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 |
|---|---|---|---|---|
1,470,849 | <p>I saw this video on Lorentz transformation and needed to refresh my memory a bit. </p>
<p>If</p>
<p>$$\frac{t}{t'}= \sqrt{1-\frac{v^2}{c^2}}$$</p>
<p>and</p>
<p>$$\gamma = \frac {t'}t $$</p>
<p>How do you make this equal ? </p>
<p>$$\gamma = \frac 1{\sqrt{1-\frac {v^2}{c^2}}}$$</p>
<p>Most of you will probabl... | Spencer | 71,045 | <p>So we have an expression for $t/t'$, $$ \frac{t}{t'} = \sqrt{1-v^2/c^2},$$</p>
<p>but we need the reciprocal, $t'/t$, in order to know what $\gamma$ is. To do this we will write the right hand side of our equation as a fraction. This can be done by writing it "over one". </p>
<p>$$ \frac{\color{red}{t}}{\color{blu... |
2,586,663 | <p>It seems to me, by trial and error, that this series converges to $1$. Is this true? How do I prove it either way?</p>
| operatorerror | 210,391 | <p>Note
$$
\sum_{k=2}^\infty\frac{\ln(k+1)}{2^k}\leq \sum_{k=2}^\infty\frac{k}{2^k}=3/2
$$
where the final equality is the usual geometric series game. Thus your series converges, since omitting finitely (here 1) many terms has no effect on convergence.</p>
|
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | farruhota | 425,072 | <p>Here is an alternative solution. See the graph for labeling:
<a href="https://i.stack.imgur.com/qKKcY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qKKcY.png" alt="enter image description here"></a></p>
<p>From the trapezium area formula we get:
$$\frac{a+x}{2}\cdot x=5+A \Rightarrow ax=10+2A-(... |
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | Deepak | 151,732 | <p>Until someone posts with "brilliant geometric insight" I was hoping exists, I might as well post my solution. Essentially similar to the OP's, except I used fewer variables.</p>
<p>I assigned $x$ to be the base of the area $2$ triangle. That meant the height of the same triangle would be $\frac 4x$. The dimensions ... |
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | Christian Blatter | 1,303 | <p>We may assume that the bounding rectangle is a square with side $s$. Guessing $s=4$ lets everything fall into place, and the resulting area of the blue triangle is $7$.</p>
<p>Now it is easily seen that making $s<4$ and keeping the areas $2$ and $4$ for the red and the yellow triangles would decrease the area of... |
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | farruhota | 425,072 | <p>The hint may assume two things: to draw extra lines 1) outside; 2) inside the box so that the solution is simple to understand.</p>
<p>Here is yet another method: draw the lines to divide the square into four rectangles indicated by $A, B, C$ and $3+3$:</p>
<p><a href="https://i.stack.imgur.com/W24st.png" rel="nof... |
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | Joe Donofrio | 590,287 | <p>Green triangle has 2x area of red triangle. Heights of both triangles are equal because all sides of square are equal. so base of green triangle = 2x base of red triangle.So r=length of base of red triangle and 2r=base of green triangle. Let s= length of side of square also height of both triangles. for red triang... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Geoff Robinson | 13,147 | <p>Another well-known proof which is somewhat related to two of the proofs by Qiaochu above
is to note that for every prime <span class="math-container">$p \leq n$</span>, the power of <span class="math-container">$p$</span> dividing <span class="math-container">$n!$</span> is at most
<span class="math-container">$p^{\... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Martin Sleziak | 8,297 | <p>This is taken from Section 1.4 of Andrew Granville's notes on <a href="https://dms.umontreal.ca/%7Eandrew/Courses/Chapter1.pdf" rel="nofollow noreferrer">Prime numbers</a>:</p>
<blockquote>
<p>We finish this section by proving that for any <span class="math-container">$f(t) \in \mathbb Z[t]$</span> of degree <span c... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | user 1591719 | 32,016 | <p>Maybe you wanna use the sum of reciprocal prime numbers. The argument for the fact that<br />
the series diverges you may find <a href="https://books.google.com/books?hl=en&id=Il64dZELHEIC&pg=PA90" rel="nofollow noreferrer">here</a> in one of Apostol's exercise.</p>
|
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Prime Leow | 116,544 | <p>Prove infinite number of primes using Wilson's Theorem and Euclid's argument:</p>
<p>Let $P$ be the maximum prime number. From Wilson's Theorem, $P$ is prime if and only if $P \mid (P - 1)! + 1$. Thus, $(P - 1)! + 1 = kP$, for some natural number $k$.</p>
<p>Let $n > P$ since there are infinite natural numbers,... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Manny Reyes | 10,968 | <p>Here's a proof in the language of ring theory. By the division algorithm, $\mathbb{Z}$ is a principal ideal domain. Thus its maximal ideals are precisely those nonzero of the form $(p)$ where $p$ is prime.</p>
<p>Assume for contradiction that there are finitely many primes $p_1, \dots, p_n$. Then the product $j =... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Trevor | 493,232 | <p>Let <span class="math-container">$P$</span> be your currently known primes (or more precisely, currently known coprime integers).</p>
<p>Find an <span class="math-container">$n$</span> such that <span class="math-container">$n+P_i$</span> is not divisible by <span class="math-container">$P_i$</span> for each <span c... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Nicky Hekster | 9,605 | <p>There is an older post from 2011 that you should check out: <a href="https://math.stackexchange.com/questions/30127/is-there-an-intuitionist-i-e-constructive-proof-of-the-infinitude-of-primes">Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?</a>.<p>
Also you should have a look at a pr... |
1,034,676 | <p>I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. </p>
<p>What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler... | Gyu Eun Lee | 52,450 | <p>Two square matrices $A$ and $B$ are said to be <em>similar</em>, or <em>conjugate</em>, if there exists an invertible square matrix $P$ such that $A = P^{-1}BP$. This is equivalent to saying that $A$ and $B$ represent the same linear transformation in different bases, with $P$ providing the change-of-basis matrix th... |
1,034,676 | <p>I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. </p>
<p>What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler... | Hao S | 202,187 | <p>Among other things the Jordan form can show that any Markov process must terminate and also helps to find the limiting state. Instructors usually motivate such processes by asking how to find equilibrium states in chemistry, but this also has applications to google's page rank system. Basically rank pages due to the... |
81,805 | <blockquote>
<p>The binomial sum
$$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$
satisfies the Fibonacci relation.</p>
</blockquote>
<p>I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions?</p>
| Martin Sleziak | 8,297 | <p>This approach was explained in the above comments by <a href="https://math.stackexchange.com/questions/81805/how-to-show-that-this-binomial-sum-satisfies-the-fibonacci-relation#comment193327_81805">N.S</a> and
and
<a href="https://math.stackexchange.com/questions/81805/how-to-show-that-this-binomial-sum-satisfies-t... |
81,805 | <blockquote>
<p>The binomial sum
$$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$
satisfies the Fibonacci relation.</p>
</blockquote>
<p>I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions?</p>
| Felix Marin | 85,343 | <p>$\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{\ds}[1]{\displaystyle{#1}}
\newcomma... |
284,848 | <p>I have this recurrence relation $T(n)=4T(\frac{n}{5})+n$ with the base case $T(x)=1$ when $x\leq5$. I want to solve it and find it's $\theta$. I think i have solved it correctly but I can't get the theta because of this term $\frac{5}{5^{log_{4}n}}$ . Any help?</p>
<p>$T(n)=4(4T(\frac{n}{5^{2}})+\frac{n}{5})+n$</... | Ishan Banerjee | 52,488 | <p>Hint :$\frac {\log{n}}{\log4}=\log_4{n}$</p>
<p>So $5^{\log_4{n}}=n^{\frac {1}{\log_5{4}}}$</p>
<p>Use this.</p>
|
120,675 | <p>Given these condition, I am seeking a proof:</p>
<p>Define a sequence of real numbers by $x_1 = 3$ and then, for $n \geq 2$,</p>
<p>$x_n = \sqrt{2 \; x_{n-1} + 1 }$</p>
<p>Prove that for all positive integers $n$, $x_n \geq x_{n+1}$.</p>
<p>I began a proof by induction, but ran out of steam. I tested the base ca... | Beni Bogosel | 7,327 | <p>Calculate $x_2$, and notice that $x_{n+1}-x_n$ has the same sign as $x_{n}-x_{n-1}$, and inductively, the same sign as $x_2-x_1$, which you just calculated.</p>
|
785,087 | <p>Off the top of my head, I can't think of a non-separable compact space. </p>
<p>Can you provide a good example?</p>
| 115465 | 113,444 | <p>A not so simple, but curious example is the Alexandroff Double Circle: </p>
<p>Consider $C_i=\{(x,y) \in \mathbb{R}^2; x^2+y^2=i\}$ for $i=1,2$. Let $X=C_1 \cup C_2$ and $f:C_1 \to C_2$ the radial homeomorphism. Define a topology on $X$ as follows: the points in $C_2$ are all isolated; for each $x \in C_1$ and $n \... |
1,818,276 | <blockquote>
<p>Evaluation of <span class="math-container">$$\lim_{x\rightarrow 1}\frac{1-x^x}{1-x^2}$$</span></p>
<p>Without using L Hospital Rule and Series expansion.</p>
</blockquote>
<p>I have solved using L Hopital Rule, and getting answer <span class="math-container">$\displaystyle = \frac{1}{2}\;$</span>. But ... | Claude Leibovici | 82,404 | <p>As usual, lab bhattacharjee provided a very good hint.</p>
<p>If you play with Taylor series around $x=1$, you have </p>
<p>$$x\log(x)=(x-1)+\frac{1}{2} (x-1)^2+O\left((x-1)^3\right)$$ $$x^x=e^{x \log(x)}=1+(x-1)+(x-1)^2+O\left((x-1)^3\right)$$</p>
<p>$$1-x^x=-(x-1)-(x-1)^2+O\left((x-1)^3\right)$$ which makes $$\... |
4,508,099 | <p>When I deal with this simple integral, I found there are several methods. Now I share one of them.</p>
<p>Letting <span class="math-container">$x\mapsto \pi-x $</span> yields <span class="math-container">$$I=\int_{0}^{\pi} \frac{d \theta}{a+b \cos \theta}=\int_{0}^{\pi} \frac{d \theta}{a-b \cos \theta}$$</span>
Addi... | Lai | 732,917 | <p><strong>By Weierstrass substitution</strong></p>
<p>Letting <span class="math-container">$t=\tan \frac{\theta}{2}$</span> transforms the integral
<span class="math-container">$$
\begin{aligned}
I&=\int_{0}^{\pi} \frac{1}{a+\frac{b\left(1-t^{2}\right)}{1+t^{2}}} \frac{2 d t}{1+t^{2}}\\
&=2 \int_{0}^{\infty} \... |
4,508,099 | <p>When I deal with this simple integral, I found there are several methods. Now I share one of them.</p>
<p>Letting <span class="math-container">$x\mapsto \pi-x $</span> yields <span class="math-container">$$I=\int_{0}^{\pi} \frac{d \theta}{a+b \cos \theta}=\int_{0}^{\pi} \frac{d \theta}{a-b \cos \theta}$$</span>
Addi... | Lai | 732,917 | <p><strong>By Contour Integration</strong></p>
<p>Considering the integral</p>
<p><span class="math-container">$$
J=\int_{0}^{2 \pi} \frac{d \theta}{a+b \cos \theta}.
$$</span></p>
<p>Let <span class="math-container">$z=e^{\theta i}$</span>, then <span class="math-container">$\cos \theta=\frac{1}{2}\left(y+\frac{1}{z}\... |
466,603 | <p>How to go about evaluating the following integral?</p>
<p>$$ I = \int \sqrt{ \dfrac {\sin(x-\alpha)} {\sin(x+\alpha)} }\,\operatorname d\!x$$</p>
<p>What I have done so far:</p>
<p>$$ I = \int \sqrt{ 1-\tan\alpha\cdot\cot x }\,\operatorname d\!x$$
Let $ t^2 = 1-\tan\alpha\cdot\cot x $</p>
<p>$$ \begin{align}
2... | Ron Gordon | 53,268 | <p>Substitute</p>
<p>$$u = \frac{\sin{(x-\alpha)}}{\sin{(x+\alpha)}}$$</p>
<p>Then, with some algebraic manipulation, we find that</p>
<p>$$dx = \frac{2 du}{\sec^2{\alpha} u^2 + 2 (\tan^2{\alpha}-1) u + \sec^2{\alpha}}$$</p>
<p>so that the integral becomes</p>
<p>$$2 \int du \frac{\sqrt{u}}{\sec^2{\alpha} \, u^2 +... |
466,603 | <p>How to go about evaluating the following integral?</p>
<p>$$ I = \int \sqrt{ \dfrac {\sin(x-\alpha)} {\sin(x+\alpha)} }\,\operatorname d\!x$$</p>
<p>What I have done so far:</p>
<p>$$ I = \int \sqrt{ 1-\tan\alpha\cdot\cot x }\,\operatorname d\!x$$
Let $ t^2 = 1-\tan\alpha\cdot\cot x $</p>
<p>$$ \begin{align}
2... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
2,151,206 | <p>Flock of sheep is walking on the path in one direction. There are 30 sheeps. Initially they all have different random speeds. </p>
<p>The main feature is that in the process of walking, if a sheep at a faster speed bumps on a sheep which has a slower speed, then the faster sheep starts moving with the speed of this... | Henry | 6,460 | <p>Providing that all orders of sheep speeds are equally likely and no pair of sheep have the same speed, the $n$th sheep (counting from the front) is the slowest of its group if and only if all $n-1$ sheep in front of it are all faster than it is, which has probability $\dfrac{1}{n}$ </p>
<p>Each group has precisely ... |
759,838 | <p>How to evaluate the integrals of</p>
<p>(a) $(\ln(x))^n$ </p>
<p>(b) $x^ne^{ax}$ where $a$ is a constant</p>
<p>By reduction formula </p>
| user71352 | 71,352 | <p>For the first question define $I_{n}=\int(\ln(x))^{n}dx$ then by integrating by parts we have:</p>
<p>$$I_{n}=x(\ln(x))^{n}-n\int(\ln(x))^{n-1}dx=x(\ln(x))^{n}-nI_{n-1}$$</p>
<p>For (b) we define $I_{n}=\int x^{n}e^{ax}dx$ then by integrating by parts we have:</p>
<p>$$I_{n}=\frac{1}{a}x^{n}e^{ax}-\frac{n}{a}\int... |
1,905,891 | <p><a href="https://i.stack.imgur.com/IDrI5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IDrI5.png" alt="enter image description here"></a></p>
<p>Doubt is that If $\gamma < \alpha$ then how does it follow that $\gamma$ doesnot belong to B and how does it follows that $\alpha \leq x $ for ever... | fleablood | 280,126 | <p>L is the set of lower bounds of B.</p>
<p>Let $l\in L;b\in B $. As $l $ is a lower bound of $B $, $l\le b $ and therefore every element of $B $ is an upper bound of $L $.</p>
<p>So if $\gamma < \sup L $, $\gamma$ is not an upper bound of $L $ and thus not an element of $B $.</p>
<p>As for the second part. $\... |
3,059,572 | <p>I've been practicing horizontal asymptotes and I came across a problem that I do not understand.</p>
<p>I understood why <span class="math-container">$\lim_{x \rightarrow \infty}\frac{x - 2}{\sqrt{x^2 + 1}} = 1$</span>, but i couldn't understand why <span class="math-container">$\lim_{x \rightarrow -\infty}\frac{x... | quid | 85,306 | <p>The core issue is that <span class="math-container">$\sqrt{x^2}$</span> does not simplify to <span class="math-container">$x$</span> for negative <span class="math-container">$x$</span>.
The symbol <span class="math-container">$\sqrt{r}$</span> for a nonnegative real number <span class="math-container">$r$</span> d... |
592,477 | <p>Is it possible that a $3$-by-$3$ matrix with integer values and determinant 1 has a real eigenvalue with algebraic multiplicity 2, that is not equal to $\pm 1$? </p>
<p>Doing some elementary computations one can rephrase the question as follows. Do there exist integers $k$ and $m$ such that $a=\frac{1}{3}(k\pm \sqr... | Ewan Delanoy | 15,381 | <p>There is no solution to your problem. The pattern of proof is familiar : we show that there cannot be a solution
with too large parameters, and then a finite numbers of cases remain to be checked
by exhaustive inspection.</p>
<p>If a solution matrix exists, it has eigenvalues $a,a,\frac{1}{a^2}$
for some real $a\no... |
38,864 | <p>Recently I was introduced to the concept of <a href="http://en.wikipedia.org/wiki/Orthogonal_polynomials">Orthogonal Polynomials</a> through the poly() function in the R programming language. These were introduced to me in the concept of polynomial transformations in order to do a linear regression. Bear in mind tha... | Piero D'Ancona | 7,294 | <p>Start <a href="http://dlmf.nist.gov/18.4" rel="nofollow">here</a></p>
|
38,864 | <p>Recently I was introduced to the concept of <a href="http://en.wikipedia.org/wiki/Orthogonal_polynomials">Orthogonal Polynomials</a> through the poly() function in the R programming language. These were introduced to me in the concept of polynomial transformations in order to do a linear regression. Bear in mind tha... | Helge | 3,983 | <p>I am not sure of your math-background, so I am trying to keep it simple, without oversimplifying some ideas. First off polynomials are nice for various reasons, e.g. a polynomial of degree $n$ has at most $n$ zeros. However, there are still many polynomials and it makes sense to choose VERY nice ones: orthogonal pol... |
118,920 | <p><a href="http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds">My stackexchange post</a> was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!</p>
<p>Let <span class="math-container">$M$</span> be a compact Riemannian manifold... | Tony | 27,736 | <p>Sometimes math problems crop up in high school competitions that are secretly linear algebra problems in disguise. For instance, #6 of USAMO 2008 (<a href="http://amc.maa.org/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo2008.pdf" rel="nofollow">http://amc.maa.org/a-activities/a7-problems/USAMO-IMO/q-usamo/-p... |
118,920 | <p><a href="http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds">My stackexchange post</a> was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!</p>
<p>Let <span class="math-container">$M$</span> be a compact Riemannian manifold... | Name | 30,062 | <p>(1) An obvious but a very conceptual application is the basic fact that the cardinality of every finite field $F$ is a power of a prime,
as $F$ can be considered as a vector space over $\mathbb{F}_p$ for $p=char F$.</p>
<p>(2) Another application is the construction of finite projective planes. Consider a vector s... |
970,160 | <p>Suppose the set </p>
<p>$$S := \left\lbrace x+i \sin \left( \frac{1}{x} \right) \Bigg\vert x \in (0,1]\right\rbrace \subseteq \mathbb{C}. $$</p>
<p>I want to show that $S$ is path-connected but $\overline{S}$ is not path connected.</p>
<p>To show that $S$ is path-connected I have to show that for all points $p,q... | Alan | 175,602 | <p>For path connected, each point p, q is characterized by $x_1,x_2\in (0,1]$ such that $p=x_1+i\sin (\frac 1 {x_1})$, $q=x_2+i\sin (\frac 1 {x_2})$. Go for the natural path, rescaling $x_1$ to 0 and $x_2$ to 1, i.e. a parameterization. So $\gamma(t)=((1-t)x_1+tx_2)+i\sin(\frac 1 {(1-t)x_1 +tx_2)})$. Continuity... |
1,696,535 | <p>I am trying to find if $f(x)=x^3-117x+53$ is irreducible or reducible. </p>
<p>I tried Eisenstein's criterion but this fails as $53$ does not divide $117$</p>
<p>After some thinking, if f(x) has any roots then they will be negative, but I have had no luck finding any.</p>
<p>Would really appreciate some help, th... | JMoravitz | 179,297 | <p>Consider the equation modulo $5$.</p>
<p>$x^3-2x+3\pmod{5}$</p>
<p>When $x\equiv 0\pmod{5}$, the above equates to $3\pmod{5}$</p>
<p>When $x\equiv 1\pmod{5}$, the above equates to $2\pmod{5}$</p>
<p>When $x\equiv 2\pmod{5}$, the above equates to $4\pmod{5}$</p>
<p>When $x\equiv 3\pmod{5}$, the above equates to ... |
2,629,164 | <p>I'm working on a calculus question, where we're asked to find $g'(z)$ for
$$g(z)=\frac{z^2-2z-8}{z-3}\cdot\frac{z^2-9}{z-4}$$
So I was thinking that first you have to use the quotient rule on each side individually and then multiply them, but I seem to keep getting stuck and it's becoming pretty frustrating.</p>
| DeepSea | 101,504 | <p>$(b\sqrt{2}+c\sqrt{3})^2 = (\sqrt{5} - a - d\sqrt{6})^2\implies 2b^2+2bc\sqrt{6}+3c^2= 5+a^2+6d^2 - 2a\sqrt{5}-2d\sqrt{30}+2ad\sqrt{6}\implies m+r\sqrt{6}= q\sqrt{5}+t\sqrt{30}\implies (r\sqrt{6} - q\sqrt{5})^2 = (t\sqrt{30}-m)^2\implies 6r^2-2qr\sqrt{30}+5q^2= 30t^2-2tm\sqrt{30}+m^2\implies \sqrt{30} \in \mathbb{Q... |
2,629,164 | <p>I'm working on a calculus question, where we're asked to find $g'(z)$ for
$$g(z)=\frac{z^2-2z-8}{z-3}\cdot\frac{z^2-9}{z-4}$$
So I was thinking that first you have to use the quotient rule on each side individually and then multiply them, but I seem to keep getting stuck and it's becoming pretty frustrating.</p>
| Jack D'Aurizio | 44,121 | <p>It is enough to exploit the existence of infinite primes $p$ such that both $2$ and $3$ are quadratic residues $\!\!\pmod{p}$ while $5$ is not. By the Chinese remainder theorem and quadratic reciprocity, there are infinite primes of the form $p=120k+47$ and anyone of them does the job. Assuming
$$ a+b\sqrt{2}+c\sqrt... |
2,264,491 | <blockquote>
<p>$$\lim_{n\to\infty}\dfrac{\ln(n+1) \sin(n)}{n \tan(n)}$$</p>
</blockquote>
<p>No idea what to do with this one! Have tried writing out as series but that method is extremely complicated. L'hopitals doesn't work here either</p>
| Paul | 202,111 | <p>Hint: $$\frac{\ln(n+1)sin(n)}{n \tan n}=\frac{\ln{(n+1)} \cos{n}}{n}$$</p>
|
2,264,491 | <blockquote>
<p>$$\lim_{n\to\infty}\dfrac{\ln(n+1) \sin(n)}{n \tan(n)}$$</p>
</blockquote>
<p>No idea what to do with this one! Have tried writing out as series but that method is extremely complicated. L'hopitals doesn't work here either</p>
| haqnatural | 247,767 | <p><strong>Hint</strong>: Use squeeze theorem $$0\le \left| \frac { \ln { \left( n+1 \right) \sin { \left( n \right) } } }{ n\tan { \left( n \right) } } \right| =\left| \frac { \ln { \left( n+1 \right) \cos { \left( n \right) } } }{ n } \right| =\left| \frac { \ln { \left( n+1 \right) } }{ n } \right| \lef... |
4,627,649 | <p>So I know how to find the complementary function of this non-homogenous equation. The complementary function is <span class="math-container">$y = A \cos x + B \sin x$</span>. But I don't understand how to find the particular solution. If we use <span class="math-container">$y_p = p \cos x + q \sin x$</span>, where <... | mathlander | 1,064,099 | <p>The elements <span class="math-container">$1$</span> and <span class="math-container">$-1$</span> are fixed. We can imagine the remaining elements as an octahedron in <span class="math-container">$3$</span>d space, where multiplying two opposite vertices gives <span class="math-container">$1$</span>, multiplying a v... |
1,030,536 | <p>how can I solve following equation analytically </p>
<blockquote>
<p>$$x^2+(3+5i)x+(7+11i)=0$$ I need the roots as follow $x=a+bi$</p>
</blockquote>
| DanielV | 97,045 | <p>Start with </p>
<p>$$P \equiv \bigg(x^2 + (3+5i)x + (7 + 11i) = 0\bigg)$$</p>
<p>Substitute $x = a+bi$ into $P$, assuming $a$ and $b$ are $\mathbb R$, expand and equate the real and complex parts of the equation to get:</p>
<p>$$\begin{cases}
P_1 \equiv \bigg(2ab + 3b + 5a + 11 = 0\bigg) \\
P_2 \equiv \bigg(-b^... |
3,342,611 | <p>A stone is thrown with a velocity of 20 ms^(-1) and at an angle of 60°.
A second stone is thrown at the same time and place,with the same velocity but an
angle of 30°.</p>
<p>(a) Find which stone finishes its flight first and by how long.</p>
<p>(b) Show that both stones land at the same place.</p>
<p>I'm hav... | Timothy Hedgeworth | 220,781 | <p>Number the people around the table <span class="math-container">$1,\ldots,12$</span>. </p>
<p>As an example, the number of ways that persons <span class="math-container">$1,\ldots,12$</span> can engage in handshakes with no arms crossing given that person <span class="math-container">$1$</span> is shaking hands wit... |
395,600 | <blockquote>
<ul>
<li>Sine <span class="math-container">$\theta$</span> = opposite/hypotenuse</li>
<li>Cosine <span class="math-container">$\theta$</span> = adjacent/hypotenuse</li>
<li>Tangent <span class="math-container">$\theta$</span> = opposite/adjacent</li>
</ul>
</blockquote>
<p>In order to calculat... | Inceptio | 63,477 | <p>I always wondered about the same thing, till I attended my first <em>calculus</em> class.</p>
<p>As Julien rightly noted. It uses power series of <span class="math-container">$\sin x, \cos x$</span> etc, to only <em>approximately</em> calculate the value of angles(in radians) you put in. You can read more about it <... |
2,416,635 | <p>I'm trying to evaluate</p>
<blockquote>
<p>$$\int_{a}^{\infty} -\frac{1}{c (1-x) x}$$
where $c, a$ are constants.</p>
</blockquote>
<p>Mathematica gives: </p>
<p>$$-\dfrac{\log \left(\dfrac{a-1}{a}\right)}{c}$$</p>
<p>where I assume that $a > 1$. How does one get this result? Since, the indefinite integra... | quasi | 400,434 | <p>Let $f(m,n,p)$ be the number of ways to order $p\;$people in a queue such that exactly $m\;$heads are visible from the front, and exactly $n\;$heads are visible from the back.
<p>
If the heights are not pairwise distinct, knowing the values of $m,n,p\;$is not sufficient to find $f(m,n,p)$.
<p>
For example, if ther... |
421,844 | <p>I wanted to know, given quadrilateral ABCD such that $AB^2+CD^2=BC^2+AD^2$ , prove that $AC⊥BD$ .</p>
<p>Help. </p>
<p>Thanks.</p>
| DonAntonio | 31,254 | <p>Hints:</p>
<p>I'll do it with <em>analytic</em> geometry: place the vertices at</p>
<p>$$D=(0,0)\;,\;C=(x,0)\;,\;A=(a,b)\;,\;B=(\alpha,\beta)$$</p>
<p>Then we get that</p>
<p>$$AB^2+CD^2=(a-\alpha)^2+(b-\beta)^2+x^2=b^2+(\alpha-x)^2+a^2+b^2=BC^2+AD^2\iff$$</p>
<p>$$a\alpha+b\beta=x\alpha\iff \frac\alpha\beta=\f... |
1,235,497 | <p>Here is the problem I'm working on (Satan's apple with bounded memory decision problem): Suppose I offer you slices of an apple, first $1/2$, then $1/4$, ... then $1/2^k$, etc. You want to eat as much of the apple as possible without eating the whole thing. For simplicity, lets have the utility $U(A_n) = n$ where $A... | Pedro Gimeno | 136,841 | <ol>
<li>The name is <a href="https://en.wikipedia.org/wiki/De_Bruijn_sequence" rel="nofollow noreferrer">De Bruijn sequences</a>.</li>
<li>They exist for all $n$. For your alphabet $\{0,1\}$, they contain $2^n$ symbols with $2^{n-1}$ 0's and the same 1's.</li>
<li>It's not easy to generate them. One way is with a shif... |
2,426,562 | <p>I'm aware of other questions regarding "increasing" mathematical intuition, but I wanted to explain my own context, since that way any advice posted on this question feed will be specific to myself.</p>
<p>I'm 16. I'm a (new) Sixth Former (roughly college level) student and has just today had my fifth Further Pure ... | B. Goddard | 362,009 | <p>The reason piano students do scales (hours and hours of scales and Hanon exercises....you have to SuperGlue your butt to the bench and go until your fingers bleed) is to develop "familiarity with one's instrument." If you watch a trained pianist, you'll see him do amazing things. His head can be looking right and... |
3,357,554 | <p>While solving a problem related to functions, I came across the following algebraic expression: </p>
<p><span class="math-container">$$\frac{a^x}{a^x+\sqrt{a}} + \frac{a^{1-x}}{a^{1-x}+\sqrt{a}}$$</span></p>
<p>Although I tried, I couldn't find an easy way to simplify it.</p>
<blockquote>
<p>Can anyone simplify... | Dr. Sonnhard Graubner | 175,066 | <p>It is <span class="math-container">$$\frac{a^x(a^{1-x}+\sqrt{a})+a^{1-x}(a^x+\sqrt{a})}{(a^x+\sqrt{a})(a^{1-x}+\sqrt{a})}$$</span>
this simplifies to
<span class="math-container">$$\frac{2a+a^{x+1/2}+a^{3/2-x}}{(a^x+\sqrt{a})(a^{1-x}+\sqrt{a})}$$</span></p>
|
220,951 | <p>Given the following arguments:</p>
<p>$ \tag A (R \to \neg S) \land (T \to \neg U)$</p>
<p>$ \tag B (V\to \neg W) \land (X \to \neg Y)$</p>
<p>$ \tag C (T \to W) \land (U \to S)$</p>
<p>$ \tag D V \lor R $</p>
<p>$$ \therefore \neg T \lor \neg U $$</p>
<p>how to prove this using the standard <a href="htt... | Hagen von Eitzen | 39,174 | <p>One expects this to follow more or less directly from $(A)$ because $p\to q\iff \neg p \lor q$.
However, there is no <em>tertium no datur</em>.
One has to find ones luch starting from (D) by case analysis:</p>
<p>By <em>simplification</em> from (A):
$$\tag{1}R\to\neg S$$
and
$$\tag2 T\to\neg U$$
By <em>simplifica... |
220,951 | <p>Given the following arguments:</p>
<p>$ \tag A (R \to \neg S) \land (T \to \neg U)$</p>
<p>$ \tag B (V\to \neg W) \land (X \to \neg Y)$</p>
<p>$ \tag C (T \to W) \land (U \to S)$</p>
<p>$ \tag D V \lor R $</p>
<p>$$ \therefore \neg T \lor \neg U $$</p>
<p>how to prove this using the standard <a href="htt... | Frank Hubeny | 312,852 | <p>Here is a proof using a <a href="http://proofs.openlogicproject.org/" rel="nofollow noreferrer">Fitch-style proof checker</a>:</p>
<p><a href="https://i.stack.imgur.com/yWak2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yWak2.png" alt="enter image description here"></a></p>
<p>The first four ... |
1,380,212 | <p>$$\lim_{y\to 0} \frac{y}{\cos\left(\frac{\pi}{2}(1+y)\right)}$$
Can anybody help me? I can use basic properties of limits, and some of those basic known limits. I know it would be easier with derivatives, but I was just wondering if it's possible without L-Hospital's rule, derivatives, Taylor series. </p>
<p>Thank... | Harish Chandra Rajpoot | 210,295 | <p>Note that $$\lim_{y\to 0}\frac{y}{\cos\left(\frac{\pi}{2}(1+y)\right)} = \lim_{y\to 0}\frac{y}{\cos\left(\frac{\pi}{2}+\frac{\pi y}{2}\right)} = \lim_{y\to 0}\frac{y}{-\sin\left(\frac{\pi y}{2}\right)} \\
= -\frac{2}{\pi}\lim_{y\to 0}\frac{\left(\frac{\pi y}{2}\right)}{\sin\left(\frac{\pi y}{2}\right)} = -\frac{2}{\... |
1,155,705 | <p>Find the last digit of $$77777^{77777}$$</p>
<p>I got a pattern going for $77777^n$ for $n=1, 2, ....$ to be:</p>
<p>$$7, 9, 3, 1$$ for $n = 1, 2, 3, 4$ respectively. </p>
<p>The idea is:</p>
<p>$$77777^{77777} \pmod{10}$$</p>
<p>I see that:</p>
<p>$$77777^n \equiv 77777^{n + 4} \pmod{10}$$</p>
<p>Using the ... | 5xum | 112,884 | <p><strong>Hint:</strong></p>
<ol>
<li>The last digit of $7777777^n$ is equal to the last digit of $7^n$.</li>
<li>The pattern for $n=1,2,3,4$ is, indeed, $7,9,3,1$. How does that pattern continue? Take a look at the last digit of $77777^n$ for $n$ from $1$ to $8$
.</li>
</ol>
|
1,968,108 | <p>Is there a name for the follow function $\frac{x}{1+x^2}$?</p>
<p>Or the integral of $\frac{x}{1+x^2}$ is a typical function?</p>
| goblin GONE | 42,339 | <p>Another fun fact about this function is that $$e^x \star \frac{x}{1+x^2} = \sin(x),$$ where the $\star$ is Hadamard product of power series, defined as follows:</p>
<p>$$\left(\sum_{i \in \mathbb{N}} a_i x^i\right) \star \left(\sum_{i \in \mathbb{N}} b_i x^i\right) := \sum_{i \in \mathbb{N}} a_i b_i x^i$$</p>
<p>... |
2,817,129 | <p>I have an system of three differential equation coupled. I put this system to matrix form. I think it should be more easy to solve.
$$
\begin{bmatrix}
\ddot x_1 \\
\ddot x_2 \\
\ddot x_3 \\
\end{bmatrix}
=
\begin{bmatrix}
0 & a_{12} & -a_{31} \\
-a_{12} & 0 & a_{23} \\
... | Math1995 | 201,497 | <p>If the interface $\Gamma$ has some regularity, and since $v=u_t$ on $\Gamma$ then $\xi \in L^2(\Omega)$. </p>
<p>Now to get the desired result, we need to show that the partial derivatives of $\xi$ are in $L^2(\Omega)$, it is sufficient to prove that the restriction of the partial derivatives of $\xi$ on $\Gamma$ ... |
612,461 | <p>How do you determine what solving method to use, for example what is the reason you solve by the quadratic formula, or by factoring, or by completing the square, or by taking a square root.</p>
| Mike | 17,976 | <p>I think the only consideration here is how hard the problem is. If it's already set up as $x^2=a$, take the square root of both sides (don't forget the negative root too of course). If you can factor it easily, do so. If factors don't come easily (may not be real or rational), completing the square can work well,... |
262,238 | <p>Five cards are dealt from a standard deck of 52. What is the probability that the 3rd card is a Queen? </p>
<p>What I dont understand here is how to factor in when one or both of the first two cards drawn are also Queens. </p>
| Ross Millikan | 1,827 | <p>You can just ignore the first two cards. Think of dealing five cards, the move the first to third position. You are in the same situation as your procedure. As long as you don't look at the first two, they don't matter.</p>
|
262,238 | <p>Five cards are dealt from a standard deck of 52. What is the probability that the 3rd card is a Queen? </p>
<p>What I dont understand here is how to factor in when one or both of the first two cards drawn are also Queens. </p>
| André Nicolas | 6,312 | <p>All orderings of the $52$ cards in the deck are <strong>equally likely</strong>. So the probability the third card in the deck is a Queen is exactly the same as the probability that the $17$-th card in the deck is a Queen, or that the first card in the deck is a Queen: They are <em>all</em> equal to $\dfrac{4}{52}$.... |
2,584,480 | <p>I'm doing a Number Theory question, and if someone could offer a hint, that would be greatly appreciated;</p>
<p>The question is: <strong>Find the sum of the perfect square divisors of the smallest integer with exactly 6 perfect square divisors.</strong></p>
<p>My reasonings: </p>
<ul>
<li>My method so far has be... | Isaac Browne | 429,987 | <p>Conversation in the comments is correct, but I would suggest the following approach. First we find the lowest number with exactly $6$ divisors, and then we square that number. This squared number will then have exactly $6$ square divisors.</p>
<p>To find the number, we use the number of divisors formula
$$d\left(\p... |
4,314,547 | <p>Edit: the problem would've gone something like (I saw it years ago): "find the speed you must run on a second lap around the same track in order to double your average speed for the 2 laps combined."</p>
<p>I have 2 approaches to this problem that give different answers, in both cases we let</p>
<p><span c... | ryang | 21,813 | <p>The average speed of the two-lap journey, i.e., the runner's average speed <em>over</em> the two laps, <span class="math-container">$$\frac{2d}{t_1+t_2}\\=\frac{t_1}{t_1+t_2}\left(\frac{d}{t_1}\right)+\frac{t_2}{t_1+t_2}\left(\frac{d}{t_2}\right)\\=\frac{t_1}{t_1+t_2}\,v_1+\frac{t_2}{t_1+t_2}\,v_2,$$</span> is the <... |
170,020 | <p>When we are given a differential operator of the form
$Lf : = \sum_\alpha a_{\alpha}(x) D^ \alpha f(x) $ , we can define the symbol associated with it to be the function:
$a(x,y) := \sum_\alpha a_\alpha (iy)^\alpha $ . </p>
<p>We then can consider an operator (in divergence form) on $L^2 (\Omega) $, given formall... | Did | 6,179 | <p>What is known is explained in C. Albanese, S. Lawi, <em>Laplace transform of an integrated gometric Brownian motion</em>, MPRF 11 (2005), 677-724, in particular in the paragraph of the Introduction beginning by <em>A separate class of models</em>...</p>
|
16,742 | <p>I have been stuck on this one for hours. </p>
<p>Let $x$, $y$, $z$ be non-negative real numbers.</p>
<p>Also we know $x + z \leq 2$.</p>
<p>Prove the following:</p>
<blockquote>
<p>$(x - 2y + z)^2 \geq 4xz - 8y$.</p>
</blockquote>
<p>Apparently this can be proven with or without AGM, which is $xy \leq \left(\... | Bill Dubuque | 242 | <p>Use $\rm\ (x-2y+z)^2 = 4xz-8y + 4y(2-x-z) + (x-z)^2 + 4y^2\:.\:$ But last 3 summands are $\ge 0\:.\ $ Note this proof requires only that $\rm\ \ y(2-x-z)\ge 0\:.$</p>
|
16,742 | <p>I have been stuck on this one for hours. </p>
<p>Let $x$, $y$, $z$ be non-negative real numbers.</p>
<p>Also we know $x + z \leq 2$.</p>
<p>Prove the following:</p>
<blockquote>
<p>$(x - 2y + z)^2 \geq 4xz - 8y$.</p>
</blockquote>
<p>Apparently this can be proven with or without AGM, which is $xy \leq \left(\... | Tomarinator | 21,832 | <p>we have to prove
$$(x - 2y + z)^2 \geq 4xz - 8y$$</p>
<p>expanding L.H.S</p>
<p>$$(x+z)^{2}+ 4y^{2} - 4y(x+z) $$</p>
<p>now,</p>
<p>$(x+z)^2 \geq 4xz$ ..............{A.M G.M .in.equality}</p>
<p>$4y(x+z) \geq 8y$ ,..........................since $x+z \leq 2$
hence,
$$(x+z)^{2}+ 4y^{2} - 4y(x+z) \geq 4xz -8y +4y... |
2,063,972 | <p>Can anybody either verify or dispute the my proof for the following argument?</p>
<p>Premise 1: (E • I) v (M •U) </p>
<p>Premise 2: ~E </p>
<p>Conclusion: ~(E v ~M)</p>
<p>Proof:</p>
<p>(1) Applying DeMorgan's Second Law to the Conclusion; The Negation of a Disjunction, it is the case that ~(E v ~M) is logicall... | Frank Hubeny | 312,852 | <p>Although the OP's proof appears correct, it may help to use a proof checker. The drawback of using such a tool is that one is forced to use the inference rules available. The benefit is the added confidence one has that one's proof is correct.</p>
<p>Here is a proof using a Fitch-style proof checker. </p>
<p><a hr... |
11,678 | <p>I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in university. We will be doing a lesson on visual proofs of the Pythagorean theorem.</p>
<p>There are at least three that are... | Tommi | 2,083 | <p>The main motivation is that different proofs generalize to different situations.</p>
<p>In addition to that</p>
<ol>
<li>Different proofs may require different tools and different theory. Sometimes, it is nice to have an "elementary" proof, which only uses simple tools, but maybe in a difficult or hard-to-understa... |
11,678 | <p>I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in university. We will be doing a lesson on visual proofs of the Pythagorean theorem.</p>
<p>There are at least three that are... | Vadim Ponomarenko | 270 | <p>Claude Monet did a famous <a href="https://en.wikipedia.org/wiki/Haystacks_(Monet_series)" rel="noreferrer">series</a> of over 20 paintings, all of one or two haystacks, all in the the same field, with various types of light and weather conditions. Here's one example:</p>
<p><a href="https://i.stack.imgur.com/QGfn... |
11,678 | <p>I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in university. We will be doing a lesson on visual proofs of the Pythagorean theorem.</p>
<p>There are at least three that are... | blackened | 4,786 | <p>It is better to solve one problem five different ways, than to solve five problems one way. —George Polya</p>
|
145,146 | <p>I want to zoom/exaggerate a portion of the following plot for <code>t</code> between <code>1</code> and <code>2</code>, such that the zoomed portion is also within the main plot.</p>
<pre><code>x[t_] = Sin[50 t]/t + Cos[t];
y[t_] = Sin[30 t]/t^2 + Cos[t^2];
P1 = Plot[x[t], {t, 0, 2 \[Pi]}, PlotStyle -> Blue];
P... | webcpu | 43,670 | <p>The highlighted region is under the plots of x[t] and y[t], so the plots are not covered by the plots like Epilog did.</p>
<p>In:</p>
<pre><code>Remove[x, y, t, highlight]
x[t_] = Sin[50 t]/t + Cos[t];
y[t_] = Sin[30 t]/t^2 + Cos[t^2];
P1 = Plot[x[t], {t, 0, 2 \[Pi]}, PlotStyle -> Blue];
P2 = Plot[y[t], {t, 0, ... |
23,113 | <p>We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and Absolute geometry where in both the inequality holds.</p>
<p>I am curious whether the triangle inequality is made to... | Gil Kalai | 1,532 | <p>A famous example of a geometry which violates the triangle inequality is $\ell_2^2$, namely the distance between two points is defined as the square of their Euclidean distance. There is also much interesrt in metric spaces that do satisfy the triangle inequality which are subsets of $\ell_2^2$. Those are called "me... |
108,737 | <p>(1) Can a small category be cocomplete? Meaning, have all small colimits? I'd be glad to see an example.</p>
<p>(2) Suppose $\mathcal C$ is a small category, with $Ob(\mathcal C)$ being of cardinality $\kappa$. May $\mathcal C$ have all small limits of cardinality $\leq \kappa$ ?
Does this now allow examples which ... | Wouter Stekelenburg | 3,603 | <p>Small (co)complete categories are posets by a theorem of Freyd.
If $C$ has all small coproducts and its class of morphisms $C_1$ is small, then $C(x,y)^{C_1}\simeq C(\coprod_{f\in C_1} x, y)\subseteq C_1$. If $C(x,y)>1$, then $C_1$ has a subset of strictly greater cardinality: contradiction.</p>
<p>A poset that ... |
2,704,575 | <p>I tried this:</p>
<p>$$\biggl (\frac{1}{8}\biggr)^{-\frac{1}{3}}= \frac{1^{-\frac{1}{3}}}{8^{-{\frac{1}{3}}}}=\frac{-\sqrt[3]{1}}{-\sqrt[3]{8}}=\frac{-1}{-2}$$</p>
<p>Also, is it possible to see what I don't understand here or are there maybe several things..</p>
| Rohan Shinde | 463,895 | <p>$$(1/8)^{-1/3}=((1/8)^{1/3})^{-1}=(1/2)^{-1}=2$$</p>
|
1,638,536 | <p>Compute the following integrals:</p>
<p>$I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$</p>
<p>$J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$</p>
<p>I do not know where to begin. I know I am supposed to use the substitution $z\rightarrow \frac{1}{t}$. I can see that $I$ has 13 poles inside the circle of radius 2.</p>
<p>Usi... | 3SAT | 203,577 | <p>For</p>
<blockquote>
<p>$$J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$$</p>
</blockquote>
<p>$$J:=\int_{|z|=10}\frac{z^3}{(z-1)(z+1)(z-i)(z+i)}dz$$</p>
<p>So the roots are: $z=1,z=-1,z=i,z=-i$</p>
<p>Now use the <a href="https://en.wikipedia.org/wiki/Residue_(complex_analysis)" rel="nofollow">Residue formula</a> for t... |
4,518,953 | <p>Given a nonnegative and convex function <span class="math-container">$f: \mathcal{D} \rightarrow \mathbb{R}_{+}$</span>, where the domain <span class="math-container">$\mathcal{D} \subset \mathbb{R}$</span>. Define the inverse of the function <span class="math-container">$g(x) = \frac{1}{f(x)}$</span>. Is there a co... | Andrew Sotomayor | 745,668 | <p>Hint: It suffices to show that there exists a real number <span class="math-container">$q$</span> such that <span class="math-container">$\lim_{n\rightarrow \infty}{{a_n}\over{n^q}}$</span> converges. Once this is done, call the value <span class="math-container">$p$</span>, and then divide both sides by <span class... |
622 | <p>Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:</p>
<ol>
<li><p>The character table of a group is packs a lot o... | Akhil Mathew | 536 | <ol>
<li><p>The representation theory of finite groups can be used to prove results about finite groups themselves that are otherwise much harder to prove by "elementary" means. For instance, the proof of <a href="http://en.wikipedia.org/wiki/Burnside%27s_theorem">Burnside's theorem</a> (that a group of order $p^a q^b$... |
622 | <p>Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:</p>
<ol>
<li><p>The character table of a group is packs a lot o... | Mariano Suárez-Álvarez | 274 | <p>Groups very, very rarely appear abstractly in (mathematical) <em>nature</em>: when we encounter them, it is almost always in the form of a representation (a linear representation, a permutation representation, a non-linear representation in the form of automorphisms of an algebra, of a variety, of a manifold, of a g... |
622 | <p>Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:</p>
<ol>
<li><p>The character table of a group is packs a lot o... | Bill Dubuque | 242 | <p>Representation theory plays a big role in the group-theoretic approach to special functions.
For example, Willard Miller showed that the powerful Infeld-Hull factorization / ladder method - widely exploited by physicists - is equivalent to the representation theory of four local Lie groups. This lie-theoretic approa... |
1,479,483 | <p>We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see <a href="https://en.wikipedia.org/wiki/Covariance_matrix">Covariance matrix</a>). We also know that every symmetric positive <em>definite</em> matrix is invertible (see <a href="ht... | martini | 15,379 | <p>If the covariance matrix is not positive definite, we have some $a \in \mathbf R^n \setminus \{0\}$ with $\def\C{\mathop{\rm Cov}}\C(X)a = 0$. Hence
\begin{align*}
0 &= a^t \C(X)a\\
&= \sum_{ij} a_j \C(X_i, X_j) a_i\\
&= \mathop{\rm Var}\left(\sum_i a_i X_i\right)
\end{align*}
So there is some... |
1,146,967 | <p>So I think I somewhat understand the type theory of the various lambda calculi in the lambda cube, from the simply typed lambda calculus to the calculus of constructions, but looking at it I'm wondering if I really need lambda abstraction to use it. I understand that you can use these lambda operators to construct q... | Samuel Schlesinger | 275,251 | <p>I think what you're after is, instead of having the following statement:</p>
<p>$(\lambda x \colon t_{dom}.e) \colon (\Pi x \colon t_{dom}. t_{cod})$</p>
<p>we have this one?</p>
<p>$(\lambda x \colon t_{dom}.e \colon t_{cod}) \colon (\lambda x \colon t_{dom}. t_{cod})$</p>
<p>It actually seems like it would alm... |
4,202,516 | <p>Prove gcd(x,y) = gcd(x-y,y) for x > y
Here is my work so far:</p>
<p>If gcd(x,y) = d, then we can denote x = ad , y = bd, so x - y = (ad - bd) = d(a - b), so (x-y) is a multiple of d, so gcd(x -y, y) = gcd(d(a - b), bd) = d, so gcd(x-y, y) = gcd(x,y).</p>
<p>My teacher challenged it by asking to show why gcd(d(a ... | Mark Saving | 798,694 | <p>Suppose <span class="math-container">$z \mid x$</span> and <span class="math-container">$z \mid y$</span>. Then write <span class="math-container">$x = az$</span>, <span class="math-container">$y = bz$</span>. Then <span class="math-container">$x - y = az - bz = (a - b) z$</span>, so <span class="math-container">$z ... |
852,476 | <p>Evaluation of $\displaystyle \int\frac{1}{\prod_{k=0}^{n}(x+k)}dx$</p>
<p>$\bf{My\; Trial::}$We can write</p>
<p>$\displaystyle \int\frac{1}{\prod_{k=0}^{n}(x+k)}dx = \int \frac{1}{(x+0)\cdot(x+1)\cdot (x+2)..........(x+n)}dx$</p>
<p>Now Using Partial fraction.......</p>
<p>$\Rightarrow \displaystyle \frac{1}{(x... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Multiply either sides by $\prod_{r=0}^n(x+r)$</p>
<p>Set $\displaystyle x+s=0\iff x=-s,0\le s\le n$ to find $$1=A_{s+1}\left(\prod_{r=0,r\ne s}^n{r+n}\right)$$</p>
|
141,978 | <p>I'm looking for a simple (or better yet, minimal) example of a planar triangulation of which dual graph (that of the faces) would be "obviously" non-Hamiltonian.</p>
<p>(NB: I once asked the same question for planar triangulations, but I'm now interested in their dual graphs.)</p>
<p>Thanks in advance!</p>
| MJD | 25,554 | <p>There are too many possible answers here because your question is so vague. Here's one possible technique: Select a random number $r$ uniformly between 0 and 1. Calculate $r' = 1+\lfloor100r^2\rfloor$. Then you get $r'$ between 1 and 50 about 70.7% of the time. (Whenver $r$ is less than ${\sqrt 2 \over 2}\approx 0.7... |
141,978 | <p>I'm looking for a simple (or better yet, minimal) example of a planar triangulation of which dual graph (that of the faces) would be "obviously" non-Hamiltonian.</p>
<p>(NB: I once asked the same question for planar triangulations, but I'm now interested in their dual graphs.)</p>
<p>Thanks in advance!</p>
| ely | 26,927 | <p>I think you are overthinking this (unless you need some sort of smoothly decaying distribution that puts more probability mass on the smaller elements). If you just need to distinguish the first 50 elements from the last 50, with no within-set distinctions, just hand-specify a discrete distribution without caring ab... |
127,359 | <p>Consider the following code:</p>
<pre><code>ListPointPlot3D[
Flatten[#, 1] &@
Table[{x, y, 1}, {x, 0, 5, 5/60}, {y, Sin[x], Cos[x] + 3, (
Cos[x] + 3 - Sin[x])/60}], PlotRange -> All]
</code></pre>
<p><a href="https://i.stack.imgur.com/6f8We.png" rel="nofollow noreferrer"><img src="https://i.stack.img... | Julien Kluge | 34,129 | <p>You can restrict your PlotRegion in Plot3D</p>
<pre><code>Plot3D[1,{x,0,5},{y,-1,4},RegionFunction->(Sin[#1]<#2<Cos[#1]+3&)]
</code></pre>
<blockquote>
<p><a href="https://i.stack.imgur.com/Cu6de.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Cu6de.png" alt="enter image descripti... |
2,585,259 | <p>I started with transforming the expression to $$\sqrt{1+\frac{72}{x-36}}$$ and I realised that $\frac{72}{x-36}$ must be one less than a square. Then I don't know what to do. Do I just try some number one less than a square and check if it's divisible by 72? Is there a smarter method I'm missing? I know there isn't ... | idok | 514,894 | <p>The number in the square root is at most $73$, and a square.
So it is one of $1,4,9,16,25,36,49,64$.<br>
Also, the number in the square root is one more from a divisor of $72$, so it is one of $4,9,25$. All those values can be achieved.</p>
|
355,972 | <p>Show that for each $n>1$</p>
<p>$$\sum\limits_{k=1}^{n-1} \frac{(n-1)!}{(k-1)!}S(n,n-k) = (n-1)^n $$</p>
<p>where $S(n,m)$ is the <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow">Stirling number of the second kind</a>. </p>
| Marko Riedel | 44,883 | <p>Suppose we are trying to show that
$$\sum_{k=1}^{n-1} \frac{(n-1)!}{(k-1)!} {n\brace n-k}
= (n-1)^n.$$</p>
<p>Recall the species for set partitions which is
$$\mathfrak{P}(\mathcal{U} \mathfrak{P}_{\ge 1}(\mathcal{Z}))$$
which gives the generating function
$$G(z, u) = \exp(u(\exp(z)-1)).$$</p>
<p>This implies for ... |
1,799,356 | <p>How do you express $-3^4$ to get an answer of $-81$. And how is $(-3)^4$ expressed to get the answer of $81$?</p>
| Jeel Shah | 38,296 | <p>The first one is $$-(3)(3)(3)(3) = -(81) = -81$$</p>
<p>and the second one is </p>
<p>$$
\begin{align}
&\underbrace{(-3)(-3)}(-3)(-3) \\
&= (9)\underbrace{(-3)(-3)} \\
&= (9)\quad(9) \\
&= 81
\end{align}
$$.</p>
<p>Does that help?</p>
|
42,378 | <p>Quick disclaimer: I'm a StackOverflow regular and completely out of my element here...so go easy on me.</p>
<p>Just wanted to ask if anyone know of a simple way to measure or calculate the volume of clothing?</p>
<p>It would be easy to get the clothing's weight (mass) and maybe a little harder to get the "true" di... | Yuval Filmus | 1,277 | <p>Consider how a graceful permutation in $S_n$ looks. The difference $n-1$ can only be realized by $1,n$ (or its reflection). The difference $n-2$ is then realized either by $n-1,1,n$ or by $1,n,2$. In order to realize $n-3$, we need to extend this to either $2,n-1,1,n$ or $n-1,1,n,3$ or $1,n,2,n-1$ or $n-2,1,n,2$.</p... |
479,616 | <p>This's an example: </p>
<p><em>For $u(z_1,z_1,\ldots,z_n)=\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |))$, where $z=(z_1,z_1,\ldots,z_n) \in \Omega=\mathbb{C}^n \setminus\{0\} $. Prove that $u \in MPSH(\Omega)$</em>.</p>
<p>And here's the solution in my textbook:</p>
<p>$\bullet... | mrf | 19,440 | <p>The statement $\partial A \subset \partial G$ makes no sense. ($A \subset \mathbb{C}$ but $G \subset \mathbb{C}^n$). But if $z \in G$ is fixed and $t\in\partial A$, then $tz \in \partial G$, which is probably what is meant.</p>
<p>By assumption, $v \le u$ on $\partial G$, so $v(tz) \le u(tz)$ for $t \in \partial A$... |
2,412,232 | <p>These seems like simple questions, but I cannot seem to find straightforward answers:</p>
<blockquote>
<p>(1) If <span class="math-container">$V$</span> and <span class="math-container">$W$</span> are <em>infinite</em> (possibly uncountable) dimensional vector spaces, does <span class="math-container">$V \bigoplus ... | Joppy | 431,940 | <p>There is no difference between the direct sum and the direct product for finitely many terms, regardless of whether the terms themselves are infinite-dimensional or not. However, they are different in the case of infinitely many terms (and drastically so).</p>
<p>A direct product $\prod_{i = 1}^\infty V_i$ can be t... |
598,265 | <p>I have a few questions about topological spaces which I am currently studying.
First some definitions that I am using:</p>
<p>Definition of subspace topology:
Given a topological space $(X,\tau)$ and a subset $S$ of $X$, the subspace topology on $S$ is defined by $\tau_{S} = \{S \cap U : U \in \tau \}$. </p>
<p>De... | Community | -1 | <p>You're using the word 'subspace' to mean two different things. When we talk about the 'subspace topology', we just mean endowing any subset of the topological <strong>space</strong> $X$ with a topology making it a topological space (hence, subspace). However, the definition of linear subspace is "subspace closed und... |
2,654,271 | <p>A certain amount is invested in scheme A for 6 years which offers simple interest at the rate of x% per annum. The same amount is invested in scheme B for 2 years which offers compound interest compounded annually at the rate of 10% per annum. Interest earned from scheme A is twice of that earned in scheme B. If the... | Parcly Taxel | 357,390 | <p>A is not correct; B is. Statistical regularity – more often called <em>independence</em> – means that</p>
<ul>
<li>the results of the three previous trials do not affect the fourth trial's outcomes</li>
<li>the four prior tosses of the coin in the fourth trial do not affect the last two tosses</li>
</ul>
<p>Theref... |
4,371,933 | <p>Given a set S, and n subsets of S, which may be overlapping or disjoint:
Find up to k elements from each of the n given subsets, such that the union of the elements selected from each subset is maximized.</p>
<p>In other words, if S = {1,2,3,4,5,6,7,8,9,10}, for n=3 the subsets given may look like {1,3,6,7,9,10}, {1... | RobPratt | 683,666 | <p>You can solve the problem via integer linear programming as follows. For <span class="math-container">$i\in [n]$</span>, let <span class="math-container">$S_i$</span> be the given subset. For <span class="math-container">$i\in [n]$</span> and <span class="math-container">$j\in S_i$</span>, let binary decision vari... |
263,233 | <p>I wanted to plot the intersections using ContourPlot3D in order to observe how do solutions look like, however, the following code does not work:</p>
<pre><code>ContourPlot3D[{Sin[a - b] + Sin[a] + Sin[a - d] == 0 &&
Sin[b - a] + Sin[b] + Sin[b - d] == 0 &&
Sin[-a] + Sin[-b] - Sin[-d] == 0 &a... | cvgmt | 72,111 | <p>We can simplify the original equations by <code>GroebnerBasis</code> method.</p>
<pre><code>Clear["Global`*"];
trigs = {Sin[-a] + Sin[-b] - Sin[-d],
Sin[a - b] + Sin[a] + Sin[a - d], Sin[b - a] + Sin[b] + Sin[b - d],
Sin[d - a] + Sin[d - b]};
polys = TrigExpand /@ trigs /. {Sin[a] -> sa, Sin[b] -... |
2,541,366 | <p>If $A=[-4,\infty),B=(-\infty,8]$ and $g:A\to B$ is defined by $g(x)=6-2x-x^2$ for every $x$ in $A$, then check whether $g$ is either one to one or onto.</p>
<p>Here $g(0)=6=g(-2)$ so $g$ is onto one-to-one but show $g$ is either onto or not</p>
| Lorenzo Riva | 506,625 | <p>$g$ is neither one-one nor onto since $6 - 2x - x^2 = 8$ has no solutions in $A,$ and like you mentioned, $g(0) = g(-2).$</p>
|
273,188 | <p>This is my first time asking a question on the Mathematica StackExchange. As the title says I'm looking for a way to add a horizontal line across my bar graph. The code I currently have is as follows:</p>
<pre><code>BarPi = BarChart[dataExperimentalPi,
ChartLabels -> {A, B, C, D, E, F}, PlotRange -> {3, 3.2... | MarcoB | 27,951 | <p>You could use grid lines:</p>
<pre><code>BarChart[
{1, 2, 3}, ChartLabels -> {"A", "B", "C"},
GridLines -> {None, {0.5}},
GridLinesStyle -> Directive[Red, Thick],
Method -> {"GridLinesInFront" -> True}
]
</code></pre>
<p><a href="https://i.stack.imgur.co... |
4,622,161 | <p>I want to solve the system:<span class="math-container">$$\begin{align}
\dfrac{dx}{dt}&= -y \sqrt{x^2+y^2} \\ \\ \dfrac{dy}{dt}&= x \sqrt{x^2+y^2}
\end{align}$$</span>I basically have no idea about how to solve this since this system is not linear. At first glance, I saw <span class="math-container">$x^2+y^... | Dstarred | 955,900 | <h1>Hint:</h1>
<p>Eliminate <span class="math-container">$dt$</span> by <span class="math-container">$\frac{dy}{dt} \div \frac{dx}{dt} = \frac{dy}{dx}$</span>. This is also convenient in removing <span class="math-container">$\sqrt{x^2 + y^2}$</span> on RHS. However, note that <span class="math-container">$\frac{dx}{dt... |
4,622,161 | <p>I want to solve the system:<span class="math-container">$$\begin{align}
\dfrac{dx}{dt}&= -y \sqrt{x^2+y^2} \\ \\ \dfrac{dy}{dt}&= x \sqrt{x^2+y^2}
\end{align}$$</span>I basically have no idea about how to solve this since this system is not linear. At first glance, I saw <span class="math-container">$x^2+y^... | Ninad Munshi | 698,724 | <p>To do polar coordinates, we can use</p>
<p><span class="math-container">$$r = \sqrt{x^2+y^2} \implies \frac{dr}{dt} = \cos\theta\frac{dx}{dt} + \sin\theta\frac{dy}{dt}$$</span></p>
<p><span class="math-container">$$\theta = \tan^{-1}\left(\frac{y}{x}\right) + C \implies \frac{d\theta}{dt} = -\frac{\sin\theta}{r}\fra... |
1,042,507 | <p>$f(x)\in C[a,b]$.For any $g(x) \in C[a,b]$ ,which has the property that $\int_a^b g(x) dx=0$,$\int_a^b f(x)g(x) dx=0$.</p>
<p>Prove:$f(x)\equiv C$,$C$ is a constant.</p>
<p>I haven't any ideas yet.
I'm thinking about using the Fourier series,would it work?</p>
| Kelenner | 159,886 | <p>Put $\displaystyle I=\int_a^b f(x)dx$. Then $\displaystyle g=f-\frac{I}{b-a}$ is such that $\displaystyle \int_a^b g(x)dx=0$. </p>
<p>Hence $\displaystyle \int_a^b f(x)g(x)dx=0$ and :</p>
<p>$$\int_a^b (f(x)-\frac{I}{b-a})^2dx=\int_a^b f(x)g(x)dx-\frac{I}{b-a}\int_a^b g(x)dx=0$$
And we are done. </p>
|
3,712,349 | <p>Knowing <span class="math-container">$P(A|B)$</span> & <span class="math-container">$P(A|C)$</span>, would it be possible to infer <span class="math-container">$P(A|B,C)$</span>?<br>
I looked at this other question: <a href="https://math.stackexchange.com/questions/1120608/any-mathematical-relation-between-these... | Especially Lime | 341,019 | <p>No. For example, let <span class="math-container">$X$</span> be a roll of a standard die, and let <span class="math-container">$B$</span> be the event <span class="math-container">$X<4$</span> and <span class="math-container">$C$</span> be the event that <span class="math-container">$X$</span> is even.</p>
<p>No... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.