ", n, " - ", twin, " - ", Length[twin]],
{n, 3, 20}]
</code></pre>
<p>The pattern breaks at $n=9$, sinc... |
363,166 | <p>For valuation rings I know examples which are Noetherian. </p>
<blockquote>
<p>I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? </p>
</blockquote>
<p>I am very eager to know. Thanks.</p>
| Hagen Knaf | 2,479 | <p>Consider the tower of domains</p>
<p>$$
K[x]\subset K[x^{1/2}]\subset \cdots \subset K[x^{1/2^k}]\subset\cdots
$$</p>
<p>where $K$ is a field and $x$ is transcendental over $K$. Every ring in the chain is a polynomial ring in one variable over $K$. Thus the localizations $O_k:=K[x^{1/2^k}]_{P_k}$, where $P_k$ is ... |
363,166 | <p>For valuation rings I know examples which are Noetherian. </p>
<blockquote>
<p>I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? </p>
</blockquote>
<p>I am very eager to know. Thanks.</p>
| mr.bigproblem | 63,621 | <p>Valuation rings that have dimension $\geq 2$ are not Noetherian. The dimension of a valuation ring is equal to the rank of its value group. </p>
<p>To get a simple example of a valuation ring that has dimension $2$, take $R = k[x,y]$, where $k$ is a field. Define the standard valuation $v: k(x,y) \rightarrow \mathb... |
363,166 | <p>For valuation rings I know examples which are Noetherian. </p>
<blockquote>
<p>I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? </p>
</blockquote>
<p>I am very eager to know. Thanks.</p>
| Uri Brezner | 91,827 | <p>In order to obtain a non Noetherian valuation ring, take $\mathbb{Z}^2$ with the lexicographic order.
Define the valuation $v:k(x,y)^* \to \mathbb{Z}^2$ as follows: for any $a \in k^*$ and $0 \le n,m \in \mathbb{Z}$ set $v(ax^ny^m)=(n,m)$.
For a polynomial $\: f=\sum f_i \in k[x,y]^*$ set $v(f)= \inf \{v(f_0),...,v(... |
70,946 | <p>I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of my dynamical system by letters and look at the sequence of letters formed by the trajectory of a point under the iter... | Gjergji Zaimi | 2,384 | <p>Intuitively this should happen for a large class of dynamical systems, but I don't know the right necessary and sufficient conditions. </p>
<p>A class of examples satisfying this is given by polyhedral billiards, where you assign a symbol to each face and correspond orbits to sequences in the obvious manner. It is ... |
1,942,364 | <p>How many squares exist in an $n \times n$ grid? There are obviously $n^2$ small squares, and $4$ squares of size $(n-1) \times (n-1)$.</p>
<p>How can I go about counting the number of squares of each size?</p>
| fleablood | 280,126 | <p>Let the vertices of the $n \times n$ grid by $\{(x,y)| 0\le x \le n; 0 \le y \le n\}$. </p>
<p>(Is that what an $n \times n$ grid is? A $1 \times 1$ has <strong>$4$</strong> vertices and an $n \times n$ grid has $(n+1)^2$ vertices? Or is a $1 \times 1$ grid a single point? I'm assuming the former.)</p>
<p>A $k... |
2,979,103 | <blockquote>
<p>Let <span class="math-container">$S$</span> be the region <span class="math-container">$\{z:0<|z|<\sqrt{2}, \ 0 < \text{arg}(z) <
\pi/4\}$</span>. Determine the image of <span class="math-container">$S$</span> under the transformation</p>
<p><span class="math-container">$$f(z)=\frac{... | TurlocTheRed | 397,318 | <p>I think this might help:</p>
<p><span class="math-container">$$f(z)=\frac{2+z^2}{1+z^2}=1+\frac{1}{1+z^2}$$</span></p>
<p>If <span class="math-container">$z=x+iy$</span>:</p>
<p><span class="math-container">$$\frac{1}{1+z^2}=\frac{1}{(x^2-y^2+1)+2ixy}=\frac{(x^2-y^2+1)-2ixy}{(x^2-y^2+1)^2+4x^2y^2}$$</span></p>
... |
2,817,507 | <p>I came across the following argument in my discrete maths textbook:</p>
<p>Since $n=O(n), 2n=O(n)$ etc., we have:
$$
S(n)=\sum_{k=1}^nkn=\sum_{k=1}^nO(n)=O(n^2)
$$</p>
<p>The accompanying question in the book is: <strong>What is wrong with the above argument?</strong></p>
<p><strong>Attempt:</strong> Performing t... | R.Jackson | 552,485 | <p>$k$ is a variable that runs from $0$ to $n$, therefore in the small cases your textbook showed you, it appears to be $O(n)$, however, the ending terms of the sum are $..., (n-2)n, (n-1)n, n^2$ which are clearly not $O(n)$. Therefore, the argument of the sum is not $O(n)$ but rather ${O(n)}^2$. This means that the ... |
2,132,936 | <p>How do you simplify this problem?
$$ \frac {\mathrm d}{\mathrm dx}\left[(3x+1)^3\sqrt{x}\right] $$
$$= \frac {(3x+1)^3}{2\sqrt {x}} + 9\sqrt{x} (3x+1)^2 $$
$$\frac{(3x+1)^2(21x+1)}{2\sqrt x} $$</p>
| Community | -1 | <p>Use the product rule of differentiation, that is $\mathrm {d}(uv) = u\mathrm {dv} + v\mathrm {du} $. We thus get, $$\frac {d}{dx}[(3x+1)^3\sqrt{x}] $$ $$= (3x+1)^3\frac {d}{dx}[\sqrt {x}] + \sqrt {x}\frac {d}{dx}[(3x+1)^3] $$ $$= \frac {(3x+1)^3}{2\sqrt {x}} + 3 (3x+1)^2 (3)\sqrt{x} $$ $$= (3x+1)^2 [\frac {3x+1}{2\s... |
1,455,348 | <p>Recently, having realized I did not properly internalize it (shame on me!), I went back to the definition of continuity in metric spaces and I found a proposition for which I was looking for a proof.</p>
<p>Here there is the result and my "proof" (in the hope to get rid of the quotation marks).</p>
<p><em>... | layman | 131,740 | <p>I'm not sure why you need an $\epsilon$-$\delta$ proof when we aren't trying to prove continuity. We are just trying to prove $\{x \mid \phi(x) \geq a \}$ is closed if $\phi$ is continuous.</p>
<p>It suffices to show the complement is open. But since $\{x \mid \phi(x) \geq a \} = \phi^{-1}( [a, \infty) )$, and $f... |
652,660 | <p>Show $\lnot(p\land q) \equiv \lnot p \lor \lnot q$</p>
<p>this is my solution . Check it please </p>
<p><img src="https://i.stack.imgur.com/1y7DB.jpg" alt="enter image description here"></p>
| Juan | 124,049 | <p>Your solution is right but needs a few adjustments.
In the second line of $(\Rightarrow)$ it should be "is" not "are" in both cases.
Showing that "$p$ is true exactly when $q$ is true" is not enough to guarantee that $p$ and $q$ are equivalent, do you know why ?</p>
|
661,026 | <p>prove or disprove this
$$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$</p>
<p>this problem is from when Find this limit
$$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=0}^{n}\binom{n}{k}^3}{\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}^3}=\dfrac{1}{8}?$$</p>
<p>first,follow I c... | Pifagor | 125,067 | <p>Find the derivative ov the left side of your first equation and use it's zeros to find the minima of that function. They will, hopefully, all be positive :)</p>
|
661,026 | <p>prove or disprove this
$$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$</p>
<p>this problem is from when Find this limit
$$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=0}^{n}\binom{n}{k}^3}{\displaystyle\sum_{k=0}^{n+1}\binom{n+1}{k}^3}=\dfrac{1}{8}?$$</p>
<p>first,follow I c... | Hagen von Eitzen | 39,174 | <p>For negative $x$, all summands are $\ge 0$ so $f(x)\ge 100$.
For $0\le x \le 4$, $f(x)\ge 100-x^3-x\ge 100-64-4>0$.
Note that
$$ f(x+1)=\frac12x^4+x^3-2x+\frac{197}2\ge (x^2-2)x+\frac{197}2>0$$
for $x>\sqrt 2$, so $f(x)>0$ for $x>\sqrt 2-1$. This covers all of $\mathbb R$.</p>
|
2,113 | <p>With respect to the stated reason for closure, I'd like to get some clarification as to what, precisely, "too localized" encompasses (at least a definition, or explanation, that is more specific and objective than the current "definition"). It just strikes me that some questions which might appear to some as being ... | Qiaochu Yuan | 232 | <blockquote>
<p>I think you're right - used for lack of "better" options. But if it doesn't satisfy "too localized", and there are no other options or available justifications for closing, then a question shouldn't be closed. It's unfair to the OP to justify closing by giving an arbitrary and uninformative reason for... |
3,043,780 | <p><a href="https://i.stack.imgur.com/h1M7D.png" rel="nofollow noreferrer">the image shows right-angled triangles in semi-circle</a></p>
<p>In Definite Integration, we know that area can be found by adding up the total area of each small divided parts.</p>
<p>So, base on the Definite Integration, we may say the area ... | David Holden | 79,543 | <p>you might like to try an alternative method that may be more transparent and robust. </p>
<p>supposing Q lies within ABC. a test probe P moving from A towards B should initially observe a decrease in the distance <span class="math-container">$PQ = r_{\lambda}$</span>. if we parameterize the position of the probe as... |
787,358 | <p>Consider the equation: ay'' +by'+cy=0</p>
<p>If the roots of the corresponding characteristic equation are real, show that a solution to the differential equation either is everywhere zero or else can take on the value zero at most once.</p>
<p>hmm I have no idea how to do this one, I think it might have to do som... | RRL | 148,510 | <p>The general solution is of the form $C_1\, \exp(r_1 x) + C_2 \, \exp(r_2x)$ where $r_1$ and $r_2$ are, in this case, the real roots of $ar^2 + br +c = 0$.</p>
<p>The roots are $ r_1 = -\alpha + \beta$ and $ r_2 = -\alpha - \beta $ where $\alpha = \frac{b}{2a}$ and $\beta = \sqrt{\alpha^2 -\frac{c}{a}}$.</p>
<p>So ... |
2,466,556 | <p><strong>Solution:</strong> </p>
<p>The vectors $\vec{AB}=(3,2,1)-(0,1,2)=3,1,-1$ and $\vec{AC}=(4,-1,0)-(0,1,2)=(4,-2,-2),$ are two direction vectors of the plane. A normal vector $\vec{n}$ to the plane is then given by $$\vec{n}=\vec{AB}\times\vec{AC}=(-4,2,-10).$$</p>
<p>Since $A$ is a point on the plane, we get... | Anurag A | 68,092 | <p>The idea of finding the equation of a plane is to have an equation that can help to locate any point $P(x,y,z)$ on the plane. In order to do that consider the vector $\vec{AP}=(x,y,z)-(0,1,2)$. This vector lies in the given plane. Now that we have a normal vector $\hat{n}$ (which by definition is orthogonal to any v... |
2,466,556 | <p><strong>Solution:</strong> </p>
<p>The vectors $\vec{AB}=(3,2,1)-(0,1,2)=3,1,-1$ and $\vec{AC}=(4,-1,0)-(0,1,2)=(4,-2,-2),$ are two direction vectors of the plane. A normal vector $\vec{n}$ to the plane is then given by $$\vec{n}=\vec{AB}\times\vec{AC}=(-4,2,-10).$$</p>
<p>Since $A$ is a point on the plane, we get... | zipirovich | 127,842 | <ol>
<li><p>The vector $\langle x-0,y-1,z-2\rangle$ is a vector connecting a generic point $P(x,y,z)$ lying in the desired plane with the point $A(0,1,2)$, which is also in this plane. Since both of them are in this plane, the vector connecting them $\overrightarrow{AP}=\langle x-0,y-1,z-2\rangle$ is in this plane.</p>... |
881,831 | <p>It is trivial that a group $G$ is abelian if and only if every subgroup of $G$ with two generators is abelian (i.e., any two elements commute).</p>
<p>If $G$ is a nilpotent group, every subgroup with two generators must be nilpotent. Is the reciprocal true? More precisely:</p>
<blockquote>
<p>Let $G$ be a group ... | Mikko Korhonen | 17,384 | <p>If we do not require that the class of a subgroup generated by two elements is bounded by some fixed constant, here is one example.</p>
<p>Consider the infinite direct sum $G = G_1 \oplus G_2 \oplus G_3 \oplus \cdots$ where $G_i$ is nilpotent of class $i$. </p>
<p>Then $G$ is not nilpotent, since it has nilpotent... |
1,903,235 | <p>According to Wikipedia, </p>
<blockquote>
<p>Hilbert space [...] extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions</p>
</blockquote>
<p>However, the article on Euclidean space states a... | Community | -1 | <p>Hilbert space: a vector space together with an inner product, which is a Banach space with respect to the norm induced by the inner product </p>
<p>Euclidean space: a subset of $\mathbb R^n$ for some whole number $n$</p>
<p>A non-euclidean Hilbert space: $\ell_2(\mathbb R)$, the space of square summable real seque... |
1,903,235 | <p>According to Wikipedia, </p>
<blockquote>
<p>Hilbert space [...] extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions</p>
</blockquote>
<p>However, the article on Euclidean space states a... | Luthier415Hz | 924,287 | <p>An Euclidean space is a normed linear space, that is, it has a norm and its elements are linear functions.</p>
<p>An Euclidean space has an inner product (scalar product):</p>
<p><span class="math-container">$$(x_\alpha,x_\beta)=0$$</span> for orthogonal elements</p>
<p>and</p>
<p><span class="math-container">$$(x_\... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Qiaochu Yuan | 232 | <p>One of the most important results you learn in a first course on abstract algebra is <a href="http://en.wikipedia.org/wiki/Burnside%27s_lemma">Burnside's lemma</a>, which has many applications in combinatorics and number theory. <a href="http://qchu.wordpress.com/2009/06/13/gila-i-group-actions-and-equivalence-relat... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Makoto Kato | 28,422 | <p>Diophantine equations of the type $m = ax^2 + bxy + c^2$ can be solved by using algebraic number theory on quadratic number fields.</p>
<p>As for connections between integral binary quadratic forms and quardratic number fields, here are some examples:</p>
<p><a href="https://math.stackexchange.com/questions/191830... |
217,291 | <p>I am trying to recreate the following image in latex (pgfplots), but in order to do so I need to figure out the mathematical expressions for the functions</p>
<p><img src="https://i.stack.imgur.com/jYGNP.png" alt="wavepacket"></p>
<p>So far I am sure that the gray line is $\sin x$, and that
the redline is some ver... | Eric O. Korman | 9 | <p>The product of all the residue classe modulo a prime $p$ is to -1 (mod p).</p>
|
1,511,078 | <p><strong>Show that the product of two upper (lower) triangular matrices is again upper (lower) triangular.</strong></p>
<p>I have problems in formulating proofs - although I am not 100% sure if this text requires one, as it uses the verb "show" instead of "prove". However, I have found on the internet the proof belo... | darkmoor | 91,474 | <p>I was interested on the same question, so allow me to exploit my logic, hopping of course to get comments for possible flaws. Suppose you have two lower triangular matrices $\mathbf{L}_1$ and $\mathbf{L}_2$ illustrated bellow</p>
<p>$$\mathbf{L}_1 =
\begin{bmatrix}
l_{11}^{(1)} & l_{12}^{(1)} & \dots ... |
3,935,811 | <p>While solving a bigger problem, I stumbeled upon a system of parametric equations
<span class="math-container">$$
\left\{
\begin{array}{ll}
\dfrac{x-a}{\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}} + \dfrac{x-c}{\sqrt{\left(x-c\right)^2+\left(y-d\right)^2}} = 0\\
\dfrac{y-b}{\sqrt{\left(x-a\right)^2+\left(y-b\right)... | Claude Leibovici | 82,404 | <p>This seems to be : find the extremum of function
<span class="math-container">$$f(x,y)=\sqrt{(a-x)^2+(b-y)^2}+\sqrt{(c-x)^2+(d-y)^2}$$</span> If you do not have any constraint, it could be any point along the line explained by @Quanto in his/her answer.</p>
|
762,651 | <p>I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this problem? It seems so general that I am not sure even how to formulate it well, let alone prove it. Intuitively, I imagine ... | evil999man | 102,285 | <p>Generally $$a>b \implies f(a)> f(b), \forall \text{ strictly increasing f(x)}$$</p>
<p>Your statement is just for $x^2$ which is true for $\Bbb R^+$. You have to state this fact while writing it down.</p>
<p>Your proof is also correct...like M.B. replied</p>
|
410,105 | <p>I have this recurrence relation:</p>
<p>$$
R(1)=1, RE(1)=0, EE(1)=0$$</p>
<p>$$a(n)=R(n) + RE(n)$$</p>
<p>$$R(n)=EE(n-1)+RE(n-1),$$$$ RE(n)=R(n-1),$$$$ EE(n)=RE(n-1)
$$</p>
<p>How do I get $a(15)$?
What kind of method do I use?</p>
| Boris Novikov | 62,565 | <p>Substitute $EE(n)$ from the last equation:</p>
<p>$$R(n)=RE(n-2)+RE(n-1).$$</p>
<p>Then substitute $RE(n)$:</p>
<p>$$R(n)=R(n-3)+RE(n-2) \ \ \ (1)$$</p>
<p>$$a(n)=R(n) + R(n-1) \ \ \ (2)$$</p>
<p>Now solve the recurrent equation ($1$) and then ($2$).</p>
|
4,402,262 | <p>A class of 24 pupils consists of 11 girls and 13 boys. To form the class committee, four of the pupils are chosen at random as "Chairperson", "Vice-Chairperson", "Treasurer", and "Secretary". Find the number of ways the committee can be formed if<br>
(i) the committee consists... | G Tony Jacobs | 92,129 | <p>As soon as you name the different positions in the committee, it's a permutation question, not a combination question. If Sally, Betty, Joe, and Mason are the four people on the committee, having Sally as Chair, Betty as Vicechair, Joe as Treasurer and Mason as Secretary is a different outcome from some other assign... |
4,402,262 | <p>A class of 24 pupils consists of 11 girls and 13 boys. To form the class committee, four of the pupils are chosen at random as "Chairperson", "Vice-Chairperson", "Treasurer", and "Secretary". Find the number of ways the committee can be formed if<br>
(i) the committee consists... | SlipEternal | 156,808 | <p>i) you are close. In each case, once you have selected the boys and girls for the committee, you need to permute their assigned roles within the committee. Because individual boys and girls are still people, they are all distinct. So, you should multiply your answer by <span class="math-container">$4!$</span></p>
<p... |
567,683 | <p>Let $F:\mathbb R^2\to \mathbb R^2$ be the force field with </p>
<p>$$F(x,y) = -\frac{(x,y)}{\sqrt{x^2 + y^2}}$$</p>
<p>the unit vector in the direction from $(x,y)$ to the origin. Calculate the work done against the force field in moving a particle from $(2a,0)$ to the origin along the top half of the circle $(x−a... | Felix Marin | 85,343 | <p>$\vec{F} = -\nabla r$ where $r \equiv \sqrt{x^{2} + y^{2}}$
$$
\color{#0000ff}{\large W}
= \int_{\left(2a,0\right)}^{\left(0,0\right)}\vec{F}\cdot{\rm d}\vec{r}
=
\int_{\left(2a,0\right)}^{\left(0,0\right)}\left(-\nabla r\right)\cdot{\rm d}\vec{r}
=
\sqrt{\left(2a\right)^{2} + 0^{2}} - \sqrt{0^{2} + 0^{2}}
=
\color{... |
1,408,036 | <p>Five points are drawn on the surface of an orange. Prove that it is possible to cut the orange in half in such a way that at least four of the points are on the same hemisphere. (Any points lying along the cut count as being on both hemispheres.)</p>
| johannesvalks | 155,865 | <p>Given a sphere with radius $1$.</p>
<p>Use the points</p>
<blockquote>
<p>$$
A_+(1,0,0), \quad A_-(-1,0,0), \quad B_+(0,1,0), \quad B_-(0,-1,0), \quad C(0,0,1)
$$</p>
</blockquote>
<p>And you <strong>cannot</strong> make such a cut.</p>
<p>As for any great circle, $(A_+,A_-)$ and $(B_+,B_-)$ are never on the s... |
2,007,176 | <p>Assume that $f:(0,∞)→ℝ$ is twice differentiable with $f(x)>0$ and $f'(x)<0$ for all $x \in (0, ∞)$. Prove that $f''(x)$ cannot always be negative. </p>
<p>I know that it intuitively makes sense because if $f$ is always decreasing with a positive derivative, then at some point, it must be concave up. However,... | Arthur | 15,500 | <p>Let $a = f'(1)$ and $b = f(1)$. Then we have $a < 0 < b$. Now assume $f''(x) \leq 0$ for all $x$. That means that $f'(x) \leq a$ for all $x > 1$, which again means that $f(x) \leq ax + b - a$ for all $x \geq 1$. Inserting some $x > \frac{a-b}{a}$ contradicts $f(x) > 0$.</p>
|
2,007,176 | <p>Assume that $f:(0,∞)→ℝ$ is twice differentiable with $f(x)>0$ and $f'(x)<0$ for all $x \in (0, ∞)$. Prove that $f''(x)$ cannot always be negative. </p>
<p>I know that it intuitively makes sense because if $f$ is always decreasing with a positive derivative, then at some point, it must be concave up. However,... | Stefano | 387,021 | <p>Another way to see this is the following. $f$ is a positive strictly decreasing function on $(0, +\infty)$. Therefore the limit of $f(x)$ for $x \to +\infty$ exists and it is finite (in particular non-negative).</p>
<p>Now, if $f''(x)$ was alays negative, then $f$ would be strictly concave on $(0,+\infty)$ and the ... |
3,105,664 | <p><span class="math-container">$$
I_n=\int_{0}^{1}\frac{(1-x)^n}{n!}e^x\,dx
$$</span></p>
<blockquote>
<p>Prove that
<span class="math-container">$$
I_n=\frac{1}{(n+1)!}+I_{n+1}
$$</span></p>
</blockquote>
<p>I tried integration by parts and still can't prove it, I appreciate any hint/answer. </p>
| Peter Foreman | 631,494 | <p>Following the proof of the recurrence relation by Zacky, we have
<span class="math-container">$$I_{n+1}=I_n-\frac{1}{(n+1)!}$$</span>
<span class="math-container">$$=I_{n-1}-\frac{1}{n!}-\frac{1}{(n+1)!}$$</span>
<span class="math-container">$$=I_0 - \sum_{k=1}^{n+1} \frac{1}{k!}$$</span>
We can calculate <span clas... |
108,331 | <p>I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the <a href="http://en.wikipedia.org/wiki/Prime_gap#Lower_bounds">prime number counting business</a>, somewhat unsettling. What is the reason for them?</p>
<p>Has it maybe to do with the series expansion of the logar... | Pedro | 23,350 | <p><strong>From A source book in Mathematics:</strong> Gauss (1971, at the age of fourteen) was the first one to suggest, <em>in a purely empiraical way</em>, the asymptotic formula $ \displaystyle \frac{x}{\log{x}}$ for $\phi(x)$.$^1$ Later on (1792-1793,1849) he suggested another formula $ \displaystyle \int_2^x \fra... |
4,415,559 | <p>I am trying to find the root of <span class="math-container">$f(x)=ln(x)-cos(x)$</span> by writing an algorithm for bisection and fixed-point iteration method. I am currently using python but whenever I'm running it using either of the two methods, it prints out "math domain error". I guess this is due to ... | emacs drives me nuts | 746,312 | <blockquote>
<p><span class="math-container">$g′(x)<1$</span> for some open interval</p>
</blockquote>
<p>What you need is the stronger <span class="math-container">$|g′(x)|<1$</span> for some interval containing the zero <span class="math-container">$x_0$</span> of <span class="math-container">$f$</span>.</p>
<p... |
1,880,090 | <p>The solution states that the ball of radius $\epsilon >0$ around a real number $x$ always contains the non-real number $x+i\epsilon/2$. </p>
<p>I don't understand the answer, for every number $x \in \mathbb{R}$ there is an open ball, right? For every $x \in \mathbb{R}$ there is an $r>0$ such that I can form a... | yago | 141,261 | <p>I think you're confusing open balls in $\mathbb{R}$ and $\mathbb{C}$. If you want to prove that $\mathbb{R}$ is open in $\mathbb{C}$, you have to prove that for any $x \in \mathbb{R}$, there exists $\epsilon > 0$ such that $B_x(\epsilon) \subseteq \mathbb{R}$, where $B_x(\epsilon)$ is a ball in $\mathbb{C}$, mean... |
206,723 | <p>Can any one explain why the probability that an integer is divisible by a prime $p$ (or any integer) is $1/p$?</p>
| Jeff P. | 18,440 | <p>See <a href="http://en.wikipedia.org/wiki/Coprime_integers#Probabilities" rel="nofollow">http://en.wikipedia.org/wiki/Coprime_integers#Probabilities</a>.</p>
|
3,933,296 | <p>What I already have,</p>
<ol>
<li>Palindrome in form XYZYX, where X can’t be 0.</li>
<li>Divisibility rule of 9: sum of digits is divisible by 9. So, we have 2(X+Y)+Z = 9M.</li>
<li>The first part is divisible by 9 if and only if X+Y is divisible by 9. So, we have 10 pairs out of 90. And each such pair the total sum... | Piquito | 219,998 | <p>The numbers are <span class="math-container">$abcba$</span> with <span class="math-container">$1\le a\le9$</span> and <span class="math-container">$0\le b,c\le9$</span> and <span class="math-container">$2(a+b)+c=9,18,27,36,45$</span>.</p>
<p><span class="math-container">$c=0$</span> gives <span class="math-container... |
1,024,068 | <p>I need to solve these two equations . </p>
<p>$ 2x + 4y + 3x^{2} + 4xy =0$</p>
<p>$ 4x + 8y + 2x^{2} + 4y^{3}$ = $0 $</p>
<p>I have added them , subtracted them . Nothing is helping here . Can anyone give hints ? Thanks</p>
| Claude Leibovici | 82,404 | <p>It is not so hard. Just eleiminate $y$ from the first equation; this gives $$y=\frac{-3 x^2-2 x}{4 (x+1)}$$ Plug it in the second equation and simplify as much as you can; you should arrive to $$(-27 x^3-22 x^2+28 x+24)\frac{x^3}{16 (x+1)^3}=0$$ So, beside the trivial solution $x=0$, remain the roots of the cubic $$... |
1,520,028 | <p>I'm struggling to figure out how to find a bound on my error for this problem:</p>
<p>Let T_{6}(x) be the Taylor polynomial of degree 6 based at a = 0 for the function f(x)=\cos(x). Suppose you approximate f(x) by T_{6}(x). If |x|\leq 1, find a bound on the error in your approximation by using the alternating serie... | abel | 9,252 | <p>the mcclaurin series $\cos x$ is $$\cos x = 1 - \frac1{2!} x^2 + \frac 1{4!}x^4 - \frac 1{6!}x^6 + \frac 1{8!} x^8 + \cdots $$ now, if you truncate this alternating series by $T_6,$ then the error committed is of the same sign as the next term and smaller in absolute value. therefore $$0 \le \cos x - T_6(x)\le \... |
1,017,989 | <p>Consider three disjoint circles not necessarily of same radii. How do you draw the smallest circle enclosing all these three circles? Where is its centre, and what is its radius? </p>
| achille hui | 59,379 | <p>If your goal is to "draw" the smallest circle and you don't really care
about the numerical values of its center and radii too much, You can construct
it in a geometrical manner.</p>
<p>Let's say the three circles are centered at $A$, $B$ and $C$ respectively.</p>
<p>There are two possibilities. </p>
<ol>
<li>The... |
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