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 |
|---|---|---|---|---|
4,019,561 | <p>If <span class="math-container">$M_{n\times n}$</span> is the set of invertible matrices with real entries. Find two matrices <span class="math-container">$A,B\in M_{n \times n}$</span> with the propriety that there not exists such a continuous function</p>
<p><span class="math-container">$$f:[0,1]\to M, \quad f(0)... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$\det(A)>0>\det(B)$</span>, then there is no such function, because otherwise the range of the map <span class="math-container">$\det\circ f$</span> would contain <span class="math-container">$\det(A)$</span> and <span class="math-container">$\det(B)$</span>, but not <span class... |
297,907 | <p>Let consider the ring $\mathbb{Z}_p$ and $\zeta$ be a $p$-th root of unity. Especially $\zeta \not \in \mathbb{Z}_p$.
Denote with $\Phi _p(x)$ the cyclotomical polynomial in $p$. Since $p$ is a prime we know that it has the shape $\Phi _p(x)= 1 + x +x^2 +... +x^{p-1}$.
This gives rise for the quotient ring</p>
<p>$... | Neil Strickland | 10,366 | <p>Put $A=\mathbb{Z}_p[\zeta]$ and $\pi=\zeta-1$. Then
$$ A/\pi=\mathbb{Z}_p[x]/(\Phi_p(x),x-1)=
(\mathbb{Z}_p[x]/(x-1))/\Phi_p(x) = \mathbb{Z}_p/\Phi_p(1) =
\mathbb{Z}/p.
$$
This is a field, so $\pi$ generates a maximal ideal. Now suppose that $u$ lies outside this maximal ideal. Let $v$ be a lift in $\... |
142,993 | <p>I'm challenging myself to figure out the mathematical expression of the number of possible combinations for certain parameters, and frankly I have no idea how.</p>
<p>The rules are these:</p>
<p>Take numbers 1...n. Given m places, and with <em>no repeated digits</em>, how many combinations of those numbers can be ... | Douglas S. Stones | 139 | <p>Here's a computational solution. We can generate all such triples using this code in <a href="http://www.gap-system.org/" rel="nofollow">GAP</a>:</p>
<pre><code>S:=Filtered(Tuples([1..8],2),c->(c[1]+c[2]) mod 2=0);
T:=Filtered(Combinations(S,3),i->i[1][1]<>i[2][1] and i[1][1]<>i[3][1] and i[2][1]&... |
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | user541686 | 22,830 | <p>Either</p>
<pre><code>Reduce[h . y == x * y, x]
</code></pre>
<p>or</p>
<pre><code>Reduce[(h - IdentityMatrix[Length[h]] x) . y == 0, x]
</code></pre>
<p>depending on whether you would rather type $y$ once or twice.</p>
|
725,602 | <p>I am trying to prove the 'second' triangle inequality:
$$||x|-|y|| \leq |x-y|$$</p>
<p>My attempt:
$$----------------$$
Proof:
$|x-y|^2 = (x-y)^2 = x^2 - 2xy + y^2 \geq |x|^2 - 2|x||y| + |y|^2 = (||x|-|y||)^2$</p>
<p>Therefore $\rightarrow |x-y| \geq ||x|-|y||$</p>
<p>$$----------------$$</p>
<p>My questions are... | Robert Israel | 8,508 | <p>If $M = \pmatrix{0 & 1\cr 1& 1\cr}$, then $M^n = \pmatrix{F_{n-1} & F_n\cr
F_n & F_{n+1}\cr}$ (easy to prove by induction). The left side of your equation is $\det(M^n) = (\det(M))^n$.</p>
|
353,480 | <p>Is $f(x)=\ln(x)$ uniformly continuous on $(1,+\infty)$? If so, how to show it?</p>
<p>I know how to show that it is not uniformly continuous on $(0,1)$, by taking $x=\frac{1}{\exp(n)}$ and $y = \frac{1}{\exp(n+1)}$.</p>
<p>Also, on which interval does $\ln(x)$ satisfy the Lipschitz condition?</p>
| Bombyx mori | 32,240 | <p>Hint: Try to show $x$ is uniformly continuous on $\mathbb{R}$. Then try to establish $\ln(x)-\ln(y)<|x-y|$ for $x,y$ large enough. </p>
|
1,885,751 | <p>An urn contains 15 Balls (5 white, 10 Black). Let's say we pick them one after the other without returning them. How many white balls are expected to have been drawn after 7 turns?</p>
<p>I can calculate it by hand with a tree model but is there a formula for this?</p>
| samerivertwice | 334,732 | <p>The probability of getting zero 1's is $(\frac{5}{6})^5=\frac{3125}{7776}$</p>
<p>There are 5 ways of getting one 1, so the probability is $5\times\frac{1}{6}\times(\frac{5}{6})^4$</p>
<p>You want the probability of neither of these events:</p>
<p>$$1-(\frac{5}{6})^5-(5\times\frac{1}{6}\times(\frac{5}{6})^4)=\fra... |
1,556,747 | <p>$$\text{a)} \ \ \sum_{k=0}^{\infty} \frac{5^{k+1}+(-3)^k}{7^{k+2}}\qquad\qquad\qquad\text{b)} \ \ \sum_{k=1}^{\infty}\log\bigg(\frac{k(k+2)}{(k+1)^2}\bigg)$$</p>
<p>I am trying to determine the convergence values. I tried with partial sums and got stuck...so I am thinking the comparison test...Help</p>
| Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\sum_{k=0}^{\infty}\frac{5^{k+1}+(-3)^k}{7^{k+2}}=\lim_{m\to\infty}\sum_{k=0}^{m}\frac{5^{k+1}+(-3)^k}{7^{k+2}}=$$
$$\lim_{m\to\infty}\frac{182+\left(-\frac{1}{7}\right)^{m}3^{1+m}-5^{3+m}7^{-m}}{490}=$$
$$\frac{1}{490}\lim_{m\to\infty}\left(182+\left(-\frac{1}{7}\right)^{m}3^{1+m}-5^{3+m}7^{-m}\righ... |
125,165 | <p>Hi friends,</p>
<p>I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic variety over $\mathbb{Q}$. If I understand correctly, the notion of critical value depends only on the Hodge number... | François Brunault | 6,506 | <p>The factor $L_\infty(M,s)$ is holomorphic and non-vanishing for $\operatorname{Re}(s)$ large enough, so it is definitely necessary to also ask that $L_\infty(\hat{M},1-s)$ has no pole at the given integer. As an example, for the Riemann zeta function $\zeta(s)$, only the even integers $n \geq 2$ are critical.</p>
<... |
3,282,206 | <p>I'm familiar with Fermat's Little Theorem and Euler's Totient, but I'm wondering whether the fact that the only shared factor of <span class="math-container">$(a,N)=1$</span> has something to do with the fact that, given the prior constraints there exists at least one <span class="math-container">$x$</span> (with <s... | İbrahim İpek | 554,493 | <p>It is because for a number to have an inverse (i.e. some element which product of both is a residue of <span class="math-container">$1$</span>), we need the number to be relatively prime with the modulus. </p>
<p><span class="math-container">$$a \cdot a^{x -1} \equiv_N 1$$</span></p>
<p>We have an inverse of <span... |
3,282,206 | <p>I'm familiar with Fermat's Little Theorem and Euler's Totient, but I'm wondering whether the fact that the only shared factor of <span class="math-container">$(a,N)=1$</span> has something to do with the fact that, given the prior constraints there exists at least one <span class="math-container">$x$</span> (with <s... | Bernard | 202,857 | <p>Not sure this fully answers you question (not very clear to me), but <span class="math-container">$\gcd(a,N)=1$</span> implies <span class="math-container">$a$</span> is a unit modulo <span class="math-container">$N$</span>, so the congruence class of <span class="math-container">$a\bmod N$</span> generates a <em>fi... |
2,648,492 | <p>I am having trouble with this problem. When they say spot I think they are essentially saying the sum, so its the probability that the sum of dice is $11$ or less.</p>
<p>I understand that there are $6^5$ combinations.</p>
<p>I found 6 ways that it can equal to $11$ $(2,3,2,2,2)(3,3,1,1,3),(4,4,1,1,1),(5,2,2,1,1),... | BruceET | 221,800 | <p>My guess is that you have been studying the Central Limit Theorem
and that you are to assume that the total $T$ on five dice is
approximately normal. Here is an outline of that method.</p>
<p>It is not difficult to show that the number $X$ on a single die has $E(X) = 7/2$ and $var(X) = 105/36.$ </p>
<p>Thus $T$ ha... |
1,293,207 | <p>A ray of light travels from the point $A$ to the point $B$ across the border between two materials. At the first material the speed is $v_1$ and at the second it is $v_2$. Show that the journey is achieved at the least possible time when Snell's law: $$\frac{\sin \theta_1}{\sin \theta_2}=\frac{v_1}{v_2}$$ holds. </p... | Demosthene | 163,662 | <p><strong>Without Lagrange multipliers</strong></p>
<p>You don't need to use Lagrange multipliers. Snell's law can be derived directly from Fermat's principle of least time, which consists in minimizing the time taken by the ray of light to travel from one point to another. Take a look at the diagram below:</p>
<p><... |
1,556,645 | <p>I am new to the axiom of choice, and currently working my way through some exercises. I am struggling with the following exercise:</p>
<p><strong>Exercise -</strong>
Prove the Axiom of Choice (every surjective $f: X \to Y$ has a section) in the following two special cases:</p>
<ol>
<li>Y is finite</li>
<li>X is co... | Asaf Karagila | 622 | <p>There is a terminological discrepancy here.</p>
<p>The Axiom of Countable Choice refers to the statement "If $S$ is a countable family of non-empty sets, then $S$ admits a choice function".</p>
<p>What you are trying to prove is that if $f\colon X\to Y$ and $X$ is countable, then $f$ admits a section. </p>
<p>Thi... |
216,031 | <p>Using image analysis, I have found the positions of a circular ring and imported them as <code>xx</code> and <code>yy</code> coordinates. I am using <code>ListInterpolation</code> to interpolate the data:</p>
<pre><code>xi = ListInterpolation[xx, {0, 1}, InterpolationOrder -> 4, PeriodicInterpolation -> True,... | MikeY | 47,314 | <p>You can use a Fourier Transform to take it into the frequency space, then truncate coming back with the inverse transform, thus putting a limit on the max frequency component and therefore max curvature.</p>
<p>Load in the data</p>
<pre><code>data = Import["testShape.csv"];
</code></pre>
<p>Center the data about ... |
2,194,376 | <p>I'm having some trouble understanding the answers to the following questions:</p>
<p><a href="https://i.stack.imgur.com/nnuu8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nnuu8.png" alt="enter image description here"></a></p>
<p>(a)</p>
<p>Why would it make sense for Eve to test out the $gcd... | Mark Viola | 218,419 | <p>Let $f_n(x)$ be given by </p>
<p>$$f_n(x)=\frac{\sin(x/n)\sin(2nx)}{x^2+4n}$$</p>
<p>Using the <a href="https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means" rel="nofollow noreferrer">AM-GM inequality</a>, we assert that</p>
<p>$$|f_n(x)|\le \frac{|x|}{n(x^2+4n)}\le \frac{1}{4n^{3/2}}$$</p>
... |
1,811,028 | <p>I want to know which of the following methods is right for graphing $f(ax-b)$ from $f(x)$ and why:</p>
<p>Method 1. First Horizontally Translate it ($f(x)$) by $b$, then Horizontally Stretch/Compress it by $a$.</p>
<p>Method 2. First Horizontally Stretch/Compress it by $a$ then Horizontally Translate it by $b/a$.<... | Tsemo Aristide | 280,301 | <p>Yes, a connected compact Lie group is the product of a semi-simple Lie group and a torus $T^n$, thus its Lies algebra $g$ is the sum of a semi-simple Lie algebra $s$ and a commutative algebra $c$, such that $[s,c]=0$, thus $[s+c,s+c]=s$.</p>
<p><a href="https://en.wikipedia.org/wiki/Compact_Lie_algebra#Definition"... |
4,537,050 | <p>Question 2 of Chapter 14 in Spivak's <em>Calculus</em> reads as follows:</p>
<blockquote>
<p>For each of the following <span class="math-container">$f$</span>, if <span class="math-container">$F(x)=\int_0^xf$</span>, at which points <span class="math-container">$x$</span> is <span class="math-container">$F'(x)=f(x)$... | S.C. | 544,640 | <p>Here is another approach using the <strong>Darboux Integral</strong> definition.</p>
<hr />
<p>Consider the function that is defined by the formula:</p>
<p><span class="math-container">$f(x)= \begin{cases}1\quad &\text {if $x=\frac{1}{n}$ for some $n \in \mathbb N$} \\0 \quad &\text{otherwise} \end{cases}$</... |
552,474 | <p>If there are,
Are there unity <strong>(but not division)</strong> rings of this kind?
Are there non-unity rings of this kind?</p>
<p>Sorry, I forgot writting the non division condition.</p>
| Community | -1 | <p>Yes, in fact much more can be said: There are rings with $1$ such that every non-zero element has a multiplicative inverse. These rings are called <a href="https://en.wikipedia.org/wiki/Division_ring">division rings</a> or skew-fields.</p>
<p>The real quaternions are an example of a non-commutative division ring.</... |
552,474 | <p>If there are,
Are there unity <strong>(but not division)</strong> rings of this kind?
Are there non-unity rings of this kind?</p>
<p>Sorry, I forgot writting the non division condition.</p>
| Matt E | 221 | <p>Another kind of example: non-commutative deformations of commutative integral domains, e.g. $\mathbb C[x,y]$ with the commutation relation $[x,y] = 1$. </p>
|
1,868,797 | <blockquote>
<p><strong>Question:-</strong></p>
<p>Three points represented by the complex numbers $a,b$ and $c$ lie on a circle with center $O$ and radius $r$. The tangent at $c$ cuts the chord joining the points $a$ and $b$ at $z$. Show that $$z=\dfrac{a^{-1}+b^{-1}-2c^{-1}}{a^{-1}b^{-1}-c^{-2}}$$</p>
</blockq... | Dietrich Burde | 83,966 | <p>An Linear Programming is degenerate if in a basic feasible solution, one of the basic variables takes on a zero value. Degeneracy is caused by redundant constraint(s), e.g. see <a href="http://optlab.mcmaster.ca/feng/4O03/LP.Degeneracy.pdf" rel="nofollow">this example</a>.</p>
|
4,429,162 | <p><a href="https://i.stack.imgur.com/9nrUn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9nrUn.png" alt="enter image description here" /></a></p>
<p>This is from Rambo's Math subject GRE book.</p>
<p>One solution to this problem is to note that the equation of the circle is <span class="math-conta... | Mark McClure | 21,361 | <p>The area swept out by a segment of length <span class="math-container">$r$</span> as it rotates about one end point through the angle <span class="math-container">$\theta$</span> is <span class="math-container">$\frac{1}{2}r^2\theta$</span>. Now, since you already know how to find the points of intersection, it's pr... |
4,429,162 | <p><a href="https://i.stack.imgur.com/9nrUn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9nrUn.png" alt="enter image description here" /></a></p>
<p>This is from Rambo's Math subject GRE book.</p>
<p>One solution to this problem is to note that the equation of the circle is <span class="math-conta... | Math Lover | 801,574 | <p><a href="https://i.stack.imgur.com/UvIdR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UvIdR.png" alt="enter image description here" /></a></p>
<p>Equation of the circle is <span class="math-container">$~x^2 + (y-k)^2 = 1$</span></p>
<p>Equation of the parabola is <span class="math-container">$~... |
76,163 | <p>Represent the position of a unit-length, oriented segment $s$ in the plane
by the location $a$ of its <em>basepoint</em> and
an orientation $\theta$: $s = (a,\theta)$. So $s$ can be
viewed as a point in $\mathbb{R^2} \times \mathbb{S^1}$.
Now I'll define a metric on this space.
Define the distance $d(s_1,s_2)$ betw... | j.c. | 353 | <p>What follows are just some illustrations, not a full answer; please refer to Anton Petrunin's answer for a nice description of the 4-dimensional geometry that the original question is embedded in. </p>
<p><a href="https://github.com/jcmathoverflow/jc-MO-nb/blob/master/segmentmetric.nb" rel="nofollow noreferrer">Her... |
76,163 | <p>Represent the position of a unit-length, oriented segment $s$ in the plane
by the location $a$ of its <em>basepoint</em> and
an orientation $\theta$: $s = (a,\theta)$. So $s$ can be
viewed as a point in $\mathbb{R^2} \times \mathbb{S^1}$.
Now I'll define a metric on this space.
Define the distance $d(s_1,s_2)$ betw... | Jean-Marc Schlenker | 9,890 | <p>A useful keyword for this problem is the Wasserstein distance, see <a href="http://en.wikipedia.org/wiki/Wasserstein_metric" rel="nofollow">wikipedia</a>. I believe that this Wasserstein distance, for $p=1$, provides a variant of the distance you're considering but for unoriented segments. There is a well-developed ... |
3,482,376 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is differentiable on <span class="math-container">$[0,\infty)$</span> and <span class="math-container">$\displaystyle \lim_{x \to \infty} \frac{f(x)}{x} = 0$</span>. Show that <span class="math-container">$\displaystyle \liminf_{x \to \infty}|f'(x)| = 0$... | Pythagoras | 701,578 | <p>What you have is for <span class="math-container">$y=2x$</span> and <span class="math-container">$c_x\in (x,2x)$</span> that <span class="math-container">$$|f'(c_x)|\leq \frac{c_x}x\left|\frac{f(2x)}{2x}-\frac{f(x)}x\right|+\left|\frac{f(c_x)}{c_x}\right|,$$</span> where <span class="math-container">$1<\frac {c_x... |
1,000,448 | <p>One of the $x$-intercepts of the function $f(x)=ax^2-3x+1$ is at $x=-1$. Determine $a$ and the other $x$-intercept.</p>
<p>I happen to know that $a=-4$ and the other $x$-intercept is at $x=\frac{1}{4}$ but I don't know how to get there. I tried substituting $x=-1$ into the quadratic formula.</p>
<p>$$
-1=\frac{-(-... | Community | -1 | <p>One approach is given in comment by Winther, here is another,<br>
$ax^2+bx+c=0$ has two roots $r_1$ and $r_2$ then $$r_1+r_2=\frac{-b}{a}$$ and $$r_1r_2=\frac{c}{a}$$</p>
|
2,668,826 | <p>I am stuck on this result, which the professor wrote as "trivial", but I don't find a way out.</p>
<p>I have the function </p>
<p>$$f_{\alpha}(t) = \frac{1}{2\pi} \sum_{k = 1}^{+\infty} \frac{1}{k}\int_0^{\pi} (\alpha(p))^k \sin^{2k}(\epsilon(p) t)\ dp$$</p>
<p>and he told use that for $t\to +\infty$ we have:</p>... | user178256 | 178,256 | <p>Let $$ I=\int_{0}^{1}\ln\left(\frac{1 + x^{11}}{1 + x^{3}}\right)
\,{\mathrm{d}x \over \left(1 + x^{2}\right)\ln\left(x\right)}+\int_{1}^{\infty}\ln\left(\frac{1 + x^{11}}{1 + x^{3}}\right)
\,{\mathrm{d}x \over \left(1 + x^{2}\right)\ln\left(x\right)}$$
$$\int_{1}^{\infty}\ln\left(\frac{1 + x^{11}}{1 + x^{3}}\righ... |
120,667 | <p>Let $V, W$ be two finite-dimensional vector spaces, $f: V\rightarrow W$ a linear map, and $U \subseteq W$ a vector subspace. I'm trying to show that $(f^{-1}(U))^0 = f^*(U^0)$, i.e. that the annihilator of the inverse image of $U$ is the image of the annihilator under the the dual $f^*$ of $f$. $(f^{-1}(U))^0 \supse... | joriki | 6,622 | <p>Given a linear functional $v^*\in(f^{-1}(U))^0$ on $V$ that annihilates all of $f^{-1}(U)$, in particular it annihilates the kernel of $f$. Thus we can define a linear functional $w^*$ on $W$ by $w^*(w)=v^*(v)$ using any preimage $v$ of $w$, since $v^*$ has the same value an all preimages of $w$. Then $w^*\in U^0$ a... |
4,025,279 | <p>I have been reading material on uniformly continuous functions. And going through the problems where we have to prove that a function is not uniformly continuous or otherwise.</p>
<p>A function defined on an interval I is said to be uniformly continuous on I if to each <span class="math-container">$\epsilon$</span> ... | Angelo | 771,461 | <p>For uniformly continuous functions, there is for each
<span class="math-container">$\varepsilon >0$</span> a <span class="math-container">$\delta >0$</span> such that when we draw a rectangle around each point of the graph with width <span class="math-container">$2\delta$</span> and height <span class="math-co... |
1,624 | <p>For example, to change the color of each pixel to the mean color of the three channels, I tried</p>
<pre><code>i = ExampleData[{"TestImage", "Lena"}];
Mean[i]
</code></pre>
<p>but it just remains unevaluated:</p>
<p><img src="https://i.stack.imgur.com/K1RRR.png" alt="enter image description here"></p>
<p>How can... | Mr.Wizard | 121 | <p>Have you looked through the included documentation?</p>
<p><a href="http://reference.wolfram.com/mathematica/guide/ImageProcessing.html" rel="nofollow">guide/ImageProcessing</a></p>
<p><a href="http://reference.wolfram.com/mathematica/tutorial/ImageProcessing.html" rel="nofollow">tutorial/ImageProcessing</a></p>
... |
935,331 | <p>Previously, to integrate functions like $x(x^2+1)^7$ I used integration by parts. Today we were introduced to a new formula in class: $$\int f'(x)f(x)^n dx = \frac{1}{n+1} {f(x)}^{n+1} +c$$
I was wondering how and why this works. Any help would be appreciated. </p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Let $u'=f'(x)$ and $v=f(x)^n$; so $u=f(x)$, $v'=n f(x)^{n-1}f'(x)$. Then $$I=\int f'(x)f(x)^n dx =f^{n+1}(x)-n\int f'(x)f(x)^ndx=f(x)^{n+1}-nI$$</p>
<p>I am sure that you can take from here.</p>
|
230,167 | <p>The following 3 matrices are useful when viewing matrices as vectors, known as commutation <span class="math-container">$K_n$</span>, symmetrizer <span class="math-container">$N_n$</span> and duplication <span class="math-container">$G_n$</span>. They are usually defined by their matrix relations below.</p>
<p><span... | Daniel Huber | 46,318 | <p>If I understand correctly, then you only need the operator "vec".
This is clear for the first line. The second line applies vec to the symmetrized version of A: (A+Transpose[A])/2. And the third line applies "vec" to a symmetric matrix, the operator is the same, only the operand is different.
The... |
230,167 | <p>The following 3 matrices are useful when viewing matrices as vectors, known as commutation <span class="math-container">$K_n$</span>, symmetrizer <span class="math-container">$N_n$</span> and duplication <span class="math-container">$G_n$</span>. They are usually defined by their matrix relations below.</p>
<p><span... | flinty | 72,682 | <p>I hope this does the trick. It's more code than yours but I've come at it from a slightly different angle - I suppose another implementation can't hurt right? I've used <code>FindPermutation</code> to get <span class="math-container">$K_n$</span> and <code>SolveAlways</code> for non-square <span class="math-containe... |
129,890 | <p>The code</p>
<pre><code>x0 = 0.25; T = 20; u1 = -0.03; u2 = 0.07; u3 = -0.04;
a = 1/100; t0 = 5; omega = 2;
a = 0.01; dis[x_] := a/(Pi (x^2 + a^2))
P[t_] := If[t <= t0, Sin[omega t], 0]
u[t_] := u1 HeavisideTheta[t - 0.8] +
u2 HeavisideTheta[t - 1.64] + u3 HeavisideTheta[t - 3.33]
pde = a D[w[x, t],... | tablecircle | 43,920 | <p>I think that maybe Mathematica can't solve the boundary condition on the second order derivative problem.(I'm not sure, waiting for other expert's answer!)
So, I tried to rewrite your code by defining <code>s[x, t] == D[w[x, t], {x, 2}]</code></p>
<p>Therefore, your pde and bc becomes:</p>
<pre><code>pde = {a D[s[... |
987,620 | <p>$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?</p>
| Ross Millikan | 1,827 | <p>If you follow through the usual proof that $\sqrt 2 $ is irrational, it goes through in this case as well. One of them (for $\sqrt 3$) is <a href="https://math.stackexchange.com/questions/131391/proving-sqrt-3-is-irrational">here</a>, but a search for irrational+sqrt will find many choices</p>
|
36,477 | <p>The length, width and height of a rectangular box are measured to be 3cm, 4cm and 5cm respectively, with a maximum error of 0.05cm in each measurement. Use differentials to approximate the maximum error in the calculated volume.</p>
<p><br><br>
Please help</p>
| Ross Millikan | 1,827 | <p>Hint: express V as a function of L, W, H. Take the partial derivative with respect to each variable to see how V changes with each.</p>
|
129,295 | <p>$$\int{\sqrt{x^2 - 2x}}$$</p>
<p>I think I should be doing trig substitution, but which? I completed the square giving </p>
<p>$$\int{\sqrt{(x-1)^2 -1}}$$</p>
<p>But the closest I found is for</p>
<p>$$\frac{1}{\sqrt{a^2 - (x+b)^2}}$$ </p>
<p>So I must add a $-$, but how? </p>
| David Mitra | 18,986 | <p>In hopes of taking advantage of the identity
$$\tan^2\theta=\sec^2\theta-1,$$ make the substitution:
$$x-1=\sec\theta, \quad dx=\sec\theta\,\tan\theta\,d\theta.$$
Then we have
$$
\int \sqrt{(x-1)^2-1}\, dx = \int\sqrt{\sec^2\theta -1}\, \sec\theta\tan\theta\,d\theta
=\int\tan^2\theta\sec\theta\,d\theta=\int (\se... |
1,422,990 | <p>How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$?</p>
<p>If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.</p>
| user236182 | 236,182 | <p>For contradiction, assume $(2^n-1)^{1/n}$ is rational. </p>
<p>Then $(2^n-1)^{1/n}=p/q$ for some $p,q\in\Bbb Z_{>0}, \gcd(p,q)=1$.</p>
<p>But then $(2^n-1)q^n=p^n\implies q^n\mid p^n\implies q=1$.</p>
<p>$2^n-1=p^n$. We can't have $p=1$, since $n\ge 2$. But then $2^n-1<2^n\le p^n$, contradiction.</p>
|
110,722 | <p>1) Many Mathematics departments ask to send a "list of publications" while applying for research postdoctoral jobs. My question is: how important is it to post my papers in arXiv. I know, posting on arXiv is always good, because people might search for the arXiv -ed papers, but how much difference is publication on ... | Delio Mugnolo | 26,039 | <p>Answer to 2): In my experience, I think an abstract is not appropriate in a list of publications.</p>
<p>Answer to 1): A submitted article that nobody can see is basically a non-existing article - the committee members do not even now whether it is a deep 300-page-essay or a 2-page-note. More generally, I believe t... |
110,722 | <p>1) Many Mathematics departments ask to send a "list of publications" while applying for research postdoctoral jobs. My question is: how important is it to post my papers in arXiv. I know, posting on arXiv is always good, because people might search for the arXiv -ed papers, but how much difference is publication on ... | Ben Webster | 66 | <p>While usually I don't like answering questions like this on MO, there is actually an important fact here specific to the mathematics community which would probably be missed on academia.stackexchange, or another non-mathematical site. </p>
<p>The answer to your 2) is:</p>
<blockquote>
<p>There is no logic behin... |
2,334,381 | <p>is strong topology on a metric space the topology that is induced by metric? Is a open set in weak topology also open in strong topology?</p>
| Subhajit Saha | 435,027 | <p>If we consider two topologies;The 1st topology is greater or bigger or stronger or finner whatever you says if it contains all the open sets of 2nd topology. So you are correct. </p>
|
173,131 | <p>Let's suppose that for the following expression:</p>
<p>$\qquad \alpha\,\beta +\alpha+\beta$</p>
<p>I know that $\alpha$ and $\beta$ are of small magnitude (e.g., 0 < $\alpha$ < 0.02 and 0 < $\beta$ < 0.02). Therefore, the magnitude of $\alpha\,\beta$ is negligible, i.e., the original expression can be... | Carl Woll | 45,431 | <p>I would follow the approach recommended by @Jens in the linked answer, which is similar to the answer by @Akku <em>( It is different since @Akku takes series of the numerator and denominator separately, and then finds the series of the ratio after normalizing)</em>. Introduce a dummy scaling variable, and then do a ... |
2,559,260 | <p>There exists a function $f$ such that $\lim_{x \rightarrow \infty} \frac{f(x)}{x^2} = 25$ and $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = 5$</p>
<p>I am confused, I do not whether it is true or not</p>
<p>I have a counter-example, but I think thre might be such function</p>
| Sri-Amirthan Theivendran | 302,692 | <p><strong>Hint</strong>
Suppose that such a function exists. Note that
$$
25=\lim_{x \to \infty} \frac{f(x)}{x^2} = \lim_{x \to \infty} \frac{f(x)}{x}\times \lim_{x \to \infty} \frac{1}{x}.
$$</p>
|
2,559,260 | <p>There exists a function $f$ such that $\lim_{x \rightarrow \infty} \frac{f(x)}{x^2} = 25$ and $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = 5$</p>
<p>I am confused, I do not whether it is true or not</p>
<p>I have a counter-example, but I think thre might be such function</p>
| oandersonm | 511,441 | <p>In this case,</p>
<p>$lim_{x\to\infty}\frac{f(x)}{x^2} = 25 = 5*5 =5* lim_{x\to \infty}\frac{f(x)}{x} $</p>
<p>so,</p>
<p>$lim_{x\to\infty}\frac1x = 5$</p>
<p>this gives a contradiction in the statement.</p>
|
2,595,612 | <p>Is true that every limit can only converge, diverge or(exclusive) not exist?</p>
<p>Can I demonstrate that it doesn't exist after I proved it doesn't converge neither diverge?</p>
<p>I've never seen this, but it makes some sort of sense to me. If a real isn't positive nor negative, it must be zero... But with limi... | Arthur | 15,500 | <p>Divergence means the limit doesn't exist. "Divergence to $\infty$" is a special case of divergence, and we sometimes say that the limit exists in those cases, but strictly speaking it doesn't (unless we're working in the extended reals, which as far as I can tell is mostly done just to indulge in this specific abuse... |
30,918 | <p>Judging by some of the posts on meta<sup>1</sup> and comments posted there it seems that there are users who try to improve the posts by correcting spelling mistakes. Of course, there are other ways to improve the posts via editing, some of them probably more important than grammar and spelling.<sup>2</sup> Still th... | GEdgar | 442 | <p>Some examples of <strong>incorrect possessive:</strong></p>
<p>Stoke's theorem (incorrect) should be Stokes' or even Stokes's since the man's name is "Stokes".</p>
<p>Similar: Baye's theorem (incorrect), which appeared somewhere in .se recently.</p>
<p>Descarte's Rule of Signs</p>
|
2,912,759 | <p>I am trying to use the characteristic function of the uniform distribution defined on (0,1) to compute the mean. I have calculated the characteristic function (correctly) and used Euler's identity to convert it to the following form:</p>
<p>$$\phi_Y(t)=\frac{\sin(t)}{t} + i \frac{1-\cos(t)}{t}$$</p>
<p>I should be... | Angina Seng | 436,618 | <p>Use Maclaurin series:
$$i\sin t-it\cos t+t\sin t+\cos t-1
=it-it+t^1+1-t^2/2-1)+O(t^3)=\frac{t^2}2+O(t^3)$$
and so
$$\lim_{t\to0}\frac{i\sin t-it\cos t+t\sin t+\cos t-1
}{t^2}=\frac12.$$</p>
|
2,816,119 | <p>Prove if $G$ is an abelian group, then $q(x) = x^2$ defines a homomorphism from $G$ into $G$. Is $q$ ever an isomorphism?</p>
<p>The first proof was no problem, I'm having trouble with the isomorphism piece though. My book has the following explanation that I don't quite understand:</p>
<p>"In order for $q$ to be ... | GSofer | 509,052 | <p>If there was an element other than the identity that is its own inverse, then the homomorphism would send it to the identity (because taking it squared would give you the identity).</p>
<p>Since a homomorphism from a group to itself is an isomorphism only if the identity is the only element in the group that's sent... |
2,816,119 | <p>Prove if $G$ is an abelian group, then $q(x) = x^2$ defines a homomorphism from $G$ into $G$. Is $q$ ever an isomorphism?</p>
<p>The first proof was no problem, I'm having trouble with the isomorphism piece though. My book has the following explanation that I don't quite understand:</p>
<p>"In order for $q$ to be ... | lhf | 589 | <p>$q$ is injective iff $G$ has no element of order $2$.</p>
<p>If $G$ is finite, this happens iff the order of $G$ is odd. In this case, $q$ is an isomorphism.</p>
<p>If $G$ is infinite, $q$ might not be surjective even if it is injective. An example is $G=\mathbb Z$.</p>
|
1,940,446 | <p>I'm working through Bona's "A Walk Through Combinatorics" and I came across this problem:</p>
<blockquote>
<p>A company has $20$ employees, $12$ male and $8$ female. How many ways are
there to form a $5$ person committee that contains at least one male and
at least one female?</p>
</blockquote>
<p>I realise ... | Parcly Taxel | 357,390 | <p>There are $\binom{20}5=15504$ ways to make <em>any</em> committee from the people available. That the incorrect answer is much larger than this upper limit suggests an error of <em>overcounting</em>, which is a common mistake in combinatorial problems.</p>
<p>For this problem, suppose the initial male and female ch... |
1,232,420 | <p>Consider the ideal $I = (13x+16y, 11x+13y)$ in the ring R = $\mathbb{Z}[x,y].$</p>
<p>Prove that $I=(x-2y, 3x+y)$ by using mutual inclusion.</p>
<p>I'm confused on how to start...do I begin by multiplying the elements in the ideal by a general element of R?</p>
| Rolf Hoyer | 228,612 | <p>You want to show that both elements of one generating set are $\Bbb Z$-linear combinations of elements from the other generating set. In general, you would use $R$-linear combinations, but due to degree considerations you can use integers. </p>
<p>As an example, you want to find relations like $-4(x-2y) + 5(3x+y)... |
15,480 | <p>Say I have two lists,</p>
<pre><code>list1 = {a, b, c}
list2 = {x, y, z}
</code></pre>
<p>and I want to map a function f over them to produce</p>
<pre><code>{f[a,x], f[a,y], f[a,z], f[b,x], f[b,y], f[b,z], f[c,x], f[c,y], f[c,d]}
</code></pre>
<p>I would assume I map the function over the first list to produce a... | halirutan | 187 | <p>What you try to achieve here is called <a href="http://en.wikipedia.org/wiki/Currying">Currying</a> which can be used in other languages like Haskell naturally. In <em>Mathematica</em> this does not work like that.</p>
<p>But what about </p>
<pre><code>Outer[f, list1, list2]
(*
{{f[a, x], f[a, y], f[a, z]},
... |
3,300,793 | <p>Having an immense amount of trouble trying to figure this problem out, and the more I think and ask about it the more confused I seem to get. I think I have finally figured it out so can someone who truly knows the correct answer justify this?</p>
<p>Problem:Let <span class="math-container">$A=\{a,b,c\}$</span>,Let... | Matthew Leingang | 2,785 | <p>With respect to JMoravitz's point of view, I think you understand the problem as intended by the instructor/problem author, and should not extend to the case that elements of <span class="math-container">$A$</span> might contain themselves as subsets.</p>
<p>I'm inferring the point of the problem is to understand t... |
2,462,297 | <p>Let $(x_1,...,x_n)$ be real numbers and M be an $n \times n$ matrix whose its column is given by the entries $x_i,x_i^2, x_i^3,...x_i^n$. Compute the determinant of M.</p>
<p>I computed the formula for the determinant of M in terms of $x_1...x_n$ but I wonder if I can find a specific value.</p>
| Ali H. | 471,756 | <p>@Tsemo Aristide is absolutely correct, you can follow that link and also <a href="https://proofwiki.org/wiki/Vandermonde_Determinant" rel="nofollow noreferrer">The Proof for your specific case here</a>. However, this is a different kind of explanation for what you have, which is not a proof but I think it might help... |
320,378 | <p>Given a number of normal distributions <span class="math-container">$N(\mu_1, \sigma^2), N(\mu_2, \sigma^2), ..., N(\mu_n, \sigma^2)$</span> with fixed variance <span class="math-container">$\sigma^2$</span>, but not necessary equal means. My question is how to approximate the variance given a number of samples of t... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand{\si}{\sigma}$</span>
Let us assume that the <span class="math-container">$n$</span> samples from the respective distributions <span class="math-container">$N(\mu_1, \sigma^2), \dots, N(\mu_n, \sigma^2)$</span> are independent. Let <span class="math-container">$X_{ij}:=X^i_j... |
2,640,477 | <p>According to <a href="https://rads.stackoverflow.com/amzn/click/0073383090" rel="nofollow noreferrer">Rosen</a>, an infinite set A is countable if $|A|= |\mathbb{Z}^+|$ which in turn can be established by finding a bijection from A to $\mathbb{Z}^+$.</p>
<p>Also, a sequence is defined as a function from $\mathbb{Z}... | fleablood | 280,126 | <p>"However, it need not be a bijection"</p>
<p>No, it doesn't.</p>
<p>"This implies that not every sequence is countable"</p>
<p>Why do you say that? </p>
<p>$f: \mathbb Z^+ \to B$. If $f$ is not surjective then there are points of $B$ that are not in the image. Those to not matter. We can restrict ourselves t... |
2,316,286 | <blockquote>
<p><strong>Theorem</strong> <em>(Cauchy-Schwarz Inequality) : If $u$ and $v$ are vectors in an inner product space $V$, then</em>
$$\langle u,v\rangle ^2\leqslant \langle u,u\rangle \langle v,v\rangle .$$</p>
</blockquote>
<p><strong>Proof</strong> : If $u=0$, then $\langle u,v\rangle = \langle u,u\ra... | alerouxlapierre | 438,519 | <p>Since you have $at^2+bt+c\geq 0$, you either have a double root or no real roots. In fact, if a polynomial of degree 2 has two distinct roots, then it is positive on a certain interval and negative on another one (you should check this). Thus, the polynomial $at^2+bt+c$ must not have two distinct roots because it is... |
2,316,286 | <blockquote>
<p><strong>Theorem</strong> <em>(Cauchy-Schwarz Inequality) : If $u$ and $v$ are vectors in an inner product space $V$, then</em>
$$\langle u,v\rangle ^2\leqslant \langle u,u\rangle \langle v,v\rangle .$$</p>
</blockquote>
<p><strong>Proof</strong> : If $u=0$, then $\langle u,v\rangle = \langle u,u\ra... | egreg | 62,967 | <p>Consider the parabola $y=ax^2+bx+c$. If the right hand side has distinct real roots, the vertex will have negative $y$-coordinate (as $a=\langle u,u\rangle>0$).</p>
<p>So the only case when no point on the parabola has negative $y$-coordinate is when $b^2-4ac\le0$.</p>
|
170,014 | <p>Determine for which $a$ values $f = x^2+ax+2$ can be divided by $g= x-3$ in $\mathbb Z_5$. </p>
<p>I don't know if there are more effective (and certainly <strong>right</strong>) ways to solve this problem, I assume there definitely are, but as I am not aware of them, I thought I could proceed like this: I have div... | DoubleTrouble | 25,409 | <p>I would be very surprised (in a good way) if there was a closed form solution of this. Even though it is straight-forward to calculate
$$E\left[\int_0^t X(s) ds\right]$$
the distribution of $$\int_0^t X(s) ds$$ is unknown. And without knowing the distribution, I believe it will be difficult to calculate the expecta... |
678,073 | <p>I am working with the multiplicative ring of integers modulo $2^{127}$.</p>
<p>Consider the set $E=\{(k,l) \mid 5^k \cdot 3^l \equiv 1\mod 2^{127}, k > 0, l> 0\}$.
I wonder if anybody knows or has an idea where to look for a result related to a lower bound for $M=\min\{k+l \mid (k,l)\in E \}$.</p>
<p>We hav... | Steven Stadnicki | 785 | <p>To flesh my 'birthday paradox' heuristic from a comment out into at least a partial answer:</p>
<p>The core concept is that among the first $m$ values of $5^k$ and the first $n$ values of $3^{-l}$, we have $mn$ different potential collisions, and if we treat these interactions as independent events then we should e... |
1,480,720 | <p>How many times do you have to flip a coin such that the probability of getting $2$ heads in a row is at least $1/2$?</p>
<p>I tried using a Negative Binomial:
$P(X=2)=$$(n-1)\choose(r-1)$$p^r\times(1-p)^{n-r} \geq 1/2$ where $r = 2$ and $p = 1/4$. However, I don't get a value of $n$ that makes sense.</p>
<p>Than... | Barry Cipra | 86,747 | <p>If you toss a coin $n$ times, the number of different ways the result can <em>aovid</em> getting two heads in a row is $F_{n+1}$, where $F_n$ is the Fibonacci sequence, with $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. So the crossover point is where $F_{n+1}/2^n\le1/2$, or $F_{n+1}\le2^{n-1}$. This occurs at $n=4$, sin... |
2,986,515 | <p>Can anyone help me with this problem? </p>
<p>Prove that for any real number <span class="math-container">$x > 0$</span> and for any <span class="math-container">$M > 0$</span> there is <span class="math-container">$N ∈ \mathbb N$</span> so that if <span class="math-container">$n > N$</span> then <span cla... | marty cohen | 13,079 | <p><span class="math-container">$\begin{array}\\
a_{n}
&= \dfrac{3n^2 - 9n + 6}{n^3 + 5n^2 + 8n + 4}\\
&= 3\dfrac{n^2 - 3n + 2}{n^3 + 5n^2 + 8n + 4}\\
&< 3\dfrac{n^2}{n^3}
\qquad\text{for } n \ge 1\\
&=\dfrac{3}{n}\\
&\le 1
\qquad\text{for } n \ge 3\\
\text{and}\\
a(n)
&\gt 0
\qquad\text{for... |
1,449,450 | <p>Is there a way to prove which one of these is bigger? $e^{(a+b)}$ or $e^a + e^b$?</p>
<p>Thanks</p>
| Rajat | 177,357 | <p>$$\frac{(e^a + e^b)}{2} \geq e^{\frac{(a+b)}{2}}\text{ (Using A.M.-G.M. inequality.)}$$
$$(e^a + e^b) > e^{\frac{(a+b+1)}{2}}\text{ (Using $4>e$).}$$
If, $e^{\frac{(a+b+1)}{2}}\leq 1$, then
$$e^{(a+b+1)}\leq e^{\frac{(a+b+1)}{2}} \text{ (Using the fact that, if $x \in [0,1]$, then } x\geq x^2).$$
So,
$$(e^a + ... |
1,449,450 | <p>Is there a way to prove which one of these is bigger? $e^{(a+b)}$ or $e^a + e^b$?</p>
<p>Thanks</p>
| hmakholm left over Monica | 14,366 | <p>Dividing through by $e^{a+b}$ gives
$$ e^a+e^b \lessgtr e^{a+b} \qquad\text{as}\qquad e^{-a}+e^{-b} \lessgtr 1$$
This is not <em>much</em> nicer, but it is <em>somewhat</em> nicer because there's now only one $a$ and one $b$. So it is now easier to solve for $a$ or $b$:
$$ e^a+e^b \lessgtr e^{a+b} \qquad\text{as}\qq... |
744,034 | <p>How do I show that for all integers $n$, $n^3+(n+1)^3+(n+2)^3$ is a multiple of $9$?
Do I use induction for showing this? If not what do I use and how? And is this question asking me to prove it or show it? How do I show it? </p>
| Mark Bennet | 2,906 | <p>Here are a some ideas to work on.</p>
<p>First the cubes mod $9$ turn out to be $0^3=0, 1^3=1, 2^3=-1, 3^3=0, 4^3=1, 5^3=-1 \dots$</p>
<p>The pattern persists because $(3n\pm1)^3=27n^3\pm 27n^2+9n\pm 1\equiv \pm 1, (3n)^3=27n^3\equiv 0$ - the sum of any consecutive three is equal to $0$ mod $9$.</p>
<p>Another wa... |
2,343,958 | <p>I am interested in a mathematical approach to quantum information theory. I have observed that several probabilists have been working in this area. What can be a suitable background and good book for this subject?</p>
| Larry B. | 364,722 | <p>There's an excellent introductory book called "Quantum Computing for Computer Scientists" by Yanofski and Manucci. It goes through the theory of quantum computation and information using simple Hermetian matrices and tensor products as the mathematical language.</p>
|
2,988,089 | <p>Let A, B, C, and D be sets. Prove or disprove the following:</p>
<pre><code> (A ∩ B) ∪ (C ∩ D)= (A ∩ D) ∪ (C ∩ B)
</code></pre>
<p>I am just wondering can i simply prove it using a membership table ( seems to easy ) or do i have to use setbuilder notation?</p>
<p>Thank you!</p>
| Martund | 609,343 | <p>This is false.
Take A = {1,2,3}
B = {3,4,5}
C = {7,8,9}
D = {9,10,11}
LHS = {3,9}
RHS = phi
Hence disproved.
Hope it helps:)</p>
|
262,745 | <p>I need to find the normal vector of the form Ax+By+C=0 of the plane that includes the point (6.82,1,5.56) and the line (7.82,6.82,6.56) +t(6,12,-6), with A=1.</p>
<p>Of course, this is easy to do by hand, using the cross product of two lines and the point. There's supposed to be an automated way of doing it, though,... | chuy | 237 | <p>Slightly different:</p>
<pre><code>p0 = {6.82, 1, 5.56};
p1 = {7.82, 6.82, 6.56};
p2 = p1 + t {6, 12, -6};
plane = InfinitePlane[{p0, p1, p2}];
SubtractSides[
Reduce[RegionMember[plane, {x, y, z}], Reals]] // TraditionalForm
(* x-0.255754 y+0.488491 z-9.28026==0. *)
</code></pre>
|
802,877 | <blockquote>
<p>Find $\displaystyle\lim_{n\to\infty} n(e^{\frac 1 n}-1)$ </p>
</blockquote>
<p>This should be solved without LHR. I tried to substitute $n=1/k$ but still get indeterminant form like $\displaystyle\lim_{k\to 0} \frac {e^k-1} k$. Is there a way to solve it without LHR nor Taylor or integrals ?</p>
<p>... | Deepak | 151,732 | <p>Put $x = \frac{1}{n}$</p>
<p>The limit becomes $\lim_{x \to 0}\frac{e^x-1}{x} \rightarrow \lim_{x \to 0}\frac{1+x-1}{x} = 1$</p>
<p>using the Maclaurin series for $e^x$</p>
<p>EDIT: Just noticed you also excluded Taylor series (which would preclude Maclaurin's) in your question. So feel free to ignore my answer.<... |
2,502,161 | <p>I'm wondering if it's valid to write the follwing: <span class="math-container">$$\lim_{x \rightarrow \infty}\frac{2}{x^r}=2\lim_{x \rightarrow \infty}\frac{1}{x^r}=2.\frac{1}{\infty}=2.0=0$$</span></p>
<p>I know it's valid to say that <span class="math-container">$\frac{1}{\infty}=0$</span> in limits but I'm not s... | Claude Leibovici | 82,404 | <p>Considering the differential equation
<span class="math-container">$$\frac{d^2y}{dx^2}\,\frac{dy}{dx}\,y=1$$</span> let us use <span class="math-container">$$\frac{dy}{dx}=\frac1 {\frac{dx}{dy}}\qquad \text{and}\qquad \frac{d^2y}{dx^2}=-\frac{\frac{d^2x}{dy^2}}{\left(\frac{dx}{dy}\right)^3 }$$</span> (see <a href="h... |
3,479,940 | <p>Say we're given a set of <span class="math-container">$d$</span> vectors <span class="math-container">$S=\{\mathbf{v}_1,\dots,\mathbf{v}_d\}$</span> in <span class="math-container">$\mathbb{R}^n$</span>, with <span class="math-container">$d\leq n$</span> (obviously). We want to test in an efficient way if S is linea... | Alex | 48,061 | <p>Let <span class="math-container">$A$</span> be not the <span class="math-container">$n\times d$</span>-matrix, but the <span class="math-container">$n\times n$</span>-matrix <span class="math-container">$$A=[v_1 \ldots v_d\ \vec{0}\ldots\ \vec 0]^t.$$</span> Reduce <span class="math-container">$A$</span> to <a hre... |
2,782,109 | <blockquote>
<p>If a positive integer $m$ was increased by $20$%, decreased by $25$%, and then increased by $60$%, the resulting number would be what percent of $m$?</p>
</blockquote>
<p>A common step-by-step calculation will take time.</p>
<p>After $20$% increase, $6m/5$.<br>
After $25$% decrease, $9m/10$.<br>
Aft... | Rhys Hughes | 487,658 | <p>$$\frac 65*\frac 34* \frac85=\frac{144}{100}$$</p>
|
2,221,033 | <p>My question is due to <a href="https://en.wikipedia.org/w/index.php?title=Imaginary_number&diff=prev&oldid=175488747" rel="noreferrer">an edit</a> to the Wikipedia article: <a href="https://en.m.wikipedia.org/wiki/Imaginary_number" rel="noreferrer">Imaginary number</a>.</p>
<p>The funny thing is, I couldn'... | Misha Lavrov | 383,078 | <p>The Wikipedia article cites <a href="https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2" rel="nofollow noreferrer">a textbook</a> that manages to confuse the issue further:</p>
<blockquote>
<p><strong>Purely imaginary (complex) number :</strong> A complex number $z = x + iy$ is called a purely imagina... |
4,288,460 | <blockquote>
<p>Suppose that <span class="math-container">$\{X_t : Ω → S := \mathbb{R}^d, t\in T\}$</span> is a stochastic
process with independent increments and let <span class="math-container">$\mathcal{B}_t :=\mathcal{B}_t^X$</span> (natural filtration) for all <span class="math-container">$t\in T$</span>. Show, fo... | Lorenzo Pompili | 884,561 | <p>The proof is not well written in that way. In particular, you have to choose <span class="math-container">$\delta_0$</span> in a suitable way that depends on <span class="math-container">$\varepsilon$</span>. Let me try to correct it below.</p>
<hr />
<p>Case 2 :
There exist some x's that satisfy h(x)=0</p>
<p>take ... |
417,896 | <p>Given a connected smooth manifold <span class="math-container">$M$</span> of dimension <span class="math-container">$m>1$</span>, points <span class="math-container">$p_1,\dots,p_n\in M$</span> and positive values <span class="math-container">$\{d_{i,j};1\leq i<j\leq n\}$</span> satisfying the strict triangle ... | Anton Petrunin | 1,441 | <p>Yes if <span class="math-container">$m\ge 2$</span>.</p>
<p>Let us construct a metric graph <span class="math-container">$\Gamma$</span> by connecting vertices <span class="math-container">$p_1,\dots,p_n$</span> by edges with lengths <span class="math-container">$m_{ij}$</span>.
Take its tiny tubular neighborhood an... |
910,070 | <p>I am working on a weighted minimization problem. Without the weights, the error function can be expressed as $e^T e$. With weights, $e$ first need to element-wise multiple by $w$, then the same formula applies: $(w \circ e)^T (w \circ e)$. How do I express it in pure matrix form (without the $\circ$). The $\circ... | Ben Grossmann | 81,360 | <p>One nice way to make your function into a matrix product is to define the diagonal matrix
$$
W = \pmatrix{w_1&&\\&\ddots&\\&&w_n}
$$
We then have
$$
(w \circ e)^T(w \circ e) = (We)^T (We) = e^T W^TWe =\\
e^T \pmatrix{|w_1|^2&&\\&\ddots&\\&&|w_n|^2} e
$$</p>
|
2,715,926 | <p>I know the definition of the limit and how it applies in this case, but I don't know how to reach the end of the proof. The limit is:</p>
<p>$\lim_{n\rightarrow \infty}{\dfrac{2n^2-3n+1}{n^2-n+7}}=2$</p>
<p>I got to:</p>
<p>$\dfrac{n+13}{n^2-n+7}<\epsilon$</p>
<p>Thank you.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>Use that $$\frac{n+13}{n^2-n+7}<\frac{13}{n}$$ since $n>0$ we get by cross multiplication
$$n^2+13n<13(n^2-n+7)$$ and this is equivalent to $$0<12n^2-26n+91$$</p>
|
2,715,926 | <p>I know the definition of the limit and how it applies in this case, but I don't know how to reach the end of the proof. The limit is:</p>
<p>$\lim_{n\rightarrow \infty}{\dfrac{2n^2-3n+1}{n^2-n+7}}=2$</p>
<p>I got to:</p>
<p>$\dfrac{n+13}{n^2-n+7}<\epsilon$</p>
<p>Thank you.</p>
| linksideal | 171,582 | <p>You could use <a href="https://en.m.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="nofollow noreferrer">L'Hopital''s Rule</a>, i.e. derive both the nominator and the denominator of your second expression to see that it converges to 0. According to L'Hopital, the same holds for the original expression!</p>
|
2,917,439 | <p>Suppose we flip two fair coins and roll one fair six-sided die.</p>
<p>What is the conditional probability that the number of heads equals the number
showing on the die, conditional on knowing that the die showed 1?</p>
<p>Let's define the following:</p>
<p>$A=\{\text{#H = # on die}\}$</p>
<p>$B=\{\text{# on die... | Ross Millikan | 1,827 | <p>You are making it <em>far</em> too complicated. If you are conditioning on the die showing $1$, you are really asking the chance that you get one head out of two flips. You can ignore all the other results from the die. It is not that the coin is $50/50$ because you have two flips. It is that if the probability ... |
2,820,623 | <p>Show that $\log(\det(H_1)) ≤ \log(\det(H_2)) + \operatorname{tr}[H^{-1}_2H_1]−N$ for all positive semidefinite matrices $H_1,H_2 \in C^N$.</p>
<p>We know that all positive semidefinite matrices are singular and so the determinant is zero and as such they are not invertible. It is clear from the expression that $\lo... | Community | -1 | <p>We consider that $H_1,H_2\in S_n^{>0}$ (they are $>0$); the other cases have not any interest.</p>
<p>Let $f:Z\in S_n^{>0}\rightarrow tr(H_2^{-1}Z)-\log(\det(Z))+\log(\det(H_2)-n$. We show that the minimum of $f$ is $0$.</p>
<p>The derivative is $Df_{Z}:K\in S_n\rightarrow tr(H_2^{-1}K)-tr(KZ^{-1})=tr(K(H... |
2,820,623 | <p>Show that $\log(\det(H_1)) ≤ \log(\det(H_2)) + \operatorname{tr}[H^{-1}_2H_1]−N$ for all positive semidefinite matrices $H_1,H_2 \in C^N$.</p>
<p>We know that all positive semidefinite matrices are singular and so the determinant is zero and as such they are not invertible. It is clear from the expression that $\lo... | nguyen0610 | 181,036 | <p>If <span class="math-container">$H_1$</span> and <span class="math-container">$H_2$</span> are positive (i.e., <span class="math-container">$>0$</span>), the inequality is equivalent to
<span class="math-container">$$\log(\det(H_2^{-1}H_1))\leq \mathrm{Tr}(H_2^{-1}H_1) - N.$$</span>
Notice that <span class="math-... |
1,006,354 | <ul>
<li>A multiple choice exam has 175 questions. </li>
<li>Each question has 4 possible answers. </li>
<li>Only 1 answer out of the 4 possible answers is correct. </li>
<li>The pass rate for the exam is 70% (123 questions must be answered correctly). </li>
<li>We know for a fact that 100 questions were answered corre... | Peter | 82,961 | <p>Hint : Use the formula for binomial distributed random variables.</p>
|
2,585,265 | <p>I understand that $\cos(\theta) = \sin(\pi/2 - \theta)$ holds true. But, </p>
<blockquote>
<p>Does $\cos(\theta) = \sin(\pi/2 +\theta)$ always hold true?</p>
</blockquote>
<p>I am asking this question because I encountered the following question in my workbook.</p>
<p>If $h(x) = \cos x$, $g(x) = \sin x$, and $h... | idok | 514,894 | <p>The number in the square root is at most $73$, and a square.
So it is one of $1,4,9,16,25,36,49,64$.<br>
Also, the number in the square root is one more from a divisor of $72$, so it is one of $4,9,25$. All those values can be achieved.</p>
|
855,570 | <p>I am having trouble with what seems like it should be a simple problem. I am trying to find intersections of connections between multiple people but I want to include any intersection of connections found between any two of the sets.</p>
<p>For example, Let</p>
<p>$A$ = {January, February, March, April, May, Augus... | JCV | 160,462 | <p>The set you are after can be formalised thus:$ \bigcup\limits_{\substack{A,B \in I \\ A \ne B}} A\cap B$, where $I$ is your set of subsets (i.e. in the example you gave $I=\{A,R,Y\}$).</p>
<p>This will give you the set of elements that are in at least two of your subsets.</p>
<p>In terms of a programming standpoin... |
402,214 | <p>I recently obtained "What is Mathematics?" by Richard Courant and I am having trouble understanding what is happening with the Prime Number Unique Factor Composition Proof (found on Page 23).</p>
<p>The first part:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/h5rCh.png" alt="enter image description he... | aram | 77,407 | <p>Take a look this is a proof by <strong>contradiction</strong> you're suposing that the two factorizations are <strong>different</strong> and getting an absurd (in this case that one prime is divisible by another prime), thus the asumption the factorizations were different is false.
m is the smallest choice because ... |
402,214 | <p>I recently obtained "What is Mathematics?" by Richard Courant and I am having trouble understanding what is happening with the Prime Number Unique Factor Composition Proof (found on Page 23).</p>
<p>The first part:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/h5rCh.png" alt="enter image description he... | André Nicolas | 6,312 | <p>We reframe the <em>logic</em> of the argument, using a variant of induction called <em>Fermat's Method of Infinite Descent</em>. </p>
<p>Call a positive integer <strong>bad</strong> if it has (apart from order) more than one factorization as a product of prime powers. Note that $1$ is not bad.</p>
<p>We want to s... |
2,323,351 | <p>I thought we take $4$ vowels and find number of arrangements $4!$ and multiply it with arrangements that can be made with consonants that is $5!/2!$. However my approach seems to be wrong. </p>
| Especially Lime | 341,019 | <p>You must have done something wrong, since for $n=7$ your expression is $35-60+20-4=-9.$ </p>
<p>To use inclusion-exclusion, your events should be something like "includes $i$ and $i+1$". Now the number of choices which include $i$ and $i+1$ is $\binom{n-2}{2}$, and there are $n-1$ possible $i$, so you should start ... |
2,323,351 | <p>I thought we take $4$ vowels and find number of arrangements $4!$ and multiply it with arrangements that can be made with consonants that is $5!/2!$. However my approach seems to be wrong. </p>
| Evargalo | 443,536 | <p>A recursive proof:</p>
<p>Let $A(k,n)$ be the number of ways to pick k integers from {1,...n} s.t. no two consecutives numbers are picked. If $n<2k-1$, then $A(k,n)=0$. $A(k,2k-1)=1$.</p>
<p>If you pick k integers from {1,...n} s.t. no two consecutives numbers are picked, there are two cases:</p>
<ul>
<li>Eith... |
119,456 | <p>I generated a 2d random array in $x-y$ plane with</p>
<pre><code>L = 10;
random = Table[{x, y, RandomReal[{-1, 1}]}, {x, 0, L, L/10}, {y, 0, L, L/10}];
</code></pre>
<p>Now I want to save it for the next using by</p>
<pre><code>iniF = Interpolation[Flatten[random, 1]];
inif[x_, y_] = c+iniF[x, y];
</code></pre>
... | Sumit | 8,070 | <p>As a general solution, you can always <code>Export</code> you data to a file.</p>
<pre><code>L = 10;
random = Table[{x, y, RandomReal[{-1, 1}]}, {x, 0, L, L/10}, {y, 0, L, L/10}];
Export[NotebookDirectory[]<>"rand.dat",Flatten[random,1]]
</code></pre>
<p>Next time you want to use this data</p>
<pre><code>r... |
3,245,223 | <p>so I am supposed to solve a proof which seems fairly easy, but the negative exponents in <span class="math-container">$$\sum_{k=0}^n \binom nk\ (\frac{(-1)^k}{k+1})= \frac{1}{n+1}$$</span> are making this question very difficult for me. I have tried using binomial theorem on the right side with <span class="math-con... | Arthur | 15,500 | <p>Hint: Consider antidifferentiating <span class="math-container">$f(x)=(1-x)^n$</span>.</p>
|
1,575,671 | <p>The whole question is that <br>
If $f(x) = -2cos^2x$, then what is $d^6y \over dx^6$ for x = $\pi/4$?</p>
<p>The key here is what does $d^6y \over dx^6$ mean?</p>
<p>I know that $d^6y \over d^6x$ means 6th derivative of y with respect to x, but I've never seen it before.</p>
| MPW | 113,214 | <p>The symbol "$\frac d{dx}$" is used to indicate a single derivative (with respect to $x$).</p>
<p>We treat repeated application of this operator symbolically as "powers" of the operator (as if it were ordinary multiplication by an ordinary fraction), writing "$\frac{d^n}{dx^n}$" to indicate $n$ successive applicatio... |
1,575,671 | <p>The whole question is that <br>
If $f(x) = -2cos^2x$, then what is $d^6y \over dx^6$ for x = $\pi/4$?</p>
<p>The key here is what does $d^6y \over dx^6$ mean?</p>
<p>I know that $d^6y \over d^6x$ means 6th derivative of y with respect to x, but I've never seen it before.</p>
| Community | -1 | <p>The correct notation for the sixth derivative is</p>
<p>$$\frac{d^6 y}{dx^6}$$</p>
<p>not $\frac{d^6 y}{d^6x}$. This notation is meant to be suggestive of taking the sixth power of the operator $d/dx$; that is,</p>
<p>$$\frac{d^6 y}{dx^6} = \underbrace{\frac{d}{dx} \cdots \frac{d}{dx}}_{6} y$$</p>
<p>Imagine $dx... |
1,738,153 | <p>I know the definition is given as follows:</p>
<p>A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means</p>
<p>Can someone help me gain some intuition for this - perhaps a basic example?</p>
<p>Thanks</p>
| John Martin | 37,200 | <p>Building off of my comment, the easiest non-trivial example I can think of is that of a finite cyclic group. Since a cyclic group is generated by a single element, we only need to say where the generator goes. Let $G$ be a cyclic group of order $2$, with generator $x$. Then $x\mapsto\begin{pmatrix}-1&0\\0&-1... |
752,517 | <p>From Wikipedia</p>
<blockquote>
<p>...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different unless their equality follows from the group axioms (e.g. $st = suu^{−1}t$, but $s ≠ t$ for $s,t,u \in... | Michael Hardy | 11,667 | <p>I don't know why I didn't say the following at the outset. I guess I was just going along with your method.</p>
<p><b>Rid your proof of trigonometric functions and instead do the following:</b></p>
<p>$$
t\mapsto (x,y) = \begin{cases} \phantom{\lim\limits_{t\to\infty}} \left( \dfrac{1-t^2}{1+t^2}, \dfrac{2t}{1+t^... |
412,642 | <p>Let $E=\mathcal{C}[0,1]$. How to prove that if $f_n\rightarrow f$ with the norm $\displaystyle{\|\cdot\|_\infty=\sup_{t\in[0, 1]}f(t)}$ then $f_n\rightarrow f$ with the norm $\displaystyle{\|\cdot\|_p=\left(\int_{0}^{1}|f(t)|^p\,dt\right)^{1/p}}$. </p>
<p>Give an example showing that the converse is not true</p>
| Ted Shifrin | 71,348 | <p>Hints: If $|f-g|<\epsilon$, what can you say about $\|f-g\|_p$?</p>
<p>Draw the graph of a continuous function $f$ with $\|f\|_1<\epsilon$ and $\|f\|_\infty=1$.</p>
|
3,946,762 | <p><a href="https://i.stack.imgur.com/JHS35.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JHS35.png" alt="enter image description here" /></a></p>
<p>In <span class="math-container">$\triangle ABC$</span> the cevians <span class="math-container">$AD$</span>,<span class="math-container">$BE$</span> ... | timon92 | 210,525 | <p>You may want to start like this: Since <span class="math-container">$EF \parallel DL$</span>, we have <span class="math-container">$\dfrac{DL}{JE}=\dfrac{DA}{AJ}$</span> and <span class="math-container">$\dfrac{DK}{JE} = \dfrac{DO}{OJ}$</span>. The problem reduces to showing that <span class="math-container">$\dfrac... |
3,946,762 | <p><a href="https://i.stack.imgur.com/JHS35.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JHS35.png" alt="enter image description here" /></a></p>
<p>In <span class="math-container">$\triangle ABC$</span> the cevians <span class="math-container">$AD$</span>,<span class="math-container">$BE$</span> ... | Solumilkyu | 297,490 | <p>Following timon92, we are going to prove <span class="math-container">$\dfrac{DA}{AJ}=\dfrac{DO}{OJ}$</span>.</p>
<p><a href="https://i.stack.imgur.com/1qJ1S.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1qJ1S.png" alt="enter image description here" /></a></p>
<p>Consider the figure (only add po... |
3,037,296 | <p>I'm confused of what <span class="math-container">$\sqrt {3 + 4i}$</span> would be after I used quadratic formula to simplify <span class="math-container">$z^2 + iz - (1 + i)$</span></p>
| TonyK | 1,508 | <p>Let <span class="math-container">$\sqrt{3+4i}=x+yi$</span>, with <span class="math-container">$x,y\in\Bbb R$</span>. Then <span class="math-container">$3+4i=(x+yi)^2=x^2-y^2+2xyi$</span>.</p>
<p>Equating real and imaginary parts gives <span class="math-container">$x^2-y^2=3$</span> and <span class="math-container">... |
1,538,496 | <p>I came across this riddle during a job interview and thought it was worth sharing with the community as I thought it was clever:</p>
<blockquote>
<p>Suppose you are sitting at a perfectly round table with an adversary about to play a game. Next to each of you is an infinitely large bag of pennies. The goal of the... | Roger Dahl | 167,445 | <p>I'm wondering if it would be possible to prove that the following solution is a winning or losing strategy:</p>
<blockquote class="spoiler">
<p> Go first and place a penny <em>near</em> the edge of the table in such a way that there isn't room for another penny on the outside of it.</p>
</blockquote>
|
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