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 |
|---|---|---|---|---|
84,076 | <p>I think computation of the Euler characteristic of a real variety is not a problem in theory.</p>
<p>There are some nice papers like <em><a href="http://blms.oxfordjournals.org/content/22/6/547.abstract" rel="nofollow">J.W. Bruce, Euler characteristics of real varieties</a></em>.</p>
<p>But suppose we have, say, a... | Igor Rivin | 11,142 | <p>This is quite nontrivial. See for example: </p>
<p>On Bounding the Betti Numbers and Computing the Euler Characteristic of Semialgebraic sets, by Saugata Basu (google has full text). </p>
<p>The canonical reference is a more recent book by Basu, Ricky Pollack and Marie-Francoise Roy, called "algorithms in real alg... |
791,372 | <p>Hi I am trying to solve this double integral
$$
I:=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2.
$$
Thank you.</p>
<p>The constant in the result is given by $\gamma\approx .577$, and is known as the Euler-Mascheroni constant. I was thinking to write
$$
I=\Re \... | Zaid Alyafeai | 87,813 | <p>Using the identity </p>
<p>$$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$</p>
<p>The integral can be written
$$
I=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\left(\cos(x)\cos(y)-\sin(x)\sin(y)\right)\,dx\,dy $$</p>
<p>Now by splitting the integrals</p>
<p>$$\int_0^\infty \int_0^\infty \frac{\log x... |
791,372 | <p>Hi I am trying to solve this double integral
$$
I:=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2.
$$
Thank you.</p>
<p>The constant in the result is given by $\gamma\approx .577$, and is known as the Euler-Mascheroni constant. I was thinking to write
$$
I=\Re \... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
1,338,999 | <p><img src="https://i.stack.imgur.com/hXfn2.png" alt="sat question"></p>
<p>My friend selected option B, I did C. We're confused. Can someone please explain this for my friend?</p>
| Michael Hardy | 11,667 | <p>I got $\dfrac{(m!)^2}{(m-k)!(m+k)!}$ where you have $\left( \dfrac{m!}{(m-k)!} \right)^2$.</p>
|
2,183,390 | <p>So, I need to solve a hard problem, which reduces to this: </p>
<blockquote>
<p>Prove that $3^{\frac{1}{3}} \notin \mathbb{Q}[13^{\frac{1}{3}}]$.</p>
</blockquote>
<p>The only thing that comes into my mind is to suppose the opposite, <em>i.e.</em>, $3^{\frac{1}{3}} \in \mathbb{Q}[13^{\frac{1}{3}}]$, and then to ... | Dietrich Burde | 83,966 | <p>Suppose that $3^{\frac{1}{3}} \in \mathbb{Q}(13^{\frac{1}{3}})$. Then $\mathbb{Q}(3^{\frac{1}{3}})=\mathbb{Q}(13^{\frac{1}{3}})$, because both field have degree $3$ over $\mathbb{Q}$. This is a contradiction, since both fields have different <a href="https://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_... |
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | peterwhy | 89,922 | <p>$$\begin{align*}
\frac{n!}{(m-1)!(n-m+1)!} + \frac{n!}{m!(n-m)!}
&= \frac{n!}{(m-1)!(n-m)!(n-m+1)} + \frac{n!}{(m-1)!(n-m)!m}\\
&= \frac{n!}{(m-1)!(n-m)!}\left[\frac1{n-m+1}+\frac1m\right]\\
&= \frac{n!}{(m-1)!(n-m)!}\cdot\frac{n+1}{m(n-m+1)}\\
&= \frac{(n+1)!}{m!(n-m+1)!}
\end{align*}$$</p>
|
1,041,226 | <p>I need to prove the following: ${n\choose m-1}+{n\choose m}={n+1\choose m}$, $1\leq m\leq n$.</p>
<p>With the definition: ${n\choose m}= \left\{
\begin{array}{ll}
\frac{n!}{m!(n-m)!} & \textrm{für \(m\leq n\)} \\
0 & \textrm{für \(m>n\)}
... | Math-fun | 195,344 | <p>Do it intuitively: assume you have n+1 objects from which you want to choose m. Now divide your n+1 objects into two groups: one that includes n objects and one group with 1 (specific) object. Choosing m from n+1 is equivalent to choosing m out of the first group (these exclude the one specific object) PLUS choosing... |
1,978,035 | <p>In the Wikipedia page about quintics, there was a list of quintics that could be solved with trigonometric roots.</p>
<p>For example:$$x^5+x^4-4x^3-3x^2+3x+1\tag1$$ has roots of the form $2\cos \frac {2k\pi}{11}$
$$x^5+x^4-16x^3+5x^2+21x-9=0\tag2$$ has roots of the form $\sum_{k=0}^{7}e^{\frac {2\pi i 3^k}{41}}$</p... | D.Matthew | 469,027 | <p>Denoted
<span class="math-container">\begin{align*} \qquad\left\{ \begin{aligned} \varepsilon_1&=\exp\left(\dfrac{\>\pi\,\!i\>}{5}\right)=\phantom{+}\frac{1+\sqrt{5}}{4}+i\sqrt{\frac{5-\sqrt{5}}{8}}\approx\phantom{+}0.809017 + 0.587785i\qquad\\ \varepsilon_2&=\exp\left(\dfrac{2\pi\,\!i}{5}\right)=\f... |
137,571 | <p>As the title, if I have a list:</p>
<pre><code>{"", "", "", "2$70", ""}
</code></pre>
<p>I will expect:</p>
<pre><code>{"", "", "", "2$70", "2$70"}
</code></pre>
<p>If I have</p>
<pre><code>{"", "", "", "3$71", "", "2$72", ""}
</code></pre>
<p>then:</p>
<pre><code>{"", "", "", "3$71", "3$71", "2$72", "2$72"}
... | kglr | 125 | <p><strong>Update:</strong></p>
<pre><code>foo = # //. {a___, p : Except["", _String], Longest[b : "" ..], c___} :>
{a, p, p, ## & @@ ConstantArray["☺", Length[{b}] - 1], c} /. "☺" -> "" &;
{"", "", "", "3$71", "", "", "2$72", "", "", "", ""} // foo
</code></pre>
<blockquote>
<p><code>{"", "", "... |
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| grok_it | 14,348 | <p>For each integer $n$ put it in $A_i$ if $i$ is the smallest integer such that $n-i$ is a square. Or you can replace square with any even-degree polynomial with integer coefficients for a whole family. Or you can say n-i must be a prime.</p>
<p>Also:
$A_i$ = all integers with exactly $i$ prime-factors, counting mult... |
184,361 | <p>I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:</p>
<blockquote>
<p><strong>DEFINITION</strong> Given $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x < z+1$$ and we d... | Aryabhata | 1,102 | <p>It is enough to prove it for $0 < x < 1$.</p>
<p>Now, let $M$ be an integer such that, $\frac{M}{n} \le x < \frac{M+1}{n}$ where $0 \le M < n$</p>
<p>Thus $[nx] = M$.</p>
<p>For $0 \le k \le n-M-1$, we have that $[x+\frac{k}{n}] = 0$. </p>
<p>For $n-1 \ge k > n-M-1$ we have that $[x + \frac{k}{n}] = ... |
1,906,146 | <p>Can the following expression be further simplified: $$a^{(\log_ab)^2}?$$</p>
<p>I know for example that $$a^{\log_ab^2}=b^2.$$</p>
| Evariste | 239,682 | <p>$$a^{(\log_ab)^2}=a^{\log_a(b)\times \log_a(b)}=b^{\log_a(b)}$$ </p>
|
2,253,645 | <p>$(1,2)$ intersection $(2,3)=\{2\}$</p>
<p>$(1,2)$ intersection $[2,3]=\{2\}$</p>
<p>$\{1,2\}$ intersection $[1,2]=[1,2]$</p>
<p>$\{1,2\}$ union $[1,2]=[1,2]$</p>
<p>$\{1,2\}$ intersection $(1,3)$ intersection $[1,3)=(1,3)$</p>
<p>$\{1,2\}$ union $(1,3)$ union $[1,3)=(1,3)$</p>
<p>Is my answer correct? I find... | rych | 73,934 | <p>Kincaid D., Cheney W. <em>Numerical analysis</em>, Chapter 6:</p>
<p>A <em>natural spline</em> of degree $2m+1$ is a function $s\in C^{2m}(\mathbb{R})$
that reduces to a polynomial of degree $\leq2m+1$ in each inner interval
and to a polynomial of degree <strong>at most $m$</strong> in $(-\infty,t_{1})$ and
$(t_{n}... |
756,735 | <blockquote>
<p>Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient exists.</p>
</blockquote>
<p>I am having trouble seeing how I can manipulate the difference quotient in order ... | Slade | 33,433 | <p>Collect fractions and use the binomial theorem.</p>
|
2,688,829 | <p>I was wondering how to find the growth rate of the function defined by the number of ways to partition $2^n$ as powers of 2. After a search through OEIS I came across <a href="https://oeis.org/A002577" rel="nofollow noreferrer">OEIS A002577</a> which is what I'm looking for. I can't seem to find any link to asymptot... | K B Dave | 534,616 | <p>Maybe this is what you're looking for: let $b(n)$ be the number of partitions of $2^n$ into powers of $2$, and assume that $2^{n+1}$ is an integer. Then de Bruijn (<a href="http://www.dwc.knaw.nl/DL/publications/PU00018536.pdf" rel="nofollow noreferrer">1948</a>) showed that there exists a function $\psi$ of period ... |
3,357,502 | <p>I'm aware that <span class="math-container">$D(n)$</span> can be calculated in O(sqrt(n)) time. Can <span class="math-container">$ D(n, k) $</span> also be calculated in O(sqrt(n)) time? What's the best algorithm?</p>
<p>For example, if <span class="math-container">$n = 8$</span> and <span class="math-container">$k... | Vo Hoang Anh | 960,369 | <p>Yes you can, I also have a <a href="https://math.stackexchange.com/questions/4230187/faster-algorithm-for-counting-non-negative-tripplea-b-c-satisfied-a-b-c">problem</a> that require this function to be calculated fast enough.</p>
<p>You can notice that there is <span class="math-container">$O(\sqrt{n})$</span> uniq... |
2,188,965 | <p>Can someone explain to me how this step done? I got a different answer than what the solution said.</p>
<p>Simplify $x(y+z)(\bar{x} + y)(\bar{y} + x + z)$</p>
<p>what the solution got </p>
<p>$x(y+z)(\bar{x} + y)(\bar{y} + x + z)$ = $x(y + z\bar{x})(\bar{y} + x + z)$ (Using distrubitive)</p>
<p>What I got</p>
<... | Kanwaljit Singh | 401,635 | <p>$y\bar{x} + y + z\bar{x} + zy$</p>
<p>= $y(\bar{x} + 1) + z\bar{x} + zy$</p>
<p>= $y + z\bar{x} + zy$</p>
<p>= $y(1 + z) + z\bar{x}$</p>
<p>= $y + z\bar{x}$</p>
<p><strong>Direct rule -</strong></p>
<p>X + YZ = (X+Y)(X+Z)</p>
<p>So we have -</p>
<p>$(y + z)(y + \bar x)$</p>
<p>= $(y + z \bar x)$</p>
|
195,150 | <p>Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?</p>
<p>I had also got a clue: it's related to <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" rel="nofollow"><code>e</code></a>.</p>
<p>Please help! (I need explanation aswell)</p>
<p><s... | kahen | 1,269 | <p>Suppose we have $a_1$, $a_2$, $\dotsc$, $a_n$ all positive and summing to $10$. Then by the <a href="http://en.wikipedia.org/wiki/AM-GM_inequality" rel="nofollow">AM–GM inequality</a> their product is maximized when $a_1 = a_2 = \dotsb = a_n$. Since for $n \geq 10$ you'd have a product of numbers less than or equal... |
195,150 | <p>Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?</p>
<p>I had also got a clue: it's related to <a href="http://en.wikipedia.org/wiki/E_%28mathematical_constant%29" rel="nofollow"><code>e</code></a>.</p>
<p>Please help! (I need explanation aswell)</p>
<p><s... | Dan Barzilay | 40,220 | <p>You were all wrong actually... i checked with my teacher and the asnwer is <code>e</code> 3.67879441 (10/e) times so: <code>e^(10/e) = 39.5986256</code></p>
|
1,431,289 | <p>Find the average rate of change of $2x^3 - 5x$ on the interval $[1,3]$.</p>
<p>I'm really confused about this problem. I keep ending up with the answer $12$, but the answer key says otherwise. Someone please help! Thanks!</p>
| MegaboofMD | 269,606 | <p>average rate of change is the difference between the function values at two points divided by the distance between the points. $$ = \frac{f(3) - f(1) }{3-1}$$ $$ = \frac{[2*(3^3)-5*(3)] -[2*(1^3)-5*(1)]}{2}$$ $$ = \frac{[54-15]-[2-5] }{2}$$ $$ = \frac{39-(-3) }{2}$$ $$ = \frac{42 }{2}$$ $$ = \frac{21} 1$$</p>
|
4,202,490 | <p>Trying to construct an example for a Business Calculus class (meaning trig functions are not necessary for the curriculum). However, I want to touch on the limit problem involved with the <span class="math-container">$\sin(1/x)$</span> function.</p>
<p>I am sure there is a simple function, or there isn't... But woul... | Ethan Bolker | 72,858 | <p>As you note, this is really outside the scope of a business calculus syllabus. I might argue that anything more than a very informal discussion of limits is too.</p>
<p>In any case I think your business calculus students could profit from understanding that functions need not come from formulas. You can convey lots... |
1,195,092 | <p>The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom.</p>
<p>I was given the following problem: </p>
<blockquote>
<p>Prove that a set $E$ is countable if and only if there is a surjection from $\mathbb{N}$ to $E$. </p>
</blockquote>
<p>I have a rough idea on how ... | owen88 | 12,981 | <p>Here is an alternative method of solution. In the following I will write $A\, \text{b.} B$ to denote the event that $A$ occurs before $B$. So you want to calculate $\textbf{P}[ 6\, \text{b.}\, 55]$.</p>
<p>Instead we work with $\textbf{P}[55 \, \text{b.} \, 6]$, and condition on the event $5\, \text{b.}\, 6$. So
\b... |
471,145 | <p>I'm reading Proofs from the Book, and I ran into following theorem:</p>
<p>Suppose all roots of polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0$ are real. Then the roots are contained in the interval:</p>
<p>$$ - \frac{a_{n-1}}{n} \pm \frac{n-1}{n}
\sqrt{a_{n-1}^2 - \frac{2n}{n-1} a_{n-2} } $$</p>
<p>So, if ... | njguliyev | 90,209 | <p>Yes.</p>
<p>Hint: Otherwise you could extend the function to larger domain using the compactness of the circle.</p>
|
3,953,674 | <p>Here is a common argument used to prove that the sum of an infinite geometric series is <span class="math-container">$\frac{a}{1-r}$</span> (where <span class="math-container">$a$</span> is the first term and <span class="math-container">$r$</span> is the common ratio):
<span class="math-container">\begin{align}
S &... | Community | -1 | <p>Let</p>
<p><span class="math-container">$$S_n=\sum_{k=0}^n ar^n.$$</span></p>
<p>Then</p>
<p><span class="math-container">$$(1-r)S_n=S_n-rS_n=a-ar^{n+1}$$</span> and</p>
<p><span class="math-container">$$S_n=a\frac{1-r^{n+1}}{1-r}.$$</span></p>
<p>This is absolutely rigorous for any <span class="math-container">$r\n... |
127,412 | <p>How to take a 3 random given name?</p>
<p>I tried:</p>
<p><a href="https://i.stack.imgur.com/BUYIT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BUYIT.png" alt="enter image description here"></a></p>
| bill s | 1,783 | <p>One approach is to use RandomChoice[ ] to get your random names:</p>
<pre><code>list = EntityList["GivenName"];
RandomChoice[list, 3]
</code></pre>
<p><a href="https://i.stack.imgur.com/VaNss.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VaNss.png" alt="enter image description here"></a></p>
|
69,208 | <p>Consider $f:\{1,\dots,n\} \to \{1,\dots,m\}$ with $m > n$.
Let $\operatorname{Im}(f) = \{f(x)|x \in \{1,\dots,n\}\}$.</p>
<p>a.)
What is the probability that a random function will be a bijection when viewed as
$$f':\{1,\dots,n\} \to \operatorname{Im}(f)?$$</p>
<p>b.)
How many different function f are the... | Brian M. Scott | 12,042 | <p>Your algebra seems to be off:</p>
<p>$$\sin^2x + \cos^2 x \approx x^2 + \left(1-\dfrac{x^2}2\right)^2 = x^2 + 1 - x^2 + \dfrac{x^4}4 = 1 + \dfrac{x^4}4,$$ which is approximately $1$ for small values of $x$.</p>
<p><strong>Added:</strong> Note that you can’t expect to get exactly $1$, since you’re using an inexact ... |
1,530,406 | <p>How to multiply Roman numerals? I need an algorithm of multiplication of numbers written in Roman numbers. Help me please. </p>
| 3SAT | 203,577 | <p>Make a table with two columns, and enter the two numbers to be multiplied into the first row.
Make the next row by halving the first number (discarding remainders) and doubling the second. Continue until there is nothing left to halve.
Cross out all the rows where the left number is even.
Add the remaining number... |
1,903,333 | <p>Let $G$ be a group. Prove that $Z(G)$ (the center of $G$) is always nonempty.</p>
<p>Can anyone give me solution of this theoretical problem? I have just started learning group theory and I am very interested in this math branch</p>
| absolute friend | 244,073 | <p>Suppose $e$ be identity element of group $G$ then </p>
<p>$$ex=xe~~~~~~~~\forall x\in G$$</p>
<p>$$\implies e\in Z(G)\implies Z(G)\neq \emptyset$$</p>
|
3,660,101 | <p>I want to determine if the series <span class="math-container">$ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $</span> converge/diverge. the sequence in the denominator is not monotinic, so I cant use Dirichlet's or Abel's tests. My intuition is that this series converge, becuase its looks cl... | Brian M. Scott | 12,042 | <p>Let </p>
<p><span class="math-container">$$s_n=\sum_{k=2}^n\frac{(-1)^k}{(-1)^k+k}=\frac13-\frac12+\frac15-\frac14+\ldots+\frac{(-1)^n}{(-1)^n+n}$$</span> </p>
<p>and </p>
<p><span class="math-container">$$s_n'=\sum_{k=2}^n\frac{(-1)^{k+1}}k=-\frac12+\frac13-\frac14+\frac15+\ldots+\frac{(-1)^{n+1}}n\;.$$</span></... |
3,660,101 | <p>I want to determine if the series <span class="math-container">$ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $</span> converge/diverge. the sequence in the denominator is not monotinic, so I cant use Dirichlet's or Abel's tests. My intuition is that this series converge, becuase its looks cl... | CHAMSI | 758,100 | <p>Let <span class="math-container">$ n $</span> be a positive integer.</p>
<p><span class="math-container">\begin{aligned}\frac{\left(-1\right)^{n}}{n+\left(-1\right)^{n}}&=\frac{\left(-1\right)^{n}}{n}\left(\frac{n}{n+\left(-1\right)^{n}}\right)\\ &=\frac{\left(-1\right)^{n}}{n}\left(1-\frac{\left(-1\right)^... |
93,458 | <blockquote>
<p>Let <span class="math-container">$n$</span> be a nonnegative integer. Show that <span class="math-container">$\lfloor (2+\sqrt{3})^n \rfloor $</span> is odd and that <span class="math-container">$2^{n+1}$</span> divides <span class="math-container">$\lfloor (1+\sqrt{3})^{2n} \rfloor+1 $</span>.</p>
</... | José Carlos Santos | 446,262 | <p>It is not hard to prove that<span class="math-container">$$(\forall n\in\mathbb N):\left(2+\sqrt3\right)^n+\left(2-\sqrt3\right)^n\in2\mathbb N.$$</span>This, together with the fact that <span class="math-container">$2-\sqrt3\in(0,1),$</span> is enough to prove that <span class="math-container">$\left\lfloor\left(2+... |
1,018,270 | <p>While I was studying about finite differences I came across an article that says "computers can't deal with limit of $\Delta x \to 0$ " in <a href="http://1drv.ms/1sB5P1B" rel="nofollow">finite differences</a>.But if computers can't deal with these equations does anybody know how they compute $ \frac {d}{dx}$ of $x^... | Community | -1 | <p>A <a href="http://en.wikipedia.org/wiki/Differentiation_rules" rel="nofollow">table of the derivatives of primitive functions combined with differentiation rules</a> yields an algorithm that allows a computer program to <em>symbolically</em> compute the derivative of any function that is a compound of primitive func... |
2,790,025 | <p><strong>GIVEN:</strong></p>
<ul>
<li>$g$ is differentiable with continuous derivative on $[a,b]$.</li>
</ul>
<p><strong>WANT TO SHOW</strong></p>
<p>$$|g(x)-g(y)| \leq \bigg(\max_{x\in [a,b]} |g'(x)|\bigg)|x-y|.$$</p>
<p><strong>USING MEAN VALUE THEOREM</strong></p>
<p>Given $x<y$ in the interval $(a,b)$, th... | Intelligenti pauca | 255,730 | <p>If $\angle OAD=\alpha$, then $\angle FOC=\alpha/2$ and:
$$
AO-AE=r(\csc\alpha-\cot\alpha)=r{1-\cos\alpha\over\sin\alpha}=
r\tan{\alpha\over2}=FC,
$$
where $r=OE=OF$ is the radius of the inscribed circle. In the same way one derives $BO-BF=ED$ and adding these two equalities we get the desired result.</p>
<p><a href... |
2,790,025 | <p><strong>GIVEN:</strong></p>
<ul>
<li>$g$ is differentiable with continuous derivative on $[a,b]$.</li>
</ul>
<p><strong>WANT TO SHOW</strong></p>
<p>$$|g(x)-g(y)| \leq \bigg(\max_{x\in [a,b]} |g'(x)|\bigg)|x-y|.$$</p>
<p><strong>USING MEAN VALUE THEOREM</strong></p>
<p>Given $x<y$ in the interval $(a,b)$, th... | Blue | 409 | <p>The angle chase ---which relies upon the fact that opposite angles of an inscribed quadrilateral are supplementary--- gives us this figure:</p>
<p><a href="https://i.stack.imgur.com/ugz0i.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ugz0i.png" alt="enter image description here"></a></p>
<p>Th... |
3,460 | <p>I asked the question "<a href="https://mathoverflow.net/questions/284824/averaging-2-omegan-over-a-region">Averaging $2^{\omega(n)}$ over a region</a>" because this is a necessary step in a research paper I am writing. The answer is detailed and does exactly what I need, and it would be convenient to directly cite t... | Chris Godsil | 1,266 | <p>You attribute it to an anonymous author and give the link. So you are not claiming credit and your readers can check the source. </p>
<p>I would see no harm in reproducing the argument as well, especially if it’s short.</p>
|
2,928,849 | <p>I have a problem understanding the following:</p>
<p><span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are independent vatiables with
<span class="math-container">$$P(X = i) = P(Y = i) = \frac{1}{2^i}. \quad i = 1, 2, \cdots$$</span>
Now the book says<span class="math-container">$ ... | Travis | 568,040 | <p>I think it's a typo.</p>
<p><span class="math-container">$P(X > i) =1-\sum_{k=1}^{k=i}P(X =k)=1-\sum_{k=1}^{k=i}{P(X = k)}
= 1 - \sum_{k=1}^{k=i}{\frac{1}{2^k}}= \frac{1}{2^i}$</span></p>
|
167,904 | <p>In one the the answers to this thread " <a href="https://mathoverflow.net/questions/119850/">Can one embedd the projectivezed tangent space of CP^2 in a projective space? </a> " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety of complete flags in the vector space $\mathbb{C^3}$ ". </p>
... | BlaCa | 14,514 | <p>Let $\pi:\mathbb{P}(T_{\mathbb{P}^2})\rightarrow\mathbb{P}^2$ be the projectivized tangent bundle. The point $x:=(p,[L])\in\mathbb{P}(T_{\mathbb{P}^2})$ corresponds to the point $p = \pi((p,[L]))\in\mathbb{P}^2$ and to the class of the line $L\subset\mathbb{P}^2$ passing through $p$. Now, the point $p$ is a line thr... |
625,608 | <p>The question is to show that the function $\phi$ given by $\phi(\lambda)=\frac{\lambda}{1+|\lambda|}$ is 1-1 on the complex plane. I would be grateful for a hint on how to start.</p>
| Henry | 6,460 | <p>Hint:</p>
<p>If $\lambda = r e^{i \theta}$ and $\mu = s e^{i \psi}$ with $r=|\lambda |$ etc.,</p>
<p>then $\phi(\lambda) = \dfrac{r e^{i \theta}}{1+r} =\left(1-\frac{1}{1+r}\right) e^{i \theta}$ and $\phi(\mu) = \left(1-\frac{1}{1+s}\right) e^{i \psi}$</p>
|
111,899 | <p>Evaluate the integral using trigonometric substitutions. </p>
<p>$$\int{ x\over \sqrt{3-2x-x^2}} \,dx$$</p>
<p>I am familiar with using the right triangle diagram and theta, but I do not know which terms would go on the hypotenuse and sides in this case. If you can determine which numbers or $x$-values go on the... | Aru Ray | 13,129 | <p>$\int \frac{x}{\sqrt{4-(x+1)^2}}dx = \int \frac{2\sin\theta-1}{\sqrt{4-4\sin^2\theta}}(2\cos\theta)d\theta$
(using the substitution $x+1=2\sin\theta$)</p>
<p>$=\int\frac{2\sin\theta-1}{2\cos\theta}2\cos\theta d\theta$</p>
<p>$= \int (2\sin\theta-1) d\theta$</p>
<p>$=-2\cos\theta-\theta +C$</p>
<p>$=-2\left(\frac... |
1,303,274 | <p>Define a sequence {$\ x_n$} recursively by</p>
<p>$$ x_{n+1} =
\sqrt{2 x_n -1}, \ and \ x_0=a \ where \ a>1
$$
Prove that {$\ x_n$} is strictly decreasing. I'm not sure where to start.</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: $$s_{2n} = 0, s_{2n+1} = 1$$. </p>
|
1,303,274 | <p>Define a sequence {$\ x_n$} recursively by</p>
<p>$$ x_{n+1} =
\sqrt{2 x_n -1}, \ and \ x_0=a \ where \ a>1
$$
Prove that {$\ x_n$} is strictly decreasing. I'm not sure where to start.</p>
| mich95 | 229,072 | <p>All you need to show is that the the sequence $x_{n}=(-1)^{n}$ does not converge. Many ways to do it : First, using the theorem saying that if a sequence converges , then any subsequence converges to the same limit. $x_{2n}$ and $x_{2n+1}$ are subsequences converging obviously to different limits. Not familiar with ... |
2,466,855 | <p>We were given this question for homework that the professor couldn't explain how to solve (even in class he had trouble working it out). I'm only aware that we should be using the law of large numbers but I'm not sure how to apply it as the book for the course provides no examples. The answer in class was 10 and the... | Simply Beautiful Art | 272,831 | <p>Denote</p>
<p>$$f(z)=\sum_{n=0}^\infty\frac{z^n}{(n!)^2}$$</p>
<p>By term-wise differentiation, we find that</p>
<p>$$f'(z)+zf''(z)=f(z)$$</p>
<p>A rather simple differential equation with the general solution</p>
<p>$$f(z)=c_1I_0(2\sqrt z)+c_2K_0(2\sqrt z)$$</p>
<p>where $I_n$ is a modified Bessel function of... |
4,600,131 | <blockquote>
<p>If <span class="math-container">$$f(x)=\binom{n}{1}(x-1)^2-\binom{n}{2}(x-2)^2+\cdots+(-1)^{n-1}\binom{n}{n}(x-n)^2$$</span>
Find the value of <span class="math-container">$$\int_0^1f(x)dx$$</span></p>
</blockquote>
<p>I rewrote this into a compact form.
<span class="math-container">$$\sum_{k=1}^n\binom... | K.defaoite | 553,081 | <p>A devious little problem indeed! I am interested in where you found it. We in fact have a very nice formula:</p>
<p><span class="math-container">$$\sum_{k=1}^n (-1)^{k-1} \binom{n}{k} (x-k)^2 =x^2$$</span></p>
<p>It follows from:</p>
<p><span class="math-container">$$F(x)=\sum_{k=1}^n (-1)^{k-1} \binom{n}{k} (x-k)^2... |
3,044,318 | <p><span class="math-container">$$\frac{e^{z^2}}{z^{2n+1}}$$</span>
Am I right that limit as z approaches infinity does not exist? So its residue at infinity is equal to <span class="math-container">$c_{-1}$</span> of Laurent series. How am I supposed to get Laurent series of this function? Where is it centered? What r... | Angina Seng | 436,618 | <p>Strictly speaking, one finds residues of differentials, not of functions.
Here the differential is
<span class="math-container">$$\alpha=\frac{\exp(z^2)}{z^{2n+1}}\,dz$$</span></p>
<p>To find the residue at <span class="math-container">$\infty$</span>, set <span class="math-container">$z=1/w$</span> and expand in p... |
250,496 | <p>Please help me out.. Is there some appropriate method to draw Hasse diagram</p>
<blockquote>
<p>My question is $L=\{1,2,3,4,5,6,10,12,15,30,60\}$</p>
</blockquote>
<p>Please explain me by step by step solution... Thanks for help..</p>
| Brian M. Scott | 12,042 | <p>I would start at the bottom. Clearly $1$ divides every member of $L$, so $1$ is the minimum element in the divisibility order on $L$. Now what is the next layer up of the Hasse diagram? It must contain those members of $L$ that are divisible by $1$ (and of course by themselves) but not by any other element of $L$. T... |
21,569 | <p>Recently I commented on a question which was about tiling a 100 by 100 grid with a 1 x 8 square. (Actually to prove this was impossible.) One of our users posted a link to a very interesting looking paper on tiling problems as a comment, and I also commented, but now looking through my comments I can't find the post... | hardmath | 3,111 | <p>If a Question is self-deleted by the poster, it will <em>not</em> show under the 10K User Tools report of <a href="https://math.stackexchange.com/tools">recently deleted items</a>. Also the items shown are limited to deletion actions in the past 30 days, if the date range selector in the upper right corner there is... |
1,699,627 | <p>Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their integral forms but $n^{-s}$ and $e^{-x}$ don't mix into $e^{-nx}$... | Graham Kemp | 135,106 | <p>The probability that you will have <em>at most 3</em> kings is the probability that you will have <em>less than 4</em>.</p>
<p>$$\mathsf P(K\leq 3) = 1 -\mathsf P(K=4)$$</p>
<p>The probability that you will have exactly all four kings is the count of ways to select 4 kings and 1 other card divided by the count of ... |
1,699,627 | <p>Why is the following equation true?$$\int_0^\infty e^{-nx}x^{s-1}dx = \frac {\Gamma(s)}{n^s}$$
I know what the Gamma function is, but why does dividing by $n^s$ turn the $e^{-x}$ in the integrand into $e^{-nx}$? I tried writing out both sides in their integral forms but $n^{-s}$ and $e^{-x}$ don't mix into $e^{-nx}$... | user247327 | 247,327 | <p>Yet another way: there are 52 cards in the deck, 4 of which are kings. The probability the first card is a king is 4/52= 1/13. There are then 51 cards left, 3 of them kings. The probability the second card is a king is 3/51= 1/17. There are then 50 cards left, 2 of them kings. The probability the third card is ... |
3,168,130 | <p><a href="https://math.stackexchange.com/questions/2690416/mathematical-proof-of-uniform-circular-motion">Here</a> is a mathematical proof that any force <span class="math-container">$F(t)$</span>, which affects a body, so that <span class="math-container">$\vec{F(t)} \cdot \vec{v(t)} = 0$</span>, where <span class="... | nonuser | 463,553 | <p>Since <span class="math-container">$$ \vec{F} = m{d\vec{v}\over dt} =m\cdot \vec{v}'$$</span></p>
<p>so <span class="math-container">$$\vec{v}'\cdot \vec{v} =0\implies (\vec{v}^2)' = 0 \implies \vec{v}^2 = constant$$</span></p>
<p>So <span class="math-container">$$|\vec{v}|^2 = constant\implies |\vec{v}| = constan... |
249,047 | <p>I have the following matrix: $$A=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1 \\
\end{bmatrix}
$$
What is the norm of $A$? I need to show the steps, should not use Matlab...<br>
I know that the answer is $\sqrt{\sqrt{5}/2+3/2}$. I am using the s... | marty cohen | 13,079 | <p>$1/(n-1)-1/(n+1) = 2/(n^2-1)
$, so
$$\sum_{n=4}^{\infty} \frac{1}{n^2-1}
= (1/2)\sum_{n=4}^{\infty} \left(\frac{1}{n-1} - \frac{1}{n+1}\right)
= \frac{1}{3} + \frac{1}{4}
$$
since all the later terms are cancelled out.</p>
<p>Whoops - as pointed out by Limitless, this should be
$\frac{1}{2}\left(\frac{1}{3} + \frac... |
962,573 | <p>I have just learned Fermat's little theorem.</p>
<p>That is,</p>
<blockquote>
<p>If $p$ is a prime and $\gcd(a,p)=1$, then $a^{p-1} \equiv 1 \mod p$</p>
</blockquote>
<p>Well, there's nothing more explanation on this theorem in my book.</p>
<p>And there are exercises of this kind</p>
<blockquote>
<p>If $\gc... | Bruce Zheng | 181,584 | <p>If $x \equiv 1 \bmod y$ and $x \equiv 1 \bmod z$, then $x \equiv 1 \bmod yz$ if $y$ and $z$ are coprime.</p>
<p>It also turns out that $a \equiv 1 \bmod 2 \Rightarrow a^2 \equiv 1 \bmod 2^3$. This is because if $a = 2k + 1$, $a^2 = 4(k^2 + k) + 1$. Since $k^2 \equiv k \bmod 2$, $k^2 + k$ is even. Thus $8 \mid 4(k^2... |
609,770 | <p>We have an empty container and $n$ cups of water and $m$ empty cups. Suppose we want to find out how many ways we can add the cups of water to the bucket and remove them with the empty cups. You can use each cup once but the cups are unique. </p>
<p>The question: In how many ways can you perform this operation.</p... | CoffeeIsLife | 97,554 | <p>The method in proving that $f(x)=\frac{1}{x}$ is not uniformly continuous is similar to the method in proving that $f(x)=\frac{1}{x^2}$. I will address $f(x)=\frac{1}{x^2}$ , and you can infer from it how to prove $f(x)=\frac{1}{x}$.</p>
<p>Assume that $f(x)=\frac{1}{x^2}$ is not uniformly continuous. This means th... |
609,770 | <p>We have an empty container and $n$ cups of water and $m$ empty cups. Suppose we want to find out how many ways we can add the cups of water to the bucket and remove them with the empty cups. You can use each cup once but the cups are unique. </p>
<p>The question: In how many ways can you perform this operation.</p... | CoffeeIsLife | 97,554 | <p>Or you could use the theorem of uniform continuity regarding Cauchy sequences. A theorem states that if $f$ is uniformly continuous on a set $S$ and $(s_n)$ is a Cauchy sequence in $S$, then $f(s_n)$ is a Cauchy sequence. </p>
<p>Choose $(s_n)=\frac{1}{n}$ (here $n\in\mathbb{N}$), which is a Cauchy sequence. Howeve... |
2,699,753 | <p>$a,b$ and $c$ are all natural numbers, and function $f(x)$ always returns a natural number. If$$
\sum_{n=b}^{a} f(n) = c,$$
in terms of $b,c$ and $f$, how would you solve for $a$? Do I require more information to solve for $a$?</p>
<p>EDIT: If $x$ increases $f(x)$ increases</p>
| saulspatz | 235,128 | <p>"The general solution to an inhomogeneous linear recurrence relation is the general solution to the associated homogeneous recurrence, plus any particular solution of the inhomogeneous recurrence."</p>
<p>Quite a mouthful, but what is means is this:</p>
<ol>
<li><p>Solve the homogeneous recurrence you get by elimi... |
2,699,753 | <p>$a,b$ and $c$ are all natural numbers, and function $f(x)$ always returns a natural number. If$$
\sum_{n=b}^{a} f(n) = c,$$
in terms of $b,c$ and $f$, how would you solve for $a$? Do I require more information to solve for $a$?</p>
<p>EDIT: If $x$ increases $f(x)$ increases</p>
| Math1000 | 38,584 | <p>Let $A(z) = \sum_{n=0}^\infty f(n)z^n$. Multiplying both sides of the recurrence by $z^n$ and summing over $n$, we have
$$
\sum_{n=1}^\infty f(n)z^n = \sum_{n=1}^\infty 9f(n-1)z^n - \sum_{n=1}^\infty 14z^n.
$$
Writing the above in terms of $A(z)$,
$$
A(z) - 3 = 9zf(z) -\frac{14z}{1-z}.
$$
Solving for $A(z)$, we have... |
2,699,753 | <p>$a,b$ and $c$ are all natural numbers, and function $f(x)$ always returns a natural number. If$$
\sum_{n=b}^{a} f(n) = c,$$
in terms of $b,c$ and $f$, how would you solve for $a$? Do I require more information to solve for $a$?</p>
<p>EDIT: If $x$ increases $f(x)$ increases</p>
| farruhota | 425,072 | <p>You can find several terms and see pattern:
$$\begin{align} &f(1)=9f(0)-14; \\
&f(2)=9f(1)-14=9(9f(0)-14)-14=9^2f(0)-9\cdot 14-14; \\
&f(3)=9f(2)-14=9(9^2f(0)-9\cdot 14-14)-14=9^3f(0)-9^2\cdot 14-9\cdot 14-14; \\
&\cdots \\
&f(n)=9^nf(0)-(9^{n-1}+9^{n-2}+\cdots+9+1)\cdot 14=9^nf(0)-\frac{9^n-1}{9... |
1,690,854 | <p>Solve the equation
$$-x^3 + x + 2 =\sqrt{3x^2 + 4x + 5.}$$
I tried. The equation equavalent to
$$\sqrt{3x^2 + 4x + 5} - 2 + x^3 - x=0.$$
$$\dfrac{3x^2+4x+1}{\sqrt{3x^2 + 4x + 5} + 2}+x^3 - x=0.$$
$$\dfrac{(x+1)(3x+1)}{\sqrt{3x^2 + 4x + 5} + 2}+ (x+1) x (x-1)=0.$$
$$(x+1)\left [\dfrac{3x+1}{\sqrt{3x^2 + 4x + 5} + 2}+... | John_dydx | 82,134 | <p>I would start by squaring both sides of the equation:</p>
<p>$$ (-x^2 + x +2)^2 = 3x^2 +4x +5$$</p>
<p>$$x^6 -2x^4-4x^3-2x^2-1 =0$$</p>
<p>As suggested by Deepak, $x = -1$ is a solution.
You can factorise fully by dividing the above polynomial by $x+1$ to obtain other factors and solutions (if any).</p>
|
2,915,735 | <p>So I just got done showing explicitly that an isomorphism exists between these two rings if the $\gcd(m, n) = 1$, and I did not have much trouble with that. For some reason I'm having a lot harder of a time showing that the result <em>is not</em> true if $m$ and $n$ are not relatively prime. Can somebody help me out... | N. S. | 9,176 | <p><strong>Hint</strong> Show that $\mbox{lcm} (m,n) \cdot (x,y) =(0,0)$ for all elements $(x,y) \in \mathbb Z_m \times \mathbb Z_n$. </p>
|
9,416 | <p>Say I pass 512 samples into my FFT</p>
<p>My microphone spits out data at 10KHz, so this represents 1/20s.</p>
<p>(So the lowest frequency FFT would pick up would be 40Hz).</p>
<p>The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc</p>
<p>So if my original sound con... | P i | 3,096 | <p>In the end I derived a formula that calculates the exact frequency from the rate at which each bin rotates.</p>
<p>My result can be seen here: <a href="https://stackoverflow.com/questions/4633203/extracting-precise-frequencies-from-fft-bins-using-phase-change-between-frames">https://stackoverflow.com/questions/4633... |
55,232 | <p>I'm looking for a concise way to show this:
$$\sum_{n=1}^{\infty}\frac{n}{10^n} = \sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right)$$
With this goal in mind:
$$\sum_{n=1}^{\infty}\left(\left(\sum_{m=0}^{\infty}{10^{-m}}\right){10^{-n}}\right) =
\sum_{n=1}^{\infty}\left(\left(\frac{... | Braindead | 2,499 | <p>Are you allowed to use derivatives? If so, then...</p>
<p>$\frac{1}{1-x} = \sum_{n=0}^\infty x^n$</p>
<p>$\frac{1}{(1-x)^2} = \sum_{n=1}^\infty n x^{n-1}$</p>
<p>$\frac{x}{(1-x)^2} = \sum_{n=1}^\infty n x^n$</p>
<p>Letting $x=\frac{1}{10}$, you get</p>
<p>$\frac{10}{81} = \sum_{n=1}^\infty \frac{n}{10^n}$</p>
|
4,536,320 | <p>Let <span class="math-container">$R$</span> be a ring and <span class="math-container">$A \subseteq R$</span> be finite, say <span class="math-container">$A = \{a\}$</span>. The set <span class="math-container">$$RaR = \{ras\:\: : r,s\in R\}$$</span> Why is this not necessarily closed under addition?</p>
<p>Take <sp... | user2661923 | 464,411 | <p>For any given <span class="math-container">$k \in \Bbb{Z}$</span> such that <span class="math-container">$1 \leq k \leq 10$</span>, <span class="math-container">$m$</span> must be some element in <span class="math-container">$\{0,1,2,\cdots,2k\}$</span>. Therefore, for each such <span class="math-container">$k$</sp... |
3,651,287 | <p>I would be very grateful if you could help me, I have a question about the Cauchy sequences, they have given me the definition that a Cauchy sequence if:</p>
<p>A sequence <span class="math-container">$(r_{n})_{n\in \mathbb{N}}$</span> is of Cauchy if:</p>
<p><span class="math-container">$\forall \epsilon>0,$</... | Kavi Rama Murthy | 142,385 | <p>They are equivalent. If the second condition is satisfied then <span class="math-container">$|r_k-r_j| \leq |r_k-r_m|+|r_m-r_j| <2 \epsilon$</span> for all <span class="math-container">$j,k >m$</span>. Do you now see the equivalence of the two definitions?</p>
|
1,043 | <p>Hi all,</p>
<p>The short-time fourier transform decomposes a signal window into a sin/cosine series.</p>
<p>How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead of sin/cos? These arbitrary basis functions are likely in my case to be very small discrete chunks of ... | Francisco Pereira | 704 | <p>You can use an arbitrary set of basis functions that can be placed or scaled at any point along your signal, if I understand what you want. The problem you'd be solving would be to find placement locations (another set of parameters) and scales (another set), to minimize squared error relative to the original signal... |
1,043 | <p>Hi all,</p>
<p>The short-time fourier transform decomposes a signal window into a sin/cosine series.</p>
<p>How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead of sin/cos? These arbitrary basis functions are likely in my case to be very small discrete chunks of ... | Vasile Moșoi | 1,093 | <p>If the signal is not periodic we talk about the integral transform of the signal (in case of Fourier transform) and it have a continuum spectrum but not discreet.
In case of polynomial decomposition the desired accuracy of approximation is valid in the finite interval in which the basis functions are orthogonal. Out... |
256,373 | <p>I've not been able to find a package which will deal with Geometric Algebra. Perhaps somebody can help?</p>
| nsap | 81,217 | <p>You might want to use FindInstance in these case:</p>
<pre><code>Table[{n,
FindInstance[x^3 + y^3 + z^3 == n && 0 < n < 100, {x, y, z},
Integers]}, {n, 1, 10}]
</code></pre>
|
4,609,236 | <p>When finding the derivative of <span class="math-container">$f(x) = \sqrt x$</span> via the limit definition, one gets</p>
<p><span class="math-container">$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt x}{h}$$</span></p>
<p>For this, I could get the answer from applying L'... | PrincessEev | 597,568 | <p>The line of reasoning is in fact circular; you need to know the derivative of the function to apply L'Hopital's rule in the first place.</p>
<p>One can (infamously) likewise apply this to</p>
<p><span class="math-container">$$\lim_{x \to 0} \frac{\sin(x)}{x}$$</span></p>
<p>but</p>
<p><span class="math-container">$$... |
4,609,236 | <p>When finding the derivative of <span class="math-container">$f(x) = \sqrt x$</span> via the limit definition, one gets</p>
<p><span class="math-container">$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt x}{h}$$</span></p>
<p>For this, I could get the answer from applying L'... | PierreCarre | 639,238 | <p>The circular use of the knowledge of <span class="math-container">$f'(x)$</span> in its computation using the definition is not reasonable and proves nothing. Your actual conclusion would just be that "If <span class="math-container">$f'(x) = \frac{1}{2 \sqrt{x}}$</span> then <span class="math-container">$f'(x... |
928,826 | <p>I have a function </p>
<p>$$f(x)=\frac{2x^2 - x - 1}{x^2 + 3x + 2}$$</p>
<p>from the interval $[0,\infty)$</p>
<p>The limit of this function is $2$. Is the range then simply from $f(0)$ to $2$, and if yes, would I write it as $[f(0],2]$ or $[f(0),2)$, i.e open brackets or closed? </p>
<p>Also, would i first nee... | Pierre Alvarez | 175,318 | <p>If you can prove that your function is monotonous, then you need to write this argument first, and then you can deduce that the range is $[f(0),2[$. </p>
|
928,826 | <p>I have a function </p>
<p>$$f(x)=\frac{2x^2 - x - 1}{x^2 + 3x + 2}$$</p>
<p>from the interval $[0,\infty)$</p>
<p>The limit of this function is $2$. Is the range then simply from $f(0)$ to $2$, and if yes, would I write it as $[f(0],2]$ or $[f(0),2)$, i.e open brackets or closed? </p>
<p>Also, would i first nee... | Jean-Claude Arbaut | 43,608 | <p>Yes, you have to prove your function is increasing. No too difficult, since the derivative is, after simplification,</p>
<p>$$f'(x)=\frac{7x^2+10x+1}{(x^2+3x+2)^2}>0$$</p>
<p>Hence the range is $[f(0), \lim_{x\to+\infty}f(x)[=[-1/2,2[$, <em>open</em> on the right.</p>
<hr>
<p>I'm not sure what you mean in you... |
1,439,920 | <blockquote>
<p>So, the question is:<br>
Calculate the probability that 10 dice give more than 2 6s.</p>
</blockquote>
<p>I've calculated that the probability for throwing 3 6s is 1/216.</p>
<p>And by that logic: 1/216 + 1/216 + .. + 1/216 = 10/216.</p>
<p>But I've been told that this isn't the proper way set it... | Lutz Lehmann | 115,115 | <p>The Newton polygon tells us that the dominant binomials are </p>
<ul>
<li>$x^5-1102x^4$ for large roots, resulting in a root close to $1102$ and</li>
<li>$-1102x^4-2015x$ for small roots, resulting in roots close to $0$ and the three third roots of $-\frac{2015}{1102}\approx (-1.22282549628)^3$.</li>
</ul>
<p>Sing... |
4,171,907 | <blockquote>
<p>If <span class="math-container">$3\sin x +5\cos x=5$</span> then prove that <span class="math-container">$5\sin x-3\cos x=3$</span></p>
</blockquote>
<p>What my teacher did in solution was as follows</p>
<p><span class="math-container">$$3\sin x +5\cos x=5 \tag1$$</span></p>
<p><span class="math-contain... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$c=\cos x$</span> and <span class="math-container">$s=\sin x$</span>, then you know that<span class="math-container">$$\left\{\begin{array}{l}3s+5c=5\\c^2+s^2=1.\end{array}\right.$$</span>This system is easy to solve. One of the solutions is <span class="math-container">$(c,s)=(1,0)$<... |
1,868,263 | <p>A Relation R on the set N of Natural numbers be defined as (x,y) $\in$R if and only if $x^2-4xy+3y^2=0$ for allx,y $\in$N then show that the relation is reflexive,transitive but not SYMMETRIC.</p>
<p>i got how this relation is reflexive or transitive but i am not able to think of any reason of why this relation is... | Robert Z | 299,698 | <p>R is not symmetric because $(3,1)\in R$ but $(1,3)\not\in R$.
Note that $x^2-4xy+3y^2=(x-y)(x-3y)$.</p>
|
1,386,307 | <p>If you consider that you have a coin, head or tails, and let's say tails equals winning the lottery. If I participate in one such event, I may not get tails. It's roughly 50%. But if a hundred people are standing with a coin and I or them get to flip it, my chances of having gotten a tail after these ten attempts... | David | 119,775 | <p>There are a few similar but different problems here, perhaps that is what is causing confusion.</p>
<ul>
<li>If you are aiming to win the lottery <em>at least once</em>, then the more times you enter, the better your chance of success. It's the same as the coin problem you described.</li>
<li>If you are aiming to ... |
2,109,347 | <p>My statistics note states that the variance of the empirical distribution is
$v= \sum_{i=1}^{n}(x_i-\bar x )^2\frac {1} {n}$ which the author then re-writes as
$v= \sum_{i=1}^{n}x_i^2 (\frac {1} {n}) - \bar x^2$. How is this achieved?</p>
| Joda | 392,273 | <p>It's just algebra</p>
<p>$$
\begin{aligned}
v&=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2=\frac{1}{n}\sum_{i=1}^n\left(x_i^2-2x_i\bar{x}+\bar{x}^2\right)\\
&=\frac{1}{n}\sum_{i=1}^nx_i^2-2\bar{x}\frac{1}{n}\sum_{i=1}^nx_i+\bar{x}^2\frac{1}{n}\sum_{i=1}^n1\\
&=\frac{1}{n}\sum_{i=1}^nx_i^2-2\bar{x}^2+\bar{x}^... |
770,544 | <p>It's clear that a system of two quadratic equations can have none, one or two solutions. </p>
<p>For example: $y = x^2 + 2$ and $y = - x^2 + 1$ have none. $y = x^2$, $2x^2 - 8x + 8$ and $y = - x^2 + 8x - 8$ have $4$ as common solution. And $2x^2 - 8x + 8 = x^2 - 4x + 5$ have $1$ and $3$ as solutions.</p>
<p>Is it ... | Mark Bennet | 2,906 | <p>It depends what you mean by a system of quadratics.</p>
<p>If you have $y=ax^2+by+c$ and $y=px^2+qx+r$ then you must have $(a-p)x^2+(b-q)x+(c-r)=0$. This equation has at most two solutions for $x$ (unless $a=p, b=q, c=r$) and each solution for $x$ gives a value for $y$.</p>
<p>However there is another way of think... |
1,176,938 | <p>How do you show that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$?</p>
<p>I don't think you can use the eisenstein criterion here</p>
| Lubin | 17,760 | <p>Another argument, special to this polynomial.</p>
<p>Any number of arguments (including multiplying the thing out) show that the polynomial is $\equiv X^4-2\pmod5$. Thus in characteristic $5$, any root is a $16$-th root of unity. But the smallest field of characteristic $5$ whose cardinality is $\equiv1\pmod{16}$ i... |
3,483,260 | <p>Given the set <span class="math-container">$\{1,2,3,4,5,6,7\}$</span>.</p>
<p>We would like to create a string of size 8 so that each of the elements of the set appears at least once in the result. How many ways are there to create such a set?</p>
<p>I think that the answer should be: order 7 elements <span class=... | Clement Yung | 620,517 | <p>For a non-negative, increasing, convex function, we have <span class="math-container">$\lim_{x \to -\infty}(f(x) - f'(x)) \geq 0$</span> exists. Consider:
<span class="math-container">$$
g(x) = (f(x) - f'(x))e^x
$$</span>
Differentiating yields:
<span class="math-container">$$
g'(x) = (f(x) - f''(x))e^x \geq 0
$$</s... |
3,249,809 | <p>What’s the explicit rule for for this number sequence?</p>
<p><span class="math-container">$$\displaystyle{1 \over 100},\ -{3 \over 95},\ {5 \over 90},\
-{7 \over 85},\ {9 \over 80}$$</span></p>
<p>The numerator changes to negative every other term, while the denominator subtracts <span class="math-container"... | beelal | 674,630 | <p>with convention we start from <span class="math-container">$n= 0$</span>, <span class="math-container">$x_n = \frac{(-1)^n(2n+1)}{100-5n}$</span> for <span class="math-container">$n=0,1, \ldots, 19$</span></p>
|
2,231,487 | <p>In [Mathematical Logic] by Chiswell and Hodges, within the context of natural deduction and the language of propositions LP (basically like <a href="http://www.cs.cornell.edu/courses/cs3110/2011sp/lectures/lec13-logic/logic.htm" rel="nofollow noreferrer">here</a>) it is asked to show, by counter-example that a certa... | Daniel Schepler | 337,888 | <p>My approach would be a bit different: if $\Gamma = \{ p \vee q \}$ where $p$ and $q$ are atomic formulae, then $\Gamma \vdash p \vee q$ so the rule would imply that either $\{ p \vee q \} \vdash p$ or $\{ p \vee q \} \vdash q$. But in the first case, assigning $p := (2 = 3)$ and $q := (3 = 3)$ would disprove that; ... |
19,285 | <p>Is anyone aware of Mathematica use/implementation of <a href="http://en.wikipedia.org/wiki/Random_forest">Random Forest</a> algorithm?</p>
| Daniel Lichtblau | 51 | <p>Disclaimer: This is not an implementation of the Random Forest Algorithm. Also, while I have on occasion used random florists, until today I had not heard of the Random Forest Algorithm.</p>
<p>I poked around a bit on the Net and learned that these take subsamples of data, subsampling the variables as well, and for... |
3,053,386 | <p>This might be a very basic question for some of you. Indeed in <span class="math-container">$\textbf Z$</span>, it's very easy. For example, <span class="math-container">$\textbf Z / \langle 2 \rangle$</span> consists of <span class="math-container">$\langle 2 \rangle$</span> and <span class="math-container">$\langl... | Kenny Wong | 301,805 | <p>For principal ideals, it's easy: if <span class="math-container">$L$</span> is a number field with ring of integers <span class="math-container">$\mathcal O_L$</span>, and if <span class="math-container">$(a)$</span> is an ideal in <span class="math-container">$\mathcal O_L$</span>, then the number of elements in <s... |
3,817,104 | <p>For <span class="math-container">$a,b,c \in \Big[\dfrac{1}{3},3\Big].$</span> Prove<span class="math-container">$:$</span></p>
<p><span class="math-container">$$(a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) \leqslant 25.$$</span></p>
<p>Assume <span class="math-container">$a\equiv \text{mid}\{a,b,c\},$</s... | nguyenhuyenag | 410,198 | <p>I found a better estimation
<span class="math-container">$$ (a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) \leqslant \frac{209}{9}.$$</span>
Equality occur when <span class="math-container">$a=b=3,\,c=\frac 13$</span> or <span class="math-container">$a=b=\frac 13,\,c=3.$</span></p>
|
3,817,104 | <p>For <span class="math-container">$a,b,c \in \Big[\dfrac{1}{3},3\Big].$</span> Prove<span class="math-container">$:$</span></p>
<p><span class="math-container">$$(a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) \leqslant 25.$$</span></p>
<p>Assume <span class="math-container">$a\equiv \text{mid}\{a,b,c\},$</s... | richrow | 633,714 | <p>Let <span class="math-container">$f(a,b,c)=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$</span>. Note that <span class="math-container">$f$</span> is concave of each variable (if other variables are fixed). Hence, since concave on <span class="math-container">$I$</span> fucntion attains its maximum at end... |
1,287,225 | <p>So I have come across a question asked by my peers. </p>
<p>Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$</p>
<p>Given that $$y_r(t)=\sqrt{t}\cdot h+k,$$ where $h$ and $k$ are independent random variables with variance $\sigma_h$ and $\sigma_k$ respectively. </p>
<p>So what is $g$ in this case? The answer key gives: $g=\s... | Michael Hardy | 11,667 | <p>$\newcommand{\E}{\operatorname{E}}\newcommand{\var}{\operatorname{var}}$
It follows from independence that $\E(hk)=\E(h)\E(k)$.</p>
<p>If $\E(h)=\E(k) = 0$ then $\E(h^2)= \var(h)$ and $\E(k^2)=\var(k)$ and we have
\begin{align}
\E\left((\sqrt{t}\, h + k)^2\right) = \E(th^2 + 2\sqrt t\,hk + k^2) & = t\E(h^2) + 2... |
1,287,225 | <p>So I have come across a question asked by my peers. </p>
<p>Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$</p>
<p>Given that $$y_r(t)=\sqrt{t}\cdot h+k,$$ where $h$ and $k$ are independent random variables with variance $\sigma_h$ and $\sigma_k$ respectively. </p>
<p>So what is $g$ in this case? The answer key gives: $g=\s... | ronno | 32,766 | <p>Building on Michael Hardy's answer. I use the notation $\bar h = E(h)$ and $\bar k = E(k)$ for readability. Since $h,k$ are independent, $E(hk) = \bar h \bar k$.</p>
<p>By definition, $E(h^2) = \sigma_h + \bar h^2$ and $E(k^2) = \sigma_k + \bar k^2$. We have,
$$\begin{align*}E((\sqrt{t}h+k)^2)& =E(th^2+2\sqrt{t... |
540,227 | <p>I was try to understand the following theorem:-</p>
<p><strong>Let $X,Y$ be two path connected spaces which are of the same homotopy type.Then their fundamental groups are isomorphic.</strong></p>
<p><strong>Proof:</strong> The fundamental groups of both the spaces $X$ and $Y$ are independent on the base points si... | Edoardo Lanari | 77,181 | <p>$f_{\#}:\pi_1(X,x_0) \to \pi_1(Y,f(x_0))$ is an isomorphism since $f$ is an homotopy equivalence; moreover $\pi_1(Y,f(x_0)) \simeq \pi_1(Y,y_0)$ since $Y$ is path-connected. </p>
|
1,494,409 | <blockquote>
<p>Let <span class="math-container">$\Bbb R^+$</span> denote the real numbers. Suppose <span class="math-container">$\phi:\Bbb R^+\to\Bbb R^+$</span> is an automorphism of the group <span class="math-container">$\Bbb R^+$</span> under multiplication with <span class="math-container">$\phi(4)=7$</span>.</p>... | CPM | 119,124 | <p>If $\phi(4)=7$, then $7=\phi(4)=\phi(2\cdot 2)=\phi(2)\phi(2) = \phi(2)^2$. Can you take it from there?</p>
|
37,804 | <p>I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are concerning <em>compact operators</em> and <em>unbounded operators</em>. Here I have the examples of $-\Delta$, the laplaci... | Paul Siegel | 4,362 | <p>I think Helge's answer cuts to the historical heart of the matter: solution operators for various differential equations tend to be bounded, non-compact operators (obtained in many cases from an unbounded differential operator via the functional calculus), and it is often quite useful from that point of view to know... |
37,804 | <p>I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are concerning <em>compact operators</em> and <em>unbounded operators</em>. Here I have the examples of $-\Delta$, the laplaci... | Pietro Majer | 6,101 | <p>In Ergodic theory unitary operators naturally arise as $U_T:f\mapsto f\circ T$ where $T$ is a measure preserving transformation on a probability space. The spectral theory of the operator $U_T$ carries some information on the dynamics of $T$. </p>
|
2,201,085 | <p>Let
$$x_{1},x_{2},x_{3},x_{5},x_{6}\ge 0$$ such that
$$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}=1$$
Find the maximum of the value of
$$\sum_{i=1}^{6}x_{i}\;x_{i+1}\;x_{i+2}\;x_{i+3}$$
where
$$x_{7}=x_{1},\quad x_{8}=x_{2},\quad x_{9}=x_{3}\,.$$</p>
| Empy2 | 81,790 | <p>Let $B=(x_2+x_6)/2,C=(x_3+x_5)/2,tR=(x_2-x_6)/2,tS=(x_3-x_5)/2$.<br>
The numbers are now $A,B(1+tR),C(1+tS),D,C(1-tS),B(1-tR)$. The sum is
$$2AB^2C+2BC^2D+2ABCD-2t^2(AB^2CR^2+BC^2DS^2-ABCDRS)$$
Let $t$ vary. The maximum is either when $t=0$ or when $t$ is as large as possible. </p>
<p>$t$ is at its largest when ... |
2,121,583 | <p>Question: Let $f,g: X \rightarrow \mathbb{R}$ continous (over $X$, and $X$ is a metric space). If $\overline{Y}\subset X $, and $f(y)=g(y)$ for every $y\in Y $, prove that $\left.f\right|_\overline{Y}= \left.g\right|_\overline{Y}$.</p>
<p>Attempt:</p>
<p>Since $\overline{Y} \subset X$, it follows that $\left.f\ri... | Momo | 384,029 | <p>HINT: Write each point of $\bar{Y}-Y$ as a limit of a sequence in $Y$, then use continuity.</p>
|
2,992,416 | <blockquote>
<p>A pendulum of length <span class="math-container">$1$</span> m and mass <span class="math-container">$100$</span> g attached to the end. Another 100 g mass move horizontally with speed 2 m/s. When collision happens this ball sticks with the pendulum and move together. Find the initial linear speed of ... | David Lui | 445,002 | <p>Consider the bijection <span class="math-container">$\{-1, 1\}^{\mathbb{Z}^2}$</span> to <span class="math-container">$P(\mathbb{Z}^2)$</span> by sending a function <span class="math-container">$f$</span> to the set <span class="math-container">$\{x \in \mathbb{Z}^2 : f(x) = 1\}$</span>. The inverse is <span class="... |
2,992,416 | <blockquote>
<p>A pendulum of length <span class="math-container">$1$</span> m and mass <span class="math-container">$100$</span> g attached to the end. Another 100 g mass move horizontally with speed 2 m/s. When collision happens this ball sticks with the pendulum and move together. Find the initial linear speed of ... | Robert Z | 299,698 | <p>Hint. Assume that <span class="math-container">$\{f_n\}_{n\in\mathbb{N}}$</span> is a countable list of ALL such functions from <span class="math-container">$\mathbb{Z}^2\to \{-1,1\}$</span>. Now define a new function <span class="math-container">$g:\mathbb{Z}^2\to \{-1,1\}$</span> such that for any <span class="mat... |
1,031,559 | <p>I'm getting ahead in my differential equations textbook (<em>Fundamentals of Differential Equations</em> by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function $\Pi_{a,b}\left(t\right)$ is given by
\begin{align}
\Pi_{a,b}\left(t\right):=u\left(t-a\right)-u\left(t-b\r... | AnonSubmitter85 | 33,383 | <p>Other than the value of $1/2$ on the edges, they're not different, just different notation. I don't know the motivation for the $1/2$'s, since it's still discontinuous, but they presumably explain why in the text. If we use the second notation, then the first definition (ignoring the $1/2$'s) is just
$$
\Pi \left( {... |
1,031,559 | <p>I'm getting ahead in my differential equations textbook (<em>Fundamentals of Differential Equations</em> by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function $\Pi_{a,b}\left(t\right)$ is given by
\begin{align}
\Pi_{a,b}\left(t\right):=u\left(t-a\right)-u\left(t-b\r... | Olli Niemitalo | 230,294 | <p>With Fourier transforms and sampled band-limited signals the concept of a <a href="https://en.wikipedia.org/wiki/Dirac_delta_function" rel="nofollow noreferrer">Dirac delta function</a> and the <a href="https://en.wikipedia.org/wiki/Dirac_comb" rel="nofollow noreferrer">Dirac comb</a> sometimes comes up. They are ze... |
132,591 | <p>Let $f(x)$ be a positive function on $[0,\infty)$ such that $f(x) \leq 100 x^2$. I want to bound $f(x) - f(x-1)$ from above. Of course, we have $$f(x) - f(x-1) \leq f(x) \leq 100 x^2.$$ This is not good for me though. I need a bound which is linear (or at worst linear-times-root) in $x$.</p>
<p>Is there an inequali... | André Nicolas | 6,312 | <p>For $2^n \le x<2^{n+1}$, let $f(x)=100(2^{n})^2$. There is an enormous jump from $f(2^{n+1}-1)$ to $f(2^{n+1})$. So even if we assume that $f$ is non-decreasing, we can have jumps of size comparable to $100x^2$. At the cost of complicating the description, we can modify the above $f(x)$ to make it strictly increas... |
3,037,291 | <p><a href="https://i.stack.imgur.com/5t6LN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5t6LN.png" alt="enter image description here"></a></p>
<p>I am trying to build a <a href="http://mathworld.wolfram.com/SnubCube.html" rel="nofollow noreferrer">snub cube</a>. I have made <span class="math-con... | Dr. Richard Klitzing | 518,676 | <p>For the mere requested values you might want to have a look <a href="https://bendwavy.org/klitzing/incmats/snic.htm" rel="nofollow noreferrer">here</a>.</p>
<p>A more descriptive way on the derivation of these values might be found already in the old German book of Max Brückner, "Vielecke und Vielflache, Theorie un... |
3,037,291 | <p><a href="https://i.stack.imgur.com/5t6LN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5t6LN.png" alt="enter image description here"></a></p>
<p>I am trying to build a <a href="http://mathworld.wolfram.com/SnubCube.html" rel="nofollow noreferrer">snub cube</a>. I have made <span class="math-con... | Maxim | 491,644 | <p>A way to obtain the coordinates of the vertices is given <a href="http://paulscottinfo.ipage.com/polyhedra/semiregular/snub-cube.html" rel="nofollow noreferrer">here</a>. To find the coordinates of <span class="math-container">$B$</span>, rotate <span class="math-container">$A = (1, v, w)$</span> by <span class="mat... |
395,618 | <p>If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into? </p>
<p>Can anybody please provide how can I approach the problem? There are numerous cases and when I go through case consideration it becomes extremely complex.</p>
| Jon Claus | 77,104 | <p>I am posting this as an answer to make it more visible. This is currently a Brilliant.org problem, in which the specific case $ 2 \times 500 $ is asked for. While it is an interesting problem meriting discussion, that should wait until after 8:00 PM EST 5/19 when the problem set is closed.</p>
|
2,419,057 | <p>Trying to assimilate the meaning of the differential I have looked for different examples of functions which:</p>
<ol>
<li>Admits all directional derivatives but are not continuous $(f: \mathbb{R}^2 \rightarrow \mathbb{R} \quad ,\quad (x,y) \mapsto
\begin{cases}
0 & \text{for } (x,y)=(0,0) \\
\frac... | zhw. | 228,045 | <p>The function $f(x,y)= \text { sgn }(x)\sqrt {x^2+y^2}$ is continuous at $(0,0),$ but $D_u f(0,0) = 1$ for all unit vectors $u$ except for $u =(0,1),(0,-1),$ in which case $D_u f(0,0) = 0.$</p>
|
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