qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,111,952 | <p><strong>My Try:</strong> </p>
<p>We substitute $y = x^{2/3}$. Therefore, $x = y^{3/2}$ and $\frac{dx}{dy} = \frac{2}{3}\frac{dy}{y^{1/3}}$</p>
<p>Hence, the integral after substitution is: </p>
<p>$$ \frac{3}{2} \int_0^\infty \sin(y)\sqrt{y} dy$$</p>
<p>Let's look at:</p>
<p>$$\int_0^\infty \left|\sin(y)\sqrt{... | mickep | 97,236 | <p>While typing, I noticed that @GEdgar already noted this, but here it goes anyway.</p>
<p>Integrating by parts, we find that
$$
\begin{align}
\int \sin(x^{2/3})\,dx &=\int -\frac{3}{2}x^{1/3}\frac{d}{dx}\cos(x^{2/3})\,dx \\
&= -\frac{3}{2}x^{1/3}\cos(x^{2/3})+\int \frac{1}{2}x^{-2/3}\cos(x^{2/3})\,dx.
\end{a... |
4,547,918 | <p>Given the torus and given the point p <span class="math-container">$\in$</span> M corresponding to the parameters <span class="math-container">$s=\frac{\pi }{4}$</span> and <span class="math-container">$t=\frac{\pi }{3}$</span>.
Determine the cartesian equation of the tangent plane to M in p.</p>
<p><span class="mat... | electrical apprentice | 912,523 | <p><span class="math-container">$$\begin{align}
\int {\cos^2 x \over 1+\sin x } \mathrm{d}x&=
\int {1-\sin^2 x \over 1 + \sin x } \mathrm{d}x
\\&=\int {(1+\sin x)(1-\sin x) \over (1+\sin x) } \mathrm{d}x\\
&=\int (1-\sin x) \mathrm d x\\&=x+\cos x + \mathrm{const}
\end{align}$$</span></p>
|
3,444,214 | <p>Let <span class="math-container">$\zeta = e^{2\pi i / 7}$</span>. I know the minimal polynomial of <span class="math-container">$\zeta$</span> over <span class="math-container">$\mathbb{Q}$</span> is <span class="math-container">$\sum_{i=0}^{6} x^{i}$</span>. But what is <span class="math-container">$[ \mathbb{Q}(\z... | Arthur | 15,500 | <p>You want a quadratic polynomial with real coefficients where <span class="math-container">$\zeta$</span> is a root. That means that its complex conjugate <span class="math-container">$\zeta^6$</span> must be the other root. Thus Vieta's formulas tells you that the polynomial you're after is
<span class="math-contain... |
492,407 | <p>I was searching for methods on how to calculate the area of a polygon and stubled across this: <a href="http://www.mathopenref.com/coordpolygonarea.html" rel="nofollow noreferrer">http://www.mathopenref.com/coordpolygonarea.html</a>.
$$
\mathop{area} =
\left\lvert\frac{(x_1y_2 − y_1x_2) + (x_2y_3 − y_2x_3) + \cdots... | Oleg567 | 47,993 | <p><img src="https://i.stack.imgur.com/wTYmV.png" alt="enter image description here"></p>
<p>Let $O$ is the origin. Denote "signed area" of triangle $OAB$: $~~S_{OAB}= \dfrac{1}{2}(x_Ay_B-x_By_A)$.<br>
It can be derived from cross product of vectors $\vec{OA}, \vec{OB}$.</p>
<p>If way $AB$ is $\circlearrowleft$ (if p... |
514,912 | <p>I have what may seem a very trivial question, but how it is answered may affect how a proof of mine is structured. It pertains to formatting and convention. When 'recursively' defining a function does it make sense to use quantifiers? </p>
<p>For example would:</p>
<p>$ 5 \in R $</p>
<p>If $ r \in R $, then $ \fo... | W.W. | 98,791 | <p>For $$\sqrt{6 + \sqrt{6 + \sqrt{6 +\dots}}}:$$</p>
<p>Let \begin{align*}
x &= \text{the given equation}\\
&= \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots}}}
\end{align*}</p>
<p>Since the series is infinite, we can write
$x = \sqrt{6 + x}$,
or $$x^2 - x - 6 = 0.$$</p>
<p>Therefore, $x = 3$ or $x = -2$</p>
<p>Sinc... |
3,521,525 | <p>How many three-digit numbers are there whose digits in the hundreds place and ones place are the same? (Assume that a nonzero digit is in the hundreds place.) </p>
<p>Please try to simplify the solution so that a child under 14 may understand this. Also, it would help if you included a formula that <em>may</em> be... | Ethan Bottomley-Mason | 657,832 | <p>The question is rather simple, so here is some advice on how to solve it. Consider the number of ways to get your statement to be true. How many ways can you get the ones and hundreds places to be the same? What are the possibilities for the tens place? To find the total number of possibilities, multiply the indepen... |
3,005,100 | <p>Given the following formula
<span class="math-container">$$
\sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}\,.
$$</span>
How can I show that this is equal to
<span class="math-container">$$
\frac{n!}{x(x+1)\cdots(x+n)}\,?
$$</span></p>
| Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
99,378 | <p>The following equation in $\mathbb{C}$:</p>
<p>$4z^2+8|z|^2-3=0$</p>
<p>is not algebraic and has 4 solutions : $\pm\frac{1}{2}$ and $\pm i\frac{\sqrt{3}}{2}$.
The Solve function in Mathematica only returns the 2 real values :</p>
<pre><code>Solve[4 z^2 + 8 Abs[z]^2 - 3 == 0, Complexes]
(* {{z -> -(1/2)}, {z -... | Suba Thomas | 5,998 | <pre><code>Solve[4 z^2 + 8 Abs[z]^2 - 3 == 0 && z \[Element] Complexes, z]
</code></pre>
<blockquote>
<p>{{z -> -(1/2)}, {z -> 1/2}, {z -> -((I Sqrt[3])/2)}, {z -> (
I Sqrt[3])/2}}</p>
</blockquote>
|
99,378 | <p>The following equation in $\mathbb{C}$:</p>
<p>$4z^2+8|z|^2-3=0$</p>
<p>is not algebraic and has 4 solutions : $\pm\frac{1}{2}$ and $\pm i\frac{\sqrt{3}}{2}$.
The Solve function in Mathematica only returns the 2 real values :</p>
<pre><code>Solve[4 z^2 + 8 Abs[z]^2 - 3 == 0, Complexes]
(* {{z -> -(1/2)}, {z -... | Daniel Lichtblau | 51 | <p>A pedestrian approach, overkill in this case, is to separate into explicit real and imaginary parts both for the expression(s) and variable(s).</p>
<pre><code>expr = 4 z^2 + 8 Abs[z]^2 - 3;
{re, im} =
ComplexExpand[{Re[expr], Im[expr]}, z] /. {Re[z] -> rez, Im[z] -> imz}
solns = Solve[{re, im} == 0];
rez +... |
2,498,359 | <p>This is a basic probability question. </p>
<p>Persons A and B decide to arrive and meet sometime between 7 and 8 pm. Whoever arrives first will wait for ten minutes for the other person. If the other person doesn't turn up inside ten minutes then the person waiting will leave. What is the probability that they will... | Ravenex | 442,239 | <p>This problem was already explained here:</p>
<p><a href="https://math.stackexchange.com/questions/1279873/basic-probability-romeo-and-juliette-meet-for-a-date?rq=1">Basic probability: Romeo and Juliette meet for a date.</a></p>
<p>It just needs to be adjusted to work in 6ths instead of 4ths (10/60 of an hour vs. 1... |
1,232,363 | <p>I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). </p>
<p>Suppose that we have a box with 100 balls and I take a random sample of size 10</p>
<p>Is a random sample of size 10 if</p>
<ul>
<li>I take... | Surb | 154,545 | <p>The approach of Mark Viola is probably the best for $2\times 2$ matrices. Note however that this generalizes to $n\times n$.</p>
<p>Indeed, the <a href="https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Positive_matrices" rel="nofollow noreferrer">Perron-Frobenius theorem</a> states (in particular) tha... |
2,965,989 | <p>Why <span class="math-container">$$p(y)=\int_0^\infty x\delta (y-x)dx=y\ \ ?$$</span></p>
<p>For me, <span class="math-container">$$p(y)=\int_0^\infty x\delta (y-x)dx=\int_{\{y\}}xdx=0.$$</span></p>
<p>If it would be written <span class="math-container">$\int_0^\infty xd\delta _y$</span>, then I would be agree wit... | Chris | 430,789 | <p>By the definition of the dirac delta function,</p>
<p><span class="math-container">$$
\int_{0}^\infty f(x) \delta(y-x) dx = f(y)
$$</span></p>
<p>for all <span class="math-container">$y \in [0, \infty)$</span> and arbitrary function <span class="math-container">$f$</span>.</p>
|
3,700,299 | <p>I want to show that <span class="math-container">$\int\limits_{-\infty}^\infty e^{-\pi x^2}dx = 1$</span>.</p>
<p>By definition
<span class="math-container">$$\int\limits_{-\infty}^\infty e^{-\pi x^2}dx = \lim\limits_{t\to\infty}\int\limits_{-t}^t e^{-\pi x^2}dx$$</span>
and since the integrand <span class="math-co... | John Hughes | 114,036 | <p>How about doing a substitution, <span class="math-container">$x = \frac{1}{\sqrt{\pi}} u; dx = \frac{1}{\sqrt{\pi}} du$</span>? </p>
<p>That'll convert your integral into an integral of <span class="math-container">$exp(-u^2)$</span>, which is just the error function, whose value "at infinity" is well known. </p>
|
3,366,781 | <blockquote>
<p>Let <span class="math-container">$(S, +, \cdot, 0)$</span> and <span class="math-container">$(S', \oplus, \otimes, 0')$</span> be two semirings. Then <span class="math-container">$f: S\rightarrow S'$</span> is said to be a homomorphism if for all <span class="math-container">$a, b\in S,$</span> <span... | kccu | 255,727 | <p>If you negate the definition of continuity, you get: "There exists <span class="math-container">$\epsilon>0$</span> such that for all <span class="math-container">$\delta>0$</span>, there exist <span class="math-container">$x,x' \in [a,b]$</span> such that <span class="math-container">$|x-x'|<\delta$</span>... |
3,908,955 | <p>Is the given series convergent or divergent? Give a reason. Show details.</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty} \frac{(-i)^n}{ln \ n}$$</span></p>
<p>So maybe I'll try using the ratio test?</p>
<p>So the series converges if <span class="math-container">$$\left| \frac{z_{n+1}}{z_n} \right| < 1$... | Community | -1 | <p>We put <span class="math-container">$f(x) = x^2 +x+1$</span></p>
<p>We can prove that :
<span class="math-container">$|f(x) - l|<\delta $</span> <span class="math-container">$ \Rightarrow $</span> <span class="math-container">$ |x-a|<\alpha $</span></p>
<p><span class="math-container">$\alpha , \delta > 0$<... |
870,030 | <p>Q: Prove that the relation given by $a\sim b\Leftrightarrow a-b\in\mathbb{Z}$ is a congruence relation on the additive group $\mathbb{Q}$.</p>
<p>A: Maybe...
<ul>
<li>$a\sim a\Leftrightarrow a-a=0\in \mathbb{Z}$ ✓
<li>$a\sim b\Leftrightarrow a-b\in \mathbb{Z}$. $a\in \mathbb{Z}\Rightarrow -a\in \mathbb{Z}$ ... | Nick | 132,027 | <p>Your proof that $a \sim b$ implies $b \sim a$ is a bit shaky. It is not necessarily true that $a \in \Bbb{Z}$ and $b \in \Bbb{Z}$ (let $a = 1/2$, $b = -1/2$, for example). You can get this more easily by simply noting that $b-a = -(a-b)$. Your proofs of reflexivity and transitivity are fine.</p>
<p>For the last pro... |
1,314,219 | <p>Is there any formula for finding the last digit of the factorials?
How to approach these type of questions?
Thanks in advance.</p>
| Deepak | 151,732 | <p>First of all, when you see this sort of seemingly intractable problem, don't despair. There's usually a very simple "trick" that makes the problem trivial.</p>
<p>In this case, you have to realise two things:</p>
<p>1) only the sum of last digits contributes to the last digit of the final sum.</p>
<p>2) factorial... |
4,069,499 | <p>If we let <span class="math-container">$x = 0$</span>.</p>
<p><span class="math-container">\begin{align*}
3(0+7)-y(2(0)+9) \\
21-9y \\
\end{align*}</span></p>
<p>Then <span class="math-container">$9y$</span> should always equal <span class="math-container">$21$</span>?
Solving for <span class="math-container">$y$</... | Raffaele | 83,382 | <p><span class="math-container">$3 (x + 7) - y (2 x + 9)=21-9y$</span> for <span class="math-container">$x=0$</span> and</p>
<p><span class="math-container">$3 (x + 7) - y (2 x + 9)=24 - 11 y$</span> for <span class="math-container">$x=1$</span></p>
<p>They are the same, so must be
<span class="math-container">$$21-9y=... |
4,069,499 | <p>If we let <span class="math-container">$x = 0$</span>.</p>
<p><span class="math-container">\begin{align*}
3(0+7)-y(2(0)+9) \\
21-9y \\
\end{align*}</span></p>
<p>Then <span class="math-container">$9y$</span> should always equal <span class="math-container">$21$</span>?
Solving for <span class="math-container">$y$</... | Joe | 623,665 | <p>If we let <span class="math-container">$x=0$</span> then we do indeed find that the expression equals <span class="math-container">$21-9y$</span>. But this does not mean that <span class="math-container">$21-9y$</span> must equal <span class="math-container">$0$</span>; it only means that <span class="math-container... |
3,880,743 | <p>If <span class="math-container">$T:\mathbb{R}^2 \rightarrow \mathbb{R}$</span> is a function such that <span class="math-container">$T(\alpha v)=\alpha T(v)$</span> <span class="math-container">$\forall \alpha \in \mathbb{R}$</span> and <span class="math-container">$v \in \mathbb{R}^2$</span>, is T necessarily a lin... | John Hughes | 114,036 | <p>As an alternative, define
<span class="math-container">$$
T(x, 0) = (x, 0)
$$</span>
and
<span class="math-container">$$
T(x, y) = (0,0)
$$</span>
for <span class="math-container">$y \ne 0$</span>. Now consider that <span class="math-container">$(1, 1) = (0, 1) + (1, 0)$</span> and apply <span class="math-container... |
2,619,638 | <p>I know a function which is not equal a.e to a continuous function is the step function or the characteristic of any interval and I also know the Dirichlet function is not an a.e continuous function but I want an example of a function with both properties.</p>
| really Nobody | 521,491 | <p>You're actually right, that should be $\sin^{-1}$ (or $\ -cos^{-1}$).</p>
<p>Indeed, since $\sin^{-1}$ and $\cos^{-1}$ are linked, as you said, by $\cos^{-1}+\sin^{-1}=\pi/2$, and since a continuous function has an infinite number of antiderivatives differing by a constant, taking $sin^{-1}$ or $cos^{-1}$ makes no ... |
7,130 | <p>I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).</p>
<p>Thank you.</p>
| Jozef | 14,829 | <p>For the directed case,</p>
<p>Given $\langle G=(V,E)\rangle$ for the Hamiltonian cycle, we can construct input $\langle G',s,t\rangle$: choose a vertex $u \in V$ and divide it into two vertices, such that the edges that go out of $u$, will go out of $s$ and the vertices that get in to $u$, will get in to $t$.</p>
|
7,130 | <p>I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).</p>
<p>Thank you.</p>
| Rotenberg | 242,055 | <p>This is a reduction from undirected Hamilton Cycle to undirected Hamilton Path. It takes a graph $G$ and returns a graph $f(G)$ such that $G$ has a Hamilton Cycle iff $f(G)$ has a Hamilton Path.</p>
<p>Given a graph $G = (V,E)$ we construct a graph $f(G)$ as follows.</p>
<p>Let $v \in V$ be a vertex of G, and let ... |
3,279,878 | <p>I got this equation while I was trying to solve a certain math Olympiad problem. I tried modulus and whatnot, but I haven't got anywhere. Is there a way to prove this?</p>
| nonuser | 463,553 | <p>Solution with infinite descent.</p>
<p>Let <span class="math-container">$(x,y,z,a)$</span> be a solution with the smallest possible <span class="math-container">$a$</span>. Say <span class="math-container">$a>0$</span>. <span class="math-container">$$x^2+2y^2+3z^2=10a^2$$</span> </p>
<p>Modulo 2 we get <span cl... |
2,779,083 | <p>Given a polynomial $f(z)\in\mathbb{C}[z]$, $\exists$ only finitely many $c$ s.t. $f(z)-c=0$ has repeated roots?
Is above true in general?
Is it true for polynomials of the form $f(z) = (z-z_1)\cdot ... \cdot (z - z_n)$ where $z_1, ... , z_n \in \mathbb{C}$are distinct?</p>
| Hagen von Eitzen | 39,174 | <p>A multiple root of $f(z)-c$ is at the same time a root of $f'(z)$. As the polynomial $f'$ has only finitely many roots, the claim follows. The only special case is when $f'\equiv 0$, and indeed then $f$ is constant and for one specific value of $c$ has roots at all.</p>
|
3,840,699 | <p>I need to calculate something of the form</p>
<p><span class="math-container">\begin{equation}
\int_{D} f(\mathbf{x}) d\mathbf{x}
\end{equation}</span></p>
<p>with <span class="math-container">$D \subseteq \mathbb{R^2}$</span>, but I only have available <span class="math-container">$f(\mathbf{x})$</span> at given sa... | Ross Millikan | 1,827 | <p>Assuming you are just given a table of values there are two approaches that come to mind.</p>
<p>One is to view each point as a sample of the value of the function. You can divide <span class="math-container">$D$</span> into regions by a <a href="https://en.wikipedia.org/wiki/Voronoi_diagram" rel="nofollow noreferr... |
1,328,909 | <p>I know how to find for which $n$ $\phi(n)=n/2$ or $\phi(n)=n/3$, my method for finding those was simply to find primes $p$ that satisfy $\Pi_p$$_|$$_n$$1-1/p$ $ = 1/2$ or $1/3$.</p>
<p>However, I don't know how to find $\Pi_p$$_|$$_n$$1-1/p = n/6$. Intuitively it seems that if I combine results for both $\phi(n) =... | Nescio | 47,988 | <p>It seemed to be popular enough in the comments so,
You can consider a function defined as $f(x)=0$ for $x\in\mathbb{R}\setminus A$ and $f(x)=1$ if $x\in\mathbb{Q}\cap{A}$ and $f(x)=2$ if $x\in\mathbb{I}\cap{A}$. It will be continuous at each point $x_0$ outside of $A$ as it will be $0$ at an environment of $x_0$ (A... |
313,437 | <p>I have to find out the convergence of the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
Any help? Thanks</p>
| Ron Gordon | 53,268 | <p>The tricky part of the integral is near $x=0$. There, note that $\sin{x} \sim x$, and consider </p>
<p>$$\int dx \frac{\log{x}}{\sqrt{x}}$$</p>
<p>Substitute $x=u^2$, $dx=2 u du$ and this integral is equal to</p>
<p>$$2 \int du u \frac{1}{u} \log{u^2} = 4\int du \log{u} = 4 (u \log{u}-u) = 2 (\sqrt{x} \log{x} - ... |
2,022,423 | <p>You are asked to <strong>permute the neighboring sub-sequence</strong> of the sequence $n,n-1,n-2,\cdots,1$ until the sequence is brought to the increasing order. </p>
<p>By <em>permute the neighboring sub-sequence</em> I mean for example:
$5,4,3,2,1 \to 5,3,4,2,1 $ or $5,4,3,2,1\to 5,2,4,3,1$ or $5,4,3,2,1\to5,2,... | Ross Millikan | 1,827 | <p>$n-1$ is an upper bound as we can exhibit an algorithm that achieves that. Take successive pairs of elements and invert them. This uses $\lfloor \frac n2 \rfloor$ swaps. Lock together the pairs you have swapped, considering the pair to be one element, and you have $n-\lfloor \frac n2 \rfloor=\lceil \frac n2 \rcei... |
2,022,423 | <p>You are asked to <strong>permute the neighboring sub-sequence</strong> of the sequence $n,n-1,n-2,\cdots,1$ until the sequence is brought to the increasing order. </p>
<p>By <em>permute the neighboring sub-sequence</em> I mean for example:
$5,4,3,2,1 \to 5,3,4,2,1 $ or $5,4,3,2,1\to 5,2,4,3,1$ or $5,4,3,2,1\to5,2,... | Brian M. Scott | 12,042 | <p>I don’t know that it’s best possible, but I can show that if $T(n)$ swaps are needed to reverse $\langle n,n-1,\ldots,1\rangle$, then $T(n)\le\left\lfloor\frac{3n}4\right\rfloor$. For convenience let $f(n)=\left\lfloor\frac{3n}4\right\rfloor$. It’s not hard to verify that $T(1)=0=f(1)$, $T(2)=1=f(2)$, $T(3)=2=f(3)$,... |
1,617,890 | <blockquote>
<p>Question: Solve $\sin(3x)=\cos(2x)$ for $0≤x≤2\pi$.</p>
</blockquote>
<p>My knowledge on the subject; I know the general identities, compound angle formulas and double angle formulas so I can only apply those.</p>
<p>With that in mind</p>
<p>\begin{align}
\cos(2x)=&~ \sin(3x)\\
\cos(2x)=&~... | Ian Miller | 278,461 | <p>You have made some errors in your calculations (or some typos here).</p>
<p>$$\sin(3x)=\cos(2x)$$</p>
<p>$$ \sin(2x+x) = \cos(2x)$$</p>
<p>$$\sin(2x)\cos(x) + \cos(2x)sin(x) = \cos(2x) $$</p>
<p>$$ 2\sin(x)\cos(x)\cos(x) + (1-2\sin^2(x))\sin(x)) = \cos(2x) $$</p>
<p>$$ 2\sin(x)\cos^2(x) + \sin(x) - 2\sin^{\bf{3... |
1,580,586 | <p>Question goes as follows:
Consider the points on a line; $A(1,3,-1)$ and $B(-1,4,-2)$. Find the point $Q$ on $L$ closest to the point $P(1,1,0)$.</p>
<p>My thinking:
Closest distance from $a$ to $b$ is always a straight line, $90$ degree angle.
Therefore:
$$
Q⋅P=0
$$</p>
<p>$$
L=
\left(\begin{array}{cc}
1\\
3\\
-1... | Deepak | 151,732 | <p>Your method is incorrect.</p>
<p>You're supposed to find the point $Q$ such that the vectors $\vec{AQ}$ and $\vec{PQ}$ are perpendicular.</p>
<p>$\vec{AQ} = \vec{OQ} - \vec{OA} = \left(\begin{array}{cc}
2t\\
-t\\
t\\
\end{array}\right)$</p>
<p>$\vec{PQ} = \vec{OQ} - \vec{OP} = \left(\begin{array}{cc}
-2t\\
2+t\\
... |
29,823 | <p>Just a random thought here: Can cohomology theories (e.g. sheaf cohomology) on the Stone space $S_n(T)$ (the space of complete n-types) of a first-order theory $T$ tell us anything interesting (e.g. the classification of theories)? Is there any result in model theory that is obtained (probably most easily) by this k... | James Freitag | 6,789 | <p>I can not really inform you about this since I don't know, but I can point you to some notes of Angus Macintyre,
<a href="http://modular.math.washington.edu/swc/notes/files/03MacintyreNotes.pdf" rel="nofollow">http://modular.math.washington.edu/swc/notes/files/03MacintyreNotes.pdf</a></p>
<p>Here are some excerpts... |
3,391,280 | <p>Prove by Induction on n that <span class="math-container">$\exists x,y,z \in Z$</span> s.t. <span class="math-container">$x\ge 2, y\ge 2, z\ge 2$</span> satisfies <span class="math-container">$x^2+y^2=z^{2n+1}$</span> </p>
<p>I'm a lot more comfortable with proving induction with <span class="math-container">$\for... | fleablood | 280,126 | <p>Well, if <span class="math-container">$x^2 + y^2 = z^{2n+1}$</span> then</p>
<p><span class="math-container">$x^2z^2 + y^2z^2 = z^{2n+1}z^2$</span></p>
|
1,053,683 | <p>How to show that
$$\sum_{n=1}^\infty\frac{1}{n^2+3n+1}=\frac{\pi\sqrt{5}}{5}\tan\frac{\pi\sqrt{5}}{2}$$
?</p>
<p><strong>My try:</strong></p>
<p>We have
$$n+3n+1=\left(n+\frac{3+\sqrt{5}}{2}\right)\left(n+\frac{3-\sqrt{5}}{2}\right),$$
so
$$\frac{1}{n^2+3n+1}=\frac{2}{\sqrt{5}}\left(\frac{1}{2n+3-\sqrt{5}}-\frac{1... | Ron Gordon | 53,268 | <p>I think we can make some use of the residue theorem. Write $n^2+3 n+1 = (n+3/2)^2-5/4$ and the sum is</p>
<p>$$\sum_{n=1}^{\infty} \frac1{\left (n+\frac{3}{2} \right )^2-\frac{5}{4}} = \frac12 \sum_{n=-\infty}^{\infty} \frac1{\left (n+\frac{3}{2} \right )^2-\frac{5}{4}} - 1 +1$$</p>
<p>(To get the doubly infinite ... |
2,264,021 | <p>Can you help me explain the basic difference between Interior Point Methods, Active Set Methods, Cutting Plane Methods and Proximal Methods.</p>
<p>What is the best method and why?
What are the pros and cons of each method?
What is the geometric intuition for each algorithm type?</p>
<p>I am not sure I understand ... | wyer33 | 33,022 | <p>Although these are all optimization algorithms, they tend to be used in different contexts. Note, you requested a lot of technical information that I don't remember off the top of my head, but perhaps the following will get you started.</p>
<h2>Interior Point and Active Set Methods</h2>
<p>Both of these algorithm... |
308,856 | <p>A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure. </p>
<p>A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesg... | Zvonimir Sikic | 94,293 | <p>Concerning Tao comment that the symmetry is broken by declaring 0⋅∞ = ∞⋅0 = 0, I would like to add that this is the reason why Lebesgue integral does not satisfy Newton-Leibniz formula. Namely, for Cantor-Lebesgue function f, f(1)–f(0) = 1 but ∫01f’ = 0 because f’ = ∞ on Cantor set C which has measure 0 (and f’ = 0 ... |
2,519,620 | <blockquote>
<p><strong>Question :</strong> Three balls are to be randomly selected without replacement from an urn containing $20$ balls numbered $1$ through $20$. If we bet that at least one of the balls that are drawn hasa number as large as or larger than $17$, what is the probability that we win the bet?</p>
</b... | Cm7F7Bb | 23,249 | <p>Let us simplify the game a little bit. Suppose that balls $1,\ldots,16$ are red and $17,\ldots,20$ are green. We win if we draw a green ball. The distribution of green balls among the three drawn balls is <a href="https://en.wikipedia.org/wiki/Hypergeometric_distribution" rel="nofollow noreferrer">hypergeometric</a>... |
2,519,620 | <blockquote>
<p><strong>Question :</strong> Three balls are to be randomly selected without replacement from an urn containing $20$ balls numbered $1$ through $20$. If we bet that at least one of the balls that are drawn hasa number as large as or larger than $17$, what is the probability that we win the bet?</p>
</b... | NewBee | 473,224 | <p>First, find out probability of not winning then 1-that.</p>
<p>Total no of outcomes(A) : 20C1 x 19C1 x 18C1</p>
<p>Favourable outcomes for not winning(B) : 16C1 x 15C1 x 14C1 (Selecting balls numbered <17)</p>
<p>So the probability of not winning(C) : B/A = 28/57</p>
<p>Thus, required probability : 1 - C = 1 ... |
3,135,386 | <p>Our teacher tells us to convert it this way <span class="math-container">$ 3^x = e^{\ln 3^x}= e^{x\cdot\ln 3}$</span> and then use the rule <span class="math-container">$e^u\cdot u'$</span> but I can't understand where <span class="math-container">$\ln$</span> comes from and how <span class="math-container">$\ln 3^x... | Sudeep Sapkota | 650,719 | <p>Any term <span class="math-container">$x$</span> can be expressed as power of <span class="math-container">$e$</span>:</p>
<p><span class="math-container">$$x=e^{\ln x}$$</span> </p>
|
129,788 | <blockquote>
<p>Let be A and B two events from the same sample set. If $\space P(A)+P(B)=1$, can one say that they are opposite events?</p>
</blockquote>
<p>In my thought:</p>
<p>$\space P(A)+P(B)=1$</p>
<p>$\space P(A)=1-P(B)$</p>
<p>So they are opposite events. But my book says no! It says that is not necessary... | Brian M. Scott | 12,042 | <p>Suppose that you roll an ordinary six-sided die. Let $A$ be the event that you roll $1,2$, or $3$, and let $B$ be the event that you roll an even number ($2,4$, or $6$). $P(A)=P(B)=\frac12$, so $P(A)+P(B)=1$; are $A$ and $B$ opposite events? (By the way, a better word is <em>complementary</em> events.)</p>
|
4,644,186 | <p>Let m be a positive integer.Find the values of <span class="math-container">$$\sum_{k=0}^n \frac{{n\choose k }}{k+1}$$</span>. Leave your answer in terms of n where appropriate.</p>
<p>Remark. There is an alternative method for computing the sums described here: make use of integration.</p>
<p>I can only list out th... | David H | 55,051 | <p><strong>Hint:</strong> You can rewrite the <span class="math-container">$\frac{1}{1+k}$</span> factor using the integral</p>
<p><span class="math-container">$$\int_0^1 x^k \,dx= \frac{1}{1+k}.$$</span></p>
<p>Then pull the summation inside the integral.</p>
|
3,016,386 | <p>Hi I am struggling with this exercise, which may be perceived as simple. so I was trying to write tangents as follows:</p>
<p><span class="math-container">$$\tan(z)=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}$$</span> and then <span class="math-container">$$z=a+bi$$</span>, which led me to <span class="math-container">... | José Carlos Santos | 446,262 | <p>It is much easier to deal with this problem using the fact that<span class="math-container">$$1+\tan^2(z)=\dfrac1{\cos^2(z)}.$$</span>So, which numbers can be written as <span class="math-container">$\dfrac1{\cos^2(z)}$</span>? Answer: all, except <span class="math-container">$0$</span>. It follows from this (and fr... |
3,939,620 | <p>Given a polynomial of the form <span class="math-container">$R(z):=\frac{P(z)}{Q(z)}$</span> such that <span class="math-container">$R(z)$</span> has no real roots and <span class="math-container">$deg(Q) \geq deg(P) + 2$</span>, then the integral can be expressed as</p>
<p><span class="math-container">$$\int_{-\inf... | Community | -1 | <p>We must have</p>
<p><span class="math-container">$$\sin x(\sin x+1)\ge0.$$</span></p>
<p>As <span class="math-container">$$\sin x+1\ge0$$</span> is always true, we seem to be left with</p>
<p><span class="math-container">$$\sin x\ge 0.$$</span></p>
<p>But there is a trap*: when <span class="math-container">$$\sin x+... |
3,995,119 | <p>I've difficulties calculating the following integral</p>
<p><span class="math-container">$$\int_z^\infty\mu\mathrm e^{-\mu y}(\mathrm e^{-\lambda z}-\mathrm e^{-\lambda y})\ \mathrm{d}y$$</span></p>
<p>I'm gonna use her to find a joint distribution of two random variables. I've try to apply the following substitutio... | Kavi Rama Murthy | 142,385 | <p>Split it into two terms. <span class="math-container">$e^{-\lambda z}$</span> is constant in the first integral. So the answer is <span class="math-container">$e^{-\lambda z}e^{-\mu z}-\frac {\mu} {\lambda+\mu}e^{-(\lambda+\mu)z}$</span></p>
|
21,156 | <p>The title says it all, is there a way to get in contact which users who consistently post answers without using <span class="math-container">$\LaTeX$</span>? I've come across a user who does that and (as I had some free time) edited about 10-15 of his posts, some of his answers were barely readable; on each post I l... | Ron Gordon | 53,268 | <p>It works like this. People can post solutions to problems all they want without MathJax/LaTeX, which is the lingua franca here. I do not have to bother reading their solution. If the OP believes it unfair that his/her solution does not get the requisite attention, then they can continue to post as they always hav... |
1,250,703 | <p>I was given the following task: define a combinatorial problem to the following equation, and say how each side of the equation solves the given problem.
The equation is:
$$ n\binom{n}{r} -r\binom{n}{r}=(r+1)\binom{n}{r+1} $$
I tried to think of a problem that both sides solve, but couldn't think of any... I don't w... | ajotatxe | 132,456 | <p>How about this?</p>
<blockquote>
<p>From a set of $n$ elements, choose one of them, and then choose $r$ more.</p>
</blockquote>
<p>(The first selected objetc must be somehow different form the others).</p>
|
1,250,703 | <p>I was given the following task: define a combinatorial problem to the following equation, and say how each side of the equation solves the given problem.
The equation is:
$$ n\binom{n}{r} -r\binom{n}{r}=(r+1)\binom{n}{r+1} $$
I tried to think of a problem that both sides solve, but couldn't think of any... I don't w... | TravisJ | 212,738 | <p>If it helps, I usually think about these types of thing like selecting a set of people from a group and then giving some of them "special titles." Also, if it helps, notice that $n=\binom{n}{1}$. For example, if I had $\binom{n}{1}\binom{n}{r}$ I might say I'm going to select $r$ people to form a committee (from $... |
1,154,763 | <p>I'm given this equation:</p>
<p>$$
u(x,y) =
\begin{cases}
\dfrac{(x^3 - 3xy^2)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\
0\quad& \text{if} \quad (x,y)=(0,0).
\end{cases}
$$</p>
<p>It seems like L'hopitals rule has been used but I'm confused because</p>
<ol>
<li>there is no limit here it's just... | Fernando | 186,454 | <p>u(0,0) can't possibly exist as it would be a division by 0. You haven't given us enough information. What is the point of this equation? What is the problem you have to solve?</p>
|
1,720,053 | <p>The PDF describes the probability of a random variable to take on a given value:</p>
<p>$f(x)=P(X=x)$</p>
<p>My question is whether this value can become greater than $1$?</p>
<p>Quote from wikipedia:</p>
<p>"Unlike a probability, a probability density function can take on values greater than one; for example, t... | H. Potter | 289,192 | <p>Discrete and continuous random variables are not defined the same way. Human mind is used to have discrete random variables (example: for a fair coin, -1 if it the coin shows tail, +1 if it's head, we have that $f(-1)=f(1)=\frac12$ and $f(x)=0$ elsewhere). As long as the probabilities of the results of a discrete ra... |
979,267 | <p>Let $a_n$ be the $n$th sequence 1, 2 , 2 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 5 , 5 , 5 , 5 , 5, . . . . . . . constructed by including the integer $k$ exactly $k$ time. Show that $a_n$ $=$ $\lfloor \frac12 + (2n+\frac14)^.5 \rfloor$</p>
<p>Let $\lvert r\rvert < 1$ be a real number. Evaluate $\sum_{i=0}^\infty i... | Sawarnik | 93,616 | <p><strong>It is.</strong></p>
<p>We are given any triangle with heights $h_a, h_b, h_c$, and we assume its sides to be $a,b,c$. To prove it is the only such triangle, we note that by the area formulae:
$$a=\frac{2\triangle}{h_a}, b=\frac{2\triangle}{h_b}, c=\frac{2\triangle}{h_c}$$</p>
<p>We now use the fact that if... |
2,255,617 | <p>I am trying to learn how to do proofs by contradiction. The proof is,</p>
<p>"Prove by Contradiction that there are no positive real roots of $x^6 + 2x^3 +4x + 5$"</p>
<p>I understand that now I am attempting to prove that there is a positive real root of this equation, so I am able to contradict myself within the... | Martin Argerami | 22,857 | <p>If we add, to the question, the reasonable requirement that the homomorphism is nonzero, the answer is still no. </p>
<p>For instance consider $R=M_2 (\mathbb R) $, $R'=M_3 (\mathbb R) $. There are many nontrivial homomorphisms $R\to R'$: for any invertible $B $, $$A\longmapsto B\,\begin{bmatrix}A&0\\0&0\e... |
1,203,269 | <p>I am trying to compute the hitting time of a linear Brownian motion on a two-sided boundary. More specifically, let $W_t$ be a (one-dimensional) Wiener process. Let $T = \inf \{t: |W_t| = a \}$ for some $ a > 0$. I want to find $\mathbb{P}\{ T > t\}$. </p>
<p>I know that probability distribution hitting time ... | Math-fun | 195,344 | <p>I will give this a try.</p>
<p>For simplicity let $T_a=\inf \{t: |W_t| = a \}$</p>
<p>\begin{align}
Pr(|W(t)|>a)&=P(|W(t)|>a|T_a<t)Pr(T_a<t)+P(|W(t)|>a|T_a>t)Pr(T_a>t)\\
\end{align}</p>
<p>$P(|W(t)|>a|T_a>t)=0$ since the time that $|W(t)|$ hits $a$ for the first time has not arrived... |
1,285,443 | <blockquote>
<p>Let us denote solution to the equation</p>
<p>$$(x+a)^{x+a}=x^{x+2a}$$</p>
<p>with $X_a$.</p>
<p>($a$ is a non-zero real number)</p>
<p>Prove that:</p>
<p>$$\lim_ {a \to 0} X_a = e$$</p>
</blockquote>
<p>This is something that I noticed while making numerical experiments for ... | zoli | 203,663 | <p>Taking the logarithm of both sides of $$(x+a)^{x+a}=x^{x+2a}$$
we get </p>
<p>$$(x+a)\ln(x+a)=x\ln(x)+2a\ln(x)$$</p>
<p>or
$$\frac{(x+a)\ln(x+a)-x\ln(x)}{a}=2\ln(x). \tag 1$$
The left hand side tends to $\frac{d(x\ln(x))}{dx}=\ln(x)+1$ if $a$ tends to zero. The right hand side does not depend on $a$. That is,</p>... |
3,433,492 | <p>I know that a function can admitted multiple series representation (according to Eugene Catalan), but I wonder if there is a proof for the fact that each analytic function has only one unique Taylor series representation. I know that Taylor series are defined by derivatives of increasing order. A function has one an... | 11101 | 723,725 | <p>You can prove that a power series is differentiable on the interior of interval of convergence, with derivative is obtained by differentiating term by term. So, you can conclude that the coefficient of <span class="math-container">$x^n$</span> must be <span class="math-container">$\frac{f^{(n)}(0)}{n!}$</span>. So, ... |
69,508 | <p>I was just wondering, when I call the <code>CopulaDistribution</code> function in Mathematica, am I calling its cumulative function or its density function?</p>
<p>I have looked up the help and am still a little bit unsure.</p>
<p>EDIT: In particular, what does it mean when I take a RandomVariate from this CopulaD... | wolfies | 898 | <p>The question is of some interest because it captures rather nicely the difference between:</p>
<p>A. <strong>mathematical statistics</strong> ... where we work with characterisations of distributions, such as starting with a pdf, or cdf, or cf ... e.g. Let $X$ be a random variable with pdf $f(x)$:</p>
<p>$$f(x) =... |
120,687 | <p>Consider the following code</p>
<pre><code>styles = {Red, Blue, {Red, Dashed}, {Blue, Dashed}}
pt1 = Plot[{x^2, 2 x^2, 1/x^2, 2/x^2}, {x, 0, 3}, Frame -> True,
PlotStyle -> styles, PlotLegends -> {"1", "2", "1", "2"}]
</code></pre>
<p>I would like the two red lines to carry the same label "1" and the two... | Mr.Wizard | 121 | <p>I got around to actually evaluating your code and I realized that <code>g</code> is <em>not</em> remembering its values; <code>DownValues[g]</code> only has a length of three. The "solution" is to restrict the function to numeric values, per <a href="https://mathematica.stackexchange.com/q/9971/121">The difference... |
761,823 | <blockquote>
<p>Suppose that $G$ is a finite abelian group that does not contain a subgroup isomorphic to $\mathbb Z_p\oplus\mathbb Z_p$ for any prime $p$. Prove that $G$ is cyclic.</p>
</blockquote>
<p><strong>Attempt</strong>: If $G$ is a finite abelian group, then let $H$ be any subgroup of $G$</p>
<p>It's given... | Kevin Arlin | 31,228 | <p>Factor $|G|$ into primes as $\prod_{i=1}^n p_i^{k_i}$ where the $p_i$ are distinct. Proceed by induction on $n$: the base case is done assuming you know prime-order groups are cyclic. Then for the induction step factor out all instances take a generator $g_1$ of a subgroup of order $p_1^{k_1}$ and $g_2$ of a subgrou... |
62,790 | <p>Among several possible definitions of ordered pairs - see below - I find Kuratowski's the least compelling: its membership graph (2) has one node more than necessary (compared to (1)), it is not as "symmetric" as possible (compared to (3) and (4)), and it is not as "intuitive" as (4) - which captures the intuition, ... | Joel David Hamkins | 1,946 | <p>Of course there are many pairing functions, and they all
have the crucial property that from the pair $(x,y)$, one
can reconstruct both $x$ and $y$. And although your
question has been answered, let me point out that all four
of the ordered pair definitions that you consider have the
property that the von Neumann ra... |
3,772,923 | <p>My child's teacher raised a quesion in class for students who are interested to prove. The teacher says that the volume of a cube is the greatest among rectangular-faced shapes of the same perimeter and asks his students to prove this proposition.</p>
<p>I considered the relationship between the length of the sides ... | Mikael Helin | 418,258 | <p>Another solution to what mjw posted, this one without use of Lagrange multipliers is as following. Fix the "perimeter" <span class="math-container">$P$</span> such that <span class="math-container">$P=4(a+b+c)$</span> is constant then the volume is</p>
<p><span class="math-container">$$
V=ab(P/4-a-b)
$$</s... |
3,772,923 | <p>My child's teacher raised a quesion in class for students who are interested to prove. The teacher says that the volume of a cube is the greatest among rectangular-faced shapes of the same perimeter and asks his students to prove this proposition.</p>
<p>I considered the relationship between the length of the sides ... | Rezha Adrian Tanuharja | 751,970 | <p>Is elementary solutions permitted?</p>
<p><span class="math-container">$$
\frac{a+b+c }{3}\geq \sqrt[3]{abc}
$$</span></p>
<p>Equality i.e. maximum volume for a given sum of side lengths is when all sides are equal</p>
|
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| Matt E | 221 | <p>The prime number theorem states that the number of primes less than or equal to
$x$ is approximately equal to $\int_2^x \dfrac{dt}{\log t}.$ The Riemann hypothesis gives a precise answer to how good this approximation is; namely, it states that the difference between the exact number of primes below $x$, and the ... |
188,102 | <p>I have the following list: </p>
<pre><code>m={{14, "extinguisher"}, {54, "virgule"}, {55, "turnoff"}, {51,
"sofa"}, {77, "beachcomber"}, {61, "stoic"}, {6,
"isomorphism"}, {34, "leftist"}, {84, "spline"}, {42,
"heartiness"}, {35, "postnatal"}, {41, "stratified"}, {66,
"silkworm"}, {95, "conformance"}, {... | kglr | 125 | <p>Using the function <code>spiral</code> from <a href="https://mathematica.stackexchange.com/a/6869/125">this answer by Heike</a> to compute the centers of disks arranged on a spiral:</p>
<pre><code>sm = SortBy[m, -#[[1]] &];
radii = Normalize[sm[[All, 1]], Max] ;
centers = spiral[radii];
labels = sm[[All, 2]];
G... |
2,508,011 | <blockquote>
<p>find the <span class="math-container">$x$</span> :</p>
<p><span class="math-container">$$x^2(x-1)^2+x^2=8(x-1)^2$$</span></p>
</blockquote>
<hr />
<p>My Try :</p>
<p><span class="math-container">$$x^2(x-1)^2+x^2=8(x-1)^2\\ x^2(x^2-2x+1)+x^2=8(x^2-2x+1)\\x^4-2x^3+x^2+x^2=8x^2-16x+8\\x^4-2x^3-6x^2+16x-8=0... | A. Goodier | 466,850 | <p>Notice that $x=2$ is a solution. So
$$x^4-2x^3-6x^2+16x-8 =(x-2)(x^3-6x+4)$$
$x=2$ is also a solution of $x^3-6x+4=0$. So
$$x^4-2x^3-6x^2+16x-8 =(x-2)^2(x^2+2x-2)=0$$
The roots are $x=2,1\pm\sqrt{3}$.</p>
|
3,615,117 | <p>I want to find the intersection of the sphere <span class="math-container">$x^2+y^2+z^2 = 1$</span> and the plane <span class="math-container">$x+y+z=0$</span>. </p>
<p><span class="math-container">$z=-(x+y)$</span> that gives <span class="math-container">$x^2+y^2+xy= \frac 12$</span></p>
<p>How do I represent thi... | Z Ahmed | 671,540 | <p>As this quadrar=tic of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> has <span class="math-container">$xy$</span> term, it cannot represent a circle.
<span class="math-container">$$x^2+y^2-xy=1/2~~~(1)$$</span>
Next, write it as quadratic of <span class="math-container">$y$</spa... |
1,407,641 | <p>If $T$ is a linear transformation and is said to be one to one or onto- this only makes sense when we specify what domain and range is right?
$T: V \rightarrow V$ may not be onto or one to one
but $T: V \rightarrow Im(T)$ is certainly onto and may or may not be one to one.
Is this right?</p>
| EPS | 133,563 | <p>Perhaps this needs a bit more clarification:</p>
<ol>
<li>Your question is really about <strong>functions</strong> in general and not related to linear algebra.</li>
<li>Any function should be thought of as a triple $(f, X, Y)$ which is normally denoted by $f\colon X\to Y$. In other words, whenever you're talking a... |
2,040,041 | <p>I was able to think that the numerator will always be positive and will overpower the denominator as well. But couldn't proceed from there.</p>
| Robert Z | 299,698 | <p>Hint. Consider the power series expansion of $e^x=\sum_{k\geq 0}\frac{x^k}{k!}$. Then
$$e^x-\frac{2(e^x-(1+x))}{x^2}=\sum_{k\geq 0}\frac{x^k}{k!}-2\sum_{k\geq 2}\frac{x^{k-2}}{k!}=\sum_{k\geq 0}\frac{x^k}{k!}-2\sum_{k\geq 0}\frac{x^{k}}{(k+2)!}\\=\sum_{k\geq 1}\frac{x^k}{k!}\left(1-\frac{2}{(k+2)(k+1)}\right).$$
S... |
2,644,910 | <p>Ali Baba is trying to enter a cave. At the entrance, there is a drum with four openings, in each of which there is a pot with a herring inside. The herring may be lying with its tail up or down. Ali Baba can put his hands into any two
openings, feel the herrings, and put any one or both of them either tail up or tai... | Davide Gallo | 511,400 | <p>Got it. Will prove by induction. We just have to show that the result holds when $n=mp$. Let $b \in \mathbb{Z}_m^*$. </p>
<p>If $b \not\equiv 0 \pmod p$ then $(b,m)=1 \land (b,p)=1 \implies (b,n)=1 \implies b \in \mathbb{Z}_n^*$. Now we have $f(b)=b$. </p>
<p>If $b \equiv 0 \pmod p$ then $m \not\equiv 0 \pmod p \i... |
4,520,388 | <p>I'm stuck on this multivariable equation:</p>
<p><span class="math-container">$$
\frac{d}{dx}\left(\int^x_af(g(b,t),t)dt\right)
$$</span></p>
<p>where a and b are just constants.</p>
<p>If this involved a single variable, it looks like one would just apply the fundamental theorem of calculus. Is there an equivalent ... | Leonid | 679,193 | <p>This doesn't actually need multiple variables, and could be deduced from the single variable FTC. You also don't want to pass the limit inside the integral since the limit of integration depends on <span class="math-container">$x$</span> (which you're trying to take the limit of).</p>
<p>Recall that if you have a si... |
1,284,938 | <p>I was revising for one of my end of year maths exams, then I came across this example on how to find lines of tangents to ellipses outside the curve. Personally, I'd use differentiation and slopes to find such lines, but the lecturer does something simpler and more elegant.</p>
<p>The question is: "Find the equatio... | doraemonpaul | 30,938 | <p>Follow the method in <a href="http://en.wikipedia.org/wiki/Method_of_characteristics#Example" rel="nofollow">http://en.wikipedia.org/wiki/Method_of_characteristics#Example</a>:</p>
<p>$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$</p>
<p>$\dfrac{dy}{dt}=2e^x-y=2e^t-y$ , we have $y=e^t+y_0e^{-t}=e^x+y_0e^{-x... |
4,312,323 | <p>I get why <span class="math-container">$\sqrt{9} = \pm 3$</span>. But (at least I think) the ± is there because there's a certain ambiguity as to which number was squared to obtain <span class="math-container">$9$</span>.</p>
<p>Does that mean that if we remove the ambiguity <span class="math-container">$\sqrt{3^2} ... | hyper-neutrino | 457,091 | <p>Your line of thinking makes sense, but it's not exactly like that - it's not that we "don't know" which value was squared to get it; rather, both are answers.</p>
<hr />
<p>In most (almost all) contexts, <span class="math-container">$\sqrt n$</span> refers to <em>only</em> the positive value of the square ... |
4,092,877 | <p>I'm trying to find the solution for the following differential equation, however, I'm not sure how to derive the answer and so I would really appreciate some support!</p>
<p><span class="math-container">$y'' - y' = x^2$</span></p>
<p>I have tried splitting this into a quadratic polynomial: <span class="math-containe... | Lukas | 844,079 | <p>You are right with your expectation. Let's first solve the homogeneous equation
<span class="math-container">$y''-y'=0$</span>. It's characteristic polynomial is <span class="math-container">$x^2-x=0$</span> which has solutions <span class="math-container">$x_1=0, x_2=1$</span>. This means the solution for the homog... |
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | Paramanand Singh | 72,031 | <p>First let me focus on the reasons behind the difficulty in assimilating the $\epsilon, \delta$ definitions.</p>
<p>For any beginner in calculus, assimilating the $\epsilon, \delta$ definition is a challenge. I have rarely seen any student for whom this definition seems natural. I don't think anyone would dispute th... |
620,756 | <p>Let $R$ be a commutative ring with $1\not =0$, and let $D\ni 1$ be a multiplicative subset of $R$. Consider the universal characterization of $D^{-1}R$:</p>
<p>There is a morphism $\pi\colon R\to D^{-1}R$ such that for all rings and morphisms $\psi\colon R\to S$ satisfying</p>
<ul>
<li>$\psi(1)=1$</li>
<li>$\psi(D... | Louis | 75,278 | <p>I've been looking at this exercise for some minutes now, and I'm deeply confused. Please correct me if I am wrong (and I am sorry for that), I'm only writing as an "answer" here because this might be too long for a comment.</p>
<p>I don't think you can solve this question purely by using the universal property give... |
620,756 | <p>Let $R$ be a commutative ring with $1\not =0$, and let $D\ni 1$ be a multiplicative subset of $R$. Consider the universal characterization of $D^{-1}R$:</p>
<p>There is a morphism $\pi\colon R\to D^{-1}R$ such that for all rings and morphisms $\psi\colon R\to S$ satisfying</p>
<ul>
<li>$\psi(1)=1$</li>
<li>$\psi(D... | Community | -1 | <p>The universal "characterisation" you provided is just a property that is only a piece of the true universal property. You can think of it as saying $\phi:R\to D^{-1}R$ is initial; but with respect to what property? Just say that every map $R\to S$ that inverts $D$ factors through $R\to D^{-1}R$ is clearly not enough... |
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | Yes | 155,328 | <p>Let $l < 0$ be the limit of $(a_{n})$. Then there is no $n \geq 1$ such that $|l-a_{n}| < |l|$, a contradiction.</p>
|
256,322 | <p>Let $A$ be an abelian group of order $n = p_1^{\alpha_1} \cdot \ldots \cdot p_k^{\alpha_k}$ (i.e., $n$'s unique prime factorization). The Primary Decomposition Theorem states that $A \cong \mathbb{Z}_{p_1^{\alpha_1}} \times \ldots \times \mathbb{Z}_{p_k^{\alpha_k}}$. On the other hand, the Fundamental Theorem of F... | Alexander Gruber | 12,952 | <p>The invariant factors (the $\mathbb{Z}_{n_i}$ in your FTFGAG decomposition) are also uniquely determined up to isomorphism. The Chinese remainder theorem gives the equivalence of these statements. I think the problem you're having is that in the primary decomposition statement the $p_k$'s don't necessarily have to... |
256,322 | <p>Let $A$ be an abelian group of order $n = p_1^{\alpha_1} \cdot \ldots \cdot p_k^{\alpha_k}$ (i.e., $n$'s unique prime factorization). The Primary Decomposition Theorem states that $A \cong \mathbb{Z}_{p_1^{\alpha_1}} \times \ldots \times \mathbb{Z}_{p_k^{\alpha_k}}$. On the other hand, the Fundamental Theorem of F... | user1770201 | 46,072 | <p>Indeed, all of the comments indicating I was misquoting the Primary Decomposition Theorem were correct. </p>
<p>Pg. 161 of Dummit & Foote states the Primary Decomposition Theorem for finite abelian groups:</p>
<blockquote>
<p>Let $G$ be an abelian group of order $n > 1$ and let the unique factorization o... |
3,066,446 | <p>Let <span class="math-container">$\overline{X}$</span> be the average of a sample of <span class="math-container">$16$</span> independent normal random variables with mean <span class="math-container">$0$</span> and variance <span class="math-container">$1$</span>. Determine c such that
<span class="math-container"... | Mike_ | 632,850 | <p>If you draw a plot of <span class="math-container">$x^2\sin x$</span>, you will see it has no minimum or maximum at <span class="math-container">$x=0$</span>. Neither <span class="math-container">$x^{2n} \sin x$</span>. However, <span class="math-container">$x^{2n+1} \sin x$</span> reaches minimum at <span class="ma... |
171,690 | <p>I am trying to make a projection on the <em>xy-plane</em> of the intersection of the surfaces from the functions: <code>1 + x^2 - y^2</code>, <code>3 Log[1 + x^2]</code>.</p>
<p><a href="https://i.stack.imgur.com/XqC1g.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XqC1g.png" alt="Intersection o... | OkkesDulgerci | 23,291 | <pre><code>ContourPlot[1+x^2-y^2==3Log[1+x^2],{x,-1.5,1.5},{y,-1.5,1.5}]
</code></pre>
|
2,778,575 | <p>Given the equation: $\sin^2{x}+\cos{x}=0$</p>
<p>How is it solved?</p>
<p>I think: $\sin^2{x}=1-\cos^2{x}$, but even if I get a quadratic equation with one function (cos), how can I solve it?</p>
| giobrach | 332,594 | <p>You are on the right track. The equation
$$\sin^2 x + \cos x = 0 $$
becomes
$$-\cos^2 x + \cos x + 1 = 0 $$
with the substitution $\sin^2 x = 1 - \cos^2 x$.
At this point, you may solve the quadratic equation in $\cos x$ to find
$$\cos x = \frac{-1 \pm \sqrt{1 + 4}}{-2} = \frac{1 \mp\sqrt 5}{2},$$
that is, formally... |
3,543,150 | <p>My question : two indefinite integrals of a function being given , how to express one indefinite integral in terms of the other? </p>
<p><a href="https://i.stack.imgur.com/VkMzJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VkMzJ.png" alt="enter image description here"></a></p>
| David G. Stork | 210,401 | <p>The function is defined for positive and negative values of <span class="math-container">$x$</span>. The graph shows the real and imaginary parts:</p>
<p><a href="https://i.stack.imgur.com/meg59.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/meg59.png" alt="Mathematica plot"></a></p>
<p><span ... |
518,627 | <p>Proof that:$ \sum\limits_{n=1}^{p} \left\lfloor \frac{n(n+1)}{p} \right\rfloor= \frac{2p^2+3p+7}{6} $
<br>
where $p$ is a prime number such that $p \equiv 7 \mod{8}$
<br>
<br>I tried to separate the sum into parts but it does not seems to go anywhere. I also tried to make a substitutions for $p$ ,but, I don't thi... | Alexander Vlasev | 11,998 | <p>This is a partial answer. By the division algorithm let $n(n+1) = q_n p + r_n$ where $0\leq r_n < p$. Then we see that</p>
<p>$$\left\lfloor\frac{n(n+1)}{p}\right\rfloor =
\left\lfloor q_n + \frac{r_n}{p}\right\rfloor = q_n +
\left\lfloor\frac{r_n}{p}\right\rfloor = q_n$$</p>
<p>So the problem transforms into... |
1,693,045 | <p>I know if $x=e^{\frac{2\pi i}{17}}$ then $x^{17}=1$ and $\Re(x)=\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>But how do I form a polynomial which has root $\cos\left(\frac{2\pi}{17}\right)$.</p>
<p>I know you can consider de Moivre's theorem and expand the LHS using binomial theorem but that will take a long time.</p... | Wojowu | 127,263 | <p>By adding equalities
$$\cos(n+1)x=\cos nx\cos x-\sin nx\sin x\\
\cos(n-1)x=\cos nx\cos x+\sin nx\sin x$$
we get an equality
$$\cos(n+1)x+\cos(n-1)x=2\cos nx\cos x$$
If we now define, by induction, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials" rel="nofollow">Chebyshev polynomials</a> $T_0(y)=1,T_1(y)=... |
3,053,975 | <p><span class="math-container">$3^6-3^3 +1$</span> factors?, 37 and 19, but how to do it using factoring, <span class="math-container">$3^3(3^3-1)+1$</span>, can't somehow put the 1 inside </p>
| Mark Bennet | 2,906 | <p><span class="math-container">$x^2-x+1$</span> factorises as <span class="math-container">$(x-\omega)(x+\omega^2)$</span> where <span class="math-container">$\omega^3=-1$</span>.</p>
<p>Here <span class="math-container">$x=27$</span> and working modulo <span class="math-container">$27$</span> the cubes are <span cla... |
274,908 | <p>I would like to plot a molecule in 3D and use different colors for the same atom type in the molecule. For example, by using:</p>
<pre><code>MoleculePlot3D[Molecule["NC(=O)C[C@H](C(=O)O)N"], ColorRules -> {"C" -> Black}]
</code></pre>
<p>all C atoms become Black. But how can I make, for exa... | Domen | 75,628 | <p>Although I was suspecting Jason B. had an undocumented option up his sleeve, I will nevertheless post an alternative solution.</p>
<pre><code>colors = {Green, Orange, Pink, Yellow};
(* Generate regular plot *)
mol = MoleculePlot3D[Molecule["NC(=O)C[C@H](C(=O)O)N"]]
(* Extract atom indices of carbons *)
i... |
2,292,520 | <p>I know that the logical negation of $$\neg(a \rightarrow b)= a \wedge \neg b $$ I am not clear what that means in the following simple setting:</p>
<p>So its clear that $$x\geq 2 \to x^2\geq 4.$$ Now I can write the logical negation of $a\to b$ as $a \wedge \neg b$, but what does that intuitively mean? </p>
<p>Sup... | Atakan Büyükoğlu | 448,764 | <p>a is $x\geq 2$ and $\neg b$ is $x^2\lt 4$.</p>
<p>So, the intuitive meaning of $a \wedge \neg b$ is that both of these cannot happen at the same time, $x\geq 2$ with $x^2\lt 4$ have no common elements in their solution sets.</p>
<p>To prove $a \wedge \neg b$, you should show that all elements in the domain of x sa... |
2,710,703 | <p>Given any non abelian group, how can I prove that every proper subgroup may be abelian? I know the definition of "abelian," but I don't know the difference between a group and a subgroup, nor do I understand how the two interconnect.</p>
| N. S. | 9,176 | <p><strong>Hint</strong> If $H$ is a proper subgroup of $G$ then $|H|$ is a proper divisor of $|G|$.</p>
<p><strong>Hint 2</strong> If all the proper divisors of $|G|$ are prime, then all the proper subgroups of $G$ are cyclic. </p>
|
1,245,775 | <p>For example, if I have the fundamental solution set $\{x^2\}$, such that $y(x)=Cx^2$ is the solution to some unknown differential equation, is it guaranteed that only one such equation exists with this solution?</p>
<p>I know I can work backwards to show that this solution satisfies $\dfrac{dy}{dx}-\dfrac{2}{x}y=0$... | Archaick | 191,173 | <p>This generative model has some interesting properties. It has a fixed average degree with each node having degree at least one. It is also my intuition that it minimizes or comes very close to minimizing clustering for a fixed number of edges. I'm not familiar with any models which behave as the one you are describi... |
1,245,775 | <p>For example, if I have the fundamental solution set $\{x^2\}$, such that $y(x)=Cx^2$ is the solution to some unknown differential equation, is it guaranteed that only one such equation exists with this solution?</p>
<p>I know I can work backwards to show that this solution satisfies $\dfrac{dy}{dx}-\dfrac{2}{x}y=0$... | D Poole | 83,727 | <p>That random graph is denoted by $\mathbb{G}_{1-out}$. A common generalization is $\mathbb{G}_{k-out}$, where in step $j$, we choose $k$ vertices out of $V\setminus\{v_j\}$ and add the $k$ edges $\{v_j, \cdot\}$. Then at the end, delete multiple edges. </p>
<p>One place that you can read about this models is Alan Fr... |
3,200,354 | <p>How can I find the maximal value in the range <span class="math-container">$[-1,1]$</span> for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> of the following expression:</p>
<p><span class="math-container">$$\sin(\Pi x)(y-3)/2.$$</span></p>
<p>I tried doing the derivative of... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$B \subset \Phi^{-1}(B)$</span> is equivalent to <span class="math-container">$\Phi (B) \subset B$</span>. Both say the same thing: whenever <span class="math-container">$b \in B$</span> we also have <span class="math-container">$\Phi (b) \in B$</span>.</p>
|
878,961 | <p>I'm asked to prove a theorem (if that is the right word) about double derivatives. I'm still struggling with understanding Leibniz notation and I could use a push in the right direction. It's easy enough for me to differentiate the function when I write it down as $f(g(x))$ but not so much with Leibniz notation.</p>... | Avitus | 80,800 | <p>Let $y=f\circ g$ and $y(x)=f(g(x))$, with $u:=g(x)$. Then</p>
<p>$$\frac{dy}{dx}:=\frac{d(f\circ g)}{dx}=\frac{df}{du}\frac{du}{dx};$$</p>
<p>whenever we write $\frac{df}{du}$ we mean $\frac{df(u)}{du}$, i.e. $\frac{dy}{du}$. Introducing the function $h(u):=\frac{df(u)}{du}$ we arrive at</p>
<p>$$\frac{d^2y}{dx^2... |
3,554,891 | <p>Let's take a look back at this familiar "Law of cosines":</p>
<blockquote>
<p>Consider the triangle <span class="math-container">$\triangle ABC$</span>. Let <span class="math-container">$a = BC, b = AC, c = AB$</span>; <span class="math-container">$\angle A, \angle B, \angle C$</span> are the angles of the ... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$\vec{BC}=\vec{a},$</span> <span class="math-container">$\vec{CD}=\vec{b},$</span> <span class="math-container">$\vec{DA}=\vec{d}$</span> and <span class="math-container">$\vec{AB}=\vec{c}.$</span></p>
<p>Thus, since <span class="math-container">$$\vec{a}+\vec{c}=-\vec{b}-\vec{d},$$... |
3,554,891 | <p>Let's take a look back at this familiar "Law of cosines":</p>
<blockquote>
<p>Consider the triangle <span class="math-container">$\triangle ABC$</span>. Let <span class="math-container">$a = BC, b = AC, c = AB$</span>; <span class="math-container">$\angle A, \angle B, \angle C$</span> are the angles of the ... | mathlove | 78,967 | <p>Let us consider convex <span class="math-container">$n$</span>-gon <span class="math-container">$A_1A_2\cdots A_n$</span> where <span class="math-container">$\overline{A_jA_{j+1}}=a_j$</span> with <span class="math-container">$\angle{A_jA_{j+1}A_{j+2}}=\theta_j$</span>.</p>
<p>Now, let us put our <span class="math-... |
129 | <p>Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not sufficient. Are there any other special homotopical properties of manifolds?</p>
| Martin O | 86 | <p>In surgery theory (which is basically a whole field of mathematics which tries to answer questions as the above), the next obstruction to the existence of a manifold in the homotopy type is that every finite complex with Poincaré duality is the base space of a certain distinguished fibration (Spivak normal fibration... |
202,699 | <p><a href="https://i.stack.imgur.com/UqPw4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UqPw4.png" alt="enter image description here"></a></p>
<p>I try to solve for "t" at the various "x" from the function of </p>
<pre><code>f[t_, x_] =
0.5 Erfc[(x - 0.0236454911650369 t)/Sqrt[4*0.010827497... | user64494 | 7,152 | <p>How about the following?</p>
<pre><code>Plot3D[Re[SphericalHarmonicY[3, 1,\[Theta],\[Phi]]] /.
{\[Phi] -> ArcSin[x*y], \[Theta]->2*ArcTan[x*Sqrt[1 - (x/4)^2-(y/2)^2]/
2/(2*(1 - (x/4)^2 - (y/2)^2) - 1)]}, {x, -4, 4}, {y, -Sqrt[1 - (x/4)^2], Sqrt[1 - (x/4)^2]},
BoxRatios -> Automatic]
</code></pre>
<p><a h... |
1,441,624 | <p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span> and <span class="math-container">$c$</span> be elements of a group <span class="math-container">$G$</span>, how can I prove that <span class="math-container">$abc$</span> and <span class="math-container">$cba$</span> do not necessar... | Chappers | 221,811 | <p>Consider the quaternion group,
$$ H_{8} = \langle \pm 1,\pm i,\pm j, \pm k \mid i^2=j^2=k^2=ijk=-1 \rangle. $$
Then $ ijk = -1 $ has order $2$ (obviously, since $(-1)^2 = 1$), but $jik = -ijk = 1 $ has order $1$.</p>
<p>(To see this, note that $ij=(ijk)(-k)=k$, but $ji = -ji(ijk) = -k = -ji $.)</p>
|
1,102,638 | <p>Let $n\in \mathbb{N}$. Can someone help me prove this by induction:</p>
<p>$$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$</p>
| k170 | 161,538 | <p>Here are the steps
$$
\lim \limits_{x \to 0}\left[{\frac{\sqrt{1 + x + x^2} - 1}{x}}\right]
$$
$$
=\lim \limits_{x \to 0}\left[{\frac{\sqrt{1 + x + x^2} - 1}{x}}\right] \left[{\frac{\sqrt{1 + x + x^2} + 1}{\sqrt{1 + x + x^2} + 1}}\right]
$$
$$
=\lim \limits_{x \to 0}\left[{\frac{1 + x + x^2 - 1}{ x\left(\sqrt{1 + x ... |
2,042,428 | <p>If I'm correct, hidden induction is when we use something along the lines of "etc..." in a proof by induction. Are there any examples of when this would be appropriate (or when it's not appropriate but used anyway)?</p>
| Mathematician 42 | 155,917 | <p>Here is an example. Suppose that $A$ is a diagonalizable matrix, i.e. $A=P^{-1}DP$ where $D$ is some diagonal matrix. Then $A^k=P^{-1}D^kP$. Indeed, we have that $$A^k=(P^{-1}DP)^k=P^{-1}D(PP^{-1})D(PP^{-1})\dots (PP^{-1})DP=P^{-1}D^kP.$$ Here we actually used induction in the dots. There are many examples of this f... |
3,065,818 | <blockquote>
<p>If <span class="math-container">$$z=\dfrac{\sqrt{3}-i}{2}$$</span> then <span class="math-container">$$(z^{95}+i^{67})^{94}=z^n$$</span> then, <span class="math-container">$\text{find the smallest positive integral value of}$</span> <span class="math-container">$n$</span> <span class="math-container">... | Bill Dubuque | 242 | <p>This is a <span class="math-container">$\rm\color{#0a0}{multiplicative}$</span> form of the following well-known <span class="math-container">$\rm\color{#90f}{additive}$</span> result about reduced fractions. The proof follows immediately by translating from additive to multiplicative form, as below.</p>
<p><span cl... |
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