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 |
|---|---|---|---|---|
8,193 | <p><strong>NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can be found above.</strong></p>
<p>As a foreigner who has spent some time in the US, it seems to me that in the U... | Bookwyrm | 8,163 | <p>Kenny LJ failed to say when he took those tests. As another person pointed out, some of the ones mentioned were possibly less subject to cultural bias in the first place but bias really surfaced as an issue in the 60's and 70's (1974 episode “The IQ Test” of the sitcom Good Times and appeared in scholarly journals ... |
2,136,734 | <p>When we divide $n^5+5$ by $n+5$, we get a remainder $-620$, i.e.,
$n^5+5=K(n+5)-620$ , now how to proceed further?</p>
| Jim Belk | 1,726 | <p>Let $G=\pi_1(X,x_0)$ and $H=\pi_1(Y,y_0)$. Since $[\pi_1(X),\pi_1(Y)]$ is a subgroup of $\pi_1(X) * \pi_1(Y)$, it is the fundamental group of a certain cover $Z$ of $X\lor Y$, with the quotient $\pi_1(X)\times \pi_1(Y)$ acting on $Z$ by Deck transformations.</p>
<p>We can describe $Z$ as follows. Let $p\colon \b... |
1,432,341 | <p>I am tasked with finding the region of the complex plane under condition:
$$\left|\frac{z-2i}{z+2}\right|\ge 1$$
I can then calculate that
$|z-2i|\ge|z+2|$. Thus, I can say I'm looking for the region where the distance from $z$ to $2i$ is greater than the distance from $z$ to $-2$. Imagining the plane, I feel as t... | null | 207,462 | <p>Given that only the absolute value is of interest</p>
<p>$$\color{red}{\left|\color{black}{\frac{z-2i}{z+2}}\right|}\ge 1$$</p>
<p>and it is of a complex number that is found by division of two other complex numbers</p>
<p>$$\left|\color{red}{\frac{\color{black}{z-2i}}{\color{black}{z+2}}}\right|\ge 1$$</p>
<p>i... |
1,921,914 | <p>How do you find the sum of $\sum \limits_{i=0}^{n-1}(1+i) $ ?</p>
<p>Actually, I am especially confused because of of the n-1. Usually, I'd start with stuff like:
$$\sum \limits_{i=0}^{0}(1+i) = ?$$
$$\sum \limits_{i=0}^{1}(1+i) = ?$$
$$\sum \limits_{i=0}^{2}(1+i) = ?$$</p>
<p>But I don't know what to do with the... | C. Falcon | 285,416 | <p><strong>Hints.</strong></p>
<ul>
<li><p>You can split the sum and you are left to compute: $$\sum_{k=0}^{n-1}1+\sum_{k=0}^{n-1}k.$$</p></li>
<li><p>The last sum can be tricky, I recommend you to define:
$$f(x)=\sum_{k=0}^{n-1}x^k.$$
Notice that the sum you are looking for is equal to $f'(1)$ and that $f(x)=\frac{x... |
3,909,307 | <p>I'm having trouble with this question. I tried to make a relation between x, y, and z using the points given, and I came up with <span class="math-container">$x=z=-y+1$</span>, which led to trying to solve the integral</p>
<p><span class="math-container">$$\int_{0}^{1}(y^2+y)\sqrt{1+(2y+1)^2}dy$$</span></p>
<p>But I... | user1551 | 1,551 | <p>This is usually proved by using <a href="https://en.wikipedia.org/wiki/Matrix_congruence" rel="nofollow noreferrer">matrix congruence</a>, but yes, you can prove the statement by using the adjugate matrix.</p>
<p>As <span class="math-container">$A$</span> has rank <span class="math-container">$n-1$</span>, it adjuga... |
2,967,291 | <p><a href="https://i.stack.imgur.com/VA5AP.png" rel="nofollow noreferrer">plot</a></p>
<p>I am trying to prove an inequality connecting the function <span class="math-container">$f(x) = \log(1+x^2)$</span> and the absolute value of its derivative i.e <span class="math-container">$ |f'(x)| =\big|\frac{(2x)}{(1+x^2)}... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Recall that we have </p>
<p><span class="math-container">$$\log(1 +x^2)< x^2$$</span></p>
<p>therefore it suffices to prove that for <span class="math-container">$x$</span> positive sufficiently small</p>
<p><span class="math-container">$$x^2<\frac{2x}{1+x^2} \iff x<\frac{2}... |
2,967,291 | <p><a href="https://i.stack.imgur.com/VA5AP.png" rel="nofollow noreferrer">plot</a></p>
<p>I am trying to prove an inequality connecting the function <span class="math-container">$f(x) = \log(1+x^2)$</span> and the absolute value of its derivative i.e <span class="math-container">$ |f'(x)| =\big|\frac{(2x)}{(1+x^2)}... | 5xum | 112,884 | <p>Clearly, both functions are even, so we can safely prove your point for positive values of <span class="math-container">$x$</span> only. We need to prove that there exists some <span class="math-container">$\epsilon >0$</span> such that <span class="math-container">$$\log(1+x^2) < \left|\frac{2x}{1+x^2}\right|... |
101,838 | <p>I am trying to factor the following polynomial: $$ 8x^3 -4x^2y -18xy^2 + 9y^3 $$</p>
<p>$$ (a-b)^3 = a^3 -3a^2b + 3ab^2 - b^3 $$
Thanks</p>
| Community | -1 | <p>Look at the following factorisation, I thought of:</p>
<p>$$\begin{align*}8x^3-4x^2y-18xy^2+9y^3&=4x^2(2x-y)-9y^2(2x-y)\\&=(4x^2-9y^2)(2x-y)\\&=(2x+3y)(2x-3y)(2x-y)\end{align*}$$</p>
<p>I also want to add that, it is natural to think of the <em>cubic identity</em> you gave us, but $9y^3$ doesn't look g... |
1,243,159 | <p>I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me. </p>
<p>Find the $3\times3$ matrix which represents a rotation clockwise through $43°$ about the point $(\frac{1}{2},1+\frac{8}{10})$</p>
<p>For example: if the rotation angle is $66°$ then th... | user1936752 | 99,503 | <p>You need three steps. </p>
<p>1) First shift your points by $(x_0,y_0)$ so that your rotation point is now at the origin. That is, for a given point $(x,y)$, we have shifted points $(x_s,y_s)=(x-x_0,y-y_0)$</p>
<p>2) Then do the rotation about zero which is a trivial. In this case we want clockwise by $\theta$</p>... |
1,911,037 | <p>So I realized that I have to prove it with the fact that $(x-y)^2+2xy=x^2+y^2$ </p>
<p>So $\frac{(x+y)^2}{xy}+2=\frac{x}{y}+\frac{y}{x}$ $\Leftrightarrow$ $\frac{(x+y)^2}{xy}=\frac{x}{y}+\frac{y}{x}-2$ </p>
<p>Due to the fact that $(x+y)^2$ is a square, it will be positive </p>
<p>$x>0$ and $y>0$ so $xy>... | haqnatural | 247,767 | <p>With AM-GM $$\frac { x }{ y } +\frac { y }{ x } \geq 2\sqrt { \frac { x }{ y } \frac { y }{ x } } = 2$$</p>
|
2,033,832 | <p>I am a math enthusiast in electrical engineering and I am planning on learning Differential Geometry for applications in Control Theory. I want to teach myself this beautiful branch of mathematics in a rigorous way.</p>
<p>I am currently going through Chapman Pugh's Real Analysis, I am then planning on studying Mun... | Fraïssé | 105,951 | <p>While the books in the comments and from the other answers are well and good, I would recommend that you also try out reading differential geometry text specifically written with engineering application in mind by an electrical engineer.</p>
<p>The book I am recommending is <strong>Shankar Sastry's Nonlinear System... |
3,087,207 | <p>In "Relational Algebra by Way of Adjunctions," found at <a href="http://www.cs.ox.ac.uk/jeremy.gibbons/" rel="nofollow noreferrer">author's page</a> (<a href="http://dx.doi.org/10.1145/3236781" rel="nofollow noreferrer">doi</a>), section 2.4, an adjunction is described using the signature:</p>
<p><span class="math-... | Community | -1 | <p>The origin of <em>tangent</em> is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point <span class="math-container">$(1,0)$</span>.</p>
|
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Andrew D. Hwang | 86,418 | <p>To make commenters' points explicit, the "degrees-mode trig functions" functions $\cos^\circ$ and $\sin^\circ$ satisfy the awkward identities
$$
(\cos^\circ)' = -\frac{\pi}{180} \sin^\circ,\qquad
(\sin^\circ)' = \frac{\pi}{180} \cos^\circ,
$$
with all that implies about every formula involving the derivative or anti... |
720,924 | <p>I think this is just something I've grown used to but can't remember any proof.</p>
<p>When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?</p>
| Michael Hardy | 11,667 | <p>\begin{align}
\lim_{x\to0} \frac {\sin x} x & = (\text{some constant that is not 0}) \\[12pt]
\frac d {dx} \sin x & = \Big( (\text{some constant that is not 0}) \cdot \cos x \Big)
\end{align}</p>
<p>The "constants" are equal to $1$ if, but only if, radians are used.</p>
|
2,092,814 | <p>How to prove the following series,
$$\sum_{k=1}^{\infty }\frac{\left ( -1 \right )^{k}}{k}\sum_{j=1}^{2k}\frac{\left ( -1 \right )^{j}}{j}=\frac{\pi ^{2}}{48}+\frac{1}{4}\ln^22$$
I know a formula which might be usful.
$$\sum_{j=1}^{n}\frac{\left ( -1 \right )^{j-1}}{j}=\ln 2+\left ( -1 \right )^{n-1}\int_{0}^{1}\fra... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
578,492 | <blockquote>
<p>How can one prove that every element $x$ of a finitely generated local commutative algebra $A$ with identity over an algebraically closed field $K$ is unit or nilpotent? </p>
</blockquote>
<p>Of course, this is equivalent to the statement that in the local algebra every prime ideal is maximal. But I ... | Tomasz Lenarcik | 100,562 | <p>This is much more complicated than the other answer, but since it gives some more insight of what is really going on here I decided to post it anyway.</p>
<p>Your statement is basically equivalent to the fact that $A$ is a zero dimensional ring, or in other words every prime ideal of $A$ is maximal. This is because... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | Chris Godsil | 1,266 | <p>(This is just an overlong comment.)</p>
<p>A basic problem is that the complete bipartite graphs $K_{1,ab}$ and $K_{a,b}$ have the same spectral radius, and these graphs would not usually be viewed as similar. And, of course, all $k$-regular bipartite graphs have the same spectral radius. </p>
<p>Also if $A$ is th... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | Kimball | 6,518 | <p>Lovasz seems to think the largest eigenvalue is not so interesting (if the graph is connected), but the first gap tells you more.</p>
<p><a href="http://www.cs.elte.hu/~lovasz/eigenvals-x.pdf" rel="noreferrer">www.cs.elte.hu/~lovasz/eigenvals-x.pdf</a></p>
<p>I believe this contains the double inequality you menti... |
63,589 | <p>A big-picture question: what "physical properties" of a graph, and in particular of a bipartite graph, are encoded by its largest eigenvalue? If $U$ and $V$ are the partite sets of the graph, with the corresponding degree sequences $d_U$ and $d_V$, then it is easy to see that the largest eigenvalue $\lambda_{\max}$ ... | Balazs | 6,107 | <p>I am not at all an expert on this, but I believe that the largest eigenvalue plays a role in several graph processes that people working in "network science" are interested in. A google search for "largest eigenvalue network" brought up <a href="http://arxiv.org/abs/0705.4503" rel="nofollow">this paper</a> whose abs... |
361,747 | <p>Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space <span class="math-container">$l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} \sum_{x\in \mathbb{Z}^{d}}|\psi(x)|^{2}<\infty\}$</span> with inner product <span cl... | Abdelmalek Abdesselam | 7,410 | <p>The source of the confusion is <em>not saying</em> explicitly what are the sets and <span class="math-container">$\sigma$</span>-algebras the measures are supposed to be on. For example, a sentence like ''By Kolmogorov's Extension Theorem, there exists a Gaussian measure <span class="math-container">$\nu_C$</span> w... |
1,379,456 | <p>i am help</p>
<p>Calculate:</p>
<p>$$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$</p>
<p>PD : use $(1+x)^{16}$ and binomio newton</p>
| David Quinn | 187,299 | <p>Put $x=i$ and find the real part of $(1+i)^{16}$</p>
<p>Can you finish this?</p>
|
1,379,456 | <p>i am help</p>
<p>Calculate:</p>
<p>$$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$</p>
<p>PD : use $(1+x)^{16}$ and binomio newton</p>
| Ángel Mario Gallegos | 67,622 | <p><strong>Alternative way:</strong></p>
<p>Note
\begin{align*}
(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})&=\text{Re}\left((1+i)^{16}\right)\\
&=\text{Re}\left((2i)^{8}\right)\\
&=\text{Re}\left((-4)^{4}\right)\\
\end{align*}</p>
|
648,809 | <p>I have a matrix $A$ given and I want to find the matrix $B$ which is closest to $A$ in the frobenius norm and is positiv definite. $B$ does not need to be symmetric.</p>
<p>I found a lot of solutions if the input matrix $A$ is symmetric. Are they any for a non-symmetric matrix $A$? Is it possible to rewrite the pro... | SriG | 258,119 | <p>I think this is a direct way to compute the closest psd matrix without using numerical optimization.</p>
<ol>
<li><p>Find the closest symmetric psd matrix (call it $S$) to $A$ as follows (see the proof of Theorem 2.1 in <a href="http://www.sciencedirect.com/science/article/pii/0024379588902236" rel="noreferrer">Hig... |
394,101 | <p>I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:</p>
<p><em>Has someone already tried this? If not, who in t... | Olaf Teschke | 100,979 | <p>Imho, mathematics has already a good culture and a relatively rich informational ecosystem to detect, document, and disseminate these issues (many options are mentioned above); what might be lacking is a kind of (semi)automated system of interlinked communication that helps spreading the information further.</p>
<p>... |
1,971,645 | <p>I tried doing this problem two ways. I am unable to get the solutions to match each other. Is one of them incorrect?</p>
<p><a href="https://i.stack.imgur.com/5vHwDxx.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5vHwDxx.jpg" alt="enter image description here"></a></p>
| B. Goddard | 362,009 | <p>On your 3rd line in the first column, $x-y$ should be $xy$.</p>
|
4,508,474 | <p>I would like to solve the ODE
<span class="math-container">$$\left(x-k_1\right)\frac{dY}{dx} + \frac{1}{a}x^2\frac{d^2 Y}{d x^2}-Y+k_2=0$$</span>
with boundary conditions <span class="math-container">$Y\left(k_3\right)=0$</span> and <span class="math-container">$\frac{d Y}{d x}\left(\infty\right)=1-b$</span> (meanin... | Tanner Swett | 13,524 | <blockquote>
<p>According to Bézout's Theorem, two polynomials of degree <span class="math-container">$m,n$</span> intersect at most at <span class="math-container">$mn$</span> points. So, two circles should intersect at most at <span class="math-container">$4$</span> points as well. But I have so far known that 2 circ... |
847 | <p>Is there any mathematical significance to the fact that the law of cosines...</p>
<p>$$
\cos(\textrm{angle between }a\textrm{ and }b) = \frac{a^2 + b^2 - c^2}{2ab}
$$</p>
<p>... for an impossible triangle yields a cosine $< -1$ (when $c > a+b$), or $> 1$ (when $c < \left|a-b\right|$)</p>
<p>For exampl... | Isaac | 72 | <p>For some a, b, and c that form a triangle: increasing the length of c increases the measure of angle C and as m∠C approaches 180°, cos C approaches -1; decreasing the length of c increases the measure of angle C and as m∠C approaches 0°, cos C approaches 1. Extending this pattern, it makes sense tha... |
847 | <p>Is there any mathematical significance to the fact that the law of cosines...</p>
<p>$$
\cos(\textrm{angle between }a\textrm{ and }b) = \frac{a^2 + b^2 - c^2}{2ab}
$$</p>
<p>... for an impossible triangle yields a cosine $< -1$ (when $c > a+b$), or $> 1$ (when $c < \left|a-b\right|$)</p>
<p>For exampl... | T.. | 467 | <p>This is not a directly a matter of hyperbolic geometry but of <em>complex Euclidean geometry</em>. The construction of "impossible" triangles is the same as the construction of square roots of negative numbers, when considering the coordinates the vertices of those triangles must have. If you calculate the coordin... |
2,853,673 | <p>I came across this as one of the shortcuts in my textbook without any proof.<br>
When $b\gt a$, </p>
<blockquote>
<p>$$\int\limits_a^b \dfrac{dx}{\sqrt{(x-a)(b-x)}}=\pi$$</p>
</blockquote>
<hr>
<p><strong>My attempt :</strong></p>
<p>I notice that the the denominator is $0$ at both the bounds. I thought of su... | Atmos | 516,446 | <p>It's called an Abel Integral ( at least in my language ). You can write that
$$
\frac{1}{\sqrt{\left(x-a\right)\left(b-x\right)}}=\frac{2}{a-b}\frac{1}{\sqrt{1-\left(\frac{2}{a-b}\left(x-\frac{b+a}{2}\right)\right)^2}}$$</p>
<p>that goes into arcsinus</p>
<blockquote>
<p>$$\int_{a}^{b}\frac{\text{d}x}{\sqrt{\lef... |
1,690 | <p>I like to ask true-false questions on exams, because I feel that they can be an efficient way to assess students' understanding of concepts and ability to apply them to somewhat unfamiliar situations. In general, I'm very happy with true-false questions, but there is one annoyance that I have never figured out how ... | TheBluegrassMathematician | 937 | <p>I would definitely say that there needs to be some part of the question asking justification. The sole purpose of the test is to see if they have sufficiently learnt the material and giving the ability to just plug and chug or even guess isn't doing mathematics. Also if the student is like me they may have made prog... |
1,139,847 | <p>I am trying to solve a differential equation, but I can't solve an integral, because I forgot which rule to apply. What rule do I use to do $$\int \frac{1}{3y-y^2}\mathrm dy\ ?$$</p>
| rubik | 2,582 | <p><strong>Hint</strong>: Make use of the fact that
$$\frac1{3y - y^2} = \frac1{3y} - \frac1{3(y - 3)}$$
by partial fractions. Then use the linearity of the integral.</p>
|
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | AsukaMinato | 68,689 | <p>A trivial way:</p>
<ol>
<li><p>select those data for which the first number of the four is less than 2</p>
</li>
<li><p>keep the counting lone integers</p>
</li>
<li><p>keep only the last three columns of the sets.</p>
</li>
</ol>
<p>translate:</p>
<pre><code>f = Which[Length @ # === 1, Sow @ #;, (* 2 *)
#... |
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | SquareOne | 19,960 | <p>Don't forget the functional way (which might be more useful/readable when things get more complicated)</p>
<pre><code>foo[x : {_}] := x
foo[{x_, b__}] := {b} /; x < 2
foo[_] := Nothing
</code></pre>
<p>Then</p>
<pre><code>Map[foo,data]// Column
</code></pre>
<blockquote>
<pre><code>{1}
{-12.7758,-0.00980164,0.000... |
278,528 | <p>Let's suppose we have the following structure of data</p>
<pre><code>data = {{1}, {0.0109, -12.7758, -0.00980164, 0.00032368},
{1.0218, -12.7764, -0.00948724, 0.00064337},
{2.0327, -12.7772, -0.00905215, 0.00095516},
{2}, {0.0109, -12.7758, -0.00980164, 0.00032368},
... | Asadullah Khaki | 89,972 | <p>You can use a combination of list comprehension and slicing to achieve your desired outcome.
Here's one way you could write the code to produce the data2 list:</p>
<p>data2 = [[i] + [sublist[1:4] for sublist in data if sublist[0] < 2 and sublist[0] == int(sublist[0])] for i in range(1,6)]
This code uses nested li... |
48,571 | <p>Hi,</p>
<p>Let $R$ be equipped with the usual Borel structure. Let $F$ be a Borel subset and $E$ be a closed subset of $R$. Then $F+E=(f+e: f\in F, e \in E \)$ is Borel? If yes, is it true for any locally compact topological group? Thanks in advance.</p>
| gowers | 1,459 | <p>At least in $R^2$ it's false, and probably in R too.</p>
<p>There exist closed subsets of $R^2$ that project to non-Borel. So if you take such a set and add it to the y-axis then you'll get a non-Borel set too.</p>
<p>There's probably some general nonsense that would allow you to transfer this result to R, but I d... |
4,033,612 | <blockquote>
<p>Seek separable solutions, <span class="math-container">$u(x,t)=X(x)T(t)$</span> of the equation, <span class="math-container">$u_{tt}=c^{2}\dfrac{1}{x}\dfrac{\partial}{\partial x}\left(x\dfrac{\partial u}{\partial x}\right)$</span>. Find the general solution assuming the separation constant is zero.</p>... | N. S. | 9,176 | <p>Your confusion seems to be the fact that you think you can choose <strong>both</strong> <span class="math-container">$a$</span> and the starting point for <span class="math-container">$n$</span> independently. (you chose <span class="math-container">$a=7$</span> but your <span class="math-container">$n$</span> start... |
4,033,612 | <blockquote>
<p>Seek separable solutions, <span class="math-container">$u(x,t)=X(x)T(t)$</span> of the equation, <span class="math-container">$u_{tt}=c^{2}\dfrac{1}{x}\dfrac{\partial}{\partial x}\left(x\dfrac{\partial u}{\partial x}\right)$</span>. Find the general solution assuming the separation constant is zero.</p>... | David K | 139,123 | <p>I think you have overlooked the fact that if <span class="math-container">$a = 7,$</span> then <span class="math-container">$7$</span> is an integer greater than or equal to <span class="math-container">$a.$</span></p>
<p>If <span class="math-container">$a = 7$</span> and if <span class="math-container">$S$</span> h... |
604,359 | <p>My experience with non commutative rings is limited to 2 by 2 matrices and the quaternions. The first of which is not a domain, and the latter is a division ring. I'm looking for an example of a domain that is not a division ring. </p>
<p>Invertible matrices do not produce an example, as they must be division rings... | rschwieb | 29,335 | <p>Wow, three answers and a comment with the same example, and not even the easiest example, IMO! (It's still a good example, though!)</p>
<blockquote>
<p><em>Similarly, is there an example within the quaternions?</em></p>
</blockquote>
<p>Absolutely! There are lots of nonzero subrings of $\Bbb H$ that would alread... |
2,526,932 | <p>So, I started out with $$f(x)=e^{-x^2}cos(x) \;\;\;at\;\;\; a=0$$ And after finding the Taylor Polynomial $T_3(x)$ for that function, I have $$T_3(x)=1-{{3x^2}\over 2}$$ Now hopefully that is correct. Next, I need to use the Taylor inequality to estimate the accuracy of the approximation $f(x) \approx T_n(x)$ where ... | Ian | 83,396 | <p>Based on the excerpt from your book, the form of the Taylor remainder that you are using is called the Lagrange remainder. It says that </p>
<p>$$f(x)-T_n(x)=\frac{f^{(n+1)}(\xi(x))}{(n+1)!} (x-a)^{n+1}$$</p>
<p>where $\xi(x)$ is an unknown number between $x$ and $a$. (Generally it is also <em>unknowable</em>, in ... |
1,639,241 | <p>I'm using gradient descent with mean squared error as error function to do linear regression. Take a look at the equations first.
<a href="https://i.stack.imgur.com/GN90y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GN90y.png" alt="enter image description here"></a>
As you can see in eq.1, the... | DanielV | 97,045 | <pre><code>Number[] a = new Number[p + 1]; // range from 1 to p
... some stuff to initialize a ...
Number[] ceps = new Number[Q + 1]; // range from 0 to Q
ceps[0] = ln(G);
for (int q = 1; q <= p; q++) {
Number sum = a[q];
for (k = 1; k <= q - 1; k++) {
sum += (k - q) / q * a[k] * ceps[q - k];
}
cep... |
991,038 | <p>Consider a probability space $\left(\Omega,\mathcal{F},P\right)
$, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F}
$. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right)
$ be the space of $\mathcal{F}
$-measurable, square integrable random variable and $L^{\infty}\left(\Omega,\mathcal{G},P\right)
$ b... | Henry Swanson | 55,540 | <p>Your suspicion is correct: you can add $-f(0)$ to both sides, and after applying some ring axioms, the $f(0)$ disappears. You can't do this for multiplication because there aren't necessarily inverses.</p>
<p>As for "why can we add/subtract/multiply/etc things to both sides of the equation", it's just a property of... |
991,038 | <p>Consider a probability space $\left(\Omega,\mathcal{F},P\right)
$, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F}
$. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right)
$ be the space of $\mathcal{F}
$-measurable, square integrable random variable and $L^{\infty}\left(\Omega,\mathcal{G},P\right)
$ b... | Mike Earnest | 177,399 | <p>The reason you can add something to both sides of an equation and have it still be true is a special case of the <a href="http://en.wikipedia.org/wiki/Equality_(mathematics)#Some_basic_logical_properties_of_equality" rel="nofollow">"substitution property of equality"</a>, which is an axiom of equality as inherent as... |
947,622 | <p><img src="https://i.stack.imgur.com/TBGYx.png" alt="">Find variance from the graph given. I know the mean is 6 but I have no idea how to find the variance using this graph</p>
| Michael Hardy | 11,667 | <p>$\mu\pm\sigma$ are the inflection points: the points where the graph stops getting steeper and starts getting less steep.</p>
|
466,271 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span>, and <span class="math-container">$d$</span> be the lengths of the sides of a quadrilateral. Show that
<span class="math-container">$$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\... | Will Nelson | 62,773 | <p>Let $T$ be the function in question:
$$
T(a,b,c,d) = a b^2(b-c) + b c^2 (c-d) + c d^2 (d-a) + d a^2 (a - b).
$$
We wish to show $T(a,b,c,d)\ge 0$ if $a,b,c,d$ are the sides of a quadrilateral. (Presumably, $a$ is the side
opposite $c$ and $b$ is opposite $d$, but it actually doesn't matter to the proof.)</p>
<p><st... |
96,720 | <p>Informally, Löb's theorem (<a href="http://en.wikipedia.org/wiki/L%25C3%25B6b%2527s_theorem" rel="noreferrer">Wikipedia</a>, <a href="http://planetmath.org/?op=getobj&from=objects&id=9381" rel="noreferrer">PlanetMath</a>) shows that:</p>
<blockquote>
<p>a mathematical system cannot assert its own soundnes... | Andreas Blass | 6,794 | <p>Löb's theorem provides the essential ingredient for a complete axiomatization of propositional provability logic. In detail: Work with the usual notation of propositional modal logic, which has propositional variables, the usual Boolean connectives, and the unary modal operator $\square$. The usual reading of $\sq... |
4,127,553 | <p>So for this problem I need to use the fact that <span class="math-container">$\frac {1-r^2}{1-2r\cos\theta+r^2}$</span>=<span class="math-container">$1+2\sum_{n=1}^{\infty} r^n\cos n\theta$</span>.</p>
<p>I replaced the term in the integral but i ended up getting <span class="math-container">$\sum_{n=1}^{\infty} r^n... | Paramanand Singh | 72,031 | <p>A much simpler way is to note that <span class="math-container">$$\int_{0}^{\pi}\frac{dx}{a+b\cos x} =\frac{\pi} {\sqrt{a^2-b^2}}, a>|b|$$</span> and differentiate it with respect to <span class="math-container">$a$</span> to get <span class="math-container">$$\int_{0}^{\pi}\frac{dx}{(a+b\cos x) ^2}=\frac{\pi a} ... |
2,948,862 | <p>So, I took one intro course in Tensor calculus and this problem reminds of that, except I can't quite recall how derivatives work with respect to components, or what those derivatives produce. Consider the following example: </p>
<p><a href="https://i.stack.imgur.com/KhN1F.png" rel="nofollow noreferrer"><img src="h... | quasi | 400,434 | <p>Yes, the kernel is the set of constant functions, and yes, there are infinitely may such functions, but they are not linearly independent.
<p>
Any nonzero constant function generates all the other constant functions, so take any one of them, and you get a basis.
<p>
As regards the eigenvalues of <span class="math-co... |
2,539,942 | <p>For any $x\in\mathbb{R}$, the series
$$ \sum_{n\geq 1}\tfrac{1}{n}\,\sin\left(\tfrac{x}{n}\right) $$
is trivially absolutely convergent. It defines a function $f(x)$ and <strong>I would like to show that $f(x)$ is unbounded over $\mathbb{R}$</strong>. Here there are my thoughts/attempts:</p>
<ol>
<li>$$(\mathcal{L}... | Dap | 467,147 | <p>[Edit: according to <a href="https://math.stackexchange.com/q/182491">https://math.stackexchange.com/q/182491</a> Hardy and Littlewood originally showed that this function was unbounded]</p>
<p>A probabilistic argument on the head $\sum_{n\leq N}\tfrac1n\sin(x/n)$ shows that it must take a $O(1)$ value in any inter... |
395,118 | <p>In a city of over $1000000$ residents, $14\%$ of the residents are senior citizens. In a simple random sample of $1200$ residents, there is about a $95\%$ chance that the percent of senior citizens is in the interval [pick the best option; even if you can provide a sharper answer than you see in the choices, please ... | amWhy | 9,003 | <blockquote>
<p>I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am stuck at $\ln{x/y} = \ln{x} + \ln{\frac{1}{y}}$, but is it okay to automatically assume $\ln{\frac{1}{y}}$ = $-\ln{y}$ from here?</p>
</blockquote>
<p>Yes, absolutely. That's how proofs work in mathemati... |
2,912,881 | <p>In an $n\times n$ board ($n\geq 3$), how many colors do we need so that we can color the cells such that no three consecutive cells (horizontal, vertical, or diagonal) are of the same color?</p>
<p>With three colors we can do it, using the pattern</p>
<p>$$131$$
$$232$$
$$312$$</p>
<p>and repeating it as necessar... | Pranav Aggarwal | 523,168 | <p>Realise that the ant in question must take 4 steps to the right and 4 upwards. Representing a step towards right as R and an upward step as U, the ant can choose paths like RRRUUUUR, URURURUR, etc.</p>
|
2,974,181 | <p>Let <span class="math-container">$E$</span> be a finite dimensional complex vector space. Let <span class="math-container">$\mathbb{P}(E)$</span> be the projective space of lines through the origin of <span class="math-container">$E$</span>. Fulton, in his book "Young Tableaux", then defines the <span class="math-co... | Joppy | 431,940 | <p>The most natural way to view <span class="math-container">$\mathbb{P}^*(E)$</span> is as equivalence classes of surjections from <span class="math-container">$E$</span> to a line, where <span class="math-container">$[f: E \twoheadrightarrow L] = [g: E \twoheadrightarrow L']$</span> if and only if there exists some i... |
3,559,942 | <p>I am trying to solve the limit:
<span class="math-container">$$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)$$</span></p>
<p>I was trying to find a way to bring it into a fraction form to apply L'Hospital's rule, and I tried using
<span class="math-... | Vasili | 469,083 | <p>I think using the difference of cubes should work:<span class="math-container">$$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)=\\
\lim_{x\to\infty}\frac{x^\frac{5}{3}\sin\frac{1}{x}}{\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{2}{3}+\left(x+\si... |
354,961 | <p>Suppose $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$. Show that $A\cap B$ is a normal subgroup of $B$.</p>
| Cameron Buie | 28,900 | <p>Let me expand on Martin's answer, since you may not have encountered the necessary results to make use of it, yet. To understand these results, you'll need to be somewhat familiar with <a href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow noreferrer">group homomorphisms</a>, the <a href="http://en.... |
15,413 | <p>I see that there's a control I can click to hide or "minimize" version 9.0's "suggestion bar". Is there a keyboard shortcut to do this?</p>
| Tuku | 4,451 | <p><a href="http://reference.wolfram.com/mathematica/tutorial/KeyboardShortcutListing.html" rel="nofollow">The <em>Mathematica</em> documentation on keyboard shortcuts</a>
shows no shortcut for the hiding the suggestion bar</p>
|
15,413 | <p>I see that there's a control I can click to hide or "minimize" version 9.0's "suggestion bar". Is there a keyboard shortcut to do this?</p>
| Vitaliy Kaurov | 13 | <p>Here is the summary:</p>
<ul>
<li><p>There is no shortcut (you can <a href="http://www.wolfram.com/support/contact/email/?topic=Feedback" rel="noreferrer">suggest here</a>)</p>
</li>
<li><p>Quick close/open labeled <strong>minimize</strong> below</p>
<p><img src="https://i.stack.imgur.com/F3ZGU.png" alt="enter image... |
1,292,500 | <p>A very, very basic question.</p>
<p>We know
$$-1 \leq \cos x \leq 1$$
However, if we square all sides we obtain
$$1 \leq \cos^2(x) \leq 1$$
which is only true for some $x$.</p>
<p>The result desired is
$$0 \leq \cos^2(x) \leq 1$$
Which is quite easily obvious anyway. </p>
<p>So, what rule of inequalities am I for... | Alberto Debernardi | 140,199 | <p>You can not deduce that if $a\leq b$ for $a,b\in \mathbb{R}$, then $a^2\leq b^2$. For example, for all $a\neq b$, both negative, such inequality is false. However, you can state your original inequality as
$$
0\leq |\cos x|\leq 1,
$$
from which you can deduce
$$
0\leq |\cos^2 x|=\cos^2 x\leq 1,
$$
since the inequali... |
796,768 | <p>i am searching for a series with this condition that $\prod 1+a_n$ converges but $\Sigma a_n$ diverges. </p>
<p>i know that if $a_n = n^{\frac{1}{2}}$ then $\Sigma a_n$ diverges but i dont know it is exactly what i want, does $\prod 1+a_n$ converges?</p>
<p>i really don't know how to check the divergence... | Bruno Joyal | 12,507 | <p>The <a href="http://en.wikipedia.org/wiki/Infinite_product" rel="nofollow">usual definition</a> of convergence for infinite products rules out such a possibility by definition.</p>
<p>In any case, if the $a_n\geq 0$, then the product converges to a finite limit if and only if $\sum a_n<\infty$.</p>
<p>If you co... |
920,732 | <p>I have been commanded on homework to find a non-bijective isomorphism in a category whose objects are sets, whose morphisms are set maps, and composition is the usual function composition. So our category is a subcategory of the category of <strong>Set</strong>. </p>
<p>But, I think that such an isomorphism is in f... | Vladhagen | 79,934 | <p>I believe that this is what we need to examine. Although we intuitively want to assume that the identity morphism is the usual "identity" map, this is actually not stated as necessary in the problem.</p>
<p>Let $\mathcal{C}$ be a category consisting of a single set $A=\{1,2\}$ as its object, morphisms as set maps, ... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | S.C.B. | 310,930 | <p>This is because Wolframalpha is plotting $y=(x-2)(x-3)$, which is a parabola. </p>
<p>As you have entered $(x-2)(x-3)=0$, it is merely indicating where the intersection is between $y=(x-2)(x-3)$ and $y=0$, which is why there are red dots on the points where the $x$-coordinates are $2$ and $3$. </p>
<p>This is not... |
2,148,389 | <p>Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?</p>
<p>Plotting: <code>x^2 - 5x + 6 = 0</code> </p>
<p>at <... | Mike | 17,976 | <p>After much discussion in comments, I have decided to interpret this as a WolframAlpha question. Many people would not plot an equation in one variable. The solution could be plotted on a number line. In $2$ dimensions, the plot would in fact be $2$ lines parallel to the $y$-axis.</p>
<p>WolframAlpha does not int... |
155,453 | <p>How can I use a list of variables (possibly subscripted) as an <a href="http://reference.wolfram.com/language/ref/AxesLabel.html" rel="noreferrer"><code>AxesLabel</code></a> without showing the braces.</p>
<p>For example, </p>
<pre><code>Plot[{x, x^2}, {x, 0, 1}, AxesLabel -> {x, {Subscript[y, 1], Subscript[y, ... | jiaoeyushushu | 35,486 | <p>Another way：</p>
<pre><code>Plot[{x, x^2}, {x, 0, 1},
AxesLabel -> {x, HoldForm[Subscript[y, 1] Subscript[y, 2]]}]
</code></pre>
<p>or</p>
<pre><code>Plot[{x, x^2}, {x, 0, 1},
AxesLabel -> {x,
Graphics[{Text[
HoldForm[Subscript[y, 1](*","*)Subscript[y, 2]], {0, 1}]},
Image... |
4,479,797 | <p>Let's say I have an urn with 10 unique objects, and I choose 3 objects from it (each choice is made without replacement). Then the probability of choosing any one object is 3/10. I calculated this probability by summing the probability the object is chosen on the first pick + probability chosen on second pick + prob... | Frank W | 552,735 | <p>Elaborating on the comment, the key is to note that the denominator can indeed be further factored</p>
<p><span class="math-container">$$\begin{align*}y^4+1 & =y^4+2y^2+1-2y^2\\ & =\left(y^2+1\right)^2-2y^2\\ & =\left(y^2+y\sqrt2+1\right)\left(y^2-y\sqrt2+1\right)\end{align*}$$</span></p>
<p>Therefore, w... |
1,397,036 | <p>A bag labeled $A$ contains $4$ red balls and $7$ green balls.
Another bag $B$ contains $6$ red and $5$ green balls.</p>
<p>A ball is transferred from bag $A$ to bag $B$, after which a ball is drawn from $B$.</p>
<p>Find the probability that it is a red ball?</p>
<p>To be honest I have no idea how to approach the ... | hvedrung | 245,555 | <p>You know that ball transferred from A to B is red with 4/11 probability.</p>
<p>Now you have 12 balls in bag B. What is probability it is red? You had 6 red and added 1 with 4/11 probability it is red.</p>
|
3,516,385 | <p>I need to apply a convergency test to</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n^4}{n!}=\sum_{n=1}^\infty \frac{n^3}{(n-1)!}$$</span></p>
<p>I can't seem to figure if any comparison test apply; those that I tried gave no useful information. Any ideas?</p>
| Especially Lime | 341,019 | <p>For a comparison-based method, note that <span class="math-container">$\frac{n^4}{n!}<\frac{n^4}{(n-5)^6}\leq\frac{2^6}{n^2}$</span> for <span class="math-container">$n\geq 10$</span>.</p>
|
228,233 | <p>I need a continuous function $f:\mathbb{Q}\rightarrow \mathbb{R}$ and discontinuous $g:\mathbb{R}\rightarrow \mathbb{R}$
s.t $f(x)=g(x)$ for all rational $x$ s. So if I say $f(x)=0$ and $g(x)=0$ for $x \in \mathbb{R}\setminus\{\sqrt2\}$ and $g(x)=1$ at $x=\sqrt 2$ .would I be right?</p>
| sperners lemma | 44,154 | <p><strong>This answer is unsalvageable wrong, I leave it so others could learn from my mistakes</strong></p>
<p>Given $f : \mathbb Q \to \mathbb R$ we can extend it to a continuous function $\bar f : \mathbb R \to \mathbb R$ in the following way:</p>
<p>For every irrational $x$ having Cauchy sequence $(x_a)$ define ... |
618,665 | <p>Show that $\sqrt{13}$ is an irrational number.</p>
<p>How to direct proof that number is irrational number. So what is the first step..... </p>
| Git Gud | 55,235 | <p>Consider the polynomial $x^2-13\color{grey}{\in \mathbb Z[x]}$. The <a href="http://en.wikipedia.org/wiki/Rational_root_theorem">rational root theorem</a> guarantees its roots aren't rational and since $\sqrt {13}$ is a root of the polynomial, it is irrational.</p>
|
618,665 | <p>Show that $\sqrt{13}$ is an irrational number.</p>
<p>How to direct proof that number is irrational number. So what is the first step..... </p>
| Warren Hill | 86,986 | <p>The standard proof that $\sqrt{p}$ is irrational for any prime $p$ is as follows</p>
<p>Let $\sqrt{p} = \frac{m}{n}$ where $m,n\in\mathbb N.$ and $m$ and $n$ have no factors in common.</p>
<p>Now $\frac{m^2}{n^2} = p \Rightarrow m^2 = p \cdot n^2$</p>
<p>Since $p$ is prime and $m^2$ is a multiple of $p$ then $m$ ... |
4,265,066 | <p>For the purpose of CG and animation, I'm looking for a function thats tends to 1 when x tends to +Infinity, and have a tangent of 1 when x = 0.</p>
<p>I found that function:</p>
<p><span class="math-container">$f\left(x\right)=2\frac{\left(\frac{2x}{p}+1\right)^{p}}{\left(\frac{2x}{p}+1\right)^{p}+1}-1$</span></p>
<... | Jean Marie | 305,862 | <p>A track, not a solution:</p>
<p>Let us first recall the Schur's complement lemma (that one finds in the "Properties" part in this <a href="https://en.wikipedia.org/wiki/Schur_complement" rel="nofollow noreferrer">reference</a>):</p>
<p>Lemma: For a block-partitioned square matrix:</p>
<p><span class="math-... |
1,008,591 | <p>I am in the middle of my proof and I want to know if the following is true, suppose $f_n$ is a Cauchy sequence, can i do this?</p>
<p>If $$\| f_n(x) - f(x) \| \to 0,$$ then can I also say this limit is true</p>
<p>$$\lim_{m \to \infty} \| f_n(x) - f_m(x) \| \to 0?$$</p>
| Robert Israel | 8,508 | <p>Of course $f'$ must be in $L^1$ because $\int_a^b f'(t)\; dt \le f(b) - f(a)$.
But $f'$ can be unbounded on every interval. You can get this by taking the
sum of a suitable series of translated and scaled "ramp" functions.</p>
|
237,227 | <p>I want to produce graphs of Fourier transforms for lectures.</p>
<p>Using the answer from
<a href="https://mathematica.stackexchange.com/questions/3506/calling-correct-function-for-plotting-diracdelta">Calling Correct Function for Plotting DiracDelta</a>
I get a problem with the code mentioned below.</p>
<p>Definiti... | azerbajdzan | 53,172 | <p>Here is a code that plots your arrows correctly:</p>
<pre><code>ca = Cases[
1.56664 DiracDelta[(4.3 + 0. I) - x] +
I Sqrt[\[Pi]/2] DiracDelta[(4.6 + 0. I) - x] +
1.56664 DiracDelta[(4.3 + 0. I) + x] -
I Sqrt[\[Pi]/2] DiracDelta[(4.6 + 0. I) + x],
a_ DiracDelta[b_] :> {{a}, x /. Solve[b == 0]... |
98,698 | <p>I have the time domain signal
$$
u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t)
$$
where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is known, but only numerically. I also have prior knowledge that $u(t)$ is a sum of a small number of sinusoids. How can I... | Douglas Zare | 2,954 | <p>A slight improvement on testing random points is to use a hill-climbing method. After you pick a random point, move it to increase the minimum distance until you hit a $(d-1)$-face of the Voronoi cell, then move within that hyperplane until you hit a $(d-2)$-face, etc. with special cases for hitting the boundary of ... |
2,203,995 | <blockquote>
<p>Show that the number of nonisomorphic groups of order $8181=3^4\cdot 101$ is equal to the number of nonisomorphic groups of order $81$. Find all the abelian groups of order $8181$ and at least one that is not abelian.</p>
</blockquote>
<p>the second part (finding all abelian groups) doesn't seem very... | tim6her | 422,444 | <p>For the first part we proof that each such group $G$ is isomorphic to a direct product of the cyclic group $\mathbb Z_{101}$ of order $101$ and a group $H$ of order $81$, i.e.,
$$ G \cong H \times \mathbb Z_{101} $$</p>
<p>By Sylow's first theorem there exists a $101$-Sylow subgroup $G_{101}$ of order $101$. Let $n... |
3,285,447 | <blockquote>
<p>Given that <span class="math-container">$a$</span> is any vector in a vector space <span class="math-container">$V$</span>, show that the set <span class="math-container">$\{xa : x \in \mathbb{R}\}$</span> of all scalar multiples of <span class="math-container">$a$</span> is a subspace of <span class=... | trisct | 669,152 | <p>Yes I think it is sufficient to exhibit the closedness under scalar multiplication and addition.</p>
<p>But your addition part is not exactly right. Let <span class="math-container">$W=\{xa:x\in\mathbb{R}\}$</span>. It should be like <span class="math-container">$xa\in W,\ ya\in W\implies xa+ya=(x+y)a\in W$</span>.... |
10,935 | <p>I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals:</p>
<p>If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,dx$.</p>
<p>In past semesters, I've shown examples in which I confirm that this theorem holds for some specif... | user52817 | 1,680 | <p>I think the best "real" application of the mean value theorem for integrals is to make a rigorous proof of the fundamental theorem of calculus. </p>
|
1,604,204 | <blockquote>
<p><span class="math-container">$a,b,c$</span> are in A.P ; <span class="math-container">$p,q,r$</span> are in H.P. And <span class="math-container">$ap,bq,cr $</span> are in G.P. Then what is the value of <span class="math-container">${p\over r} + {r\over p}\ \ ?$</span></p>
<p><span class="math-container... | Empy2 | 81,790 | <p>The symmetry is easier to see if you keep $b$ and $y=1/q$, rather than $a$ and $x=1/p$.<br>
$$\frac{b-d}{y-k},\frac by,\frac{b+d}{y+k}\text{ are in GP}\\
(b^2-d^2)y^2=b^2(y^2-k^2)\\ \frac db=\pm \frac ky$$
You have some manipulation to go. Good luck.</p>
|
1,604,204 | <blockquote>
<p><span class="math-container">$a,b,c$</span> are in A.P ; <span class="math-container">$p,q,r$</span> are in H.P. And <span class="math-container">$ap,bq,cr $</span> are in G.P. Then what is the value of <span class="math-container">${p\over r} + {r\over p}\ \ ?$</span></p>
<p><span class="math-container... | mathlove | 78,967 | <p>Using
$$a+c=2b$$
$$\frac 1p+\frac 1r=\frac 2q\quad\Rightarrow\quad \frac{p+r}{pr}=\frac 2q\quad\Rightarrow\quad p+r=\frac{2pr}{q}$$
$$apcr=(bq)^2\quad\Rightarrow\quad \frac{pr}{q^2}=\frac{b^2}{ac}$$</p>
<p>we have
$$\begin{align}\frac pr+\frac rp&=\frac{(p+r)^2-2pr}{pr}\\\\&=\frac{\left(\frac{2pr}{q}\right)... |
78,279 | <p>In another question (<a href="https://mathematica.stackexchange.com/questions/77229/using-timeseriesforecast-for-forecasting-the-traffic-growth/78275">Using TimeSeriesForecast for forecasting the traffic growth</a>) I asked to use the TimeSeriesForecast on <a href="https://docs.google.com/spreadsheets/d/1dmv_C2_J7uG... | KennyColnago | 3,246 | <p>Try checking the parallel kernel settings by clicking on the menu bar:
Evaluation > Parallel Kernel Configuration</p>
<p>Click the tab Parallel in the window that pops up.</p>
<p>Uncheck Automatic as it may have fewer kernels than you want, subject to the limit imposed by your license.</p>
<p>Then click Manual se... |
4,105,190 | <p>Just been trying to prove the following by mathematical deduction for research but having some issues. Mind helping out?</p>
<p>Prove that A = ∅ if and only if B = A∆B.</p>
<p>What I have so far...
A∆B = (A-B)∪(B-A)
= (A∩B^Compliment)∪(B∩A^Compliment)</p>
<p>But not too sure how to explain or go from here...</p>
| hamam_Abdallah | 369,188 | <p>Let
<span class="math-container">$$S_n=\frac{\sum_{k=0}^n(-1)^k2}{n+1}$$</span></p>
<p>then</p>
<p><span class="math-container">$$S_{2n+1}=0\;\;\text{ and } \;S_{2n}=\frac{2}{2n+1}$$</span></p>
<p><span class="math-container">$$\lim_{n\to\infty}S_{2n}=\lim_{n\to\infty}S_{2n+1}=0$$</span></p>
<p><span class="math-con... |
1,446,797 | <p>A sequence is non-decreasing if $k_1 \leq k_2 \leq k_3$.</p>
<p>Now <strong>I need to find the number of non-decreasing sequences of length-$n$ sequences from $\{1,2,....m\}$</strong></p>
<p>I basically see it as sum of the numbers of strictly increasing sequences plus other sequences.</p>
<p>The number of stric... | user84413 | 84,413 | <p>Let <span class="math-container">$x_i$</span> be the number of times the digit <span class="math-container">$i$</span> appears in the sequence, for <span class="math-container">$1\le i\le m$</span>.</p>
<p>Then the sequence is determined by the values of the <span class="math-container">$x_i$</span> since it is non... |
1,446,797 | <p>A sequence is non-decreasing if $k_1 \leq k_2 \leq k_3$.</p>
<p>Now <strong>I need to find the number of non-decreasing sequences of length-$n$ sequences from $\{1,2,....m\}$</strong></p>
<p>I basically see it as sum of the numbers of strictly increasing sequences plus other sequences.</p>
<p>The number of stric... | Hugo Peyron | 633,577 | <p>You can use the bars and stars method to solve this problem. As a matter of fact, you deal with a n bits number (this will correspond to the stars). The bars will correspond to a changement of the value that we place. For example *** | ** |* correspond to 111223. As you can pick up m different values for your seque... |
3,831,510 | <p>If <span class="math-container">$a$</span> is a constant, what is the name of a curve of the form <span class="math-container">$a*(x+y) = x*y$</span>? And how is this equation related to more this curve's more general equations/characteristics? Plotting this curve, I would risk calling it a hyperbola, but I'm not su... | Crostul | 160,300 | <p><span class="math-container">$$xy=a(x+y)$$</span>
<span class="math-container">$$xy-ax-ay=0$$</span>
<span class="math-container">$$xy-ax-ay+a^2=a^2$$</span>
<span class="math-container">$$(x-a)(y-a)=a^2$$</span>
The curve is an hyperbola.</p>
|
1,861,662 | <p>Consider the collection of those $\mathbb{R}$-valued functions on an interval $I\subseteq\mathbb{R}$, which have a dense set of points of continuity. I would expect this collection to be closed under taking uniform limits.</p>
<p>What are good ways to prove this? (Or in fact, what is a counterexample)?</p>
| Aweygan | 234,668 | <p>Firstly, to answer your question, if the limit $\lim\limits_{n\to\infty}a_n$ does not exist, it makes no sense to say that $\lim\limits_{n\to\infty}a_n=a$, where $a$ does not exist. We would say that the sequence $\{a_n\}_{n=1}^\infty$ is <em>divergent</em>. In short, no, it is not good notation to say that $a\neq... |
1,861,662 | <p>Consider the collection of those $\mathbb{R}$-valued functions on an interval $I\subseteq\mathbb{R}$, which have a dense set of points of continuity. I would expect this collection to be closed under taking uniform limits.</p>
<p>What are good ways to prove this? (Or in fact, what is a counterexample)?</p>
| fleablood | 280,126 | <p>Suppose I wrote something meaningless and terrible such as </p>
<p>$\sqrt{\mathbb R} + 27 \ne 13 \implies $</p>
<p>$\sqrt{\mathbb R} \ne -14 \implies $</p>
<p>$\mathbb R \ne 196$</p>
<p>Your assumption ought to be that I clearly don't have the foggiest idea what I'm talking about and I am clearly ... lost.</p>
... |
550,817 | <p><img src="https://i.stack.imgur.com/d4kwN.png" alt="enter image description here"></p>
<p>I know that if I substitute the first matrix for $T(M)$ I see what T does to each of the basis vectors. I don't understand how that creates a $3\times 3$ matrix though.</p>
<p>I was looking at this question for a hint, but I... | Berci | 41,488 | <p>The matrix of a transformation $T$ in a basis $(e_i)_i$ can be written with column vectors $[..T(e_i)..]$, all coordinated in the given basis.</p>
<p>In this particular case, you should calculate the $3$ matrix products: $T(e_1),\ T(e_2),\ T(e_3)$, then express these three results as the linear combination of $e_1,... |
1,168,862 | <p>In a case such as $x^x=27$ we can solve by inspection that $x=3$.</p>
<p>But how can we solve this algorithmically in general, given that $N\in Q$ where $Q\subset\Bbb Z$ and $Q=[1,10^{10}]$ and $x\in\Bbb R$?</p>
| 5xum | 112,884 | <p>To find $x$, you have to solve the equation</p>
<p>$$x\ln(x) = \ln(N)$$</p>
<p>This equation does not have a closed form solution using only the functions we usually use. However, using <a href="http://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow">Lambert's function</a>, which is the inverse of the func... |
496,544 | <p>The question asks:</p>
<p>Find the smallest value (for real $x$ and $y$) of: $$x^4+2x^2+y^4-2y^2+3$$</p>
<p>I don't think I understand this question, it is in a completing the square exercise and I don't really know where to start. I can factorise parts and mess around with it but it does not help at all. Any help... | Rebecca J. Stones | 91,818 | <p>We observe that $$f(x,y):=x^4+2x^2+y^4-2y^2+3=(x^2+1)^2+(y^2-1)^2+1.$$</p>
<p>The minimum of $f(x,y)$ therefore occurs when $|x^2+1|$ is minimized and when $|y^2-1|$ is minimized.</p>
|
1,077,248 | <p>Prove:
For every positive integer $n, n^2 + 4n + 3$ is not a prime.</p>
<p>I tried to disprove the statement, which I could not using several number examples with constructive proof.</p>
<p>However I am not sure how to correctly step by step prove it.</p>
<p>Thank you in advance!</p>
| MathPassenger | 933,765 | <p>Recall that <span class="math-container">$n^2 + 4n + 3 = (n + 1)(n + 3)$</span>. Since <span class="math-container">$n$</span> is necessarily a positive integer, it is obvious that <span class="math-container">$n^2 + 4n + 3$</span> is composite.</p>
|
4,093,001 | <p>Show that if <span class="math-container">$X$</span> is path connected then every path <span class="math-container">$f:I\to X$</span> is homotopic to a constant path <span class="math-container">$g(t)=x$</span>. Does this mean show that there is an <span class="math-container">$x$</span> (or even for all <span class... | Meissner | 844,908 | <p>Well, I think that the quastion could be wrong, because "path connected" is not enough to ensure that any path is homotopic to a point. As a counterexample, a loop in the circle <span class="math-container">$\mathbb{S}^1$</span> (with its usual topology) is not homotopic to a point.</p>
<p>The question tha... |
158,687 | <p>I have 3 matrices, each of size $101 \times 101$.</p>
<pre><code>List6 = Table[{x1[[i, j]]}, {i, 101}, {j, 101}]
List7 = Table[{y1[[i, j]]}, {i, 101}, {j, 101}]
List8 = Table[{xDisp1[[i, j]]}, {i, 101}, {j, 101}]
</code></pre>
<p>and I want to use <code>ListContourPlot</code></p>
<pre><code>ListContourPlot[{{x1[[... | David G. Stork | 9,735 | <p>I don't think it makes much sense to create a <code>ListContourPlot</code> with two or more separate lists. What do you think it should look like? How will you display three different values at a given location?</p>
<p>I'd use <code>ListPlot3D</code> to keep the data sets visible:</p>
<pre><code>list1 = Table[Ra... |
4,351,777 | <p>I'm studying about tensors and have already understood the following theorem:</p>
<blockquote>
<p><span class="math-container">$C^1_1$</span> is the unique linear function such that <span class="math-container">$C_1^1(v\otimes\eta)=\eta(v)=v(\eta)$</span> for all <span class="math-container">$v\in V$</span> and <spa... | Community | -1 | <p>The existence part is fine: you basically define <span class="math-container">$C_l^k$</span> on a basis of <span class="math-container">$\mathcal{T}_s^r(V)$</span> in the way it is supposed to be and then extend it linearly.</p>
<p>The problem is in the "uniqueness" part. Unlike the book's proof, it does n... |
1,980,909 | <p>I was playing around with square roots today when I "discovered" this. </p>
<p>$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$</p>
<p>$\sqrt{1 + x} = x$</p>
<p>$1 + x = x^2$ </p>
<p>Which, via the quadratic formula, leads me to the golden ratio. </p>
<p>Is there any significance to this or is it just a random coin... | yurnero | 178,464 | <p>Let $a_n$ include $n$'s 1. Then $a_{n+1}=\sqrt{1+a_n}$. One can show by induction that $a_n$ is increasing and is bounded from above by $3$. Induction steps:
$$
a_{n+1}=\sqrt{1+a_n}\geq\sqrt{1+a_{n-1}}=a_n; a_{n+1}=\sqrt{1+a_n}<\sqrt{1+3}=2<3.
$$
So there exists $L=\lim a_n$ satisfying $L=\sqrt{1+L}$, which im... |
1,980,909 | <p>I was playing around with square roots today when I "discovered" this. </p>
<p>$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$</p>
<p>$\sqrt{1 + x} = x$</p>
<p>$1 + x = x^2$ </p>
<p>Which, via the quadratic formula, leads me to the golden ratio. </p>
<p>Is there any significance to this or is it just a random coin... | Mark Fischler | 150,362 | <p>Any expression of the form $\sqrt{n+\sqrt{n+\sqrt{n+\ldots}}}$ will have a valueof the form
$$
\frac{-1\pm\sqrt{1+4n}}{2}
$$
so it is no coincidence that you get a simple answer like you got.</p>
<p>As to whether it is a coincidence that this is the golden ratio, pretty much the defining expression for the golden ... |
4,383,159 | <p>I am currently working through my course text and one of the examples for a unital, commutative Banach algebra is the space of continuous functions <span class="math-container">$C([a,b])$</span>. Further, in the text, there is a theorem stating:</p>
<blockquote>
<p><span class="math-container">$\sigma(a) \neq 0$</sp... | RaGon | 1,021,410 | <p>The reason for my misunderstanding was supposing that we look for the inverse regarding the composition of functions when it was rather the multiplicative inverse we were looking for.</p>
|
132,980 | <p>Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).</p>
<p>One of the most important objects that are associated to an algebraic variety is its derived category. So I'm wond... | Sasha | 4,428 | <p>Derived category of a toric variety has a full exceptional collection which was constructed by Kawamata using toric Minimal Model Program. As far as I know no good direct relation between the collection and the fan (polyhedron) of the variety is known. </p>
|
1,099,885 | <p>How can you calculate</p>
<p>$$\lim_{x\rightarrow \infty}\left(1+\sin\frac{1}{x}\right)^x?$$</p>
<p>In general, what would be the strategy to solving a limit problem with a power?</p>
| Zubin Mukerjee | 111,946 | <p>Take a logarithm. Let $A$ be the limit. Then</p>
<p>\begin{align}
A &= \lim_{x\to\infty} \left(1+\sin\frac{1}{x}\right)^x \\\\
&=\lim_{x\to\infty} \,\,\exp\left(x\ln\left(1+\sin\frac{1}{x}\right)\right)\\\\
&=\lim_{y\to0} \,\,\exp\left(\frac{\ln\left(1+\sin y\right)}{y}\right)\\\\
\end{align}</p>
<p>Us... |
4,473,931 | <p>Swedish Lotto rules can be found <a href="https://cdn1.svenskaspel.net/content/cms/documents/779afe3f-0363-4c36-b79c-1881549a8cbc/1.13/spelregler-lotto-och-joker.pdf" rel="nofollow noreferrer">here</a>. As far as I know these are the most recent rules. In several places the probability of winning Drömvinsten is quot... | JMoravitz | 179,297 | <p>Your attempted calculations seem to ignore the possibility of the ending digits outperforming the leading digits as well as your attempted calculations seem to ignore the possibility of the ending digits performing just as well as the leading digits. For exactly two leading or ending to match, we need one of the fol... |
295,130 | <p>I am taking Abstract Algebra right now and working on the exercises in the introductory section on Set Theory. I am having trouble proving the following question although it makes intuitive sense to me due to the intersection either being occupied by an even number or odd number of sets in the collection, leading to... | Brian M. Scott | 12,042 | <p>This is the <a href="http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow">inclusion-exclusion theorem</a>; you’ll find two proofs in the linked article.</p>
|
2,215,409 | <p>$$T(0)=1$$
$$(N\gt0)\;T(N)\;=\;\sum_{k=0}^{N-1}(k+1)T(k)$$</p>
<p>How can I find the closed representation of this function?
I don't need to know the answer, I just need someone to point me in the right direction.</p>
| lioumens | 371,514 | <p>Start writing out what the sequence is, and see if you can notice a pattern... that's normally a good first step for simple problems.</p>
<p>Think about breaking up the sum on the right into something more convenient.</p>
<p>Then you can find the ratio of consecutive terms, which will give more insight to the clos... |
2,411,812 | <blockquote>
<p>When <span class="math-container">$c$</span> is real and in the interval <span class="math-container">$[-1,1]$</span>, the roots <span class="math-container">$z$</span> of <span class="math-container">$z^2-2cz+1=0$</span> have <span class="math-container">$|z|=1$</span>; when <span class="math-container... | Michael Rozenberg | 190,319 | <p>If $|c|\leq1$ then $z=c\pm\sqrt{1-c^2}i$.</p>
<p>Thus, we see that $$|z|=\sqrt{c^2+\left(\sqrt{1-c^2}\right)^2}=1.$$</p>
<p>If $|c|>1$ then for $c>1$ we have $|c+\sqrt{c^2+1}|>1$ and since $z_1z_2=1$,</p>
<p>for the second root $z_2$ we have $|z_2|<1$.</p>
<p>For $c<-1$ we have $|c-\sqrt{c^2-1}|&g... |
2,102,326 | <blockquote>
<p>Let $a,n$ be positive integers. Find all $a$ such that for some $n$ the largest power of $2$ dividing $(n+1)(n+2) \cdots (an)$ is greater than $2^{(a-1)n}$.</p>
</blockquote>
<p>Since I thought there were no such $a$, I thought about proving this by contradiction. That is, assume that $2^{(a-1)n+1}$ ... | rtybase | 22,583 | <p>Since $$(n+1)(n+2)...(an)=\frac{(an)!}{n!}$$
According to <a href="http://www.cut-the-knot.org/blue/LegendresTheorem.shtml" rel="nofollow noreferrer">Legendre's Theorem</a>
$n!=2^{n-r_1}\cdot q_1$ where $r_1$-the number of $1$'s in binary expansion of $n$ and similarly $(an)!=2^{an-r_2}\cdot q_2$ and
$$\frac{(an)!}... |
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