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 |
|---|---|---|---|---|
58,024 | <p><img src="https://i.stack.imgur.com/cTpA2.jpg" alt="Show that..."></p>
<p>The picture says it all. "Vis at" means "show that". My first thought was that h is 2x, which is not correct. Maybe the formulas for area size is useful? </p>
<p>EDIT: (To make the question less dependent from the <a href="https://math.meta.... | Michael Hardy | 11,667 | <p>You can split the triangle into one triangle with base $g$ and height $x$, and another with base $h$ and height $x$. Just draw the diagonal line from the right angle to the opposite vertex of the $x\times x$ square. One of those triangles has area $gx/2$; the other has area $hx/2$. But they must add up to $gh/2$.... |
3,455,009 | <p>In the proof of the expectation of the binomial distribution,</p>
<p><span class="math-container">$$E[X]=\sum_{k=0}^{n}k \binom{n}{k}p^kq^{n-k}=p\frac{d}{dp}(p+q)^n=pn(p+q)^{n-1}=np$$</span></p>
<p>Why is <span class="math-container">$\sum_{k=0}^{n}k \binom{n}{k}p^kq^{n-k}= p\frac{d}{dp}(p+q)^n$</span>?</p>
<p>I ... | Steve Kass | 60,500 | <p>Equivalently, you want to find all complex numbers <span class="math-container">$z$</span> for which <span class="math-container">$z$</span> and <span class="math-container">$z+1$</span> are cube roots of the same complex number <span class="math-container">$c$</span>. In the <a href="https://en.wikipedia.org/wiki/C... |
3,444,673 | <p>How to evaluate this double integral
<span class="math-container">$$\int_{0}^1\int_{x^2}^x \frac{x}{y}e^{-\frac{x^2}{y}}dydx.$$</span></p>
<p>It seems like I am evaluating the double integral of a non-elementary function. I tried substitutions but the integral is growing. </p>
<hr>
<p>Following Fred's suggestion,... | Stefan Lafon | 582,769 | <p>We'll need to use the fact that, by Cauchy-Schwartz:
<span class="math-container">$$1+\alpha\beta = 1\times 1 +\beta\times\gamma
\leq\sqrt{1+\beta^2}\sqrt{1+\gamma^2}\tag{1}$$</span>
First inequality:
<span class="math-container">$$\begin{split}
1+\bigg(\int|f|d\mu\bigg)^2&=\int\bigg[1\bigg(\int|f(x)|d\mu(x)\b... |
3,444,673 | <p>How to evaluate this double integral
<span class="math-container">$$\int_{0}^1\int_{x^2}^x \frac{x}{y}e^{-\frac{x^2}{y}}dydx.$$</span></p>
<p>It seems like I am evaluating the double integral of a non-elementary function. I tried substitutions but the integral is growing. </p>
<hr>
<p>Following Fred's suggestion,... | Samrat Mukhopadhyay | 83,973 | <p>The second inequality is fine.</p>
<p>For the first inequality, first note that the desired inequality can be expressed as <span class="math-container">$\sqrt{1+(\mathbb{E}(|f|))^2}\le \mathbb{E}\sqrt{1+|f|^2}$</span>, which then can be seen to be a simple consequence of the Jensen's inequality since the function <... |
3,620,612 | <p>I had a question in the exercises of a complex analysis course I couldn't solve, It asked me to evaluate this integral <span class="math-container">$$\int_{-\pi}^{\pi}\frac{dx}{\cos^2(x) + 1}$$</span></p>
<p>I tried to evaluate it without using residues, the antiderivative of this function contains tan, which is no... | mathlover123 | 761,688 | <p>First start by:
<span class="math-container">$$\int _{-\pi }^{\pi }\:\frac{dx}{\left(\cos \left(x\right)\right)^2+1}=4\int _{0\:}^{\frac{\pi }{2}\:}\:\frac{dx}{\left(\cos \:\left(x\right)\right)^2+1}=4\int _{0\:}^{\frac{\pi \:}{2}\:}\:\frac{\left(\sec \left(x\right)\right)^2dx}{2+\left(\tan \left(x\right)\right)^2}$... |
19,880 | <p>I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously.</p>
<p>The known expansions of $\ln(1+x)$ and $\cos(x)$ gives:</p>
<p>$$\forall x \gt -1,\ \ln(1+x)=\sum_{n=1}^{k} (-1)^{n-1}\frac{x^n}{n} + R_{k}(x... | Derek Jennings | 1,301 | <p>The MacLaurin series of $f(x)$ is given by</p>
<p>$$f(x)= f(0)+f^{\prime}(0)x + \frac{ f^{\prime \prime }(0)}{2!}x^2 +
\frac{ f^{\prime \prime \prime}(0)}{3!}x^3 + \cdots $$</p>
<p>and so you need to calculate the values of $f(0),f^{\prime}(0),\ldots$ for your function
$f(x)=\log( \cos x).$</p>
<p>We have
$$\beg... |
1,973,686 | <p>I am stuck on two questions :</p>
<ol>
<li>If $f,g\in C[0,1]$ where $C[0,1]$ is the set of all continuous functions in $[0,1]$ then is the mapping $id:(C[0,1],d_2)\to (C[0,1],d_1)$ continuous ? where $id$ denotes the identity mapping.</li>
</ol>
<p>where $d_2(f,g)=(\int _0^1 |f(t)-g(t)|^2dt )^{\frac{1}{2}} $ and ... | marwalix | 441 | <p>Take $f(x)=\sqrt{x}$ one has $f\in L^2([0,1])$ and obviously $f\notin L^1([0,1])$ for the integral diverges at $0$.</p>
<p>Now consider a sequence of functions in $L^2$ that converges to the above $f$. If the identity were continuous the sequence should converge in $L^1$ but we have seen that $f\notin L^1$ so the i... |
610,672 | <p>Could anyone help me with homework or give me a hint? Any help would be highly appreciated.</p>
<p>Given a set of N distinct objects:</p>
<p>How many ways are there to pick any number of them to be in a pile while the rest are in anotherpile? If your answer is written in terms of binomial coecients, use the Binom... | Eric Auld | 76,333 | <p>Think about going to each object and flipping a switch on it, L or R, to decide which pile it goes in. How many choices do you have to make in this process? Now make sure to divide by two because we don't care to distinguish between the left and the right piles.</p>
|
2,571,031 | <p>I need to solve the quadratic programming problem $$ \text{minimize}\,\, \sum_{j=1}^{n}(x_{j})^{2} \\ \text{subject to}\,\,\, \sum_{j=1}^{n}x_{j}=1,\\ 0 \leq x_{j}\leq u_{j}, \, \, j=1,\cdots , n $$</p>
<p>I know that the first thing I need to do is form the Lagrangian. </p>
<p>Now, for a problem in standard form... | copper.hat | 27,978 | <p>The primal problem is $\inf_x \sup_{\mu, \lambda \ge 0, \nu \ge 0 } L(x,\lambda, \nu, \mu)$, the dual is $ \sup_{\mu, \lambda \ge 0, \nu \ge 0 }\inf_x L(x,\lambda, \nu, \mu)$.</p>
<p>Since ${\partial L(x,\lambda, \nu, \mu) \over \partial x} = 2x + \lambda - \nu + \mu e$, where $e=(1,1,...)^T$, we can compute an ex... |
2,571,031 | <p>I need to solve the quadratic programming problem $$ \text{minimize}\,\, \sum_{j=1}^{n}(x_{j})^{2} \\ \text{subject to}\,\,\, \sum_{j=1}^{n}x_{j}=1,\\ 0 \leq x_{j}\leq u_{j}, \, \, j=1,\cdots , n $$</p>
<p>I know that the first thing I need to do is form the Lagrangian. </p>
<p>Now, for a problem in standard form... | robjohn | 13,854 | <p><strong>Basic Variational Approach</strong></p>
<p>Since
$$
\sum_{j=1}^nx_j=1\tag1
$$
any variation of the $x_j$'s must satisfy
$$
\sum_{j=1}^n\delta x_j=0\tag2
$$
At an interior critical point of
$$
\sum_{j=1}^nx_j^2\tag3
$$
we will have
$$
\sum_{j=1}^n2x_j\delta x_j=0\tag4
$$
At an interior critical point, any ch... |
3,700,367 | <p><strong>What is the <em>average</em> distance from any point on a unit square's perimeter to its center?</strong></p>
<p>The distance from a square's corner to its center is <span class="math-container">$\dfrac{\sqrt{2}}{2}$</span> and from a point in the middle of a square's side length is <span class="math-contai... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Proving <span class="math-container">$M^{n-1} = 0 \implies$</span> the set <span class="math-container">$\{I,M,\dots,M^{n-1}\}$</span> is linearly dependent (not linearly independent) is easy: simply note that any set that includes the zero-vector (or in this case the zero matrix, since our ve... |
4,558,460 | <p>According to the implicit function theorem(on <span class="math-container">$\mathbb R^2$</span> for simplicity), if <span class="math-container">$\displaystyle\frac{\partial f}{\partial y}\ne 0$</span> at <span class="math-container">$(x_0, y_0)$</span>, then on a neighborhood of <span class="math-container">$(x_0, ... | Andrew D. Hwang | 86,418 | <p>tl; dr: There is No Hope of a converse along the suggested lines.</p>
<hr />
<p>Arguably the simplest, most devastating counterexample is <span class="math-container">$f(x, y) = y^{3}$</span>, whose zero level is the <span class="math-container">$x$</span>-axis, the graph of a constant function, but whose differenti... |
839,124 | <p>This is similar to an exercise I just posted. The necessary part is easy, but the sufficient condition I'm having trouble seeing.</p>
<p>$\Rightarrow$. Since $(x,y)=g,$ there exist integers $x_1, y_1$ such that $x=gx_1, y=gy_1$. Since $[x,y]=l$, there exist integers $x_2, y_2$ such that $l=xx_2=yy_2$. Then
$$l=... | Bill Dubuque | 242 | <p>$\begin{eqnarray}{\bf Hint}\quad\ g\mid \ell &\iff&\!\!\! (g,\,\ell)\, =\, g\quad\ &\rm or&\ \quad g\le \ell &\!\iff&\!\! g\wedge\ell\, =\, g\quad\text{in lattice language} \\
&\iff&\!\!\! {\bf [\,}g,\,\ell\,{\bf ]}\, =\, \ell\ \ \ & &\ \ \ \ \phantom{g\le \ell} &\!\iff... |
44,868 | <p><strong>Bug introduced in version 8 or earlier and fixed in 10.0</strong></p>
<hr>
<p>I have created a notebook with two cells. This is the content of the first:</p>
<pre><code>g = Graph[{1 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 3, 1 \[UndirectedEdge] 3, 1 \[UndirectedEdge] 4, 4 \[UndirectedEdge] 5, 4 \[Undirec... | István Zachar | 89 | <p>This is a bug in <code>Pick</code> caused by <code>SparseArray</code>, has nothing to do with <code>Graph</code>. Minimal example (<code>SparseArray</code> object is the fullform version of your <code>vLM</code>):</p>
<pre><code>x = {1, 2, 3, 4, 5, 6};
Pick[x, SparseArray[Automatic, {6}, 0, {1, {{0, 3}, {{2}, {3}, ... |
897,633 | <p><strong>First question:</strong></p>
<p>Let's say we have a hypothesis test:</p>
<p>${ H }_{ 0 }:u=100$
and ${ H }_{ 1 }:u\neq 100$.</p>
<p>The sample has a size of 10 and gives an average $u=103$ and a p-value = 0.08.
The level of significance is 0.05.</p>
<p>I'm asked the following question (exam):</p>
<p>A) ... | heropup | 118,193 | <p>A hypothesis test in the frequentist sense is a procedure by which one arrives at a decision about whether the data contains sufficient evidence to accept the alternative hypothesis. In other words, there are two choices: either you reject the null $H_0$, or the test is inconclusive.</p>
<p>The reason why you can... |
3,193,305 | <p>A random variable <span class="math-container">$x$</span> from the set <span class="math-container">$\{1, 2, ... ,n\}. $</span> Let <span class="math-container">$x$</span> has distribution function <span class="math-container">$f(k) = Y(n) · g^k$</span> where <span class="math-container">$g$</span> is a fixed numbe... | Lutz Lehmann | 115,115 | <p>You should have found out that the Wronskian is constant as the coefficient of the first derivative term is zero.</p>
<p>After that, it is just a matter of re-scaling one or both of the solutions to get the Wronski-determinant to have the value 1 at one and thus every point.</p>
<hr>
<p>(<em>Add</em>) Interpretin... |
389,750 | <p>Given A(1,4) and B (3,-5) use the dot product to find point C so that triangle ABC is a right angle triangle.</p>
| user77528 | 77,528 | <p>Just put $(C-A).(B-A)=0$, that should solve it. Note that there are infinitely many such triangles.</p>
|
2,290,395 | <p>What if in Graham’s Number every “3” was replaced by “tree(3)” instead? How big is this number? Greater than Rayo’s number? Greater than every current named number?</p>
| Daniela Bellachioma | 919,701 | <p>No, Rayo's Number is just too big, imagine a Googol symbols in the first order set theory, you cannot express it, why? because even writing down a symbol per Planck time (5.39 x 10^-44 seconds) it would still take about 10^48 years, and another problem is the space, the number of particles in the observable is about... |
3,145,973 | <p>Show that for every integer <span class="math-container">$n ≥ 3$</span>, the number <span class="math-container">$n!e$</span> is not an integer.</p>
<p>I have shown the inequality <span class="math-container">$\displaystyle0< \sum_{m=n+1} \frac{1}{m!} < \frac{1}{n!}$</span> for <span class="math-container">$n... | Rylee Lyman | 447,318 | <p>Maybe let's just talk through why restrictions is the way to go.</p>
<p>Suppose we have <span class="math-container">$\phi\colon G \to G$</span> an automorphism. We want to show that <span class="math-container">$\phi(N) = N$</span>. Since <span class="math-container">$K$</span> is characteristic, we know that <spa... |
2,632,696 | <p>I have this equation: $x^2y'+y^2-1=0$. It's an equation with separable variable. When I calculate the solution do I have to consider the absolute value for the argument of the log? </p>
| mordecai iwazuki | 167,818 | <p>Sorry I meant to write $\tanh^{-1}(y)$ in my comment, so the answer can be found as follows,</p>
<p>$$\frac{dy}{1-y^2} = \frac{dx}{x^2}\\
\implies\tanh^{-1}(y) = -\frac{1}{x} + c\\
\implies y = \tanh(c-\frac{1}{x})$$</p>
|
537,965 | <p><span class="math-container">$X_0:\Omega\rightarrow I$</span> is a random variable where <span class="math-container">$I$</span> is countable. Also <span class="math-container">$Y_1,Y_2,\dots$</span> are i.i.d. <span class="math-container">$\text{Unif}[0,1]$</span> random variables. </p>
<p>Define a sequence <span ... | José Luis León | 551,176 | <p>The question is a particular case of a more general theorem:</p>
<blockquote>
<p>Let <span class="math-container">$S$</span> be a numerable set and let <span class="math-container">$X_0:\Omega \to S$</span> be a random variable on <span class="math-container">$(\Omega, \mathcal{F}, P)$</span>. Let <span class="ma... |
177,209 | <p>I found the following problem while working through Richard Stanley's <a href="http://www-math.mit.edu/~rstan/bij.pdf">Bijective Proof Problems</a> (Page 5, Problem 16). It asks for a combinatorial proof of the following:
$$ \sum_{i+j+k=n} \binom{i+j}{i}\binom{j+k}{j}\binom{k+i}{k} = \sum_{r=0}^{n} \binom{2r}{r}$$
w... | Sasha | 11,069 | <p>Restating your question, you are seeking to find the generating function of the left-hand-side:
$$
g(x) = \sum_{n=0}^\infty x^n \sum_{i+j+k=n}\binom{i+j}{i} \binom{j+k}{j} \binom{k+i}{k} = \sum_{i=0}^\infty \sum_{j=0}^\infty \sum_{k=0}^\infty x^{i+j+k} \frac{(i+j)! (i+k)! (j+k)!}{i!^2 j!^2 k!^2}
$$
First, carry o... |
177,209 | <p>I found the following problem while working through Richard Stanley's <a href="http://www-math.mit.edu/~rstan/bij.pdf">Bijective Proof Problems</a> (Page 5, Problem 16). It asks for a combinatorial proof of the following:
$$ \sum_{i+j+k=n} \binom{i+j}{i}\binom{j+k}{j}\binom{k+i}{k} = \sum_{r=0}^{n} \binom{2r}{r}$$
w... | Rijul Saini | 27,729 | <p>Recall that <span class="math-container">$\sum_{n \ge 0} \binom nm x^n = \frac{x^m}{(1-x)^{m+1}}$</span>.</p>
<p>We have <span class="math-container">\begin{align*}
\sum_{i+j+k=n}\binom{i+j}{i} \binom{j+k}{j} \binom{k+i}{k} &= \sum_{i,j}\binom{i+j}{i} \binom{n-i}{j} \binom{n-j}{i} \\ &= \sum_{i,j}\binom{i+j}... |
1,640,285 | <p>A single-celled spherical organism contains $70$% water by volume. If it loses $10$% of its water content, how much would its surface area change by approximately?</p>
<ol>
<li>$3\text{%}$</li>
<li>$5\text{%}$</li>
<li><p>$6\text{%}$</p></li>
<li><p>$7\text{%}$</p></li>
</ol>
| Jack's wasted life | 117,135 | <p>If $V$ is its volume and $S$ is its surface area, $S$ is proportional to $V^{2\over3}$ so
$$
{dS\over S}={2\over3}{dV\over V}={2\over3}{0.1\times0.7V\over V}\approx0.05
$$</p>
|
1,089,078 | <p>Suppose we have a deck of cards, shuffled in a random configuration. We would like to find a $k$-bit code in which we explain the current order of the cards. This would be easy to do for $k=51 \cdot 6=306$, since we could encode our deck card-by-card, using $2$ bits for the coloring and $4$ bits for the number on ea... | Aryabhata | 1,102 | <p>Order the permutations of $\{1,2,\dots, 52\}$ in lexicographic order.</p>
<p>Encode the $r^{th}$ permutation in that list as $r$ (requiring no more than $226$ bits).</p>
<p>Find a way to get to and get back the permutation, given $r$.</p>
|
1,089,078 | <p>Suppose we have a deck of cards, shuffled in a random configuration. We would like to find a $k$-bit code in which we explain the current order of the cards. This would be easy to do for $k=51 \cdot 6=306$, since we could encode our deck card-by-card, using $2$ bits for the coloring and $4$ bits for the number on ea... | Loren Pechtel | 23,794 | <p>To extend upon Arthur's solution:</p>
<p>There is a fair amount of space that goes unused between the highest available card and the end of the bit field. A fairly simple way to recover a bit of this is to encode two cards at a time. Using his same basic system I get 240 bits needed, only 14 bits above the theore... |
990,796 | <p>I have a homework question in a discrete mathematics class that asks me to determine how many 7-digit id numbers <strong>do not</strong> contain three consecutive sixes. </p>
<p>It seems clear that I should approach this by determining the number that <strong>do</strong> have three consecutive sixes and subtracting... | Steve Kass | 60,500 | <p>You started out just fine by counting the following kinds of numbers:</p>
<pre><code>___ ___ ___ ___ ___ ___ ___
6 6 6 1
6 6 6 2
6 6 6 3
6 6 6 4
6 6 6... |
990,796 | <p>I have a homework question in a discrete mathematics class that asks me to determine how many 7-digit id numbers <strong>do not</strong> contain three consecutive sixes. </p>
<p>It seems clear that I should approach this by determining the number that <strong>do</strong> have three consecutive sixes and subtracting... | Masacroso | 173,262 | <p>You can see it as a sequence of X numbers different than 6 where you put between them, or in the final of the string, a group of six of any cardinality between 0 and 2.</p>
<p>At maximum, because digit length is 7, you can have a string of 7 different numbers with gaps/ends with cardinality 0.</p>
<p>For any gap w... |
1,357,638 | <p>Here is the problem:</p>
<p>Suppose $n$ people are at a party, and some number of them shake hands. At the end of the party, each guest $G_i$, $1 \leq i \leq n$ shares that they shook hands $x_i$ times. Assume there were a total of $h \geq 0$ handshakes at the party. Use induction on $h$ to prove that:</p>
<p>$x_i... | André Nicolas | 6,312 | <p>Assume that the handshakes occur <em>sequentially</em>: At time $1$ there is a handshake, then there is a handshake at time $2$, and so on. Let us suppose that when $h=k$, then $x_1+x_2+\cdots +x_n=2k$. Now at time $k+1$, a new handshake takes place, say between person $i$ and person $j$. Then $x_i$ is incremented b... |
2,972,950 | <p>Everything on this question is in complex plane.</p>
<p>As the book describes a property of a winding number, it says that:</p>
<blockquote>
<p>Outside of the [line segment from <span class="math-container">$a$</span> to <span class="math-container">$b$</span>] the function <span class="math-container">$(z-a) / ... | Alexander Gruber | 12,952 | <p>The imaginary part of <span class="math-container">$u=\frac{z-a}{z-b}$</span> is <span class="math-container">$$\frac{u-\overline{u}}{2i}=\frac{1}{2i}\left(\frac{z-a}{z-b}-\overline{\frac{z-a}{z-b}}\right)=\frac{1}{2i}\left(\frac{z-a}{z-b}-\frac{\overline{z}-\overline{a}}{\overline{z}-\overline{b}}\right)$$</span></... |
3,965,455 | <p>Find <span class="math-container">$E(X^3)$</span> given <span class="math-container">$X$</span> is in <span class="math-container">$Exp(2)$</span></p>
<p>My idea is that we can use <span class="math-container">$f_X(x)=2e^{-2x}$</span> and integrate <span class="math-container">$\int_{0}^{a}x^3f_X(x)dx$</span>. But f... | Tan | 814,070 | <p>Observe that, <span class="math-container">$$f(x)=\frac{10x}{x-10}=\frac{10x-100+100}{x-10}=10+\frac{100}{x-10}$$</span> So, you should find all <span class="math-container">$x \in \mathbb Z$</span> such that <span class="math-container">$x-10$</span> divides <span class="math-container">$100$</span>, which can be d... |
2,408,223 | <p>Compute $\int_0^2 \lfloor x^2 \rfloor\,dx$.</p>
<p>The challenging part isn't the problem itself, but the notation around the x^2. I don't know what it is. If someone could clarify, that would be great!</p>
<p>Edit: Clarified that it represents the floor function, can anyone give me a hint on how to start working ... | Xander Henderson | 468,350 | <p>That is the greatest integer, or floor, function. The notation $\lfloor x \rfloor$ stands for the greatest integer less than or equal to $x$. Think of it as "rounding down" to the next integer.</p>
|
1,301,116 | <p>We know that if $f \in \mathcal R[a,b]$ and if $a = c_0 < c_1<\cdots<c_m =b$, then the restrictions of $f$ to each subinterval $[c_{i-1},c_i]$ are Riemann integrable.</p>
<p>Is the converse true, i.e if $f ; [a,b] \to \Bbb R$, and $a = c_0 < c_1<\cdots<c_m =b$ and that the restrictions of $f$ to e... | user1337 | 62,839 | <p>As $f$ is Riemann integrable on $[c_{i-1},c_i]$, for any $\epsilon>0$ there exists a partition $P_i$ of it such that $U(f,P_i)-L(f,P_i)< \epsilon/m$. If you use the partitions $\{P_i\}_{i=1}^m$ to create a partition of $[a,b]$ in the obvious way, you're done.</p>
|
43,611 | <p>I posted this on Stack Exchange and got a lot of interest, but no answer.</p>
<p>A recent <a href="http://people.missouristate.edu/lesreid/POW12_0910.html" rel="nofollow">Missouri State problem</a> stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing i... | Community | -1 | <p>Decompose a line without a point into a union of disjoint half-open intervals. Put copies of this line on the plane so that the distinguished point is $(0,0)$ and the lines point in all possible directions. You have covered the plane without one point, $(0,0)$. Now take one of these lines and replace it by a line co... |
43,611 | <p>I posted this on Stack Exchange and got a lot of interest, but no answer.</p>
<p>A recent <a href="http://people.missouristate.edu/lesreid/POW12_0910.html" rel="nofollow">Missouri State problem</a> stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing i... | rpotrie | 5,753 | <p>If you consider upper semicontinuous decompositions on compact connected sets, then, in <a href="https://www.ams.org/journals/bull/1968-74-01/S0002-9904-1968-11919-6/S0002-9904-1968-11919-6.pdf" rel="nofollow noreferrer">this paper</a> it is proved that it is not possible to fill any euclidean space in such a way.</... |
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Peter Taylor | 46,140 | <p><a href="http://openproblemgarden.org/" rel="noreferrer">Open Problem Garden</a></p>
<p>I occasionally stumble across this in my search results. It's currently skewed heavily to graph theory, which suggests that the user base also skews that way. Con: it doesn't appear to be very active; the most recent addition is ... |
438,263 | <p>Is there a concrete example of a <span class="math-container">$4$</span> tensor <span class="math-container">$R_{ijkl}$</span> with the same symmetries as the Riemannian curvature tensor, i.e.
<span class="math-container">\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} = R_{klij}, \\
R... | Gerry Myerson | 3,684 | <p>If it's a problem in Number Theory, the annual West Coast Number Theory meetings have a problem session, and the problems get collected & edited & posted to <a href="https://westcoastnumbertheory.org/problem-sets/" rel="noreferrer">https://westcoastnumbertheory.org/problem-sets/</a></p>
<p>If you can't come ... |
143,070 | <p>Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case. </p>
<p><img src="https://i.stack.imgur.com/yhrcd.jpg" alt="square"></p>
<p>Now we seek some addition condition on $X\to Y$ that forces the right squar... | john | 8,751 | <p>A sufficient condition in a category with pullbacks is that $X \to Y$ be a pullback stable regular epimorphism: a regular epimorphism all of whose pullbacks are also regular epimorphisms. This property holds in any regular category.</p>
<p>Stability implies that $A \to B$ and $A \to P$ are (pullback stable) regula... |
143,070 | <p>Suppose whole square and the left square in the diagram below are pullbacks, then we may wonder whether the right square is a pullback. It is usually not the case. </p>
<p><img src="https://i.stack.imgur.com/yhrcd.jpg" alt="square"></p>
<p>Now we seek some addition condition on $X\to Y$ that forces the right squar... | Michal R. Przybylek | 13,480 | <p>I tried to write the explanation of the comment about a dozen of times, but was never satisfied with the result. Finally, I decided to write a full note (I will try to put it on arXiv in a few minutes; <a href="http://arxiv.org/abs/1311.2974" rel="nofollow">here it is</a>) describing the natural setting for such que... |
36,774 | <p>Do asymmetric random walks also return to the origin infinitely?</p>
| Did | 6,179 | <p>This is a consequence of the law of large numbers. The position $S_n$ at time $n$ is the sum of $S_0$ and of $n$ i.i.d. displacements, each with expectation $m\ne0$, hence $S_n/n\to m$ almost surely. In particular, $|S_n|\ge |m|n/2$ for every $n\ge N$ where $N$ is random and almost surely finite, which implies $S_n\... |
36,774 | <p>Do asymmetric random walks also return to the origin infinitely?</p>
| Conrado Costa | 226,425 | <p>It depends. If you consider a Random Walk in a Random Environment, it may be asymmetric and recurrent. See <a href="https://arxiv.org/pdf/0707.3160.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/0707.3160.pdf</a></p>
<p>Also, if your walk is homogeneous,</p>
<p>$$ X_i = \begin{cases} +1 \text{ with probabil... |
296,727 | <p><b>Assuming that G is a finite cyclic group, let "a" be the product of all the elements in the group.</b> </p>
<p>i. <b> If G has odd order, then a=e.</b> Is this because there are an even number of non-trivial elements must have their inverses within the non-trivial factors within the product?</p>
<p>ii. <b> If G... | DonAntonio | 31,254 | <p>Let $\,G=\langle x\rangle =\{1\,,\,x\,,\,x^2\,,\,\ldots\,,\,x^{n-1}\}\,$:</p>
<p>$$x\cdot x^2\cdot\ldots\cdot x^{n-1}=x^{\frac{n(n-1)}{2}}=\begin{cases}x^{(n-1)k}\neq 1&\,,\,\,\text{ if}\,\;\;\;n=2k\,\,\,\text{ is even}\\{}\\x^{kn}=1&\,,\,\text{ if}\,\;\;n-1=2k\,\,\,\text{is even}\end{cases}$$</p>
|
3,841,542 | <p>I am trying to show that <span class="math-container">$\sqrt{\sqrt{2}+5}$</span> is constructible through a diagram.</p>
<p>I know how to show something of the form <span class="math-container">$\sqrt[n]{a}$</span> is constructible through a diagram, but I am really having a difficult time with this one.</p>
<p>Any ... | Abhijeet Vats | 426,261 | <p><strong>Hint:</strong></p>
<p>Have you considered differentiating:</p>
<p><span class="math-container">$$\int_{0}^{\infty} e^{-ax} \sin(kx) \ dx = \frac{k}{a^2+k^2}$$</span></p>
<p>with respect to <span class="math-container">$k$</span> instead? :-)</p>
|
3,841,542 | <p>I am trying to show that <span class="math-container">$\sqrt{\sqrt{2}+5}$</span> is constructible through a diagram.</p>
<p>I know how to show something of the form <span class="math-container">$\sqrt[n]{a}$</span> is constructible through a diagram, but I am really having a difficult time with this one.</p>
<p>Any ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
234,866 | <p>Problem: when I draw a rectangle and put a coloured edge around it, the displayed edge is centred along the nominal edge and if it follows the same course as one of the axes then it does not show up. For example:</p>
<pre><code>Graphics[{EdgeForm[Red], FaceForm[], Rectangle[{0, 0}, {4, 3}]}, Axes -> True]
</code>... | Jagra | 571 | <p>You can change the <code>AxisOrgin</code>.</p>
<pre><code>Graphics[{EdgeForm[Red], FaceForm[], Rectangle[{0, 0}, {4, 3}]},
Axes -> True, AxesOrigin -> {-1, -1}]
</code></pre>
<p><a href="https://i.stack.imgur.com/sJPqE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sJPqE.png" alt="enter i... |
234,866 | <p>Problem: when I draw a rectangle and put a coloured edge around it, the displayed edge is centred along the nominal edge and if it follows the same course as one of the axes then it does not show up. For example:</p>
<pre><code>Graphics[{EdgeForm[Red], FaceForm[], Rectangle[{0, 0}, {4, 3}]}, Axes -> True]
</code>... | Brett Champion | 69 | <p>The first way that comes to mind for me is to use <code>Offset</code> to "shrink" the rectangles by the width of the edges:</p>
<pre><code>Graphics[Table[{
EdgeForm[{AbsoluteThickness[1], RandomChoice[{Red, Blue}]}], FaceForm[],
Rectangle[Offset[{1/2, 1/2}, {i, j}], Offset[{-1/2, -1/2}, {i + 1, j ... |
445,816 | <p>I have to show that</p>
<blockquote>
<blockquote>
<p>$\mathbb{C}=\overline{\mathbb{C}\setminus\left\{0\right\}}$,</p>
</blockquote>
</blockquote>
<p>what is very probably an easy task; nevertheless I have some problems.</p>
<p>In words this means: $\mathbb{C}$ is the smallest closed superset of $\mathbb{C... | Cameron Buie | 28,900 | <p><strong>Hint</strong>: (I assume you're using the usual topology on $\Bbb C$.) Note that $\{0\}$ is not an open set, so $\Bbb C\setminus\{0\}$ is not a closed set. $\Bbb C$ is closed, however. Is $\Bbb C$ a superset of $\Bbb C\setminus\{0\}$? If so, how many extra points does it have? What can you conclude?</p>
|
2,423,569 | <p>I am asked to show that if $T(z) = \dfrac{az+b}{cz+d}$ is a mobius transformation such that $T(\mathbb{R})=\mathbb{R}$ and that $ad-bc=1$ then $a,b,c,d$ are all real numbers or they all are purely imaginary numbers. </p>
<p>So far I've tried multiplying by the conjugate of $cz+d$ numerator and denominator and see i... | Wojowu | 127,263 | <p>Hints: if we assume $c\neq 0$, $T(z)=\frac{a}{c}-\frac{1}{c(cz+d)}$. Letting $z$ go to infinity shows $\frac{a}{c}$ is real, hence $c(cz+d)$ is real for all $z\in\mathbb R$. From there conclude $cd$ and $c^2$ are real. The rest should be easy. The case $c=0$ is also not difficult.</p>
|
997,463 | <p>For example, a complex number like $z=1$ can be written as $z=1+0i=|z|e^{i Arg z}=1e^{0i} = e^{i(0+2\pi k)}$.</p>
<p>$f(z) = \cos z$ has period $2\pi$ and $\cosh z$ has period $2\pi i$.</p>
<p>Given a complex function, how can we tell if it is periodic or not, and further, how would we calculate the period? For ex... | lab bhattacharjee | 33,337 | <p>If Trigonometric substitution is not mandatory, write $$x^3=\frac{x(4x^2+9)-9x}4$$</p>
<p>$$\implies\frac{x^3}{(4x^2+9)^{\frac32}}=\frac14\cdot\frac x{\sqrt{4x^2+9}}-\frac94\cdot\frac x{(4x^2+9)^{\frac32}}$$</p>
<p>Now write $4x^2+9=v$ or $\sqrt{4x^2+9}=u\implies4x^2+9=u^2$</p>
|
3,140,696 | <p>I am trying to figure out the proper definition of a small circle on a biaxial ellipsoid of revolution. One definition is the intersection of the ellipsoid with a cone emanating from the center of the ellipsoid.</p>
<p>The other way I can imagine to define it is a plane intersecting the ellipsoid in which the plane... | Community | -1 | <p>The only small circles that you can get are found by intersection with a plane orthogonal to the revolution axis, which are also the intersections with a coaxial cone with apex at the center. This can be sketched in 2D:</p>
<p>Intersection with other planes yield ellipses, and intersection with non-coaxial cones ar... |
1,523,392 | <p>This is question 2.4 in Hartshorne. Let $A$ be a ring and $(X,\mathcal{O}_X)$ a scheme. We have the associated map of sheaves $f^\#: \mathcal{O}_{\text{Spec } A} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A \rightarrow \Gamma(X,\mathcal{O}_X)$. Thus there is a natural map $\alp... | Shuhang | 239,526 | <p>You have the restriction map: $r_i: \Gamma(X)\longrightarrow\Gamma(U_i)$,
This gives you $Spec\Gamma(U_i)\longrightarrow SpecA$. Gluing works because the restriction maps are compatible to each other.</p>
|
1,523,392 | <p>This is question 2.4 in Hartshorne. Let $A$ be a ring and $(X,\mathcal{O}_X)$ a scheme. We have the associated map of sheaves $f^\#: \mathcal{O}_{\text{Spec } A} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A \rightarrow \Gamma(X,\mathcal{O}_X)$. Thus there is a natural map $\alp... | Takumi Murayama | 116,766 | <p><strong>EDIT:</strong> I want to add that the relevant parts of EGA to compare are [<a href="http://www.numdam.org/item?id=PMIHES_1960__4__5_0" rel="noreferrer">EGAI</a>, Thm. 1.7.3], which is the analogue of [Hartshorne, II, Prop. 2.3(c)], and [<a href="http://www.numdam.org/item?id=PMIHES_1960__4__5_0" rel="norefe... |
165,489 | <p>I have problem solving this equation, smallest n such that $1355297$ divides $10^{6n+5}-54n-46$. I tried everything using my scientific calculator, but I never got the correct results(!).and finally I gave up!. Could you help me find the first 2 solutions for this equation ? (thanks.)</p>
| KennyColnago | 3,246 | <p>There is no need for a brute force search. There is also no need for compilation, which may run into the maximum compiled integer limit. A faster solution follows from a bit of theory. You can get the first 225000 solutions (up to $n\approx 306$ billion) in less than a second.</p>
<p>The original equation is ${\rm ... |
3,238,563 | <p>I have a question about a proof I saw in a book about basic algeba rules. The rule to prove is:
<span class="math-container">\begin{eqnarray*}
\frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0}
\end{eqnarray*}</span></p>
<p>And the proof: </p>
<p><span class="math-container">\begin{eqnarray*}
1 = a \frac{... | LarrySnyder610 | 663,638 | <p>I don't think they are replacing <span class="math-container">$a \to \frac1a$</span>. I think the logic in the first implication is they are taking 1 over both sides. So <span class="math-container">$1/1\to 1$</span> on the LHS, and on the RHS,
<span class="math-container">$$a \to \frac1a \text{ and } \frac1a \to \... |
3,238,563 | <p>I have a question about a proof I saw in a book about basic algeba rules. The rule to prove is:
<span class="math-container">\begin{eqnarray*}
\frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0}
\end{eqnarray*}</span></p>
<p>And the proof: </p>
<p><span class="math-container">\begin{eqnarray*}
1 = a \frac{... | zwim | 399,263 | <p>I'm not terribly fond of this proof.</p>
<p>I would rather go on defining the inverse:</p>
<ul>
<li><span class="math-container">$y$</span> is the inverse of <span class="math-container">$x\iff xy=1$</span> then we write it <span class="math-container">$y=\frac 1x$</span>.</li>
<li>since everything is symmetrical ... |
2,929,094 | <p>Differentiation of
<span class="math-container">$\int_{a(x)}^{b(x)} f(x,t)\,\text{d}t$</span> is done by Leibniz's integral rule:
<span class="math-container">$$\frac{\text{d}}{\text{d}x} \left (\int_{a(x)}^{b(x)} f(x,t)\,\text{d}t \right )= f\big(x,b(x)\big)\cdot \frac{\text{d}}{\text{d}x} b(x) - f\big(x,a(x)\big)\... | nonuser | 463,553 | <p><em>Nonstandard, simple, a bit over powered, but most creative solution:</em></p>
<hr>
<p>Consider a homothety <span class="math-container">$H_1$</span> with center at <span class="math-container">$O$</span> which takes <span class="math-container">$A\mapsto C$</span>. Then it takes <span class="math-container">$F... |
187,432 | <p>Can we evaluate the integral using <a href="http://en.wikipedia.org/wiki/Jordan%27s_lemma#Application_of_Jordan.27s_lemma">Jordan lemma</a>?
$$ \int_{-\infty}^{\infty} {\sin ^2 (x) \over x^2 (x^2 + 1)}\:dx$$</p>
<p>What de we do if removeable singularity occurs at the path of integration?</p>
| robjohn | 13,854 | <p>Using $\sin^2(z)=\frac12(1-\cos(2z))$, you should be able to handle this in much the same way as <a href="https://math.stackexchange.com/questions/160022/integration-by-means-of-complex-analysis/160099#160099">this answer</a>.</p>
<hr>
<p><strong>Details</strong> (modified from the answer mentioned above)</p>
<p>... |
4,247,268 | <p><strong>Q:</strong></p>
<blockquote>
<p>If <span class="math-container">$f\left(x\right)=-\frac{x\left|x\right|}{1+x^{2}}$</span> then find <span class="math-container">$f^{-1}\left(x\right)$</span></p>
</blockquote>
<p>My approach:</p>
<ol>
<li>Dividing the cases when <span class="math-container">$x\ge0$</span> and... | Kavi Rama Murthy | 142,385 | <p>What you have done is correct. All you have to do is switch <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. You writing <span class="math-container">$f^{-1}(y)$</span> in terms of <span class="math-container">$y$</span> so change <span class="math-container">$y$</span> to <span c... |
2,716,363 | <p>I understand the core principles of how to prove by induction and how series summations work. However I am struggling to rearrange the equation during the final (induction step).</p>
<p>Prove by induction for all positive integers n,</p>
<p>$$\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2$$</p>
<p>After both proving fo... | ℋolo | 471,959 | <p>$$\frac{1}{4}k^2(k+1)^2+(k+1)^3=\frac{1}{4}k^2(k+1)^2+(k+1)(k+1)^2=(k+1)^2\left(\frac14k^2+k+1\right)=(k+1)^2\left(\frac14k^2+\frac44(k+1)\right)=(k+1)^2\left(\frac14(k^2+4(k+1))\right)=\frac14(k+1)^2\left(k^2+4(k+1))\right)$$</p>
|
96,864 | <p>I don't think that this is the case. I am reading over one of my professor's proof, and he seems to use this fact. Here is the proof:
Let $B$ be a Boolean algebra, and suppose that $X$ is a dense subset of $B$ in the sense that every nonzero element of $B$ is above a nonzero element of $X$. Let $p$ be an element in ... | Qiaochu Yuan | 232 | <p>The set of upper bounds is closed under intersection, so $p \cap q$ is an upper bound less than $p$. </p>
|
96,864 | <p>I don't think that this is the case. I am reading over one of my professor's proof, and he seems to use this fact. Here is the proof:
Let $B$ be a Boolean algebra, and suppose that $X$ is a dense subset of $B$ in the sense that every nonzero element of $B$ is above a nonzero element of $X$. Let $p$ be an element in ... | Michael Greinecker | 21,674 | <p>Let $p$ and $q$ be upper bounds of $Y$. Then $p\wedge q$ is an upper bound of $Y$ and $p\wedge q\leq p$ and $p\wedge q\leq q$. Now if for all upper bounds $q$ of $Y$, $p\leq p\wedge q\leq q$, $p$ must be the least upper bound. Otherwise $p\wedge q<p$ for some $q$ and $p\wedge q$ is a strictly lower upper bound of... |
5,528 | <p>Let H be a subgroup of G. (We can assume G finite if it helps.) A complement of H in G is a subgroup K of G such that HK = G and |H∩K|=1. Equivalently, a complement is a transversal of H (a set containing one representative from each coset of H) that happens to be a group.</p>
<p>Contrary to my initial naive... | Noah Snyder | 22 | <p>There's an excellent online resource for group theory definitions and theorems called <a href="http://groupprops.subwiki.org/wiki/Main_Page" rel="nofollow">"Groupprops, The Group Properties Wiki."</a> It's still in pre-alpha, but it has a lot of stuff and hopefully will continue to improve.</p>
<p>In particular, t... |
960,865 | <p>How can i prove 2 is a primitive root mod 37, without calculating all powers of 2 mod 37?</p>
| Peter Taylor | 5,676 | <p>The order of any element in an order 36 group is a factor of 36 (Lagrange's theorem), so it suffices to check $2^a \not\equiv 1\pmod{37}$ for $a \in \{1, 2, 3, 4, 6, 9, 12, 18\}$.</p>
<p>But in fact that can be reduced further: if $2^a \equiv 1\pmod{37}$ then $2^{ab} \equiv 1\pmod{37}$, so the tests can be reduced ... |
636,089 | <p>Let the function
$$
f(x) = \begin{cases} ax^2 & \text{for } x\in [ 0, 1], \\0 & \text{for } x\notin [0,1].\end{cases}
$$ Find $a$, such that the function can describe a probability density function. Calculate the expected value, standard deviation and CDF of a random variable X of such distribution.</p>
<p... | NasuSama | 67,036 | <p><strong>Expected Value</strong></p>
<p>In general, the expected value is determined by this following expression</p>
<p>$$\mathrm{E}(X) = \int_{-\infty}^{\infty} xf(x)\,dx$$</p>
<p>where $f(x)$ is the probability density function. For your problem, the expected value is</p>
<p>$$\begin{aligned}
\mathrm{E}(X) &a... |
3,514,659 | <p>Let <span class="math-container">$f:[a, b]\rightarrow \mathbb{R}$</span> be continuous and let <span class="math-container">$F:[a, b]\rightarrow \mathbb{R}$</span> be defined by <span class="math-container">$F(x)=\int_a^x f(t) \,dt$</span>. Then <span class="math-container">$F$</span> is differentiable whose derivat... | Lee Mosher | 26,501 | <p>Here's a hint: start with the fact that if <span class="math-container">$a \le x \le b$</span> then
<span class="math-container">$$\int_a^b f(t) \, dt = \int_a^x f(t) \, dt + \int_x^b f(t) \, dt
$$</span>
which is a simple consequence of the definition of definite integrals.</p>
|
3,514,659 | <p>Let <span class="math-container">$f:[a, b]\rightarrow \mathbb{R}$</span> be continuous and let <span class="math-container">$F:[a, b]\rightarrow \mathbb{R}$</span> be defined by <span class="math-container">$F(x)=\int_a^x f(t) \,dt$</span>. Then <span class="math-container">$F$</span> is differentiable whose derivat... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Note that
<span class="math-container">$$
A = \int_a^b f(t) dt
$$</span>
is a constant and
<span class="math-container">$$
G(x) = \int_x^b f(t) dt = \int_a^b f(t) dt - \int_a^x f(t) dt = A - F(x)
$$</span></p>
|
343,281 | <p>Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:</p>
<blockquote>
<p>The axioms of arithmetic are obviously correct, and the principles of proof obviously preserve correctness. Why cannot one simply conclude consistency, i.e., w... | Thomas Benjamin | 20,597 | <p>In his answer to David Roberts' mathoverflow question, "<span class="math-container">$Z_2$</span> versus second-order <span class="math-container">$PA$</span>" (question 97077), Prof. Ali Enayat writes (Under the subheading, "Regarding the second question)":</p>
<blockquote>
<p>One way to see this is based on an ... |
120,525 | <p>I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities". </p>
<p>Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" divisor on $X$, say $D$ has strictly normal crossings. (Maybe we could just take $D$ to be a closed subscheme?... | David E Speyer | 297 | <p>Here is a heuristic argument that there is nothing to explain:</p>
<p>The probability that <span class="math-container">$p$</span> divides the sum of the preceding primes is <span class="math-container">$1/p$</span>. So the expected number of primes less than <span class="math-container">$10^9$</span> with this pro... |
120,525 | <p>I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities". </p>
<p>Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" divisor on $X$, say $D$ has strictly normal crossings. (Maybe we could just take $D$ to be a closed subscheme?... | Javier | 31,020 | <p>Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.</p>
<p>I studied with FLorian Luca [1] a related problem that could help to answer question Q2:</p>
<p>Let $A$ the set of integers $n$ such that the sum of t... |
1,068,609 | <p>My prof has taught us that we can express the proposition $⟦$there are exactly two entities characterized by $P$$⟧$ thus:</p>
<p><img src="https://i.stack.imgur.com/aIJbL.jpg" alt="enter image description here"></p>
<p>That proposition looks verbose, despite the fact that it references just two entities. It seems ... | Spenser | 39,285 | <p>Let $G(s)$ be an anti-derivative of
$$g(s)=\frac{\sqrt{1+s^2}}{s}.$$
By the Fundamental Theorem of Calculus,
$$F(t)=\int_1^{t^2}g(s)ds=G(t^2)-G(1)$$
so
$$F'(t)=2tg(t^2)=\frac{2\sqrt{1+t^4}}{t}.$$</p>
|
1,068,609 | <p>My prof has taught us that we can express the proposition $⟦$there are exactly two entities characterized by $P$$⟧$ thus:</p>
<p><img src="https://i.stack.imgur.com/aIJbL.jpg" alt="enter image description here"></p>
<p>That proposition looks verbose, despite the fact that it references just two entities. It seems ... | Cookie | 111,793 | <p>$$F'(t)=\frac d{dt} \left(\int_1^{t^2} \frac{\sqrt{1+s^2}}{s} ds \right)=\frac{\sqrt{1+t^2}}{t} \cdot \frac d{dt}(t^2)$$</p>
|
3,285,036 | <p>Obviously this cannot happen in a right rectangle, but otherwise - as Sin(0) or 180 or 360 equals 0, I guess there is no way to find out what the original angle was?</p>
| guitarphish | 687,096 | <p>This is the inverse problem. The inverse sin function <span class="math-container">$\arcsin(x)$</span> is not one-to-one <a href="https://en.wikipedia.org/wiki/Injective_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Injective_function</a> as you pointed out. In general this can type of situation ... |
25,284 | <p>I recently worked my way through Walter Warwick Sawyer's book, <em>Mathematician's Delight</em>, which has opened my eyes to Maths. I used to fear maths, feeling I was incapable. Sawyer (among other authors) has a gift for teaching the subject. I now feel much more confident tackling Maths problems, I have a better ... | Sue VanHattum | 60 | <p>I already loved math when I encountered his books. But yes, I was also inspired. <em><strong>Mathematician's Delight</strong></em> might be the one I put dozens of page markers in, so I could find all the great ideas again. He helped me think about how I might want to teach differently, especially in beginning algeb... |
25,284 | <p>I recently worked my way through Walter Warwick Sawyer's book, <em>Mathematician's Delight</em>, which has opened my eyes to Maths. I used to fear maths, feeling I was incapable. Sawyer (among other authors) has a gift for teaching the subject. I now feel much more confident tackling Maths problems, I have a better ... | JRN | 77 | <p>No, I was never inspired by him because I had never heard of him before you mentioned it.</p>
<hr />
<p><sup>Note: My answer is for the original version of the question. Since then, the question has been edited so my answer is no longer appropriate.</sup></p>
|
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | J. M.'s persistent exhaustion | 50 | <p>Here's one method, using a slightly modified example:</p>
<pre><code>mylist = {"[a]", "a", "[b]", "b", "b", "b", "[ c ]", "c", "c"};
pos = Append[Flatten[Position[mylist,
s_String /; StringMatchQ[s, "[" ~~ ___]]], Length[mylist] + 1]
{1, 3, 7, 10}
Take[mylist, {#1, #2 - 1}] & @@@ Partition[pos... |
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | Leonid Shifrin | 81 | <p>At the risk of being annoying, I will pitch the linked lists again. Here is the code using linked lists:</p>
<pre><code>ClearAll[split];
split[{}] = {};
split[l_List] :=
Reap[split[{}, Fold[{#2, #1} &, {}, Reverse@l]]][[2, 1]];
split[accum_, {h_, tail : {_?sectionQ, _} | {}}] :=
split[Sow[Flatten[{accum, h... |
11,973 | <p>I have a list of strings called <code>mylist</code>:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
</code></pre>
<p>I would like to split <code>mylist</code> by "section headers." Strings that begin with the character <code>[</code> are section headers in my application. Thus, I ... | Murta | 2,266 | <p>Here's my suggestion:</p>
<pre><code>mylist = {"[a]", "a", "a", "[b]", "b", "b", "[ c ]", "c", "c"};
Split[mylist, ! StringMatchQ[#2, "[*"] &]
</code></pre>
<p>and we get:</p>
<pre><code>{{"[a]", "a", "a"}, {"[b]", "b", "b"}, {"[ c ]", "c", "c"}}
</code></pre>
|
316,601 | <p>Can anyone tell me what I am doing wrong? need to prove for $k\ge2$
$$(5-\frac5k )(1+\frac{1}{(k+1)^2}) \le 5 - \frac{5}{k+1}$$$$(5-\frac5k )(1+\frac{1}{(k+1)^2})= 5(1-\frac1k)(1+\frac1{(k+1)^2})$$
$$=5(1+\frac1{k+1)^2}-\frac1k-\frac1{k(k+1)^2})$$
$$= 5(1-\frac{k^2+k+2}{k(k+1)^2})$$
$$=5(1-\frac{k(k+1)}{k(k+1)^2}+\f... | Chrisuu | 63,178 | <p>Your process is correct until the 4th step. There is a sign error in the 5th step. Gigili's answer provides the correct solution, but if you are still stumped about where the minus sign is coming from, here is a more detailed look at all the manipulations and properties involved to get from the 4th step in your proc... |
2,755,213 | <p><strong>Question.</strong> Find, with proof, the possible values of a rational number $q$ for which $q+\sqrt{2}$ is a reduced quadratic irrational.</p>
<p>So, by definition a <em>quadratic irrational</em> is one of the form $u+v\sqrt{d}$ where $u,v\in\Bbb Q, v\neq 0$ and $d$ being square-free. Then, it is said to b... | Kirk Fox | 551,926 | <p>Since we have $-1 < q-\sqrt{2} < 0$ and $q+\sqrt{2} > 1$, we can simply move the square roots to get the inequalities
$$-1 + \sqrt{2} < q < \sqrt{2} \text{ and } q > 1 - \sqrt{2}$$
Because $-1 + \sqrt{2} > 1 - \sqrt{2}$, our first inequality is all that is necessary. This means we can define a s... |
2,162,452 | <p>Question: Find the slope of the tangent line to the graph of $r = e^\theta - 4$ at $\theta = \frac{\pi}{4}$.</p>
<p>$$x = r\cos \theta = (e^\theta - 4)\cos\theta$$</p>
<p>$$y = r\sin \theta = (e^\theta - 4)\sin\theta$$</p>
<p>$$\frac{dx}{d\theta} = -e^\theta\sin\theta + e^\theta\cos\theta + 4\sin\theta$$
$$\frac{... | Dr. Sonnhard Graubner | 175,066 | <p>solving the quadratic equation we get
$$x_1=m+\sqrt{-2m-3}$$
$$x_2=m-\sqrt{-2m-3}$$ then we get
$$x_1^2+x_2^2=2m^2-4m-6$$
can you proceed?
i have got $$min(2m^2-4m-6)$$ under the condition $$-2m-3\geq 0$$ is equal to $$\frac{9}{2}$$ and $$m_{min}=-\frac{3}{2}$$</p>
|
2,162,452 | <p>Question: Find the slope of the tangent line to the graph of $r = e^\theta - 4$ at $\theta = \frac{\pi}{4}$.</p>
<p>$$x = r\cos \theta = (e^\theta - 4)\cos\theta$$</p>
<p>$$y = r\sin \theta = (e^\theta - 4)\sin\theta$$</p>
<p>$$\frac{dx}{d\theta} = -e^\theta\sin\theta + e^\theta\cos\theta + 4\sin\theta$$
$$\frac{... | Bernard | 202,857 | <p>You don't need to use the quadratic formula; $r^s+s^2$ is a symmetric polynomial in $r$ and $s$, hence it can be expressed as a function of the <em>elementary symmetric functions</em>:
$$S=r+s=2m,\quad P=rs=m^2+2m+3.$$
Indeed $\;r^2+s^2=S^2-2P=2m^2-4m-6$. This is a quadratic polynomial in $m$, and its minimum is at... |
2,601,851 | <p>I have a binary variable $y_{t}$ that is equal to $1$ iff the job is scheduled at slot $t$. I need to write constraints that guarantee that if the job is scheduled somewhere, then it must be scheduled for a period of $A$ consecutive slots. I tried to write it this way:</p>
<p>$\sum_{t'=t}^{t+A-1}y_{t'}\geqslant A y... | Mark | 147,256 | <p>Recall that $\epsilon$ is an arbitrary positive number. In other words, we know that $\inf\{U(f, P')\} - \sup\{L(f, P')\}$ is smaller than every positive number. Also, $\sup\{L(f, P')\} \leq \inf\{U(f, P')\}$. The only way we can satisfy both these properties is if $\inf\{U(f, P')\} - \sup\{L(f, P')\} = 0$.</p>
|
2,601,851 | <p>I have a binary variable $y_{t}$ that is equal to $1$ iff the job is scheduled at slot $t$. I need to write constraints that guarantee that if the job is scheduled somewhere, then it must be scheduled for a period of $A$ consecutive slots. I tried to write it this way:</p>
<p>$\sum_{t'=t}^{t+A-1}y_{t'}\geqslant A y... | Brian Borchers | 6,310 | <p>What's critical here is the quantifier "for every $\epsilon > 0$." </p>
<p>Spivak has shown that the difference is less than $\epsilon$ for every $\epsilon>0$. That means (for example) that the difference is less than 0.1, and 0.01, and 1.0e-300, and 1.0e-3000, or any other tiny quantity you want to pick. ... |
3,511,660 | <p>Can you help me please I could not figure this out.</p>
<p>Given: </p>
<p><span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span>, <span class="math-container">$f'(0)$</span> exists, <span class="math-container">$f(x)\neq0$</span> and for all <span class="math-container">$a, b\in\mathbb{R}$</span>, <span ... | azif00 | 680,927 | <p>First note that <span class="math-container">$f(0) = 1$</span>. Next, for any real number <span class="math-container">$x$</span>,
<span class="math-container">$$
\begin{align}
\frac{f(x+h) - f(x)}{h} &= \frac{f(x)f(h) - f(x)}{h} \\
&= f(x)\frac{f(h) - 1}{h} \\
&= f(x)\frac{f(h) - f(0)}{h}
\end{align}
$$... |
1,970,458 | <p>Consider a stock that will pay out a dividends over the next 3 years of $1.15, $1.8, and 2.35 respectively. The price of the stock will be $48.42 at time 3. The interest rate is 9%. What is the current price of the stock?</p>
| alexjo | 103,399 | <p>$D_1=1.15,\, D_2=1.8,\,D_3=2.35, \,P_3=48.42,\,r=9\%$.
$$
P_0=\frac{D_1+P_1}{1+r}=\frac{D_1}{1+r}+\frac{D_2+P_2}{(1+r)^2}=\frac{D_1}{1+r}+\frac{D_2}{(1+r)^2}+\frac{D_3+P_3}{(1+r)^3}
$$</p>
|
2,991,366 | <blockquote>
<p>Consider a point <span class="math-container">$Q$</span> inside the <span class="math-container">$\triangle ABC$</span> triangle, and <span class="math-container">$M$</span>, <span class="math-container">$N$</span>, <span class="math-container">$P$</span> the intersections of <span class="math-container... | Phil H | 554,494 | <p>Any isosceles triangle can comply with your stated conditions. Consider the sketch below. The positions of <span class="math-container">$E$</span> and <span class="math-container">$F$</span> can be adjusted between <span class="math-container">$E_1$</span> and <span class="math-container">$E_2$</span> and <span clas... |
4,399,371 | <p>According to my textbook, the formula for the distance between 2 parallel lines has been given as below:</p>
<p><a href="https://i.stack.imgur.com/ZQtQk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZQtQk.png" alt="enter image description here" /></a></p>
<p>Where PT is a vector from the first l... | TomKern | 908,546 | <p>The length of the cross product of two vectors is <span class="math-container">$|a \times b| = |a| |b| \sin(\theta)$</span> where <span class="math-container">$\theta$</span> is the angle between them.</p>
|
3,525,814 | <p>One reasonably well-known property of the Thue-Morse sequence is that it can be used to provide solutions to the <a href="https://en.wikipedia.org/wiki/Prouhet%E2%80%93Tarry%E2%80%93Escott_problem" rel="nofollow noreferrer">Prouhet–Tarry–Escott problem</a> - for example, splitting the first eight nonnegative integer... | Chen Shuwen | 954,936 | <p>The question you ask is also call "Multigrade Chains".
You may see plenty of such solutions on my below website:
<a href="http://eslpower.org/chains.htm" rel="nofollow noreferrer">Multigrade Chains</a></p>
<p>Example:</p>
<p><span class="math-container">$ \\\ \ \ \ 0^k+ 567^k+644^k+1778^k+1855^k+2422^k \... |
3,858,517 | <p>Is it possible to count exactly the number of binary strings of length <span class="math-container">$n$</span> that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as <span class="math-container">$0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$</s... | BillyJoe | 573,047 | <p>Here I am going to use generating functions like in <a href="https://math.stackexchange.com/a/1956058/573047">this answer to a related problem</a> to compute columns of @RobPratt table for <span class="math-container">$k \ge 3$</span>.</p>
<p>We can define:</p>
<p><span class="math-container">$$S_y(k,i) = \left\{\te... |
2,416,671 | <p>Here is the problem: Let $K$ be a compact subset of $ \mathbb{R}^{m} $ ($m>1$) with empty interior and such that $\mathbb{R}^{m}\setminus K $ has no bounded component. For $n=1,2,...$, we define
$$K_{n}=\lbrace x\in \mathbb{R^{m}}: distance (x,K)=1/n\rbrace.$$ Prove that for all $x\in K$, there is a sequence $(y... | H. H. Rugh | 355,946 | <p>We fix $x\in K$. The following is a bit along the spirit in your attempt.</p>
<p>The function ${\rm dist} (z,K)$ is (1-Lipschitz) continuous in $z$.
It follows that also
$$M(r)= \sup \{ {\rm dist}(z,K): \|z-x\|\leq r \}, \; \; r\geq 0$$
is (1-Lipschitz) continuous in $r\geq 0$. </p>
<p>We have $M(0)=0$ (since $... |
3,520,354 | <p>In the problem <span class="math-container">$\frac{8.01-7.50}{3.002}$</span></p>
<p>Why would the answer be <span class="math-container">$0.17$</span> and not <span class="math-container">$0.170$</span>? My least amount of <em>sig figs</em> is <span class="math-container">$3$</span> in the original equation. The o... | Community | -1 | <p>Unless the 0 in <span class="math-container">$7.50$</span> is an exact figure, you can't deem it significant.</p>
<blockquote>
<p>The significant figures (also known as the significant digits and decimal places) of a number are digits that carry meaning contributing to its measurement resolution. This includes all d... |
226,097 | <p>I am having a problem with the following exercise. Can someone help me please.</p>
<p>Find all functions $f$ for which $f'(x)=f(x)+\int_{0}^1 f(t)dt$</p>
<p>Thank you in advance</p>
| Beni Bogosel | 7,327 | <p>Your differential equation is $f'=f+c$ where $c$ is a constant. This is a first order linear differential equation and the solution has a simple formula. Here you can deduce it:</p>
<p>$f'-f=c$ is equivalent to $(e^{-x}f(x))'=ce^{-x}$ and therefore
$$ f(x)=e^x(-ce^{-x}+d)=-c+de^x$$</p>
<p>Now you have to impose th... |
2,637,812 | <p>Here is dice game question about probability.</p>
<p>Play a game with $2$ die. What is the probability of getting a sum greater than $7$?</p>
<p>I know how the probability for this one is easy, $\cfrac{1+2+3+4+5}{36}=\cfrac 5{12}$.</p>
<p>I don't know how to solve the follow-up question:</p>
<p>Play a game with ... | kiyomi | 527,262 | <p>$$\lim_\limits{x\to0}\frac{x^2-x^2+\frac{x^4}{2}+\mathcal{O}\left(x^5\right)}{x^4+\mathcal{O}\left(x^5\right)}=\lim_\limits{x\to0}\frac12\cdot\frac{x^4+\mathcal{O}\left(x^5\right)}{x^4+\mathcal{O}\left(x^5\right)}=\frac12.$$</p>
|
2,637,812 | <p>Here is dice game question about probability.</p>
<p>Play a game with $2$ die. What is the probability of getting a sum greater than $7$?</p>
<p>I know how the probability for this one is easy, $\cfrac{1+2+3+4+5}{36}=\cfrac 5{12}$.</p>
<p>I don't know how to solve the follow-up question:</p>
<p>Play a game with ... | user | 505,767 | <p>As an alternative</p>
<p>$$\frac{x^2-\log(1+x^2)}{x^2\sin^2x}=\frac{x^2-\log(1+x^2)}{x^4}\cdot\frac{x^2}{\sin^2x}\to \frac12$$</p>
<p>indeed</p>
<p>$\frac{x^2}{\sin^2x}\to 1$ by standard limit</p>
<p>and let $y=x^2\to 0$</p>
<p>$$\frac{x^2-\log(1+x^2)}{x^4}=\frac{y-\log(1+y)}{y^2}\stackrel{HR}\implies\frac{1-\f... |
396,085 | <p>The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is</p>
<p>a) $48$</p>
<p>b) $144$</p>
<p>c) $24$</p>
<p>d) $72$</p>
<p>I don't want whole solution just give me the hint how can I solve it.Thanks.</p>
| Manuj Khullar | 100,314 | <p>There is a direct formula:</p>
<p>Let
$$s = (m_1+m_2+m_3)/2,$$</p>
<p>Then
$$\text{area} = \frac{4}{3}\sqrt{s(s-m_1)(s-m_2)(s-m_3)}.$$</p>
<p>This gives answer of above question as $72$.</p>
|
2,012,020 | <p>Would it have sense to defined Cauchy sequence in non metric space ? (for example, if $(X,T)$ is a topology space :
$$\forall U\in T, 0\in U, \exists N\in \mathbb N: x_n-x_m\in U.$$</p>
<p>And if yes, would it be interesting ?</p>
| Lee Mosher | 26,501 | <p>As you wrote it, no, this does not make sense, because there is no subtraction operation $x_n - x_m$ in a metric space.</p>
<p>However, there is a theory of <a href="https://en.wikipedia.org/wiki/Uniform_space" rel="nofollow noreferrer">uniform spaces</a>, which is a special kind of topological space $X$ which need... |
2,012,020 | <p>Would it have sense to defined Cauchy sequence in non metric space ? (for example, if $(X,T)$ is a topology space :
$$\forall U\in T, 0\in U, \exists N\in \mathbb N: x_n-x_m\in U.$$</p>
<p>And if yes, would it be interesting ?</p>
| Bargabbiati | 352,078 | <p>If (xα) is a sequence from $\mathbb N$ into X, and if Y is a subset of X, then we say that (xα) is eventually in Y (or residually in Y) if there exists an α in A so that for every β in A with β ≥ α, the point xβ lies in Y.</p>
<p>If (xα) is a sequence in the topological space X, and x is an element of X, we say tha... |
397,040 | <p>What is the domain for $$\dfrac{1}{x}\leq\dfrac{1}{2}$$</p>
<p>according to the rules of taking the reciprocals, $A\leq B \Leftrightarrow \dfrac{1}{A}\geq \dfrac{1}{B}$, then the domain should be simply $$x\geq2$$</p>
<p>however negative numbers less than $-2$ also satisfy the original inequality. When am I missin... | egreg | 62,967 | <p>Inequalities with fractions require some care. If you write yours in the form
$$
\frac{1}{x}-\frac{1}{2}\le0
$$
you see that it's equivalent to
$$
\frac{2-x}{2x}\le0
$$
or to
$$
\frac{x(2-x)}{2x^2}\le0
$$
Since $2x^2>0$, we can remove the denominator (but keeping the condition that $x\ne0$). So we have the standa... |
1,097,658 | <p>I read in a notes: A semi-function is a relation (not a function) with of the form $y^2=f(x)$. </p>
<p>It seems that we can get more that one values for $f(x)$ for a single value of $x$. </p>
<p>Could any-one please help me to understand this notion.</p>
<p>The link of the note is <a href="http://www.google.co.in... | KittyL | 206,286 | <p>The notes say $f(x)$ is a function, so we can get exactly one value for $f(x)$ with a single value of $x$. However, we can get two values of $y$ given a single value of $x$, since it could be plus or minus.</p>
|
1,143,200 | <p>The jump diffusion model is defined as
$$dS_t = \mu S_t dt + \sigma S_t dW_t + S_t d \left(\sum^{N_t}_{i=1}(V_i - 1)\right)\;\;\;\;\;\;\;(1)$$
, where ${V_i}$ is a sequence of iid non-negative random variables and it is independent of $W_t$. In the Merton's jump diffusion model ,
$log(V) \sim N(\mu_J, \sigma^2_J)$ ... | user48672 | 138,298 | <p>In your case </p>
<p>the quadratic variation can be obtained by formally squaring the SDE
$
d[S,S]_t = dS_t \cdot dS_t =\left( \mu S_t dt + \sigma S_t dW_t + S_t d \left(\sum^{N_t}_{i=1}(V_i - 1)\right) \right)^2
$
using the fact that
$
(\mu S_t)^2 dt^2 = 0
$
and
$
\mu S_t dt \cdot \sigma S_t dW_t =0
$
we get</p>... |
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