qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,896,399 | <p>Im asked to solve find the derivative of: $$ \frac{\ln x}{e^x}$$
<strong>my attempt</strong>
$$D\frac{\ln x}{e^x} = \frac{\frac{1}{x}e^x + \ln (x) e^x}{e^x} = e^x \frac{\frac{1}{x}+\ln x}{e^{2x}} = \frac{\frac{1}{x}+\ln x}{e^x}$$</p>
<p>But this is apparently wrong and the correct answer is:
$$\frac{\frac{1}{x} - ... | Community | -1 | <p>Recall the formula:</p>
<p>$$\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}$$</p>
<p>You forgot the minus sign!</p>
|
1,896,399 | <p>Im asked to solve find the derivative of: $$ \frac{\ln x}{e^x}$$
<strong>my attempt</strong>
$$D\frac{\ln x}{e^x} = \frac{\frac{1}{x}e^x + \ln (x) e^x}{e^x} = e^x \frac{\frac{1}{x}+\ln x}{e^{2x}} = \frac{\frac{1}{x}+\ln x}{e^x}$$</p>
<p>But this is apparently wrong and the correct answer is:
$$\frac{\frac{1}{x} - ... | Dr. Sonnhard Graubner | 175,066 | <p>you will need the quotient rule and we get $$\frac{\frac{1}{x}e^{x}-\ln(x)e^{x}}{(e^x)^2}$$</p>
|
1,896,399 | <p>Im asked to solve find the derivative of: $$ \frac{\ln x}{e^x}$$
<strong>my attempt</strong>
$$D\frac{\ln x}{e^x} = \frac{\frac{1}{x}e^x + \ln (x) e^x}{e^x} = e^x \frac{\frac{1}{x}+\ln x}{e^{2x}} = \frac{\frac{1}{x}+\ln x}{e^x}$$</p>
<p>But this is apparently wrong and the correct answer is:
$$\frac{\frac{1}{x} - ... | Bobson Dugnutt | 259,085 | <p>Instead of carrying around the quotient rule in your brain, I'd stick with just the product rule. In this case</p>
<p>$$\left(\ln(x)e^{-x}\right)'=\frac{1}{x} e^{-x}+\ln(x)(-e^{-x})=e^{-x}\left(\frac{1}{x}-\ln(x)\right)$$</p>
|
4,062,300 | <p>Given 16 Trials equally distributed to 4 individuals ( 4 Trials per person).</p>
<p>How many combinations are there where these conditions are met:<br />
Condition A: 5 Total Successes<br />
Condition B: 2 or Fewer Successes per Individual.</p>
<p>I understand Condition A in exclusivity would be:<br />
<span class="... | spectralblue | 826,334 | <p>What tools are you allowed to use to solve this problem? There are multiple ways to solve this based on your background. If it's just basic algebra and trig identities, list all the values of <span class="math-container">$x$</span> for which <span class="math-container">$\sin(\frac{\pi}{9}x) = c$</span> where <span ... |
20,817 | <p>I am referencing to this question on mathunderflow:<br>
<a href="https://math.stackexchange.com/questions/223238/what-is-the-average-of-rolling-two-dice-and-only-taking-the-value-of-the-higher">What is the average of rolling two dice and only taking the value of the higher dice roll?</a></p>
<p>Now I tried:</p>
<p... | bill s | 1,783 | <p>It is possible to do the calculation analytically in Mathematica as well. Here's one way to define the desired distribution:</p>
<pre><code> maxDist =
TransformedDistribution[Max[x, y],
{x \[Distributed] DiscreteUniformDistribution[{1, 6}],
y \[Distributed] DiscreteUniformDistribution[{... |
3,807,293 | <p>I do apologise if this question has been asked before, but I am having trouble trying to get from <code>!A & B || A & !B</code> to <code>A XOR B</code>.</p>
<p>I've tried doing it in reverse by starting with <code>A XOR B</code> (<code>(A || B) & !(A & B)</code>), and I reached <code>(A || B) & (... | Integrand | 207,050 | <p>If you cut the semicircle in half vertically and translate it down, you get a <span class="math-container">$10\times 5$</span> rectangle. Your approach doesn't work because the 'spiky' part below the semicircle is outside it; therefore, you should add it to the semicircle, which means you add the rectangle below it ... |
2,717,786 | <p>Problem: I am reading through Rudin's text on functional analysis and came across the proof for the following theorem: Suppose $X, Y$ are Banach spaces, and $T$ is a bounded linear operator from $X$ to $Y$. Then $T$ is compact $\iff$ $T^*$ is compact. In his proof of the theorem (picture below)<a href="https://i.sta... | szw1710 | 130,298 | <p>This is an ordinary definition of the operator norm - a supremum of its values on a unit ball.</p>
|
551,686 | <p>The question asks to:</p>
<p>construct a DFA which accepts exactly $\frac{n(n-1)(n-2)}{6} + \frac{n(n-1)}{2}+1$ many members of $\{0, 1\}^n$ for every n.</p>
<p>I have no idea where to start to constructing the DFA, could you give some directions?
By the way, how many states should this DFA have?</p>
| Brian M. Scott | 12,042 | <p>HINT: Every $w\in\{0,1\}^n$ has $n$ symbols. There are $$\binom{n}3=\frac{n(n-1)(n-2)}6=\binom{n}3$$ ways to choose $3$ of those places to be $1$, leaving the rest $0$. $\dfrac{n(n-1)}2$ is also a binomial coefficient and can be interpreted similarly. Use these ideas to identify which $n$-symbol words to include in ... |
551,686 | <p>The question asks to:</p>
<p>construct a DFA which accepts exactly $\frac{n(n-1)(n-2)}{6} + \frac{n(n-1)}{2}+1$ many members of $\{0, 1\}^n$ for every n.</p>
<p>I have no idea where to start to constructing the DFA, could you give some directions?
By the way, how many states should this DFA have?</p>
| Rafe Kettler | 20,774 | <p>First you might want to ask yourself how many members there are in $\{0, 1\}^n$. Since there are 2 elements in $\{0, 1\}$, clearly there are $2^n$ in total for each $n$. Turns out this doesn't get us anywhere in the problem, but when you have to do combinatorial reasoning it's always useful to think about how big th... |
241,394 | <p>The convergence in the Central Limit Theorem is weak convergence, which is weaker than convergence in probability. I set it as an exercise to find an example that convergence in distribution does not imply convergence in probability:</p>
<blockquote>
<p>Let $(X_j)_{j\geq 1}$ be i.i.d. with $E[X_1]=0$ and $\sigma_... | J.Mike | 383,241 | <p>Assume that $\sigma=1$. Suppose $S_n/\sqrt{n}\overset{P}\to Z$, then $|S_n/\sqrt{n}-S_{2n}/\sqrt{2n}|\overset{P}\to 0$$\quad(*)$.</p>
<p>Next we want to deduce a contradiction with equation (*). Notice that</p>
<p>$$(S_{2n}-S_n)/[\sqrt{n}\cdot(\sqrt{2}-1)]=(\sqrt{2}\cdot S_{2n}/\sqrt{2n}-S_n/\sqrt{n})/(\sqrt{2}-1)... |
938,791 | <p>I want to check, whether $$\sum\limits_{n=0}^{\infty }{\frac{n!}{(a+1)(a+2)...(a+n)}}$$</p>
<p>converges or diverges.</p>
<p>$a$ is a constant number</p>
<p>Ratio test</p>
<p>$$\begin{align}
& \frac{a_{n}}{a_{n-1}}=\frac{n!}{(a+1)(a+2)...(a+(n-1))(a+n)}\cdot \frac{(a+1)(a+2)...(a+(n-1))}{(n-1)!}=\frac{n}{a... | Jack D'Aurizio | 44,121 | <p>Since:
$$(a+1)(a+2)\cdot\ldots\cdot(a+n)=\frac{\Gamma(a+n+1)}{\Gamma(a+1)}$$
assuming $\Re(a)>1$ we can write the original series as:
$$ S = \sum_{n\geq 1}\frac{\Gamma(n+1)\Gamma(a+1)}{\Gamma(a+n+1)}=\sum_{n\geq 1}n\cdot B(n,a+1)=\sum_{n\geq 1}n\int_{0}^{1}x^{n-1}(1-x)^a\,dx
\tag{1}$$
but $\sum_{n\geq 1}n x^{n-1}... |
901,549 | <p>I am struggling with part c of this question. Could someone please tell me how to approach and solve this type of questions?</p>
<p><img src="https://i.stack.imgur.com/SIpJ8.png" alt="enter image description here"></p>
| Tom | 103,715 | <p><b>Hint:</b> You know that $X$ takes on the values $-1, 0,$ and $2$; you know $Y$ takes on the values of $-2, 1,$ and $2$. So, the values of $X+Y$ are ...? After you answer this, suppose you wanted find $P(X+Y=-2)$, the only way this can happen is if $Y=-2$ and $X=0$, so $P(X+Y=-2) = P(X=0,Y=-2) = .2$</p>
<p>Can yo... |
21,310 | <p>I would like to do a For loop with a jump 2*Pi instead of 1.</p>
<pre><code>For[j = 0, j <= nmax, (incr = 2 Pi), ...]
</code></pre>
<p>Actually what I want to do is <code>j = n 2 Pi</code>, where <code>n</code> takes values from 0 to <code>n-1</code>, so the iteration stops at <code>nmax = (n - 1) 2 Pi</code>.<... | bill s | 1,783 | <p>All the iterators in Mathematica work pretty much the same way.</p>
<pre><code>(i, a, b, c}
</code></pre>
<p>means the variable is $i$, the starting value is $a$, the ending value is $b$ and the interval is $c$. So in your case, </p>
<pre><code> {j, 0, nmax, 2 Pi}
</code></pre>
<p>would do it. Honestly though, y... |
2,155,201 | <p>By a real ODE I mean an ordinary differential equation with only real coefficients and the resulting function is a function of a real argument. If such a solution exists, can you give an example?</p>
<p>Edit: To add to this, is it still possible if the initial conditions must also be real?</p>
| KCd | 619 | <p>There is always a basis of real solutions for a real <em>linear</em> ODE with constant coefficients having degree greater than 1, thus ruling out the weird examples like a polynomial equation having no derivatives at all. See Theorem 4.1 of <a href="http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/diffeqdim.pd... |
3,009,919 | <p>Give <span class="math-container">$n$</span>-th derivatives, can we find a function ?<br>
For example, if <span class="math-container">$f^{(i)}(x)=i,i=1,2,\cdots,n.$</span> , can we construct a function such that it satisfies the condition.<br>
It just occurs in my mind. I didn't know if it has a solution.</p>
| Jimmy R. | 128,037 | <p>Assume that <span class="math-container">$f$</span> is such that it satisfies this condition. In particular, this implies that <span class="math-container">$f^{(1)}(x)=1$</span> for all <span class="math-container">$x$</span> and hence <span class="math-container">$f^{(2)}(x)=0$</span> for all <span class="math-cont... |
3,009,919 | <p>Give <span class="math-container">$n$</span>-th derivatives, can we find a function ?<br>
For example, if <span class="math-container">$f^{(i)}(x)=i,i=1,2,\cdots,n.$</span> , can we construct a function such that it satisfies the condition.<br>
It just occurs in my mind. I didn't know if it has a solution.</p>
| Q.Reindeerson | 584,654 | <p>If I understand correctly you are asking if we can always construct a function such that its first derivative is equal to the constant function <span class="math-container">$x \mapsto 1$</span> , it second derivative equal to <span class="math-container">$ x \mapsto 2 $</span>, etc.</p>
<p>Well this is impossible b... |
4,183,945 | <p>I have the following question:
Let <span class="math-container">$A$</span> be a symmetric positive definite real matrix. Then we are asked to show that there exits two matrices, <span class="math-container">$C$</span> and <span class="math-container">$D$</span> where <span class="math-container">$C$</span> is lower ... | Naren | 696,689 | <p>Substitute <span class="math-container">$x=\sin y$</span>, we have <span class="math-container">$$\int_0^{\frac{\pi}{2}} \operatorname{Li}_2\left(-\sin^2 y\right)dy\stackrel{\rm dilog.}{=}\sum_{k=1}^{\infty}\frac{(-1)^k}{k^2}\int_0^{\frac{\pi}{2}}\sin^{2k}dy$$</span> Since the last integral is Well-know <a href="htt... |
1,942,342 | <p>This is not a homework, just something I am curious about. There are two independent variables in this problem: $x$, and $n$, the number of $x$'s. I am wondering if it is possible to express Graham's number with "reasonable" values for $x$ and $n$, say, $x \le$ a trillion, and $n\le 6$? If no, how much loser do the ... | Simply Beautiful Art | 272,831 | <p>The answer is "no", where we define "reasonable" as something with</p>
<p>$$x,n<10^{10}$$</p>
<p>which is more than you asked for.</p>
<p>Let's start with a conversation on <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation" rel="nofollow noreferrer">Knuth's up-arrow notation</a>. It's rather ... |
2,899,422 | <p>I need some hints for solving $yy''-(y')^2=xy^2$. </p>
<p>I noticed that the left hand side is close to $(yy')'$:</p>
<p>$yy''-(y')^2=xy^2\ \Leftrightarrow\ yy''+(y')^2-2(y')^2=xy^2\ \Leftrightarrow\ (yy')'-2(y')^2=xy^2$.</p>
<p>But I don't know how to continue expressing the terms as derivatives of some function... | MRobinson | 587,882 | <p>Try integrating by parts; </p>
<p>$\int x^2 e^{x^2}dx = \int uv' - \int u'v$</p>
<p>$u = x, v' = xe^{x^2}$</p>
<p>$u'=1, v = \frac{e^{x^2}}{2}$</p>
<p>So your integral becomes </p>
<p>$= \frac{xe^{x^2}}{2} - \int\frac{e^{x^2}}{2}dx$</p>
<p>Now you need to use properties of the Error function to get</p>
<p>$... |
2,423,846 | <blockquote>
<p>a) Parametrize the line $L$ through $P = (2, 1, 2)$ that intersects the
line </p>
<p>$x = 1 + t$, $y = 1 − t$, $z = 2t$ </p>
<p>perpendicularly.</p>
<p>b) Parametrize the $z$ axis.</p>
<p>c) What is the distance from this line $L$ to the $z$-axis?</p>
</blockquote>
<p><strong>My... | Robert Z | 299,698 | <p>A parametrization of the $z$ axis is $x=0$, $y=0$, $z=s$ with $s\in\mathbb{R}$. </p>
<p>Therefore the squared distance between $P$ along the line $L$, and $Q$ along the $z$-axis, is the $2$-variable function
$$f(t,s)=(2 + (1/6)t-0)^2+(1 + (5/6)t-0)^2+( 2 + (1/3)t-s)^2.$$
Note that the minimum distance will be att... |
3,403,664 | <p>I have been asked to integrate <span class="math-container">$\int Ae^{at} dt$</span> and I know that the integral is <span class="math-container">$\frac Aae^{at} + c$</span>.</p>
<p>Although this is simple it goes against my understanding that <span class="math-container">$\int a^{n} da = \frac a{n+1}^{n+1} + c$</... | Community | -1 | <p>All the indefinite integrals involved in the discussion:</p>
<p><span class="math-container">$$
\int Ae^{at}\, dt=\frac{A}{a}e^{at}+C,\quad (a\neq 0);\tag{1}
$$</span></p>
<p><span class="math-container">$$
\int a^n\, da=\frac{1}{n+1}a^{n+1}+C,\quad (n\neq -1).\tag{2}
$$</span></p>
<p>In (1), <span class="math-co... |
1,770,858 | <p>I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?</p>
| Sri-Amirthan Theivendran | 302,692 | <p>Use the <a href="https://en.wikipedia.org/wiki/Cauchy_condensation_test" rel="nofollow">Cauchy condensation test</a>. Note that
$$2^na_{2^n}=\frac{2^n}{(\ln 2^n)^2}=\frac{2^n}{n^2(\ln 2)^2}$$so that $\sum 2^n a_{2^n}$ does not converge (terms don't go to zero) and hence the original series diverges.</p>
|
142,180 | <p>First take a question as an example:</p>
<p>Let $f:L\to M$ be an irreducible morphism in $\mathrm{mod}-A$ and $X$ be a right $A$-module.
Show that $\mathrm{Ext}_A(X,f):\mathrm{Ext}_A(X,L)\to\mathrm{Ext}_A(X,M)$ is a monomorphism,if $\mathrm{Hom}_A(M,X)=0$. </p>
<p>What is the special meaning of $\mathrm{Ext}$ func... | Aaron | 28,841 | <p>In answer to your additional question about Example VI.3.11 in [ASS]:</p>
<p>The way too see how the module manifest after tilting is probably not so apparent from the text. The main thing is to note that now the elements in your new algebra $B=End_A(T)$ is given by homomorphisms. The good thing of computation wi... |
3,460,598 | <p>Suppose that <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> have the same distribution. Then, <span class="math-container">$$\phi_{t X_1} (1) = \phi_{X_1}(t) = \phi_{X_2}(t) = \phi_{t X_2} (1).$$</span></p>
<p>Why do <span class="math-container">$tX_1$</span> and <span class... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> have the same distribution. so do <span class="math-container">$f(X_1)$</span> and <span class="math-container">$f(X_2)$</span> for any Borel measurable function <span class="math-container">$f$</span> on the real line. In par... |
3,229,556 | <p><strong>Problem:</strong></p>
<p>Let <span class="math-container">$T$</span> be a tree that has <span class="math-container">$i\ge1$</span> branch nodes, all of which have the same degree<sup>[1]</sup> <span class="math-container">$d$</span>. Show that the number of leaves <span class="math-container">$(l)$</span> ... | Lee Mosher | 26,501 | <p>There is a slick proof using the Euler characteristic formula: if a tree has finitely many nodes then the number of nodes minus the number of edges equals <span class="math-container">$1$</span>.</p>
<p>You've used <span class="math-container">$i$</span> for the number of nodes of degree <span class="math-container... |
3,229,556 | <p><strong>Problem:</strong></p>
<p>Let <span class="math-container">$T$</span> be a tree that has <span class="math-container">$i\ge1$</span> branch nodes, all of which have the same degree<sup>[1]</sup> <span class="math-container">$d$</span>. Show that the number of leaves <span class="math-container">$(l)$</span> ... | Shubham Johri | 551,962 | <p>I'd say the simplest way is to recall that the sum of degrees of all vertices is twice the number of edges, which is <span class="math-container">$n-1=i+l-1$</span> for the tree in question. Thus,<span class="math-container">$$2(i+l-1)=l+di\\\implies l=2+(d-2)i$$</span></p>
|
22,720 | <p>Let the heat kernel on $(0,\infty)\times \mathbb R^n$ be given by </p>
<p>$\Psi(x,t) = (4\pi t )^{-\frac{n}{2}} e^{ -\dfrac{|x|^2}{4t} }$</p>
<p>for $t>0$, otherwise $0$ except at the origin of space-time.</p>
<p>It is clear that the heat kernel is smooth everywhere except at the origin. Let $\delta > 0$. H... | Ross Millikan | 1,827 | <p>Even with the exponent on $t$ being negaitve I don't think it is true. At the origin, you just need to show that the $\exp(\frac{-1}{t})$ term dominates every polynomial. But for $t$ a little bit positive, $\frac {d}{dt}t^{-n}\exp(\frac{-1}{t})\approx t^{-n-2}\exp(\frac{-1}{t})$ which grows unboundedly with $n$. ... |
535,102 | <p>Does anyone know an irreducible polynomial $f \in K[x,y]$ such that the quotient $K[x,y]/(f)$ is not a UFD? Is it known when this quotient is a UFD?</p>
<p>Thanks.</p>
| Lubin | 17,760 | <p>Yes, I think that the standard example is to take $f(x,y)=x^3-y^2$. Then when you look at it this is the same as the set of polynomials in $K[t]$ with no degree-one term. That is, things that look like $c_0+\sum_{i>1}c_it^i$, the sum being finite, of course. You prove this by mapping $K[x,y]$ to $K[t]$ by sending... |
535,102 | <p>Does anyone know an irreducible polynomial $f \in K[x,y]$ such that the quotient $K[x,y]/(f)$ is not a UFD? Is it known when this quotient is a UFD?</p>
<p>Thanks.</p>
| Theon Alexander | 165,460 | <p>The thing is: a UFD <span class="math-container">$A$</span> is always normal, i.e. every element in its field of quotients is integral over <span class="math-container">$A$</span> if and only if it belongs to <span class="math-container">$A$</span>. The above example, <span class="math-container">$f(x,y)=x^3-y^2$</s... |
191,618 | <p>Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then
$$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$
I can prove these identies under the condition $\mathbb{E}T<+\infty$. But what about this one?</p>
| yangmengqh | 57,639 | <p>Well, I provide a proof using Burkholder-Davis-Gundy(BDG) inequality.(<a href="http://en.wikipedia.org/wiki/Quadratic_variation#Martingales" rel="nofollow noreferrer">http://en.wikipedia.org/wiki/Quadratic_variation#Martingales</a>)</p>
<p>Let $M=\left\{M_t=W_{t\wedge T};0\le t<+\infty\right\}$, then $M$ is a ma... |
2,832,802 | <blockquote>
<p>Let $F = \mathbb{Q}$ and $K$ be the splitting field of $x^4 - 7$ over $F$. Determine $Gal(K/F)$ and find all the intermediate subfields of $K/F$.</p>
</blockquote>
<p>We have $x^4 - 7 = (x-\sqrt[4]{7})(x + \sqrt[4]{7})(x - i\sqrt[4]{7})(x + i\sqrt[4]{7})$ and so, $K = \mathbb{Q}(\sqrt[4]{7},i)$. Now,... | Angina Seng | 436,618 | <p>There's an element of order $4$ (shall we call it $\alpha$?)
fixing $i$ and taking $\sqrt[4]7$ to $i\sqrt[4]7$ (and so cycling
the $i^k\sqrt[4]7$).</p>
<p>There's also complex conjugation (shall we call that $\beta$?)
which fixes each of $\pm\sqrt[4]7$ and swaps $\pm i\sqrt[4]7$.</p>
|
1,501,988 | <p>How do I solve the following equation?
$$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$</p>
<p>I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way to solve this?</p>
| Thomas | 26,188 | <p>If $x<-2$ then $x + 2 < 0$. So $\lvert x \rvert - \lvert 2 + x \rvert = - x + (2+x)$. So you really just want to solve
$$
-x + (2+x) = x.
$$
Here the solution is $x = 2$. This contradicts $x < -2$. So there are not solutions for $x<-2$.</p>
<p>Now if $-2\leq x < 0$, then $\lvert x \rvert = -x$ and $... |
3,769,681 | <p>What's the name for sequences sitting between geometric and arithmetic?</p>
<p>E.g. let <span class="math-container">$x_{n+1}=ax_n+b$</span></p>
<p>I can't find a general name for these.</p>
<p>These sequences may also be a <a href="https://en.wikipedia.org/wiki/Lucas_sequence" rel="nofollow noreferrer">Lucas Sequen... | ama | 24,827 | <p>In a french context they are called "suites arithmético-géométriques".</p>
|
1,822,314 | <p>I think of two numbers, $x$ and $y$.
When I add them together I get $5$ and when I find the difference I get $13$. What numbers did I think of?</p>
<p>I need to know how to write this down in simultaneous equation form</p>
<p>many thanks</p>
| Olivier Oloa | 118,798 | <p>Let's call it $x$,$y$. </p>
<p>Sum: $x+y$</p>
<p>Difference: $x-y$</p>
<p>Can you take it from here?</p>
|
1,709,798 | <p>If $f$ is a bounded function on an interval E, and E has measure $0$, Is $f$ measurable? What is the value of $\int_E f$?</p>
<p>I have the question above in Royden Analysis 4e. </p>
<p>Intuition suggests that f is measurable because E is , and that $\int_E f = 0$ because the lebesgue integral ignores intervals o... | Henricus V. | 239,207 | <p>I'm going to remove the restriction that $E$ is an interval, because otherwise it will be trivial.</p>
<p>Since $f$ is bounded on $E$, there exists $y \geq 0$ such that $|f\restriction_E(x)| < y$, so $|f\chi_E| < y \chi_E$, where $\chi_E$ is the characteristic function of $E$. Then
$$ \left| \int_E f\,d\lambd... |
1,709,798 | <p>If $f$ is a bounded function on an interval E, and E has measure $0$, Is $f$ measurable? What is the value of $\int_E f$?</p>
<p>I have the question above in Royden Analysis 4e. </p>
<p>Intuition suggests that f is measurable because E is , and that $\int_E f = 0$ because the lebesgue integral ignores intervals o... | user217285 | 217,285 | <p>The original exercise from Royden is as follows:</p>
<blockquote>
<p>Let $E$ have measure zero. Show that if $f$ is a bounded function on $E$, then $f$ is measurable and $\int_E f = 0.$</p>
</blockquote>
<p>Let $F$ be a measurable subset of $\mathbb R$. Then $f^{-1}(F)$ is a subset of $E$, and since subsets of m... |
3,576,675 | <p>Normally, I would solve the integral and then evaluate the limit, but the integral seems particularly difficult, and I'm wondering if there's some sort of trick involved. The solution said to just use L'Hospital and the Fundamental Theorem of Calculus, but I don't know where to begin. I've tried moving the <span c... | Bernard | 202,857 | <p><strong>Hint</strong>:
Set <span class="math-container">$f(x)=\int_0^x\frac{t+t^2}{1+\sin t}\,\mathrm dt$</span>.
You have to find the limit of <span class="math-container">$\;\frac{f(x)}{x^2}$</span> as <span class="math-container">$x\to 0$</span>. Apply L'Hospital's rule, after you've calculated <span class="math... |
3,576,675 | <p>Normally, I would solve the integral and then evaluate the limit, but the integral seems particularly difficult, and I'm wondering if there's some sort of trick involved. The solution said to just use L'Hospital and the Fundamental Theorem of Calculus, but I don't know where to begin. I've tried moving the <span c... | Paramanand Singh | 72,031 | <p>You can use the substitution <span class="math-container">$z=t^2$</span> so that <span class="math-container">$dt=\frac{dz}{2\sqrt{z}}$</span> and the desired limit is <span class="math-container">$$\frac{1}{2}\lim_{x\to 0}\frac{1}{x^2}\int_{0}^{x^2}\frac{1+\sqrt{z}}{1+\sin\sqrt{z}}\,dz$$</span> The integrand tends ... |
2,154,351 | <p>I've recently (yesterday) started to studying complex analysis on my own. So i came across this question at the end of the first chapter
" Let $G$ be an open set in $ \mathbb{C} $ . Is $\{ z : z \in G \lor \bar z \in G \}$ open?</p>
<p>My attempt :</p>
<p>By the definition a set is open if $ A \subseteq \mathbb{C... | skyking | 265,767 | <p>What you then do is that if $\bar z\in G$ that $\mathcal D(\bar z,\delta)\subset G$. Now consider $w \in \mathcal D(z, \delta)$ instead, that is $|w-z|<\delta$ this means that $|\bar w-\bar z|<\delta$ so $\bar w\in\mathcal D(\bar z, \delta)$. So since $\bar w\in G$ we have that $w$ is in the greater set and th... |
2,438,594 | <p>Let $\left({z_n}\right)$ be a sequence of distinct points in $D\left(0,1\right)$=$\{{z \in \mathbb{C} :\vert{z}\vert \lt{1}} \}$ with $\lim_{n\to\mathbb{\infty}} {z_n}=0$. Consider the following statements P and Q.</p>
<p>P: There exists a unique analytic function $f$ on $D\left(0,1\right)$ such that $f\left({z_n}\... | quasi | 400,434 | <p>Using your equations . . .</p>
<ol>
<li>$\;a + b = 4$</li>
<li>$\;ab = k$</li>
<li>$\;a + c = (-k)$</li>
<li>$\;ac= (-4)$</li>
</ol>
<p>Adding equations $(1)$ and $(4)$, we get
$$a + b + ac=0\tag{5}$$
Adding equations $(2)$ and $(3)$, we get
$$a + c + ab=0\tag{6}$$
Subtracting equation $(6)$ from equation $(5)$, w... |
2,805,136 | <p>sadly, i don't know how to solve this question, so i cannot give my input on what i did, only my thoughts.</p>
<p>let $φ$ be a formula with a free variable $x$. for another variable, let's say $y$: $φ(y)=sub(φ,x,y)$ is $φ$ where each occurrence of $x$ is replaced with $y$.</p>
<p>1)prove that if $y$ is not in $φ$,... | Mauro ALLEGRANZA | 108,274 | <p><em>Hint</em></p>
<blockquote>
<p>Let <span class="math-container">$φ$</span> be a formula with a free variable <span class="math-container">$x$</span>. Let's say : <span class="math-container">$φ(y)=sub(φ,x,y)$</span> is <span class="math-container">$φ$</span> where each occurrence of <span class="math-container">$... |
2,556,993 | <p>I've come across two limits which are reported to take on a range of values rather than a unique one.</p>
<p>They are:</p>
<p>$$\lim \limits_{x \to \infty} \space\frac{1+\cos x}{1-\sin x} = 0 \space to\space \infty$$</p>
<p>$$\lim \limits_{x \to \infty} \space \frac{2+2x+\sin(2x)}{(2x+\sin(2x))e^{\sin x}} = \fra... | ayeayemaung | 249,040 | <p>The limits <strong>definitely do not</strong> exist for these expressions. Don't let any resources on the web convince you otherwise. However, that said, the <a href="https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior" rel="nofollow noreferrer">limits inferior and superior</a> may exist. So, a more accu... |
2,973,278 | <p>In the book of <em>The theory of Functions</em> by Titchmarsh, at page 83, in the proof of the fact that analytic functions can be expressed as power series, it is stated that</p>
<p><span class="math-container">$$\frac{1}{w-z} = \frac{1}{w-a} + \frac{z-a}{(w-a)^2} + \dots \frac{(z-a)^n}{(w-a)^{n+1}} \dots ,$$</s... | Mark Bennet | 2,906 | <p>Suppose we have <span class="math-container">$S=\frac 1p+\frac {q}{p^2}+\frac {q^2}{p^3}+\dots$</span> and we multiply by <span class="math-container">$\frac pq$</span> then <span class="math-container">$$\frac pq S=\frac 1q+\frac 1p+\frac q{p^2}+\dots=\frac 1q+S$$</span> so that <span class="math-container">$$S=\fr... |
574,536 | <p>I tried this problem by counting the number of 5-digit numbers and then dividing it by 2 just like we do for three digits. Please help. Thank you.</p>
| P Vanchinathan | 28,915 | <p>Let $S$ be the set in question. Let $E$ be its complement in the set of all 5 digit numbers. $S $ and $E$ partition the collection of all 5 digit numbers and $S$ does not contain 99999. Adding 1 to elements of S gives elements of E. This function is from S to E and is in fact a bijection and so both sets contain eq... |
2,288,402 | <p>I tried simplifying this first:</p>
<p>$$\frac{\pi}{3}\le \arg(-z+3+i)\le \frac{5\pi}{3} \Leftrightarrow \\
\frac{\pi}{3}\le \arg(-(z+(3-i)))\le \frac{5\pi}{3} \Leftrightarrow \\
\frac{\pi}{3}\le \arg(z+(3-i))+\pi \le \frac{5\pi}{3} \Leftrightarrow \\
???$$</p>
<p>My book does the following:</p>
<p>$$\frac{\pi}{3... | DonAntonio | 31,254 | <p>The answer is already given, in a rather nice and clear way, in the (first so far) comment below your question: observe that</p>
<p>$$\arg(-z+3+i)=\arg(-(z-(3+i))=e^{-\pi i}\arg(z-(3+i))$$</p>
<p>and you have thus "added" $\;-\pi\;$ to the argument...that's all!</p>
|
4,336,337 | <p>I've come across this problem:
<span class="math-container">$$
S=\sum_{i=0}^{\infty}2^i \\
S= 1+ 2\sum_{i=0}^{\infty}2^i \\
S=1+2S \\
S=-1
$$</span>
I know that you cannot rearrange this series because it is not <a href="https://en.wikipedia.org/wiki/Absolute_convergence#Rearrangements_and_unconditional_convergence"... | razivo | 789,052 | <p>As Mark raised in the comments, doing any sort of arithmetic with divergent series is meaningless, we often do arithmetic with limits and then justify why it was fine after the fact by showing they were convergent, another example of this going wrong:<br />
<span class="math-container">$$\lim_{n\to\infty} 1 = \lim_{... |
347,088 | <p>In a famous paper Rademacher used the circle theorem to give a formula for the fourier coefficients of the partition function <span class="math-container">$1/f(q)$</span> where <span class="math-container">$f(q) = \prod_{n=1}(1-q^n)$</span>, and in another paper he gave the coefficients of the j-invariant function ... | jah | 148,022 | <p>You can find a nice review of Radmemacher sums and series for both modular and mock modular forms here: <a href="https://arxiv.org/abs/1210.3066" rel="nofollow noreferrer">https://arxiv.org/abs/1210.3066</a></p>
<p>This review includes formulae for the coefficients of modular functions of weight <span class="math-c... |
3,173,766 | <p>If you have the option to roll 2 separate 10-sided dice or a single 10-sided die that you add 2 to the result with a goal of at least one die getting a specific number or higher (say 6 for the purposes of this question) is one of these options inherently better?</p>
<p>It seems that the single die with a modifier i... | RandyF | 305,487 | <p>So, with one die, you can get 6 or greater by rolling a 4, 5, 6, 7, 8, 9, 10. This is a 70% chance. With two dice, you can get a six or greater if the first dice is six or greater (50%) or the first dice is less than 6 (50%) but the second is greater than six (50%). So, the total in this scenario, since the two dice... |
32,091 | <p>I have a <code>list</code> of points (e.g. <code>{1.2,0.4}</code>) which I export to a file using</p>
<pre><code>Export[fileName, list];
</code></pre>
<p>I want to add a line before the end of file. For instance instead of ending the file with </p>
<pre><code>1.236677 4.64567567
1.236677 4.64567567
1.236676 4.645... | halirutan | 187 | <p>Here, on my OSX the following works</p>
<pre><code>Export["tmp/foo.txt", Append[Range[10], "\n"]]
</code></pre>
<blockquote>
<p><img src="https://i.stack.imgur.com/8o6r1.png" alt="Mathematica graphics"></p>
</blockquote>
|
984,154 | <p>Let say my girlfriend makes $2000$ Euros per month and I make $3400$ Euros, let say all our living costs sum up to $1540$ Euros.
How can i calculate how much each of us must pay (off the $1540$ Euros) so as our participation is proportional to our income?</p>
<p>Thanks</p>
| layman | 131,740 | <p>The total income is $2000 + 3400 = 5400$ euros. Since your girlfriend makes only $2000$ euros (sexist!), she makes $\frac{2000}{5400} = .37...$ (roughly $37 \% $) of the income, which means she should only pay $37 \% $ of the rent.</p>
<p>Then, she should only pay $1540 *.37 = 569.80 $ euros, roughly.</p>
|
2,257,601 | <p>Let $$x^4-4x^3+mx^2+nx+1=0$$ have four roots positive real number. Then : what is $n,m \in \Bbb R$</p>
<p>My try </p>
<p>Let four roots be $x=a,b,c,d \in \Bbb R^+$ so, $$x^4-4x^3+mx^2+nx+1=0\\(x-a)(x-b)(x-c)(x-d)=0$$
Next?</p>
| hamam_Abdallah | 369,188 | <p><strong>hint by Rolle</strong></p>
<p>$$f'(x)=4x^3-12x^2+2mx+n $$
has 3 roots then </p>
<p>$$f''(x)=12x^2-24x+2m $$</p>
<p>has 2 roots. So $\delta'=144-24m>0$.</p>
<p>or $m <6$.</p>
|
271,846 | <p>In his <em>"Some topological properties..." (1955)</em>, Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ in the coordinate $\mathbb{R}^n$-subspace to any other homeomorphic compact $K'$ set in this subspace. </p>
... | Igor Belegradek | 1,573 | <p>Klee's trick easily follows from Tietze extension theorem, and there is a generalization when $\mathbb R^n$ is replaced by a locally convex linear space $E$, which is based on Dugundji's extension theorem. </p>
<p>Incidentally, the sets $K$, $K^\prime$ need only be closed. Compactness is not required.</p>
<p>A com... |
4,049,205 | <p>Let <span class="math-container">$\{a_n\}:\mathbb{N}\to\mathbb{R}$</span>, and we define <span class="math-container">$E$</span> to be the set of all sub-sequential limits of <span class="math-container">$\{a_n\}$</span> as well as possibly the symbols <span class="math-container">$\infty$</span> and <span class="ma... | JLinsta | 895,235 | <p>Every sequence of real numbers has a monotone subsequence. This monotone subsequence converges to something in <span class="math-container">$\mathbb{R} \cup \{-\infty,\infty\}$</span>, i.e. <span class="math-container">$E \neq \emptyset$</span>.</p>
<p>If you would like to see a proof of this fact, look up the Monot... |
1,505,859 | <p>Some days ago I presented some exercises to a bunch of engineers. One of the exercises involved was: Use $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x}$ to show that $\lim_{n \to \infty} \left(1 + \frac{\sqrt{n}}{n}\right)^{-n} = 0$. I've done this by showing that $\left(1 + \frac{\sqrt{n}}{n}\right)^... | Paul Sinclair | 258,282 | <p>No, $\{\frac{1}{n}(y^\ast)^{-1}(A)\}_{n\in\mathbb{N}}$ is not a local basis of neighborhoods of $0$ in $Y$. $\ker y^\ast$ is a subspace of $Y$ of codimension $1$, and the entire kernal is contained in every set of that collection.</p>
<p>That is why you need more than one $y^\ast$.</p>
|
1,605,864 | <p>Definition $1$: </p>
<p>A poset $P=(X, \prec)$ is called Interval order if there exists a function $f$ which maps each $x \in X$ to an interval $I_x$ of real line such that $x \prec y$ iff $I_x \prec I_y$ ($I_x$ lies totally to the left of $I_y$).</p>
<p>Definition $2$:</p>
<p>Interval order dimension of a poset ... | Brian M. Scott | 12,042 | <p>The following example may help a bit with the notion of interval order dimension. Consider the poset $P$ whose Hasse diagram is shown below.</p>
<pre><code> b d
| |
a c
</code></pre>
<p>Suppose that $P$ is an interval order, with intervals $I_a=[x_a,y_a],I_b=[x_b,y_b],I_c=[x_... |
1,135,416 | <p>I have two arrays with 16 unsigned integers. I compute the sum of the first array = x and the sum of the second array = y. What is the chance they will be the same?
Also, how many combinations out of the total combinations are the same?</p>
<p>An example:
Lets say an array1 contains these numbers (low values just f... | user2566092 | 87,313 | <p>If the board has odd width (any length is fine) then yes, there is trivially such a sequence. Just assume all pieces are the $2 \times 2$ squares. So maybe you should insist that the board width is even?</p>
|
154,573 | <p>How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?</p>
| Brian M. Scott | 12,042 | <p>Both $\sqrt x$ and $\ln x$ are increasing functions of $x$, so $\sqrt x+\ln x=1$ can have at most one solution. As you note, it does have one, namely $x=1$, but that must be the only one: $\sqrt x+\ln x<1$ when $0<x<1$, and $\sqrt x+\ln x>1$ when $x>1$.</p>
|
3,276,351 | <blockquote>
<p>A particle is allowed to move in the <span class="math-container">$\mathbb{Z}\times \mathbb{Z}$</span> grid by choosing any of the two jumps:</p>
<p>1) Move two units to right and one unit up</p>
<p>2) Move two units up and one unit to right.</p>
<p>Let <span class="math-container">$P=(... | Robert Z | 299,698 | <p>Note that starting from <span class="math-container">$(x,y)$</span> we can reach <span class="math-container">$(X,Y)\in\{(x+2,y+1),(x+1,y+2)\}$</span>. In any case if <span class="math-container">$X+Y-(x+y)$</span> is divisible by <span class="math-container">$3$</span>. This means that if we start from the origin <... |
3,276,351 | <blockquote>
<p>A particle is allowed to move in the <span class="math-container">$\mathbb{Z}\times \mathbb{Z}$</span> grid by choosing any of the two jumps:</p>
<p>1) Move two units to right and one unit up</p>
<p>2) Move two units up and one unit to right.</p>
<p>Let <span class="math-container">$P=(... | Charles Hudgins | 647,305 | <p>The above posts are certainly sufficient answers, but for fun I thought I would precisely characterize the set of points that can be reached from the origin.</p>
<p>Note that performing a move of type (1) amounts to adding <span class="math-container">$(2,1)$</span> to the particle's position vector, and, likewise,... |
211,625 | <p>The problem here is that the matrix
$$\begin{pmatrix}
2 & 3 \\
1 & -1 \\
\end{pmatrix}$$
is invertible but not unimodular and hence the elements $2x+3y$ and $x-y$ generate a free abelian group of rank 2 but still a proper subgroup of $\langle x,y\rangle$.
But the question was to prove that $2x+3y$ and $x-y... | Scott Carter | 722 | <p>Think of the torus. Its fundamental group is ${\mathbb Z} \oplus {\mathbb Z}$. Now draw a $(2,3)$ curve and a $(1,-1)$ curve on it. You can do this using a square with the edges identified. These two curves should intersect transversely once. Thus they, too, are a symplectic basis for $H_1(T^2)$ and hence for $\pi_1... |
1,619,208 | <blockquote>
<p>A person walks downhill at 10 km/h, uphill at 6 km/h and on the plane at 7.5 km/h. If the person takes 3 hours to go from a place A to another place B, and 1 hour on the way back, the distance between A and B is
$\begin{array}1 1. 15 km && 2. 23.5 km \\ 3. 16 km && 4. \text{Given dat... | N. F. Taussig | 173,070 | <p>The number $n!$ can be defined recursively as follows:</p>
<ol>
<li>$1! = 1$</li>
<li>$n! = n(n - 1)!$, $n \geq 1$</li>
</ol>
<p>Using the definition to compute $5!$ yields
\begin{align*}
5! & = 5 \cdot 4!\\
& = 5 \cdot 4 \cdot 3!\\
& = 5 \cdot 4 \cdot 3 \cdot 2!\\
& = 5 \cdot 4 \cdot 3 \c... |
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | Arthur | 15,500 | <p>In the first one, there is a difference between the two resulting teams. One is playing, the other is sitting on the sidelines. So swapping the two teams around makes for a different choice of starting five.</p>
<p>In the second case, there is no difference. The only thing that matters is who is on team with who, a... |
1,480,103 | <p>I have an assignment, that beside other things, asks about a basis of a group, more precisely, it have been asked to "give an example of a basis for $T_d$" (tetrahedron group).</p>
<p>But I been on all lectures and searched internet for such a definition, but found nothing.</p>
<p>Any idea what "Group Basis" means... | algbr | 587,554 | <p>It means a set of generators, such that every element of the generated group has a unique representation in terms of the generators.</p>
|
3,187,856 | <p>Find all complex-valued <span class="math-container">$n \times n$</span> matrices <span class="math-container">$A$</span> such that <span class="math-container">$\det A = 0$</span> and <span class="math-container">$\text{rank}(AB) = \text{rank}(BA)$</span> for any <span class="math-container">$n \times n$</span> com... | A.Γ. | 253,273 | <p>Another way to do it: assume <span class="math-container">$\operatorname{rank}A=r$</span>.</p>
<p>Since both conditions (rank and <span class="math-container">$\det$</span>) are invariant under pre- and postmultiplication of <span class="math-container">$A$</span> by an invertible transformation <span class="math-c... |
273,020 | <p>I want to reproduce the effect in the figure below, approximating a surface with tangent planes,
<a href="https://i.stack.imgur.com/sfjgz.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/sfjgz.gif" alt="enter image description here" /></a><br />
Now I can only use polygons to approximate inside the surface,... | cvgmt | 72,111 | <ul>
<li><p>Thanks @Michael E2 point out that quadratic paraboloid work. Here we use the definiton of differential of function to get the tangent plane for 2-variables function,see <code>df</code> as in the code.</p>
</li>
<li><p>And we treat the general form of quadratic paraboloid <code>f[x,y]=a-b*x^2-c*y^2;</code> o... |
116,196 | <p>Simple Question from a Stumped Rookie: </p>
<p>Using ListPlot, I can't get PlotStyle to look below the first level of a nested list to assign individual saturation values for each point; it assigns the first three saturation values to the first three groups of points. </p>
<pre><code>pts = {{{0.10, 485}, {0.22, 4... | kglr | 125 | <pre><code>options={VertexLabels -> Placed["Name",Center],
VertexShapeFunction->"Square", VertexSize->.8, VertexStyle->Orange};
g1= Graph[Range[0,20], input, ##&@@options]
</code></pre>
<p><img src="https://i.stack.imgur.com/SlqvR.png" alt="Mathematica graphics"></p>
<pre><code>junctions ... |
3,399,424 | <p>This question is from the proof of Durrett Theorem 2.3.8 and my post here: <a href="https://math.stackexchange.com/questions/3397637/using-the-first-and-second-borel-cantelli-lemma-to-find-necessary-and-sufficient">Using the first and second Borel-Cantelli Lemma to find necessary and sufficient condition for converg... | JacobsonRadical | 502,975 | <p>Okay I am now going to answer my own post. Note that in my edit, the first part is correct but second one is not, since I remembered the inclusion of <span class="math-container">$L_{p}$</span> space in an opposite direction. It is actually that for finite measure space, and <span class="math-container">$1\leq p<... |
3,399,424 | <p>This question is from the proof of Durrett Theorem 2.3.8 and my post here: <a href="https://math.stackexchange.com/questions/3397637/using-the-first-and-second-borel-cantelli-lemma-to-find-necessary-and-sufficient">Using the first and second Borel-Cantelli Lemma to find necessary and sufficient condition for converg... | Community | -1 | <p>Combining
<span class="math-container">$$
\int_{0}^{\infty}\mathsf{P}(|X_{1}|>x)\,dx=\sum_{n\ge 0}\int_{n}^{n+1}\mathsf{P}(|X_{1}|>x)\,dx
$$</span>
and
<span class="math-container">$$
\mathsf{P}(|X_1|>n+1)\le\int_{n}^{n+1}\mathsf{P}(|X_{1}|>x)\,dx\le \mathsf{P}(|X_1|>n),
$$</span>
one gets
<span class... |
2,350,358 | <blockquote>
<p>Let $(X_n)_{n\in\mathbb N}$ denote a simple, symmetric, random walk, defined by $X_0=0$ and $X_n = \sum\limits_{j=1}^n\xi_j$ where $(\xi_i)_{i \in \mathbb N}$ is i.i.d. with $\mathbb{P}(\xi_i = 1)= \mathbb{P}(\xi_i = -1) = \frac{1}{2}$. The maximum process of $(X_n)_{n\in\mathbb N}$ is defined by $M_... | Did | 6,179 | <p>Fix $1\leqslant t<T$ and consider the atom $A=\{\forall k\in[t],\xi_k=+1\}$ where $[t]=\{1,2,\ldots,t\}$.</p>
<p>Then $A$ is in the sigma-algebra $\mathcal F_t=\sigma(X_k;k\in[t])=\sigma(\xi_k;k\in[t])$ and $A\subseteq\{M_t=X_t\}$.
Furthermore, $\{\tau_T=t\}\cap A=\{M^{(t)}_T\leqslant0\}\cap A$ where $M^{(t)}_T... |
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | Joe Goldiamond | 516,719 | <p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}dt}$$
Use the hint:: $t=\cos u$ to obtain:
$$\pi = 2\int{ {\sqrt{(1-\cos^2u}}) du} = -2\int \sin (u)\sin (u)du= -2\int \sin^2(u) du$$
Can you continue from here?</p>
|
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | Jack D'Aurizio | 44,121 | <p>The definition $\pi=2\int_{-1}^{1}\sqrt{1-x^2}\,dx$ is actually equivalent to $\pi=\Gamma\left(\frac{1}{2}\right)^2$, or to $\pi=2\arcsin(1)$, or to $\pi=6\arcsin\frac{1}{2}$, or to
$$ \pi = \sum_{n\geq 0}\frac{3\binom{2n}{n}}{16^n(2n+1)}=\color{red}{3}+\frac{1}{8}+\frac{9}{640}+\frac{15}{7168}+\ldots $$
hence $\pi&... |
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | Community | -1 | <p><strong>Hint:</strong></p>
<p>It can be shown rigorously that </p>
<p>$$\sqrt{1-t^2}\ge 1-t^2+\frac{t^4(1-t^2)}{0.65}$$</p>
<p>and the integral of the polynomial is $\frac{412}{273}>\frac32$.</p>
<p><a href="https://i.stack.imgur.com/QFqKI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QFq... |
820,685 | <p>Another question from a beginner:</p>
<p>We want to prove that ⊨ is not symmetric by finidng concrete formulas φ and ψ for which we can show that φ ⊨ ψ and ψ ⊭ φ.</p>
<p>Thank you for your help!</p>
| Jon | 124,068 | <p>If $\models$ means classical consequence counterexamples are easy to find. Let $R, S$ be one place relation symbols. We have $\exists x (Rx \wedge Sx) \models \exists x Rx \wedge \exists x Sx$, but not vice versa. </p>
<p>For the first claim let $\mathfrak{A} = \langle D, I\rangle$ be a first-order model and $\beta... |
820,685 | <p>Another question from a beginner:</p>
<p>We want to prove that ⊨ is not symmetric by finidng concrete formulas φ and ψ for which we can show that φ ⊨ ψ and ψ ⊭ φ.</p>
<p>Thank you for your help!</p>
| Mauro ALLEGRANZA | 108,274 | <p>We have that :</p>
<blockquote>
<p>$p \land q \vDash p$</p>
</blockquote>
<p>but :</p>
<blockquote>
<p>$p \nvDash p \land q$.</p>
</blockquote>
<p>Use truth-table to verify it. </p>
|
3,948,378 | <p>For <span class="math-container">$x\in(-\frac{\pi}{2}, -\frac{\pi}{2})\setminus \{0\}$</span> we can say that since the area of sector of circle is greater than the triangle (ABC) area I have <span class="math-container">$2|\sin{x}|<2|x|$</span>. But what's about x in the 2nd and 3rd quadrant?
Look at the figure ... | VIVID | 752,069 | <p>Area of the triangle is <span class="math-container">$$S_\triangle = \frac{1\cdot 1\cdot \sin x}{2}$$</span>
Area of the sector is <span class="math-container">$$S_\text{sec} = \frac{1^2\cdot x}{2}$$</span>
Since it is vivid from the picture below that <span class="math-container">$S_\text{sec} > S_\triangle$</sp... |
3,555,007 | <p>I want to find out the function to generate the sequence with the following pattern.
<code>1 2 3 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3</code> ....
Where the lower bound number is 1 upper number bound number is 3. Each time numbers start from 1 and each number repeats <code>2 ^</code> n times, with n starting with 0.</... | NazimJ | 533,809 | <p>Im not sure if you want a closed form solution or not. The following solution is programmable with while loops</p>
<p>Let
<span class="math-container">$$ 1,2,3,1,1,2,2,3,3,...=a_1 , a_2, a_3, a_4, ... $$</span>
Define <span class="math-container">$n_0 = 0$</span> to be the initiation of the sequence. The sequence... |
3,555,007 | <p>I want to find out the function to generate the sequence with the following pattern.
<code>1 2 3 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3</code> ....
Where the lower bound number is 1 upper number bound number is 3. Each time numbers start from 1 and each number repeats <code>2 ^</code> n times, with n starting with 0.</... | Somos | 438,089 | <p>The following <a href="http://pari.math.u-bordeaux.fr/" rel="nofollow noreferrer">PARI/GP</a> code will do what you want</p>
<pre><code>a(n) = (n-3*2^(k=exponent((n-1)\3+1))+2)\2^k+1;
</code></pre>
<p>Note that <code>exponent(n)</code> is the binary exponent of <span class="math-container">$n$</span>. In other wor... |
1,038,219 | <ol>
<li>(n) people put their name in a hat. </li>
<li>Each person picks a name out of the hat to buy a gift for. </li>
<li>If a person picks out themselves they put the name back into the hat.</li>
<li>If the last person can only pick themselves then the loop is invalid and either<br>
. start again<br>
. or step bac... | Laars Helenius | 112,790 | <p>In order to count the number of perfect loops with $n$ people, we need to count the number of distinct labelings of a cycle on $n$ vertices, say $c(n)$. While I don't have a proof yet, I think the number is given by
$$
c(n)=\frac{1}{2n-3}\binom{3n-6}{n-2}.
$$
On the other hand, to count the number of valid loops we ... |
547,184 | <p>Determine values of the constants $A$ and $B$ such that $a_n=An+B$ is a solution of the recurrence relation $a_n=2a_{n-1}+n+5$.</p>
<p>I know that the characteristic equation is $r-2 = 0$ which has the root $r = 2$.
Usually I find constants $A$ and $B$ by $a_n=Ar^n+Br^n$. It
looks like I cannot apply this here beca... | 1233dfv | 102,540 | <p>Consider the homogeneous recurrence relation $a_n=2a_{n-1}$ where $n\geq1$. The characteristic equation is $x-2=0$, where $x=2$ is the root of the characteristic root. Thus the general solution is $a_n=a2^n$ where $a$ is a constant. Let $a_n=bn+c$ where $b$ and $c$ are constants. We do this because $b_n=n+5$, so a l... |
251,607 | <p>The simplest Laguerre polynomials are
$$
L_k(x)=(\frac{d}{dx}-1)^k\left(\frac{x^k}{k!}\right).
$$
I would like to find a simple reference for proving or disproving the following assertions.</p>
<p>(1) All the $k$ zeroes of $L_k$ are simple and located on the positive half-line.</p>
<p>(2) The largest zero of $L_k... | T. Amdeberhan | 66,131 | <p>Both assertions hold true, in fact the roots lie in the interval
$$(0,k+(k-1)\sqrt{k}).$$ </p>
<p>There are many books for references, one among which is</p>
<p>"Basic Hypergeometric Series", by George Gasper, Mizan Rahman, Encyclopedia of Mathematics and its applications.</p>
|
361,143 | <p>Suppose I have a set containing $n$ elements and I consider the number of partitions of it in which all the constituent classes are even. I let $R_n$ denote this number ($R_0=1$) and proceed as follows to obtain the exponential generating function for $R_n$.</p>
<p>Since $$R_n=\sum_{k=1}^{\lfloor n/2\rfloor}(\text{... | Marko Riedel | 44,883 | <p>It is easiest to approach this problem using symbolic combinatorics as formalized by Flajolet and Sedgewick. Let $\mathcal{R}$ be the combinatorial class in question. Then we have straightforwardly the following combinatorial class equation
$$\mathcal{R} =
\mathfrak{P}(\mathfrak{P}_2(\mathcal{Z}) + \mathfrak{P}_4(\... |
2,316,076 | <p>I was given the following permutation
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
6 & 5 & 2 & 4 & 3 & 1 \\
\end{pmatrix}</p>
<p>and I was asked to write it as the product of disjoint cycles.</p>
<p>The disjoint cycles I found were $(1, 6)$ and $(2, 5, 3)$, which my ... | Wuestenfux | 417,848 | <p>Indeed, the cycles are (1,6), (2,5,3), and (4) where (4) is a fixed point. By convention, fixed points can be left out if you know the degree of the permutation, here $n=6$.</p>
|
2,493,123 | <p>I am pretty much stuck, since smaller examples are impossible to find, nor does this seem like an induction question. I am thinking of using proof by contradiction, but don't know where to find the contradiction. I also struggle to recall the theorem that may help me answer this problem.</p>
| alden | 496,504 | <p>Take the points of the plane $(0,0)$ and $(n,1/n)$ for $n=1,2,...$. Its convex hull gets arbitrarily close to the all the positive part of the X axis, but it is not going to include any of its points except $(0,0)$.</p>
<p>Your argument is not correct, since the convex hull doesn't have to be open.
For example the ... |
3,070,292 | <p>I'm trying to prove or disprove the following statement:</p>
<blockquote>
<p>Number of even natural solutions of <span class="math-container">$x_1+x_2+x_3+x _4=14$</span> is equal to the number of odd natural solutions of <span class="math-container">$x_1+x_2+x_3+x _4=14$</span>.</p>
</blockquote>
<p>I think tha... | Daniel Mathias | 627,720 | <p>Assuming 'natural' means 'positive integer':</p>
<p>The total number of solutions is:<span class="math-container">$$\binom{14-1}{4-1}=286$$</span></p>
<p>Even solutions are positive integer solutions to <span class="math-container">$y_1+y_2+y_3+y_4=7$</span>, where <span class="math-container">$y_i=\frac{x_i}{2}$<... |
629,756 | <p>Consider the eigenvalue problem
\begin{equation}
\left\{ \begin{array}{l}
\Phi \in C^{2}(\mathbb{R}) \ \text{and bounded }\\
-\Phi^{''}(x)=\lambda\Phi(x), \ x\in \mathbb{R}.
\end{array} \right.
\end{equation}</p>
<p>and assume that the solution is given by $\Phi(x)=Ae^{i\sqrt{\lambda}x}+Be^{-i\sqrt{\lambda}x}$.<... | x tang | 97,573 | <p>If $\lambda<0,$ then $\sqrt{\lambda}=i \Im{\sqrt{\lambda}} + \Re{\sqrt{\lambda}},$ then $\Phi(x)=Ae^{i\sqrt{\lambda}x}+Be^{-i\sqrt{\lambda}x} $ can be written by $\Phi(x)=Ae^{-\Im{\sqrt{\lambda}}x + i\Re{\sqrt{\lambda}}x }+B e^{-\Im{\sqrt{\lambda}} x+ i\Re{\sqrt{\lambda}}x }.$ But $e^{-\Im{\sqrt{\lambda}}x}$ is n... |
10,537 | <p>In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.</p>
<p><strong>As a teacher</strong>, to what extent is it important and useful for me to ... | marco trevi | 6,286 | <p>My point of view on the issue is that it depends on the purpose of the drawing. In general I follow the rule of drawing "almost-everywhere" correct drawings. I see essentially two scenarios:</p>
<p><em>1) Getting an intuition of a geometry problem</em></p>
<p>When trying to solve a geometry problem (i.e. finding c... |
10,537 | <p>In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also geometric constructions.</p>
<p><strong>As a teacher</strong>, to what extent is it important and useful for me to ... | Tommi | 2,083 | <p>Reasons why drawing precise pictures is important:</p>
<ul>
<li>Students need to learn how to draw with ruler and compass, as it is a necessary (or at least useful) skill in other subjects and in the world outside school. As mentioned by marco trevi, precise drawings are also important for geometrical constructions... |
190,914 | <p>In <a href="https://math.stackexchange.com/questions/38350/n-lines-cannot-divide-a-plane-region-into-x-regions-finding-x-for-n">this</a> post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. </p>
<p>What happens if we use circles instead of lin... | John | 30,229 | <p>One may deduce the formula $n^{2}-n+2$ as follows: Start with $m$ circles already drawn on the plane with no two of them tangent, none of them being inside another and no three of them concurrent. Then draw the $m+1$ circle $C$ so that is does not violate the propeties stated before and see how it helps increase the... |
3,990,110 | <p>I was seeking the other day for intuition about the Hölder inequality and more precisely the geometric intuition behind the relation <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>.</p>
<p>I found <a href="http://www.tricki.org/article/Geometric_view_of_H%C3%B6lders_inequality" rel="nofollow nore... | John Dawkins | 189,130 | <p>But you do have something like that: Young's inequality says that for <span class="math-container">$u,v\ge 0$</span> and conjugate exponents <span class="math-container">$p$</span> and <span class="math-container">$q$</span>,
<span class="math-container">$$
uv\le p^{-1}u^p+q^{-1}v^q.
$$</span>
(This can be proved wi... |
3,672,378 | <p>Consider the following Primal LP </p>
<p><span class="math-container">$$
\min_{x_1,x_2,x_3} x_1+x_2+x_3 $$</span>
subject to constraints,
<span class="math-container">$$
x_1+x_2 \ge a \\
x_1+x_3 \ge b \\
x_2+x_3 \ge c\\
x_1 \ge d \\
x_2 \ge e \\
x_3 \ge f
$$</span></p>
<p>Here <span class="math-container">$$a,b,c... | gsandhu | 327,522 | <p>Can check if LP is bounded by seeing if primal has a feasible solution and dual has a feasible solution, then primal is bounded.</p>
|
77,958 | <p>I would like to ask the following: if for a group <span class="math-container">$G$</span> the homology <span class="math-container">$H_n(G,\mathbb{Z})$</span> is <span class="math-container">$\mathbb{Z}$</span>-torsion for every <span class="math-container">$n\geq n_0$</span>, then what can be said concerning <span ... | Ian Agol | 1,345 | <p>There exists an acyclic group <span class="math-container">$G$</span> which has the property that there exists a finite-index normal subgroup <span class="math-container">$H\lhd G$</span> such that <span class="math-container">$\mathbb{Z} \leq H_1(H;\mathbb{Z})$</span>. In particular, then <span class="math-contain... |
94,209 | <p>Consider the following region:</p>
<pre><code>eq1 = 4 x;
eq2 = 1/x;
eq3 = 1;
cover = y^2/x^2;
bottom = eq3 < y < eq1 && eq3 < y < eq2;
reg3D = ImplicitRegion[bottom && 0 <= z <= cover, {{x, 1/4, 1}, {y, 1, 2}, {z, 0, 15}}];
RegionPlot3D[bottom && z < cover, {x, 1/4, 1},... | Alexei Boulbitch | 788 | <p>Here is an alternative:</p>
<pre><code>eq1 = 4 x;
eq2 = 1/x;
eq3 = 1;
cover = y^2/x^2;
bottom = eq3 < y < eq1 && eq3 < y < eq2;
new3D = ImplicitRegion[
bottom && 0 <= z <= cover, {{x, 1/4, 1}, {y, 1, 2}, {z, 0, 9.7}}];
Needs["NDSolve`FEM`"];
mesh = ToElementMesh[new3D, "MaxCel... |
8,263 | <p>1.What should be added to $15^{81}$ to make it divisible by 7 ? Suggest a general technique to counter problems like this.</p>
<p>2.For how many two-digit numbers, the sum of the digits is greater than the product of the digits?</p>
| Community | -1 | <p><strong>HINT:</strong> For the first one use the fact that $15^{2} \equiv 1 \ (\text{mod} \ 7)$</p>
|
8,263 | <p>1.What should be added to $15^{81}$ to make it divisible by 7 ? Suggest a general technique to counter problems like this.</p>
<p>2.For how many two-digit numbers, the sum of the digits is greater than the product of the digits?</p>
| Ross Millikan | 1,827 | <p>For the second question, for single digits $a+b\geq ab$ when a=0 or 1, b=0 or 1, or a=b=2.</p>
|
4,361,604 | <p>I am studying the <strong>the basic theorem of field extensions</strong> with <em>Charles C. Pinter</em>'s book: <em>A book of Abstract Algebra</em>. The very same theorem that is being discussed here: <a href="https://math.stackexchange.com/questions/2272238/example-of-basic-theorem-of-field-extensions">Example of ... | Lubin | 17,760 | <p>It occurs to me that you may be looking at the offending equation
<span class="math-container">$$
(J+a)(J+b)=(J+ab)
$$</span>
and making too much of it. In particular, how is “<span class="math-container">$(J+a)(J+b)$</span>” to be understood? To multiply these two cosets, you don’t take all possible products, one f... |
1,897,814 | <blockquote>
<p>Find the co-efficient of $x^3y^3zw^2$ in the expression of $(x-y+2z-2w)^9$.</p>
</blockquote>
<p>I have clearly understood how to use the bin theorem for $(a+b)^n$, but am unable to extend the theorem to $4$ terms .</p>
| epi163sqrt | 132,007 | <p>It is also possible to apply the binomial theorem <em>iteratively</em>. We focus on one term and keep the other terms together. This way we reduce the number of terms step by step by one, till we have only two left at the end.</p>
<p>To keep the calculations manageable it is convenient to use the <em>coefficient... |
15,508 | <p>In probability/statistics, there is a notion of two things being "independent", which would basically mean that any information we can get about one thing has no effect on our (probabilistic) knowledge of the other.</p>
<p>What are the possible notions of "independent" for the natural numbers? Un... | Michael Lugo | 143 | <p>There are some ways to assign probability measures to the set of natural numbers. Consider the probability measure <span class="math-container">$P_s$</span> on the positive integers which assigns "probability" <span class="math-container">$n^{-s}/\zeta(s)$</span> to the integer <span class="math-container... |
1,783,841 | <p>If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$.
And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.</p>
<p>I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?</p>
<p>I am referring to a case li... | peter.petrov | 116,591 | <p>If $f(x)=f'(x)$, then $f(x) = C \cdot e^x$.<br>
That's the important application/consequence here. </p>
|
375,348 | <p>Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.</p>
<p>This is the converse of the Chinese remainder theorem in abstract algebra. Any help would be appreciated.</p>
<p>Thanks!</p>
| Coffee_Table | 67,062 | <p><strong>Approach:</strong> Contrapositively, $(m,n)>1 \implies \mathbb{Z}_m\times\mathbb{Z}_n$ not cyclic $\implies \mathbb{Z}_m\times\mathbb{Z}_n \ncong \mathbb{Z}_{nm}$ since the latter is cyclic.</p>
<p>Basically, the contrapositive is much easier to verify (which is fine for our purposes, since it is equival... |
3,555,250 | <p>I was learning differential calculus when I came with a doubt that why do we need to find derivative of polynomial. How is it useful?</p>
| g.kov | 122,782 | <p><span class="math-container">$\require{begingroup} \begingroup$</span>
<span class="math-container">$\def\i{\mathbf{i}}$</span></p>
<p>One of the beautiful and somewhat unexpected
application of the derivative of a polynomial
is <a href="https://en.wikipedia.org/wiki/Marden%27s_theorem" rel="nofollow noreferrer">M... |
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