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 |
|---|---|---|---|---|
173,107 | <p>are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\forall p\in(a,b)$$<br>
This question came up when thinking about $L_p$ norms as functions of $p$ and thus as a mappi... | Matthew Daws | 406 | <p><strong>Edit:</strong> This is correct, except for the actual application to the question at hand, which is wrong; see below for details.</p>
<p>I learnt about this technique from:</p>
<p>MR2526788 (2010m:46098)<br/>
Haagerup, Uffe(DK-SU-CS); Musat, Magdalena(1-MEMP)<br/>
Classification of hyperfinite factors up t... |
2,216,723 | <p>A player at draw poker holds the seven of spades and the eight, nine,
ten, and ace of diamonds. Aware that all the other players are drawing three cards, he figures that any hand he could win with a flush he could also win with a straight. For which should he draw? </p>
| Misha Lavrov | 383,078 | <p>The determinant is the product of the eigenvalues. We know that $\frac12$ is an eigenvalue with multiplicity at least $m-2$, because $M - \frac12I$ has rank $1$. In addition, the sum of the eigenvalues is the trace of $M$, which is $m-1$. So if $x$ is the remaining eigenvalue, then $x + \frac12(m-2) = m-1$, which me... |
2,680,744 | <p>If image of a simple loop is contained in an open connected set $D\subseteq R^2$ such that this loop is homotopic to a point in $D$, will interior of curve be contained in $D$.</p>
<p>What if we take closed contour instead of simple loop?</p>
<p>Definition: Simple loop is homeomorphic to $S^1$.</p>
<p>A similar q... | Mauro ALLEGRANZA | 108,274 | <p>For the second task, you need <a href="http://integral-table.com/downloads/logic.pdf" rel="nofollow noreferrer">logical equivalence rules</a>, like e.g. <a href="https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Informal_proof" rel="nofollow noreferrer">De Morgan's laws</a> and the equivalence between $(p \to q)$ and... |
2,680,744 | <p>If image of a simple loop is contained in an open connected set $D\subseteq R^2$ such that this loop is homotopic to a point in $D$, will interior of curve be contained in $D$.</p>
<p>What if we take closed contour instead of simple loop?</p>
<p>Definition: Simple loop is homeomorphic to $S^1$.</p>
<p>A similar q... | Bram28 | 256,001 | <p>I am not sure what you mean by 'contradiction theory' ... but I am guessing the idea is to show the statement is a tautology by showing that the assumption that it is not leads to a contradiction. So, below I'll show you a semi-formal technique to do something like this; it's known as a 'short truth-table'.</p>
<p... |
1,703,100 | <p>A car starts from rest and accelerates in $a = \frac{2\cdot m}{3\cdot s^3}t$,</p>
<p>After $3$ seconds, The car will be $27$ metres from beginning.</p>
<p>Find distance as function of time.</p>
<p>I know i have to integral the acceleration,But i don't know how.</p>
<p>I found the Equation is $x_{t} = 24 + \frac{... | Cm7F7Bb | 23,249 | <p>The result of the first throw can be $1,2,3,4,5,6$ with equal probabilities. If the result of the first throw is $1$, then we throw the dice again and can obtain $1,2,3,4,5,6$ with equal probabilities. The random variable can be equal to $2,3,4,5,6$ with the probabilities $1/6$ after the first throw or it can be equ... |
42,053 | <p>I have this problem to solve:</p>
<blockquote>
<p>Hockey teams 1 and 2 score goals at times of Poisson process with rates 1 and 2. Suppose that $N_1(0)=3$ and $N_2(0)=1$.
What is the probability that $N_1(t)$ will reach 5 before $N_2(t)$ does?</p>
</blockquote>
<p>I've re-worded this to: What is the prob that ... | Robert Israel | 8,508 | <p>You must assume that the two Poisson processes are independent. Goal-scoring (without regard to which team scores) is then a Poisson process $N(t) = N_1(t) + N_2(t)$ with rate $1+2=3$, and each individual goal that is scored has probability $1/3$ of coming from team 1.</p>
|
42,053 | <p>I have this problem to solve:</p>
<blockquote>
<p>Hockey teams 1 and 2 score goals at times of Poisson process with rates 1 and 2. Suppose that $N_1(0)=3$ and $N_2(0)=1$.
What is the probability that $N_1(t)$ will reach 5 before $N_2(t)$ does?</p>
</blockquote>
<p>I've re-worded this to: What is the prob that ... | Did | 6,179 | <p>Yes, each scored goal has probability 1/3 of being scored by team 1 and probability 2/3 of being scored by team 2, and by the magics of Poisson proceses, each goal is <em>colored</em> independently from the others. </p>
<p>Hence you look for the probability of a random walk starting from (3,1) and with step (+1,0) ... |
237,496 | <p>I have a complex data set and I want to fit with a mathematical model using <code>NonlinearModelFit</code>, but an error appears. Following is the data I'm using and the code that I have tried.</p>
<pre><code> dataset = {{0.1, 78.2356 - 0.8683 I}, {0.1998, 78.4897 - 0.5794 I}, {0.2996,
78.1675 - 1.0737 I}, {0... | Anton Antonov | 34,008 | <p>Can you use model fitting that is done separately for the real and imaginary parts?</p>
<p>Example is given below.</p>
<h2>Separate fitting of real and imaginary parts</h2>
<p>Fit for the real parts:</p>
<pre><code>e0 = 8.854*10^-12;
modelfitRe = NonlinearModelFit[Re /@ dataset, Re[eInf + (e1 - eInf)/(1 + I*2*\[Pi]*... |
2,316,246 | <p>I was pondering on how to answer this question. It looks like this:</p>
<p>In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the d... | Michael Hardy | 11,667 | <p>This is the sort of situation for which the normal approximation was first introduced (by Abraham de Moivre in the 18th century). I have up-voted the answer by Big Agness, but I want to give some detail here that was not in that answer.</p>
<p>The difficulty is that computing the exact answer arithmetically is proh... |
886,337 | <p>I need to find $$\int x\sqrt{(a^2 - x^2)}dx$$</p>
<p>I tried putting $x=a cos(t)$ but I ended up getting a very complicated expression, so any tips?</p>
| Shine | 157,361 | <p>$-\frac{1}{3}(a^2-x^2)^\frac{3}{2}$ is one of the origional function because $\frac{d}{dx}\bigg(-\frac{1}{3}(a^2-x^2)^\frac{3}{2}\bigg)=x\sqrt{a^2-x^2}.$</p>
<p>Why did I come up with it because $\int x\sqrt{a^2-x^2}dx=\int\sqrt{a^2-x^2}d(x^2)=-\frac{1}{3}(a^2-x^2)^\frac{3}{2}$+C.</p>
|
4,315,926 | <p>This question is in the context of a robotics problem. The goal is to track a robot using both its onboard odometry system and a VR system (HTC Vive Pro) using a VR controller mounted to the robot.</p>
<p>What is <strong>known</strong> is the transformation between odometry origin and the robot (measurements <span c... | Bertrand87 | 953,938 | <p>One way using complex analysis:</p>
<p>Since the integrand is even:
<span class="math-container">$$
I=\int_{0}^{\pi} \frac{d x}{(a-\cos x)^n} = \frac{1}{2} \int_{-\pi}^{\pi} \frac{d x}{(a-\cos x)^n}
$$</span></p>
<p>Since <span class="math-container">$a>1$</span>, we can expand the integrand with the generalized ... |
1,514,028 | <p>It's pretty obvious to me as I see the plot of the two functions but how can I prove it with some algebra?</p>
| Yes | 155,328 | <p>By mean-value theorem, if $-1 \leq x < 0$, then
$$
|\sin x - \sin 0| = |\sin x| \leq |x|\sup_{0 < t < x}|\cos t| \leq |x|.
$$</p>
|
4,149,316 | <p>Is <span class="math-container">$f(x,y) = x^2 - 2e^y$</span> concave or convex?</p>
<p>My thought process is as such:</p>
<p><span class="math-container">$f'(x,y)=2x - 2e^y$</span>;</p>
<p><span class="math-container">$f''(x,y)=2 - 2e^y$</span></p>
<p>Thus when <span class="math-container">$y<0$</span> (<span cla... | Siong Thye Goh | 306,553 | <p>The hessian is <span class="math-container">$\begin{bmatrix} 2 & 0 \\ 0 & -2e^{-y}\end{bmatrix}$</span>. The eigenvalues are <span class="math-container">$2$</span> (positive) and <span class="math-container">$-2e^{-y}$</span> (negative). Hence it is indefinite.</p>
<p>If you fix <span class="math-container"... |
2,079,246 | <p>Is (A vector +B vector) always perpendicular to (A vector - B vector)?</p>
| Vidyanshu Mishra | 363,566 | <p>Try to get resultant from the parallelogram law, and you will see that $(\vec A + \vec B)$ and $(\vec A - \vec B)$ are perpendicular only when their magnitude is equal.</p>
<p>Resultant of $(\vec A + \vec B)=\sqrt{a^2+b^2+2AB\cos\theta}$ and resultant of $(\vec A - \vec B)=\sqrt{a^2+b^2-2AB\cos\theta}$ where $\the... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Timothy Shields | 49,879 | <p>The question is missing the concept of <a href="https://en.wikipedia.org/wiki/Utility" rel="noreferrer">utility</a>, a function that specifies how much you value each possible outcome. In the game, the utility of ending the game with a certain score would be the value you place on that outcome. Although you could ce... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | user381133 | 381,133 | <p>I interpreted this such that you're in competition against another player.</p>
<p>If playing second your strategy is simple - play until you've won.</p>
<p>If playing first, you must accumulate enough points such that the next player is statistically more likely to lose than win - that means making them roll more ... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Mike Earnest | 177,399 | <p>It's not hard to see that the optimal strategy in this game is "stop on $k$ or higher," for some positive integer $k$. This is because what is optimal is only determined by your current score, not your history of rolls, and if rolling again on $k$ is unwise, then rolling again on $k+1$ is unwise as well (you stand t... |
1,977,081 | <p>You play a game using a standard six-sided die. You start with 0 points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled 6, then you lose everything and the game ends. Otherwise, add the score from the die to your total points and con... | Adrian | 98,081 | <p>Here's an answer using value function iteration:</p>
<p>Let $v(x)$ denote the value of state $x$. The Bellman equation for this problem is $$v(x) = \max\left\{u(x), \frac{1}{6} \left[u(0) + \sum^5_{k=1}v(x+k)\right]\right\},$$
where $u$ is the utility of ending the game with $x$.</p>
<p>Several of the answers abov... |
113,266 | <p>One of the problems of my Statistics assignment requires me to calculate the expected value of a minimum. Here's the situation:</p>
<p>The following density function is given: $f(x) = \frac{\theta}{x^2}$ where $x\ge\theta$ and $\theta>0$</p>
<p>I have to calculate: E[min{$X_i$}]</p>
<p>My initial guess was tha... | Did | 6,179 | <blockquote>
<p>Minima and expectations of nonnegative random variables are both well suited to complementary CDF, two remarks which together make for a painless solution. </p>
</blockquote>
<p>In the present case, $\mathrm P(X_i\geqslant x)=\theta/x$ for every $i$ and every $x\geqslant\theta$ hence $M_n=\min\limits... |
2,130,527 | <p>I am trying to plot the function $(-2(\alpha-1)/\alpha)^{(-2(\alpha+2)/\alpha)}$ in maple,
but I got the following warning</p>
<pre><code>plot( (-2*(alpha-1)/alpha)^(-2*(alpha+2)/alpha) ,alpha=5..10);
</code></pre>
<blockquote>
<p>Warning, unable to evaluate the function to numeric values in the
region; comple... | Community | -1 | <p><strong>Hint:</strong> Show that $(a_n)_n$ is a Cauchy sequence. Therefore:</p>
<ol>
<li>show $|a_{n+2}-a_{n+1}|\leq c^{n+1}|a_1-a_0|$</li>
<li>estimate $|a_n-a_m|$ by using step 1 and dont forget about the geometric sum and series, where you have to use $0<c<1$.</li>
</ol>
|
1,011 | <p>None of these are critical, just polishing stuff.</p>
<hr>
<p>When breaking out the up/down votes on a post, the top number crashes into the separator line:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/OApgh.png" alt="vote crash"></p>
</blockquote>
<hr>
<p>For some badges, e.g. taxonomy on the pare... | theTuxRacer | 2,843 | <p><a href="http://i56.tinypic.com/2mweedv.jpg" rel="nofollow">http://i56.tinypic.com/2mweedv.jpg</a></p>
<p>there are two glitches in the image, so i ddnt need to post two images. the other glitch in the upper right corner.</p>
|
1,011 | <p>None of these are critical, just polishing stuff.</p>
<hr>
<p>When breaking out the up/down votes on a post, the top number crashes into the separator line:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/OApgh.png" alt="vote crash"></p>
</blockquote>
<hr>
<p>For some badges, e.g. taxonomy on the pare... | Jonas Meyer | 1,424 | <p>Apparently these have been fixed, so we probably don't need this on the unanswered list anymore. (I'm posting this because the question was just bumped by <a href="https://math.meta.stackexchange.com/users/-1/community">Community</a>. Anyone who agrees with me can upvote either of the answers until at least one of... |
1,044,096 | <p>Let $\mathscr { S } $ the class of all closed intervals $[a,b]$, where $a$ and $b$ are rational , i.e. $a,b\in \mathbb{Q}$ and $a<b$ . Show that $\mathscr{S}\cup \{\{p\}:p\in\mathbb{ Q }\}$ is a basis of the topology $\tau$ of the real line $\mathbb{R}$ generated by $\mathscr {S}$ .</p>
<p>Having problems... | egreg | 62,967 | <p>The class $\mathscr{S}'=\mathscr{S}\cup\{\{p\}:p\in\mathbb{Q}\}\cup\{\emptyset\}$ is closed under binary intersections, because the intersection of two closed proper intervals is either a closed proper interval or a singleton, unless it is empty.</p>
<p>Every element of $\mathscr{S}'$ is an intersection of elements... |
5,197 | <p>Is it possible to split a list into two lists at a specific position? The main list is for example: </p>
<pre><code>data={{xa,ya},{xb,yb},{xc,yc},...,{xz,yz}}.
</code></pre>
<p>I want to split this list into two new lists: </p>
<pre><code>data1={{xa,ya},{xb,yb},...,{xi,yi}}
</code></pre>
<p>and </p>
<pre><cod... | Dr. belisarius | 193 | <p>This can be accomplished easily using <code>Part</code> (<code>[[ ]]</code>) and <a href="http://reference.wolfram.com/mathematica/ref/Span.html"><code>Span</code></a> (<code>;;</code>), as follows:</p>
<pre><code>data = {{x1, y1}, {x2, y2}, {x3, y3}, {x4, y4}, {xz, yz}};
data[[;; 3]]
data[[4 ;;]]
(* ->
{{x1, y... |
751,293 | <p>I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it doesn't require two semesters of undergrad analysis that are prerequisite to the complex variables class for math ma... | Stephen Montgomery-Smith | 22,016 | <p>If you take a circular piece of wire, and dip it into a soap solution to make a bubble surface across the circle, and then place the circular piece of wire in the $xy$-plane, if the circular wire has small perturbations of order $\epsilon$, then the first order solutions of the $z$-coordinate as functions of $x$ and... |
999,227 | <p>I have convinced myself that this true, however I'm at a loss of where I should start with this proof. Looking at a similar proof with 3 instead of 9, I saw the use of the basis representation theorem, but I'm not 100% comfortable with that, so it was hard for me to follow. Is there a way to do this with mod?</p>
| Clayton | 43,239 | <p><strong>Hint:</strong> $10\equiv1\pmod{9} $.</p>
|
2,483,254 | <p>I have to compute the limit</p>
<p>$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as</p>
<p>$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively... | the_candyman | 51,370 | <p>Let's pose $z = \frac{1}{\sqrt{N+1}}$. Then, when $N$ goes to $+\infty$, then $z$ goes to $0$. You can rewrite your limit as follows:</p>
<p>$$\lim_{z \to 0} \frac{1}{z} \log\left(1 + xz^2\right).$$</p>
<p>It is well known that:</p>
<p>$$\lim_{a \to 0} \frac{\log\left(1 + a\right)}{a} = 1.$$</p>
<p>Starting from... |
879,051 | <p>How do topologists prove continuity of a function with the usual topology at the endpoints of a closed interval? For instance, how would a topologist prove continuity for $f(x)=x^2$ on the closed interval $[0,1]$ at the point $x = 1$ using open sets (not using calculus)?</p>
| Pedro | 23,350 | <p>It is simply a matter of translating the $\varepsilon$-$\delta$ definition to a metric language. Saying that given any $\varepsilon >0$ there exists a $\delta >0$ such that $|x-a|<\delta\implies |f(x)-f(a)|<\varepsilon$ is saying that, given any open ball $V=B(f(a),\varepsilon)$ there is an open ball $U=... |
1,055,229 | <p>Consider the following exercise from a book I'm reading:</p>
<p>If $n$ is odd show that
$$ f: O(n) \to SO(n) \times \{1,-1\}, A \mapsto (A \operatorname{det}{A}, \operatorname{det}{A})$$</p>
<p>is an isomomorphism.</p>
<p>But why does $n$ have to be odd? I can easily show that it's a group homo., injective and s... | Saibal | 191,019 | <p>If $n$ is even, multiplying $A\in O(n)\backslash SO(n)$ by $\det(A)=-1$ should not give a matrix in $SO(n)$.</p>
|
518,794 | <p>$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$.<br>
Prove (formally) that $f$ is continuous. </p>
<p>I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of image, has a pre-image that belongs to an open ball around (x,y), and since image and pre-image are both o... | rfauffar | 12,158 | <p>You can use limits to show it if you want or open balls. Using limits, the following facts is easy: The quotient of two continuous functions that go to $\mathbb{R}$ is continuous, wherever the functions are defined and the denominator is not 0.</p>
<p>After proving that, it is easy to see that the projections $\pi_... |
4,458,244 | <p>To factorize <span class="math-container">$xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$</span> I used the fact that <span class="math-container">$x=-y$</span> and <span class="math-container">$y=z$</span> and <span class="math-container">$x=-z$</span> make the expression zero. Hence it factorize to <spa... | Ivan Kaznacheyeu | 955,514 | <p>I think OP approach is the simplest one because it can be done just in mind without paper writing. One of possible another ways</p>
<p><span class="math-container">$$A=xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)=x^2(y-z)+x(y^2-z^2)+yz(y-z)$$</span></p>
<p><span class="math-container">$(y-z)$</span> is c... |
4,458,244 | <p>To factorize <span class="math-container">$xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$</span> I used the fact that <span class="math-container">$x=-y$</span> and <span class="math-container">$y=z$</span> and <span class="math-container">$x=-z$</span> make the expression zero. Hence it factorize to <spa... | User | 1,044,391 | <p>This method can be faster.</p>
<ul>
<li>
<ol>
<li>Write the expression as a general quadatic respect to <span class="math-container">$y$</span> ( or <span class="math-container">$x$</span>, or <span class="math-container">$z$</span>)</li>
</ol>
</li>
</ul>
<p>You get,</p>
<p><span class="math-container">$$A=(x+z)y^2... |
3,350,074 | <p>Say, I have a circle </p>
<p><span class="math-container">$$\left(x – \frac 12 \right)^2 + \left( y – \frac 12 \right)^2 = \frac 14$$</span></p>
<p>If I have a point <span class="math-container">$(0,0)$</span> and a point on the circumference of the circle <span class="math-container">$(x_1,y_1)$</span>, how woul... | Con | 682,304 | <p>Take <span class="math-container">$B = \lbrace b \rbrace$</span> and <span class="math-container">$C = \lbrace c \rbrace$</span> for some integers <span class="math-container">$b \neq c$</span>. Now define a function <span class="math-container">$f \colon \mathbb{Z} \rightarrow \mathbb{Z}$</span>, such that <span cl... |
299,156 | <p>We say that paths $\alpha, \beta: I \to X$ with common initial point $\alpha(0)=\beta(0)$ and common terminal point $\alpha(1)=\beta(1)$ are homotopic provided that there is a continuous function $H:I \times I \to X$ such that $H(t,0)=\alpha(t)$ and $H(t,1)=\beta(t)$ for $t \in I$ and $H(0,s)=\alpha(0)=\beta(0)$ and... | Dylan Wilson | 423 | <p>Of course.</p>
<p>The inclusion map $I \times \{t\} \rightarrow I \times I$ is continuous, therefore the composite
$$
I \times \{t\} {\rightarrow} I \times I \stackrel{H}{\rightarrow} X
$$</p>
<p>is continuous. The assumption on endpoints shows this is a path from $\alpha(0)$ to $\alpha(1)$. </p>
|
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | chepner | 78,700 | <p>I'm surprised no one has mentioned composition. You've identified that you need to shift the range</p>
<p>$$f_1(x) = x - 1$$</p>
<p>then take the value <em>modulo</em> $n$</p>
<p>$$f_2(x) = x \mod n$$</p>
<p>and finally shift the result again</p>
<p>$$f_3(x) = x + 1$$</p>
<p>The function you want is just the c... |
231,027 | <p>I have the following complicated expression:</p>
<pre><code>(3 (Sqrt[6] a^13 + a^12 e0 + Sqrt[6] a^11 e0^2 + a^10 e0^3 + Sqrt[6] a^9 e0^4 -
a^8 e0^5 - Sqrt[6] a^7 e0^6 - a^6 e0^7 + Sqrt[6] a^5 e0^8 - a^4 e0^9 -
Sqrt[6] a^3 e0^10 + a^2 e0^11 -Sqrt[6] a e0^12 + e0^13)) /
(256 e0 (a^6 + Sqrt[6] a^5 e0 + a^4 e0^2... | Daniel Huber | 46,318 | <p>We give your expression a name: <code>y== expression</code>, then we add a second equation <code>x== a/e0</code> and eliminate a. Next, we solve for y to get the original function back. Finally we have an expression in x that we can expand around zero.</p>
<pre><code>ex = (3 (Sqrt[6] a^13 + a^12 e0 + Sqrt[6] a^11 e0... |
1,349,720 | <p>Say we have two irreducible polynomials in $Q [x] $. We call them $f, g$. Say one of the roots of $f $ is $a$. Is it possible that $g$ satisfies a root of the form $a^n$ for some natural number $n $? </p>
<p>Thanks </p>
| Lucian | 93,448 | <p><strong>Hint:</strong> Take a root of the form $A\sqrt\alpha+B\sqrt\beta$, where both <em>a</em> and <em>b</em> are square-free. For instance, for $A=B=1$, $\alpha=2$, $\beta=3$, and $n=2$, we have $x^2-10x+1$ and $x^4-10x^2+1$.</p>
|
3,005,576 | <p>I want to show explicitly that <span class="math-container">$L^2[0,1]$</span> subspace of <span class="math-container">$L^1[0,1]$</span> has no internal points. </p>
<p>If S subspace of X, I have the following result:</p>
<p>S = X if and only if the interior of S <span class="math-container">$\neq \emptyset$</s... | uniquesolution | 265,735 | <p>No, you are not missing anything. If <span class="math-container">$X,Y$</span> are normed spaces, and <span class="math-container">$X$</span> is a proper subspace of <span class="math-container">$Y$</span>, then <span class="math-container">$X$</span> has empty interior relative to <span class="math-container">$Y$</... |
3,005,576 | <p>I want to show explicitly that <span class="math-container">$L^2[0,1]$</span> subspace of <span class="math-container">$L^1[0,1]$</span> has no internal points. </p>
<p>If S subspace of X, I have the following result:</p>
<p>S = X if and only if the interior of S <span class="math-container">$\neq \emptyset$</s... | Kavi Rama Murthy | 142,385 | <p>I think 'explicitly' here means you have to show that if <span class="math-container">$f \in L^{2}$</span> then <span class="math-container">$B(f,\epsilon)$</span> is not contained in <span class="math-container">$L^{1}$</span> for any <span class="math-container">$\epsilon >0$</span>. General theory is not suppo... |
1,064,534 | <p>$$x^3-3y^2x=-1$$
$$3yx^2 -y^3=1$$</p>
<p>This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was just a system unrelated to the problem, how would one solve it? I seem to have trouble when the terms are lik... | Ishfaaq | 109,161 | <ul>
<li>When $x = y$ the system reduces to $ x = y= t $ and $ 2t^3 = 1 $ hence the solution is $x = y= \dfrac{1}{\sqrt[3]2}$</li>
<li>Suppose $x \neq y$ then the system reduces to (after adding both equations) $$ x^3 - y^3 = 3y^2x - 3yx^2 $$</li>
</ul>
<p>$$ \require{cancel} \cancel {(x - y)} (x^2 + xy + y^2) = - ... |
1,254,275 | <p>We are given a number $$K(n) = (n+3) (n^2 + 6n + 8)$$ defined for integers n. The options suggest that the number K(n) should either always be divisible by 4, 5 or 6. </p>
<p>Factorizing the second bracket term, we get $$ K(n) = (n+3) (n + 2) (n+4)$$</p>
<p>i.e the number is always divisible by -4 -2 and -3. IMO t... | AlexR | 86,940 | <p>$K(n)$ is the product of three consecutive integers. Therefor it's automatically divisible by $2$ and $3$, thus by $6$. $K(n)$ need not be divisible by $4$. Consider
$$K(3) = 5\cdot6\cdot7 = 210$$</p>
<p>In general, the product of $k$ consecutive integers is divisible by all natural numbers $n\le k$.</p>
|
1,254,275 | <p>We are given a number $$K(n) = (n+3) (n^2 + 6n + 8)$$ defined for integers n. The options suggest that the number K(n) should either always be divisible by 4, 5 or 6. </p>
<p>Factorizing the second bracket term, we get $$ K(n) = (n+3) (n + 2) (n+4)$$</p>
<p>i.e the number is always divisible by -4 -2 and -3. IMO t... | ajotatxe | 132,456 | <p>$K(n)$ is the product of three consecutive numbers. If the first and the third are odd and the second is even (of course) but not a multiple of $4$, then $K(n)$ will not be a multiple of $4$ either.</p>
|
3,104,936 | <p>I was wondering if anyone could explain to me why this holds true?</p>
<p><span class="math-container">$$C\left(n, k\right) = \frac{C\left(n, k-1\right) \cdot C(n - (k - 1), 1)}k$$</span></p>
<p>or</p>
<p><span class="math-container">$$C\left(3, 2\right) = \frac{C\left(3, 1\right) \cdot C(2, 1)}2$$</span></p>
<p... | Community | -1 | <p>Expanded with factorials,</p>
<p><span class="math-container">$$\frac{n!}{k!(n-k)!}=\frac1k\frac{n!}{(k-1)!(n-k+1)!}\frac{(n-k+1)!}{1!(n-k)!}.$$</span></p>
<p>The simplifications are obvious.</p>
|
4,529,873 | <p>I am currently dealing with the two terms "Gateaux-derivation" and "Frechet-derivation" and would like to know if there is a Frechet differentiable function <span class="math-container">$f$</span> and Gateaux differentiable function <span class="math-container">$g$</span>, so that <span class="ma... | daw | 136,544 | <p>Take
<span class="math-container">$$
g(x,y) = \begin{cases}
1 & \text{ if } x\ne0 \text{ and } y=x^2\\
0 & \text{ else }
\end{cases}
$$</span>
and
<span class="math-container">$$
f(x) = (x,x^2).
$$</span>
Then <span class="math-container">$g:\mathbb R^2 \to \mathbb R$</span> is Gateaux, <span class="math-con... |
3,118,410 | <p>I've been trying to use the change of base property but I'm not having much luck. Can anyone give me any ideas on how I should approach this problem? The answer is 49/5</p>
<p>Thanks.</p>
| Haris Gušić | 450,231 | <p><strong>Hint:</strong>
<span class="math-container">$$\log_x(5x)=\frac{2}{\log_7 x} = \frac{\log_7 49}{\log_7 x}$$</span>
I hope that you have tested the conditions for the logarithms to be defined.</p>
|
4,441,421 | <p>I am studying PDEs, in particular hyperbolic conservation laws.
In particular we are using the method of characteristic to solve some of the problems.</p>
<h1>Setting</h1>
<p>Given the quite general PDE
<span class="math-container">$$
U_t + \partial_x \big( a(x,t)U(x,t)\big) = 0\iff U_t + a\cdot U_x = - a_x\cdot U
$... | Alan Cai | 966,080 | <p>I don't think the statement you are trying to prove is true for any right triangle. Here is why.</p>
<p>Let the two adjacent sides of the right triangle be represented by vectors, A = (0,a), B = (b,0), then the hypotenuse of the triangle would be B-A = (b,-a) by the parallelogram law. Let the vector V = (1,1)(in fac... |
750,468 | <p>Show that if $f: U \to \mathbb{C}$ is analytic on an open set $U$ in $\mathbb{C}$, then the Cauchy Riemann equations for $f$ hold in $U$. </p>
<p>My question is, how do the Cauchy-Riemann equations differ on an open set $U$ versus a function $f$ on $\mathbb{C}$? Not sure how to prove this.</p>
| Jair Taylor | 28,545 | <p>This is one of those cases where the proof strategy is just to write down your conclusion and your premise, and at every step just write down the first thing you can do with the tools you are given. </p>
<p>Always start by writing down the premise: </p>
<blockquote>
<p>Suppose that $g \circ f = id$ and $f$ is ... |
1,896,399 | <p>Im asked to solve find the derivative of: $$ \frac{\ln x}{e^x}$$
<strong>my attempt</strong>
$$D\frac{\ln x}{e^x} = \frac{\frac{1}{x}e^x + \ln (x) e^x}{e^x} = e^x \frac{\frac{1}{x}+\ln x}{e^{2x}} = \frac{\frac{1}{x}+\ln x}{e^x}$$</p>
<p>But this is apparently wrong and the correct answer is:
$$\frac{\frac{1}{x} - ... | Uri Goren | 203,575 | <p>Note that:
$$ \frac{\ln x}{e^x}=e^{-x}\ln {x}$$
And
$$(uv)'=u'v+uv'$$
Thus
$$ (\frac{\ln x}{e^x})'=(e^{-x}\ln {x})'=(-e^{-x}\ln {x})+(\frac{e^{-x}}{x})=\frac{\frac{1}{x} - \ln x}{e^x}$$</p>
|
4,062,300 | <p>Given 16 Trials equally distributed to 4 individuals ( 4 Trials per person).</p>
<p>How many combinations are there where these conditions are met:<br />
Condition A: 5 Total Successes<br />
Condition B: 2 or Fewer Successes per Individual.</p>
<p>I understand Condition A in exclusivity would be:<br />
<span class="... | Sid | 428,996 | <p>uhm the following way isn't that easy... you have to consider several properties of functions...</p>
<p>first off: <span class="math-container">$\sin(\dfrac{\pi x}{9}) = \dfrac{2x}{9} \iff \sin^{-1}(\dfrac{2x}{9}) = \dfrac{\pi x}{9}$</span></p>
<p>Let: <span class="math-container">$f(x) =\sin^{-1}(\dfrac{2x}{9}) - \... |
3,807,293 | <p>I do apologise if this question has been asked before, but I am having trouble trying to get from <code>!A & B || A & !B</code> to <code>A XOR B</code>.</p>
<p>I've tried doing it in reverse by starting with <code>A XOR B</code> (<code>(A || B) & !(A & B)</code>), and I reached <code>(A || B) & (... | Paco Adajar | 477,061 | <p>To answer the question as to why it doesn't work: draw the entire circle. Your cuts are not actually removing the entirety of two quarter circles, but only a section thereof.</p>
<p><a href="https://i.stack.imgur.com/i6ALW.png" rel="noreferrer"><img src="https://i.stack.imgur.com/i6ALW.png" alt="Whole circle with tw... |
682,057 | <p>I'm seeing a physics paper about percolation (<a href="http://arxiv.org/abs/cond-mat/0202259" rel="nofollow">http://arxiv.org/abs/cond-mat/0202259</a>). In the paper the following asymptotic relation is used without derivation.
$$
\sum_{k=0}^{\infty} k P(k) (1 - \epsilon)^{k-1} \sim \left<k\right> - \left<k... | Kai Sikorski | 127,462 | <p>$$
\sum_{k=0}^{\infty} k P(k) (1 - \epsilon)^{k-1} = \sum_{k=0}^\infty k P(k) \left(\sum_{j=0}^{k-1} {k-1 \choose j}(-\epsilon)^j \right)\\
= \sum_{k=0}^\infty k P(k) { k-1 \choose 0} - \sum_{k=0}^\infty k P(k) { k-1 \choose 1} \epsilon + \ldots\\
= \langle k \rangle - \langle k(k-1) \rangle\epsilon + \ldots
$$</p>
|
328,021 | <p><span class="math-container">$f(x_1,x_2,\ldots x_n)$</span> is polynomial with integer coefficients.
<span class="math-container">$f$</span> is rather large to be computed explicitly, but an algorithm can
compute it efficiently at integers and complex number and "lazy" form using
parenthesis.
The degree of... | Aaron Meyerowitz | 8,008 | <p>I think these comments may be helpful.</p>
<p>I will assume that we have some polynomial or multinomial (perhaps with monomials from some restricted set) and an oracle which instantly returns the value at any given point (perhaps from a restricted domain). We wish to find a specific coefficient and wonder how many ... |
3,618,340 | <p>I read a set of notes on projective curves, where it introduced discrete valuations on a field <span class="math-container">$K$</span> as a surjective homomorphism <span class="math-container">$v:K^\times \to \mathbb Z$</span> where <span class="math-container">$v(x+y) \geq \min(v(x), v(y))$</span> for all <span cla... | Hagen Knaf | 2,479 | <p>Historically one/the(?) reason for the prominent appearance of discrete valuations in the case of algebraic curves is that one gets a theory that possesses strong similarities to algebraic number theory. Here are some points of the story:</p>
<ul>
<li>In algebraic number theory the <span class="math-container">$p$<... |
1,604,226 | <p>Let $a,b,c$ be positive real numbers. Prove that </p>
<p>$\frac{bc}{a^2+bc}+\frac{ca}{b^2+ca}+\frac{ab}{c^2+ab}\le{\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}}$.</p>
<p>I have no idea how to solve this question.</p>
| Michael Rozenberg | 190,319 | <p>We can use SOS here.
$\sum\limits_{cyc}\left(\frac{a}{b+c}-\frac{bc}{a^2+bc}\right)=\sum\limits_{cyc}\frac{a^3+abc-b^2c-bc^2}{(b+c)(a^2+bc)}=\sum\limits_{cyc}\frac{a^3-abc+2abc-b^2c-bc^2}{(b+c)(a^2+bc)}=$
$=\frac{1}{2}\sum\limits_{cyc}\frac{2a(a^2-bc)+2bc(2a-b-c)}{(b+c)(a^2+bc)}=\frac{1}{2}\sum\limits_{cyc}\frac{(a-... |
1,917,852 | <p>Suppose $(\Omega, \mathcal{F}, \mu)$ is a probability space and $f$ is integrable function. Then, is it true that $\mu \{ \omega: |f(w)|=\alpha\}=0$ for all but countably many $\alpha$ ? \</p>
| detnvvp | 85,818 | <p>Suppose that there exists an uncountable set $A$ such that, if $a\in A$, then $\mu\left(\{\omega\in\Omega\Big{|}|f(\omega)|=a \}\right)>0$. For every $n\in\mathbb N$, set $A_n$ to be the set of $a\in A$ such that $$\frac{1}{n}\leq\mu\left(\{\omega\in\Omega\Big{|}|f(\omega)|=a \}\right)<\frac{1}{n-1},$$ then $\... |
3,335,850 | <p>I know that if <span class="math-container">$x$</span> is an element of A by definition of subsets then it is an element of <span class="math-container">$B$</span>, but I don't know how to use the complement to prove the latter half.</p>
| Jean Lagace | 91,358 | <p>Pick an element <span class="math-container">$x \in C - B$</span>. By definition, it is in <span class="math-container">$C$</span>, and also it is not in <span class="math-container">$B$</span>. Since <span class="math-container">$A \subseteq B$</span>, <span class="math-container">$x$</span> is not in <span class="... |
824,255 | <p>Suppose I have a polynomial (of any order) and I'm not able to calculate the roots. Is there a way to get at least some information about the roots such as how many of them are complex, negative or positive? For example, I can safely identify the behavior (and therefore roots' character) of $f(x)=ax+b$ or even a qua... | Tom | 103,715 | <p>Let $g(x) = x$, $h(x) = \sin(x)$ and $k(x) = \frac{1}{x}$ defined for all $x \neq 0$. Then what you have is that $f(x) = g(x) \cdot h(k(x))$. In other words, $f$ is created by the product and composition of "nice" functions. So, if you know that $g, h,$ and $k$ are differentiable (when $x \neq 0$) and if you know th... |
901,549 | <p>I am struggling with part c of this question. Could someone please tell me how to approach and solve this type of questions?</p>
<p><img src="https://i.stack.imgur.com/SIpJ8.png" alt="enter image description here"></p>
| callculus42 | 144,421 | <p>ad a)</p>
<p>$P(Y=y_j)=\sum_{i \in I} \ p(X=x_i,Y=y_j) $</p>
<p>$x_i \in \{-1,0,2 \}; \ y_j \in \{-2,1,2 \}$ </p>
<p>$P(Y=-2)=0.1+0.2+0.2=0.5$ </p>
<p>Sum of the first row. For the whole marginal probability function you have additional calculate $P(Y=1)$ and $P(Y=2)$</p>
<p>ad b)</p>
<p>$E(x\cdot y)=\sum_{i \... |
2,155,201 | <p>By a real ODE I mean an ordinary differential equation with only real coefficients and the resulting function is a function of a real argument. If such a solution exists, can you give an example?</p>
<p>Edit: To add to this, is it still possible if the initial conditions must also be real?</p>
| John Hughes | 114,036 | <p>$$
f(x)^2 + 1 = 0
$$
That's a differential equation where the "derivative" coefficient is zero; as it happens, the solution is one of the constant functions $f(x) = \pm i$. </p>
<p>If you want one with a derivative term, consider
$$
f(x)^2 + 1 = xf'(x)
$$
At $x = 0$, any solution must have $f(0) = \pm i$. </p>
|
3,009,919 | <p>Give <span class="math-container">$n$</span>-th derivatives, can we find a function ?<br>
For example, if <span class="math-container">$f^{(i)}(x)=i,i=1,2,\cdots,n.$</span> , can we construct a function such that it satisfies the condition.<br>
It just occurs in my mind. I didn't know if it has a solution.</p>
| ShapeOfMatter | 618,435 | <p>It sounds like you're asking for a function <span class="math-container">$f$</span>, such that
<span class="math-container">$$f'=1$$</span>
<span class="math-container">$$f''=2$$</span>
<span class="math-container">$$f'''=3$$</span>
<span class="math-container">$$...$$</span>
But if <span class="math-container">$f'$... |
623,873 | <p>I'm given with the following function:</p>
<p>$f(x) = x^n \arctan\left(\dfrac{1}{x}\right) , x\ne0 $ , and $f(0)=0$ . </p>
<p>I'm asked to:</p>
<ol>
<li>determine when is it continuous in $x=0$ </li>
<li>determine when is it differentiable in $x=0$</li>
<li>determine when is it continuously differentiable in $x=0... | Charles | 1,778 | <p>$$
\sum_{p\le x}p = \frac{x^2}{2\log x} + \frac{x^2}{2\log^2x} + \frac{x^2}{4\log^3x} + \frac{3x^2}{8\log^4x} + O\left(\frac{x}{\log^5x}\right).
$$</p>
|
623,873 | <p>I'm given with the following function:</p>
<p>$f(x) = x^n \arctan\left(\dfrac{1}{x}\right) , x\ne0 $ , and $f(0)=0$ . </p>
<p>I'm asked to:</p>
<ol>
<li>determine when is it continuous in $x=0$ </li>
<li>determine when is it differentiable in $x=0$</li>
<li>determine when is it continuously differentiable in $x=0... | Kusavil | 78,968 | <p>There are formulas for computing prime numbers, the problem lies in the costs and time needed for that and there is often (or we don't know) no closed formula for that, like <a href="http://www.matematyka.pl/159928.htm#p624705" rel="nofollow noreferrer">these ones:</a></p>
<p>1.
<span class="math-container">$$p_n= ... |
1,562,416 | <p>Suppose $32$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so no two of the three chosen objects are adjacent nor diametrically opposite.
This is again a problem in a math contest in India and this is how I tried it:</p>
<p><strong>Number of ways of... | Smayan das | 296,724 | <p>Suppose I denote the objects as 1-32.
Then suppose I select object 1 first. Then there would be 2 cases.</p>
<p>CASE 1 :-If I choose 3 or 31 (as I can't select 32 or 2 but I can choose those adjacent to 2 and 32) then there would be 25 possibilities for each .
Thus there being a total of 25×2 = 50 possibil... |
2,899,422 | <p>I need some hints for solving $yy''-(y')^2=xy^2$. </p>
<p>I noticed that the left hand side is close to $(yy')'$:</p>
<p>$yy''-(y')^2=xy^2\ \Leftrightarrow\ yy''+(y')^2-2(y')^2=xy^2\ \Leftrightarrow\ (yy')'-2(y')^2=xy^2$.</p>
<p>But I don't know how to continue expressing the terms as derivatives of some function... | Community | -1 | <p><strong>By indeterminate coefficients:</strong></p>
<p>Assume the solution to be of the form $P(x)e^{x^2}$ for some polynomial $P$. Then</p>
<p>$$(P(x)e^{x^2})'=(P'(x)+2xP(x))e^{x^2}.$$</p>
<p>For the left factor to equal $x^2$, $P$ must be of the first degree, let $ax+b$.</p>
<p>Now</p>
<p>$$P'(x)+2xP(x)=a+2x(... |
2,200,672 | <p>Find the roots of $f(x)=x^2+x+1$ modulo $7$, and modulo $13$, and modulo $91$ </p>
<p>I think for mod $7$ and $13$, it can be done by trial and error. But how about mod $91$?</p>
| fleablood | 280,126 | <p>To find the solutions to $x^2 + x +1 \mod k$ we can factor $x^2 + x + 1$ and to factor that we need to find the $nm \equiv 1 \mod k$ so that $n + m \equiv 1 \mod k$.</p>
<p>For $k = 7$, 7 is prime so every term has a multiplicative inverse. $1^{-1} = 1$ and $1 +1 \equiv 2$ so that won't do. $2^{-1}=4$ but $2+4 \eq... |
3,230,922 | <p>I was working through some textbook problems for my Number Theory class and needed some help with the following question:</p>
<p>Describe the set of quadratic integers α in Q[sqrt−3] for which α ̄ and α are associates.</p>
<p>I'd really like some help with this problem. Thank you!</p>
| José Carlos Santos | 446,262 | <p>As a first step, you can search for an element <span class="math-container">$at+bu\in K$</span> such that<span class="math-container">$$(\forall ct+du\in K):(at+bu)(ct+du)=(ct+du)(at+bu)ct+du.$$</span>After a few computations (Linear Algebra stuff), you will get <span class="math-container">$a=b=\frac12$</span>. So,... |
2,458,259 | <p>Are there any sequences s.t. $\{a_{n}\}^{\infty}_{n=1}$ and $\{b_{n}\}^{\infty}_{n=1}, b_{n}\neq0, \forall n$ which diverge, but $\{\frac{a_{n}}{b_{n}}\}^{\infty}_{n=1}$ converges to $0$.</p>
<p>I have tried to plug in a bunch of examples, but failing to come up with one. </p>
| Randall | 464,495 | <p>$a_n = (-1)^n$ and $b_n =n$ should work. </p>
|
1,770,858 | <p>I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?</p>
| Vik78 | 304,290 | <p>The sum is clearly bounded below by the harmonic series starting from 2 (since $\ln(n)^2 < n$), so it diverges.</p>
|
1,770,858 | <p>I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?</p>
| Olivier Oloa | 118,798 | <p>One may onserve that
$$
0<\ln x<\sqrt{x} ,\qquad x>2,
$$ giving
$$
0<(\ln x)^2<x, \qquad x>2,
$$ then</p>
<blockquote>
<p>$$
\int_2^M \frac{dx}x < \int_2^M \frac{dx}{(\ln x)^2}, \qquad M>2,
$$ </p>
</blockquote>
<p>thus, by letting $M \to \infty$, the initial integral <strong>diverges</st... |
142,180 | <p>First take a question as an example:</p>
<p>Let $f:L\to M$ be an irreducible morphism in $\mathrm{mod}-A$ and $X$ be a right $A$-module.
Show that $\mathrm{Ext}_A(X,f):\mathrm{Ext}_A(X,L)\to\mathrm{Ext}_A(X,M)$ is a monomorphism,if $\mathrm{Hom}_A(M,X)=0$. </p>
<p>What is the special meaning of $\mathrm{Ext}$ func... | Aaron | 28,841 | <p>As Matt has said in the comment, representation of quivers, and modules over the corresponding path algebra are really the same thing. Since you said you understand what it means to take Ext in homological algebra, i.e. taking Ext of modules over an algebra. So what you want for the answer is just $Rep(kQ/I) \sime... |
637,099 | <p>I'm having trouble verifying my proof, would appreciate some input on this one.</p>
<p>Let $(X,d)$ be a metric space with $E\subset X$.</p>
<p>Suppose $E$ is closed in $X$, which means that $E=\overline{E}$.
By defintion of the <em>boundary</em> of $E$, denoted $\partial E$ we have that</p>
<p>$\partial E=\overli... | gerw | 58,577 | <p>One possibility is to take
$$f_n(x) = a_n \, \chi_{[0,1/n]}(x)$$
with suitable chosen $a_n$.</p>
|
2,033,470 | <blockquote>
<p>Assume that an integer <span class="math-container">$a$</span> has a multiplicative inverse modulo an integer <span class="math-container">$n$</span>. Prove that this inverse is unique modulo <span class="math-container">$n$</span>.</p>
</blockquote>
<p>I was given a hint that proving this Lemma:
<sp... | Bill Dubuque | 242 | <p>If <span class="math-container">$\,b,\bar b$</span> are inverses of <span class="math-container">$\,a\,$</span> then <span class="math-container">$\ b\equiv b(a\bar b)\equiv (ba)\bar b\equiv \bar b\, $</span> by <a href="https://math.stackexchange.com/a/879262/242">Congruence Product Rule</a>. We used only commutati... |
1,928,142 | <p>I need help with this geometric problem. </p>
<p>$ \overline {AB}$ is a diameter in circle $M$ , $C \in$ $\overrightarrow{AB}$ lying outside the circle , $\overline{CD}$ is drawn tangent to the circle at point $D$ , then $\overrightarrow{DH} \bot \overrightarrow{AB} $ to intersect it at $H$.</p>
<p><strong>Prove t... | operatorerror | 210,391 | <p>Shortcut: In the first quadrant take $\arctan(y/x)$ </p>
|
257,368 | <p>I have the following system of ODEs:</p>
<p>dx/dt = a/(1 + z) - Q*z</p>
<p>dy/dt = Qx - qy</p>
<p>dz/dt = qy - cz/(K + z).</p>
<p>Assuming K = 1, Q = q < c, and a = c*(Sqrt[c/Q] - 1), is there a way to use Mathematica to test whether a Hopf bifurcation exists?</p>
<p>Any help would be greatly appreciated!</p>
<p>... | E. Chan-López | 53,427 | <p>The following answer is just a guide to detect the Hopf bifurcation in your system:</p>
<p>Considering <span class="math-container">$a=\frac{3}{8}$</span>, <span class="math-container">$c=\frac{4}{3}$</span>, <span class="math-container">$k=\frac{3}{2}$</span>, <span class="math-container">$Q=\frac{1}{2}$</span> and... |
3,924,376 | <p>Given <span class="math-container">$i,a_i,b_i\in\{1...n\},\space a_i\neq a_j,b_i\neq b_j,\forall i\neq j$</span> prove that
<span class="math-container">$$\sum_{i=1}^n|a_i-b_i|\le\big\lfloor \frac{n^2}{2}\big\rfloor$$</span>
This problem was proposed by the new contributor @user3458994 and it was closed by five use... | Neat Math | 843,178 | <p>I have already given an answer in that post. I will post it again here. It's somewhat similar to the rearrangement inequality: When <span class="math-container">$\{a_i\}$</span> and <span class="math-container">$\{b_i=i\}$</span> have opposite order, the sum of the absolute difference reaches maximum (there could be... |
105,577 | <p>The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an intractable problem.</p>
<p>I'm posting this question in the hope that someone may have investigated this curve already ... | Igor Rivin | 11,142 | <p>Take a look at <a href="https://mathoverflow.net/questions/42016/algorithms-for-finding-rational-points-on-an-elliptic-curve">this question and answers.</a> I am conjecturing that Cremona's mwrank (which, I believe, is available in sage) will find what you are looking for.</p>
<p><strong>EDIT</strong> Yes, <a href=... |
2,244,321 | <p>Let $f, g$ absolutely continuous on $[a,b]$ with $f'(x)=g'(x)$ a.e. on $[a,b]$. Prove if $f=g$ at a single point in $[a,b]$ then $f=g$ everywhere</p>
<p>This is not homework, this is part of a list of problems that it would be advisable to solve before a general exam. Any suggestion is welcome, I have tackled it on... | Rafael | 303,887 | <p>Let $M'$ be a finitely generated submodule of $M$ and $v:M'\rightarrow M$ the canonical inclusion. Then there are $y'_1,y'_2,...,y'_n\in M$ such that $M'=<y'_1,...,y'_n>$ and, since $M$ is the direct limit of $M_i$, if $u_i:M_i\rightarrow M$ is the canonical map, there are $l_1,...,l_n\in I$ such that $y'_i=u_... |
1,822,314 | <p>I think of two numbers, $x$ and $y$.
When I add them together I get $5$ and when I find the difference I get $13$. What numbers did I think of?</p>
<p>I need to know how to write this down in simultaneous equation form</p>
<p>many thanks</p>
| Soham | 242,402 | <p>Let the two numbers be $a$ and $b$.From the question,we have </p>
<p>$$a+b=5$$ and $$a-b=13$$.</p>
<p>Adding them we get,$$(a+b)+(a-b)=18$$ </p>
<p>$$\implies2a=18$$</p>
<p>$$\implies a=9$$</p>
<p>Substituting $a=9$ in the first equation,we have $9+b=5$ or $b=-4$.</p>
<p>So,$a=9,b=-4$ is the solution.</p>
|
417,855 | <p>Let <span class="math-container">$a>1$</span> and define <span class="math-container">$G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$</span> where <span class="math-container">$\Gamma$</span> is the Gamma function. This series is convergent on <span class="ma... | Jorge Zuniga | 141,375 | <p>For arbitrary real <span class="math-container">$a > 0$</span> this is a special case of the generalized <span class="math-container">$_p\Psi_q(A;B;ζ)$</span> Fox-Wright function, where <span class="math-container">$A=[(a_1,\alpha_1),(a_2,\alpha_2),...,(a_p,\alpha_p)]$</span> and <span class="math-container">$B=[... |
9,719 | <p>I have proved this using Venn diagram but when I am trying to prove this using the rule that <strong><em>"If $ A \subset B \text{ and } B \subset A $ then $ A = B $"</em></strong>, I am having some problems with my understanding of the same,here is how I did so far:</p>
<p>Let $x \in A \cap (B-C) \Rightarrow x \in... | Aryabhata | 1,102 | <p>Hint: $x \notin C \Rightarrow x \notin D \cap C$ for any set $D$.</p>
|
335 | <p>I looked at the “accept rate” of some of the site’s top users and found that it varies widely in the range 45%–100%. I was surprised with that, as it may be taken as an indicator that either:</p>
<ol>
<li>hard questions don’t tend to get very good answers (let's assume that top users ask difficult questions), or</l... | David | 103 | <p>When I don't get a satisfying answer to my question I don't accept. I don't see how that could possibly be negotiable.</p>
|
2,556,993 | <p>I've come across two limits which are reported to take on a range of values rather than a unique one.</p>
<p>They are:</p>
<p>$$\lim \limits_{x \to \infty} \space\frac{1+\cos x}{1-\sin x} = 0 \space to\space \infty$$</p>
<p>$$\lim \limits_{x \to \infty} \space \frac{2+2x+\sin(2x)}{(2x+\sin(2x))e^{\sin x}} = \fra... | StackTD | 159,845 | <p>In the usual sense (and common definition), these limits <strong>do not</strong> exist (since, loosely speaking, the function values don't come arbitrarily close to a specific value for sufficiently large values of $x$).</p>
<p>For a simpler example, consider:
$$\lim_{x \to +\infty} \sin x$$
The fact that $\sin x$ ... |
3,753,788 | <p>In his book Serge Lang provides the following proof that every bounded set that is closed is also compact.</p>
<p>" Let S be a closed and bounded set. This implies that there exists <span class="math-container">$B$</span> s.t. <span class="math-container">$|z|\leq B$</span> where <span class="math-container">$z... | Mark | 470,733 | <p>This is indeed a mistake. You should take a convergent subsequence of <span class="math-container">$y_{n_1}$</span> which converges, not just a subsequence of <span class="math-container">$y_n$</span>. In that case <span class="math-container">$x_{n_2}$</span> is a subsequence of <span class="math-container">$x_{n_... |
3,753,788 | <p>In his book Serge Lang provides the following proof that every bounded set that is closed is also compact.</p>
<p>" Let S be a closed and bounded set. This implies that there exists <span class="math-container">$B$</span> s.t. <span class="math-container">$|z|\leq B$</span> where <span class="math-container">$z... | Moe Sarah | 787,944 | <p>As <span class="math-container">$\mathbb{R}^2=\mathbb{C}$</span>, we may proceed by noting that:</p>
<p><strong>Every closed and bounded set in <span class="math-container">$\mathbb{R}^2$</span> is compact.</strong></p>
<p>Here is a proof:</p>
<p>Let <span class="math-container">$F\subseteq \mathbb{R}^2$</span> be b... |
3,997,676 | <p>Let <span class="math-container">$1\leq n\in \mathbb{N}$</span>, <span class="math-container">$E=\left (e_1, \ldots , e_n\right )$</span> the standardbasis of <span class="math-container">$\mathbb{R}^n$</span>, <span class="math-container">$B=\left (b_1, \ldots , b_n\right )$</span> a basis of <span class="math-cont... | Adam Rubinson | 29,156 | <p><span class="math-container">$0 \notin \mathbb{N}$</span>.</p>
<p>Let <span class="math-container">$f:\mathbb{N} \to \mathbb{Z}$</span> be a surjective and injective function. Define:</p>
<p><span class="math-container">$$g(n) = \cases {f(1),\quad n =1\\
f(1),\quad n =2\\
f(n-1),\qu... |
2,934,472 | <p>Let <span class="math-container">$X_1,X_2,\dots$</span> denote a sequence of identically distributed, exchangeable, but not independent, Bernoulli<span class="math-container">$(p)$</span> random variables. If there exists <span class="math-container">$n>0$</span> such that</p>
<p><span class="math-container">$$
... | Did | 6,179 | <p>First, some notations: for every <span class="math-container">$n$</span>, let <span class="math-container">$X_{1:n}=(X_1,X_2,\ldots,X_n)$</span>, and, for every word <span class="math-container">$w=(w_1,w_2,\ldots,w_n)$</span> in <span class="math-container">$\{0,1\}^n$</span>, let <span class="math-container">$|w|=... |
2,910,643 | <p>This question was in my maths paper . I tried to prove that the series converges but failed to do so. Any hint are welcomed .</p>
| b00n heT | 119,285 | <p>Hint: $$(n+1)\cdot(n+2)\cdots(n+k)=\frac{(n+k)!}{n!}$$
from here you only need to get to $\displaystyle \sum \frac1{k!}=e$, which is easy by adding some missing terms.</p>
|
3,173,766 | <p>If you have the option to roll 2 separate 10-sided dice or a single 10-sided die that you add 2 to the result with a goal of at least one die getting a specific number or higher (say 6 for the purposes of this question) is one of these options inherently better?</p>
<p>It seems that the single die with a modifier i... | Brian Tung | 224,454 | <p>Suppose in option (a), you have <span class="math-container">$k$</span> <span class="math-container">$n$</span>-sided dice, and in option (b), you have a single <span class="math-container">$n$</span>-sided die, and you add a modifier <span class="math-container">$m$</span>. Furthermore, suppose that you are aiming... |
201,103 | <p>When comparing datasets using <code>DistributionChart</code>, is it possible to add a horizontal line to the chart that lies over all of the individual distributions within the chart?</p>
| MelaGo | 63,360 | <p>You can also use <code>Show</code>:</p>
<pre><code>dc = DistributionChart[
RandomVariate[NormalDistribution[2, .5], {2, 100}]];
Show[dc, Graphics[Line[{{.5, 2}, {2.5, 2}}]]];
</code></pre>
<p><a href="https://i.stack.imgur.com/i7cAi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/i7cAi.png" a... |
3,517,376 | <p>A friend of mine presented a problem I found interesting:</p>
<blockquote>
<p>Compute the following: <span class="math-container">$$\sum_{n=0}^\infty\left(\prod_{k=1}^j(1+x^k)\right)[x^{mn}]$$</span> where <span class="math-container">$P(x)[x^n]$</span> denotes the <span class="math-container">$x^n$</span> coeffi... | Simply Beautiful Art | 272,831 | <p>Since the only criterion for a term to contribute to the sum is for the power to be a multiple of <span class="math-container">$m$</span>, we can consider all exponents mod <span class="math-container">$m$</span> i.e. <span class="math-container">$\mathbb R[x]/(x^m-1)$</span>. Thus if <span class="math-container">$m... |
984,154 | <p>Let say my girlfriend makes $2000$ Euros per month and I make $3400$ Euros, let say all our living costs sum up to $1540$ Euros.
How can i calculate how much each of us must pay (off the $1540$ Euros) so as our participation is proportional to our income?</p>
<p>Thanks</p>
| 5xum | 112,884 | <p>Write down the equations that must be satisfied. Let $x$ be the amount you pay and $y$ the amount your girlfriend pays.</p>
<p>One equation is simple: $x+y=1540$, since you must pay the whole costs. The other equation comes from the proportions that must be satisfied:</p>
<p>$$x:y = 3400 : 2000$$
Which can also be... |
679,343 | <p>I have been researching this for the past hour or so, but I have not really found a sufficient answer. What I have found is that vaguely, math education PhDs set policies for math education. However, universities almost uniformly have math professors teach mathematics courses. I believe that usually the math prof... | Fly by Night | 38,495 | <p>Having a PhD in anything doesn't suddenly make you into anything. Lots of people with Mathematics PhDs don't work in universities and don't do mathematical research. </p>
<p>People who have a PhD in mathematical education will have spent time researching mathematical pedagogy, more often than not, at pre-university... |
1,740,221 | <p>If I have $x^{\lfloor\frac{c}{a-b}\rfloor}$ is this equivalent to $(x^{\lfloor\frac{1}{a-b}\rfloor})^{\lfloor c \rfloor}$ if a,b,c are all integers and x is between 0 and 1?</p>
<p>I'm concerned with this expression in the context of a geometric series. Normally if we have $\sum_{i=0}^{\infty} r^{q}$ with |r| < ... | Graham Kemp | 135,106 | <p>No. Witness:</p>
<p>$$\left\lfloor \tfrac{4}{5-2}\right\rfloor = 1 \\ \lfloor 4\rfloor\left\lfloor \tfrac 1{5-2} \right\rfloor=0$$</p>
|
1,869,843 | <p>We know that:
$$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3 $$ </p>
<p>My question is there other examples that satisfies:
$$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b $$</p>
| Rosie F | 344,044 | <p>Put $n=2.$ Then the RHS is$$S=\sum k^b=2^b+1=\frac{2^{2b}-1}{2^b-1}.$$ By <a href="http://mathworld.wolfram.com/ZsigmondyTheorem.html" rel="nofollow noreferrer">Zsigmondy's Theorem</a>, if $b\ne 3$, there is a prime $p$ where the order of 2 mod $p$ is $2b$, and so $p\mid S$. (If $b=3$, the RHS is $9=3^2$ and there i... |
271,846 | <p>In his <em>"Some topological properties..." (1955)</em>, Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ in the coordinate $\mathbb{R}^n$-subspace to any other homeomorphic compact $K'$ set in this subspace. </p>
... | Dan Ramras | 4,042 | <p>This answer is a correction to my comment above. In <a href="http://math.iupui.edu/%7Edramras/M-Cohen-Klee-trick-notes.pdf" rel="noreferrer">these notes</a> from Marshall Cohen's course (from Spring 2001), he uses the Klee trick to give a simple proof that if $f, g: X \to S^n$ are non-surjective embeddings with $X$ ... |
3,837,815 | <blockquote>
<p>Prove <span class="math-container">$\log_{4}6$</span> is irrational.</p>
</blockquote>
<p>I've seen proofs of this which boil down to:
<span class="math-container">$$4^{m} = 6^{n}$$</span></p>
<p>But how does this prove that it is irrational?
For example it is possible to have <span class="math-containe... | redroid | 411,951 | <p>It proves it's irrational because for it to be rational would require integers <span class="math-container">$n,m$</span> that make the equation you've written true. However, that equation decomposes into:</p>
<p><span class="math-container">$$ 2^{2m} = 3^n 2^n $$</span></p>
<p>or if you like:</p>
<p><span class="mat... |
12,239 | <p>I know this is a trivial problem but am stuck with this thing.</p>
<p>$A \in \mathbb{R}^{m \times m}$ is a symmetric positive definite matrix and $C \in \mathbb{R}^{m \times p}$
</p>
<p>How do I show that the matrix $$B = \begin{bmatrix} A & C \\ C^T & 0 \end{bmatrix}$$ is indefinite?</p>
<p>I can show th... | Hans Lundmark | 1,242 | <p>Hint for a less convoluted argument: Unless $C$ is zero (in which case the statement is false), there are vectors $x$ and $y$ such that $x^T C y$ is a nonzero $1\times 1$ matrix. Now consider $v=[x,ky]^T$ for large $k$ (positive or negative, depending on the sign of $x^T C y$).</p>
|
3,638,206 | <p>Is it true that if <span class="math-container">$m$</span> is a Lebesgue measure on <span class="math-container">$[0,1]$</span> and that <span class="math-container">$\lambda$</span> is the counting measure on <span class="math-container">$[0,1]$</span>, and that both defined on the Lebesgue <span class="math-contai... | Physical Mathematics | 592,278 | <p>Fix <span class="math-container">$E \subseteq \mathbb{R}$</span> Lebesgue measurable. Let <span class="math-container">$h = 1_{\{0\}} m(E)$</span> (i.e. <span class="math-container">$h(x)=0$</span> for all <span class="math-container">$x \neq 0$</span> and <span class="math-container">$h(0) = m(E)$</span>). Then <sp... |
221,867 | <p>For a Hermitian nonnegative-definite matrix $A$, if $Ax$ is always real for any real vector $x$, can we conclude that $A$ is also real?</p>
| copper.hat | 27,978 | <p>Since $e_i^T A e_j = [A]_{i,j}$, and $Ax$ is real for real $x$, then $A$ must be real.</p>
|
619,519 | <p>The equation $\log_3(\sqrt{x+1}+1)=(3^{x+1}-1)^2\,$
has two solutions, but I can't solve the equation.</p>
| Claude Leibovici | 82,404 | <p>After the nice remark from Constructor and the simpler formulation given by labuwx, let see how to solve the equation </p>
<p>3^t - Sqrt[t] - 1 = 0 </p>
<p>I think that Newton iterative scheme is very simple. Starting with a guess (say t_old), the iterates are given by </p>
<p>t_new = t_old - f(t_old) / f... |
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