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 |
|---|---|---|---|---|
1,998,938 | <p>How can I solve \begin{cases}
u_t-u_{xx}=0,&\text{if $0<x<1, t>0$}\\
u(0,t)=u(1,t)=0, & \text{if $t>0$}\\u(x,0)=u_0(x), &\text{if $x\in(0,1)$} \end{cases}</p>
<p>where $$u_0=min(x,1-x)$$</p>
| robjohn | 13,854 | <p>We can derive a recursion for $f_n$ which is valid for $n\ge2$ and $1\lt s\lt n$:
$$
\begin{align}
f_n(s)
&=\int_0^\infty\left\{\frac1t\right\}^nt^{s-1}\,\mathrm{d}t\\
&=\int_0^\infty\{t\}^nt^{-s-1}\,\mathrm{d}t\\
&=-\frac1s\int_0^\infty\{t\}^n\,\mathrm{d}t^{-s}\\
&=\frac ns\int_0^\infty\{t\}^{n-1}t^... |
3,913,732 | <p>The following was asked by a high school student which I could not answer. Please help</p>
<p>In the figure below, show that the bisector of <span class="math-container">$\angle AEB$</span> and <span class="math-container">$\angle AFD$</span> intersect at perpendicular <img src="https://i.stack.imgur.com/76tDO.jpg" ... | Aman Kumar | 663,536 | <p>One way to solve it is as follows:</p>
<p><span class="math-container">$$\cos{(1.3\pi\cos{\theta})} = \cos{1.3\pi}$$</span></p>
<p>Using the identity I gave above,</p>
<p><span class="math-container">$$1.3\pi\cos{\theta} = 2n\pi\pm\ 1.3\pi.....n\in Z$$</span></p>
<p>Dividing both sides by <span class="math-container... |
132,862 | <p>Is it true that given a matrix $A_{m\times n}$, $A$ is regular / invertible if and only if $m=n$ and $A$ is a basis in $\mathbb{R}^n$?</p>
<p>Seems so to me, but I haven't seen anything in my book yet that says it directly.</p>
| Henry | 6,460 | <p>Informally, $\{0,1,2,3,\ldots\}\subseteq\mathbb{N}$ comes from the first and second axioms.</p>
<p>Of course you would need to define what $\{0,1,2,3,\ldots\}$ is. Perhaps writing it $\{0,S(0),S(S(0)),S(S(S(0))),\ldots\}$ makes it clearer. Then you can take a particular member of this set and use the first and se... |
33,817 | <p>It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.</p>
<ul>
<li>What are some of the important results leading toward proving this?</li>
<li>What are the most promising theories and approaches for this problem?</li>
</ul>
| muad | 4,361 | <p>There is a proof of the algebraic independence of $\pi$ and $e^\pi$ in <a href="http://www.springer.com/mathematics/numbers/book/978-3-540-41496-4">Introduction to Algebraic Independence Theory</a> and <em>a detailed exposition of methods created in last the 25 years</em> although I have not read it.</p>
|
1,665,833 | <p>Given that A $\in$ M $_{mxn}$ (<strong>R</strong>). Assume that {$v_1$...$v_n$} is a basis for $R^n$ such that {$v_1$...$v_k$} is a basis for Null(A). </p>
<p>How would I prove that {A$v_{k+1}$...A$v_n$} spans Col(A)?</p>
| Michael James Kali Galarnyk | 522,520 | <p>Here is some python code to make your own z-table (in case you find it useful)</p>
<pre><code>from scipy.integrate import quad
import numpy as np
import pandas as pd
def normalProbabilityDensity(x):
constant = 1.0 / np.sqrt(2*np.pi)
return(constant * np.exp((-x**2) / 2.0) )
standard_normal_table = pd.Data... |
3,620,375 | <p>I am asked to calculate the integral <span class="math-container">$$\int_C \frac{1}{z-a}dz$$</span> where <span class="math-container">$C$</span> is the circle centered at the origin with radius <span class="math-container">$r$</span> and <span class="math-container">$|a|\neq r$</span></p>
<p>I parametrized the cir... | Surb | 154,545 | <p>If <span class="math-container">$r<|a|$</span> the integral is indeed <span class="math-container">$0$</span>, but if <span class="math-container">$r>|a|$</span>, then using <a href="https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula" rel="nofollow noreferrer">Cauchy integral formula</a> (or more genera... |
3,079,493 | <p>Let <span class="math-container">$$D_6=\langle a,b| a^6=b^2=1, ab=ba^{-1}\rangle$$</span> <span class="math-container">$$D_6=\{1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}$$</span></p>
<p>I would like to compute its character table and its irreducible representations.</p>
<p>I will explain what I have done so fa... | Angina Seng | 436,618 | <p>In general, <span class="math-container">$D_n$</span> is a group of order <span class="math-container">$2n$</span> with a cyclic subgroup <span class="math-container">$C_n$</span>
of order <span class="math-container">$n$</span> generated by <span class="math-container">$a$</span> say. Also <span class="math-contain... |
4,486,594 | <p>Let <span class="math-container">$X$</span> be the Riemann surface of <span class="math-container">$w^{2} \ =\text{sin} \ z$</span> in <span class="math-container">$ \mathbb{C}^{2}$</span>, i.e. let <span class="math-container">$X = \{(z,w): w^2 = \text{sin} \ z\}$</span>.</p>
<p>The Riemann surface structure on <sp... | Moishe Kohan | 84,907 | <p>The thing to prove is that the surface <span class="math-container">$X$</span> has infinite topological type. Instead of your function <span class="math-container">$F$</span>, I will consider the function <span class="math-container">$g(z,w)=z$</span> on <span class="math-container">$X$</span>, which is (generically... |
704,680 | <p>We have $$\sqrt{x -2} = 3 -2\sqrt{x}$$.</p>
<p>I am to find whether a real number exists for this relation, and the real number that satisfies.</p>
<p>I start by squaring both sides, which yields: </p>
<p>$$x - 2 = 4x - 12\sqrt{x} + 9$$.</p>
<p>Whence:</p>
<p>$$ -3x = -12\sqrt{x} + 11 \\
\sqrt{x} = \frac{x}{4} ... | TheBridge | 4,437 | <p>Hi I think it is true that the operator $T$ that you defined is such that the image of any measurable set of $\mathcal{B}([0,+\infty))$ is a measurable set or as claimed that $T$ preserves measurability. </p>
<p>The idea is to view things in a "topological" way first. If you "examine" carefully $T$ I think that you... |
704,680 | <p>We have $$\sqrt{x -2} = 3 -2\sqrt{x}$$.</p>
<p>I am to find whether a real number exists for this relation, and the real number that satisfies.</p>
<p>I start by squaring both sides, which yields: </p>
<p>$$x - 2 = 4x - 12\sqrt{x} + 9$$.</p>
<p>Whence:</p>
<p>$$ -3x = -12\sqrt{x} + 11 \\
\sqrt{x} = \frac{x}{4} ... | Evan Aad | 37,058 | <p>The following is an addendum to <a href="https://math.stackexchange.com/users/442/gedgar">GEdgar</a>'s <a href="https://math.stackexchange.com/a/705320/37058">answer</a>, aimed to clarify some points for my future reference, as well as for the sake of other readers who, like me, do not find these points self evident... |
704,680 | <p>We have $$\sqrt{x -2} = 3 -2\sqrt{x}$$.</p>
<p>I am to find whether a real number exists for this relation, and the real number that satisfies.</p>
<p>I start by squaring both sides, which yields: </p>
<p>$$x - 2 = 4x - 12\sqrt{x} + 9$$.</p>
<p>Whence:</p>
<p>$$ -3x = -12\sqrt{x} + 11 \\
\sqrt{x} = \frac{x}{4} ... | GEdgar | 442 | <p>Let's try this for the tail field.</p>
<p>Notation<br>
$\mathcal{B}_1$ the Borel sets for $\mathbb R$, a Polish space,<br>
$\mathcal{B}_2$ the Borel sets for $\mathbb R^2$, a Polish space,<br>
$\mathcal{B}_{\left[0,\infty\right)} = \sigma\big(\pi_t \colon t \in [0,\infty)\big)$ the Borel sets for $\mathbf{C}_{\left... |
3,872,750 | <p>Suppose we have a series <span class="math-container">$$\sum_{n=2}^\infty (-1)^n \frac{n^2}{10^n} = \sum_{n=2}^\infty (-1)^n b_n$$</span>.</p>
<p>I want to apply the alternating series test to see if it converges.</p>
<p>I need to show that:</p>
<p><span class="math-container">$$\lim_{n \rightarrow \infty} \frac{n^2... | Arthur | 15,500 | <p>For an equation
<span class="math-container">$$
[a]x=[b]
$$</span>
(where <span class="math-container">$a,b\in\Bbb Z$</span> and <span class="math-container">$x\in\Bbb Z_n$</span> for some natural <span class="math-container">$n$</span>), there are either <span class="math-container">$\gcd(a,n)$</span> solutions, or... |
73,785 | <p>I am new to Mathematica, and I have read this <a href="https://mathematica.stackexchange.com/questions/29203/determine-the-2d-fourier-transform-of-an-image">post</a> to understand how to perform Fourier transform on an image. My mission is to extract information on the typical distance between the black patches in t... | bill s | 1,783 | <p>It looks like random blobs, and that's what the FFT suggests...</p>
<pre><code>img = Import["http://i.stack.imgur.com/hALsH.jpg"];
imgBW = ImageData@ColorConvert[img, "Grayscale"];
imgZ = imgBW - Mean@Mean[imgBW];
xf = Abs[Fourier[imgZ, FourierParameters -> {1, -1}]];
{d1, d2} = Ceiling[Dimensions[xf]/2];
xCent... |
73,785 | <p>I am new to Mathematica, and I have read this <a href="https://mathematica.stackexchange.com/questions/29203/determine-the-2d-fourier-transform-of-an-image">post</a> to understand how to perform Fourier transform on an image. My mission is to extract information on the typical distance between the black patches in t... | bill s | 1,783 | <p>One approach is to locate the black components and then measure some properties of them. Here we locate them using <code>MorphologicalComponents</code>, find the centroids using <code>ComponentMeasurements</code> and then calculate the distance between the centroids using <code>Nearest</code>.</p>
<pre><code>img = ... |
1,617,698 | <p>While I was trying to find the formula of something by my own means I came across this sum which I need to solve, however I don't know if there is a solution for it, maybe it doesn't mean anything and I made a mistake. However if there's an equation which can replace this sum I will appreciate it a lot if you show m... | JimmyK4542 | 155,509 | <p>Let $S = \displaystyle\sum_{k = 1}^{n}\sin\dfrac{k\pi}{2n}$. (The $k = 0$ term is $0$, so we can ignore it). Then, by using the product to difference identity $\sin A \sin B = \dfrac{1}{2}\left(\cos(A-B)-\cos(A+B)\right)$, we have:</p>
<p>$S\sin \dfrac{\pi}{4n}$ $= \displaystyle\sum_{k = 1}^{n}\sin\dfrac{k\pi}{2n}\... |
2,921,439 | <p>I got this summation from the book <a href="https://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow noreferrer">Concrete Mathematics</a> which I didn't exactly understand:</p>
<p>$$
\begin{align}
Sn &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}} \\
&= \sum_{1 \leq... | mvw | 86,776 | <p>There is an algorithm to handle such integrals: <a href="https://en.wikipedia.org/wiki/Partial_fraction_decomposition" rel="nofollow noreferrer">Partial fraction decomposition</a></p>
|
2,921,439 | <p>I got this summation from the book <a href="https://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow noreferrer">Concrete Mathematics</a> which I didn't exactly understand:</p>
<p>$$
\begin{align}
Sn &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}} \\
&= \sum_{1 \leq... | 5xum | 112,884 | <p>$$-x^3 - x^2 + 4x + 4 = -x^2(x+1) +4(x+1) = (x+1)(4-x^2)$$</p>
<p>meaning that</p>
<p>$$\frac{x+1}{-x^2-x^2+4x+4} = \frac{x+1}{(x+1)(-x^2+4)} = \frac{1}{-x^2+4}$$</p>
<p>which is also the derivative you are looking for</p>
|
2,921,439 | <p>I got this summation from the book <a href="https://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow noreferrer">Concrete Mathematics</a> which I didn't exactly understand:</p>
<p>$$
\begin{align}
Sn &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}} \\
&= \sum_{1 \leq... | Dr. Sonnhard Graubner | 175,066 | <p>Use that$$\frac{x+1}{-x^3-x^2+4x-4}=\frac{1}{4(x+2)}-\frac{1}{4(x-2)}$$</p>
|
132,226 | <p>After edit:</p>
<p>How do we show that for every (holomorphic) vector bundle over a curve, is it possible to deform it to another one which is decomposable (into line bundles)? </p>
<p>Before edit:</p>
<p>I am not sure how much obvious or wrong is the following question:</p>
<p>For every (holomorphic) vector bun... | David E Speyer | 297 | <p>Francesco, in comments, shows that any vector bundle on a curve degenerates to a direct sum of a line bundles. (By the way, I observe the convention that "degeneration" means moving towards the special fiber and "deformation" means moving away from it; you are doing degeneration.)</p>
<p>In the comments, the OP ask... |
2,087,724 | <p>Let $\Omega $ a smooth domain of $\mathbb R^d$ ($d\geq 2$), $f\in \mathcal C(\overline{\Omega })$. Let $u\in \mathcal C^2(\overline{\Omega })$ solution of $$-\Delta u(x)+f(x)u(x)=0\ \ in\ \ \Omega .$$
Assume that $f(x)\geq 0$ for $x\in \Omega $. Prove that $$\int_{B(x,r)}|\nabla u|^2\leq \frac{C}{r^2}\int_{B(x,2r)}|... | Glitch | 74,045 | <p>Suppose that $\varphi : \bar{B}(x,2r) \to [0,\infty)$ is Lipschitz and $\varphi =0$ on $\partial B(x,2r)$. Note that since $\varphi$ is Lipschitz, it is differentiable almost everywhere by Rademacher's theorem. Then
$$
0 \le \int_{B(x,2r)} \varphi^2 f u^2 = \int_{B(x,2r)} \varphi^2 u \Delta u = \int_{B(x,2r)} - \... |
1,002,777 | <p>I want to convert this polynomoial to partial fraction.</p>
<p>$$
\frac{x^2-2x+2}{x(x-1)}
$$</p>
<p>I proceed like this:
$$
\frac{x^2-2x+2}{x(x-1)} = \frac{A}{x} + \frac{B}{x-1}
$$
Solving,
$$
A=-2,B=1
$$
But this does not make sense. What is going wrong?</p>
| lab bhattacharjee | 33,337 | <p>As the highest power of $x$ in the numerator & the denominator are same,</p>
<p>using <a href="http://en.wikipedia.org/wiki/Partial_fraction_decomposition" rel="nofollow">Partial Fraction Decomposition</a>,
$$\frac{x^2-2x+2}{x(x-1)}=1+\frac Ax+\frac B{x-1}$$</p>
<p>$1$ is found by $$\frac{\text{The coefficient... |
2,293,162 | <p>$$f_n(x)=\begin{cases} 1-nx,&\text{for }x\in[0,1/n]\\
0 ,&\text{for }x \in [1/n,1]
\end{cases}$$ </p>
<p>Then which is correct option?</p>
<p>1.$\lim\limits_{n\to\infty }f_n(x)$ defines a continuous function on $[0,1]$. </p>
<p>2.$\lim\limits_{n\to\infty }f_n(x)$ exists for all $x\in [0,1]$. $f_n(0)=1$... | Community | -1 | <p>The function cannot be continuous at $x=0$, because its value is $1$ and the values at all $x>0$ are $0$. (Because for $x>0$, there is always an $n$ such that $1/n<x$).</p>
<p>The option 2. is correct because the limit is defined for all $x$ (as above) and the last two equalities come from the definition.<... |
3,948,418 | <p>Ok so on doing a whole lot of Geometry Problems, since I am weak at Trigonometry, I am now focused on <span class="math-container">$2$</span> main questions :-</p>
<p><span class="math-container">$1)$</span> <strong>How to calculate the <span class="math-container">$\sin,\cos,\tan$</span> of any angle?</strong></p>
... | Tyma Gaidash | 905,886 | <p>This will be my attempt at answering your question about finding <span class="math-container">$\sin(\frac{143°}{3})=\sin(\frac x3)$</span>.</p>
<p>Let θ=<span class="math-container">$\frac x3$</span> and using <a href="https://mathworld.wolfram.com/Multiple-AngleFormulas.html" rel="nofollow noreferrer">this website<... |
2,612,794 | <p>I have a very elementar question but I do not see where my mistake is. </p>
<p>Suppose we have a sequence $(x_n)$ with
$\lim_{n\to\infty}x_n=1$. Moreover, suppose that the sequence $({x_n}^c)$ for some constant $c>1$ has limit $\lim_{n\to\infty}{x_{n}}^c=c$.</p>
<p>Then
$$
\lim_{n\to\infty}\log({x_n}^c)=\log(c)... | Michael Hardy | 11,667 | <p>If $\lim\limits_{n\,\to\,\infty} x_n =1$ then $\lim\limits_{n\,\to\,\infty} x_n^c = 1.$
\begin{align}
\lim_{n\,\to\,\infty} (x_n^c) & = \left( \lim_{n\,\to\,\infty} x_n \right)^c & & \text{because } x \mapsto x^c \text{ is a continuous function} \\[10pt]
& = 1^c = 1.
\end{align}</p>
|
2,165,213 | <p><strong>The Problem</strong></p>
<p>Let $V=k^3$ for some field $k$. Let $W$ be the subspace spanned by $(1,0,0)$ and let $U$ be the subspace spanned by $(1,1,0)$ and $(0,1,1)$. Show that $V= W \oplus U$. Explain your argument in detail.</p>
<hr>
<p><strong>What I Know</strong></p>
<ol>
<li><p>I know that a fi... | InsideOut | 235,392 | <p>Let $q(x)$ be an element in $\Bbb R[x]$, and let $n=\deg q(x)$. Then
$$q(x)=(x-5)g(x)+r(x),$$ but $r(x)\in \Bbb R$, so it is a constant because $\deg g(x)=n-1$. </p>
<p>Thus $\Bbb R[x]/( x-5 ) \cong \Bbb R$, because $\ker\phi=\{p(x) : (x-5)/ p(x)\}\cong \Bbb R_{\ge 1}[x]$.</p>
<p>EDIT: the last isomorphism is the ... |
3,060,742 | <p><span class="math-container">$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = 1.644934$</span> or <span class="math-container">$\frac{\pi^2}{6}$</span></p>
<p>What if we take every 3rd term and add them up? </p>
<p>A = <span class="math-container">$ \frac{1}{3^2} + \fra... | robjohn | 13,854 | <p><strong>Polylogarithms</strong></p>
<p>A useful formula that can be applied here is
<span class="math-container">$$
\frac13\sum_{k=0}^2e^{2\pi ijk/3}=[3\mid j]\tag1
$$</span>
So
<span class="math-container">$$
\begin{align}\newcommand{\Li}{\operatorname{Li}}
\sum_{j=0}^\infty\frac1{(3j+1)^2}
&=\frac13\sum_{k=0}... |
2,154,960 | <p>Here is the question and i dont really understand</p>
<p>Point $(a,b)$ is on the function $f(x)=\frac{2}{x}$ $x>0$. Show that the area of the triangle formed by the tangent line at $(a,b)$ , the $x$ axis and $y$ axis is equals to $4$.</p>
<p>What is the question asking?</p>
<p>I used the first principle to fin... | Paul Sundheim | 88,038 | <ol>
<li>Find $b$ in terms of $a$ using $f(x)=2/x$. This gives you the point $(a,b)$ in terms of $a$.</li>
<li>Find the slope of the tangent line at $(a,b)$ using the derivative of $f$, in terms of $a$.</li>
<li>Find the $y$-intercept of the line using the point and the slope.</li>
<li>Use the equation of the line to g... |
2,154,960 | <p>Here is the question and i dont really understand</p>
<p>Point $(a,b)$ is on the function $f(x)=\frac{2}{x}$ $x>0$. Show that the area of the triangle formed by the tangent line at $(a,b)$ , the $x$ axis and $y$ axis is equals to $4$.</p>
<p>What is the question asking?</p>
<p>I used the first principle to fin... | David G. Stork | 210,401 | <p>The questioner asked for help understanding the question itself, and here a picture is worth $10^3$ words:</p>
<p><a href="https://i.stack.imgur.com/RibfO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RibfO.png" alt="enter image description here"></a></p>
<p>The goal is then to find the value ... |
355,296 | <p>How can we evaluate $$\displaystyle\int \frac{x^2 + x+3}{x^2+2x+5} dx$$ </p>
<p>To be honest, I'm embarrassed. I decomposed it and know what the answer should be but<br>
I can't get the right answer. </p>
| Christopher A. Wong | 22,059 | <p>You can decompose your integrand as follows:</p>
<p>$$ \frac{ x^2 + 2x + 5 - x - 1 - 1}{x^2 + 2x + 5} = 1 - \frac{x + 1}{x^2 + 2x + 5} - \frac{1}{(x+1)^2 + 4}$$</p>
<p>You can integrate the first term directly, the second term after the substitution $u = x^2 + 2x + 5$, and the third term by recalling that $(\arcta... |
355,296 | <p>How can we evaluate $$\displaystyle\int \frac{x^2 + x+3}{x^2+2x+5} dx$$ </p>
<p>To be honest, I'm embarrassed. I decomposed it and know what the answer should be but<br>
I can't get the right answer. </p>
| Community | -1 | <p><strong>Hint</strong> Use the decomposition
$$\frac{x^2 + x+3}{x^2+2x+5}=1-\frac{ x+2}{x^2+2x+5}=1-\frac{1}{2}\frac{ 2x+2}{x^2+2x+5}-\frac{ 1}{x^2+2x+5}$$
and
$$\frac{ 1}{x^2+2x+5}=\frac{ 1}{(x+1)^2+4}=\frac{1}{4}\frac{ 1}{(\frac{x+1}{2})^2+1}$$
the first fraction is on the form $\frac{f'}{f}$ and the second have t... |
642,443 | <p>Let $\{a_n\}_{n\ge1}^{\infty}=\bigg\{\cfrac{1}{1\cdot3}+\cfrac{1}{2\cdot4}+\dots+\cfrac{1}{n\cdot(n+2)}\bigg\}$. Find $\lim_{n\to \infty}{a_n}$.</p>
<p>I write:
$$\lim_{n\to \infty}{a_n}=\sum_{n=1}^{\infty}{\frac{1}{n\cdot(n+2)}}=\sum_{n=1}^{\infty}{\frac{1}{n^2+2n}}\approx\sum_{n=1}^{\infty}{\cfrac{1}{n^2}}$$</p>
... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}... |
8,997 | <p>I have a set of data points in two columns in a spreadsheet (OpenOffice Calc):</p>
<p><img src="https://i.stack.imgur.com/IPNz9.png" alt="enter image description here"></p>
<p>I would like to get these into <em>Mathematica</em> in this format:</p>
<pre><code>data = {{1, 3.3}, {2, 5.6}, {3, 7.1}, {4, 11.4}, {5, 14... | WReach | 142 | <p>Here is a manual method using copy-and-paste that is suitable for small volumes of data on an ad hoc basis...</p>
<p>1) Enter the following expression into a notebook, but don't evaluate it:</p>
<pre><code>data = ImportString["", "TSV"]
</code></pre>
<p>2) Copy the cells from the source spreadsheet onto the clipb... |
481,167 | <p>Let $V$ be a $\mathbb{R}$-vector space.
Let $\Phi:V^n\to\mathbb{R}$ a multilinear symmetric operator.</p>
<p>Is it true and how do we show that for any $v_1,\ldots,v_n\in V$, we have:</p>
<p>$$\Phi[v_1,\ldots,v_n]=\frac{1}{n!} \sum_{k=1}^n \sum_{1\leq j_1<\cdots<j_k\leq n} (-1)^{n-k}\phi (v_{j_1}+\cdots+v_{j... | Gilles Bonnet | 60,457 | <p><strong>This in not an answer</strong>, but an incomplete attempt of induction proof.</p>
<p>First, we will consider the following notation:
$$\Phi_v[v_1,\ldots,v_{n-1}]=\Phi[v_1,\ldots,v_{n-1},v],$$ so $\Phi_v:V^{n-1}\to\mathbb{R}$ is the multinear symmetric operator we obtain when we fix a variable in $\Phi$.
We ... |
204,592 | <p>The matrix exponential is a well know thing but when I see online it is provided for matrices. Does it the same expansion for a linear operator? That is if $A$ is a linear operator then $$e^A=I+A+\frac{1}{2}A^2+\cdots+\frac{1}{k!}A^k+\cdots$$</p>
| kalvotom | 38,469 | <p>Yes, you can define an exponential of any linear BOUNDED operator by this series. If the operator is unbounded then it is not always possible. </p>
|
1,285,014 | <p>Let $R,S$ be commutative rings with identity.</p>
<p>Proving that $X \sqcup Y$ is an affine scheme is the same as proving that $Spec(R) \sqcup Spec(S) = Spec(R \times S)$.</p>
<p>I proved that if $R,S$ are rings, then the ideals of $R \times S$ are exactly of the form $P \times Q$, where $P$ is an ideal of $R$ and... | Demosthene | 163,662 | <p>Note that you can rewrite $p\to\neg(q\lor r)$ as $\neg p\lor(\neg(q\lor r))$, i.e. $\neg p\lor(\neg q\land\neg r)$. This fits exactly the truth table (as you would expect), and shows that the proposition is true whenever $p$ is false or both $q$ and $r$ are false.</p>
<p>This would usually enough be enough to answe... |
1,285,014 | <p>Let $R,S$ be commutative rings with identity.</p>
<p>Proving that $X \sqcup Y$ is an affine scheme is the same as proving that $Spec(R) \sqcup Spec(S) = Spec(R \times S)$.</p>
<p>I proved that if $R,S$ are rings, then the ideals of $R \times S$ are exactly of the form $P \times Q$, where $P$ is an ideal of $R$ and... | Theuth | 240,068 | <p>See also K-map:
<a href="http://en.wikipedia.org/wiki/Karnaugh_map" rel="nofollow">http://en.wikipedia.org/wiki/Karnaugh_map</a>
It is a very clever and fast method to derive DNF and other useful things</p>
|
1,828,097 | <p>If we contruct two strainght lines as shown:<a href="https://i.stack.imgur.com/8K5Eo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8K5Eo.png" alt="enter image description here"></a></p>
<p>Then join them such that to complete a triangle. <a href="https://i.stack.imgur.com/Uvtnw.png" rel="nofoll... | abcabc123 | 347,408 | <p>Well first of all it is because $\infty$ is not a number, you can't do what you usually do with numbers but let's suppose here that it is, let's suppose $\infty$ is a number, the bigest number. Then you can imagine a finite constant $k$ added to $\infty$ which has to be $\infty$ also i.e. $k+\infty=\infty$. That the... |
2,227,027 | <p>If $f(x)$ is defined everywhere except at $x=x_0$, would $f'(x_0)$ be undefined at $x=x_0$ as well?</p>
<p>One example is: $$f(x)=\ln(x)\rightarrow f'(x)=\frac{1}{x}$$</p>
<p>In this particular case, both $f(x)$ and $f'(x)$ are undefined at $x=0$. I wonder if this always holds true.</p>
<p>Thank you.</p>
| The Count | 348,072 | <p>This depends on some conventions, but the typical answer is <em>yes</em>, because if a function is not defined somewhere, it cannot have a slope there! In other words, we certainly don't have a slope where there is no function value!</p>
<p>We need a value $f(x_0)$ to plug into the limit definition of the derivativ... |
3,273,756 | <blockquote>
<p>I am supposed to give a 9-dimensional irreducible representation of <span class="math-container">$\mathfrak{so}(4)$</span>.</p>
</blockquote>
<p>I know that <span class="math-container">$\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$</span> and hence I have a 6-dimensional reducible rep... | user10354138 | 592,552 | <p><strong>Hint</strong>: The space of <span class="math-container">$4\times 4$</span> symmetric traceless matrices is of dimension <span class="math-container">$9$</span>.</p>
|
1,861,890 | <p>Given that $$s_{n}=\frac{(-1)^{n}}{n},$$ I want to show $$\lim_{n\to\infty}{s_{n}}=0$$ in the metric space $X=\mathbb{C}.$ However, it seems to me that <strong>Archimedean Property is not applicable to the case above</strong>, because $s_{n}$ is not always positive for each $n$. Then, how can I do that?</p>
| George Law | 141,584 | <p>You don’t need $s_n$ to be always positive; all you need is to show that for every real $\varepsilon>0$, $|s_n-0|<\varepsilon$ for all $n$ greater than some $N$.</p>
|
3,260,530 | <p>I read from wikipedia that a neighbourhood of a point <span class="math-container">$p$</span> is a subset <span class="math-container">$V$</span> of a topological space <span class="math-container">$\{X,\tau\}$</span> that includes an open set <span class="math-container">$U$</span> such that <span class="math-conta... | Theo Bendit | 248,286 | <p>A subset <span class="math-container">$V$</span> of a topological space <span class="math-container">$X$</span> is a neighbourhood of point <span class="math-container">$p \in X$</span> if some open set <span class="math-container">$U$</span> exists such that
<span class="math-container">$$p \in U \subseteq V.$$</sp... |
2,400,336 | <p>My first try was to set the whole expression equal to $a$ and square both sides. $$\sqrt{6-\sqrt{20}}=a \Longleftrightarrow a^2=6-\sqrt{20}=6-\sqrt{4\cdot5}=6-2\sqrt{5}.$$</p>
<p>Multiplying by conjugate I get $$a^2=\frac{(6-2\sqrt{5})(6+2\sqrt{5})}{6+2\sqrt{5}}=\frac{16}{2+\sqrt{5}}.$$</p>
<p>But I still end up w... | Frank | 332,250 | <p>There's actually a general formula for these kinds of expressions. Namely$$\sqrt{X\pm Y}=\sqrt{\frac {X+\sqrt{X^2-Y^2}}2}\pm\sqrt{\frac {X-\sqrt{X^2-Y^2}}2}$$
Where $X,Y$ are real numbers. Simply substituting $X=6$ and $Y=\sqrt{20}$ gives the proper denesting. The proof of this is quite simple. Assume that$$X\pm Y=\... |
1,261,504 | <p>I am trying to proof $ab = \gcd(a,b)\mathrm{lcm}(a,b)$.</p>
<p>The definition of $\mathrm{lcm}(a,b)$ is as follows:</p>
<p>$t$ is the lowest common multiple of $a$ and $b$ if it satisfies the following:</p>
<p>i) $a | t$ and $b | t$ </p>
<p>ii) If $a | c$ and $b | c$, then $t | c$.</p>
<p>Similiarly for the $\g... | BruceET | 221,800 | <p>(a) If you are using at a normal table, first determine whether it
gives areas $P(Z \le z)$ [or $P(0 < Z \le z)].$ In the first
instance look in the <em>body</em> of the table to find the closest
probability to .9500; in this case, probably a tie between .9496 and .0505. Then find the corresponding value of $z$ ... |
163,917 | <p>Suppose $B_{\epsilon}$ are closed subsets of a compact space and $B_{\epsilon} \supset B_{\epsilon'} \quad \forall \epsilon > \epsilon'$. Furthermore, $B_0 = \bigcap_{\epsilon>0} B_{\epsilon}$. For a continuous function $f$ can we conclude that
$$f(B_0) = \bigcap_{\epsilon>0} f(B_{\epsilon})?$$</p>
<p>I ... | kaba | 51,161 | <p>The following answer is a generalization of the one given by Asaf Karagila.</p>
<h4>Theorem</h4>
<p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be spaces, <span class="math-container">$I$</span> be an ordinal, <span class="math-container">$V : I \to \mathcal{P}(X)$</span> ... |
2,631,284 | <p>I'm trying to find all $n \in \mathbb{N}$ such that</p>
<p>$(n+2) \mid (n^2+5)$ </p>
<p>as the title says, I've tried numbers up to $20$ and found that $1, 7$ are solutions and I suspect that those are the only $2$ solutions, however I have no idea how to show that.</p>
<p>I've done nothing but basic transformati... | PNT | 873,280 | <p><span class="math-container">$(n+2) \mid (n^2+5)\implies \exists m \in \mathbb N$</span> such that :
<span class="math-container">$$\frac{n^2+5}{n+2}= m \iff n^2-mn+(5-2m)=0$$</span></p>
<p>By Solving the quadratic equation in <span class="math-container">$n$</span>, We get:
<span class="math-container">$$n = \frac{... |
104,626 | <p>I encountered the following differential equation when I tried to derive the equation of motion of a simple pendulum:</p>
<p>$\frac{\mathrm d^2 \theta}{\mathrm dt^2}+g\sin\theta=0$</p>
<p>How can I solve the above equation?</p>
| yoyo | 6,925 | <p>replacing $\sin\theta$ by $\theta$ (physically assuming small angle deflection) gives you a homogeneous second order linear differential equation with constant coefficients, whose general solution can be found in most introductory diff eq texts (or a google search). this new equation represents a simple harmonic os... |
2,403,404 | <p>I would be thankful if anyone can answer my question. This is a very basic question. Let's say we wish to minimise the quantity</p>
<p>$$\hat{h}= \|h-h_i\|+\lambda\|h-u\|,$$</p>
<p>where:</p>
<p>$$h=[13,17,20, 17, 20, 14, 17, 18, 16, 15, 15, 12, 19, 13, 17, 13]^\top,\\ h_i=[18, 17, 14, 13, 17, ... | Vasili | 469,083 | <p>Let's consider this triangle on the coordinate plane. Let A has coordinates (0,0) and B has coordinates (6,0). Point C will belong either to line y=4 or y=-4 but it's irrelevant, both cases will produce the same result. Let x be the abscissa of point C. We can define CA+CB as the following function $f(x)=\sqrt{x^2+4... |
4,048,785 | <p>Show that <span class="math-container">$2r^2-3$</span> is never a square, <span class="math-container">$r=2,3,...$</span></p>
<p>I know that no perfect square can have <span class="math-container">$2, 3, 7$</span>, or <span class="math-container">$8$</span> as its last digit. I'm not sure how to do this with congrue... | abiessu | 86,846 | <p><span class="math-container">$2r^2-3$</span> is odd, so let <span class="math-container">$(2y+1)^2=4y^2+4y+1=2r^2-3$</span>, then <span class="math-container">$4y^2+4y=2r^2-4$</span> so <span class="math-container">$2\mid r$</span> giving <span class="math-container">$r=2s\to y^2+y=2s^2-1$</span>. But <span class="... |
3,893,908 | <p>I want to compute <span class="math-container">$$\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx$$</span></p>
<p>My approach is <span class="math-container">$$\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx=\int_{-\infty}^{\infty} \frac{1}{(x -\pi)^2}dx+\int_{-\infty}^{\infty} \frac{\cos(x)}{(x -\pi)^2... | Franklin Pezzuti Dyer | 438,055 | <p><strong>HINT:</strong> With the substitution <span class="math-container">$x\to x+\pi$</span>, we get</p>
<p><span class="math-container">$$\int_{-\infty}^\infty \frac{1-\cos x}{x^2}dx$$</span></p>
<p>then integrate by parts with <span class="math-container">$u=1-\cos x$</span> and <span class="math-container">$dv=d... |
3,893,908 | <p>I want to compute <span class="math-container">$$\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx$$</span></p>
<p>My approach is <span class="math-container">$$\int_{-\infty}^{\infty} \frac{1+\cos(x)}{(x -\pi)^2}dx=\int_{-\infty}^{\infty} \frac{1}{(x -\pi)^2}dx+\int_{-\infty}^{\infty} \frac{\cos(x)}{(x -\pi)^2... | Henry Lee | 541,220 | <p><span class="math-container">$u=x-\pi,du=dx$</span>
<span class="math-container">$$\int_{-\infty}^\infty\frac{1+\cos(x)}{(x-\pi)^2}dx=\int_{-\infty}^\infty\frac{1+\cos(u+\pi)}{u^2}du=\int_{-\infty}^\infty\frac{1-\cos(u)}{u^2}du$$</span>
and now if we look at this for <span class="math-container">$u\to\infty$</span>:... |
1,638,051 | <p>$$\int\frac{dx}{(x^{2}-36)^{3/2}}$$</p>
<p>My attempt:</p>
<p>the factor in the denominator implies</p>
<p>$$x^{2}-36=x^{2}-6^{2}$$</p>
<p>substituting $x=6\sec\theta$, noting that $dx=6\tan\theta \sec\theta$ </p>
<p>$$x^{2}-6^{2}=6^{2}\sec^{2}\theta-6^{2}=6^{2}\tan^{2}\theta$$</p>
<p>$$\int\frac{dx}{(x^{2}-36... | helpmeh | 200,346 | <p>attempt 2:</p>
<p>$$\int\frac{dx}{(x^{2}-36)^{3/2}}$$</p>
<p>the factor in the denominator implies</p>
<p>$$(x^{2}-36)^{3/2}=({x^{2}-6^{2}})^{3/2}$$</p>
<p>substituting $x=6sec\theta$, noting that $dx=6tan\theta sec\theta d\theta$ </p>
<p>$$(x^{2}-6^{2})^{3/2}=(6^{2}sec^{2}\theta-6^{2})^{3/2}d\theta=(36tan^{2}\... |
23,994 | <p>For which values of m does the equation:
$$3 \ln x+m x^3 = 17$$
have $1$ solution? $2$ solutions? $0$ solution?</p>
<p>Thanks.</p>
| I. J. Kennedy | 130 | <p>Can't improve on Jonas Meyer's answer, but playing with <a href="http://omnium-gatherum.appspot.com/pages/so23994.html" rel="nofollow">this</a> geogebra applet might help you get a feel for the equation. Move the slider to change $m$ in increments of 1/1000000.</p>
|
2,683,326 | <p>I have a function $f(x)$ whose second order Taylor expansion is represented by $f_2(x)$. Is it true that $$f(x)>f_2(x)$$ for all $x$? Any help in this regard will be much appreciated. Thanks in advance.</p>
| BruceET | 221,800 | <p>For the case where the breaks are five independent observations from
$\mathsf{Unif}(0,1),$ an argument similar to @GrahamKemp's can be used
to show that the longest piece is longer than $1/2$ with probability $6/32 = 3/16 = 0.1875.$ </p>
<p>If the intervals are arranged in a circle, the probability that the clockwi... |
2,837,683 | <p>I have to solve the integral $$\int_D \sqrt{x^2+y^2} dx dy$$ where $D=\{(x,y)\in\mathbb{R^2}: x^{2/3}+y^{2/3}\le1\}$.</p>
<p>I am not able to find a parameterization that suits the integrand.
I tried with $$\cases{x=(r\cos t)^3\\y=(r\sin t)^3}$$ in order to reduce the domain to a circle but then the integral become... | Jack D'Aurizio | 44,121 | <p>We may exhibit a parametrization of the integration domain through $x=\rho\cos^3\theta, y=\rho\sin^3\theta$ with $\rho\in[0,1]$ and $\theta\in[0,2\pi)$. By considering the Jacobian of this transformation and symmetry, we get that the original integral equals</p>
<p>$$ 4 \int_{0}^{1}\int_{0}^{\pi/2}3\rho^2\sin^2\the... |
2,828,472 | <p>This question is regarding property of little o notation given in Apostol Calculus. The property is given on page 288 and stated as:</p>
<blockquote>
<p>Theorem 7.8 (c) As $x\to a$ we have $f(x)\cdot o (g(x)) = o(f(x)g(x))$.</p>
</blockquote>
<p>Here say $h(x) = o(g(x))$ then we have $f(x) \lim_{x\to a} \frac{h(... | epi163sqrt | 132,007 | <p>This answer is based upon Apostol's settings. First of all we should be aware that the equality sign $=$ here is used to indicate a subset relation $\subseteq$ between two sets
\begin{align*}
f(x)o(g(x))=o(f(x)g(x)\qquad\qquad \text{as }x\to a
\end{align*}
meaning that whenever a function $h=h(x)$ is an element o... |
2,952,392 | <p>Revisit the following discussion: </p>
<p><a href="https://math.stackexchange.com/questions/843909/prove-that-the-inverse-image-of-an-open-set-is-open">Prove that the inverse image of an open set is open</a></p>
<p>Obviously, the above discussion is based on Euclidean space (which is also a metric space, so the pr... | Robert Lewis | 67,071 | <p>Look at</p>
<p><span class="math-container">$p(x, y) = x^2 + y^2 - 1 \in \Bbb R[x, y]; \tag 1$</span></p>
<p>this polynomial has an uncountable infinity of zeroes</p>
<p><span class="math-container">$(x, y) = (\cos \theta, \sin \theta) \in \Bbb R^2, \; \theta \in [0, 2\pi). \tag 2$</span></p>
<p>The problem is t... |
2,952,392 | <p>Revisit the following discussion: </p>
<p><a href="https://math.stackexchange.com/questions/843909/prove-that-the-inverse-image-of-an-open-set-is-open">Prove that the inverse image of an open set is open</a></p>
<p>Obviously, the above discussion is based on Euclidean space (which is also a metric space, so the pr... | Wuestenfux | 417,848 | <p>There is a characterization of zero-dimensional ideals in the polynomial ring <span class="math-container">$R={\Bbb K}[x_1,\ldots,x_n]$</span> using Gröbner bases. An ideal <span class="math-container">$I$</span> is zero-dimensional if its zero locus <span class="math-container">$V_{\Bbb K}(I)$</span> is finite.
Equ... |
10,601 | <p>It sometimes happens that the same user posts <strong>exactly</strong> the same question twice in a row.</p>
<p>Examples: </p>
<ul>
<li><a href="https://math.stackexchange.com/questions/446622/drawing-at-least-90-of-colors-from-urn-with-large-populations">1</a> <a href="https://math.stackexchange.com/questions/446... | robjohn | 13,854 | <p>I think the first thing to do is to vote to close if you have the reputation to do so.</p>
<p>Flagging the moderators about a duplicate is okay if it is definitely a duplicate. If there is some question, then it would be better to allow the community to decide than have a moderator unilaterally decide for the commu... |
10,601 | <p>It sometimes happens that the same user posts <strong>exactly</strong> the same question twice in a row.</p>
<p>Examples: </p>
<ul>
<li><a href="https://math.stackexchange.com/questions/446622/drawing-at-least-90-of-colors-from-urn-with-large-populations">1</a> <a href="https://math.stackexchange.com/questions/446... | Willie Wong | 1,543 | <p>I look through the few cases where your flag was declined. In the ones that I saw, my instinct would have been to decline too: the OP generally have rephrased the question so that at first glance it is not entire obvious that the questions are exact duplicates. (Moving sentences around and such.) </p>
<p>It would h... |
2,912,376 | <p>I understood why he chose the positive square root in the sin but why the tan is also positive ? Isn't the tan positive and negative in this interval ?
<a href="https://i.stack.imgur.com/2rBFs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2rBFs.png" alt="enter image description here"></a></p>
| Math Lover | 348,257 | <p>Note that when $-1 <x < 0$, $$\tan(\cos^{-1}(x)) = \frac{\sqrt{1-x^2}}{x} < 0.$$</p>
|
340,575 | <p>I got my exam on Thursday, and just got a few questions left. Anyway I would aprreciate help a lot! Can anyone please help me to solve this task? You can see the picture below. The need is to finde the size of the two radius. I thought about working with cords, like the cord AC is the same size like another one. Sti... | Vincent Tjeng | 52,208 | <p>The key insight is that $\angle ACB=90°$.</p>
<p>To show this, we draw a line passing through $C$ that is tangent to both the circles $k_1,k_2$ at $C$. This is possible since the two circles are tangent to each other. Let this tangent intersect the line $AB$ at $D$.</p>
<p>Now, we have $DA=DC$ since $DA,DC$ are li... |
4,061,536 | <p>We know that in a finite group of order say <span class="math-container">$g$</span>, an element of the group will have order of element <span class="math-container">$m\leq g$</span>. However, is it necessarily true that at least one element in the group <span class="math-container">$\textbf{must}$</span> have order ... | José Carlos Santos | 446,262 | <p>No. If <span class="math-container">$G$</span> has order <span class="math-container">$m$</span> and <span class="math-container">$H$</span> has order <span class="math-container">$n$</span>, then <span class="math-container">$G\times H$</span> has order <span class="math-container">$mn$</span>. But if <span class="... |
4,061,536 | <p>We know that in a finite group of order say <span class="math-container">$g$</span>, an element of the group will have order of element <span class="math-container">$m\leq g$</span>. However, is it necessarily true that at least one element in the group <span class="math-container">$\textbf{must}$</span> have order ... | kabenyuk | 528,593 | <p>Here is the complete answer to your question.
A group of order <span class="math-container">$n$</span> has an element of order <span class="math-container">$n$</span> if and only if the group is cyclic. This statement is an easy exercise.</p>
|
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | JMP | 210,189 | <p>If $s>t$ then we have $s-t>0$. If $s$ and $t$ are complex, and $s-t=u$ (u<>0), then we need $u>0$. However as u lies on a circle with +ve radius $r, u$ is always greater than $0$. This means that $-u=t-s>0$, implying $t>s$, a contradiction, so there is no order over the complex numbers.</p>
|
1,116,022 | <p>I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.</p>
<p>But does it ever make sense to compare a real number and a complex/imaginary one?</p>
<p>For example, could one say that $5+2i> 3$ because the real part of $5+2i
$ is bigger than the real part of $3$? Or i... | Daniel W. Farlow | 191,378 | <p><strong>Observation:</strong> Many of the properties of the real number system $\mathbb{R}$ hold in the complex number system $\mathbb{C}$, but there are some rather interesting differences as well--one of them is the concept of <em>order</em>. The concept of order used in $\mathbb{R}$ does not carry over to $\mathb... |
4,554,231 | <p>How do you evaluate this limit? I can't manage to do it, even after manipulating the limit expression in several different ways, and using L'Hôpital's rule.</p>
<p><span class="math-container">$$
\lim_{h\,\to\, 0^{+}}\,
\left(\frac{{\rm e}^{-1/h^{2}}\,}{h}\right)
$$</span></p>
| Robin | 602,386 | <p>We wish to evaluate</p>
<p><span class="math-container">$$L =\lim _{h \to 0^+} \left(\frac{e^{-{1/h^2}}}{h}\right).$$</span></p>
<p>Substitute <span class="math-container">$u = \frac{1}{h}$</span> to get</p>
<p><span class="math-container">$$L =\lim _{u \to \infty} (ue^{-u^2}).$$</span></p>
<p>We have <span class="m... |
2,651,394 | <p>I am attempting to create a function in Matlab which turns all matrix elements in a matrix to '0' if the element is not symmetrical. However, the element appears to not be reassigning.</p>
<pre><code>function [output_ting] = maker(a)
[i,j] = size(a);
if i ~= j
disp('improper input!')
else
end
c = 1;
b = a.';
... | lab bhattacharjee | 33,337 | <p>$$\ln\dfrac{1+\dfrac{\sin(x+h)-\sin x}{\sin x}}h$$</p>
<p>$$=\ln\dfrac{\left(1+\dfrac{\sin(x+h)-\sin x}{\sin x}\right)}{\dfrac{\sin(x+h)-\sin x}{\sin x}}\cdot\dfrac{\sin(x+h)-\sin x}{h\sin x}$$</p>
<p>Now $\lim_{u\to0}\dfrac{\ln(1+u)}u=1$</p>
<p>and $$\lim_{h\to0}\dfrac{\sin(x+h)-\sin x}h=\lim_{h\to0}\dfrac{\sin\... |
3,005,208 | <p>I want to solve this polynomial analytically. I know the useful answer is between 0 and 1. Is there any way I can write the answer based on a, b, and c?
<span class="math-container">$$
6\cdot a \cdot x^4 + 2 \cdot b \cdot x^3-b \cdot c=0
$$</span>
Also, an approximate answer is acceptable, for example, an answer wit... | Bayesian guy | 594,523 | <p>You could use Ferrari's method for solve in general.
<a href="https://proofwiki.org/wiki/Ferrari%27s_Method" rel="nofollow noreferrer">https://proofwiki.org/wiki/Ferrari%27s_Method</a> this is an easy algorithmic way to do it.</p>
|
86,067 | <p>So I am having an issue using <code>NDSolve</code> and plotting the function. So I have two different <code>NDSolve</code> calls in my plotting function. (They are technically the same, just have different names; but that can be changed back if at all possible because I want them to be the same.) But the second one ... | BlacKow | 8,114 | <p>Not a real answer. But you can try to put your plot in <code>Inset[]</code>, then add another <code>Inset[]</code> for x-axis and yet another <code>Inset[]</code> for y-axis
and then stitch all scales together…
Something like this (nothing is stitched)</p>
<pre><code>Graphics[{Transparent, Rectangle[],
Inset[Lis... |
246,817 | <p>We have the succession and its formula:
$$
1^2+4^2+\cdots+ (3k-2)^2 = \dfrac{k(6k^2-3k-1)}{2}
$$</p>
<p>Now we need to apply it for $k+1$:
$$
1^2+4^2+\cdots+ (3n-2)^2 +(3(k+1)-2)^2 = \\
\dfrac{k(6k^2-3k-1)}{2} + (3(k+1)-2)^2
$$</p>
<p>I know that the result must be $\frac{1}{2}(k+1)(6(k+1)^2-3(k+1)-1)$ but I wasn'... | Brian M. Scott | 12,042 | <p>First combine everything into a single fraction:</p>
<p>$$\begin{align*}
\frac{k(6k^2-3k-1)}{2} + \big(3(k+1)-2\big)^2&=\frac{k(6k^2-3k-1)}2+(3k+1)^2\\
&=\frac{k(6k^2-3k-1)+2(3k+1)^2}2\\
&=\frac{6k^3-3k^2-k+18k^2+12k+2}2\\
&=\frac{6k^3+15k^2+11k+2}2\;.\tag{1}
\end{align*}$$</p>
<p>At this point the... |
1,860,267 | <blockquote>
<p>Prove the convergence of</p>
<p><span class="math-container">$$\int\limits_1^{\infty} \frac{\cos(x)}{x} \, \mathrm{d}x$$</span></p>
</blockquote>
<p>First I thought the integral does not converge because</p>
<p><span class="math-container">$$\int\limits_1^{\infty} -\frac{1}{x} \,\mathrm{d}x \le \int\lim... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\left|\,\int_1^\infty\frac{\cos(x)}{x}\,\mathrm{d}x\,\right|
&\le\left|\,\int_1^\pi\frac{\cos(x)}{x}\,\mathrm{d}x\,\right|+\sum_{k=1}^\infty\left|\,\int_{(2k-1)\pi}^{(2k+1)\pi}\frac{\cos(x)}{x}\,\mathrm{d}x\,\right|\\
&=\left|\,\int_1^\pi\frac{\cos(x)}{x}\,\mathr... |
135,159 | <p>Slow morning.
Can someone help me figure it out? I have a feeling it is trivially easy and not worthy of a thread.
$$
3^{n+1} + 3^n = 4\cdot3^n
$$</p>
<p>Thanks.</p>
| Community | -1 | <p><strong>Answers this version of the question</strong></p>
<p>The previous version of the question claimed that for all <span class="math-container">$n \in \Bbb N$</span>, <span class="math-container">$$3^{n+1}+3n=4\cdot 3n \tag{1}$$</span></p>
<p>However <span class="math-container">$(1)$</span> is not true for all ... |
4,469,583 | <p>How can I construct/define arbitrary semi-computable (but not computable) sets?</p>
<p>Recall that a set A is semi-computable if it is domain of a computable function f. Recall also that a set A is computable if and only if both A and the complement set (A<sup>c</sup>) are semi-computable.</p>
<p>In particular, I am... | Wuestenfux | 417,848 | <p>Let <span class="math-container">$\phi_0,\phi_1,\phi_2,\ldots$</span> be an enumeration of all <span class="math-container">$n$</span>-ary partial recursive functions.</p>
<p>It is well-known that the set <span class="math-container">$K=\{x\mid x\in dom\,\phi_x\}$</span> is semi-computable. Moreover, the set <span c... |
1,904,767 | <p>I'm trying to understand regularization in machine learning. one way of regularization is adding a l1 norm to the error function. This is said to produce sparsity. But I can't understand.</p>
<p>sparsity is defined as "only few out of all parameters are non-zero". But if you look at the l1 norm equation, it is the ... | Robert Israel | 8,508 | <p>Perhaps this is discussing a situation where the possible parameter values are a discrete set. If every nonzero parameter has absolute value at least $\epsilon$, the number of nonzero parameters is at most $1/\epsilon$ times the $\ell^1$ norm of the parameter vector.</p>
|
70,728 | <p>I've started taking an <a href="http://www.ml-class.org/" rel="noreferrer">online machine learning class</a>, and the first learning algorithm that we are going to be using is a form of linear regression using gradient descent. I don't have much of a background in high level math, but here is what I understand so fa... | Christian Sykes | 322,386 | <p>Despite the popularity of the top answer, it has some <strong>major</strong> errors. The most fundamental problem is that $g(f^{(i)}(\theta_0, \theta_1))$ isn't even <em>defined</em>, much less equal to the original function. The focus on the <a href="https://en.wikipedia.org/wiki/Chain_rule" rel="noreferrer">chain ... |
2,702,060 | <p>A little confusion on my part. Study of multi variable calculus and we are using the formula for length of a parameterized curve. The equation makes intuitive sense and I can work it OK. But I also recall using the same integral with out the parameterizing to find the length of a curve where the first term of the sq... | Arian | 172,588 | <p>For $n$ even i.e. $n=2m$ for some $m\in\mathbb{N}$. Let $S:=1+2+...+2m$ then</p>
<p>$$2S=S+S=(1+2+...+(2m-1)+2m)+(2m+(2m-1)+...+2+1)=((1+2m)+(2+(2m-1))+...+((2m-1)+2)+(2m+1))=\underbrace{((2m+1)+...+(2m+1))}_{2m-times}=(2m+1)\cdot 2m$$
Therefore $$S=\frac{(2m+1)2m}{2}=m(2m+1)$$
Analogue for $n$ odd. The formula is ... |
2,702,060 | <p>A little confusion on my part. Study of multi variable calculus and we are using the formula for length of a parameterized curve. The equation makes intuitive sense and I can work it OK. But I also recall using the same integral with out the parameterizing to find the length of a curve where the first term of the sq... | user | 505,767 | <p>This classical result can be also easily proved by the following trick</p>
<p><a href="https://i.stack.imgur.com/3mlDm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3mlDm.png" alt="enter image description here"></a></p>
<p>An extended discussion also for more general cases here <a href="https:... |
9,111 | <p>What function can I use to evaluate $(x+y)^2$ to $x^2 + 2xy + y^2$? </p>
<p>I want to evaluate It and I've tried to use the most obvious way: simply typing and evaluating $(x+y)^2$, But it gives me only $(x+y)^2$ as output. I've been searching for it in the last minutes but I still got no clue, can you help me?</p>... | image_doctor | 776 | <p>You might also try:</p>
<pre><code>Apart[(x + y)^2]
</code></pre>
<blockquote>
<p>x^2 + 2 x y + y^2</p>
</blockquote>
|
4,014,554 | <p>A simple heuristic of the first million primes shows that no prime number can be bigger than the sum of adding the previous twin primes.</p>
<p>Massive update:
@mathlove made a comment that leaves me completely embarrassed. <span class="math-container">$13 > 7 + 5$</span> I don’t know how I missed it and I deeply... | Barry Cipra | 86,747 | <p>Let's modify the OP's observation as follows:</p>
<blockquote>
<p>Let <span class="math-container">$(p,p+2)$</span> and <span class="math-container">$(q,q+2)$</span> be <em>consecutive</em> pairs of twin primes,
e.g., <span class="math-container">$(107,109)$</span> and <span class="math-container">$(137,139)$</span>... |
644,057 | <p>I am having trouble with this problem:</p>
<p>Let $a_n$ be sequence of positive terms with $$\frac{a_{n+1}}{a_n}\lt \frac{n^2}{(n+1)^2}.$$
Then is the series $\sum a_n$ convergent?</p>
<p>Thanks for any help.</p>
| Yiorgos S. Smyrlis | 57,021 | <p>Note that
$$
a_n=a_1\prod_{k=1}^{n-1}\frac{a_{k+1}}{a_k}\le a_1\prod_{k=1}^{n-1}\frac{k^2}{(k+1)^2}=\frac{a_1}{n^2},
$$
and as the series $\displaystyle\sum_{n=1}^\infty\frac{1}{n^2}$ converges so does the series $\displaystyle\sum_{n=1}^\infty a_n$, as a consequence of the comparison test.</p>
|
2,336,988 | <blockquote>
<p>Let $a,b,c>0 ,2b+2c-a\ge 0,2c+2a-b\ge 0,2a+2b-c\ge 0$ show that
$$\sqrt{\dfrac{2b+2c}{a}-1}+\sqrt{\dfrac{2c+2a}{b}-1}+\sqrt{\dfrac{2a+2b}{c}-1}\ge 3\sqrt{3}$$</p>
</blockquote>
<p>I try use AM-GM and Cauchy-Schwarz inequality and from here I don't see what to do</p>
| Michael Rozenberg | 190,319 | <p>By AM-GM
$$\sum_{cyc}\sqrt{\frac{2b+2c}{a}-1}=\sum_{cyc}\frac{2\sqrt3(2b+2c-a)}{2\sqrt{3a(2b+2c-a)}}\geq$$
$$\geq\sum_{cyc}\frac{2\sqrt3(2b+2c-a)}{3a+2b+2c-a}=\sum_{cyc}\frac{2\sqrt3(2b+2c-a)}{2(a+b+c)}=3\sqrt3$$</p>
|
2,379,405 | <blockquote>
<p>Determine convergence or divergence of $$ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx$$</p>
</blockquote>
<p>As the graph of the function suggests convergence, Let's find an upper bound that converges.</p>
<p>$$ \int_0^{\infty} \frac{1 + \cos^2x}{\sqrt{1+x^2}} dx \leq
\int_0^{\infty} \frac{2... | user0102 | 322,814 | <p>Here it is an alternative approach:
\begin{align*}
\int_{0}^{\infty}\frac{1+\cos^{2}(x)}{\sqrt{1+x^{2}}}dx \geq \int_{0}^{\infty}\frac{dx}{\sqrt{1+x^{2}}} = \int_{0}^{\infty}\frac{\cosh(u)}{\sqrt{1+\sinh^{2}(u)}}du = \int_{0}^{\infty}du = +\infty
\end{align*}</p>
|
149,872 | <p>How would I show that $|\sin(x+iy)|^2=\sin^2x+\sinh^2y$? </p>
<p>Im not sure how to begin, does it involve using $\sinh z=\frac{e^{z}-e^{-z}}{2}$ and $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$?</p>
| Javier | 2,757 | <p>If you use the sine addition formula, the pythagorean identities, and the fact that $\sin(ix)=i\sinh (x)$ and $\cos(ix) = \cosh(x)$, then you get this:</p>
<p>$$
\begin{align}
\sin(x+iy) &= \sin x \cos (iy)+\cos x \sin(iy) \\
&= \sin x \cosh y + i \cos x \sinh y
\end{align}
$$</p>
<p>$$
\begin{align}
|\sin... |
1,610,055 | <p>I feel rather silly for having to ask this question in specific and am by no means looking for a flat out step by step answer. I understand the definition for the euclidean norm in an n-dimensional space (as defined <a href="https://en.wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm" rel="nofollow">here</a>). I... | Nicolás Giossa | 303,222 | <p>First observe that $\ \|x-z\| < 2$ and $\ \|y-z\| < 3 \Rightarrow \ \|x-z\| + \ \|y-z\| < 5$. </p>
<p>So, using triangle inequality, we have $\ \|x-y\| = \ \|x-z+z-y\| \leq \ \|x-z\| + \ \|y-z\| < 5 $.</p>
|
2,436,167 | <p>I appear to be misunderstanding a basic probability concept. The question is: you flip four coins. At least two are tails. What is the probability that exactly three are tails? </p>
<p>I know the answer isn't 1/2, but I don't know why that's so. Isn't the probability of just getting 1 tail in the remaining two coin... | RideTheWavelet | 394,393 | <p>I believe the source of your confusion is arising from the fact that the specific tosses on which the two tails took place are not specified. So if we compute the conditional probability which is being described, letting $A$ be the event that we get at least $2$ tails, and $B$ be the event that we get exactly $3$ ta... |
3,121,103 | <p>The integral surface of the first order partial differential equation <span class="math-container">$$2y(z-3)\frac{\partial z}{\partial x}+(2x-z)\frac{\partial z}{\partial y} = y(2x-3)$$</span> passing through the curve <span class="math-container">$x^2+y^2=2x, z = 0$</span> is</p>
<ol>
<li><span class="math-contain... | Aleksas Domarkas | 562,074 | <p>All surfaces passing through the curve <span class="math-container">$x^2+y^2=2x, z = 0$</span>. Bat only
first surface <span class="math-container">$$x^2+y^2-z^2-2x+4z=0\tag{1}$$</span> is solution of pde. We check it</p>
<p>Let <span class="math-container">$z=z(x,y)$</span>. We differentiate equation <span class=... |
2,524,487 | <p>I can not find a way to prove that the abelian group ($\mathbb{Q}_{>0}$,*) is a free abelian group with countable basis. Is is even true?</p>
| lisyarus | 135,314 | <p>By the fundamental theorem of arithmetic, any natural number can be expressed as $p_1^{k_1}\cdot \dots \cdot p_n^{k_n}$, where $p_i$ are primes and $k_i \in \mathbb{N}$.</p>
<p>Extending this to positive rationals, any positive rational can be expressed in the form $p_1^{k_1}\cdot \dots \cdot p_n^{k_n}$, where $p_i... |
1,619,292 | <p>Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. </p>
<p>Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ and put $Y_i:=p_i(Y)$ where $p_i:\bigoplus_{i\in I}X_i\longrightarrow X$
is the obvious projection. </p>
<p>I... | matt | 118,474 | <p>Here are my suggestions:</p>
<ul>
<li>Stephen Strogatz <b>Nonlinear Dynamics and Chaos</b></li>
<li>Online SOOC <b>chaosbook.org</b></li>
<li>Edward Ott <b>Chaos in Dynamical Systems</b></li>
</ul>
<p>If you know nothing about nonlinear dynamics, then Strogatz is the best place to start. If you want to jump straig... |
1,346,073 | <p>$$100\frac{dy^2}{dx^2} + y = 0$$</p>
<p>Is this worked out by using the auxillary equation such that:</p>
<p>$$100m^2 + 1 = 0$$</p>
<p>so $m = \pm i\sqrt{1/100}$ ?</p>
<p>So the general solution would be
$y(x) = A cos (1/10) + B sin(1/10)$?</p>
<p>I am not sure if I've gone about this the right way.</p>
| orangeskid | 168,051 | <p>Also see <a href="http://www.jstor.org/stable/2372705?" rel="nofollow">Horn's theorem</a>. It says that two real $n$ vectors $\bf{x}$, $\bf{y}$ are the eigenvalues and the diagonal of a (real) symmetric matrix if
and only if $\bf{x}\succ \bf{y}$ in the <a href="https://en.wikipedia.org/wiki/Majorization" rel="nofol... |
4,520,506 | <p>I know that <span class="math-container">$ x \gt 0 $</span> because of logarithm precondition, and I can see that <span class="math-container">$ x \neq 1 $</span> because otherwise it would lead to <span class="math-container">$ 0^0$</span> which is problematic, but when I checked the graph of the function I have di... | José Carlos Santos | 446,262 | <p>The expression <span class="math-container">$(k+1)(k+2)\ldots n$</span> is the product of the numbers <span class="math-container">$k+1$</span>, <span class="math-container">$k+2$</span>, …, <span class="math-container">$n$</span>. When <span class="math-container">$n=3$</span> and <span class="math-container">$k=2$... |
2,179,289 | <p>Every valuation ring is an integrally closed local domain, and the integral closure of a local ring is the intersection of all valuation rings containing it. It would be useful for me to know when integrally closed local domains are valuation rings.</p>
<p>To be more specific,</p>
<blockquote>
<p>is there a prop... | Badam Baplan | 164,860 | <p>A very late answer, but I think there are some other connections and references that merit mention and maybe will interest someone down the line.</p>
<p>As mentioned by M. Zafrullah, in the presence of the local condition we can reduce the search to a property making integrally closed domains Prüfer.</p>
<p>The firs... |
276,987 | <p>I want to visualize the following set in Maple:</p>
<blockquote>
<p>$\lbrace (x+y,x-y) \vert (x,y)\in (-\frac{1}{2},\frac{1}{2})^{2} \rbrace$ </p>
</blockquote>
<p>Which commands should I use? Is it even possible?</p>
| Michael Greinecker | 21,674 | <p>Let $X$ be a set with at leas two elements $x$ and $y$. Let the topology consist of arbitrary unions of singletons from $X\backslash \{y\}$ and the doubleton $\{x,y\}$. The result is a fairly boring topology such that any finer topology is discrete.</p>
|
1,618,411 | <p>I'm learning the fundamentals of <em>discrete mathematics</em>, and I have been requested to solve this problem:</p>
<p>According to the set of natural numbers</p>
<p>$$
\mathbb{N} = {0, 1, 2, 3, ...}
$$</p>
<p>write a definition for the less than relation.</p>
<p>I wrote this:</p>
<p>$a < b$ if $a + 1 <... | Andres Mejia | 297,998 | <p>$a<b \iff \exists p \in \mathbb{N_{>0}}$: $b=a+p$.</p>
|
1,618,411 | <p>I'm learning the fundamentals of <em>discrete mathematics</em>, and I have been requested to solve this problem:</p>
<p>According to the set of natural numbers</p>
<p>$$
\mathbb{N} = {0, 1, 2, 3, ...}
$$</p>
<p>write a definition for the less than relation.</p>
<p>I wrote this:</p>
<p>$a < b$ if $a + 1 <... | fleablood | 280,126 | <p>You can either have a direct definition or a recursive definition. If you have a recursive definition you need a base case from which all cases arrive.</p>
<p>Your function appears to be recursive but it has no base case.</p>
<p>a < b if a + 1< b + 1 which raises the question what is the definition of a + 1... |
4,403,081 | <p><span class="math-container">$$\int \dfrac{dx}{x\sqrt{x^4-1}}$$</span></p>
<p>I need to solve this integration.
I solved and got <span class="math-container">$\dfrac12\tan^{-1}(\sqrt{x^4-1}) + C$</span>, however the answer given in my textbook is <span class="math-container">$\dfrac12\sec^{-1}(x^2) + C$</span></p>
<... | Zaragosa | 691,503 | <p>The answer was already given by B. Goddard, here I try to give a visual answer:</p>
<p><a href="https://i.stack.imgur.com/jQJGX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jQJGX.png" alt="enter image description here" /></a></p>
|
4,403,081 | <p><span class="math-container">$$\int \dfrac{dx}{x\sqrt{x^4-1}}$$</span></p>
<p>I need to solve this integration.
I solved and got <span class="math-container">$\dfrac12\tan^{-1}(\sqrt{x^4-1}) + C$</span>, however the answer given in my textbook is <span class="math-container">$\dfrac12\sec^{-1}(x^2) + C$</span></p>
<... | tryingtobeastoic | 541,298 | <p>There is nothing wrong with your answer. The antiderivative you found is <a href="https://www.integral-calculator.com/#expr=%5Cfrac%7B1%7D%7Bx%5Csqrt%7Bx%5E4-1%7D%7D" rel="nofollow noreferrer">correct</a>.</p>
<p>Your book's answer is also <a href="https://www.integral-calculator.com/#expr=%5Cfrac%7B1%7D%7Bx%5Csqrt%... |
1,046,066 | <p>Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent?</p>
<p>My attempt was to use the comparison test, but I'm stuck at finding the behaviour of $\displaystyle \prod_1^n k^k$ as $n$ goes to infinity. Thanks in advance.</p>
| Milly | 182,459 | <p>Hint:
$$\log\left(\prod_{k=1}^n k^k\right) = \sum_{k=1}^n k\log k\cdots $$</p>
|
1,296,420 | <p>I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?</p>
| wlad | 228,274 | <p>Let $G = \mathbb Z ^ \mathbb N$ (with pointwise addition as the product). Then let $f:G \times G \longrightarrow G$ be $$f(g,h)(n) = \begin{cases} g(k), &n = 2k \\ h(k), &n = 2k+1 \end{cases}$$
You can verify $f$ is an isomorphism.</p>
|
1,296,420 | <p>I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?</p>
| Cameron Buie | 28,900 | <p>Let $G$ be the trivial group, for the only finite example.</p>
|
3,986,831 | <p>the question is:
true or false: if <span class="math-container">$f_n(x)'$</span> converges uniformly to <span class="math-container">$f(x)'$</span> then <span class="math-container">$f_n(x)$</span> converges uniformly to <span class="math-container">$f(x)$</span>. I tried many examples and they all confirmed the sta... | Yuval Peres | 360,408 | <p>True on a finite interval as long as convergence holds at one point. Simply integrate.</p>
|
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