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 |
|---|---|---|---|---|
1,892,669 | <p>Let $F=\left \langle a,b \right \rangle$ be the rank 2 free group. Then, a map $f:\left \langle a \right \rangle \rightarrow \left \langle b \right \rangle$ with $f(a):=b$ is an isomorphism with $\left \langle x,f(x) \right \rangle$ is a rank 2 free group for any $x\in\left\langle a\right\rangle - \{ 1\}.$</p>
<p>C... | Giulio Bresciani | 118,028 | <p>Let $p$ be a prime of $R$, and $q$ it's restriction to $D$. We have $D_q\subseteq R_p \subseteq K$ with $D_q$ a DVR. It is immediate to see that there are no proper intermediate rings between a DVR and its fraction field, hence if $p$ is nonzero then $D_q=R_p\neq K$, if $p=(0)$ then $D_q=R_p= K$. Hence all localizat... |
3,353,952 | <p>True or not: <span class="math-container">$~f:[0,3] \to \mathbb R~$</span> can never be surjective</p>
<p>Is the <span class="math-container">$~f~$</span> continuous? And if its an closed interval that means it has a fixed point in <span class="math-container">$~0~$</span> and <span class="math-container">$~3~$</sp... | Lutz Lehmann | 115,115 | <p>The task now actually is
<span class="math-container">$$ y^4dx+2xy^3dy=\dfrac{ydx-xdy}{x^3y^3} $$</span>
then the same transformations as below for the former reproduction of the equation apply, only the residual factors will have to account for the moved factor <span class="math-container">$y^3$</span>. This gives
... |
3,353,952 | <p>True or not: <span class="math-container">$~f:[0,3] \to \mathbb R~$</span> can never be surjective</p>
<p>Is the <span class="math-container">$~f~$</span> continuous? And if its an closed interval that means it has a fixed point in <span class="math-container">$~0~$</span> and <span class="math-container">$~3~$</sp... | Certainly not a dog | 691,550 | <p>It is probably done by inspection rather than intuition. </p>
<p>See that the RHS is <span class="math-container">$\frac{y^3}{x}$</span> of <span class="math-container">$-d(\frac{y}{x})$</span>. Dividing both sides by that will leave us with: <span class="math-container">$$-d\frac{y}{x} = yxdx + 2x^2dy$$</span></p>... |
188,465 | <p>This question has multiple parts to it. The setup is that I have a matrix that is a function of two parameters a and b. I wish to plot the eigenvalues of this matrix along a general path in the a-b plane and I want these two branches to have the correct coloring. For example, for the simple path for {a,b} from {0,0}... | Henrik Schumacher | 38,178 | <p>The problem is that at the time of the call to <code>Plot</code>, it is not clear that it is about two function that are to plot. Actually, you tell <em>Mathematica</em>'s <code>Plot</code> command to plot an <span class="math-container">$\mathbb{R}^2$</span>-valued function. You can circumvent this issue, e.g. with... |
188,465 | <p>This question has multiple parts to it. The setup is that I have a matrix that is a function of two parameters a and b. I wish to plot the eigenvalues of this matrix along a general path in the a-b plane and I want these two branches to have the correct coloring. For example, for the simple path for {a,b} from {0,0}... | bbgodfrey | 1,063 | <p>1) Basically, Mathematica has no way of knowing whether to treat the two curves as having the same or distinct colors. Using <code>Evaluate</code> tells it to use distinct colors. (The underlying reasons relate to the order of evaluation.)</p>
<p>2) <code>Evaluate</code> has no effect for <code>testfunc</code>, ... |
1,432,776 | <blockquote>
<p>For a real symmetric $2\times2$ matrix $A$, we define a $2\times2$ matrix $\sqrt{A}$ that satisfies $\sqrt{A}r_1=\sqrt{\lambda_1}r_1,\ \sqrt{A}r_2=\sqrt{\lambda_2}r_2$ where $r_1, r_2$ are the eigenvectors of $A$, and $\lambda_1, \lambda_2$ are the corresponding eigenvalues. Show that $(\sqrt{A})^2$ i... | Antitheos | 261,163 | <p><strong>Hints:</strong> </p>
<ul>
<li>If you show that $A v_i= Bv_i$ for all elements of a basis$ \{v_1,v_2,...,v_n \}$ of $\mathbb{R}^n$, you habe proven that $A=B$ by linearity of those functions. </li>
<li>$\{r_1,r_2\}$ is a basis of $\mathbb{R}^2$.</li>
</ul>
|
58,901 | <p>I had previously asked:
<a href="https://mathoverflow.net/questions/47943/narratives-in-modular-curves">Narratives in Modular Curves</a></p>
<p>Since then, I've read quite a bit more (but not nearly enough) and I have a few follow up questions about the big picture. As you will soon see, I'm confused about how to t... | Franz Lemmermeyer | 3,503 | <p>This is an answer to the part of your question concerning the case of dimension $1$. I'll omit any details in higher dimension; even if you're only vaguely familiar with elliptic curves you will see the bigger picture.</p>
<ol>
<li><p>A Pell conic is an affine curve of the form $C_N: Q_0(X,Y) = 1$, where $Q_0$ is t... |
1,635,188 | <p>I am trying to solve $$\lim_{x\to 0} \frac{\int_{0}^{x^2} x^2 e^{-t^2} dt}{-1+e^{-x^4}} $$ using The Fundamental Theorem of Calculus (FTC).
I already know that the answer is -1, </p>
<p>Using FTC (correct me if I am wrong) we get:
$$
\lim_{x\to 0} \frac{x^2 e^{-x^4}}{-1+e^{-x^4}}
$$</p>
<p>Which has the result of ... | Future Algebraist | 209,594 | <p>Since this is a $\frac{0}{0}$ limit, you can use L'Hospital's Rule. To do so, factor the numerator as $x^2 g(x)$, where $$g(x) = \int_{0}^{x^2} e^{-t^2} dt$$</p>
<p>You are allowed to pull the $x^2$ outside of the integrand as the integral is with respect to $t$. Use FTC (and the product rule) to calculate the deri... |
2,636,131 | <blockquote>
<p>Suppose that each square of a $4 \times 7$ chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color?</p>
</blockquote>
... | tdugan | 529,865 | <p>isnt there a trivial case?</p>
<p>"the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color?"</p>
<p>Each square of the board is a rectangle and every corner of the square must be the same color because it is onl... |
287,976 | <p>I want to prove that the sum of the fourth powers of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $6n$. I consider the distance from 1 to the other $n$th roots of unity given by $\omega^k$, $k=1,2,\dots, (n-1)$. So basically my working is
$$\sum_1^{n-1}|1-\omega^k|^4=\sum_1^{n-1}(|1-\o... | Did | 6,179 | <p>Let $\omega_k=\omega^k$. Recall that $(\omega_k)_{1\leqslant k\leqslant n}$ is the set of roots of the polynomial $x^n-1$. The sum
$$
\sum_{k=1}^n\omega_k=\sum_{k=1}^{n}\omega^k=1+\sum_{k=1}^{n-1}\omega^k
$$
is the sum of the roots of $x^n-1$, hence its opposite equals the coefficient of $x^{n-1}$ in this polynomia... |
2,737,823 | <p><strong>Background</strong></p>
<p>Hey everyone. I'm absolutely stumped on an exercise I am working on out of Axler's <em>Linear Algebra Done Right</em>, 3rd edition. Funnily enough, <a href="http://linear.axler.net/InnerProduct.pdf" rel="nofollow noreferrer">the sample chapter</a> available on his website is the... | Eric Wofsey | 86,856 | <p>This is not really a linear algebra problem. It is an analysis problem that involves definitions from linear algebra. So in order to solve it, you're mainly going to have to do analysis; the linear algebra is just a matter of writing down what the definitions mean.</p>
<p>So, let's write down the definition of $U... |
2,762,447 | <p>Consider the following non-linear differential equation,
$$
\dot{x}(t)=a-b\sin(x(t)), \ \ x(0)=x_0\in\mathbb{R},
$$
and assume that $a$ and $b$ are positive real numbers with $a>b$.
Note that the solution $x(t)$ exists and can be analytically computed (<a href="http://www.wolframalpha.com/input/?i=dx(t)%2Fdt%20%3... | anomaly | 156,999 | <p>Why are you computing $\phi$ explicitly? Just note that $\phi$ fixes $1^\perp\subset \mathbb{H}$ and clearly leaves the inherited norm invariant; that is, it lies in $SO(1^\perp) = SO(3)$.</p>
|
1,662,958 | <blockquote>
<p>Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice. </p>
</blockquote>
<p>The way I thought of it is this : if Bob gets $0$ heads... | user2357112 | 91,416 | <p>Get out some red paint. Paint all the heads sides on Bob's coins, and paint all the <em>tails</em> sides of Alice's coins. Bob wins if and only if at least $n + 1$ coins out of $2n + 1$ land red side up. By symmetry, the probability of this happening is $1/2$.</p>
|
181,532 | <blockquote>
<p>Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field.</p>
</blockquote>
<p>How can one justify the answer in the shortest number of lines?</p>
| Abhishek Parab | 645 | <p>$I$ must clearly be a prime ideal, since $R/I$ must be a field. Every prime ideal of $\mathbb Z[i]$ is maximal since its a PID. So $I$ can be taken to be any nonzero prime ideal. </p>
|
1,753,901 | <p>The question says:
Let $U$ be a set $ \{ 1, 2, . . . , n \}$ for an arbitrary positive integer $n$. How many subsets are there? How many possible relations of the form $A \subseteq B$ are there? Can you make an informed guess as to how many of these relations are true?</p>
<p>Since the given set has $n$ elements, t... | Nuvishramun | 612,546 | <p>This post is quite old, but I thought it might be useful to leave this here.</p>
<p>Consider the relation <span class="math-container">$A \subseteq B$</span>. For each element <span class="math-container">$x \in U$</span>, there are three ways in which that relation is true, namely, </p>
<ol>
<li><span class="math... |
2,565,880 | <p>Number of ways in which 5 boys and 4 girls can be seated around a circular table such that no two girls sit together and two particular boys are always together ?</p>
<p>The answer to this question is $3!2!4!$ . It is done by considering $2$ boys as one unit and the the number of units (of boys) is $4$ so they can ... | dxiv | 291,201 | <p>Alternative geometric solution: let $G$ be the centroid of the triangle with vertices $A(0,0)$, $B(0,3)$, $C(3,3)\,$. By the definition of the <a href="https://en.wikipedia.org/wiki/Centroid" rel="nofollow noreferrer">centroid</a>, the coordinates of $G$ will be $(\frac{0+0+3}{3}, \frac{0+3+3}{3})=(1,2)\,$.</p>
<p>... |
200,242 | <p>I saw the name $p$-adic group on a book I was reading, so I tried to find some related documents. Although I've found something on this topic, there is no definition.</p>
<p>Would anyone please explain the definition for a $p$-adic group to me? Thanks very much.</p>
| M Turgeon | 19,379 | <p>Since the only tag you put is group-theory, it may be the case that you do not have the appropriate background for understanding the following definition. Nonetheless, here is an attempt.</p>
<p>First, let $\mathbb{Q}_p$ be the field of $p$-adic numbers, and let $\overline{\mathbb{Q}}_p$ be an algebraic closure. Le... |
2,987,877 | <p>Find the inverse Laplace of <span class="math-container">$ \ \frac{2}{(s-1)^3(s-2)^2}$</span>. </p>
<p><strong>Answer:</strong></p>
<p>To do this we have to make partial fractions as follows:</p>
<p><span class="math-container">$ \frac{2}{(s-1)^3(s-2)^2}=\frac{A}{S-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-1)}+\frac{J}{(... | Nosrati | 108,128 | <p>Another way is using convolution
<span class="math-container">\begin{align}
\frac{2}{(s-1)^3(s-2)^2}
&= \frac{2}{(s-1)^3}\cdot\frac{1}{(s-2)^2} \\
&= {\cal L}\left(t^2e^t\right){\cal L}\left(te^{2t}\right) \\
&= \int_0^xt^2e^t(x-t)e^{2x-2t}\ dt \\
&= e^{2x}\int_0^xe^{-t}(xt^2-t^3)\ dt \\
&= \colo... |
4,187,052 | <p>I have a problem that looks like a typical problem of maximizing functions in a compact interval. However, I am not being able to prove the bound I need.</p>
<blockquote>
<p>Let <span class="math-container">$n\geq 6$</span> be an integer number. Consider the function: <span class="math-container">$$f(t) = \frac{n^2}... | Luis Ferroni | 312,208 | <p>Ok, with some clever separation into cases I managed to do it.</p>
<ul>
<li><p>If <span class="math-container">$1-t^2 < \frac{2}{3n}$</span> then just bound:
<span class="math-container">$$f(t) < \frac{n^2}{2}\cdot 1^{n-4}\cdot \frac{2}{3n} \cdot \left( 1 - \frac{n-3}{n}\right) =1.$$</span></p>
</li>
<li><p>If... |
312,249 | <p>I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given equation to:</p>
<p>$(n-2)! + (n-3)! + (n-4)! = 272n - 1088$ </p>
<p>However, I am not sure how I would solve for $\... | André Nicolas | 6,312 | <p>It is best to leave factorials out of this: $\dbinom{x}{2}=\dfrac{x(x-1)}{2}$.</p>
<p><strong>Remarks:</strong> $1.$ The binomial coefficients were simplified incorrectly. </p>
<p>$2.$ We don't even need "algebra" to solve the problem. Evaluate your expression, for $n=4, 5, \dots$. After a short while your sum is ... |
312,249 | <p>I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given equation to:</p>
<p>$(n-2)! + (n-3)! + (n-4)! = 272n - 1088$ </p>
<p>However, I am not sure how I would solve for $\... | Ramez Hindi | 89,017 | <p>$n=13$ after solving this equation $3n^2-21n-234=0
$</p>
|
4,321,742 | <p>In Spivak's Calculus, Ch. 5 on Limits, there is the following theorem about the uniqueness of a limit of a function near a point:</p>
<blockquote>
<p>A function cannot approach two different limits near <span class="math-container">$a$</span>. In other
words, if <span class="math-container">$f$</span> approaches <sp... | Claude Leibovici | 82,404 | <p>Similar to the <a href="https://math.stackexchange.com/questions/4178129/a-tough-integral-int-0-infty-frac-operatornamesech-pi-x14x2-ma">question</a> already mentioned by @Laxmi Narayan Bhandari, considering the <a href="https://functions.wolfram.com/ElementaryFunctions/Sech/06/ShowAll.html" rel="nofollow noreferrer... |
1,534,693 | <p>I am a little stuck on coming up with geometrical explanation for why the following equalities are true. I tried arguing the $\cos(\theta)$ is the projection to the x-axis of a vector $r$ inside a unit circle, so as it goes around by $2 \pi$, the projections on both the positive and negative part of the x-axis cance... | Eric Naslund | 6,075 | <p><strong>First set of equations:</strong> Here's a geometric reason. Imagine we place $1$ kg weights at the unit circle corresponding to angles of $\theta$, $\theta+\frac{2\pi}{3}$ and $\theta+\frac{4\pi}{3}$. These points form a configuration that is a rotation of </p>
<p><a href="https://i.stack.imgur.com/khv2h.jp... |
1,534,693 | <p>I am a little stuck on coming up with geometrical explanation for why the following equalities are true. I tried arguing the $\cos(\theta)$ is the projection to the x-axis of a vector $r$ inside a unit circle, so as it goes around by $2 \pi$, the projections on both the positive and negative part of the x-axis cance... | Mihir | 237,754 | <p>The answer by @Eric Naslund gives really good insight to first bit the problem. My answer is in the direction of the second bit, but doesnt present it as elegantly (as I cant seem to find a physical phenomena that makes use of those equations). It uses the first bit in the explaination and proof so hopefully, you sh... |
803,792 | <p>I have just taken calculus quiz but I could not find $\displaystyle \int_2^\infty\frac{\log^3(x-1)}{x^2}dx$? Any help would be appreciated. Thanks in advance.</p>
<p>EDIT:</p>
<p>Forgot to mention, my tutor gave us hints about this question.</p>
<ol>
<li>Use Taylor series</li>
<li>$\displaystyle \zeta(3)=\sum_{n... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
3,341,059 | <p>In design controller for a first order system such as:
<span class="math-container">$$\dot{x}=-ax+bu$$</span>
they assume that value of <span class="math-container">$a>0$</span>. I need to know is it possibile to neglect this assume an design the controller even if <span class="math-container">$a=0$</span> as sa... | Dr. Sonnhard Graubner | 175,066 | <p>Since we have the factor<span class="math-container">$$(x-3)^{\sqrt{2}}$$</span> it must be <span class="math-container">$$x>3$$</span> then it is <span class="math-container">$$(x-3)^{\sqrt{2}}$$</span> a real number.</p>
|
1,254,175 | <blockquote>
<p>Prove that $D_{12}\cong S_3 \times C_2$.</p>
</blockquote>
<p>I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how this will lead necessarily to a unique solutions.</p>
<p>No full solutions please hi... | hunter | 108,129 | <p>You know that $D_{12}$ is the group of symmetries of a Hexagon. Draw the three longest diagonals of the hexagon, and label them $a$, $b$, and $c$. Using this, can you describe a function from $D_{12}$ to $S_3$? What is the kernel of this function? Can you put a copy of $S_3$ back in $D_{12}$?</p>
|
4,490,862 | <p>Prove, using a combinatorial argument, that the numbers below are integers for any natural <span class="math-container">$n$</span>: <span class="math-container">$$\frac{(2n)!}{2^n}$$</span></p>
<p><strong>Attempt:</strong>
Suppose we organize a queue with <span class="math-container">$2n$</span> people, this can be ... | true blue anil | 22,388 | <ul>
<li>Line up <span class="math-container">$2n$</span> people in <span class="math-container">$(2n)!$</span> ways,</li>
<li>Take out two at a time serially to get <span class="math-container">$n$</span> pairs</li>
<li>Two ways to assign a leader from each such pair</li>
</ul>
<p>This represents <span class="math-con... |
432,416 | <p>Question is simple:</p>
<ol>
<li><p>Does the elliptical shape of the earth affect its radius? (Yes!!?)</p></li>
<li><p>If it is true: How?</p></li>
<li><p>How can I determine the exact distance between two points on the earth with this influence?</p></li>
</ol>
<p><strong>Notice: When I measure the distance betwee... | cffk | 36,388 | <p>In answer to question 3. Computing distance on an ellipsoid of revolution (oblate or prolate) is addressed in <a href="http://dx.doi.org/10.1007/s00190-012-0578-z" rel="nofollow">"Algorithms for geodesics"</a>. The algorithms given there are available in several different languages using
<a href="http://geographic... |
1,120,826 | <blockquote>
<p>Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$.</p>
</blockquote>
<p><strong>Work of OP:</strong> The thing is I have no idea, everything I tried ended up on nothing.</p>
<p><strong>Third party commentary:</strong> Standard ideas to attack such problems include induction and congruenc... | quid | 85,306 | <p>Note that </p>
<ul>
<li>$2^3$ is $-1$ modulo $9$ </li>
<li>$2^{3^n} = (2^{3})^{3^{n-1}}$ </li>
<li>$3^{n-1}$ is odd.</li>
</ul>
|
1,120,826 | <blockquote>
<p>Prove that $2^{3^n} + 1$ can be divided by $9$ for $n\ge 1$.</p>
</blockquote>
<p><strong>Work of OP:</strong> The thing is I have no idea, everything I tried ended up on nothing.</p>
<p><strong>Third party commentary:</strong> Standard ideas to attack such problems include induction and congruenc... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\rm \,\ {\rm mod}\,\ A^{\large B} + 1\!:\,\ \color{#c00}{A^B\equiv -1}\ \Rightarrow\ \color{}{A}^{\large BC}\equiv (\color{#c00}{A^{\large B}})^{\large C}\equiv (\color{#c00}{-1})^{\large C} $ </p>
|
4,155,766 | <p>The area of a triangle is <span class="math-container">$14\sqrt{3}$</span> <span class="math-container">$cm^2$</span>. The lengths of two sides of the triangle are <span class="math-container">$7$</span> <span class="math-container">$cm$</span> and <span class="math-container">$8$</span> <span class="math-container"... | g.kov | 122,782 | <p>Wlog let <span class="math-container">$a=7,\ b=8$</span> and <span class="math-container">$S=14\sqrt3$</span>.
It is more convenient to work with the squares,
<span class="math-container">$a^2=49,\ b^2=64,\ S^2=588$</span>.</p>
<p>From Heron’s formula for the area in terms of square sides,</p>
<p><span class="math-c... |
3,960,527 | <p>Let <span class="math-container">$X$</span> be a topological space and <span class="math-container">$A\subset X$</span>. Show that <span class="math-container">$X = int(A) \cup Fr(A)\cup int(X-A)$</span>, this being a union disjointed.</p>
<p>To show this equality I must show the inclusions:
<span class="math-contai... | Azur | 656,302 | <p>Yes.</p>
<p>The second inclusion comes for free, since <span class="math-container">$X$</span> is the topological space you're working in, so every set you're treating is a subset of <span class="math-container">$X$</span>.</p>
<p>The other needs a little more work. If you don't know where to start, try drawing <sp... |
1,069,120 | <p>I have my discrete structures exam tomorrow, and right now i am practicing mathematical induction, specially proofs. while proving, i just get confused because i don't understand what should i add or subtract to prove the inductive step. i was wondering if there is any tip or trick to know what should we add or subt... | Mauro ALLEGRANZA | 108,274 | <p>Consider the <a href="http://en.wikipedia.org/wiki/Tower_of_Hanoi" rel="nofollow">Tower of Hanoi</a> game and try to "see" the general procedure in it ...</p>
<p>We want to prove that :</p>
<blockquote>
<p>The minimum number of moves required to solve it is $2^n - 1$, where $n$ is the number of disks.</p>
</bloc... |
4,251,726 | <p>I posted <a href="https://math.stackexchange.com/questions/4247878/if-the-radius-of-convergence-of-the-series-sum-a-n-zn-is-r-whats-the-rad?noredirect=1#comment8830417_4247878">If the radius of convergence of the series $\sum a_n z^n$ is $R$, what's the radius of convergence of $\sum s_n z^n$, where $s_n$ partia... | PM 2Ring | 207,316 | <p>This is <em>not</em> an efficient way to compute zillions of digits of the reciprocal subfactorial series, but it's ok for computing a few thousand digits.</p>
<p>It's just a quick hack of <a href="https://math.stackexchange.com/a/1295561/207316">my <span class="math-container">$e$</span> calculator</a> which conver... |
3,603,312 | <p>there are two urns with White balls and Black balls. first urn has 21 whites and 5 blacks, second one has 8 whites and 9 blacks. we take 7 balls from first urn and put them into the second one. afterwards, out of the second urn we take one ball. what is the probability that it's white?</p>
<p>I've been struggling o... | Mano Prakash P | 765,546 | <p>I am about to give a detailed explanation in another way,
When we are picking 7 balls at a time from First Urn (Say Urn A), it is trivial that, we could transfer a minimum of 2 white balls and a maximum of 7 white balls to the second urn (Say Urn B).</p>
<p>So, on the whole, the number of white balls picked fro... |
2,133,984 | <p>If $f(s) = (1+s)^{(1+s)^{(1+s)/s}/s}$, show that $\lim_{s \to \infty} f(s)/s = 1$.</p>
<p>This function comes up
in the parameterization
of the solutions to
$x^y = y^x$.
See for example, here:
<a href="https://math.stackexchange.com/questions/1664284/are-there-real-solutions-to-xy-yx-3-where-y-neq-x">Are there real... | Brevan Ellefsen | 269,764 | <p>$$\frac{f(s)}{s} = (1+s)^{(1+s)^{(1+s)/s}/s}/s$$
$$\log(\frac{f(s)}{s}) = \frac{(1+s)^{(1+s)/s}\log(1+s)}{s}-\log(s)$$
We are now claiming that
$$\lim_{s \to \infty} \left(\frac{(1+s)^{(1+s)/s}\log(1+s)}{s}-\log(s)\right)=0$$</p>
<hr>
<p>We now see that the LHS can be changed into
$$ \lim_{s \to \infty}\left(\frac... |
4,645,763 | <p>Recently I played a little bit around with GeoGebra and I constructed the in- and circumcircle of a <span class="math-container">$\triangle ABC$</span> with <span class="math-container">$A=(0,0)$</span> and <span class="math-container">$B=(1,0)$</span> and I asked myself if it is possible to construct the area where... | robjohn | 13,854 | <p>In <a href="https://math.stackexchange.com/a/171641">this answer</a>, it is shown that the distance <span class="math-container">$d$</span> between the incenter and circumcenter is
<span class="math-container">$$
d^2=R(R-2r)\tag1
$$</span>
where <span class="math-container">$R$</span> is the circumradius and <span c... |
3,222,253 | <blockquote>
<p>Given <span class="math-container">$$ P = \left \{ (a,b,c,d) \in \mathbb R^4 \mid a + b + c + d = 0 \right \} $$</span> find <span class="math-container">$ P^\perp $</span>.</p>
</blockquote>
<p>Am I right if I multiply <span class="math-container">$$ P^T P = 0$$</span></p>
<p><span class="math-cont... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\dim P=3$</span>, <span class="math-container">$\dim P^\perp=1$</span>. Actually, <span class="math-container">$P^\perp=\mathbb R(1,1,1,1)^T$</span>, since, <em>by definition</em>, <span class="math-container">$P=\bigl((1,1,1,1)^T\bigr)^\perp$</span>.</p>
|
1,826,711 | <p>How can I find the smallest positive integer $n$ such that
$$(1-0.03)^n<0.03$$</p>
<p>without the help of a computer? </p>
| Ross Millikan | 1,827 | <p>I would start with $n \log(1-0.03) \lt \log 0.03, n \gt \frac {\log 0.03}{\log(1-0.03)}$. If you can use log tables, this is an easy divide. If not, I would hope that the Taylor series for the denominator, $-0.03 -\frac 12 \cdot 0.03^2 \approx \log (1-0.03)$ is accurate enough. You might even get by with the firs... |
1,826,711 | <p>How can I find the smallest positive integer $n$ such that
$$(1-0.03)^n<0.03$$</p>
<p>without the help of a computer? </p>
| Community | -1 | <p>I don't think there is a shortcut as the numbers $0.97$ and $0.03$ don't appear to have particular properties and the solution is the ratio of their logarithms, not especially easy to compute.</p>
<p>A simple but tedious possibility is by the use of exponential search followed by dichotomy. You start from $0.97^1$ ... |
2,373,109 | <blockquote>
<p>Let $\{x,y,z\}\subset[0,+\infty)$,and $x+y+z=6$. Show that:
$$xyz(x-y)(x-z)(y-z)\le 27$$</p>
</blockquote>
<p>I tried AM -GM but without success.
$$xyz\le\left(\dfrac{x+y+z}{3}\right)^3=8$$
maybe $$(x-y)(x-z)(y-z)\le \dfrac{27}{8}$$ it doesn't always true。</p>
| Michael Rozenberg | 190,319 | <p>We can assume that $(x-y)(x-z)(y-z)\geq0$.</p>
<p>Let $x+y+z=3u$, $xy+xz+yz=3v^2$, $xyz=w^3$ and $u=tw$.</p>
<p>Hence, we need to prove that
$$(x+y+z)^6\geq1728xyz(x-y)(x-z)(y-z)$$ or
$$27u^6\geq64w^3(x-y)(x-z)(y-z)$$ or
$$729u^{12}\geq4096w^6(x-y)^2(x-z)^2(y-z)^2$$ or
$$27u^{12}\geq4096w^6(3u^2v^4-4v^6-4u^3w^3+6... |
2,959,862 | <p>I'm having trouble finding the Fourier transform of <span class="math-container">$g(t) = \cos^2{a x}$</span>. </p>
<p>I know the answer has to be a summation of <span class="math-container">$3$</span> dirac delta functions, but I'm having trouble showing this. I'll show you where my work ran into a problem.</p>
<p... | Brightsun | 118,300 | <p>The problem you encounter essentially boils down to proving
<span class="math-container">$$
\int_{-\infty}^{+\infty}e^{-ikx}dx = 2\pi \delta(k)\,.
$$</span>
There are many ways to prove this fact. For instance, one can first prove that the Fourier transform extends in an invertible way to tempered distribution (to ... |
3,881,711 | <p>Sorry in advance, it is probably a stupid question.
I encountered it when I was thinking about the birthday problem. The probability of having at least one pair of the same birthday is
<span class="math-container">$$ 1- \frac{365\cdot364\cdot\ldots\cdot(365-n+1)}{365^n}$$</span> and it is above 0.5 for n>22.
Howe... | TheSilverDoe | 594,484 | <p>You have
<span class="math-container">$$e^{1/n}-1 \sim \frac{1}{n}$$</span>
so
<span class="math-container">$$\frac{\sqrt[n]{e}-1}{n} \sim \frac{1}{n^2}$$</span></p>
<p>so the series is convergent by comparison.</p>
|
2,182 | <p>I wondered if it is appropriate to ask on mathematica stack exchange a question about what they think about the ergonomy of mathematica in comparison to other softwars (matlab etc).</p>
<p>Because I find mathematica very unfriendly in comparison of everything I learnt but I would have to have other point of view to... | Kuba | 5,478 | <p><strong>Alternative Partial Merge Option</strong>:</p>
<p>I understand arguments for having <a href="https://mathematica.stackexchange.com/questions/tagged/geographics" class="post-tag" title="show questions tagged 'geographics'" rel="tag">geographics</a> but I think that considering a big picutre we should... |
1,249,248 | <p>Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function that is differentiable at the point $x_0$. Prove that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the Fourier series. </p>
<p>Note: We only covered the Riemann Lebesgue Lemma for the hypotheses that $f$ is continuous.... | zhw. | 228,045 | <p>Note $\lim_{y\to 0}\,(f(x_0-y)-f(x_0))/\sin(y/2)$ exists since $f'(x_0)$ exists. So we can regard this function as continuous on $[-\pi,\pi].$ We're integrating this against the numerator in $D_n(y),$ which is $\sin (n+1/2)y.\,$ Apply Riemann-Lebesgue.</p>
|
351,626 | <p>Given $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous and a fixed $\delta$,
define
$$f_{\delta}(x):=\int_{x-\delta}^{x+\delta}f(\xi)\,\mathrm{d}\xi.$$</p>
<p>$f_{\delta}$ behaves like the average of $f$ in a short interval $(x-\delta,x+\delta)$. </p>
<p>Apparently it is not linear, but it should be Lipschitz conti... | Chris Janjigian | 23,078 | <p>Take $f(x) = 3x^2$ so that $f_\delta (x) = (x + \delta)^3 - (x-\delta)^3 = 6 \delta x^2 + 2\delta^3$ which is not even uniformly continuous on $\mathbb{R}$, so definitely not Lipschitz.</p>
|
1,492,350 | <blockquote>
<p><strong>Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$.</strong></p>
</blockquote>
<p>Attempt:
We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$</p>
<p>Given $\Delta f(x)=e^x$ i.e $f(x)=\Delta^{-1}e^x=(E-1)^{-1}f(x)$... | Community | -1 | <p>Taking the first order forward difference of the exponential, you get</p>
<p>$$\Delta e^x=(e-1)e^x.$$</p>
|
1,625,306 | <p>I have to evaluate an integral $I(a) = \sin(ax)\cos(x)$ from $0$ to $\pi/2$.The variable of $a$ is not is greater than $1$:</p>
<p>$$\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$$
I attempted to change the function to $[\sin(ax+x)+\sin(ax-x)]/2$ and then integrate, but I am left with (-)cosines with a zero in the denominato... | zz20s | 213,842 | <p>$I(a)=\int_0^{\pi/2} \sin (ax) \cos (x) dx= \frac{1}{2}\int_0^{\pi/2} \sin(x(a+1))+\sin(x(a-1)) dx=$</p>
<p>$$\frac{1}{2}\left[\int_0^{\pi/2} \sin(x(a+1))dx+\int_0^{\pi/2} \sin(x(a-1)) dx\right]$$</p>
<p>I'm going to start by calculating antiderivatives, and apply the bounds later. For the first integral, let $u=x... |
3,137,295 | <p>I am doing tasks from Concrete Mathematics by Knuth, Graham, Patashnik for trainning, but there are a lot of really tricky sums like that: <br><br>
Calculate sum <span class="math-container">$$S_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor $$</span></p>
<h2>My idea</h2>
<p>I had th... | Qurultay | 338,156 | <p>From <span class="math-container">$\left\lfloor x+\frac{1}{2}\right\rfloor=\lfloor 2x\rfloor-\lfloor x\rfloor$</span> we have</p>
<p><span class="math-container">$$\left\lfloor \frac{n}{2^k}+\frac{1}{2}\right\rfloor=\left\lfloor \frac{n}{2^{k-1}}\right\rfloor-\left\lfloor \frac{n}{2^k}\right\rfloor$$</span>
therefo... |
278,669 | <p>$f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable, $\lim_{x\rightarrow\infty}f(x)=1$, $\lim_{x\rightarrow\infty}f'(x)=c$ we need to show $c=0$</p>
<p>well, I tried like this $|f(x)-1|<\epsilon\forall x>M$, where $M$ is very large, $|f'(x)-c|<\epsilon\forall x>M$, what more I can say?thank you.</p... | Ishan Banerjee | 52,488 | <p>If c isn't 0 then f is increasing/decreasing and can't converge to a limit.</p>
|
3,117,312 | <p>For the equation:</p>
<p><span class="math-container">$$y^2 = \frac{\ln(1 - xe^{xy^2})}{1 - xe^{xy^2}}$$</span></p>
<p>How might one go about solving for <span class="math-container">$x$</span> in terms of <span class="math-container">$y$</span>? First, I attempted solving the equation by first performing the subs... | little o | 543,867 | <p><strong>Hint</strong> <span class="math-container">$:$</span> <span class="math-container">$2^{qb}-1 = (2^b -1) (2^{b(q-1)} + 2^{b(q-2)} + \cdots + 1).$</span></p>
<p>The above hint is enough to prove that <span class="math-container">$b \mid a \implies 2^b-1 \mid 2^a - 1.$</span> </p>
<p>To prove that <span clas... |
3,117,312 | <p>For the equation:</p>
<p><span class="math-container">$$y^2 = \frac{\ln(1 - xe^{xy^2})}{1 - xe^{xy^2}}$$</span></p>
<p>How might one go about solving for <span class="math-container">$x$</span> in terms of <span class="math-container">$y$</span>? First, I attempted solving the equation by first performing the subs... | Wuestenfux | 417,848 | <p>Well, if <span class="math-container">$m=kn$</span>, then <span class="math-container">$(2^m-1)/(2^n-1) = ((2^n)^k-1)/(2^n-1) = (2^n)^{k-1} +\ldots+2^n+1$</span>, i.e., <span class="math-container">$2^n-1$</span> divides <span class="math-container">$2^m-1$</span>.</p>
<p>Conversely, if <span class="math-container"... |
1,865,735 | <p>I'm not sure about the use of the Theorem.
I have:</p>
<p>$$f(x)=\int_0^{x^2}(t-1)g(t)dt$$
I need the derivative of $f$. I know i have to apply the chain rules, but i'm not sure about the results.
My result is:
$$f'(x)=(4x*g(x^2))+((x^2-1)*g'(x^2))$$
Is the correct way?</p>
| Jan Eerland | 226,665 | <p>$$\frac{\partial}{\partial x}\left[\int_0^{y(x)}(t-1)g(t)\space\text{d}t\right]=(y(x)-1)g(y(x))y'(x)$$</p>
<p>So:</p>
<p>$$f'(x)=\frac{\partial}{\partial x}\left[\int_0^{x^2}(t-1)g(t)\space\text{d}t\right]=2x(x^2-1)g(x^2)$$</p>
|
1,865,735 | <p>I'm not sure about the use of the Theorem.
I have:</p>
<p>$$f(x)=\int_0^{x^2}(t-1)g(t)dt$$
I need the derivative of $f$. I know i have to apply the chain rules, but i'm not sure about the results.
My result is:
$$f'(x)=(4x*g(x^2))+((x^2-1)*g'(x^2))$$
Is the correct way?</p>
| user247327 | 247,327 | <p>The "Fundamental Theorem of Calculus" says that $\frac{d}{du}\int_a^u f(t)dt= f(u)$. Combine that with the "chain rule": if u is a function of x then $\frac{dF}{dx}= \frac{dF}{du}\frac{du}{dx}$. Here we are taking $u= x^2$ so that $\frac{du}{dx}= 2x$. Then $\frac{d}{dx}\int_a^{x^2} (t- 1)g(t)dt= \frac{d}{du}\int_a... |
3,439,090 | <ul>
<li>The set of rational numbers <span class="math-container">$\mathbb{Q}$</span> is not a connected topological space</li>
</ul>
<p>My Attempt. Let <span class="math-container">$\alpha \in \mathbb{R}$</span> be an irrational number.
By definition, <span class="math-container">$\alpha \notin \mathbb{Q}$</span>.
Co... | Community | -1 | <p><span class="math-container">$\mathbb Q$</span> is trivially open on <span class="math-container">$\mathbb Q$</span>. So <span class="math-container">$S$</span> and <span class="math-container">$T$</span> are simply each the intersection of two open sets on <span class="math-container">$\mathbb Q$</span>, which is ... |
3,439,090 | <ul>
<li>The set of rational numbers <span class="math-container">$\mathbb{Q}$</span> is not a connected topological space</li>
</ul>
<p>My Attempt. Let <span class="math-container">$\alpha \in \mathbb{R}$</span> be an irrational number.
By definition, <span class="math-container">$\alpha \notin \mathbb{Q}$</span>.
Co... | José Carlos Santos | 446,262 | <p>Because a subset <span class="math-container">$A$</span> of <span class="math-container">$\mathbb Q$</span> is an open subset of <span class="math-container">$\mathbb Q$</span> if and only if there is an open subset <span class="math-container">$A^\ast$</span> of <span class="math-container">$\mathbb R$</span> such ... |
1,262 | <p>Beside the fact that I would like to see more posts in our <a href="http://mathematica.blogoverflow.com/" rel="nofollow noreferrer"><em>Mathematica</em> Stack Exchange Blog</a>, I have serious concerns that the majority of the people here is able to find it at all.</p>
<p>There seems to be no direct link from the m... | Emilio Pisanty | 1,000 | <p>I'm not completely sure, but I believe new blog posts will appear for a while in the Community Bulletin. Since the last blog post is six months old I'm not terribly surprised it doesn't.</p>
<p>One way to promote the blog, if the community wants to, is to use a <a href="https://mathematica.meta.stackexchange.com/qu... |
468,807 | <p>I want to use Chebyshev's inequality to calculate the an upper bound on the probability that X lies outside the range $[6, 14]$.</p>
<p>X has mean $\mu = 10$.</p>
<p>So I must find $\alpha$ such that $\vert X - 10 \vert \geq \alpha$.</p>
<p>I also know that $X < 6 \lor X > 14$.</p>
<p>How can I solve for ... | Robert Israel | 8,508 | <p>Hint: what are $|6 - 10|$ and $|14 - 10|$?</p>
|
468,807 | <p>I want to use Chebyshev's inequality to calculate the an upper bound on the probability that X lies outside the range $[6, 14]$.</p>
<p>X has mean $\mu = 10$.</p>
<p>So I must find $\alpha$ such that $\vert X - 10 \vert \geq \alpha$.</p>
<p>I also know that $X < 6 \lor X > 14$.</p>
<p>How can I solve for ... | Adriano | 76,987 | <p><strong>Hint:</strong> Think of $\mu = 10$ as the "centre" or midpoint of the interval, and think of $\alpha$ as the "radius" or half the width of the interval.</p>
|
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| Roman Y. Andronov | 102,191 | <p>My previous construction took 11 steps to accomplish. Here's an 8-step one. My reasoning is here:</p>
<p><a href="http://romanyandronov.elementfx.com/pse/ryapserac04.html" rel="nofollow noreferrer">http://romanyandronov.elementfx.com/pse/ryapserac04.html</a></p>
<p>Invariant this time - <em>an equilateral triangle... |
2,746,388 | <p>A common tangent to two curves is a line that is tangent to the two curves, but not necessarily at the same point.</p>
<p>Find, in terms of $a$ and $b$, the explicit equation of the common tangent to the two curves $y = x^2 + ax + b$ and $y = x^2 + bx + a$, where $a$ is not equal to $b$.</p>
<p>Also find, in terms ... | Ng Chung Tak | 299,599 | <p>$$y_1=x_1^2+ax_1+b \tag{1}$$</p>
<p>$$y_2=x_2^2+bx_2+a \tag{2}$$</p>
<p>Equating the slope of the common tangent:</p>
<p>$$m=2x_1+a=2x_2+b$$</p>
<p>$$x_1=\frac{m-a}{2} \tag{3}$$</p>
<p>$$x_2=\frac{m-b}{2} \tag{4}$$</p>
<p>Now
\begin{align}
m &= \frac{y_1-y_2}{x_1-x_2} \\
&= \frac{x_1^2+ax_1+b-x_2^2... |
282,122 | <p>Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? </p>
<p>(a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$.<br>
(b) $S$ is connected in $\mathbb{M}_n(\mathbb{R})$.<br>
(c) $S$ is compact in $\mathbb{M}_n(\mathbb{R})$. </p>
<p>O... | Balbichi | 24,690 | <p>It is not just connected, in fact it is path connected. for $A,B$ such matrices we have $x^TAx\ge 0$, $x^TBx\ge 0$ so for $\lambda \in [0,1]$ we get $x^T[\lambda A+(1-\lambda)B]x\ge 0$.</p>
|
3,667,131 | <p>I have this indefinite integral , with <span class="math-container">$a\in \Bbb R, \: a\neq 0$</span></p>
<p><span class="math-container">$$\int \frac{dx}{\sqrt{a^2+x^2}}, \tag 1$$</span></p>
<p>I solve the integral <span class="math-container">$(1)$</span> with <span class="math-container">$x=at$</span>, and using... | Aditya Dwivedi | 697,953 | <p>Substitute <span class="math-container">$ x = ai\sin(\theta)$</span> where <span class="math-container">$i^2 = -1$</span></p>
|
1,224,085 | <p>A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would <em>say that</em> the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the <em>same... | Sam Clearman | 230,539 | <p>The best way to think about your question is that you are considering the set of all sequences $(a_1,a_2,\dotsc)$. A priori, two sequences are equal if and only if each of their terms is equal.</p>
<p>However, we can define various <em>equivalence relations</em> on sequences. In particular, if we are considering ... |
1,224,085 | <p>A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would <em>say that</em> the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the <em>same... | epi163sqrt | 132,007 | <blockquote>
<p>In order to answer OP's question we first have to clearly state the meaning(s) of the symbol
\begin{align*}
\sum_{n=k}^{\infty}a_n\tag{1}
\end{align*}
with $a_n$ being real numbers.</p>
</blockquote>
<p>$$$$</p>
<blockquote>
<p><strong>First meaning:</strong> Infinite series are sequences ... |
1,600,911 | <p>So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$: </p>
<p>$Q_a(a) = f(a)$</p>
<p>$Q_a'(a) = f '(a)$</p>
<p>$Q_a''(a) = f ''(a)$</p>
<p>But then how do you derive the function $Q_a(x) = f(a) + f '(a)(x-a) + f ''(a) (x-a)^2/2$?</p>
<p>Or can you esti... | KittyL | 206,286 | <p>What you wrote is the formula for quadratic approximation, which is derived from Taylor series. </p>
<p>In your case, you need to set $f(x)=\ln x$ and $a=1$, then use the formula.</p>
|
2,145,549 | <p>I'd like to ask a question about what can I possibly do wrong with determining asymptotes of the function</p>
<p>$$x \mapsto x-2\sqrt{x^2+1} $$</p>
<p>OK, so when it comes to vertical asymptotes function, we can't have any because domain of the function is the set of all real numbers.</p>
<p>Now, I'm trying to de... | Community | -1 | <p>You can reason on this problem by means of the asymptotic behavior of the function (see <a href="https://en.wikipedia.org/wiki/Asymptotic_analysis" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Asymptotic_analysis</a>).</p>
<p>For large $x$, we have </p>
<p>$$f(x)\approx x-2\sqrt{x^2}=x-2|x|$$</p>
<p>so... |
2,871,589 | <p>This may be a stupid question, but I was learning some set and group theory and it just made me think. Clearly the continuum is an infinite quantity $\mathfrak{c}$, but the set of all reals is also infinitely long. Or is it that $\sup(\mathbb{R})=\mathfrak{c}$ and $\mathfrak{c}\notin\mathbb{R}$. Regardless this is a... | Community | -1 | <p>I am just going to approach one aspect of this because I actually wrote code for something like this around a week ago on how to generate a matrix like <a href="https://www.quora.com/How-can-I-fill-a-Boolean-matrix-knowing-how-many-1-are-there-for-each-row-and-for-each-column/answer/Ryan-Howe-1" rel="nofollow norefe... |
3,412,063 | <p>I'm studying the Lorenz dynamical system, and I'm asking myself if the critical points are unstable critical points. </p>
<p>Considering the theory they are unstable - one eigenvalue <span class="math-container">$\in \mathbb{R}$</span> which is negative and 2 complex eigenvalues with a negative real part. But when ... | DonAntonio | 31,254 | <p>Induction can work here fine: suppose <span class="math-container">$\;a_0,...,a_{n-1}\;$</span> are scalars such that (with <span class="math-container">$\;T^0=I$</span>)</p>
<p><span class="math-container">$$0=\sum_{k=0}^{n-1}a_kT^kv\stackrel{\text{apply}\;T}\implies0=T0=\sum_{k=0}^{n-1}a_kT^{k+1}v\;,\;\;\text{but... |
3,412,063 | <p>I'm studying the Lorenz dynamical system, and I'm asking myself if the critical points are unstable critical points. </p>
<p>Considering the theory they are unstable - one eigenvalue <span class="math-container">$\in \mathbb{R}$</span> which is negative and 2 complex eigenvalues with a negative real part. But when ... | José Carlos Santos | 446,262 | <p>Suppose that<span class="math-container">$$\alpha_0v+\alpha_1Tv+\cdots+\alpha_{n-1}T^{n-1}v=0.\tag1$$</span>Then<span class="math-container">$$T^{n-1}(\alpha_0v+\alpha_1Tv+\cdots+\alpha_{n-1}T^{n-1}v)=0,$$</span>which means that <span class="math-container">$\alpha_0T^{n-1}v=0$</span>. Therefore, <span class="math-c... |
1,563,429 | <p>Working through some questions and I'm stuck on the following:</p>
<p>Find an example of vector spaces $V$ and $W$, and linear transformations $T:V \to W$
and $S: W \to V$, such that $T\circ S$ = $I_w$ but $S\circ T \neq I_V.$ </p>
| learner | 228,313 | <p>Take $V=\Bbb R^3$ and $W=\Bbb R^2$. Consider the linear transformations $T\colon V\to W$ and $S\colon W\to V$ defined by,</p>
<p>$$S(x,y)=(x,y,0)\quad, \quad T(x,y,z)=(x,y)$$</p>
<p>Then, note that we have,</p>
<p>$$(T\circ S)(x,y)=T(S(x,y))=T(x,y,0)=(x,y)=I_W(x,y)$$</p>
<p>But,</p>
<p>$$(S\circ T)(x,y,z)=S(T(x... |
587,950 | <p>There are 2 planes.</p>
<p>plane 1: $2x+3y-4z=15$
plane 2: $x+y-4z=17$</p>
<p>How can I find the acute angle between 2 planes with those 2 equations?
thanks ;)</p>
| Henry Swanson | 55,540 | <p>Find the normal vectors to both planes, $\hat{n}_1$ and $\hat{n}_2$.</p>
<p>Next, find the angle between them by using the fact that $\vec{u} \cdot \vec{v} = |u| |v| \cos{\theta}$.</p>
<p>How is the angle between the normal vectors related to the angle between the planes? (<a href="http://www.netcomuk.co.uk/~jenol... |
3,123,120 | <p>Prove that given sequence <span class="math-container">$$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$</span> </p>
<p>is a Cauchy sequence </p>
<p>My attempt :
<span class="math-container">$|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\dfrac{(-1)^{m+1}}{m+2}\cdots\dots+\dfra... | greg | 357,854 | <p>Define the scalar variable and its differential
<span class="math-container">$$\eqalign{
\alpha &= w^Tx = x^Tw \cr d\alpha &= x^Tdw
}$$</span>
The derivative of the <a href="https://en.wikipedia.org/wiki/Logistic_function#Derivative" rel="nofollow noreferrer">logistic function</a> for a scalar variable is s... |
2,759,636 | <p>I attempted to use Pascal's triangle identity to help out, but I do not know how to deal with $\frac{1}{n+1}$.</p>
| robjohn | 13,854 | <p><strong>Approach 1:</strong></p>
<p>Note that
$$
\frac{2n+1}{n+1}\overbrace{\ \ \binom{2n}{n}\ \ }^{\large\frac{(2n)!}{n!\,n!}}=\overbrace{\binom{2n+1}{n+1}}^{\large\frac{(2n+1)!}{(n+1)!\,n!}}
$$
Then, because $\frac1{n+1}=2-\frac{2n+1}{n+1}$, we have
$$
\begin{align}
\frac1{n+1}\binom{2n}{n}
&=2\binom{2n}{n}-\... |
636,094 | <p>The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind.</p>
<p>I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one?</p>
<p>If it cannot be done in closed form, I wonder what a good asymptotic is.</p>
<p>I considered using Taylor series both for solving the int... | AlexR | 86,940 | <p>Maple seems to only have a difficulty with $$\int_0^1\ln(x-\ln x) dx = 0.3224577\ldots =: C$$
The taylor series about $y=1$ is the given by
$$I(y+1) = C + \frac16 y^3 - \frac1{12} y^4 + \frac1{40} y^5 - \frac1{180} y^6 + \mathcal O(y^7)$$
<img src="https://i.stack.imgur.com/6xbAe.png" alt="Plot of $I(y)$ for $y \in ... |
193,481 | <p>$\#2^{\Omega} = 2^{\#\Omega}$</p>
<p>So far I know that when the size of $\Omega$ is 0, we have the $\emptyset$ and the size of the power set for $\Omega$ will be 1, or $2^{0} = 1 $ </p>
<p>How do I start by proving this for any size $n$ of $(\Omega)$. </p>
<p>Also does $\Omega$ have to be finite for this to be t... | Brian M. Scott | 12,042 | <p>HINT: Suppose that you know that $\{1,\dots,n\}$ has $k$ subsets. Start with one of these $k$ sets. To get a subset of $\{1,\dots,n,n+1\}$ you can either stick with the set that you have, or you can add $n+1$ to it. In this way each subset of $\{1,\dots,n\}$ gives rise to two subsets of $\{1,\dots,n+1\}$. Can you se... |
4,129,851 | <p>This was a problem in my textbook.</p>
<p>Suppose we had a bag with <span class="math-container">$2$</span> balls, an orange and a blue ball. If we pick a blue ball, we simply put it back. If we select an orange ball, we put it back but add <strong>another</strong> orange ball. Suppose we do this <span class="math-c... | ncmathsadist | 4,154 | <p>These are the Fourier coefficents of the function <span class="math-container">$x \mapsto e^{x^2}I_{[0,1]}(x)$</span>. Invoke Riemann-Lebesgue: these converge to zero. Like the Ginzu knife man said on TV, there's more. By Parseval's theorem they are square-summable, too!</p>
|
2,674,043 | <p>A linear model has been fitted under the usual assumptions, i.e. Y = Xβ + ε, with $ε ∼ N(0,σ^2I)$. How would the sketch of a residual plot look for residuals from an exponential distribution with expectation 0?</p>
| V. Vancak | 230,329 | <p>Here is a model with $\beta_0 = -8$ and $\beta_1$. As said @BruceET, in order to maintain expectation of $0$, I have to set $\lambda = 1/8$ in the exponential noise. In this case you can view $\epsilon_i \sim \mathcal{E}xp(\lambda) -\beta_0$, thus $\mathbb{E}{\epsilon_i}=0$. What you see on both plots is a random sa... |
4,666 | <p>It looks to me like a number in a base other than base 10 gets evaluated before the evaluator ever gets a chance to be tweaked.</p>
<p>For example, <code>FullForm[16^^abcdef]</code> or even <code>FullForm[HoldAll[16^^abcdef]]</code> both produce <code>11259375</code>.</p>
<p>Am I missing a trick that would get me ... | celtschk | 129 | <p>What is important here is to distinguish between data and representation. When you input an integer, you actually input a <em>representation</em> of the integer. That is, even without specifying the base, you don't enter the integer 42 (you would be hard-pressed to do that), but the decimal representation of the int... |
3,781,193 | <p>For example, if we divide 100 by 50, then 100 by 49.8, then 49.8, etc. down to 100 divided by 1, we will have a list of 491 quotients, 10 of which are integers (2, 4, 5, 8, 10, 20, 25, 40, 50, 100). For the first 250 divisors (50.0 through 25.1), there is only one integer quotient (2). For the last 41 divisors (5.0 ... | Steve Kass | 60,500 | <p>Think about what you’re asking graphically. You are looking for values of <span class="math-container">$100\over d$</span> that are integers (or near-integers), where <span class="math-container">$d$</span> is a multiple of <span class="math-container">$0.1$</span> between <span class="math-container">$1$</span> and... |
3,781,193 | <p>For example, if we divide 100 by 50, then 100 by 49.8, then 49.8, etc. down to 100 divided by 1, we will have a list of 491 quotients, 10 of which are integers (2, 4, 5, 8, 10, 20, 25, 40, 50, 100). For the first 250 divisors (50.0 through 25.1), there is only one integer quotient (2). For the last 41 divisors (5.0 ... | fleablood | 280,126 | <p>Using <span class="math-container">$0.1$</span> as your step value and <span class="math-container">$q$</span> as your quotient is the <em>exact</em> same thing as using <span class="math-container">$1$</span> as your step value and <span class="math-container">$10q$</span> as your quotient.</p>
<p>And the number of... |
626,700 | <p>Evaluate $\int\int (x+y+z)dS$, where $S$ is the boundary of the unit ball $B$; i.e $S$ is the set of $(x,y,z)$ with $x^2+y^2+z^2=1$.</p>
<p>I parametrized usually using $x=sin\theta cos\phi,y=sin\theta sin\phi,z=cos\theta$, where $0\le \theta \le \pi$, and $0\le \phi \le 2\pi$.</p>
<p>But in this way the integral ... | arthur | 119,007 | <p>I think that this integral is in fact zero. Look at the problem this way: For any point on the unit ball where $x + y + z$ is positive, you can find another point on the ball where the integrand is negative, and of the the same magnitude, simply by rotating the ball 180 degrees. Hence, the integral should be zero si... |
1,503,957 | <p>Okay, maybe I am just really bad with exponents or forgot how exponents work but how do you do these 2 problems, here's what I got so far. I need to state whether thee sequence is increasing, decreasing, and use the ratio rule and difference rule to figure it out. </p>
<blockquote>
<p><strong>Ratio rule</strong>:... | Michael Grant | 52,878 | <p>Define $S=\sum_i y_i$; then
$$\nabla^2 f(x) = S^{-2} ( S \mathop{\textrm{diag}}(y) - yy^T )$$
Now let's show that $\nabla^2 f(x) \succeq 0$; or, equivalently, that $v^T\left(\nabla^2 f(x)\right) v \geq 0$ for all $v\in\mathbb{R}^n$. We have
$$ S^2 v^T (\nabla^2 f(x) ) v = v^T (S \mathop{\textrm{diag}}(y) - yy^T) v ... |
696,859 | <p>Is there a closed form for $k$ in the expression
$$am^k + bn^k = c$$
where $a, b, c, m, n$ are fixed real numbers?</p>
<p>If there is no closed form, what other ways are there of finding $k$?</p>
<p>Motivation: It came up when trying to apply an entropy model to allele distribution in genetics. The initial populat... | Lucian | 93,448 | <p>A closed form solution can only exist if <em>m</em> is a rational power of <em>n</em>, and/or $abc=0$. If such is not the case, let $\gamma=\dfrac1{\ln m-\ln n},\quad\alpha=\dfrac cb,\quad\beta=-\dfrac ab$ . Then $k=-x$, where <em>x</em> is the solution to the recursive equation $x=\gamma\ln(\alpha m^x+\beta)$, whic... |
1,631,535 | <p>In the lecture notes for a course I'm taking, the definition of a convex function is given as follows:</p>
<p>"a function $f$ is convex if, for any $x_1$ and $x_2$, and for any $\alpha$ $\in$ [0,1], $\alpha f(x_1) + (1-\alpha)f(x_2) \ge f(\alpha x_1 + (1-\alpha ) x_2)$" </p>
<p>That is, if you draw a line segment ... | Mark Viola | 218,419 | <p>The <a href="https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means#Weighted_AM.E2.80.93GM_inequality" rel="nofollow">Weighted AM-GM</a> states</p>
<p>$$\lambda x+(1-\lambda)y\ge x^{\lambda}y^{1-\lambda}$$</p>
<p>Therefore, we have</p>
<p>$$\log(\lambda x+(1-\lambda)y)\ge \log(x^{\lambda}y^{1-... |
1,808,222 | <p>While solving PhD entrance exams I have faced the following problem:</p>
<blockquote>
<p>Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n x_i =1$ and $x_i \ge0$.</p>
</blockquote>
<p>I was trying to use <a href="https://en.wikipedia.org... | Vaneet | 335,303 | <p>This is the standard water-filling problem (As an example, see <a href="http://www.comm.utoronto.ca/~weiyu/loading_icc.pdf" rel="noreferrer">http://www.comm.utoronto.ca/~weiyu/loading_icc.pdf</a> ). You would see this problem in any communications textbook including <a href="https://people.eecs.berkeley.edu/~dtse/bo... |
2,129,281 | <p>Here is the problem I am stuck with: Is it true that for every positive integer $n > 1$,
$$\sum\limits_{k=1}^n \cos \left(\frac {2 \pi k}{n} \right) =0= \sum \limits_{k=1}^n \sin \left(\frac {2 \pi k}{n} \right)$$
I'm imagining the unit circle and adding up the value of both trig functions separately but I canno... | Alex Ortiz | 305,215 | <p>It's a little bit easier with complex numbers. Set $\zeta = e^{2\pi i/n}$ and note that the real part of the sum
$$
S = \sum_{k=1}^n\zeta^k
$$
is precisely the sum $\sum_{k=1}^n\cos(\frac{2\pi k}{n})$ and the imaginary part of $S$ is precisely the sum $\sum_{k=1}^n\sin(\frac{2\pi k}{n})$. Therefore, if we show that ... |
1,672,509 | <p>I missed a couple of my Linear algebra classes, so I'm a little lost on this question...</p>
<p>Given $S_1$, $S_2$, $S_3 : \mathbf{R}^2\to \mathbf{R}^2$ are linear mappings defined by:</p>
<p>$S_1(x_1, x_2) = (x_1-x_2, -2x_1+x_2)$</p>
<p>$S_2(x_1,x_2) = (2x_1- x_2, -4x_1+ x_2)$</p>
<p>$S_3(x_1,x_2) = (-x_1+ x_2 ... | Cristian Q | 335,466 | <p>This is exercise 26 from Chapter 2 , Peressini.
Solution is solve Dual geometric Programing: Maximize $$V(\delta)=(\frac{1000}{\delta_1})^{\delta_1}(\frac{2}{\delta_2})^{\delta_2}(\frac{2}{\delta_3})^{\delta_3}(\frac{1}{\delta_4})^{\delta_4}$$
Where $\delta = (\delta_1 ,\delta_2 ,\delta_3 ,\delta_4)$
with the condit... |
473,446 | <p>Let $X$ be connected cubic $s$-regular graph then $|Aut(X)| = 2^{s-1}\cdot 3\cdot |V(X)|$. I want a reference for proof.</p>
| Chris Godsil | 16,143 | <p>Biggs's "Algebraic Graph Theory". I would assume it's used in Tutte's paper on arc-transitive cubic graphs. </p>
<p>But all this result says if that if $\mathrm{Aut}(X)$ is $s$-regular then $|\mathrm{Aut}(X)|$ is equal to the number of $s$-arcs. This would not normally need a reference.</p>
|
3,516,494 | <p><span class="math-container">$$2x^2 + 3x + 1$$</span></p>
<p>applying quadratic formula:</p>
<p><span class="math-container">$$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$</span></p>
<p><span class="math-container">$$a=2, b=3, c=1$$</span></p>
<p><span class="math-container">$$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \c... | Brian | 646,141 | <p>You are not "solving" <span class="math-container">$$2x^2+3x+1$$</span>
but <span class="math-container">$$
2x^2+3x+1 \mathbf{=0}.
$$</span></p>
<p>For the <em>specific</em> values of <span class="math-container">$x$</span> you found, it is indeed true that
<span class="math-container">$$
2x^2+3x+1 = x^2+\frac{3}{... |
2,485,482 | <p>$4*3^x - 9*2^x = 5* 3^\frac x2 * 2^\frac x2$ </p>
<p>I did not understand this equality how to solve it for $x$?</p>
| user577215664 | 475,762 | <p>$$4*3^x - 9*2^x = 5* 3^\frac x2 * 2^\frac x2$$</p>
<p>$$4 - 9*\frac {2^x}{3^x} = 5* \frac {3^\frac x2}{3^x} * 2^\frac x2$$</p>
<p>$$4 - 9*\bigg(\frac 2 3\bigg)^x = 5* \frac{2^\frac x2}{3^\frac x2} $$</p>
<p>Substitute $a=(\frac 23 )^\frac x2$</p>
<p>$$4 - 9a^2 = 5a$$</p>
<p>Solve for a...</p>
<p>$$ 9a^2 +5a-4... |
3,271,414 | <p>This article says <a href="https://en.wikipedia.org/wiki/Set-theoretic_limit#Almost_sure_convergence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Set-theoretic_limit#Almost_sure_convergence</a>.</p>
<blockquote>
<p>The event that a sequence of random variables <span class="math-container">$Y_1, Y_2, \... | user10354138 | 592,552 | <p>Remember the event
<span class="math-container">$$
\{\limsup_{n\to\infty}\lvert Y_n-Y\rvert=0\}
$$</span>
is
<span class="math-container">$$
\{\omega\in\Omega : \limsup_{n\to\infty}\lvert Y_n(\omega)-Y(\omega)\rvert=0\}.
$$</span>
But the limsup of events
<span class="math-container">$$
\limsup_{n\to\infty}\{\lvert ... |
3,271,414 | <p>This article says <a href="https://en.wikipedia.org/wiki/Set-theoretic_limit#Almost_sure_convergence" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Set-theoretic_limit#Almost_sure_convergence</a>.</p>
<blockquote>
<p>The event that a sequence of random variables <span class="math-container">$Y_1, Y_2, \... | drhab | 75,923 | <p>The first expression concerns a <span class="math-container">$\limsup$</span> of functions. </p>
<p>The second expression concerns a <span class="math-container">$\limsup$</span> of sets (which again is a set).</p>
<hr>
<p><span class="math-container">$$\omega\in\{\limsup_{n\to\infty}|Y_n-Y|=0\}\iff\limsup_{n\to\... |
1,906,342 | <p>15 people are randomly assigned to three cars, each holding 4, 5, and 6 people respectively. The owners of the cars are among the 15, and will be randomly assigned to one of the cars. What is the probability that each owner gets assigned to his or her own car? </p>
<p>My thought process:
The total number of ways t... | Ross Millikan | 1,827 | <p>You are assuming that the owners are one per car, but the problem statement does not say that. You could have all three owners in the six passenger car, for example. Your calculation does not include that possibility.</p>
|
740,397 | <p>I've just proved that for <span class="math-container">$|G|$</span> finite, the number of elements of order <span class="math-container">$2$</span> is odd. But how about the case <span class="math-container">$|G|$</span> infinite?</p>
<p>(Here, there is assumption that the set of elements of order <span class="math-... | Derek Holt | 2,820 | <p>Let $X$ be the set of elements of $G$ of order $2$ and suppose that $X$ is finite. Then the subgroup $H=\langle X \rangle$ of $G$ generated by $X$ is finite, and so if $X$ is nonempty, then $|X|$ must be odd.</p>
<p>To see this, note that, since the conjugacy class of any $x \in X$ is finite, the index $|G:C_G(x)|$... |
740,397 | <p>I've just proved that for <span class="math-container">$|G|$</span> finite, the number of elements of order <span class="math-container">$2$</span> is odd. But how about the case <span class="math-container">$|G|$</span> infinite?</p>
<p>(Here, there is assumption that the set of elements of order <span class="math-... | Mikko Korhonen | 17,384 | <p>Let $t$ be an element of order $2$ in $G$, and assume that the number of elements of order $2$ is finite. It is enough to prove that $\{x \in G: x^2 = 1\}$ contains an even number of elements. To do that, you can pair each $x \in C_G(t)$ with $xt$ and each $x \not\in C_G(t)$ with $txt^{-1}$.</p>
|
2,080,644 | <p>I know that a function is odd when
$$f(-x) = -f(x)$$
Therefore I can say that if for a function $$-f(x) + f(x) = f(-x) + f(x) = 0$$</p>
<p>Then the function is odd!</p>
<p>I tried to use this <em>trick</em> to prove that $f(x) = \ln\left(x+\sqrt{x^2 + 4}\right) - \ln2$ is odd.</p>
<p>However, I would want to prov... | Community | -1 | <p>$$\left(x+\sqrt{x^2 + 4}\right)\left(-x+\sqrt{x^2 + 4}\right)=4=2\cdot2$$ so that, taking the logarithm,</p>
<p>$$f(x)+f(-x)=0.$$</p>
|
4,142,944 | <p><span class="math-container">$$ \int_0^t \frac{x \cos u - x^2}{1 - 2x \cos u + x^2} \, du\quad\text{with}\quad x \in ]-1,1[ $$</span></p>
<blockquote class="spoiler">
<p> <span class="math-container">$$ \int_0^t \frac{x \cos u - x^2}{1 - 2x \cos u + x^2} \, du = \arctan \left( \frac{x \sin t}{1- x \cos t} \right) $... | eyeballfrog | 395,748 | <p>I also don't see a way to spot that antiderivative, so here's what I would call the "systematic" way of doing it. That is, simply applying general purpose integration techniques and identities without trying to find antiderivatives by inspection.</p>
<p>When working with integrals involving rational functi... |
377,169 | <p>How does one calculate the value within range <span class="math-container">$-1.0$</span> to <span class="math-container">$1.0$</span> to be a number within the range of e.g. <span class="math-container">$0$</span> to <span class="math-container">$200$</span>, or <span class="math-container">$0$</span> to <span class... | llf | 298,781 | <p>A short proof of Matt L.'s answer:</p>
<p>We want a function $f: [a, b] \rightarrow [c, d]$ such that</p>
<p>$$
\begin{alignat}{2}
f&(&a) &= c \\
f&(&b) &= d.
\end{alignat}
$$</p>
<p>If we assume the function is to be linear (that is, the output scales as the input does), then</p>
<p>$$\d... |
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