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 |
|---|---|---|---|---|
1,217,654 | <p>I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ added to the $e^{-s}$ is messing me up. The answer I got was $y(t)= \frac{1}{8} u_1(t)e^{-2t}$ but I'm not sure if th... | Daniel W. Farlow | 191,378 | <p>Consider the algebraic representation of $\mathrm{floor}(x)=\lfloor x\rfloor$ (usually the floor of $x$ is represented by $\lfloor x\rfloor$, not $[x]$):
$$
x-1<\color{blue}{\lfloor x\rfloor\leq x}.
$$
Hence, for arbitrary $a$ and $b$, we have
$$
\lfloor a\rfloor\leq a\tag{1}
$$
and
$$
\lfloor b\rfloor\leq b.\tag... |
1,217,654 | <p>I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ added to the $e^{-s}$ is messing me up. The answer I got was $y(t)= \frac{1}{8} u_1(t)e^{-2t}$ but I'm not sure if th... | Marco Cantarini | 171,547 | <p>Define $\left\{ a\right\} ,\left\{ b\right\}$
the fractional part of $a$
and $b$
and assume $\left\{ a\right\} +\left\{ b\right\} <1$
. Then$$\left\{ a+b\right\} =\left\{ \left[a\right]+\left[b\right]+\left\{ a\right\} +\left\{ b\right\} \right\} =\left\{ \left\{ a\right\} +\left\{ b\right\} \right\} =\le... |
245,083 | <p>I'm new to integral calculus, I started literally 15 minutes ago, and I need help with this question:</p>
<p>$$\int \dfrac{\ln(x)^2}{x} dx $$</p>
<p>My first step was:</p>
<p>$$\int \dfrac{1}{x}\ln(x)^2 dx $$</p>
<p>However, what to do next, how to solve this using the reverse chain rule? </p>
| DonAntonio | 31,254 | <p>Hint: directly</p>
<p>$$\int f(x)^nf'(x)dx=\frac{f(x)^{n+1}}{n+1}+K$$</p>
|
245,083 | <p>I'm new to integral calculus, I started literally 15 minutes ago, and I need help with this question:</p>
<p>$$\int \dfrac{\ln(x)^2}{x} dx $$</p>
<p>My first step was:</p>
<p>$$\int \dfrac{1}{x}\ln(x)^2 dx $$</p>
<p>However, what to do next, how to solve this using the reverse chain rule? </p>
| Jijbentlekker | 50,763 | <p><em>This is an attempt at answering my own question</em>:</p>
<p>$$\int \dfrac{\ln(x)^2}{x} dx $$</p>
<p>We can rewrite that as:</p>
<p>$$\int \dfrac{1}{x} . \ln(x)^2 dx $$</p>
<p>Let $u = \ln(x)$,
$\dfrac{du}{dx} = \dfrac{1}{x}$</p>
<p>$\dfrac{1}{x}.dx=du$ </p>
<p>We get $\int u^2 du = \dfrac{1}{3}u^3 + c = \... |
1,835,414 | <p>What is the fastest method to find which number is bigger manually?</p>
<p>$\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $</p>
| Claude Leibovici | 82,404 | <p>I would multiply by the conjugate of denominator $$a=\frac{3 \sqrt{3}-4}{7-2 \sqrt{3}}=\frac{1}{37} \left(13 \sqrt{3}-10\right)$$ $$b=\frac{3 \sqrt{3}-8}{1-2 \sqrt{3}}=\frac{1}{11} \left(13 \sqrt{3}-10\right)$$ this makes $a<b$.</p>
|
211,379 | <p>The numbers 7, 8, 9, apart from being part of a really lame math joke, also have a unique property. Consecutively, they are a prime number, followed by the cube of a prime, followed by the square of a prime. Firstly, does this occurence happen with any other triplet of consecutive numbers? More importantly, is there... | only | 28,849 | <p><a href="http://en.wikipedia.org/wiki/Catalan%27s_conjecture">Catalan's conjecture</a> implies that a square and a cube cannot be consecutive unless they are (0,1) or (8,9). So this is the only triplet satisfying this property.</p>
|
3,254,273 | <p>While I was studying Propositional Calculus from Elliott Mendelson's book of Introduction to Mathematical Logic, in the section of Formal Theory I came across a notation <span class="math-container">$\Gamma$</span> that represents a set of well-formed formulas (wfs) in a statement that is phrased as follows:</p>
<p... | hmakholm left over Monica | 14,366 | <p>As far as this concrete definition of "consequence" goes, <em>logical axioms</em> and members of <span class="math-container">$\Gamma$</span> indeed play the same role.</p>
<p>The reason for distinguishing between them is that there are <em>other contexts</em> where we're only interested in <span class="math-contai... |
3,724,383 | <p>I encountered this SAT type Math question and do not know how to progress.</p>
<p>Before a plum is dried to become a prune, it is 92% water. A prune is just 20% water. If only water is evaporated in the drying process, how many pounds of prunes can be made with 100 pounds of plums?</p>
<p>My attempt:</p>
<p>Since a ... | Dave | 334,366 | <p>The coefficient of <span class="math-container">$x^k$</span> in <span class="math-container">$\prod_{n=0}^\infty (1+x^{2^n})$</span> is the number of ways of writing <span class="math-container">$k$</span> as a sum of distinct powers of <span class="math-container">$2$</span>. There is a unique such way of writing a... |
1,139,377 | <p>I am trying to evaluate:
$$I = \int_{-\infty}^{\infty} \frac{\sin^2(x)}{x^2} \, dx.$$
Using a contour semi-circle (upper plane), I can get:
$$ \oint_{C} f(z) \,dz = \oint_{C} \frac{1 - e^{2iz}}{z^2} \, dz.$$
The whole issue is the $z^2$. I cannot use the residue theory, because it lies <em>on</em> the contour.</p>
... | Asier Calbet | 166,157 | <p>How about good old integration by parts?
$$ \int_{-\infty}^{+\infty} \frac{\sin^2(x)}{x^2} dx = \Big|_{-\infty}^{+\infty} -\frac{\sin^2(x)}{x} + \int_{-\infty}^{+\infty} \frac {2 \sin(x)\cos(x)}{x} dx = 0+ \int_{-\infty}^{+\infty} \frac{\sin(2x)}{x} dx = \int_{-\infty}^{+\infty} \frac{\sin(x)}{x} dx = \pi $$ where ... |
2,630,628 | <pre><code>2/(x+y) + 2/(y+z) + 2/(z+x) >= 9/(x+y+z)
</code></pre>
<p>I am using the following vectors:</p>
<p>(x+y y+z z+x) and (1 1 1)</p>
<p>I obtain something resembling the aforementioned inequality, but maybe my choice of vectors is totally wrong. Let me know if I should post a picture of my work!</p>
<p>Th... | Claude Leibovici | 82,404 | <p>In the same spirit as Keith McClary's answer, consider
$$\epsilon^{-1}x^3=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\tanh(x)=x-\frac{x^3}{3}+O\left(x^5\right)$$ making the non trivial root to be
$$x=\pm\frac{1}{\sqrt{\frac{1}{\epsilon }+\frac{1}{3}}}$$</p>
<p>For illustration purposes, let $\epsilon=10^{-k}$
$$\left(
\begin{a... |
49,633 | <p>I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has to choose (at least if one is not willing to deal with forms of odd parity). </p>
<p>While I was wondering how to ext... | Bugs Bunny | 5,301 | <p>I would say the orientation of $V$ is a $(k^*)^2$-orbit on $\det (V) \setminus \{0\}$. It generalizes to rings. I don't think it gives Hodge-star but I may be wrong.</p>
<p>If you need Hodge-star, you don't monkey around, choose a basis and define Hodge-star using this basis. Now your "orientation" is an equivalenc... |
759,760 | <p>How do I quantify this statement?</p>
<p>Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$
for some $n \in \mathbb{N}$.</p>
<p>When I am trying to prove this, I am led (in the course of proving the reverse implication) to an existential implication: </p>
<p>There exists an $n \in \mathb... | Lee Mosher | 26,501 | <p>You put "if" in the wrong place:</p>
<ul>
<li>There exists an n ∈ ℕ such that if 1≤x≤1+1/n then 1≤x≤2</li>
</ul>
<p>should be </p>
<ul>
<li>If there exists an n ∈ ℕ such that 1≤x≤1+1/n then 1≤x≤2.</li>
</ul>
|
66,738 | <p>I've used <code>TreeForm</code>, and I appreciate that the syntax is fairly short, especially in comparison with <code>TreeGraph</code>.</p>
<p>Is it possible specify a different colors in <code>TreePlot</code> for coloring a section of the nodes and the paths to those nodes?</p>
| Pegasus Roe | 37,497 | <pre><code>framedWithColor[color_] := Function[{position, label},
{Text[Framed[label, Background -> color], position]}
];
TreeForm[
{{1, 2}, {3, 4}},
VertexRenderingFunction -> framedWithColor[Pink]
]
</code></pre>
<p><a href="https://i.stack.imgur.com/CohAj.png" rel="nofollow noreferrer"><img src="https:... |
2,160,944 | <blockquote>
<p>Prove by induction:
$$\sin{x}+\sin{3x}+\dots+\sin{(2n-1)x}=\frac{\sin^2{nx}}{\sin{x}}$$</p>
</blockquote>
<p>I tried the problem using the normal rule of induction(the first principle), but I failed.I failed to make the form $\sin^2{(m+1)x}$. Somebody help me.</p>
| egreg | 62,967 | <p>You know that
$$
\sin y=\frac{e^{iy}-e^{-iy}}{2i}
$$
so you have
$$
\frac{\sin^2nx}{\sin x }+\sin(2n+1)x=
\left(\frac{e^{inx}-e^{-inx}}{2i}\right)^2\frac{2i}{e^{ix}-e^{-ix}}+
\frac{e^{i(2n+1)x}-e^{-i(2n+1)x}}{2i}
$$
This becomes
$$
\frac{e^{2inx}-2+e^{-2inx}+(e^{i(2n+1)x}-e^{-i(2n+1)x})(e^{ix}-e^{-ix})}
{2i(e^{ix}-e... |
742,577 | <p>It's been a while since I've had to do math and I've been stuck around a problem for two good hours. I hate asking questions but I can't figure it out. </p>
<p>I have the following problem: </p>
<p>Find the zero/domain of </p>
<p>$$ f(x) = \frac{9x^3-4x}{(x-3)(x^2-2x+1)} $$</p>
<p>So far, I've been able to find ... | Community | -1 | <p>Let $N$ be the number of dice thrown randomly over a square of area $R\times R$. The largest circle inscribed has area $\pi R^2/4$. Let $N=R\times R$. The number of darts that fall inside the circle is the area of the circle: $N_{in}=\pi R^2/4$, i.e.,
$$\pi=\dfrac{4N_{in}}{N}$$
<strong>Edit</strong> What I did seems... |
94,014 | <p>Is there any reasonable approach, essentially different from Wightman's axioms and Algebraic Quantum Field Theory, aimed at obtaining rigorous models for realistic Quantum Field Theories? (such as Quantum Electrodynamics).</p>
<p>EDIT: the reason for asking "essentially different" is the following. It is possible t... | Community | -1 | <p>There is a nice formulation of the geometry of QFT, available at:</p>
<p><a href="http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics-impa-v01-2011.pdf" rel="nofollow noreferrer">Towards the mathematics of quantum field theory (Jan 10 2011)</a>. URL originally posted.</p>
<p><a h... |
3,722,748 | <p>I would appreciate it if someone could help me with the following problem. I can not understand how a delta function <span class="math-container">$\delta(x)$</span> is integrated from zero to infinity. Because the integration interval should contain zero.</p>
| LL 3.14 | 731,946 | <p>So first the Dirac delta is not a function, it is a distribution of order <span class="math-container">$0$</span>, and so can be also interpreted as a bounded measure. As a measure, it is defined as acting on a set <span class="math-container">$A$</span> by
<span class="math-container">$$
\delta_0(A) = \left\{
\begi... |
2,694,390 | <p>I am looking for $x\in\mathbb{R}$ and $y\in\mathbb{R}$ such that:</p>
<ul>
<li>$x^2+y^2=1$</li>
<li>$x^2+xy$ is maximized.</li>
</ul>
<p>How can I find them?</p>
<p>Thank you!</p>
| Doug M | 317,162 | <p>$\begin{bmatrix}x&y\end{bmatrix} \begin{bmatrix} 1&\frac 12\\\frac 12&0\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}$</p>
<p>The egeinvalues of that matrix are $\frac {1+\sqrt{2}}2$ and $\frac {1-\sqrt{2}}2$</p>
<p>$\lambda_1 \|\mathbf x\|<\mathbf x^T A\mathbf x \le \lambda_2 \|\mathbf x\|$</p>
<p... |
2,694,390 | <p>I am looking for $x\in\mathbb{R}$ and $y\in\mathbb{R}$ such that:</p>
<ul>
<li>$x^2+y^2=1$</li>
<li>$x^2+xy$ is maximized.</li>
</ul>
<p>How can I find them?</p>
<p>Thank you!</p>
| Bumblebee | 156,886 | <p>Let $A=x^2+xy$ and $1=x^2+y^2.$ Since here we have at most quadratic terms, we can use the following completely ad hoc method: Lets determine $a\in\Bbb{R}$ such that $$aA+1=(a+1)x^2+axy+y^2$$ is a perfect square. For that we need the discriminant $\Delta=a^2-4(a+1)=(a-2)^2-8=0.$ Thus $a=2(1\pm\sqrt2).$ Then $2(1\pm\... |
6,292 | <p>I am trying to calculate the average distance between all vertices of a complete $k$-ary tree. A complete $k$-ary tree is a tree such that all vertices have $k$ children except for the leafs of the tree. </p>
<p>Suppose the tree has $r$ levels, where level 0 is the root. So the height of the tree is $r-1$. First i ... | Steven Stadnicki | 785 | <p>Here's another approach to trying to find the total distance: assume that we have some 'initial' node on level $m$, and we're trying to find the distance to the nodes at equal or lower levels. Start with the equal level: there are $(k-1)$ siblings, at distance 2; then $k(k-1)$ 'first cousins' (as each of our parent... |
2,137,807 | <blockquote>
<p>Two balls are drawn at random from a box containing ten balls numbered 0, 1, ..., 9. Let the
random variable X be the maximum of the two numbers drawn. <strong>Find</strong> the
probability function $f(x)$.</p>
</blockquote>
<p>This is <strong>hypergeometric</strong> isn't it?</p>
<p>So we have... | Brevan Ellefsen | 269,764 | <p>Tangent is periodic with period $\pi$, so we have
$$\tan(a)=\tan(b) \iff b=n\pi+a$$
In this case we find that $2x = n\pi + x \implies x=n\pi$</p>
|
547,151 | <p>I am attempting to show that if $R$ is commutative, local notherian ring with $J$ nilpotent then $R$ is Artinian.</p>
<p>Clearly as $\exists k$ such that $J^k=0$ then if there is no other ideal other than $R$ and $J^i$ then we are done but how do I show this?</p>
<p>That is how do I show that if $I$ is an ideal of... | cgonagu | 89,570 | <p>Just set $f(x) = 0$, that is,</p>
<p>$\tan(2x) - 1 = 0 \implies \tan(2x) = 1 \implies 2x = \arctan(1) \implies x = \frac{\arctan(1)}{2} \approx 0.3927$</p>
|
547,151 | <p>I am attempting to show that if $R$ is commutative, local notherian ring with $J$ nilpotent then $R$ is Artinian.</p>
<p>Clearly as $\exists k$ such that $J^k=0$ then if there is no other ideal other than $R$ and $J^i$ then we are done but how do I show this?</p>
<p>That is how do I show that if $I$ is an ideal of... | AlexR | 86,940 | <p>$\tan$ is a $\pi$-periodic function and $\tan \frac\pi 4 = 1$. Therefor
$$\begin{align*}
\tan(2x) & = 0 \\
\Leftrightarrow 2x & = \frac \pi 4 + k\pi & k\in\mathbb Z\\
\Leftrightarrow x & = \frac \pi 8 + k\frac\pi 2 & k\in\mathbb Z
\end{align*}$$
For $x\in[0,3]$ you get $k\in\{0, 1\}$ and thus
$$x... |
11,178 | <p>I have the trigonometric equation
\begin{equation*}
\sin^8 x + 2\cos^8 x -\dfrac{1}{2}\cos^2 2x + 4\sin^2 x= 0.
\end{equation*}
By putting $t = \cos 2x$, I have
\begin{equation*}
\dfrac{3}{16} t^4+ \dfrac{1}{4}t^3 + \dfrac{5}{8}t^2 -\dfrac{7}{4}t + \dfrac{35}{16} = 0.
\end{equation*}
How do I tell Mathematica to ... | TheDoctor | 2,263 | <p>First plot the trigonometric equation:</p>
<pre><code>Plot[Sin[x]^8 + 2 Cos[x]^8 - 1/2 Cos[2 x]^2 + 4 Sin[x]^2 == 0, {x, 0, 2 Pi}]
</code></pre>
<p>You will see that there are no (real) solutions. Is this what you expect?</p>
|
284,322 | <p>Let <span class="math-container">$f$</span> be a continuous and integrable function over <span class="math-container">$[a,b]$</span>. Prove or disprove that</p>
<p><span class="math-container">$$\int_a^b |f(x)|\ \mathrm{d}x\geq \left | \int_a^b f(x)\ \mathrm{d}x\right|
$$</span></p>
| User 17670 | 40,125 | <p>Think about what your inequality is saying:</p>
<ol>
<li>Imagine coordinates $y$ and $x$.</li>
<li>Imagine some arbitrary curve on that plane $f(x)$.</li>
<li>Imagine integrating $f(x)$ by taking the area under the $x$-axis away from the area above the $x$-axis.</li>
<li>Imagine repeating the integration for $\vert... |
1,815,344 | <p>We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all roots of $g(x)$. Let $F=\{a\in E:a^q=a\}$.</p>
<p>I understand that $F$ is closed under addition (since $E$ has cha... | Qingzhong Liang | 340,337 | <p>When $q=2^n$, the characteristic is $2$. So, $2=0\Rightarrow 1=-1$. Hence $(-a)^q=a^q=a=-a$.</p>
<p>In general, if $a,b\in F$, then $(ab)^{p^n}=a^{p^n}b^{p^n}=a\cdot b\Rightarrow ab\in F$. Also, $(a+b)^{p^n}=\sum_{k=0}^{p^n}\binom{p^n}{k}a^{p^n-k}b^k=a^{p^n}+b^{p^n}=a+b$, since $p|\binom{p^n}{k}$ for $1\leqslant k\... |
117,664 | <p>I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$:
$$ u_t = (\ln u)_{xx}$$
which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions
$$ u(a,t) = u(b,t) $$
$$ u_x(a,t) = u_x(b,t) $$
for $t \geq 0$ and an ... | guacho | 33,135 | <p>Maybe this book is a good choice: <a href="http://ukcatalogue.oup.com/product/9780198569039.do#.UaC1zPER0uU" rel="nofollow">http://ukcatalogue.oup.com/product/9780198569039.do#.UaC1zPER0uU</a></p>
<p>In the web of the author (<a href="http://www.uam.es/personal_pdi/ciencias/jvazquez/coursejlv.html" rel="nofollow">h... |
3,657,887 | <p>I have been doing some practice questions for an upcoming Maths Challenge. There's one question I can't seem to grasp. I'm not sure entirely sure where to start. I don't know how to approach this one. Any help would be appreciated</p>
<p><span class="math-container">$n$</span> in the form <span class="math-containe... | antimeme | 566,725 | <p>There are many ways to do this, but my preference is to use Euler's solution to the Basel problem (<a href="https://en.wikipedia.org/wiki/Basel_problem" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Basel_problem</a>). Here's how I would define the lower set of the Dedekind cut:</p>
<p><span class="math-c... |
3,281,820 | <p>Let <span class="math-container">$X,Y$</span> be random variables <span class="math-container">$\Omega \to \mathbb{R}$</span> (where <span class="math-container">$(\Omega, \mathcal{F})$</span> is a measurable space). Let further <span class="math-container">$f,g:\Omega \to \mathbb{R}$</span> be two functions such th... | user3257842 | 365,433 | <ol>
<li><span class="math-container">$¬a \implies 1 - a$</span></li>
<li><span class="math-container">$a \wedge b \implies ab$</span></li>
<li><span class="math-container">$a \vee b \implies a + b - ab$</span></li>
<li><span class="math-container">$a \rightarrow b \implies ab - a + 1$</span> </li>
<li><span class=... |
225,182 | <p>Is the following true? I cannot see a counterexample and it seems very intuitively clear, at least in the embedded case.</p>
<p>Claim:
Consider the set $S$ of closed immersed Riemann surfaces $\Sigma \subset (X,g)$ (I am particularly interested in spheres), with the magnitude of the mean curvature bounded from abov... | Rbega | 26,801 | <p>Assuming I am interpreting your question correctly, this is true when the ambient space is Euclidean and the submanifold is closed (i.e. compact and without boundary). To see this one may invoke a result of Topping that bounds the (intrinsic) diameter of a closed, immersed submanifold of dimension $m$ by the $L^{m-... |
2,752,558 | <p>I'm trying to model a problem in GLPK but it turned out to be non linear.</p>
<p>A simplified version of the model is written below. Basically it is a weighted average of a set of features of all enabled points substracting a cost associated to enabling those points, provided there are exactly P enabled points.</p>... | Erwin Kalvelagen | 295,867 | <p>This can be linearized but with some effort. The ratio
$$y=\frac{\sum_i f_i w_i x_i}{\sum_i w_i x_i}$$ with $x_i \in \{0,1\}$
can be written as a (nonlinear) constraint:
$$ y \sum_i w_i x_i = \sum_i f_i w_i x_i$$ where $y$ is an additional continuous variable.
The non-linear expression $(y x_i)$ is a continuous var... |
25,163 | <p>You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually <span class="math-container">$\log$</span> or a trig function, and you watch them write this:
<span class="math-container">$$
\log x = 42 \qquad\text{so}\qqu... | Chris Cunningham | 11 | <p>Ask the student to critique this work:</p>
<blockquote>
<p>Solve for <span class="math-container">$x$</span>: <span class="math-container">$\sqrt{x} = 3$</span></p>
<p>Easy: <span class="math-container">$x = \frac{3}{\sqrt{\phantom{x}}}$</span>.</p>
</blockquote>
<p>I have tried this a small number of times, and it ... |
145,255 | <p>I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing to find the answer to the following embarrassingly easy (I think) question.</p>
<blockquote>
<p>What is the area o... | Dan Asimov | 55,515 | <p>As an equilateral spherical triangle gets arbitrarily small, its angles all approach π/3. So one might say there is a degenerate spherical triangle whose angles are in fact all π/3 and whose area is 0.</p>
|
24,608 | <p>Hi. Are there nice/simple examples of cyclic extensions $L/K$ (that is, Galois extensions with cyclic Galois group) for which $L$ cannot be written as $K(a)$ with $a^n\in K$?</p>
<p>Thanks.</p>
| David Loeffler | 2,481 | <p>An example is given by any cubic Galois extension of $\mathbb{Q}$, e.g. the extension $\mathbb{Q}(\alpha)$ where $\alpha = \cos \frac{2\pi}{9}$ is a root of $x^3 - 3x + 1$. </p>
<p>This works because if the extension were of the form $\mathbb{Q}(\beta)$ with $\beta^3 \in \mathbb{Q}$, then since it is Galois it woul... |
3,597,611 | <p>I know that <span class="math-container">$\{a_i\}=R(pq)$</span>, and the title is step <span class="math-container">$(b)$</span>, here is step <span class="math-container">$(a)$</span> (maybe a hint?): </p>
<p>Let <span class="math-container">$p$</span> and <span class="math-container">$q$</span> be two distinct... | Bill Dubuque | 242 | <p>As <a href="https://math.stackexchange.com/a/2372358/242">explained here</a>, by pairing up inverses the product reduces to the product of all self-inverse <span class="math-container">$a_i$</span> (roots of <span class="math-container">$\,x^2\equiv 1).\, $</span> <a href="https://math.stackexchange.com/a/2041335/24... |
2,207,862 | <p>Reading a paper, I came across the statement that there are no transversely orientable codimension-one foliations on even dimensional spheres $\mathbb{S}^{2n}$. Could anyone explain why?</p>
| David | 342,780 | <p>To address your last question directly, if you continue to add (strictly positive) numbers together, they can converge, but the terms need to go to zero sufficiently fast. Two very standard examples are $$\sum_{n=1}^\infty \frac{1}{n} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots = \infty,$$ but $$\sum_{n=1}^\infty \... |
1,974,310 | <p>How can I prove that for all $n\in\mathbf{N}$ that $6 | n^5 + 5n$?</p>
<p>I tested for $n = 2$ and got $6 | 32 + 10 = 42$. </p>
| fleablood | 280,126 | <p>1) Show that $2|n^5+5n$.</p>
<p>2) Show that $3|n^5 + 5n$.</p>
<p>If we can show both, separately, it has to follow that $6=2*3|n^5 + 5n$.</p>
<p>1)$n^5 + 5n = n(n^4+5)$. If $n$ is even so is $n(n^4+5)$. If $n$ is odd then $n^5$ is odd and $n^5 + 5$ is even and $n(n^4+5)$ is even.</p>
<p>So $n^5 + 5n$ is even.... |
4,521,068 | <p>If I write sequence <span class="math-container">$2^n$</span> exponential sequence, "1,2,4,8,16,..." until I reach <span class="math-container">$2^{1000}$</span>, how many digit number I have to write ?</p>
<p>I try using the method that work for linear sequence by writing some first sequence and find the ... | user2661923 | 464,411 | <p>Addendum added to give an exact answer.</p>
<hr />
<p>Let <span class="math-container">$\displaystyle M = \log_{10} (2) \approx 0.301029995.$</span> <br>
This implies that for any non-negative integer <span class="math-container">$k$</span>, <br>
<span class="math-container">$\displaystyle \log_{10}\left(2^k\right) ... |
104,733 | <p>The most straight forward way for me to think about is like this:</p>
<pre><code>H[N_] := RandomReal[{0,10},{N,N}]
symmetryH[N_]:=(H[N]+H[N]\[Transpose])/2
symmetryH[4]//MatrixForm
</code></pre>
<p>However, the above code does not work because the second call of <code>H[N]</code> will generate a different matrix f... | an offer can't refuse | 7,414 | <p>One can use the <code>Module</code> to declare a local variable:</p>
<pre><code>H[N_] := RandomReal[{0,10},{N,N}]
symH[N_]:=Module[{h=H[N]},(h+h\[Transpose])/2];
symH[4]//MatrixForm
</code></pre>
<p>This should work.</p>
|
1,103,702 | <p>Let $ V$ be a n-dimensional space and T a linear operator diagonalizable on $V$. If $c_{1}, ..., c_{k}$ are characteristics values of vector space V, $W_{i}$ are the characteristic space associated the characteristic value $c_{i}$ and $f(x)=(x-c_{1})^{d_{1}}...(x-c_{k})^{d_{k}}$ is the polynomial characteristic of $... | Community | -1 | <p>We have </p>
<p>$$W_i=\ker((T-c_i\operatorname{id})^{d_i})$$
and by the primary decomposition theorem</p>
<p>$$V=\oplus_{i=1}^k W_i$$
and let $\alpha_i=\dim W_i$ so the restriction $T_i$ of $T$ to $W_i$ is trigonalizable and we see easily that $$\chi_{T_i}(x)=(x-c_i)^{\alpha_i}$$</p>
<p>so since </p>
<p>$$f(x)=\... |
1,103,702 | <p>Let $ V$ be a n-dimensional space and T a linear operator diagonalizable on $V$. If $c_{1}, ..., c_{k}$ are characteristics values of vector space V, $W_{i}$ are the characteristic space associated the characteristic value $c_{i}$ and $f(x)=(x-c_{1})^{d_{1}}...(x-c_{k})^{d_{k}}$ is the polynomial characteristic of $... | Marc van Leeuwen | 18,880 | <p>This is obvious, considering a basis$~B$ of eigenvectors which is assumed to exist: $W_i$ is spanned by those vectors of$~B$ with eigenvalue$~c_i$, and since the corresponding entries of the diagonal matrix of$~T$ with respect to the basis$~B$ are all $c_i$, one gets a factor $(X-c_i)^{\dim W_i}$ in the characterist... |
1,625,858 | <p>I have this question in my notebook.A Drunk person performs a random walk over positions $0,\pm1,\pm2,\dots$ He starts at 0, he takes successive 1 unit steps going to the right with probability p and to the left with probability $1 - p$, his steps are independent let $X$ denote his position after $n$ steps. Find the... | YoTengoUnLCD | 193,752 | <p>Hint</p>
<p>$$\frac {e^z+e^{-z}}{2}=0 \iff e^{2z}=-1$$</p>
<p>Added</p>
<p>Let $z=x+iy$. Let $w:=e^{2z}$ then $w=e^{2z}=e^{2x+2yi}=e^{2x}(\cos2y+i\sin2y)$.</p>
<p>Now, just compare radius and argument with $-1$.</p>
<p>We have $|w|=e^{2x}=1=|-1|$ so $x=0$.</p>
<p>Lastly,
\begin{align}
\arg(w)&=2y\\ \arg(-... |
1,650,131 | <p>Suppose we are given a finitely generated free module $M$ over a ring $R$. Assume $N \subseteq M$ is a free submodule of $M$.</p>
<p>If $B$ is a basis for $M$, does it follow that there exists a subset $A \subseteq B$ such that $A$ is a basis for $N$?</p>
<p>As stated, I'm pretty sure the question is incorrect: Ju... | Eric Wofsey | 86,856 | <p>This never holds if $R$ is a nonzero ring and $B$ has more than one element: just take any two distinct elements $a,b\in B$, and let $N$ be the submodule generated by $a+b$.</p>
|
2,246,271 | <p><a href="https://i.stack.imgur.com/ylBiy.png" rel="nofollow noreferrer">I have this picture</a>, and I have to explain why the lower part of the sphere corresponds to the representative surface of a function. I don't manage to understand the goal of the question, anybody can help me?</p>
<p>A second question is to ... | A.Γ. | 253,273 | <p>I guess, the second question is meant to be a straightforward calculation of the integral
$$
\iint_{x^2+y^2\le R^2}\left(R-\sqrt{R^2-x^2-y^2}\right)\,dxdy.
$$
(It has nothing to do with the surface of the sphere, but is the volume between the hemisphere and the $xy$-plane.) In the polar coordinates
$$
\begin{cases}... |
2,566,633 | <p>We say that the sequense $\{T_\alpha\}_{\alpha<\lambda}\subseteq[\omega]^\omega$ (where $\lambda$ is a ordinal) is a tower iff</p>
<ul>
<li>$\alpha<\beta<\lambda\rightarrow T_\beta\subseteq^*T_\alpha$.</li>
<li>$\neg\exists K\in[\omega]^\omega\forall\alpha<\lambda(K\subseteq^*T_\alpha)$.</li>
</ul>
<p>... | Stefan Mesken | 217,623 | <p>In order for $\mathfrak{t}$ to be well-defined, you only need to see that a tower exists. Consider, for sequences $(T_{\alpha} \mid \alpha < \lambda)$, $T_\alpha \subseteq [\omega]^{\omega}$, the property</p>
<p>$$
\alpha < \beta < \lambda \implies T_{\beta} \subseteq^* T_{\alpha} \wedge T_\beta \neq T_\al... |
3,626,214 | <p>Is there any simple way to know how to reverse a percentage?</p>
<p>For example if I have 100 and it goes down by 10% I end up with 90. If I then add to it by 10% I end up with 99, not the 100 that you would think of. Is there a simple trick to quickly work out the reverse of a percentage change (even if it only wo... | user159888 | 159,888 | <p><span class="math-container">$A\subset Y$</span> by assumption and <span class="math-container">$A\subset \overline{A}$</span> by definition implies the result.</p>
|
3,429,338 | <p>Please help. I've missed some lectures, and now I'm stuck (my fault!). The lectures notes don't explain elaborately, and I can't find good tutorials online. I've somehow managed to arrive at <span class="math-container">$(Q \lor P) \land P$</span>. If this is correct, can this be simplified further? Thanks heaps.</p... | Angela Pretorius | 15,624 | <p>This is a birds-eye view of a torus with <span class="math-container">$K_6$</span> embedded on it's surface. You can only see the three vertices on the top of the torus, but I'm sure you can imagine how they connect to the three vertices on the bottom of the torus.
<a href="https://i.stack.imgur.com/zF2V9.png" rel="... |
1,444,652 | <p>If A$,B$ and $C$ are random events in a sample space and if $A,
B$ and $C$ are pairwise independent and $A$ is independent of $(B \cup C)$, then is it true that $A,B$ and $C$ are mutually independent.</p>
<p><strong>My Attempt :</strong> (with questions of this type at my college the answer is usually in the affirm... | Samrat Mukhopadhyay | 83,973 | <p>$$P(A\cap (B\cup C))=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)\\\implies P(A)(P(B)+P(C)-P(B\cap C))=P(A)(P(B)+P(C))-P(A\cap B\cap C)\\\implies P(A)P(B)P(C)=P(A\cap B\cap C)$$</p>
|
224,559 | <p>Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as <strong>relative information</strong>, <strong>relative entropy</strong>, <strong>information gain</strong> or <strong>Kullback–Leibler divergence</strong> is defined to be </p>
<p>$$ D_{KL}(p\|q) = \sum_{i... | Steve Huntsman | 1,847 | <p>The relative entropy is frequently framed as the inverse temperature times the difference between the Helmholtz and variational/Bethe free energies, the latter of which is minimized as a proxy objective function in mean field theory and Bayesian estimation via the well-known expectation maximization algorithm. <a hr... |
224,559 | <p>Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as <strong>relative information</strong>, <strong>relative entropy</strong>, <strong>information gain</strong> or <strong>Kullback–Leibler divergence</strong> is defined to be </p>
<p>$$ D_{KL}(p\|q) = \sum_{i... | Tom Leinster | 586 | <p>I don't have Hobson's book, but I do have a paper by Hobson in which he proves the theorem that Wikipedia is presumably referring to:</p>
<blockquote>
<p>Arthur Hobson, A new theorem in information theory. <em>Journal of Statistical Physics</em> 1 (1969), 383-391.</p>
</blockquote>
<p>(What a title. Long gone ... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Andrew Critch | 84,526 | <p>In short, I'd tell your friend: <em>"If you believe a ring can be understood geometrically as functions its spectrum, then modules help you by providing more functions with which to measure and characterize its spectrum."</em></p>
<p>Elements of a module over a ring $R$ are like generalized functions on $Spec(R)$.... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Orbicular | 1,734 | <p>I always thought one should regard issues like
ring theory/module theory or theory of (abstract) groups/ representation theory of groups
in an analogous manner to theory of abstract manifolds/embeddings of manifolds. So you can disentangle "mixed" notions and work out the concepts more clearly. It's not like embeddi... |
1,070,699 | <p>Consider the matrix</p>
<p>$$
A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\
\frac{-5}{8\sqrt{3}} & 11/8
\end{bmatrix}
$$
which of the following transformations of the coordinate axis will make the matrix $A$ diagonal?</p>
<p>Rotation in $-60$ counter clock wise, $-30$ ccw, $30$ ccw, $60... | Community | -1 | <p><strong>Hint:</strong> We want <span class="math-container">$\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}A\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$</span> to be diagonal. So we need the <span class="math-container">$(1,2)$</span> and <sp... |
155,365 | <p>It is well-known that for any presheaf $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Set}$, the category of elements (obtained by the so-called Grothendieck construction) of $F$ is a comma category $y/\ulcorner F \urcorner$ in the category $\mathrm{CAT}$ of </p>
<p>$$\mathcal{C} \xrightarrow{y} \widehat{\mathcal{... | Adam Gal | 4,477 | <p>Possibly the comma 2-category gives what you want.</p>
<p>If you think of $C^{op}\rightarrow Gpd$ as a functor of 2-categories (e.g. pass to the nerve for some model of $(\infty,2)$ categories) and then take the comma object
$$
\matrix{(r\downarrow F)&\rightarrow& C \\
\downarrow & & \downarrow _{r... |
431,783 | <p>I am taking an <a href="https://class.coursera.org/calcsing-002/class/index" rel="noreferrer">online course</a> and we are currently learning Integration and this is my first time experiencing intergration, though I have some knowledge of it. I am having some difficulty understanding what a differential equation act... | Robert Mastragostino | 28,869 | <p>The important thing to note is that the unknown in a differential equation is the <em>function</em>, not some particular value of the variable. You're defining a relationship that some collection of functions should satisfy and then finding out what those functions are.</p>
<p>$$\frac{dx}{dt}=f(t)$$</p>
<p>is sayi... |
708,694 | <p>Find values for $a$ and $b$ so that $z=a+bi$ satisfies $\displaystyle \frac{z+i}{z+2}=i$. Below are my workings:</p>
<p>so far i simplify $\displaystyle \frac{z+i}{z+2}=i$ to $z=zi+i$ </p>
<p>which $a=i$, $b=z$ </p>
| DanielWainfleet | 254,665 | <p>Let $A$ be a dense $G_{\delta}$ set. Let $A=\cap_{n\in \Bbb N}U_n$ where each $U_n$ is open. Each $U_n$ is dense because $U_n\supset A$ and $A$ is dense. </p>
<p>Let $B=\{b_n:n\in \Bbb N\}$ be any non-empty countable subset of $\Bbb R.$ (It does not matter whether $b_m=b_n$ for some distinct $m,n.$) Let $V_n=U_n\s... |
773,601 | <p>Can you suggest to me an example of two non-abelian subgroups of a group which are conjugate?</p>
<p>And are all abelian subgroups are self conjugate?</p>
<p>I mean if $H$ is an abelian subgroup of a group $G$, then does $g^{-1} H g = H$ hold for all $g\in G$?</p>
| Thomas | 26,188 | <p>I am not 100% sure that I understand what you are looking for. </p>
<p>First: If $H$ is a normal subgroup of $G$, then $g^{-1}Hg = H$ for all $g\in G$. So for an example you can pick your favorite normal subgroup of a non-abelian group. Concretely, you could consider the <a href="http://en.wikipedia.org/wiki/Symmet... |
773,601 | <p>Can you suggest to me an example of two non-abelian subgroups of a group which are conjugate?</p>
<p>And are all abelian subgroups are self conjugate?</p>
<p>I mean if $H$ is an abelian subgroup of a group $G$, then does $g^{-1} H g = H$ hold for all $g\in G$?</p>
| WWK | 115,619 | <p>Example:</p>
<p>Let $G=S_3$, $H_1=\{(1),(12)\}$, $g_1=(13)$, $g_2=(23)$, then
$$g_1^{-1}H_1g_1= \{(1),(23)\} \neq H_1$$
$$g_2^{-1}H_1g_2= \{(1),(13)\} \neq H_1$$</p>
|
4,379,228 | <p>I came across an example of an integral by change of variable that reads as follows</p>
<p>Find <span class="math-container">$$\int_a^{b} \frac{1}{\sqrt[2]{(x-a)(b-x)}}dx$$</span></p>
<p>What they do is to assume that <span class="math-container">$a<b$</span>. They claim that the convenient change of variable con... | mio | 995,003 | <p>Here provides a new aproach.</p>
<p>Note the identity<span class="math-container">$$\dfrac{x-a}{b-a}+\dfrac{b-x}{b-a}=1$$</span>which inspires us to do this change of variable:<span class="math-container">$$\dfrac{x-a}{b-a}=\sin^{2}t,\ t\in(0,\dfrac{\pi}{2}).$$</span>Now,<span class="math-container">$\dfrac{b-x}{b-a... |
191,738 | <p>I have the following limit:</p>
<p>$$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$</p>
<p>where $\alpha>0$.</p>
<p>Evaluating this in Mathematica suggests that this converges, but I don't know how to evaluate it. Any he... | BaronVT | 39,526 | <p>Maybe start by doing something along the lines of noting that</p>
<p>$$\sum_{m=0}^{\infty}\frac{(\alpha\sqrt{n})^m}{m!} = e^{\alpha \sqrt{n}}$$</p>
<p>so that the final sum is</p>
<p>$$e^{\alpha \sqrt{n}} - \sum_{m=n-k}^{\infty}\frac{(\alpha\sqrt{n})^m}{m!}$$</p>
|
4,245,883 | <p>Here's a problem from my probability textbook:</p>
<blockquote>
<p>Of three independent events the chance that the first <em>only</em> should happen is <em>a</em>; the chance of the second <em>only</em> is <span class="math-container">$b$</span>; the chance of the third <em>only</em> is <span class="math-container">... | true blue anil | 22,388 | <p>I have used a different method for confirmation, converting a straight line to a <span class="math-container">$2D$</span> lattice path.</p>
<ul>
<li><p>Number of tosses taken has to be of the form <span class="math-container">$(2k+1)$</span>, i.e. <em>odd</em></p>
</li>
<li><p><span class="math-container">$k$</span... |
779,585 | <p>A spinner has 4 sectors of area 10%, 20%, 30% and 40%.</p>
<p>What is the expected # of spins for the spinner to stop on each sector at least once ?</p>
<p><strong>edit</strong></p>
<p>If areas were equal, it'd be 4/4 +4/3 +4/2 +4/1 = 8.33</p>
<p>and with the unequal probabilities above, it'd be > 10 as pointed ... | Daniel R | 83,553 | <p>Analytic continuation of the zeta function to $\mathrm{Re}(s) > 0$ gives (see (16)-(20) <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" rel="nofollow">here</a>)</p>
<p>$$\zeta(s)=\frac1{1-2^{1-s}}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}$$</p>
<p>The sum is the Dirichlet $\eta$ function, which ca... |
868,582 | <p>Suppose that $u$ and $w$ are defined as follows: </p>
<p>$u(x) = x^2 + 9$</p>
<p>$w(x) = \sqrt{x + 8}$</p>
<p>What is: </p>
<p>$(u \circ w)(8) = $</p>
<p>$(w \circ u)(8) = $</p>
<p>I missed this in math class. Any help?</p>
| Ivo Terek | 118,056 | <p>Look at the expression for $u(x)$. When we write $u(w(8))$, it means to use $w(8)$ everywhere that had "$x$" in the expression for $u(x)$. However, $w(8)$ is just a number, you can calculate separately using the expression for $w(x)$. And finally, $u(w(8))$ and $w(u(8))$ need <strong>not</strong> be equal. </p>
|
4,088,277 | <blockquote>
<p>The set <span class="math-container">$\{\frac{1}{x^2-3}: x\in\mathbb{Q}\}$</span> is bounded? Explain.</p>
</blockquote>
<p>I got this question some weeks ago in an <strong>Introduction to Real Analysis</strong> exam. In the exam review, the professor mentioned it could be solved with <em>supremum and i... | lhf | 589 | <p><em>Hint:</em> <a href="https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method" rel="nofollow noreferrer">Heron's method</a> starting at <span class="math-container">$x_0=1$</span> produces a sequence of rationals converging to <span class="math-container">$\sqrt3$</span>.</p>
|
11,149 | <p>I could sample a set of m elements from the uniform distribution over a universe $U$ of n >> m elements. Alternately, I could select a random probability distribution $\mathcal{D}$, and sample $m$ elements from $\mathcal{D}$. </p>
<p>EDIT (per Michael Lugo): When I say "select a random probability distribution", I ... | Pete L. Clark | 1,149 | <p>From a MathSciNet search:</p>
<hr>
<p>Földes, Stéphane; Steinberg, Richard
A topological space for which graph embeddability is undecidable.
J. Combin. Theory Ser. B 29 (1980), no. 3, 342--344.</p>
<p>From the introduction: ``From Edmonds' permutation theorem and a generalization due to Stahl, it follows that gra... |
11,149 | <p>I could sample a set of m elements from the uniform distribution over a universe $U$ of n >> m elements. Alternately, I could select a random probability distribution $\mathcal{D}$, and sample $m$ elements from $\mathcal{D}$. </p>
<p>EDIT (per Michael Lugo): When I say "select a random probability distribution", I ... | gowers | 1,459 | <p>With a different notion of undecidability, the following is another example. Ramsey's theorem says that if you 2-colour the edges of a countably infinite graph then there will be an infinite set of vertices such that all the edges between them have the same colour. A natural question one might then ask is whether if... |
172,715 | <p>Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.</p>
<p>I am interested in what can be said about the Ehrhart polynomial when
$P$ has any of the properties</p>
<ul>
<li>is integrally closed</li>
<li>has a unimodular triangulation</li>
<li>has a unimodular pulling triangulation</li... | Per Alexandersson | 1,056 | <p>To add a partial answer, something I should have thought of before:</p>
<p>Every lattice polytope $P$ can be made into an integral polytope, by maginfying it with a sufficiently large factor. It is known that the dimension of $P$ suffices, to $dP$ is integrally closed. However, if $f(x)$ is the Ehrhart polynomial f... |
2,481,021 | <p>Given a (limit) ordinal $\alpha$ we could define its cofinality $\text{cf}(\alpha)$ as the least ordinal $\beta$ such that there's an unbounded function from $\beta$ to $\alpha$. It's not difficult to prove that $\text{cf}(\alpha)$ is a cardinal. However, for $\alpha$ cardinal we could define its cofinality to be th... | Asaf Karagila | 622 | <p>Since we are talking about partitions, the order type matters less, so might as well assume $\alpha$ is a cardinal, since it makes things simpler.</p>
<p>Now given a partition of $\alpha$, such that all the parts are small, this means that each part is a bounded set. So fixing a minimal order type of a well orderin... |
1,754,820 | <p>$\frac{n^4}{\binom{4n}{4}}$</p>
<p>$= \frac{n^4 4! (4n-4)!}{(4n)!}$</p>
<p>$= \frac{24n^4}{(4n-1)(4n-2)(4n-3)}$</p>
<p>$\rightarrow \infty$ as $n \rightarrow \infty$</p>
<p>However, the answer key says that</p>
<p>$\frac{n^4}{\binom{4n}{4}}$</p>
<p>$= \frac{6n^3}{(4n-1)(4n-2)(4n-3)}$ this is the part I don't u... | robjohn | 13,854 | <p><strong>Hint:</strong>
$$
\begin{align}
\frac{n^4}{\binom{4n}{4}}
&=\frac{n^4}{\frac{4n(4n-1)(4n-2)(4n-3)}{4!}}\\
&=\frac{n^4}{\frac{4^4n^4\left(1-\frac1{4n}\right)\left(1-\frac2{4n}\right)\left(1-\frac3{4n}\right)}{4!}}\\
&=\frac{4!}{4^4}\frac1{\left(1-\frac1{4n}\right)\left(1-\frac2{4n}\right)\left(1-\... |
1,227,180 | <p>I would appreciate if somebody could help me with the following problem:</p>
<blockquote>
<p>Let $f(a)$ area of S, $A(a,a^2)$, $B(b,b^2)$ and $\overline{AB}=1$, given that:</p>
</blockquote>
<p><img src="https://i.stack.imgur.com/nyWcH.png" alt="enter image description here"></p>
<p>Find that $$\lim_{a\to \inft... | RE60K | 67,609 | <p>$$\sqrt{(b-a)^2+(b^2-a^2)^2}=1\implies \sqrt{(b-a)^2(1+(b+a)^2)}=1\\
\implies (b-a)^2(1+(b+a)^2)=1$$
$$\lim_{a\to\infty}a^3f(a)=\lim_{a\to\infty}a^3\int_a^{b}\left(\frac{b^2-a^2}{b-a}(x-a)-x^2\right){\rm d}x$$
These two relations can get you started.</p>
|
2,580,214 | <p><strong><em>Edit</em></strong> : I'm ''derivation'' and ''integration'' beginner. So I don't have any techniques to solve equations like that.This why I think there have to be a clever trick to solve this. a), b), c) was okay. You can see my solution below. I want to emphasize that I don't want to see the solution. ... | Guy Fsone | 385,707 | <p>$$f′(x)=e^x+f(x)\Longleftrightarrow f'-f =e^x\Longleftrightarrow(fe^{-x})'=1\Longleftrightarrow fe^{-x}=x-c\Longleftrightarrow f(x)= e^x(x+c)$$</p>
|
2,875,361 | <p><strong>If on a circumference they are marked $n$ equally spaced points, those points
can be joined by line segments contiguous (without lifting the pencil). If you join the consecutive points, you get a polygon regular of n sides (that's not funny). But if you join non-contiguous points (skipping from one, or two o... | Dr. Richard Klitzing | 518,676 | <p>In fact it is not the "number you skip", as @EthanBolker said, it is more the $k$-th vertex you'd visit next in a total sequance of $n$ vertices, which has or has not a common factor. That is, whenever $\gcd(n,k)=1$, you could trace a complete polygon, which finally visits all of the $n$ vertices.</p>
<p>It is this... |
3,857,471 | <p>Consider the pdf of of <span class="math-container">$Gamma(\alpha,\beta)$</span></p>
<p><span class="math-container">\begin{align}
f(x;\alpha,\beta) & = \frac{ \beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)} \quad \text{ for } x > 0 \quad \alpha, \beta > 0, \\[6pt]
\end{align}</span></p>
<p>My goal... | Fawkes4494d3 | 260,674 | <p>Following my second comment, <span class="math-container">$$\begin{aligned} &P(a \leq X_{\alpha,\beta} \leq b)-P(a < X_{\alpha-1,\beta} < b) \\ =& \int_a^b \dfrac{ \beta^{\alpha} w^{\alpha-1}e^{-\beta w} }{\Gamma (\alpha)} dw - \int_a^b \dfrac{ \beta^{\alpha-1} w^{\alpha-2}e^{-\beta w} }{\Gamma (\alpha... |
3,820,722 | <p>I found a problem in calculus lecture notes where it is asked to find the supremum for the follownig set <span class="math-container">$X = [0, \sqrt{2}] ∩ Q$</span></p>
<p>I assumed that the <span class="math-container">$\sup X = \sqrt{2}$</span></p>
<p>Then I wrote (according to definition of supremum)</p>
<ol>
<li... | 1123581321 | 482,390 | <p>HINT</p>
<p><span class="math-container">$\exists (q_n)\subseteq \mathbb{Q}$</span> s.t it is increasing and <span class="math-container">$q_n\to\sqrt{2}$</span></p>
|
824,149 | <p>How can I find the number of Pythagorean triplets <span class="math-container">$(a, b, c)$</span> such that:</p>
<p><span class="math-container">$1\le a \le b \le c \le N$</span>,</p>
<p><span class="math-container">$
a, b, c, N \in \mathbb {N}
$</span></p>
<p><span class="math-container">$
1 \le N\le10^6
$</span>... | Greg Martin | 16,078 | <p>I doubt there is a simple closed form for this quantity. There is an exact formula
$$
\sum_{n\ge1} \sum_{\substack{m>n \\ \gcd(m,n)=1 \\ m\not\equiv n\pmod 2}} \bigg\lfloor \frac N{m^2+n^2} \bigg\rfloor,
$$
which follows from <a href="http://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple" rel="nofol... |
824,149 | <p>How can I find the number of Pythagorean triplets <span class="math-container">$(a, b, c)$</span> such that:</p>
<p><span class="math-container">$1\le a \le b \le c \le N$</span>,</p>
<p><span class="math-container">$
a, b, c, N \in \mathbb {N}
$</span></p>
<p><span class="math-container">$
1 \le N\le10^6
$</span>... | poetasis | 546,655 | <p>The number of primitive Pythagorean triples for <span class="math-container">$A\le N$</span> is over <span class="math-container">$50\%$</span> because there are one-or-more triples for every odd number greater than one. As <span class="math-container">$A$</span> increases, the chance of having more than one [primit... |
2,128,697 | <p>The integral is: $\int^{\pi}_0\int^{\pi}_x \frac{\sin(y)} {y} dy dx$.
I don't know if in this case I can change the order of the integral, but if so, I would have to integrate $\frac{\sin(x)} {x}$ in any case, so I don't know how to solve this integral.</p>
| user247327 | 247,327 | <p>Your thinking that "I would have to integrate $\frac{sin(x)}{x}$ in any case" is wrong. Changing the order of integration gives $\int_0^\pi\int_0^y \frac{sin(y)}{y}dxdy= \int_0^\pi \left[\frac{sin(y)}{y}x\right]_{x=0}^y dy= \int_0^\pi \frac{sin(y)}{y}(y- 0)dy= \int_0^\pi sin(y)dy$.</p>
|
3,708,953 | <p>Suppose you have two random variables </p>
<p><span class="math-container">$X:\Omega \to \mathbb{R}$</span> and <span class="math-container">$Y:\Omega \to \mathbb{R}$</span> </p>
<p>which are not necessarily independent. </p>
<p>How is the product <span class="math-container">$XY$</span> defined and how do I calc... | angryavian | 43,949 | <p>From your last comment, I think your confusion stems from some misleading notation from your source material. (Without further context it is hard to clarify.)</p>
<p>If <span class="math-container">$X(\omega_1) = x_1$</span> and <span class="math-container">$X(\omega_2) = x_2$</span> and <span class="math-container... |
1,918,181 | <p>Show that any bijection from $[0,1)$ to $(0,1)$ has infinitely many discontinuities.</p>
<p>I have thought about this question but I have no any idea. Any idea is valuable for me, thanks.</p>
| Slade | 33,433 | <p>Suppose that <span class="math-container">$f$</span> is discontinuous only at a finite set <span class="math-container">$A$</span>, and let <span class="math-container">$B = A \cup \{0\}$</span>. Set <span class="math-container">$|B| = n$</span>.</p>
<p><span class="math-container">$[0,1)\setminus B$</span> is a d... |
2,453,754 | <p><a href="https://i.stack.imgur.com/PqMXe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PqMXe.png" alt="enter image description here"></a></p>
<p>Given a group table, determine the order of $x_5, x_3, x_5x_3, (x_5)^2(x_3), x_5^{-1}x_3$.</p>
<p>So the order of an element is the least positive in... | GAVD | 255,061 | <p>One has $x_5x_3 = x_6$, then $ord(x_5x_3) = ord(x_6) = 2$.</p>
<p>$x_5^2x_3 = x_3$, then $ord(x_5^2x_3) = ord(x_3) = 2$.</p>
<p>$x_5^{-1}x_3 = x_5x_3$, then $ord(x_5^{-1}x_3) = ord(x_5x_3) = 2$.</p>
|
883,903 | <p>I came across this problem whilst studying for a comprehensive exam in real analysis; for reference, see Exercise 1.24(A) in Folland's <em>Real Analysis</em>; it's a modification of that.</p>
<p>Consider the unit interval $I:=\left[0,1\right]$, and let $\mathcal{M}$ be the $\sigma$-algebra of all Lebesgue measurabl... | Rene Schipperus | 149,912 | <p>Let $D$ be borel such that $E\subseteq D$ and $m(D)=1$. Let $F \subseteq D-E$ be measurable, then $D-F$ is measurable and $m(D)=m(D-F)+m(F)$ now $E \subseteq D-F$ so
$m(D-F)=1$ and we have $m(F)=0$. Also $m(I-D)=0$ is obvious. Thus we have if $F\subseteq I-E$ is measurable then $m(F)=0$.</p>
|
2,723,763 | <p>The integral is </p>
<p>$$\int^3_1\sqrt{16-x^2}dx$$</p>
<p>I've used the trig substitution method, replacing $x$ with $4\sin\theta$:</p>
<p>$$x=4\sin\theta, \quad \theta=\arcsin\left(\frac x4\right), \quad dx=4\cos\theta \ d\theta$$
(I've excluded the intervals for the definite integral for now)
\begin{align}
I&a... | Dr. Sonnhard Graubner | 175,066 | <p>Two things must be observed: $$\sqrt{\cos^2(t)}=|\cos(t)|$$ and $$\cos^2(t)=\frac{1}{2}\left(\cos(2t)+1\right)$$
Form your control, the result should be $$-1/2\,\sqrt {15}-8\,\arcsin \left( 1/4 \right) +3/2\,\sqrt {7}+8\,
\arcsin \left( 3/4 \right)
$$</p>
|
4,365,928 | <p>I have the problem <span class="math-container">$2\frac{dy}{dt} = y(y-2), y(0)=3$</span>, which I get how to solve up until the intial conditions. The solution is <span class="math-container">$y(t)=\frac{2}{1-Ce^t}$</span>. The solution then plugs in three initial conditions and solves for C. They do <span class="... | Angel | 109,318 | <p>The only initial condition given by the problem is <span class="math-container">$y(0)=3,$</span> so there is no reason why the other ones should be relevant. Regarding the equilibrium solutions, notice that this equation is of the form <span class="math-container">$$y'=f(y).$$</span> The equilibrium solutions are gi... |
1,253,887 | <p>Evaluate the integral:</p>
<p>$\displaystyle \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$</p>
<p>(using substitution)</p>
<p>Here's my attempt at solution:</p>
<p>u = $\sin^5(x)$</p>
<p>$du = 5\sin^4(x) \cdot \cos(x) \cdot dx$</p>
<p>$ \frac {1}{5\sin^4(x)} du = \cos(x) \cdot dx $</p>
<p>Also, low... | Olivier Oloa | 118,798 | <p>You may just perform the change of variable $$v=\sin x, \qquad dv=\cos x\:dx,$$ giving</p>
<blockquote>
<p>$$
\int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx=\int_{1/2}^{1} \frac{dv}{v^{5/7}}=\left[\frac72 v^{2/7}\right]_{1/2}^1=\frac{7}{2}\left(1-\frac{1}{2^{2/7}}\right).
$$</p>
</blockquote>
|
4,309,551 | <p>Circle formula that touches <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> axes and passes <span class="math-container">$P = (2, -1)$</span>.</p>
<p>Firstly I tried to solve the problem using perpendicular bisector conjunctions. P.B of <span class="math-container">$XP$</span> and... | Darshan P. | 992,496 | <p>The equation of circle with <span class="math-container">$(h, k) $</span> as centre of the circle: <span class="math-container">$(x-h)^2 + (y-k)^2 = r^2$</span></p>
<p>The point is in the fourth quadrant <span class="math-container">$P(2, -1)$</span>
<span class="math-container">$\implies h = \pm k$</span> also cir... |
349,482 | <p>I came across this function in a context in which I need to know if it is injective. The function is $f:\mathbb{R}^2\backslash\{(0,0)\} \longrightarrow \mathbb{R}^2\backslash\{(0,0)\}$ and defined by $$f(x,y) = \Biggl( \frac{x}{x^2+y^2},\frac{2y}{x^2+y^2} \Biggl).$$</p>
<p>I guess it is injective. I have failed to ... | Berci | 41,488 | <p><strong>Hint</strong> Express $x$ and $y$ from $u=\frac{x}{x^2+y^2}$ and $v=\frac{2y}{x^2+y^2}$. </p>
<p>One more hint: for start, calculate $u^2+(v/2)^2$.</p>
|
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| Mikasa | 8,581 | <p>If someone wants me to know a great book as you wanted, I'll suggest him "Modern Calculus", an old book written by R.A.Silverman. I don't know why; but I've got much more basic concepts in Calculus from the old books. This book makes an student a solid root in Calculus.</p>
|
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| amWhy | 9,003 | <p>As a computer science student, you might also want to check out Knuth, Graham, and Patashnik <a href="http://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow"><strong>Concrete Mathematics</strong></a>, which I consider to be absolutely indispensable for comp sci students. It's a very thorough book! Not ne... |
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| rraallvv | 44,838 | <p>More often than not Amazon's search has got the best book listed first. For precalculus it's <a href="http://www.amazon.com/s/ref=sr_st?keywords=precalculus&qid=1356359830&rh=n%3A283155%2Ck%3Aprecalculus&sort=relevanceexprank" rel="nofollow">"The Complete Idiot's Guide to Precalculus"</a>. I guess the ti... |
286,636 | <p>I understood up to now that the term 'law of composition' was an older alternative to 'binary operation'. But according to <a href="http://en.wikipedia.org/wiki/Composition" rel="nofollow">this Wikipedia disambiguation page</a> it is an alternative to 'binary function'. Which is correct? Also, which term is preferre... | Vishal | 511,047 | <p>A law of composition on a set $S$ is any rule for combining pairs $a, b$ of elements of $S$ to get another element say, $p$ of $S$. Some models of this concept are addition and multiplication </p>
|
465,631 | <blockquote>
<p>Is it possible to embed $\Bbb{C}(x)$ (the field of rational functions over the complex numbers) in $\Bbb{C}$ ?</p>
</blockquote>
<p>Thank you!</p>
| Alex Youcis | 16,497 | <p>Yes. Note that $\overline{\mathbb{C}(X)}$ is of characteristic zero, and still of the same cardinality as $\mathbb{C}$. So, there is an isomorphism $\overline{\mathbb{C}(X)}\xrightarrow{\approx}\mathbb{C}$. </p>
|
1,553,754 | <p>There is a highly believable theorem:</p>
<p>Let $A, B$ be disjoint sets of generators and let $F(A), F(B)$ be the corresponding free groups. Let $R_1 \subset F(A)$, $R_2 \subset F(B)$ be sets of relations and consider the quotient groups $\langle A | R_1 \rangle$ and $\langle B | R_2 \rangle$.</p>
<p>Then $\lang... | Derek Holt | 2,820 | <p>The Universal Property of the free product <span class="math-container">$G*H$</span> of groups <span class="math-container">$G$</span> and <span class="math-container">$H$</span> is that there exist homomorphisms <span class="math-container">$i_G:G \to G*H$</span> and <span class="math-container">$i_H:H \to G*H$</sp... |
3,733,247 | <p>I need to find the points in the Cartesian plane that make <span class="math-container">$x+y+\sqrt{(x-y)^2-4}$</span> positive. I got a little progress but then I get stuck:</p>
<p>The problem is equivalent to solving
<span class="math-container">$$-(x+y)<\sqrt{(x-y)^2-4}$$</span>
If <span class="math-container">... | Riemann'sPointyNose | 794,524 | <p>As pointed out by Josb B. in the comments, indeed if <span class="math-container">${-(x+y) < 0}$</span> then because the square root is always going to be positive (the square root always spits out the principle root) - then the inequality automatically holds <strong>so long as the argument inside the root is pos... |
4,188,257 | <p>I am looking to solve:</p>
<p>Show that <span class="math-container">$$\sum_{n \ge 0} \sum_{k \ge 0} {n\choose k} {2k \choose k} y^k x^n = \frac{1}{\sqrt{(1-x)(1-x(1+4y))}}$$</span></p>
<p>and then use that to show that</p>
<p><span class="math-container">$$\sum_{k \ge 0} {n \choose k} {2k \choose k} (-2)^{-k} = \be... | Marko Riedel | 44,883 | <p>Here is a different evaluation of the sum by way of enrichment. We
seek to show that for <span class="math-container">$n\ge 0$</span></p>
<p><span class="math-container">$$\sum_{k=0}^n {n\choose k} {2k\choose k} \frac{(-1)^k}{2^k}
= \begin{cases}
{n\choose n/2} \frac{1}{2^n}, & n \quad\text{even} \\
0 & n \q... |
2,849,851 | <blockquote>
<p>Prove that $(|x-1|^{\frac{6}{2}})^n = |x-1|^{3n}$ for every $x \in \mathbb{R}$ and $n \in \mathbb{N}$. </p>
</blockquote>
<p>I have some trouble and hard days to finish this problem. It seems so easy to think the way by using the exponentiation properties there, but this is one of real analysis probl... | Dr. Sonnhard Graubner | 175,066 | <p>Note that $$(\frac{6}{2})\cdot n=3n$$</p>
|
3,753,563 | <p>In the proof of the Stone-Weierstrass theorem (7.26), Rudin claims <span class="math-container">$Q_n \to 0$</span> uniformly. Can someone explain why this is the case? I don't see how that immediately follows from the bound.</p>
<p><a href="https://i.stack.imgur.com/vTCvf.png" rel="nofollow noreferrer"><img src="ht... | User5678 | 632,875 | <p>Hint: show that for any real number <span class="math-container">$\epsilon>0$</span> you can find an integer <span class="math-container">$N_{\epsilon}>0$</span> such that <span class="math-container">$x_n<\epsilon\;\forall n > N_{\epsilon}$</span></p>
|
4,533,830 | <p>How do we plot the implicit function <span class="math-container">$z=f(x,y)$</span><br />
which is defined by the equation <span class="math-container">$x+y+z = e^z$</span>?</p>
<p>I kind of know WolframAlpha and SymPy but I am open to other suggestions too.</p>
<p>I am curious to see how this function looks like.<b... | Dan | 398,708 | <p>In your diagram and equations, where it shows <span class="math-container">$S$</span>, it should be <span class="math-container">$S_2$</span>, right?</p>
<p>Draw point <span class="math-container">$P'$</span> a small distance further up the hyperbola.</p>
<p>According to the <a href="https://www.cuemath.com/geometry... |
1,022,268 | <p>I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm talking about the Lorentzian manifolds and Lorentz-Minkowski spaces (some notations of it are $\Bbb L^n$, $\Bbb E^n_1$, et... | Ivo Terek | 118,056 | <p>List of books and texts on Lorentzian Geometry: </p>
<ul>
<li><p><a href="http://www.iejgeo.com/matder/dosyalar/makale-155/2014-1-5.pdf" rel="nofollow noreferrer">Differential Geometry of Curves and Surfaces in Lorentz-Minkowski
space</a> (<a href="https://arxiv.org/pdf/0810.3351v1.pdf" rel="nofollow noreferrer">al... |
1,022,268 | <p>I find it very hard to find books on Lorentzian Geometry, more focused on the geometry behind it, instead of books that go for the physics and General Relativity approach. More specifically, I'm talking about the Lorentzian manifolds and Lorentz-Minkowski spaces (some notations of it are $\Bbb L^n$, $\Bbb E^n_1$, et... | Ivo Terek | 118,056 | <p>Initially I had asked this question because I was having trouble finding material on Lorentz geometry, say, in the same level as do Carmo's book on curves and surfaces. Bear in mind that I had little to no knowledge about manifolds at the time. I was so frustrated about not finding such a book... that I wrote one. N... |
84,308 | <p>As we know, if $f$ is a computable function, then every pre-image of a r.e. set under $f$ is also a r.e. set, i.e. $f^{-1}(X)$ is a r.e.set if $X$ is a r.e. set. So I want to know that if a function satisfies that every pre-image of a r.e. set is also a r.e. set, then can we conclude that $f$ is a computable functi... | Joel David Hamkins | 1,946 | <p>If your hypothesis has a degree of uniformity, so that we may uniformly enumerate $f^{-1}X$ from an enumeration of $X$, that is, if given an index $e$ for a c.e. se $W_e$, then we may compute an index for $f^{-1}W_e$, then the answer is yes. To compute $f(n)$, simply consider indices for the singleton sets $W_{e_m}=... |
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