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 |
|---|---|---|---|---|
588,228 | <p>A,B,C are distinct digits of a three digit number such that </p>
<pre><code> A B C
B A C
+ C A B
____________
A B B C
</code></pre>
<p>Find the value of A+B+C.</p>
<p>a) 16 b) 17 c) 18 d) 19</p>
<... | md arif sarkar | 462,036 | <p>Let, $A=1, B=2, C=3$</p>
<p>The digit $A$ has $3$ times $=1\cdot 3=3$</p>
<p>The digit $B$ has $3$ times$=2\cdot 3=6$</p>
<p>The digit $C$ has $3$ times $=3\cdot 3=9$<br>
Now:</p>
<p>$A+B+C=3+6+9=18$</p>
|
1,515,645 | <p>The question is to find the values of a real number $\lambda$ for which the following equation is satisfied for all real values of $\alpha$ which are not integral multiples of $\pi/2$
$${\sin\lambda\alpha\over \sin\alpha}-{\cos\lambda\alpha\over \cos\alpha}=\lambda-1$$</p>
<p>All I could do was to guess some values... | fleablood | 280,126 | <p>Let <span class="math-container">$1111111111111111 \equiv k \pmod {17}$</span>.</p>
<p>Then <span class="math-container">$9999999999999999 \equiv 9k\pmod {17}$</span></p>
<p>And <span class="math-container">$10000000000000000 =10^{16} \equiv 9k + 1\pmod {17}$</span>.</p>
<p>ANd but Fermat's little theorem:</p>
<p><s... |
259,431 | <p>In the book of Richard Hammack, I come accross with the following question:</p>
<blockquote>
<p>There are two different equivalence relations on the set $A = \{a,b\}$.
Describe them.</p>
</blockquote>
<p>OK, I found that the solution is,</p>
<p>$$R_1 = \{(a,a),(b,b),(a,b),(b,a)$$
and
$$R_2 = \{(a,a),(b,b)\}$$<... | smackcrane | 12,039 | <p>$R_3 = \{(a, a)\}$ and $R_4 = \{(b, b)\}$ are not equivalence relations because they are not reflexive; $b$ must be related to itself.</p>
|
653,887 | <p>Why is the following set linearly independent for all x on ($-\infty$, $\infty$)?</p>
<p>$$\{1+x, 1-x, 1-3x\}$$</p>
<p>The Wronskian is $0$, but Wolfram Alpha says it is still linear independent? Why is this?</p>
<p>Thanks!</p>
<p><a href="http://www.wolframalpha.com/input/?i=Is+%7B1-x%2C+1%2Bx%2C+1-3x%7D+linea... | Amzoti | 38,839 | <p>Hint:</p>
<p>Since the Wronskian is zero, no conclusion can be drawn about linear independence.</p>
<p>For linear independence, we want to go back to the basic definitions.</p>
<p>Can you proceed?</p>
|
1,600,266 | <p>I'm trying to work out sum of this series $$1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} + \ldots$$</p>
<p>I know one method is to do substitutions and getting the series into a form of a known series. So far I've converted the series into $$ 1 + \frac{2}{x} + \frac{3}{x^2} + \frac{4}{x^3} + \ldots $$ where $x=... | Brian Tung | 224,454 | <p>Not sure what the book has in mind there. Perhaps there is a line missing? Is it possible to give a bit more context than you have so far?</p>
<p>Anyway, some remarks about divisibility for binary numbers: A binary number is divisible by $3$ (i.e., $11_2$) iff the number of even-indexed $1$'s and the number of od... |
835,454 | <p>I was struggling with the following problem (from linear algebra):</p>
<blockquote>
<p>Let $V$ be the vector space of the $2 \times 2$ matrices with real coefficients. Consider the action of the group $SL_2(\mathbb{R})$ on $V$, namely for any matrix $T \in SL_2(\mathbb{R})$ and for every matrx $A \in V$ we define... | ThePunisher | 88,132 | <p>I have tried to show the actual homomorphism. I found the matrix in $GL_4(\mathbb{R})$ associated to the general matrix in $SL_2(\mathbb{R})$, namely:
$$ \left( \begin{matrix} a & b \\ c & d\end{matrix} \right) \quad \text{with} \ ad-bc\neq0 $$
And it should be the following:
$$ \left( \begin{matrix}ad &... |
2,278,120 | <p>In one of my algorithm courses, there is this: </p>
<blockquote>
<p>A subset $S$ of vertices in a directed graph $G$ is
strongly connected if for each pair of distinct
vertices ($v_i$, $v_j$) in $S$, $v_i$ is connected to $v_j$ and
$v_j$ is connected to $v_i$.</p>
</blockquote>
<p>And then the following ex... | Jose Arnaldo Bebita Dris | 28,816 | <p>I think "for each pair of distinct vertices $(v_i, v_j)$ in $S \times S$" means</p>
<blockquote>
<p>"for each $(v_i, v_j) \in S \times S$, $v_i \neq v_j$."</p>
</blockquote>
|
2,785,449 | <p>So, it is obvious that the empty set is a subset of every set, such that <strong>∀A:∅⊆A</strong>.</p>
<p>Therefore, if the null set is an element of some set A, such that <strong>A = {∅}</strong>, is the empty set considered a singleton element of A?</p>
<p>Thank you.</p>
| D. Brogan | 404,162 | <p>Yes, the set $\{\varnothing\}$ is a set whose only element is the empty set $\varnothing$.</p>
|
2,161,913 | <p>Consider the angle $\theta$ with $$0<\theta<\frac{\pi}{2}$$ Suppose that $$cos\theta=\frac{l}{n} \space and \space \space sin\theta=\frac{m}{n}$$
are rational numbers, with postive integers $l,m$ and $n$. Show that $l$ and $m$ cannot both odd integers.<br>
MY ATTEMPT: Assume that $l$ and $m$ are both odd int... | Claude Leibovici | 82,404 | <p><em>Much less elegant than RRL'answer.</em></p>
<p>Consider $$P_n=\left(\prod _{i=1}^n \sec \left(\frac{1}{i}\right)\right){}^{\frac{1}{n}}$$ $$\log(P_n)=\frac 1n \sum _{i=1}^n \log\left(\sec \left(\frac{1}{i}\right)\right)$$ Now, by Taylor for large values of $i$ $$\sec \left(\frac{1}{i}\right)=1+\frac{1}{2 i^2}+\... |
4,037,206 | <p><a href="https://i.stack.imgur.com/VfORl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VfORl.png" alt="enter image description here" /></a></p>
<p>I have done the following for the median <br/>
<span class="math-container">$2(x-1) = .50$</span> <br/>
results in <span class="math-container">$x = ... | BruceET | 221,800 | <p><strong>An approach using a <a href="https://en.wikipedia.org/wiki/Beta_distribution" rel="nofollow noreferrer">beta distribution</a>.</strong> Let <span class="math-container">$Y \sim \mathsf{Beta}(2,1).$</span>
Then <span class="math-container">$X = Y+1.$</span> Results are the same as in @heropup's Answer (+1).<... |
1,609,446 | <p>I read the definition that $f$ is in $O(x^n)$ if $|f(x)|<C|x^n|$ for some $C$.</p>
<p>I'm struggling to understand how to check this. For example, supposedly $f(x) = 5x+3x^2$ is in $O(x)$ but not $O(x^2)$?</p>
<p>If I plot $f(x) = 5x + 3x^2$ and $g(x)=x$ I see that the first goes to infinity much quicker.</p>
... | ml0105 | 135,298 | <p>The function $f(x) = 5x + 3x^{2} \not \in O(x)$, but it is in $O(x^{2})$. The way you show that $f(x) \in O(x^{2})$ is by picking a constant $C$ and appropriate constant $k$ such that $|f(x)| \leq Cx^{2}$ for all $x \geq k$.</p>
<p>Big-O can often be ascertained using limits. If the following condition holds:</p>
... |
1,335,734 | <blockquote>
<p>A library leases its photocopier. One monthly bill was 750 dollars for
12,000 copies. Another month, the bill was $862.50 for 16,500 copies.
How much does the library pay for each copy?</p>
</blockquote>
<p>How should I start set up the equation for this problem?
How exactly do I find the linear... | TomMonTom | 237,366 | <p>Thank you Sky, I see now that upon one part of the chair rule becoming a complete polynomial with a coefficient it can then be distributed amongst the single variables of y<em>dy/dx and x respectively without having to foil as if it were (2x+2x)</em>(x^2+y^2).</p>
|
187,511 | <p>There are two questions:</p>
<ol>
<li><p>How to prove that in general </p>
<p>$[\hat{A}(\mathbb HP^m)]_{4m} = 0$</p>
<p>It is possible to verify it for low values of $m$.</p></li>
<li><p>How to prove that in general</p>
<p>$\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$</p>
<p... | Juan Ospina | 61,860 | <p>It is possible to obtain a more direct proof using other idea due to Professor Ebert.</p>
<p>It is well known that </p>
<p>$\hat{A} = \prod _ i{\frac {x_{{i}}/2}{\sinh \left( x_{{i}}/2 \right) }}= 1+\hat{A}_{1}+\hat{A}_{2}+\hat{A}_{3}+....$</p>
<p>where </p>
<p>$\hat{A}_{1}=-\frac{1}{24}p_{{1}}$</p>
<p>$\hat{A}... |
3,190,781 | <blockquote>
<p>Suppose the polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfies the following conditions:<br>
(A) If <span class="math-container">$P(x)$</span> is divided by <span class="math-container">$x^2 − 4x + 3$</span>, the remainder is <span class="math-container">$65x −... | trancelocation | 467,003 | <p>You need only <span class="math-container">$P(1)$</span>, since</p>
<ul>
<li><span class="math-container">$x^2-4x+3 = (x-1)(x-3)$</span> and</li>
<li><span class="math-container">$x^2+6x-7 = (x-1)(x+7)$</span></li>
</ul>
<p>Hence,</p>
<ul>
<li><span class="math-container">$P(1) = 65-68 = -3$</span></li>
<li><span... |
3,411,982 | <p>I know the answer using
Total-none of it is <span class="math-container">$6 = 900−648=252$</span></p>
<p>My doubt is that if we you it like</p>
<p>when unit digit is <span class="math-container">$6 = 9\cdot10\cdot1 = 90$</span></p>
<p>when tens place is <span class="math-container">$6 = 9\cdot1\cdot10 = 90$</span... | Axion004 | 258,202 | <p>Suppose that 'three-digit' means <span class="math-container">$abc$</span>, where <span class="math-container">$a>0$</span>. </p>
<p>Now, we first count that there are <span class="math-container">$900$</span> of these numbers.</p>
<p>Of numbers without a <span class="math-container">$6$</span>, then it's <spa... |
1,007,529 | <p>Suppose that the function $f:[-1,1]\rightarrow[-1,1]$ is continuous. Use the intermediate value theorem to prove that there exists a $c \in [-1,1]$ such that $f(c)=c^5$. You should carefully justify each of the hypothesis of the theorem. Question is to be done very formally.</p>
<p>First can I just say, shouldn't $... | Ben Grossmann | 81,360 | <p>To your first question, $c$ is not necessarily in $(-1,1)$. For example, what if $f(x) = x$?</p>
<p>Also, the question asks about an arbitrary function $f(x)$, so you shouldn't define $f(x) = x^5$.</p>
<p>Here's a hint: consider the function $g(x) = f(x) - x^5$. If $g(-1)$ or $g(1)$ is equal to $0$, the we're do... |
4,155,453 | <p>I am trying to evaluate <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^\infty \frac{n}{n^2+k^2}.$$</span> Now I am aware that clearly <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^n \frac{n}{n^2+k^2} = \int_0^1 \frac{1}{1+x^2}dx = \tan^{-1}(1) = \frac{\pi}{4},$$</span> but I do not know ... | NoName | 649,394 | <p>Denote <span class="math-container">$\displaystyle \mathcal{S}(n) = \sum_{k \ge 1} \frac{n}{n^2+k^2}$</span>. We have</p>
<p><span class="math-container">\begin{aligned} \mathcal{S}(n) & = \sum_{k\geq 1}\frac{n}{ n^2+k^2} \\& = \frac{1}{2}\sum_{k\geq 1}\frac{2n }{ n^2+k^2} \\& = \frac{1}{2}\sum_{k\geq 1... |
3,133,152 | <blockquote>
<p><span class="math-container">$g(x)=|\sin{x}-1|+|3-\cos{x}-\sin{x}|+2\sin{x}$</span></p>
<p>Answer: Above equality is simplified to <span class="math-container">$$1-\sin{x}+3-\cos{x}-\sin{x}+2\sin{x}=4-\cos{x}$$</span></p>
</blockquote>
<p><span class="math-container">$$-1 \le\sin{x}\le1$$</span></p>
<p>... | farruhota | 425,072 | <p>Yes, usually, the absolute value is defined as (see <a href="https://en.wikipedia.org/wiki/Absolute_value#Real_numbers" rel="nofollow noreferrer">Wikipedia</a>):
<span class="math-container">$$|x|=\begin{cases}\ \ \ x,\ x\ge 0\\ -x,\ x<0\end{cases}$$</span>
However, it can be adjusted (see the opening paragraph i... |
2,335,334 | <p>I'm trying to solve the following ODE:
$$3xy''+(3x+1)y'+y=0$$
So I started by assuming a solution(and its derivates) with the form:</p>
<p>$$
\tag{1}y = \sum_{n=0}^{\infty } a_{n} x^{n + r}$$</p>
<p>$$y' = \sum_{n=0}^{\infty } {(n+r)}a_{n} x^{n + r-1}$$</p>
<p>$$y'' = \sum_{n=0}^{\infty } {(n+r-1)(n+r)}a_{n} x^{n... | Tom Himler | 457,359 | <p>So the most direct way of telling you what the answer is is this:</p>
<p>For each question you have 1/2 chance of getting the right answer.</p>
<p>Since there are four questions, each has a 1/2 chance of getting the right answer. So to find it, you do (1/2)x(1/2)x(1/2)x(1/2). Therefore, for this case, there is a 1... |
3,579,457 | <p>Last Christmas, a kid in my neighborhood dressed in a pirate suit and knocked my door for "trick or treat". He explained to me that Dec 25, which is 25 written in base 10, is equal to 31 in base 8, which is Oct 31. So ingeniously, Halloween coincides with Christmas. </p>
<p>I really wonder which other two dates wou... | Hagen von Eitzen | 39,174 | <p>The point in the original "date identity" is that the abbreviations Dec and Oct for the month names are also interpreted as abbreviations for the base, i.e., "decimal 25 = octal 31" becomes "Dec 25 = Oct 31". There is no month name abbreviated as Bin or Hex, for example, so binary and hexadecimal can be excluded. I... |
245,880 | <p>A box contains 100 balls. Each ball has a number from 1 to 10. How many balls should I draw (ball is put back in box after drawing) to predict the number of balls for each number with 95% certainty.</p>
<p>A prediction is correct if for each of the 10 numbers, the number of balls with that number is correctly predi... | Tunococ | 12,594 | <p>If I understand correctly, your problem is estimating the parameter of a Bernoulli distribution. I will attempt to find an approximate answer to the problem using CLT. (This is not an exact solution.)</p>
<p>Suppose we're interested in estimating the number of balls with number 1 written on them. Consider a draw a ... |
245,880 | <p>A box contains 100 balls. Each ball has a number from 1 to 10. How many balls should I draw (ball is put back in box after drawing) to predict the number of balls for each number with 95% certainty.</p>
<p>A prediction is correct if for each of the 10 numbers, the number of balls with that number is correctly predi... | Danica | 19,147 | <p>You're basically asking for confidence intervals on the probabilities of a <a href="http://en.wikipedia.org/wiki/Multinomial_distribution" rel="nofollow">multinomial distribution</a>.</p>
<p>Let $Q$ be the number of balls in the box (you said 100), of which there are distinct labels 1 to $m$ (you said 10). Let's sa... |
77,466 | <p>There is a standard syllabus for a first graduate course in algebra. One teaches groups,
rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of
category theory, perhaps a little bit of representation theory, all this a little bit superficially, to give an idea of the fundamental
algebraic st... | David White | 11,540 | <p>As a graduate student in algebraic topology, but one who has taken many "second year" graduate courses in algebra, the one I think I would have enjoyed the most had it ever been offered (and the one which would have been most useful for me personally) would go something like this:</p>
<p>Textbook: <em><a href="http... |
77,466 | <p>There is a standard syllabus for a first graduate course in algebra. One teaches groups,
rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of
category theory, perhaps a little bit of representation theory, all this a little bit superficially, to give an idea of the fundamental
algebraic st... | Joe Johnson | 10,206 | <p>When I took the second graduate algebra course we spent the first half on noncommutative ring theory. We covered topics like Jacobson radicals, artinian/noetherian rings, semi-simple algebras. We used Herstein and Jacobson's algebra books. The second half was all the ground work for complex representations of fin... |
1,016,585 | <p>I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$</p>
<p>How do I do this?</p>
<p>It looks like it is close to being an $\operatorname{arcsin}$?</p>
<hr>
<p>I would have thought I could just do:
$$\int \left(\sqrt{x^2 - c}\right)^{-\frac12}\, dx=\frac{2\sqrt{c+x^2}}{2x}\text{??... | 5xum | 112,884 | <p><strong>HINT</strong>:
$$\frac{1}{\sqrt{x^2-c}} = \frac{1}{\sqrt{c}\sqrt{\frac{x^2}{c} - 1}} = \frac{1}{\sqrt{c}}\cdot \frac{1}{\sqrt{\left(\frac{x}{\sqrt{c}}\right)^2 - 1}}$$</p>
|
30,970 | <p>I try to filter sublists of a list which match a pattern. </p>
<pre><code>test = {{"String1", "a"}, {"String2", "b"}, {"String3",
"a"}, {"String4", "a"}};
</code></pre>
<p>The result should be:</p>
<pre><code>result = {{"String1", "a"}, {"String3", "a"}, {"String4", "a"}}
</code></pre>
<p>That means the first e... | halirutan | 187 | <p>If you use <code>Cases</code>, then you can give the pattern directly</p>
<pre><code>Cases[test, {_String, "a"}]
</code></pre>
<p>If you want to stick with <code>Select</code> you can do this as well but you have to transform your pattern match into a <em>test function</em></p>
<pre><code>Select[test, MatchQ[#, {... |
1,782,796 | <p>Is there a particular name given to a matrix of <strong><em>m</em></strong> rows and <strong><em>n</em></strong> columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance:</p>
<pre><code>0 1 0 0 0 0 0
0 0 0 0 1 0 0
0 1 0 0 0 0 0
0 0 0 0 0 0 1
0 0 1 0 0 0 0
</code></pre>
<p>Note: I... | L. Fermín | 523,546 | <p>Permutation Matrix
<a href="https://en.wikipedia.org/wiki/Permutation_matrix" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Permutation_matrix</a></p>
<p>It applies to squared matrices, but you can generalize for nonsquare matrices</p>
|
2,170,941 | <p>I know the derivative of position vector function $r(t)$ is velocity, and its unit vector is tangent vector, therefore, the derivative of tangent vector is the unit vector of acceleration and normal vector. <strong>Why isn't that the case?</strong></p>
| Daniel Robert-Nicoud | 60,713 | <p>By definition, acceleration is the variation of velocity in time, and thus the derivative of the velocity function.</p>
<p>Write $x(t)$ for the position, $v(t):=\dot x(t)$ for the velocity, and $a(t):=\dot v(t)$ for the acceleration. The direction vector is
$$d(t):=\frac{v(t)}{\|v(t)\|}\ .$$
If we differentiate it,... |
2,170,941 | <p>I know the derivative of position vector function $r(t)$ is velocity, and its unit vector is tangent vector, therefore, the derivative of tangent vector is the unit vector of acceleration and normal vector. <strong>Why isn't that the case?</strong></p>
| Doug M | 317,162 | <p>If speed $(\|r'(t)\|)$ is constant, then the acceleration must be normal to the direction of travel.</p>
<p>But if speed is variable then acceleration can be broken into a component parallel to the direction of travel and a component that is perpendicular to the direction of travel.</p>
|
2,211,461 | <p>Given that $f$ is an integrable function on $X$ and $\{E_k\}_{k=1}^\infty$ where each $E_k$ is a measurable set such that $\lim_{k\rightarrow \infty} \mu(E_k) = 0$</p>
<p>Can we show that $$\lim_{k\rightarrow \infty} \int_{E_k} fd\mu = 0$$ </p>
<p>I want to prove like this:
$$|\int_{E_k} fd\mu| \leq sup|f|\cdot \m... | RiezFrechetKolmogorov | 390,584 | <p>Write integral over $E_k$ as integral of f*indicator ($E_k$) = $f_k$ , it should work</p>
|
1,381,017 | <p>Could you have me to find the ferivative of $$f(t)=\frac {1}{\rho}\log (1+\rho t)$$ with repsect to $t$?</p>
<p>And Is it
$$\lim_{t\to \infty} \frac {f'(t)}{t}=1?$$
<em>Update:</em>
Based on Hint of Surb:</p>
<p>$$f(t)'=\frac {1}{1+\rho t}$$</p>
<p>Then
$$\lim_{t\to \infty} \frac {f'(t)}{t}=0$$</p>
<p>Is it co... | Peter | 82,961 | <p>Wolfram gives the following result :</p>
<p><a href="http://www.wolframalpha.com/input/?i=integral+sqrt%28sin%28x%29cos%28x%29%29" rel="nofollow">http://www.wolframalpha.com/input/?i=integral+sqrt%28sin%28x%29cos%28x%29%29</a></p>
<p>As you can see, there is no "nice" antiderivate to $\sqrt{sin(x)cos(x)}$</p>
|
27,896 | <p>How would I solve these differential equations? Thanks so much for the help!</p>
<p>$$P'_0(t) = \alpha P_1(t) - \beta P_0(t)$$
$$P'_1(t) = \beta P_0(t) - \alpha P_1(t)$$</p>
<p>We also know $P_0(t)+P_1(t)=1$</p>
| Community | -1 | <p>Note that from the equation you have $$P'_0(t) = \alpha P_1(t) - \beta P_0(t) = -P'_1(t)$$ which gives us $P'_0(t) + P'_1(t) = 0$ which gives us $P_0(t) + P_1(t) = c$. We are given that $c=1$. Use this now to eliminate one in terms of the other.</p>
<p>For instance, $P_1(t) = 1-P_0(t)$ and hence we ... |
800,802 | <p>Can someone give me a reference in which I can find the following result </p>
<p>Let $C$ be a curve, then $$g(C)=\frac{(n-1)(n-2)}{2}-s.$$</p>
<p>where $g=$ genus of $C,$ $n=$ degree of curve, $s=$ number of singular points.</p>
<p>Any help would be appreciated.</p>
| Marco | 149,606 | <p>You can try books on algebraic geometry, and look for the genus formula, or Plücker formulas. </p>
|
3,912,076 | <p>A particle of mass m slides down on the curve <span class="math-container">$z=1+\frac{x^2}{2}$</span> in the <span class="math-container">$xz$</span>-plane without friction under the action of constant gravity.Suppose <span class="math-container">$z$</span>-axis points vertically upward.Find the Lagrangian of given ... | Narasimham | 95,860 | <p>The Lagrangian is set up with Mechanics energy formulation and DE set up using Euler Lagrange Equation.</p>
<p>Angles <span class="math-container">$ \phi$</span> are tangent or normal rotations positive counterclockwise.</p>
<p>Parabola trough coordinates are taken wlog <span class="math-container">$ R=1, z_{min}... |
978,202 | <p>When I use $\sin x \sim x$ , the answer is $1$ , is the answer correct?</p>
| deepesh kumar singh | 185,074 | <p>in the limit $x \to \infty$, the numerator is bounded between $[-1,1]$ while the denominator approach infinity so the limit must be zero. </p>
|
3,787,576 | <p><span class="math-container">$$
\mbox{Prove}\quad
\int_{0}^{1}{\mathrm{d}x \over
\left(\,{x - 2}\,\right)\,
\sqrt[\Large 5]{\,x^{2}\,\left(\,{1 - x}\,\right)^{3}\,}\,}
=
-\,{2^{11/10}\,\pi \over \,\sqrt{\,{5 + \,\sqrt{\,{5}\,}}\,}\,}
$$</span></p>
<ul>
<li>Being honest I havent got a clue where to start. I dont thi... | Felix Marin | 85,343 | <p><span class="math-container">$\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{\exp... |
1,373,759 | <p>Given the natural number $n$,who is in the form $p^2 \cdot q^2$,with $p$,$q$ prime numbers.Also $φ(n)$ is given.Describe a fast algorithm(polynomial time) that calculates the $p$ and $q$.Apply your algorithm to calculate the $p$,$q$ when $n=34969$ and $φ(n)=29920$</p>
<p>I found this problem on a mathematical compe... | rpf | 256,834 | <p>Recall $\sqrt{gT} = \sqrt{g} * \sqrt{T}$.
As the book indicates, when calculating partial derivatives, we let only one variable change while the others are constant. </p>
<p>Say you're looking for $\frac{dn}{dT}$, then think of $n$ as </p>
<p>$n = constant * \sqrt{T}$. </p>
<p>Now calculating $\frac{dn}{dT}$ sho... |
3,669,266 | <p>The equation <span class="math-container">$(1-x)^n = x$</span> has a solution in <span class="math-container">$x' \in (0,1)$</span> and indeed the solution <span class="math-container">$x' \to 0$</span> as <span class="math-container">$n \to \infty$</span>. (Consider <span class="math-container">$f(x) = (1-x)^n$</s... | Sangchul Lee | 9,340 | <p>Let <span class="math-container">$x_n$</span> be the unique solution of the equation <span class="math-container">$(1-x)^n=x$</span> on <span class="math-container">$(0, 1)$</span>.</p>
<ol>
<li><p>Let <span class="math-container">$f(x) = (1 - x)^n - x $</span>. Then by using the inequality <span class="math-contai... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | José Figueroa-O'Farrill | 394 | <p>The <a href="http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture" rel="nofollow">Poincaré conjecture</a> is an example of a purely topological statement which apparently cannot be proved only by topological means.</p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | José Figueroa-O'Farrill | 394 | <p>Many geometric problems cannot be solved without hard analysis. Perhaps the best known example is the <a href="http://en.wikipedia.org/wiki/Calabi_conjecture" rel="nofollow">Calabi Conjecture</a> proved by Yau.</p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Darsh Ranjan | 302 | <p>Mathematical logic can be motivated by other areas of math in at least two different ways: </p>
<p>[1] It allows you to formulate (and prove) results about the unsolvability of certain problems. These are obviously essential, since they tell you that you shouldn't spend too much time trying to solve those problems,... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Douglas Zare | 2,954 | <p><strong>Problem:</strong> You need to multiply large numbers, with $10^9$ digits (or take products of power series). Your computer doesn't have the ability to do $10^{18}$ calculations. </p>
<p><strong>Solution:</strong> Recognize multiplication of a power series as a convolution. Take a discrete Fourier transform ... |
1,072,427 | <p>Prove using contradiction that any prime number greater than $3$ is of the form $6n \pm 1$.</p>
<p>Thanks for any help</p>
| turkeyhundt | 115,823 | <p>Can you divide $(6n+2)$ by anything? What about $(6n+3)$?</p>
|
128,935 | <p>Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper <em>Über die nichtarchimedischen Grössensysteme</em> (Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse 116 (Abteilung IIa), 1... | Asaf Karagila | 7,206 | <p>I don't know the answer to (1), and would be glad to give it some thought later this week. Regardless to (1) the answer to (2) is semi-negative.</p>
<p>There are two conjectures which seem to be slightly older (although not by much) than this of Hahn, although both were not explicitly stated as conjectures but sinc... |
128,935 | <p>Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper <em>Über die nichtarchimedischen Grössensysteme</em> (Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse 116 (Abteilung IIa), 1... | Vladimir Kanovei | 34,207 | <p>Regarding </p>
<ol>
<li>In 1902 Beppo Levi introduced the Partition Principle stating that if S is a partition of A then |S|≤|A|. Although he coined this principle to argue against its use by Bernstein, the latter rejected the criticism and claimed that this is one of the more important principles of set theory. </... |
443,013 | <p>If a linear map $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$ is injective, then there exists $c>0$ such that $|Ax|\geq c|x|$ for all $x\in\mathbb{R}^m$</p>
<p>Could someone give any solution or hint?</p>
<p>Thanks.</p>
| copper.hat | 27,978 | <p>Since $\ker A = \ker (A^T A)$, we see that $A$ is injective iff $A A^T$ is invertible.
It is easy to see that $A^T A$ is symmetric and positive semi-definite. Since $A^T A$ is invertible, we see that $A^T A$ is positive definite, hence $A^T A \ge \lambda I$, where $\lambda$ is the smallest eigenvalue of $A^TA$ (and ... |
942,738 | <p>First of all I must state that I am not a mathematician, so please correct me if I use wrong terminology.</p>
<p>I am building a web application which needs to calculate the rating for each entity based on both the quantity and score of the reviews for that entity.</p>
<p>In other words, I don't want to just calcu... | Marc Bogaerts | 118,955 | <p>What you can do, for instance, is take the rate of reviews (w weighted mean), divide it by two (to reduce the scoring to a scale of <span class="math-container">$[0,5]$</span> and add this value to <span class="math-container">$5(1-e^{-q})$</span>. So the formula becomes <span class="math-container">$$\text{score}=5... |
3,808,111 | <p>Let <span class="math-container">$(X_n)_{n \geq 1}$</span> be a sequence of pairwise independent random variables such that :</p>
<p><span class="math-container">$$\sum_{n=1}^{\infty} n^{-1} P\left\{\max _{1 \leq m \leq n}\left|\sum_{k=1}^{m}\left(X_{k}-E X_{k}\right)\right|>\varepsilon n\right\}<\infty$$</spa... | Fawkes4494d3 | 260,674 | <p>Take <span class="math-container">$f(x)=x^4$</span>. Note that <span class="math-container">$f''(0)=0$</span> and <span class="math-container">$0$</span> is a point of global minima of <span class="math-container">$f$</span>.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Derivative_test#Higher-order_derivative_t... |
3,069,369 | <p>I know how to perform polynomial regression. But is there any method to use for estimating the degree of the polynomial that is best suited? Some kind of meta-regression.</p>
<p>With best suited I mean the grade that has the highest probability of being the true degree of the source for the data.</p>
<p>For exampl... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\ $</span> It is <span class="math-container">$\ (2x^4-7x^2+5) + (x^3-x)\, $</span> and both clearly have roots <span class="math-container">$\,x = \pm 1$</span> </p>
|
3,718,255 | <p>So i solved the first part of this question, that there's 588 ways for choosing 5 digit integers, but i can't solve "Find the number of integers in (i) which satisfies "if there's 8, it's followed by 6"
i tried to solve it using different cases:</p>
<p>Case 1) there's two 8s and two 6s => (3 * 3!)/... | user-492177 | 492,177 | <p>If there is no <span class="math-container">$8$</span> then</p>
<p>Total <span class="math-container">$5$</span> digit distinct numbers=<span class="math-container">$\frac{5!}{2!}-\frac{4!}{2!}$</span>=<span class="math-container">$60-12=48$</span></p>
<p>If there is one <span class="math-container">$8$</span>, it m... |
607,324 | <blockquote>
<p><span class="math-container">$d \in \mathbb{Z}$</span> is a square-free integer (<span class="math-container">$d \ne 1$</span>, and <span class="math-container">$d$</span> has no factors of the form <span class="math-container">$c^2$</span> except <span class="math-container">$c = \pm 1$</span>), and ... | Brian Rushton | 51,970 | <p>Yes, you idea is right. When doing formal power series with matrices, you usually think of the constants as multiples of the identity.</p>
<p>One reason is that the ring of matrices is an $\mathbb{R}$ algebra, meaning that it contains a copy of the real numbers (i.e multiples of the identity) which commutes with al... |
4,415,254 | <p>When is <span class="math-container">$n!>x^n$</span>? assuming that x is a fixed positive. I know we can take the log of both sides and use the following formula:</p>
<p><span class="math-container">$$n\log(x) = \log(x^n) < \log(n!) = \sum_{i = 1}^n\log(i)$$</span>.
But this still gives me difficulty trying to... | Ivan Kaznacheyeu | 955,514 | <p>If you need only estimate for <span class="math-container">$n$</span>, you can use following claim, which is simple to prove: <span class="math-container">$n! \geq (\sqrt{n})^n$</span>.</p>
<p><span class="math-container">$$(n!)^2=(1\cdot 2\cdot ...\cdot n)\cdot(n\cdot(n-1)\cdot...\cdot 1)
=(1\cdot n)\cdot(2\cdot(n-... |
4,113,744 | <p>The number of real values <span class="math-container">$(x,y)$</span> for which <span class="math-container">$$2^{x+1}+3^y=3^{ y+2}-2^x$$</span> is ?
I went like, <span class="math-container">$$2^{x+1}+2^x=3^{y+2}-3^y$$</span>
after solving which I got; <span class="math-container">$(x,y)=(3,1)$</span>
Is there any ... | whatever124125 | 919,871 | <p><span class="math-container">$ 2^{x+1}+3^y=3^{y+2}-2^x \Leftrightarrow 2^{x+1}+2^x=3^{y+2}-3^y\Leftrightarrow 3\cdot 2^x= 8\cdot 3^y \Leftrightarrow 2^{x-3}=3^{y-1}$</span>, take <span class="math-container">$\log_3\implies
\log_32^{x-3}=\log_33^{y-1}\Leftrightarrow (x-3)\log_32=y-1 \implies y=(x-3)\log_32+1,~$</sp... |
3,631,714 | <p>If <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R} $</span> is a function on the Schwartz space <span class="math-container">$\mathcal{S} (\mathbb{R}^n) $</span> we define the Fourier transform of <span class="math-container">$f$</span> as the function
<span class="math-container">$$ \hat f(\xi)= \int_{... | Amit Bendkhale | 440,224 | <p>It is the National Council of Educational Research and Training, that decides the syllabus, and publishes content of secondary and senior-secondary students.
You can check their site for E-Books <a href="http://ncert.nic.in/textbook/textbook.htm?lemh1=0-6" rel="nofollow noreferrer">http://ncert.nic.in/textbook/textb... |
1,401,661 | <blockquote>
<p>Dice are cubes with pips (small dots) on their sides, representing
numbers 1 through 6. Two dice are considered the same if they can be
rotated and placed in such a way that they present matching numbers on
the top, bottom, left, right, front, and back sides.</p>
<p>Below is an example of t... | wltrup | 232,040 | <p>I won't give you an actual answer, and this post is not a hint to solve the given question, but I will suggest a way that you can simplify the problem in order to understand better what's being asked.</p>
<p>(By the way, that's often a <em>very</em> useful approach, that is, simplifying the problem to get a better ... |
364,059 | <p>While reading up on "Glivenko Cantelli Theorem" from Probability Models by K.B Athreya, the author used 2 lemmae to prove it. One was called Scheffe's lemma, the other Polya's theorem.</p>
<p>Scheffe's Lemma is stated as follows:</p>
<p>Let $f_n, f$ be non negative $\mu$ integrable functions. If $f_n \to f$ a.e an... | Bombyx mori | 32,240 | <p>I don't think your "proof" works. Let $f_{n}=n\chi[0,\frac{1}{n}]$, $f=0$. Then $f_{n}\rightarrow f$ almost everywhere. But you cannot conclude that $$\int f_{n}d\mu\rightarrow 0$$ as it is constant $1$. As the commenter pointed out to apply dominated convergence theorem you need a function $g$ such that $|f_{n}|\le... |
4,014,144 | <p>Let <span class="math-container">$f(x) = \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$</span>.
where <span class="math-container">$x, c_1, c_2, \dots, c_k \in \mathbb R^n$</span>.
What fast iterative methods are available for finding the (approximate) min of <span class="math-container">$f$</span> with the constraint <span c... | Alt | 123,685 | <p>Convert it to a quadratically constrained linear program by the following trick: Add an auxiliary variable <span class="math-container">$t$</span> such that: <span class="math-container">$\max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)\leq t$</span>
Then, this will imply <span class="math-container">$c_i^Tx\leq t~~ \forall i$</... |
2,501,450 | <p>I was trying for a while to prove non-existence of theu following limit:</p>
<p>$$\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}$$</p>
<p>Unfortunately, with no results.</p>
<p>My hope was to show, that:</p>
<p>$$\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}\not=0$$</p>
<p>But showing that wa... | Yiorgos S. Smyrlis | 57,021 | <p>Note first that
$$
\lim_{n\to\infty}n(\sqrt[n]{5}-1)=\ln 5,
$$
Hence
$$
(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}=n\cdot(-1)^n\cdot\frac{1+\frac{4n}{2^n}+\frac{6}{2^n}}{n(\sqrt[n]{5}-1)}.
$$
Clearly
$$
\frac{1+\frac{4n}{2^n}+\frac{6}{2^n}}{n(\sqrt[n]{5}-1)}\to \frac{1}{\ln 5}
$$
while
$$
n\cdot(-1)^n\,\,\,\text{dive... |
2,501,450 | <p>I was trying for a while to prove non-existence of theu following limit:</p>
<p>$$\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}$$</p>
<p>Unfortunately, with no results.</p>
<p>My hope was to show, that:</p>
<p>$$\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}\not=0$$</p>
<p>But showing that wa... | stanly | 382,677 | <p>Try to show difference bwtween $\lim_{n \rightarrow \infty} \sup a_{n}$ and $\lim_{n \rightarrow \infty} \inf a_{n}$.</p>
|
235,425 | <p>Ok, here is what I have for the proof of this conjecture. Let me know if I'm on the right path? all input appreciated.</p>
<p>There exist integers $j$, $k$, and $m$, such that,
$b = aj $ and $ c = ajk.$ Then $c = ajk $ (substituting $aj$ for $b$)
let $m = jk$, then $c = ma, => a|c.$ </p>
| Mhenni Benghorbal | 35,472 | <p>Note that,</p>
<p>$$a|b \implies b=ma \,,$$</p>
<p>and </p>
<p>$$ b|c \implies c =nb \implies c = nma \implies c = q a \implies a|c \,,$$</p>
<p>for $m,n,q \in \mathbb{Z}$ and $q=nm.$ </p>
|
594,482 | <p>If a,b and c are elements of a ring, does the equation ax+b=c always have a solution x? If it does, must the solution be unique? </p>
| Boris Novikov | 62,565 | <p>It is enough to consider the equation $ax=b$.</p>
<p>If $b=0$ then the set of solutions coincides with the right annulator of $a$: $N_a=\{x|ax=0\}$. For arbitrary $b$ let $x_0$ is a solution of $ax=b$ (if exists!). Then the set of solutions coincides with the coset $N_a+x_0$.</p>
<p>This is all what one can say ab... |
591,014 | <p>I need help with this proof:</p>
<p>$f: X\rightarrow Y$</p>
<p>$C,D\subseteq Y$</p>
<p>$f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$</p>
<p>Thanks.</p>
| angryavian | 43,949 | <p>To solve these kinds of questions, you want to show that $$f^{-1}(C\cap D) \subseteq f^{-1}(C) \cap f^{-1}(D)$$ and $$f^{-1}(C\cap D) \supseteq f^{-1}(C) \cap f^{-1}(D).$$</p>
<p>I will show $\subseteq$ here; you should try the other direction.</p>
<p>Suppose $x \in f^{-1} (C \cap D)$. This means $f(x) \in (C\cap ... |
690,729 | <p>Vancouver is 300 km away from Seattle. Two friends, one leaving from each city on a bike want to find a campsite between the two cities. One cyclist starts from vancouver at 25 km/h. His friend will start 2.0h later from seattle at 32 km/h. How far from Vancouver do the friends meet?</p>
<p>Attempt:</p>
<p>dA + dB... | Ross Millikan | 1,827 | <p>Your direction is fine. It is good to define your variables, like $dA$ is the distance traveled by the one from Vancouver, etc. The vB should be multiplied by $(t-2)$ as the one from Seattle leaves two hours later, so that cyclist is traveling two hours less. You dropped multiplying the $2$ by vB, so it should be... |
903,831 | <p>One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in parallel to ring of integers, I think converting a polynomial ring to a numerical (i.e. integer) ring is useful.</p>
... | lhf | 589 | <p>If you accept subrings of the complex numbers as numerical rings, then just take a homomorphism $\mathbb Z[X] \to \mathbb C$ with nontrivial kernel. This reduces to sending $X$ to an algebraic number.
A prime example is $X \mapsto i$, which gives you the numerical ring $\mathbb Z[i]$.</p>
<p>If you mean encoding th... |
2,873,309 | <p>Let's say each month a utility bill comes and it needs to be split among a variable number of roommates. Let's say each person occupied the house for a certain percentage of the month: person 1 100%, person 2 100%, person 3, 75%, person 4, 50%. How can I calculate how much each person has to pay? (The number of peop... | Bram28 | 256,001 | <p>Another method (it's certainly simpler) is to just add up the occupancy rates for the people, and then figure out the percentage of the total bill that each person has to pay as their share.</p>
<p>For your example, we add up:</p>
<p>$$100+100+75+50=325$$</p>
<p>And now we can say that persons 1 and 2 are each re... |
2,897,374 | <p>So today in my Algebraic Topology class we were trying to construct a homotopy and at one point we needed to basically construct a (continuous) bijection between some intervals $[a, b]$ and $[c, d]$ sending $a \mapsto c$ and $b \mapsto d$. </p>
<p>My professor said that a bijection $\varphi : [a, b] \to [c, d]$ wou... | amsmath | 487,169 | <p>The projection that you are looking for is $Px = x - \frac 1{\|v\|^2}\langle x,v\rangle v$, i.e.,
$$
P = I - \frac{vv^T}{v^Tv},
$$
where $v = (1,1,\ldots,1)^T$.</p>
|
2,661,718 | <p>My basic understanding is that each time
I have 30% chance of winning the prize
so between 3 and 4 tries I should win it</p>
<p>cause .3 +.3 +.3 = 90% as I need to win only once with 3 try
.3+.3+.3+.3 = 120% with 4 try</p>
<p>I do remember a formula saying 1-(0.7)^3 = 65.7% chance but I dont remember what that ... | user | 505,767 | <p>Let $y=x+1$ then</p>
<p>$$x > -1 \implies x^2 \ge (1+x) \{\log(1+x)\}^2$$</p>
<p>is equivalent to</p>
<p>$$y > 0 \implies (y-1)^2 \ge y \log^2 y$$</p>
<p>$$\iff -\frac{|y-1|}{\sqrt y}\le\log y\le\frac{|y-1|}{\sqrt y} \iff e^{-\frac{|y-1|}{\sqrt y}}\le y\le e^{\frac{|y-1|}{\sqrt y}}\iff 1\le ye^{\frac{|y-1|... |
2,661,718 | <p>My basic understanding is that each time
I have 30% chance of winning the prize
so between 3 and 4 tries I should win it</p>
<p>cause .3 +.3 +.3 = 90% as I need to win only once with 3 try
.3+.3+.3+.3 = 120% with 4 try</p>
<p>I do remember a formula saying 1-(0.7)^3 = 65.7% chance but I dont remember what that ... | Barry Cipra | 86,747 | <p>Let $x=u^2-1$ with $u\gt0$. The inequality to prove becomes $(u^2-1)^2\ge u^2(\log(u^2))^2$, or</p>
<p>$$\left| u-{1\over u}\right|\ge2|\log u|$$</p>
<p>Since each side is now invariant under $u\to1/u$, it suffices to prove the inequality for $u\ge1$, in which case we can remove the absolute value signs and write ... |
1,825,392 | <p>Let $f(z)$ be an entire function so that,</p>
<p>$$ \int \frac{|f(z)|}{1 + |z|^3} dA(z) < \infty$$</p>
<p>where the integral is taken over the entire complex plane. Show that $f$ is a constant.</p>
<p>I believe that the idea is to use the mean value property; that is:</p>
<p>$$f(z) = \frac{1}{\pi\delta^2}\in... | Jonas Meyer | 1,424 | <p>You can use Cauchy's integral formula to show that $f^{(2)}=0$. Given $w\in \mathbb C$, take $r\geq\max\{1,2|w|\}$. Let $C_r$ be the circle with radius $r$ centered at $0$. Then $$f^{(2)}(w)=\frac{1}{\pi i}\int_{C_r}\frac{f(z)}{(z-w)^3}\,dz,$$</p>
<p>so $$|f^{(2)}(w)|\leq \frac1\pi \int_0^{2\pi}\frac{|f(re^{i\th... |
686,167 | <p>I'm working on an assignment where part of it is showing that $S_k=0$ for even $k$ and $S_k=1$ for odd $k$, where</p>
<blockquote>
<p>$$S_k:=\sum_{j=0}^{n}\cos(k\pi x_j)= \frac{1}{2}\sum_{j=0}^{n}(e^{ik\pi x_{j}}+e^{-ik\pi x_{j}}) $$</p>
<p>Here $x_j=j/(n+1)$.</p>
</blockquote>
<p>So, working through the al... | Sandeep Thilakan | 124,957 | <p>Using trigonometric identities</p>
<p>\begin{align}
\cos (k \pi x_j) &= \frac{\sin (k \pi x_{j+1}) - \sin (k \pi x_{j-1})}{2 \sin (\frac{k \pi}{n+1})} \\
\sum_{j=0}^{n} \cos (k \pi x_j) &= 1+ \frac{1}{2 \sin (\frac{k \pi}{n+1})} \sum_{j=1}^{n} {\sin (k \pi x_{j+1}) - \sin (k \pi x_{j-1})} \\
\end{align}
In ... |
1,219,851 | <p>If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ? Trivially $n$ cannot be even , so this leaves us only with the possibilities $n \equiv1,3,5( \mod 6) $ , but I cannot reduce the cases . Please help , Thanks in advance . </p>
| ajotatxe | 132,456 | <p>If $n\equiv1,5\pmod 6$ then $n^2\equiv 1\pmod 6$.</p>
<p>On the other hand, $2^n\equiv 2$ or $4\pmod 6$ depending on $n$ is odd or even. So if $n^2\equiv 1\pmod 6$, then
$$2^n+n^2\equiv 2+1\equiv 3\pmod 6$$
therefore $2^n+n^2$ is a multiple of three.</p>
<p>Notice the case $1^2+2^1$, that is the only exception to ... |
3,520,893 | <p>As in group theory, there is a concept of isomorphism between metric spaces called isometry.
Two metric spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are isometric if there is a function that perserves the distance of two elements. That function is called isometry.</p>
<... | Giuseppe Negro | 8,157 | <p>That's too ambitious, there is an enormous variety of metric spaces. But if you restrict to specific classes, then a lot has been done. I can think of two examples: Hilbert spaces and geodesic surfaces. These are completely classified. </p>
<p>Hilbert spaces are vector spaces equipped with a scalar product, which i... |
2,330,496 | <p>Players Ruby and Bob are given an undirected graph and a number $N$. First Ruby colors $N$ vertices red, then Bob colors $N$ vertices blue (they must be distinct from Ruby's choices). Afterward, all other points on the graph are given the color of whichever color they are closest to (shortest path) with ties left bl... | Gregory J. Puleo | 183,812 | <p>I believe that the first player cannot always guarantee a tie, even when $N=1$. Some observations:</p>
<ul>
<li><p>Given a graph $G$ and a vertex subset $X \subset V(G)$, we can always modify $G$ so that the players are <em>forced</em> to choose a vertex in $X$: add a huge independent set $I$ to $G$ and make all th... |
1,083,141 | <p>According to Zeidler, 1995, in his book "Applied Functional Analysis: Application to Mathematical Physics".</p>
<p>Dirichlet problem is a problem to minimize $$F(u)=\frac{1}{2}\int_G(u')^2\ dx-\int_G fu\ dx,\qquad u=g \textrm{ in } bd(G).$$
Then, He says</p>
<blockquote>
<p>If $G$ is open, bounded, nonempty subs... | Quickbeam2k1 | 63,384 | <p>I'll try to hint on answers to your first and second question.</p>
<p>Ad 1. As my comment suggested, the $C^2$ regularity of $u$ is needed to give sense to the term $-u''=f$. If you consider the generalized boundary value problem only first order derivatives appear. Essentially, this is the reason why only $W^{1,2}... |
1,360,910 | <p>My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here.</p>
<p>Isn't the statement absolutely obvious for sets $A,B,C$ with $A\subseteq C$? You just need to draw a diagram and it b... | rschwieb | 29,335 | <p>Judging from your diagram, it looks like (?) you are interpreting $+$ as set union. Certainly, if you are working with sets using union and intersection, your picture is accurate. </p>
<p>But using using the $+$ to join submodules of a module does not behave the same way. If $A$ and $B$ are submodules of a module, ... |
1,576,007 | <p>I just completed an exam and was curious as to the following question:</p>
<blockquote>
<p>How many surjections are there from $\{1, 2, 3, ..., n\}$ to $\{1, 2\}$ if $n \geq 3$?</p>
</blockquote>
<p>My assumption was the following:</p>
<p>The first element of the domain has the possibility to map to either 1 or... | ultrainstinct | 177,777 | <p>Basically you can think of it as a binary string, $0$ means the element in the domain maps to the first element in the codomain, $1$ means the element maps to the second element in the codomain. There are $2^n$ ways to make a binary string of length $n$, but with the rules of surjectivity we need to take away the ca... |
2,604,660 | <p>In a box there are $16$ ice-creams: $6$ lemon flavor,$4$ mint flavor and $6$ strawberry flavor.When we extract two ice-creams,what's the probability of getting two different flavors,given that at least one is strawberry flavor.</p>
<p>That's my solution : </p>
<p>$P(A)$ = "Different flavors" = $\frac{\binom{16}{2}... | true blue anil | 22,388 | <p>Assuming that each ice-cream has the same probability of being drawn,</p>
<p>you could compute it as the ratio of the ways of getting</p>
<p>$\text{ (one strawberry, one other) / (one strawberry, one other + two strawberries)}$</p>
<p>= $\dfrac{\binom61\binom{10}1}{\binom61\binom{10}1+\binom62}$ </p>
|
701,677 | <ol>
<li><p>If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb C)$? E.g. branching rules</p></li>
<li><p>Similarly for the groups $\mathrm{SO}_n(\mathbb C) \subset \mathrm O_... | Mauro ALLEGRANZA | 108,274 | <p>See the comment on the original question :</p>
<blockquote>
<p>if $H$ is invertible and the arguments of $H^{-1}$ are $f_A$ and $f_B$, then for $f$ to exists we need $f_A$ and $f_B$ to agree at $A \cap B$, that is $\forall x \in A \cap B$, $f_A(x) = f_B(x)$.</p>
</blockquote>
<p>But the definition of Brian Scott... |
1,218,912 | <p>My problem is following:
$$\binom{n}{r} + \binom{n+1}{r+1} + \binom{n+2}{r+2} + \dots + \binom{n+M}{r+M}$$</p>
<p>how can we reduce it to a more short solution</p>
<p>Here $\dbinom{n}{r} = \dfrac{n!}{r! (n-r)!}$ and thus same as regular.Please help me in solving the above expression</p>
| Marko Riedel | 44,883 | <p>This question can be treated using basic complex variables, which is
an instructive exercise.</p>
<p><P>Suppose we seek to compute
$$\sum_{q=0}^M {n+q\choose r+q}$$
with $n\ge r.$</p>
<p><P>Introduce the integral representation
$${n+q\choose r+q}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon}
\frac{(1+z)^{n+q}}{z^{r+q+1... |
4,474,803 | <p>Let <span class="math-container">$f$</span> be a differentiate function and consider two operators
<span class="math-container">$$
A(f(x))=\int_0^1 \frac{ d}{dx}f(x t^\mu) dt,\\
B(f(x))=\mu xf(x)+\int_0^x f(t) dt,
$$</span>
where <span class="math-container">$\mu $</span> is a parameter.</p>
<p>I need to prove that<... | Chris Eagle | 693,182 | <p>The answer to your general question is very much "no", because moving a quantifier past a connective can change the quantifier. For instance, <span class="math-container">$\neg \forall x \phi(x)$</span> is equivalent to <span class="math-container">$\exists x \neg\phi(x)$</span> and in the vast majority of... |
966,267 | <p>Original PDE
$$T_t=\alpha T_{xx}$$
I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. </p>
<p>Given the initial condition
$$T(x,0)=sin(\frac{\pi x}{L})$$
where $$L=1$$</p>
<p>I would like to find the exact sol... | Leucippus | 148,155 | <p>Given the pde $u_{t} = \alpha u_{xx}$, $u(0,t) = u(L,t) = 0$ and $u(x,0) = \sin\left(\frac{\pi x}{L} \right)$ : the equation can be separated by use of $u(x,t) = F(t) G(t)$ and leads to the equations
\begin{align}
F' + \lambda^{2} \alpha F &= 0 \\
G'' + \lambda^{2} G &= 0
\end{align}
which have solutions
\b... |
252,679 | <p>I simply want to be able to assign the subscript from something like Subscript[x, 1] to a new variable. How should I do this?</p>
| Hans Olo | 61,267 | <p>If you have a variable</p>
<pre><code>y=Subscript[x, 1];
</code></pre>
<p>then to extract the index it suffices to do</p>
<pre><code>y[[2]]
</code></pre>
<p>which in this case returns 1 as expected.</p>
<p>For more than one indices, this works</p>
<pre><code>In[1]:= y=Subscript[x,1,2,3]
Out[1]= Subscript[x,1,2,3]
... |
252,679 | <p>I simply want to be able to assign the subscript from something like Subscript[x, 1] to a new variable. How should I do this?</p>
| bbgodfrey | 1,063 | <p>For <code>y</code> with any number of subscripts ,</p>
<pre><code>Rest[List @@ y]
</code></pre>
<p>also works. For instance</p>
<pre><code>y = Subscript[x, 1, 2];
Rest[List @@ y]
(* {1, 2} *)
</code></pre>
<p>As a word of advice, use <code>Subscript</code> and the like for formatting expressions for display but not... |
4,009,001 | <p>This problem was came to a Facebook post of mathigon ......the problem seems trickier than I expected...can you help me to find the distance??<a href="https://i.stack.imgur.com/CNQiX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CNQiX.png" alt="enter image description here" /></a></p>
| PTDS | 277,299 | <p><a href="https://i.stack.imgur.com/jmroO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jmroO.png" alt="enter image description here" /></a></p>
<p>Let <span class="math-container">$\angle CAB = \angle ABC = \angle BAD = \angle DBA = \theta$</span></p>
<p>Area of the <span class="math-container">... |
4,009,001 | <p>This problem was came to a Facebook post of mathigon ......the problem seems trickier than I expected...can you help me to find the distance??<a href="https://i.stack.imgur.com/CNQiX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CNQiX.png" alt="enter image description here" /></a></p>
| Quanto | 686,284 | <p>The area of the lens formed by two overlapping circle of radius <span class="math-container">$r$</span> and at distance <span class="math-container">$d$</span> apart is
given by <a href="https://mathworld.wolfram.com/Lens.html" rel="nofollow noreferrer">https://mathworld.wolfram.com/Lens.html</a></p>
<p><span class=... |
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | Arian | 172,588 | <p>Take the set $V:=\{v\in\mathbb{R}^2: v=(x,0), v=(0,y), x,y\in\mathbb{R}\}$ Clearly if $v\in V$ then for any $\alpha\in\mathbb{R}$ we have $\alpha v\in V$, but $v_1+v_2\notin V$ where $v_1=(x,0),v_2=(0,y)$ for any such pairs and $x\neq0, y\neq 0$. </p>
|
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | John Coleman | 294,695 | <p>You could just take the empty set. Since it is empty it isn't a vector space, but it is vacuously closed under scalar multiplication.</p>
<p>Obviously this is a somewhat cheap example, but it does solve the problem as stated, and underscores that existential properties (e.g. the existence of a 0 vector) can't be de... |
281,325 | <p>I am not very familiar with Stone-Čech compactification, but I would like to understand why the remainder $\beta\mathbb{R}\backslash\mathbb{R}$ has exactly two connected components.</p>
| Seirios | 36,434 | <p>I finally found something:</p>
<p>For convenience, let $X= \mathbb{R} \backslash (-1,1)$. </p>
<p>First, we show that $(\beta \mathbb{R})\backslash (-1,1)= \beta (\mathbb{R} \backslash (-1,1))$. Let $f_0 : X \to [0,1]$ be a continuous function. Obviously, $f_0$ can be extend to a continuous function $f : \mathbb{R... |
4,574,446 | <p>This is question 13 on page 294 of Vector Calculus by Marsden and Tromba.</p>
<blockquote>
<p>Find the volume of the region determined by <span class="math-container">$x^2 + y^2 + z^2 \leq 10, z \geq 2.$</span></p>
</blockquote>
<p>I have attempted it as follows.</p>
<p>The region can be described by using spherical... | Peter Phipps | 15,984 | <p>Let <span class="math-container">$n+2 = k^2+r$</span> where <span class="math-container">$0 \le r \lt 2k+1$</span>, so <span class="math-container">$k = \lfloor \sqrt{n+2}\rfloor$</span> is the largest integer less than <span class="math-container">$\sqrt{n+2}$</span>.</p>
<p>The requirement <span class="math-contai... |
3,828,982 | <blockquote>
<p><strong>Statement</strong></p>
<p>If <span class="math-container">$U$</span> is open and <span class="math-container">$\text{int}(S)\neq\emptyset$</span> then <span class="math-container">$\text{int}(U\cap S)\neq\emptyset$</span> too when <span class="math-container">$U\cap S\neq\emptyset$</span> and wh... | Henno Brandsma | 4,280 | <p>Connectedness problems often degenerate in <span class="math-container">$\Bbb R$</span>, it's the plane where stuff gets interesting.</p>
<p><span class="math-container">$U=\{x\mid \|x\|< 1\}$</span> which is open, path-connected.</p>
<p><span class="math-container">$S= \{x \mid \|x\|\ge 1\} \cup \{(x,y):xy=0\}$<... |
3,025,375 | <p>What is<span class="math-container">$$\lim_{n→∞}n^3(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt2)?$$</span>So it is<span class="math-container">$$\lim_{n→∞}\frac{n^3(\sqrt{n^2+\sqrt{n^4+1}})^2-(n\sqrt{2})^2}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}=\lim_{n→∞}\frac{n^3(n^2+\sqrt{n^4+1}-2n^2)}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}.$$</span... | lab bhattacharjee | 33,337 | <p>First replace <span class="math-container">$1/x=h$</span> to find </p>
<p><span class="math-container">$$L=\lim_{n→∞}n^3(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt2)=\lim_{h\to0^+}\dfrac{\sqrt{1+\sqrt{1+h^4}}-\sqrt2}{h^4}$$</span></p>
<p>Let <span class="math-container">$\sqrt{1+\sqrt{1+h^4}}=y+\sqrt2\implies1+\sqrt{1+h^4}=(\... |
2,888,913 | <p>4 Boys & 4 Girls are to be seated in a line find number of ways , so that Boys & Girls are in alternate seats.</p>
<p>My approach:</p>
<p>If boys are seated in B$1$,B$2$,B$3$,B$4$ positions than at each gap between two consecutive boys a girl can sit so, there will be <strong>C$(5,4)$</strong> ways for gir... | BallBoy | 512,865 | <p>One of the $C(5,4)$ ways you count to seat the girls fills gaps 1,3,4,5. But then boys B1 and B2 are seated next to each other, so boys and girls don't alternate. So there are not $C(5,4)$ ways to choose the spots for girls. </p>
|
2,888,913 | <p>4 Boys & 4 Girls are to be seated in a line find number of ways , so that Boys & Girls are in alternate seats.</p>
<p>My approach:</p>
<p>If boys are seated in B$1$,B$2$,B$3$,B$4$ positions than at each gap between two consecutive boys a girl can sit so, there will be <strong>C$(5,4)$</strong> ways for gir... | WaveX | 323,744 | <p>Look at it this way, we have$$BGBGBGBG$$</p>
<p>We have $4$ ways for first boy, $4$ ways for first girl, $3$ ways for second boy, $3$ ways for second girl, and so on.</p>
<p>So we have $$4*4*3*3*2*2*1*1 = 4!*4!$$</p>
<p>But note that we could have also started with the pattern $$GBGBGBGB$$ so we need to double ou... |
4,192,879 | <p>This question is about the <em>provided solution</em> to Question 11d from Chapter 8 of Spivak's Calculus. The relevant background material (i.e. 11a-11c) is as follows:</p>
<blockquote>
<p><span class="math-container">$(a)$</span> If <span class="math-container">$\{a_n\}$</span> is a sequence of positive terms such... | Ben | 754,927 | <p>Recall the lemmas used in the proofs concerning limits for products, and quotients:</p>
<blockquote>
<p>(2) If
<span class="math-container">$$|x - x_0| < \min\left(1,\frac{\varepsilon}{2(|y_0|+ 1)}\right) \text{ and } |y - y_0| < \frac{\varepsilon}{2(|x_0|+ 1)},$$</span>
then
<span class="math-container">$$ |x... |
2,738,023 | <p>We know we can check if two vectors are 'orthogonal' by doing an inner product.</p>
<p>$a*b=0$</p>
<p>tells us that these two vectors are orthogonal</p>
<p>here comes the question:</p>
<p>if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction. </p>
| Michael Hoppe | 93,935 | <p>Two vectors are parallel iff the absolute value of their dot product equals the product of their lengths.</p>
<p>Iff their dot product equals the product of their lengths, then they “point in the same direction”.</p>
|
2,064,403 | <p>Example. Let $V$ be a finite dim vector space with two different bases</p>
<p>$S = \{ u_1,u_2 \} = \{ (1,2),(3,5) \}$ and $S' = \{ v_1, v_2 \} = \{ (1,-1), (1,-2) \} $</p>
<p>You can check that $v_1 = -8u_1 + 3u_2$ and $v_2 = -11u_1+4u_2$ and $P = \begin{bmatrix}
-8 &-11 \\
3& 4
\end{bmatrix}$ is the ... | levap | 32,262 | <p>This is a good question. It might be comforting to know that there is always some arbitrary choices involved in the whole issue of representing vectors of an abstract vector space as an array of numbers, linear maps as a double array of numbers (matrices) and what one means by changing a basis. The most important th... |
21,257 | <p>I am interested in examples where the <a href="http://en.wikipedia.org/wiki/Shooting_method" rel="nofollow">Shooting Method</a> has been used to find solutions to systems of ordinary differential equations that are either </p>
<ul>
<li>reasonably large systems, or </li>
<li>the search algorithm in the shooting para... | Jitse Niesen | 2,610 | <p>Another shameless plug ... Coworkers and I used the Evans function formalism, which is a variant of the shooting method to deal with unstable directions (probably the same problem as mentioned by yfarjoun), on a boundary value problem of the form $y'(t) = (\lambda A_1 + A_2(t)) y(t)$ with $y$ specified as $t \to \pm... |
3,579,739 | <p>Its also mentioned that this question is an example of <strong>Sampling without replacement</strong></p>
<p>My question is , by the general method of how I do these kind of problems , I would assume that there are two possibilities</p>
<ol>
<li><p>You chose a defective bottle and then a non defective which will h... | Jaap Scherphuis | 362,967 | <p>Here is another viewpoint.</p>
<p>Suppose you don't know there is a flawed bottle. You choose two of the five bottles. You are then told that one of the five bottles contains a crack. It is equally likely to be any of the five bottles, so each bottle has a probability of <span class="math-container">$\frac{1}{5}$</... |
3,287,067 | <p>I'm reading a solution to the following exercise:</p>
<p>"Assume that <span class="math-container">$\lim_{x\to c}f\left(x\right)=L$</span>, where <span class="math-container">$L\ne0$</span>, and assume <span class="math-container">$\lim_{x\to c}g\left(x\right)=0.$</span> Show that <span class="math-container">$\li... | J.G. | 56,861 | <p>In the spirit of the cosine example, take <span class="math-container">$f(x)=g\left(|x|-\frac12 a\right)$</span> for odd <span class="math-container">$g$</span> with <span class="math-container">$g\left(\frac12a\right)\ne0$</span>.</p>
|
3,930,466 | <p>I want to prove by induction that</p>
<p><span class="math-container">$$\left(\tfrac{1}{n}\sum_{k=1}^{n} x_k\right)^2\leq\tfrac{1}{n}\sum_{k=1}^{n} x_k^2$$</span></p>
<p>But I have no idea how. For n=1 it is trivial that
<span class="math-container">$$\left(\tfrac{1}{1}\sum_{k=1}^{1} x_k\right)^2\leq\tfrac{1}{1}\sum... | G Cab | 317,234 | <p>Hint:</p>
<p>you can resolve the <span class="math-container">$1/n+1$</span> hurdle by considering <em>weighted averages</em> of previous <span class="math-container">$n$</span> points and new added one.</p>
<p>Then consider that <span class="math-container">$f(x)=x^2$</span> is convex</p>
|
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