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 |
|---|---|---|---|---|
1,392,858 | <p>Is is known that the space of symmetric matrices $\mathbb{R}_{sym}^{n \times n}$ has $\binom{n}{2}$ dimensions.</p>
<p>And according to the spectral theorem every symmetric matrix $A \in \mathbb{R}_{sym}^{n \times n}$ has a spectral decomposition in terms of 1-rank matrices.</p>
<p>A = $\sum_{i=1}^n \lambda_i v_i... | InsideOut | 235,392 | <p>The dimension space of $n$-symmetric matrices is $\frac{n(n+1)}{2}$. By the spectral theorem every symmetric matrix, with real coefficient, is diagonalizable; in other words every symmetric matrix is similar to a diagonal matrix. But this not means that the dimension of $n$-symmetric matrices in $n$. </p>
<p>The se... |
563,499 | <p>What's the summation of the following expression;</p>
<p>$$\sum_{k=1}^{n+3}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{n-k}$$
The solution is said to $$2\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\right)$$</p>
<p>But I'm getting $$\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\right).$$
How is this possi... | pi37 | 46,271 | <p>Note that
$$
\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{n-k}=\left(\frac{1}{4}\right)^n 2^k
$$
So the sum in question simplifies to
$$
\left(\frac{1}{4}\right)^n\left(\sum_{k=1}^{n+3} 2^k\right)
$$
Now by the geometric series formula
$$
\sum_{k=1}^{n+3} 2^k=2\sum_{k=0}^{n+2} 2^k=2(2^{n+3}-1)
$$
So the to... |
555,239 | <p>Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?</p>
| N. S. | 9,176 | <p>$$x-y = x^2 + y^2 - xy \Leftrightarrow \\
2x-2y = 2x^2 + 2y^2 - 2xy \Leftrightarrow \\
0= 2x^2 + 2y^2 - 2xy-2x+ 2y \Leftrightarrow \\
(x-y)^2+(x-1)^2+(y+1)^2=2$$</p>
<p>As $x,y$ are integers, there are only 2 possibilities for each bracket: $0$ or $1$. So two of the squares have to be $1$ and the third one must be ... |
590,817 | <p>(I'm a software developer so excuse me)</p>
<p>I'm building an application for a client and one of the formulas that has been provided in the spec is <code>value1 = value2 * (1 + 5%)</code>. When I asked about it I was told that it's some kind of notation for <code>value1 = value2 * 0.15</code>.</p>
<p>Also, they ... | Rachit Bhargava | 113,115 | <p>According to me, it seems for first:</p>
<p>value1 = value2 * 1.05</p>
<p>And, the second one is alright!</p>
|
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | David Richerby | 97,480 | <p>Matrices are used widely in computer graphics. If you have the coordinates of an object in 3d space, then scaling, stretching and rotating the object can all be done by considering the coordinates to be vectors and multiplying them by the appropriate matrix. When you want to display that object on-screen, the <a hre... |
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | alephzero | 223,485 | <p>Determinants are of great theoretical significance in mathematics, since in general "the determinant of something <span class="math-container">$= 0$</span>" means something very special is going on, which may be either good news of bad news depending on the situation.</p>
<p>On the other hand determinants have very... |
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | G Cab | 317,234 | <p>Besides the applications already mentioned in the previous answers, just consider that matrices are the fundamental basis for <a href="https://en.wikipedia.org/wiki/Finite_element_method" rel="nofollow noreferrer">Finite Elements</a> design, today widely used in every sector of engineering. </p>
<p>Actually a <a hr... |
2,623,735 | <p><a href="https://i.stack.imgur.com/5QfOQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5QfOQ.png" alt="enter image description here"></a></p>
<p>I have proven that $S$ is also a basis. But I am not sure about the second one. Is it just identity matrix $3\times 3$ as we don't change anything?</p... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>If $A=\tan (\frac {x}{2}) $ then</p>
<p>$$\cos (x)=\frac {1-A^2}{1+A^2} $$
and
$$\sin (x)=\frac {2A}{1+A^2} $$</p>
|
4,414,843 | <p>For reference: Show that the area of triangle <span class="math-container">$ABC = R\times MN(R=BO)$</span></p>
<p>I can't demonstrate this relationship</p>
<p><a href="https://i.stack.imgur.com/A3ci3.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A3ci3.jpg" alt="enter image description here" />... | sirous | 346,566 | <p>Hints for a geometric solution:</p>
<p>-Draw a circle radius <span class="math-container">$R$</span> center on B, it intersect altitude BH at E.</p>
<p>-Connect N to E and extend it to meet a circle center on N and radius MN at F. It can be seen that NF is parallel with AC, so it is perpendicular on BH at point E. S... |
677,785 | <p>I have to evaluate this integral:</p>
<p>$$
\int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} \operatorname d\!x \operatorname d\!y\,
$$</p>
<p>I have no idea what to do with $\;{e}^{x^7}$.</p>
<p>I have even <a href="http://www.wolframalpha.com/input/?i=int+e%5Ex%5E7+dx" rel="nofollow">tried $\int{e}^{x^7} dx$ with Wolfram... | Community | -1 | <p>The interchange of the order of integration is justified by Fubini's theorem:</p>
<p>$$
\int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} dxdy=\int_0^2\int_0^{x^2}y^2 {e}^{x^7} dydx=\frac 1 3\int_0^2 x^6{e}^{x^7} dx=\frac1{21}{e}^{x^7} \bigg|_0^2=\frac {{e}^{2^7}-1}{21}
$$</p>
|
2,101,750 | <p>The WP article on general topology has a section titled "<a href="https://en.wikipedia.org/wiki/General_topology#Defining_topologies_via_continuous_functions">Defining topologies via continuous functions</a>," which says,</p>
<blockquote>
<p>given a set S, specifying the set of continuous functions $S \rightarrow... | Daron | 53,993 | <p>The first statement can be patched up using functions into the Sierpinski Space $\{0,1\}$ with topology $\{\varnothing , \{1\}, \{0,1\}\}$. Since a continuous function $X \to \{0,1\}$ can be identified with the open set $f^{-1}(1)$ we see the continuous functions into the Sierpinski space are the same thing as the o... |
1,777,901 | <p>My task is this:</p>
<p>Show that $$\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$$</p>
<p>My work so far: </p>
<p>If we approximate $\ln(x)$ around $x = 1$, we get:</p>
<p>$\ln(x) = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + ...$</p>
<p>Substituting $x = 2$ then gives us:</p>
<p>$... | Solumilkyu | 297,490 | <p>Note that
\begin{align}
\frac{1}{1-x}&=1+x+x^2+\cdots\qquad(-1<x<1).
\end{align}
Thus
\begin{align}
-\ln(1-x)&=\int\frac{1}{1-x}{\rm d}x\\
&=x+\frac{1}{2}x^2+\frac{1}{3}x^3+\cdots\\
&=\sum_{n=1}^\infty\frac{1}{n}x^n.
\end{align}
By taking $x=\frac{1}{2}$ into the above equation, we have
$$\sum_... |
1,777,901 | <p>My task is this:</p>
<p>Show that $$\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$$</p>
<p>My work so far: </p>
<p>If we approximate $\ln(x)$ around $x = 1$, we get:</p>
<p>$\ln(x) = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + ...$</p>
<p>Substituting $x = 2$ then gives us:</p>
<p>$... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may write</p>
<blockquote>
<p>$$
\begin{align}
\sum \limits_{n=1}^\infty \frac{1}{n2^n}&=\sum \limits_{n=1}^\infty \int_0^{1/2}x^{n-1}dx
\\\\&=\int_0^{1/2}\sum \limits_{n=1}^\infty x^{n-1}\:dx
\\\\&=\int_0^{1/2}\frac1{1-x}\:dx
\\\\&=\left[-\ln(1-x)\right]_0^{1/2}
\\\\&am... |
3,064,501 | <p>So I was trying to find the <strong>time complexity</strong> of an algorithm to find the <span class="math-container">$N$</span>th prime number (where <span class="math-container">$N$</span> could be any positive integer).</p>
<p>So is there any way to exactly determine how far <span class="math-container">$(N+1)$<... | DanaJ | 117,584 | <p>You're got what looks like two questions.</p>
<ol>
<li><p>The time complexity of the nth prime. In practice it is <span class="math-container">$O\big(\frac{n^{2/3}}{\log^2n}\big)$</span> using fast prime count algorithms. In theory this could be lowered to something on the order of <span class="math-container">$O... |
268,676 | <p>It is not hard to check that the three roots of $x^3-2=0$ is $\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}$, hence the splitting field for $x^3-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}]$. However, since $\sqrt[3]{2}\zeta_3^{2}$ can be compute through $\sq... | Hagen von Eitzen | 39,174 | <p>Note that $(\alpha\zeta_n^k)^n = \alpha^n\zeta_n^{nk}=\alpha^n=a$, $0\le k<n$.</p>
|
268,676 | <p>It is not hard to check that the three roots of $x^3-2=0$ is $\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}$, hence the splitting field for $x^3-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \sqrt[3]{2}\zeta_3, \sqrt[3]{2}\zeta_3^{2}]$. However, since $\sqrt[3]{2}\zeta_3^{2}$ can be compute through $\sq... | Alfonso Fernandez | 54,227 | <ol>
<li><p>Assuming that by $\zeta_5$ you mean the fifth primitive root of unity, the roots of $x^5-2$ are simply $\sqrt[5]{2}, \sqrt[5]{2} \zeta_5, \sqrt[5]{2} \zeta_5^2, \sqrt[5]{2} \zeta_5^3, \sqrt[5]{2} \zeta_5^4$.</p></li>
<li><p>The roots of $x^n - a$ are given when $x^n=a$. Let $x=re^{i\theta}$ be such a root, ... |
3,364,316 | <p>While I'm reading E. Landau's <em>Grundlagen der Analysis</em> (tr. <em>Foundations of Analysis</em>, 1966), I couldn't understand the proof of <em>Theorem 3</em> at the segment of <em>Natural Numbers</em> which I've quoted below.</p>
<blockquote>
<p><strong>Theorem 3:</strong> <em>If</em><br>
<span class="math... | Claude Leibovici | 82,404 | <p><em>Siminar to Omnomnomnom's answer.</em></p>
<p>Your idea was good
<span class="math-container">$$a_n= \frac{p_{n}}{p_{n}+p_{n+1}} \sim \frac{n \log (n)}{n \log (n)+(n+1) \log (n+1)}$$</span> Now, using Taylor expansion for large values of <span class="math-container">$n$</span>
<span class="math-container">$$a_n... |
2,130,397 | <p>If I want to find the power series representation of the following function:</p>
<p>$$ \ln \frac{1+x}{1-x} $$</p>
<p>I understand that it can be written as </p>
<p>$$ \ln (1+x) - \ln(1-x) $$</p>
<p>And I understand that if I now write in the power series representations for $ln(1+x)$ and $ln(1-x)$:</p>
<p>$$\su... | Bernard | 202,857 | <p><em>Bioche's rules</em> prompt you to set $i=\tan x$, $\mathrm d t=(1+t^2)\mathrm d x$, so you finally get the integral of the rational function
$$\int_0^\infty\frac{\mathrm dt}{2+t^2}.$$</p>
|
1,525,660 | <p><a href="https://i.stack.imgur.com/w2y9k.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w2y9k.png" alt="image"></a></p>
<p>Problem above. (Sorry I can't embed yet and the link seems to be removed when hyperlinked)</p>
<p>Hello,</p>
<p>Fairly simple (I imagine) question that I am stuck on.
The ... | E.H.E | 187,799 | <p>$$\frac{1}{1-x}=\sum_{n=0}^{\infty }x^n$$
$$\frac{d^3}{dx^3}(\frac{1}{1-x})=\frac{6}{(1-x)^4}=\sum_{n=3}^{\infty }n(n-1)(n-2)x^{n-3}$$
$$\frac{1}{(1-x)^4}=\frac{1}{6}\sum_{n=3}^{\infty }n(n-1)(n-2)x^{n-3}$$
now replace $x$ by $x^2$
$$\frac{1}{(1-x^2)^4}=\frac{1}{6}\sum_{n=3}^{\infty }n(n-1)(n-2)x^{2n-6}=1+4x^2+10x^4... |
112,651 | <p>What is known about the set of well orderings of $\aleph_0$ in set theory without choice? I do not mean the set of countable well-order types, but the set of all subsets of $\aleph_0$ which (relative to a pairing function) code well orderings. And I would be interested in an answer in, say, ZF without choice. My ... | Amit Kumar Gupta | 7,521 | <p>Consider the tree of finite partial attempts to build a well-ordering, and notice that it has size continuum.</p>
<p>More rigorously, let:</p>
<p>$$T = \{ f : n \to \omega\ |\ n \in \omega, f \mbox{ injective } \}$$</p>
<p>ordered by extension. This is clearly an $\omega$ branching tree of height $\omega$, and i... |
4,192,687 | <p>Let <span class="math-container">$f: [0,1] \rightarrow \mathbb{R}$</span> be a continuous function. <br />
How can I show that <span class="math-container">$ \lim_{s\to\infty} \int_0^1 f(x^s) \, dx$</span> exists?<br />
It is difficult for me to calculate a limit, as no concrete function or function values are given... | Vladimir | 154,757 | <p><span class="math-container">$\lim_{s\to\infty} f(x^s)=f(0)$</span> if <span class="math-container">$x<1$</span> and <span class="math-container">$f(1)$</span> if <span class="math-container">$x=1$</span>. Thus, <span class="math-container">$\lim_{s\to\infty}\int_0^1f(x^s)dx=f(0)$</span>.</p>
|
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | Henno Brandsma | 4,280 | <p>There are reasonably accessible proofs that are purely general topology. First one needs to show Brouwer's fixed point theorem (which has an elementary proof, using barycentric subdivion and Sperner's lemma), or some result of similar hardness. Then one defines a topological dimension function (there are 3 that all ... |
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | Qiaochu Yuan | 232 | <blockquote>
<p>is there intuition for why a proof is so difficult?</p>
</blockquote>
<p>Sure: the topological category is horrible. A generic continuous function is bizarre and will <a href="http://en.wikipedia.org/wiki/Space-filling_curve">violate your geometric intuitions</a>. When we prove that $\mathbb{R}^n$ is... |
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | Community | -1 | <p>Consider the one point compactifications, <span class="math-container">$S^n$</span> and <span class="math-container">$S^m$</span>, respectively. If <span class="math-container">$\mathbb R^n$</span> is homeomorphic to <span class="math-container">$R^m$</span>, their one-point compactifications would be, as well. But... |
1,283,325 | <p>I got two questions about $p$-adic numbers:</p>
<blockquote>
<ol>
<li>I often read that the field $\mathbb Q_p$ is much different than the field $\mathbb R$.</li>
</ol>
</blockquote>
<p>An element of $\mathbb Q_p$ is of the form $\sum_{i=-k}^{\infty}a_ip^i$ where $a_i\in \{0,...,p-1\}$.</p>
<p>But isn't thi... | Dietrich Burde | 83,966 | <ol>
<li><p>No, for example $x=\sqrt{-1}=i\in \mathbb{Q}_5$, but $x\not\in \mathbb{R}$.</p></li>
<li><p>See <a href="http://en.wikipedia.org/wiki/P-adic_number">here</a>. </p></li>
</ol>
|
1,626,821 | <p>Is there a name for a vector with all equal elements? If so, what is it?</p>
<p>For example,</p>
<p>$$ (7, 7, 7, 7, 7) $$</p>
| Narasimham | 95,860 | <p><em>Diagonal vector</em> which makes equal angle to co-ordinates axes.</p>
|
875,644 | <p>I have a parabolic basin which i am trying to find the equation for so I can reproduce it. I have taken $3$ points along one line of it to find the equation of the parabola, and I'm wondering if there is a way I can go from this to the equation of the parabolic basin.
The equation I have for the parabola is:</p>
<... | Shine | 157,361 | <p>In your first step, it should be: $m=y_1 -y_2$</p>
<p>$$\int_0^1\int_0^1g(y_1-y_2)\Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_0^1\int_{y_2}^{1}g(y_1-y_2)dy_1dy_2=\int_0^1\int_0^{1-y_2}g(m)dm dy_2=\int_0^1\int_0^1 g(m)I_{(0<m<1-y_2)}dmdy_2=\int_0^1\int_0^1 g(m)I_{(0<m<1-y_2)}dy_2 dm \quad=\int_0^1\int_0^1 g(... |
1,613,645 | <p>Let's get started:</p>
<p>$$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$</p>
<p>since $|x|$ is an even function:</p>
<p>$$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$</p>
<p>Integration by parts yields:</p>
<p>$$e^{-inx}\Big|_0^{\pi} + \frac{1}{in} \int_0^\pi e^{-inx} dx = (-1)^n - 1 + \frac{1}{in} \l... | Arthur | 15,500 | <p>It has moved $\frac{35}{60}$ of a full circle (a full circle consists of $60$ minutes, and it has moved $35$ of those). A full circle is $2\pi\cdot15cm=30\pi cm$. It has therefore moved a total of
$$
\frac{35}{60}\cdot30\pi cm=\frac{35\pi}{2}cm
$$</p>
|
1,913,873 | <p>If $a,b,c,d,e>1$, then prove that:
$$\frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{d^2}{e-1} + \frac{c^2}{a-1} + \frac{e^2}{d-1} \ge 20. $$</p>
<p>I don't know how to begin. What should be the approach?</p>
| levap | 32,262 | <p>Note that we can rewrite your expression as</p>
<p>$$ \left( \frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{c^2}{a-1} \right) + \left( \frac{d^2}{e-1} + \frac{e^2}{d-1} \right) \geq 20. $$</p>
<p>Let us consider first the baby problem of finding the minimum value of $f(x) = \frac{x^2}{x-1}$ assuming $x > 1$. The min... |
1,913,873 | <p>If $a,b,c,d,e>1$, then prove that:
$$\frac{a^2}{b-1} + \frac{b^2}{c-1} + \frac{d^2}{e-1} + \frac{c^2}{a-1} + \frac{e^2}{d-1} \ge 20. $$</p>
<p>I don't know how to begin. What should be the approach?</p>
| Community | -1 | <p>Using Cauchy-Scwarz to
\begin{align*}
\frac{a_1}{\sqrt{x_1}}, \frac{a_2}{\sqrt{x_2}}, \ldots, \frac{a_n}{\sqrt{x_n}} \\
\sqrt{x_1}, \sqrt{x_2}, \ldots, \sqrt{x_n}
\end{align*}
we get
\begin{align*}
\frac{a_1^2}{x_1}+\frac{a_2^2}{x_2}+\cdots + \frac{a_n^2}{x_n} \geq \frac{(a_1+a_2+\cdots + a_n)^2}{x_1+x_2+\cdots+x_n... |
185,766 | <p>After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course? </p>
<p>And would an advanced linear algebra course be taught in graduate schools?</p>
| Mathemagician1234 | 7,012 | <p>First of all, it's not clear what an advanced course in linear algebra at either the undergraduate or graduate level consists of. It really depends on what the first course consists of and this varies enormously from university to university depending not only on the background and career paths of the students, but ... |
185,766 | <p>After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course? </p>
<p>And would an advanced linear algebra course be taught in graduate schools?</p>
| Paul Siegel | 1,509 | <p>To answer your question succinctly, a first course on linear algebra should cover the basic computational tools: row reduction, determinants, and eigenvalues. A more advanced course should force the students to come to terms with more abstract language (vector spaces over an arbitrary field), and it should contain ... |
111,425 | <p>If $R$ is a unital integral ring, then its characteristic is either $0$ or prime. If $R$ is a ring without unit, then the char of $R$ is defined to be the smallest positive integer $p$ s.t. $ pa = 0 $ for some nonzero element $a \in R$. I am not sure how to prove that the characteristic of an integral domain without... | Mariano Suárez-Álvarez | 274 | <p>Suppose $p$ is the characteristic of $R$ and not prime, so that $p=mn$ for some positive integers $m$,~$n>1$. In particular, $p>n$ and $p>m$. <em>According to the definition you are using</em>, $p$ is the least positive number such that there exists a non-zero $a\in R$ with $pa=0$: it follows that $na\neq0$, a... |
300,460 | <p>How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$</p>
| Community | -1 | <p>$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n(n+1)}$</p>
<p>$=(1-\frac12)+(\frac12-\frac13)+\ldots+(\frac{1}{n}-\frac{1}{n+1})$</p>
<p>$=1-\frac{1}{n+1}$</p>
<p>$=\frac{n}{n+1}$</p>
|
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| Joseph Nelson Pulikkottil | 415,350 | <p>There are uncountable number of positive real numbers, take
{An}=cn where c is a positive real number
Do this with every positive numbers it is increasing and uncountable.</p>
|
4,164,650 | <p>The outline of the exercise is that: Fix <span class="math-container">$b > 1, y > 0$</span> and show that there is a unique real <span class="math-container">$x$</span> such that <span class="math-container">$b^x = y$</span>. (For further specification see e.g. <a href="https://math.stackexchange.com/questions... | Blue | 409 | <p>The second-degree equation <span class="math-container">$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$</span> corresponds to a conic in <em>general position</em>. Moving its focus to the origin imposes certain relations on the coefficients that can be difficult to describe.</p>
<p>It's easier to start with the definition that a (non-cir... |
1,850,258 | <p>From where can I learn mathematics from the basic blocks up? I feel like I have a lot of holes in the mathematics that I know and I would like to see where all those concepts come from. I would like to see what are the ideas that are took from granted, as foundation, and which ideas are made from this foundation.</p... | Mathmo123 | 154,802 | <p>I think all four of your questions can be answered by looking at the following question:</p>
<blockquote>
<p>Which prime numbers $p$ can be written as a sum of two squares?</p>
</blockquote>
<p>i.e. when do there exist integers $a,b$ such that
$$p = a^2+b^2.$$</p>
<p>This is certainly a natural, number theoreti... |
3,715,522 | <p>I am trying to understand fully how drug half-life works. So I derived this relationship: </p>
<p><span class="math-container">$$\ U_{r} = \frac{1+\ U_{r-1}}{2}$$</span>
Where <span class="math-container">$\ U_{0}=0$</span> and r is a set of natural numbers.</p>
<p>My issue to how to deduce a relationship for the... | aritracb | 782,875 | <p><span class="math-container">$U_r=\frac{1}{2}+\frac{U_{r-1}}{2}=\frac{1}{2}+\frac{1}{4}+\frac{U_{r-2}}{4}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{U_{r-3}}{8}=...=\sum_{i=1}^{r}\frac{1}{2^i}+\frac{U_0}{2^r}=1-\frac{1}{2^r}$</span></p>
<p>and hence <span class="math-container">$\sum_{1}^{n}U_r=n-\sum_1^n\frac{1}{2^... |
663,563 | <p>it seems obvious that this integral is zero and so is the limit but what theorem we are using here?</p>
<p>I see it's connected to Riemann sums with an interval=zero Right ?</p>
<p>The function $\mathrm{f}$ is continuous.</p>
<p>$$\lim_{x \to 0}\int_0^x\mathrm{f}(x)\ \mathrm{d}x= \ ?$$</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Another approach valid when $f$ is continuos: the integral is a function $F$ s.t. $F′=f$ and $F(0)=0$. By continuity, $\lim F=F(0)=0$.</p>
|
4,315,572 | <p>exercise:</p>
<p>Let us assume that the function f has derivatives of all orders.</p>
<p>Suppose that all zeros of <span class="math-container">$f$</span> have finite multiplicity. Let <span class="math-container">$a$</span> and <span class="math-container">$b$</span> be points of <span class="math-container">$A$</s... | emil agazade | 801,252 | <p>Considering very useful comments and the answer of @TonyK and after long thoughts I have made conclusion:</p>
<p>What we know:</p>
<ol>
<li><p>All zeros are isolated.</p>
</li>
<li><p>Given open bounded interval but the end points is in the domain of the function.</p>
</li>
</ol>
<p>So to find a contradiction we ass... |
1,700,246 | <p>Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an element $g\in SO_{3}(q)$ defined by: $g: e\mapsto -e$, $f\mapsto \frac{1}{2}e -f +d$, $d\mapsto e+d$. I would like to de... | Joe | 107,639 | <p>Obviously $r^2\ge0$ so if $y\le0$ the required inequality can't be reached.</p>
<p>If otherwise $y>0$ you can consider $\sqrt y$ which is again $>0$; then you have two cases:</p>
<ul>
<li>$x\ge0$; in this case you have $\sqrt x<\sqrt y$ and since $\Bbb Q$ is dense in $\Bbb R$, you can find an $r\in\Bbb Q$... |
1,182,684 | <p>Let $\mathbf{F}(x,y,z) = y \hat{i} + x \hat{j} + z^2 \hat{k}$ be a vector field. Determine if its conservative, and find a potential if it is.</p>
<p><strong>Attempt at solution:</strong></p>
<p>We have that $\frac{\partial F_1}{\partial y} = 1 = \frac{\partial F_2}{\partial x} $, $\frac{\partial F_1}{\partial z} ... | Mark Viola | 218,419 | <p>You don't have to find the integration constant immediately. Keep proceeding as follows. </p>
<p>After you determined that $f(x,y,z) = xy+g(y,z)$, differentiate with respect to $y$. </p>
<p>This gives $\frac{\partial f}{\partial y}=x+\frac{\partial g}{\partial y}=F_y=x$. </p>
<p>Thus, $\frac{\partial g}{\part... |
165,853 | <blockquote>
<p>Schauder's conjecture: "<em>Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point.</em>" [Problem 54 in The Scottish Book]</p>
</blockquote>
<p>I wonder whether this conjecture is resolved. I know R. Cauty [So... | Mohammad Golshani | 11,115 | <p>I have taken the following from the review of the following paper "<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2778675" rel="nofollow noreferrer">Schauder's conjecture on convex metric spaces</a>" written in 2010 :</p>
<blockquote>
<p>One of the most resistant open problems in the theory of n... |
165,853 | <blockquote>
<p>Schauder's conjecture: "<em>Every continuous function, from a nonempty
compact and convex set in a (Hausdorff) topological vector space into
itself, has a fixed point.</em>" [Problem 54 in The Scottish Book]</p>
</blockquote>
<p>I wonder whether this conjecture is resolved. I know R. Cauty [So... | jaco | 49,821 | <p>There is R. Cauty paper from 2012 titled 'Un theoreme de Lefschetz-Hopf pour les fonctions a iterees compactes' which from what I heard was reviewed to establish a correct proof of Schauder's conjecture ( or rather a generalization for iterates of f ), and will appear in an international journal.</p>
<p>Edit: publ... |
173,286 | <p>I have these two functions <code>fun</code> and <code>microstep</code>.Fun makes use of a Module construct within which I define the <code>Array</code> I need to store the values of magnetization for different temperatures (each case stored in a different row).
<code>microstep</code> is the function that store the... | enano9314 | 32,571 | <p>Just give your function <code>microstep[tindex_, temp_, matrix_, mcindex_]</code> an extra arg, so make it <code>microstep[tindex_, temp_, matrix_, mcindex_, magnetization_]</code> and feed it in as an argument.</p>
<p>By default <code>microstep</code> is looking for <code>magnetization</code> in the global scope, ... |
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Q the Platypus | 264,438 | <p>The major motivation for classifying infinities (other then the intrinsic enjoyment of mathematics) is that different infinities permit different properties.</p>
<p>Countable infinities can be reasoned about using inductive proofs. On the other hand many of the properties analysis makes use of requires uncountable... |
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Chill2Macht | 327,486 | <p>If it's any consolation, in practice you won't encounter more than two different types of infinity: that corresponding to the natural numbers, and that corresponding to the real numbers (cardinality of the continuum).</p>
<p>The reason why it's necessary to differentiate between those two is that the "size" (cardin... |
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Community | -1 | <blockquote>
<p>infinity is just... infinity?</p>
</blockquote>
<p>This is your problem; this is simply wrong. Now, to be fair, it's a fairly well entrenched wrong idea because mankind has spent thousands of years trying to reason about the infinite, and we've only really figured out <em>how</em> to do so in the pas... |
1,165,207 | <p>From browsing the internet so far I've came to the conclusion that an ordered tuple is something in which there is no repettion of the elments
Eg: An ordered 4-tuple is (1,2,3,4) or (5,3,1,7) i.e. no elements are repeated</p>
<p>But an unordered 4-tuple is something like this (1,2,2,3) or (7,1,4,1)</p>
<p>Is this ... | Robert Israel | 8,508 | <p>No, that's completely wrong. Whether ordered or unordered, there may or may not be repetitions. The correct distinction is that in an ordered tuple the order counts, and in an unordered tuple it doesn't. So $(1,2,2,3)$ and $(2,1,2,3)$ are different as ordered $4$-tuples, but they are the same as unordered $4$-tup... |
3,920,812 | <p>Please excuse if the formatting of this post is wrong.</p>
<p>There's a question that asks for the 2nd derivative of <span class="math-container">$y-2x-3xy=2$</span></p>
<p>From what I know, I have to use implicit differentiation, using which I get: <span class="math-container">$$\frac{12+18y}{(1-3x)^{2}}$$</span>
B... | boojum | 882,145 | <p>(This was too long to be a comment.)</p>
<p>You can substitute <span class="math-container">$ \ y \ $</span> into the implicit differentiation calculation when it is a single function suggested by the original equation. Often, though, that equation may be describing two or more implicit functions and it is not nece... |
3,961,131 | <p>Suppose we are given 2 predicates
<span class="math-container">$A(x)$</span> and <span class="math-container">$B(x)$</span> with domain <span class="math-container">$M$</span>.</p>
<p>Suppose next we are given the following predicate
<span class="math-container">$$\neg (A(x) \land B(x)) \land (\forall x(A(x) \righta... | Ross Millikan | 1,827 | <p>You are getting hung up on truth sets when there is only one variable in the problem. Focus on one element <span class="math-container">$x$</span> of <span class="math-container">$M$</span> and ask whether <span class="math-container">$A(x)$</span> and <span class="math-container">$B(x)$</span> can be true because ... |
401,389 | <p>I worked through some examples of Bayes' Theorem and now was reading the proof.</p>
<p>Bayes' Theorem states the following:</p>
<blockquote>
<p>Suppose that the sample space S is partitioned into disjoint subsets <span class="math-container">$B_1, B_2,...,B_n$</span>. That is, <span class="math-container">$S = B_1 ... | amWhy | 9,003 | <p>As Tharsis pointed out, and was clarified in the comments, it is all of sets of the given by $\;(A \cap B_i),\; 1 \leq i \leq n\;$ that are pairwise disjoint. </p>
<p>$$\;(A \cap B_i)\cap (A \cap B_j) = \varnothing,\;\;\;\forall i, j,\;\;\text{s.t.}\;\;1 \leq i, j\leq n\;\;\text{and}\;\;i\neq j$$</p>
<p>e.g., $(A\... |
401,389 | <p>I worked through some examples of Bayes' Theorem and now was reading the proof.</p>
<p>Bayes' Theorem states the following:</p>
<blockquote>
<p>Suppose that the sample space S is partitioned into disjoint subsets <span class="math-container">$B_1, B_2,...,B_n$</span>. That is, <span class="math-container">$S = B_1 ... | broccoli | 50,577 | <p><a href="http://bayesianthink.blogspot.com/2012/08/understanding-bayesian-inference.html" rel="nofollow">Here</a> is a write that describes Bayes theorem in detail along with a bunch of examples on how to use it (different write ups)</p>
|
1,647,673 | <p>Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$</p>
<p>Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$.</p>
<p>EDIT: I was hoping there is a way without using a known inequality, ie to prove that $RHS-LHS\geq 0$</p>
| Michael Rozenberg | 190,319 | <p>I think you mean $RHS=\sqrt{n(a_1^2+a_2^2+...+a_n^2)}$. </p>
<p>If so we have:</p>
<p>$RHS-LHS=\frac{\sum\limits_{1\leq i<j\leq n}\left(|a_i|-|a_j|\right)^2}{RHS+LHS}\geq0$, which you wished. </p>
|
760,195 | <p>I've seen this proof in a text. I have an issue with it and wanted to check its validity. </p>
<p>Let $X\sim B(n,p)$, we seek the expectation. We let $q=1-p$
\begin{equation}
E(X)=\sum_{j=0}^{n} j {n\choose j} p^{j}q^{n-j}=p\partial_{p}\sum_{j=0}^{n} {n\choose j} p^{j}q^{n-j}=\underline{p\partial_{p} (p+q)^{n}} \q... | DanielV | 97,045 | <p>There are a few common notations in mathematics where an object is used as a function, but it hides a declaration of an assumption.</p>
<p>One example is the line integral, consider (A0), the proposition $X = \oint_{\omega} y\,dz$ . This is actually two propositions, one hiding in the other:</p>
<p>$$X = \int_\om... |
236,927 | <p>I thought this would be a hard problem but I found a link that seems to ask the answer to this question as a homework problem? Can somone help me out here, are there an infinite number of prime powers that differ by 1? or are there a finite number of them? If so which are they?</p>
| André Nicolas | 6,312 | <p><a href="http://en.wikipedia.org/wiki/Catalan%27s_conjecture" rel="nofollow">Catalan's Conjecture,</a> a theorem since $2002$, shows that the only examples where the exponents are $\ge 2$ are $3^2-2^3$.</p>
<p>If we allow exponent equal to $1$, the answer is not known. Perhaps there are infinitely many Fermat prime... |
236,927 | <p>I thought this would be a hard problem but I found a link that seems to ask the answer to this question as a homework problem? Can somone help me out here, are there an infinite number of prime powers that differ by 1? or are there a finite number of them? If so which are they?</p>
| Paolo Leonetti | 45,736 | <p>In number theory, Mihailescu theorem is the solution to a famous and old conjecture formulated by French mathematician Eugene Charles Catalan in $1844$ (see "Note extraite d'une lettre adressè à l'èditeur). Although it was proved in April $2002$, it appeared for first time in Crelle's Jounal in $2004$. His formulati... |
348,324 | <p>In an interview the interviewer asked me the following but I failed to give the answer. </p>
<p>$\{0,1\}^\mathbb{N}$ with product topology is homeomorphic to which subset of $\mathbb{R}$?</p>
<p>Can anyone give me the answer and explain me please? thanks for your kind help.</p>
| Fly by Night | 38,495 | <p>This idea was dealt with by Roger Penrose in page 370 of his book "<em><a href="http://www.amazon.co.uk/Road-Reality-Complete-Guide-Universe/dp/0099440687" rel="nofollow">The Road to Reality: A Complete Guide to the Laws of the Universe</a></em>".</p>
<p>The set is called the <a href="http://en.wikipedia.org/wiki/C... |
1,412,869 | <p>Does it make sense to talk about Christoffel symbols in flat space time? Do they have non-zero values? I understand that the Christoffel symbols appear as an indication of curvature in space. So, are they non-existent in flat space-time?</p>
| Victor Caran | 529,004 | <p>As frakbak explained, one has a notion of Christoffel symbols in flat spacetime, as they basically record information about derivatives of the metric tensor with respect to different indices, each. It makes kind of sense, since changes of the metric tensor describe local changes of a scalar product which encodes inf... |
3,746,630 | <p>So I am solving some probability/finance books and I've gone through two similar problems that conflict in their answers.</p>
<h2>Paul Wilmott</h2>
<p>The first book is Paul Wilmott's <a href="https://smile.amazon.com/Frequently-Asked-Questions-Quantitative-Finance/dp/0470748753" rel="nofollow noreferrer">Frequently... | Ingix | 393,096 | <p>The answer by Brian M. Scott shows the main effect at work: In the Wilmot scenario, the <strong>expected value</strong> after 260 trading days is enourmous, but the probability of it actually being <strong>higher than the starting value</strong> is small. That's not at odds with each other, it'
s how the expected va... |
1,884,852 | <p>Suppose there are <em>k</em> dice thrown. Let <em>M</em> denote the minimum of the <em>k</em> numbers rolled. </p>
<p>I've learned that finding the individual probability is:</p>
<p>$$P(M = m) = P(M \ge m) - P(M \ge m + 1) $$</p>
<p>Can someone please explain this to me? I've tried plugging in values for $m = 1, ... | Sarvesh Ravichandran Iyer | 316,409 | <p>Note that:</p>
<p>$$
\Pr[M \geq m] = \sum_{i=m+1}^\infty Pr[M = i]
$$
because if $M \geq m$, then either $M=m$ or $M=m+1$ or $M=m+2$ or $M=m+3$ or ...
(by the way, the infinite sum exists because it is bounded above by 1)</p>
<p>Similarly,
$$
\Pr[M \geq (m+1)] = \sum_{i=(m+1)+1}^\infty Pr[M = i]
$$
because if $M ... |
4,036,896 | <p>Let <span class="math-container">$\boldsymbol{A}$</span> is a real symmetric matrix, <span class="math-container">$\boldsymbol{B}$</span> is a real antisymmetric matrix, <span class="math-container">$\boldsymbol{A}^2 = \boldsymbol{B}^2$</span>, prove <span class="math-container">$\boldsymbol{A} = \boldsymbol{B} = \b... | user8675309 | 735,806 | <p>note the orthogonality<br />
<span class="math-container">$-1\cdot \text{trace}\big(BA\big)= \text{trace}\big(B^TA\big)= \text{trace}\big(B^TA\big)^T= \text{trace}\big(A^TB\big)= \text{trace}\big(AB\big)= \text{trace}\big(BA\big)$</span><br />
<span class="math-container">$\implies \text{trace}\big(BA\big)= 0$</spa... |
2,948,327 | <p>Being an undergraduate student I find difficult to understand the perfect differences between normal and partial differential equations. Elaborate the answer </p>
| gandalf61 | 424,513 | <p>Let's look at some examples of ODEs and PDEs in physics:</p>
<p>1) A particle moves under the influence of gravity, electromagnetic forces, viscosity or other forces. The position of the particle is a function of a single independent variable (time) so we can represent the equation of motion of the particle by an O... |
2,246,025 | <p>How do I solve this?
$$y^\prime = y^2 -4$$
I think I am supposed to use the separable equations method and then use partial fractions.</p>
| badjohn | 332,763 | <p>Try studying infinite numbers. They will be quite different from anything in software. (Though they aren't particularly new.)</p>
<p>You could look at octonians.</p>
|
104,818 | <p>Can anyone help me with this? I want to know how to solve it.</p>
<blockquote>
<p>Let $f:\mathbb R \longrightarrow \mathbb R$ be a continuous function with period $P$. Also suppose that $$\frac{1}{P}\int_0^Pf(x)dx=N.$$ Show that $$\lim_{x\to 0^+}\frac 1x\int_0^x f\left(\frac{1}{t}\right)dt=N.$$</p>
</blockquote>... | Davide Giraudo | 9,849 | <p>Using the function $g(t)=f\left(\frac P{2\pi}t\right)$, we can assume that $f$ is $2\pi$-periodic. If $P_n$ is a sequence of trigonometric polynomial which converges uniformly to $f$, then
$$\left|\frac 1x\int_0^xf\left(\frac 1t\right)dt-\frac 1{2\pi}\int_0^{2\pi}f(t)dt\right|\leq 2\sup_{s\in\mathbb R}|f(s)-P_n(s)|... |
85,052 | <p>A housemate of mine and I disagree on the following question: </p>
<p>Let's say that we play a game of yahtzee. Of the five dice you throw, two dice obtain the value 1, two other dice obtain the value 2, and one die shows you six dots on the top side. Given the fact that you haven't thrown a "full house" yet, you s... | David Mitra | 18,986 | <p>Imagine throwing that die twice. Repeat this 999 times. </p>
<p>Out of the 999 trials, how many of them would have the first throw showing 1 or 2?
Well, around 1/3 of them. 333 of the 1000 trials have the first throw resulting in 1 or 2.</p>
<p>Now, how many of the trials will have the second throw resulting in 1 ... |
1,884,958 | <p>How do I show that <span class="math-container">$$\text{Var}(aX+b)=a^2\text{Var}(X).$$</span>
Since I am reading statistics for the first time, I don't have any idea how to start.</p>
<p>Thanks for helping me.</p>
| RandomGuy | 153,631 | <p>Directly from the definition: $Var(aX)=E[(aX)^2] - E[(aX)]^2=E[a^2X^2]-E[(aX)]^2=a^2E[X^2]-(aE[X])^2=a^2E[X^2]-a^2E[X]^2=a^2(E[X^2]-E[X]^2)=a^2Var(X),$ where in the third and fourth equality, I have applied the linearity of Expectation, in the form $E[cX]=cE[X]$.</p>
<p>Next, observe $Var[Y+b]=Var(Y)$, with a simil... |
121,909 | <p>I came across this question while studying primitive roots. I know it has something to do with the fact that if the order of $a$ is $m$ then for every $k \in \mathbb{Z}$, the order of $a^k$ is $m/(m,k)$. The question is as follows: </p>
<blockquote>
<p>Let $p$ be an odd prime. Prove that $a^2$ is never a primi... | JavaMan | 6,491 | <p>The order of $a^2$ is no more than $\frac{p-1}{2}$ since $a^{p-1} \equiv 1\pmod{p}$ by <a href="http://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">Fermat's Little Theorem</a>.</p>
|
121,909 | <p>I came across this question while studying primitive roots. I know it has something to do with the fact that if the order of $a$ is $m$ then for every $k \in \mathbb{Z}$, the order of $a^k$ is $m/(m,k)$. The question is as follows: </p>
<blockquote>
<p>Let $p$ be an odd prime. Prove that $a^2$ is never a primi... | Arturo Magidin | 742 | <p>$g$ is a primitive root modulo $p$ if and only if the order of $g$ modulo $p$ is $p-1$.</p>
<p>If the order of $a$ is not $p-1$, then $a^2$ has order less than or equal to the order of $a$, hence is not a primitive root.</p>
<p>If the order of $a$ <em>is</em> $p-1$ then what is the order of $a^2$?</p>
|
389,675 | <p>I'm trying to use a program to find the largest prime factor of 600851475143.
This is for Project Euler here: <a href="http://projecteuler.net/problem=3">http://projecteuler.net/problem=3</a></p>
<p>I first attempted this with the code that goes through every number up to 600851475143, tests its divisibility, and a... | apnorton | 23,353 | <p>Project Euler problems (at least the ones that I have done) tend to deal with a lot of number theory topics. So, reading an introductory number theory book could be helpful.</p>
<p>With regards to your particular situation, I suggest finding primes first, then testing the primes for divisibility. That is, to find... |
138,723 | <p>By cleaning up a notebook, I mean how can I hide all the codes in the notebook so that the end-users can't see it? I saw Eric Schulz's famous interactive calculus textbook, the users can't see the code, and there is no cell brackets on the right hand side of the CDF. </p>
| Carl Woll | 45,431 | <p>Another possibility is to modify the notebooks style sheet by defining a new "Screen Environment" that hides cell brackets and closes input cells:</p>
<pre><code>SetOptions[
EvaluationNotebook[],
StyleDefinitions -> Notebook[{
Cell[StyleData[StyleDefinitions->"Default.nb"]],
Cell[Style... |
1,321,544 | <p>How do you evaluate the following?</p>
<p>$$\cos\left[\cos^{-1}\left(\frac{3}{4}\right)\right]$$</p>
<p>To me the cosine of an arc cosine is just the value, which would be $3/4$.</p>
| Mythomorphic | 152,277 | <p>One possible way is to draw a right-angled triangle with hypotenuse$=4$, base$=3$.</p>
<p>So we have angle between hypotenuse and base, $\theta=\arccos\frac34$.</p>
<p>So in the same triangle, $\cos\left[\cos^{-1}\left(\frac{3}{4}\right)\right]=\cos\theta=\frac34$</p>
|
1,457,838 | <p>I am given position vectors: $\vec{OA} = i - 3j$ and $\vec{OC}=3i-j$.
And asked to find a position vector of the point that divides the line $\vec{AC}$ in the ratio $-2:3$. </p>
<p>So I found the vector $\vec{AC}$, and it is $2i+2j$. Then, if the point of interest is $L$, position vector $\vec{OL} = \vec{OA} + \lam... | Samrat Mukhopadhyay | 83,973 | <p>If $p$ is prime then the later statement is obvious. Now, let $p>1$ and $\forall 1<n\le \sqrt{p},\ n\not| p$. If $p$ is not prime, then $\exists\ 1<a\le b<p$ such that $p=ab\implies p\ge a^2\implies a\le \sqrt{p}$ and $a|p$ which is a contradiction. </p>
|
3,187,451 | <p>Can you help me find a function <span class="math-container">$f(X,Y)$</span>, such that <span class="math-container">$f(1,x) = f(x,1) = f(\ln x, \ln x)$</span>?</p>
<p>Either always, for all <span class="math-container">$x$</span> or in the limit <span class="math-container">$x$</span> tends to infinity, all these ... | Botond | 281,471 | <p>What do you think about
<span class="math-container">$$f(x,y)=(a_1,...,a_n)$$</span>
For as many constants (<span class="math-container">$n$</span>) as you'd like: <span class="math-container">$a_1,...,a_n \in \mathbb{R}$</span>?</p>
|
3,187,451 | <p>Can you help me find a function <span class="math-container">$f(X,Y)$</span>, such that <span class="math-container">$f(1,x) = f(x,1) = f(\ln x, \ln x)$</span>?</p>
<p>Either always, for all <span class="math-container">$x$</span> or in the limit <span class="math-container">$x$</span> tends to infinity, all these ... | Empy2 | 81,790 | <p><span class="math-container">$$\exp(\sqrt{(X-1)(Y-1)}+1)+|X-Y|$$</span>
<span class="math-container">$f(x,1)=f(1,x)=x+e-1; f(\ln(x),\ln(x))=x$</span></p>
|
3,270,856 | <p>So i have to calculate this triple integral:</p>
<p><span class="math-container">$$\iiint_GzdV$$</span>
Where G is defined as: <span class="math-container">$$x^2+y^2-z^2 \geq 6R^2, x^2+y^2+z^2\leq12R^2, z\geq0$$</span></p>
<p>So with drawing it it gives this:</p>
<p><a href="https://i.stack.imgur.com/60hok.jpg" r... | user10354138 | 592,552 | <p>The condition
<span class="math-container">$$
x^2+y^2-z^2\geq 6R^2
$$</span>
gives
<span class="math-container">$$
r^2(\sin^2\theta-\cos^2\theta)\geq 6R^2
$$</span>
So you want <span class="math-container">$(r,\theta)\in(0,\infty)\times(0,\frac\pi2)$</span> such that
<span class="math-container">$$
r^2\leq 12R^2 \te... |
3,033,812 | <p>My problem: If there are 5 different candies in a jar and a child wants to take out one or more candies, how many ways can this be done? </p>
<p>I said it is <span class="math-container">$^5C_1 -\; ^5C_0 = 5-1 = 4$</span> ways. The <span class="math-container">$-1$</span> for the unwanted case using this trick:</p>... | Especially Lime | 341,019 | <p>Generally you use a power whenever you have several independent choices to make, and each choice has the same options.</p>
<p>Here, for each candy in the jar you can choose to take it, or not (<span class="math-container">$2$</span> options). All these decisions are independent, and there are <span class="math-cont... |
198,995 | <p>From Barbeau's <em>Polynomials</em>:</p>
<blockquote>
<ul>
<li>(a) Is it possible to find a polynomial, apart from the constant $0$ itself, which is identically equal to $0$ (i.e. a polynomial
$P(t)$ with some nonzero coefficient such that $P(c)=0$ for each
number $c$)?</li>
</ul>
</blockquote>
<p>And t... | Arthur | 15,500 | <p>I don't know if this will answer all your questions, but what the author is doing is to assume there is a non-zero polynomial which evaluates to zero at all points, then reach a contradiction. He does this by analyzing the hypothetical polynomial in a number of different ways:</p>
<hr>
<p>Assume the polynomial $p(... |
202,041 | <p>I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, and is two dimensional for the torus, but the higher dimensional cases stump me, and I am unable to find the result. An... | Sam Nead | 1,650 | <p>This is the geometric version of Riemann's "problem of moduli". A connected, closed, oriented surface of genus $g$ has a moduli space of dimension $6g - 6$. </p>
<p>One way to count the dimension of hyperbolic structures is via <a href="http://en.wikipedia.org/wiki/Fenchel%E2%80%93Nielsen_coordinates" rel="nofoll... |
390,129 | <p>Let <span class="math-container">$O$</span> be a <span class="math-container">$d$</span>-dimensional rotation matrix (i.e., it has real entries and <span class="math-container">$OO^T = O^TO = I$</span>). Let <span class="math-container">$\mathbf{x}$</span> be a uniformly random bitstring of length <span class="math-... | Marco | 143,536 | <p>Adding more detail to Mikael's point, the result seems to hold <strong>on average</strong> over <span class="math-container">$O$</span> because of the following:</p>
<ol>
<li><p>Using a <a href="https://en.wikipedia.org/wiki/Chernoff_bound" rel="nofollow noreferrer">Chernoff bound</a>, we can see that the probabilit... |
970,270 | <p><strong>Problem</strong></p>
<p>On the one hand, a complex measure decomposes into:
$$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$</p>
<p>This gives rise to the integrability condition:
$$f\in L(\mu)\iff f\in L(\mu_\alpha)\quad(\alpha=1,\ldots,3)$$</p>
<p>On the other hand... | Disintegrating By Parts | 112,478 | <p>If we have a signed measure $\mu$ on a measurable space $(\Omega,\Sigma)$, then $\mu$ may be allowed to assume values of $\pm \infty$, but not both. A complex measure $\mu$ is not allowed to assume any type of infinite value. For either type of measure $\mu$ which is not allowed to assume infinite values, the variat... |
970,270 | <p><strong>Problem</strong></p>
<p>On the one hand, a complex measure decomposes into:
$$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$</p>
<p>This gives rise to the integrability condition:
$$f\in L(\mu)\iff f\in L(\mu_\alpha)\quad(\alpha=1,\ldots,3)$$</p>
<p>On the other hand... | C-star-W-star | 79,762 | <blockquote>
<p>In fact, the very very heart of the whole story here are the two inequalities:
$$|\mu(E)|\leq|\mu|(E)<\infty\quad(E\in\Sigma)$$</p>
</blockquote>
<p>For positive measures given as derivative:
$$\kappa(E)=\int_Eh\mathrm{d}\lambda\quad(h\geq0)$$
the positive integrals agree:
$$\int f\mathrm{d}\kap... |
3,163,697 | <p>So, I have to approximate a value of square root of e: <span class="math-container">$\sqrt{e}$</span> with a precision of <span class="math-container">$10^{-3}$</span>.
I have calculated the first and second derivative:
So instead of <span class="math-container">$\sqrt{e}$</span> I need to approximate the value for... | Carl Christian | 307,944 | <p>There are some fundamental issues which have yet to be addressed. </p>
<p>In general, our goal is to compute a target value <span class="math-container">$T \in \mathbb{R}$</span>. Typically, we cannot compute the exact value, so we settle for an approximation <span class="math-container">$A$</span>. The error is by... |
2,900 | <p>I saved an <code>InterpolationFunction</code> in a ".mx" files using <code>DumpSave</code> on a variable that was scoped by a <code>Module</code>. Here is a stripped-down example:</p>
<pre><code>Module[{interpolation},
interpolation=Interpolation[Range[10]];
DumpSave["interpolation.mx", interpolation];
]
</co... | Sjoerd C. de Vries | 57 | <p>You can open the MX file in an ASCII editor. The variable name is there in plain text (interpolation$511 in my case). The rest is binary gibberish.</p>
<p>So given an <code>MX</code> file with a single variable with the <code>$</code> suffix, the following expression can be used to access that variable directly:</p... |
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | Zalnd | 478,981 | <p><span class="math-container">$$\frac{4\tan(x)}{1-\tan^2(x)} = \frac{3}{\tan(x)}$$</span></p>
<p><span class="math-container">$$\frac{4\tan(x)}{1-\tan^2(x)} - \frac{3}{\tan(x)} = 0$$</span></p>
<p><span class="math-container">$$\frac{4\tan^2(x)-3[1-\tan^2(x)]}{\tan(x)[1-\tan^2(x)]} = 0$$</span></p>
<p><span class="ma... |
1,453,067 | <p>My friend's professor raised this question in a coaching and he and I tried everything we could think of. But later I thought that since $\sin (2x) $ can have values only between -1 & +1 and anything but +1 makes the equation complex ( keeping in mind that the integral is meant to be non-complex), there is no so... | Brevan Ellefsen | 269,764 | <p>$$\int \sqrt {\sin(2x)-1}dx$$
Substitute $u=2x$ and $du=2dx$
$$= \frac{1}{2}\int \sqrt {\sin(u)-1}du$$
Substitute $s=\sin u -1$ and $ds = \cos u \,du$ and notice that $\cos u = \sqrt{1-\sin^2u} = \sqrt{1-(s+1)^2} = \sqrt{1-(s^2 + 2s + 1)} = \sqrt{1- s^2 - 2s - 1} = \sqrt{-s^2 - 2s} = \sqrt{-s(s+2)}$
$$= \frac{1}{... |
2,529,262 | <p>I have five real numbers $a,b,c,d,e$ and their arithmetic mean is $2$. I also know that the arithmetic mean of $a^2, b^2,c^2,d^2$, and $e^2$ is $4$. Is there a way by which I can prove that the range of $e$ (or any ONE of the numbers) is $[0,16/5]$. I ran across this problem in a book and am stuck on it. Any help w... | quasi | 400,434 | <p>\begin{align*}
&(a - 2)^2 + (b-2)^2 + (c-2)^2 + (d-2)^2 + (e - 2)^2\\[4pt]
&= (a^2 + b^2 + c^2 + d^2 +e^2) - 4(a + b + c + d +e) + 20\\[4pt]
&= 20 - 4(10) + 20\\[4pt]
&= 0\\[10pt]
&\;\text{hence}\\[10pt]
&\;a = b = c = d = e = 2\\[4pt]
\end{align*}</p>
|
3,791,936 | <p>An advanced sum <a href="https://www.facebook.com/photo.php?fbid=3190290677734375&set=a.222846247812181&type=3&theater" rel="noreferrer">proposed</a> by Cornel Valean:</p>
<blockquote>
<p><span class="math-container">$$S=\sum_{n=1}^\infty\frac{2^{2n}H_{n+1}}{(n+1)^2{2n\choose n}}$$</span>
<span class="ma... | Ali Shadhar | 432,085 | <p>Following @Felix's idea above:</p>
<p><span class="math-container">$$S=\sum_{n=1}^\infty\frac{2^{2n}H_{n+1}}{(n+1)^2{2n\choose n}}=\sum_{n=2}^\infty\frac{2^{2n-2}H_n}{n^2{2n-2\choose n-1}}$$</span></p>
<p>Note that</p>
<p><span class="math-container">$$\frac{{2n+2\choose n+1}}{{2n\choose n}}=\frac{\frac{\Gamma(2n+3)... |
258,132 | <p>Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by many as a new axiom (or some stronger principle which implied it). This is because we intuitively think that if we ... | Maxime Ramzi | 102,343 | <p>By definition, if a sentence $\phi$ is independent from the theory $T$, then both $T + \phi$ and $T+\neg \phi$ are coherent. Gödel's completeness theorem shows that if a theory is coherent, then it is consistent, meaning there is a model that satisfies it. In less technical terms, if you cannot derive a contradictio... |
263,650 | <p>As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation problem" a differential graded Lie algebra or $L_{\infty}$-algebra that controls it. </p>
<p>I've seen this idea re... | Justin Hilburn | 333 | <p>I haven't thought about Einstein metrics but, as AHusain mentioned, Kevin Costello has written down many examples. The keyword is elliptic moduli problem. Look at <a href="https://arxiv.org/abs/1111.4234" rel="nofollow noreferrer">https://arxiv.org/abs/1111.4234</a></p>
|
263,650 | <p>As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation problem" a differential graded Lie algebra or $L_{\infty}$-algebra that controls it. </p>
<p>I've seen this idea re... | domenico fiorenza | 8,320 | <p>The Quillen-Drinfeld-Deligne-etc. philosopy should not be looked at as something too mysterious.</p>
<p>Namely, it reduces to the fact that if the set of objects one is interesting in the infinitesimal deformations of is not too wild, then it can be described in the form $f(v)+Q(v)=0$, where $f:V\to W$ is a linear ... |
2,945,913 | <p>I have a quick question about simplifying these exponents and then comparing them:</p>
<p><span class="math-container">$8^{\log_2 n}, 2^{3log_2(log_2n)}$</span> and <span class="math-container">$2^{(log_2(n))^2} $</span></p>
<p>I know the third one evaluates to <span class="math-container">$n^{log_2(n)}$</span>, b... | Matthew Hunt | 257,736 | <p>The usual method for integrals of this form is to use</p>
<p><span class="math-container">$$t=\tan\frac{u}{2}$$</span></p>
|
1,822,562 | <p>Please explain what method you used to prove so.
$$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$</p>
| Mark Viola | 218,419 | <p>In <a href="https://math.stackexchange.com/questions/1517704/limit-lim-x-to0-1-tan9x-frac1-arcsin5x/1520012#1520012">THIS ANSWER</a>, I showed using elementary inequalities from geometry that the tangent function satisfies the inequalities </p>
<p>$$x\le \tan(x)\le x\sec(x) \tag 1$$</p>
<p>for $0\le x<\pi/2$. ... |
3,066,967 | <p>Prove that <span class="math-container">$k^2+k+1$</span> is not divisible by <span class="math-container">$101$</span> for any natural <span class="math-container">$k.$</span></p>
| Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$$k^2+k+1\equiv0(\mod101).$$</span>
<span class="math-container">$k\equiv0(\mod101)$</span> is impossible, which says that we can assume that <span class="math-container">$k$</span> is not divisible by <span class="math-container">$101$</span>.</p>
<p>Now, <span class="math-containe... |
199,738 | <p>It is known that, if a function $f$ from a planar domain $D$ to a Banach space $A$ is weakly analytic [i.e. $l(f)$ is analytic for every $l$ in $A^*$], then $f$ is strongly analytic [i.e. $\lim_{h \to 0} h^{-1}[f(z+h)-f(z)]$ exists in norm for every $z$ in $D$].</p>
<p>Now the question is, if above $f$ is assumed t... | Rudy the Reindeer | 5,798 | <p>Let $X$ be an infinite dimensional Banach space. Let $T_w$ denote the weak topology and let $T$ denote the norm topology. Then $\mathrm{id}: (X, T_w) \to (X, T)$ is not continuous but $\mathrm{id}: (X, T_w) \to (X, T_w)$ is. </p>
<p>Note that every map $f: X \to Y$ that is weakly continuous where weakly means $f: (... |
2,972,938 | <p>When is it possible to make a change of variables in the limit?</p>
<p>For example <span class="math-container">$\lim_{x \to \infty}(\ln x/x)$</span>, can I change <span class="math-container">$x=e^{y}$</span>?</p>
<p>Then <span class="math-container">$\lim_{x \to \infty}(\ln x/x)= \lim_{y \to \infty}(y/e^{y})$</s... | user247327 | 247,327 | <p>What?? If "<span class="math-container">$A^2$</span> is the identity matrix" then <span class="math-container">$A^3= A(A^2)= A$</span>. No "calculation" at all required!</p>
|
222,639 | <p>It seems to be true that multiply transitive permutation groups have been classified completely (using CFSG), but I am having trouble finding a reference where this classification is actually stated. Is there a canonical reference?</p>
| Michael Zieve | 30,412 | <p>This is a bit absurd as a reference, but if you just want an explicit list that has the dubious merit of having gotten past a referee, the <span class="math-container">$3$</span>-transitive groups are listed on pp. 86-87 of Abhyankar's paper <a href="https://www.ams.org/journals/bull/1992-27-01/S0273-0979-1992-00270... |
2,205,042 | <p>I want to show that there exists some $M$ such that for any $n$ and any $x \in [\epsilon, 2\pi - \epsilon]$ we have $\left |\sum_{m = 1}^n e^{imx} \right|\leq M$. Geometrically, it is like starting at the origin facing east, then turning left by $x$ degrees and moving forward by 1 unit of distance, and repeating thi... | mrp | 134,447 | <p>Well, you are missing the crucial parts. First of all, the set of accepting states should be $F_1 \cup F_2$, not the intersection. The idea is that we start in $M_1$, and when we have accepted a word $w_1$ in $M_1$, then we non-deterministically choose between either stopping there, or continue reading a word $w_2$ ... |
510,633 | <p>A independent variable is "the input" and the dependent variable is the "output", atleast thats how it was explained to us.</p>
<p>But if you have some random function can't both variables be seen as "affecting" the other variable?</p>
<p>For example, in $ y = 1/x$, "x" could be seen as an input and "y" the output... | Brian M. Scott | 12,042 | <p>HINT: Use the binomial theorem to expand both sides of the desired inequality. For example,</p>
<p>$$\left(1+\frac1n\right)^n=\sum_{k=0}^n\binom{n}k\left(\frac1n\right)^k\;.$$</p>
<p>Note that</p>
<p>$$\binom{n}k=\frac{n(n-1)(n-2)\ldots(n-k+1)}{k!}\;.$$</p>
|
1,136,060 | <p>What identity would I need to use to solve for $\theta$? </p>
<p>$5 + \cos(\theta) = 7\sin(\theta)$ </p>
<p>By plugging this into a calculator, I was able to get $\theta \approx 53.13^\circ$. </p>
| copper.hat | 27,978 | <p>Write $7 \sin \theta - \cos \theta = 5$, then
${7 \over \sqrt{50}} \sin \theta - {1 \over \sqrt{50}}\cos \theta = {5 \over \sqrt{50}}$.</p>
<p>Now note that $({7 \over \sqrt{50}})^2 + ({1 \over \sqrt{50}})^2 = 1$. Now find
$\psi$ such that $\cos \psi = {7 \over \sqrt{50}}$, $\sin \psi = {1 \over \sqrt{50}}$, then t... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.