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 |
|---|---|---|---|---|
342,491 | <p>How to prove the following:</p>
<p>$a_n = \left\{\left(1+\frac{1}{n}\right)^n\right\}$ is bounded sequence, $ n\in\mathbb{N}$</p>
| André Nicolas | 6,312 | <p>Below is the same answer as the one by Ju'x, in slightly different language. We want to show that $\left(1+\dfrac{1}{n}\right)^n \lt e$, or equivalently that $1+\frac{1}{n}\lt e^{1/n}$. The series expansion of $e^{1/n}$ is a sum of positive terms, the first two of which are $1$ and $\dfrac{1}{n}$.</p>
<p>The downsi... |
3,932,279 | <p>Here, I am taking Lebesgue measure. There is a notion that a set <span class="math-container">$E$</span> is measruable iff <span class="math-container">$m_*(E) = m^*(E)$</span>. Let <span class="math-container">$E' = [0,1] \cap \mathbb I$</span>, where <span class="math-container">$\mathbb I$</span> denote the set o... | Measure me | 854,564 | <p>You actually know that there exists by other routes, in fact you know that <span class="math-container">$|[0,1]\cap \mathbb{Q}|=0$</span> so it is measurable in the Lebesgue complete measure and it implies that <span class="math-container">$|[0,1]\cap \mathbb{I}|=1$</span>.</p>
<p>So you know it is possible, the onl... |
2,321,850 | <p>The base is the semicircle $$y=\sqrt{16−x^2},$$
where -4 $\le$ $x$ $\le$ 4. The cross-sections perpendicular to the $x$-axis are squares.
$$\\$$
So far this is what I have:</p>
<p>$\int (Area)\,dx$</p>
<p>$\implies$ $\int \frac{π}{2} ($$r^2$$)\ dx$.</p>
<p>$r$ = $\frac{\sqrt{16−x^2}}{2}$</p>
<p>$\implies$ $ \fra... | BBot | 454,930 | <p>You must use the area of a square.</p>
<p>Remember that the area of a square is $s^2$.</p>
<p>For this problem, your side length is $y=\sqrt{16−x^2}$, which means your area for the square is $y=\sqrt{16−x^2}^2.$
$$\\$$ </p>
<p>$$\int_{-4}^{4}{16 +x^2 } \,dx$$
$$\\$$
Can you continue from there?</p>
|
1,341,771 | <p>I'm trying to show that given three distinct points $z_1,z_2,z_3\in\mathbb C$, the rational function
$$
f(z) = \frac{(z-z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(z_2 - z_3)z + (z_1z_3 - z_1z_2)}{(z_2 - z_1)z + (z_1z_3 - z_2z_3)} = \frac{az + b}{cz + d}
$$
is a Möbius transformation. That is, I must show that $... | Travis Willse | 155,629 | <p>We can factor $z_3$ from the expression for $d$ and $z_1$ from the expression for $c$ and then group:
\begin{align}
ad-bc
&= (z_2 - z_3) (z_1 z_3 - z_2 z_3) - (z_2 - z_1) (z_1 z_3 - z_1 z_2) \\
&= (z_1 - z_2) (z_2 - z_3) z_3 - (z_1 - z_2) (z_2 - z_3) z_1 \\
&= (z_1 - z_2) (z_2 - z_3) (z_3 - z_1) .
\end{a... |
11,172 | <p>As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$</span> because <span class="math-container">$\in... | Daniel R. Collins | 5,563 | <p>The first thing I would say is that principally you want to stick with strategy (a), and not veer into (b) or (c). You definitely want to dig into, and spend time on, these trouble spots with proper mathematical reasoning; certainly not "Because I said to do it this way", as that doesn't actually count as math at al... |
467,279 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>I'm in the section titled "Compact Metric Spaces".</p>
<p>The problem is in the title.</p>
<p>My attempt at the proof is as follows:</p>
<p>Let $\{a_n\}_{n=1}^\infty$ be a Cauchy sequence in $X$. We will show that $\{a_n\}_{n=1}^\infty$ converges to a point in $... | Henno Brandsma | 4,280 | <p>In the cases you handle you only can construct a subsequence of the original sequence that converges to some $a$. In the case where $S$ is finite, so finitely many values $a_n$ occur, we can conclude (pigeon hole principle) that there exists $a \in S$ and infinitely many $n$ (say all $n \in M \subset \mathbb{N}$ tha... |
2,895,923 | <p>I am trying to solve this simple geometry problem but I am always tangled in so many equations it makes my head spin. I tried solving it via similar triangles but i cant seem to eliminate all the unwanted variables. Please help.</p>
<p>I have to prove $ r_1\times r_3=(r_2)^2$</p>
<p><a href="https://i.stack.imgur... | Narasimham | 95,860 | <p><a href="https://i.stack.imgur.com/xZxzS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xZxzS.png" alt="Contacting Circles"></a></p>
<p>We have $h'$s of contacting circles from common apex A, the series of similar triangles:</p>
<p>$$ h_2-h_1=2r_1, h_3-h_2=2r_2, h_4-h_3=2r_3 \tag1$$</p>
<p>... |
830,755 | <p>I would like to compute the following,
$$
\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy
$$
It is obvious that we can rewrite the integral above to,
$$
\int_0^{\infty}\int_0^{\infty}e^{-(x+y)^2}\ dx\,dy
$$
so we are ending up with something looking like a gaussian integral. I think that a smart substitution ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{... |
2,190,508 | <p>Let $f:[a,b] \to \mathbb R$ be nice differentiable function. By Fundamental Theorem of Calculus (FTC), we have $$\int_a^{b} f'(t) dt = f(b) -f(a).$$</p>
<p>By Cauchy-Schwartz inequality, we have, $\int_{a}^{b} |f(t)| dt \leq (b-a)^{1/2}(\int_a^b |f(t)|^2 dt)^{1/2} .$ Using this, and FTC, we have $\int_{a}^{b} (... | Andrew D. Hwang | 86,418 | <p>In reverse order: If
$$
f(x) = f(a) + \underbrace{\frac{f(b) - f(a)}{b - a}}_{m}(x - a),\quad a \leq x \leq b,
$$
then $f'(t) \equiv m$, so
$$
\int_{a}^{b} f'(t)^{2}\, dt = m^{2} (b - a)
= \frac{\bigl(f(b) - f(a)\bigr)^{2}}{b - a}.
$$
In words, the stated inequality is saturated for affine functions (as would be exp... |
35,327 | <p>This is a bit of a follow-up to a previous question: <a href="https://mathematica.stackexchange.com/questions/35256/how-can-i-merge-multiple-sets-of-morphological-components-perhaps-selected-usin">How can I merge multiple sets of morphological components (perhaps selected using different metrics)?</a></p>
<p>I've r... | Dr. belisarius | 193 | <p>Based on nikie's comment:</p>
<pre><code>i = Import["http://i.stack.imgur.com/gSXIj.png"];
k = Image[MorphologicalComponents[i]];
DynamicModule[{pts = {{-1, 1}/2}},
{LocatorPane[Dynamic[pts], Image[k, ImageSize -> 300], LocatorAutoCreate -> True,
Appearance -> Gra... |
2,764,447 | <p>In the following question:</p>
<p><a href="https://i.stack.imgur.com/Rgzjr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rgzjr.jpg" alt="enter image description here" /></a>
I'm getting the answer as LHL=-infinity while RHL= infinity</p>
<p>The answer given to this question is D.</p>
<p>However ... | Dylan Zammit | 400,332 | <p>The formal definition of the existence of a limit at a point $a$:
$$\forall\epsilon>0\exists\delta>0, |x-a|<\delta\implies|f(x)-L|<\epsilon \text{ where } L\in\mathbb{R}$$</p>
<p>So no, the limit must be a constant.</p>
|
3,266,705 | <p>I know that if <span class="math-container">$f(x)$</span> is irreducible then <span class="math-container">$\langle f(x) \rangle$</span> is a prime ideal. Then I thought: is it maximal? Then I search about it, I find that it is not maximal ideal but cannot find any proof. </p>
<p>Any help is appreciated. Thanks.... | Daniel Schepler | 337,888 | <p>If <span class="math-container">$f$</span> has degree 0, i.e. <span class="math-container">$f$</span> is a constant polynomial with a prime value <span class="math-container">$p$</span>, then <span class="math-container">$\langle p \rangle \subsetneqq \langle p, x \rangle \subsetneqq \langle 1 \rangle$</span>.</p>
... |
322,448 | <p>I'm supposed to show that: </p>
<p>$$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ </p>
<p>and the required answers are: $$ P=5, Q=4, R=-4 $$ </p>
<p>I tried to solve this with partial fractions like so: </p>
<p>$$5(x-1)(x+2) = A(x+3) + B(x-2)$$ </p>
<p>$\implies$ $A$=4, $B$=-4... | Archy Will He 何魏奇 | 65,082 | <p>You are given the wrong expression. After some calculation I figure out that in order for these two expressions to be equivalent,</p>
<p>$$\frac{5(x-1)(x-2)}{(x-2)(x+3)} = 5 + \frac{4}{(x-2)} + \frac{-4}{(x+3)}$$ </p>
<p>instead of having $\frac{5(x-1)(x-2)}{(x-2)(x+3)}$, it should be</p>
<p>$$\frac{5(x-1)(x+2)}... |
2,207,260 | <p>I have to find the asymptotics of the following integral</p>
<p>$$\int^1_0 \frac{\sin(x)}{x(1+x)^n} dx$$</p>
<p>as $n\to\infty$.</p>
<p>I know I am supposed to use the lebesgue dominated convergence theorem and create a sequence $b_n$ and aim for $\lim_{n\to\infty} a_n/b_n =1$ but I can't seem to get my head roun... | Mark Viola | 218,419 | <p>Let $I_n$ be given by</p>
<p>$$\begin{align}
I_n&=\int_0^1 \frac{\sin(x)}{x(1+x)^n}\,dx\\\\
&=\int_0^1 (1+x)^{-n}\,dx+\sum_{k=1}^\infty\frac{(-1)^{k}}{(2k+1)!}\int_0^1x^{2k}(1+x)^{-n}\,dx\\\\
&=\frac{1}{n-1}-\frac{2}{2^n(n-1)}+\sum_{k=1}^\infty\frac{(-1)^{k}}{(2k+1)!}\int_0^1 x^{2k}(1+x)^{-n}\,dx\tag 1
... |
3,374,024 | <p>Basicly, i am bad at mathematics very hard to me got score 70. I try learn trigonometry now, starting from the very basic.
I practice to find the value of <span class="math-container">$$\sin(135^\circ)$$</span> using my knowing about <span class="math-container">$\sin(45^\circ)={\sqrt2\over2}$</span>, <span class="m... | ComFreek | 85,341 | <p>I think the authors just want to make a difference between the theory of sequents and the meta level. The meta level here is talking about the theory of sequents.</p>
<p>I previously had a paragraph in this answer elaborating on possible differences in connotations, but I've come to the conclusion that I cannot sep... |
4,527,455 | <p>My goal is to find all values of "a" so that the circle <span class="math-container">$x^2 - ax + y^2 + 2y = a$</span> has the radius 2</p>
<p>The correct answer is: <span class="math-container">$a = -6$</span> and <span class="math-container">$a = 2$</span></p>
<p>I tried solving it by doing this:<br/>
<sp... | eMathHelp | 166,193 | <p><span class="math-container">$\frac{a^2+4a+4}{4}$</span> should equal <span class="math-container">$2^2=4$</span>, not <span class="math-container">$2$</span>.</p>
|
56,804 | <p>I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. </p>
<p>I know that if you have a 3-cycle and a 2-cycle operating on three elements, ... | Adam Karlson | 301,192 | <p>I am not sure if this is too late. But the symmetry can be seen by plotting on the complex plane. The roots form the vertices of the equilateral triangle. So, you can rotate all of them and preserve symmetry. Hand-wavy but I guess it works.</p>
|
1,939,107 | <p>An unbiased coin is tossed repeatedly and outcomes are recorded. What is the expected no of toss to get HT ( one head and one tail consecutively) ?</p>
<p>I tried to solve above question using the following probability tree diagram.</p>
<p><a href="https://i.stack.imgur.com/10cO3.png" rel="nofollow noreferrer"><im... | epi163sqrt | 132,007 | <p>We can describe all valid tosses as follows:</p>
<blockquote>
<p>The valid tosses are</p>
<ul>
<li>zero or more tosses of T followed by </li>
<li>zero or more tossed of H followed by</li>
<li>HT</li>
</ul>
<p>This corresponds to the tree diagram where we can see following valid paths starting... |
3,238,306 | <p>Hello I have the differential equation <span class="math-container">$$x'=\frac{8t+10x}{17t+x}$$</span>
Brought it in a eqation where I can substitute <span class="math-container">$$ u=\frac{x}{t}$$</span>
and after some transformations I got the equation
<span class="math-container">$$ \frac{17+u}{8-7u-u^2}du=\frac... | Dr. Sonnhard Graubner | 175,066 | <p>I would write <span class="math-container">$$a(b-1)\geq b$$</span> and this is <span class="math-container">$$a(b-1)\geq b-1+1$$</span> or
<span class="math-container">$$(b-1)(a-1)\geq 1$$</span> this is true since <span class="math-container">$$a-1\geq 1$$</span> and <span class="math-container">$$b-1\geq 1$$</spa... |
2,845,001 | <p>If I know the length of a chord of a circle and the length of the corresponding arc, but do not know the circle's radius, is there a formula by which I can calculate the length of the sagitta?</p>
| Claude Leibovici | 82,404 | <p>Starting from Ted Shifrin's answer, I think that we can have a reasonable estimate of $\theta$ using the Padé approximant built at $\theta=0$
$$\dfrac{\sin(\theta)}{\theta}=\frac{1-\frac{53 }{396}\theta^2+\frac{551 }{166320}\theta^4 } {1+\frac{13 }{396}\theta^2+\frac{5 }{11088}\theta^4 }$$ which is quite good for $0... |
2,871,084 | <p>Consider that we have a circle drawn a round the <strong>origin (0,0)</strong>. That circle has some points drawn on its circumference. Each of those points has range and azimuth $(r,\theta)$, the <strong>r</strong> and <strong>$\theta$</strong> values of these points are calculate with responding to the <strong>ori... | zhw. | 228,045 | <p>It's easy to see every function in the range is Lipschitz, but there are plenty of functions in $C[a,b]$ that are not Lipschitz. For example, $g(x)=\sqrt {x-a}.$ Hence the integral operator is not surjective.</p>
|
1,050,141 | <p>Find the roots of the equation $3^{x+2}$+$3^{-x}$=10 . By inspection the roots are $x=0$ and $x=-2$. But how can I solve this equation otherwise? </p>
| Dr. Sonnhard Graubner | 175,066 | <p>your equation is equivalent to $3^{2x}-\frac{10}{9}3^x+\frac{1}{9}=0$ solve this as a quadratic equation.</p>
|
85,622 | <p>I'm trying to understand how this works in terms of Galois theory and local class field theory. Assume we have an extension of local fields $E/L/K$ s.t. $E/L$ and $L/K$ are abelian. I'm interested in recognizing when $E/K$ is Galois. Clearly, $E/K$ is Galois if and only if $E$ is always fixed by an extension of an $... | Franz Lemmermeyer | 3,503 | <p>If $L/K$ is abelian and $K/k$ cyclic, then $L/k$ is going to be normal if the Galois group at the gottom acts on the class group attached to $L/K$ via class field theory, and it will be abelian if action fixes the classes. This is due to Hasse and follows easily from the standard results in class field theory. This ... |
380,696 | <p>Show that for any odd $n$ it follows that $n^2 \equiv 1 \mod 8$ and for uneven primes $p\neq 3$ we have $p^2 \equiv 1\mod 24$.</p>
<p>My workings so far: I proceeded by induction. Obviously $1^2 \equiv 1 \mod 8$. Then assume that for a certain uneven number $k$ we have $k^2 \equiv 1 \mod 8$. Then the the next uneve... | lab bhattacharjee | 33,337 | <p>For the second question, any prime $>3,$ can be expressed as $6a\pm1$ where $a$ is a positive integer</p>
<p>So, $p^2=(6a\pm1)^2=36a^2\pm12a+1=24a^2+24\cdot\frac{a(a\pm1)}2+1\equiv1\pmod{24}$ as $a(a\pm1)$ is divisible by $2$</p>
<p>Evidently, we don't need prime, any odd number not divisible by $3$ will satisf... |
306,178 | <p>Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decreasing sequence. Anyone can help ? I consider the ratio $\frac{y_{n+1}}{y_n}$ but I got stuck.</p>
| robjohn | 13,854 | <p>Note that
<span class="math-container">$$
\begin{align}
\frac{\left(1+\frac1n\right)^{n+1}}{\left(1+\frac1{n+1}\right)^{n+2}}
&=\left(\frac{n+1}{n}\right)^{n+1}\left(\frac{n+1}{n+2}\right)^{n+2}\\
&=\frac{n}{n+1}\left(\frac{(n+1)^2}{n(n+2)}\right)^{n+2}\\
&=\frac{n}{n+1}\left(1+\frac1{n(n+2)}\right)^{n+2... |
2,213,047 | <p>Let $F$ be the set of all functions $f : \mathbb{R} \to \mathbb{R}$. A relation $c$ is defined on $F$ by
$f c g$ if and only if $f(x) ≤ g(x)$ for all $x ∈ \mathbb{R} $.
Prove that '$c$' is a partial order.</p>
<p>I have proved that the relation is <em>reflexive</em>. Now I must prove that it is <em>antisymmetric</e... | R.W | 253,359 | <p>I think you are confuse about your own notation. You define the relation as</p>
<blockquote>
<p>Let <span class="math-container">$\mathcal{F}(\mathbb{R},\mathbb{R})$</span> be the set of all functions <span class="math-container">$f: \mathbb{R}\to \mathbb{R}$</span>. So define <span class="math-container">$\colo... |
2,673,078 | <p>Generalize the Monty Hall problem where there are $n \geq 3$ doors, of
which Monty opens $m$ goat doors, with $1 \leq m \leq n$.<br><strong>Original Monty Hall Problem:</strong> There are $3$ doors, behind one of which there is a car (which you want), and behind the other two of which there are goats (which you don’... | manisar | 843,108 | <p>While the original question has been answered, I'll post another solution to the actual problem at hand.</p>
<p>It will greatly simplify the calculations if we assume that the host momentarily identified a set <span class="math-container">$y > m$</span> doors (just in his mind) and opened <span class="math-contai... |
2,462,852 | <p>I want to prove that $f(x)=x^3+x+2$, $f: \mathbb R \rightarrow \mathbb R$ is bijective without calculus. My attempts at showing to prove that it' injective and surjective are written below:</p>
<p>$1)$ Injectivity: </p>
<p>I want to show that $\forall a,b \in \mathbb R$ $f(a)=f(b) \implies a=b$.</p>
<p>I started ... | Guy | 206,544 | <p>As for the injectivity, if $a>b$, then $a^3>b^3$, so $f(a)=a^3+a+2>b^3+b+2=f(b)$.</p>
<p>As for the surjectivity, there is a formula to find a real solution of an equation of degree 3; it is not very nice, but you can use it. You can find it <a href="https://en.wikipedia.org/wiki/Cubic_function#Roots:_Natu... |
688,711 | <p>Assume the desity of air <span class="math-container">$\rho$</span> is given by</p>
<p><span class="math-container">$\rho(r)=\rho_0$$e^{-(r-R_0)/h_0}$</span> for <span class="math-container">$r\ge R_0$</span></p>
<p>where <span class="math-container">$r$</span> is the distance from the centre of the earth, <span c... | cffk | 36,388 | <p>You don't have to do integrals! Divide atmospheric pressure A = 101.3 kPa by g = 9.8 m/s<sup>2</sup> to give the mass per unit area (kg/m<sup>2</sup>). Multiply this by the area of the earth and you're done. (Assumptions: g is a constant over the height of the atmosphere; g independent of latitude; neglect the ma... |
165,235 | <p>I have a weighted, directed graph with 100 vertices and the maximal number of edges, 9900. Are there <em>Mathematica</em> tools or packages available to visualize which edges have large weights? (If you know of non-<em>Mathematica</em> software, that's fine, too.)</p>
<p>Background: My institution has 100 majors... | Szabolcs | 12 | <p>I will use <a href="http://szhorvat.net/mathematica/IGraphM" rel="noreferrer">IGraph/M</a> for the following answer.</p>
<p>Here's a complete directed graph on 100 with weighted edges. There are lots of edges with small weights and a few with large weights.</p>
<pre><code>graph = CompleteGraph[100, DirectedEdges -... |
213,533 | <p>I am trying to play a sequence of sounds in Mathematica using <code>Table</code>. My real example is much more complicated so that is why I got to using <code>Table</code>, </p>
<p>Why doesn't the follow code work? </p>
<pre><code>frequencies = {{100, 200}, {200, 300}, {400, 600}};
Sound[
Table[
Sound[Play[... | A little mouse on the pampas | 42,417 | <p>This code is written by Shenlan.</p>
<pre><code> Block[{s1 = "C", s2 = "D", s3 = "E", s4 = "F", s5 = "G", s6 = "A",
s7 = "B", t1 = 0.25, t2 = 0.5},
Sound[SoundNote @@@ {{s3, t1}, {s3, t1}, {s4, t1}, {s5,
t1},(*||*){s5, t1}, {s4, t1}, {s3, t1}, {s2, t1}, {s1, t1}, {s1,
t1}, {s2, t1},... |
1,212,177 | <p>Does $i\arg(e^{2z})=2iy?$ If it does I have solved my problem, and hence it seems like it must be the case, but I don't see it.</p>
<p>$$i\arg(e^{2z})=i\arg(e^{2x+2iy})=i\arg(e^{2x}e^{2iy})\implies \theta=2y(?)$$ Why does the $2x$ get 'ignored'?</p>
| AlexR | 86,940 | <p>Maybe it becomes a little clearer when looking at the principal branch of the logarithm:</p>
<p>$$\log z = \ln |z| + i\arg z$$
So $\arg (e^{2z}) = \Im \log(e^{2z}) = \Im 2z = 2\Im z = 2y$. Note that we have assumed $y\in (-\pi, \pi)$ and your equality is only valid in this case. What you can always say is that
$$i\... |
1,997,341 | <p>We know that the binomial theorem and expansion extends to powers which are non-integers. </p>
<p>For integer powers the expansion can be proven easily as the expansion is finite. However what is the proof that the expansion also holds for fractional powers? </p>
<p>A simple an intuitive approach would be apprecia... | Cheerful Parsnip | 2,941 | <p>You could calculate, for example, $(1+x)^{1/2}=a_0+a_1x+a_2x^2+\cdots$ by squaring both sides and comparing coefficients. For example we can get the first three coefficients by ignoring all degree $3$ terms and higher:
$$1+x=a_0^2+2a_0a_1x+2a_0a_2x^2+a_1^2x^2+\cdots$$</p>
<p>From here we can conclude that $a_0=\pm1... |
1,883,765 | <p>In the following situation I know how to find the <code>red</code> distance as <code>(diagonal - diamater) / 2</code> however I'm not sure how to find the <code>yellow</code> and <code>green</code> distances.</p>
<p><a href="https://i.stack.imgur.com/RMNLh.png" rel="nofollow noreferrer"><img src="https://i.stack.im... | Joshua Lochner | 351,844 | <p>In the above diagram: </p>
<p>Construct a square, such that the diagonal (as shown by the red line) connects the outside square and the circle in the shortest length...</p>
<p>Now, we can see that the green and yellow lines are equal (sides of a square)... Let the red line $= red$, Therefore the green and yellow l... |
2,939,605 | <p>The given task goes as follows:</p>
<blockquote>
<p>Show that <span class="math-container">$ f: \mathbb{R} \longrightarrow \mathbb{R}$</span> defined by <span class="math-container">$f(x) = \sqrt{1 + x^2} $</span> is not a polynomial function.</p>
</blockquote>
<p>I tried this approach - if <span class="math-con... | Chinnapparaj R | 378,881 | <p>Suppose <span class="math-container">$f$</span> is a polynomial,namely <span class="math-container">$$f(x)=\sqrt{1+x^2}={a_0+a_1x+\cdots+a_nx^n}=p(x)$$</span> for some <span class="math-container">$n$</span>. Then <span class="math-container">$$\frac{f(x)}{x}=\frac{p(x)}{x}$$</span> for all non zero <span class="m... |
2,939,605 | <p>The given task goes as follows:</p>
<blockquote>
<p>Show that <span class="math-container">$ f: \mathbb{R} \longrightarrow \mathbb{R}$</span> defined by <span class="math-container">$f(x) = \sqrt{1 + x^2} $</span> is not a polynomial function.</p>
</blockquote>
<p>I tried this approach - if <span class="math-con... | Angel Moreno | 327,493 | <p>The limit of the function <span class="math-container">$f(x)$</span> when <span class="math-container">$x$</span> tends to infinity is <span class="math-container">$x$</span>.
<span class="math-container">$$\lim_{x\to\infty} f(x) = x$$</span></p>
<p>Therefore, either the function <span class="math-container">$f(x)$... |
2,448,745 | <p>I am trying to derive the general form of the equation of the circle given two points $(x_1,y_1)$, $(x_2,y_2)$and the angle $\theta$ subtended by the chord joining the two points. </p>
<p>So, $\theta$ is basically the angle between two lines
$\implies \tan \theta = |\dfrac{\dfrac{y-y_1}{x-x_1}-\dfrac{y-y_2}{x-x_2}... | Tobias Kildetoft | 2,538 | <p>If we do this in two dimensions, like you suggest, then we get something that starts looking like a $2$-category (possibly with just one object, i.e. a monoidal category).</p>
<p>A common way to describe "nice" monoidal categories is in terms of diagrams, and these diagrams can then be composed both horizontally, b... |
72,098 | <p>This is my first time using Stack Exchange, but it looks like a good resource. I am here to ask a couple questions about my homework. We're working from the latest edition of <em>Abstract Algebra</em> by Herstein. This is problem #3 from p. 73 of that book, and it reads as follows:</p>
<blockquote>
<p>Let $G$ be ... | Arturo Magidin | 742 | <p>I'm somewhat concerned with what you write, as I am not sure what it is you are trying to say.</p>
<p>You seem to be trying to show that if $f\in A(G)$, then $f=L_a$ for some $a$; first, <em>nobody is asking you to prove that</em>. And second, you don't seem to be doing that. And third, what you write, $L_a=A(G)$, ... |
2,735,854 | <p>How many permutations of the word $STRESSLESSNESS$ begin OR end with an $E$? </p>
<p>Correct me if I'm wrong, but you would have to subtract the permutations where $E$ begins AND ends the permutation?</p>
| Michael Hardy | 11,667 | <p>One way of doing it is this:</p>
<p>number that begins with an E, plus number than ends with an E, minus number that both begins and ends with an E.</p>
<p>Another way is this:</p>
<p>First find the total number of permutations, then subtract the number that begin and end with a non-E.</p>
|
104,328 | <p>This question is related to <a href="https://mathematica.stackexchange.com/questions/104310/export-bug-in-combination-with-paralleltable">Export bug in combination with ParallelTable?</a>.</p>
<p>The following code shows an even stranger bug than in the upper question.</p>
<p><strong>The y-axis should always be lo... | mrz | 19,892 | <p>This is a workaround, which I found at </p>
<blockquote>
<p>example/OutOfCoreImageHistogramComputation</p>
</blockquote>
<pre><code>counts = ConstantArray[0, 256];
f[x_] := Block[{b = Clip[Floor[255*x + .5], {0, 255}]}, counts[[b + 1]]++];
image = ColorConvert[ExampleData[{"TestImage", "Lena"}], "Grayscale"];
... |
2,698,284 | <p>Here is the problem: </p>
<p>Let A be a set with n elements. Find an expression for S(n, 2), the number of partitions of A into exactly two subsets. You can either start with the general recurrence for S(n, k), or count S(n, 2) directly. </p>
<p>I'm having trouble understanding what exactly it wants me to do. So f... | Michael Rees | 881,353 | <p>I think that you need S(n, 2)=2^(n-1)+1, assuming that both sub-sets are required to not be empty.</p>
<p>For example n=3 gives 3 subsets (i.e {{1,2}, {3}}, {{1,3}, {2}}, and {{2,3}, {1}}).</p>
<p>There is a recurrence relation: S(n+1,2)=2*S(n,2)+1</p>
<p>This can be interpreted as saying that when you and a new ele... |
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | Narasimham | 95,860 | <p>To determine a conic we need to solve 5 equations with 5 given points.</p>
<p>Likewise here we are given 4 points and 4 simultaneous equations, not involving any $xy $ term.. so solve it by Cramer's determinants. </p>
|
2,329,003 | <p>Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3 $ in the plane $ x+y+z=1 $ with positive orientation . $$ $$ I have thought the parametriation: </p>
<p>\begin{align} x(t)=1+ 3 \cos (t) \hat j +3 \sin (t) \hat k \\ y(t)=1+3 \cos (t) \hat i+3 \sin (t) \hat k \\ z(t)=-1+3 \cos (t) \hat i... | Donald Splutterwit | 404,247 | <p>The plane can be parameterised as follows
\begin{eqnarray*}
x=1+t \\
y=1+s \\
z=-1-t-s.
\end{eqnarray*}
Now substitute this into the equation for the sphere $(x-1)^2+(y-1)^2+(z+1)^2=9$. We have
\begin{eqnarray*}
t^2+s^2+(t+s)^2=9 \\
\end{eqnarray*}
Rearrange this to</p>
<p>\begin{eqnarray*}
(2t+s)^2+3s^2=18
\end{eq... |
212,466 | <p>Consider the list <code>list={1,2,3,4,5,6,7,8}</code> and imagine I wanted to change it to <code>{{1,2,3},{4,5,6},{7,8}}</code>. How can I do this?</p>
<p>Using <code>ArrayReshape</code> in the usual manner</p>
<pre><code>list = {1, 2, 3, 4, 5, 6, 7, 8};
ArrayReshape[list, {3, 3}]
</code></pre>
<p>yields <code>{{... | OkkesDulgerci | 23,291 | <pre><code>list = {1, 2, 3, 4, 5, 6, 7, 8};
Partition[list, UpTo[3]]
</code></pre>
<blockquote>
<p>{{1, 2, 3}, {4, 5, 6}, {7, 8}}</p>
</blockquote>
|
4,585,589 | <p>In the 4th edition of "Matrix Computations", Golub and Van Loan present "Algorithm 5.1.1 (Householder Vector)". The first couple of lines (translated into MATLAB-syntax) read:</p>
<pre><code>m = length(x); sigma = x(2:m)'*x(2:m); v = [1; x(2:m)];
if sigma == 0 && x(1) >= 0
beta = 0;
... | Lutz Lehmann | 115,115 | <p>You are correct, this is another error in this algorithm (complete code in <a href="https://github.com/Jiangtang/Matrix-Computations/blob/master/Archives/matlab_GVL/Chapter5/Functions/House.m" rel="nofollow noreferrer">a github archive</a>, are there more official online sources?). It is clearly stated in the header... |
1,098,587 | <blockquote>
<p>Prove that for a bipartite graph $G$ on $n$ vertices the number of edges in $G$ is at most $\frac{n^2}{4}$. </p>
</blockquote>
<p>I used induction on $n$. </p>
<p>Induction hypothesis: Suppose for a bipartite graph with less than $n$ vertices the result holds true. </p>
<p>Now take a bipartite g... | Matt Samuel | 187,867 | <p>There is no need to use induction here. A bipartite graph is divided into two pieces, say of size $p$ and $q$, where $p+q=n$. Then the maximum number of edges is $pq$. Using calculus we can deduce that this product is maximal when $p=q$, in which case it is equal to $n^2/4$.</p>
<p>To show the product is maximal wh... |
1,050,664 | <p>Can someone explain why the inverse $4$ modulo $9$ is $7$? What am I missing?
$$9 = 2\cdot4 + 1$$
$$1 = 9-4\cdot2$$
$$1 = -2\cdot4 + 1\cdot9$$</p>
<p>Isn't then $-2$ inverse of $4$ modulo $9$?</p>
| John Butnor | 185,327 | <p>To find the modular multiplicative inverse x of 4(mod 9), we must solve the equation </p>
<p>x = 4^(-1)(mod 9)</p>
<p>4x = 1(mod 9)</p>
<p>4x = 28(mod 9)</p>
<p>x = 7(mod 9)</p>
<p>The multiplicative inverse is 7. If you multiply a number by its inverse, you get 1.</p>
<p>7*4 = 28, but 28(mod 9) = 1.</p>
|
2,100,793 | <p>Let us say that I have a set $A=\{1,2, 3\}$. Now I need to access, say, the element $3$ of $A$. How do I achieve this?</p>
<p>I know that sets are unordered list of elements but I need to access the elements of a set. Can I achieve this with a tuple? Like $A=(1, 2, 3)$, should I write $A(i)$ to access the i-th elem... | Milo Brandt | 174,927 | <p>You are speaking of the index operator one often finds in programming when using arrays. Given that sets are unordered, it does not make sense to do this with sets, as $\{1,2\}=\{2,1\}$. </p>
<p>You probably want to use a <em>tuple</em>, which is just a (finite) sequence of elements, and is denoted as you wrote $A$... |
356,279 | <p><strong>Definitions</strong> (<a href="https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467" rel="nofollow noreferrer">https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467</a>):</p>
<p>Given a planar convex region <span class="math-container">$C$</span> (could be smooth or polygonal), an <strong>area... | Nandakumar R | 142,600 | <p>An answer to the question 2 (written with K Sheshadri)</p>
<p>The guess made above that needs to be proved: For a general convex polygonal region with no symmetries, for every orientation, we have a unique line with that orientation that separates out the same fraction of both area and outer boundary length. The va... |
4,232,796 | <p>Consider <span class="math-container">$\sqrt{x^2+y^2}+2\sqrt{x^2+y^2-2x+1}+\sqrt{x^2+y^2-6x-8y+25}$</span>. I need to find global or local minima of this function, but Wolfram Alpha doesn't seem to find one; the answer is that <span class="math-container">$1 + 2\sqrt{5}$</span> is its global minimum</p>
<p>Am I doin... | Giorgos Giapitzakis | 907,711 | <p>Consider the set <span class="math-container">$$D = \left\{\frac{1}{k+1}: k \in \mathbb{N}\right\}$$</span>
Since <span class="math-container">$f^{(n)}|_D\equiv 0$</span>, <span class="math-container">$f^{(n)}$</span> is holomorphic and <span class="math-container">$0$</span> is an accumulation point of <span class=... |
3,575,015 | <p>This question is taken out of the text book High Dimensional Probability by Roman. I am unsure from the context if it means locally or globally Lipschitz. But the question is as follows:</p>
<p>Every differentiable function <span class="math-container">$f:\mathbb{R}^n\rightarrow{\mathbb{R}}$</span> is Lipschitz, an... | lonza leggiera | 632,373 | <p>From the comments, I gather that if <span class="math-container">$\ f_t\ $</span> is a solution to the initial value problem
<span class="math-container">$$
a_tf''(x) + b_tf'(x) + c_tf(x)=0\ ,\\
f(0)=k, f'(0)=0
$$</span>
where <span class="math-container">$\ a_t, b_t, c_t>0\ $</span> and <span class="math-contain... |
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | David | 117,084 | <p>$ \clubsuit$ Try to view every thing to one dimension or two dimension where we can see geometrically.. </p>
<p>$\quad$ It is quite often helps to realize the things and to generalize it.</p>
<p>$\clubsuit$ Aim to guess the answer first. If we realize that the answer is correct then proving </p>
<p>$\quad$ is no... |
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | Michael | 155,065 | <p>Here is a late answer (several years late it seems): </p>
<p>1) One minor piece of advice is to have respect for others. (Well, perhaps that is a major piece of advice.) </p>
<p>Here is a minor example:
People on stackexchange that
answer a question two hours after a major hint
has been given, using the basic ... |
549,411 | <p>How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for</p>
<p>$$
\frac{d}{dx} \prod_{j = 1}^n \psi_j(x)
$$</p>
| Steven Stadnicki | 785 | <p>These are known as <a href="http://en.wikipedia.org/wiki/Beatty_sequence" rel="nofollow"><em>Beatty sequences</em></a> and they've been quite thoroughly studied (you can find one of your two sequences at <a href="http://oeis.org/A001951" rel="nofollow">http://oeis.org/A001951</a> ). The most interesting property (a... |
488,287 | <p>I'm trying to solve the following problem : </p>
<p>$ABCD$ is a square of side $4$ units. Find the area of the shaded region as shown in the figure.</p>
<p><img src="https://i.stack.imgur.com/S095z.png" alt="Figure"></p>
<p>The area of square is obviously $16$ , but what after that?</p>
| Hagen von Eitzen | 39,174 | <p>Let $P(x)\equiv (x=1)$ and $Q(x)\equiv (x=0)$ and use the fact that $1\ne 0$</p>
|
488,287 | <p>I'm trying to solve the following problem : </p>
<p>$ABCD$ is a square of side $4$ units. Find the area of the shaded region as shown in the figure.</p>
<p><img src="https://i.stack.imgur.com/S095z.png" alt="Figure"></p>
<p>The area of square is obviously $16$ , but what after that?</p>
| Ittay Weiss | 30,953 | <p>Let $P(x)$ be the claim that $x$ is an odd integer. Let $Q(x)$ be the claim that $x$ is an even integer. Then $\exists x(P(x)\wedge Q(x))$ is false while $\exists x P(X) \wedge \exists x Q(x)$ is true. </p>
|
278,637 | <p>After spending more than 1 hr on this, and looking at many questions, I give up as I am not able to figure a solution.</p>
<p>I have my current package in the standard location given by <code>FileNameJoin[{$UserBaseDirectory, "Applications"}]</code> which on windows is</p>
<pre><code>C:\Users\Owner\AppData... | Michael E2 | 4,999 | <p>Just a refactoring of the OP's idea:</p>
<pre><code>scalarize // ClearAll;
scalarize[x_List] /; MatchQ[MinMax@Dimensions@x, {1, 1}] :=
First@Flatten[x];
scalarize[x_] := x;
</code></pre>
<p>Another way:</p>
<pre><code>scalarize // ClearAll;
scalarize[x_List] :=
CheckAbort[
x /. {Except[{_}, _List] :> Abor... |
718,609 | <p>This theorem is the converse of Wilson's theorem:</p>
<blockquote>
<p>If $n$ is composite and $n>4$, then $(n-1)! \equiv 0 \pmod n$</p>
</blockquote>
<p>The question holds up for all the composites I have tried but I'm struggling to form a proof for all composites greater than $4$.</p>
| André Nicolas | 6,312 | <p>Suppose first that there exist integers $a$ and $b$, such that $1\lt a\lt b\le n-1$ and $ab=n$. Then since $b\le n-1$, it follows that $ab$ divides $(n-1)!$. For each of $a$ and $b$ appears among the numbers from $1$ to $n-1$.</p>
<p>But $n$ can be composite without meeting the above condition: It could be that the... |
47,753 | <p>What are the Method options Solve command accepts? Solve has the Method option, however documentation contains no methods that it accepts...</p>
| János Tóth | 13,216 | <p>With <code>Method -> Reduce</code>, <code>Solve</code> uses only equivalent transformations and finds all solutions.</p>
<pre><code>Solve[x E^x == 1/2, x, Method -> Reduce]
</code></pre>
<blockquote>
<pre><code>{{x -> ConditionalExpression[ProductLog[C[1], 1/2], C[1] ∈ Integers]}}
</code></pre>
</blockquo... |
2,226,386 | <p>Solve $$\frac{dx}{y+z}=\frac{dy}{z+x}=\frac{dz}{x+y}.$$ Solve the similtaneous equation by using method of multipliers. How can we choose these multipliers? Is there any specific method to know these multipliers?</p>
| Andrei | 331,661 | <p>It all comes from the invariance with respect to the reference frame. What I mean is that if you draw the same triangle a little to the left, or right, or up, or down, the problem should not change. In fact you can even rotate the whole figure, and the answer should be the same. That means that you can choose any or... |
2,996,775 | <p><span class="math-container">$$ \frac {1}{\log_2(x-2)^2} + \frac{1}{\log_2(x+2) ^2} =\frac5{12}.$$</span></p>
<p>I made the graph using wolfram alpha it is giving answer as
6. But how to solve it algebraically?
base of logarithm is 2.</p>
<p>Tried using taking Lcm but then two different log terms
are getting for... | Batominovski | 72,152 | <p>Define <span class="math-container">$f:\mathbb{R}\setminus\{\pm1,\pm3\}\to\mathbb{R}$</span> by <span class="math-container">$$f(x):=\frac{1}{\log_2\big((x-2)^2\big)}+\frac{1}{\log_2\big((x+2)^2\big)}$$</span> for all real numbers <span class="math-container">$x\neq \pm1,\pm2,\pm3$</span>, and <span class="math-cont... |
3,370,061 | <p>Could anyone clarify why this Boolean expression AB'+AB'AC' = AB'? I did not understand what happened to the C'</p>
| Bram28 | 256,001 | <p><span class="math-container">$$AB'+AB'AC'=AB'(1+AC')=AB'1=AB'$$</span></p>
|
1,463,258 | <p>Given a cumulative distribution function of the form $P(X\leq x) = 1- e^{-\lambda x^3}$, is there any way to represent it in terms of an exponential or any other distribution? I've thought about exponentials, but don't know how to deal with the cubed term. Thanks!</p>
| Clarinetist | 81,560 | <p>The cumulative distribution function $$F_{X}(x) = 1-e^{-\lambda x^3}\text{, }x \geq 0$$
implies that $X$ follows a (two-parameter) <a href="https://en.wikipedia.org/wiki/Weibull_distribution#Distribution_function" rel="nofollow">Weibull</a> distribution. Using the parametrization in the link, the <a href="https://en... |
947 | <p>I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:</p>
<p>Maximum minimum distance to another person — that is, the person for whom the closest other person is farthest away.</p>
<p>Assume a (admittedly miraculous) input o... | j.c. | 353 | <p>It may be possible to reformulate your problem so that "<a href="https://doi.org/10.1007/s007780050006" rel="nofollow noreferrer">distance-based outlier</a>" algorithms will apply.</p>
|
3,790,932 | <p>If <span class="math-container">$f : \bar{\mathbb{Q}} \to \mathbb{Q}$</span> is a continuous function, where <span class="math-container">$\bar{\mathbb{Q}}$</span> denotes the set of algebraic numbers, does such function have to be constant?</p>
| freakish | 340,986 | <blockquote>
<p>can such function be constant?</p>
</blockquote>
<p>I assume you mean "does it have to be constant?". No, it doesn't.</p>
<blockquote>
<p><strong>Lemma.</strong> There is a non-constant continuous function <span class="math-container">$f:X\to Y$</span> between any countable metrizable space <s... |
1,160,699 | <p>In my discrete math book, I was tasked with finding a counterexample for this:</p>
<blockquote>
<p>If $n$ is prime, then $2^n-1$ is prime.</p>
</blockquote>
<p>Does there exist a counterexample for such a statement? Also, am I wrong in thinking that when something asks for a counterexample, is it looking for som... | Tim Raczkowski | 192,581 | <p>The statement says $2^n-1$ is prime for <strong>all</strong> primes $n$. A counterexample would be a prime number $n$ such that $2^n-1$ is not prime.</p>
|
1,696,713 | <p>I am solving exact differential equation, but I am stuck on the step on how to simplify this term or how to rewrite it. </p>
<p>$e^{-2\ln{\sin{x}}}$</p>
| Marco Disce | 17,010 | <p>I would say the best mathematical setting to represent physical causation are <a href="https://en.wikipedia.org/wiki/Dynamical_system_(definition)" rel="nofollow">dynamical systems</a>:</p>
<ul>
<li>You have a global "state" $x(t)$ (a vector with many coordinates) with flour, milk and eggs represented by set of var... |
484,550 | <p>The problem is as follows: let $n_1, n_2,..., n_t$ be positive integers. Prove that if $n_1+n_2+...+n_t-t+1$ objects are placed into $t$ boxes, then for some $i, i=1, 2, ..., t$, the $i$th box contains at least $n_i$ objects. </p>
<p>I'm having difficulty getting started in developing a proof because I have no in... | Barry Cipra | 86,747 | <p>A way to think about it is to rewrite $n_1+n_2+\cdots n_t - t+1$ as </p>
<p>$$(n_1-1)+(n_2-1)+\cdots+(n_t-1)+1$$</p>
<p>and then think that if you want try to avoid putting $n_i$ balls into the $i$th box, you can put at most $n_1-1$ balls in the first box, $n_2-1$ in the second, etc., which will leave $1$ ball lef... |
209,760 | <p>Is it possible to transform this equation to give <span class="math-container">$R$</span>?
<span class="math-container">$$y=x\left[\frac{\left(1+\frac{R}{12}\right)^{12\times{25}}}{\frac{R}{12}}-1\right]$$</span> </p>
| draks ... | 19,341 | <p>You could rewrite like $\frac{R}{12}\left(\frac yx+1\right)-\sum_{k=0}^{12\times 25}\binom{12\times 25}{k}\left(\frac R{12}\right)^k=0$, but to my knownledge, there is no general way to solve that other than numerical.</p>
|
2,489,429 | <p>$\int \frac{x}{x+1}\,dx$ so I set substitution to $x+1=t$ then I differentiate to get $dx=dt$ and then proceed to get $\int 1\,dt - \int \frac{1\,dt}{t}$ to get $t-\ln(|t|)+C$ but when I replace back the substitution I get $x+1$ in the final answer instead of just the $x$ , so my question is why was it a mistake to ... | Jaideep Khare | 421,580 | <p>There is nothing wrong in your solution .</p>
<p>$x+1-\ln|x+1| + C$</p>
<p>Now let $C+1=C'$</p>
<p>$x-\ln|x+1|+C'$</p>
<p>You can always adjust the constant of integration.</p>
|
689,921 | <p>There are two teams.Two games were played.There are three possible outcomes which are win, lose or draw. how many permutations are there?</p>
| Guy | 127,574 | <p>The first outcome can either be a win, lose or draw. Let's assume that it was a win. In this case, the second outcome can also have $3$ possibilities. If the first outcome was instead either a lose or draw, even then in both cases, the second outcome can either be win, lose or draw. </p>
<p>Thus the total permuatio... |
1,012,985 | <p>Let $A = \left[ \begin{matrix} 3 & 2 & 1\\ 5 & 0 &1\end{matrix}\right]$, </p>
<p>how can I know if there is a matrix $N$ , st. $AN=0$ (N is not a zero matrix) </p>
| Robert Israel | 8,508 | <p>Hint: what could you say about the columns of such a matrix?</p>
|
1,552,747 | <p>While we were introduced to integration, we were told about some basic concepts that, as we were told, could not be proved based on our level of sophistication. They are as follows:</p>
<ol>
<li>$$\int_a^b \! f(x) \, \mathrm{d}x=\phi(b)-\phi(a),$$ where $\phi$ is a primitive of $f$ in $[a,b]$</li>
<li>$$\int_b^a \!... | User2956 | 283,118 | <p>$\int_a^b \! f(x) \, \mathrm{d}x=g(b)-g(a)$ is a result obtained directly from the fundamental theorem of calculus.</p>
<p>Say there's a function $f(x)$</p>
<p>Now the area under the curve from $0$ up til $x$ can be found using Reimann sums. Let A(x) = $\int_0^x f(t) dt$ where A(x) is the area function that gives ... |
1,037,530 | <p>what is the basic difference between a mapping and a function? many say they are same but the opposite views are also seen. is mapping a restricted version of a function?</p>
| Ittay Weiss | 30,953 | <p>Probably most modern texts would agree that there is no mathematical difference between function and mapping. Historically, and not a very long time ago, and in difference disciplines the two were used to mean different things. For instance, in topology and metric space theory sometimes 'mapping' stands for 'continu... |
1,928,892 | <p>I'm solving a graduate entrance examination problem.
We are required to establish the inequality using the following result:</p>
<p>for $x,y > 0$, $\frac{x}{y} + \frac{y}{x} > 2$ (1), which is easy to prove as it is equivalent to $(x - y)^2 > 0$.</p>
<p>But when it comes to an inequality combining $x, y,... | Ian Miller | 278,461 | <p>Apply your formula three times:</p>
<p>$$\frac{x+y}{x+z}+\frac{x+z}{x+y}>2$$</p>
<p>$$\frac{y+x}{y+z}+\frac{y+z}{y+x}>2$$</p>
<p>$$\frac{z+x}{z+y}+\frac{z+y}{z+x}>2$$</p>
<p>Combining gives:</p>
<p>$$\frac{x+y}{x+z}+\frac{x+z}{x+y}+\frac{y+x}{y+z}+\frac{y+z}{y+x}+\frac{z+x}{z+y}+\frac{z+y}{z+x}>6$$<... |
2,238,165 | <p>Can anyone help with this limit?</p>
<p>\begin{equation*}
\lim_{x \rightarrow 4}
\frac{16\sqrt{x-\sqrt{x}}-3\sqrt{2}x-4\sqrt{2}}{16(x-4)^2}
\end{equation*}</p>
<p>I've tried a variable change of \begin{equation*} y=\sqrt{x} \end{equation*} but this didn't help.</p>
| Kanwaljit Singh | 401,635 | <p>Hint -</p>
<p>You have to take cases where 5 repeated at least twice and 7 repeated at least thrice.</p>
<p>Alternate way is find the number of ways number with length N can be arranged and then subtract cases with at most 5 repeating only once and 7 repeating at most twice.</p>
|
2,661,004 | <p>I have the following given as an answer to a questions, but I don't understand how I get $1+ \log n$ in the second step. I think I am to use L'Hospital's, but deriving the logarithm is $1/(n\ln(b))$, so I don't know why I'm keeping $\log(n)$. I know it's probably something basic, but what am I missing about that s... | The Phenotype | 514,183 | <p>An open ball in $(1,7]$ is of the form $(1,7]\cap (a,b)$ by definition of <a href="https://en.wikipedia.org/wiki/Subspace_topology" rel="nofollow noreferrer">subspace topology</a>. So for example the open ball $(1,7]\cap (2,8)=(2,7]$ in $(1,7]$ (a subspace of $\mathbb{R}$) includes $7$.</p>
|
131,583 | <p>My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any <strong>algorithm</strong>) that we can use to rewrite it using language of differential forms, tensor, exterior calculus, Hodg... | Igor Khavkine | 2,622 | <p>This is rather standard, though, unfortunately this knowledge is confined to a rather narrow segment of mathematicians working on PDEs. The main reason is that, most of the time, this is not necessary for practical problems. In fact there are multiple ways converting a PDE into invariant form.</p>
<p>In mathematica... |
1,348,763 | <p>Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or smaller than another function must be proven.</p>
| Community | -1 | <p>First, why is there any question of convergence or divergence? Check that the integrand is continuous and defined for all real numbers, so there's no vertical asymptotes to be concerned about; that leaves only the upper bound of $\infty$.</p>
<p>Next, let's guess whether it converges or not. For this, we try to u... |
194,128 | <p>I am trying to solve this problem by induction. The sad part is that I don't have a very strong grasp on solving by inductive proving methods. I understand that there is a base case and that I need an inductive step that will set $k = n$ and then one more step that basically sets $k = n + 1$. </p>
<p>Here is the pr... | Raul Gonzalez | 89,717 | <p>To prove by induction, you need to prove two things. First, you need to prove that your statement is valid for $n=1$. Second, you have to show that the validity of the statement for $n=k$ implies the validity of the statement for $n=k+1$. Putting these two bits of information together, you effectively show that your... |
27,865 | <p>Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.</p>
<p>Are there deep connections between these two transforms? <a href="http... | JAYSHREE RATHOD | 307,432 | <p>Fourier transform does not exist for every signal application.So by introducing the region of convergence in Fourier transform which is known as Laplace Transform one may have indirectly the Fourier transform of signal. </p>
|
2,254,694 | <p>I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end:</p>
<p>Attempt 1:</p>
<p>Write out the terms of $b_n$:</p>
<p>$$b_1=a_{2}-\frac{a_{1}}{2}$$
$$b_2=a_{3}-\frac{a_{2}}{2}$$
$$b_3=a_{4}-\frac{a_{3}}{2}$$
$$\cdots$$
$$... | mercio | 17,445 | <p>Let $\epsilon > 0$.</p>
<p>Since $a_{n+1}-a_n/2$ converges to $0$, there is an integer $m$ such that
for any $n \ge m$, $|a_{n+1}-a_n/2| \le \epsilon/4$. </p>
<p>For such an $n$, you have<br>
$|a_{n+1}| - \epsilon/2 \\
\le |a_{n+1} - a_n/2| + |a_n/2| - \epsilon/2 \\
\le \epsilon/4 + |a_n|/2 - \epsilon/2 \\
= |a... |
4,556,193 | <p>I meet a series of the form</p>
<p><span class="math-container">$$\sum_{n=0}^{\infty} \frac{x^n}{(2n-1)!!}$$</span>
where <span class="math-container">$(-1)!! = 1$</span>.</p>
<p>I guess it is a Taylor expansion of a function but I don't know what it is. Could anyone here help me?</p>
<p>Remark: The problem comes fr... | robjohn | 13,854 | <p>To compute
<span class="math-container">$$
f(x)=\sum_{n=0}^\infty\frac{x^n}{(2n-1)!!}\tag1
$$</span>
consider
<span class="math-container">$$
\begin{align}
g(x)
&=\frac{f(2x)}{\sqrt{2x}}\tag{2a}\\
&=\sum_{n=0}^\infty\frac{(2x)^{n-1/2}}{(2n-1)!!}\tag{2b}
\end{align}
$$</span>
where
<span class="math-container... |
1,517,245 | <p>I would appreciate if somebody could help me with the following problem</p>
<p>Q: Arrangements of $1,2,..,81$ in an $9 \times 9$ matrix such that each row and each column is increasing. How many possible $X$ ?</p>
<p><a href="https://i.stack.imgur.com/Te1TB.png" rel="nofollow noreferrer"><img src="https://i.stack.... | J.-E. Pin | 89,374 | <p>A monoid with an distinguished absorbing element is called a <em>monoid with zero</em> in the literature.
A monoid with zero in which the units are exactly the nonzero elements is called a <em>group with zero</em> [1, p. 5; 2, Def 1.3.1, p. 34; 4] or a <em>$0$-group</em> [3].</p>
<p>[1] A. H. Clifford, G. B. Presto... |
2,303,339 | <p>How does this proof work?</p>
<p><strong>Theorem.</strong>$\quad$Let $G$ be a group. Then $G$ has a unique identity. </p>
<p><strong>Proof.</strong>$\quad$Assume that $e$ and $f$ are two identities in $G$. Since $e$ is
an identity, $ef=f$; and since $f$ is an identity, $ef=e$. Thus $e=ef=f$. </p>
<p>I think ne... | Kyle Miller | 172,988 | <p>In general, a matrix <span class="math-container">$X\in \mathbb{M}_n(\mathbb{F})$</span> is determined by the linear map <span class="math-container">$\mathbb{M}_n(\mathbb{F})\to \mathbb{F}$</span> defined by <span class="math-container">$Y\mapsto \operatorname{tr}(XY)$</span>. This is because if <span class="math-... |
3,971,220 | <p>Let <span class="math-container">$G$</span> be a group (not necessary abelian) generated by <span class="math-container">$a, b$</span> that has a group presentation: <span class="math-container">$$\langle a, b \mid a^m = 1, b^n = 1, ba = a^rb\rangle. $$</span>Then how can I prove that <span class="math-container">$(... | José Carlos Santos | 446,262 | <p>No, it is not correct. If <span class="math-container">$G$</span> is generated by <span class="math-container">$a$</span>, then its quotient groups are:</p>
<ul>
<li><span class="math-container">$G/\{e\}\simeq G$</span>;</li>
<li><span class="math-container">$G/\{e,a^5\}\simeq\Bbb Z_5$</span>;</li>
<li><span class="... |
3,971,220 | <p>Let <span class="math-container">$G$</span> be a group (not necessary abelian) generated by <span class="math-container">$a, b$</span> that has a group presentation: <span class="math-container">$$\langle a, b \mid a^m = 1, b^n = 1, ba = a^rb\rangle. $$</span>Then how can I prove that <span class="math-container">$(... | Community | -1 | <p>Any quotient of a cyclic group is cyclic. One way to see it is that it is a homomorphic image (of the cyclic group).</p>
<p>Furthermore, any homomorphic image has order dividing the order of the group (first isomorphism theorem).</p>
<p>Finally it is easy to see that all the cyclic groups of orders dividing, in thi... |
1,279,947 | <p>In physics, 1m is defined 'the length which light travels in certain time'.
And i've thought 2m has twice many elements than 1m.
But today i've realized intervals (0,1) and (0,2) are equipotent.
I was shocked because i've thought length and cardinality are similar concepts.
Cardinality is clearly defined in set theo... | Simone | 77,622 | <p>Just talking in an heuristic way, you can think that the idea of length is similar to the idea of area or volume, which is generalized in mathematics to measure theory. In some sense, you'd like to associate a positive real number to a subset of a set such that if you consider the union of disjoint pieces you get th... |
1,279,947 | <p>In physics, 1m is defined 'the length which light travels in certain time'.
And i've thought 2m has twice many elements than 1m.
But today i've realized intervals (0,1) and (0,2) are equipotent.
I was shocked because i've thought length and cardinality are similar concepts.
Cardinality is clearly defined in set theo... | Chris Culter | 87,023 | <p>See 5xum's answer! Regarding this part:</p>
<blockquote>
<p>Cardinality is clearly defined in set theory. But what is length?</p>
</blockquote>
<p>Cardinality is meaningful for all sets, which is why it's typically encountered in a first course on set theory. Measure is a more complicated concept, which is typic... |
3,556,334 | <p>Given a rectangle with dimensions <span class="math-container">$10$</span>cm and <span class="math-container">$6$</span>cm, show that for every <span class="math-container">$3$</span> points in the interior of the rectangle, the area of the triangle is less than <span class="math-container">$30$</span> cm<span clas... | Andrei | 331,661 | <p>Let's have the rectangle <span class="math-container">$ABCD$</span> and <span class="math-container">$P$</span>, <span class="math-container">$Q$</span>, and <span class="math-container">$R$</span> three points inside. Let's use <span class="math-container">$A=(0,0)$</span>, <span class="math-container">$B=(a,0)$</s... |
2,924,549 | <p>Given that the three fundemental geometries are euclidean geometry (zero curvature), riemannian geometry (positive curvature) and lobachevskian geometry (negative curvature), I am curious as to what the link is between these curvatures and the conic sections of an ellipse, an hyperbole and a circle. </p>
<p>I have... | Zeno Rogue | 265,219 | <p>AFAIK these names come from far-fetched analogies, and other meanings of these three Greek words, rather than any specific similarities to the similarly named curves.</p>
<p>The name "elliptic geometry" is especially confusing to people who hear it for the first time, because it sounds like it would describe ellips... |
2,924,549 | <p>Given that the three fundemental geometries are euclidean geometry (zero curvature), riemannian geometry (positive curvature) and lobachevskian geometry (negative curvature), I am curious as to what the link is between these curvatures and the conic sections of an ellipse, an hyperbole and a circle. </p>
<p>I have... | brainjam | 1,257 | <p>From Greenberg's <em>Euclidean and Non-Euclidean Geometries</em>:</p>
<blockquote>
<p>[I]n the early nineteenth century two alternative geometries were
proposed. In hyperbolic geometry (from the Greek <em>hyperballein</em>, "to
exceed") the distance between the rays increases. In elliptic geometry
(from t... |
2,390,804 | <p>How does one compute the following integral?
$$\int_{0}^{\infty} \frac{\sqrt{x}\sin(x)}{1+x^2} dx$$</p>
<hr>
<p>I have tried extending $x$ to the complex plane then evaluating the following contour integral
$$\oint_C \frac{\sqrt{x}e^{ix}}{1+x^2} dx$$
with the contour $C$ running along the whole real axis and then... | Hans | 64,809 | <p>My goal is to prove
$$\int_{0}^{\infty} \frac{\sqrt{x}\sin(x)}{1+x^2} dx =\frac{\pi}{2\sqrt{2}\:e}\left(-e^2\text{erfc}(1)+\text{erf}(1) +1\right)$$
as stated in @JJacquelin's answer. My following attempt falls short of that. But it does provide an equivalent integral with much faster convergence.</p>
<p>Take the ... |
13,467 | <p>I will present two problems alongside solutions, student is doing problems of type I like a cakewalk but has several issues with the problems of type 2;</p>
<p><strong>Type I</strong></p>
<blockquote>
<p>Consider an experiment of rolling two dice:</p>
<p>Sample space $$ S = \{(1,1),(1,2),(1,3), \cdots, (6,6... | Nate Bade | 9,019 | <p>To use set builder notation you need to carefully state to which set every element you want to discuss belongs. In general it goes </p>
<p>$P = \{x\in\text{some set} : x\text{ satisfies some truth evaluable proposition}\}$. </p>
<p>Here's how I would do it in your case:</p>
<p>Let $P$ be the set of all humans, le... |
2,993,958 | <blockquote>
<p>Do there exist three non null vectors <span class="math-container">$a,b,c$</span> with <span class="math-container">$$a.b=a.c$$</span> such that <span class="math-container">$b\ne c$</span>?</p>
</blockquote>
<p>My attempt:</p>
<p>a.(b-c)=0</p>
<p>But is it possible to say b-c=0 since a is non null... | edm | 356,114 | <p>Consider <span class="math-container">$$a=\begin{bmatrix}1\\0\end{bmatrix}, b=\begin{bmatrix}0\\1\end{bmatrix},c=\begin{bmatrix}0\\2\end{bmatrix}.$$</span> As long as <span class="math-container">$a$</span> is perpendicular to <span class="math-container">$b-c$</span>, you will get the equality <span class="math-con... |
3,879,366 | <p>This might or might not be author specific, but in Algebraic Statistics by Seth Sullivant the author uses a notation of <span class="math-container">$S_{K, K}$</span> for example. So far I have not been able to find out what it means. Here is an <a href="https://i.stack.imgur.com/oEJHw.png" rel="nofollow noreferrer"... | Claude Leibovici | 82,404 | <p><span class="math-container">$$g(x) = \Phi(x) = {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt=\frac{1}{2} \left(\text{erf}\left(\frac{x}{\sqrt{2}}\right)+1\right)$$</span>
<span class="math-container">$$f(x,g(x)) = x(1-g(x)-g(x)^2)=-\frac{1}{4} x \left(\text{erf}\left(\frac{x}{\sqrt{2}}\right)
\... |
4,141,812 | <p>Considering the vector space <span class="math-container">$\mathbb{R}^3$</span>, find all values of <span class="math-container">$k$</span> such that
<span class="math-container">\begin{align*}
\begin{bmatrix}
2k^2 \\
-3k \\
1
\end{bmatrix} \in \text{ span }
\left\{
... | nir shahar | 890,149 | <p>Notice that the function <span class="math-container">$f(x):=e^x+x$</span> is continuous, and <span class="math-container">$\lim_{x\rightarrow \infty} f(x) = \infty$</span>, but also <span class="math-container">$\lim_{x\rightarrow -\infty} f(x)=-\infty$</span>.</p>
<p>In fact, we only need to know that there are tw... |
216,040 | <p>A model for the movement of a stock price supposes that if the present price is S then after one period, say one second, it will either go up to uS with probability p or go down to dS with probability q = 1 - p. Assuming that successive movements are independent,approximate the probability that the stock will be up ... | Ben | 27,132 | <p>The probability to win when $k$ = 3:
$$p = 6/6^3 = 1/36$$</p>
<p>The probability to win when $k$ = 2:</p>
<p>There are 3 ways to select 2 dice from a set of 3 dice($_3C_2$). For each of these, there is a 1/6 chance that the 2 dice will show the same number:
$$p = 3*{1 \over\ 6} = 1/2$$</p>
<p>To calculate our av... |
1,148,674 | <p>I have always asked myself why this happens. </p>
<p>If $x = 4$, then $\sqrt{x} = 2$, but if I search for the $\sqrt{4}$, I get $2$ & $-2$.</p>
| Vladimir Vargas | 187,578 | <p>$\sqrt{\cdot}$ is a function. As a function it cannot return $2$ values. Actually mathematicians defined that function to show the positive root. What does it mean?</p>
<p>Suppose $x^2=b$. Then $x$ could take two vales: $\sqrt b$ and $-\sqrt b$. Both of them satisfy the equation. Meanwhile, if we have $x=\sqrt c$, ... |
1,148,674 | <p>I have always asked myself why this happens. </p>
<p>If $x = 4$, then $\sqrt{x} = 2$, but if I search for the $\sqrt{4}$, I get $2$ & $-2$.</p>
| justsomeguy90 | 174,221 | <p>The map z->z^2 is taken to be ramification of order 2. Setting w=z^2 we have that the inverse function z=sqrt(w) has a branch point at zero, so that there is nontrivial monodromy about the origin in the sense of riemann surface theory. In this sense and modern algebraic senses, the square root fuction is a multivalu... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.