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 |
|---|---|---|---|---|
3,849,311 | <blockquote>
<p>What is the remainder when dividing the polynomial
<span class="math-container">$$P(x)=x^n+x^{n-1}+\cdots+x+1$$</span> with the polynomial
<span class="math-container">$$x^3-x$$</span> if <span class="math-container">$n$</span> is a natural odd number?</p>
</blockquote>
<p>So, what I know so far is:</p>... | R. J. Mathar | 805,678 | <p>The remainder is <span class="math-container">$$\left(\frac{n+1}{2}\frac{1}{x-1}-\frac{1}{x}\right)(x^3-x) = \frac{(x+1)(2+(n-1)x)}{2}$$</span></p>
|
16,831 | <p>As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presu... | Elias Costa | 19,266 | <p>There is a proof, using the Weierstrass approximation theorem, that if $f$ is continuous then $f$ is necessarily zero! ... |
2,463,421 | <p>The question is:</p>
<p>Nadir Airways offers three types of tickets on their Boston-New York flights. First-class tickets are \$140, second-class tickets are \$110, and stand-by tickets are \$78. If 69 passengers pay a total of $6548 for their tickets on a particular flight, how many of each type of ticket
were sol... | Colm Bhandal | 252,983 | <p>Edit: Thanks to commenters.</p>
<p>Presumably what's being asked here is how many four digit hexadecimal numbers are there whose digits are all in increasing order. We also have the restriction that $0$ can't be chosen, because, given digits of increasing order, this would imply $0$ as the first digit, which isn't ... |
140,294 | <p>Generative adversarial networks (GAN) is regarded as one of "the most interesting idea in the last ten years in machine learning" by Yann LeCun. It can be used to generate photo-realistic images that are almost indistinguishable from the real ones.</p>
<p>GAN trains two competing neural networks: a generator networ... | partida | 15,961 | <p>This is a GAN using more accurate loss function,Thanks for @Michael Curry</p>
<pre><code>mnist = ResourceData["MNIST"];
mnistDigits = First /@ mnist;
randomDim = 10;
generator = NetChain[{128, Ramp, 128, Ramp, 28*28, LogisticSigmoid,
ReshapeLayer[{1, 28, 28}]}, "Input" -> randomDim];
disc... |
382,504 | <p>Let <span class="math-container">$n \geq 2$</span> be an integer, and let <span class="math-container">$f(x) = \prod\limits_{k = 1}^n(x - \alpha_k)$</span> be a monic irreducible polynomial in <span class="math-container">$\mathbb Z[x]$</span>, with the property that <span class="math-container">$f(-\alpha_k) \neq 0... | Anton | 22,733 | <p>See the discussion <a href="https://mathoverflow.net/a/382552">above</a>.</p>
<p>Also, if <span class="math-container">$f(x) = x^n + \dotsb + a_{n-1}x + a_n$</span>, then</p>
<ul>
<li><p>for <span class="math-container">$n = 2$</span>, <span class="math-container">$\operatorname{Res}(f(x), f(-x)) = 2^2a_2a_1^2$</spa... |
2,215,087 | <p>I'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$</p>
<p>By multiplicativity, it suffices to show that $\forall x \in \mathbb{Q}(\sqrt{11}) \exists n \in \mathbb{Z}(\sqrt{11}):|N(n-x)| < 1$</p>
<p>For the analogous s... | Chan Tai Man | 876,234 | <p><strong>Division algorithm for Euclidean domain <span class="math-container">$\mathbb{Z}[\sqrt{11}]$</span></strong></p>
<p>While Oppenheim (1934) proved that there is a division algorithm for <span class="math-container">$\mathbb{Z}[\sqrt{11}]$</span>. No examples was given explicitly. Inspired by his proof, one su... |
2,215,087 | <p>I'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$</p>
<p>By multiplicativity, it suffices to show that $\forall x \in \mathbb{Q}(\sqrt{11}) \exists n \in \mathbb{Z}(\sqrt{11}):|N(n-x)| < 1$</p>
<p>For the analogous s... | Chan Tai Man | 876,234 | <p>Another way to graph it for showing that <span class="math-container">$\mathbb{Z}[{\sqrt{11}}]$</span> is norm-euclidean. However, this method fails for <span class="math-container">$\mathbb{Z}[{\sqrt{19}}]$</span>. There are a few small gaps that this elementary method fails to cover.</p>
<p>Z[sqrt(11)] with max a,... |
2,444,196 | <p>The center of a group $G$ is defined as $Z(G):=\{ z\in G : gz = zg, \; \forall g \in G\}$.</p>
<p>The goal is to show that if $\vert G\vert = pq$, where $p$ and $q$ are not necessarily distinct primes then either $G$ is abelian or $Z(G) = \{ e\}$.</p>
<p>I want to suppose that $Z(G) \neq \{ e\}$ and then use the f... | Community | -1 | <p>This argument admittedly doesn't use the result <span class="math-container">$G/Z(G) \space\text{cyclic}\Rightarrow G\space\text{abelian}$</span>.</p>
<p>If <span class="math-container">$q=p$</span>, it is well known that <span class="math-container">$G$</span> of order <span class="math-container">$p^2$</span> is a... |
4,331,081 | <p>Suppose <span class="math-container">$a, b >0$</span>. I'm looking for closed expressions for the following integral:
<span class="math-container">$$\int_{-\pi}^{\pi}\sqrt{a^{2}-2ab\cos(x)+b^{2}}dx $$</span>
I tried to solve this by myself and got nowhere and even wolfram alpha couldn't get me an answer so maybe ... | Eldar Sultanow | 993,738 | <p>Mathematica yields the result shown below when executig:</p>
<pre><code>Integrate[Sqrt[a^2 - 2 ab*Cos[x] + b^2], {x, -Pi, Pi}]
</code></pre>
<p><span class="math-container">$$4 \sqrt{a^2+2 \text{ab}+b^2} E\left(\frac{4 \text{ab}}{a^2+b^2+2 \text{ab}}\right)$$</span></p>
<p>where <span class="math-container">$E(\ldo... |
2,583,454 | <p>Consider for instance the linear system:</p>
<p>$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\right)$$</p>
<p>This is over determined and thus has no solution. Y... | Ian | 83,396 | <p>Consider an analogy with the 1D case: $1=0$ is false but $0(1)=0(0)$ is true. Multiplying both sides by zero turned a false equation into a true one.</p>
<p>In the case of an overdetermined system $Ax=b$, you are effectively multiplying certain <em>components</em> of the space by zero when you multiply by $A^T$. Th... |
2,583,454 | <p>Consider for instance the linear system:</p>
<p>$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\right)$$</p>
<p>This is over determined and thus has no solution. Y... | Kajelad | 354,840 | <p>One intuition for this is thinking about it geometrically. In your case, $A$ is a linearly independent $3\times 2$ matrix, so its column space is a plane in $\mathbb R^3$, and $A\vec x$ is a point in this plane. If we choose a new point $\vec b\in\mathbb R^3$, it generically won't be on the plane of $\text{col}\ A$.... |
3,769,843 | <p>This is a multiple choice question from my Text Book</p>
<p>Let <span class="math-container">$A=\{1,2,3\}$</span>. The no. of relations containing <span class="math-container">$(1,2)$</span> and <span class="math-container">$(1,3)$</span> which are reflexive and Symmetric but not transitive is</p>
<p>(A) <span class... | Luke Hutchison | 365,886 | <p>The answer by "None" is the same as the answer I arrived at. I'm posting my working here:</p>
<p>To map <span class="math-container">$X_i$</span>, the local coordinates of a point <span class="math-container">$X$</span> captured in the coordinate system of camera position <span class="math-container">$L_i$... |
4,382,317 | <p>Show that <span class="math-container">$|z+1| = 2\cos(\frac{\theta}{2})$</span>, <span class="math-container">$z = cis(\theta)$</span> and <span class="math-container">$z \in C$</span></p>
<p>Here is what I have managed to do:</p>
<ul>
<li><span class="math-container">$r=1$</span></li>
<li><span class="math-containe... | Amaan M | 860,916 | <p>Hints:</p>
<ol>
<li><p>Given a point <span class="math-container">$z = a + bi$</span> in the complex plane, the magnitude of <span class="math-container">$z$</span> is <span class="math-container">$|z| = \sqrt{a^2 + b^2}$</span>, from the Pythagorean theorem. Plot the point <span class="math-container">$z + 1 = a + ... |
3,890,382 | <blockquote>
<p>Find the locus of <span class="math-container">$z$</span> such that <span class="math-container">$\arg \frac{z-z_1}{z-z_2} = \alpha$</span>.
Use and draw <span class="math-container">$w = \frac{z-z_1}{z-z_2}$</span>.</p>
</blockquote>
<p>This exercise was discussed many times -- <a href="https://math.st... | Raffaele | 83,382 | <p>Let <span class="math-container">$z=x+iy;\;z_1=x_1+iy_1;\;z_2=x_2+iy_2$</span>
<span class="math-container">$$\frac{z-z_1}{z-z_2}= \frac{x^2-x (x_1+x_2)+x_1 x_2+(y-y_1) (y-y_2)}{(x-x_2)^2+(y-y_2)^2}+i\frac{x (y_2-y_1)+x_1 (y-y_2)+x_2 (y_1-y)}{(x-x_2)^2+(y-y_2)^2}$$</span>
<span class="math-container">$$\text{arg}\le... |
3,984,480 | <p>Show that <span class="math-container">$-\vec{0} = \vec{0}$</span> in any vector space.</p>
<p>I know this is a seemingly obvious statement but is the following justification correct:</p>
<p>Assume <span class="math-container">$-\vec{0} \neq \vec{0}$</span>.</p>
<p><span class="math-container">$$(4): \vec{u} + \vec{... | bonsoon | 48,280 | <p>Somewhat related: All subgroups of finitely generated abelian groups are finitely generated, however this is not necessarily so for subgroups of finitely generated groups.</p>
<p>Perhaps the classic example of the free group of 2 generators has subgroups that are not finitely generated, and its relation to algebraic... |
2,797,329 | <p>The function $y_1 = x^2$ is a solution of
$x^2y'' − 3xy' + 4y = 0$.
Find the general solution of the nonhomogeneous linear differential equation
$x^2y'' − 3xy' + 4y = x^2$</p>
<p>I know the equation $x^2y'' − 3xy' + 4y = 0$ is a Euler-Cauchy equation but I'm not sure how to proceed with this question; any help is ... | Rebellos | 335,894 | <p><strong>General way :</strong></p>
<p>Working over the homogeneous equation :</p>
<p>$$x^2y'' − 3xy' + 4y = 0$$</p>
<p>Since this is an Euler-Cauchy equation, assume that a solution will be proportional to $x^\lambda$ for some constant $\lambda$. Thus, this means that :</p>
<p>$$x^2 \cdot (x^\lambda)'' - 3x \cdo... |
3,206,730 | <blockquote>
<p>Let <span class="math-container">$f : (-1,1)\to (-\pi/2,\pi/2)$</span> be the function defined by <span class="math-container">$f(x)= \tan^{-1}\left(\frac{2x}{1-x^2}\right)$</span> the verify that <span class="math-container">$f$</span> is bijective</p>
</blockquote>
<p>To check objectivity I assumed... | egreg | 62,967 | <p>Let
<span class="math-container">$$
f(x)=\arctan\frac{2x}{1-x^2},\qquad g(x)=\frac{2x}{1-x^2}
$$</span></p>
<p>The derivative is
<span class="math-container">$$
f'(x)=\frac{1}{1+g(x)^2}g'(x)
$$</span>
Now
<span class="math-container">$$
1+g(x)^2=1+\frac{4x^2}{(1-x^2)^2}=\frac{(1+x^2)^2}{(1-x^2)^2}
$$</span>
and
<sp... |
2,528,306 | <p>The answer is 648 but I tried to solve this problem in reverse, so I ended up with 630. Theee are 10 ways to pick the third digit, 9 ways to pick the second digit, and 7 ways to pick the first digit. So why do these answers differ. Please do not close this question as I am trying to learn mathematics and I have stum... | Air Conditioner | 504,810 | <p>Let's start by picking the 1st digit. There are 9 choices, as it can be any digit but 0. The 2nd digit can be any digit but the first digit, so you have 9 choices. The 3rd digit can be any digit but digit 1 and digit 2, so you have 8 choices. So you have $9*9*8=648$. </p>
<p>If you start by picking the 3rd digit, i... |
1,915,366 | <p>Suppose we have two Gaussian distributed random variable $X$~$N(0,\sigma^2)$ and $Y$~$N(0,\sigma^2)$. These variables are not independent. What will be the expected value of product of square of this random variables</p>
<p>$E[X^2Y^2]$ = ??</p>
<p>Edit 1: They are jointly Gaussian distributed with correlation coef... | Snufsan | 122,989 | <p>Try using the <a href="https://en.wikipedia.org/wiki/Law_of_total_expectation" rel="nofollow">Law of total expectation</a> - set $Z = X^2Y^2$ and use:</p>
<p>$$\mathbb{E}[X^2Y^2]=\mathbb{E}[Z]=\int_{y}\mathbb{E}[Z\mid Y=y]\cdot\Pr[Y=y]$$</p>
|
1,441,603 | <p>Solve the PDE for $u(x,y)$ $$\frac{\partial^2 u}{\partial x \, \partial y} = 0$$
I was thinking to integrated both sides in respect to $x$ first to get $$x= c(x)$$ then i will have $$c(x)-x=0$$ then i will integrate in respect to y but i think this wrong because it does not making any sense to me. </p>
| u184 | 218,127 | <p>If you integrate both sides w.r.t. x, you should get $$\frac{\partial u}{ \partial y} = c(y)$$ for some arbitrary function c. Integrating again w.r.t. y now gives $$u=a(y)+b(x)$$ where $b$ is an arbitrary function of $x$ and $a$ is the function you get when integrating $c$ which is again arbitrary since $c$ was. ... |
980,941 | <p>How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?</p>
| mookid | 131,738 | <p><strong>Hint:</strong> use also that
$$
1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}6
$$</p>
<p>$$
1 + (1+2) + \dots + (1 +2+\dots +n) =
\frac{1(1+1)}2 + \frac{2(2+1)}2 + \dots + \frac{n(n+1)}2
\\=\frac 12 \left[
(1^2 + 1) + (2^2 + 2 ) + \dots + (n^2 + n)
\right]
\\=\frac 12 \left[
(1^2 + 2^2 + \dots + n^2) + (1 ... |
47,603 | <p>Is it possible to express the functions $S(x)=x+1$ and $Pd(x)=x\dot{-}1$ in terms of the functions $f_1$, $f_2$, $f_3$ and $f_4$, where $f_1(x)=0$ if $x$ is even or $1$ if $x$ is odd, $f_2(x)=\mbox{quot}(x,2)$, $f_3(x)=2x$ and $f_4(x)=2x+1$? For example, $S(x)=f_4(f_2(x))$ if x is even. Is there a similar formula if... | Guillaume Brunerie | 10,217 | <p>I assume that the problem is: does there exists a sequence $(u_1,\dots,u_n)\in{}\{1,2,3,4\}$ such that for all $x\in\mathbb{N}$, $S(x) = f_{u_n}(\dots (f_{u_1}(x))\dots)$ (and similarly for $Pd$)</p>
<p>Let’s prove by induction on $n$ that every such function either is of the form $f(x)=2^kx+l$ where $k\ge 0$ and $... |
487,084 | <p>I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in $\langle g \rangle$ that has order $p$?</p>
| Mauricio Tec | 64,684 | <p>There are some results that are much stronger than that. <a href="http://en.wikipedia.org/wiki/Cauchy%27s_theorem_%28group_theory%29" rel="nofollow noreferrer">Cauchy's theorem</a> states that every finite group whose order is divisible by some prime <span class="math-container">$p$</span> has a subgroup of order <s... |
3,342,094 | <p>I am asked to prove following proposition:</p>
<blockquote>
<p><strong>Proposition 1.</strong> If an invertible matrix <span class="math-container">$\mathbf A$</span> has a left inverse <span class="math-container">$\mathbf{B}$</span> and a right inverse <span class="math-container">$\mathbf{C}$</span>, then <spa... | Sambo | 454,855 | <p><strong>EDIT</strong>: In the assumption of the theorem, we have that <span class="math-container">$A$</span> is invertible, so assuming the existence of a (left-and-right) inverse <span class="math-container">$A^{-1}$</span> is reasonable. Therefore, the original proof is correct.</p>
<p>The reason why I originall... |
138,698 | <p>I want to evaluate the following double summation</p>
<pre><code>Sum[(-1)^(i + j + i*j)*Exp[-Pi/2*( i^2 + j^2)], {i, -Infinity,
Infinity}, {j, -Infinity, Infinity}]
</code></pre>
<p>I am really new both in using Mathematica and in doing mathematics using computer. I don't know if there is some special technics ... | Szabolcs | 12 | <p><a href="https://en.wikipedia.org/wiki/Array_programming" rel="noreferrer">Vectorization</a> is one of the most effective ways to increase performance. The <code>numpy</code> code you show is fast because it uses vectorization. </p>
<p>Vectorization means working with entire arrays instead of element by element: ... |
976,881 | <p>Simple and quick question. These two have to do with 90 degree angles.</p>
<p>This is the picture of the two words I have.</p>
<blockquote>
<p>Perpendicular is strictly restricted to lines.</p>
</blockquote>
<ul>
<li><p>"Line A and B are perpendicular to each other."</p></li>
<li><p>"v=(1, 1) and w=(-1, 1) -> c... | rogerl | 27,542 | <p>Usually the term "orthogonal matrix" is reserved for matrices whose columns are not only mutually perpendicular, but also unit vectors. So if you were to divide each entry in your matrix above by $2$, it would be an orthogonal matrix.</p>
<p>There appears to be no standard term for a matrix whose columns are just o... |
976,881 | <p>Simple and quick question. These two have to do with 90 degree angles.</p>
<p>This is the picture of the two words I have.</p>
<blockquote>
<p>Perpendicular is strictly restricted to lines.</p>
</blockquote>
<ul>
<li><p>"Line A and B are perpendicular to each other."</p></li>
<li><p>"v=(1, 1) and w=(-1, 1) -> c... | Roland Puntaier | 390,013 | <blockquote>
<p>Perpendicular is strictly restricted to lines.</p>
</blockquote>
<p>"perp, short for perpendicular complement" can have more dimensions.
This in an article named <a href="https://en.wikipedia.org/wiki/Orthogonal_complement" rel="nofollow noreferrer">orthogonal component</a>.</p>
<p>So it is no... |
4,166,894 | <p>Let <span class="math-container">$N\triangleleft G$</span> and <span class="math-container">$K\leqslant G$</span>. Consider <span class="math-container">$\phi :G \mapsto G/N$</span> onto group homomorphism. Show that <span class="math-container">$\phi(K)=KN/N$</span>.</p>
<p>I thought using the equality <span class=... | Misha Lavrov | 383,078 | <p>This is a special case of the <a href="https://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem" rel="nofollow noreferrer">Kruskal–Katona theorem</a>.</p>
<p>For a set <span class="math-container">$S$</span>, let <span class="math-container">$\binom Sk$</span> denote all the <span class="math-container">$k$</spa... |
1,950,809 | <p>I'm fairly certain that the probability of both dice returning an even number is $1/4$.</p>
<p>I got this by saying that since these are independent events, with each die returning an even number being $1/2$, then the probability of both being even is $1/2 \times 1/2 = 1/4$.</p>
<p>Further, there are 36 outcomes, ... | Graham Kemp | 135,106 | <blockquote>
<p>What I can't seem to get over, is that there are an equal number of odd and even numbers, so, why is the answer not 1/2 ?</p>
</blockquote>
<p>Because they are not complementary events. There is another possibility.</p>
<p>The probability that both dice show even numbers is: $1/4$</p>
<p>The... |
270,624 | <p>For a polynomial $f(x) = \sum_{i=0}^dc_ix^i \in \mathbb Z[x]$ of degree $d$, let</p>
<p>$$
H(f):=\max\limits_{i=0,1,\ldots, d}\{|c_i|\}
$$</p>
<p>denote the naive height. Further, define</p>
<p>$$
R(M, r, d) := \#\{f(x) \colon \text{$H(f) \leq M$, $\deg f = d$ and $f(x)$ has extactly $r$ real roots}\}.
$$</p>
<p... | Stanley Yao Xiao | 10,898 | <p>The quadratic case can be dealt with as follows. A quadratic polynomial $f(x) = ax^2 + bx + c \in \mathbb{Z}[x]$ has two distinct real roots if and only if $\Delta(f) = b^2 - 4ac > 0$, and a pair of complex conjugate roots if and only if $\Delta(f) < 0$. </p>
<p>We now let $a,b,c$ vary in the box $[-X,X]^3$. ... |
2,724,744 | <p>I have a basic math question.</p>
<p>If I have the following inequality:
$$-a-b > -1$$
and I want to flip (or reverse) the sign. What is the correct way of the following? And why?</p>
<p>i) $a+b \le 1$<br>
ii) $a+b < 1$</p>
<p>Many thanks! (:</p>
| Dr. Sonnhard Graubner | 175,066 | <p>Multiplying the given inequality by $-1$ we get $$a+b<1$$</p>
|
33,005 | <p>Let n and p be any positive integer, make $p$ the subject of the equation: $(3n + p)\bmod4 = 0$. How is it done?</p>
<p>I've worked out that the only values for p are 1, 2, 3 and 0.</p>
<p>This formula is for calculating the amount of padding required in a bitmap's pixel array:</p>
<blockquote>
<p>Padding bytes... | Bill Dubuque | 242 | <p>$\rm\: mod\ 4\::\ \ 3\equiv -1\ $ so $\rm\ 0\ \equiv\ p + 3\:n\ \equiv\ p - n\ \iff\ p\ \equiv\ n\ \ $ by basic <a href="http://en.wikipedia.org/wiki/Modular_arithmetic" rel="nofollow">modular arithmetic.</a> </p>
<p>Alternatively $\rm\ 4\ |\ p + 3\:n\ \iff\ 4\ |\ p - n + 4\:n\ \iff\ 4\ |\ p - n\ \ $ where $\rm\ a... |
1,757,092 | <p>I want to find an explicit formula for $\sum_{n=0}^\infty n^3x^n$ for $|x|\le1$.Is the idea that first to show that this series is convergent and then we can find the number that it converges to? I tried to use ratio test, but it didn't work. Any suggestion? Thanks!</p>
| Nicholas Stull | 28,997 | <p>The Ratio Test will tell you for what $x$ the series converges:</p>
<p>$$L = \lim_{n\to\infty} \left|\frac{(n+1)^3x^{n+1}}{n^3x^n}\right| = |x|\lim_{n\to\infty} \frac{(n+1)^3}{n^3} = |x|$$</p>
<p>And the ratio test tells us that the series converges absolutely if $|x|<1$ (you should check that it diverges at th... |
6,562 | <p>I want to make some button shaped graphics that would essentially be a rectangular shape with curved edges. In the example below I have used <code>Polygon</code> rather than <code>Rectangle</code> so as to take advantage of <code>VertexColors</code> and have a gradient fill. The code below illustrates the sort of th... | Alec Titterton | 65,882 | <p>I've evolved the vector-based approach.
Rather than converting the object being rounded to an image, I needed to interact with maps and 3D graphics so have used an Overlay with a transparent interior. This means the object in question can exist as its original head and behaves the same, it just has rounded edges ove... |
388,225 | <p>If we have a random graph $G \in g(n,\frac{1}{2})$ how do we show that the expected number of edges is $\frac{1}{2} {{n}\choose{2}}$</p>
<p>Thanks in advance</p>
| Community | -1 | <p>Let $\mathbb{I}_k$ be the indicator function for the edge $k$, i.e.,
$$\mathbb{I}_k = \begin{cases}1 & \text{ if $k^{th}$ edge is present}\\ 0 & \text{ if $k^{th}$ edge is not present} \end{cases}$$
The quantity you are interested in is
\begin{align}
\mathbb{E}\left(\sum_{k=1}^{\binom{n}2} \mathbb{I}_k\right... |
2,694,525 | <p>I came across this exercise</p>
<p>$f(x,y)= \lim_{y\to\infty}{{1-y\sin{\pi x\over y}}\over \arctan x}$</p>
<p>The result I get is ${1-\pi x \over \arctan x}$, which depends on the value of $x$.</p>
<p>However, the question I have is that whatever $x$ is, since it's in the $\sin()$, which is a bounded function, sh... | Community | -1 | <p>You appear to be under the delusion that the correct way to apply the change of variable $y=\frac1x$ to $\lim_{y\to \infty} y\sin\frac{a}{y}$ is $$\lim_{x\to\infty}\frac1x\sin(ax)$$ while it should rather be $$\lim_{x\to 0}\frac1x\sin(ax)$$ (if we wanted to be pedantic, it should actually be $x\to 0^+$, but whatever... |
1,641,922 | <p>I've came accros this excersize:<br>
Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$
suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where $\bar z$ is the complex conjugate of $z$. Prove that $f$ cannot be analytic in $D$.<br>
I started with assuming th... | DeepSea | 101,504 | <p><strong>hint</strong>: Let $C,J$ be Kathy's and Jason's ages as of now. Thus: $C = 2J , C-6 = 5(J-6)$. Can you finish it off?</p>
|
1,266,210 | <p>Hello everybody my query is regarding the number of positive integral solution.</p>
<blockquote>
<p>In the sport of cricket, find the number of ways in which a batsman can score $14$ runs in $6$ balls not scoring more than $4$ runs in any ball.</p>
</blockquote>
| Brian M. Scott | 12,042 | <p>If they’re the $6$ balls of a single over, and we’re talking about one batsman, either he scores nothing but $0,2$, and $4$, or he scores $3$ twice and $4$ twice, with the other batsman taking strike twice and scoring an odd number of runs each time. I suspect, though, that you’re intended to assume that the same ba... |
332,603 | <p>I've passed by this article:
<a href="http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/" rel="noreferrer">http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/</a></p>
<p>and this paper:
<a href="http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf" rel="no... | Inceptio | 63,477 | <p>If formulae for computing primes existed, there wouldn't be a a thing called 'Largest known primes' . And moreover, there are few primes like Mersenne primes and Fermat Prime. But eventually, their converse isn't true <strong>ALWAYS</strong>.</p>
<p>For eg: Mersenne Prime.</p>
<p>$q$ is $prime$ which is equal to $... |
332,603 | <p>I've passed by this article:
<a href="http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/" rel="noreferrer">http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/</a></p>
<p>and this paper:
<a href="http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf" rel="no... | Barry K | 212,888 | <p>Well there have been a # of functions developed all of which-I believe-have started at 1 & gone up. The problem is: they are sloooooow! If it takes l000 steps to obtain all primes up to #7 ... well! The thing about them though is they show weird relationships to other things: One uses pentagonal #s; one only pi ... |
333,360 | <p>I know the series for $\cos(x)$ it is $\sum \limits_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}$ </p>
<p>which will result in $\sum \limits_{n=0}^\infty \dfrac{\left(-1\right)^ n x^{2n+1}}{\left(2n\right)!}$ </p>
<p>Which is great when you already know the series; however, my question is how does one find Maclaurin ... | Henrik Finsberg | 67,299 | <p>If you know that your'e function satisfies a differential equation you can use Picard iteration to find the McLaurin series. If your functions satisfy $y' = f(t,y)$, you can use the formula $ y(x) = y(0) + \int_0^x f(t,y(t)) dt$ to calculate the series.</p>
|
1,803,205 | <p>I am trying to find $\int_0^{\infty} \frac{dx}{1 + x^n}$ using contour integration. I did the computation by taking the contour $[0,R] \cup \gamma_R \cup [R e^{2i\pi/n}, 0]$, with $\gamma_R$ the arc joining $R$ to $Re^{2i\pi/n}$, and found that (for $n \ge 2$):</p>
<p>$$\int_0^{\infty} \frac{dx}{1 + x^n} = \frac{2\... | reuns | 276,986 | <p>$$\int_{[0,Re^{2i \pi /n}]} \frac{1}{1+z^n} dz = \int_0^R \frac{1}{1+(e^{2i \pi /n}x)^n} d(e^{2i \pi /n}x) = e^{2i \pi /n} \int_0^R \frac{1}{1+x^n} dx$$ the poles of $\frac{1}{1+z^n}$ are at $e^{i\pi (2k+1)/ n}, k = 0 \ldots n-1$. the only one inside the whole contour $\Gamma_R : [0,R] \cup [R, R e^{2 i \pi /n}] \cu... |
3,366,158 | <blockquote>
<p>find a linear transformation such that <span class="math-container">$M^2 (1- M) =0$</span> but <span class="math-container">$M$</span> is not idempotent ?</p>
</blockquote>
<p>My attempt : i take take the vector space generated by the base <span class="math-container">$\{e_1, e_2\}$</span> and def... | pancini | 252,495 | <p>We know <span class="math-container">$M$</span> is idempotent if and only if <span class="math-container">$M^2=M$</span>, and the assumption is that <span class="math-container">$M^3=M^2$</span>. So is it possible that <span class="math-container">$M^2\neq M$</span> but <span class="math-container">$M^3=M^2$</span>?... |
1,257,900 | <p>This is a question about cyclotomic polynomials and I have already shown that $x^n-1 =\Pi\Phi_d(x)$, taking the product over all divisors d of n. </p>
| quid | 85,306 | <p>Each of $\zeta^j$ for $j=0, \dots, n-1$ is a root of $X^n -1$. Thus $(X-\zeta^j)$ must divide it. Since all the $\zeta^j$ are distinct, the linear factors are co-prime and thus the product divides $X^n - 1$ too. </p>
<p>As the degree of both is $n$ and they are both normed, equality follows.</p>
|
1,257,900 | <p>This is a question about cyclotomic polynomials and I have already shown that $x^n-1 =\Pi\Phi_d(x)$, taking the product over all divisors d of n. </p>
| Timbuc | 118,527 | <p>Hints:</p>
<p>1) Show that for $\;1\le j,\,k\le n\;,\;\;\zeta^j=\zeta^k\iff j=k\;$ </p>
<p>2) Show that $\;\left(\zeta^k\right)^n=1\;,\;\;\forall\,k=1,2,...,n\;$</p>
<p>3) Show that all the roots of $\;x^n-1\;$ are different</p>
<p>There you go...</p>
|
779,042 | <p>Any pointers on how should I start?</p>
<p>$$I:=\int_ {0}^{\infty} \frac {\cos(ax)} {(x^2 + b^2)^n} \ \mathrm{d}x$$</p>
| Bennett Gardiner | 78,722 | <p>I will leave some details to the reader. W.L.o.G. assume $a,b>0$. Consider
$$
J(a,b) = \int_0^{\infty} \frac{\cos(ax)}{x^2+b^2} \ \mathrm{d}x,
$$
then
$$
\frac{\partial^{n-1} J}{\partial b^{n-1}} = (-2b)^{n-1}(n-1)!\int_0^{\infty} \frac{\cos(ax)}{(x^2+b^2)^n} \ \mathrm{d}x.
$$
The integral $J$ can be calculate... |
399,948 | <p>How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or $\csc(50)$, etc.. I tried looking for how to do it but neither my textbook or any other place has a tutorial for it. I w... | Peter Košinár | 77,812 | <p>For some angles (such as 15 or 75 degrees), you can apply the formulas for <a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle.2C_triple-angle.2C_and_half-angle_formulae" rel="nofollow">doubling and halving the angles</a> and those for <a href="http://en.wikipedia.org/wiki/List_of_tri... |
2,718,495 | <p>$$\lim_{n \to \infty}\frac{1}{n} \xi_{\big| \Bbb{N}} (A \cap[1,n]),$$</p>
<p>where $\xi_{\big| \Bbb{N}}$ is the counting measure on $\Bbb{N}$.</p>
<p>I am looking for $A \subset \Bbb{N}$ for which $\lim_{n \to \infty}\frac{1}{n} \xi_{\big| \Bbb{N}} (A \cap[1,n])$ is not defined. So I need to find $A$ so that the l... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Let <a href="https://en.wikipedia.org/wiki/Natural_density#Properties_and_examples" rel="nofollow noreferrer">$A=\bigcup\limits_{k=0}^\infty \{2^{2k},\ldots,2^{2k+1}-1\}$</a> (be a set of integers whose binary expansion contains an odd number of digits).</p>
<p>\begin{align}
n &= 2^{2m+1} - 1,
\frac{1}{n} \xi_{... |
2,543,123 | <p>Let $\gcd(a, 11) = 1$. If $3a^7 \equiv 5 \pmod{11}$, show that $a \equiv 3 \pmod{11}$.</p>
<p>My first approach was to use Euler's theorem:</p>
<p>$a^{10} \equiv 1 \pmod{11}$</p>
<p>$3a^7 \equiv 5 \pmod{11}$ implies that $a^{-3} \equiv 9 \pmod{11}$ </p>
<p>I feel i'm not on the right track, hints are appreciate... | JayTuma | 506,755 | <p>If you sum the number you get then the answer is no, since</p>
<ol>
<li>there is no way you can achieve $1$ (the least possible value is $2$)</li>
<li>$P(X+Y = 2) = \frac{1}{36}$ while $P(X+Y = 7) = \frac{1}{6}$ as you can easily check, where $X$ is a random variable representing the result of one dice roll</li>
</... |
1,368,455 | <p>"I take a journey and, due to heavy traffic, crawl along the first half of the complete distance of my journey at an average speed of $10$ mph. How fast would I have to travel over the second half of the journey to bring my average speed to $20$ mph?"</p>
<p>At work, this has been a topic of a long debate.<br>
Prop... | Ángel Mario Gallegos | 67,622 | <p>Let $t$ hours the time spent in the first half of the distance, namely $d$ miles, it is known that $$d/2=10t$$
The time needed in order to travel the remaining half of distance, with a speed $v$, is $t_2=(d/2)/v$ and must satisfy
\begin{align*}
\frac{d}{\frac{d}{20}+\frac{d}{2v}}&=20\\
\frac{20vd}{vd+10d}&=2... |
1,220,800 | <blockquote>
<p>Calculation of x real root values from $ y(x)=\sqrt{x+1}-\sqrt{x-1}-\sqrt{4x-1} $</p>
</blockquote>
<p>$\bf{My\; Solution::}$ Here domain of equation is $\displaystyle x\geq 1$. So squaring both sides we get</p>
<p>$\displaystyle (x+1)+(x-1)-2\sqrt{x^2-1}=(4x-1)$.</p>
<p>$\displaystyle (1-2x)^2=4(... | Hagen von Eitzen | 39,174 | <p>For $x\ge1$ we have $$\sqrt{4x-1}\ge \sqrt {3x} $$
and $$\sqrt{x+1}\le \sqrt {2x}$$
hence
$$\sqrt{x+1}-\sqrt{x-1}\le \sqrt{2x}<\sqrt{3x}\le\sqrt{4x-1} $$</p>
|
3,170,871 | <p>Could anyone please give me a hint on how to compute the following integral?</p>
<p><span class="math-container">$$\int \sqrt{\frac{x-2}{x^7}} \, \mathrm d x$$</span></p>
<p>I'm not required to use hyperbolic/ inverse trigonometric functions.</p>
| logo | 587,007 | <p>Try the substitution
<span class="math-container">$$
u=\frac{x-2}{x}
$$</span> </p>
<p>or equivalently</p>
<p><span class="math-container">$$
x=\frac{2}{1-u}
$$</span></p>
|
1,507,190 | <p>Prove that if the vector $v$ belongs both to the nullspace and the row space of a matrix $A$, then $v = 0$.</p>
<p>I did this solution but I don't know if it's right.</p>
<p>$V=A^Tb$ and $Av=0\Rightarrow AV=AA^Tv\Rightarrow A^2b=0\Rightarrow b=0 \Rightarrow v=0$</p>
| Michael Biro | 29,356 | <p>Hint: If $A\mathbf{v} = \mathbf{0}$ and $\mathbf{x}^TA = \mathbf{v}^T$, then what is $\mathbf{v} \cdot \mathbf{v}$?</p>
|
1,507,190 | <p>Prove that if the vector $v$ belongs both to the nullspace and the row space of a matrix $A$, then $v = 0$.</p>
<p>I did this solution but I don't know if it's right.</p>
<p>$V=A^Tb$ and $Av=0\Rightarrow AV=AA^Tv\Rightarrow A^2b=0\Rightarrow b=0 \Rightarrow v=0$</p>
| Ned | 67,710 | <p>Since $v$ is in Nul A, $<v,r> = 0$ for each row $r$ of A. So for any linear combination $w$ of the rows, $<w,v>=0$. But $v$, being in Row A, is itself such a linear combination $w$. So $<v,v>=0$.</p>
|
71,952 | <p><strong>Background</strong></p>
<p>Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a real number $V(A_1, \ldots, A_n)$, measured in $\mathrm{metres}^n$. </p>
<p>As I understand it... | Tom Leinster | 586 | <p>Sorry to answer my own question, but asking this in public seems to have spurred me into thought.</p>
<p>As auniket suspected, the answer is "yes" in the strongest sense I'd hoped: properties 1-3 do characterize mixed volume. In fact, something slightly stronger is true: $V$ is the unique function $(\mathscr{K}_n)... |
3,377,031 | <p>I am stuck on a homework question and I don't know how to approach this. Please help! The question is as follows:</p>
<p>"Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be two events. </p>
<p><span class="math-container">$P(A)= 0.3$</span><br>
<span class="math-container">... | Community | -1 | <p>Hint: <span class="math-container">$P(A'\cap(A\cup B))$</span> represents the probability that <span class="math-container">$A$</span> doesn't happen and also that either <span class="math-container">$A$</span> or <span class="math-container">$B$</span> happens. How can you simplify that statement?</p>
|
3,377,031 | <p>I am stuck on a homework question and I don't know how to approach this. Please help! The question is as follows:</p>
<p>"Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be two events. </p>
<p><span class="math-container">$P(A)= 0.3$</span><br>
<span class="math-container">... | Bram28 | 256,001 | <p>Just as in logic you have that <span class="math-container">$A'(A+B)=A'B$</span>, in set theory you have that <span class="math-container">$A'\cap(A\cup B)=A'\cap B$</span></p>
<p>And I assume you can find <span class="math-container">$P(A'\cap B)$</span> yourself</p>
|
1,083,841 | <p>I have extracted the below passage from the wikipedia webpage - Point (geometry): </p>
<blockquote>
<p>In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. </p>
</blockquote>
<p>I think the above passage imply\ies that the point is zero dimensional. If... | James Herndon | 811,806 | <p>The simplest answer I believe is that a line is not "made up" of points, in the sense that a chain is made of links. Rather, points are best thought of as positions, and positions have no size. It is possible to define a set of possible positions in such a way that one can derive from this set all the prop... |
96,110 | <p><span class="math-container">$A = \begin{pmatrix}
0 & 1 &1 \\
1 & 0 &1 \\
1& 1 &0
\end{pmatrix} $</span></p>
<p>The matrix <span class="math-container">$(A+I)$</span> has rank <span class="math-container">$1$</span> , so <span class="math-container">$-1$</span> is an eigenvalue with an al... | Student | 19,118 | <p>An other way to see that is the following (maybe less clear, but I try anyway) : $A+I$ is a $3\times 3$ matrix with rank one, and thus has for eigenvalues $0$ with multiplicity two and an other one which is given by ${\rm Tr}(A+I)$ [is that clear for you ?]. </p>
<p>Then, since $A+I$ and $I$ trivially commutes, the... |
3,277,171 | <p>I am a ninth grader and I would like to learn mathematics on my own. I have already learned algebra, geometry, trigonometry, some precalculus, number theory and tried to understand some calculus. Apart from those I learnt a bit from other areas of mathematics but not enough to be worth mentioning. </p>
<p>I have le... | runway44 | 681,431 | <p>Your classification is missing spirals: images of one-parameter homomorphisms <span class="math-container">$\exp(zt)$</span> when the complex number <span class="math-container">$z$</span> is not purely real or purely imaginary. Or discrete spirals, such as the image of the integers under such a homomorphism, or "du... |
3,277,171 | <p>I am a ninth grader and I would like to learn mathematics on my own. I have already learned algebra, geometry, trigonometry, some precalculus, number theory and tried to understand some calculus. Apart from those I learnt a bit from other areas of mathematics but not enough to be worth mentioning. </p>
<p>I have le... | P Vanchinathan | 28,915 | <p>More groups can be identified by taking a subring of <span class="math-container">$A\subset\mathbf{C}$</span> and then the group <span class="math-container">$A^*$</span> of invertible elements of that ring. For example <span class="math-container">$A$</span> could be the ring of integers of some algebraic number fi... |
2,894,376 | <blockquote>
<p>$2$ different History books, $3$ different Geography books and $2$ different Science books are placed on a book shelf. How many different ways can they be arranged? How many ways can they be arranged if books of the same subject must be placed together?</p>
</blockquote>
<p>For the first part of the ... | Key Flex | 568,718 | <p>All the books can be arranged in $(2+3+2)!=7!$ ways</p>
<p>There are $3$ branches, three units of books: $\{$History$\}$,$\{$Geography$\}$,$\{$Science$\}$- Arranging branches $=3!$ ways.</p>
<p>Arranging the books within the branches:</p>
<p>History: $2!$</p>
<p>Geography: $3!$</p>
<p>Science:$2!$</p>
<p>Total... |
2,297,421 | <p>Probably a really simple question, but I am trynig to fit an air bed in a tent.</p>
<p>Circular tent with a diameter of $3$m and a central vertical pole in the middle.</p>
<p>The air bed measures $1.41$ m $\times$ $1.9$ m.</p>
<p>Will the air bed fit fully inside the tent without being obstructed by the central p... | Ben G. | 334,171 | <p>I believe the simplest way to solve this problem is to simply count the number of distinct polynomials of that form that factor, and see that it is less than $n$. Thus some polynomials of that form do not factor.</p>
<p>For every odd n, 2 is a unit modulo $n$. Thus it is sufficient to show that some $x^2+x+i$ is ir... |
2,297,421 | <p>Probably a really simple question, but I am trynig to fit an air bed in a tent.</p>
<p>Circular tent with a diameter of $3$m and a central vertical pole in the middle.</p>
<p>The air bed measures $1.41$ m $\times$ $1.9$ m.</p>
<p>Will the air bed fit fully inside the tent without being obstructed by the central p... | Arpan1729 | 444,208 | <p>Say, no such $k$, exists.</p>
<p>So there exists numbers $x_1,x_2\dots ,x_{n} \in \{0,1,2,3...n\}$, such that</p>
<p>$x_1^2+x_1=0$ <strong>(mod n)</strong></p>
<p>$x_2^2+x_2=2$ <strong>(mod n)</strong></p>
<p>$x_3^2+x_3=4$ <strong>(mod n)</strong>
.</p>
<p>.</p>
<p>.</p>
<p>.
$x_{n}^2+x_{n}=2(n-1)$ <strong>(... |
35,230 | <p>This happens frequently, both on the main site and on meta: An old question pops back up on the front page, I open it, the text under the title says "Modified today", but when I check the timeline the last event is years ago, even if I show vote summaries.</p>
<p>Here's the <a href="https://math.stackexcha... | Martin Sleziak | 8,297 | <p>As already mentioned in the other answer, a question can be bumped for various reasons - in particular, it might be caused by activity on an answer (rather than a question). See the <a href="https://math.meta.stackexchange.com/tags/bumping/info">corresponding tag-info</a> and the FAQ post: <a href="https://meta.stac... |
136,453 | <p>For every $k\in\mathbb{N}$, let
$$
x_k=\sum_{n=1}^{\infty}\frac{1}{n^2}\left(1-\frac{1}{2n}+\frac{1}{4n^2}\right)^{2k}.
$$
Calculate the limit $\displaystyle\lim_{k\rightarrow\infty}x_k$.</p>
| lhf | 589 | <p>The additive structure of the integers is trivial: it's generated by 1. This is in essence the Peano axioms.</p>
<p>The multiplicative structure of the integers is not trivial: it's generated by prime numbers. In other words, prime numbers are the multiplicative building blocks of the integers in the sense that eve... |
4,011,581 | <p>I know that a function is continuous at a point if the limit from left and right side exists and are equal and for a function to be continuous, the function should be continuous at all points. My question is that if I want to check continuity of a function, I cannot practically check continuity at each and every poi... | GBA | 791,969 | <p>One way is to take an arbitrary point <span class="math-container">$x_0$</span> with no prior assumptions and show that for every <span class="math-container">$\varepsilon>0$</span> there exists <span class="math-container">$\delta>0$</span> (which may depend on <span class="math-container">$x_0$</span>) s.t f... |
178,666 | <p>For every natural number n, let:</p>
<ul>
<li><p>Gn be the number of distinct group structures with at most n elements;</p></li>
<li><p>An be the number of distinct abelian group structures wit at most n elements;</p></li>
<li><p>Sn be the number of distinct solvable group structures with at most n elements.</p></l... | Emil Jeřábek | 12,705 | <p>The number of abelian groups of order at most $n$ is $O(n)$, whereas if $n=2^k$, the number of class $2$ nilpotent groups of order $n$ is $2^{(2/27)k^3+O(k^{8/3})}=n^{\Omega(\log^2n)}$ by a result of Sims, hence the answer to question 1 is $0$. It is conjectured that the global asymptotic density of $2$-groups of ni... |
52,480 | <p>The question comes from a statement in Concrete Mathematics by Graham, Knuth, and Patashnik on page 465.</p>
<p>$$\sum_{k \geq n} \frac{(\log k)^2}{k^2} = O \left(\frac{(\log n)^2}{n} \right).$$</p>
<p>How is this calculated?</p>
| davidlowryduda | 9,754 | <p>I think the easiest way here is to simply find $\displaystyle \int_k ^{\infty} \frac {(\log x)^2}{x^2}dx$. After some work, it turns out to be $\dfrac{\log k(\log k + 2) + 2}{k}$. Oh - but I'm also somewhat confident that Qiaochu's suggestion to Sum by Parts would work as well (and give almost the same answer).</p>
|
4,251,161 | <p><strong>Objective</strong><br />
I need to find roots of <span class="math-container">$$f(x)=c$$</span> in interval <span class="math-container">$[a,b]$</span>, where</p>
<ul>
<li><span class="math-container">$f(a)=0$</span> and <span class="math-container">$c<f(b)<1$</span></li>
<li><span class="math-containe... | Simply Beautiful Art | 272,831 | <p>The largest bottleneck is the function evaluations, but luckily for us the root can be bracketed with something like bisection, ensuring we can avoid interpolation iterations if we know they will likely be unhelpful.</p>
<p>This is exactly where <a href="http://dl.acm.org/citation.cfm?id=248563" rel="nofollow norefe... |
2,992,127 | <p>I know that <span class="math-container">$ad\neq bc $</span> is sufficient for <span class="math-container">$z$</span> irrational because if <span class="math-container">$ad = bc$</span> then <span class="math-container">$\frac{ax+b}{cx+d} = \frac{ax+b}{cx+d} \frac{cb}{ad} = \frac{cax+cb}{cax+da}\frac{b}{d}$</span> ... | Community | -1 | <p>Assume by contradiction that</p>
<p><span class="math-container">$$\frac{ax+b}{cx+d}=\frac pq,$$</span> or</p>
<p><span class="math-container">$$(qa-pc)x+qb-pd=0,$$</span> and <span class="math-container">$x$</span> is rational !</p>
<p>Note that this doesn't work when <span class="math-container">$qa-pc=qb-pd=0$... |
3,471,684 | <p>Is always correct statement that if natural numbers <span class="math-container">$a,b \in \Bbb N$</span> for which LCM<span class="math-container">$(a,b)=16\cdot(a,b)$</span>, then <span class="math-container">$a|b$</span> or <span class="math-container">$b|a$</span>?</p>
<p>I used formula that LCM<span class="math... | nonuser | 463,553 | <p>Let <span class="math-container">$d= (a,b)$</span>. We know that <span class="math-container">$LCM (a,b) ={ab\over d}$</span>. Then as you noted we have <span class="math-container">$$ab =16d^2$$</span> We can also write <span class="math-container">$a=dx$</span> and <span class="math-container">$b=dy$</span> where ... |
3,990,195 | <p>I need some assistance solving what seems to be a very <a href="https://i.stack.imgur.com/arMiv.png" rel="nofollow noreferrer">intuitive problem</a>, but becomes tough when only using strict natural deduction and not assuming De Morgan laws.</p>
<p>Laws allowed: Implication, And, Or, MT, PBC, Copy Rule, Negation, Do... | algebruh | 875,254 | <p><a href="https://i.stack.imgur.com/jPvEk.png" rel="nofollow noreferrer">My proof here</a>. Let me know if you see anything wrong with it, or if there's a more efficient way</p>
|
1,195,175 | <p>In <a href="http://en.wikipedia.org/wiki/Solid_angle" rel="nofollow">wikipedia</a> the solid angle is defined as follows:</p>
<blockquote>
<p>In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point. </p>
</blockquote>
<p>Why solid angle is... | Mark Bennet | 2,906 | <p>You should look at it as follows:</p>
<p>$$\sqrt {x+\delta x}-\sqrt x =\left(\sqrt {x+\delta x}-\sqrt x\right) \cdot \frac {\sqrt {x+\delta x}+\sqrt x }{\sqrt {x+\delta x}+\sqrt x}=\frac {\delta x}{\sqrt {x+\delta x}+\sqrt x }$$</p>
<p>Now the denominator is "large" compared with $\delta x$ and you don't need to g... |
105,535 | <p>In a thread in <a href="https://math.stackexchange.com/questions/186292/derivatives-of-the-riemann-zeta-function-at-s-0">MSE</a> I proposed an older routine of mine for the efficient computation of coefficients; I use a very similar routine for the quick&dirty computation of the Stieltjes-constants. </p>
<p>... | Fredrik Johansson | 4,854 | <p>You could compare with the output from mpmath:</p>
<pre><code>sage: import mpmath
sage: mpmath.mp.dps = 1000
sage: %time mpmath.stieltjes(511)
CPU times: user 123.17 s, sys: 0.02 s, total: 123.19 s
Wall time: 123.40 s
mpf('673581492593841075447052270498937988033439947306384442967711559788996269245614412865378751092... |
1,731,382 | <p>Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?</p>
| J. M. ain't a mathematician | 498 | <p>As I've shown in <a href="https://math.stackexchange.com/a/63075">a previous answer</a>, the focus of a parabola rolling on a straight line traces a catenary. Similarly, the directrix of the same rolling parabola will envelope another catenary, a reflection of the one being traced by the focus.</p>
<p>Here is a mod... |
878,517 | <p>Is there any more solutions to this functional equation $f(f(x))=x$?</p>
<p>I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.</p>
| doraemonpaul | 30,938 | <p>In fact this belongs to a functional equation of the form <a href="http://eqworld.ipmnet.ru/en/solutions/fe/fe2315.pdf" rel="nofollow">http://eqworld.ipmnet.ru/en/solutions/fe/fe2315.pdf</a>.</p>
<p>Let $\begin{cases}x=u(t)\\f=u(t+1)\end{cases}$ ,</p>
<p>Then $u(t+2)=u(t)$</p>
<p>$u(t)=\theta(t)$ , where $\theta(... |
4,643,832 | <p><strong>Question</strong></p>
<p><a href="https://i.stack.imgur.com/XSeNR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XSeNR.png" alt="enter image description here" /></a></p>
<p>I am trying to prove this using balls (that is what we use in my school). The definition is that a subset <span clas... | Mastrem | 253,433 | <p>With Bernoulli's inequality:
<span class="math-container">$$
\left(1+\frac1y\right)^{x-y}\ge 1+(x-y)\cdot\frac 1y=1+\frac xy-1=\frac xy.
$$</span></p>
|
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | WGroleau | 203,965 | <p>If you can define a function to determine whether a given integer is in the set, then I can write a program that can print integers in the set until it runs out of storage.</p>
<p>Your "question" is actually a statement, but assuming you ask whether the statement is true, the answer is "yes." For example, I know o... |
370,599 | <p>If A is an invertible $nxn$ matrix prove that:$ adj(adjA)=(A)(detA)^{n-2}$
I have done this but it somewhere went wrong:
$ adj(adjA)=adj(A^{-1} detA)=(A^{-1}detA)^{-1} det(A^{-1}detA)=AdetA det(A^{-1}detA)= Adet(AA^{-1}detA)=A (detA)^n $ </p>
| Ross Millikan | 1,827 | <p>How about $x=12,y=15,n=3?$</p>
<p>For the update, $x=27,y=189,n=6$</p>
|
1,341,385 | <p>I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my own. My dad wants me to memorize about 30 trig identities (though some of them are very similar) since I'm missing tri... | Plutoro | 108,709 | <p>Usually, yes, though I prefer Euler's identity. Pretty much every trig identity can be derived from
$$e^{ix}=\cos(x)+i\sin(x).$$
However, it is useful to memorize some of the common ones because they will help you a lot in calculus and beyond to quickly identify when an expression can be simplified. I would start w... |
2,674,799 | <blockquote>
<p>Let $X_{2n}$ be the group whose presentation is$\langle x,y\,|\,x^n=y^2=1, xy=yx^2\rangle$. From $x=xy^2$, it is seen that $x^3=1$, hence $X_{2n}$ has at most $6$ elements. I have to show that if $n=3k$, then $X_{2n}$ has exactly $6$ elements. </p>
</blockquote>
<p>I can't see where I am having probl... | Derek Holt | 2,820 | <p>The elements $x=(1,2,3)$ and $y=(2,3)$ of $S_3$ satisfy the three relations when $n=3k$, and $S_3$ has order $6$ and is generated by $x$ and $y$. So $|X_{2n}| \ge 6$ when $n=3k$.</p>
<p>Also, the relations can be used to write any element of $X_{2n}$ in the form $x^ay^b$, the fact that $x^3=y^2=1$ in $X_{2n}$ prov... |
65,002 | <p>I am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are often dropped. This strikes me as very sloppy notation if $1 \times 1$ matrices are not at least <em>functionally equivale... | Xander Henderson | 468,350 | <h2>Some Background</h2>
<p>There are three basic kinds of mathematical spaces that are being asked about in the original question: scalars, vectors, and matrices. In order to answer the question, it is perhaps useful to develop a better understanding of what each of these objects is, or how they should be interpret... |
3,567,662 | <p>I have just learned how to convert a plane in R3 from Cartesian to parametric form, by setting 2 variables to 0 and solving for the 3rd one in order to obtain 3 points on the plane, and solve from there. However, this does not work when 1 or 2 of the variables are 0, as it is not possible to find 3 points on the pla... | infinity | 719,182 | <p>Denote by <span class="math-container">$f:\Bbb N \to X$</span> , <span class="math-container">$g:\Bbb N \to Y$</span> the bijections from <span class="math-container">$\Bbb N$</span> to <span class="math-container">$X,Y$</span>.</p>
<p>Then <span class="math-container">$h : \Bbb N \times \Bbb N \to X \times Y$</spa... |
2,895,382 | <p>Let $J_n=\{1,\dots,n\}$. How do I show that the set of all functions $J_n\to \mathbb N$ is countable? Any function is given by specifying the images of $1,\dots,n$. There are $|\mathbb N|$ options for the image of each $i=1,
\dots, n$. So intuitively, the set of such functions is the union of $n$ copies of $\mathbb ... | Robert Israel | 8,508 | <p>It's not the union, it's the Cartesian product.</p>
|
2,660,316 | <p>$$\frac{\mathrm{d}}{\mathrm{d}y}\left(\frac 2{\sqrt{2\pi}}\int_0^{\sqrt y} \exp\left(-{\frac{x^2}{2}}\right) \,\mathrm{d}x\right).$$</p>
<p>I try to integrate first and then do the differentiation but it's not easy. I want to know other way to do it. Thank you.</p>
| Lubin | 17,760 | <p>I think the most useful way of looking at this situation is that the series
$$
H(x)=1+\sum_1^\infty\binom{1/2}nx^n
$$
is the Binomial Series for $(1+x)^{1/2}$, as @Hurkyl has already noted. Please note that the only denominators in the coefficients are powers of $2$: they are $p$-integral for all $p\ne2$. This means... |
1,761,857 | <p>I need to show that, for $f:X\to \mathbb{R}$ bounded, we have:</p>
<p>$$\sup\{|f(x)-f(y)|, x,y\in X\}= \sup f - \inf f$$</p>
<p>Well, I know that </p>
<p>$$\sup\{|f(x)-f(y)|, x,y\in X\}\ge |f(x)-f(y)|$$ but in what this helps? I really have no idea in how to prove this one</p>
| RRL | 148,510 | <p>We have</p>
<p>$$\inf f \leqslant f(x) \leqslant \sup f, \\ -\inf f \geqslant -f(y) \geqslant -\sup f.$$</p>
<p>Hence, </p>
<p>$$f(x) - f(y) \leqslant \sup f - \inf f.$$</p>
<p>Now interchange $x$ and $y$.</p>
<p>$$f(x) - f(y) \geqslant -(\sup f - \inf f).$$</p>
|
2,135,228 | <p>Find</p>
<p>(a) $P\{A \cup B\}$</p>
<p>(b) $P\{A^c\}$</p>
<p>(c) $P\{A^c \cap B\}$</p>
<p>This is what I have right now:</p>
<p>(a) $P\{A \cup B\}=0.4+0.5=0.90$</p>
<p>(b) $P\{A^c\}= 1-0.4=0.60$</p>
<p>(c) $P\{A^c \cap B\}= (0.6)\cdot(0.5)=0.30$</p>
<p>Am I doing it correctly?</p>
| Community | -1 | <p>When I have issues like this, I often like to completely change the variables I'm using. $u$ and $v$ are the dummy variables you use in the pointwise definition of $\oplus$? Then use dummy variables $a,b,c$ in the formulation of associativity.</p>
<p>When you're trying to compute $(a \oplus b) \oplus c$, the fact y... |
108,200 | <p>If you are to calculate the hypotenuse of a triangle, the formula is:</p>
<p>$h = \sqrt{x^2 + y^2}$</p>
<p>If you don't have any units for the numbers, replacing x and y is pretty straightforward:
$h = \sqrt{4^2 + 6^2}$</p>
<p>But what if the numbers are in meters?<br />
$h = \sqrt{4^2m + 6^2m}$ <em>(wrong, would... | Jose Garcia | 17,341 | <p>if you know the Hamiltonian of the system then</p>
<p>$ Tr(exp(-t \Delta)\sim \int dq \int dp exp(-t\sum_{ab}p^{2}_{a}p^{2}_{b}-tV(q))$</p>
<p>the integeral is taken over ALL the p's and the q's momenta and position of the particle.</p>
<p>if the operator is a Laplacian operator then </p>
<p>$ Tr(exp(-t \Delta)\... |
2,965,193 | <p>Basically the question is asking us to prove that given any integers <span class="math-container">$$x_1,x_2,x_3,x_4,x_5$$</span> Prove that 3 of the integers from the set above, suppose <span class="math-container">$$x_a,x_b,x_c$$</span> satisfy this equation: <span class="math-container">$$x_a^2 + x_b^2 + x_c^2 = 3... | Ricky Tensor | 583,074 | <p>Any square integer must be congruent to either 0 or 1 mod 3. So for each of the 5 squares, we put it into hole 0 if it is congruent to 0 and into hole 1 if it is congruent to 1. Then take three squares from the hole with at least 3 squares and add them together. You will get either: <span class="math-container">$0+0... |
2,390,670 | <p>For an array with range $n$ filled with random numbers ranging from 0 (inclusive) to $n$ (exclusive), what percent of the array contains unique numbers?</p>
<p>I was able to make a program that tries to calculate this with repeated trials and ended up with ~63.212%.</p>
<p>My Question is what equation could calcul... | Henno Brandsma | 4,280 | <p>If my interpretation is correct there are $n^n$ equiprobable array fillings, of which $n!$ have all numbers occurring at most once.</p>
<p>So you get as the answer: $$\frac{n!}{n^n}$$ and <a href="http://mathworld.wolfram.com/StirlingsApproximation.html" rel="nofollow noreferrer">Stirling's approximation formula</a... |
4,193,578 | <p>This is maybe a very easy one, but I can't find a solution...</p>
<p>I'm looking for a sequence <span class="math-container">$a_1,...,a_n$</span> such that <span class="math-container">$0\leq a_1<\cdots<a_n<1$</span> and <span class="math-container">$\sum_{k=1}^na_k=1$</span>. Of course, this should work fo... | Alan | 175,602 | <p>Assuming you are okay with a piecewise rule, this is simply done with a modification of the geometric sequence with base <span class="math-container">$\frac 1 2$</span>. First note that this will be impossible for <span class="math-container">$n=1$</span> no matter what sequence you choose. Since you want the sequ... |
182,316 | <p>I am trying to find an example of a separable Hausdorff space which has a non-separable subspace. This led me to ask the question in the title: is the set of irrationals, regarded as a subspace of the real line, separable or non-separable?</p>
<p>A space is separable if it contains a countable dense subset. A su... | Asaf Karagila | 622 | <p>Every subspace of a separable metric space is separable.</p>
<p>For the irrationals, take the irrational algebraic numbers, those are dense in $\mathbb R$ and therefore in the irrationals too. As Jacob remarks below, if $\alpha$ is irrational, then $\{\alpha+q\mid q\in\mathbb Q\}$ is also dense.</p>
<p>Generally s... |
3,911,221 | <p>I am working on a probability exercice and I am trying to calculate E(Y) which comes down to this expression :</p>
<p><span class="math-container">$$ E(Y) = \int_{-∞}^{+∞} y\frac{e^{-y}}{(1+e^{-y})^{2}} \, \mathrm{d}y $$</span></p>
<p>I tried to use integrals by part but it diverges and I can't find a good change of... | DatBoi | 734,160 | <p>Note that the integrand is an odd function<span class="math-container">$\big(f(y)=-f(-y)\big)$</span> <span class="math-container">$\to$</span>which means that integral is <span class="math-container">$0$</span></p>
|
3,911,221 | <p>I am working on a probability exercice and I am trying to calculate E(Y) which comes down to this expression :</p>
<p><span class="math-container">$$ E(Y) = \int_{-∞}^{+∞} y\frac{e^{-y}}{(1+e^{-y})^{2}} \, \mathrm{d}y $$</span></p>
<p>I tried to use integrals by part but it diverges and I can't find a good change of... | Z Ahmed | 671,540 | <p><span class="math-container">$$E=\int_{-\infty}^{\infty} y\frac{e^{-y}}{(1+e^{y})^2}dy=\int_{-\infty}^{\infty} \frac{y}{4} \text{sech}^2(y/2) dy=0.$$</span>
As the integrand is an odd function.</p>
<p>However, <span class="math-container">$$\int_{0}^{\infty} \frac{y}{4} \text{sech}^2(y/2) dy$$</span>
is convergent ... |
1,782,558 | <p><strong>Problem</strong></p>
<p>I have two differential equations</p>
<p>$ \frac{dx}{dt} + \frac{dy}{dt} + x + y = 0$</p>
<p>$ 2 \frac{dx}{dt} + \frac{dy}{dt} + x = 0 $</p>
<p>initial conditions: $y(0) = 1$ and $x(0) = 0$</p>
<p><strong>Attempt</strong></p>
<p>I've solved the system via the Matrix method of s... | JJacquelin | 108,514 | <p>In a so simple case, the method with matrix appears a bit overmuch.
$$\begin{cases}
x'+y'+x+y=0 \\
2x'+y'+x=0
\end{cases} \quad\to\quad (2x'+y'+x)-(x'+y'+x+y)=0 \quad\to\quad y=x'$$
$2x'+y'+x=2x'+x''+x=0\quad\to\quad x=Ae^{-t}+Bte^{-t}\quad\to\quad y=(-A+B)e^{-t}-Bte^{-t}$</p>
<p>Conditions : $\begin{cases}
x(0)=A=... |
664,152 | <p>I have a homework problem that I'm very stuck on. The problem statement is as follows:</p>
<p>"Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then $\mu^*(E \cup F) = \mu^*(E) + \mu^*(F)$. Prove that every open set is a splitting set. (Recall that the distance b... | Quinn Culver | 11,030 | <p>Here's a strategy for making your second attempt work: since $E = (E\cap U) \cup (E \cap U^c)$, you can conclude that $d(E\cap U, E \cap U^c)=0$. Now you should be able to shrink $U$ just a little, yielding $U'$ so that $d(E \cap U', E \cap U^c)>0$ but still $\mu^{*}(E)<\mu^{*}(E\cap U')+ \mu^{*}(E\cap U^c)$.<... |
1,801,946 | <p>I need to find the equation of tangent line passing $(2,3)$ and perpendicular to $3x+4y=8$. Need help in this and also show me how you got the answer. I will be very thankful.</p>
| StackTD | 159,845 | <p>Your approach my depend on what you have seen about slopes and/or normal vectors and their relations with respect to being perpendicular.</p>
<p><strong>Slope approach</strong></p>
<p>A line through $(x_1,y_1)$ with slope $m$ has the following equation:
$$y = m(x-x_1)+y_1$$
If $m_1$ and $m_2$ are slopes correspond... |
788,245 | <p>$$\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$$ I know this series does not converge. Can someone show me how to prove that? Should i use criteria of Dalamber or any other criteria?</p>
| Zook | 135,276 | <p>The Ratio Test is usually the best way to approach factorials in series, however with this one you can just use the Test for Divergence by showing that the terms of the series do not approach $0$. In fact, the terms should all be greater than $1$, because the numerator is greater than the denominator. That's pretty ... |
3,773,933 | <p>Question goes like this:</p>
<p>In a box containing <span class="math-container">$36$</span> strawberries, <span class="math-container">$2$</span> of them are rotten. Kyle randomly picked <span class="math-container">$5$</span> of these strawberries.<br>
a. What is the probability of having at least 1 rotten strawbe... | Robert Shore | 640,080 | <p>It's often easier to approach a problem like this backwards. Let's compute the probability that you picked no rotten strawberries.</p>
<p>There are <span class="math-container">$\binom{34}{5}$</span> ways to choose <span class="math-container">$5$</span> good strawberries and <span class="math-container">$\binom{36... |
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