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 |
|---|---|---|---|---|
245,049 | <p>I am trying to do a n-round of convolution of a function. The code is posted as below. But it is not working. Is there a solution?</p>
<pre><code>p[x_] := 1/(x + 1)*UnitStep[x]
p1[x_] := Convolve[p[y], p[y], y, x]
p2[x_] := Convolve[p[y], p1[y], y, x]
</code></pre>
<p>p1 succeeded. But the output of p2 only repeats ... | bill s | 1,783 | <p>There are two issues: do you have the form right, and is the integral (the convolution) solvable. Let's do the first one by picking a really simple example, like p[x]:=UnitStep[x]. You can see that your code fails. Because of the way things are defined, you need to change the variables, or they get all messed up. No... |
4,277,616 | <blockquote>
<p>In how many different ways can we arrange <span class="math-container">$120$</span> students into <span class="math-container">$6$</span> groups for <span class="math-container">$6$</span> different classes so that the largest group has at most <span class="math-container">$2$</span> members more than t... | Salmon Fish | 955,791 | <p>For an another approach : When we check over the question , we see that it is distributing distingusiable objects into distinguishable boxes question. By that reason , using exponential generating function is more suitable than using ordinary generating functions.</p>
<p>It is given that the largest class can have a... |
2,838,312 | <p>What is the Fourier transform of $\mathrm{e}^{ik|x|}$? Here, $k > 0$ is real.</p>
<p>I use the definition $$ F(\omega) = \int_{-\infty}^\infty \mathrm{e}^{-i\omega x} f(x) \mathrm{d}x.$$</p>
<p>Thanks!</p>
| José Carlos Santos | 446,262 | <p>Note that if $|z|=r$, then\begin{align}|a_{n-1}z^{n-1}+\cdots+a_1z+a_0|&\leqslant|a_{n-1}|r^{n-1}+\cdots+|a_1|r+|a_0|\\&\leqslant(r-1)(r^{n-1}+\cdots+r+1)\\&=r^n-1\\&<r^n\\&=|z|^n.\end{align}Therefore, by Rouché's theorem, $z^n$ and $p(z)$ have the same number of zeros in that region. But $z^n... |
1,281,627 | <p>Today I completed the chapter of '**Limits **' in my school, and I found this chapter very fascinating. But the only problem I have with limits and Derivatives is that I don't know How can I use it in my daily life. (Any Book Recommendation?)</p>
| user157227 | 157,227 | <p>You shouldn't view limits as a tool to solve problems. Instead, you should view limits as a way to describe situations (or ask more interesting problems).</p>
<p>The derivative is a perfect example of this. If you want to express the idea of "instantaneous rate of change," you are going to use limits to do this.</p... |
997,999 | <p>I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can't understand how these things combine together and actually an integral can be the same time those two things. If the integration is the opposite of differentiation, then t... | Ivo Terek | 118,056 | <p>Integration is not exactly the opposite as differentiation. And the interpretation of the integral as the area under a curve is good for positive functions only, otherwise you must keep your attention for signs. These things combine together by the fundamental theorem of calculus, which states that if $f$ is a conti... |
2,214,236 | <p>The question:</p>
<blockquote>
<p>An object is dropped from a cliff. How far does the object fall in the 3rd second?"</p>
</blockquote>
<p>I calculated that a ball dropped from rest from a cliff will fall $45\text{ m}$ in $3 \text{ s}$, assuming $g$ is $10\text{ m/s}^2$.</p>
<p>$$s = (0 \times 3) + \frac{1}{2}\... | Chen Mo | 431,584 | <p>You are correct. If the question is exactly how you phrased it, the displacement should be 45m down. </p>
<p>We know that:</p>
<p>a (acceleration) = $10m/s^2$</p>
<p>t (time) = 3 seconds</p>
<p>u (initial velocity) = $0m/s$</p>
<p>Hence, using the formula $s=ut-(1/2)at^2$ the answer should be 45 m.</p>
|
30,918 | <p>Judging by some of the posts on meta<sup>1</sup> and comments posted there it seems that there are users who try to improve the posts by correcting spelling mistakes. Of course, there are other ways to improve the posts via editing, some of them probably more important than grammar and spelling.<sup>2</sup> Still th... | Martin Sleziak | 8,297 | <p>Strings which most likely appear in the words which are
misspelled:</p>
<ul>
<li>alegb*; SEDE <a href="//data.stackexchange.com/math/query/1157659/posts-containing-a-given-text-case-insensitive?num=5000&text=alegb" rel="nofollow noreferrer">anywhere</a>, <a href="//data.stackexchange.com/math/query/1157688/sear... |
948,862 | <p>Consider the <a href="http://en.wikipedia.org/wiki/Fenchel's_duality_theorem" rel="noreferrer">Fenchel dual</a> and the <a href="http://en.wikipedia.org/wiki/Duality_(optimization)" rel="noreferrer">Lagrangian dual</a>. </p>
<p>Are these duals equivalent? In other words, is using one of the these duals (say for... | Pushpendre | 52,858 | <p>Let the fenchel dual of a function $f : \mathbb{R}^n \to \mathbb{R}$ be $D_f$. Consider the convex program $P^{+} = \min f(x)$ s.t. $x \ge \boldsymbol{0}$ and let $D_l$ be this program's lagrangian dual then for $\boldsymbol{\lambda} \ge \boldsymbol{0}$ $D_f(\lambda) = -D_l(\lambda)$. So on the positive orthant the ... |
948,862 | <p>Consider the <a href="http://en.wikipedia.org/wiki/Fenchel's_duality_theorem" rel="noreferrer">Fenchel dual</a> and the <a href="http://en.wikipedia.org/wiki/Duality_(optimization)" rel="noreferrer">Lagrangian dual</a>. </p>
<p>Are these duals equivalent? In other words, is using one of the these duals (say for... | theV0ID | 82,092 | <p>From your comments, I understand that your confusion is about the relation of Fenchel and Lagrange duality. So I will focus on that relation. Particularly, I would like to contradict the comment which stated, that Lagrange and Fenchel duality are two categorically different concepts.</p>
<p>Consider the <em>primal ... |
2,831,199 | <p>What is the probability of getting $6$ $K$ times in a row when rolling a dice N times?</p>
<p>I thought it's $(1/6)^k*(5/6)^{n-k}$ and that times $N-K+1$ since there are $N-K+1$ ways to place an array of consecutive elements to $N$ places.</p>
| drhab | 75,923 | <p>In the following for convenience I write $p$ for $\frac{1}{6}$ and $q$ for $\frac{5}{6}$</p>
<p>Let $A_{K,N}$ denote the event that among the first $N$ throws there
is a consecutive row of $K$ sixes.</p>
<p>Let it be that the $F$-th throw is the first throw that does not
give a six, so that $P\left(F=r\right)=p^{r... |
320,378 | <p>Given a number of normal distributions <span class="math-container">$N(\mu_1, \sigma^2), N(\mu_2, \sigma^2), ..., N(\mu_n, \sigma^2)$</span> with fixed variance <span class="math-container">$\sigma^2$</span>, but not necessary equal means. My question is how to approximate the variance given a number of samples of t... | David G. Stork | 89,654 | <p>You can solve this precisely computing the basic integrals. </p>
<p>For two normal distributions <span class="math-container">$N(\mu_1, \sigma_1)$</span> and <span class="math-container">$N(\mu_2, \sigma_2)$</span> the variance is:</p>
<p><span class="math-container">$$\frac{1}{4} \left(\text{$\mu $1}^2-2 \text{$... |
2,951,242 | <p>1) <span class="math-container">$cl(\mathbb R) $</span></p>
<p>2) <span class="math-container">$int ([1, \infty) \cup $</span> {3})</p>
<p>3) <span class="math-container">$ \partial (-1,\infty ) \cap $</span> {-3}
it’s a boundary</p>
<p>My solution:
1) it’s same <span class="math-container">$\mathbb R $</span... | dmtri | 482,116 | <p>In case 2) why do you write that the interior is the intersection of interiors? This may lead you to false , if for example instead of <span class="math-container">$3$</span> were <span class="math-container">$-2$</span>.</p>
|
2,640,477 | <p>According to <a href="https://rads.stackoverflow.com/amzn/click/0073383090" rel="nofollow noreferrer">Rosen</a>, an infinite set A is countable if $|A|= |\mathbb{Z}^+|$ which in turn can be established by finding a bijection from A to $\mathbb{Z}^+$.</p>
<p>Also, a sequence is defined as a function from $\mathbb{Z}... | Community | -1 | <p>As a sequence is a set indexed by the natural numbers, there exists a surjection from the naturals to the set. Let <span class="math-container">$A$</span> be the set and g: <span class="math-container">$\mathbb{N}\rightarrow A$</span> the surjection. Then we can define a map <span class="math-container">$f:A \righta... |
678,073 | <p>I am working with the multiplicative ring of integers modulo $2^{127}$.</p>
<p>Consider the set $E=\{(k,l) \mid 5^k \cdot 3^l \equiv 1\mod 2^{127}, k > 0, l> 0\}$.
I wonder if anybody knows or has an idea where to look for a result related to a lower bound for $M=\min\{k+l \mid (k,l)\in E \}$.</p>
<p>We hav... | coffeemath | 30,316 | <p>Let $k=l=n$ so that $5^k3^l=15^n=(16-1)^n=(2^4-1)^n$. Now take $n=2^{123}$ and apply the binomial theorem, which gives all but the last two terms clearly divisible by $2^{127}$ and the last two terms are
$$-2^{123}\cdot 2^4 +1,$$
which is $1$ mod $2^{127}.$ So here $k+l=2^{124},$ which is a fourth of the value ord(3... |
3,464,282 | <p>I have a heat type equation
<span class="math-container">$$\frac{d}{dt}V + \frac{1}{2} \sigma^{2} S^{2} \frac{d^{2}}{dS^{2}}V + (r-D) S \frac{d}{ds} V - rV = 0$$</span></p>
<p>Am asked to prove the solution is separable
<span class="math-container">$$V=A(t) B(s)$$</span>
and that A(t) is 1st order diff eq and B(S) ... | Donald Splutterwit | 404,247 | <p>We have
<span class="math-container">\begin{eqnarray*}
-\frac{\frac{d}{dt} A(t)}{A(t)} = \frac{1}{2} \sigma^{2} S^{2} \frac{\frac{d^{2}}{dS^{2}} B(S)}{B(S)} + (r-D) S \frac{\frac{d}{dS} B(S)}{B(S)} - r.
\end{eqnarray*}</span>
The RHS is a function of <span class="math-container">$t$</span> only and the LHS is a fun... |
1,439,850 | <p>So the problem states that the centre of the circle is in the first quadrant and that circle passes through $x$ axis, $y$ axis and the following line: $3x-4y=12$. I have only one question. The answer denotes $r$ as the radius of the circle and then assumes that centre is at $(r,r)$ because of the fact that the circl... | haqnatural | 247,767 | <p>$${ x }^{ 2 }\frac { dy }{ dx } +3xy=1\\ \\ { x }^{ 2 }\frac { dy }{ dx } +3xy=0\\ \frac { dy }{ dx } =-3\frac { y }{ x } \\ \int { \frac { dy }{ y } } =-3\int { \frac { 1 }{ x } dx } \\ \ln { \left| y \right| =-3\ln { \left| cx \right| } } \\ y=\frac { C }{ { x }^{ 3 } } \\ y=\frac { C\left( x \right) }{ { x }^... |
2,986,515 | <p>Can anyone help me with this problem? </p>
<p>Prove that for any real number <span class="math-container">$x > 0$</span> and for any <span class="math-container">$M > 0$</span> there is <span class="math-container">$N ∈ \mathbb N$</span> so that if <span class="math-container">$n > N$</span> then <span cla... | Cesareo | 397,348 | <p>Hint.</p>
<p><span class="math-container">$$
\left(a_{n}\right)_{n \in \mathbb{N}} = \dfrac{3n^2 - 9n + 6}{n^3 + 5n^2 + 8n + 4} = \frac{3(n-2)(n-1)}{(n+1)(n+2)^2}
$$</span></p>
|
1,821,186 | <p>Why is the solution of $|1+3x|<6x$ only $x>1/3$? After applying the properties of modulus, I get $-6x<1+3x<6x$. And after solving each inequality, I get $x>-1/9$ and $x>1/3$, but why is $x>-1/9$ rejected? </p>
| Robert Israel | 8,508 | <p>Try balls of radius $1/m$ centred at the points $x_n$ of your dense countable subset.</p>
<p>Edit: to see how this works, consider a ball of radius $r$ around any point $p$. What conditions on $m$ and $x_n$ would guarantee that the ball of radius $1/m$ around $x_n$ contains $p$ and is contained in this ball?</p>
|
428,415 | <p>I tried using integration by parts twice, the same way we do for $\int \sin {(\sqrt{x})}$
but in the second integral, I'm not getting an expression that is equal to $\int x\sin {(\sqrt{x})}$.</p>
<p>I let $\sqrt x = t$ thus,
$$\int t^2 \cdot \sin({t})\cdot 2t dt = 2\int t^3\sin(t)dt = 2[(-\cos(t)\cdot t^3 + \int... | Sri Krishna | 74,110 | <p>you have done till this - $ 2[-\cos(t)\cdot t^3+(\sin(t)\cdot 3t^3 - \int 6t \cdot \sin(t))]]$</p>
<p>again use parts to get </p>
<p>$ 2[-\cos(t)\cdot t^3+(\sin(t)\cdot 3t^3 - 6(t(-\cos t) +\sin(t))$ = </p>
<p>$ 2[-\cos(t)\cdot t^3+(\sin(t)\cdot 3t^3 + 6(t(\cos t) -6\sin(t))$ .
giving you your answer .</p>
<p>... |
428,415 | <p>I tried using integration by parts twice, the same way we do for $\int \sin {(\sqrt{x})}$
but in the second integral, I'm not getting an expression that is equal to $\int x\sin {(\sqrt{x})}$.</p>
<p>I let $\sqrt x = t$ thus,
$$\int t^2 \cdot \sin({t})\cdot 2t dt = 2\int t^3\sin(t)dt = 2[(-\cos(t)\cdot t^3 + \int... | amWhy | 9,003 | <p>Yes, indeed, continue as you did in the comments, treating $\int 6t\sin t \,dt\;$ as a separate integral, use integration by parts, and add (or subtract, if appropriate) that result to your earlier work, and you will end with an expression with no integrals remaining!:</p>
<p>$$\int t^2 \cdot \sin({t})\cdot 2t dt =... |
2,988,089 | <p>Let A, B, C, and D be sets. Prove or disprove the following:</p>
<pre><code> (A ∩ B) ∪ (C ∩ D)= (A ∩ D) ∪ (C ∩ B)
</code></pre>
<p>I am just wondering can i simply prove it using a membership table ( seems to easy ) or do i have to use setbuilder notation?</p>
<p>Thank you!</p>
| Melody | 598,521 | <p>Well it's sometimes true. Example, let all of the sets be <span class="math-container">$\{1\}$</span>.</p>
<p>To show it doesn't always hold we could consider <span class="math-container">$A,B=\{1\}$</span> and <span class="math-container">$C,D=\{0\}$</span>. Then <span class="math-container">$A\cap B=\{1\}$</span>... |
262,745 | <p>I need to find the normal vector of the form Ax+By+C=0 of the plane that includes the point (6.82,1,5.56) and the line (7.82,6.82,6.56) +t(6,12,-6), with A=1.</p>
<p>Of course, this is easy to do by hand, using the cross product of two lines and the point. There's supposed to be an automated way of doing it, though,... | cvgmt | 72,111 | <p><code>RegionWithin</code> can be use to express "the plane that includes the point and the line".</p>
<pre><code>Clear["Global`*"];
pt = {6.82, 1, 5.56};
lineeq = {7.82, 6.82, 6.56} + t {6, 12, -6};
line = ParametricRegion[lineeq // Rationalize, t];
plane = Hyperplane[{a, b, c}, d];
eqs = Region... |
3,789,676 | <p>I am try to calculate the derivative of cross-entropy, when the softmax layer has the temperature T. That is:
<span class="math-container">\begin{equation}
p_j = \frac{e^{o_j/T}}{\sum_k e^{o_k/T}}
\end{equation}</span></p>
<p>This question here was answered at T=1: <a href="https://math.stackexchange.com/questions/9... | samirzach | 262,548 | <p>The cross-entropy loss for softmax outputs assumes that the set of target values are one-hot encoded rather than a fully defined probability distribution at <span class="math-container">$T=1$</span>, which is why the usual derivation does not include the second <span class="math-container">$1/T$</span> term.</p>
<p>... |
4,243,030 | <p>I tried to evaluate the integral <span class="math-container">$$ \oint_c\dfrac{dz}{\sin^2 z}$$</span> where <span class="math-container">$c$</span> is a circle <span class="math-container">$|z|=1/2$</span>. The only pole within <span class="math-container">$c$</span> is <span class="math-container">$z=0$</span> and ... | Greg Martin | 16,078 | <p>You seem to be confusing Cauchy's theorem with its converse. Your example shows that the converse of Cauchy's theorem is false, but that doesn't refute Cauchy's theorem itself.</p>
|
2,502,161 | <p>I'm wondering if it's valid to write the follwing: <span class="math-container">$$\lim_{x \rightarrow \infty}\frac{2}{x^r}=2\lim_{x \rightarrow \infty}\frac{1}{x^r}=2.\frac{1}{\infty}=2.0=0$$</span></p>
<p>I know it's valid to say that <span class="math-container">$\frac{1}{\infty}=0$</span> in limits but I'm not s... | Hans | 64,809 | <p>$$y''=\frac{dy'}{dy}\frac{dy}{dx},$$
$$y''y'y=y'^2\frac{dy'}{dy}y=\frac13\frac{dy'^3}{d\ln y}.$$
You should be able to finish the rest.</p>
|
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | agt | 12,617 | <p>In a first course aiming to introduce differentiable manifolds as the spaces on which do calculus, you could give to the students the notion of connection at least on vector bundles.</p>
<p>In order to reflect on the reason for this choice, I report the words of S.S.Chern closing the introduction of Global Differen... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | David MJC | 10,106 | <p>Differential forms.</p>
<p>Books by Darling (Differential forms and connections) and Madsen-Tornehave (From calculus to cohomology: de Rham cohomology and characteristic classes) may help.</p>
|
857,801 | <p>"If in the obvious equalities $(k+1)^3−k^3=3k^2+3k+1$, for the different values $k=1,2,…,n−1$, we add the left and the right sides separately, we obtain the equation $n^3−1=3σ_n+\frac{3(n−1)n}{2}+n−1$, where $σ_n=1^2+2^2+…+(n−1)^2$."</p>
<p>I'm stuck trying to understand what the author has done in the paragraph be... | Surb | 154,545 | <p>This equality </p>
<p>$$(k+1)^3−k^3=3k^2+3k+1,$$</p>
<p>is true for every $k$, in particular it is true when $k=1$,$k=2$,etc... Writing all these equations, we get</p>
<p>$$\begin{array}{rcl}
(1+1)^3−1^3&=&3\cdot 1^2+3\cdot 1+1\\
(2+1)^3−2^3&=&3\cdot 2^2+3\cdot 2+1\\
&\vdots &\\
((n-2)+1)^... |
1,877,558 | <p>For instance, let $(\mathbb{R}, \mathfrak{T})$ be $\mathbb{R}$ with the usual topology. </p>
<p>Why is that $\mathfrak{T} \times \mathfrak{T}$ is a basis on $\mathbb{R} \times \mathbb{R}$ instead of topology?</p>
<p>It seems that people just take $\mathfrak{T} \times \mathfrak{T}$ as a basis by definition. There m... | drhab | 75,923 | <p>If $X=(|X|,\tau)$ denotes a topological space with underlying set $|X|$ and topology $\tau$ then the collection $\tau\times\tau:=\{U\times V\mid U,V\in\tau\}$ <strong>is by definition</strong> a <a href="https://en.wikipedia.org/wiki/Base_(topology)#Definition_and_basic_properties" rel="nofollow">base</a> for a topo... |
2,782,109 | <blockquote>
<p>If a positive integer $m$ was increased by $20$%, decreased by $25$%, and then increased by $60$%, the resulting number would be what percent of $m$?</p>
</blockquote>
<p>A common step-by-step calculation will take time.</p>
<p>After $20$% increase, $6m/5$.<br>
After $25$% decrease, $9m/10$.<br>
Aft... | Love Invariants | 551,019 | <p>There is no shorter method. Only thing you can do to make it short is do all calculations in the end. Log would've worked but it is for only even increases/decreases and for a long period of time<br>
OR<br>
For an alternative take the number as 1, it doesn't change anything except making expression simpler.</p>
|
2,782,109 | <blockquote>
<p>If a positive integer $m$ was increased by $20$%, decreased by $25$%, and then increased by $60$%, the resulting number would be what percent of $m$?</p>
</blockquote>
<p>A common step-by-step calculation will take time.</p>
<p>After $20$% increase, $6m/5$.<br>
After $25$% decrease, $9m/10$.<br>
Aft... | G Tony Jacobs | 92,129 | <p>The quickest way to do this is to dispense with algebra, and instead of calling your original amount $m$, call it $100$.</p>
<p>A $20\%$ increase turns $100$ into $120$.</p>
<p>A $25\%$ decrease takes off $1/4$ of that, leaving $90$.</p>
<p>A $60\%$ increase adds $6/10$ of $90$, which is $6\times 9$, which is $54... |
2,221,033 | <p>My question is due to <a href="https://en.wikipedia.org/w/index.php?title=Imaginary_number&diff=prev&oldid=175488747" rel="noreferrer">an edit</a> to the Wikipedia article: <a href="https://en.m.wikipedia.org/wiki/Imaginary_number" rel="noreferrer">Imaginary number</a>.</p>
<p>The funny thing is, I couldn'... | Ethan Bolker | 72,858 | <p>I don't think there is a </p>
<blockquote>
<p>complete and formal definition of "imaginary number"</p>
</blockquote>
<p>It's a useful term sometimes. It's an author's responsibility to make clear what he or she means in any particular context where precision matters. If $0$ should count, or not, then the text mu... |
947,358 | <p>Okay $g(x)= \sqrt{x^2-9}$</p>
<p>thus, $x^2 -9 \ge 0$</p>
<p>equals $x \ge +3$ and $x \ge -3$</p>
<p>thus the domains should be $[3,+\infty) \cup [-3,\infty)$ how come the answer key in my book is stating $(−\infty, −3] \cup[3,\infty)$. </p>
| layman | 131,740 | <p>When you get to the point of $x^{2} - 9 \geq 0$, you are definitely allowed to add $9$ to both sides to get $x^{2} \geq 9$, but you are <strong>not</strong> allowed to then take the square root of both sides.</p>
<p>When you have inequalities (like $< , \le, >, \ge$), you can't take the square root of both si... |
1,320,469 | <p>I am working on the following problem</p>
<blockquote>
<p>[R. Vakil] Exercise 19.8.B: Suppose $C$ is a curve of genus $g>1$ over
a field $k$ that is not algebraically closed. Show that $C$ has a
closed point of degree at most $2g-2$ over the base field.</p>
</blockquote>
<p>I have no idea how to do this q... | Bananeen | 165,356 | <p>Pick a point $p\in C$. Consider invertible sheaf $\omega_C (-p)$. In the process of exercise 19.8.A one shows using Riemann-Roch that $h^0 (C, \omega(-p))=g-1$ and so under assumption $g>1$ it has a global section $s$. By the description of exercise 14.2.J, $s$ corresponds to a rational section $t$ of $\omega_C ... |
417,896 | <p>Given a connected smooth manifold <span class="math-container">$M$</span> of dimension <span class="math-container">$m>1$</span>, points <span class="math-container">$p_1,\dots,p_n\in M$</span> and positive values <span class="math-container">$\{d_{i,j};1\leq i<j\leq n\}$</span> satisfying the strict triangle ... | André Henriques | 5,690 | <p>It is not possible to find <span class="math-container">$5$</span> points <span class="math-container">$x_1,\ldots,x_5$</span> on a genus zero Riemannian 2-manifold (a sphere) such that <span class="math-container">$d(x_i,x_j)=1$</span> for all <span class="math-container">$i,j$</span>.</p>
<p>The reason is that the... |
216,021 | <p>Suppose $A$ and $B$ are $n \times n$ matrices. Assume $AB=I$. Prove that $A$ and $B$ are invertible and that $B=A^{-1}$.</p>
<p>Please let me know whether my proof is correct and if there are any improvements to be made.</p>
<p>Assume $AB=I$. Then $(AB)A=IA=A$. So, $A(BA)=AI=A$. Then $BA=I$. Therefore $AB=BA=I... | Brian M. Scott | 12,042 | <p>A good first step would be to look at some of the answers to <a href="https://math.stackexchange.com/questions/3852/if-ab-i-then-ba-i">this question</a>. The accepted one, by Davidac897, is pretty elementary and is probably the place to start. You’re almost certainly not yet ready for Martin Brandenberg’s answer, an... |
216,021 | <p>Suppose $A$ and $B$ are $n \times n$ matrices. Assume $AB=I$. Prove that $A$ and $B$ are invertible and that $B=A^{-1}$.</p>
<p>Please let me know whether my proof is correct and if there are any improvements to be made.</p>
<p>Assume $AB=I$. Then $(AB)A=IA=A$. So, $A(BA)=AI=A$. Then $BA=I$. Therefore $AB=BA=I... | Martin Argerami | 22,857 | <p><strong>Proof #1:</strong> (along the lines mentioned in the comments)</p>
<p>As $AB=I$, you know that $A$ is onto as a linear transformation, because $x=Ix=ABx=A(Bx)$ for any $x\in\mathbb{R}^n$. This implies that $A$ is bijective, being a surjective linear transformation in a finite-dimensional space. So there exi... |
1,596 | <p><a href="https://en.wikipedia.org/wiki/Lenna" rel="nofollow noreferrer">Lenna</a> is commonly used as an example placeholder image. I also recently used it in an <a href="https://mathematica.stackexchange.com/questions/87693/how-to-put-an-imported-image-in-a-disk/87699#87699">answer on the site</a>. However, as the ... | Niki Estner | 242 | <p>First of all, I see @Lightness Races in Orbit's point. (If you don't, imagine you're the only male person in e.g. a painting class, so you feel somewhat isolated and awkward anyway, when you find out what you'll be painting is a nude shot of Burt Reynolds. Wearing a feather boa ;-) You probably wouldn't complain, bu... |
1,596 | <p><a href="https://en.wikipedia.org/wiki/Lenna" rel="nofollow noreferrer">Lenna</a> is commonly used as an example placeholder image. I also recently used it in an <a href="https://mathematica.stackexchange.com/questions/87693/how-to-put-an-imported-image-in-a-disk/87699#87699">answer on the site</a>. However, as the ... | Mr.Wizard | 121 | <p>Seeking the original for context I conclude:</p>
<ol>
<li><p>It seems to me a fairly tasteful photograph. There is similar nudity in many pieces of classical artwork that ostensibly have broad appeal. Not only is the cropped headshot only that but it does not allude to anything that a "reasonable observer" would ... |
2,170,278 | <p>The answer is $\binom {13}1 \binom42 \binom{12}3 \binom 41^3$</p>
<p>I want to break the last term and see what happens. [Struggling with the concept so trying to work with it as much as possible].</p>
<p>$\binom 41^3$ means that $\heartsuit \diamondsuit \spadesuit$ is different from $\diamondsuit \heartsuit \spad... | Fabio Somenzi | 123,852 | <p>I think the problem is that in writing $P(k+1) \rightarrow P(k)$ you are implicitly assuming that $k+1 > 0$. That is, a full statement could be written as</p>
<p>$$ \forall k \in \mathbb{N} \,.\, k > 0 \rightarrow (P(k) \rightarrow P(k-1)) \enspace. $$</p>
<p>Once you get to $0$, you are "off the hook." Fo... |
178,028 | <p>I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.</p>
| Jacob | 37,079 | <p>Here's a hint: try using the difference of squares formula on the left hand side of the equation $x^2 -7y^2 =1$.</p>
|
178,028 | <p>I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.</p>
| Clive Newstead | 19,542 | <p>For $a+b\sqrt{7}$ we seek $x+y\sqrt{7}$ such that $(a+b\sqrt{7})(x+y\sqrt{7})=1$. Expanding these brackets and comparing coefficients gives</p>
<p>$$\begin{align} ax+7by &= 1 \\ bx+ay &= 0 \end{align}$$</p>
<p>It is then just a task of solving for $x$ and $y$.</p>
<p>This can be done using matrices:
$$\be... |
1,261,825 | <p>How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail. </p>
| wythagoras | 236,048 | <p>You have to use the Lambert W function, the inverse of $x e^x$. Using this function, one can find that the inverse of $y=x^x$ is $x=e^{W(\ln(y))}$</p>
|
1,445,702 | <p>i'm a little confused. </p>
<p>1)Which axis is which in 3 Dimensional system?</p>
<p>2)Does it matter if I switch the x-axis to y-axis?</p>
| Senex Ægypti Parvi | 89,020 | <p>The way I understand it is:<br>
$(0\mid 0\mid 0)$ is said to be "the origin."</p>
<p>[b is any non-zero number]
$(b\mid 0\mid 0),(0\mid b\mid 0),(0\mid 0\mid b)$ are said to be on the x-axis, the y-axis and the z-axis, respectively. </p>
<p>[b and c are any non-zero numbers]<br>
$(0\mid b\mid c),(b\mid 0\mid c)... |
366,401 | <p>Let <span class="math-container">$\nu$</span> be the uniform measure on the unit circle <span class="math-container">$\mathbb{S}^1 \subset \mathbb{R}^2$</span>, normalised so that <span class="math-container">$\nu(\mathbb{S}^1) = 1$</span>. Suppose <span class="math-container">$\mu$</span> is a Borel probability mea... | Pierre PC | 129,074 | <p>Here is a second example where a given basis fails to do the job. If <span class="math-container">$\mu$</span> has, say, a continuous positive density, then there a homeomorphism <span class="math-container">$h$</span> that sends <span class="math-container">$(\mathbb S^1,\mu)$</span> to <span class="math-container"... |
88,145 | <p>A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{Z}$ (implicitly as a group scheme) especially when $n$ is a prime power: <a href="https://mathoverflow.net/question... | A Stasinski | 2,381 | <p>Let $G$ be a smooth group scheme of finite type over $\mathbb{Z}/p^{r}$
(one can also do this over any Artinian local ring with perfect residue
field, but then some details below change). For example, $G$ could
be a reductive group scheme over $\mathbb{Z}/p^{r}$. This also includes
Chevalley groups over $\mathbb{Z}$... |
1,738,153 | <p>I know the definition is given as follows:</p>
<p>A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means</p>
<p>Can someone help me gain some intuition for this - perhaps a basic example?</p>
<p>Thanks</p>
| Nate | 91,364 | <p>I have a favorite example, or rather pair of examples that I like to use to motivate looking at representations assuming a bit of familiarity with groups.</p>
<p>The first thing I want to consider is the full symmetry group of a regular tetrahedron, where by full I mean "allowing reflections". In this case it's eas... |
151,937 | <p>In <code>FindGraphCommunities</code>, how can one find the vertices associated with the edges that are found to connect one or more communities?</p>
| Szabolcs | 12 | <p>We'll use this example graph:</p>
<pre><code>g = ExampleData[{"NetworkGraph", "LesMiserables"}]
</code></pre>
<p>Find the edges which are <em>within</em> communities:</p>
<pre><code>subgraphs = EdgeList@Subgraph[g, #] & /@ FindGraphCommunities[g];
</code></pre>
<p>Visualize them:</p>
<pre><code>HighlightGra... |
922,731 | <p>I am stuck on this question, not sure what I am doing wrong:
\begin{align*}
f(x) &=(5-x^2)(\sqrt{x}) \\
\\
f'(x) &=(5-x^2)(x)^{1/2} \\
&=(-2x)(x)^{1/2}+(-x^2+5)(\dfrac{1}{2})(x)^{-1/2}\\
&=-2x^{3/2}+\dfrac{1}{2}(-x^{3/2}+5x^{-1/2})\\
&=\dfrac{1}{2}x^{-1/2}\bigg(x^2-2x^2+10\bigg)\\
&=\dfrac{... | Satish Ramanathan | 99,745 | <p>Answer</p>
<p>In your fourth step:</p>
<p>$f'(x) = \frac{-5}{2}x^{\frac{3}{2}} + \frac{5}{2}x^{\frac{-1}{2}}$</p>
<p>$f'(x) =-\frac{5}{2}\left(\frac{x^2 -1}{\sqrt{x}}\right)$</p>
|
922,731 | <p>I am stuck on this question, not sure what I am doing wrong:
\begin{align*}
f(x) &=(5-x^2)(\sqrt{x}) \\
\\
f'(x) &=(5-x^2)(x)^{1/2} \\
&=(-2x)(x)^{1/2}+(-x^2+5)(\dfrac{1}{2})(x)^{-1/2}\\
&=-2x^{3/2}+\dfrac{1}{2}(-x^{3/2}+5x^{-1/2})\\
&=\dfrac{1}{2}x^{-1/2}\bigg(x^2-2x^2+10\bigg)\\
&=\dfrac{... | mathlove | 78,967 | <p>Note that
$$-2x^{3/2}+\dfrac{1}{2}(-x^{3/2}+5x^{-1/2})\not =\dfrac{1}{2}x^{-1/2}\bigg(x^2-2x^2+10\bigg).$$</p>
<p>We have
$$f'(x)=\cdots=-2x^{3/2}+\dfrac{1}{2}(-x^{3/2}+5x^{-1/2})=\frac 12x^{-\frac 12}(-4x^2-x^2+5)=\frac{5(1-x^2)}{2\sqrt x}.$$</p>
|
3,037,296 | <p>I'm confused of what <span class="math-container">$\sqrt {3 + 4i}$</span> would be after I used quadratic formula to simplify <span class="math-container">$z^2 + iz - (1 + i)$</span></p>
| KCd | 619 | <p>If you want to understand <em>where</em> a formula for the real and imaginary parts of square roots comes from, set <span class="math-container">$(x+yi)^2 = a+bi$</span>, where we know <span class="math-container">$a$</span> and <span class="math-container">$b$</span> we want to find <span class="math-container">$x$... |
1,538,496 | <p>I came across this riddle during a job interview and thought it was worth sharing with the community as I thought it was clever:</p>
<blockquote>
<p>Suppose you are sitting at a perfectly round table with an adversary about to play a game. Next to each of you is an infinitely large bag of pennies. The goal of the... | mvw | 86,776 | <p>Update: OP changed rules at <a href="https://math.stackexchange.com/posts/1538496/revisions">revision 3</a>.</p>
<p>My attempt as spoiler below.</p>
<blockquote class="spoiler">
<p>I move first and cover the whole table with pennies.</p>
</blockquote>
|
1,335,842 | <p>The smallest solution to the above equation for various primes are:</p>
<p>$(p=2)$ $3^2 = 2*2^2 +1$</p>
<p>$(p=3)$ $2^2 = 2*1^2 +1$</p>
<p>$(p=5)$ $9^2 = 5*4^2 +1$</p>
<p>$(p=7)$ $8^2 = 7*3^2 +1$</p>
<p>Is there at least one solution for each prime?
If there is one solution, there are infinite.</p>
| orangeskid | 168,051 | <p>Here is the recipe for finding a (all the ) solution(s) of the equation
$$a^2 - d b^2 = 1$$
where $d>0$ is a square free integer. </p>
<p>Consider the continued fraction expansion of $\sqrt{d}$. It will be of the form
$$\sqrt{d} = [q,\overline{q_1, q_2 , \ldots q_r}]$$
with $q = [\sqrt{d}]$, with the sequence $q... |
1,961,727 | <p>As far as I understood <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram–Schmidt orthogonalization</a> starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.</p>
<p>I have no problem understand... | erfink | 376,021 | <p>If the entire vector space we wanted a basis for all of $\mathbb{R}^n$ or a basis for, say, the $xy$-plane in $\mathbb{R}^3$, then we could certainly do this. The problem arises when we're looking for an orthonormal basis for subspaces that are more complicated.</p>
<p>For example, let take our subspace to be the p... |
1,961,727 | <p>As far as I understood <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram–Schmidt orthogonalization</a> starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.</p>
<p>I have no problem understand... | Wrzlprmft | 65,502 | <p>To give a somewhat practical example, let’s consider the following iterative sequence:</p>
<p>$$ x_{i+1} = \frac{A x_i}{\left |A x_i \right|},$$</p>
<p>Where the $x_i$ are vectors and $A$ is a quadratic matrix with a matching dimension. The first vector $x_1$ shall be random¹.</p>
<p>When $i$ is increased, $x_i$ ... |
3,238,670 | <p>Could someone explain <strong>how to get from: <span class="math-container">$x-\frac{1}{x}=A$</span> to <span class="math-container">$x+\frac{1}{x}=\sqrt{A^2+4}$</span></strong> ? It is one of the Algebra II tricks.</p>
<p>Thanks.</p>
| BladeofDagger | 676,262 | <p>Ok, I got it!</p>
<p><span class="math-container">$x-\frac{1}{x}=A$</span></p>
<p><span class="math-container">$\Bigl(x-\frac{1}{x}\Bigl)^2=A^2$</span></p>
<p><span class="math-container">$x^2+\frac{1}{x^2}-2=A^2$</span></p>
<p><span class="math-container">$x^2+\frac{1}{x^2}=A^2+2$</span></p>
<p><span class="ma... |
3,238,670 | <p>Could someone explain <strong>how to get from: <span class="math-container">$x-\frac{1}{x}=A$</span> to <span class="math-container">$x+\frac{1}{x}=\sqrt{A^2+4}$</span></strong> ? It is one of the Algebra II tricks.</p>
<p>Thanks.</p>
| zwim | 399,263 | <p>Remark that <span class="math-container">$x-\frac 1x=A\iff x^2-Ax-1=0$</span></p>
<p>Which is a quadratics with roots <span class="math-container">$x$</span> and <span class="math-container">$-\frac 1x$</span>.</p>
<p>Calculating <span class="math-container">$\Delta=A^2+4$</span> and roots <span class="math-contai... |
2,939,585 | <p>I want to prove that if <span class="math-container">$ \gamma$</span> is a closed path and <span class="math-container">$\gamma\subseteq B_R(0) $</span> then <span class="math-container">$\mathbb{C}\setminus B_R(0)\subseteq \operatorname{Ext}_\gamma$</span> where <span class="math-container">$ \operatorname{Ext}... | Paul Frost | 349,785 | <p>It is well-known that the winding number is <em>homotopy invariant</em>, i.e. if <span class="math-container">$\gamma_0, \gamma_1 : [0,1] \to \mathbb{C} \setminus \{ a\}$</span> are homotopic closed paths (which means that there exists a continuous map <span class="math-container">$\Gamma : [0,1] \times [0,1] \to \... |
999,147 | <p>I'm looking to gain a better understanding of how the cofinite topology applies to R.
I know the definition for this topology but I'm specifically looking to find some properties such as the closure, interior, set of limit points, or the boundary set and how these change based on whether a subset A in R is closed, ... | Clive Newstead | 19,542 | <p>In the cofinite topology, '$A$ is closed' means '$A$ is finite or is $\mathbb{R}$' and '$A$ is open' means '$A$ has finite complement or is empty'. In particular...</p>
<ul>
<li>If $A \subseteq \mathbb{R}$ is infinite then the only closed set containing $A$ is $\mathbb{R}$, and hence $\operatorname{cl} A = \mathbb{... |
9,934 | <p>I have been given some code with the following line</p>
<pre><code>PeriodicExtension[g_, x_] := If[Abs[x] < Pi, g[x], PeriodicExtension[g, x - 2 Sign[x] Pi]]
</code></pre>
<p>I do not understand the syntax. I would appreciate if someone can explain what this
code does for different values of <code>x</code>.</p>... | J. M.'s persistent exhaustion | 50 | <p>I'm not terribly fond of the use of a recursive solution when a non-recursive approach is easier to look at, so here's a more compact and more general implementation based on <a href="https://math.stackexchange.com/a/63463">this math.SE answer</a> I wrote:</p>
<pre><code>PeriodicExtension[g_, {a_?NumericQ, b_?Numer... |
3,534,364 | <blockquote>
<p><span class="math-container">$x^2y'^2 + 3xyy' +2y^2 = 0 $</span></p>
</blockquote>
<p>Usually, to solve an ODE with respect to <span class="math-container">$y'=p$</span>, we first isolate the <span class="math-container">$y$</span>, to get <span class="math-container">$y = f(x,p)$</span> and then dif... | user577215664 | 475,762 | <p><span class="math-container">$$x^2y'^2 + 3xyy' +2y^2 = 0$$</span>
I suppose <span class="math-container">$y \ne 0$</span>
<span class="math-container">$$x^2 \frac {y'^2}{y^2} + 3x\frac {y'}{y} +2 = 0$$</span>
<span class="math-container">$$x^2 \left ( \frac {y'}{y} \right )^2 + 3x \left ( \frac {y'}{y} \right ) +2 ... |
345,094 | <p>If $f(x-1)+f(x-2) = 5x^2 - 2x + 9$</p>
<p>and</p>
<p>$f(x)= ax^2 + bx + c$</p>
<p>what would be the value of $a+b+c$?</p>
<p>I was doing</p>
<p>$f(x-1)+f(x-2)= f(x-3)$
then
$f(x)$</p>
<pre><code>a = 5
b = -2
c = 9
</code></pre>
<p>$(5-3)+(-2-3)+(9-3)$</p>
<p>But do not think is is correct</p>
<p>What would... | Clayton | 43,239 | <p>We know $$f(x-1)=a(x-1)^2+b(x-1)+c\quad\text{and}\quad f(x-2)=a(x-2)^2+b(x-2)+c.$$ Expand these terms, add them, and combine like terms via powers of $x$. Now you can get three equations in three variables by equating the coefficients of the left with the right since you know $$f(x-1)+f(x-2)=5x^2-2x+9.$$</p>
|
345,094 | <p>If $f(x-1)+f(x-2) = 5x^2 - 2x + 9$</p>
<p>and</p>
<p>$f(x)= ax^2 + bx + c$</p>
<p>what would be the value of $a+b+c$?</p>
<p>I was doing</p>
<p>$f(x-1)+f(x-2)= f(x-3)$
then
$f(x)$</p>
<pre><code>a = 5
b = -2
c = 9
</code></pre>
<p>$(5-3)+(-2-3)+(9-3)$</p>
<p>But do not think is is correct</p>
<p>What would... | cloned | 53,024 | <p>$$f''(x)=2a, 4a=f''(x-1)+f''(x-2)=10, a=2.5$$</p>
<p>$$f'(x)=2ax+b=5x+b, 5(x-1)+5(x-2)+2b$$ </p>
<p>$$f'(x-1)+f'(x-2)=10x-2, -15+2b=-2, b=6.5$$</p>
<p>$$f(x)=ax^2+bx+c, 2.5(x-1)^2+6.5(x-1)+2.5(x-2)^2+6.5(x-2)+2c=f(x-1)+f(x-2)=5x^2-2x+9$$</p>
<p>let $x=0$ on both side, $2.5-6.5+10-13+2c=9, c=8$</p>
<p>so $a+b+... |
3,568,693 | <p>I am trying to solve <span class="math-container">$n! = 10^6$</span> for <span class="math-container">$n$</span>. I thought to do this using the <a href="https://en.wikipedia.org/wiki/Gamma_function" rel="nofollow noreferrer">gamma function</a>:</p>
<p><span class="math-container">$$(n - 1)! = \Gamma(n) = \int_0^\i... | Claude Leibovici | 82,404 | <p>If you look at <a href="https://math.stackexchange.com/questions/430167/is-there-an-inverse-to-stirlings-approximation/461207#461207">this question</a>, you will find a very good approximation of the inverse of the factorial function.</p>
<p>Applied to your case
<span class="math-container">$$n! =y$$</span>it write... |
1,644,845 | <blockquote>
<p>Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist.</p>
</blockquote>
<p>Let $z=r(\cos(\theta)+i \sin(\theta))$. So $\frac{\Re(z)}{z} =\cos ^2(\theta) - i \cos(\theta)\sin(\theta) $, and $$\lim_{z \to 0} \frac{\Re(z)}{z} = \lim_{r \to 0} (\cos ^2(\theta) - i \cos(\theta)\sin(\theta)) = (\cos ... | blizzard22 | 549,349 | <p>I believe you have the denominators switched. Also, say $c_1 = \chi^2_{1, 1-\alpha/2} $ and $c_2 =\chi^2_{1, \alpha/2 } $
then for (c), the expected width of the interval is
$$ \mathbb{E} \left[ n (\bar{X} - \mu )^2 ( \frac{1}{c_2} - \frac{1}{c_1}) \right] =
n ( \frac{1}{c_2} - \frac{1}{c_1}) \times \mat... |
3,356,544 | <p>A lot of calculators actually agree with me saying that it is defined and the result equals 1, which makes sense to me because:</p>
<p><span class="math-container">$$ (-1)^{2.16} = (-1)^2 \cdot (-1)^{0.16} = (-1)^2\cdot\sqrt[100]{(-1)^{16}}\\
= (-1)^2 \cdot \sqrt[100]{1} = (-1)^2 \cdot 1 = 1$$</span></p>
<p>Howev... | zwim | 399,263 | <p>If we stay strictly in the real domain, then exponentiation is defined by the formula <span class="math-container">$x^y=\exp(y\ln(x))$</span>.</p>
<p>Since the logarithm is defined only on positive numbers, exponentiation is also well defined only on positive bases <span class="math-container">$x$</span>.</p>
<p>W... |
1,998,938 | <p>How can I solve \begin{cases}
u_t-u_{xx}=0,&\text{if $0<x<1, t>0$}\\
u(0,t)=u(1,t)=0, & \text{if $t>0$}\\u(x,0)=u_0(x), &\text{if $x\in(0,1)$} \end{cases}</p>
<p>where $$u_0=min(x,1-x)$$</p>
| reuns | 276,986 | <p>Let $$F_k(s) = s\int_0^\infty \{x\}^k x^{-s-1}dx = s \int_0^\infty \left\{\frac1t\right\}^k t^{s-1}dt$$</p>
<p>Show that $$\int_0^x \{t\}^kdt = \frac{\lfloor x \rfloor +\{x\}^{k+1}}{k+1}$$</p>
<p><a href="https://en.wikipedia.org/wiki/Abel%27s_summation_formula" rel="nofollow noreferrer">And that (where it converg... |
1,998,938 | <p>How can I solve \begin{cases}
u_t-u_{xx}=0,&\text{if $0<x<1, t>0$}\\
u(0,t)=u(1,t)=0, & \text{if $t>0$}\\u(x,0)=u_0(x), &\text{if $x\in(0,1)$} \end{cases}</p>
<p>where $$u_0=min(x,1-x)$$</p>
| Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic... |
3,366,064 | <p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. </p>
<p>My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with van... | J. W. Tanner | 615,567 | <p>Assuming the spoon is a hemisphere with radius <span class="math-container">$R$</span>, </p>
<p>let <span class="math-container">$x$</span> be the height from the bottom of the spoon, and let <span class="math-container">$h$</span> range from <span class="math-container">$0$</span> to <span class="math-container">$... |
3,366,064 | <p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. </p>
<p>My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with van... | Mohammad Riazi-Kermani | 514,496 | <p>Without loss of generality we assume the radius of the sphere to be <span class="math-container">$1$</span></p>
<p>The volume of the liquid is found by an integral <span class="math-container">$$V= \int _{-1}^{-1+h} \pi (1-y^2 )dy$$</span></p>
<p>and you want the volume of the liquid to be half of the hemisphere w... |
3,987,470 | <p>I am reading about how a wrong formulation of the tower of Hanoi and the inductive hypothesis can lead to a dead-end.<br />
The example I am reading states the following:</p>
<blockquote>
<p>The task is to move N discs from a <em>specific</em> pole to another
<em>specific</em> pole. Assume there are poles <span clas... | Brian M. Scott | 12,042 | <p>You are correct: the author’s explanation of the problem with the second part is rubbish. He should instead have pointed to the problems that you saw:</p>
<ul>
<li>The induction hypothesis is incomplete, since it pretty clearly must include a requirement that pole <span class="math-container">$C$</span> be available... |
4,227,536 | <blockquote>
<p>Let <span class="math-container">$X$</span> be the product space <span class="math-container">$\Bbb R^{\Bbb R}$</span>. Let <span class="math-container">$A \subset X $</span> be the set of all characteristic functions of finite sets. Show that the constant map <span class="math-container">$g, g(x) = 1$<... | cohomonoid | 586,016 | <p>Let's assume the question is in the context of the product topology on the Cartesian product <span class="math-container">$\mathbb{R}^\mathbb{R} = \prod_{x\in\mathbb{R}} \mathbb{R}$</span> based on the usual metric topology on <span class="math-container">$\mathbb{R}$</span>. Then convergence in <span class="math-c... |
2,912,570 | <p>Let $X$ and $Y$ be two standard normal distributions with correlation $-0.72$. Compute $E(3X+Y\mid X-Y=1)$.</p>
<p>My solution: Conditioning on $X-Y=1$, we have $E(3X+Y\mid X-Y=1) = E(4Y+3\mid X-Y=1) = 3+4E(Y\mid X-Y=1) = 3$.</p>
<p>(1) Is my solution correct? My intuition is that the conditional density of $Y$ re... | grand_chat | 215,011 | <p>A hint for your question (2): Define variables $U:=X+Y$ and $V:=X-Y$. Check that $\operatorname{Cov}(U,V)=0$. Since $U$ and $V$ are jointly Gaussian, a covariance of zero implies that $U$ and $V$ are independent. Now express $E(Y\mid X-Y)$ in terms of $U$ and $V$: $$E(Y\mid X-Y)=E\left(\textstyle\frac12(U-V)\mid V\... |
627,258 | <p>Helly everybody,<br>
I'm trying to find another approach to topology in order to justify the axiomatization of topology. My idea was as follows:</p>
<p>Given an <strong>arbitrary</strong> collection of subsets of some space: $\mathcal{C}\in\mathcal{P}^2(\Omega)$<br>
Define a closure operator by: $\overline{A}:=\big... | user119107 | 119,107 | <p>If the whole space $X$ is not open, then constant functions might not be continuous, depending on the definition chosen for "continuous."</p>
|
496,255 | <p>Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact? </p>
| lhf | 589 | <p>No, integers of the form $4n+3$ cannot be written as a sum of two squares.</p>
<p>To prove this, consider $z=x^2+y^2$ modulo $4$ and you'll see that you cannot get $3$.</p>
|
1,985,402 | <p>I wrote down $$12 \times 0 = 0$$ Then, I divided both sides by $0$ like so:
$$12 = \frac {0}{0}$$ I know that $$ \frac {0}{x} = 0, x \in R$$
Therefore, $$12=0$$ which is a false statement. Where did I go wrong?</p>
| Lolas-Poaras | 382,736 | <p>You cannot divide anything by 0. That is just not possible unless you want an undefined answer.</p>
|
1,218,140 | <p>I am reading Hartshorne's proof of $\mathbb{P}^1$ being simply connected as a scheme. It seems one ingredient of the proof is that if $X\rightarrow\mathbb{P}^1$ is an étale covering, then X has only finitely many connected components. But I do not see why.</p>
<p>Thanks in advance.</p>
| Keenan Kidwell | 628 | <p>Since $f$ is finite, in particular of finite type, $X$ is a Noetherian scheme. Noetherian schemes are locally connected (see <a href="http://stacks.math.columbia.edu/tag/04MF" rel="nofollow">http://stacks.math.columbia.edu/tag/04MF</a>), so the connected components of $X$ are open. They therefore form a covering of ... |
3,079,493 | <p>Let <span class="math-container">$$D_6=\langle a,b| a^6=b^2=1, ab=ba^{-1}\rangle$$</span> <span class="math-container">$$D_6=\{1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}$$</span></p>
<p>I would like to compute its character table and its irreducible representations.</p>
<p>I will explain what I have done so fa... | user3482749 | 226,174 | <ol>
<li><p>It has four elements, so yes. </p></li>
<li><p>Get the characters by messing around with orthogonality, then come up with a representation that does that. </p></li>
<li><p>Label the vertices of your hexagon <span class="math-container">$a_1$</span> through <span class="math-container">$a_6$</span>. Then tak... |
3,265,835 | <p>I came across some equation in physics which had a different kind of integration. Like it should have <span class="math-container">$dx2$</span> but had <span class="math-container">$d2x$</span> . And I did some substitution for solving it like putting <span class="math-container">$x= u^2$</span> and then double diff... | user66081 | 66,081 | <p>I guess we are dealing here with <span class="math-container">$\int \int \mathbb{x} \, d^2 \mathbb{x}$</span>, where <span class="math-container">$\mathbb{x}$</span> is a vector. Another notation for this would be
<span class="math-container">$$
\int \int \left( x \atop y \right) dx dy
.
$$</span></p>
<p>This quant... |
3,750,747 | <blockquote>
<p><span class="math-container">$$\frac{\cos^2\left(\dfrac\pi2 \cos\theta\right)}{\sin^2\theta} = 0.5$$</span></p>
</blockquote>
<p>I want to solve the above equation for <span class="math-container">$\theta$</span> in order to find its value, but I am stuck.</p>
<p>Could anyone enlighten me by a method to... | Claude Leibovici | 82,404 | <p>This is a transcendental equation; then no analytical solutions and numerical methods are required.</p>
<p>Making the problem more general, you want to solve for <span class="math-container">$x$</span> the equation
<span class="math-container">$$y=\cos ^2\left(\frac{\pi}{2} \cos (x)\right) \csc ^2(x)\qquad \text{w... |
3,750,747 | <blockquote>
<p><span class="math-container">$$\frac{\cos^2\left(\dfrac\pi2 \cos\theta\right)}{\sin^2\theta} = 0.5$$</span></p>
</blockquote>
<p>I want to solve the above equation for <span class="math-container">$\theta$</span> in order to find its value, but I am stuck.</p>
<p>Could anyone enlighten me by a method to... | Quanto | 686,284 | <p>Let <span class="math-container">$x =\cos\theta$</span> to simplify the equation to</p>
<p><span class="math-container">$$\cos(\pi x) +x^2=0$$</span></p>
<p>which has the trivial roots <span class="math-container">$\pm 1$</span> (excluded due to <span class="math-container">$\sin\theta \ne 0$</span>), as well as the... |
4,321,604 | <p>I have come across an expression like this,</p>
<p><span class="math-container">$$ \frac{f(x) + f(a)}{2\sqrt{f(x)f(a)}}\,\delta(x-a), $$</span></p>
<p>where I expected to find just <span class="math-container">$\delta(x-a)$</span>. When I thought about it, though, I realised maybe... they are identical? Because both... | PrincessEev | 597,568 | <p>Not quite. Bear in mind that</p>
<p><span class="math-container">$$\frac{f(x) + f(a)}{2\sqrt{f(x)f(a)}} \Bigg|_{x=a} = \frac{2 f(a)}{2 \sqrt{ f(a)^2 }} = \frac{f(a)}{|f(a)|} = \begin{cases}
1 & f(a) > 0 \\
-1 & f(a) < 0\end{cases}$$</span></p>
<p>since <span class="math-container">$\sqrt{z^2} = |z|$</s... |
2,877,833 | <p><a href="https://i.stack.imgur.com/9wqM5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9wqM5.png" alt="enter image description here"></a></p>
<p>Look at this part:</p>
<blockquote>
<p>Define the vector $p = -\nabla f(x^*)$ and note that $p^T\nabla f(x^*)
= -||\nabla f(x^*)||^2 <0$. Becau... | Theoretical Economist | 388,944 | <p>Let $p \in \mathbb R^n$, and $g:\mathbb R^n \to \mathbb R$, so that we can write $g(x) = (g_1(x),g_2(x),\ldots,g_n(x))$. Write $$G(x) = p^Tg(x) = p_1g_1(x) + p_2g_2(x) + \cdots + p_ng_n(x).$$</p>
<p>If $g$ is continuous, then so each each $g_i$. Hence, $G$ inherits continuity from $G$. Now take $p$ as defined in yo... |
313,030 | <p>I often find myself writing a definition which requires a proof. You are defining a term and, contextually, need to prove that the definition makes sense. </p>
<p>How can you express that? What about a definition with a proof?</p>
<p>Sometime one can write the definition and then the theorem. But often happens tha... | Nik Weaver | 23,141 | <p>I disagree with the implication that it's always necessary, when trying to write something to be understood, that every sentence has to be a logical consequence of previous sentences or assumed background knowledge.</p>
<p>If you're careful about it, and you say this is what you're doing, I think it can be pedagogi... |
356,353 | <p>I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units in $\mathcal{O}_{\mathbb{Q}(\sqrt{n})}$, and their norms satisfy the desired equation. Although this is a nice connect... | Will Jagy | 10,400 | <p>EDIT: there have been many comments on my answers asking about the use of the word <strong>automorph</strong>. This is a real thing! I did not just make up a word. For a fixed quadratic form, you get a <strong>group of integral automorphs</strong>. In just two variables, there is a good recipe for finding all. In th... |
1,315,744 | <p>Already I know that harmonic series, $$\sum_{k=1}^n\frac1k $$ is divergent series.</p>
<p>And, it is also divergent by Abel Sum or Cesaro Sum.</p>
<p>However, I do not know how to prove it is divergent by concept of Abel or Cesaro.</p>
<p>Abel Sum or Cesaro Sum do not exist in this problem.</p>
<p>But, how can I... | JignEsh Shingod | 887,397 | <p>Since the term of the Harmonic series are of O(1/n). Suppose, if the Harmonic series is Cesaro summable, then by Hardy's theorem, the series is convergent, but Harmonic series is divergent. So, Harmonic series is not Cesaro summable.</p>
|
2,781,153 | <p>I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother.</p>
<blockquote>
<p>Prove that when you divide the area of the circle by the area of the triangle that you will ... | Love Invariants | 551,019 | <p>Hint: Find the side of the triangle using pythagoras theorem.<br>
$hypotenuse=2r$</p>
|
2,781,153 | <p>I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother.</p>
<blockquote>
<p>Prove that when you divide the area of the circle by the area of the triangle that you will ... | Henry | 6,460 | <p>Hints:</p>
<ul>
<li><p>You need to calculate the height and width in terms of $r$: there are two ways of doing this which will give the same area</p></li>
<li><p>You need to divide the area of the circle by the area of the triangle </p></li>
</ul>
<p><a href="https://i.stack.imgur.com/ZGA1C.png" rel="nofollow nore... |
2,781,153 | <p>I've a right triangle that is inscribed in a circle with radius $r$ the hypotunese of the triangle is equal to the diameter of the circle and the two other sides of the triangle are equal to eachother.</p>
<blockquote>
<p>Prove that when you divide the area of the circle by the area of the triangle that you will ... | Jan Eerland | 226,665 | <p>Well, first of all let's summarise the things we can say about this problem and after that we can set up a system of equations.</p>
<blockquote>
<ol>
<li>By the <a href="https://en.wikipedia.org/wiki/Pythagorean_theorem" rel="nofollow noreferrer">Pythagorean theorem</a>, we can write:
<span class="math-contai... |
1,137,079 | <p>I'm new to the concept of complex plane. I found this exercise:</p>
<blockquote>
<p>Let $z,z_1,z_2\in\mathbb C$ such that $z=z_1/z_2$. Show that the length of $z$ is the quotient of the length of $z_1$ and $z_2$.</p>
</blockquote>
<p>If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ then $|z_1|=\sqrt{x_1^2+y_1^2}$ and $|z_2|... | JMP | 210,189 | <p>Let $z_1=r.e^{i\theta}$ and $z_2=R.e^{i\alpha}$. The result follows easily.</p>
|
757,049 | <p>as the title suggests, I need help proving that the cardinality of $(0,1)$ and $[0,1]$ are the same. </p>
<p>Here is my work: </p>
<p>$f:[0,1] \rightarrow (0,1)$</p>
<p>Let $n\in N$</p>
<p>Let $A=\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4}....\}\cup \{0\}$</p>
<p>On $[0,1]\in A: f(x)=x$</p>
<p>On $A: f(0)=\frac{1}... | Michael Weiss | 79,741 | <p>I think you're getting a little too bogged down in the details. The basic idea is that the sets $$A_0=\{a_0,a_1,a_2\ldots,a_n,\ldots\}\text{ and }A_1=\{a_1,a_2,a_3\ldots,a_n,\ldots\}$$ have the same cardinality, where $\{a_0,a_1,a_2\ldots\}$ is any sequence of distinct numbers. So $$B\cup A_0\text{ and }B\cup A_1$$ ... |
1,141,632 | <p>I need to answer a question on fractals from the book <em>Fractals Everywhere</em> by M. Barsley and I have been struggling with it for a while:</p>
<p>Use collage theorem to help you find an IFS consisting of two affine maps in $\mathbb{R}^2$ whose attractor is close to this set:
<img src="https://i.stack.imgur.co... | Mark McClure | 21,361 | <p>If you have Mathematica, you can generate a good approximation to the set like so:</p>
<pre><code>Import["https://sites.google.com/a/unca.edu/mark-mcclure/IteratedFunctionSystems.m"]
IFS = {
{{{0.980019, -0.153438}, {0.25526, 0.88033}}, {54.4695, -50.3965}},
{{{-0.131103, -0.0796567}, {0.123349, -0.136077}}, {9... |
3,464,291 | <blockquote>
<p>If <span class="math-container">$x,y,z>0.$</span> Then minimum value of</p>
<p><span class="math-container">$x^{\ln(y)-\ln(z)}+y^{\ln(z)-\ln(x)}+z^{\ln(x)-\ln(y)}$</span></p>
</blockquote>
<p>what i try</p>
<p>Let <span class="math-container">$\ln(x)=a,\ln(y)=b.\ln(z)=c$</span></p>
<p>So <span class=... | gt6989b | 16,192 | <p>After your substitution you get what seems like a more manageable
<span class="math-container">$$
f(a,b,c) = e^{a(b-c)} + e^{b(c-a)} + e^{c(a-b)}
$$</span>
and you can now minimize easily using the standard techniques.</p>
|
2,794,715 | <p>Is it right that</p>
<p><strong>$$\sqrt[a]{2^{2^n}+1}$$</strong></p>
<p>for every $$a>1,n \in \mathbb N $$ </p>
<p>is always irrational?</p>
| Rhys Hughes | 487,658 | <p>$$a=1\to 2^{2^n}\in \Bbb Q$$
$$a=2\to \sqrt{2^{2^n}} \in \Bbb Q$$</p>
|
3,991,105 | <p>I understand the concept of one coordinate moving while the rest don't change, however I can't make up the exact mapping that would prove this. Can anyone give me the concrete mapping?</p>
| Anonymath | 875,272 | <p>Let <span class="math-container">$(a,b,c)$</span>, <span class="math-container">$(d,e,f)$</span> be two points in <span class="math-container">$\mathbb{R}^3\setminus\mathbb{Q}^3$</span>. Then at least one of <span class="math-container">$a,b,c$</span> is not rational. By symmetry, we may assume WLOG that it is <span... |
3,991,105 | <p>I understand the concept of one coordinate moving while the rest don't change, however I can't make up the exact mapping that would prove this. Can anyone give me the concrete mapping?</p>
| José Carlos Santos | 446,262 | <p>Take <span class="math-container">$(a_1,a_2,a_3),(b_1,b_2,b_3)\in\Bbb R^3\setminus\Bbb Q^3$</span>. Assume, for instance, that <span class="math-container">$a_1,b_3\notin\Bbb Q$</span>. Then consider<span class="math-container">$$\begin{array}{rccc}\gamma\colon&[0,2]&\longrightarrow&\Bbb R^3\\&t&... |
2,261,410 | <blockquote>
<p>The generating function for a Bessel equation is:</p>
<p>$$g(x,t) = e^{(x/2)(t-1/t))}$$</p>
<p>Using the product $g(x,t)\cdot g(x,-t)$ show that:</p>
<p>a) $$[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$$</p>
<p>and consequently:</p>
<p>b)</p>
<p>$$|J_0(x)|\le 1, \... | Paramanand Singh | 72,031 | <p>I gave some comments on the accepted answer to help clarify some doubts raised by OP. I think it is proper to avoid too much discussion in comments and hence I put all that explanation into an answer.</p>
<p>The generating function $g(x, t) = e^{(x/2)(t-(1/t))}$ has the property that $g(x, - t) =g(x, 1/t)$ and this... |
4,309,247 | <p>My question comes from an exercise in Shilov's <em>Linear Algebra</em>. His hint is to use induction, but I'm struggling to get anywhere. I looked through the book and couldn't find any theorem that seemed useful, so I'm guessing there is some sort of manipulation I must be missing? A good first step to take would b... | DreamAR | 983,565 | <p><span class="math-container">$AB=I+BA,$</span> so <span class="math-container">$A^2B=A+(AB)A=2A+BA^2.$</span> This gives you the proof when <span class="math-container">$m=2.$</span></p>
<p>Assume you has the proposition when <span class="math-container">$m=k-1,$</span> try to prove the case when <span class="math-c... |
3,060,742 | <p><span class="math-container">$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = 1.644934$</span> or <span class="math-container">$\frac{\pi^2}{6}$</span></p>
<p>What if we take every 3rd term and add them up? </p>
<p>A = <span class="math-container">$ \frac{1}{3^2} + \fra... | User3910 | 9,235 | <p>KM101 deleted his hint... not sure why.</p>
<p><span class="math-container">$\frac{1}{3^2}+\frac{1}{6^2}+\frac{1}{9^2}+\dots=\frac{1}{3^2 1^2}+\frac{1}{3^2 2^2}+\frac{1}{3^2 3^3}+\dots=\frac{1}{9}(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots)= \frac{\pi^2}{54}$</span></p>
<p><span class="math-container">$\frac{... |
642,443 | <p>Let $\{a_n\}_{n\ge1}^{\infty}=\bigg\{\cfrac{1}{1\cdot3}+\cfrac{1}{2\cdot4}+\dots+\cfrac{1}{n\cdot(n+2)}\bigg\}$. Find $\lim_{n\to \infty}{a_n}$.</p>
<p>I write:
$$\lim_{n\to \infty}{a_n}=\sum_{n=1}^{\infty}{\frac{1}{n\cdot(n+2)}}=\sum_{n=1}^{\infty}{\frac{1}{n^2+2n}}\approx\sum_{n=1}^{\infty}{\cfrac{1}{n^2}}$$</p>
... | Hagen von Eitzen | 39,174 | <p>Telescope! Note that $$ \frac1{n(n+2)}=\frac12\cdot\left(\frac1n-\frac1{n+2}\right)$$</p>
|
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