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 |
|---|---|---|---|---|
186,830 | <p>Let $x_1, \dots, x_k \in \mathbb{R}^n$ be distinct points and let $A$ be the matrix defined by $A_{ij} = d(x_i, x_j)$, where $d$ is the Euclidean distance. Is $A$ always nonsingular?</p>
<p>I have a feeling this should be well known (or, at least a reference should exists), on the other hand, this fact fails for ge... | joriki | 6,622 | <p>I think it should be possible to show that your distance matrix is always nonsingular by showing that it is always a Euclidean distance matrix (in the usual sense of the term) for a non-degenerate set of points. I don't give a full proof but sketch some ideas that I think can be fleshed out into a proof.</p>
<p>Two... |
3,810,399 | <p>Let <span class="math-container">$p>2$</span> be a real number and consider the sum <span class="math-container">$J=\sum_{d=0}^{\infty}|\binom{p-2}{d}|$</span>.</p>
<p>I want to know whether <span class="math-container">$J$</span> is a finite quantity or not?</p>
<p>Indeed, if we consider <span class="math-conta... | Martin R | 42,969 | <p>It suffices to investigate the case that <span class="math-container">$x = p-2 > 0$</span> is <em>not</em> an integer, because the sum is finite otherwise.</p>
<p>Let <span class="math-container">$n \ge 0$</span> be an integer with <span class="math-container">$n < x < n+1$</span>. For <span class="math-con... |
3,810,399 | <p>Let <span class="math-container">$p>2$</span> be a real number and consider the sum <span class="math-container">$J=\sum_{d=0}^{\infty}|\binom{p-2}{d}|$</span>.</p>
<p>I want to know whether <span class="math-container">$J$</span> is a finite quantity or not?</p>
<p>Indeed, if we consider <span class="math-conta... | Martin R | 42,969 | <p>If <span class="math-container">$x = p-2 > 0$</span> is not an integer then one can apply <a href="https://en.wikipedia.org/wiki/Ratio_test#2._Raabe%27s_test" rel="nofollow noreferrer">Raabe's test</a> to <span class="math-container">$\sum_{d=0}^\infty b_d$</span> with <span class="math-container">$b_d = \left| \... |
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | Akiva Weinberger | 166,353 | <p>This is equivalent to the well-known fact that the sum of the first $n$ odd numbers is $n^2$. For example, $1+3+5+7+9+11=36$. Why are they equivalent? Because of this:
\begin{align}
1+2+3+4+5+\phantom16&\\
{}+1+2+3+4+\phantom15&\\
-----------&\\
1+3+5+7+9+11&
\end{align}</p>
|
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | bartgol | 33,868 | <p>Legend wants that Carl Friedrich Gauss discover the formula</p>
<p>$$
\sum_{i=1}^n i = \dfrac{n(n+1)}{2}
$$
when he was six. Not surprising, since gaussing, ehm, guessing "Gauss" when trying to remember who found a certain result has a non trivial probability of success...</p>
|
235,315 | <p>How many different six digit positive integers are there (duplicates allowed), where each digit is between 0 and 7 (inclusive), and the sum equals 20?</p>
<p>I know that there are 229,376 total possible 6 digit numbers which do not have an 8 or a 9. But what process do I use to eliminate those numbers whose digital... | Robert Israel | 8,508 | <p>You could use dynamical programming. Let $a(m,n)$ be the number of $m$-digit nonnegative integers with each digit 0 to 7 and the sum of digits is $n$. Then $a(1,n) = 1$ for $0 \le n \le 7$, $0$ otherwise, and
$$a(m+1,n) = \sum_{d=0}^{\min(n,7)} a(m,n-d)$$</p>
<p>EDIT: That solution allows leading zeros. If lead... |
235,315 | <p>How many different six digit positive integers are there (duplicates allowed), where each digit is between 0 and 7 (inclusive), and the sum equals 20?</p>
<p>I know that there are 229,376 total possible 6 digit numbers which do not have an 8 or a 9. But what process do I use to eliminate those numbers whose digital... | Gerry Myerson | 8,269 | <p>The answer is the coefficient of $x^{20}$ in $$(x+x^2+\cdots+x^7)(1+x+\cdots+x^7)^5$$ Do you see why? Then the first bracket is $x(1-x^7)/(1-x)$, and then second is the 5th power of $(1-x^8)/(1-x)$. OK? So now you need to be able to expand $x(1-x^7)(1-x^8)^5(1-x)^{-6}$, and pick out the coefficient of $x^{20}$. Can ... |
3,637,653 | <p>Suppose <span class="math-container">$A,B,C\in M_2(\mathbb{C})$</span>and they are linearly independent.Try to show that there exist <span class="math-container">$x_1,x_2,x_3\in \mathbb{C}$</span> such that the matrix <span class="math-container">$x_1A+x_2B+x_3C$</span>is invertible.</p>
<p>I know the span <span cl... | Angina Seng | 436,618 | <p>Let's assume the contrary, that a <span class="math-container">$\Bbb C$</span>-linear subspace of <span class="math-container">$M_2(\Bbb C)$</span>
consists entirely of invertible matrices. If we write a generic matrix
<span class="math-container">$$M=\pmatrix{x&y\\z&t}$$</span>
then this subspace is given b... |
2,000,626 | <blockquote>
<p>Is it possible that $P(A\cap B)$ is greater than $P(A)$ or $P(B)$?</p>
</blockquote>
<p>I think not.<br>
Let's assume WLOG that $P(A\cap B) > P(A)$. Then,
$$P(B|A)=\frac{P(B\cap A)}{P(A)} > \frac{P(A)}{P(A)} = 1$$</p>
<p>Contradiction.</p>
<p>Is my proof valid?</p>
| Jimmy R. | 128,037 | <p><strong>Hint:</strong> $A\cap B\subseteq A$ and $A\cap B\subseteq B$. </p>
<p>(What you did is correct, but is not the shortest way to do it.)</p>
|
3,597,812 | <p><span class="math-container">$f: \mathbb C \to \mathbb C$</span> is entire function such that</p>
<blockquote>
<blockquote>
<p><span class="math-container">$f(1/n)=1/n^2$</span> for all <span class="math-container">$n \in \mathbb N$</span>, Then to show <span class="math-container">$f(z)=z^2$</span></p>
</b... | Mihály András Csirik | 764,204 | <p>How about the function <span class="math-container">$g(z)=f(z)-z^2$</span>?</p>
|
1,230,037 | <p>For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$.</p>
<p>Any idea to solving it?</p>
| Beni Bogosel | 7,327 | <p>Note that <a href="https://math.stackexchange.com/questions/487032/subgroups-of-s-n-of-index-n-are-isomorphic-to-s-n-1">this question</a> proves that a subgroup of $S_n$ of index $n$ is isomorphic to $S_{n-1}$.</p>
<p>Let $H$ be a subgroup of $A_n$ of index $n$. Then $H \rtimes \Bbb{Z}_2$ is a subgroup of $S_n$ of ... |
3,807,296 | <p>Suppose you have a triangulated region in the plane, the triangulation consisting of <span class="math-container">$n$</span> triangles. Take an arbitrary triangle of this triangulation and call it <span class="math-container">$\Delta_i$</span> with <span class="math-container">$1\leq i\leq n$</span>.</p>
<p>The neig... | Brandon du Preez | 628,921 | <p>Trebor has already given an answer that more than suffices to show to that we shouldn't expect this method to work for map colouring, but it is worth pointing that similar difficulties arise even for general graph colouring.</p>
<p>The problems of colouring a graph optimally, and `extending' a colouring are quite di... |
1,490,051 | <p>As stated on the title, my question is: (a) represent the function $ f(x) = 1/x $ as a power series around $ x = 1 $. (b) represent the function $ f(x) = \ln (x) $ as a power series around $ x = 1 $. </p>
<p>Here's what I tried:</p>
<p>(a) We can rewrite $ 1/x $ as $ \frac{1}{1 - (1-x)} $ and thus using the series... | robjohn | 13,854 | <p>$$
\begin{align}
\frac1x
&=\frac1{1+(x-1)}\\
&=1-(x-1)+(x-1)^2-(x-1)^3+\dots\\
&=\sum_{k=0}^\infty(-1)^k(x-1)^k
\end{align}
$$
(a) You are correct; your series is the same as mine, however, usually we expand in powers of $(x-a)^n$.</p>
<p>(b) integrating $\frac1t$ between $t=1$ and $t=x$ gives
$$
\begin... |
3,198,705 | <p>I have this problem:</p>
<blockquote>
<p>In a game, the probability of win is <span class="math-container">$1/3$</span> and of lose is <span class="math-container">$2/3$</span>, ¿What
is the probabilty of win at least 1 prize playing 3 times?</p>
</blockquote>
<p>The probability of win at least 1 prize, playin... | Jason Swanson | 11,867 | <p>Continuing on from Tom Chen's answer, let <span class="math-container">$X\sim t_{n-1}$</span> and
<span class="math-container">$$
f(x) = \frac x{(1-x^2)^{1/2}}.
$$</span>
Then <span class="math-container">$f(P)=_d (n-1)^{-1/2}X$</span>, so that <span class="math-container">$\|U\|=_d g(X)$</span>, where
<span class="... |
2,002,014 | <p>Is the following set of vectors independent or dependent:</p>
<p>$$(1,-2,5,-3)^T, \;\;(2,3,1,-4)^T \;\; and \;\; (3,8,-3,5)^T \;\;?$$</p>
<p>How do I determine whether this set is dependent or independent ?</p>
| Hypergeometricx | 168,053 | <p>Change the coordinate system to map $A$ to the origin in order to facilitate algebraic manipulations.
$$\begin{align}
(\;\ x,\;y)&\rightarrow \;\;(x+1,y-3)\\
A(-1,3)&\rightarrow A'(\;\;\;\;\;0,\;\;0)\\
B(\;\;2,7)&\rightarrow B'(\;\;\;\;\;3,\;\;4)\\
P(\;\;x,y)&\rightarrow P'(x+1, y-3)=P(x', y')
\\\\
|... |
2,194,490 | <blockquote>
<p>If $n$ is a natural number such that $ n \geq 2$, then the numbers $n! + 2, n! + 3, n! + 4... n! + n$ are all composite.
(Thus, for any n greater than or equal to 2, one can find n consecutive composite numbers)</p>
</blockquote>
<p>I started with just plugging in numbers to see if they were compos... | wckronholm | 10,449 | <p>The statement you have to prove is that if $ 2 \leq k \leq n$ then $n! + k$ is composite.</p>
<p>Isn't $k$ a factor of $n!$?</p>
|
123,844 | <p>Show that the solution to
$$T(n) = 2T\left(\biggl\lfloor \frac n 2 \biggr\rfloor+17\right)+n$$
is $\Theta(n \log n)$?</p>
<p>So the induction hypothesis is
$$ T \left( \frac n 2 \right) = c\cdot \frac n2 \cdot \log \frac n2.$$
Hence,
$$ T(n) = 2c \cdot \frac n2 \cdot \log \frac n2 + 17 + n $$</p>
<p>but how do ... | Ross Millikan | 1,827 | <p>Hint: Now you want to prove that the right side is less than $cn \log n$. The $2$'s cancel nicely. Now write $\log \frac n2 = \log n - \log 2$. If $c$ is large enough you can take care of the $n$ term, and if $n$ is large enough the $17$ won't matter.</p>
|
954,377 | <p>Let $M\in GL_n(\mathbb{R})$ such that all its coefficients are non zero.</p>
<p>How can one show that $M^{-1}$ has at most $n^2-2n$ coefficients equal to zero ?</p>
<p>I have no idea how to tackle that problem, I've tried drawing some contradiction if $M^{-1}$ had $n^2+1-2n$ zero coefficients but couldn't find any... | abnry | 34,692 | <p><strong>Hint:</strong> Show that if there are at least $n^2-2n+1$ zeros, there will be a column of $M^{-1}$ with exactly 1 nonzero entry. Convince yourself that without loss of generality this might as well be the first column. Then convince yourself that $M M^{-1}$ will have the first column a multiple of one of th... |
2,240,781 | <p>Suppose $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{f}(\mathbf{x})=\mathbf{L}$ and $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{g}(\mathbf{x})=\mathbf{K}$. I want to prove that $\lim_{\mathbf{x} \to \mathbf{c}} \mathbf{f}(\mathbf{x})\bullet \mathbf{g}(\mathbf{x})=\mathbf{L}\bullet \mathbf{K}$, where $\bullet$ denotes the... | Ted Shifrin | 71,348 | <p><strong>HINT</strong>: Modify your argument slightly by choosing $\delta_1$ so that $0<\|\mathbf x-\mathbf c\|<\delta_1\implies \|\mathbf f(\mathbf x)-\mathbf L\|<\min(\epsilon,1)$. This will tell that you that $\|\mathbf f(\mathbf x)\|<\|\mathbf L\|+1$ when $0<\|\mathbf x-\mathbf c\|<\delta_1$. Ca... |
864,568 | <p>I am trying to figure out how to take the modulo of a fraction. </p>
<p>For example: 1/2 mod 3. </p>
<p>When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?</p>
| vadim123 | 73,324 | <p>One natural way to define the modular function is $$a \pmod b = a-b\left\lfloor \frac{a}{b}\right\rfloor$$
where $\lfloor \cdot \rfloor$ denotes the <a href="http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="noreferrer">floor function</a>. This is the approach used in the influential book <a href="http... |
2,205,087 | <p>Could you please give me an intuitive explanation why the dot product is defined this way?</p>
| AlexT | 428,860 | <p>In general, you could say that a function is a triplet of objects $f, A, B$ such that $A$ and $B$ are sets and $f$ is a relation such that $\forall a\in A \exists ! b\in B | f(a)=b$, that is to say that $\textit{the image}$ of an element $a$ is one and only element $f(a)=b$ in B. </p>
<p>As a consequence, if you we... |
1,703,491 | <p>I have the following problem:
$$\int\left(\frac{x+2}{x^2+x+1}\right)dx$$
I received this by simplification of another integral. But my question is how to procede from here. Is there a way to simplify this?</p>
| Doug M | 317,162 | <p>The denominator doesn't factor. In fact, the roots are complex. Partial fractions aren't going to get you anywhere. Complete the square, and do a tan substitution.</p>
|
1,599,840 | <p>Please help me write down a step by step solution to this question:</p>
<p>Let $T$ be a linear transformation from $V$ to $V$. Prove that there exists a non-zero linear transformation from $S$ from $V$ to $V$ such that $TS=0$ if and only if there exists a non-zero vector $v\in V$ such that $T(v)=0$. </p>
<p>Assume... | Tryss | 216,059 | <p>As the assumtion <em>$V$ is finite dimensional</em> was initially not specified in the question, here is idea of a possible proof in the infinite dimensional case :</p>
<ul>
<li>Suppose there exists a non-zero vector $v \in V$ such that $T(v) = 0$.</li>
</ul>
<p>Take $\{v\}$, and complete it to a basis $B$ of $V$.... |
157,024 | <p>I was searching through the web to find out any webpage which contains the consolidated list for all mathematical cheat sheets in various topics in one place. I couldnt find one. It would be good if we can share the known cheat sheet links in this post for various mathematical topics.</p>
| Joe | 24,942 | <p>I think <a href="http://tutorial.math.lamar.edu/cheat_table.aspx">this</a> will be your best bet. I have been using it for years.</p>
|
157,024 | <p>I was searching through the web to find out any webpage which contains the consolidated list for all mathematical cheat sheets in various topics in one place. I couldnt find one. It would be good if we can share the known cheat sheet links in this post for various mathematical topics.</p>
| Arif | 58,321 | <p>Here you'll get some LaTeX formatted math cheat sheets <a href="http://thinkprime.wordpress.com/" rel="nofollow">http://thinkprime.wordpress.com/</a></p>
<p>~Arif</p>
|
2,842,177 | <blockquote>
<p>$n| 3^n +1$</p>
</blockquote>
<p>My progress so far:</p>
<ol>
<li><p>$3^n + 1$ is even , thus $n$ is also even</p></li>
<li><p>$3^n + 1 \equiv n \equiv 2 \mod 10$ or $3^n + 1 \equiv n \equiv 0 \mod 10$</p></li>
<li><p>$3^n + 1 \equiv n \equiv 1 \mod 3$ / <em>I'm not quite sure about this</em></p></l... | Batominovski | 72,152 | <p>There are infinitely many such numbers $n$. In fact, you can demand that $n$ have a given number of prime divisors.</p>
<p><strong>Claim:</strong> Fix an integer $a>1$. Let $k$ be a nonnegative integer. There exists a positive integer $n_k$ such that exactly $k$ distinct prime natural numbers divide $n_k$ an... |
2,546,222 | <blockquote>
<p>If $p(x)$ is an irreducible polynomial of degree n in $F[x]$ then $F[x]/\langle p(x)\rangle \cong F(c)$ where c is a root of $p(x)$. Prove every element of $F(c)$ can be written uniquely as $a_0+a_1c+...+a_{n-}c^{n-1}$ for some $a_0,...,a_{n-1}\in F$.</p>
</blockquote>
<p>So to be unique there is onl... | Mark Bennet | 2,906 | <p>So if $q(x)\in F[x]$ we can use the division algorithm to write $$q(x)=p(x)d(x)+r(x)$$ where the degree of $r$ is strictly less than the degree of $p$.</p>
<p>Now we have $q(c)=r(c)$ as required (existence). If $q(c)=r(c)=r_1(c)$ are two representatives, consider $r(c)-r_1(c)$ as a polynomial function of $c$ which ... |
1,141,357 | <p>When deriving the integration by parts formula, you can use the product rule to do so,<br>
i.e. $\{uv\}' = uv' + vu'$</p>
<p>$\Rightarrow \int \{uv\}' = \int udv + \int vdu$</p>
<p>hence $uv = \int udv + \int vdu$. </p>
<p>If $uv$ is the integral of $\{uv\}'$ then why is the formula rearranged to read</p>
<p>$... | user2566092 | 87,313 | <p>The main reason the integration by parts formula is written that way is because it makes it clear that if you can compute the antiderivative $v$ of $dv$, and if $du$ becomes "simpler", e.g. if $u$ is a polynomial, then you can typically reduce the overall "complexity" of the integral by using integration by parts, a... |
1,316,861 | <p>Please help me to solve this question:</p>
<p>Let $H$ be a hyperelliptic curve over $\mathbb{F}_{103}$ given by the equation $ y^2 = x^5+1$. let $J$ be the jacobian of $H$ defined over $\mathbb{F}_{103}$. Show that # H($\mathbb{F}_{103^2}$) = # J($\mathbb{F}_{103}$).</p>
| Ivan Pogildiakov | 558,309 | <p>Let us consider a more general problem. Let <span class="math-container">$p$</span> be an odd prime, <span class="math-container">$p \not\equiv 1$</span> modulo <span class="math-container">${5}$</span>, and let <span class="math-container">$H$</span> be the hyperelliptic curve defined by <span class="math-container... |
177,102 | <p>I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real solutions. The equations were derived as the gradients of a sum-of-squares cost function, which I am attempting to find a... | Neil Hoffman | 27,453 | <p>I believe INTLAB, an interval analysis package in MATLAB might be worth considering. It is explained <a href="http://www.ti3.tuhh.de/rump/intlab/narep416.pdf" rel="nofollow">here</a>. Specifically, section 7.4 All Solutions of a Nonlinear System (with an implementation in Appendix A.3) is worth taking a look at.</p>... |
3,674,370 | <blockquote>
<p>The vertices <span class="math-container">$B$</span> and <span class="math-container">$C$</span> of a <span class="math-container">$\triangle ABC$</span> lie on the line, <span class="math-container">$\frac{x-2}{3}=\frac{y–1}{0}=\frac{z}{4}$</span> such that <span class="math-container">$BC = 5$</span... | Community | -1 | <p>This is a pure mathematical question, and the answer is simple: with just a compass you can not even make sure to draw a circle. At best, a convex spiral (not necessarily closed).</p>
<hr>
<p>If you add the condition that you are able to swim in a perfectly regular way (?), you can indeed follow a circle. Of unkno... |
1,360,394 | <p>For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.</p>
| Robert Israel | 8,508 | <p>If $n$ is odd, it can be written as $i + (n-i)$ for $i$ from $1$ to $(n-1)/2$, thus $(n-1)/2$ ways. If $n$ is even, it can be written as $i + (n-i)$ for $i$ from $1$ to $n/2$, thus $n/2$ ways. The case $i=n/2$, i.e. $n = (n/2) + (n/2)$, is the sum of two natural numbers but not the sum of two <strong>distinct</str... |
1,360,394 | <p>For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.</p>
| pancini | 252,495 | <p>For any number $n\in \mathbb N$, we could write $n=1+(n-1)=2+(n-2)=\cdots=n-1+(1)$.</p>
<p>That is $n-1$ different ways. However, since many ways are the same — the first and last, for example — we want to exclude these.</p>
<p>If $n$ is odd, then the $n-1$ ways we found is an even number of ways and half of them ... |
382,170 | <blockquote>
<p>Show that every $2\times 2$ matrix $A$, for which $A^2=-I$, is similar to $\begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix}$.</p>
</blockquote>
<p>I need help proving this. Don't have any idea how to proceed. Obviously look at the matrix as a linear operator...</p>
| Librecoin | 53,846 | <p><strong>Proof over the real matrices</strong>:</p>
<p>Let $A=\pmatrix{a&b\\c&d}$ be a real matrix such that $A^2=-I$. Then</p>
<p>$$A^2=\pmatrix{a&b\\c&d}\pmatrix{a&b\\c&d}=\pmatrix{a^2+bc&b(a+d)\\c(a+d)&d^2+bc}=\pmatrix{-1&0\\0&-1}=-I$$</p>
<p>and</p>
<p>$\left\{\begin{ar... |
754,057 | <p>I have the problem: $\sin(2x)=\tan(x)$</p>
<p>I used the double angle formula to get $2\sin(x)\cos(x)=\tan(x)$.</p>
<p>But after that step, I do not know whether or not to subtract $\tan(x)$ or to set $2\sin(x)\cos(x)$ to $u$ and solve for $U$.</p>
| mookid | 131,738 | <p>$$
2\sin(x)\cos(x)=\tan(x)
$$</p>
<p>Here, two possibilities: $\sin(x) = 0 \implies x \in \{ 0, \pi, 2\pi\}$</p>
<p>or
$$
2\cos(x)=\frac 1{\cos x}
\iff \cos^2 x = \frac 12
\iff x \in \left\{ \frac \pi 4, \frac {3\pi} 4, \frac {5\pi} 4, \frac {7\pi} 4
\right\}
$$</p>
|
26,019 | <p>This is a little bit of a niche topic.</p>
<p>I've dealt with a pretty bad dose of long COVID that has caused some serious gaps in my mathematics (basically causing terrible arithmetic skills and a really shaky foundation). Here's essentially what I'm dealing with. I've seen the mathematics in old courses (Ross' Ana... | guest advisor | 20,884 | <ol>
<li><p>If you want to go down this path: I would go with a review text instead, like <strong>Frank Ayres First Year College Math</strong>. (Schaum's Outline, I'm familiar with the first edition, available for cheaps on the Intertubes.) I have a strong feeling that if you work all the problems--you do know you ne... |
1,158,666 | <p>I know that every closed and bounded set in $\mathbb{R}$ is compact (like $[a,b]$)</p>
<p>so i can conclude that every bounded set in $\mathbb{R}$ is relatively compact, by contradiction i say that let $A\subset\mathbb{R}$ be a bounded set but not relatively compact it means that $\overline{A}$ is not compact, but ... | Uncountable | 215,630 | <p>Since A is bounded, it has a supremum and infimum. But $\sup A=\sup \bar {A} $ and $\inf A=\inf \bar {A}$. Hence $\bar A $ is bounded. </p>
<p>To see why this claim is true. Suppose $\sup \bar {A} $ is bigger than $\sup A $ by $ \epsilon$. Since $\bar {A} $ is closed, $\sup\bar {A}\in \bar {A} $. Now the ball of... |
2,298,032 | <p>A factory manufactures products in two grades of quality. 90% of its revenues comes from manufacturing the first-rate product. How will the revenues change if production of the first-rate product is increased by 10% while the production of second-rate product is decreased by 10%?</p>
| Jay Zha | 379,853 | <p><strong>HINT</strong></p>
<p>Say current total amount is $m$.
Let $x$ for first-rate product, $y$ for the rest.</p>
<p>Step 1: could you show what $x,y$ are in terms of $m$?</p>
<p>Step 2: what is $x$ increasing by $10\%$, and what is $y$ decreasing by $10\%$?</p>
<p>Step 3: what is the new total amount in term... |
2,398,193 | <blockquote>
<p>Let $f: \Bbb N \to \Bbb N$ via $f(n) = n^3+1$. Is the function bijective?</p>
</blockquote>
<p>Injective: Suppose $f(a) = f(b) $</p>
<p>$$\implies a^3 +1 = b^3 +1 \implies a^3 =b^3\implies a=b$$
Hence, $f$ is injective.</p>
<p>Surjective: We know that $n \ge 1 \implies n^3 \ge 1$
How do I prove it... | Sahiba Arora | 266,110 | <p>You're on the right track. </p>
<p>$$f(n)=n^3+1\geq 1+1=2$$ Thus, $1$ doesn't have a pre-image. Hence, $f$ is not surjective.</p>
|
2,398,193 | <blockquote>
<p>Let $f: \Bbb N \to \Bbb N$ via $f(n) = n^3+1$. Is the function bijective?</p>
</blockquote>
<p>Injective: Suppose $f(a) = f(b) $</p>
<p>$$\implies a^3 +1 = b^3 +1 \implies a^3 =b^3\implies a=b$$
Hence, $f$ is injective.</p>
<p>Surjective: We know that $n \ge 1 \implies n^3 \ge 1$
How do I prove it... | Bumblebee | 156,886 | <blockquote>
<p>HINT:<br>
Cube of any natural number is of the form $7k$ or $7k\pm1.$ </p>
</blockquote>
|
317,294 | <p>Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$.</p>
<p>Dimension theorem states:
$$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$</p>
<p>My question is:</p>
<p>Why is $U \cap W$ necessary in this theorem?</p>
| Henfe | 33,565 | <p>You can think in $A=\{a_1,...,a_n\}$ as base of $U$ and $B=\{b_1,...,b_m\}$ as a base of $V$, we know that the dimension of a vectorial space is the number of elements of some base. If $b_1$ it isn't linear combination of the elements of $A$ so $C_1=A \cup \{b_1$}. If $b_p$ it isn't linear combination of the elemen... |
4,156,469 | <p>Let's say a random variable is supported on a semi-infinite interval (say <span class="math-container">$(0, \infty)$</span> or all real numbers). We take a finite interval within the support. We then consider the distribution of this random variable conditional on it lying within the finite interval. Without loss of... | Sonu | 932,415 | <p>n=2m and m is odd, here Z(D_n) consists of two elements that is {R_0,R_180} and since m is odd then no other elements such R_90,R_270 can exist in Z(D_n)
And since the reflection are distinct then the no of elements of order 2 is just n/3=m.</p>
|
2,308,430 | <p>I have information about 2 points and an arc. In this example, point 1 (x1,y1) and point 2(x2,y2) and I know the arc for example 90 degree or 180 degrees.</p>
<p>From this information, I want to calculate the center of the circle. Which is (x,y) in this case.</p>
<p><a href="https://i.stack.imgur.com/JQ5sA.png" re... | Oiler | 270,500 | <p>Plot a third point on the arc $(x_{3}, y_{3})$. Then the center of the circle is the intersection of the perpendicular bisectors of the segments connecting $(x_{1}, y_{1})$, $(x_{3}, y_{3})$ and $(x_{2}, y_{2})$, $(x_{3}, y_{3})$. I'll leave it to you to prove it!</p>
|
2,939,028 | <blockquote>
<p>Find<span class="math-container">$$\lim_{x→0}\frac{\ln\cos3x}{\ln\cos2x}.$$</span></p>
</blockquote>
<p>Can anyone give me a hint about finding this limit without using L'Hopital?</p>
| egreg | 62,967 | <p>What you want to compute is
<span class="math-container">$$
\lim_{x\to0}\frac{\ln\cos x}{x^2}=\lim_{x\to0}\frac{1}{2}\frac{\ln(1-\sin^2x)}{x^2}=
\frac{1}{2}\lim_{x\to0}\frac{\ln(1-\sin^2x)}{\sin^2x}\frac{\sin^2x}{x^2}=-\frac{1}{2}
$$</span>
because
<span class="math-container">$$
\lim_{x\to0}\frac{\ln(1-x)}{x}=-1
$$... |
2,805,007 | <p>I need to solve this:</p>
<blockquote>
<p>$$f:\mathbb R\to\mathbb R,\mathcal C^1\text{-function and } a,b\in\mathbb R \text{ such that } a\lt b.$$ </p>
</blockquote>
<p>1) Probe that there is $M\gt 0$ such that $\forall x\in[a,b]$ it's verified that $ \vert f'(x)\vert \le M$</p>
<p>2) Conclude that $\forall x, ... | Tengu | 58,951 | <p><strong>Goal:</strong> (Negate the definition of uniformly continuity) We want to show that for some $\varepsilon$ then for any $\delta>0$, there exists $x,y$ so $|x-y|<\delta$ but $|f(x)-f(y)|>\varepsilon$.</p>
<p><strong>Hint:</strong> Note that $|f(x)-f(y)|=|x^2-y^2|=|x-y| \cdot |x+y|$. This implies tha... |
75,355 | <p>A groupoid is a category in which all morphisms are invertible.(*) The groupoids form a very nice subclass of categories. The inclusion of the groupoids into the 2-category of small categories admits both left and right (weak) adjoints. So you can localize (or <em>complete</em>) a category to a groupoid. If E denote... | Benjamin Steinberg | 15,934 | <p>I believe the following is an example of a category whose leanification is discrete but which is not a groupoid. It should be possible to simplify it. There may be details to work out. Let $B$ be the bicyclic monoid. It is generated by $a,b$ subject to the relations $ab=1$. (Note $ba\neq 1$). Let $K(B)$ be the... |
2,059,571 | <blockquote>
<p>Let $$V=\{(x_1,x_2,x_3,\dots,x_{100})\in\mathbb{R}^{100}\,|\, x_1=x_2=x_3 \text{ and } x_{51}=x_{52}=x_{53}= \dots=x_{100}\}$$ What is $\dim V$?</p>
</blockquote>
<p>If W is a subspace of vector space $V$ then $$\dim W = \dim V - \text{number of linearly independent restrictions}$$
In our case $\dim ... | Leox | 97,339 | <p>$V=<v_1, v_2, e_4, e_5, \ldots, e_{50}>$, where
$v_1=(1,1,1,0, \ldots,0)$, $v_2=(\underbrace{0, \ldots, 0}_{\text{$50$ zeros}}, 1,1,\ldots 1)$ and $e_i=(0, 0, \ldots \underbrace{1}_{i}, \ldots ,0)).$.
So, $ \dim V=49.$</p>
|
4,488,353 | <blockquote>
<p>Let <span class="math-container">$\alpha: I \subset \mathbb R \to \mathbb R^3$</span> be a regular curve in space (not necessarily Parameterized by arc length). Show that:
<span class="math-container">$$k(r) = \frac{\vert \alpha’(r) \times \alpha’’(r) \vert}{\vert \alpha’(r)\vert^3}$$</span>
<span class... | paul garrett | 12,291 | <p>EDIT: My earlier "answer" was toooooo glib, and not accurate! EDIT-EDIT: some precise details filled in...</p>
<p>In general, we should tend to expect that a semi-simple algebraic group's points over a localization of the defining ring (here <span class="math-container">$\mathbb Z$</span>, actually), is in... |
4,488,353 | <blockquote>
<p>Let <span class="math-container">$\alpha: I \subset \mathbb R \to \mathbb R^3$</span> be a regular curve in space (not necessarily Parameterized by arc length). Show that:
<span class="math-container">$$k(r) = \frac{\vert \alpha’(r) \times \alpha’’(r) \vert}{\vert \alpha’(r)\vert^3}$$</span>
<span class... | Just a user | 977,740 | <p>It's not finite. This is closely related to the Golden Gates in quantum computation. Roughly, not only the group is not finite, its elements can actually be used to approximate any element in <span class="math-container">$PU(2)$</span>. Details can be found in <a href="https://publications.ias.edu/sites/default/file... |
928,047 | <p>In a round-robin tournament, each team plays every other team exactly once. Show that if no games end in ties, then no matter what the outcomes of the games, there will be some way to number the teams so that team 1 beat team 2, and team 2 beat team 3, and team 3 beat team 4,
and so on.</p>
<p>I have the base case ... | Moshe Rosenfeld | 650,294 | <p>A simple inductive proof. Easy for 2 teams. Assume you can do it for n team and consider n+1 team. Set aside Team "X". Arrange the n teams so that team k--> k+1. If X beats 1 then X --> 1 -->2...-->n. If not look for the first team j that X beats. Then:
1-->2-->...j-1 -->X -->j -->...n is your "inductive answer."</p... |
124,530 | <p>I want to have a function value of an expression where some variables are solutions to some set of equations, with some values of parameters. I had an idea to use pure functions for that. </p>
<p>However, since both the expression and the equations are lengthy I'd like to place them in a separate expressions define... | Szabolcs | 12 | <h3>Why doesn't it work?</h3>
<p><code>Evaluate</code> only has effect at the first level in a held expression.</p>
<pre><code>{Hold[1 + 1], Hold[Evaluate[1 + 1]], Hold[f[1 + 1]]}
(* {Hold[1 + 1], Hold[2], Hold[f[1 + 1]]} *)
</code></pre>
<h3>Why does <code>Evaluate</code> only work at the first level?</h3>
<p>Beca... |
124,530 | <p>I want to have a function value of an expression where some variables are solutions to some set of equations, with some values of parameters. I had an idea to use pure functions for that. </p>
<p>However, since both the expression and the equations are lengthy I'd like to place them in a separate expressions define... | Sascha | 4,597 | <p>Another method would be to use delayed evaluation to work around the issue of variable renaming. </p>
<pre><code>MWE1[a_, b_][x_, y_][δ_] := b x^2 + a y^2 + 3 + δ;
Dx[a_, b_][x_, y_][δ_] := D[MWE1[a, b][x, y][δ], x];
Dy[a_, b_][x_, y_][δ_] := D[MWE1[a, b][x, y][δ], y];
</code></pre>
<p>Note that one has to introd... |
1,134,234 | <p>Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $M$ be an $A$-module.</p>
<p>I want to turn the following object into an $A/\mathfrak{m}$-module: $$A/\mathfrak{m} \otimes_A M$$</p>
<p>I attempt to multiply in this way:</p>
<p>$$\left( [a],\sum_{i = 1}^n [b_i] \otimes_A m_i \right) \mapsto \sum... | Eric Wofsey | 86,856 | <p>If $\alpha$ and $\beta$ are automorphisms of $G$, then the cosets $\alpha X$ and $\beta X$ are the same iff $\alpha(K)=\beta(K)$. Thus $X$ will fail to have finite index if there are infinitely many different subgroups of $G$ that are conjugate to $K$ under automorphisms of $G$. For instance, if $G$ is an infinite... |
1,134,234 | <p>Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $M$ be an $A$-module.</p>
<p>I want to turn the following object into an $A/\mathfrak{m}$-module: $$A/\mathfrak{m} \otimes_A M$$</p>
<p>I attempt to multiply in this way:</p>
<p>$$\left( [a],\sum_{i = 1}^n [b_i] \otimes_A m_i \right) \mapsto \sum... | ahulpke | 159,739 | <p>I don't think this is true without some further finiteness condition that limits the number of images of $K$:</p>
<p>Take $G$ the (two-sided) infinite sequences with entries in 0,1 and as operation component-wise addition (so its an infinite direct product of $C_2$ with itself), and as $K$ the kernel of the project... |
3,044,271 | <p>If <span class="math-container">$x^2+y^2=1$</span>. then the range of expression <span class="math-container">$3x^2-2xy$</span> without trigonometric substitution method</p>
<p>what i have done try here is use arithmetic geometric inequality</p>
<p><span class="math-container">$\displaystyle x^2+y^2\geq 2xy$</sp... | Shubham Johri | 551,962 | <p>Since <span class="math-container">$x^2+y^2=1$</span>, you can substitute for <span class="math-container">$y=\pm\sqrt{1-x^2}$</span> in <span class="math-container">$f(x,y)=3x^2-2xy$</span> to get <span class="math-container">$g(x)=f(x,\pm\sqrt{1-x^2})=3x^2\pm2x\sqrt{1-x^2},-1\le x\le1$</span> and find the extrema ... |
4,622,026 | <p>I am a bit confused about <span class="math-container">$\int e^{e^x+x}dx$</span>. If we made a <span class="math-container">$u$</span>-sub of <span class="math-container">$e^x$</span> then the derivative is <span class="math-container">$e^x$</span> and so we have <span class="math-container">$\int e^udu$</span>. But... | Robert Israel | 8,508 | <p>The numerator <span class="math-container">$3! \times {}^{12}C_3$</span> is correct for the number of outcomes where the three birthdays are in different months, but the denominator is incorrect. You want the number of outcomes with no restriction: each birthday can be in any month. So <span class="math-container">... |
1,663,244 | <p>How to represent a graph in a function?</p>
<p>For example, I used 3 functions : </p>
<p>$$f(x)=x^2$$
$$g(x)=x$$
$$h(x)=3$$</p>
<p>These 3 functions were plotted on the same graph and the result (after edit) is as given below</p>
<p>How would you represent the below graph in a function, lets say $k(x)$ ?</p>
<p... | DylanSp | 308,461 | <p>You'd use a piecewise function. In this case, we have
$$
k(x) =
\begin{cases}
x^2 & 0 \leq x \lt 1 \\
x & 1 \leq x \lt 3 \\
3 & x \geq 3
\end{cases}
$$</p>
|
32,797 | <p>Let $\phi\colon F\to G$ be a homomorphism of finitely generated abelian groups. If $F$ is free, then $\ker(\phi)$ is also free and thus admits a basis.</p>
<p>Question: <em>Is there a general procedure to find a basis for $\ker(\phi)$?</em></p>
<hr>
<p>As an example, consider the homomorphism $\phi\colon\mathbb Z... | lhf | 589 | <p>Try also the <a href="http://en.wikipedia.org/wiki/Smith_normal_form" rel="nofollow">Smith normal form</a>.</p>
|
4,344,014 | <p>Please, give advise or reference for solving first order ODE: <span class="math-container">$a(y)y^\prime + y = b(t)$</span>, where <span class="math-container">$a$</span>, <span class="math-container">$b$</span> are known function. It would be better to find just one solution.</p>
| Essaidi | 708,306 | <p>There isn't a general form for the solution but one can study some special cases :</p>
<ol>
<li> Cas <span class="math-container">$b$</span> is constant the the equation become with separated variables : If :
<span class="math-container">$$a(y) y' + y = b(t) = \lambda$$</span>
then :
<span class="math-container">$... |
2,484,034 | <p>I want to prove that the intersection between the cone $z^2=x^2+y^2$ and the plane $x+y+2z=2$ is an elispe in this plane. </p>
<p>My work:</p>
<p>I suppose to prove it I have to see that the equation $x+y+2\sqrt{x^2+y^2}=2$ can be rewritten as $\frac{x^2}{a}+\frac{y^2}{b}=c$. To do so I have tried to completing t... | Dan Uznanski | 167,895 | <p>Kind of a trick here: the quadratic object is the standard double cone. Any plane that intersects it must be a conic section; this plane does not cross the origin so it's not degenerate, and its maximum slope of $\sqrt 2 / 2$ is between 0 and 1 so it's an ellipse.</p>
|
812,642 | <p>I want some intuitive understanding of the trigonometric functions. One way is to understand ways they can be computed when just an angle in degrees or radians is given. The sine of an angle is defined as the ratio of two sides of a right triangle. If this is the case then I can't find a way to compute it without r... | Lutz Lehmann | 115,115 | <p>If one reduces it to the most simple form, an angle is a point on the unit circle. In general it is an equivalence class of pairs of rays originating from the same point, after rotation, the first ray can be made horizontal, the originating point the origin and the second ray is defined by one point on it, which can... |
812,642 | <p>I want some intuitive understanding of the trigonometric functions. One way is to understand ways they can be computed when just an angle in degrees or radians is given. The sine of an angle is defined as the ratio of two sides of a right triangle. If this is the case then I can't find a way to compute it without r... | Mark Bennet | 2,906 | <p>If the angle $x$ is in radians, $$\sin x = x-\frac {x^3}{3!}+\frac {x^5}{5!}-\frac {x^7}{7!}+\dots$$with odd powers and alternating signs. This converges pretty quickly, especially for small $x$.</p>
|
812,642 | <p>I want some intuitive understanding of the trigonometric functions. One way is to understand ways they can be computed when just an angle in degrees or radians is given. The sine of an angle is defined as the ratio of two sides of a right triangle. If this is the case then I can't find a way to compute it without r... | Jesse Madnick | 640 | <p>The OP's question is as follows: given an angle <em>measure</em> (like, say, 57 degrees), how does one actually construct an angle with that measure? The OP would then like to use this angle to create a right triangle in order to find the sine of the angle. The OP is interested in <em>exact values</em>, not approx... |
4,928 | <p>Consider the following list of countries which I would like to highlight on a world map:</p>
<pre><code>MyCountries={"Germany","Hungary","Mexico","Austria","Bosnia","Turkey","SouthKorea","China"};
</code></pre>
<p>From the documentation center <a href="http://reference.wolfram.com/mathematica/ref/CountryData.html"... | Brett Champion | 69 | <p>Alternate approach:</p>
<pre><code>Graphics[{
{Orange, CountryData[#, "SchematicPolygon"] & /@ MyCountries},
{LightBrown,
CountryData[#, "SchematicPolygon"] & /@ Complement[CountryData[], MyCountries]}
}]
</code></pre>
<p>which draws all of "your countries" in one color, and then the rest of ... |
301,393 | <p>The question I am working on is:</p>
<blockquote>
<p>Prove that if $m+n$ and $n+p$ are even integers, where
$m$, $n$,and $p$ are integers, then $m+p$ is even. What kind
of proof did you use?</p>
</blockquote>
<p>I was thinking--and I aware that this may not be the most efficient method--of proving four diffe... | Math Gems | 75,092 | <p><strong>Hint</strong> $\, $ If $\rm\,m\,$ and $\rm\,p\,$ have same parity as $\rm\,n\,$ then $\rm\,m+p\,$ has same parity as $\rm\,n+n,\,$ which is even. </p>
<p>Said in congruence language: $\rm\ mod\ 2\!:\ m\equiv n,\ p\equiv n\:\Rightarrow\: m+p\equiv n+n\equiv 2n\equiv 0$</p>
|
679,807 | <p>I am trying to find the value of $2^{-1-i}$. </p>
<p>I rewrite it like this, $2^{-1-i}=e^{\ln(2)(-1-i)}={1\over{e^{\ln2}e^{i\ln2}}}=1/2$</p>
<p>Since $e^{i\ln2}=e^{Re(i\ln2)}=e^0=1$. </p>
<p>This looks way nicer than it should be, I think, can anyone tell me where I go wrong, and maybe a good way to do this probl... | lab bhattacharjee | 33,337 | <p>$$2^{-1-i}=2^{-1}\cdot2^{-i}=\frac12\cdot(e^{\text{Log}2})^{-i}=\frac{e^{i(-\text{Log}2)}}2$$</p>
<p>where $\displaystyle\text{Log}_ez=\log_ez+2n\pi i$</p>
<p>Now, use <a href="http://mathworld.wolfram.com/EulerFormula.html" rel="nofollow">Euler Formula</a></p>
|
329,791 | <p>I am looking for an explicit expression for the Cantor function for points in the cantor set. Does anyone know of one?</p>
<p>Thanks</p>
| amWhy | 9,003 | <p>As Brian was alluding to, the canonical <a href="http://en.wikipedia.org/wiki/Cantor_function" rel="nofollow">Cantor Function</a> is discussed/defined at the linked Wikipedia entry. You'll find the following algorithm for the function:</p>
<blockquote>
<p>Formally, the Cantor function $c : [0,1] → [0,1]$ is defin... |
329,791 | <p>I am looking for an explicit expression for the Cantor function for points in the cantor set. Does anyone know of one?</p>
<p>Thanks</p>
| ASB | 111,607 | <p>Define the Cantor function $ f:[0,1]\rightarrow [0,1] $ as follows.</p>
<p>Let $ x\in [0,1] $.</p>
<p>Then $$ x= \sum\limits_{n = 1}^\infty \frac{a_n}{3^n}\text{ ; where }a_{n}\in \{0,1,2\}\text{ for each } n\in \mathbb{N}. $$</p>
<p>If $ x\in C $ then $ x $ can be written as of the form $$ \sum\limits_{n = 1}^\... |
1,963,885 | <p>Find the sum
$$\sum_{n=1}^{\infty} \dfrac{4n}{n^4+2n^2+9}.$$</p>
<p>By calculator, we can predict that its sum is equal to $\dfrac{5}{6}$ so I think we should use inequalities to prove it. And I found that</p>
<p>$\dfrac{5}{6(n^4+n^2)} < \dfrac{4n}{n^4+2n^2+9}< \dfrac{5}{6(n^2+n)}$ for all $n\ge n_0$, $n_0$ ... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic... |
3,031,174 | <p><span class="math-container">$$\lim_{x\to2}\left(\frac{1}{x(x-2)^2}-\frac{1}{x^2-3x+2}\right)$$</span></p>
<p><strong>What I tried:</strong> Got the fraction to the same denominator
<span class="math-container">$$\begin{align}
\lim_{x\to2}{\frac{x^2-3x+2-x(x-2)^2}{x(x-2)^2(x^2-3x+2)}}&=\lim_{x\to2}{\frac{(x-2)(... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Use that <span class="math-container">$$x^2-3x+2-x(x-2)^2=-(x-2)(x^2-3x+1)$$</span> and <span class="math-container">$$x^2-3x+2=(x-1)(x-2)$$</span></p>
|
3,031,174 | <p><span class="math-container">$$\lim_{x\to2}\left(\frac{1}{x(x-2)^2}-\frac{1}{x^2-3x+2}\right)$$</span></p>
<p><strong>What I tried:</strong> Got the fraction to the same denominator
<span class="math-container">$$\begin{align}
\lim_{x\to2}{\frac{x^2-3x+2-x(x-2)^2}{x(x-2)^2(x^2-3x+2)}}&=\lim_{x\to2}{\frac{(x-2)(... | Siong Thye Goh | 306,553 | <p>The numerator tend to a positive number, the denominator goes to <span class="math-container">$0$</span> from the positive directione. The sequence actually goes to infinity. </p>
<p><a href="https://i.stack.imgur.com/fSg8u.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fSg8u.png" alt="enter ima... |
4,257,836 | <p>Let <span class="math-container">$G$</span> be a tree on <span class="math-container">$n$</span> vertices when <span class="math-container">$n$</span> is even. Then for each vertex, the sum of distances from it to all other vertex is computed. It is interesting to note that all of them have the same parity.</p>
<p>I... | Subhajit Ghosh | 966,227 | <p>If <span class="math-container">$f(x)$</span> has roots <span class="math-container">$x_1,...x_k$</span> then <span class="math-container">$g(x)=f(x-3)$</span> has roots <span class="math-container">$x_1+3,...x_k+3$</span>. If <span class="math-container">$h(x)=g(1/x)$</span> has roots <span class="math-container">$... |
4,257,836 | <p>Let <span class="math-container">$G$</span> be a tree on <span class="math-container">$n$</span> vertices when <span class="math-container">$n$</span> is even. Then for each vertex, the sum of distances from it to all other vertex is computed. It is interesting to note that all of them have the same parity.</p>
<p>I... | David | 971,259 | <p>Let <span class="math-container">$$k={1\over a+3}+{1\over b+3}+{1\over c+3}$$</span>
Since <span class="math-container">$k$</span> is symmetric in <span class="math-container">$a, b \text{ and }c,$</span> it can be expressed
in terms of the elementary symmetric functions for <span class="math-container">$a, b \text{... |
4,257,836 | <p>Let <span class="math-container">$G$</span> be a tree on <span class="math-container">$n$</span> vertices when <span class="math-container">$n$</span> is even. Then for each vertex, the sum of distances from it to all other vertex is computed. It is interesting to note that all of them have the same parity.</p>
<p>I... | user2661923 | 464,411 | <p><span class="math-container">$f(x-3) = (3)(x-3)^3 + (-14)(x-3)^2 + (1)(x-3) + (62)$</span>.</p>
<p>Term by term you have that <span class="math-container">$f(x-3)$</span></p>
<p><span class="math-container">$= ~~(3)[x^3 + 3(x^2)(-3) + 3(x)(-3)^2 + (-3)^3]$</span></p>
<p><span class="math-container">$+ ~~(-14)[x^2 + ... |
80,078 | <p>Given $f$:</p>
<p>$$
f(x) = \begin{cases}
\frac1{x} - \frac1{e^x-1} & \text{if } x \neq 0 \\
\frac1{2} & \text{if } x = 0
\end{cases}
$$</p>
<p>I have to find $f'(0)$ using the definition of derivative (i.e., limits). I already know how to differentiate and stuff, b... | Gottfried Helms | 1,714 | <p>Hmm, another approach, which seems simpler to me. However I'm not sure whether it is formally correct, so possibly someone else can also comment on this. </p>
<p>The key here is that the expression $\small {1 \over e^x-1 } $ is a very well known generation function for the bernoulli-numbers<br>
$$\small {1 \ove... |
1,892,376 | <p>The question was: </p>
<blockquote>
<p>From the letters in MAGOOSH, we are going to make three-letter
"words." Any set of three letters counts as a word, and different
arrangements of the same three letters (such as "MAG" and "AGM") count
as different words. How many different three-letter words can be made... | Eric Wofsey | 86,856 | <p>There are two different ways you can set this up in the language of metric spaces. If $X$ and $Y$ are metric spaces, say that a bijection $f:X\to Y$ is a <em>similarity</em> if there exists a constant $r>0$ such that $d(f(x),f(y))=rd(x,y)$ for all $x,y\in X$. There are then two definitions we could make:</p>
<... |
2,743,099 | <p>So I have two points lets say <code>A(x1,y1)</code> and <code>B(x2,y2)</code>. I want to find a point <code>C</code> (there will be two points) in which if you connect the points you will have an equilateral triangle. I know that if I draw a circle from each point with radius of equal to <code>AB</code> I will find ... | Anastassis Kapetanakis | 342,024 | <p>Let $C(x_o, y_o)$
$$AB=\sqrt{((x_1-x_2)^2+(y_1-y_2)^2}$$
The following must happen:
$$AC=BC=AB$$
Or else:
$$AC=AB \ and \ BC=AB$$
And you solve the two equations.</p>
|
2,757,743 | <p>It is clear that the Galois group of $\mathbb{Q}(\omega):\mathbb{Q}$ where $\omega$ is a primitive root of unity is a cyclic group of order 10. Let $\sigma$ be a primitive element then $\sigma^2$ has order 5. </p>
<p>I want to find an $\alpha$ such that $Fix(<\sigma^2>)=\mathbb{Q}(\alpha)$.</p>
<p>Using the ... | Jonathan Dunay | 538,622 | <p>Consider the orbit of $\omega$ under $<\sigma>$, which is $\{\omega^i|1\leq i\leq10\}$. In cycle notation, we can write $\sigma^2$ as $(\omega^\text{ } \text{ } \omega^4\text{ } \omega^5\text{ } \omega^9\text{ } \omega^3)(\omega^2 \text{ } \omega^8\text{ } \omega^{10}\text{ } \omega^7\text{ } \omega^6)$. From ... |
1,805,431 | <blockquote>
<p>The simplest of the logical paradoxes is <em>Russell's paradox</em>, which can be described as follows: </p>
<p>Let $S$ denote the <em>set of all sets that are not elements of themselves</em>. Is $S$ an element of itself? </p>
<ul>
<li><p>Well, if $S$ is an element of $S$, then - by the v... | goblin GONE | 42,339 | <p>You're misreading the structure of the argument. The structure is:</p>
<ul>
<li><p>Suppose $S$ has the property $\forall x(x \in S \iff x \notin x)$.</p></li>
<li><p>Then $S \in S \iff S \notin S$, a contradiction.</p></li>
<li><p>Therefore, the assumption that $S$ has the property $\forall x(x \in S \iff x \notin ... |
19,067 | <p>I'm trying to modify <code>Plus</code> but am running into trouble with it being <code>Listable</code>:</p>
<pre><code>ClearAll[f, g, h]
Attributes[f] = {Listable};
h /: f[x_h, l_List] := 0
h /: g[x_h, l_List] := 0
f[h[1], {1, 1}] (* {f[h[1], 1], f[h[1], 1]}, not OK I want 0 *)
g[h[1], {1,1}] (* 0 as expected ... | whuber | 91 | <p>Geometrically, the locus of $(x,y)$ for which $a x - b y - 1 = 0$ is a line orthogonal to the vector $(a, -b)$, because the equation can be written $(a,-b)\cdot (x,y)=1$. This invites us to pose the problem as a geometric intersection:</p>
<blockquote>
<p>Find all points of intersection of the line $a x - b y - ... |
2,446,000 | <p>I' ve tried with $x^2 = {[x]+1\over 2}$ so $x$ is a square root of half integer. And know? What to do with that?</p>
| Community | -1 | <p>Rewrite</p>
<p>$$x=\pm\sqrt{\frac{[x]+1}2}$$ and try increasing values for the integer $[x]$.</p>
<ul>
<li><p>$[x]=-1\to x=0$: incompabible;</p></li>
<li><p>$[x]=0\to x=\pm\dfrac1{\sqrt 2}$: $x=\color{green}{\dfrac1{\sqrt2}}$ is a solution;</p></li>
<li><p>$[x]=1\to x=\pm1$: $x=\color{green}{1}$ is a solution;</p>... |
4,409,499 | <p>Suppose that <span class="math-container">$f:[0,1]\rightarrow [0,1]$</span> is a continuous function such that <span class="math-container">$f(f(x))=x$</span> for all <span class="math-container">$x\in [0,1]$</span>.</p>
<p>We know f is one to one and onto. Morover, it has a fixed point.</p>
<p>If we assume further ... | TheSilverDoe | 594,484 | <p><span class="math-container">$x \mapsto 1-x$</span> is certainly not the only example. Another one is
<span class="math-container">$$x \mapsto 1 - \sqrt{2x-x^2}$$</span></p>
|
320,194 | <p>I have 5 types of symptoms, I want to know all kind of combinations a patient could have:</p>
<p>The set is $(vomit, excrement, urine, dizzyness, convulsion)$</p>
<p>As patient can show only one, or even 5 of them I am listing them as:</p>
<p>So</p>
<h1>Possibilities with one symptom</h1>
<pre><code>vomit
excre... | Greg | 64,901 | <p>I haven't taken a combinatorics class thus far, but the question you are asking is "What are all of the potential combinations?" </p>
<p>You have answered your own question here by listing all potential combinations. Maybe I miss the point here.</p>
<p>edit: A general formula that encompasses many situations in ma... |
3,670,384 | <p>I’ve read just the basics of some introductory analysis books and sometimes they show that we can characterize things like limits, continuity, compactness, etc. in terms of sequences. </p>
<p>I’ve heard that these sequential criteria hold for general metric spaces, but that in topology for example one encounters si... | Brian M. Scott | 12,042 | <p>Many of these hold for <a href="https://en.wikipedia.org/wiki/Sequential_space" rel="noreferrer"><em>sequential spaces</em></a>. These can be defined in a variety of equivalent ways. One simple way that uses no new terminology is that <span class="math-container">$X$</span> is sequential iff for each non-closed <spa... |
897,973 | <p>Suppose we found a proof that "<a href="http://simple.wikipedia.org/wiki/Twin_Prime_Conjecture" rel="nofollow">The Twin Prime Conjecture</a> cannot be proven", without any conclusion as to the conjecture itself being true or false.</p>
<p>Is it then possible for the conjecture to be true? or, must the conjecture t... | Sidharth Ghoshal | 58,294 | <p>If a conjecture X is unprovable that can be translated to the following:</p>
<p>if X is true no contradiction can be derived</p>
<p>if X is false no contradiction can be derived</p>
<p>Both of these must be true because otherwise X would not be unprovable (we could merely choose one of the appropriate routes and ... |
897,973 | <p>Suppose we found a proof that "<a href="http://simple.wikipedia.org/wiki/Twin_Prime_Conjecture" rel="nofollow">The Twin Prime Conjecture</a> cannot be proven", without any conclusion as to the conjecture itself being true or false.</p>
<p>Is it then possible for the conjecture to be true? or, must the conjecture t... | Carl Mummert | 630 | <p>There is an important issue with statements of the form "$X$ is unprovable": <em>in what system</em> is $X$ unprovable? We assume here that $X$ is some consistent statement written in some formal language. </p>
<p>Statements provable with no non-logical axioms are known as logical validities. Few interesting mathem... |
2,396,287 | <p>The product has only positive factors so it has zero as lower bound. Also the product is decreasing as all its factors are less than one. In conclusion the series must have a limit. I also compute the first 150 values of the product and I got around 0.297. I believe that the product converges very, very, slowly to z... | Jack D'Aurizio | 44,121 | <p>The given product equals
$$\exp\sum_{n\geq 1}\log\frac{\log(2n)}{\log(2n+1)}=\exp\sum_{n\geq 1}-\int_{2n}^{2n+1}\frac{dx}{x\log x}\\=\exp\left[C-\sum_{n\geq 2}\frac{1}{(2n+\eta_n)\log(2n+\eta_n)}\right]$$
with $\eta_n\in(0,1)$ by the mean value theorem. Since both the series $\sum_{n\geq 2}\frac{1}{(2n+1)\log(2n+1)}... |
1,320,365 | <p>I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to $\mathbb Z_5$. </p>
<p>The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \} \to \mathbb N$ defined as $N(a... | Alex M. | 164,025 | <p>Sometimes it is better to give elementary proofs to elementary problems, and a constructive, explicit proof gives us the feeling that we are in control. Therefore, if we denote by $\widehat {x + y \Bbb i}$ elements of $\Bbb Z [\Bbb i] / (1 + 2 \Bbb i)$ and by $\bar n$ elements of $\Bbb Z _5$, it is really easy to ch... |
2,311,583 | <p>$$
\int_{0}^{1}\sqrt{\,1 + x^{4}\,}\,\,\mathrm{d}x
$$ </p>
<p>I used substitution of tanx=z but it was not fruitful. Then i used $ (x-1/x)= z$ and
$(x)^2-1/(x)^2=z $ but no helpful expression was derived.
I also used property $\int_0^a f(a-x)=\int_0^a f(x) $
Please help me out</p>
| Jack D'Aurizio | 44,121 | <p>We can do better than elliptic integrals or $\Gamma$ / Beta / hypergeometric functions:
$$ \int_{0}^{1}\sqrt{1+t^4}\,dt = \frac{\sqrt{2}}{3}+\frac{\pi}{6\,\text{AGM}\left(1,\frac{1}{\sqrt{2}}\right)} \tag{1}$$
due to integration by parts and what is shown in <a href="https://math.stackexchange.com/a/2721116/44121">t... |
1,336,419 | <p>What are the three final numbers of $2003^{2003}$ and $2003^{2003^{2003}}$? </p>
<p>Do I use the Chinese Remainder Theorem here, and if so, how?</p>
| lab bhattacharjee | 33,337 | <p>As $\displaystyle2003\equiv3\pmod{1000}$</p>
<p>$$2003^{2003^{2003}}\equiv3^{2003^{2003}}\pmod{1000}$$</p>
<p>Now Carmichael function $\lambda(1000)=\cdots=100$</p>
<p>$$\implies3^{2003^{2003}}\equiv3^{2003^{2003}\pmod{100}}\pmod{1000}$$</p>
<p>As $\displaystyle2003\equiv3\pmod{100},2003^{2003}\equiv3^{2003}\pm... |
69,839 | <p>Is it possible to get all points on a Polyhedron surface using two surface parameters, say </p>
<p>$ \phi,\theta $ spherical co-ordinates?</p>
<p>Just like in <code>ParametricPlot3D</code>, can we start with <code>PolyhedronData["Tetrahedron"]</code> to obtain spatial point positions?. The tip of position vector s... | cvgmt | 72,111 | <pre><code>RegionPlot3D[PolyhedronData["Octahedron", "Region"],
MeshFunctions -> {Function[{x, y, z}, ArcTan[z, Sqrt[x^2 + y^2]]],
Function[{x, y, z}, Arg[x + I*y]]}, Mesh -> 20, Boxed -> False,
PlotPoints -> 50]
</code></pre>
<p><a href="https://i.stack.imgur.com/wJNb9.png" rel=... |
19,399 | <p>I am to give the following for an interview:</p>
<p>"a short 7–10-minute teaching demonstration on logarithms. Please consider this as your first 10 minutes of introducing logarithms as if you have not previously mentioned the word to this class. Treat this as closely to what you would do during an in-person cl... | Chris Cunningham | 11 | <p>Explain what you explained here to start the interview. Do a 3-5 minute intro that introduces logarithms the way your video would introduce it. Then spend the rest of the time doing whatever else you would do in class.</p>
<p>Bring the activity with you. Hand it out to the committee.</p>
<p>Basically I don't see any... |
4,226,480 | <p>Assume <span class="math-container">$ u_{0}:\mathbb{R}\to\mathbb{R} $</span> is continuous and such that <span class="math-container">$ \lim_{t\to-\infty}u_{0}\left(t\right)=\lim_{t\to\infty}u\left(t\right)=L $</span>.</p>
<p>Prove that there exists unique <span class="math-container">$ u:\left\{ Im\left(z\right)\ge... | Martin R | 42,969 | <p><span class="math-container">$g(z)=\frac{z-i}{z+i}$</span> maps the upper half-plane onto the unit disk <span class="math-container">$\Bbb D$</span>, and <span class="math-container">$\Bbb R \cup \{ \infty \}$</span> to its boundary <span class="math-container">$\partial \Bbb D$</span>, with <span class="math-contai... |
93,383 | <blockquote>
<p>A hotel can accommodate 50 customers, experiences show that $0.1$ of those who make a reservation will not show up. Suppose that the hotel accepts 55 reservations. Calculate the probability that the hotel will be able to accommodate all of the customers that show up. </p>
</blockquote>
<p>I only ... | Artem | 29,547 | <p>Here is how you can use Poisson distribution:
$$
\mathbf P(X=k)=\frac{\lambda^k}{k!}e^{-\lambda}.
$$
In you case you should take the "success" as "person does not appear", probability of success is $p=0.1$. The probability that you have more than 5 successes (i.e., you can accommodate all the people) is
\begin{align... |
1,537,761 | <blockquote>
<p>Let $n$ be positive number, if $a \equiv b \pmod{2n}$, prove that
$a^2 \equiv b^2 \pmod{4n}$.</p>
</blockquote>
<p>By the congruence in hypothesis, we have $a-b = 2nk$ where $k$ is an integer.
Then $a = b+2nk$ and $a^2 = b^2+4n^2k^2+4knb$. From this we get $a^2-b^2 = 4kn(kn+b)$.</p>
<p>Now I have ... | marty cohen | 13,079 | <p>Suppose
$a \equiv b \bmod 2kn
$.
($k=1$ in this case.)</p>
<p>Then
$a
=b+2jkn
$
for some $j$,
so
$a^2
=b^2+4bkjn+4j^2k^2n^2
=b^2+4kn(bj+j^2kn)
$
so
$a^2
\equiv b^2 \bmod 4kn
$.</p>
|
1,722,587 | <p>Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very interesting properties.</p>
<p>One of the properties of $\phi$ is: $$\phi^2=\phi+1$$ Is there a constant like $\phi$ ... | KKZiomek | 318,370 | <p>This is weird, but I'm answering my own question.</p>
<p>Actually just because $\phi$ is a constant kind of misled me, and I forgot that this is normal quadratic equation.</p>
<p>You can solve it like this:</p>
<p>$$-x^2+x+1=0$$
$$\frac{-1\,\pm\,\sqrt{1-4(-1)(1)}}{2(-1)}$$
$$\frac{-1\,\pm\,\sqrt{5}}{-2}$$
$$\frac... |
1,305,151 | <p>I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ is a root.</p>
<p>I know roots of unity and De Moivre's theorem is clearly going to be important here but I just ca... | Zev Chonoles | 264 | <p>Are you familiar with the Chebyshev polynomials $T_n$ (<a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials" rel="noreferrer">Wikipedia link</a>)? One property is that
$$T_n(\cos(\theta))=\cos(n\theta)$$
(see <a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials#Trigonometric_definition" rel="noreferrer... |
1,305,151 | <p>I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ is a root.</p>
<p>I know roots of unity and De Moivre's theorem is clearly going to be important here but I just ca... | mathreadler | 213,607 | <p>The fourier transform together with higher frequencies are polynomials in lower frequencies should be enough to write any of them as radicals of polynomials in the other frequencies. And of course that radicals of polynomials are algebraic.</p>
|
1,234,500 | <p>I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often don't know how to go about tackling it. When I see the solution, often I understand it perfectly but arriving at that s... | tomi | 215,986 | <p>The "three solutions" referred to are three distinct values of $x$ that make the equation true. As you will see later, one of those values must be a repeated root, so arguably there are four roots, but let that go for the moment...</p>
<p>This equation can be restated as $|x^2+4x+3|=m(x-2)$. Writing it this way, it... |
1,234,500 | <p>I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often don't know how to go about tackling it. When I see the solution, often I understand it perfectly but arriving at that s... | Piquito | 219,998 | <p><img src="https://i.stack.imgur.com/dWWiV.png" alt="enter image description here">
From the figure it is clear that the only possibility is when the red straight line of slope m and passing through the point (2, 0) is tangent to |f(x)|in a point of the interval [-3, -1] (otherwise there are none or two or four solut... |
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