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 |
|---|---|---|---|---|
23,471 | <p>I'm trying to find an explanation for the different sizes I'm seeing for fonts added to graphics in different ways, and haven't yet located an easy to understand explanation. Here's a minimal example:</p>
<pre><code>Graphics[
{LightGray,
Rectangle[{0, 0}, {72, 72}],
Red,
Style[
Text["Hig", {0, 0... | Sjoerd C. de Vries | 57 | <p>I will not duplicate geordie's explanation of the scaling-with-graphics-resize part of the question.</p>
<p>The reason the displayed font looks too small is the setting of the "ScreenResolution" option (part of <code>FontProperties</code>) to 72, which used to be the default for decades, but is incorrect for most s... |
216,031 | <p>Using image analysis, I have found the positions of a circular ring and imported them as <code>xx</code> and <code>yy</code> coordinates. I am using <code>ListInterpolation</code> to interpolate the data:</p>
<pre><code>xi = ListInterpolation[xx, {0, 1}, InterpolationOrder -> 4, PeriodicInterpolation -> True,... | Michael E2 | 4,999 | <p>An approach to this problem may be found by searching for "person curve" on this site. In particular, I will adapt <a href="https://mathematica.stackexchange.com/a/17780/4999">this answer by @SimonWoods</a> and <a href="https://mathematica.stackexchange.com/a/19200/4999">this answer by @J.M.</a> to <a href="https://... |
2,278,018 | <p>Let $\;\;\displaystyle \sum_{n=1}^\infty U_n\;$ be a divergent series of positive real numbers.</p>
<p>Then, show that the series $\;\displaystyle\sum_{n=1}^\infty \dfrac{U_n}{1+U_n}\;$ is divergent.</p>
<p>Is there is any easy method to prove it? </p>
| hamam_Abdallah | 369,188 | <p>Suppose that $U_n\geq 0$ and that $\sum \frac {U_n}{1+U_n}$ is convergent.</p>
<p>$$\begin{aligned}\sum\left(1-\frac {1}{1+U_n}\right)& convergent\\
&\Rightarrow\lim_{n\to+\infty}\left(1-\frac {1}{1+U_n}\right)=0\\
&\Rightarrow\lim_{n\to+\infty} U_n=0\\
\implies \frac {U_n}{1+U_n}\sim \frac {U_n }{1}
\e... |
1,868,797 | <blockquote>
<p><strong>Question:-</strong></p>
<p>Three points represented by the complex numbers $a,b$ and $c$ lie on a circle with center $O$ and radius $r$. The tangent at $c$ cuts the chord joining the points $a$ and $b$ at $z$. Show that $$z=\dfrac{a^{-1}+b^{-1}-2c^{-1}}{a^{-1}b^{-1}-c^{-2}}$$</p>
</blockq... | Kemei Kimutai | 1,042,350 | <p>The above question is not degenerate since if you work out this by simplex method, <span class="math-container">$X_1$</span> is not a basic variable so it can take a value <span class="math-container">$>=0$</span>. In the final tablue, the solution could be degenerate if either <span class="math-container">$X_2$<... |
103,545 | <p>It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$. </p>
<p>In their 1995 <a href="http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf">paper</a> "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the tim... | Stopple | 6,756 | <p>It's not true for any old modular form. Since the forms live in a vector space over $\mathbb C$, you can achieve any complex number as a coefficient.</p>
<p>Here's a partial answer to what is true. You need to have a cusp form that is an eigenfunction of the Hecke operators, normalized so the leading coefficient ... |
176,893 | <p>Suppose I have a polynomial
$$
p(x)=\sum_{i=0}^n p_ix^i.
$$
For simplicity furthermore assume $p_n=1$. </p>
<p>As it is well known we may use Gershgorin circles to give an upper bound for the absolute values of the roots of $p(x)$. The theorem states that all roots are contained within a circle with radius
$$
r=\ma... | Robert Israel | 13,650 | <p>Since the coefficients are $\pm 1$ times sums of products of the roots, this is obvious: for a polynomial of degree $d$ with all roots of absolute value $\le r$,</p>
<p>$$ |p_i| \le {d \choose i} r^{d-i} $$</p>
<p>The maximum of the right side is at $i = \left\lfloor \dfrac{d-r}{1+r} \right\rfloor$ or $\left\lceil... |
1,225,655 | <p>Find the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form.</p>
<p>I don't know how can I get the tangent line, without a given equation!!, this is part of cal1 classes.</p>
| Emilio Novati | 187,568 | <p>If we suppose that your curve is the graph of a function $y=f(x)$ such that $f(0) = \sqrt{\pi/2}$, than the equation of the tangent at $x=0$ is:</p>
<p>$
y-\sqrt{\pi/2}=f'(0)(x-0)
$</p>
<p>i.e.</p>
<p>$y=f'(0) x+\sqrt{\pi/2}$</p>
|
813,825 | <p>In strong Induction for the induction hypothesis you assume for all K, p(k) for k
<p>If for example I am working with trees and not natural numbers can I still use this style of proof?</p>
<p>For example if I want my induction hypothesis to be that p(k) for k < n where n is a node in the tree and everything sma... | Robert Israel | 8,508 | <p>Yes, strong induction will work on a partially ordered set such as a tree, as long as there are no infinite downward chains
$a_1 > a_2 > a_3 > \ldots$.</p>
|
25,285 | <p>I am just looking for a basic introduction to the Podles sphere and its topology. All I know is that it's a q-deformation of $S^2$. </p>
| mathphysicist | 2,149 | <p>I am not an expert but perhaps the paper <a href="http://www.fsr.ac.ma/GNPHE/ajmpvolume3-2006/ajmp0607.pdf" rel="nofollow">Spectral geometry: case of quantum spheres</a> by Andrzej Sitarz could serve as a reasonably good starting point.</p>
|
1,000,448 | <p>One of the $x$-intercepts of the function $f(x)=ax^2-3x+1$ is at $x=-1$. Determine $a$ and the other $x$-intercept.</p>
<p>I happen to know that $a=-4$ and the other $x$-intercept is at $x=\frac{1}{4}$ but I don't know how to get there. I tried substituting $x=-1$ into the quadratic formula.</p>
<p>$$
-1=\frac{-(-... | Sammy Black | 6,509 | <p>If $r$ and $s$ are roots of $f(x) = ax^2 + bx + c$ then $x-r$ and $x-s$ are <a href="http://en.wikipedia.org/wiki/Factor_theorem" rel="nofollow">factors</a> of $f(x)$. Therefore,
$$
f(x) = a(x - r)(x - s) = a \bigl( x^2 - (r + s)x + rs \bigr),
$$
so
$$
\left\{
\begin{align}
-a(r + s) &= b \\
ars &= c
\end{a... |
481,173 | <p>The most common way to find inverse matrix is $M^{-1}=\frac1{\det(M)}\mathrm{adj}(M)$. However it is very trouble to find when the matrix is large.</p>
<p>I found a very interesting way to get inverse matrix and I want to know why it can be done like this. For example if you want to find the inverse of $$M=\begin{b... | Brian M. Scott | 12,042 | <p>This is a very standard method; if you discovered it on your own, congratulations! It works because each of the elementary row operations that you’re performing is equivalent to multiplication by an <a href="http://en.wikipedia.org/wiki/Elementary_matrix">elementary matrix</a>. To convert $A$ to $I$, you perform som... |
3,196,797 | <p>Suppose <span class="math-container">$gcd(m,n)=1$</span>, and let <span class="math-container">$F :Z_n→Z_n$</span> be defined by <span class="math-container">$F([a])=m[a]$</span>. Prove that <span class="math-container">$F$</span> is an automorphism of the additive group <span class="math-container">$Z_n$</span>. I ... | Lada Dudnikova | 477,927 | <p>I shall prove two things</p>
<ol>
<li>The solution of <span class="math-container">$am = 0 \ (mod \ n)$</span> is trivial </li>
<li>(1) means that the map is injective.</li>
</ol>
<hr>
<p>Let there a non-zero element s.t. <span class="math-container">$am = 0$</span> modulo <span class="math-container">$n$</span>... |
2,668,826 | <p>I am stuck on this result, which the professor wrote as "trivial", but I don't find a way out.</p>
<p>I have the function </p>
<p>$$f_{\alpha}(t) = \frac{1}{2\pi} \sum_{k = 1}^{+\infty} \frac{1}{k}\int_0^{\pi} (\alpha(p))^k \sin^{2k}(\epsilon(p) t)\ dp$$</p>
<p>and he told use that for $t\to +\infty$ we have:</p>... | Jack D'Aurizio | 44,121 | <p>We may consider that for any $\alpha>0$
$$ \frac{d}{d\alpha}\int_{0}^{+\infty}\frac{\log(1+x^\alpha)}{(1+x^2)\log x}\,dx =\int_{0}^{+\infty}\frac{x^{\alpha}}{(1+x^2)(1+x^{\alpha})}\,dx=\frac{\pi}{2}-\int_{0}^{+\infty}\frac{dx}{(1+x^2)(1+x^{\alpha})}$$
and with or without the Beta function it is well-known that $\... |
2,707,514 | <p>I'm reading <em>A First Course in Modular Forms</em> by Diamond and Shurman and am confused on a small point in Chapter 2. Let <span class="math-container">$\Gamma$</span> be a congruence subgroup of <span class="math-container">$\operatorname{SL}_2(\mathbb Z)$</span>. <span class="math-container">$\gamma \in \ma... | Lee Mosher | 26,501 | <p>Since each elliptic fixed point of an element of $\Gamma$ is an elliptic fixed point of an element of $SL_2(\mathbb{Z})$, it suffices to prove the stronger statement that the elliptic fixed points of $SL_2(\mathbb{Z})$ form a discrete set. And in fact what I'll prove is even stronger, they form a discrete <em>closed... |
1,624 | <p>For example, to change the color of each pixel to the mean color of the three channels, I tried</p>
<pre><code>i = ExampleData[{"TestImage", "Lena"}];
Mean[i]
</code></pre>
<p>but it just remains unevaluated:</p>
<p><img src="https://i.stack.imgur.com/K1RRR.png" alt="enter image description here"></p>
<p>How can... | acl | 16 | <p>Certainly. For instance, here's how to reduce the number of colours to 10 (randomly chosen in RGB space):</p>
<pre><code>i = Import["ExampleData/lena.tif"]
</code></pre>
<p><img src="https://i.stack.imgur.com/4uKUl.png" alt="Mathematica graphics"></p>
<p>You can try <code>ImageData[i]</code> to see the actual RGB... |
90,263 | <p>Let $\mathcal{E} = \lbrace v^1 ,v^2, \dotsm, v^m \rbrace$ be the set of right
eigenvectors of $P$ and let $\mathcal{E^*} = \lbrace \omega^1 ,\omega^2,
\dotsm, \omega^m \rbrace$ be the set of left eigenvectors of $P.$ Given any two
vectors $v \in \mathcal{E}$ and $ \omega \in \mathcal{E^*}$ which correspond to
t... | BDH | 24,315 | <p>With a completely eclectic sense of culture within a school, there's a question that can be brought up, being, "What is the purpose of this course for each and every student that walks into my classroom?" Many answers will arise, whether it is a pure fascination of the subject or because it can be seen as a small st... |
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 ... | Roman | 26,598 | <p>Using <a href="https://mathematica.stackexchange.com/a/201001/26598">partial memoization</a>:</p>
<pre><code>Clear[p];
p[0] = Function[x, 1/(x + 1)*UnitStep[x]];
p[n_Integer?Positive] := p[n] =
Function[x, Evaluate@Convolve[p[n - 1][y], p[0][y], y, x]]
p[0][x]
(* UnitStep[x]/(1 + x) *)
p[1][x]
(* ((-I... |
84,711 | <p>This is a homework question I was asked to do</p>
<p>Of a twice differentiable function $ f : \mathbb{R} \to \mathbb{R} $ it is given that $f(2) = 3, f'(2) = 1$ and $f''(x) = \frac{e^{-x}}{x^2+1}$ . Now I have to prove that $$ \frac{7}{2} \leq f\left(\frac{5}{2}\right) \leq \frac{7}{2} + \frac{e^{-2}}{... | Community | -1 | <p>The calculation can be done in the following way:
$f'''(x)=(f''(x))'= -\frac{e^{-x}}{x^2+1} -\frac{2xe^{-x}}{(x^2+1)^2}$ which at $x=2$ yields $f'''(2)=-\frac{9e^{-2}}{25}$ and so \begin{eqnarray*}
f(5/2) & = & f(2)+f'(2)(5/2-2) + f''(2)(5/2-2)^2/2! +f'''(2)(5/2-2)^3/3!+... |
84,711 | <p>This is a homework question I was asked to do</p>
<p>Of a twice differentiable function $ f : \mathbb{R} \to \mathbb{R} $ it is given that $f(2) = 3, f'(2) = 1$ and $f''(x) = \frac{e^{-x}}{x^2+1}$ . Now I have to prove that $$ \frac{7}{2} \leq f\left(\frac{5}{2}\right) \leq \frac{7}{2} + \frac{e^{-2}}{... | Sasha | 11,069 | <p>Start with $f^\prime(x) = f^\prime(2) + \int_2^x f^{\prime\prime}(y) \mathrm{d} y = 1 + \int_2^x \frac{\exp(-u)}{1+u^2} \mathrm{d} u$. Then
$$ \begin{eqnarray}
f(x) &=& f(2) + \int_2^x f^\prime(z) \mathrm{d} z = 3 + \int_2^x \left( 1 + \int_2^z \frac{\exp(-u)}{1+u^2} \mathrm{d} u \right) \mathrm{d} z \\ ... |
1,892 | <p>Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.</p>
<p>Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conv... | Casebash | 123 | <p>Actually $P \ne NP$ <em>does</em> mean that our current NP-hard problems have no polynomials time solutions. NP-complete problems are the hardest problems in NP and NP-hard problems are at least as hard as this. So if $P \ne NP$, then all these NP-hard problems must be harder than P.</p>
<p>Whether the proof helps ... |
1,892 | <p>Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.</p>
<p>Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conv... | JDH | 413 | <p>Many of the problems we know to be in NP or NP-complete are problems that we actually want to solve, problems that arise, say, in circuit design or in other industrial design applications. Furthermore, since the diverse NP-complete problems are all polynomial time related to one another, if we should ever learn a fe... |
1,892 | <p>Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.</p>
<p>Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conv... | Charles Stewart | 100 | <p>Currently, if a manager asks their software engineering team to look at implementing some utility, and the team says that requirements are NP hard, that's a reason that the project requirements need to be changed before work on implementation can begin. That's because no-one knows how to give feasible solutions to ... |
1,892 | <p>Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.</p>
<p>Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conv... | n0vakovic | 956 | <p>Not directly related to the question, but definitely relevant. </p>
<p>Three days ago <a href="http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_preliminary.pdf" rel="nofollow">proof</a> for P != NP is published. Community thinks it looks serious.</p>
|
3,274,807 | <p>Given an <span class="math-container">$n \times n$</span> symmetric matrix with real coefficients it has <span class="math-container">$n$</span> eigenvalues. I was wondering are the eigenvalues continuous with respect to the coefficients of the matrices? I have seen somewhere that the eigenvalues of matrices are co... | cangrejo | 86,383 | <p>The set of real numbers is a subset of the set of complex numbers, if we consider that real numbers are complex numbers with imaginary part equal to zero. Therefore, whatever holds for all complex numbers holds for real numbers.</p>
<p>As you observe, a polynomial might not have real roots. However, all the eigenva... |
3,274,807 | <p>Given an <span class="math-container">$n \times n$</span> symmetric matrix with real coefficients it has <span class="math-container">$n$</span> eigenvalues. I was wondering are the eigenvalues continuous with respect to the coefficients of the matrices? I have seen somewhere that the eigenvalues of matrices are co... | Nitin Uniyal | 246,221 | <p>For order <span class="math-container">$2$</span> matrix <span class="math-container">$A=\begin{pmatrix}a&b\\b&c\\\end{pmatrix}$</span>, eigenvalues are <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> s.t. <span class="math-container">$\alpha+\beta=a+c$</span> and... |
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>
| wythagoras | 236,048 | <p>Derivatives have some applications in economics, for example marginal cost, and I believe that marginal cost is used a lot in business applications.</p>
|
3,929,063 | <p>I always want to apply category theory to structures that involve "time" or "stepping"/"increment"(discrete "time").</p>
<p>I visualize it as a sequence of categories that are somehow connected(generally the objects are the same and only the morphisms will change, not sure if ... | Mozibur Ullah | 26,254 | <p>Time is a physical concept and it's a question of how it is to be modelled. It's quite possible that it's an artifact of our modelling that we get time invariance since causal nets, a model in quantum gravity allow, for modelling of time as open - a relatively recent result.</p>
<p>This to me seems a prime suspect i... |
3,929,063 | <p>I always want to apply category theory to structures that involve "time" or "stepping"/"increment"(discrete "time").</p>
<p>I visualize it as a sequence of categories that are somehow connected(generally the objects are the same and only the morphisms will change, not sure if ... | N. Virgo | 27,193 | <p>I don't know if the following are exactly what you're looking for, since they are about modelling dynamical systems <em>within</em> category theory, rather than having a dynamical system <em>of</em> categories, but they are likely to be useful for inspiration, if nothing else.</p>
<p>There's a classic way to look at... |
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... | 5xum | 112,884 | <p>What you are asking is, esencially, the fundamental theorem of calculus.</p>
<p>The process in analysis is such that first, for continuous functions (actually, a slightly larger class of functions, not only continuous, but that isn't all that important), we can calculate the area under their graphs by taking the li... |
173,131 | <p>Let's suppose that for the following expression:</p>
<p>$\qquad \alpha\,\beta +\alpha+\beta$</p>
<p>I know that $\alpha$ and $\beta$ are of small magnitude (e.g., 0 < $\alpha$ < 0.02 and 0 < $\beta$ < 0.02). Therefore, the magnitude of $\alpha\,\beta$ is negligible, i.e., the original expression can be... | Akku14 | 34,287 | <p>Let me give a slightly different form, that is - as far as I see quickly - equivalent to that of Henrik Schumacher, and show, why there appears a q[t]^2 term.</p>
<pre><code>f[a_, b_] := a b + a + b
</code></pre>
<p>Take series for small eps, and fix the result with eps->1</p>
<pre><code>(Series[f[a eps, b eps],... |
2,559,260 | <p>There exists a function $f$ such that $\lim_{x \rightarrow \infty} \frac{f(x)}{x^2} = 25$ and $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = 5$</p>
<p>I am confused, I do not whether it is true or not</p>
<p>I have a counter-example, but I think thre might be such function</p>
| Contestosis | 462,389 | <p>There is no such a function.
To prove that, let $f$ be a solution.
The conditions on the limits impose that $f$ does not vanish near $+ \infty $.
Therefore, the ratio of the too quantities is $x$ and the limit of $x$ near $ + \infty $ is $ +\infty$ and not $ 5 / 25 $.
Another way of seeing that :
If $ f(x)/x $ has 5... |
302,243 | <p>Let $f:[0,1]\to\mathbb{R}$ be a Lipschitz function, and $\pi f$ be its piecewise linear interpolant on an equispaced grid with $n$ points.</p>
<p>It should be true (if I am not making mistakes with the constant) that
$$
\int_0^1 |f - \pi f| \leq \frac{1}{4n} \operatorname{Lip}(f).
$$</p>
<p>Do you have a reference... | Jason Starr | 13,265 | <p>That is false. I am taking a break from something else, so I will mostly refer to other MO answers. I gather from your example that you allow $X$ to be singular. Then Simpson proved that every finitely presented group $G$ is isomorphic to the fundamental group of a (usually singular) complex projective variety $X... |
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>$$s = (0 \times 3) + \frac{1}{2}\... | Sophie | 431,586 | <p>Using the formula $s=ut+1/2at^2$</p>
<p>Where $a$ is acceleration, $u$ is the initial velocity, $t$ is the time and $s$ the the displacement. </p>
<p>We can deduce that the displacement will be $45m$. </p>
<p>You are indeed correct!</p>
|
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Beni Bogosel | 7,327 | <p>Define $I_n =\displaystyle \int_0^1 \frac{x^n}{1+x}$. Then you can obtain immediately that $I_{n+1}+I_n = \displaystyle \frac{1}{n+1}$. Next note that $0\leq I_{n+1}\leq I_n$ since for $0\leq x \leq 1$ the inequality $0\leq \frac{x^{n+1}}{1+x} \leq \frac{x^n}{1+x}$ holds.</p>
<p>Therefore $I_n \to 0$ as $n \to \inf... |
377,152 | <p>Let n be a fixed natural number. Show that:
$$\sum_{r=0}^m \binom {n+r-1}r = \binom {n+m}{m}$$</p>
<p>(A): using a combinatorial argument and (B): by induction on $m$?</p>
| Brian M. Scott | 12,042 | <p>Finding this combinatorial argument isn’t altogether straightforward if you’ve not yet had much experience. The righthand side is easy to interpret: it’s the number of ways of choosing $m$ numbers from the set $[n+m]=\{1,2,\dots,n+m\}$. Similarly, each term on the lefthand side is easy to interpret: $\binom{n+r-1}r$... |
1,456,411 | <p>When we're introduced to $\mathbb{R}^3$ in multivariable calculus, we first think of it as a collection of points. Then we're taught that you can have these things called <em>vectors</em>, which are (equivalence classes of) arrows that start at one point and end up at another.</p>
<p>At this point $\mathbb{R}^3$ is... | Ivo Terek | 118,056 | <p>In differential geometry, we deal with tangent spaces to manifolds. Let's make a precise distinction in our case. There are more than one definition of tangent space, amd one can show certain equivalences between them.</p>
<p>Take $p\in \Bbb R^3$ and define: $$T_p(\Bbb R^3) = \{(p,v) \mid v \in \Bbb R^3\}.$$ So far... |
141,423 | <p>Let $V \subset H \subset V'$ be a Hilbert triple.</p>
<p>We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying
$$\int_0^T u(t)\varphi'(t)=-\int_0^T u'(t)\varphi(t)$$
for all $\varphi \in C_c^\infty(0,T)$.</p>
<p>Then we define the space $W = \{u \in L^2(0,T;V) : u' \i... | Igor Khavkine | 2,622 | <p>The main property that you would want in your weak derivative is that it defines a closed, unbounded operator on $L^2$. Suppose you have an unbounded operator $A$ (for you it would be the derivative) defined on a dense domain $D(A)$ in $L^2$. As such, $A$ need not be closed (it's graph in $L^2\times L^2$ is not a cl... |
62,581 | <p>I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?</p>
| J. M. ain't a mathematician | 498 | <p>I'll make the assumption that: </p>
<ol>
<li><p>The <em>oblique coordinate</em> system <span class="math-container">$(u,v)$</span> with angle <span class="math-container">$\varphi$</span> and the Cartesian system <span class="math-container">$(x,y)$</span> share an origin.</p></li>
<li><p>Axis <span class="math-con... |
62,581 | <p>I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?</p>
| Jiri Kriz | 12,741 | <p>Let us denote by "old" the usual cartesian system with orthogonal axes and by "new" the system with the skew axes $(\alpha_1, \alpha_2)^T, (\beta_1, \beta_2)^T$ (expressed in the old system). An old vector $(x,y)^T$ can be expressed as a linear combination of the skew vectors:
$$
\left( \begin{array}{c} x \\ y \end... |
3,300,793 | <p>Having an immense amount of trouble trying to figure this problem out, and the more I think and ask about it the more confused I seem to get. I think I have finally figured it out so can someone who truly knows the correct answer justify this?</p>
<p>Problem:Let <span class="math-container">$A=\{a,b,c\}$</span>,Let... | JMoravitz | 179,297 | <p>To summarize my comments above, all that a set needs to be to be called a relation is to be a subset of a cartesian product of sets. Nothing more, nothing less. With regards to Q1, after some rewording the question asks if the set of ordered pairs <span class="math-container">$\{(x,y)\in A\times A~:~x\subseteq y\}... |
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>
| Lazar Šćekić | 519,090 | <p>So, let's say that X is a random variable for tracing the number of 6s.
Let's say that C is the condition for the 6s to be consecutive.
Since the task is to find the probability of gaining at least K subsequent 6s, we can look for probability of event A-at least K subsequent 6s fell this way:
P(A)=P(X=K|C)+P(X=... |
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 $</span... | José Carlos Santos | 446,262 | <ol>
<li>Correct: it is <span class="math-container">$\mathbb R$</span>.</li>
<li>Correct, but why didn't you just write that the interior is <span class="math-container">$(-1,\infty)$</span>?</li>
<li>Wrong. Since <span class="math-container">$\partial(-1,\infty)=\{-1\}$</span>, <span class="math-container">$\bigl(\pa... |
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}... | Netchaiev | 517,746 | <p>Every sequence has a countable or a finite set of values. </p>
<p>Besides, you are mixing two ideas : a sequence $(u_n)_n$ is a function $n\mapsto u_n\in F$ ($F$ being any possible set) and almost never a bijection, but the set of all its values are finite or countable.</p>
|
1,480,720 | <p>How many times do you have to flip a coin such that the probability of getting $2$ heads in a row is at least $1/2$?</p>
<p>I tried using a Negative Binomial:
$P(X=2)=$$(n-1)\choose(r-1)$$p^r\times(1-p)^{n-r} \geq 1/2$ where $r = 2$ and $p = 1/4$. However, I don't get a value of $n$ that makes sense.</p>
<p>Than... | OFRBG | 42,793 | <p>Well, I think you may think of it as a binary tree. The tree either splits towns $H$ or toward $T$. We want to find the level of the tree where $2^{n-1}$ nodes have at least double heads. There might be a neater way of doing this, but this works:</p>
<ul>
<li>Take the first trial. You get either $T$ or $H$.</li>
<l... |
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) ... | gt6989b | 16,192 | <p>For simpler notation, denote <span class="math-container">$F_x(x,t)$</span> the partial of <span class="math-container">$F$</span> wrt <span class="math-container">$x$</span>. Your PDE is then
<span class="math-container">$$V_t + \frac{\sigma^2 S^2}{2} V_{SS} + (r-D)s V_S - rV = 0,$$</span>
which looks like the Blac... |
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... | E.H.E | 187,799 | <p>$$x^2y'+3xy=1$$
divide by $x$
$$xy'+3y=\frac{1}{x}$$
we can solve this O.D.E by Euler-Cauchy Method</p>
<p>1- to find the complementary solution
$$xy'+3y=0$$
assume
$$y_c=x^m$$
$$y'=mx^{m-1}$$
substitute it to get
$$m=-3$$
hence
$$y_c=C_1x^{-3}=\frac{C_1}{x^3}$$ </p>
<p>2- to find the particular solution
... |
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>
| MPW | 113,214 | <p><strong>Hint:</strong> Well, you have a countable collection of points to use as centers of balls, and you have a countable basis $\{B(x,\tfrac1n):n\in\mathbb N\}$ at each such point $x$, so...</p>
|
2,900,454 | <p>There are so many different methods I've found on SE and through Matlab, and they're all giving me different results.</p>
<p>Specifically, I have {v1} = (1,2,1) and {v2} = (2,1,0) in set S. What is the method to find {v3} vectors that are orthogonal to both v1 and v2?</p>
<p>I'm preparing for a final and I'm tryin... | PackSciences | 588,260 | <p>What I would do:</p>
<ul>
<li>Compute the planes orthogonal to both of your vectors.</li>
</ul>
<p>A plane orthogonal to the vector $(a,b,c)$ has the equation $ax + by + cz + d = 0, \forall d \in \mathbb{R}$</p>
<ul>
<li>Compute the intersection of the two planes by replacing the first plane equation in the secon... |
2,901,783 | <p>I am having trouble solving a multi part question.</p>
<p>Express $ \frac x{x^2-3x + 2} $ in the partial fraction form.</p>
<p>The answer I got was $\frac2{x-2}-\frac1{x-1}$ .</p>
<p>The problem comes when they asked:</p>
<p>Show that, if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then... | Ennar | 122,131 | <p>I believe you are already done and just have linguistic issue; I think that "if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then [...]" should be interpreted as "if $x$ has property $P(x)$, then [...]" where $P(x)\equiv$ "$x^3$ and higher powers can be neglected". That is, you are not suppo... |
628,681 | <p>The computed moments of log normal distribution can be found <a href="http://en.wikipedia.org/wiki/Log-normal_distribution#Arithmetic_moments">here</a>. How to compute them?</p>
| heropup | 118,193 | <p>If $X$ is lognormal, then $Y = \log X$ is normal. So consider $${\rm E}[X^k] = {\rm E}[e^{kY}] = \int_{y=-\infty}^\infty e^{ky} \frac{1}{\sqrt{2\pi}\sigma} e^{-(y-\mu)^2/(2\sigma^2)} \, dy. $$ Now observe that $$\begin{align*} ky - \frac{(y-\mu)^2}{2\sigma^2} &= - \frac{-2k\sigma^2 y + y^2 - 2\mu y + \mu^2}{2\... |
628,681 | <p>The computed moments of log normal distribution can be found <a href="http://en.wikipedia.org/wiki/Log-normal_distribution#Arithmetic_moments">here</a>. How to compute them?</p>
| Machinato | 240,067 | <p>First let us write <span class="math-container">$X = \exp Y$</span> where <span class="math-container">$Y$</span> is normal. Let us denote <span class="math-container">$\operatorname{E}\left[Y\right]=\mu$</span> and <span class="math-container">$\operatorname{Var}\left[Y\right]=\sigma^2$</span>. We can write <span c... |
2,988,089 | <p>Let A, B, C, and D be sets. Prove or disprove the following:</p>
<pre><code> (A ∩ B) ∪ (C ∩ D)= (A ∩ D) ∪ (C ∩ B)
</code></pre>
<p>I am just wondering can i simply prove it using a membership table ( seems to easy ) or do i have to use setbuilder notation?</p>
<p>Thank you!</p>
| Mohammad Riazi-Kermani | 514,496 | <p>Let <span class="math-container">$$A=B=\{1,2,3,4,5\}$$</span> and <span class="math-container">$$C=D=\{6,7,8,9,10\}$$</span> We have <span class="math-container">$$(A ∩ B) ∪ (C ∩ D)= \{1,2,3,4,5,6,7,8,9,10\}$$</span></p>
<p>while <span class="math-container">$$ (A ∩ D) ∪ (C ∩ B) =\emptyset $$</span></p>
|
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,... | Michael E2 | 4,999 | <p>An algebraic approach to add to the mix of answers:</p>
<pre><code>n = {1, b, c}; (* unknown normal with a=1 *)
p = {x, y, z}; (* free point on the plane *)
coeff = SolveAlways[n.(p-pt) == 0 /. {Thread[p -> pt], Thread[p -> lineeq]}, t]
(* {{b -> -0.255754, c -> 0.488491}} *)
n . (p - pt) == 0 /. Firs... |
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... | greg | 357,854 | <p><span class="math-container">$
\def\o{{\tt1}}\def\p{\partial}
\def\F{{\cal L}}
\def\L{\left}\def\R{\right}
\def\LR#1{\L(#1\R)}
\def\fracLR#1#2{\L(\frac{#1}{#2}\R)}
\def\Diag#1{\operatorname{Diag}\LR{#1}}
\def\trace#1{\operatorname{Tr}\LR{#1}}
\def\qiq{\quad\implies\quad}
\def\grad#1#2{\frac{\p #1}{\p #2}}
$</span>Be... |
574,041 | <p>Consider a set of linear equations described by $A\vec{X}=\vec{B}$ is given, where $A$ is an $n\times n$ matrix and $\vec{X}$ and $\vec{B}$ are n-row vectors. Also suppose that this system of equations have a unique solution and this solution is given.</p>
<p>Imagine a new set of linear equations $A'\vec{X}=\vec{B}... | D Left Adjoint to U | 26,327 | <p>Let's look at the $3\times 3$ case when $a_{21}$ is changed by $+k$. The determinant $\det(A)$ is in every entry in $A^{-1}$, so let's look at that. Expand along the second row to see that $\det(A + k e_{21}) = -(a_{21} + k)|C_{21}| + a_{22} |C_{22}| - a_{23}|C_{23}|$ where $|C_{ik}|$ is the determinant of the mat... |
802,877 | <blockquote>
<p>Find $\displaystyle\lim_{n\to\infty} n(e^{\frac 1 n}-1)$ </p>
</blockquote>
<p>This should be solved without LHR. I tried to substitute $n=1/k$ but still get indeterminant form like $\displaystyle\lim_{k\to 0} \frac {e^k-1} k$. Is there a way to solve it without LHR nor Taylor or integrals ?</p>
<p>... | Тимофей Ломоносов | 54,117 | <p>Why should one use Taylor where we don't need it at all?</p>
<p>$$L = \lim\limits_{n\to\infty}n\left(e^\frac{1}{n}-1\right)=\lim\limits_{x\to0}\frac{e^x-1}{x}$$</p>
<p>Substitute $u=e^x-1$. Then $x=\ln(u+1)$</p>
<p>$$L=\lim\limits_{u \to 0}\frac{u}{\ln(1+u)}=\lim\limits_{u\to0}\frac{1}{\frac{1}{u}\ln(1+u)}=\lim\l... |
2,945,367 | <p><a href="https://i.stack.imgur.com/MGzHc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/MGzHc.png" alt="enter image description here"></a></p>
<p>We were given a couple formulas, but the one that immediately stood out to me was the Vfinal = Vinitial + at</p>
<p>so we know the patrol will consta... | Keen-ameteur | 421,273 | <p>Observe that if <span class="math-container">$P:=\{ \xi_i \}_{i=0}^n$</span> is a partition of <span class="math-container">$[a,b]$</span>, and <span class="math-container">$\xi_j= \zeta_0 < \zeta_1<...< \zeta_{l_j}=\xi_{j+1}$</span>, then by (a) you know that:</p>
<p><span class="math-container">$\underse... |
2,717,821 | <p>Since we have 4 digits there is a total of 10000 Password combinations possible.</p>
<p>Now after each trial the chance for a successful guess increases by a slight percentage because we just tried one password and now we remove that password from the "guessing set". That being said I am struggling with the actual ... | Siong Thye Goh | 306,553 | <p>In the second trial, there are $9998$ ways out of $9999$ ways we can miss the right pin.</p>
<p>$$1-\frac{9999}{10000}\cdot \frac{9998}{9999}\cdot \frac{9997}{9998}$$</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 ... | John Sidles | 11,394 | <p><b>Update:</b> it may be Spivak's new book <a href="http://rads.stackoverflow.com/amzn/click/0914098322" rel="nofollow"><i>Physics for Mathematicians: Mechanics I</i></a> covers most of the material that this answer had in mind. I've just ordered a copy, and will report on it when it arrives.</p>
<hr>
<p>Neither ... |
195,832 | <p>I want to download the content of the website(contains text) and only a few lines from the content (from a specific number of the line up to the last line minus specific offset). Unfortunately, I do not know how to get the number of line in the content. For example, I want to replace 59 with the length of the conten... | dionys | 20,144 | <p>One approach is to split the imported data on newlines using <code>StringSplit</code> so the length of the resulting list is the line count. Then we can use <code>Riffle</code> and <code>StringJoin</code> to get the selected lines:</p>
<pre><code>Module[{start = 51, offset = 85, raw, data, linecount, stop},
raw = ... |
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... | Matti P. | 432,405 | <p>This is what I would put into my calculator:
$$
1.2 \times \underbrace{0.75}_{=1-0.25} \times 1.6 = 1.44
$$</p>
|
2,187,765 | <p>This is part of Exercise 2.7.9 of F. M. Goodman's <em>"Algebra: Abstract and Concrete"</em>.</p>
<blockquote>
<p>Let $C$ be the commutator subgroup of a group $G$. Show that if $H$ is a normal subgroup of $G$ with $G/H$ abelian, then $C\subseteq H$.</p>
</blockquote>
<p>The following seems to be wrong.</p>
<h2>... | Adam Hughes | 58,831 | <p>$G/H=\{gH: g\in G\}$ by definition. this is only a group under $(gH)(g'H) = (gg')H$ if $Hg' = g'H$. But this is just another way of stating the definition of $H$ being normal. In your proof you just neglected to note that $xyx^{-1}y^{-1}H$ is only relevant because it is equal to $(xH)(yH)(x^{-1}H)(y^{-1}H)$ because ... |
2,187,765 | <p>This is part of Exercise 2.7.9 of F. M. Goodman's <em>"Algebra: Abstract and Concrete"</em>.</p>
<blockquote>
<p>Let $C$ be the commutator subgroup of a group $G$. Show that if $H$ is a normal subgroup of $G$ with $G/H$ abelian, then $C\subseteq H$.</p>
</blockquote>
<p>The following seems to be wrong.</p>
<h2>... | drhab | 75,923 | <p>The fact that $G/H$ is abelian gives us the second equality of:$$xyH=(xH)(yH)=(yH)(xH)=yxH$$
Consequently we find: $$x^{-1}y^{-1}xy=(yx)^{-1}xy\in H$$</p>
<p>This for every $x,y\in G$ so we are allowed to conclude that $H$ contains the commutator subgroup.</p>
|
1,910,983 | <p>What conditions are equivalent to <strong>singularity</strong> of matrix $A\in \mathbb{R}^{n,n}$.<br>
<strong>a.</strong> $\dim(ker A) \ge 0$<br>
<strong>b.</strong> There is exist vector $b$ such that $Ax=b$ is contradictory.<br>
<strong>c.</strong> $rank(A^T) < n$ </p>
<p><strong>a.</strong> is true for each ... | Drew N | 178,098 | <p>A bit of modular arithmetic reveals a simple way to compute the answer, at least for the following special case:</p>
<p>Assume that the non-periodic string has no elements in common with the periodic part, and also that the repeated string has no duplicates e.g. your periodic part can be {4,7,0} but not {4,7,4,0}. ... |
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>
| Kushashwa Ravi Shrimali | 232,558 | <p>As the other user mentioned, it is basically the application of <a href="https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf" rel="noreferrer">Lambert W Function</a>.</p>
<p>Say, $x^x = z$ which implies, $x \ln x = \ln z$.</p>
<p>Now, I can write: $x = e^{\ln x} $ using the properties of logarithms and exponential f... |
1,006,354 | <ul>
<li>A multiple choice exam has 175 questions. </li>
<li>Each question has 4 possible answers. </li>
<li>Only 1 answer out of the 4 possible answers is correct. </li>
<li>The pass rate for the exam is 70% (123 questions must be answered correctly). </li>
<li>We know for a fact that 100 questions were answered corre... | user4568 | 173,559 | <p>We have already answered 100 questions, so there are only 75 questions left to answer.
Since we are guessing our way through the multiple choice questions, our probability of success in each question will be $\frac{1}{4}$
Since the pass mark is $\frac{123}{175}$, we need at least $\frac{23}{75}$ in the final 75 que... |
812,778 | <p>Prove that $(4/5)^{\frac{4}{5}}$ is irrational.</p>
<p><strong>My proof so far:</strong></p>
<p>Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational.</p>
<p>Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are integers.</p>
<p>Then $\dfrac{4^4}{5^4}=\dfrac{p^5}{q^5}$</p>
<p>$\therefore$ $4^4... | André Nicolas | 6,312 | <p><strong>Outline:</strong> If the number $\alpha$ is rational, there exist integers $p$ and $q$ which are <strong>relatively prime</strong> such that $\alpha=\frac{p}{q}$.</p>
<p>From your $4^4q^5=5^4p^5$, argue that $5$ divides $q$, and then that $5$ divides $p$. </p>
|
812,778 | <p>Prove that $(4/5)^{\frac{4}{5}}$ is irrational.</p>
<p><strong>My proof so far:</strong></p>
<p>Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational.</p>
<p>Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are integers.</p>
<p>Then $\dfrac{4^4}{5^4}=\dfrac{p^5}{q^5}$</p>
<p>$\therefore$ $4^4... | user3294068 | 140,502 | <p>This is essentially the same as the standard proof that $\sqrt 2$ is irrational.</p>
<p>Let $z = (4/5)^{(4/5)}$. Now to calculate $z^5$:</p>
<p>$$
z^5 = \left(\left({4\over 5}\right)^{4\over 5}\right)^5 = \left({4\over 5}\right)^{4}
= {4^4\over 5^4}
$$</p>
<p>Clearly $z \neq 0$ as $0^{(5/4)} = 0 \neq 4/5$.</p>
... |
119,456 | <p>I generated a 2d random array in $x-y$ plane with</p>
<pre><code>L = 10;
random = Table[{x, y, RandomReal[{-1, 1}]}, {x, 0, L, L/10}, {y, 0, L, L/10}];
</code></pre>
<p>Now I want to save it for the next using by</p>
<pre><code>iniF = Interpolation[Flatten[random, 1]];
inif[x_, y_] = c+iniF[x, y];
</code></pre>
... | Wjx | 6,084 | <p>Add this simple code before u run everything will let Random generate exactly the same result every time you run. If you're careful enough, you may find this in lot's of posts with randomly generated input.</p>
<pre><code>SeedRandom["Whatever you write here, keep it the same in multiple runs!"]
</code></pre>
<p>Ho... |
2,231,388 | <blockquote>
<p>Consider a ring map $B \rightarrow A$. Consider the map $f:A \otimes_{B}A \rightarrow A$, where $x \otimes y$ goes to $xy$. Let $I$ be the kernel of $f$. Why is it true that $I/I^2$ is isomorphic to $I \otimes_{A \otimes_{B}A} A$?</p>
</blockquote>
<p>This is what I've been able to prove till now:</p... | Georges Elencwajg | 3,217 | <p>Given an $R$-module $M$ and an ideal $I\subset R$ we have $R$-module morphisms $$M\otimes_R R/I\to M/IM: m\otimes \tilde r\mapsto \overline {rm} \operatorname \quad {and} \quad M/IM \to M\otimes_R R/I:\overline {m}\mapsto m\otimes \tilde 1$$ which are easily seen to be inverse to each other and yield an isomorphi... |
746 | <p>There have been a number of questions in the Close part of Review lately which were basically asking for help creating an algorithm to do some mundane task (see <a href="https://mathoverflow.net/questions/140585/how-to-perform-divide-step-of-in-place-quicksort#comment362909_140585">here</a>, <a href="https://mathove... | François G. Dorais | 2,000 | <p>Yes, it's possible, but as Anna Lear <a href="https://meta.mathoverflow.net/a/163">explained in an earlier answer</a>, there are some requirements. There needs to be a clear pattern of questions to be migrated and the migrated posts need to have low rejection rate on the target site. In the mean time, if you need to... |
1,575,671 | <p>The whole question is that <br>
If $f(x) = -2cos^2x$, then what is $d^6y \over dx^6$ for x = $\pi/4$?</p>
<p>The key here is what does $d^6y \over dx^6$ mean?</p>
<p>I know that $d^6y \over d^6x$ means 6th derivative of y with respect to x, but I've never seen it before.</p>
| Community | -1 | <p>For convenience, first transform</p>
<p>$$-2\cos^2(x)=-1-\cos(2x).$$</p>
<p>Then the sixth derivative is $$2^6\cos(2x),$$ because $\cos(x)''=-\cos(x)$ and because of the scaling of the variable .</p>
<p>At $x=\dfrac\pi4$, $$0.$$</p>
|
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>
| kglr | 125 | <p><strong>Update:</strong> Functions for finding the edges that connect communities and for tabulating the results:</p>
<pre><code>ClearAll[connectingEdgesF, tabulateF]
connectingEdgesF = Module[{g = #}, Complement[EdgeList[#],
Flatten[EdgeList[Subgraph[g, #]] & /@ FindGraphCommunities[g]]]] &;
tabulat... |
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>
| user | 505,767 | <p>Recall that</p>
<p><span class="math-container">$$z=x+iy=|z|(\cos \theta+i\sin \theta)$$</span><span class="math-container">$$\implies \sqrt z=\sqrt{|z|}\left(\cos \left(\frac{\theta}2+k\pi\right)+i\sin \left(\frac{\theta}2+k\pi\right)\right),\,k=0,1$$</span></p>
|
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>
| Dave | 334,366 | <p>Start by squaring both sides:
<span class="math-container">$$\begin{align}x-\frac{1}{x}&=A\\\left(x-\frac{1}{x}\right)^2&=A^2\\x^2-2+\frac{1}{x^2}&=A^2.\end{align}$$</span>
Then try adding <span class="math-container">$4$</span> to both sides and "reversing" the processes above.</p>
|
1,913,689 | <blockquote>
<p>Let $f: X \rightarrow Y$ be a function. $A \subset X$ and $B \subset Y$.
Prove $A \subset f^{-1}(f(A))$.</p>
</blockquote>
<p>Here is my approach. </p>
<p>Let $x \in A$. Then there exists some $y \in f(A)$ such that $y = f(x)$. By the definition of inverse function, $f^{-1}(f(x)) = \{ x \in X$ suc... | drhab | 75,923 | <p>A nice mnemonic on preimages is:$$x\in f^{-1}(C)\iff f(x)\in C\tag1$$
It is evident that: $$\forall x\left[x\in A\implies f\left(x\right)\in f\left(A\right)\right]$$</p>
<p>According to $(1)$ here $f(x)\in f(A)$ can be replaced by $x\in f^{-1}(f(A))$.</p>
<p>This results in:$$\forall x\left[x\in A\implies x\in f^{... |
2,007,373 | <p>At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. </p>
<p>For me the first instance that comes to memory was in 7th grade in a inner city USA school district. </p>
<p>Getting to the point, my geometry teacher taught,</p>
<p>"a p... | Henricus V. | 239,207 | <p>Viewpoint from measure theory:</p>
<p>The length/area/volume/hypervolume of a set $S \subseteq \mathbb{R}^n$ is merely its Lebesgue measure $\lambda(S)$.</p>
<p>Since $\lambda$ is a continuous measure, the measure of any single point is $0$, so $\lambda(\{x\}) = 0$ for all $x$, but uncountable unions of points, su... |
2,939,585 | <p>I want to prove that if <span class="math-container">$ \gamma$</span> is a closed path and <span class="math-container">$\gamma\subseteq B_R(0) $</span> then <span class="math-container">$\mathbb{C}\setminus B_R(0)\subseteq \operatorname{Ext}_\gamma$</span> where <span class="math-container">$ \operatorname{Ext}... | zhw. | 228,045 | <p>The function</p>
<p><span class="math-container">$$\text { Ind}_\gamma(z) = \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w-z}\,dw$$</span></p>
<p>is a continuous integer valued function defined on <span class="math-container">$\Omega =\mathbb C\setminus \gamma^*,$</span> where <span class="math-container">$\gamma^*$</s... |
2,109,832 | <p>This is for beginners in probability!</p>
<p>Could someone give me a step by step on how to find the MGF of the binomial distribution?</p>
| spaceisdarkgreen | 397,125 | <p>You can use the double angle formula to write $$\begin{eqnarray}\sin(2\arcsin(3/5)) &=& 2\sin(\arcsin(3/5))\cos(\arcsin(3/5))\\ &=& 2\sin(\arcsin(3/5))\sqrt{1-\sin^2(\arcsin(3/5))}\end{eqnarray}$$</p>
<p>We have $\sin(\arcsin(3/5)) =3/5$, so can plug this in to get the answer.</p>
|
2,109,832 | <p>This is for beginners in probability!</p>
<p>Could someone give me a step by step on how to find the MGF of the binomial distribution?</p>
| marty cohen | 13,079 | <p>$\begin{array}\\
\sin(2\arcsin(x))
&=2\sin(\arcsin(x))\cos(\arcsin(x))\\
&=2x\sqrt{1-\sin^2(\arcsin(x))}\\
&=2x\sqrt{1-x^2}\\
\end{array}
$</p>
<p>Putting $x = 3/5$,
since
$\sqrt{1-x^2} = 4/5$,
I get
$2\dfrac35 \dfrac45
=\dfrac{24}{25}
$.</p>
|
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, ... | Community | -1 | <p>So to answer your first question if $O$ is an open set that isn't empty in the co-finite topology then by definition $\bar{O}$ is the smallest closed set that contains $O$. Now we know that $O$ has infinitly many elements and the only closed set that doesn't only have finitely many elements is $\mathbb{R}$ so it mus... |
3,982,103 | <p>Let <span class="math-container">$(X,\tau)$</span> be a topological space. Prove that <span class="math-container">$\tau$</span> is the finite-closed topology on <span class="math-container">$X$</span> if and only if (i)<span class="math-container">$(X,\tau)$</span> is a <span class="math-container">$T_1$</span>-spa... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$O$</span> be an open set in <span class="math-container">$(X,\tau)$</span>, so <span class="math-container">$O \in \tau$</span>.</p>
<ul>
<li>If <span class="math-container">$O=\emptyset$</span>, <span class="math-container">$O \in \tau_{fc}$</span>, as required.</li>
<li>If <span c... |
2,305,656 | <p>I solved this problem on my own, months ago, but the solution seems to me completely forgotten, a little help on it would be appreciated:</p>
<p>Suppose $\alpha= \alpha(t)$ on an interval $I$ is a smooth (of class C$^1$) parametric representation of the curve $C$, and for any $t \in I$ we have $\space\space\frac{d}... | Michael Rozenberg | 190,319 | <p>I think you mean $a=b$.
Thus, $2a>c$ or $\frac{a}{c}>\frac{1}{2}$ and
$$k=\frac{r}{R}=\frac{\frac{2S}{a+b+c}}{\frac{abc}{4S}}=\frac{16S^2}{2abc(a+b+c)}=$$
$$=\frac{(a+b-c)(a+c-b)(b+c-a)}{2abc}=\frac{(2a-c)c^2}{2a^2c}=\frac{\frac{2a}{c}-1}{\frac{2a^2}{c^2}}.$$
Hence,
$$\frac{2ka^2}{c^2}-\frac{2a}{c}+1=0,$$
whic... |
189,014 | <p>Ok, I know the simple answer is to set some form of Hold attribute to the function but bear with me for a bit while I explain my motivation and why that is not quite what I want.</p>
<hr>
<p>I have a collection of data that is naturally grouped together and some functions that operate on that data. To me, the obvi... | Mr.Wizard | 121 | <p>You might consider putting the "type" label inside an Association itself. This will complicate key addressing but simplify other handling.</p>
<pre><code>asc = <|"atomData" ->
<|"Atom Names" -> {"N", "C", "O", "C", "H", "H"},
"Atom Nr" -> {56, 23, 117, 81, 211, 5},
"Resname" -> {A... |
1,791,146 | <p>I know that a set G with a binary operation $*$ is a group, if:</p>
<ol>
<li><p>$a*b\in G$, for all $a, b \in G$.</p></li>
<li><p>$*$ is associative:</p></li>
</ol>
<p>$$(a*b)*c=a*(b*c) \\ \text{for all }a, b, c\in G.$$</p>
<ol start="3">
<li>An identity element $e \in G$ exists, such that</li>
</ol>
<p>$$a*e = ... | goblin GONE | 42,339 | <p>As BrianO says, <span class="math-container">$\emptyset$</span> is not a group, because every group has an identity element. This also means that <span class="math-container">$\emptyset$</span> is not a vector space, it's not a ring, it's not a module, and it's not a boolean algebra. However, <span class="math-conta... |
2,393,525 | <p>I have two questions which I think both concern the same problem I am having. Is $...121212.0$ a rational number and is $....12121212....$ a rational number? The reason I was thinking it could be a number is when you take the number $x=0.9999...$, then $10x=9.999...$ . Therefore, we conclude $9x=9$ which means $x=1... | fleablood | 280,126 | <p>$0.a_0a_1a_2..... = \sum\limits_{k=0}^{\infty} a_k*10^{-k} = \lim\limits_{n\rightarrow \infty }\sum\limits_{k=0}^{n} a_k*10^{-k}$. This limit exists. For one thing the terms $a_k*10^{-k}$ get <em>small</em> and approach infinitesimal. (But more importantly, the difference between the finite sums becomes infinitesi... |
1,042,227 | <p>I want to verify that the solution to the difference equation</p>
<p>$m_x - 2pqm_{x-2} = p^2 + q^2$</p>
<p>with boundary conditions</p>
<p>$m_0 = 0$</p>
<p>$m_1 = 0$</p>
<p>is</p>
<p>$$m_x = -\frac{1}{2}(\frac{1}{\sqrt{2pq}} +1)(\sqrt{2pq})^x + \frac{1}{2}(\frac{1}{\sqrt{2pq}} - 1)(-\sqrt{2pq})^x + 1$$</p>
<p... | Bumblebee | 156,886 | <p><strong>HINT:</strong> Since you have the solution we can use the mathematical induction for verify the solution. Also $(-1)^x=(-1)^{x-2}, \forall x\in\mathbb{N}.$</p>
|
25,162 | <p>Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs).
Assume that:</p>
<p>(1) The system has an absorbing ball, that is every trajectory eventually enters this ball
and stays in it. </p>
<p>(2) The system has a unique stationary point, and this stationary point is locally
asymptotically... | coudy | 6,129 | <p>No. You could have in the ball a compact attractor K containing no periodic orbits. In fact there are attractors on which the dynamic is minimal (all trajectories are dense in K) and conjuguated to
(the suspension of) an adding machine. </p>
<p>Examples of such attractors even appear in the unidimensional setting,... |
25,162 | <p>Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs).
Assume that:</p>
<p>(1) The system has an absorbing ball, that is every trajectory eventually enters this ball
and stays in it. </p>
<p>(2) The system has a unique stationary point, and this stationary point is locally
asymptotically... | Martin M. W. | 1,227 | <p>As the questioner notes in a comment, the answer is Yes for n<3. </p>
<p>One way to create counterexamples for larger n is to use the work on the Seifert Conjecture. Start with a vector field pointing inward to the origin, and replace a little piece of it with an "aperiodic plug." This "plug" looks from the outs... |
3,545,250 | <p>Being new to calculus, I'm trying to understand Part 1 of the Fundamental Theorem of Calculus. </p>
<p>Ordinarily, this first part is stated using an " area function" <em>F</em> mapping every <em>x</em> in the domain of <em>f</em> to the number " integral from a to x of f(t)dt". </p>
<p>However, I encounter <stro... | Paramanand Singh | 72,031 | <p>I think the key issue here is that you are unable to understand how integration and differentiation are reverse processes.</p>
<p>In order to understand and appreciate it fully you need to know the definition of derivative (easy) and that of integral (difficult and mostly avoided in beginner's calculus texts).</p>
... |
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... | Conrad | 298,272 | <p><span class="math-container">$(-1)^{2.16}=(-1)^{\frac{54}{25}}=\exp({\frac{54}{25}(2k+1)i\pi})=\exp({\frac{4}{25}(2k+1)i\pi})$</span> </p>
<p>is a set of <span class="math-container">$25$</span> numbers corresponding to <span class="math-container">$k=0,...24$</span> as the exponential above has period <span class=... |
385,789 | <p>Can anybody please help me this problem?</p>
<p>Let $K = \mathbb{F}_p$ be the field of integers module an odd prime $p$, and $G = \mathcal{M}^*_n(\mathbb{F}_p)$ the set of $n\times n$ invertible matrices with components in $\mathbb{F}_p$. Based on the linear (in)dependence of the columns of a matrix $M\in G$, get t... | anon | 11,763 | <p>The process for creating an arbitrary invertible matrix is as follows:</p>
<ul>
<li>The first column of an invertible matrix can be selected to be any nonzero vector.</li>
<li>Second column can be picked to be any vector not in the span of the first.</li>
<li>Third column can be picked to be any vector not in the s... |
273,798 | <p>I am writing a large numerical code where I care a lot about performance, so I am trying to write compiled functions that are as fast as possible.</p>
<p>I need to write a function that does the following. Consider a list of positive integers, for example {5,3}, take its flattened binary form (with a given number of... | Henrik Schumacher | 38,178 | <p>The post linked by MarcoB contains a link to this Wikipedia page which is very illuminating:</p>
<p><a href="https://en.wikipedia.org/wiki/Hamming_weight" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Hamming_weight</a></p>
<p>There I found also the very useful remark that a population count function has b... |
3,069,987 | <p>I know that whatever numbers you choose for x and y and their sum equals to 1 will satisfy the equation <span class="math-container">$x^2 + y = y^2 + x$</span></p>
<p>Algebraic proof: </p>
<p>Given: <span class="math-container">$x + y = 1$</span></p>
<p><span class="math-container">$$LS = x^2+ y
= (1-y)^2 + y ... | Michael Rozenberg | 190,319 | <p>Because <span class="math-container">$$0=x^2+y-(y^2+x)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1).$$</span></p>
|
1,121,205 | <p>Can we find an bijective continuous map $f:X\to Y$ from a disconnected topological space $X$ to a connected topological space $Y$?</p>
<p>It seems counter intuitive for me, but I am not able to prove that $f(X)$ will be disconnected. I cannot think of any counterexample either. Can someone help?</p>
| Forever Mozart | 21,137 | <p>Let $X=Y=\{0,1\}$. Give $X$ the discrete topology and give $Y$ the indiscrete topology. The identity function from $X$ to $Y$ is a continuous bijection, $X$ is not connected, and $Y$ is connected.</p>
|
132,862 | <p>Is it true that given a matrix $A_{m\times n}$, $A$ is regular / invertible if and only if $m=n$ and $A$ is a basis in $\mathbb{R}^n$?</p>
<p>Seems so to me, but I haven't seen anything in my book yet that says it directly.</p>
| Frank | 460,691 | <p>Note: I'm still learning about the Peano Axioms, so if any part of my answer is inaccurate, please let me know.</p>
<p>First, consider why we even bother thinking about the natural numbers. Of course, it's because they obey unique properties; namely, the natural numbers are "natural" in the sense that they... |
179,581 | <p><strong>Problem:</strong></p>
<p>(a). If $f$ is continuous on $[a,b]$ and $\int_a^x f(t) dt = 0$ for all $x \in [a,b]$, show that $f(x) = 0$ for all $x \in [a,b]$.</p>
<p>(b). If $f$ is continuous on $[a,b]$ and $\int_a^x f(t)dt = \int_x^b f(t)dt$ for all $x \in [a,b]$, show that $f(x)=0$ for all $x\in [a,b]$.</p>... | EuYu | 9,246 | <p>Some hints:</p>
<p>a) Recall the first part of the Fundamental Theorem of Calculus
$$f(x) = \frac{d}{dx}\int_a^x f(t)\ dt$$</p>
<p>b) Write $$\int_{a}^{x}f(t)\ dt + \int_{b}^{x}f(t)\ dt = 0$$ and do something similar to part a.</p>
|
3,366,064 | <p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. </p>
<p>My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with van... | TonyK | 1,508 | <p>It makes things a bit simpler if we turn your measuring spoon upside down, and model it as the set of points <span class="math-container">$\{(x,y,z):x^2+y^2+z^2=1, z\ge 0\}$</span>. The area of a cross-section at height <span class="math-container">$z$</span> is then <span class="math-container">$\pi(1-z^2)$</span>,... |
3,366,064 | <p>I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere. </p>
<p>My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with van... | Community | -1 | <p>Alternative: use two teaspoons.</p>
<p>Use water as you develop your skill. Fill tsp A, and pour into tsp B until the contents appear equal. Each now contains half a tsp. And now you know what half a tsp looks like in practice.</p>
<p>And you don't have to calculate cosines against thumb-sized hardware.</p>
|
2,912,570 | <p>Let $X$ and $Y$ be two standard normal distributions with correlation $-0.72$. Compute $E(3X+Y\mid X-Y=1)$.</p>
<p>My solution: Conditioning on $X-Y=1$, we have $E(3X+Y\mid X-Y=1) = E(4Y+3\mid X-Y=1) = 3+4E(Y\mid X-Y=1) = 3$.</p>
<p>(1) Is my solution correct? My intuition is that the conditional density of $Y$ re... | heropup | 118,193 | <p>Unfortunately, $$\operatorname{E}[Y \mid X-Y = 1] \ne 0.$$ You can see this if you look at this picture:</p>
<p><a href="https://i.stack.imgur.com/YMhJv.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YMhJv.jpg" alt="enter image description here"></a></p>
<p>Here, the ellipses represent curves ... |
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