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 |
|---|---|---|---|---|
162,324 | <p>Let $x_n$ be a sequence in a Hilbert space such that
$\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$.</p>
<p>Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $.</p>
<p>I need to show that $K$ is compact, $\operatorname{co}(K)$ is bounded, but not closed and finally find ... | hearse | 71,768 | <p>I will show you an example if the formulation of the minimization problem was of a single variable as: $f(x)+\lambda g(x)$</p>
<p>Now to find the lambda's first solve a closed form for x by setting the gradient w.r.t x as zero. You will have a closed form for x, containing the lambda's. </p>
<p>Now consider this t... |
594,498 | <p>Let $U$ be a linear subspace of $V$. Show that $\dim_K(U)\leq \dim_K(V)$ and conclude, that $\dim_K(U)= \dim_K(V) \Leftrightarrow U = V$. Is the equivalence $\dim_K(U)= \dim_K(V) \Leftrightarrow U = V$ also true for $\dim_K(V)= \infty$?</p>
<p>I've proved so far that $\dim_K(U)= \dim_K(V) \Leftrightarrow U = V$,but... | Community | -1 | <p><strong>Hint</strong></p>
<p>Let the $V$ the subspace of $\mathbb R[x]$ defined by
$$V=\mathrm{span}\left\{x^{2n},\ n\in\mathbb N\right\}$$
Prove that $V\varsubsetneq \mathbb R[x]$. What's $\dim V?$</p>
|
2,249,036 | <p>I was watching a video on PDEs and when arriving at the part of Fourier Series, the professor said:</p>
<blockquote>
<p>And one of the most fascinating reads I ever had was a paper by Riemann on the history of this [Fourier Series].</p>
</blockquote>
<p>I tried looking for it but didn't succeed, and I was wonder... | Chappers | 221,811 | <p>The scalar product of vectors $a=(a_1,a_2)$ and $b=(b_1,b_2)$ is given by the two formulae (provable equivalent using the cosine rule, see <a href="https://math.stackexchange.com/a/2227712/221811">here</a>)
$$ a \cdot b = a_1b_1+a_2b_2 = \sqrt{a_1^2+a_2^2}\sqrt{b_1^2+b_2^2} \cos{\theta}, $$
where $\theta$ is the ang... |
121,631 | <p>How to prove most simply that if a polyonmial $f$, has only real coefficients and $f(c)=0$, and $k$ is the complex conjugate of $c$, then $f(k)=0$?</p>
| WimC | 25,313 | <p>Look at $\overline{f(c)}$ and use that conjugation is a homomorphism of $\mathbb{C}$. That is, $\overline{a+b} = \overline{a}+\overline{b}$ and $\overline{a\cdot b} = \overline{a} \cdot \overline{b}$.</p>
|
2,696,097 | <p>$ \lim_{n\to \infty} ( \lim_{x\to0} (1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} $</p>
<p>The answer should be $ {e}^\frac{1}{3} $</p>
<p>I haven't encountered problems like this before and I'm pretty confused, thank you.</p>
<p>I guess we must use the remarkable limit of $ \frac{\tan(x)}{x} ... | celtschk | 34,930 | <p>“Let $K$ be the set of elements in $G$ not in $H$, also including the identity.” — that's a contradiction, as the identity is always in $H$. I'll assume you mean $K = (G\setminus H)\cup \{e\}$ where $e$ is the identity.</p>
<p>$K$ cannot be a subgroup.</p>
<p><strong>Proof:</strong> Take $h\in H$ and $k\in K$, bot... |
1,267,644 | <p>Having a circle of radius <span class="math-container">$R$</span> with the center in <span class="math-container">$O(0, 0)$</span>, a starting point on the circle (e.g. <span class="math-container">$(0, R)$</span>) and an angle <span class="math-container">$\alpha$</span>, how can I move the point on the circle with... | Community | -1 | <p>well you can consider the following linear transformation given by:</p>
<p>T : <span class="math-container">$R^2$</span> <span class="math-container">$\rightarrow$</span> <span class="math-container">$R^2$</span></p>
<p>given by T(x,y) = (cos<span class="math-container">$\theta$</span>x - sin<span class="math-cont... |
1,610,800 | <p>When dividing $f(x)$ by $g(x)$: $f(x)=g(x)Q(x)+R(x)$.
How to find the quotient $Q(x)$ and the remainder $R(x)$?
For example: $f(x)=\ 2x^4+13x^3+18x^2+x-4 \ $
, $g(x)=\ x^2+5x+2 \ $</p>
<blockquote>
<p>At first $g(x)= (x+4.56)(x+0.44)$ then we use synthetic division : $Q(x)=
\ 2x^2+3x-1 \ $ , $R(x)= \ 0.04x-1.98... | Community | -1 | <p><strong>Hint</strong>:</p>
<p>If two polynomials have common roots, they have a common factor (which is the product of the binomials $z-r_i$).</p>
<p>This common factor is their $\gcd$, for which the Euclidean algorithm can be used (divide $p$ by $q$; if there is a remainder, let $r$, divide $q$ by $r$, and so on)... |
1,598,335 | <p>Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$.
I have tried: </p>
<p>Let
$y=3^x$ to $y=e^{x'}$ </p>
<p>$$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$</p>
<p>$$x\log_{e}(3)=x'$$</p>
<p>Gives the transformation as dilate by $\log_e(3)$
And also this:</p>
<p>$$... | Angelo Mark | 280,637 | <p>Clearly </p>
<p>$1) \to$ $$2x=\cos(\theta)+\sqrt{2}\sin(\theta)$$</p>
<p>$2) \to $ $$2y=-\cos(\theta)+\sqrt{2}\sin(\theta)$$</p>
<p>Thus by $1)+2)$ we get ,</p>
<p>$$2x+2y=2\sqrt{2}\sin(\theta) $$</p>
<p>So $$\frac{x+y}{\sqrt{2}}=\sin(\theta)$$</p>
<p>Thus by $1)-2)$ we get ,</p>
<p>$$2x-2y=2\cos(\theta) $$</... |
3,503,175 | <p>How to prove the following formulas</p>
<p><span class="math-container">$$
\sum_{n= 0}^{\infty} \frac{\cos(nx)}{n!} = e^{\cos(x)} \cos(\sin x) \\
\sum_{n= 0}^{\infty} \frac{\sin(nx)}{n!} = e^{\cos(x)} \sin(\sin x)
$$</span></p>
<p><strong>without</strong> using complex numbers ? </p>
<p>These summations can be do... | pathfinder | 23,431 | <p>For the second identity, you can use the Chebyshev polynomial of the second kind <span class="math-container">$U_n(x)$</span> since
<span class="math-container">$$\sin(nx)=\sin xU_{n-1}(\cos x).$$</span> Hence,
<span class="math-container">$$\sum_{n=0}^\infty\frac{\sin(n x)}{n!}=\sin x\sum_{n=0}^\infty\frac{ U_{n-1}... |
3,503,175 | <p>How to prove the following formulas</p>
<p><span class="math-container">$$
\sum_{n= 0}^{\infty} \frac{\cos(nx)}{n!} = e^{\cos(x)} \cos(\sin x) \\
\sum_{n= 0}^{\infty} \frac{\sin(nx)}{n!} = e^{\cos(x)} \sin(\sin x)
$$</span></p>
<p><strong>without</strong> using complex numbers ? </p>
<p>These summations can be do... | QC_QAOA | 364,346 | <p>Define</p>
<p><span class="math-container">$$g(x)=\sum_{n= 0}^{\infty} \frac{\cos(nx)}{n!}$$</span></p>
<p><span class="math-container">$$f(x)=\sum_{n= 0}^{\infty} \frac{\sin(nx)}{n!}$$</span></p>
<p>We can differentiate <span class="math-container">$g(x)$</span> and <span class="math-container">$f(x)$</span> ter... |
1,587,007 | <p>I have the following matrix $\mathbf{U}$ which is in echelon form. The strange to me is that I havent met a matrix with first column zero. </p>
<p>$$\mathbf{U} = \begin{bmatrix}
0 & 5 & 4 & 3 \\
0 & 0 & 2 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}$$</p>
<... | Dave L | 300,365 | <p>Any of $B_1$, $B_2$ or $B_3$ is a basis for the column space of $\mathbf{U}$. It is pretty clear that the last column is a linear combination of either $B_2$ or $B_3$, with the coefficients 0 and 1. There is a little arithmetic to do to find the coefficients to express the last column in terms of $B_1$, but you ca... |
4,005,463 | <p>In the problem I am solving, table values are given for function and it says to find <span class="math-container">$f''(0.5)$</span> using second order central difference formula. I know the formula which is
<span class="math-container">$\frac{f(x+\triangle x)-2f(x)+f(x-\triangle x)}{{\triangle x}^2}$</span>. But the... | AlvinL | 229,673 | <p>It's not said that <span class="math-container">$Y$</span> is a complete metric space, but we can work in the completion <span class="math-container">$\overline{Y}$</span>.</p>
<p>If I'm reading this right, we want to prove</p>
<blockquote>
<p>If <span class="math-container">$\mathscr F \subseteq C(X,Y)$</span> is e... |
1,627,381 | <p>There is a subset of sigma field $G_2$, say $G_1 \subset G_2$. $G_1$ is proven to be a sigma field.
Does this necessarily imply that $G_1 = G_2$?</p>
| Artem Mavrin | 225,635 | <p>No: If $X$ is nonempty, $\mathcal{G}_1 = \{\emptyset, X\}$, and $\mathcal{G}_2 = \mathcal{P}(X)$, then $\mathcal{G}_1$ and $\mathcal{G}_2$ are $\sigma$-fields with $\mathcal{G}_1 \subseteq \mathcal{G}_2$, but $\mathcal{G}_1 \neq \mathcal{G}_2$.</p>
|
286,932 | <p>For convenience we work with commutative rings instead of commutative algebras.</p>
<hr>
<p>Fix a commutative ring $R$. Consider the functor $\mathsf{Mod}\longrightarrow \mathsf{CRing}$ defined by taking an $R$-module $M$ to $R\ltimes M$ (with dual number multiplication). Following the <a href="https://ncatlab.org... | Vladimir Sotirov | 75,650 | <p>Some time ago I worked out how to define Kähler differentials and derive their cotangent and conormal exact sequences in any protomodular category with pullbacks and pushouts. Based on the same idea, I think the following argument works to show that a morphism being unramified is equivalent to the vanihing of its mo... |
87,544 | <p>For teaching purposes, I want to create a <em>Mathematica</em> notebook that will "notice" when the user defines or redefines a variable or function of a particular name, so that it can check the value and take some appropriate action. For example, the notebook might be monitoring the symbol "foo", so that if the us... | Karsten 7. | 18,476 | <p>One can use <a href="http://reference.wolfram.com/language/ref/$Pre.html?q=%24Pre" rel="nofollow noreferrer"><code>$Pre</code></a> to check if an input expression defines the correct variable and is doing so using the correct value.</p>
<pre><code>SetAttributes[check, HoldAll]
check[new_Set] := (Print["You guessed ... |
1,654,354 | <p>Why is $A=\{(x_1,x_2,...,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set?</p>
<p>This claim was shown in a solution I ran into, and I don't see how it holds. I try to follow the formal definition of nullity, which is having the possibility to be covered by a collection of open cubes, whose volume is as small as desire... | skyking | 265,767 | <p>The trick is to find a cover of $A_{j;k} = \{x: x_j = x_k\}$ then we use that $A = \bigcup A_{j;k}$ (so a cover for each of $A_{j;k}$ will together cover $A$).</p>
<p>Now to cover it with open cubes the trick is that we first don't have to use cubes of the same size. It should also be understood that we're allowed ... |
84,977 | <p>In how many ways 3 flags of colors black, purple & yellow can be arranged at the corners of an equilateral triangle?</p>
| Listing | 3,123 | <p>There is only a very finite amount of possibilities... Try to find out the distinct cases:</p>
<p><img src="https://i.stack.imgur.com/pO6u3.jpg" alt="a"> <img src="https://i.stack.imgur.com/yXjQ6.jpg" alt="b"> <img src="https://i.stack.imgur.com/Wagot.jpg" alt="c"></p>
<p><img src="https://i.stack.imgur.com/AXXBP.... |
4,249,789 | <p>My understanding is that the span of a set is a set of all vectors that can be obtained from the linear combination of all the vectors in the original set as shown in image #<a href="https://i.stack.imgur.com/3v6g4.png" rel="nofollow noreferrer">1</a>.</p>
<p><img src="https://i.stack.imgur.com/3v6g4.png" alt="Image... | Mason | 752,243 | <p>The rank of a matrix <span class="math-container">$A \in M(m \times n, \mathbb{R})$</span> is defined as the dimension of its range, i.e. <span class="math-container">$\dim(R(A))$</span>. We will show that <span class="math-container">$\dim(R(A)) = \dim(R(A^T))$</span>.</p>
<p>Using the identity <span class="math-co... |
504,997 | <p>I have a dynamic equation,
$$ \frac{\dot{k}}{k} = s k^{\alpha - 1} + \delta + n$$
Where $\dot{k}/k$ is the capital growth rate as a function of savings $s$, capital $k$, capital depreciation rate $\delta$, and population growth rate $n$.</p>
<p>I have been asked to find the change in the growth rate as $k$ increase... | Community | -1 | <p>Notation for partial derivatives is inherently awkward. e.g. "growth rate as $k$ increases" doesn't <em>actually</em> make sense: there's implicit context "... while holding $s$, $\alpha$, $n$, and $\delta$ constant".</p>
<p>Differentials become cleaner when notation gets confusing. We have</p>
<p>$$ d \frac{\dot{... |
2,851,609 | <p>I need to find the solution to the inequality $(x - y)(x + y -1) > z$, where $x,y,z \geq 0$ and $x,y,z \leq 1$. As $z$ is positive, then the inequality holds whenever (i) $x - y > 0$ and $x + y - 1 > 0$ OR (ii) $x - y < 0$ and $x + y - 1 < 0$. I can solve the cases (i) and (ii) on their own, but I don... | Dr. Sonnhard Graubner | 175,066 | <p>Hint; It is $$x^2-x-y^2+y-z>0$$ you can solve it now for $x$ or $y$.</p>
|
1,382,507 | <p>$$A= \left\{\frac{m}{n}+\frac{4n}{m}:m,n\in\mathbb{N}\right\}$$</p>
<hr>
<p>Since $m,n\in \mathbb{N}$, infimum is zero because $m,n$ both are increasing to infinity. Then the supremum is $5$ when $m,n$ are equal to $1$. </p>
<p>But I don't think my approach is right. Can someone give a hit or suggestion to get t... | wltrup | 232,040 | <p>$$\frac{\frac{1}{x+3} - \frac{1}{3}}{x} =
\frac{3 - (x+3)}{3x\,(x+3)} =
\frac{-x}{3x\,(x+3)} =
\frac{-1}{3(x+3)}
$$</p>
<p>And you can continue from there.</p>
|
1,382,507 | <p>$$A= \left\{\frac{m}{n}+\frac{4n}{m}:m,n\in\mathbb{N}\right\}$$</p>
<hr>
<p>Since $m,n\in \mathbb{N}$, infimum is zero because $m,n$ both are increasing to infinity. Then the supremum is $5$ when $m,n$ are equal to $1$. </p>
<p>But I don't think my approach is right. Can someone give a hit or suggestion to get t... | Mark Viola | 218,419 | <p>The solution presented by @wltrup is solid and efficient. I thought that it would be instructive to present another way forward. </p>
<p>Here, straightforward application of L'Hospital's Rule reveals</p>
<p>$$\begin{align}
\lim_{x\to 0}\frac{\frac{1}{x+3}-\frac13}{x}&=\lim_{x\to 0}\left(-\frac{1}{(x+3)^2}\ri... |
3,002,767 | <p><span class="math-container">$l^p$</span> appears frequently in undergrad real analysis courses, I wonder if there is any strong connection between <span class="math-container">$l^p$</span> and <span class="math-container">$L^p$</span> space? (Other than they look similar)</p>
<p>I give one definition of <span clas... | Paul | 396,004 | <p><span class="math-container">$L^p$</span> and <span class="math-container">$\ell^p$</span> spaces both come from the same definition in measure theory. Given a measure space <span class="math-container">$(\Omega,\Sigma,\mu)$</span> you can define <span class="math-container">$L^p(\Omega,\Sigma,\mu)$</span> as the se... |
289,367 | <p>Given a positive definite matrix $Q\in\mathbb{R}^{n \times n}$, I want to find a diagnonal matrix $D$ such that $rank(Q-D) \leq k < n$.</p>
<p>I think this can be regarded as a generalization of eigenvalue problem, which is basically problem of finding a diagonal matrix $\lambda I$ such that $rank(Q-\lambda I) &... | Alexandre Eremenko | 25,510 | <p>Your problem can be restated as follows: To a given symmetric matxix, can
you add a diagonal matrix so that the result has eigenvalue $0$ with high
multiplicity?</p>
<p>This belongs to the theory which is called Additive Inverse Eigenvalue Problems. See, for example this paper, which seems to treat a very similar p... |
2,607,668 | <p>I am trying to prove/disprove $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)$. Apparently $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)+2k\pi$ where $k=0,1,\text{ or }-1$, but I have no idea why. I keep on finding that answer online. I am very lost on how to prove this stat... | Jan Eerland | 226,665 | <p>Let's say we have two complex numbers $\text{z}_1$ and $\text{z}_2$ we can write:</p>
<ul>
<li>$$\text{z}_1=\left|\text{z}_1\right|\cdot\exp\left(\left(\arg\left(\text{z}_1\right)+2\pi\cdot\text{k}_1\right)\cdot i\right)\tag1$$</li>
</ul>
<p>Where $0\le\arg\left(\text{z}_1\right)<2\pi$ and $\text{k}_1\in\mathbb... |
2,607,668 | <p>I am trying to prove/disprove $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)$. Apparently $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)+2k\pi$ where $k=0,1,\text{ or }-1$, but I have no idea why. I keep on finding that answer online. I am very lost on how to prove this stat... | user | 505,767 | <p>It can be shown in many ways. The simplest is to consider the exponential form of complex numbers $$z=\rho e^{i\theta}$$</p>
<p>with $|z|=\rho$ and $Arg(z)=\theta$</p>
|
1,157,877 | <p>We have $n$ bags of sand, with volume $$v_1,...,v_n, \forall i: \space 0 < v_i < 1$$ but not essentially sorted. we want to place all bag to boxes with volumes 1. We propose one algorithm:</p>
<blockquote>
<p>At first we place all bags in the original order. Then we select one
box and place on it, bag $1,... | user121049 | 121,049 | <p>Write $g(x)=\int_0^x{f(\alpha)}$
The equation becomes something like $\frac{d(xg)}{dx}=-\frac{d Ln(1-g)}{dx}$ This gives you an equation for g which is unfortunately not solvable.</p>
|
3,957,972 | <p>As part of a longer problem, I need to find the number of elements of the multiplicative group of <span class="math-container">$\mathbb{Z}_2[x] / (x^3 + x^2 + 1)$</span>.</p>
<p>I have no idea where to start with this. I understand that these polynomials have coefficients in the finite field <span class="math-contai... | Lubin | 17,760 | <p>I suspect that you have all the ingredients already, maybe have already put them together to get your desired degree of understanding. But let me lay it all out:</p>
<p>Any time you have a field <span class="math-container">$k$</span> and an irreducible <span class="math-container">$k$</span>-polynomial <span class=... |
3,957,972 | <p>As part of a longer problem, I need to find the number of elements of the multiplicative group of <span class="math-container">$\mathbb{Z}_2[x] / (x^3 + x^2 + 1)$</span>.</p>
<p>I have no idea where to start with this. I understand that these polynomials have coefficients in the finite field <span class="math-contai... | Community | -1 | <p>The polynomial <span class="math-container">$x^3+x^2+1$</span> is irreducible, because neither <span class="math-container">$0$</span> nor <span class="math-container">$1$</span> is a root <span class="math-container">$\bmod2$</span>. So we get a field, an extension of <span class="math-container">$\Bbb Z_2$</span>... |
437,053 | <p>I'm struggling with this nonhomogeneous second order differential equation</p>
<p><span class="math-container">$$y'' - 2y = 2\tan^3x$$</span></p>
<p>I assumed that the form of the solution would be <span class="math-container">$A\tan^3x$</span> where A was some constant, but this results in a mess when solving. The ... | Sean Boyd | 85,168 | <p>Both $a$ and $a^3$ generate the cyclic subgroup of order 5 to which they both belong. Try writing $a$ in terms of $a^3$; i.e., if $b=a^3$, express $a$ in terms of $b$. We can do this because the powers of $a$ in the problem statement are relatively prime to $5$ (and thus generate the cyclic subgroup $\langle a \rang... |
437,053 | <p>I'm struggling with this nonhomogeneous second order differential equation</p>
<p><span class="math-container">$$y'' - 2y = 2\tan^3x$$</span></p>
<p>I assumed that the form of the solution would be <span class="math-container">$A\tan^3x$</span> where A was some constant, but this results in a mess when solving. The ... | Lucas Willhelm | 761,431 | <p>I will complete the proof, by proving two separate statements. The first being, <span class="math-container">$C(a)\subseteq C(a^3)$</span>. Naturally, the second statement is <span class="math-container">$C(a^3) \subseteq C(a)$</span>. First, suppose some <span class="math-container">$x \in C(a)$</span>. This implie... |
3,391,118 | <p>let <span class="math-container">$x , y$</span> be a real numbers with <span class="math-container">$-3\leq x \leq5$</span> and <span class="math-container">$-2\leq y\leq -1$</span> , I ask if the range of <span class="math-container">$x-y$</span> is <span class="math-container">$[-1,6]$</span> or <span class="math... | JMP | 210,189 | <p>You should end up with:</p>
<p><span class="math-container">$$-2\le x-y \le 7$$</span></p>
<p>When you negate the <span class="math-container">$y$</span> inequality, you have:</p>
<p><span class="math-container">$$2\le -y \le 1$$</span></p>
<p>which is obviously wrong, you should have:</p>
<p><span class="math-... |
3,030,332 | <p>I have started complex analysis and I am stuck on one definition 'extended complex plane'.Book is saying 'To visualize point at infinity,think of complex plane passing through the equator of a unit sphere centred at 0.To each point z in plane there corresponds exactly one point P on surface of sphere which is obtain... | user | 505,767 | <p>The points inside the unit circle are projected onto the emisphere under the plane (that is <span class="math-container">$Z<0$</span>):</p>
<p><a href="https://i.stack.imgur.com/SC6vO.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SC6vO.jpg" alt="enter image description here"></a></p>
<p>(cr... |
231,403 | <p>Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1, 1}(0, 1)?$$</p>
| Romain Gicquaud | 24,271 | <p>Yes, this is the Sobolev injection $W^{1, 1}(]0, 1[) \to C^0([0, 1])$ (see e.g. Brezis' book on functional analysis)
$$
\|u\|_{L^\infty} \leq C \left( \|u'\|_{L^1} + \|u\|_{L^1}\right)
$$</p>
<p>followed by the $\epsilon$-Young inequality:</p>
<p>\begin{align*}
\|u\|_{L^p} & \leq \left(\|u\|_{L^1} \|u\|_{L^\... |
2,234,478 | <p>Given 3 points, A (x1, y1), B (x2, y2) and C (x3, y3), what is the best way to tell if all 3 points lie within a circle of a given radius <strong>r</strong>?</p>
<p>The best I could come up with was to find the Fermat point F (x4, y4) for the triangle ABC, and then check if the distance from F to each A, B and C is... | Claude Leibovici | 82,404 | <p>Starting using egreg's solution, consider that you look for the zero of function $$f(t)=2^t\log t+\log 2$$ $$f'(t)=\frac{2^t}{t}+2^t \log (2) \log (t)$$ and, starting from a "reasonable" guess $t_0$, Newton method will update it according to $$t_{n+1}=t_n-\frac{f(t_n)}{f'(t_n)}$$ Starting with $t_0=1$, the successi... |
446,948 | <blockquote>
<p>Suppose that <span class="math-container">$X$</span> is a geometric random variable with parameter (probability of success) <span class="math-container">$p$</span>.</p>
<p>Show that <span class="math-container">$\Pr(X > a+b \mid X>a) = \Pr(X>b)$</span></p>
</blockquote>
<p>First I thought I'd s... | Michael Hardy | 11,667 | <p>Here's how to handle an infinite geometric series when the index starts at $n$ instead of $0$:
\begin{align}
\sum_{k=n}^\infty ax^k & = ax^n + ax^{n+1} + ax^{n+2} + ax^{n+3}+\cdots \\[10pt]
& = (ax^n) + (ax^n)x +(ax^n)x^2 + (ax^n)x^3+\cdots \\[10pt]
& = b + bx + bx^2 + bx^3 + \cdots
\end{align}</p>
<p>N... |
2,783,930 | <p>Here's what the authors of a textbook that I've been following argue:</p>
<blockquote>
<p>Let $x,y\in \mathbb{R}$. Assume that $x \le y$ and $y ≤ x$. We claim
that $x = y$. If false, then either, $x < y$ or $y < x$ by law of
<em>trichotomy</em>. Assume that we have $x < y$. Since $y ≤ x$, either $x =... | Bram28 | 256,001 | <blockquote>
<p>I need a clarification on proof done by the authors. They say "Assume that we have $x < y$" and on another line they say "hence concluded $x < y$ from the first inequality $x ≤ y$". Did they deduce it or assume it? </p>
</blockquote>
<p>They <em>deduce</em> it from $x \le y$ and the assumption ... |
2,783,930 | <p>Here's what the authors of a textbook that I've been following argue:</p>
<blockquote>
<p>Let $x,y\in \mathbb{R}$. Assume that $x \le y$ and $y ≤ x$. We claim
that $x = y$. If false, then either, $x < y$ or $y < x$ by law of
<em>trichotomy</em>. Assume that we have $x < y$. Since $y ≤ x$, either $x =... | farruhota | 425,072 | <blockquote>
<p>They say "Assume that we have $x<y$"</p>
</blockquote>
<p>You are taking it out of context. The full statement is:</p>
<blockquote>
<p>We claim that $x=y$. If false, then either, $x<y$ or $y<x$ by law of trichotomy. Assume that we have $x<y$.</p>
</blockquote>
<p>"If false, then..." i... |
2,783,930 | <p>Here's what the authors of a textbook that I've been following argue:</p>
<blockquote>
<p>Let $x,y\in \mathbb{R}$. Assume that $x \le y$ and $y ≤ x$. We claim
that $x = y$. If false, then either, $x < y$ or $y < x$ by law of
<em>trichotomy</em>. Assume that we have $x < y$. Since $y ≤ x$, either $x =... | fleablood | 280,126 | <p>Their argument:</p>
<p>Given $x \le y$ and $y \le x$.</p>
<p>Assume $x \ne y$. </p>
<p>$x\le y$ means $x=y$ or $x < y$. We ruled out $x=y$ so that leaves $x < y$. Stick a pin in that.</p>
<p>$y \le x$ means that $y=x$ or $y < x$. Remove the pin and compare with $x < y$. Neither $y =x$ nor $y <... |
48,237 | <p>I have found by a numerical experiment that first such primes are:
$2,5,13,17,29,37,41$. But I cannot work out the general formula for it.<br>
Please share any your ideas on the subject.</p>
| jspecter | 11,844 | <p>As the ideal $(p)$ is maximal in $\mathbb{Z}.$ The ring $\mathbb{Z}/pZ$ is a field. It follows that the nonzero elements form a group (the group of units $(\mathbb{Z}/p\mathbb{Z})^{\times})$ and this group is cyclic. Note $-1\in(\mathbb{Z}/p\mathbb{Z})^{\times}$ and for $p>2$ the order of $-1$ in this group is 2.... |
2,409,312 | <p>In <a href="https://math.stackexchange.com/a/1999967/272831">this previous answer</a>, MV showed that for $n\in\Bbb N$,</p>
<p>$$\int\frac1{1+x^n}~dx=C-\frac1n\sum_{k=1}^n\left(\frac12 x_{kr}\log(x^2-2x_{kr}x+1)-x_{ki}\arctan\left(\frac{x-x_{kr}}{x_{ki}}\right)\right)$$</p>
<p>where</p>
<p>$$x_{kr}=\cos \left(\fr... | Peter | 220,102 | <p>One can use <a href="https://math.stackexchange.com/questions/1354106/what-is-the-integration-of-int-1-x2n-1dx/1354485#1354485">this answer</a> for evaluating $\dfrac1{1+x^a}$ in the following form
$$\frac1{1+x^a}=\sum_{k=1}^aa_k(x-x_k)^{-1} \tag {2}$$
where $a_k=\frac{-x_k}{n}$ and $x_k=e^{i(2k-1)\pi/n}$, $k=1, \cd... |
3,275,966 | <p>During the drawing lottery falls six balls with numbers from <span class="math-container">$1$</span> to <span class="math-container">$36$</span>. The player buys the ticket and writes in it the numbers of six balls, which in his opinion will fall out during the drawing lottery. The player wants to buy several lotter... | RobPratt | 683,666 | <p>In fact, 9 tickets are enough. See <a href="http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=8782FF45BF67A451E558E80711D4CAFC?doi=10.1.1.70.8377&rep=rep1&type=pdf" rel="nofollow noreferrer">this paper</a>.</p>
<pre><code>
1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
1... |
1,042,285 | <p>How can i prove boundedness of the sequence
$$a_n=\frac{\sin (n)}{8+\sqrt{n}}$$ without using its convergence to $0$? I know since it is convergent then it is bounded.</p>
| DeepSea | 101,504 | <p>$|\sin (n)| \leq 1$, and $8+\sqrt{n} > 8 \Rightarrow |a_n| < \dfrac{1}{8}$</p>
|
2,045,812 | <p>(gcd; greatest common divisor) I am pulling a night shift because I have trouble understanding the following task.</p>
<p>Fibonacci is defined by this in our lectures:<br>
I) $F_0 := 1$ and $F_1 := 1$</p>
<p>II) For $n\in\mathbb{N}$, $n \gt 1$ do $F_n=F_{n-1}+F_{n-2}$</p>
<p><strong>Task</strong><br>
Be for n>0 t... | eepperly16 | 239,046 | <p>What you need here is the so-called <em>Extended Euclidean Algorithm</em> where you back substitute to calculate numbers $a$ and $b$ such that $aF_n + bF_{n+1} = \gcd(F_n,F_{n+1})$. For example, let's use $F_3 = 3$ and $F_4 = 5$, Then</p>
<p>$$ 5 = 1\cdot3 + 2 $$
$$ 3 = 1\cdot2 + 1 $$</p>
<p>Rearranging and substi... |
4,415,037 | <p>I would like to prove upper and lower bounds on <span class="math-container">$|\cos(x) - \cos(y)|$</span> in terms of <span class="math-container">$|x-y|$</span>. I was able to show that <span class="math-container">$|\cos(x) - \cos(y)| \leq |x - y|$</span>. I'm stuck on the lower bound. Does anyone know how to appr... | Community | -1 | <p>I don't have enough reputation to comment so I apologize that this had to be an answer. I know that's probably not what you are looking for maybe because it's so easy, but <span class="math-container">$-\left|x-y\right|$</span> works because:</p>
<p><span class="math-container">\begin{eqnarray}
-\left|x-y\right| \le... |
1,833,406 | <p>Now I have a topological space $X$ that is $C_2$ and $T_4$, and $U$ is an open set in it, I want to show that $U$ can be expressed as $\cup_{i\in\Bbb Z_+} F_i$ where $F_i$ are closed sets, <strong>without the aid of any metrisation theorem</strong> (but Uryshon's theorem about normal spaces and the equivalent Tietze... | Behrouz Maleki | 343,616 | <p>$$f'(x)=2x\tan^{-1}x+1$$
$$f'(x)=1+2x\sum\limits_{n=1}^{\infty }{\frac{{{(-1)}^{n+1}}}{(2n-1)}}\,{{x}^{2n-1}}=1+2\sum\limits_{n=1}^{\infty }{\frac{{{(-1)}^{n+1}}}{(2n-1)}}\,{{x}^{2n}}$$ therefore
$$f(x)=c+x+2\sum\limits_{n=1}^{\infty }{\frac{{{(-1)}^{n+1}}}{(2n+1)(2n-1)}}\,{{x}^{2n+1}}$$
we have $f(0)=0$, thus $c=0... |
651,174 | <p>I've got a complex equation with 4 roots that I am solving. In my calculations it seems like I am going through hell and back to find these roots (and I'm not even sure I am doing it right) but if I let a computer calculate it, it just seems like it finds the form and then multiplies by $i$ and negative $i$. Have a ... | Ben Grossmann | 81,360 | <p>Here's a way of doing things:
$$
\begin{align*}
\frac{8\sqrt 3}{z^4 + 8} &= i \implies\\
z^4 + 8 &= \frac{8\sqrt 3}{i} = -i\,8\sqrt 3\\
z^4 &= -8-i8\sqrt{3}
\\&=
16\left(\cos\left(\frac{4\pi}{3}\right) +i\,\sin \left(\frac{4\pi}{3} \right) \right)
\\&=
16\left(\cos\left(\frac{4\pi}{3} + 2 \pi... |
2,988,987 | <blockquote>
<p>For example, we have <span class="math-container">$f(x)=\frac{1}{x^2-1}$</span></p>
</blockquote>
<p>Would the domain be <span class="math-container">$$\mathcal D(f)=\{x\in\mathbb{R}\mid x\neq(1,-1)\}$$</span> or rather
<span class="math-container">$$\mathcal D(f)=\{x\in\mathbb{R}\mid x\neq \{1,-1\}\... | Doesbaddel | 587,094 | <blockquote>
<p>For example, we have <span class="math-container">$f(x)=\frac{1}{x^2-1}$</span></p>
</blockquote>
<p>The domain <span class="math-container">$\mathcal D$</span> would be <em>(suggested from Mauro ALLEGRANZA)</em>:</p>
<p><span class="math-container">$$\mathcal{D}(f)=\left\{x\in\mathbb{R}\mid x\neq 1... |
2,249,929 | <p>Is there a more general form for the answer to this <a href="https://math.stackexchange.com/q/1314460/438622">question</a> where a random number within any range can be generated from a source with any range, while preserving uniform distribution? </p>
<p><a href="https://stackoverflow.com/q/137783/866502">This</a>... | Ross Millikan | 1,827 | <p>The questions you linked to have strategies that are easily generalized. If $M \gt N$ you can just roll the die, accept any number $\le M$ and roll again if the number is $\gt M$. This is very simple, but may lead to a lot of rolling if $M$ is rather larger than $N$. If $M \gt 2N$ you can take the largest multipl... |
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | Dr. Sonnhard Graubner | 175,066 | <p>prove with induction that $$\sum_{n=0}^k \frac{n^2+3n+2}{4^n}=\frac{1}{27} 4^{-k} \left(-9 k^2-51 k+2^{2 k+7}-74\right)$$</p>
|
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | Hazem Orabi | 367,051 | <p>$$
\begin{aligned}
& S_{0} = \sum_{n=0}^{\infty} \frac{n^{0}}{4^{n}} = \sum_{n=0}^{\infty} \frac{1}{4^{n}} = \sum_{n=0}^{\infty} (1/4)^{n} = \frac{1}{1 - (1/4)} \Rightarrow \color{red}{S_{0} = \frac{4}{3}} \\ \\
& S_{1} = \sum_{n=0}^{\infty} \frac{n^{1}}{4^{n}} = \sum_{n=0}^{\infty} \frac{n + 1 - 1}{4^{n}} =... |
4,597,679 | <p>My textbook is <em>A Mathematical Introduction to Logic, 2nd Edition</em> by Enderton.</p>
<p>The question initially comes when I was trying to prove Exercise 4. on pg.99</p>
<p><span class="math-container">$$\text{Show that if }x \text{ does not occur free in }\alpha,\text{ then }\alpha \vDash \forall x \alpha$$</s... | georgy_d | 251,394 | <p>I don't really like to talk about bound variables, because it is rather about bound occurences of variables.</p>
<p>The main idea is that everything you derived are valid formulas.(Theorem of correctness)</p>
<p>Valid formulas are precisely those which are true under any interpretation of variables, functional and p... |
3,440,732 | <p>How can I show that <span class="math-container">$P=\{\{2k-1,2k\},k\in \mathbb {N}\}$</span> is the basis of <span class="math-container">$\mathbb N$</span>.
Obviously we have to show that basis's orders do work.
But the problem is if I take two subsets that belong to <span class="math-container">$P$</span> , for ex... | ajotatxe | 132,456 | <p>Well, if <span class="math-container">$n$</span> is even, then <span class="math-container">$n\in \{n-1,n\}\in P$</span>. If it is odd then <span class="math-container">$n\in\{n,n+1\}\in P$</span>.</p>
|
263,961 | <p>Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from algebraic structures (theories of abstract groups, rings, fields) and real and complex analysis (theories of expansion... | Seirios | 36,434 | <p>Ax found the following application in complex analysis:</p>
<blockquote>
<p><strong>Theorem:</strong> If $f : \mathbb{C}^n \to \mathbb{C}^n$ is an injective polynomial function, that is there exist $f_1,...,f_n \in \mathbb{C}[X_1,...,X_n]$ such that $f=(f_1,...,f_n)$, then $f$ is surjective.</p>
</blockquote>
<... |
923,871 | <p>Let $Y=\{(a,b)∈ \Bbb R\times \Bbb R∣ a≠0\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)*(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$.</p>
<p>I already do the proof of ∗ is an operation on Y. And proof is associative like this:
$$\{(A,B)*(C,D)\}*(E,F)=(A,B)*\{(C,D)\}*(E,F)\}$$</p>
<p>$$(AC,AD+B)(... | Cameron Buie | 28,900 | <p>There are four things you need to accomplish, here:</p>
<ul>
<li>You need to show that $*$ is an operation on $Y.$ In particular, you must show that if $(a,b),(c,d)\in Y,$ then $(a,b)*(c,d)\in Y.$ This should be straightforward from the definition of $Y.$</li>
<li>You need to show that $*$ is an <em>associative</em... |
4,007,450 | <p>Let <span class="math-container">$y=f(x)$</span> the graph of a real-valued function. We define its curvature by : <span class="math-container">$$curv(f) = \frac{|f''|}{(1+(f')^2)^{3/2}}$$</span></p>
<p>I would like to know if there is any function (apart from the trivial anwser <span class="math-container">$f(x)=0$... | Shaun | 104,041 | <p>You're correct.</p>
<p>If <span class="math-container">$a^2=e$</span> for all <span class="math-container">$a\in G$</span>, then for any <span class="math-container">$g,h\in G$</span>,</p>
<p><span class="math-container">$$\begin{align}
(gh)^2&=ghgh\\
&=e\\
&=ee\\
&=g^2h^2\\
&=gghh,
\end{align}$$... |
524,568 | <p>I'm working on a recursive function task which i'm a bit stuck at. I've tried to google it on how I can solve this task, but with no luck</p>
<p>Here is the task:</p>
<blockquote>
<p><em>Provide a recursive function $r$ on $A$</em>* <em>which gives the number of
characters in the string</em></p>
</blockquote>
... | mihirj | 100,576 | <p>Kindly see if this helps you.</p>
<p>r = f(A)</p>
<p>f(A) = 0 if f(A) = ∅</p>
<pre><code> = 1 + f(A-{c}) otherwise
</code></pre>
<p>where {c} is the character extracted from the string.</p>
|
3,409,342 | <p>Let <span class="math-container">$X$</span> be a set containing <span class="math-container">$A$</span>.</p>
<p>Proof:
<span class="math-container">$y\in A \cup (X \setminus A) \Rightarrow y\in A$</span> or <span class="math-container">$y \in (X \setminus A)$</span></p>
<p>If <span class="math-container">$y \in ... | fleablood | 280,126 | <p>The inverse is really trivial.</p>
<p>Proposition: For any sets <span class="math-container">$A, B$</span> than <span class="math-container">$A \subset A\cup B$</span>.</p>
<p>Pf: If <span class="math-container">$x \in A$</span> then (<span class="math-container">$x\in A$</span> or <span class="math-container">$x... |
877,477 | <p>I'm stuck with the following question, which looks quite innocent.</p>
<p>I'd like to show that if a covering space map $f:\tilde{X}\to X$ between cell complexes is null-homotopic, then the covering space $\tilde{X}$ must be contractible.</p>
<p>Since $f$ is null-homotopic there exists a homotopy $H_t:\tilde{X}\to... | Quang Hoang | 91,708 | <p>Since $f$ is nullhomotopic, $f_*:\pi_n(\tilde X)\to \pi_n(X)$ are trivial for all $n$. Consequently $\pi_n(\tilde X)$ are all trivial. Whitehead theorem implies $\tilde X$ is contractible.</p>
|
1,361,948 | <p>$\frac{df}{dx} = 2xe^{y^2-x^2}(1-x^2-y^2) = 0.$</p>
<p>$\frac{df}{dy} = 2ye^{y^2-x^2}(1+x^2+y^2) = 0.$</p>
<p>So, $2xe^{y^2-x^2}(1-x^2-y^2) = 2ye^{y^2-x^2}(1+x^2+y^2)$.</p>
<p>$x(1-x^2-y^2) = y(1+x^2+y^2)$</p>
<p>$x-x^3-xy^2 = y + x^2y + y^3$</p>
<p>Is the guessing the values of the variables the only way of so... | Budenn | 250,821 | <p>$$\mathrm{Re} \left[\frac{1 + i}{\sigma \delta \left( 1 - e^{-(1 + i)t/\delta} \right) }\right]
= \mathrm{Re}\left[\frac{(1 + i)\left( 1 - e^{-(1 - i)t/\delta} \right)}{\sigma \delta \left( 1 - e^{-(1 + i)t/\delta} \right) \left( 1 - e^{-(1 - i)t/\delta} \right)} \right]
= \frac{1 - \mathrm{Re}\left[e^{-(1 - i)t/\d... |
2,841,640 | <p>What is a vector space? I can see two different formulations, and between them there is one difference: commutativity. </p>
<blockquote>
<p><strong>DEFINITION 1</strong> (See <a href="https://proofwiki.org/wiki/Definition:Vector_Space" rel="noreferrer">here</a>)</p>
<p>Let $(F, +_F, \times_F)$ be a division ... | AnalysisStudent0414 | 97,327 | <p>Usually, over a field.</p>
<p>On <a href="https://en.wikipedia.org/wiki/Vector_space" rel="nofollow noreferrer">Wikipedia</a> (I know, I know) I read that "Some authors use the term vector space to mean modules over a division ring" (<a href="https://en.wikipedia.org/wiki/Vector_space#cite_note-120" rel="nofollow n... |
2,400,110 | <p>Let's say that we have a matrix of transfer functions:
$$G(s) = C(sI-A)^{-1}B + D$$</p>
<p>And we create the sensitivity matrix transfer function:</p>
<p>$$S(s) = (I+GK)^{-1}$$</p>
<p>Where $K$ is our controller gain matrix.</p>
<p>We also create the complementary sensitivity transfer function matrix:</p>
<p>$$... | hzh | 528,159 | <p>I think you started with some wrong definitions.</p>
<p>The right definition for stabilizability and detectability should be:</p>
<blockquote>
<p>A system is stabilizable if the uncontrollable states are stable.</p>
<p>A system is detectable if the unobservable states are asymptotically stable.</p>
</blockquote>
<p>... |
3,789,873 | <p>Let <span class="math-container">$X, Y, X_n's$</span> be random variables for which <span class="math-container">$X_n+\tau Y\to_D X+\tau Y$</span> for every fixed positive constant <span class="math-container">$\tau$</span>. Show that <span class="math-container">$X_n \to_D X$</span>.</p>
<p>I dont think we can let ... | Oliver Díaz | 121,671 | <p>This is just to complement the answer by Shalop and to address a comment posted by the OP.</p>
<p>The first and third are bounded by a small term:
<span class="math-container">$$
\begin{align}
\Big|\Bbb E[e^{it X_{n}}]-\Bbb E[e^{it X_{n} + i t \epsilon Y}]\Big|&\leq \Bbb E\big[\big|e^{itX_n}\big(1-e^{it\varepsi... |
4,095,715 | <p>I know how to do these in a very tedious way using a binomial distribution, but is there a clever way to solve this without doing 31 binomial coefficients (with some equivalents)?</p>
| trancelocation | 467,003 | <p>Here is just a standard trick from generating functions using</p>
<p><span class="math-container">$$\frac 1{(1-y)^n} = \sum_{k=0}^{\infty}\binom{k+n-1}{n-1}y^k$$</span>.</p>
<p>To simplify the expressions set</p>
<p><span class="math-container">$$y=x^2\Rightarrow \text{ we look for }[y^6]\frac{(1-y^4)^n}{(1-y)^n}$$<... |
2,811,991 | <p>$$\int_{-3}^5 f(x)\,dx$$<br>
for
$$ f(x) =\begin{cases}
1/x^2, & \text{if }x \neq 0 \\
-10^{17}, & \text{if }x=0
\end{cases}
$$</p>
<p>I tried with Newton Leibniz formula, is this correct ?</p>
<p>$\int_{-3}^0 f(x)dx$ + $\int_{0}^5 f(x)dx$ =</p>
<p>$1/x^2 |_{-3}^{0} $ $ + $ $1/x^2 |_0^5$=</p>
<p>$3/(-... | Greg | 495,411 | <p>$\frac1{x^2}$ is not the integral of $\frac1{x^2}$.</p>
|
851,459 | <p>hello $$$$ I am trying to find explanation how to derive cahn hilliard equation:</p>
<p>$$ u_t =\Delta (w'(u)-\epsilon ^2 \Delta u)$$ </p>
<p>as gradient flow of energy functional $$ : E[u]=\int w(u)+\epsilon ^2 |\nabla u|^2 ) $$
I tried to follow the definition of gradient flow from :
<a href="http://anhngq... | Student | 124,626 | <p>to get first a Cahn-Hilliard system first we write the mass conservation i.e. $\dfrac{\partial u}{\partial t}=-h_x$, where $h$ is the mass flux which is related to the chemical potential $\mu$ by a constitutive relation $h=-\mu_x$, and that the chemical potential $\mu$ is a variational derivative of $\Psi$ with resp... |
730,253 | <p>let $x,y$ such
$$2^x+5^y=2^y+5^x=\dfrac{7}{10}$$</p>
<p>prove or disprove $x=y=-1$ is the only solution for the system.</p>
<p>My try: since
$$2^x-2^y=5^x-5^y$$</p>
<p>But How can prove or disprove $x=y$?</p>
| Greg Martin | 16,078 | <p>If $(x,y)$ is a solution, then we have both $2^y=\frac7{10}-5^x$ and
$$
2^y = (5^y)^{\log2/\log5} = \big(\tfrac7{10}-2^x\big)^{\log2/\log5}.
$$
So define $f(x)=\frac7{10}-5^x$ and $g(x)=\big(\tfrac7{10}-2^x\big)^{\log2/\log5}$. We want to show that the only place $f$ and $g$ are equal is at $x=-1$; it suffices to sh... |
1,923,298 | <p>I am really struggling with this question and it isn't quite making sense. Please help and if you don't mind answering quickly.</p>
<p>Reflection across $x = −1$</p>
<p>$H(−3, −1), F(2, 1), E(−1, −3)$.</p>
| Alberto Takase | 146,817 | <p>You could argue by induction on the size of the finite set $B$. If $|B|=0$, then $B=\varnothing$. Therefore $A\cup B$ is finite (completing the base case). Now assume that $A\cup B$ is finite for $|B|\le k$. Then you have to show that $A\cup B$ is finite for $|B|=k+1$. [Hint: $B=\{x\}\cup(B\setminus\{x\})$ for some ... |
3,561,860 | <p>Given <span class="math-container">$x_i, x_j \in [0,1]$</span>, and a payoff function <span class="math-container">$u_i(x_i, x_j) = (\theta_i + 3x_j - 4x_i)x_i$</span> if <span class="math-container">$x_j < 2/3$</span>, and <span class="math-container">$= (3x_j-2)x_i$</span> if <span class="math-container">$x_j \... | Claude Leibovici | 82,404 | <p>It could be easier to use the fundamental theorem of calculus
<span class="math-container">$$F(x) = \int_{0}^{x} (1+t^2)\cos(t^2)\,dt \implies F'(x)=(1+x^2)\cos(x^2)$$</span> Now, let <span class="math-container">$y=x^2$</span> and you should arrive at
<span class="math-container">$$F'(x)=1+\sum_{n=1}^\infty\frac{n ... |
3,565,727 | <blockquote>
<p>Show that the antiderivatives of <span class="math-container">$x \mapsto e^{-x^2}$</span> are uniformly continuous in <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>So we know that for a function to be uniformly continuous there has to exists <span class="math-container">$\var... | Peter Szilas | 408,605 | <p>MVT for integrals:</p>
<p><span class="math-container">$f(x)=\displaystyle{\int_{a}^{x}}e^{-t^2}dt$</span></p>
<p><span class="math-container">$|f(x)-f(y)|=|\displaystyle{\int_{y}^{x}}e^{-t^2}dt|=$</span></p>
<p><span class="math-container">$e^{-s^2}|x-y| \le |x-y|,$</span></p>
<p>where <span class="math-contain... |
975,210 | <p>\begin{align}
\left| f(b)-f(a)\right|&=\left| \int_a^b \frac{df}{dx} dx\right|\\ \ \\
&\leq\left| \int_a^b \left|\frac{df}{dx}\right|\ dx\right|.
\end{align}</p>
<p>I do not understand why the second line is greater or equal than the top equation. Can anyone explain please?</p>
| DeepSea | 101,504 | <p>$\left|\displaystyle \int_a^b \dfrac{df}{dx} dx\right| \leq \displaystyle \int_a^b \left|\dfrac{df}{dx}\right|dx \leq \left|\displaystyle \int_a^b \left|\dfrac{df}{dx}\right|dx\right|$</p>
|
3,770,391 | <p>According to <span class="math-container">${\tt Mathematica}$</span>, the following integral converges if
<span class="math-container">$\beta < 1$</span>.</p>
<p><span class="math-container">$$
\int_{0}^{1 - \beta}\mathrm{d}x_{1}
\int_{1 -x_{\large 1}}^{1 - \beta}\mathrm{d}x_{2}\, \frac{x_{1}^{2} + x_{2}^{2}}{\l... | RRL | 148,510 | <p>We can prove this integral converges for <span class="math-container">$0 < \beta < 1$</span> without evaluation.</p>
<p>Write this as</p>
<p><span class="math-container">$$\int_0^{1-\beta}\int_{1-x_1}^{1-\beta} \frac{x_1^2 + x_2^2}{(1-x_1)(1-x_2)}\, dx_2 \, dx_1\\ = \underbrace{\int_0^{\beta}\int_{1-x_1}^{1-\b... |
3,837,745 | <p>The two equations are</p>
<p><span class="math-container">$$\begin{aligned}3x^2-12y=& \ 0\\
3y^2-12x=& \ 0
\end{aligned}$$</span></p>
<p>Using system of equations find all of the solutions?</p>
<p>I found the first one to be <span class="math-container">$(0,0).$</span> The answer key says <span class="math-c... | Gteal | 725,635 | <p>We have <span class="math-container">$3x^2-12y=0$</span> and <span class="math-container">$3y^2-12x=0$</span>.</p>
<p>Solving for <span class="math-container">$x$</span> in the second equation gives us <span class="math-container">$x=\frac{1}{4}y^2$</span>.</p>
<p>Substituting this into the first equation we have</p... |
3,273,830 | <p>Okay so I've started to study derivatives and there is this idea of continuity. The book says <em>"a real valued function is considered continuous at a point iff the graph of a function has no break at the point of consideration, which is so iff the values of the function at the neighbouring points are close enough ... | mlchristians | 681,917 | <p>I hope your textbook also provides a more formal definition of continuity at a point:</p>
<p><span class="math-container">$f(x)$</span> is continuous at a point <span class="math-container">$a$</span> if and only if all three of the following hold---</p>
<p>(1) <span class="math-container">$f(a)$</span> is defined... |
3,273,830 | <p>Okay so I've started to study derivatives and there is this idea of continuity. The book says <em>"a real valued function is considered continuous at a point iff the graph of a function has no break at the point of consideration, which is so iff the values of the function at the neighbouring points are close enough ... | Adam Latosiński | 653,715 | <p>If the values of the function in points <span class="math-container">$x$</span> close enough to a specific point <span class="math-container">$x_0$</span> are not arbitrarily close to <span class="math-container">$f(x_0)$</span> the graph will be (usually) broken.</p>
<p>This definition is not completely rigorous, ... |
523,376 | <p>Suppose $a_i$ is a sequence of positive integers. Define $a_1 = 1$, $a_2 = 2$ and $a_{n+1} = 2a_n + a_{n-1}$. Does it follow that </p>
<p>$$ \gcd(a_{2n+1} , 4 ) = 1 $$ ???</p>
<p>Im trying to see this by induction assuming above holds, we need to see that $\gcd(a_{2n+3} , 4 ) = 1$.</p>
<p>But, $\gcd(a_{2n+3} , 4 ... | ulilaka | 42,323 | <p>In order to get a parametrization defined on an open subset of $\mathbf{R}^{2}$ you need to restrict your angular parameter $u$ to the open interval $(0, 2\pi)$. But the image of $x(u,v)$ in this case is not precisely the Möbius band, but the band minus one vertical line where it should "close". If you take $u$ belo... |
310,669 | <p>This is related to a course I'm taking in computer science theory. </p>
<p>Let $\sum$ be an alphabet. Then the set of all strings over $\sum$, denoted as $\sum^*$ has the operation of concatenation (adjoining two strings end to end). Clearly, concatenation is associative, $\sum^*$ is closed under concatenation, a... | Ittay Weiss | 30,953 | <p>No, $\Sigma^*$ is not a group (unless $\Sigma = \emptyset $, in which case $\Sigma ^*$ is a trivial group with one element). The reason is that the only element having an inverse is the empty word. So if $a\in \Sigma$, then $a$ as an element of $\Sigma ^*$ does not have an inverse (rigorously, note that the length o... |
522,714 |
<p>$$
dz_t \sim O\left(\sqrt{dt}\,\right)
$$</p>
<p>$z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation on wikipedia because I thought it might be related, but I can't see the link.</p>
| Mark Bennet | 2,906 | <p>Note that $\cfrac 1x=\cfrac 1{\sqrt 3+\sqrt 2}=\cfrac {\sqrt 3-\sqrt 2}{\sqrt 3-\sqrt 2}\cdot\cfrac 1{\sqrt 3+\sqrt 2}=\sqrt 3-\sqrt 2$</p>
<p>So you need to find $(\sqrt 3+\sqrt 2)^4-(\sqrt 3-\sqrt 2)^4$</p>
<p>Now note that the terms which have even powers of $\sqrt 2$ will cancel, and the odd powers will double... |
3,540,593 | <p>Are <span class="math-container">$n$</span> vectors are orthogonal if performing the inner product of all <span class="math-container">$n$</span> vectors at once yields zero?</p>
<p>In other words, could I say that <span class="math-container">$\hat{i} \perp \hat{j} \perp \hat{k}$</span>?</p>
<p>For example, suppo... | Eduline | 743,749 | <p>Given that <span class="math-container">$p$</span> is an odd prime. We need to find the number of positive integers k with <span class="math-container">$1<k<p$</span> such that <span class="math-container">$k^2 \equiv 1 \pmod {p}\implies p|k^2-1\implies p|(k-1)(k+1)\implies p|k-1$</span> or <span class="math-c... |
162,863 | <p>if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation?</p>
<p>Thanks in advance!</p>
| rschwieb | 29,335 | <p>I can't be sure what formula for a general rotation you have, but it <em>should</em> depend upon an angle through which you are rotating. Doesn't your formula look something like $R(\Theta, u)$ where $\Theta$ is the angle of rotation, and $u$ is a unit vector which tells you the axis of rotation?</p>
<p>If so, you ... |
162,863 | <p>if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation?</p>
<p>Thanks in advance!</p>
| Kallus | 32,086 | <p>I believe what you're looking for are exponent and logarithm formulas for quaternions, which can be found on the Wikipedia page on <a href="https://en.wikipedia.org/wiki/Quaternion#Exponential.2C_logarithm.2C_and_power" rel="nofollow">quaternions</a>. The Wikipedia page even gives a formula for raising a quaternion ... |
4,496,772 | <p>Let <span class="math-container">$V$</span> be a vector space of dimension <span class="math-container">$n$</span> over a finite field <span class="math-container">$F$</span> with <span class="math-container">$q$</span> elements, and let <span class="math-container">$U\subseteq V$</span> be a subspace of dimension <... | cigar | 1,070,376 | <p>Given a basis <span class="math-container">$\{v_1,v_2,\dots, v_k\}$</span> for <span class="math-container">$U$</span>, how many ways can you extend the basis to a larger linearly independent set?</p>
<p>So, the next vector can't be a linear combination of the <span class="math-container">$v_1,\dots, v_k$</span>. H... |
1,422,859 | <p>$$\sqrt{1000}-30.0047 \approx \varphi $$
$$[(\sqrt{1000}-30.0047)^2-(\sqrt{1000}-30.0047)]^{5050.3535}\approx \varphi $$
Simplifying Above expression we get<br>
$$1.0000952872327798^{5050.3535}=1.1618033..... $$
Is this really true that
$$[\varphi^2-\varphi]^{5050.3535}=\varphi $$</p>
| Claude Leibovici | 82,404 | <p><em>This is not an answer but it is too long for a comment.</em></p>
<p>Working with illimited precision, let $$a=\sqrt{1000}-\frac{300047}{10000}\approx 1.6180766016837933200$$ $$(a^2-a)^{\frac {50503535}{10000}}\approx 1.6180331121536741389$$ $$\phi\approx 1.6180339887498948482$$ Using as exponent $5050.3592$ (sa... |
769,272 | <p>So, I'm trying to solve the wave equation with the Fourier transform, and I'm struggling to figure out how to apply the BC's. Here's the problem I considered:</p>
<p>$$\frac{d^2u}{dt^2}=c^2\frac{d^2u}{x^2}$$
$$u(x,0)=g(x)$$
$$\frac{du}{dt}=0$$ at t = 0</p>
<p>Running through computations, I find that the Fourier ... | doppz | 48,746 | <p>There's a remarkable theorem by Gauss (that's actually what it's called: the <a href="http://en.wikipedia.org/wiki/Theorema_Egregium" rel="nofollow">Theorema Egregerium</a>) that says Guassian curvature $K$ can be found using only the first fundamental form. There are many different ways to then compute it, one of w... |
2,399,216 | <p>How to find the coefficient of $x^4$ in the expansion of $(1+x+x^2)^{20}$?</p>
| Community | -1 | <p><strong>HINT</strong></p>
<p>Let $f(x)=(1+x+x^2)^{20}$. Use the forth derivative for $x=0$ , that is $f^{''''}(0)$</p>
<p>If $c$ is the wanted coefficient, then $f^{''''}(x)=4! \cdot c + x \cdot g(x)$. Now making $x=0$ one gets $c = \frac {f^{''''}(0)}{4!}$ </p>
|
2,399,216 | <p>How to find the coefficient of $x^4$ in the expansion of $(1+x+x^2)^{20}$?</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\left[x^n\right]\left(1+x+x^2\right)^{20}
&=\left[x^n\right]\left(\frac{1-x^3}{1-x}\right)^{20}\\
&=\left[x^n\right]\overbrace{\sum_{j=0}^{20}\binom{20}{j}\left(-x^3\right)^j}^{\left(1-x^3\right)^{20}}\overbrace{\sum_{k=0}^\infty\binom{-20}{k}(-x)^k\vphantom{\sum_{j=0}^{20}}}^{(1-x)^{-20}}\\... |
3,791,350 | <p>An example question is:</p>
<p>In radian measure, what is <span class="math-container">$\arcsin \left(\frac{1}{2}\right)$</span>?</p>
<p>Select one:</p>
<p>a. <span class="math-container">$0$</span></p>
<p>b. <span class="math-container">$\frac{\pi}{6}$</span></p>
<p>c. <span class="math-container">$\frac{\pi}{4}$</... | B. Goddard | 362,009 | <p>There's a sort of silly way to keep the sines of common angles in your head. The common angles are:</p>
<p><span class="math-container">$$0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}.$$</span></p>
<p>The sine of each of these, in order is:</p>
<p><span class="math-container">$$\frac{\sqrt{0}}{2}, \... |
618,823 | <p>Reading a book I encountered the following claim, which I don't understand. Let $X$ be a smooth projective curve over $\mathbb{C}$, and $q\in X$ a rational point.
Denote by $\pi_i: X^n\to X$ the $i$-th projection of the cartesian $n$-product of the curve onto $X$ itself. The claim is that</p>
<blockquote>
<p>The ... | Andrew D. Hwang | 86,418 | <p>Geometrically the tensor product $L$ is ample because a sufficiently high power $L^N$ is the tensor product of pullbacks of very ample bundles on $X$.</p>
<p>In more detail, say the $N$-fold tensor power $\mathcal{O}_X(q)^N$ is very ample, and that the sections $s_1, \dots, s_m$ projectively embed $X$. For each $i ... |
3,414,009 | <p>Given functions
<span class="math-container">$$
f(x)=\sum_{i=0}^\infty a_ix^i\,\,\,\text{ and }\,\,\,g(x)=\sum_{j=0}^\infty b_jx^j
$$</span>
the following simplification
<span class="math-container">$$
f(x)g(x)=\left(\sum_{i=0}^\infty a_ix^i\right)\left(\sum_{j=0}^\infty b_jx^j \right)=\sum_{i=0}^\infty\sum_{j=0}^\i... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$f(x)=\sum_{n=0}^\infty a_nx^n$</span>, then <span class="math-container">$a_n=\frac{f^{(n)}(0)}{n!}$</span> and the same thing applies to <span class="math-container">$g$</span>. But then, if <span class="math-container">$f(x)g(x)=\sum_{n=0}^\infty c_nx^n$</span>:</p>
<p><span class... |
1,577,838 | <p>Let $p$ and $q$ be primes such that $p=4q+1$. Then $2$ is a primitive root modulo $p$.</p>
<p>Proof. </p>
<p>Note that $q\not=2$ since $4\cdot2+1=9$ is not prime. $\mathrm{ord}_p(2)\vert p-1=4q$, so $\mathrm{ord}_p(2)=1,\;2,\;4,\;q,\;2q,\;\mathrm{or}\;4q$.</p>
<p>Clearly $\mathrm{ord}_p(2) \not= 1$, and $\mathrm{... | Jyrki Lahtonen | 11,619 | <p>Looks good to me. As you observed the key for ruling out the possible orders $q$ and $2q$ is that in either case $2$ would end up being a quadratic residue modulo $p$ - in violation of (an extension of) the law of quadratic reciprocity.</p>
<p>You can actually combine those two cases. Observe that irrespective of w... |
2,541,131 | <p>$$ S = \sum_{n=1}^{99} \frac{(5)^{100}}{(25)^{n} + (5)^{100}}$$</p>
<p>I tried writing first and end terms to make a similar face in the denominator, but in vain. The denominators are getting same in alternate terms.
I tried adding and subtracting by 1 to look after a v(n) and v(n-1) pair of terms also, but that j... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$f(n)=\dfrac{a^m}{a^n+a^m}$$</p>
<p>$$f(2m-n)=\dfrac{a^m}{a^{2m-n}+a^m}=\dfrac1{a^{m-n}+1}=\dfrac{a^n}{a^m+a^n}=1-f(n)$$</p>
<p>Here $2m=100,a=25$</p>
<p>Set $m=1,50$ and add</p>
|
2,541,131 | <p>$$ S = \sum_{n=1}^{99} \frac{(5)^{100}}{(25)^{n} + (5)^{100}}$$</p>
<p>I tried writing first and end terms to make a similar face in the denominator, but in vain. The denominators are getting same in alternate terms.
I tried adding and subtracting by 1 to look after a v(n) and v(n-1) pair of terms also, but that j... | Gribouillis | 398,505 | <p>Hint:</p>
<p>$$\frac{1}{q + 1} + \frac{1}{\frac{1}{q}+1} =\ ?$$</p>
|
992,487 | <p>Consider the following problem.</p>
<p>A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the support of a proportion of the panel of at least $A$, taking into account voting weights. Each country $C_i$ has a p... | Marco | 234,614 | <p>Forgive me if this solution may appear to be too "simplistic", maybe I misunderstood something in your original question, but what I gathered is that you have to set the various votes of countries in a manner that you maximize the probability of a resolution passing.
Strangely enough for all the assumptions you ment... |
2,662,717 | <p>Let $(f_k)_{k=m}^\infty$ be a sequence of differentiable functions $f_k:[a,b]\rightarrow R$ whose derivatives are continuous. Suppose there exists a sequence $(M_k)_{k=m}^\infty$ in $R$ with $|f_k'|\le M_k$ for all $x\in X, k\geq m,$ and such that $\sum_{k=m}^\infty M_k$ converges. Assume also that there is some $x_... | JonathanZ supports MonicaC | 275,313 | <p>Yes, you can do such a "double induction". There are multiple different ways you can go about it, and yours looks to be valid.</p>
<p>However, often you don't need to do two inductions but it's merely enough to say "Fix an arbitrary $m$" and then do an induction on $n$. Essentially you are doing an induction (on $... |
2,662,717 | <p>Let $(f_k)_{k=m}^\infty$ be a sequence of differentiable functions $f_k:[a,b]\rightarrow R$ whose derivatives are continuous. Suppose there exists a sequence $(M_k)_{k=m}^\infty$ in $R$ with $|f_k'|\le M_k$ for all $x\in X, k\geq m,$ and such that $\sum_{k=m}^\infty M_k$ converges. Assume also that there is some $x_... | Bram28 | 256,001 | <p>Your double induction is not sufficient the way you have set it up. You end up proving only the cases with $n=0$ and arbitrary $m$, and with $m=1$ and arbitrary $n$. </p>
<p>Also, while I'm not sure you really are trying to do this, but it almost sounds like you want to go from the $(m,n)$ case directly to the $(m... |
3,896,327 | <p>For the first part of this question, I was asked to find the either/or version and the contrapositive of this statement, which I found as follows:</p>
<p>i) either <span class="math-container">$n \leq 7$</span>, or <span class="math-container">$n^2-8n+12$</span> is composite</p>
<p>ii) if <span class="math-container... | Parcly Taxel | 357,390 | <p><span class="math-container">$$n^2-8n+12=(n-2)(n-6)$$</span>
Now if <span class="math-container">$n>7$</span> then both factors are at least <span class="math-container">$2$</span>, which means <span class="math-container">$n^2-8n-12$</span> is composite. This settles the matter.</p>
|
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | Gerry Myerson | 8,269 | <p>There are many general approaches. One of the simplest and best is Newton's Method, which you will find in any calculus text, or by searching the web. </p>
<p>By the way, I think your first formula should be $f(x)=(1/2)(x+(3/x))$, no?</p>
|
69,472 | <blockquote>
<p><strong>Theorem 1</strong><br>
If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$<br>
If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with
$$|g'(x)| \leq k, \text{ for all } \in (a, b)$$
then the fixed... | Fixee | 7,162 | <p>Somewhat off-topic, but...</p>
<p>When I was a kid, we didn't have calculators yet, so we learned to compute sqrts on paper using the <a href="http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Digit-by-digit_calculation" rel="nofollow">digit-at-a-time</a> method. This cannot compete with Newton-Raphson... |
132,618 | <p>Ogg characterized the finitely many N such that $X_0(N)_{\mathbb{Q}}$ is hyperelliptic, and Poonen proved in "Gonality of modular curves in characteristic p" that for large enough N, $X_0(N)_{\mathbb{F}_p}$ is not hyperelliptic.</p>
<p><strong>Question</strong>: Are there any N such that $X_0(N)_{\mathbb{Q}}$ is no... | JSE | 431 | <p>This is sort of a cheap answer, but I would look at values of N where X_0(N) has genus 3 and is not hyperelliptic (over Q). In genus 3, the locus of hyperelliptic curves has codimension 1 in the whole moduli space of curves, so one would expect a non-hyperelliptic curve to reduce to a hyperelliptic curve mod p for ... |
132,618 | <p>Ogg characterized the finitely many N such that $X_0(N)_{\mathbb{Q}}$ is hyperelliptic, and Poonen proved in "Gonality of modular curves in characteristic p" that for large enough N, $X_0(N)_{\mathbb{F}_p}$ is not hyperelliptic.</p>
<p><strong>Question</strong>: Are there any N such that $X_0(N)_{\mathbb{Q}}$ is no... | Maarten Derickx | 23,501 | <p>A curve $C$ is hyperelliptic if and only if the canonical map $C \to \mathbb P^*(\Omega^1(C))$, which sends a point $p$ to the codimension 1 subspace $V_p \subset\Omega^1(C)$ of all one forms vanishing at $p$, is a two to one map. In the hyperelliptic case its image will be a $\mathbb P^1$, and the degree two map wi... |
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