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 |
|---|---|---|---|---|
1,786,403 | <p>Consider an ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ with $O$ as the origin. $n$ points denoted as $P_1,P_2,\cdots$ are taken on the ellipse such that angle $P_iOP_{i+1}=\frac{\pi}{n}$ where $i\in(1,n-1)$. Find the value of: $$\sum_{i=1}^n\frac{1}{OP_i^2}$$</p>
<p>I took $P_1$ as $(3,0)$. Let other points be $P_i(3\... | mrprottolo | 84,266 | <p>Your expression is equivalent to
$$x \frac{x^{16}-1}{1+x^2}=7.$$
Then
$$x^{16}=7 \frac{1+x^2}{x}+1$$
notice that $x$ must be positive and that the min of $7 \frac{1+x^2}{x}+1$ on the positive real line is $15$.</p>
|
439,820 | <blockquote>
<p>Is $\large i^{i^i}$ real ? How to find it?</p>
</blockquote>
<p>Thank You!</p>
| Sujaan Kunalan | 77,862 | <p>Wolfram Alpha gives the answer to be:</p>
<p>$0.94715899... + 0.320764449... i$.</p>
<p>Therefore it is not a real number, as the answer has an imaginary component to it.</p>
|
439,820 | <blockquote>
<p>Is $\large i^{i^i}$ real ? How to find it?</p>
</blockquote>
<p>Thank You!</p>
| Empy2 | 81,790 | <p>First find all the values of $i^i$. Then, if $\beta$ is one of those values, a possible value of $i^{i^i}$ is $(e^{i\pi/2})^{\beta}$. Or $(e^{-3i\pi/2})^{\beta}$.</p>
|
1,672,276 | <p>I'm preparing to an exam and trying to solve <span class="math-container">$a_n=2 a_{n-1} - a_{n-2} + 2^n$</span>, where <span class="math-container">$a_0=0$</span> and <span class="math-container">$a_1=1$</span>.</p>
<p>This is my approach:</p>
<blockquote>
<p>Let <span class="math-container">$A(z)=\sum_{n \geq 0} a... | Étienne Bézout | 92,466 | <p>You have to be more careful with your indexing. I prefer to always cast the equation so that it is defined for $n \geq 0$ for greater consistency, but of course it doesn't matter as long as you keep track of your indices.</p>
<p>Anyway, the correct equation would be </p>
<p>$$A(z)(z^2-2z+1) = z + z^2 \frac{4}{1-2z... |
1,672,276 | <p>I'm preparing to an exam and trying to solve <span class="math-container">$a_n=2 a_{n-1} - a_{n-2} + 2^n$</span>, where <span class="math-container">$a_0=0$</span> and <span class="math-container">$a_1=1$</span>.</p>
<p>This is my approach:</p>
<blockquote>
<p>Let <span class="math-container">$A(z)=\sum_{n \geq 0} a... | Semiclassical | 137,524 | <p>While the generating function approach works systematically, you can solve this example by hand directly without too much trouble. First I'll define $a_{-1}=1$ (so that the recurrence holds down to $n=1$) and define $b_n=a_n-a_{n-1}$. Then your recurrence becomes $b_n=b_{n-1}+2^n$ for all positive $n$, and from the ... |
1,080,847 | <p>Let $s(x)=\int_0^xa(t)\,dt$. I wonder:</p>
<p>If $\displaystyle \lim_{x\to\infty}s(x)=L$ is it true that $\displaystyle\lim_{x\to\infty}a(x)=0?$</p>
| Martin R | 42,969 | <p>The statement is false even for continuous non-negative functions.
As an example, consider the function $a: [0, \infty) \to \mathbb R$ whose
graph consists of triangles with baseline $[n, n + 1/n^2]$ and height $2$
for each $n \in \mathbb N$ and is zero otherwise.
Then $\int_0^\infty a(x)dx = \sum_{n=1}^\infty 1/n^2... |
2,687,167 | <p>Is there an inverse function of $f(x) = x^2 + \pi\cos x$?</p>
<p>I don't think there is because of the $x^2$, but I don't know how to prove it.</p>
| user | 505,767 | <p>It should be declared over which interval we are considering the function.</p>
<p>Indeed f(x) is not invertible as function from $\mathbb{R}\to\mathbb{R}$ but it is invertible, for example, if we assume a restriction $[a,+\infty)\to[b, +\infty)$ indeed</p>
<ul>
<li>$f’(x)=2x-\pi\sin x>0$ for some $x\ge a$ and $... |
2,075,664 | <p>In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits.</p>
<p><strong>My attempt</strong>:We should find number of two pairs that can be digits of this number for choosing the place of digits we have $*8$ (We should notice we cant have $0$ in the beginning.But the biggest ... | Joffan | 206,402 | <p>The two digit sum $T$ can be a number between $0$ and $18$ (we can worry about zeroes presently). For values of $T$ up to $9$, we can $T$ between digits in $T{+}1$ ways, and for values from $9$ to $18$ we can split them in $18-T+1=19-T$ ways. This applies independently to the first two digits and the last, except th... |
21,048 | <p>When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.</p>
<p>Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which "feel" ve... | Dave L Renfro | 745 | <p>Below are two elementary/expository papers that I know for generating strange examples, although how interesting they are will vary from person to person (however, a connection with the special theory of relativity does exist), and unless some additional conditions are imposed I don't believe they're finite-dimensio... |
21,048 | <p>When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.</p>
<p>Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which "feel" ve... | Michael Seifert | 13,965 | <p>The set of solutions to a system of linear homogeneous ODEs is a vector space, and the dimension of this vector space is equal to the total order of the system. The idea that every solution is the linear combination of some "special" set of solutions is just a statement that we can always find a basis of ... |
21,048 | <p>When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.</p>
<p>Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which "feel" ve... | Pteromys | 16,332 | <p>This is the example that made me realize the importance of abstract linear algebra as an undergraduate: the edge spaces and the vertex space of a graph (<a href="https://en.wikipedia.org/wiki/Edge_and_vertex_spaces" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Edge_and_vertex_spaces</a>). The edge graph ... |
293,444 | <p>I have a Boolean expression. we'll call it <code>F</code>.</p>
<p>for instance, <code>F = ab' + ad + c'd + d'</code>.</p>
<p>Assuming I did all the necessary steps too get <code>F</code> complement , i.e. <code>F'</code>.</p>
<p>I got: <code>F' = b'd + ac'd'</code>.</p>
<p>How do I get the <code>Product of sums<... | KhasSravya | 323,090 | <p>If the boolean expression is AB+BC.Then the product of sum expression is (A+B)(B+C).To find this, the complement of sum product expression is the product of sum expression. (AB+BC)'=A'B'+B'C'
By applying DEMORGANS LAW
=(A'+... |
1,145,378 | <p>I believe a function has to be bijective to be invertible, in fact Wikipedia says that an invertible function is another name for a bijection.
<a href="https://en.wikipedia.org/wiki/Bijection" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Bijection</a></p>
<p>But in answers to the question <a href="https:/... | Community | -1 | <p>Hint: if function $ f : A \rightarrow B $ was not surjective, how would we define $ f^{-1} : B \rightarrow A $ for an element that was not in the image of $ f $?</p>
|
1,217,316 | <p>Prove that equation
$$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots</p>
<p>and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$
has two real roots</p>
<blockquote>
<p>I think that:</p>
<p>$$x^{4k+2}+x^{4k+1}-x^{4k}-x^{4k-1}+x^{4k-2}+x^{4k-3}-..+x^2+x-1=0$$</p>
<p>and
$$x^{4k+2}-x^{4k+1}+x^{4k}+x^{4k-1}-x^{4k-2}-x^{4... | abel | 9,252 | <p>since $ f(\pm \infty) = \infty, f(0) = -1$ shows that $f$ has one positive root and one negative root. so $f$ has at least two real roots.</p>
|
1,217,316 | <p>Prove that equation
$$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots</p>
<p>and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$
has two real roots</p>
<blockquote>
<p>I think that:</p>
<p>$$x^{4k+2}+x^{4k+1}-x^{4k}-x^{4k-1}+x^{4k-2}+x^{4k-3}-..+x^2+x-1=0$$</p>
<p>and
$$x^{4k+2}-x^{4k+1}+x^{4k}+x^{4k-1}-x^{4k-2}-x^{4... | Domenico Vuono | 227,073 | <p>HINT: you can write this equation:
$x^6+x^5-x^4-x^3=-x^2-x+1$
$y=x^6+x^5-x^4-x^3$
$y=-x^2-x+1$
and draw two functions noting that there are two points where two fnctions intersect.</p>
|
175,764 | <p>I have the latest version of Wolfram Workbench plugin installed on Eclipse. But the <code>Documentation Tools</code> palette contained in this does not have all the styles that are part of the standard documentation. Is there a way to get the latest version of <code>Documentation Tools</code> in use by people at Wol... | Szabolcs | 12 | <p>The current version of the Workbench / Documentation Tools still produces the style used in Mathematica 11.0 and earlier.</p>
<p>This style can also be displayed (mostly fine) in Mathematica 11.1 and later. The few problems (such as large font in See Also links) can be corrected by post-processing the generated doc... |
1,190,505 | <p>I am trying to find the points at which two circles intersect. The circles I am working with are: $$(1) \qquad x^2+y^2 = \frac{9}{4}$$ $$(2) \qquad (x-2)^2+y^2=\frac9 4$$ I am following this answer: <a href="https://math.stackexchange.com/a/418932/136870">https://math.stackexchange.com/a/418932/136870</a>. $$$$ W... | Brian M. Scott | 12,042 | <p>Instead of following a recipe, look at the geometry: in this problem it’s exceptionally simple.</p>
<p>We can tell immediately from the equations that the centre $C_1$ of the first circle is the origin, $\langle 0,0\rangle$, and the centre $C_2$ of the second is $\langle 2,0\rangle$. Moreover, each circle has radiu... |
2,586,663 | <p>It seems to me, by trial and error, that this series converges to $1$. Is this true? How do I prove it either way?</p>
| user | 505,767 | <p>You can rewrite as</p>
<p>$$\sum_{n=2}^{+\infty} \frac{\log n}{2^{n-1}}$$</p>
<p>which clearly converges e.g. for ratio comparison with $\sum\frac{1}{n^2}$.</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="... | rb1957 | 472,677 | <p>if the square side is x, then the width of area 2 is 4/x and the reminder of that side is (x^2-4)/x. Similiarly the width of the 4 area is 8/x, and the other side of the 3 area is (x^2-8)/x.
Then from the 3 area (x^2-4)/x * (x^2-8)/x = 3*2. Let A = x^2, then (A-4)(A-8) = 6A. A^2 -18A +32 = 0 ... A = 16, and blue ... |
16,946 | <p>Polya Enumeration Formula gives us 6 equivalence classes of 2-colorings of a square. But, in the Polya coloring, the following 2 colorings belong to 2 different equivalence classes:</p>
<pre><code>00
01
</code></pre>
<p>and</p>
<pre><code>11
10
</code></pre>
<p>(0 and 1 are the 2 colors.)</p>
<p>What is the the... | Kristal Cantwell | 1,098 | <p>There is a paper <a href="http://alexandria.tue.nl/repository/freearticles/597547.pdf" rel="noreferrer">A survey of generalizations of Pólya's enumeration theorem</a> which discusses a generalization of Polya's theorem involving colors. It gave <a href="http://alexandria.tue.nl/repository/freearticles/597529.pdf" re... |
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... | Qiaochu Yuan | 232 | <p>The following proof can be extracted from <a href="https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate" rel="nofollow noreferrer">Erdős' proof of Bertrand's postulate</a> (although perhaps this argument should be credited to Chebyshev). We need the following two lemmas from that page.</p>
<p><strong>Lemma ... |
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... | Pete L. Clark | 299 | <p>When I taught undergraduate number theory I subjected my students to a barrage of proofs of the infinitude of the prime numbers: see <a href="http://alpha.math.uga.edu/%7Epete/4400primes.pdf" rel="nofollow noreferrer">these lecture notes</a>. I gave eight proofs altogether. Of course by now the list which has been... |
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... | Milo Brandt | 174,927 | <p>One can prove directly that <a href="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes" rel="nofollow noreferrer">the Sieve of Eratosthenes</a> produces an infinite sequence of primes (and produces every prime). This is a worthwhile proof to record since, while it lacks much novel content, it is reassuring that th... |
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... | Sundara Narasimhan | 630,304 | <p>Here is a proof which essentially is in the same spirit as that of Euclid, but I have not seen this written up somewhere. So I mention it here.</p>
<p>Let <span class="math-container">$p_1,p_2,\ldots p_n$</span> be the only prime numbers. Then consider the number <span class="math-container">$$s=p_2p_3\cdots p_n+p_1... |
43,841 | <p>I have a very large data set (around 3000) that looks like:</p>
<pre><code>{{1749.08, 580.}, {1749.17, 626.}, {1749.25, 700.}, {1749.33,
557.}, {1749.42, 850.}, {1749.5, 835.}, {1749.58, 948.}, {1749.67,
663.}....
</code></pre>
<p>The first number is the year plus the month/12. So I want to iterate over all ... | halirutan | 187 | <p>One way is to gather the years by using <code>Floor</code> and then adding up the values for each year. </p>
<pre><code>data = {{1749.08, 580.}, {1749.17, 626.}, {1749.25,
700.}, {1749.33, 557.}, {1749.42, 850.}, {1749.5, 835.}, {1749.58,
948.}, {1749.67, 663.}, {1750.00, 653.}};
Total[Part[#, All, 2]] &a... |
3,731,916 | <p>Can somebody find an elementary proof of the following identity:</p>
<p><span class="math-container">$$
\arcsin ( 1 - 2x) + 2 \arcsin(\sqrt{x}) = \frac\pi2
$$</span></p>
<p>I noticed it while solving the following integral:</p>
<p><span class="math-container">$$ I = \int \frac{\sqrt{x}}{\sqrt{1 - x}} = -2 \sqrt{1 - ... | Franklin Pezzuti Dyer | 438,055 | <p>Here’s an elementary proof. Let <span class="math-container">$x=\sin^2\phi$</span>. Then, since <span class="math-container">$\cos(2\phi)=1-2\sin^2\phi$</span>, we have
<span class="math-container">$$\arcsin(1-2x)=\arcsin(1-2\sin^2\phi)=\arcsin(\cos(2\phi))=\frac{\pi}{2}-2\phi$$</span>
and
<span class="math-containe... |
751,986 | <p>I've done a bit of fiddling and I believe the answer to be $(p-1)$ but I don't know how to prove it. The first part of the question asks what $1^p + 2^p + \cdots + (p-1)^p$ is modulo $p$ and it comes out as $1 + 2 + \dots + (p-1) \equiv 0 \ (mod \ p)$ but I'm not sure if it's related or not. Other than this, all I h... | DonAntonio | 31,254 | <p>Hints:</p>
<p>$$1+2+\ldots+p-1=\frac{(p-1)p}2\implies (p-1)!=p-1\pmod{\frac{(p-1)p}2}\iff$$</p>
<p>$$(p-1)\left((p-2)!-1\right)=0\pmod{\frac{(p-1)p}2}$$</p>
<p>But Wilson's Theorem gives us</p>
<p>$$(p-1)!=-1\pmod p\implies(p-2)!=\frac{(p-1)!}{p-1}=\frac{-1}{-1}=1\pmod p\implies\ldots$$</p>
|
4,230,752 | <p>My textbook has two problems: <span class="math-container">$$\frac{d}{dx}\int^x_2\sin(t^2)dt$$</span> and <span class="math-container">$$\frac{d}{dx}\int^{\cos(x)}_0e^{t^2}dt$$</span></p>
<p>It says that the first derivative is (obviously) equal to <span class="math-container">$\sin(x^2)$</span> and then gives the m... | Kman3 | 641,945 | <p>Both integrals should be differentiated by <span class="math-container">$x$</span> and not by <span class="math-container">$t$</span>. With this in mind:</p>
<p>We know from the fundamental theorem of calculus (FTC) that if
<span class="math-container">$$F(x)=\int_a^x f(t) \ dt$$</span></p>
<p>and <span class="math-... |
1,818,276 | <blockquote>
<p>Evaluation of <span class="math-container">$$\lim_{x\rightarrow 1}\frac{1-x^x}{1-x^2}$$</span></p>
<p>Without using L Hospital Rule and Series expansion.</p>
</blockquote>
<p>I have solved using L Hopital Rule, and getting answer <span class="math-container">$\displaystyle = \frac{1}{2}\;$</span>. But ... | Community | -1 | <p>By the generalized binomial theorem,</p>
<p>$$(1+t)^{1+t}=1+(1+t)t+(1+t)t\frac{t^2}2+(1+t)t(t-1)\frac{t^3}{3!}+\cdots.$$</p>
<p>Then</p>
<p>$$\lim_{t\to0}\frac{(1+t)t+(1+t)t\dfrac{t^2}2+(1+t)t(t-1)\dfrac{t^3}{3!}+\cdots}{2t+t^2}=\lim_{t\to0}\frac{1+t(\cdots)}{2+t}=\frac12.$$</p>
|
1,101,865 | <p>Let $f: \mathbb{Q} \to \mathbb{Q}$ be defined by $f(x) = 3x - 1$.</p>
<blockquote>
<p>(a) Find a binary operation $\ast$ on $\mathbb{Q}$ such that $f:(\mathbb{Q},+) \to (\mathbb{Q},\ast)$ is an isomorphism.</p>
</blockquote>
<p>Here's my work. Is it correct?</p>
<p>Since $f(x)$ is bijective, we just need to fin... | egreg | 62,967 | <p>As Meelo says, your proof is involved, but correct. However, it all boils down to the following general statement.</p>
<p>In your case, $f(x)=3x-1$ is a bijection and you don't need to actually use it. The new operation is obtained by doing the original one on the preimages of the elements and then going back with ... |
759,838 | <p>How to evaluate the integrals of</p>
<p>(a) $(\ln(x))^n$ </p>
<p>(b) $x^ne^{ax}$ where $a$ is a constant</p>
<p>By reduction formula </p>
| Community | -1 | <ul>
<li><p>$$\int(\ln(x))^ndx=x(\ln(x))^n-\int xd((\ln(x))^n)=x(\ln(x))^n-\int x\left(\dfrac{n(\ln(x))^{n-1}}{x}\right)dx\\
\implies \int(\ln(x))^ndx=x(\ln(x))^n-\int {n(\ln(x))^{n-1}}dx$$</p></li>
<li><p>$$\int x^ne^{ax}dx=\dfrac{1}{a}\int x^nd(e^{ax})=\dfrac{1}{a}e^{ax}x^n-\dfrac{1}{a}\int e^{ax}(nx^{n-1})dx$$</p></... |
1,905,891 | <p><a href="https://i.stack.imgur.com/IDrI5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IDrI5.png" alt="enter image description here"></a></p>
<p>Doubt is that If $\gamma < \alpha$ then how does it follow that $\gamma$ doesnot belong to B and how does it follows that $\alpha \leq x $ for ever... | lEm | 319,071 | <p>By definition of supremum, if $\gamma < \alpha$, then $\gamma$ is not an upper bound of $L$. Since every element of $B$ is an upper bound of $L$, so $\gamma$ must not be in $B$.</p>
<p>Since $\alpha$ is an upper bound of $L$, if $\beta \in L$, then we have a contradiction that $\beta \leq \alpha $. So $\beta$ is... |
3,059,572 | <p>I've been practicing horizontal asymptotes and I came across a problem that I do not understand.</p>
<p>I understood why <span class="math-container">$\lim_{x \rightarrow \infty}\frac{x - 2}{\sqrt{x^2 + 1}} = 1$</span>, but i couldn't understand why <span class="math-container">$\lim_{x \rightarrow -\infty}\frac{x... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Use that <span class="math-container">$$\frac{x}{\sqrt{x^2}}=\frac{x}{|x|}$$</span></p>
|
2,026,020 | <p>In how many 7-digit numbers $5$ threes came one after another?</p>
<p>I put one $3$ for all $5$ threes after thiswe know that this is the case of three digit numbers that a $3$ always exist now For easy solving I calculate the case that never a $3$ exist.Now by calculating I get this answer:</p>
<p>$900-8*9*9=900-... | John | 7,163 | <p>I'm thinking the book is answering a different question. (The $648$ in your answer is very close to $651$, the answer the book gives, so maybe something is getting lost in translation.)</p>
<p>However, the $252$ you have doesn't quite account for everything. You're using a single $3$ to represent the five $3$s in... |
2,026,020 | <p>In how many 7-digit numbers $5$ threes came one after another?</p>
<p>I put one $3$ for all $5$ threes after thiswe know that this is the case of three digit numbers that a $3$ always exist now For easy solving I calculate the case that never a $3$ exist.Now by calculating I get this answer:</p>
<p>$900-8*9*9=900-... | Barry Cipra | 86,747 | <p>The string of five consecutive $3$'s can start with the first, second, or third digit. In the first case, the final two digits can be anything, so it accounts for $10\cdot10=100$ numbers. In the second case, the first digit can be anything but a $0$ or a $3$ while the final digit can be anything, so it accounts fo... |
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... | J. M. isn't a mathematician | 7,934 | <p>Helge presented the <em>continuous</em> case in his answer; for the purposes of data fitting in statistics, one usually deals with <em>discrete</em> orthogonal polynomials. Associated with a set of abscissas $x_i$, $i=1\dots n$ is the discrete inner product</p>
<p>$$\langle f,g\rangle=\sum_{i=1}^n w(x_i)f(x_i)g(x_i... |
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... | Sam Hopkins | 25,028 | <p>You can use linear algebra over $\mathbb{F}_2$ to solve the game "Lights Out": <a href="http://en.wikipedia.org/wiki/Lights_Out_%28game%29">http://en.wikipedia.org/wiki/Lights_Out_%28game%29</a></p>
|
2,639,340 | <p>Let $\mathcal{L}$ be the Laplacian matrix of a connected, undirected and unweighted graph $\mathcal{G}$ with $n$ vertices, and let $\Delta \in \mathbb{R}^{n\times n}$ be a diagonal matrix with at least one nonzero entry (for instance $\Delta=\mbox{diag}\{1,0,\cdots,0\}$).</p>
<p>Numerically, I can see that</p>
<p>... | orangeskid | 168,051 | <p>More or less like the solution above, but with less theory. </p>
<p>If $a_{ij}$ is the weight between the vertices $i$, $j$
$$x^t \cdot L \cdot x = \sum_{i<j}a_{ij}(x_i-x_j)^2 \ge 0$$
(we knew that).
We can also see for what vectors $x$ we have equality: they lie in the subspace generated by $(x^C)$, where for ... |
641,490 | <p>Can someone please explain how to complete the square? Is there a specific formula you have to put the given equation in, or something? Every time I search how to complete the square each equation seems to employ a different method and it just confuses me. I'm not even understanding the concept at all. Can someone p... | egreg | 62,967 | <p>A slightly different view, without dividing by $a$, but multiplying by $4a$, so the degree two term is easily a square and we also get the degree one term in good form to be twice a product:
\begin{gather}
ax^2+bx+c=0\\[2ex]
4a^2x^2+4abx+4ac=0\\[2ex]
4a^2x^2+4abx+b^2-b^2+4ac=0\\[2ex]
(2ax+b)^2=b^2-4ac
\end{gather}</... |
3,598,294 | <p>How to prove that </p>
<p><span class="math-container">$$\large\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+ 1)}\right)= H_{\frac1a}, \quad a\in \mathbb{R},\ |a|>1$$</span></p>
<p>This question is a formulated form of <a href="https://math.stackexchange.com/questions/3597699/find-the-limit-of-an-infi... | Dr. Wolfgang Hintze | 198,592 | <p>The direct way.</p>
<p>Renaming <span class="math-container">$x\to m$</span> we can write</p>
<p><span class="math-container">$$m \sum _{n=1}^m \log \left(\frac{1}{m n (a n+1)}+1\right)=m \log \left(\prod _{n=1}^m \left(\frac{1}{m n (a n+1)}+1\right)\right)$$</span></p>
<p>Now</p>
<p><span class="math-container"... |
4,623,057 | <p>Given the primal
<span class="math-container">$$\max z= 5x_1-4x_2+3x_3$$</span>
subject to</p>
<p><span class="math-container">$2x_1+x_2-6x_3=20$</span></p>
<p><span class="math-container">$ 6x_1+5x_2+10x_3\leq 76$</span></p>
<p><span class="math-container">$8x_1-3x_2+6x_3\geq 50$</span></p>
<p>with <span class="mat... | FD_bfa | 1,007,603 | <p>The concept was <a href="https://mathshistory.st-andrews.ac.uk/Miller/mathword/a/" rel="nofollow noreferrer">first used</a> by the German Mathematician E.Harnack in 1884 and it was named by the Italian Mathematician G.Vitali in 1905. It is not explicitly stated as to why Vitali chose this particular name, although t... |
4,623,057 | <p>Given the primal
<span class="math-container">$$\max z= 5x_1-4x_2+3x_3$$</span>
subject to</p>
<p><span class="math-container">$2x_1+x_2-6x_3=20$</span></p>
<p><span class="math-container">$ 6x_1+5x_2+10x_3\leq 76$</span></p>
<p><span class="math-container">$8x_1-3x_2+6x_3\geq 50$</span></p>
<p>with <span class="mat... | Alp Uzman | 169,085 | <p>There is also a more algebraic perspective, that at least contextualizes the notation it seems to me. Although admittedly there is no new content in what I write below.</p>
<hr />
<p>Namely, that given a <span class="math-container">$\sigma$</span>-algebra of subsets, the collection of its <span class="math-containe... |
281,047 | <p>Given an integer $n$, is there a <em>deterministic algorithm</em> to find in poly$(\log n)$ time an integer $q$, $n < q< n^{c}$, such that $gcd(q,n!)=1$? Here $c>1$ is some fixed constant. </p>
<p>Obviously, a small prime $n< q < 2n$ works. But those are <a href="https://terrytao.wordpress.com/2010... | Dan Brumleve | 2,003 | <p>If we could find a number co-prime to $2^n!$ in $n^{O(1)}$ time, or even just a number divisible by at least one prime greater than $2^n$, we could factor it to find such a prime. This would constitute a solution to the <a href="http://michaelnielsen.org/polymath1/index.php?title=Finding_primes" rel="nofollow noref... |
3,321,963 | <blockquote>
<p>Let <span class="math-container">$x\geq 1, n\in N$</span>
Show that
<span class="math-container">$$
\prod_{i=1}^n(x+n-i)\geq x^{n+1}
$$</span></p>
</blockquote>
<p>I can see that the product is greater or equal to <span class="math-container">$x^n$</span>, but I cannot understand where power <spa... | Arthur | 15,500 | <p>If <span class="math-container">$n$</span> and <span class="math-container">$a$</span> have any common factors, then <span class="math-container">$n$</span> and <span class="math-container">$ab$</span> have the same factors, and possibly more, in common. Say <span class="math-container">$\gcd(n, ab) = d>1$</span>... |
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>
| pathfinder | 23,431 | <p>Consider the more general equation:</p>
<p>$$x^2+bx+c=0.$$</p>
<p>The quadratic formula gives:</p>
<p>$$x=\frac{-b\pm\sqrt{b^2-4c}}{2}.$$</p>
<p>Now substitute $b=3+5i$ and $c=7+11i$ and simplify. This gives</p>
<p>$$x=\frac{-(3+5i)\pm(-44-14i)^{1/2}}{2}.$$</p>
<p>We can write this as</p>
<p>$$x=\frac{-(3+5i)... |
1,035,983 | <p>The identity is: </p>
<p>$$\sum_{k=0}^{m} (-1)^{k} {{n} \choose {k}}{{n-k}\choose{m-k}} = 0$$</p>
<p>I'm not even sure where to begin. Does anyone have any suggestions?</p>
| arindam mitra | 30,981 | <p>First see that the identity holds only when $m>0$. For $m=0$, L.H.S equates to $1$. </p>
<hr>
<p>Proof by <strong>Double Counting</strong></p>
<p>Q: Count the number of ways of selecting m red balls from a set of n blue balls and r red balls. Balls are distinct and numbered 1 to n and 1 to r respectively.</p>
... |
850,260 | <p>Are there books or papers where economics is formalized and studied very rigorously? I am very interested in this topic. I would preferably like free online books and/or papers, but that is not necessary.</p>
| Bradley Anderson | 174,403 | <p>Axiomatic Theory of Economics, 1999, by Victor Aguilar</p>
<p>This is a short 26-page simplified exposition written at the undergraduate level:</p>
<p><a href="http://axiomaticeconomics.com/Axiomatic_Economics_Article_by_Aguilar.pdf" rel="nofollow">http://axiomaticeconomics.com/Axiomatic_Economics_Article_by_Aguil... |
3,595,184 | <p>I'm being asked to prove the following equality</p>
<p><span class="math-container">$$\sin(\alpha)=\frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}$$</span></p>
<p>and I support the idea that they are <strong>not</strong> equal (the rest of my class seems to desagree with me which made me doubt a little).</p>
<p>The ... | Behnam Esmayli | 283,487 | <p>For angles <span class="math-container">$0 < \alpha < \pi/2 $</span> here is a geometric proof. The pictured right-angled triangle exists. Now what is the sine of <span class="math-container">$\alpha$</span> in that picture?
<a href="https://i.stack.imgur.com/Bkh1Y.png" rel="nofollow noreferrer"><img src="http... |
395,600 | <blockquote>
<ul>
<li>Sine <span class="math-container">$\theta$</span> = opposite/hypotenuse</li>
<li>Cosine <span class="math-container">$\theta$</span> = adjacent/hypotenuse</li>
<li>Tangent <span class="math-container">$\theta$</span> = opposite/adjacent</li>
</ul>
</blockquote>
<p>In order to calculat... | George V. Williams | 54,806 | <p>Calculators either use the Taylor Series for <span class="math-container">$\sin / \cos$</span> or the CORDIC algorithm. A lot of information is available on Taylor Series, so I'll explain CORDIC instead.</p>
<p>The input required is a number in radians <span class="math-container">$\theta$</span>, which is between <... |
421,844 | <p>I wanted to know, given quadrilateral ABCD such that $AB^2+CD^2=BC^2+AD^2$ , prove that $AC⊥BD$ .</p>
<p>Help. </p>
<p>Thanks.</p>
| Shaswata | 68,110 | <p>Here's a solution using vector geometry.</p>
<p>According to the Problem,
$$(\vec{B}-\vec{A})^{2}+(\vec{D}-\vec{C})^{2}=(\vec{C}-\vec{B})^{2}+(\vec{D}-\vec{A})^{2}$$</p>
<p>$$\rightarrow A^{2}+B^{2}+C^{2}+D^{2}-2\vec{A}\cdot \vec{B}-2\vec{C}\cdot\vec{D}=A^{2}+B^{2}+C^{2}+D^{2}-2\vec{C}\cdot \vec{B}-2\vec{D}\cdot\v... |
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 $A... | kodlu | 66,512 | <p>Cyclic Gray codes will work, not the only solution, though.</p>
<p>0,1</p>
<p>0,1,1,0 [reverse] and augment by first zero and one on the left 00,01,11,10</p>
<p>write as sequence (I wrote 2 periods since you need to slide windows of length 2 along to see all tuples</p>
<p>0011,0011</p>
<p>000
001
010
011
111
11... |
2,426,562 | <p>I'm aware of other questions regarding "increasing" mathematical intuition, but I wanted to explain my own context, since that way any advice posted on this question feed will be specific to myself.</p>
<p>I'm 16. I'm a (new) Sixth Former (roughly college level) student and has just today had my fifth Further Pure ... | Ethan Bolker | 72,858 | <p>I think that particular question isn't a good one to trigger the need for developing intuition, but yours is a good question.</p>
<p>I strongly recommend George Polya's <em>Mathematics and Plausible Reasoning, Volume 1:
Induction and Analogy in Mathematics</em> . I first encountered it when I was at about your age.... |
3,616,484 | <p>Let <span class="math-container">$I = \{f(x) \in \mathbb{Z}[x] : \text{$f(n) \equiv 0 \pmod2$ for all $n \in \mathbb{Z}$}\}$</span>.
Prove that <span class="math-container">$I$</span> is a finitely-generated ideal of <span class="math-container">$\mathbb{Z}[x]$</span> by finding a finite set of generators for <span... | Doug M | 317,162 | <p><span class="math-container">$P\pmatrix {2\\-1}$</span> projects the vector onto itself.</p>
<p>Since <span class="math-container">$\pmatrix {1\\2}$</span> is orthogonal to <span class="math-container">$\pmatrix {2\\-1}$</span> </p>
<p><span class="math-container">$P\pmatrix {1\\2} = 0$</span></p>
<p>In terms of... |
914,349 | <p>Find the equation of the line that satisfies the given conditions:</p>
<p>Through $(-1,2)$; parallel to the line $x=5$</p>
<p>I know that the equation of this line is $x=-1$ because $x=5$ is a vertical line and any line parallel to it would be vertical as well. So the line runs vertically along $x=-1$. </p>
<p>B... | John Joy | 140,156 | <p>Consider the line$$x=5$$
$$0=x-5$$
$$0y=x-5$$
$$y=\frac{x}{0}-\frac{5}{0}$$
$$y=\infty x-5\infty$$
Now try to determine the slope of a line perpendicular to this line.
$$m_\perp = -\frac{1}{m}=-\frac{1}{\infty}=0$$
Then try to find a line that passes through $(-1,2)$ with a slope of $m_\perp=0$
$$m_\perp=0=\frac{y-2... |
493,756 | <p>I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's <em>Understanding Analysis</em>. In section 2.7, he has the following exercise:</p>
<p><strong>Exercise 2.7.5</strong> (a) Show that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ also converges absolutely. Does this proposition hold w... | MathBS | 515,312 | <p>Apply limit test, observe <span class="math-container">${a_n^2\over |a_n|}\to0$</span>. Now <span class="math-container">$\sum |a_n|$</span> converges <span class="math-container">$\implies\sum a_n^2$</span> converges.</p>
|
2,531,089 | <p>So I have the following values:</p>
<p>$X_i ~ N(\mu=9.6, \sigma=1.6)$</p>
<p>Samples = 8.4, 7.8, 10.3, 8.7, 10.7, 10.0, 8.8, 12.1</p>
<p>X̄ = 9.6</p>
<p>α = 0.05</p>
<p>I should also mentioned that I have the following hypothesis test: </p>
<p>$H_0: μ \le 8$</p>
<p>$H_1: μ > 8$</p>
<p>I want to find the n... | jgsmath | 455,126 | <p>Write $2x + 1 = (x+1)^2 - x^2$ and use the factorization for $a^2 - b^2$ repeatedly. </p>
|
2,531,089 | <p>So I have the following values:</p>
<p>$X_i ~ N(\mu=9.6, \sigma=1.6)$</p>
<p>Samples = 8.4, 7.8, 10.3, 8.7, 10.7, 10.0, 8.8, 12.1</p>
<p>X̄ = 9.6</p>
<p>α = 0.05</p>
<p>I should also mentioned that I have the following hypothesis test: </p>
<p>$H_0: μ \le 8$</p>
<p>$H_1: μ > 8$</p>
<p>I want to find the n... | achille hui | 59,379 | <p>We can rewrite the product as a telescoping one. The end result is:
$$\prod_{k=0}^6 \left((1+x)^{2^k} + x^{2^k}\right) =
\prod_{k=0}^6 \frac{(1+x)^{2^{k+1}} - x^{2^{k+1}}}{(1+x)^{2^k} - x^{2^k}}
= \frac{(1+x)^{2^7} - x^{2^7}}{(1+x)^{2^0} - x^{2^0}} = (x+1)^{128} - x^{128}$$</p>
|
3,749,518 | <p>A very quick question. I was looking at the limit <span class="math-container">$$\lim_{x\to1} \frac{\ln(1-x)}{\tan\left(\frac{\pi x}{2}\right)}$$</span> and started to think about how this limit exists. I know by using L'Hôpital's rule the answer is <span class="math-container">$0$</span>.</p>
<p>But how does the li... | DodoDuQuercy | 718,230 | <p>When considering the limit as <span class="math-container">$x$</span> goes to <span class="math-container">$1$</span> of a quantity containing <span class="math-container">$\ln (1-x)$</span>, it is tacitly understood that <span class="math-container">$x$</span> is below <span class="math-container">$1$</span> when t... |
3,999,029 | <p>Hi guys I came across a few neat formulas to calculate the number of intergers in an interval divisable by another integer!
I was wondering why does it work? what's the mathematics behind them?</p>
<p>If for example I want to find the number of integers between 99 and 993 that are divisable by 2
I can do the followi... | DatBoi | 734,160 | <p>This formula is based on arithmetic progression.</p>
<p>Consider the interval <span class="math-container">$[a,b]$</span> and suppose <span class="math-container">$x,y$</span> are respectively the smallest and largest numbers divisible by <span class="math-container">$k$</span> in this interval.</p>
<p>Note that the... |
727,064 | <p>I've been given a problem regarding the solution set of a matrix:</p>
<pre><code> Let A = 1 2 -5 1
-1 3 0 , b = 0
1 -2 k h
</code></pre>
<p>Find the value(s) of <em>k</em> and <em>h</em> such that the matrix equation Ax = b has</p>
<p>(a) a unique solution
(b) infin... | Brad S. | 133,841 | <p>Let's start by forming the augmented matrix $[A|b]$ and doing elementary row operations to get the $A$ part into row echelon form.
$$\left[\begin{array}{ccc|c}1&2&-5&1\\-1&3&0&0\\1&-2&k&h\end{array}\right] \Rightarrow_{1} \left[\begin{array}{ccc|c}1&2&-5&1\\0&5&am... |
1,376,285 | <p>Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.</p>
| bob the pie | 196,035 | <p>Given any irrational number r, and rational number k, since k=r+(k−r), the sum of two irrational numbers, for any rational k, f(x)=f(x+k). As this property applies to all real numbers, rational or not, the function must be equal for all real numbers, and henceforth, it is constant. </p>
|
4,286,907 | <blockquote>
<p>Find Distance between a Point <span class="math-container">$(-4,2)$</span> and a Line having equation <span class="math-container">$3x−y−10=0$</span></p>
</blockquote>
<p>My attempt:</p>
<p>I used a graphing calculator to graph the equation <span class="math-container">$3x−y−10=0$</span> and the point <... | huzaifa abedeen | 845,875 | <p><a href="https://i.stack.imgur.com/kM7to.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kM7to.png" alt="enter image description here" /></a></p>
<p>By the distance formula, I found the length of <span class="math-container">$AB$</span> as <span class="math-container">$\dfrac{10}{3} \sqrt{10}$</sp... |
3,753,748 | <p>If a set F is closed and <span class="math-container">$E \subset F$</span>, then <span class="math-container">$\bar{E} \subset F$</span>. Note <span class="math-container">$\bar{E}$</span> is the closure of <span class="math-container">$E$</span>. My question is why is <span class="math-container">$\bar{E}$</span> s... | hamam_Abdallah | 369,188 | <p>Take <span class="math-container">$$F=[-1,1]$$</span>
<span class="math-container">$$E_1=(-1,1)$$</span>
<span class="math-container">$$E_2=(0,1)$$</span>
<span class="math-container">$$E_3=(-1,1)\cap \Bbb Q$$</span></p>
<p>then</p>
<p><span class="math-container">$$\bar{E_1}=[-1,1]=F$$</span>
<span class="math-cont... |
239,450 | <p>I am trying to evaluate a function, which has simplified form as below.</p>
<pre><code>f[a_] := 5*a
X[a_, b_]:=Module[{A= f[a], g[b] = 6*b, x = A + g[b]}, x];
Table[ParallelTable[X[a, b], {b, 1, 6400}], {a, 1, 40}]
</code></pre>
<p>Now, my problem is that for 6400 runs over "b", f[a] is calculated every ... | Filipe Miguel | 63,733 | <p>I am not sure if this is what you are looking for, but this may help. You can use memoization to define f such that every time it is evaluated, it stores the value in memory:</p>
<pre><code>f[a_]:=f[a]=5*a
</code></pre>
<p>Now if you call <code>f[3]</code> twice, the first time he will compute 5*3 = 15, and the seco... |
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... | Brevan Ellefsen | 7,464 | <p>Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could per... |
1,786,382 | <blockquote>
<p>If <span class="math-container">$\cos \alpha\;,\cos \beta\;,\cos \gamma$</span> are the roots of the equation <span class="math-container">$9x^3-9x^2-x+1=0\;, \alpha,\beta,\gamma \in \left[0,\pi\right]$</span></p>
<p>Then value of <span class="math-container">$ \left(\sum \alpha\;,\sum \cos \alpha\right... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Clearly, $$0=9x^2(x-1)-(x-1)=(x-1)(9x^2-1)$$</p>
<p>Use <a href="https://math.stackexchange.com/questions/1224415/how-do-i-prove-that-arccosx-arccos-x-pi-when-x-in-1-1">How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$?</a></p>
|
144,395 | <p>Suppose I have an image containing several morphological components and I wish to extract them as individual images (with sizes equal to their bounding boxes; these individual images shouldn't contain parts of neighboring components), then apply some image-processing functions to each of them <em>individually</em>, ... | Alexey Popkov | 280 | <p>Here is a <code>SparceArray</code> version of <a href="https://mathematica.stackexchange.com/users/27331/ali-hashmi">Ali</a>'s <a href="https://mathematica.stackexchange.com/a/144402/280">second</a> method which is expected to be more memory-efficient (at least for images of type <code>"Real"</code>):</p>
<pre><cod... |
2,396,650 | <blockquote>
<p>Suppose <strong>X</strong> is a random variable with probability density function $f$ and cumulative density function $F$. Show that
$$P(a \leq \textbf{X} \leq b) = f(a) + F(b) - F(a)$$</p>
</blockquote>
<p>$$F(x)=P(\textbf{X} \leq x)=\int\limits_{-\infty}^x f(t)dt$$</p>
<p>$$\begin{align}
P(a \... | Eduardo Sebastian | 451,521 | <p>Ok, the cumulative distribution function is defined as</p>
<p><span class="math-container">$F(x_i) = P(X \leq x_i) = f(1) + f(2) + ... + f(x_i)$</span></p>
<p>But what happens if we want the cumulative distribution function, within a range?</p>
<p>Per example, i have this cumulative distribution function</p>
<p>... |
23,113 | <p>We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and Absolute geometry where in both the inequality holds.</p>
<p>I am curious whether the triangle inequality is made to... | David Sprehn | 5,762 | <p>There are a number of good answers from differential geometry, analysis, etc. But I suspect the questioner had axiomatic geometries in mind. So here's another take from that perspective.</p>
<p>There are two statements of the triangle inequality in plane geometry.
(1) If A,B,C are noncollinear points, then $AC\lt... |
2,704,575 | <p>I tried this:</p>
<p>$$\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}$$</p>
<p>Also, is it possible to see what I don't understand here or are there maybe several things..</p>
| Jaideep Khare | 421,580 | <p>You need to use $$a^{-x} = \frac 1{a^x} \tag1$$ and $$a^{1/n}= \sqrt[n]{a} \tag 2$$</p>
<p>Therefore $$\left(\frac 18 \right)^{- 1/3} = \frac{1^{-1/3}}{8^{-1/3}}=\frac{1/1^{1/3}}{1/8^{1/3}}= \frac{1/\sqrt[3]1}{1/\sqrt[3]8} = \frac{1}{1/2}=2$$</p>
|
1,638,536 | <p>Compute the following integrals:</p>
<p>$I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$</p>
<p>$J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$</p>
<p>I do not know where to begin. I know I am supposed to use the substitution $z\rightarrow \frac{1}{t}$. I can see that $I$ has 13 poles inside the circle of radius 2.</p>
<p>Usi... | pathfinder | 23,431 | <p>For $$J=\int_{|z|=10}\frac{z^3}{z^4-1}dz,$$
we see there are poles at $r^4e^{i4\theta}-e^{i2\pi k}=0,$ so that $r=1$ and $$4\theta=2\pi k\implies\theta=\frac{\pi k}{2}\implies\theta\in\left\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\}.$$
Since $\left|e^{i\theta}\right|<10$ for all $\theta$ then the residue theorem... |
1,599,002 | <p>As I am no mathematician, I have been struggling to find an equation to accurately predict points spaced along a curve separated by distance d. Given two points, assume a string with a length equal to 125% of the distance between the two points. If the string is "hooked" at its ends to the two points, what is the e... | Andrew D. Hwang | 86,418 | <p>Call your known points $(0, y_{1})$ and $(x_{2}, y_{2})$. There exist constants $a$, $b$, and $c$ such that the desired catenary has equation
$$
y = c + a\cosh \frac{x - b}{a}.
\tag{1}
$$
(This is the standard catenary of Wikipedia translated horizontally by $b$ and vertically by $c$.)</p>
<p><a href="https://i.sta... |
622 | <p>Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:</p>
<ol>
<li><p>The character table of a group is packs a lot o... | Ricardo Restrepo | 1,295 | <p>Certainly, group actions used to model dynamical systems, for instance, Markov Chains, find extremely handy the results from representation theory.
For example, suppose you have a deck of cards, and every time you choose two cards at random and transpose them. How long you have to wait for the deck to be randomized?... |
3,637,285 | <p>I have a question about how this theorem interacts with the definition of continuity in an example <span class="math-container">$f(x) = x^2$</span>.</p>
<p><strong>Theorem:</strong> If <span class="math-container">$a \in \mathbb{R}$</span> is such that <span class="math-container">$0 \leq a < \epsilon$</span> fo... | Physical Mathematics | 592,278 | <p><span class="math-container">$$w:= -81i = 81 e^{-i\pi/2},81 e^{3i\pi/2}, 81e^{7i\pi/2},81 e^{11i\pi/2},$$</span>
So the 4th roots of <span class="math-container">$w$</span> are:
<span class="math-container">$$\sqrt[4]{81} e^{-i\pi/8}, \sqrt[4]{81}e^{3i\pi/8}, \sqrt[4]{81} e^{7i\pi/8}, \sqrt[4]{81} e^{11i\pi/8},$$</... |
1,479,483 | <p>We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see <a href="https://en.wikipedia.org/wiki/Covariance_matrix">Covariance matrix</a>). We also know that every symmetric positive <em>definite</em> matrix is invertible (see <a href="ht... | uniquesolution | 265,735 | <p>As is nicely explained here <a href="https://math.stackexchange.com/questions/1258863/whats-the-best-way-to-think-about-the-covariance-matrix?rq=1">What's the best way to think about the covariance matrix?</a></p>
<p>if $A$ is the covariance matrix of some random vector $X\in\mathbb{R}^n$, then for every fixed... |
1,146,967 | <p>So I think I somewhat understand the type theory of the various lambda calculi in the lambda cube, from the simply typed lambda calculus to the calculus of constructions, but looking at it I'm wondering if I really need lambda abstraction to use it. I understand that you can use these lambda operators to construct q... | Bruno Bentzen | 174,538 | <p>Unlike set theory, where sets taken as the most fundamental objects, type theory takes functions as the first-class mathematical objects. Then we obviously need something like a <strong>function type</strong> to work on. But how to construct inhabitants of this type? By <strong><span class="math-container">$\lambda$... |
2,430,179 | <p>I understand this question has been asked before but I couldn't really find the answer I really need. I attempted this myself but I find that there is a contradiction in my answer. </p>
<p>Here is my attempt:</p>
<p>Let $Y=Xv$ and $Y>0$ where $X$ is a linearly independent matrix and $v$ is a vector.
Then $$Y^{T... | Michael Hardy | 11,667 | <p>You wrote "If $X$ is linearly independent", but the assumption you actually need is that the <b>columns</b> of $X$ are linearly independent. That of course means there are at least as many rows as columns.</p>
<p>Suppose $X\in \mathbb R^{n\times k}.$ Then $X^T X \in \mathbb R^{k\times k}.$ Then for any $v\in\mathbb... |
4,202,516 | <p>Prove gcd(x,y) = gcd(x-y,y) for x > y
Here is my work so far:</p>
<p>If gcd(x,y) = d, then we can denote x = ad , y = bd, so x - y = (ad - bd) = d(a - b), so (x-y) is a multiple of d, so gcd(x -y, y) = gcd(d(a - b), bd) = d, so gcd(x-y, y) = gcd(x,y).</p>
<p>My teacher challenged it by asking to show why gcd(d(a ... | Michael Hardy | 11,667 | <ul>
<li>First prove that every divisor that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> have in common is a divisor of <span class="math-container">$x-y.$</span></li>
<li>Then prove that every divisor that <span class="math-container">$x-y$</span> and <span class="math-container... |
273,459 | <p>I have the polynomial</p>
<p><code>5x^4 - 6x^3*y - 3x^2*y^2 - 3x*y^3 - y^4 + x^8 + x^7 + x^7*y + 4x^6*y^2 + x^6*y^2 + x^6*y + x^6 + x^5*y^3 + 4x^5*y^3 + 4x^5*y^2 - x^5*y^2 + x^5*y + x^4*y^4 + x^4*y^4 + x^4*y^4 + x^4*y^3 - x^4*y^3 + x^4*y^2 + 3x^4*y^2 + 3x^3*y^5 + x^3*y^5 + x^3*y^4 - x^3*y^4 + x^3*y^3 + x^3*y^3 + ... | Nasser | 70 | <blockquote>
<p>I get a list with all the pairs of its exponents?</p>
</blockquote>
<p>One way could be</p>
<pre><code>p = 5 x^4 - 6 x^3*y - 3 x^2*y^2 - 3 x*y^3 - y^4 + x^8 + x^7 + x^7*y +
4 x^6*y^2 + x^6*y^2 + x^6*y + x^6 + x^5*y^3 + 4 x^5*y^3 +
4 x^5*y^2 - x^5*y^2 + x^5*y + x^4*y^4 + x^4*y^4 + x^4*y^4 +
x^4*... |
4,257,878 | <p>If <span class="math-container">$P(1,2)$</span>, <span class="math-container">$Q(−3,2)$</span> and <span class="math-container">$R(3,−2)$</span> are the vertices of the triangle <span class="math-container">$PQR$</span>, then find the value of <span class="math-container">$\tan Q$</span>.</p>
<blockquote class="spoi... | Alessio K | 702,692 | <p>The slope of the line which passes through the points <span class="math-container">$Q(-3,2)$</span> and <span class="math-container">$R(3,-2)$</span> is</p>
<p><span class="math-container">$$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-2-2}{3+3}=-\frac{2}{3}$$</span></p>
<p>Then the inclination angle, with the horizontal line <... |
1,232,336 | <p>Suppose $G$ is an abelian group such that all non-identity elements in $G$ has order $2$.</p>
<p>Prove that $G$ is a vector space over $\mathbb Z_2$</p>
<p>Since $G$ is an abelian group only thing to show is to find an external composition .define $\circ :G\times \mathbb Z_2\rightarrow G$ by $g\circ 0=0$ and $g \c... | David | 119,775 | <p>"Is this the solution?" - well if I were your instructor I would want lots more detail. But I'm not, so you should ask your instructor.</p>
<p>"Where is $g^2=e$ used?" - first of all, the vector "addition" will be the group operation. So it would be much less confusing to write the group operation as addition: wr... |
3,516,840 | <p>Let <span class="math-container">$N(A^\top)$</span> be the left null space of <span class="math-container">$A$</span></p>
<p><span class="math-container">$N(A^\top) = \{x| A^\top x = 0\}$</span></p>
<p>Let <span class="math-container">$C(A)^\perp$</span> be the orthogonal complement of the column space of <span cl... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong></p>
<p>Using the formula for the difference of cubes,</p>
<p><span class="math-container">$x-8=(\sqrt[3]x-2)(x^{2/3}+2\sqrt[3]{x}+4)$</span>.</p>
|
3,516,840 | <p>Let <span class="math-container">$N(A^\top)$</span> be the left null space of <span class="math-container">$A$</span></p>
<p><span class="math-container">$N(A^\top) = \{x| A^\top x = 0\}$</span></p>
<p>Let <span class="math-container">$C(A)^\perp$</span> be the orthogonal complement of the column space of <span cl... | Michael Hardy | 11,667 | <p>The denominator is already <span class="math-container">$0$</span> in the limit. That doesn't mean the limit doesn't exist.</p>
<p>Recall the factorization of a difference of two cubes:
<span class="math-container">$$
a^3 - b^3 = (a-b)(a^2+ab+b^2).
$$</span>
In particular
<span class="math-container">$$
x-8 = \big(... |
3,516,840 | <p>Let <span class="math-container">$N(A^\top)$</span> be the left null space of <span class="math-container">$A$</span></p>
<p><span class="math-container">$N(A^\top) = \{x| A^\top x = 0\}$</span></p>
<p>Let <span class="math-container">$C(A)^\perp$</span> be the orthogonal complement of the column space of <span cl... | Quanto | 686,284 | <p>Take the steps below<span class="math-container">$$\lim_{x \to 8} \frac{\sqrt[3]{x} -2}{x-8}
=\lim_{x \to 8} \frac{(\sqrt[3]{x} -2)((\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4)}{(x-8)((\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4)}$$</span>
<span class="math-container">$$=\lim_{x \to 8}\frac{x-8}{(x-8)((\sqrt[3]{x})^2 +2\sqrt[3]{x} + 4)}... |
1,257,184 | <p>I never understand where to start with proofs, but whenever I see them done I understand them. </p>
<p>My attempt:</p>
<p>For this one could I just use the property of inner products to prove this?</p>
<p>That being $v\cdot (cu)=c(u\cdot v)$, for $c$ a scalar and $u,v$ vectors. And if so then how would I prove it... | Quality | 153,357 | <p>You can show this as follows;</p>
<p>expanding $||x+y||^2=\langle x+y,x+y \rangle=||x||^2+2\langle x, y \rangle + || y||^2$</p>
<p>if you want to say why this is, then it is just from the properties of inner product. i.e., $$\langle x+y , x+y \rangle= \langle x, x+y \rangle + \langle y, x+y \rangle$$ (linearity)</... |
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 ... | Yanko | 426,577 | <p>The definition of finite direct sum and the definition of finite direct product is exactly the same definition. (Unless you are working in categories and then the definitions aren't the same, but as vector spaces they are isomorphic).</p>
<p>Infinite direct sum of $\{V_i\}_{i\in I}$ is the set of all <strong>finite... |
2,214,706 | <p>\begin{array}{c|lcr}
x & y\\
\hline
1 & 20\\
2 & 16\\
3 & 13\\
4 & 11\\
5 & 10
\end{array}</p>
<p>How do you start a solution when you have these type of problem? I was used to given an equation but not like all just numbers. It was state that the answer is $y=\frac{1}{2}x^2$ -$\frac{11}{2}x... | Moo | 295,756 | <p>We can use a <a href="https://en.wikipedia.org/wiki/Least_squares" rel="nofollow noreferrer"><em>Least Squares</em></a> fit and will use the example on page $3$ of these <a href="https://math.la.asu.edu/~checkman/242/242LN1.6_5.3.pdf" rel="nofollow noreferrer"><em>notes</em></a> as a guide. We setup the system of eq... |
1,576,142 | <p>The Oxford Concise Dictionary of Mathematics defines a pseudoprime as any number $n$ that satisfies the condition $a^n = a \pmod n$ "for all integers $a$". To find a pseudoprime, we have to go about iterating through each integer less than $n$, at least until we get to the point where we can be sure none of the numb... | paw88789 | 147,810 | <p>Let's say we had a number $n$ of size roughly $10^{100}$. To test for primality by trial division, we would have to divide by all the primes up to about $10^{50}$ (if the number is in fact prime). According to the Prime Number Theorem there are about $\frac{10^{50}}{\ln\left(10^{50}\right)}\approx 8.7\times10^{47}... |
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... | user532578 | 532,578 | <p>B is correct. It is the probability of two outcomes out of four equally likely outcomes and equals 1/2. The others are the probability of one outcome and equal 1/4.</p>
|
1,602,137 | <p>There are a few questions on this topic already. However, none of them really answer my question.
The most relevant are these:</p>
<p><a href="https://math.stackexchange.com/questions/1097796/quadratic-optimisation-with-quadratic-equality-constraints">Quadratic optimisation with quadratic equality constraints</a></... | Johan Löfberg | 37,404 | <p>Quadratic equalities are never convex. Simply consider the trivial scalar case $x^2=1$, which has two distinct feasible points $-1$ and $1$.</p>
<p>As it is nonconvex, you cannot convert it to an SOCP, so your final question doesn't really make any sense.</p>
|
4,115,343 | <p>I read about the construction of a function
<span class="math-container">$G : D_n → ∂D_n$</span> for an <span class="math-container">$n$</span>-dimensional disk <span class="math-container">$D_n$</span> using <span class="math-container">$F(p)$</span> and <span class="math-container">$p$</span> in the proof of Brouw... | Thomas Andrews | 7,933 | <p>If <span class="math-container">$F(x,y)=(F_1(x,y),F_2(x,y))$</span> then you need to find the <span class="math-container">$t\leq 0$</span> so that:</p>
<p><span class="math-container">$$(x,y)+t(F_1(x,y)-x,F_2(x,y)-y)$$</span> is on the boundary.</p>
<p>This requires <span class="math-container">$$\left(tF_1+(1-t)x\... |
2,270,476 | <blockquote>
<p>Knowing that $x^2+y^2 = 2 $ prove that $x+y \le 2$</p>
</blockquote>
<p>I rewrite this as
$$x^2+y^2 \ge x+y$$
Now, multiply both sides by 2
$$x^2 -2x + 1 + y^2-2y + 1 -2 +x^2 + y^2 \ge 0$$
I substitue $2$ for $x^2+y^2$
$$(x-1)^2+(y-1)^2\ge0$$
Which is true</p>
| Takahiro Waki | 268,226 | <p>Your proof is insecure and accidentally.
When a line $x+y=t$ is tangent to circle, t is max or minimum.</p>
|
129,854 | <p>I have a four points in plane and need to test (based on point coordinates) whether they are able to form a convex quadrilateral:</p>
<p><img src="https://i.stack.imgur.com/qMiCH.png" alt="convex quadrilateral"></p>
<p>Of course, the test should avoid configurations like these:</p>
<p><img src="https://i.stack.im... | Esam Bustaty | 186,431 | <p>Maybe it's too late for an answer to be useful to the OP (7 yrs too late!), but it still might be useful to someone.</p>
<p>My solution would be to get the <a href="http://mathworld.wolfram.com/BarycentricCoordinates.html" rel="nofollow noreferrer">barycentric coordinates</a> <a href="https://gamedev.stackexchange.c... |
2,949,936 | <p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> are different complex numbers. Prove that the points <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> form an equi... | Community | -1 | <p>We have <span class="math-container">$a^2+b^2+c^2-ab-ac-bc=(a+\xi b+\xi^2 c)(a+\xi^2 b+\xi c)$</span>, where <span class="math-container">$\xi$</span> is the primitive <span class="math-container">$3$</span>rd root of unity <span class="math-container">$\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right... |
1,775,820 | <p>Determine whether the set </p>
<p>$$A=\{z\in\mathbb{C}\,\colon \Re{(z)}\ge0,\,\Im{(z)}\ge0,\, \vert z\vert\le1\}$$</p>
<p>is closed.</p>
<hr>
<p>This is true if its complement is open. We have that</p>
<p>$$\overline{A} = \mathbb{C}\setminus\{z\in\mathbb{C}\,\colon \Re{(z)}\ge0,\,\Im{(z)}\ge0,\, \vert z\vert\le... | Alex M. | 164,025 | <p>You needlessly complicate the problem: denoting by $re(z) = \Re z, \ im(z) = \Im z, \ abs(z) = |z|$, notice that $A = B \cap C \cap D$ where $B = re^{-1} ([0, \infty)), \ C = im^{-1} ([0, \infty)), \ D = abs^{-1} ([0,1])$. Since the functions $re, im, abs$ are continuous and the intervals $[0,\infty)$ and $[0,1]$ ar... |
3,280,104 | <p><a href="https://i.stack.imgur.com/Rwzmt.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rwzmt.jpg" alt="enter image description here"></a></p>
<p>The diagram above is distilled from an engineering problem I'm working on. The goal is to compute <span class="math-container">$R$</span> in terms of ... | Claude Leibovici | 82,404 | <p>Except if you want to play with the quartic equation in <span class="math-container">$\sqrt R$</span>, numerical methods will work nicely.</p>
<p>With regard to your second plot, if instead of plotting <span class="math-container">$R(\theta)$</span>, you plot <span class="math-container">$\frac 1{R(\theta)}$</span... |
253,386 | <p>I have a function for the variable <span class="math-container">$x>0$</span> and constant <span class="math-container">$a\in\mathbb{R}$</span>. I write it in its two alternative forms</p>
<p><span class="math-container">$$f(x)=\cos a\pi-\cos x\pi =-2\sin\frac{(a-x)\pi}{2}\sin\frac{(a+x)\pi}{2} $$</span></p>
<p>Th... | E. Chan-López | 53,427 | <p>The Discrete System:</p>
<p><span class="math-container">$$
\begin{align}
x_{n+1}&=c \left(1-(x_n+y_n)\right)+x_n \left(1-p y_n\right)\\
y_{n+1}&=\left(p x_n+b\right)y_n
\end{align}
$$</span></p>
<p>The Discrete System code:</p>
<pre><code>F[{x_, y_}, {b_, c_, p_}] := Evaluate[{(1 - p*y)*x + c*(1 - x - y), (... |
3,263,900 | <p>Request help in understanding the telescoping sum for the given series. <br>
For <span class="math-container">$n\geq1$</span>,
<span class="math-container">\begin{align}
(a-b)\sum_{i=0}^{n-1}a^ib^{n-1-i}&=\sum_{i=0}^{n-1}a^{i+1}b^{n-1-i}-\sum_{i=0}^{n-1}a^ib^{n-i}\\
&=\sum_{i=0}^{n-1}(a^{i+1}b^{n-(i+1)}-a^ib... | J. W. Tanner | 615,567 | <p>Given<span class="math-container">$$ \sum_{i=0}^{n-1}(a^{i+1}b^{n-(i+1)}-a^ib^{n-i})\\$$</span>
Change the index of summation on the first sum:
<span class="math-container">$$=\sum_{i=1}^na^ib^{n-i} - \sum_{i=0}^{n-1}a^ib^{n-i}$$</span>
Take out the <span class="math-container">$i=n$</span> term from the first sum a... |
52,113 | <h2>Description</h2>
<p>I want to add vertical line and horizonal lines between data to make every point have an independent rectangle area. In addtion, every independent rectangle has a number <span class="math-container">$1,2,3,4,5,\ldots$</span></p>
<h2>Algorithm</h2>
<p>I have a method that adds a line between two ... | xyz | 9,627 | <p>I think <code>GridLineData</code> should be optimized by adding a condition as below:</p>
<blockquote>
<p>Classify the data that has been sorted by using this condition that the numerical difference is small (i.e., two data is very close to) (classification interval is set to δ),
In each class of data, the min... |
907,620 | <p>We will call a matrix <em>positive matrix</em> if all elements in the matrix are positive, and we will denote the <em>largest eigenvalue</em> with <span class="math-container">$\lambda_{\max}$</span>, what is exist because of the <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem" rel="nofollow n... | Ewan Delanoy | 15,381 | <p>Your property follows easily the Collatz-Wieland formula, which is proved
on <a href="http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Collatz.E2.80.93Wielandt_formula" rel="nofollow">wikipedia</a>.
In fact, that wikipedia page proves everything from scratch and provides references. So it should conta... |
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