qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,678,908 | <p>Let</p>
<p><span class="math-container">$$
V = \{ax^3+bx^2+cx+d|a,b,c,d \in Z_7\}
$$</span></p>
<p>Let the subspace <span class="math-container">$U$</span> Be a subspace of <span class="math-container">$V$</span>:</p>
<p><span class="math-container">$$
U = \{p(x) \in V | p(3)=p(5)=0\}
$$</span></p>
<p>The answer... | Anton Vrdoljak | 744,799 | <p><strong>Hints:</strong></p>
<p><span class="math-container">$|2x+3|=\begin{cases}2x+3, & 2x+3 \ge 0 \iff x \ge -\frac{3}{2} \\-2x-3, & 2x+3 < 0 \iff x < -\frac{3}{2}\end{cases}$</span></p>
<p><span class="math-container">$|x-1|=\begin{cases}x-1, & x-1 \ge 0 \iff x \ge 1 \\-x+1, & x-1 < 0 \... |
4,142,970 | <p>Find the cardinality of the set <span class="math-container">$\{ X \subset \mathbb{N} : |X| = \aleph_0\} $</span>.</p>
<p>First off all I could easily prove that the set given above, say <span class="math-container">$S$</span>, is uncountable.</p>
<p>Because <span class="math-container">$P(\mathbb{N})$</span> is unc... | TonyK | 1,508 | <p>Your set <span class="math-container">$S$</span> is the power set of <span class="math-container">$\aleph_0$</span> (cardinality <span class="math-container">$2^{\aleph_0}=\mathfrak{c}$</span>) minus the set of finite subests of <span class="math-container">$\aleph_0$</span> (cardinality <span class="math-container"... |
3,335,257 | <p>This is the full question:</p>
<blockquote>
<p>show that if <span class="math-container">$p$</span> is an odd prime then the number of ordered pair solutions of the congruence<span class="math-container">$x^2-y^2 \equiv a \pmod p,$</span> is <span class="math-container">$p-1$</span> unless <span class="math-conta... | Alessandro Cigna | 641,678 | <p>Trying to find integer solutions <span class="math-container">$0\le x,y< p$</span> of <span class="math-container">$x^2-y^2\equiv 0 mod p$</span> as you observed implicates that <span class="math-container">$p\mid (x-y)(x+y)$</span>, thus you see <span class="math-container">$p\mid x+y$</span> or <span class="mat... |
557,146 | <p>I have a clock time in form of HH.MM.SS (hours, minutes, sec), I need to create a way to convert it to one number so that I can determent which time is bigger.
for example
00.00.00 is the smallest
23.59.59 is the biggest</p>
<p>11.03.50 > 11.02.57
etc..</p>
<p>I thought to do the following</p>
<p>hours*10000+min... | Trevor Wilson | 39,378 | <p>If all you need to do is compare two times in HH:MM:SS format, say $(h,m,s)$ and $(h',m',s')$, then you do not need to convert to seconds. You can just use the lexicographical ordering:</p>
<p>\begin{align*}
(h,m,s) <_{\text{lex}} (h',m',s')
\iff &h < h', \text{ or } \\
&h = h' \mathbin{\And} m < ... |
3,294,446 | <p>I'm currently working on a definite integral and am hoping to find alternative methods to evaluate. Here I will to address the integral:
<span class="math-container">\begin{equation}
I_n = \int_0^\frac{\pi}{2}\ln^n\left(\tan(x)\right)\:dx
\end{equation}</span>
Where <span class="math-container">$n \in \mathbb{N}$</s... | Stefan Lafon | 582,769 | <p><span class="math-container">$$\begin{split}
I_{2k} &= \int_0^\infty\frac{\ln^{2k}u}{u^2 + 1}du \\
&= \int_0^1\frac{\ln^{2k}u}{u^2 + 1}du +\int_1^{+\infty}\frac{\ln^{2k}\left(u\right)}{u^2 + 1}du \\
&=\int_0^1\frac{\ln^{2k}u}{u^2 + 1}du +\int_0^{1}\frac{\ln^{2k}t}{t^2 + 1}dt \,\,\,\left(\text{by } u\righ... |
3,294,446 | <p>I'm currently working on a definite integral and am hoping to find alternative methods to evaluate. Here I will to address the integral:
<span class="math-container">\begin{equation}
I_n = \int_0^\frac{\pi}{2}\ln^n\left(\tan(x)\right)\:dx
\end{equation}</span>
Where <span class="math-container">$n \in \mathbb{N}$</s... | Lai | 732,917 | <p>Using the Reflection property of Beta function, we have
<span class="math-container">$$
\begin{aligned}
\int_0^{\frac{\pi}{2}} \tan ^a x d x=\int_0^{\frac{\pi}{2}} \sin ^a x \cos ^{-a} x d x =& \frac{1}{2} B\left(\frac{a+1}{2}, \frac{1-a}{2}\right)=\frac{\pi}{2 \sin \left(\frac{a+1}{2} \pi\right)}=\frac{1}{2} \s... |
1,141,412 | <p>Prove that the ring of integers of $\mathbb Q (\sqrt{-5})$ does not have unique factorisation.</p>
<p>Since $-5\equiv 3\pmod 4$, I know that the ring of integers of $\mathbb Q (\sqrt{-5})$ is $\mathbb Z [\sqrt{-5}]$. </p>
<p>I assume the way to prove that this does not have a unique factorisation is to give two di... | Robert Cardona | 29,193 | <p>In a UFD, an element is irreducible if and only if it is prime.</p>
<p>Observe that <span class="math-container">$2$</span> is irreducible in <span class="math-container">$\mathbb Z + \mathbb Z\sqrt{-5}$</span>: Suppose <span class="math-container">$$2 = (a + b \sqrt{-5})(c + b \sqrt{-5}),$$</span> taking the norm ... |
475,221 | <p>I am working on problem #25 of <em>Linear Algebra and its Applications</em> and the question asks:</p>
<blockquote>
<p>Find an equation involving $g$, $h$, and $k$ that makes this augmented matrix correspond to a consistent system:
$$\left(\begin{array}{ccc|c}
1& -4& 7& g \\
0& 3& -5& h ... | Eric Auld | 76,333 | <p>$$\frac{y^2}{y^2+d^2} = \frac{y^2 + d^2}{y^2+d^2} - \frac{d^2}{y^2+d^2}$$</p>
<p>The latter is an arctangent integral.</p>
|
475,221 | <p>I am working on problem #25 of <em>Linear Algebra and its Applications</em> and the question asks:</p>
<blockquote>
<p>Find an equation involving $g$, $h$, and $k$ that makes this augmented matrix correspond to a consistent system:
$$\left(\begin{array}{ccc|c}
1& -4& 7& g \\
0& 3& -5& h ... | user64494 | 64,494 | <p>Maple code with(Student[Calculus1]):IntTutor$(y^2/(y^2 +d^2),y$); produced the output which can be seen <a href="http://rapidshare.com/files/1020936765/screen2.24.08.13.docx" rel="nofollow">here</a>. It coincides with the excellent answer by Eric Auld.</p>
|
2,751,417 | <p>I want to prove the following for real invertible $2 \times 2$ matrices $A$ and $B$:</p>
<p>$\det(ABA^{-1}B^{-1})=1$</p>
<p>I tried to write it out for random numbers, and that seems to work out well. But when I tried to write it out in general, it became far too much paperwork. So I think there must be a simpler/... | Fred | 380,717 | <p>Hints: $det(XY)=\det(X) \det(Y)$, hence, if $X$ is invertible: $1=\det(X) \det(X^{-1})$.</p>
|
2,546,792 | <blockquote>
<p><strong>Definition.</strong> A metric space $M$ is connected if there is no other open disjoint $A$ and $B$ in $M$ such that $M=A\cup B$ than the empty set and the total space $M$. A subset $C\subset M$ is connected if the subspace $C$ is connected.</p>
<p><strong>Definition</strong>. A connected... | MathsLearner | 451,337 | <p>We divide the shaded portion into six equal parts by three lines, each of them joining the centroid and one vertex.</p>
<p><a href="https://i.stack.imgur.com/dpxlG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dpxlG.jpg" alt="enter image description here"></a></p>
<p>So area of shaded area(A) ... |
820,490 | <p>I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?</p>
| Fly by Night | 38,495 | <p>The proof is based simply on similar triangles. If a right-angled triangle has an angle $\theta$ then the other two angles are $90^{\circ}$ and $(90-\theta)^{\circ}$. If two triangles have the same angles then they are similar.</p>
<p>My picture shows two similar triangles: $\triangle OAB$ and $\triangle OA'B'$. </... |
820,490 | <p>I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?</p>
| mursalin | 154,755 | <p>If you want to verify this intuitively for yourself, then this answer could prove to be helpful. But remember, <em>this is not a proof</em>.</p>
<p>First try to visualize what <span class="math-container">$\sin\theta$</span> looks like. If you need some help, this is what it looks like.</p>
<p><a href="https://i.sta... |
99,750 | <p>Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).</p>
<p>I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd ... | Robert Bryant | 13,972 | <p>It is not always clear what one means by 'the simplest description' of one of the exceptional Lie groups. In the examples you've given above, you quote descriptions of these groups as automorphisms of algebraic structures, and that's certainly a good way to do it, but that's not the only way, and one can argue that... |
99,750 | <p>Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).</p>
<p>I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd ... | Skip | 6,486 | <p>Here is a description that is new and you can judge whether it is beautiful. <em>Given any simple complex Lie group $G$ and almost any irreducible representation $V$, the stabilizer of almost any $G$-invariant polynomial $f$ on $V$ has identity component $G$.</em></p>
<p>Cartan's examples </p>
<ul>
<li>$G = E_6$,... |
1,336,209 | <p>I need to understand very good how the properties of this formula</p>
<p>$\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $</p>
<p>Taken from the paper "Radian Reduction for Trigonometric Function" (Hanek Payne Algorithm)</p>
<p>Some remarkable... | math110 | 58,742 | <p>Hint: since
$$\lim_{n\to\infty}\dfrac{1^4+2^4+\cdots+n^4}{n^5}=\int_{0}^{1}x^4dx=\dfrac{1}{5}$$</p>
|
2,829,362 | <p>Suppose the random variable $T$ which represents the time needed for one person to travel from city A to city B ( in minutes). $T$ is normally distributed with mean $60$ minutes and variance $20$ minutes. Also, suppose $600$ people depart at the exact same time with each of their travel time being independent from o... | José Carlos Santos | 446,262 | <p>This is an improper integral. It is defined as$$\lim_{\varepsilon\to0^+}\int_\varepsilon^1\frac1x\,\mathrm dx+\lim_{\varepsilon\to0^-}\int_{-1}^\varepsilon\frac1x\,\mathrm dx$$if both limits exist. In this case, none of the limits exist (in $\mathbb R$).</p>
|
3,752,676 | <p>what is <span class="math-container">$P(P(P(333^{333})))$</span>, where P is sum of digit of a number. for an example <span class="math-container">$P(35)=3+5=8$</span></p>
<p>a)18</p>
<p>b)9</p>
<p>c)33</p>
<p>d)333</p>
<p>f)5</p>
<p>I tried to find this but I couldn't. I started to find a pattern for an example the... | Ayman Hourieh | 4,583 | <p>One option is to use <a href="https://en.wikipedia.org/wiki/Tietze_transformations" rel="nofollow noreferrer">Tietze transformations</a>.</p>
<p>Let's start with the presentation found in Batominovski's answer,
<span class="math-container">$$
A = \langle \alpha, \beta \mid 4\alpha, 4\beta, 2(\alpha + \beta) \rangle.... |
3,752,676 | <p>what is <span class="math-container">$P(P(P(333^{333})))$</span>, where P is sum of digit of a number. for an example <span class="math-container">$P(35)=3+5=8$</span></p>
<p>a)18</p>
<p>b)9</p>
<p>c)33</p>
<p>d)333</p>
<p>f)5</p>
<p>I tried to find this but I couldn't. I started to find a pattern for an example the... | jjuma1992 | 715,368 | <p>Let <span class="math-container">$G=Z_4 \oplus Z_{12}$</span> and <span class="math-container">$M=\langle (2,2) \rangle.$</span> You can actually consider the matrix <span class="math-container">$[2\,\,\, 2]$</span> and reduce this matrix to smith normal form to get <span class="math-container">$[0 \,\,\, 2].$</span... |
308,251 | <p>I asked this question on Mathematics Stackexchange (<a href="https://math.stackexchange.com/q/2863312/660">link</a>), but got no answer.</p>
<p>Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$. </p>
<p>Recall that $A$ can be defined as the set of expressi... | Uriya First | 86,006 | <p>[Edit: The lemma was revised and proved, changed the point of view from series to sequences.]</p>
<p>[2nd Edit: The proof of the lemma was improved, and now the argument can show that Cauchy series such as $x_1+(x_{1^3+1}^2+\dots x_{2^3}^2)+(x_{2^3+1}^{3}+\dots+x_{3^3}^3)+\dots$, diverge in the $\mathfrak{m}$-... |
2,815 | <p>This is a final exam question in my algorithms class:</p>
<p>$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, and $a,b,c,d$ are distinct positive integers. Find all taxicab numbers $k$ such that $a,b,c,d < n$ in $O(n)$ time.</p>
<p>I don't know if the problem had a typo or not, because $O(n^3)$ seems more reas... | rgrig | 512 | <p>For $O(n^2)$ (randomized) time, you can also use a hashtable of size $\Theta(n^2)$. Looking up will be constant time because the number of taxicab numbers is $O(n^2)$.</p>
|
2,815 | <p>This is a final exam question in my algorithms class:</p>
<p>$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, and $a,b,c,d$ are distinct positive integers. Find all taxicab numbers $k$ such that $a,b,c,d < n$ in $O(n)$ time.</p>
<p>I don't know if the problem had a typo or not, because $O(n^3)$ seems more reas... | phil_20686 | 150,039 | <p>I think you can do better, imagine the (a,b) pairs as forming a matrix filled in with a^3+b^3 in the upper right triangle. e.g. (for squares due to limited mental arithmetic)</p>
<pre><code>2, 5, 10, 17, 26
- 8, 13, 20, 29
- -, 18, 25, 34
- - - 32, 41
- - - - 50
</code></pre>
<p>So it should be clear f... |
2,353,062 | <p>is there a simple and vivid example of paradoxical decomposition specifically on the banach-tarski paradox? is there a math software that can be used as an application of the paradox?</p>
| Alex Ortiz | 305,215 | <p>To negate the definition properly, we must show that <em>there exists an $\epsilon>0$</em> such that <em>for all $\delta>0$</em>, <em>there exists an $x$</em> such that $0<|x-2|<\delta$ but $|f(x)-8|\ge \epsilon$. Pay attention to the order of the quantifiers here: the limit definition is of the form
$$
... |
2,353,062 | <p>is there a simple and vivid example of paradoxical decomposition specifically on the banach-tarski paradox? is there a math software that can be used as an application of the paradox?</p>
| Siong Thye Goh | 306,553 | <p>To prove that the statement is false, I just have to pick a particular $\epsilon$ value, let me pick $\epsilon = 1$.</p>
<p>Suppose on the contrary that there is an $\delta> 0$, such that $|x-2| < \delta \implies |2x-3|<1$.</p>
<p>This is equivalent to $2-\delta < x < 2+\delta \implies 1 < x <... |
2,459,323 | <p>Q) I have to show that if every vertex in graph $G$ has an even degree , then
$G$ has no bridge. <br>
So by contradiction ,<br> If every vertex in graph $G$ has an even degree , then
$G$ has a bridge , which means $G - e$ will result in a disconnected graph.<br>
We know that in this graph, 1 vertex is adjacent t... | JMoravitz | 179,297 | <p>When beginning your proof by contradiction, what you should have rather than "If every vertex has even degree <strong>then</strong> $G$ has a bridge" is:</p>
<blockquote>
<p>Suppose every vertex in $G$ has even degree <strong>and</strong> $G$ has a bridge.</p>
</blockquote>
<p>We then look at what happens to the... |
646,596 | <p>I need to calculate the described squared circle... with only given the length of the side (a).</p>
<p><img src="https://i.stack.imgur.com/VZ8Yn.png" alt="enter image description here"></p>
<p>I need to calculate the area of described squared circle. How exactly is it done with only a side given?</p>
| amWhy | 9,003 | <p>Use the Pythagorean Theorem to find the length of the diagonal $2R$ ($2R$ is the length of the diagonal of the square and the circle), then solve for the radius of the circle: $R$: $$a^2 + a^2 = (2R)^2\iff R = \frac{a}{\sqrt 2}$$</p>
<p>Then use the formula for the area of a circle: $A = \pi R^2$.</p>
|
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Michael Biro | 29,356 | <p>How precise do you need the calculation to be?</p>
<p>As a quick and dirty approximation, we know that $2^3 = 8$ and $e^2 \approx 2.7^2 = 7.29$, and so $\ln(2)$ should be just over $\frac{2}{3} \approx 0.67$. Contiuing to match powers, we find $2^{10} = 1024$, and
$e^7 \approx (2.7)^7 = (3 - 0.3)^7 = 3^7 -7(3)^6(.... |
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Jaume Oliver Lafont | 134,791 | <p>$$\log2=\frac{2}{3}\left(1+\frac{1}{27}+\frac{1}{405}+\frac{1}{5103}+\frac{1}{59049}+\frac{1}{649539}+...\right)$$</p>
<p>The denominator is $(2k+1)9^k$.</p>
<p><a href="http://oeis.org/A155988" rel="nofollow">http://oeis.org/A155988</a></p>
<p>Gourdon and Sebah discuss the efficiency of this formula in
<a href="... |
544,673 | <p>let $\alpha$ be a cycle of length $s$, say $\alpha = (a_1, a_2, \ldots, a_s)$</p>
<p>Prove $\alpha^2$ is a cycle if and only if $s$ is odd.</p>
<p>Let me start off by saying I am in my 5th week of Group Theory. I often have trouble getting these problems started. This is my first proof based course. </p>
<p>I ... | Brian M. Scott | 12,042 | <p>HINT: Look at a couple of examples: $(1234)$ sends $1$ to $2$ and $2$ to $3$, so $(1234)^2$ sends $1$ to $3$. $(1234)$ sends $2$ to $3$ and $3$ to $4$, so $(1234)^2$ sends $2$ to $4$. $(1234)$ sends $3$ to $4$ and $4$ to $1$, so $(1234)^2$ sends $3$ to $1$. Finally, $(1234)$ sends $4$ to $1$ and $1$ to $2$, so $(123... |
3,671,675 | <blockquote>
<p>Question: Suppose <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> is a twice differentiable function with <span class="math-container">$f(0)=1$</span>, <span class="math-container">$f'(0)=0$</span> and satisfies <span class="math-container">$f''(x)-5f'(x)+6f(x)\ge 0$</span> for every <... | jvc | 686,748 | <p>When you want to compare two "solutions" of an two order EDO, it is often useful to consider the Wronskien. To avoid problem of division by 0, let's consider :
<span class="math-container">$$u = f + 2 e^{3x}$$</span>
and
<span class="math-container">$$v = 3 e^{2x}$$</span>
Define the Wronskien between <span class="... |
2,428,881 | <p><img src="https://i.stack.imgur.com/0STsn.png" alt="Trig Limits"></p>
<blockquote>
<p>I am confuse with the part of sin^2. I am not sure if there is a trig identity to simplify. I was thinking of trying to rearranging the equation so that it would be similar to sin(x)/x to solve it. Which direction do I go since ... | Community | -1 | <p>Drop the cosine and use a change of variable to absorb the $3$ under the sine:</p>
<p>$$\lim_{x\to0}\frac{\sin^2(3x)}{x^2\cos x}=\lim_{u\to0}\frac{\sin^2(u)}{\left(\dfrac u3\right)^2}=9\left(\lim_{u\to0}\frac{\sin u}u\right)^2.$$</p>
|
849,336 | <p>$$\eqalign{\tan^2\theta-\sec^2\theta &=\tan^2\theta-\dfrac1{\cos^2\theta}\\&=\dfrac{\sin^2\theta}{\cos^2\theta}-\dfrac1{\cos^2\theta}\\&=-\dfrac{\cos^2\theta}{\cos^2\theta}\\&=-1.}$$
note:$\theta≠ (2k+1)\frac{\pi}{2}$, $k$ : integer number </p>
<p>May i know if my answer is correct or not..</p>
| John Joy | 140,156 | <p>You seem to have a penchant for expressing everything in terms of sines and cosines when simplifying trigonometric expressions. I used to do the same (because that's the way that Teacher did it), but I soon found it to be too cumbersome.</p>
<p>After a while, I started using 3 different triangles to help me out. P... |
1,771,961 | <p>Suppose $U = \{(x,x,y,y) \in \mathbb{F}^4 : x, y \in \mathbb{F}\}$ </p>
<p>Find a subspace W of $\mathbb{F}^4$ such that $\mathbb{F}^4 = U\oplus W$</p>
<p>Attempt: Now from what I understand I would think that an element of $\mathbb{F}^4$ would look like $$(w,x,y,z) \quad\mbox{such that}\quad
w,x,y,z \in \mat... | Solumilkyu | 297,490 | <p>Let $W=\{(0,w,z,0)\in\mathbb{F}^4:w,z\in\mathbb{F}\}$, then we check that such $W$ is the desired subspace, as the following two steps.</p>
<ol>
<li>Given $(a,b,c,d)\in\mathbb{F}^4$, it is easy to decompose the vector
as below.
\begin{align}
(a,b,c,d)
&=(a,b-a+a,c-d+d,d)\\
&=(a,a,d,d)+(0,b-a,c-d,0),
\end{a... |
24,671 | <p>Adding to the <a href="https://math.stackexchange.com/search?q=%22for+dummies%22">for dummies</a>.</p>
<p><a href="http://www.paulsprojects.net/opengl/sh/technical.html" rel="noreferrer">The real spherical harmonics are orthonormal basis functions on the surface of a sphere.</a></p>
<p>I'd like to fully understand... | Ross Millikan | 1,827 | <p>$\theta$ and $\phi$ the coordinates of a spherical surface. They are similar to latitude ($\theta$) and longitude ($\phi$) except that $\theta$ goes from $0$ to $\pi$ and $\phi$ goes from $0$ to $2\pi$. Each harmonic has a value at every point, for example $Y_1^{-1}(\theta,\phi)=\frac{1}{2}\sqrt{\frac{3}{2\pi}}\si... |
24,671 | <p>Adding to the <a href="https://math.stackexchange.com/search?q=%22for+dummies%22">for dummies</a>.</p>
<p><a href="http://www.paulsprojects.net/opengl/sh/technical.html" rel="noreferrer">The real spherical harmonics are orthonormal basis functions on the surface of a sphere.</a></p>
<p>I'd like to fully understand... | bobobobo | 1,069 | <p>I think the point that was confusing me/missing link was that spherical harmonics functions <strong>are the solution of</strong> the Laplace's differential equation:</p>
<p>$$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0$$</p>
<p>Orthogonal means the functions... |
2,895,343 | <blockquote>
<p>Differentiate $\tan^3(x^2)$</p>
</blockquote>
<p>I first applied the chain rule and made $u=x^2$ and $g=\tan^3u$. I then calculated the derivative of $u$, which is $$u'=2x$$ and the derivative of $g$, which is
$$g'=3\tan^2u$$</p>
<p>I then applied the chain rule and multiplied them together, which ... | salvarico | 379,130 | <p>Your answer is not correct because $g' \neq 3\tan^2u$. For getting $g'$ you need to apply the chain rule once more.
$$g(u) = f(h(u)) = f(\tan u) = \tan^3u$$
where $f(x) = x^3$ and $h(u) = \tan u$.</p>
<p>So $g'(u) = f'(h(u))*h'(u) = 3(h(u))^2*\sec u = 3\tan^2u * \sec u$</p>
|
360,889 | <p>What can be said about publishing mathematical papers on e.g. viXra if the motivation is its low barriers and lack of experience with publishing papers and the idea behind it is to build up a reputation, provided the content of the publication suffices that purpose. </p>
<p>Can that way of getting a foot into the ... | Alexandre Eremenko | 25,510 | <p>If you want to build a reputation as a mathematician, post your preprint on the arXiv (this is a preprint server, not counted as publication, the posts are not refereed, and there are essentially no "barriers").</p>
<p>Then send your paper to a mainstream mathematical journal. Avoid those journals which are not rev... |
76,457 | <p>I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.</p>
<p>How do I determine if a point $(x,y)$ is within the area bounded by the ellipse? </p>
| marty cohen | 13,079 | <p>Another way uses the definition of the ellipse
as the points whose sum of distances to the foci is constant.</p>
<p>Get the foci at $(h+f, k)$
and $(h-f, k)$,
where $f = \sqrt{r_x^2 - r_y^2}$.</p>
<p>The sum of the distances
(by looking at the lines from
$(h, k+r_y)$ to the foci)
is
$2\sqrt{f^2 + r_y^2}
= 2 r_x
... |
211,123 | <p>I am writing some tests and would like to introduce large chunks of random alphabetic text (just characters a-z) to the input. The way I am generating the text now is like this:</p>
<pre><code>RandomString[length_Integer] :=
StringJoin[
Table[
FromLetterNumber[RandomInteger[{1, 26}]],
... | Alucard | 18,859 | <p>just in case someone is not interested in real words and wants something faster</p>
<pre><code>StringJoin@RandomChoice[Flatten@{Alphabet[], " "}, 100000]
</code></pre>
|
1,116,202 | <p>I am taking a class in which knowledge of gradients is a prerequisite. I am familiar with gradients but don't have too much experience, so I am having trouble understanding the following example.</p>
<p>$\theta, x \in \mathbb R^d$.</p>
<p>Define $\nabla J(\theta) = \begin{pmatrix} \frac{\partial}{\partial \theta_1... | velut luna | 139,981 | <ol>
<li><p>$$\frac{\partial}{\partial\theta_i}(\theta \cdot x)=\frac{\partial}{\partial\theta_i}(\theta_1x_1+\theta_2x_2+\cdots+\theta_ix_i+\cdots)=x_i$$</p></li>
<li><p>Gradient of a scalar $\it{is}$ a vector.</p></li>
</ol>
|
3,304,800 | <p>Given that <span class="math-container">$x^2-4cx+b^2 \gt 0$</span> <span class="math-container">$\:$</span> <span class="math-container">$\forall$</span> <span class="math-container">$x \in \mathbb{R}$</span> and <span class="math-container">$a^2+c^2-ab \lt 0$</span></p>
<p>Then find the Range of <span class="math-... | Sen47 | 684,888 | <p>This is not a complete solution. Just my opinion on the possible solution.</p>
<p>Cross multiplying and subtracting <span class="math-container">$x+a$</span> gives an equation:</p>
<p><span class="math-container">$x^2y+x(by-1)+(yc^2-a)=0$</span></p>
<p>The discriminant turns out to be:
<span class="math-container... |
1,110,122 | <p>Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd.</p>
<p>Computing this in wolfram says that it is uncomputable, which leads me to believe that the only way to find this would be methods for solving definite integrals. I am wondering if it is possible with any othe... | mrf | 19,440 | <p>Another silly answer, using complex analytic methods (similar to sos440's answer, but avoiding use of Bessel functions):</p>
<p>Rewrite the integrand using Euler's formulas and put $z = e^{ix}$, thus mapping $[-\pi,\pi]$ to the unit circle (some algebra omitted):
$$
\int_{-\pi}^\pi \sin \sin x \, dx =
-\frac1{2} \i... |
1,616,460 | <blockquote>
<p>If $p(x)$ be a quadratic equation with real coefficient satisfying $$x^2-2x+2\leq p(x)\leq 2x^2-4x+3\;\forall x\in \mathbb{R}$$ and $p(11) = 181.$ Then $p(16)= $ </p>
</blockquote>
<p>$\bf{My\; Try::}$ We can Write Inequality as $$(x-1)^2+1\leq p(x)\leq 2(x-1)^2+1$$</p>
<p>Here from above inequalit... | Eric Wofsey | 86,856 | <p>The second sentence of the proof of Corollary 9 is kind of poorly phrased. Here's a clearer phrasing:</p>
<blockquote>
<p>Suppose now that $L$ is an $S$-module, $K\subseteq N$ is a submodule, and $\psi:N/K\to L$ is an injective $R$-module homomorphism. Write $\varphi:N\to L$ for the composition of $\psi$ with t... |
690,822 | <p>Let <span class="math-container">$G$</span> be a graph of order <span class="math-container">$n$</span> and let <span class="math-container">$k$</span> be an integer with <span class="math-container">$1\leq k\leq n-1$</span>. Prove that if <span class="math-container">$\delta(G)\geq (n+k-2)/2$</span>, then <span cla... | bof | 111,012 | <p>Assume for a contradiction that $G$ is not $k$-connected. Since $n\gt k$, this means there is a set $S\subseteq V(G)$ such that $|S|=k-1$ and $G-S$ is disconnected. Then $G-S$ has $n-k+1$ vertices, and so the smallest component of $G-S$, call it $H$, has at most $\frac{n-k+1}2$ vertices. Let $v$ be any vertex in $H$... |
1,393,311 | <blockquote>
<p>Let $f:[0,1]\to[0,1]$ be a continuous function. Define $h:(0,1)\to[0,1]$ such that, $$h(x)=f(x)-\left\lfloor f(x)\right\rfloor$$Is $h$ continuous? Here $\left\lfloor x\right\rfloor$ is the <a href="https://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&cad=rja&ua... | DanielWainfleet | 254,665 | <p>This is Dominik's answer,with details: The set A is a compact subspace of some space S. The space A is compact iff any non-empty family F of closed subsets of A with the F.I.P. ...( Finite Intersection Property: i.e. if $G \subset F$ and $\phi \not =G$, then $\bigcap G \not = \phi$.) ... satisfies $\bigcap F \not =... |
251,799 | <p>I have a PDE in the form of
$$
\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + u = \delta(x-1),
$$
with initial condition $u(x,0)=100$. I'm trying to solve it numerically, but I have no idea on which method should I use. Most of the examples that I refer to has no delta function in the PDE.</p>
<p>... | doraemonpaul | 30,938 | <p>Note that this PDE just belongs to the PDE of the form <a href="http://eqworld.ipmnet.ru/en/solutions/fpde/fpde1302.pdf" rel="nofollow">http://eqworld.ipmnet.ru/en/solutions/fpde/fpde1302.pdf</a> so that exact solution can find easily, so you have not necessarily to solve it numerically.</p>
<p>The general solution... |
1,595,909 | <p>I am studying Measure Theory and I am stuck in some concepts about continuity of a measure.</p>
<p>Let $(S_{1}, \Sigma,\mu)$ be a measurable space, where $\mu$ is a probability mesure such that $\mu (S_{1}) = 1.$ Let also $S_{\delta} \subset \Sigma, 0\leq \delta \leq 1,$ be a family of sets such that $S_{\delta_{1}... | Jendrik Stelzner | 300,783 | <p>I see no problem in this.</p>
<p>By the theorem you have $\mu(\bigcup_{j=1}^\infty E_j) = c$ with $\bigcup_{j=1}^\infty E_j = \bigcup_{j=1}^\infty S_{1-1/j} \subseteq S_1$. That $c < 1 = \mu(S_1)$ just tells us that $\bigcup_{0 \leq \delta < 1} S_\delta = \bigcup_{j=1}^\infty S_{1-1/j} \subsetneq S_1$ is a pr... |
516,501 | <p>Find limit: $x_n=\dfrac{1+\frac12+...+\frac1{2^n}}{1+\frac14+...+\frac1{4^n}}$</p>
<p>as $n \rightarrow \infty$</p>
<p>My "intuition" says that it should be $\frac34$ but I don't know how to proove it with rigour.</p>
| Zarrax | 3,035 | <p>By induction you can prove the following formula for the sum of terms of a geometric progression:
$$a + ar + ar^2 + ... + ar^n = {a - ar^{n+1} \over 1 - r}$$
So you can apply this in the numerator with $a = 1, r = {1 \over 2}$, and in the denominator with $a = 1, r = {1 \over 4}$. You obtain
$$x_n = {1 - {1 \over 2^... |
516,501 | <p>Find limit: $x_n=\dfrac{1+\frac12+...+\frac1{2^n}}{1+\frac14+...+\frac1{4^n}}$</p>
<p>as $n \rightarrow \infty$</p>
<p>My "intuition" says that it should be $\frac34$ but I don't know how to proove it with rigour.</p>
| DonAntonio | 31,254 | <p>Geometric series</p>
<p>$$\sum_{n=k}^m r^n=r^k\frac{(r^{m+1}-1)}{r-1}$$</p>
<p>and then a little arithmetic of limits:</p>
<p>$$x_n=\frac{\frac{1-\frac1{2^n}}{1-\frac12}}{\frac{1-\frac1{4^n}}{1-\frac14}}=\frac{2-\frac{1}{2^{n-1}}}{\frac{4-\frac{1}{4^{n-1}}}{3}}\xrightarrow[n\to\infty]{}\frac{2-0}{\frac{4-0}3}=\fr... |
618,288 | <p>I know it's been answered before (at least to the case with $n$ different eigenvalues) but I didn't find a proof for the general case, and I would like some help with this question.</p>
<p>We are given linear transforms $S,T: V\to V$ where $V$ is some vector space.</p>
<p>We are given that $S$ and $T$ commute, $ST... | user44197 | 117,158 | <p>If $S$ and $T$ commute and both are diagonalizable then there is a common transform that will <em>simultaneously</em> diagonalize both.</p>
<p>First find a transform $M$ so that $M^{-1}SM$ is diagonal. Suppose that the eigenvalues of $S$ are sorted so repeated eigenvalues of $S$ are grouped together. </p>
<p>Let u... |
123,502 | <p>Suppose $A \subseteq \{1,\dots,n\}$ does not contain any arithmetic progressions of length $k+1$. What is the largest number of $k$-term arithmetic progressions that $A$ can have? (one may also wish to put some lower or upper on the size of $A$) We can work over $\mathbb{Z}_p$ if it makes the answer any easier. The ... | Gerhard Paseman | 3,528 | <p>I can offer some general common sense that might apply; it is derived from life
experience and not from the specific setting of my mentoring undergraduates
or being mentored as an undergraduate.</p>
<p>I find that practice is one way to develop ability in a particular skill set. I note
that many of the more respec... |
3,170,742 | <p>I have a system of recurrence relations in the following form:</p>
<p><span class="math-container">$$
\begin{pmatrix}
f(n+1)\\
g(n+1)\\
\end{pmatrix}
= \textbf{A}
\begin{pmatrix}
f(n)\\
g(n)\\
\end{pmatrix} +\vec{b}
$$</span></p>
<p>which hold for all <span class="math-container">$n \in \{ 0,1,2,...,N\}.$</span> ... | Vasili | 469,083 | <p>Hint: <span class="math-container">$\sin(\frac{\pi}{3}-x)=\sin \frac{\pi}{3}\cos x-\cos \frac{\pi}{3}\sin x$</span></p>
|
158,944 | <p>Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last?</p>
<p>I mean, not like a trig function where each intersect is the same distance from the last. (2,0); (4,0); (6,0); (8,0). And not like a spiral... | prajwal | 245,977 | <p>you can find the curves using matrix method by taking the columns as the x polynomials and multiply with the constant matrix which finally equated to given y.
but on which does the degree of x which have to be taken. on what does that depend.</p>
|
237,464 | <p>Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.</p>
<p>Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty f_k(a;b)$$
QUESTIONs: </p>
<p>(1) Does $f(a;b)=1$ have any solution with $a\neq 0$?</p>
<p>(2) If yes: Sing... | Iosif Pinelis | 36,721 | <p>Let us show that for each $a\in[-\frac12,\frac12]\setminus\{0\}$ there is a unique $b\in(0,\infty)$ such that $f(a;b)=0$.
By factoring $(2k+\frac{1}{2}\pm a)^2+b$
as $4 [k + \frac14\, (1 \pm 2 a - 2 i\sqrt b)][k + \frac14\, (1 \pm 2 a + 2 i\sqrt b)]$,
noting that $\prod_{k=1}^n(k+c)=\Gamma(n+1+c)/\Gamma(1+c)$ for... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | dtldarek | 26,306 | <p>I've tried to put it into words, but I guess the following picture is much better (of course, please alter the text in your mind so that it fits the current context).</p>
<p><a href="http://www.calamitiesofnature.com/archive/?c=323" rel="noreferrer"><img src="https://i.stack.imgur.com/owX5F.jpg" alt="Calamities of ... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | Ganesh | 1,585 | <p>A bit of background first: my area is Computer Science, and I've been a student as well as a professional software engineer. As such, my answer is tailored to this profession. As a CS student I'd learnt a fair amount of mathematics : discrete maths, logic, plus the standard engineering stuff: ODE, PDE, Transforms, O... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | b.sahu | 60,209 | <p>I have specialzed in High Voltage Engg.I have worked for about 8 years in Industry and more than 30 years in Academics. I used One percent of whatever Maths I have learnt in Industry and about Ten percent in Academics.</p>
<p>I cannot blame the designers of University Courses ,or the Industry or the Govt. for this ... |
289,757 | <p>I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician would say I have not heard a lot of math, for myself I would say that I... | Alex | 38,873 | <p>There's one (excellent!) real-life application of derivatives, the one that should be explained to people as early as in high school, I guess: neural classifiers aka ANNs. </p>
<p>My doctoral degree was, like, 80% mathematical and 20% computational, but my postdoc jobs are the exact opposite (probably even 90% comp... |
2,466,022 | <p>In Real and Complex Analysis, 3rd Edition, Walter Rudin advances the following:</p>
<p><a href="https://i.stack.imgur.com/XEVwK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XEVwK.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/dTwbg.png" rel="nofollow noreferrer... | Thorgott | 422,019 | <p>Hint: $\forall z\in\mathbb{C}:0\cdot z=0$</p>
|
3,203,346 | <p>I know that <span class="math-container">$f(0)=2$</span>, <span class="math-container">$f'(0)=3$</span> and <span class="math-container">$g=f^{-1}$</span>.</p>
<p>But how can I find the value of <span class="math-container">$g'(2)$</span>?</p>
| Olivier Roche | 649,615 | <p>Use the fact that <span class="math-container">$g \circ f = Id$</span>, and derive both sides in 0, you get:
<span class="math-container">$$
f'(0) \cdot g'(f(0)) = 1 \\
\textrm{thus }g'(f(0)) = \frac{1}{f'(0)}
$$</span>
Hence, <span class="math-container">$g'(2) = \frac{1}{3}$</span> .</p>
<p>Btw that's how you get... |
4,370,989 | <p>I have looked in some of my old books and found an exericse that I do not know how to solve. It seems pretty simple though.</p>
<p>The question is as follows:</p>
<pre><code>Which of these integers are prime?:
111
111.1
111.111
111.111.11
111.111.111
111.111.111.1
111.111.111.111
</code></pre>
<p>I remember one rul... | Jamie Radcliffe | 25,795 | <p>The group elements are <strong>not</strong> words in <span class="math-container">$a,b,c, \dots$</span>, but instead equivalence classes of words in <span class="math-container">$a,b,c,\dots,a^{-1}, b^{-1}, c^{-1},\dots$</span>. The equivalence relation is generated by all equivalences <span class="math-container">$... |
128,651 | <p>Let $M$ be a smooth manifold (maybe compact, if that helps). Denote by $\operatorname{Diff}(M)$ the group of diffeomorphisms $M\to M$ and by $R(M)$ the space of Riemannian metrics on $M$. We obtain a canonical group action
$$ R(M) \times \operatorname{Diff}(M) \to R(M), (g,F) \mapsto F^*g, $$
where $F^*g$ denotes th... | Eric O. Korman | 9 | <p>This map will not be transitive in general. For example, if $g$ is a metric and $\phi \in Diff(M)$ then the curvature of $\phi^* g$ is going to be the pullback of the curvature of $g$. So there's no way for a metric with zero curvature to be diffeomorphic to a manifold with non-zero curvature. Or for example, if ... |
3,741,122 | <p>Recently I've tried to find the difference between partial differentiation and total differentiation.
I've heard the total derivative is defined on single value functions, while the partial derivative by contrast is defined on multivariate functions.
My problem is, that total differentiation is used on multivariate ... | 5xum | 112,884 | <p>It simplifies to <span class="math-container">$\frac{\partial f}{\partial x_i}$</span>, because <span class="math-container">$\frac{\partial x_k}{\partial x_i}=0$</span> if <span class="math-container">$i\neq k$</span>.</p>
|
1,515,900 | <p>In many books and papers on analysis I met this equality without proof:</p>
<p>$$\sup \limits_{t\in[a,b]}f(t)-\inf \limits_{t\in[a,b]}f(t)=\sup \limits_{t,s\in[a,b]}|f(t)-f(s)|$$</p>
<p>Can anyone show strict and nice proof of that equality?</p>
<p>I would really grateful foe your help!</p>
| Nicolas | 213,738 | <p>We have
$$\sup_{t\in[a,b]}f(t)\geq f(t)\quad\quad\quad\forall t\in[a,b]$$
and
$$\inf_{t\in[a,b]}f(t)\leq f(t)\quad\quad\quad\forall t\in[a,b]$$
so
$$-\inf_{t\in[a,b]}f(t)\geq -f(t)\quad\quad\quad\forall t\in[a,b]$$
whence
$$\sup_{t\in[a,b]}f(t)-\inf_{t\in[a,b]}f(t)\geq f(t)-f(s)\quad\quad\quad\forall s,t\in[a,b].$$
... |
830,599 | <p>The function $f$ is defined as follows:
$$f(x):=\sum_{j=1}^{\infty} \frac{x^j}{j!} e^{-x}$$</p>
<p>It's easy to see that $f(0)=0$. But I am interested in the value
$$\lim_{x \rightarrow 0^+} f(x).$$</p>
<p>Even <a href="https://www.wolframalpha.com/input/?i=lim_%28x-%3E0%29+%28sum_%28i%3D1%29%5Einfinity+x%5Ej%2F%... | Eric Towers | 123,905 | <p>Wolfram Alpha can <a href="http://www.wolframalpha.com/input/?i=Limit%5BSum%5Bx%5Ej/j!%20E%5E-x,%20%7Bj,%201,%20%5C%5BInfinity%5D%7D%5D,%20x%20-%3E%200,%20Direction%20-%3E%20-1%5D" rel="nofollow">do this limit</a>. "Direction->-1" means to approach the limit from larger values.</p>
|
13,829 | <p>I was trying to understand the notion of a connection. I have heard in seminars that a connection is more or less a differential equation. I read the definition of Kozsul connection and I am trying to assimilate it. So far I cannot see why a connection is a differential equation. Please help me with some clarificati... | daniel.jackson | 3,237 | <p>There is a standard technique we learned for dealing with integrals of the form $(*) \int_{a}^{\infty}f(x)\cos (\alpha x)dx$ or $\int_{a}^{\infty}f(x)\sin (\alpha x)dx$ when $f(x)$ is continuous and positive in $[a, \infty)$.</p>
<ol>
<li><p>If $\int_{a}^{\infty}f(x)dx$ converges, then (*) converges absolutely by t... |
2,439,324 | <blockquote>
<p>Does there exist an uncountable set $A \subset \mathbb{R}$, such that for every $a \in A$ and every $\epsilon>0$, $(a-\epsilon,a+\epsilon)\not\subset A$?</p>
</blockquote>
<p>I am not sure what the answer is, but I am having trouble trying to construct such a set. Any hints?</p>
| Levon Haykazyan | 11,753 | <p>Hint: The set of rational numbers is dense and countable.</p>
|
1,371,549 | <p>Differentiate the Function : $y=\log_2(e^{-x} \cos(\pi x))$</p>
<p>Here is my work. What I have I done wrong?
<a href="https://i.stack.imgur.com/w9RSN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w9RSN.jpg" alt="enter image description here"></a></p>
| Bort | 256,093 | <p>Your result seems to be okay, you could and should however simplify it. Also, your calculation would have been simpler if you had said something like</p>
<p>$$y=\log_2 e^{-x} \cos\pi x \\
= \frac{-x + \log\cos\pi x}{\log 2} $$</p>
<p>and thus</p>
<p>$$y' \log 2 = -1-\pi\tan\pi x$$</p>
|
3,351,012 | <p><span class="math-container">$\int e^{2\theta}\sin(3\theta)d\theta$</span> seems to be leading me in circles. The integral I get when I use integration by parts, <span class="math-container">$\int e^{2\theta}\cos(3\theta)d\theta$</span> just leads me back to <span class="math-container">$\int e^{2\theta}\sin(3\theta... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>In case integration by parts is not mandatory,</p>
<p>find
<span class="math-container">$$\dfrac{e^{2x}(a\cos3x+b\sin3x)}{dx}$$</span> and compare with <span class="math-container">$e^{2x}\sin3x$</span> to find the values of <span class="math-container">$a,b$</span></p>
|
3,351,012 | <p><span class="math-container">$\int e^{2\theta}\sin(3\theta)d\theta$</span> seems to be leading me in circles. The integral I get when I use integration by parts, <span class="math-container">$\int e^{2\theta}\cos(3\theta)d\theta$</span> just leads me back to <span class="math-container">$\int e^{2\theta}\sin(3\theta... | Claude Leibovici | 82,404 | <p>As lab bhattacharjee answered, <em>in case integration by parts is not mandatory</em>, you can make life easier considering that what you need is the imaginary part of
<span class="math-container">$$I=\int e^{2\theta} e^{3i \theta}\,d\theta=\int e^{(2+3i)\theta}\,d\theta=\frac {e^{(2+3i)\theta}}{(2+3i)}=\frac{2-3i}... |
2,423,055 | <p>I am sort of baffled by this thing, already real number has every thing in it why is this concept of $\Bbb R^2$ ? What does it mean? What is its advantage?</p>
| Dave | 334,366 | <p>$\Bbb R$ is the set of real numbers. That is, $\Bbb R=\{x:\text{$x$ is a real number}\}$.</p>
<p>$\Bbb R^2$ is the set of pairs of real numbers. That is, $\Bbb R^2=\Bbb R\times\Bbb R=\{(x,y):\text{$x$ and $y$ are real numbers}\}$.</p>
|
801,680 | <p>Is it true that every point in a space equipped with the Hausdorff topology is closed?</p>
<p>I am assuming this is true but am having a difficult time proving it. Fix $a \in X$ where $X$ is equipped with the Hausdorff topology. If I can show that $X \setminus \{a\}$ is open then I am finished.</p>
<p>I know that ... | Ferra | 83,471 | <p>More generally, points are closed iff the topology is T1, and Hausdorff implies T1. A topology is T1 if for every pair of points $x,y$ there are open sets $U,V$ s.t. $U$ contains $x$ but not $y$ and $V$ contains $y$ but not $x$.</p>
<p>If points are closed, clearly the topology is T1. Conversely, take a point $x$ o... |
1,743,465 | <p>If $a=(1,2,3,4,5)$ is an example of a vector in $\mathbb R^5$, what could be an example of a vector in $\mathbb C^5$ ? is it $a=(1,2,3,4,i5)$ ?</p>
<p>Also, $x=a+ib$ is $2$ dimensional, can a complex number be one dimensional? like when $a=0$ or $b=0$, but if $b=0$ then it is a real number, so can we say that all... | Marc | 132,141 | <p>A vector in $\mathbb{C}^5$ is a vector of 5 elements $(x_1,\ldots,x_5)$, where each $x_i\in\mathbb{C}$. </p>
<p>About the dimensionality. This all depends on the base field you choose. It is true that $\mathbb{C}$ is two dimensional over $\mathbb{R}$, thus $\mathbb{C}^5$ is 10 dimensional over $\mathbb{R}$. However... |
482,030 | <p>I'm reading a proof of the irrationality of <span class="math-container">$\sqrt 2$</span>. In a step it states that <span class="math-container">$2d^2=n^2$</span> implies that <span class="math-container">$n$</span> is multiple of <span class="math-container">$2$</span>. How?</p>
| DonAntonio | 31,254 | <p>$\;2\;$ is a prime and divides the left side in $\,2d^2=n^2\;$ ,
so by the fundamental theorem of arithmetic it also divides the right hand side...</p>
|
482,030 | <p>I'm reading a proof of the irrationality of <span class="math-container">$\sqrt 2$</span>. In a step it states that <span class="math-container">$2d^2=n^2$</span> implies that <span class="math-container">$n$</span> is multiple of <span class="math-container">$2$</span>. How?</p>
| Glen O | 67,842 | <p>Since $d$ is an integer, $2d^2$ must be even. For any odd integer $n$, $n^2$ must also be odd. Therefore, $n$ must be even, and thus a multiple of 2.</p>
|
482,030 | <p>I'm reading a proof of the irrationality of <span class="math-container">$\sqrt 2$</span>. In a step it states that <span class="math-container">$2d^2=n^2$</span> implies that <span class="math-container">$n$</span> is multiple of <span class="math-container">$2$</span>. How?</p>
| Benjamin Dickman | 37,122 | <p>If $n^2 = 2d^2$, then $n^2$ is a multiple of $2$ hence even.</p>
<p>We now prove: $n^2$ even implies $n$ even.</p>
<p><strong>Proof:</strong> We tackle the contrapositive, i.e., $n$ odd implies $n^2$ odd.</p>
<p>Since $n$ is odd, we can write $n = 2k+1$ for some integer $k$.</p>
<p>Then $n^2 = (2k+1)^2 = 4k^2 + ... |
1,573,947 | <p>I'm trying to see the relationship between the sample variance equation </p>
<p>$\sum(X_i- \bar X)^2/(n-1)$ and the variance estimate, $\bar X(1-\bar X),$ in case of binary samples. </p>
<p>I wonder if the outputs are the same, or if not, what is the relationship between the two??</p>
<p>I'm trying to prove their... | BruceET | 221,800 | <p>I suppose your question is whether the two formulas give the
same answer for binary data. Here is an example to illustrate
that they are almost the same, but not exactly.</p>
<p>Suppose I have a sample of a thousand zeros and ones in which
there are 283 ones. Then $\bar X = 283/1000 = 0.283.$ Thus,
$\bar X(1-\bar X... |
1,901,170 | <blockquote>
<p>Consider the region $D$ bounded by the positive $y$-axis, the line $y=8$ and the line $y=x^3$. Evaluate the following integral between $x=0$ and $x=2$.
$$\iint_D \frac{x^3y^2}4\ dx\ dy$$</p>
</blockquote>
<p>What are the limits and what is the solution? I've asked people and they all give conflicti... | Andres Mejia | 297,998 | <p>This is not formal, but emphasizes a geometric intuition: <a href="http://rads.stackoverflow.com/amzn/click/0393925161" rel="nofollow">Div, Grad, Curl</a>.</p>
<p>Alternatively, there is a somewhat standard book, depending on what level of vector calculus you are looking for:
<a href="http://rads.stackoverflow.com/... |
1,558,210 | <p>In a linear algebra book I'm reading now, there was the following exercise:</p>
<blockquote>
<p>Let <span class="math-container">$W\subseteq V$</span> be a subspace of vector space <span class="math-container">$V$</span>. Do there always exist two subspaces <span class="math-container">$W_1,W_2\subseteq V$</span> su... | Daron | 53,993 | <p>Your lemma is true when suitable interpreted. </p>
<p>Of course you need $\phi$ to be a continuous function on $[a,b]$. </p>
<p>Then you define $\displaystyle \lim _{f(x) \to f(t)} \phi(x) = L$ to mean that for any $\epsilon >0$ there is $\delta > 0$ such that $|f(x)-f(t)|< \delta \implies |\phi(x)-L|<... |
3,654,315 | <blockquote>
<p>Let <span class="math-container">$m$</span> be an odd positive integer. Prove that</p>
<p><span class="math-container">$$ \dfrac{ \sin (mx) }{\sin x } = (-4)^{\frac{m-1}{2}} \prod_{1 \leq j
\leq \frac{(m-1)}{2} } \left( \sin^2 x - \sin^2 \left( \dfrac{ 2 \pi
j }{m } \right) \right) $$</span></p>
</bl... | dezdichado | 152,744 | <p>Too long for a comment: Look up <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials" rel="nofollow noreferrer">Chebyshev polynomial of second kind</a>. They are literally what you are dealing with:
<span class="math-container">$$U_n(\cos x) = \dfrac{\sin((n+1)x)}{\sin x}.$$</span></p>
<p>Your attempt at in... |
3,654,315 | <blockquote>
<p>Let <span class="math-container">$m$</span> be an odd positive integer. Prove that</p>
<p><span class="math-container">$$ \dfrac{ \sin (mx) }{\sin x } = (-4)^{\frac{m-1}{2}} \prod_{1 \leq j
\leq \frac{(m-1)}{2} } \left( \sin^2 x - \sin^2 \left( \dfrac{ 2 \pi
j }{m } \right) \right) $$</span></p>
</bl... | epi163sqrt | 132,007 | <p>This answer can be seen as supplement of @Conrads answer providing some more details.</p>
<blockquote>
<p>We start with the right-hand side of OPs identity. Letting <span class="math-container">$m=2k+1$</span> we obtain:
<span class="math-container">\begin{align*}
\color{blue}{(-4)^k}&\color{blue}{\prod_{j=1}^k\... |
893,959 | <p>I have a series of problems in inequalities that I can not solve,please help me if you can.</p>
<p>problem 1 :$a,b,c \geq 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $
prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$</p>
<p>problem 2 : $a,b,c\geq0$ and $a+b+c = 1$ prove that </p>
<p>1, $ \sqrt{a+\frac{(b... | Ahaan S. Rungta | 85,039 | <p><em>Sketch to #3</em></p>
<p>We can use the method of [Lagrange Multipliers]. Let $ f(a,b,c) = a + b + c $. This is our restraint function, where the restraint is $ f(a,b,c) = 1 $. Let $$ g(a,b,c) = \dfrac {1 + a^2}{1 + b^2} + \frac {1 + b^2}{1 + c^2} + \frac {1+c^2}{1 + a^2}. $$Then, we get $$ \nabla f = \left<... |
1,657,720 | <blockquote>
<p>Two tennis players, $A$ and $B$, are rendered equal in a game. It
takes two point lead for the winner. A player who has one point is
said to benefit. Assuming that $A$ has a probability $p$ of winning
each point and $B$ a probability $1-p$, independently of all other
points, determine: $(a)$ ... | carmichael561 | 314,708 | <p>After two points, either $A$ has won, $B$ has won, or the players are back where they started. These three events have probabilities $p^2,(1-p)^2$, and $2p(1-p)$, respectively.</p>
<p>Let $W$ be the event that $A$ wins, then from the above we have
$$ \mathbb{P}(W)=p^2+2p(1-p)\mathbb{P}(W)$$
hence
$$ \mathbb{P}(W)=\... |
1,657,720 | <blockquote>
<p>Two tennis players, $A$ and $B$, are rendered equal in a game. It
takes two point lead for the winner. A player who has one point is
said to benefit. Assuming that $A$ has a probability $p$ of winning
each point and $B$ a probability $1-p$, independently of all other
points, determine: $(a)$ ... | chandu1729 | 64,736 | <p>There are 5 possible states for the Markov Chain.</p>
<p>State 1: A and B are equal </p>
<p>State 2 : A benefit</p>
<p>State 3 : B benefit</p>
<p>State 4 : A Win</p>
<p>State 5 : B Win</p>
<p>Let the probability that A wins starting from the state i be $a_i$. It's clear that $a_4 = 1$ and $a_5 = 0$. We have th... |
11,435 | <p>I know what they are and how to define them. They have to serve some purpose - else why include them? But I never used them and all examples on this site and the docs never show a practical use of <code>UpValues</code>.</p>
<p>Can someone give me an example to see when they are actually needed?</p>
| Leonid Shifrin | 81 | <p>Oh yes. <code>UpValues</code> are used quite a bit. There are several common uses, and you may have a look at <a href="https://mathematica.stackexchange.com/questions/96/what-is-the-distinction-between-downvalues-upvalues-subvalues-and-ownvalues/">this</a> and especially <a href="https://stackoverflow.com/questions... |
1,606,023 | <blockquote>
<p>Given that $U_{n+1} = U_n-U_n^2$ and $0 < U_0 < 1$, show that $0 < U_n \leq 1/4$ for all $n \in \mathbb{N}^*$.</p>
</blockquote>
<p>Given that $S_n$ is the sum of $\frac{1}{1-U_k}$ from k=0 to n</p>
<p>Calculate $S_n$ in function of n </p>
<p>I see that $\frac{1}{1-U_{n}}=\frac{1}{U_{n+1}}... | Kim Jong Un | 136,641 | <p>Observe that $\frac{1}{4}-U_{n+1}=(U_n-\frac{1}{2})^2>0$. The inequality is strict because by the induction hypothesis $U_n\in(0,\frac{1}{4}]$ and so $U_n\neq\frac{1}{2}$. Similarly, $U_{n+1}=U_n(1-U_n)>0$ also because $U_n\in(0,\frac{1}{4}]$.</p>
<p>The base case is identical:
$$
\frac{1}{4}-U_1=(U_0-\frac{1... |
626,920 | <p>$a_n=\sum_{k=1}^{n} \frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}$</p>
<p>How to find $\lim a_n$?</p>
| Claude Leibovici | 82,404 | <p><strong>HINT</strong> </p>
<p>As Daniel Fisher suggested, rewrite 1/(n+k) as (1/n) 1/(1+k/n) and you are done through integration. I am sure you can take from here.</p>
<hr>
<p><em>Below is a reformatted version of this answer</em></p>
<p><strong>HINT</strong> </p>
<p>As Daniel Fisher suggested, rewrite $\d... |
3,805,452 | <p>Given a stick and break it randomly at two places, what is the probability that you can form a triangle from the pieces?</p>
<p>Here is my attempt and the answer does not match, so I am confused what went wrong with this argument.</p>
<p>I first denote the two randomly chosen positions by <span class="math-container... | Varun Vejalla | 595,055 | <p>The lengths of the sides of the stick will be <span class="math-container">$1-A, A-B$</span>, and <span class="math-container">$B$</span>. For the stick to form a valid triangle, the following three conditions must hold by the triangle inequality:</p>
<p><span class="math-container">$$\begin{align*}
1-A + A-B>B \... |
3,816,140 | <p>The points A, B, C, D have position vectors <strong>a</strong>, <strong>b</strong>, <strong>c</strong> and <strong>d</strong> respectively. Prove that the lines joining the midpoints of opposite edges of the tetrahedron ABCD bisect each other and give the position vector of the point of intersection.</p>
<p>I have s... | Robert Z | 299,698 | <p>Note that by <a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity" rel="nofollow noreferrer">Vandermonde's_identity</a>,
<span class="math-container">\begin{align*}
\sum_{i=1}^{n-3}\frac{\binom{n-3}{i}\binom{n+i-1}{i}}{i+1}\cdot(-1)^{i+1}&=\frac{1}{n-1}\sum_{i=1}^{n-3}\binom{n-3}{i}\binom{n+i-1}{i+1}\... |
764,437 | <p>I am trying to prove that $\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)$.</p>
<p>My initial plan was to use the first isomorphism theorem. I showed that there is a map $\phi: \mathbb{Z}[x] \rightarrow \mathbb{Z}[i]$, given by $\phi(f)=f(i).$ This map is onto and homorphic. The part I have a question on is showing th... | Community | -1 | <p>You can define your isomorphism as follows</p>
<p>$\phi:\mathbb{Z}[i]\rightarrow \mathbb{Z}[x]/\langle x^2+1 \rangle$ </p>
<p>$\phi(a+bi)=(a+bx)+\langle x^2+1 \rangle$ </p>
<p>If you compute $[a+bx+\langle x^2+1 \rangle]\times [c+dx+\langle x^2+1 \rangle]$</p>
<p>We get $ac+(ad+bc)x+bdx^2+\langle x^2+1 \rangle$.... |
1,191,740 | <p>$a_{n+1} - a_n = 3n^2 - n$ ;$a_0=3$</p>
<p>I need help for solving the particular solution.
Based on a chart in my textbook if you get $n^2$ the particular solution would be
$A_2n^2 + A_1n + A_0$ and $n$ has the particular solution of $A_1n+A_0$.<br>
So given $3n^2 - n$, my first thought was that if the equation w... | abel | 9,252 | <p>take a particular solution in the form $$a_n = An^3 + Bn^2+Cn,\\
3n^2 - n = a_{n+1} - a_n = A(n+1)^3 - An^3 + B(n+1)^2 - Bn^2+ C\\= 3An^2 + n(3A+2B) + A+B+C$$ equating the coefficients gives $$A = 1, B = -2, C=1. $$</p>
<p>the homogeneous solution is $ a_n = D $ so the general solution is $$a_n = n^3 -2n^2 +n + D... |
1,291,657 | <p>Is Bezout's lemma enough to confirm the GCD of a number?</p>
<p>So suppose we have <span class="math-container">$$ax+by=z$$</span> does this mean <span class="math-container">$$\gcd(a,b)=z$$</span></p>
| AlexR | 86,940 | <p>I note that $\mathrm{hcf}$ is usually denoted as $\gcd$ (<em>greatest common divisor</em>). Now from
$$ax+by = z$$
we can only see that $\gcd(a,b) \mid z$ (i.e. $z$ is a multiple of $\gcd(a,b)$) and by Bezout's lemma there exist <em>some</em> $(x,y)$ such that $z=\gcd(a,b)$.<br>
You can't infer anything else about $... |
99,617 | <p>How can I animate a point on a polar curve? I have used <code>Animate</code> and <code>Show</code> together before in order to get the curve and the moving point together on the same plot, but combining the polar plot and the point doesn't seem to be working because point only works with Cartesian coordinates.</p>
... | ubpdqn | 1,997 | <pre><code>Animate[
ParametricPlot[f[a, s, u], {u, 0, 15},
PlotRange -> {{0, 50}, {0, 30}},
Epilog -> {Red, PointSize[0.05], Point[f[a, s, t]]}], {a, 1,
20}, {s, 0, Pi/2}, {t, 0, 5},
Initialization :> (f[a_, s_, t_] :=
a t { Cos[s] , Sin[s]} - {0, 4.9 t^2})]
</code></pre>
<p><a href="https://... |
1,404,960 | <p>Say you have a function $f(x)$ and a line $g(x)=ax+b$. How do you reflect $f$ about $g$?</p>
<p>I am apparently supposed to write more text, but the line above is all I am after, hence I wrote this sentence as well.</p>
| user84413 | 84,413 | <p>Represent the graph of f parametrically by $x=t, y=f(t)$. </p>
<p>If we reflect the point $(t, f(t))$ in the line $y=ax+b$ to get the point $(x,y)$, then
$\color{red}{y-f(t)=-\frac{1}{a}(x-t)}$ </p>
<p>since the line and the line segment between $(t, f(t))$ and $(x,y)$ are perpendicular to each other.</p>
<p>We ... |
2,262,846 | <p>This question is somewhat naive. Please see the following proof before reading the question itself. </p>
<p>Let $L$ be a vector space over a field $\Bbb{F}.$ Assume there is an associative multiplication on $L$ so that $L$ is an associative ring, and denote this multiplication simply as $xy$ for $x,y\in L$. Further... | StackTD | 159,845 | <p><strong>Hint</strong>: you could use $\left(x^3-y^3\right)^2 \ge 0 \implies x^6+y^6 \ge 2x^3y^3$, so:
$$\left| \frac{x^3 y^4}{x^6 + y^6}\right| \le \left| \frac{x^3 y^4}{2x^3y^3}\right| = \ldots$$</p>
|
2,592,581 | <p>Lets define the following series
$$S_n =\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}.$$
I know that $S_0$ does not converge so let's suppose $n \in N$ and define $S_n$ for some $n$. We have $S_1=1$ , $S_2=\frac1 4$ , $S_3=\frac 1 {18}$ , $S_4=\frac 1{96}$ , $S_5=\frac 1{600}$ etc..<br/><br/>
the numerator in al... | Rohan Shinde | 463,895 | <p>$$\frac{1}{i(i+1)(i+2)\cdots(i+n)}$$
$$=\frac{1}{n}\left(\frac{1}{i(i+1)\cdots(i+n-1)}-\frac{1}{(i+1)(i+2)\cdots(i+n)}.\right)$$ </p>
<p>The first term of this series becomes
$$=\frac{1}{n}\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right) $$</p>
<p>The second term becomes
$$=\frac{1}{n}\left(\frac{1}{(n+1)!}-\frac{2}{(... |
2,592,581 | <p>Lets define the following series
$$S_n =\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}.$$
I know that $S_0$ does not converge so let's suppose $n \in N$ and define $S_n$ for some $n$. We have $S_1=1$ , $S_2=\frac1 4$ , $S_3=\frac 1 {18}$ , $S_4=\frac 1{96}$ , $S_5=\frac 1{600}$ etc..<br/><br/>
the numerator in al... | omegadot | 128,913 | <p>If you failed to see the telescoping nature of the series, here is an alternative approach that makes use of the <a href="https://en.wikipedia.org/wiki/Beta_function" rel="noreferrer">Beta function</a>.</p>
<p>\begin{align*}
S_n &= \sum_{i = 1}^\infty \frac{1}{i (i + 1) (i + 2) \cdots (i + n)} \quad n \in \math... |
2,300,386 | <blockquote>
<p>Consider the initial value problem:</p>
<p>\begin{cases}
u_{tt} &= c^2 u_{xx} \ \ & \text{for} \ -\infty < x < \infty, \ 0 \leq t < \infty\\
u(x,0) &= \phi(x) \ \ & \text{for} \ -\infty < x < \infty\\
u_t(x,0) &= \psi(x) \ \ & \text{for} \ -\infty < x &... | Winther | 147,873 | <p>Define $E_k(t) \equiv \frac{1}{2}\int_{\mathbb{R}}u_t^2(x,t){\rm d}x$ and $E_p(t) \equiv \frac{1}{2}\int_{\mathbb{R}}c^2u_x^2(x,t){\rm d}x$. Then taking the derivative and using integration by parts we get
$$\frac{d}{dt}(E_k(t) + E_p(t)) = \int_{\mathbb{R}}u_t[u_{tt} - c^2u_{xx}]{\rm d}x = 0$$
so the total energy is... |
2,946,408 | <p>Here is the question:
Find the volume of the region bounded by the <span class="math-container">$x$</span>-axis, <span class="math-container">$x=4$</span>, and <span class="math-container">$y=sqrt(x)$</span>. (rotated about x-axis)</p>
<p>I understand that we can find the cross sectional area at each point with res... | Phil H | 554,494 | <p>Your <span class="math-container">$\pi x$</span> just so happens to equal <span class="math-container">$\pi (\sqrt{x})^2$</span> for the disc method. For <span class="math-container">$\pi y^2$</span>, it does not equal <span class="math-container">$2\pi y(4 - y^2)$</span> for the shell method.</p>
|
2,245,010 | <p>Does there exist a topological group which can be covered by (nontrivial and proper) open subgroups of itself? If so, what are groups of these types called and is this a nice property for a topological group to have? Or is this just impossible?</p>
| Angina Seng | 436,618 | <p>There's $\Bbb Q_p$, the $p$-adic numbers under addition. This is an infinite, locally compact and totally disconnected group. The subgroups
$p^n \Bbb Z_p$ for $n\in\Bbb Z$ cover $\Bbb Q_p$ and are open therein
(and are compact to book).</p>
|
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