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 |
|---|---|---|---|---|
3,370,076 | <p>The total mechanical energy is conserved when a ball is dropped from a height of 4.00 <span class="math-container">$\mathit{m}$</span>, and it makes a elastic collision with the ground. Assuming no non-conservative forces are acting find the period of the ball. g of course is 9.81.</p>
<p><span class="math-containe... | Nicolas | 498,847 | <p>Suppose <span class="math-container">$ax = ay$</span>, then <span class="math-container">$x = a^{-1}ax = a^{-1}ay = y$</span>. Also, for any <span class="math-container">$y \in G$</span>, let <span class="math-container">$x := a^{-1}y$</span>, then <span class="math-container">$ax = aa^{-1}y = y$</span>.</p>
|
114,733 | <p>Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?</p>
| 138 Aspen | 909,868 | <p>I wrote a simple demo with Mathematica to demonstrate it.</p>
<pre><code>f[z_] := (z - 1)/(z + 1);
Manipulate[
ComplexListPlot[Table[f[re + I*im], {im, -50, 50, 0.1}],
PlotRange -> {{-5, 5}, {-5, 5}}], {{re, 0}, -3, 3}]
</code></pre>
<p><a href="https://i.stack.imgur.com/7VICt.gif" rel="nofollow noreferrer"><... |
4,286,983 | <p>Let <span class="math-container">$p,q$</span> be two odd primes. Prove that <span class="math-container">$p$</span> is a primitive root of <span class="math-container">$q$</span> if and only if <span class="math-container">$\frac{x^q-1}{x-1}$</span> is an irreducible polynomial on <span class="math-container">$\math... | WhatsUp | 256,378 | <p>Here is a "fake" elementary proof, as it is essentially a translation of some results on finite fields (as will any answer probably be). A "real proof" is attached at the end.</p>
<hr />
<p><strong>The "fake" proof.</strong></p>
<p>Let <span class="math-container">$f(x) \in \Bbb F_p[x]$... |
2,235,610 | <p>I need some help for the proof of the uniformization theorem (Silverman's Advanced Topics ...).</p>
<p>If we have $G_{4}(\Lambda_{1})=G_{4}(\Lambda_{2}) $ and $ G_{6}(\Lambda_{1})=G_{6}(\Lambda_{2})$ (with $\Lambda_{1},\Lambda_{2}$ two lattices and $G_{n}$: Einsenstein serie).</p>
<p>Why we have $\Lambda_{1}=\Lam... | Angina Seng | 436,618 | <p>There is a nice analytic proof of this. The Weierstrass function $\wp(z)$ associated to $\Lambda$ satisfies a differential equation with coefficients derived from $G_4(\Lambda)$ and $G_6(\Lambda)$. It is the unique even function with principal part $1/z^2$ satisfying this. It has poles at
the points of $\Lambda$. So... |
102,383 | <p>I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP. </p>
<p><strong>The Problem</strong><br>
Let $A$ be a $n\times x$ possibly complex matrix and $B$ a diagonal, real $n\times n$ matrix with m... | Robert Israel | 13,650 | <p>Let's write this in block matrices: $$B = \pmatrix{B_{11} & 0\cr 0 & 0\cr}, \ A = \pmatrix{A_{11} & A_{12}\cr A_{21} & A_{22}\cr}, u = \pmatrix{u_1 \cr u_2\cr}$$ where $B_{11}$ has full rank.
Then the eigenvector equations
$A u = \lambda B u$ become $A_{11} u_1 + A_{12} u_2 = \lambda B_{11} u_1$ an... |
2,564,217 | <p>For a project I'm doing, I'm wrapping an led strip light around a tube. The tube is 19mm in diameter and 915mm tall. I'm going to coil the led strip around the tube from top to bottom and the strip is 8mm wide, so the coils will be 8mm apart. How long does the led strip need to be to fully cover the tube?</p>
<p>Th... | user326210 | 326,210 | <ul>
<li>Your approach seems correct. Let the rows of the table correspond to the 6 outcomes on the black die, and let the columns of the table correspond to the 6 outcomes on the red die.</li>
<li>For entry in the table, you can fill in the value of $X$ (outcome of the red die) and the value of $Y$ (absolute differenc... |
1,610,700 | <blockquote>
<p>$$\int \frac{x-3}{\sqrt{1-x^2}} \mathrm dx$$</p>
</blockquote>
<p>I know that $\int \frac{1}{1-x^2}\mathrm dx=\arcsin(\frac{x}{1})$ but how can I continue from here? </p>
| zz20s | 213,842 | <p>Write $\int \frac{x-3}{\sqrt{1-x^2}}\mathrm dx=\int \frac{x}{\sqrt{1-x^2}}\mathrm dx-\int \frac{3}{\sqrt{1-x^2}} \mathrm dx$.</p>
<p>For the first term, let $u=1-x^2$, leading to $\mathrm du=-2x \mathrm dx$. If you're still having trouble, write $\frac{-du}{2}=x \mathrm dx$, which appears in the numerator of your f... |
2,956,791 | <p>There is an equation here:
<span class="math-container">$$\sqrt{x+1}-x^2+1=0$$</span>
Now we want to write the equation <span class="math-container">$f(x)$</span> like <span class="math-container">$h(x)=g(x)$</span> in a way that we know how to draw h and g functions diagram.
Then we draw the h and g function diagra... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<ul>
<li><p>First draw <span class="math-container">$\sqrt{x}$</span>.</p></li>
<li><p>Now think of having drawn <span class="math-container">$h(x)$</span>, how would you draw <span class="math-container">$h(x\color{red}+1)$</span>.</p></li>
</ul>
|
2,142,042 | <p>how would you use induction to prove this:</p>
<p>$\sin(x)-sin(3x)+sin(5x)-...+(-1)^{(n+1)}sin[(2n-1)x] = \frac{(-1)^{(n+1)}sin2nx}{2cosx} $</p>
<p>I know how you assume its true for n=k, and then prove for n=k+1, but I get to </p>
<p>Left Hand Side: $\frac{(-1)^{(k+1)}sin2kx}{2cosx}+(-1)^{k+2}sin[(2k+1)x]$ but I... | Pierpaolo Vivo | 302,446 | <p>$$
\int_0^{\infty} \frac{x^{2p-1} dx}{(ax^2+b)^{p+q}}=\frac{1}{b^{p+q}}\int_0^{\infty} \frac{x^{2p-1}dx}{((a/b)x^2+1)^{p+q}}\ ,
$$
then change variables $(a/b)x^2=t\Rightarrow 2(a/b)xdx=dt$ to obtain
$$
\frac{1}{b^{p+q}}\frac{b}{2a}(b/a)^{p-1}\int_0^{\infty} \frac{t^{p-1}dt}{(t+1)^{p+q}}\ ,
$$
and then use the ident... |
1,050,917 | <p>I have a problem that I have to solve. I need to find center of the circle containing the point $(x,y)$. The point is $x=2,y=3$ with radius $r=3$. I need to find the center of circle. Is there equation for that? I use this equation.<br>
$$(x-h)^2+(y-k)^2=r^2$$
How I can find $h$ and $k$ for the center of circle if ... | Ross Millikan | 1,827 | <p>You have one equation in two unknowns, so should not expect a unique solution. Draw a circle around $(2,3)$ with radius $3$. Any of the points on this circle could be the center of the circle you seek.</p>
|
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | skoh | 10,200 | <p>$y$ is a value.</p>
<p>The difference between the two would essentially boil down to <em>dimensions</em> of $y$, with y-as-value being uni-dimensional and y-as-point being n-dimensional. Now, even with multiple $x$'s, all information about the intercept (think about difference in the intercepts of two models) is ca... |
428,408 | <p>Consider a norm on <span class="math-container">$\mathbb C^2$</span> as <span class="math-container">$\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$</span></p>
<p><em>Question.</em> Is <span class="math-container">$(\mathbb C^2,\|.\|)$</span> linearly isometric to <span class="math-container">$(\m... | Christian Remling | 48,839 | <p>There is no such map <span class="math-container">$f$</span>. Let's try to map from the second space (with the funny norm, which I'll denote simply by <span class="math-container">$\|\cdot\|$</span>) back to <span class="math-container">$(\mathbb C^2, \|\cdot \|_{\infty})$</span>. Let <span class="math-container">$u... |
271,105 | <p><strong>tl;dr</strong> What are some good workflows for developing and running data processing pipelines with Mathematica?</p>
<hr />
<p>I sometimes develop data processing pipelines with Mathematica. I load some data, transform it, and derive some summary results. I tend to experiment quite a bit when doing this, c... | David Keith | 44,700 | <p>I have frequently encountered a need to do the same thing. An example is a complex notebook which loads an image and analyzes it as scientific data. I want to run the same analysis on a large number of images to obtain a result for each. But the notebook is long and complex. Trying to merge it into a single cell for... |
1,290,363 | <p>So I already proved Closure and Associativity, now I'm trying to find the identity element of this operation defined as:
$$
a * b = a + b - ab
$$</p>
<p>But my identity element gets cancelled...</p>
<p>(The set defined in this exercise is the real numbers.)</p>
<p><img src="https://i.stack.imgur.com/ZchjC.jpg" al... | Sammy Black | 6,509 | <p>Your calculation was good up to
$$
e = ae.
$$
Remember that if $e$ is to be the identity, then you want this equation to hold <strong>for all $a$</strong>. If you like, you can rewrite the equation as
$$
0 = (a-1)e,
$$
The only solution is $e = 0$.</p>
<hr>
<p>By the way, there's a neat way to understand this ope... |
4,150,320 | <p>I need to prove <span class="math-container">$\displaystyle \lim _{x\to 2-} \left(\frac{|x-2|}{x^2-4}\right)=\frac{-1}{4}$</span></p>
<p>I know the definition <span class="math-container">$\forall \varepsilon >0, \exists \delta >0, 0>2-x>\delta$</span> then <span class="math-container">$\left|\left(\dfra... | miracle173 | 11,206 | <p>I wrote in a comment that one should do a check for the equations and that may reveal a way to deduce these equation. So for the equation <span class="math-container">$(6)$</span>
<span class="math-container">$$Q_{1} =\frac{p_{22} - p_{12} }{p_{11}+ p_{22} -2 p_{12} } Q+ \frac{\left( p_{23} -p_{13} \right) Q_{3} +... |
3,243,733 | <p><strong>Use induction to show that the Fibonacci numbers satisfy F(n) <span class="math-container">$\ge$</span> <span class="math-container">$(2 ^ {(n-1) / 2})$</span> for all n <span class="math-container">$\ge$</span> 3</strong></p>
<p>My work thus far:</p>
<blockquote>
<p>Base Case: F(3) <span class="math-co... | Anurag Singh | 466,382 | <p>Clearly, <span class="math-container">$F(n) \geq F(n-1)$</span> for all <span class="math-container">$n\geq 2.$</span> From inductive step we have that <span class="math-container">$F(n-1) \geq 2^{(n-2)/2}$</span>.</p>
<p>Then observe that, for all <span class="math-container">$n\geq 3$</span></p>
<p><span class="... |
177,574 | <p>Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional <em>real</em> vector, and define
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} \cdot (x_i - p)^2.$$
Experiments show that for small values of $k$
$$\exists x \in \mathbb{R}^{k+1} \,.\, \fo... | John Mount | 56,665 | <p>This is not a solution but some background to the question.</p>
<p>Define $$S(k,p,x) = \sum_{i=0}^k {k \choose i} p^i (1-p)^{k-i} (x_i-p)^2.$$
Define $$f(k) = \mathrm{argmin}_x \max_p S(k,p,x).$$
Then $f(k)$ is the minimax square-loss solution to trying to estimate the win rate of a random process by observing $k$ ... |
177,574 | <p>Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional <em>real</em> vector, and define
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} \cdot (x_i - p)^2.$$
Experiments show that for small values of $k$
$$\exists x \in \mathbb{R}^{k+1} \,.\, \fo... | Vladimir Dotsenko | 1,306 | <p>The solutions described via the link <a href="http://winvector.github.io/freq/explicitSolution.html" rel="nofollow">http://winvector.github.io/freq/explicitSolution.html</a> (posted in one of the earlier answers) can be given by the following formula:
$$
x_i=\frac{(k-2i)\sqrt{k}+(2i-1)k}{2k(k-1)}=\frac{1}{2(1+\sqrt... |
95,598 | <p>I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants.</p>
<p>I am trying to find the probability density, so I need to find the product of $\psi$ with it's complex conjugate. The problem is, im not sure what is it's complex conjugate, I know the complex conj... | Sasha | 11,069 | <p>The <a href="http://en.wikipedia.org/wiki/Complex_conjugate" rel="nofollow">complex conjugation</a> will map $A \to \bar{A}$ and $B \to \bar{B}$. </p>
<p>If, say $A= 5 + 4 i$, then $\bar{A} = 5 - 4 i$, as you noted. So
$$
\bar{\psi} = \bar{A} \mathrm{e}^{-i (k x - \omega t)} + \bar{B} \mathrm{e}^{i (k x + \omega... |
2,507,328 | <p>A bit of a beginner question, but I've been told that between 2 x-intercepts, for any polynomial of degree 2 or higher. That is true. But the controversy here is that apparently, it has to be exactly in the middle between the 2. I'm not talking about quadratics; the only turning point is at $x = -b/2a$. I am talking... | fonini | 113,664 | <p>If you want to use just the fact that "there is a Cauchy sequence", then indeed you're going to have a hard time. It's much easier if instead you <em>construct</em> that sequence yourself. Tip: do something like $\pi = \mbox{limit of $3; 3.1; 3.14; \ldots$}$</p>
|
1,344,161 | <p>Suppose $k\geq 2$ is an integer. I want to show $$\frac{1+k+k(k-2)}{1+\frac{k-1}{k}+\frac{(-1-\sqrt{k-1} )^2}{k(k-2)}}$$ is not an integer. It is equal to $$\frac{(k-2) k (k^2-k+1)}{2 (k^2-2 k+\sqrt{k-1}+1)}.$$</p>
<p>If I can show this then I will be able to finish my proof of the <a href="https://en.wikipedia.org... | Bill Dubuque | 242 | <p>It is a special case of the following</p>
<p><strong>Theorem</strong> <span class="math-container">$\ $</span> Suppose <span class="math-container">$\,f,g\in \Bbb Z[x]\,$</span> are polynomials, and <span class="math-container">$\,j\neq 0\,$</span> and <span class="math-container">$\,k,a\,$</span> are all integers.<... |
3,436,219 | <p><img src="https://i.stack.imgur.com/VplT3.jpg" alt="enter image description here"></p>
<p>I could use gaussian elimination if I make some assumptions or does any one have another suggestion?</p>
| Jolly Llama | 599,716 | <p>Andrew Chin's answer definitely works. Define <span class="math-container">$x\boxdot y$</span> by
<span class="math-container">$$x \boxdot y = x - y - 2,$$</span>
and it fits the data given. </p>
|
2,470,958 | <p>Let's say that I've got the following IVP:</p>
<p>$\frac{dy}{dx} = f(x,y)$</p>
<p>$y(x_0) = y_0$</p>
<p>And I want conditions that guarantee existence and uniqueness of its solution.</p>
<p>On the one hand I've got the Picard–Lindelöf theorem. It asks that there exists a rectangle $R = [a,b] \times [c,d]$, conta... | Hans Lundmark | 1,242 | <p>Suppose $\frac{\partial f}{\partial y}$ is continuous. Then $|\frac{\partial f}{\partial y}|$ has a greatest value $K$ on the rectangle in question (by the extreme value theorem). The mean value theorem for derivatives says that
$$
f(x,y_2)-f(x,y_1) = \frac{\partial f}{\partial y}(x,\eta) \, (y_2-y_1)
$$
for some $\... |
2,548,942 | <p>What would be the best approach to calculate the following limits </p>
<p>$$ \lim_{x \rightarrow 0} \left (1+\frac {1} {\arctan x} \right)^{\sin x}, \qquad \lim_{x \rightarrow 0} \frac {\tan ^7 x} {\ln (7x+1)} $$
in a basic way, using some special limits, without L'Hospital's rule? </p>
| user | 505,767 | <p>A solution for the first <strong>by Taylor series</strong>:</p>
<p>we can write the limit as follow:
$$\left (1+\frac {1} {\arctan x} \right)^{\sin x}=e^{sinx \ \log{\left (1+\frac {1} {\arctan x} \right)}}$$</p>
<p>Calculate Taylor series expansion for each term at the first order:
$$\sin x = x+o(x)$$</p>
<p>$$\... |
117,500 | <p>How would you go about finding the conjugacy classes of the nonabelian group of order 21, $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$?</p>
| Mikko Korhonen | 17,384 | <p>If $G$ is a nonabelian group of order $21$, then $G$ has trivial center. Otherwise $G/Z(G)$ would be cyclic and $G$ would be abelian.</p>
<p>Thus any element of order $3$ has its centralizer of order $3$ and thus has $7$ elements in its conjugacy class. By the same argument, an element of order $7$ has $3$ elements... |
514,338 | <p>Okay so my algebra knowledge is pretty guff..</p>
<p>I am taking a control systems class and pretty much all the questions I am expected to revise, are about doing this algebraic manipulation and I don't know what steps the tutor is taking to do it..</p>
<p>Okay here goes..</p>
<p>If the transfer function of a sy... | Michael Hoppe | 93,935 | <p>Multiply nominator and denominator by $20s+1$.</p>
|
1,295,259 | <p>How to prove that;</p>
<blockquote>
<p>$a^{|G|}=e$ if a $\in G $</p>
</blockquote>
<p>if $G$ is a finite group and $e$ is its identity.</p>
<p>I think this could be done through pigeonhole principle but I don't want to use the Lagrange theorem.</p>
<p>How should I start?</p>
| npatrat | 220,440 | <p>Let $d=ord(a)$. Then $a^d=e$ and $H=\{ e,a,a^2,...,a^{d-1} \}$ is a subgroup of $G$. By Lagrange, we have: $|G| \vdots |H|$, so $|G| \vdots d$; then, there is an $k \in \mathbb{N}$ such that $kd=|G|$. From this and $a^d=e$ we get that $a^{dk}=e^k$ or $a^{|G|}=e$.</p>
|
687,352 | <p>How many experiments should we conduct so that we could state that with more than $0.9$ probability the event occurs at least once. The probability that the event occurs is $0.7$. </p>
<p>I have tried the following:</p>
<p>Let's say the number of experiments is equal to $n$.
The opposite of 'occurs at least once'... | AlexR | 86,940 | <p>Note that
$$\left.\frac{|\sin x|}{|x|}\right|_{[n, n+1]} \ge\frac1{n+1}|\sin(x)|\tag 1$$
And that
$$\int_{\alpha}^{\alpha +1}|\sin(x)| dx \ge \int_{-\frac12}^{\frac12} |\sin x| dx = 2\int_0^{\frac12} \sin x dx = 2(1-\cos(\frac12)) =: C > 0\tag 2$$
So we have
$$\int_1^\infty \left|\frac{\sin x}x\right| dx =\sum_{n... |
3,047,241 | <blockquote>
<p>Let <span class="math-container">$X_1, X_2, \cdots, X_n$</span> be i.i.d. <span class="math-container">$\sim \text{Bernoulli}(p)$</span>. Then <span class="math-container">$\bar{x}$</span> is an unbiased estimator of <span class="math-container">$p$</span>.</p>
</blockquote>
<p>How should I approach ... | Ankit Seth | 393,189 | <p>You know that <span class="math-container">$E(X) = p$</span> or, for any <span class="math-container">$i$</span>, <span class="math-container">$E(X_i) = p$</span>.</p>
<p>So,</p>
<p><span class="math-container">$$E(\bar X) = E\Bigl(\frac {\sum_{i=1}^n X_i}{n}\Bigl)$$</span>
<span class="math-container">$$=\frac{1}... |
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | Toby Bartels | 8,508 | <p>In the theory of electromagnetism, the classical Stokes Theorem moves between the differential and integral forms of two of Maxwell's four equations; see <a href="https://en.wikipedia.org/wiki/Stokes%27_theorem#In_electromagnetism" rel="nofollow">https://en.wikipedia.org/wiki/Stokes%27_theorem#In_electromagnetism</a... |
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | Buschi Sergio | 6,262 | <p>I find interesting that divergence theorem (that is a corollary of the general Stokes theorem on differentiable varieties) in a vectorial form (one integral for any cathesian cohordinate) give a proof of the Archimedes' principle of buoyancy on the fluid .</p>
<p>given a body immersed on a (incompressible) fluid ... |
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | yael fregier | 13,742 | <p>You can tell your students that a clever use of Stokes theorem can give you a Fields medal. Indeed, the proof that the formality map given by M. Kontsevich is a $L_\infty$-morphism, is nothing else than Stokes theorem. A detailed account of this can be found in <code>Deformation quantization of Poisson manifolds. L... |
2,704,394 | <p>Here is the formal statement:</p>
<blockquote>
<p>Let $\lambda_1, \lambda_2, \lambda_3$ be distinct eigenvalues of $n\times n$ matrix $A$. Let $S=\{v_1, v_2, v_3\}$, where $Av_i = \lambda_i v_i$ for $1\leq i\leq 3$. Prove $S$ is linearly independent. </p>
</blockquote>
<p>Many resources online state the general ... | Robert Lewis | 67,071 | <p>Here's the $n$-eigenvector proof:</p>
<p>We assume</p>
<p>$A\vec v_i = \lambda_i \vec v_i, \; 1 \le i \le n, \tag 1$</p>
<p>with </p>
<p>$\lambda_i \ne \lambda_j, \; 1 \le i, j \le n; \tag 2$</p>
<p>assume there is a linear dependence between the eigenvectors:</p>
<p>$\displaystyle \sum_1^n a_i \vec v_i = 0, \... |
3,754,030 | <p>Let <span class="math-container">$I^n$</span> be the <span class="math-container">$n$</span>-cube <span class="math-container">$[0,1]^n$</span>. Also define two subsets of <span class="math-container">$\partial I^n$</span>:</p>
<ul>
<li><span class="math-container">$A=\{(x_1,\ldots,x_n)\mid x_1=0\}$</span></li>
<li>... | Greg Martin | 16,078 | <p>One way that is conceptually simple, and that could be turned into an explicit formula with enough effort, is:</p>
<ul>
<li>Define <span class="math-container">$g\colon I^n \to U$</span>, where <span class="math-container">$U$</span> is the closed ball whose boundary sphere circumscribes <span class="math-container"... |
3,203,607 | <p>"Each cell of a 100 × 100 table is painted either black or white and all
the cells adjacent to the border of the table are black. It is known that in every
2 × 2 square there are cells of both colours. Prove that in the table there is 2 × 2
square that is coloured in the chessboard manner."</p>
<p><a href="https://... | Mike Earnest | 177,399 | <p><strong>Hint:</strong> Stick <span class="math-container">$99\times 99$</span> needles on this grid, each at a place where four cells meet in a corner. For each pair of needles at distance one apart, connect them with a piece of string if the the two squares touching the edge between them have different colors. </p>... |
1,079,493 | <blockquote>
<p>Prove that <span class="math-container">$f(x) = x^3 + 3x - 1$</span> is irreducible in <span class="math-container">$\mathbb Q[X]$</span>.<br />
Let <span class="math-container">$\theta$</span> be a root of <span class="math-container">$f(x)$</span>. Compute <span class="math-container">$\frac{1}{\thet... | Tim Raczkowski | 192,581 | <p>Another idea is that if $f(x)$ is factorable over $\Bbb Q[x]$ it must have at least one rational zero. However, by the rational zero theorem, $\pm 1$ are the only possible rational zeros, but neither one is a zero.</p>
|
148,807 | <p>I'm not sure if these types of questions are accepted here or not (I'm very sorry if it's not), but it would be great if anyone could explain me this.</p>
<blockquote>
<p><strong>Question:</strong>
Using his bike, Daniel can complete a paper route in 20 minutes. Francisco, who walks the route, can complete it i... | Unreasonable Sin | 592 | <p>Suppose the paper route is 1 mile in length. Then Daniel is traveling at 3 miles an hour and Fransico is traveling at 2 miles an hour. Imagine the paper route is a straight line running left to right. Daniel starts his route at the far left side of the line traveling towards the right, and Fransico starts his route ... |
3,033,943 | <blockquote>
<p><span class="math-container">$\textbf{Problem}$</span> Let <span class="math-container">$\Omega$</span> be an open, bounded and connected subset of <span class="math-container">$\mathbb{R}^n$</span>. Suppose that <span class="math-container">$\partial \Omega$</span> is <span class="math-container">$C^... | Enkidu | 455,216 | <p>If you use the sequence definition of continuity, you take an arbitrary sequence <span class="math-container">$x_n\xrightarrow{\to \infty } x$</span> and want to prove that the image also converges.
Observe that any converging sequence <span class="math-container">$x_n\xrightarrow{\to \infty } x$</span> defines a se... |
122,945 | <p>Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following</p>
<p>Theorem: Fix $k \geq 0, 0<\alpha<1$. Let $f\in C^{k,\alpha}(S^1)$. Then its harmonic extension $H(f)$, which is ... | timur | 824 | <p>It follows from the Schauder theory. You can also establish Kellog's theorem directly. One approach is given in DiBenedetto's PDE book, where he uses Kellog's theorem in the proof of Schauder estimates.</p>
|
1,134,215 | <p>How can I determine whether {$\frac{z}{1+z^2}$; z $\in$ $\mathbb{C}$ \ {-i, i}} is bounded? My textbook is very poor at describing boundedness for complex functions. Thanks for the help!</p>
| Eric Wofsey | 86,856 | <p>If $\alpha$ and $\beta$ are automorphisms of $G$, then the cosets $\alpha X$ and $\beta X$ are the same iff $\alpha(K)=\beta(K)$. Thus $X$ will fail to have finite index if there are infinitely many different subgroups of $G$ that are conjugate to $K$ under automorphisms of $G$. For instance, if $G$ is an infinite... |
487,123 | <p>How to evaluate the following limit?
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$</p>
<p>For this problem I have two methods. But I'd like to know if there are better methods.</p>
<p><strong>My solution 1:</strong></p>
<p>Using Stolz-Cesaro Theorem, we have
$$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}=\lim_... | robjohn | 13,854 | <p>If we let
$$
a_n=\frac1{n!}\sum_{k=1}^nk!
$$
then obviously, $a_n\ge1$. Furthermore, we get that
$$
a_{n+1}=1+\frac{a_n}{n+1}
$$
Suppose that for some $n\ge1$, $a_n\le2$, then
$$
\begin{align}
a_{n+1}
&=1+\frac{a_n}{n+1}\\
&\le1+\frac{2}{n+1}\\
&\le2
\end{align}
$$
Since $a_1=1$, we have that $a_n\le2$ f... |
227,109 | <p>I keep mixing them up, because they are very similar.</p>
<p>Some contrapositives resemble some contradictions.</p>
| amWhy | 9,003 | <p>When one speaks of a <strong>contrapositive</strong> or proving a contrapositive, one is speaking about the contrapositive of an <em>implication</em> (an "if...then" statement), and as pointed out in the earlier answers, if one wants to prove that $$P \implies Q\tag{1}$$ one can choose, instead, to prove $$\lnot Q \... |
2,959,686 | <p>I'm trying to see if I can find a bijection between two groups that are infinite of which one in the subset of the other. If I find the inverse <span class="math-container">$\phi^{-1}(x)=\frac{1}{5}x$</span> since it doesn't work for <span class="math-container">$x \in \mathbb{Z}$</span> (because I will have values ... | Tsemo Aristide | 280,301 | <p>The inverse is not defined on the whole set, but only on the subset so it is a good argument. You can define <span class="math-container">$\phi:5\mathbb{Z}\rightarrow\mathbb{Z}$</span> by <span class="math-container">$\phi(z)={1\over 5}z$</span>.</p>
|
892,114 | <p>i have three number
1 2 3 which will always be in this order {123}, i want to find out number of cases can be made,
like {1},{2},{23},{13},{12},{123}{3},{}. but each number has two states like "a" "b", i.e, each one will become different entity,like 2a,2b,3a,3b,1a,
with only exception i.e. 1 will have only one stat... | evinda | 75,843 | <p>$$(x-2)^2=x^2-4x+4$$</p>
<p>$$(x-2)^2-12=x^2-4x+4-12=x^2-4x-8$$</p>
|
187,975 | <p>Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states that for every $t \in [0,1]$, there exists a set $S \in \Sigma$ with $\mu(S) = t$.</p>
<p>I would like to use the foll... | Ramiro de la Vega | 17,836 | <p>I would say this is folklore (I proved it and used it many years ago on my undergrad thesis), but here is a concrete reference:</p>
<p>Such a family of measurable sets is called a $[0,1]$-family in <em>On the Skorokhod representation theorem</em> by Jean Carlos Cortissoz, PAMS, Vol.135, No. 12, 2007 (see Definition... |
331,962 | <p>We have an first order ODE : </p>
<p>Equation1 : $y' + y = x$ ?
We can view the left-hand side as an operator acting on $y$. </p>
<p>In that case $L=(d/dx + 1)$ </p>
<p>$L(y_1) = x$<br>
$L(y_2)=x$<br>
$L(y_1+y_2)=x$<br>
So, clearly $L(y_1+y_2) = x \neq L(y_1)+L(y_2) = 2x$ </p>
<p>So why is $y'+y=x$ ... | Julien | 38,053 | <p>As we discussed earlier in another thread, the following
$$
L(y):=y'+y
$$
is a linear operator. Note it is not $y'+1$. And note that linear means
$$
L(\alpha y+ \beta z)=\alpha L(y)+ \beta L(z)
$$
for every scalars $\alpha,\beta$, and every differentiable functions $y,z$. The fact that $L$ is linear is merely the fa... |
1,474,867 | <p>I was trying to prove </p>
<p>$$\left|\int_{0}^{a}{\frac{1-\cos{x}}{x^2}}dx-\frac{\pi}{2}\right|\leq \frac{3}{a}$$ or $\leq \frac{2}{a}$. My work: I would like to use Fubini's theorem to prove it. </p>
<p>I notice that $\frac{1}{x^2}=\int^{\infty}_{0}{ue^{-xu}}du$. </p>
<p>Then, I got $\int_{0}^{a}{\frac{1-\cos{x... | Julian Rosen | 28,372 | <p>There appears to be a minor mistake in your computation. We have:
$$\begin{align*}
\int_0^a \frac{1-\cos(x)}{x^2}dx&=\int_0^a(1-\cos x)\int_0^\infty u e^{-xu}\,du\,dx\\
&=\int_0^\infty u\int_0^a (1-\cos x)e^{-xu}dx\,du\\
&=\int_0^\infty (1-e^{-au})-\frac{u^2}{1+u^2}+\frac{u^2\cos a-u\sin a}{1+u^2}\\
&... |
1,108,832 | <p>Q: A team of $11$ is to be chosen out of $15$ cricketers of whom $5$ are bowlers and $2$ others are wicket keepers. In how many ways can this be done so that the team contains at least $4$ bowlers and at least $1$ wicket keeper?</p>
| idm | 167,226 | <p>$$\binom{5}{4}\binom{2}{1}\binom{8}{6}+\binom{5}{5}\binom{2}{1}\binom{8}{5}+\binom{5}{4}\binom{2}{2}\binom{8}{5}+\binom{5}{5}\binom{2}{2}\binom{8}{4}$$</p>
|
2,510,322 | <p>$\left( f(x) \right ) =\min_{t<x}\left(t^2\right)$</p>
<p>How do I sketch this function for all real x? I don't get what minimum means in this context how do I sketch such a function when t is in the function but x isn't the square term? </p>
| Community | -1 | <p>$t^2$ is a decreasing function from $-\infty$ to $0$, so that its smallest value for $t\le x\le0$ is achieved at the bound $x$ and is $x^2$.</p>
<p>$t^2$ has a global minimum at $x=0$, with value $0$, so that its smallest value for $t,0\le x$ is $0$. Hence</p>
<p>$$f(x)=\begin{cases}x\le 0\to x^2,\\x\ge 0\to 0.\en... |
2,887,440 | <p>We were asked in our Calculus class to prove that,</p>
<blockquote>
<p>$f(x+y) - f(x) = \frac {\sec^2(x) \tan(y)} {1 - \tan(x) \tan(y)}$ given that $f(x) = \tan(x)$</p>
</blockquote>
<p>I have gotten so far as:</p>
<p>$$f(x+y) - f(x)$$</p>
<p>$$\tan(x+y) - \tan(x)$$</p>
<p>$$\frac{\tan(x)+\tan(y)}{1-\tan(x)\t... | tarit goswami | 579,780 | <p>For your first query: "how $f(x+y)$ became $tan(x+y)$?" - Inside the bracket, we write the parameters of a function. As, $f(x)=tan(x)$ for all $ x\in \mathbb{R} $, hence, $f(z)=tan(z)$ also with $z=x+y$, as reals are closed under addition(means for any two real $x$ and $y$ , $x+y$ is also a real number). Writing $x+... |
2,756,798 | <p>Consider the sequence space $l^2:=\{(x_n)_n\mid \sum^\infty_{n=0}x_n<\infty\}$ together with the norm
$$
||(x_n)_n||=(\sum^\infty_{n=0}|x_n|^2)^{1/2}
$$
How can I show that the triangle inequality holds for $||\cdot||$?</p>
| N. S. | 9,176 | <p><strong>Hint</strong> Cauchy-Schwartz. You can either show that $l^2$ is an inner product space, or use the fact that for each $N$ you have by C-S in $\mathbb R^N$:
$$(\sum^N_{n=0}|x_n+y_n|^2)^{1/2} \leq (\sum^N_{n=0}|x_n|^2)^{1/2}+(\sum^N_{n=0}|y_n|^2)^{1/2}\leq (\sum^\infty_{n=0}|x_n|^2)^{1/2}+(\sum^\infty_{n=0}|y... |
759,087 | <p>I'm busy writing my thesis, and I'm looking for some concise notation to denote the supremum of the matrix entries of, say $A \in M_n(\mathbb{R})$. How should I do this? </p>
<p>Looking for something like
$$\sup_{a_{i,j} \in A}|a_{i,j}|$$
but the notation $a_{i,j} \in A$ in reality doesn't make much sense in my opi... | Algebraic Pavel | 90,996 | <p>The defined quantity is not a "norm", it <strong>is</strong> a norm (not an operator norm though and not sub-multiplicative). I'm not aware of a standard notation for this quantity, but $\|\cdot\|_M$ or $\|\cdot\|_{\max}$ look suitable.</p>
|
1,754,931 | <p>If a sequence has a pattern where +2 is the pattern at the start, but 1 is added each time, like the sequence below, is there a formula to find the 125th number in this sequence? It would also need to work with patterns similar to this. For example if the pattern started as +4, and 5 was added each time.</p>
<block... | fleablood | 280,126 | <p>Let a be the first term. c be the added term. Then you add m more each term.</p>
<p>The kth term is a + c +(c+m)+... +(c + (k-2)m).</p>
<p>That is the kth term is $a + (k-1)c + m\sum_{i=0}^{k-2}i= a + (k-1)c +m\frac {(k-1)(k-2)}{2}$</p>
|
95,819 | <p>I think I have solved a problem in <em>Topology</em> by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight what is troubling me.</p>
<p>We prove by contradiction: Assume $X$ is not Hausdorff. Then there exist points... | Martin Sleziak | 8,297 | <p>Your claim that $(U\times V)\cap \Delta=(U\cap X)\times(V\cap X)$ is incorrect. Since $U,V\subseteq X$, this is the same as claiming
$(U\times V)\cap \Delta=U\times V$.</p>
<p>If you use
$(U\times V)\cap \Delta = \{(x,x); x\in U\cap V\}$ instead, the rest of your proof should work fine, but there are still several ... |
7,575 | <p>How could I display text that flashed red for a half second or so and then reverted to black? (Or was put in bold and reverted to normal, etc.)</p>
| kglr | 125 | <pre><code> PrintTemporary[Style["text", Red]]; Pause[2]; "text"
</code></pre>
<p><strong>EDIT:</strong> This looks too plain in comparison to all the cool effects that can be achieved with methods used in other answers. The following is an attempt to arm-twist <code>PrintTemporary</code> to perform similar tricks:</p... |
4,008,152 | <p>Question itself: Throw a coin one million times. What is the expected number of sequences of six tails, if we <strong>do not allow overlap</strong>?</p>
<p>I know when overlap is allowed, the answer is (1,000,000-5)/(2^6). Not sure if we can just do (1,000,000-5)/(2^6) divided by 6 if overlap is not allowed?</p>
<p>... | Rana | 43,899 | <p>If you are considering non-overlapping occurrence of <span class="math-container">$6$</span> consecutive Tails, then the occurrence of <span class="math-container">$6$</span> consecutive Tails is a renewal event. So, the whole might of renewal theory may be applied. See for more details in Feller Vol I.</p>
<p>I am ... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | Willie Wong | 125 | <p>Many of the disadvantages of allowing aids can be, in principle, resolved by </p>
<ol>
<li>requiring that the only aids the students have are handwritten by themselves and </li>
<li>setting a length limit. (I've seen somewhere between one index card [for non Americans: a piece of paper around 10 x 15 cm squared] an... |
98,700 | <blockquote>
<p>Suppose you wanted to write the number 100000. If you type it in ASCII, this would take 6 characters (which is 6 bytes). However, if you represent it as unsigned binary, you can write it out using 4 bytes.</p>
</blockquote>
<p>(from <a href="http://www.cs.umd.edu/class/sum2003/cmsc311/Notes/BitOp/asc... | Per Alexandersson | 934 | <p>As Joey tels you, the reason is that numbers are usually stored in the data type "integer", which (almost) always comes in 32 bit variants.
The processor is taylormade to add/subtract/multiply integers of exactly this size,
otherwise, you'll need 32*32*(number of operations) different circuits for every combination ... |
663,736 | <p>For a very large number n, how many divisibility tests are required to establish if its prime?</p>
<p>I know this has something to do with the Golden Number, but I can't figure out what. I did try searching for an answer but not much luck.</p>
<hr>
<p>!!EDIT!!
(It wont let me answer my own question for upto 8hour... | CorBrand | 126,846 | <p>Why would you not just check any given number to see if it is a prime number by simply attempting to divide it by preceding prime numbers? No matter how big the number is there can not be that many you would need to check it against.
Other than that unless you know a formula that can tell you what any given prime n... |
3,104,890 | <p>I'm trying to solve the following problem:</p>
<blockquote>
<p>Ten people are sitting around a round table. Three of them are chosen
at random to give a presentation. What is the probability that the
three chosen people were sitting in consecutive seats?</p>
</blockquote>
<p>I got the wrong answer but cannot... | drhab | 75,923 | <p>For your mistake see the answer of Arthur.</p>
<p>A bit more concise solution:</p>
<p>If the first person has been chosen then yet <span class="math-container">$2$</span> out of <span class="math-container">$9$</span> must be chosen. </p>
<p>In <span class="math-container">$3$</span> of these cases the three chos... |
629,275 | <p>A function $f$ is defined on an open set $D$ of $\mathbb R^{2}$ is called a differentiable at a point $x\in D$ if there is a vector $m \in \mathbb R^{2} $ such that
$$\lim_{h\to 0} \frac{f(x+h)-f(x)-m\cdot h}{|h|}=0.$$</p>
<p><strong>My questions are</strong>:
(1) What is a geometric interpretation of $f:\mathbb R... | Matheman | 117,904 | <p>(1) A function $f: D \rightarrow \mathbb{R}$ is differentiable at an interior point $x_0$ of $D$ if $\nabla f(x_0)$ exists and $$\lim_{x \rightarrow x_0} (f(x_0)+ \nabla f(x_0)(x-x_0) + o\left( \| x-x_0 \right \|)=0$$
holds (here $x,x_0$ are vectors of two or more dimensions).
Geometrically, for one variable functio... |
1,130,142 | <p><img src="https://i.stack.imgur.com/NXr1V.png" alt="enter image description here"></p>
<p>This is how I solved this problem but I have some reservations regarding my answer.</p>
<p>1st house = x ; 2nd house = 3x ; 3rd house = [3x + x] - 2610</p>
<p>12(x) + 12(3x) + 12(4x - 2610) = 186,390</p>
<p>96x = 155,070</p... | Kapoios | 37,324 | <p>Your system of equations is:</p>
<p>$y=3x$,
$z=4x-2,610$,
$6x+12y+12z=186,390$,</p>
<p>where $x$ is the first house's monthly rent and $y$, $z$ are the monthly rents for the second and the third house respectively.</p>
<p>Putting the first two equations into third and doing the calculations, gives
$x=2,419$.</p>
... |
363,391 | <p>In <a href="https://math.stackexchange.com/q/2602271/682690">this MathSE question</a>,
classification of finite simple groups with Abelian Sylow 2-subgroups,
credit is rightly given to John Walter. But in the introduction to his paper, Walter explicitly states that "It seems to be a very difficult problem to sh... | JCA | 159,448 | <p>It is described in Gorenstein's book on finite simple groups.</p>
|
254,253 | <blockquote>
<p>If the only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on... | André Nicolas | 6,312 | <p>There are $\dbinom{10}{4}$ equally likely ways to pick $4$ numbers. The number of ways to pick $1$, $8$, and two from the $6$ numbers from $2$ to $7$ inclusive is $\dbinom{6}{2}$. </p>
<p>Then multiply by $3$.</p>
<p><strong>Remark:</strong> In your solution, implicitly the numbers are being obtained in some speci... |
2,174,061 | <p>in $\Delta ABC$ if the $AD\perp BC$,$D\in BC$,and such $$|BC|=2|AD|$$
show that
$$\dfrac{|AB|}{|AC|}\le\sqrt{2}+1$$
<a href="https://i.stack.imgur.com/SXDvI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SXDvI.png" alt="enter image description here"></a></p>
<p>since
$$\cot{B}+\cot{C}=\dfrac{BD}... | Mick | 42,351 | <p>First of all, I must point out that the picture is wrongly drawn. The reason is if AB is shorter than AC, the LHS of the inequality-to-be-proved is less than 1 and obviously is less than $\sqrt 2 + 1$, the RHS. This means there is nothing to be proved. Therefore, we must have $AB \ge AC$ ….. (1).</p>
<p><a href="ht... |
3,363,944 | <p>A group consisting of <span class="math-container">$3$</span> men and <span class="math-container">$6$</span> women attends a prizegiving ceremony. If <span class="math-container">$ 5$</span> prizes are awarded at random to members of the group, find the probability that exactly <span class="math-container">$3 $</sp... | Oliver Kayende | 704,766 | <p>Part b) : Assuming the prizes are identical there are <span class="math-container">$${9\choose 1}+4*{9\choose 2}+6*{9\choose 3}+4*{9\choose 4}+{9\choose 5}=c={13\choose 5}$$</span> total ways of distributing them ; the nth term the count when there are n winners. Exactly <span class="math-container">$$({6\choose 3}+... |
2,261,927 | <p>How to get alternative form from equation 1)</p>
<p>$$ 1) -a^2 + a + b^2 -b $$</p>
<p>to equation 2)</p>
<p>$$ 2) (a-b)(a+b-1)$$</p>
| Ian Miller | 278,461 | <p>$$-a^2+a+b^2-b=-(a^2-b^2)+(a-b)$$</p>
<p>$$=-(a-b)(a+b)+(a-b)$$</p>
<p>To make it more obvious let $C=a-b$</p>
<p>$$=-C(a+b)+C$$</p>
<p>$$=C\big(-(a+b)+1\big)$$</p>
<p>$$=-C\big((a+b)-1\big)$$</p>
<p>$$=-(a-b)(a+b-1)$$</p>
|
2,261,927 | <p>How to get alternative form from equation 1)</p>
<p>$$ 1) -a^2 + a + b^2 -b $$</p>
<p>to equation 2)</p>
<p>$$ 2) (a-b)(a+b-1)$$</p>
| gue | 354,959 | <p>Another way would be polynomial division. </p>
<p>$-a^2 + a + b^2 - b : (a-b) = -a -b +1$</p>
<p>$a^2 - ab $</p>
<hr>
<p>$ -ab +a +b^2 -b$</p>
<p>$ ab -b^2 $</p>
<hr>
<p>$ a - b$</p>
|
4,131,747 | <p>I am having trouble with this problem in my Linear Algebra review:</p>
<blockquote>
<p>Find an equation for the plane parallel to <span class="math-container">$2x-y+2z=4 $</span> such that the
point <span class="math-container">$(3,2,-1) $</span> is equidistant from both planes.</p>
</blockquote>
<p>The answer is <s... | David K | 139,123 | <p>In order for the point <span class="math-container">$(3,2,−1)$</span> to be equidistant from two distinct parallel planes, it must be midway between them. Furthermore, one plane is a reflection of the other plane through the point <span class="math-container">$(3,2,−1).$</span></p>
<p>The point <span class="math-con... |
4,317,945 | <p>A function <span class="math-container">$h : A → \mathbb{R}$</span> is Lipschitz continuous if <span class="math-container">$\exists K$</span> s.t.</p>
<p><span class="math-container">$$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$</span></p>
<p>Suppose that <span class="math-container">$I = [a, b]$</span... | TheSilverDoe | 594,484 | <p>Let <span class="math-container">$f(x)=x^2(x+1)^n$</span>. One has <span class="math-container">$$f(x)=x^2(x+1)^n = x^{n+2} + nx^{n+1} + \frac{n(n-1)}{2}x^n + P(x)$$</span></p>
<p>where <span class="math-container">$P$</span> is a polynomial of degree <span class="math-container">$n-1$</span>. Hence
<span class="mat... |
600,404 | <p>I'm trying to study line bundle over $S^2$. <a href="https://mathoverflow.net/questions/113924/line-bundle-on-s2">In this post</a> was outlined the method based on clutching functions. But now I'm interesting in another approach. </p>
<p>For the sphere there is two maps : upper hemisphere and lower hemisphere with ... | DonAntonio | 31,254 | <p>Recognize the function evaluated at some partition of some interval...</p>
<p>$$\lim_{n\to\infty}\;\frac1n\sum_{k=1}^n\left(\frac kn\right)^p=\int\limits_0^1x^p\,dx=1\;\ldots$$</p>
<p>The first equality above stems from the fact that we <strong>know</strong> that $\;x^p\;$ is integrable on $\;[0,1]\;$ and thus we ... |
2,781,801 | <p>When asked to evaluate $g$ at the point specified above we would get $\dfrac{1}{e} \cdot \log_e(\frac{1}{\sqrt e})$ and that evaluates to some -0.18393... but the correct answer is -1/2e. How does it get simplified to that?</p>
| AHusain | 277,089 | <p>$$
\begin{eqnarray*}
\frac{1}{e} * \ln ( \frac{1}{\sqrt{e}}) &=& \frac{1}{e} * \ln ( e^{-1/2})\\
&=& \frac{1}{e} * \frac{-1}{2} \ln ( e)\\
&=& \frac{1}{e} * \frac{-1}{2} \\
&=& \frac{-1}{2e} \\
\end{eqnarray*}
$$</p>
|
2,567,332 | <p>A Greek urn contains a red, blue, yellow, and orange ball. A ball is drawn from the urn at random and then replaced. If one does this $4$ times, what is the probability that all $4$ colors were selected?</p>
<p>I approached this questions by doing $(1/4)^4$ because there's always a $1/4$ chance of selected a specif... | visitor | 401,140 | <p>The existing solutions provide the correct probability, but do not directly answer the question "What am I doing wrong?"</p>
<p>$(1/4)^4$ is the probability of a <em>specific</em> sequence of draws such as:</p>
<p>red, blue, yellow, orange</p>
<p>blue, yellow, orange, red</p>
<p>yellow, orange, blue, red</p>
<p... |
2,413,891 | <blockquote>
<p><strong>Question :</strong> Evaluate - $$\int_{0}^{1}2^{x^2+x}\mathrm dx$$</p>
</blockquote>
<p><strong>My Attempt :</strong> First I tried to evaluate the indefinite integral of $2^{x^2+x}$ in order to put the limits $0$ and $1$ later on, but couldn't integrate it. Then I checked on WA and came to k... | Claude Leibovici | 82,404 | <p>If you cannot use special functions, then either numerical integration or approximation would be required.</p>
<p>For example, consider the Taylor expansion built around $x=\frac 12$ (mid point of the integration interval selected in order to tvoid promoting one of the bounds). You would get
$$2^{x^2+x}=2^{3/4}+2... |
1,566,471 | <p>Hi can someone please help?</p>
<p>I need to evaluate this indefinite integral:</p>
<p>$$\int \frac{(\ln x)^5}x dx$$</p>
<p>I know I need to use substitution, so if I let <em>u= x</em> but I can't figure out the antiderivative for the top portion.</p>
<p>Thank you!</p>
| spandan madan | 296,493 | <p>A mistake in your argument as far as I understand-</p>
<p>If you are picking a random number in a continues range of [0,1], the probability of getting an exact number is zero as the number of options is infinite, so your sample space is infinite. While it is mathematically correct, it makes no sense as such.</p>
<... |
3,914,626 | <p>Let <span class="math-container">$A$</span> be a <span class="math-container">$k$</span>-dimensional non singualar matrix with integer coefficients. Is it true that <span class="math-container">$\|A^{-1}\|_\infty \leq 1$</span>? How can I show that? Could you give me a counterexample?It is clear that <span class="ma... | Mauro ALLEGRANZA | 108,274 | <p>The <a href="https://iep.utm.edu/nat-ded/#H7" rel="nofollow noreferrer"><span class="math-container">$(\forall \text I)$</span> rule</a> is:</p>
<blockquote>
<p>if <span class="math-container">$\Gamma \vdash \varphi[x/a]$</span>, then <span class="math-container">$\Gamma \vdash \forall x \varphi$</span>, provided t... |
2,110,286 | <p>Show that if $A$ and $B$ are subsets of a set $S$, then $\overline{A \cap B}=\overline{A}\cup \overline{B}$.</p>
<p>I tried to prove that $A \cap B=A \cup B$ because I didn't realize that the overline meant to prove it for the <em>closure</em> of the sets.</p>
<p>So, now I am confused about how to prove for closur... | Alex Mathers | 227,652 | <p>Like I said in my comment, I'm pretty sure that $\overline A$ is referring to the complement of $A$ in $S$. The way to prove this problem is to just blindly "chase elements":</p>
<p>Let $x\in\overline{A\cap B}$. Then $x\in S$ but $x\notin A\cap B$. Therefore $x\notin A$ or $x\notin B$. This precisely means $x\in\ov... |
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | Justin | 72,616 | <p>See: <a href="http://forums.xkcd.com/viewtopic.php?f=17&t=48391" rel="nofollow">$\sum_{n=1}^\infty(2^{\frac1{n}}-1)$</a> </p>
<p>There are a few methods listed there, one being writing ${2^{\frac1n}}$ as a power series. The easiest to understand is probably the limit comparison test where $b_n = \frac1n$.</p>
... |
690,331 | <p>Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? </p>
<p>For example, I am looking at computing an integral $\int_\gamma {1\over{z+4}}dz$ where $\gamma$ is the circle of radius $1$, centered at $-4$, oriented <strong>counterclockwise.</strong>... | anon | 11,763 | <p>$~$«$\displaystyle\sum_{i=1}^k E_i$ <em>is direct $\,\Leftrightarrow\,$ the $E_i$s intersect trivially pairwise</em> »$~$ is true for $V$ iff $k<3$ or $\dim V=1$.</p>
<p><em>Proof exercise</em>: Suffices to consider $k=3$, $\dim V=2$. Show if $V=\langle v,w\rangle$ then $\langle v\rangle,\langle w\rangle,\langle... |
2,282,818 | <p>I'm getting $f(x)=2x+f(0)$ and $f(x)=f(0)-2x$ by setting $y=0$, but I'd like to verify. Am I right?</p>
| Just_to_Answer | 439,212 | <p>Another way to look at it is perhaps that the condition is equivalent to<br>
$$\left|\dfrac{f(x)-f(y)}{x-y}\right| = 2 \quad \quad \text{for } x \neq y$$
which says that absolute values of the slopes of the secant lines at any pair of points $x$ and $y$ are always 2. That is, the possible slopes of the secant lines ... |
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | tomasz | 30,222 | <p>Note that sequential compactness also implies pseudocompactness, so any sequentially compact space which is not compact will work as well (the particular point topology is not sequentially compact, either, so it's different kind).</p>
<p>For example, the Corson space $\Sigma([0,1]^\kappa)$ of sequences of length $\... |
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | Austin Mohr | 11,245 | <p><a href="http://topology.jdabbs.com" rel="nofollow">$\pi$-Base</a>, a searchable version of Steen and Seebach's <a href="http://books.google.com/books/about/Counterexamples_in_Topology.html?id=DkEuGkOtSrUC" rel="nofollow"><em>Counterexamples in Topology</em></a>, gives the following examples of pseudocompact spaces ... |
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | Brian M. Scott | 12,042 | <p>An example that does not depend on countable compactness is Mrówka’s space $\Psi$. Subsets of $\omega$ are said to be <em>almost disjoint</em> if their intersection is finite. Let $\mathscr{A}$ be a maximal almost disjoint family of subsets of $\omega$, and let $\Psi=\omega\cup\mathscr{A}$. Points of $\omega$ are is... |
2,060,156 | <p>First thing I want to mention is that this is not a topic about why $1+2+3+... = -1/12$ but rather the connection between this summation and $\zeta$.</p>
<p>I perfectly understand that the definition using the summation $\sum_{k=1}^\infty k^{-s}$ of the zeta function is only valid for $Re(s) > 1$ and that the fu... | mfl | 148,513 | <p>We have that $$\frac{7k-5}{5k-3}=\frac{6l-1}{4l-3}\iff kl+8k+l=6.$$ That is, if $k\ne -1,$</p>
<p>$$l=2\frac{3-4k}{k+1}=-2\left(4-\frac{7}{k+1}\right)=-8+\frac{14}{k+1}.$$ Since $l$ has to be an integer $k+1$ must divide $14.$ So, we have that $k\in\{-15,-8,-3,-2,0,1,6,13\}.$ </p>
<p>Note that $k\ne -1$ since if $... |
1,384,735 | <p>What is the ODE satisfied by $y=y(x)$ </p>
<p>given that $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$</p>
<p>I understand that I need to get it in some form of $\int \cdots \;dy = \int \cdots \; dx$, but am not sure how to go about it.</p>
| Harish Chandra Rajpoot | 210,295 | <p>We have, $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$
Let $y=ux\implies \frac{dy}{dx}=x\frac{du}{dx}+u$ $$u+x\frac{du}{dx}=\frac{-x-2ux}{ux-2x}$$
$$u+x\frac{du}{dx}=\frac{2u+1}{2-u}$$ $$x\frac{du}{dx}=\frac{2u+1}{2-u}-u$$ $$x\frac{du}{dx}=\frac{1+u^2}{2-u}$$ $$\frac{(2-u)du}{1+u^2}=\frac{dx}{x}$$ Integrating both the side... |
503,589 | <ol>
<li><p>Let $\epsilon>0$. Prove that the set of those $x\in [0,1]$ such that there exist infinitely many fractions $p/q$, with relatively prime integers $p$ and $q$ such that
$$\bigg |x-\frac{p}{q}\bigg|\leq \frac{1}{q^{2+\epsilon}}$$
is a set of measure zero.</p></li>
<li><p>Let $(a_n)$ be a sequence of real nu... | Robert Israel | 8,508 | <p>Hint for (a): if $X$ is uniform on $[0,1]$, consider the random variables $Y_q = 1$ if $|X - p/q| \le 1/q^{2+\epsilon}$ for some $p$ relatively prime to $q$, $0$ otherwise. </p>
|
3,115,168 | <p>I've converted <span class="math-container">$\cos^3(x)$</span> into <span class="math-container">$\cos^2(x)\cos(x)$</span> but still have not gotten the answer. </p>
<p>The answer is <span class="math-container">$\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$</span></p>
<p>My answer was the same except I did not have a ... | Michael Rybkin | 350,247 | <p><span class="math-container">$$
\begin{align}
\int\cos^3{x}\,dx
&=\int\cos^2{x}\cdot\cos{x}\,dx\\
&=\int\cos^2{x}(\sin{x})'\,dx\\
&=\cos^2{x}\sin{x}+2\int\sin^2{x}\cos{x}\,dx\\
&=\cos^2{x}\sin{x}+2\int(1-\cos^2{x})\cos{x}\,dx\\
&=\cos^2{x}\sin{x}+2\int\cos{x}\,dx-2\int\cos^3{x}\,dx\\
&=\cos^2... |
2,083,127 | <p>How to show that $\lim_{n \rightarrow \infty} \frac{[a^{n+1}]}{[a^n]}=a$, where
$[a]$ = integer part of a?<br>
Here $a>1$. But I suspect it is true for all $a \ne 0$. </p>
| Στέλιος | 403,502 | <p>For $x>1$, we have the trivial inequalities $0<x-1<[x]\leq x$. We apply them and get:</p>
<p>$\frac{a^{n+1}-1}{a^n}\leq \frac{[a^{n+1}]}{[a^n]}\leq \frac{a^{n+1}}{a^n-1}$</p>
<p>But it is easy to check that for $a>1$, we have:</p>
<p>$\frac{a^{n+1}-1}{a^n}=a-a^{-n}\rightarrow a-0=a$,</p>
<p>$\frac{a^... |
2,889,835 | <p>If I have random lag times from <code>a=0.1</code> to <code>b=0.3</code> and a time to live (TTL) of <code>x=0.25</code>, what would be the packet loss in per cent?</p>
<p>Ok so basically I have packets that arrive in <code>Random [a,b]</code> time, if that random value is greater than <code>x</code> the packet get... | Kusma | 514,933 | <p>In complete metric spaces, Cauchy sequences are the same thing as convergent sequences, so in complete spaces, the statement contradicts the definition of continuity. </p>
<p>However, in your example, the space $(0,1)$ that your function is defined on is not complete. Hence there are Cauchy sequences that do not co... |
1,924,568 | <p>This is a question that a friend asked me (has the final answer too).</p>
<p>The pdf of a random variable $X$ is</p>
<p>$$ f(x) = 0.5,\quad -1 < x < 1 $$</p>
<p>The random variable Y is defined as </p>
<p>$$ Y = \begin{cases}
-2X, & -1 < X < 0 \\
X+1, & 0 < X <1
\end{cases}$$</p>
<p>I... | Graham Kemp | 135,106 | <p>Yes. It's called <em>folding</em>; when two disjoint intervals of the support of $X$ fold into the same interval in the support of $Y$, then the <em>change of variables transformation</em> folds the combined influence. $$\begin{align}f_Y(y)~=~& \Big\lvert\dfrac{\mathrm d x_1(y)}{\mathrm d y}\Big\rvert~f_X(x_... |
517,282 | <p>Suppose $a,n \in \mathbb{Z}$, and $n>a>0$. How do I prove that $\nexists x \in \mathbb{Z}$ s.t. $nx = a$ ? I'm really not sure where to start on this one. I'd be happy if someone could give me a hint.</p>
<p>Edit: I've solved this by contradiction, but I will not be 'accepting' an answer from below because I ... | Community | -1 | <p><strong>Hint</strong>: Note that $|nx| = |n| |x|$, and consider the cases $x = 0$ and $|x| \ge1$ separately. Start by noting that $a < n$, so how are $|n| |x|$ and $a$ related?</p>
|
149,558 | <p>I always use <code>InputForm</code> to check the result object,such as <code>Dataset</code> or <code>Graphics</code> or other objects.But if you are in the result of <code>InputForm</code>,you cannot use the Front-End function of balance the bracket. Note this gif</p>
<p><a href="https://i.stack.imgur.com/51OYd.gif"... | Kuba | 5,478 | <p>Carl's tip seems to be the best quick solution. </p>
<p>Very often I find syntax/style highlighting very useful too so I use:</p>
<pre><code>CellPrint[ExpressionCell[InputForm@#, "Input"]] &
</code></pre>
<p>to get everything what Input cells offer:</p>
<pre><code>Plot[x, {x, 0, 1}, PlotPoints -> 10, MaxR... |
6,931 | <p>One of the key steps in <a href="http://en.wikipedia.org/wiki/Merge_sort">merge sort</a> is the merging step. Given two sorted lists</p>
<pre><code>sorted1={2,6,10,13,16,17,19};
sorted2={1,3,4,5,7,8,9,11,12,14,15,18,20};
</code></pre>
<p>of integers, we want to produce a new list as follows:</p>
<ol>
<li>Start w... | Rojo | 109 | <p>Not to different to Heike's I think, because I haven't followed it line by line. Please let me know if it's too similar to be a separate answer</p>
<pre><code>merge[l1_, l2_, f_] := Block[{mergeAux},
mergeAux[list1_, list2_] :=
mergeAux[list2,
Function[fr,
Drop[list1, Length@Sow[TakeWhile[list1, f... |
2,305 | <p>I need an algorithm to produce all strings with the following property. Here capital letter refer to strings, and small letter refer to characters. $XY$ means the concatenation of string $X$ and $Y$.</p>
<p>Let $\Sigma = \{a_0, a_1,\ldots,a_n,a_0^{-1},a_1^{-1},\ldots,a_n^{-1}\}$ be the set of usable characters. Eve... | Sam Nead | 1,307 | <p>First of all, you might be interested in the work of Chas and Phillips: "Self-intersection of curves on the punctured torus". I've only skimmed their paper, but they seem to be doing something closely related to what you want.</p>
<p>Second I want to guess, for some reason, that the average time to compute self-in... |
1,821,800 | <p>Consider the system of ODE in $\Bbb R^2 $ </p>
<p>$\dfrac{dY}{dt}=AY$ where $Y(0)=$ \begin{bmatrix} 0 \\ 1\end{bmatrix} $t>0$ </p>
<p>where $ A=$ \begin{bmatrix} -1 & 1 \\ 0 & -1\end{bmatrix}</p>
<p>and $Y(t)=$\begin{bmatrix} y_1(t) \\ y_2(t)\end{bmatrix}</p>
<p><strong>My try</strong>:
$dy_1(t)=-y_1(... | mathlove | 78,967 | <p>Suppose that there exists such a positive rational number.</p>
<p>We have
$$x^2-\lfloor x^2\rfloor+x-\lfloor x\rfloor =1,$$
i.e.
$$x^2+x=\lfloor x^2\rfloor +\lfloor x\rfloor +1$$
We can set $x:=p/q$ where $p,q$ are positive integer with $\gcd(p,q)=1$, then
$$x^2+x=\frac{p}{q}\left(\frac pq+1\right)=m\tag1$$
where $... |
1,821,800 | <p>Consider the system of ODE in $\Bbb R^2 $ </p>
<p>$\dfrac{dY}{dt}=AY$ where $Y(0)=$ \begin{bmatrix} 0 \\ 1\end{bmatrix} $t>0$ </p>
<p>where $ A=$ \begin{bmatrix} -1 & 1 \\ 0 & -1\end{bmatrix}</p>
<p>and $Y(t)=$\begin{bmatrix} y_1(t) \\ y_2(t)\end{bmatrix}</p>
<p><strong>My try</strong>:
$dy_1(t)=-y_1(... | mathreadler | 213,607 | <p><strong>EDIT</strong> I accidentally tried solving the wrong problem. I did not know about the $\{\cdot\}$ notation for fractional part. This is an attempt to show that $x^2+x=1$ has no solutions for $x\in\mathbb{Q}$.</p>
<hr>
<p>Here's my attempt:
Assume there are $p,q \in {\mathbb Z}$ so that they are relative p... |
191,548 | <p>Say I have a list:</p>
<pre><code>{{Line[{{-Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, 1}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]},
{Line[{{-Sqrt[5/8 + Sqrt[5]/8],1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, ... | amator2357 | 61,985 | <p>As bad as it may look, this seems to be working fine:</p>
<p><code>Partition[Partition[Flatten[Position[points, #] & /@ Catenate @ Cases[lines, Line[pts : {{_, _} ..}] :> pts, Infinity]],2],Length[l]]</code></p>
|
237,031 | <p>The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the strongest large cardinal not known to be inconsistent with choice, as I understand)? This is implicit in the ordering of t... | Joel David Hamkins | 1,946 | <p>Regarding the edit, one can easily show some simple lower bounds for a Reinhardt cardinal that are far stronger than an inaccessible cardinal. For example, if $\kappa$ is a Reinhardt cardinal, assuming ZF only, then it is clear that $\kappa$ is inaccessible and weakly compact and much more in $L$, because it is the ... |
3,663,054 | <p>In my introductory abstract algebra course, the quotient group <span class="math-container">$G/H$</span> was defined as
<span class="math-container">$$G/H=\{gH:g\in G\}$$</span>
which is a <strong>set of sets</strong>. In an exercise, I should show that for the group of invertible matrices <span class="math-contain... | mag | 750,434 | <p>Regarding the distribution: It <a href="https://quant.stackexchange.com/questions/18646/distribution-of-stochastic-integral">holds</a> <span class="math-container">$$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau),$$</span> since <span class="math-container">$f(t)=e^{-t}$</span> is a square inte... |
2,357,115 | <blockquote>
<p>$A$ is invertible matrix over $\mathbb{R}$, prove that $AA^t+A^tA$ is invertible</p>
</blockquote>
<p>It seems to be a trivial question, but it's not.
I tried using determinants, i.e $|A| \ne 0 \to |AA^t+A^tA|\ne0$, but calculating $|AA^t+A^tA|$ is not easy.</p>
| José Carlos Santos | 446,262 | <p>The matrix $A$ is similar to an upper triangular matrix $T$ (over $\mathbb C$, although perhaps not over $\mathbb R$). The entries of the main diagonal of $T$ are all non zero, since $A$ is invertible. The entries of both matrices $TT^t$ and $T^tT$ are the squares of the entries of the main diagonal of $T$, and ther... |
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