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 |
|---|---|---|---|---|
2,806,858 | <p>There is an equation
$$\sin2\theta=\sin\theta$$
We need to show when the right-hand side is equal to the left-hand side for $[0,2\pi]$.
<hr>
Let's rewrite it as
$$2\sin\theta\cos\theta=\sin\theta$$
Let's divide both sides by $\sin\theta$ (then $\sin\theta \neq 0 \leftrightarrow \theta \notin \{0,\pi,2\pi\}$)
$$2\cos... | poyea | 498,637 | <p>You only consider solutions for $\cos\theta=\frac{1}{2}$ in the first case. In fact if you perform such cancellation, you should consider also solutions given by $\sin\theta=0$. </p>
<p>Why? Multiply both sides by zero:$$2\cos\theta=1,\,\,2\cos\theta\,\cdot0=1\,\cdot0$$ Bare in mind $\sin\theta$ could be $0$, you h... |
3,891,336 | <p>I have a problem with this question:</p>
<p>we have a language with alphabet {a, b, c}, all strings in this language have even length and does not contain any substring "ab" and "ba" for example these strings acceptable: "accb", "aa", "bb", "bcac", and thes... | saulspatz | 235,128 | <p>We need <span class="math-container">$7$</span> non-terminals. <span class="math-container">$S$</span> is the initial state, <span class="math-container">$T,\ U,$</span> and <span class="math-container">$V$</span> all indicate that an odd number of characters have been read, and that the last character read was <sp... |
1,323,317 | <p>I have to compute the following quantity:</p>
<p>$$
1) \sum\limits_{k=0}^{n} \binom{n}{k}k2^{n-k}
$$</p>
<p>Moreover, I have to give an upper bound for the following quantity:</p>
<p>$$
2) \sum\limits_{k=1}^{n-2} \binom{n}{k}\frac{k}{n-k}
$$</p>
<p>As regards 1), I see that $\binom{n}{k}k2^{n-k}=\frac{n! ... | André Nicolas | 6,312 | <p>I prefer the combinatorial approach, but we can do it by manipulation. First note that our sum is
$$\sum_{k=1}^n k\binom{n}{k}2^{n-k}\tag{1}$$
since the $k=0$ term makes no contribution to the sum.
Then use the fact that $\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}$ to rewrite (1) as
$$n\sum_{k=1}^n \binom{n-1}{k-1}2^... |
3,416,600 | <p>Show that <span class="math-container">$|{\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}|\le|b-c|$</span> where <span class="math-container">$a,b,c\in\mathbb{R}$</span></p>
<p>I'd like to get an hint on how to get started. What I thought to do so far is dividing to cases to get rid of the absolute value. <span class="math-containe... | Ris | 318,407 | <p>It will be easy if you think <span class="math-container">$\sqrt{a^2 + b^2}$</span> as the euclidean distance. Consider the three points <span class="math-container">$A(a, 0), B(0, b), C(0, c)$</span>. Then the inequality can be transformed into the triangular inequality <span class="math-container">$\lvert \overlin... |
165,900 | <p>Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) \in k p$. My question: is $G$ some well known algebraic group? </p>
| Jim Humphreys | 4,231 | <p>The best known situation of this type involving an algebraic group would occur in type $E_6$, where there is a long history and quite a bit of literature. Is this the "well known algebraic group" you have in mind? In case you have access to MathSciNet, you can find a list of literature cited by a fairly recent a... |
165,900 | <p>Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) \in k p$. My question: is $G$ some well known algebraic group? </p>
| Jérémy Blanc | 23,758 | <p>If the field is $\mathbb{C}$ (or algebraically closed of characteristic $\not=2,3$), then you can put any smooth cubic into the Hessian form:
$$X^3+Y^3+Z^3+\lambda XYZ=0$$
for some $\lambda\in \mathbb{C}$. This corresponds to put the nine inflection onto the intersection of the curve with $XYZ=0$.</p>
<p>Then, the ... |
4,436,210 | <p>I have been given this exercise: Calculate the double integral:</p>
<blockquote>
<p><span class="math-container">$$\iint_D\frac{\sin(y)}{y}dxdy$$</span>
Where <span class="math-container">$D$</span> is the area enclosed by the lines: <span class="math-container">$y=2$</span>, <span class="math-container">$y=1$</span... | A. P. | 1,027,216 | <p>This type of problem is easier to analyse directly from the matrix avoiding going back to the system of linear equations as follows, this avoids some confusion.</p>
<p>The system can be reduced by row as
<span class="math-container">$$\begin{bmatrix} 1 & 2 & -3 & | & 4\\ 3 & -1 & 5 & | &a... |
3,722,407 | <p>I am struggling with this problem:</p>
<blockquote>
<p>Let n be an even number, and denote <span class="math-container">$[n]=\{1,2,...,n\}$</span>. A sequence of
sets <span class="math-container">$S_1 , S_2 , \cdots , S_m \subseteq [n]$</span> is considered <em>graceful</em>
if:</p>
<ol>
<li><span class="math-contai... | metamorphy | 543,769 | <p>Here's an idea to get a <strong>complete</strong> asymptotics (obviously not in <em>fixed</em> powers of <span class="math-container">$n$</span>, too).</p>
<p>For a fixed <span class="math-container">$n>1$</span>, the solution <span class="math-container">$w=w_n(z)$</span> of <span class="math-container">$w=1+zw^... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Daniel Moskovich | 2,051 | <p><a href="http://en.wikipedia.org/wiki/Smale_conjecture" rel="nofollow noreferrer">The Smale Conjecture</a>.</p>
<hr>
<p>This was <a href="http://www.jstor.org/discover/10.2307/2007035?uid=3738992&uid=2129&uid=2&uid=70&uid=4&sid=21103240235653" rel="nofollow noreferrer">proven by Hatcher in 1983... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Dietrich Burde | 32,332 | <p>The proof of the <a href="http://en.wikipedia.org/wiki/Oppenheim_conjecture"><em>Oppenheim conjecture</em></a> by G. A. Margulis in $1986$ may qualify.
It is a famous result, $27$ years ago, has a hard proof, which has not been dramatically simplified (if I am not mistaken, the simplification of Dani and Margulis n... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Alexandre Eremenko | 25,510 | <p>A major 19th century result is the general Uniformization theorem: Every simply connected Riemann surface is conformally equivalent either to the plane or to the unit disc or to the sphere. There were improvements of the proof, and many different proofs, but simplifications are not "dramatic". It is still difficult ... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Yiftach Barnea | 5,034 | <p>The <a href="https://en.wikipedia.org/wiki/Burnside%27s_problem" rel="nofollow noreferrer">Restricted Burnside Problem</a> asked whether there is a bound on the size of a finite group with <span class="math-container">$d$</span> generators and exponent <span class="math-container">$n$</span>. In the 1950s, Kostrikin... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Lasse Rempe | 3,651 | <p>The <a href="http://www.jstor.org/stable/2944326?seq=1#page_scan_tab_contents" rel="nofollow">Benedicks-Carleson theorem</a> on the existence of strange attractors for Hénon maps is an example, I would say. Over the years, there have been some attempts to give improved presentations of the proof, but I don't believe... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Larry B. | 141,260 | <p>Chazelle's <a href="https://link.springer.com/article/10.1007/BF02574703" rel="noreferrer">linear time algorithm for the triangulation of a polygon</a> has not been improved upon since its creation in 1991.</p>
<p>Technically, this is a computer science theorem, but I think it belongs here for a couple reasons. It'... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | MoebiusCorzer | 283,812 | <p>I want to give a more analytic (and probably less intuitive) way of seeing why this is not true. Let's fix $A$, the area of a triangle. Let $a,b>0$ be the respective lengths of two sides of a triangle. Now, let $x$ be the angle between the side of length $a$ and the one of length $b$.</p>
<p>Then, we know that t... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | tim | 401,692 | <p>A rectangle, by definition has $4$ right angles (the clue is in rect and angle). So, no, no need for all the clever workings. All <em>rectangles</em> are parallelograms, but not all <em>parallelograms</em> can be rectangles. Only the ones with all four corners being $90$ degrees each. </p>
|
1,043,266 | <p>Carefully see this problem(I have solved them on my own, I'm only talking about the magical coincidence):</p>
<blockquote>
<p>A bag contains 6 notes of 100 Rs.,2 notes of 500 Rs., 3 notes of 1000 Rs..Mr. A draws two notes from the bag then Mr. B draws 2 notes from the bag.<br>
(i)Find the probability that A has... | turkeyhundt | 115,823 | <p>In the case where A takes some of the bills that will make 600, yes, the probability that B can also get 600 goes down, but in the cases where A takes some bills that do not contribute to a combination of 600, the probability B can get 600 goes up! So it balances out to be the same.</p>
<p>You can also see why the... |
668,664 | <p>Solve $\dfrac{\partial u}{\partial t}+u\dfrac{\partial u}{\partial x}=x$ subject to the initial condition $u(x,0)=f(x)$.</p>
<p>I let $\dfrac{dt}{ds}=1$ , $\dfrac{dx}{ds}=u$ , $\dfrac{du}{ds}=x$ and the initial conditions become: $t=0$ , $x=\xi$ and $u=f(\xi)$ when $s=0$ .</p>
<p>I believe this leads to $t=s$ , bu... | doraemonpaul | 30,938 | <p>Follow the method in <a href="http://en.wikipedia.org/wiki/Method_of_characteristics#Example" rel="nofollow">http://en.wikipedia.org/wiki/Method_of_characteristics#Example</a>:</p>
<p>$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$</p>
<p>$\begin{cases}\dfrac{dx}{ds}=u\\\dfrac{du}{ds}=x\end{cases}$</p>
<p>$... |
192,072 | <p>Bonjour!<br>
I'm trying this number-theory problem, but i don't have any idea how to solve it.<br>
Can you give me some hints ?</p>
<p>We have got any $\mathbb{Z_+}$ number. Let it be $n$.<br>
Then we must proof that $2 \nmid \sigma(n) \implies n = k^2 \vee n = 2k^2$.<br>
Thanks for any help </p>
| robjohn | 13,854 | <p><strong>Hint:</strong> If the prime factorization of $n$ is
$$
n=\prod_k p_k^{e_k}\tag{1}
$$
then
$$
\begin{align}
\sigma(n)
&=\prod_k\frac{p_k^{e_k+1}-1}{p_k-1}\\
&=\prod_k\left(1+p_k+p_k^2+\dots+p_k^{e_k}\right)\tag{2}
\end{align}
$$
and count the number of summands in $(2)$.</p>
|
3,712,699 | <p>Let <span class="math-container">$K$</span> be a number field with ring of integers <span class="math-container">$\mathcal{O}_K$</span> and let <span class="math-container">$p$</span> be a rational prime. Let <span class="math-container">$(p) = \mathfrak{p}_1^{e_1}\ldots\mathfrak{p}_r^{e_r}$</span> be the prime fact... | GreginGre | 447,764 | <p>It is known that the different ideal <span class="math-container">$D_K$</span> is divisible by <span class="math-container">$\mathfrak{p}_1^{e_1-1}\cdots \mathfrak{p}_r^{e_r-1}$</span>, hence contained in <span class="math-container">$\mathfrak{p}_1^{e_1-1}\cdots \mathfrak{p}_r^{e_r-1}$</span> . Therefore <span clas... |
188,900 | <p><strong>Bug introduced in 10.0 and persisting through 11.3 or later</strong></p>
<hr>
<p>In <code>11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)</code> writing:</p>
<pre><code>f[w_, x_, y_, z_] := w*x^2*y^3 - z*(w^2 + x^2 + y^2 - 1)
eqn = {D[f[w, x, y, z], w] == 0,
D[f[w, x, y, z], x] == 0,
... | OkkesDulgerci | 23,291 | <p>You can use <code>Reduce</code></p>
<pre><code>f[w_, x_, y_, z_] := w*x^2*y^3 - z*(w^2 + x^2 + y^2 - 1)
eqn = {D[f[w, x, y, z], w] == 0, D[f[w, x, y, z], x] == 0,
D[f[w, x, y, z], y] == 0, D[f[w, x, y, z], z] == 0};
red = Reduce[eqn, Backsubstitution -> True]
</code></pre>
<blockquote>
<p><span class="mat... |
1,624,221 | <p>For the former one, I am aware that if let $F(x)=\int_a^x f(t)dt$, then it also equals $\int_0^x f(t)dt-\int_0^a f(t)dt$, so $F'(x)= f(x)-0=f(x)$. But who can tell me why $\int_0^a f(t)dt$ is $0$?</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: $\dfrac{1}{n^2-1} = \dfrac{1}{2}\cdot \left(\dfrac{1}{n-1}-\dfrac{1}{n+1}\right) = \dfrac{1}{2}\left(\dfrac{1}{n-1}-\dfrac{1}{n}\right) + \dfrac{1}{2}\left(\dfrac{1}{n}-\dfrac{1}{n+1}\right)$. From this you see that there are $2$ sums you calculate, and using telescoping the first sum is $\dfr... |
3,189,303 | <p>What is the point of constant symbols in a language?</p>
<p>For example we take the language of rings <span class="math-container">$(0,1,+,-,\cdot)$</span>.
What is so special about <span class="math-container">$0,1$</span> now? What is the difference between 0 and 1 besides some other element of the ring?</p>
<p>... | Mark Kamsma | 661,457 | <p>Clive's answer is already a good one, I just wanted to add another important point about constants. They give us the power to say infinitely many things about one element.</p>
<p>For example, if we consider <a href="https://en.wikipedia.org/wiki/Peano_axioms" rel="noreferrer">Peano Arithmetic</a> then obviously <sp... |
4,032,969 | <p>I have an integral that depends on two parameters <span class="math-container">$a\pm\delta a$</span> and <span class="math-container">$b\pm \delta b$</span>. I am doing this integral numerically and no python function can calculate the integral with uncertainties.</p>
<p>So I have calculated the integral for each mi... | Claude Leibovici | 82,404 | <p>What is inside the square root is
<span class="math-container">$$\gamma ^2+ (3 a+4 \eta )x+ 3( a+2 \eta )x^2+ (a+4 \eta )+\eta x^4\tag 1$$</span> Write it as
<span class="math-container">$$\eta\, (x-r_1 ) (x-r_2 ) (x-r_3 ) (x-r_4)$$</span> where the <span class="math-container">$r_i$</span> are the roots of the q... |
2,648,370 | <p>$$\int\frac{x^2}{\sqrt{2x-x^2}}dx$$
This is the farthest I've got:
$$=\int\frac{x^2}{\sqrt{1-(x-1)^2}}dx$$</p>
| haqnatural | 247,767 | <p><strong>Hint</strong>: substitute $$x-1= \sin t $$ so as $$\\ x-1= \sin t \\ x=\sin t +1\\ dx = \cos t\,dt \\ \int \frac { x^2 }{ \sqrt { 1-(x-1)^2 } } \, dx=\int \frac { \cos t (\sin t +1)^2 }{ \sqrt { 1-\sin^2{t} } } \, dt=\int (\sin t +1)^2 \, dt \\ $$</p>
|
4,044,654 | <p>We say that a continuous function <span class="math-container">$u:\mathbb{R}^d\to \mathbb{R}$</span> is subharmonic if it satisfies the mean value property
<span class="math-container">$$u(x)\leq \frac{1}{|\partial
B_r(x)|}\int_{\partial B_r(x)}u(y)\,\mathrm{d}y \qquad (\star)$$</span> for
any ball <span class="ma... | Martin R | 42,969 | <p>Using that <span class="math-container">$u$</span> is midpoint-convex works in higher dimensions as well.</p>
<p><span class="math-container">$y \mapsto x - (y-x) = 2x-y$</span> maps the sphere <span class="math-container">$\partial B_r(x)$</span> bijectively onto itself (each point is mapped to the “opposite” point... |
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | user52817 | 1,680 | <p>The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing the graph of the function with an everyday pencil, you would slide right across the discontinuity without even noticin... |
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | Stephan Kulla | 5,097 | <p>To give a partial answer to my question: In "A radical approach to real analysis" David Bressoud gives a good explanation why the intermediate value property (IVP) is no good choice for continuity (pp. 91 ff):</p>
<ol>
<li>The image of a closed interval might not be bounded.</li>
<li>The sum of two functions with t... |
10,600 | <p>As mentioned in <a href="https://matheducators.stackexchange.com/questions/1538/counterintuitive-consequences-of-standard-definitions">this question</a> students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \t... | John | 6,433 | <p>There is an entirely different perspective on this entire problem which is revealed by asking: Is continuity what we should teach students? That is, before we think about motivating a formal definition of continuity, we might wish to question whether continuity is the concept that we really need students to know.</p... |
631,214 | <p>Two kids starts to run from the same point and the same direction of circled running area with perimeter 400m. The velocity of each kid is constant. The first kid run each circle in 20 sec less than his friend. They met in the first time after 400 sec from the start. Q: Find their velocity.</p>
<p>I came with one e... | Siméon | 51,594 | <p>Using the integral formula $\ln(x)=\int_1^x \frac{dt}{t}$ and $\ln(y^2)=2\ln(y)$, we have for all $x \in (0,1)$,
$$
|x\ln(x)| =2\left|x\ln(x^{1/2})\right| \leq 2\int_{\sqrt{x}}^1\frac{x}{t}\,dt\leq 2 \int_{\sqrt{x}}^1\frac{x}{\sqrt{x}}\,dt \leq 2\sqrt{x}(1-\sqrt{x}).
$$
The conclusion follows by squeezing since the ... |
186,890 | <p>Working with other software called SolidWorks I was able to get a plot with a curve very close to my data points:</p>
<p><a href="https://i.stack.imgur.com/DooKo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DooKo.png" alt="enter image description here"></a></p>
<p>I tried to get a plot as acc... | Bob Hanlon | 9,362 | <p>Use <a href="https://reference.wolfram.com/language/ref/Interpolation.html" rel="nofollow noreferrer"><code>Interpolation</code></a></p>
<pre><code>Clear["Global`*"]
dados = {{0, 0}, {1, 1000}, {2, -750}, {3, 250}, {4, -1000}, {5, 0}};
{xmin, xmax} = MinMax[dados[[All, 1]]]
(* {0, 5} *)
f = Interpolation[dados, ... |
3,628,374 | <p>We have that <span class="math-container">$W \in \mathbb{R}^{n \times m}$</span> and we want to find <span class="math-container">$$\text{prox}(W) = \arg\min_Z\Big[\frac{1}{2} \langle W-Z, W-Z \rangle+\lambda ||Z||_* \Big]$$</span></p>
<p>Here, <span class="math-container">$||Z||_*$</span> represents the trace nor... | Community | -1 | <p>You have shown that all elements of order <span class="math-container">$p$</span> are in the same conjugacy class. But <span class="math-container">$\begin{bmatrix}1&1\\0&1\end{bmatrix}$</span> has order <span class="math-container">$p$</span>.</p>
|
3,131,516 | <p>I would like to know if this differential equation can be transformed into the hypergeometric differential equation</p>
<p><span class="math-container">$ 4 (u-1) u \left((u-1) u \text{$\varphi $1}''(u)+(u-2) \text{$\varphi $1}'(u)\right)+\text{$\varphi $1}(u) \left((u-1) u \omega ^2-u (u+4)+8\right)=0$</span></p>
| JMoravitz | 179,297 | <p>My thought process as I go to factor this expression goes something like the following:</p>
<p><span class="math-container">$$2x^3+3x^2-2x$$</span></p>
<p>"Oh, there is no constant term and everything is a multiple of <span class="math-container">$x$</span>... so that means I can safely factor that out by itsel... |
1,512,515 | <p>I have tested all the primes up to 50,000,000 and did not find a single prime which satisfies the condition "sum of digits of prime number written in base7 divides by 3". E.g. </p>
<ul>
<li>13 (Base10) = 16 (Base7) --> 7 (sum of digits in base 7)</li>
<li>1021 (Base10) = 2656 (Base7) --> 19</li>
<li>823541 (Base10)... | Matthias | 164,923 | <p>You know: If and only if a number can be devided by three than the sum of its digits to the base of then can be devided by three. Therefore each prime number larger than three has a sum of digits which could not be devided by three.</p>
<p>So what is about the base of $7$:
Now when switching from base $10$ to base ... |
1,512,515 | <p>I have tested all the primes up to 50,000,000 and did not find a single prime which satisfies the condition "sum of digits of prime number written in base7 divides by 3". E.g. </p>
<ul>
<li>13 (Base10) = 16 (Base7) --> 7 (sum of digits in base 7)</li>
<li>1021 (Base10) = 2656 (Base7) --> 19</li>
<li>823541 (Base10)... | Matthias Klupsch | 19,700 | <p>Let $a = \sum_{i = 0}^n a_i 7^i$ be a number written in base $7$, that is, $0 \leq a_i \leq 6$. Note that $7^i = (1 + 3 \cdot 2)^i = 1 + 3 b_i$ for some $b_i \geq 0$. Hence $a = \sum_{i = 0}^n a_i + 3 \sum_{i = 0}^n a_i b_i$ is divisible by $3$ if and only if its sum of digits is divisible by $3$. Hence the sum of d... |
1,049,841 | <p>Out of interest </p>
<p>If i have the map $\phi: R \longrightarrow R/I $ where $R$ is a ring and $I$ is a nilpotent ideal ?</p>
<p>then would i be right in saying that if i were to apply this map to the jacobson radical of $R$ it would take me to the jacobson radical of $R/I$ </p>
<p>i.e. is the following true:
$... | egreg | 62,967 | <p>A nil ideal (in particular a nilpotent ideal) is contained in every maximal right ideal. Indeed, if $I$ is a nil ideal and $\mathfrak{m}$ is a maximal right ideal with $I\not\subseteq\mathfrak{m}$, we have $r+x=1$ with $r\in I$ and $x\in\mathfrak{m}$. But, as $r$ is nilpotent, say $r^n=0$, we have
$$
(1-r)(1+r+r^2+\... |
12,281 | <p>In propositional logic, for example:
$$\neg p \vee q.$$
</p>
<p>If $p$ is true at the outset, does that mean it must be considered false when comparing with q in the disjunction?</p>
<p>P.S. I am unsure about tags for this question.</p>
| Arturo Magidin | 742 | <p>If $p$ is true, then $\neg p$ is false. To evaluate $\neg p \vee q$, you must evaluate $\neg p$ and you must evaluate $q$. If either $\neg p$ is true or $q$ is true, then $\neg p\vee q$ is true. </p>
<p>In other words, you really need to figure out $(\neg p)\vee q$, performing first the operation inside the parenth... |
12,281 | <p>In propositional logic, for example:
$$\neg p \vee q.$$
</p>
<p>If $p$ is true at the outset, does that mean it must be considered false when comparing with q in the disjunction?</p>
<p>P.S. I am unsure about tags for this question.</p>
| Yuval Filmus | 1,277 | <p>If $p$ is true, then $\lnot p \lor q \Leftrightarrow q$.</p>
<p>In general, $p$ and $\lnot p$ have the opposite value: if one is true then the other is false, and vice versa.</p>
<p>You can think of $p$ as some proposition, say "today is Sunday". Then $\lnot p$ stands for "today is <i>not</i> Sunday".</p>
|
12,281 | <p>In propositional logic, for example:
$$\neg p \vee q.$$
</p>
<p>If $p$ is true at the outset, does that mean it must be considered false when comparing with q in the disjunction?</p>
<p>P.S. I am unsure about tags for this question.</p>
| Dan Christensen | 3,515 | <p>I'm not sure I understand your question, but this may help.</p>
<p>Truth Table for ~p v q:</p>
<p>~ p v q<br>
F T T T<br>
F T F F<br>
T F T T<br>
T F T F</p>
<p>If p is true, and ~p v q is true (first line only), then q is true.</p>
<p>Note that ~p v q is logically equivalent to p => q. </p>
|
1,761,668 | <p>Wikipedia says about logical consequence:</p>
<blockquote>
<p>A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.</p>
</blockquote>
<p>But if φ and ψ are both true under some interpretations, then... | joy | 101,393 | <p>Consider the complement of $M$ and take a sequence there to show that it is closed. Since we are working in a metric space, this approach works.</p>
|
1,910,927 | <p>$$x^2y+y^2z+xz^2-yz^2-x^2z-xy^2=(x-y)(x-z)(y-z)$$</p>
<p>I would like to know if there is any method by which you can have like this result. </p>
| Jean Marie | 305,862 | <p>Let us denote by lower case letters $a,b,c,d,e$ the abscissas of points $A,B,C,D,E$ resp.</p>
<p>Two basic observations about this issue:</p>
<ul>
<li><p>it is "up to a translation" which allows to take $E$ as the origin, i.e., $e=0$.</p></li>
<li><p>it is "up to a symmetry", which allows $D$ to be set at abscissa... |
1,910,927 | <p>$$x^2y+y^2z+xz^2-yz^2-x^2z-xy^2=(x-y)(x-z)(y-z)$$</p>
<p>I would like to know if there is any method by which you can have like this result. </p>
| user491617 | 491,617 | <p>given
equation1 $b-a = 2$</p>
<p>equation2 $c-b = 3$</p>
<p>equation3 $d-c = 5$</p>
<p>equation4 $e-d = 4$</p>
<p>need to find $e - c$ and $e - a$
solution:</p>
<p>add equation3 and equation4 and you get $e-c = 9$</p>
<p>add all of them and you get $e - a = 14$</p>
|
44,552 | <p>I was pushing my way through a physics book when the author separated the variables of the Schrödinger equation and I lost the plot:</p>
<p>$$\Psi (x, t) = \psi (x) T(t)$$</p>
<p>can someone please explain how this technique works and is used? It can be in general maths or in the context of this problem. Thanks </... | Bill Dubuque | 242 | <p>Regarding your question about the generality of separation of variables, there is an extremely beautiful Lie-theoretic approach to symmetry, separation of variables and special functions,
e.g. see Willard Miller's <a href="http://www.ima.umn.edu/~miller/separationofvariables.html" rel="noreferrer">book [1]</a>. I qu... |
2,480,528 | <blockquote>
<p>Find a formula for $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)$ then prove it. </p>
</blockquote>
<p>I assumed that $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)=\frac{2n}{2n-1}$ after doing a few cases from above then I tried to prove it with induction would this be a fair approach or ... | Andreas Blass | 48,510 | <p>You want $|1+z|=|1-i\bar z|$. The left side here is the distance from $z$ to $-1$. The right side equals $|i+\bar z|$ which in turn equals $|-i+z|$, the distance from $z$ to $i$. So $z$ satisfies your equation iff it is equidistant from $-1$ and $i$. These $z$'s form a straight line in the complex plane, whose equ... |
2,480,528 | <blockquote>
<p>Find a formula for $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)$ then prove it. </p>
</blockquote>
<p>I assumed that $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)=\frac{2n}{2n-1}$ after doing a few cases from above then I tried to prove it with induction would this be a fair approach or ... | nonuser | 463,553 | <p>Since $|1+z|=|1-i\overline{z}|$ and $|w| = |\overline{w}|$, you can rewrite like this $$|z-(-1)|=|1+z|=|1-i\overline{z}| = |\overline{1-i\overline{z}}| =|1+iz| =|i||-i+z| = |z-i| $$</p>
<p>So $z$ is at equal distance from $-1$ and $i$. So $z$ is on perpendicular bisector of segment between $-1$ and $i$. </p>
|
1,801,112 | <p>Find the simplest solution:</p>
<p>$y' + 2y = z' + 2z$ I think proper notation is not sure, y' means first derivate of y. ($\frac{dy}{dt}+ 2y = \frac{dz}{dt} + 2z$)</p>
<p>$y(0)=1$</p>
<p>I got kind of confused, is $y=z=1$ a proper solution here? Or is disqualified because a constant is not reliant on time and so... | SchrodingersCat | 278,967 | <p>$$\frac{dy}{dt}+ 2y = \frac{dz}{dt} + 2z$$
$$\frac{dy}{dt}-\frac{dz}{dt}=-2(y-z)$$
$$\frac{d(y-z)}{dt}=-2(y-z)$$
$$\frac{d(y-z)}{(y-z)}=-2dt$$</p>
<p>Integrating both sides, we get
$$\ln|y-z|=-2t+c$$ where $c$ is a constant of integration.</p>
<p>Using the given condition, we have
$$\ln|1-z(0)|=c$$</p>
<p>So we h... |
145,286 | <p>Yesterday I got into an argument with @UnchartedWorks over <a href="https://mathematica.stackexchange.com/a/145207/26956">in the comment thread here</a>. At first glance, he posted a duplicate of <a href="https://mathematica.stackexchange.com/a/145202/26956">Marius' answer</a>, but with some unnecessary memoization:... | webcpu | 43,670 | <p>What if let pick and unitize run before Pick and Unitize? pick is still faster than Pick.
<strong>In:</strong></p>
<pre><code>Clear[unitize, pick, n, data]
SeedRandom[1];
n = -1;
data = RandomChoice[Range[0, 10], {10^8, 6}];
unitize[x_] := unitize[x] = Unitize[x];
pick[xs_, sel_, patt_] := pick[xs, sel, patt] = Pi... |
145,286 | <p>Yesterday I got into an argument with @UnchartedWorks over <a href="https://mathematica.stackexchange.com/a/145207/26956">in the comment thread here</a>. At first glance, he posted a duplicate of <a href="https://mathematica.stackexchange.com/a/145202/26956">Marius' answer</a>, but with some unnecessary memoization:... | Shadowray | 47,416 | <p><strong>Cause of speed up</strong></p>
<p>This is definitely not memoization. The reason for the observed speed up is that for large arrays (e.g. 10^8 elements), the memory clean up operations may take noticeable time. If one doesn't free memory, one can perform some operations a bit faster.</p>
<p>Here is a simpl... |
3,382,241 | <p>I am trying to find the smallest <span class="math-container">$n \in \mathbb{N}\setminus \{ 0 \}$</span>, such that <span class="math-container">$n = 2 x^2 = 3y^3 = 5 z^5$</span>, for <span class="math-container">$x,y,z \in \mathbb{Z}$</span>. Is there a way to prove this by the Chinese Remainder Theorem?</p>
| David G. Stork | 210,401 | <p>There are <span class="math-container">$5$</span> boxes. Put a ball in each. So all that remains to calculate is how to place the <span class="math-container">$20$</span> remaining balls in the <span class="math-container">$5$</span> boxes.</p>
<p>Can you take it from here?</p>
|
2,322,294 | <p>I am trying to follow K.P. Hart's course <a href="http://fa.its.tudelft.nl/~hart/37/onderwijs/old-courses/settop" rel="nofollow noreferrer">Set-theoretic methods in general topology</a>. In <a href="http://fa.its.tudelft.nl/~hart/37/onderwijs/old-courses/settop/rudin.pdf" rel="nofollow noreferrer">Chapter 6</a>, Rud... | Mike V.D.C. | 114,534 | <p>The ring you are looking for is the Laurent Polynomial ring over $\mathbb{Z}$, namely $\mathbb{Z}[x,x^{-1}]$. This can be <strong>constructed</strong> as follows:
$$\frac{\mathbb{Z}[x][y]}{\langle xy-1\rangle}$$</p>
<p>This you can compute in almost all CASs.</p>
<p>Hope this helps.</p>
<p>-- Mike</p>
|
78,143 | <p>I don't know the meaning of geometrically injective morphism f of schemes. </p>
<p>What's the definition of "geometrically injective"?</p>
<p>I can't find it. I hope your answer.</p>
<p>Thanks.</p>
| Sanjay | 18,579 | <p>I don't find link to add comment. You can find the various equivalent condition for radicial morphism and its proof in "Altman & Kleiman, Introduction to Grothendieck Duality Theory"
on page 119.</p>
|
308,117 | <p>I have the matrix
$$A := \begin{bmatrix}6& 9& 15\\-5& -10& -21\\ 2& 5& 11\end{bmatrix}.$$ Can anyone please tell me how to both find the eigenspaces by hand and also by using the Nullspace command on maple? Thanks.</p>
| Mhenni Benghorbal | 35,472 | <p>Here are the maple commands</p>
<p>with(LinearAlgebra):</p>
<p>A := <<6,9,15>|<-5,-10,-21>|<2,5,11>>;</p>
<p>NS := NullSpace(A);</p>
<p>ES := Eigenvectors(A);</p>
|
4,136,248 | <p>Let <span class="math-container">$a,b\in\mathbb{R}^+$</span>.
Suppose that <span class="math-container">$\{x_n\}_{n=0}^\infty$</span> is a sequence satisfying
<span class="math-container">$$|x_n|\leq a|x_{n-1}|+b|x_{n-1}|^2, $$</span>
for all <span class="math-container">$n\in\mathbb{N}$</span>. How can we bound <sp... | user6247850 | 472,694 | <p>Here is a way to bound the <span class="math-container">$x_n$</span> for the sequence <span class="math-container">$x_n = a x_{n-1}+bx_{n-1}^2$</span>, assuming <span class="math-container">$x_n \ge 0$</span> for all <span class="math-container">$n$</span> (which is the case if <span class="math-container">$x_0 \ge ... |
2,099,516 | <p>For independent Gamma random variables $G_1, G_2 \sim \Gamma(n,1)$, $\frac{G_1}{G_1+G_2}$ is independent of $G_1+G_2$. Does this imply that $G_1+G_2$ is independent of $G_1-G_2$? Thanks!</p>
| BruceET | 221,800 | <p>No, not independent. Here is a quick simulation in R statisticsl
software of 100,000 realizations of $X = G_1 + G_2$ and $Y = G_1 - G_2,$
for the case $n = 4.$ You should be able to turn the
central point of it into a proof. (Notice that $G_1$ and $G_2$ take only
nonnegative values.)</p>
<pre><code>m = 10^5; n = 4... |
2,403,608 | <p>I was asked to solve for the <span class="math-container">$\theta$</span> shown in the figure below.</p>
<p><a href="https://i.stack.imgur.com/3Yxqv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3Yxqv.png" alt="enter image description here" /></a></p>
<p>My work:</p>
<p>The <span class="math-con... | amWhy | 9,003 | <p>Hints:</p>
<p>Triangle HIG, and triangle DCE are isosceles triangles, with $\angle HIG = \angle DCE = 90^\circ - 60^\circ = 30^\circ$. </p>
<p>In isosceles triangles, the base angles are congruent, so $\angle IHG = \angle HGI = \dfrac{180^\circ - 30^\circ}{2} = 75^\circ.$</p>
<p>Similarly, for $\angle CED$ and ... |
4,398,207 | <p>I have the following exercise:</p>
<blockquote>
<p>Prove if the functor that sends an abelian group to it's <span class="math-container">$n$</span>-torsion subgroup for <span class="math-container">$n\geq 2$</span> is exact.</p>
</blockquote>
<p>I know that I need to take <span class="math-container">$f\colon M\to N... | Berci | 41,488 | <p>With Abelian groups one usually applies additive notation.</p>
<p>Yes, <span class="math-container">$F(A)=\{a\in A:\,n\cdot a=0\}$</span>.<br>
Then, <span class="math-container">$F$</span> sends a homomorphism <span class="math-container">$f:A\to B$</span> to its restriction to <span class="math-container">$F(A)$</s... |
1,292,759 | <blockquote>
<p>Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that
$$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$</p>
</blockquote>
<p>This isn't hard problem. I have already solved it in following way:<br/>
Let $x=\frac1a,y=\frac1b,z=\frac1c$, then $xyz=1$. Now, it is enought to prove that... | Dr. Sonnhard Graubner | 175,066 | <p>why must you use derivatives? the proof is simple with Cauchy Schwarz:
we have
$$\frac{1}{a^3(b+c)}=\frac{\frac{1}{a^2}}{a(b+c)}=\frac{\frac{1}{a^2}}{\frac{b+c}{bc}}$$ thus we have
$$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\geq $$
$$\frac{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}{\frac{2}{a... |
2,110,561 | <p>so I want to find the volume of the body D defined as the region under a sphere with radius 1 with the center in (0, 0, 1) and above the cone given by $z = \sqrt{x^2+y^2}$. The answer should be $\pi$. A hint is included that you should use spherical coordinates. I've started by making a equation for the sphere, $x^2... | Martin Argerami | 22,857 | <p>If you try spherical coordinates, the following happens. For the cone, you have <span class="math-container">$$\rho\cos\phi=z=\sqrt{x^2+y^2}=\sqrt{\rho^2\sin^2\phi\cos^2\theta+\rho^2\sin^2\phi\sin^2\theta}=\rho\sin\phi.$$</span>(this assumes that you chose <span class="math-container">$\phi$</span> so that <span cla... |
2,518,305 | <p><a href="http://www.mit.edu/%7Esame/pdf/qualifying_round_2017_answers.pdf" rel="noreferrer">This is a problem from MIT integration bee 2017.</a></p>
<p><span class="math-container">$$\int_0^{\pi/2} \frac 1 {1+\tan^{2017} x} \, dx$$</span></p>
<p>I have tried substitution method, multiplying numerator and denominator... | Guy Fsone | 385,707 | <p>Setting the change of variable: $u=\frac\pi2-x $ and since, $\tan x =\cot(\frac\pi2 -x)$ we have,
\begin{align}
& \int_0^{\frac\pi2}\frac{1}{1+\tan^{2017} x} \, dx = \int_0^{\frac\pi2}\frac{1}{1+\tan^{2017} (\frac\pi2-u) } \, du \\[10pt]
= {} & \int_0^{\frac\pi2}\frac{1}{1+\cot^{2017}u} \, du = \int_0^{\fra... |
260,865 | <p>I am fairly new to mathematica and I am trying to plot a 3D curve defined by multiple formulas. I have the curve <span class="math-container">$K$</span> from the point <span class="math-container">$(\frac{1}{2},-\frac{1}{2}\sqrt{3},0)$</span> to <span class="math-container">$(\frac{1}{2},\frac{1}{2}\sqrt{3},2\sqrt{3... | Ulrich Neumann | 53,677 | <p>The three conditions define a region</p>
<pre><code>reg = ImplicitRegion[{x^2 + y^2 == 1, z == y/x + Sqrt [3],x > 1/2}, {x, y, z}]
</code></pre>
<p>which is plotted with <code>Region</code></p>
<pre><code>Region[reg, Axes -> True, BoxRatios -> {1, 1, 1}, Boxed -> True]
</code></pre>
<p><a href="https://i... |
365,128 | <p>How does one find the Inverse Laplace transform of $$\frac{6s^2 + 4s + 9}{(s^2 - 12s + 52)(s^2 + 36)}$$ where $s > 6$?</p>
| Ron Gordon | 53,268 | <p>You need to find the poles of the expression; in your case, you have poles at $s=6 \pm 4 i$ and $s=\pm 6 i$. You then find what are called the residues of the LT times $e^{s t}$ at the poles. The residue at a pole $s_k$ is </p>
<p>$$\lim_{s \rightarrow s_k} \left [ (s-s_k) \frac{6 s^2 + 4 s+9}{(s^2-12 s+52)(s^2+3... |
2,970,234 | <p>So, I'm given the following query:</p>
<p>Write the Taylor series centered at <span class="math-container">$z_0 = 0$</span> for each of the following complex-valued functions:</p>
<p><span class="math-container">$$f(z) = z^2\sin(z),\quad g(z) = z\sin(z^2)$$</span></p>
<p>Then, use these series to help you compute... | Stockfish | 362,664 | <p>Note that for every power series <span class="math-container">$P(z)$</span> with constant term <span class="math-container">$c$</span> we have <span class="math-container">$\lim_{z \to 0} P(z) = c$</span>. So the question is: in the entire limit represented as series, do we have a constant term? Do we have a summand... |
903,049 | <p>I have to find the series expansion and interval of convergence for the function $\ln(1 - x)$.</p>
<p>For the expansion, I have gone through the process and obtained the series:</p>
<p>$-x - (x^2/2) - (x^3/3) - . . . - (-1)^k((-x)^k)/k$</p>
<p>I know that the interval of convergence will be $(-1,1)$, but am havin... | cjferes | 89,603 | <p>You already know that
$$\log(1-x)=-\sum_{k=1}^{\infty} \frac{x^k}{k}=\sum_{k=1}^{\infty} a_kx^k$$</p>
<p>Then,
$$a_k=-\frac{1}{k}$$</p>
<p>The ratio test, then, is:
$$\biggl|{\frac{a_{k+1}}{a_k}}\biggr|=\frac{\frac{1}{k+1}}{\frac{1}{k}}=\frac{k}{k+1}$$</p>
<p>The convergence radius $R$ is given by:
$$\lim_{k\righ... |
1,840,159 | <blockquote>
<p>Question: Prove that a group of order 12 must have an element of order 2.</p>
</blockquote>
<p>I believe I've made great stride in my attempt.</p>
<p>By corollary to Lagrange's theorem, the order of any element $g$ in a group $G$ divides the order of a group $G$.</p>
<p>So, $ \left | g \right | \mi... | Micapps | 300,392 | <p>Approach without Sylow's theorem: By what you've shown, all you need to do is discount the possibility that all group elements have order $3$ or $1$. The only element with order $1$ is the identity. What can you say about a group that consists of the identity, and $11$ elements of order $3$?</p>
<p>Hint: the elemen... |
2,239,192 | <p>Let $P_n$ be the polynomials of degree no more than n with basis $Z_n=(1, x, x^2,\dotsc,x^n)$. The derivative transformation $D$ goes from $P_n$ to $P_{n-1}$. Write out the matrix for $D$ from $(P_4, Z_4)$ to $(P_3, Z_3)$.</p>
<p>I haven't done a problem similar to this so I'm not sure how to go about doing this. ... | Chappers | 221,811 | <p>The matrix of a linear map $L:(V,B) \to (U,C)$ is found by writing $L(b_j)$ in terms of $c_i$ for each $b_j \in B$. In this case,
$$ D(1)=0, \qquad D(x) = 1, \qquad D(x^2)=2x, \qquad D(x^3) = 3x^2, \qquad D(x^4) = 4x^3, $$
so the matrix is
$$ \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 2 &a... |
786,655 | <p>Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$</p>
<p>What are the different methods to numerically compute the distribution X+Y, X*Y, X/Y, min(X,Y) etc...?</p>
<p>I read abou... | M.X | 66,726 | <p>you can use Mathematica for such calculations.
For example TransformedDistribution[] can help you solve the problem if it can be solved in closed form at all.</p>
|
32,294 | <p>I sort of asked a version of this question before and it was unclear; try I will now to make an honest attempt to state everything clerly.</p>
<p>I am trying to evaluate the following, namely $\nabla w = \nabla |\vec{a} \times \vec{r}|^n$, where $\vec{a}$ is a constant vector and $\vec{r}$ is the vector $<x_1,x_... | Community | -1 | <p>Say I differentiate $|\mathbf{a} \times \mathbf{r}|^2$. Then for the second term it agrees with $\mathbf{a} \times (\mathbf{r} \times \mathbf{a})$, but for the first term there is some confusion. Say I look at the first term of $\nabla |\mathbf{a}|^2|\mathbf{r}|^2$, namely $|\mathbf{a}|^2|\mathbf{r}|^2_{x_1} \mathbf... |
3,571,047 | <p>Here's what I have so far:</p>
<p><span class="math-container">$$\frac{\partial f}{\partial y}|_{(a,b)} = \lim\limits_{t\to 0} \frac{\sin(a^2 + b^2 + 2tb + t^2) - \sin(a^2 + b^2)}{t} = \lim\limits_{t\to 0} \frac{\sin(a^2 + b^2)[\cos(2tb + t^2) - 1] + \cos(a^2 + b^2)\sin(2tb + t^2)}{t}$$</span>
I can see that side l... | Toby Mak | 285,313 | <p>You can also proceed with the substitution <span class="math-container">$x = t^2, dx = 2t \ \mathrm{d} t$</span>:</p>
<p><span class="math-container">$$ \int \dfrac{t^2}{1+t} \ 2t \ \mathrm{d} t = 2 \int \dfrac{t^3}{1+t} \mathrm{d} t = 2 \int \dfrac{t^2(t+1)-t(t+1)+(t+1)-1}{1+t} \ \mathrm{d} t$$</span>
<span class="... |
4,020,986 | <blockquote>
<p>For every <span class="math-container">$n \in \mathbb{N}$</span> denote <span class="math-container">$x_n=(n,0) \in \mathbb{R^2}.$</span> Show that the set <span class="math-container">$\mathbb{R^2} \setminus \{x_n \mid n \in \mathbb{N} \}$</span> is an open subset of the plane.</p>
</blockquote>
<p>The... | Actually Fritz | 879,727 | <p>First of all, the most direct way to proceed in these situations is exactly what hamam_Abdallah proposed in his answer. Recall that closed sets are by definition complements of open sets, so if you manage to do that, it will immediately follow that <span class="math-container">$\mathbb{R}^2 \setminus (x_n)_{n \in \m... |
135,663 | <p>It is a problem for a Hatcher's book, and it is my homework problem.</p>
<p>It is a section 2.2 problem 3, stating:</p>
<p>Let $f:S^n\to S^n$ be a map of degree zero. Show that there exist points $x,y \in S^n$ with $f(x)=x$ and $f(y)=-y$. Use this to show that if $F$ is a continuous vector filed defined on the uni... | Community | -1 | <p><strong>Hint</strong></p>
<ul>
<li><p>$A_5$ is simple. </p></li>
<li><p>What is the index of such a group? Let $A_5$, a simple group act on left cosets of this proper subgroup? What can you say about the kernel of the homomorphism that comes with this action? </p></li>
<li><p>So, now apply first isomorphism theorem... |
135,663 | <p>It is a problem for a Hatcher's book, and it is my homework problem.</p>
<p>It is a section 2.2 problem 3, stating:</p>
<p>Let $f:S^n\to S^n$ be a map of degree zero. Show that there exist points $x,y \in S^n$ with $f(x)=x$ and $f(y)=-y$. Use this to show that if $F$ is a continuous vector filed defined on the uni... | Mikko Korhonen | 17,384 | <p>Show that every group of order $15$ is cyclic. The result follows since there is no element of order $15$ in $A_5$.</p>
|
57,213 | <p>Let <span class="math-container">$A \in \mathbb{Q}^{6 \times 6}$</span> be the block matrix below:</p>
<p><span class="math-container">$$A=\left(\begin{array}{rrrr|rr}
-3 &3 &2 &2 & 0 & 0\\
-1 &0 &1 &1 & 0 & 0\\
-1&0 &0 &1 & 0 & 0\\
-4&6 ... | Andrea | 14,351 | <p><strong>Theorem.</strong> Let $V$ be a finite $\mathbb{K}$-vector space and let $f \in \mathrm{End}(V)$ an endomorphism with minimal polynomial $m_f(t) \in \mathbb{K}[t]$. If $a(t) \in \mathbb{K}[t]$, then $a(f) \in \mathrm{GL}(V)$ if and only if $\gcd(a,m_f)=1$.</p>
<p><em>Proof.</em> $\Leftarrow$) Since Bezout's ... |
57,213 | <p>Let <span class="math-container">$A \in \mathbb{Q}^{6 \times 6}$</span> be the block matrix below:</p>
<p><span class="math-container">$$A=\left(\begin{array}{rrrr|rr}
-3 &3 &2 &2 & 0 & 0\\
-1 &0 &1 &1 & 0 & 0\\
-1&0 &0 &1 & 0 & 0\\
-4&6 ... | Did | 6,179 | <p>Put $A$ in Jordan form. The diagonal is made of $3$s and $-1$s. In this vector basis, $f(A)$ is also upper triangular and its diagonal is made of $f(3)$s and $f(-1)$s. Hence $f(A)$ is invertible if and only if there is no zero on the diagonal of its Jordan form if and only if $f(3)$ and $f(-1)$ are nonzero (and this... |
2,372,171 | <p>Let $T$ be a bounded linear operator on a Hilbert space $H$. I have to show that the following are equivalent:</p>
<p>(i) $T$ is unitary</p>
<p>(ii) For every orthonormal basis $\{u_{\alpha}:\alpha\in \Lambda\}$, $\{T(u_{\alpha}):\alpha\in \Lambda\}$ is an orthonormal basis.</p>
<p>(iii) For some orthonormal basi... | Nina Simone | 467,038 | <p>It is enough to compute $T^*T(u_\alpha)$ for all $\alpha$, since the operator is determined by its values at a basis.</p>
<p>We can compute the coordinates of this vector in the basis $u_\alpha$. So, we do</p>
<p>$$T^*T(u_\alpha)\cdot u_{\beta}=T(u_\alpha)\cdot T(u_\beta)=\delta_{\alpha,\beta}$$</p>
<p>Therefore ... |
81,811 | <p>I heard this example was given in Whitehead's paper A CERTAIN OPEN MANIFOLD WHOSE GROUP IS UNITY.( <a href="http://qjmath.oxfordjournals.org/content/os-6/1/268.full.pdf" rel="nofollow">http://qjmath.oxfordjournals.org/content/os-6/1/268.full.pdf</a> ) But I was confused by his term. Thus I'm looking for an explanati... | Marco Golla | 13,119 | <p>It's discussed in Kirby's "The topology of 4-manifolds", around page 80, and at a glance the argument looks "modern".</p>
|
283,747 | <p>Let $BG$ denote the classifying space of a finite group $G$. For which group cohomology classes $c\in H^2(G;\mathbb{Z}/2)$ does there exist a real vector bundle $E$ over $BG$ such that $w_2(E)=c$?</p>
| Neil Strickland | 10,366 | <p>Put
$$A=\{c\in H^2(BG;\mathbb{Z}/2): c = w_2(V) \text{ for some } V\}$$
Here are some observations:</p>
<ol>
<li>Let $V$ be a real representation with determinant $L$, so $w_1(V)=w_1(L)$. Put $W=V\oplus L\oplus L\oplus L$. We find that $\det(W)=1$ and $w_2(W)=w_2(V)$. It follows that $A=\{w_2(W):\det(W)=1\}$. ... |
65,480 | <p>The example question is </p>
<blockquote>
<p>Find the remainder when $8x^4+3x-1$ is divided by $2x^2+1$</p>
</blockquote>
<p>The answer did something like</p>
<p>$$8x^4+3x-1=(2x^2+1)(Ax^2+Bx+C)+(Dx+E)$$</p>
<p>Where $(Ax^2+Bx+C)$ is the Quotient and $(Dx+E)$ the remainder. I believe the degree of Quotient is d... | Peđa | 15,660 | <p>Polynomial division allows for a polynomial to be written in a divisor–quotient form:</p>
<p>$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$, where degree(D) < degree(P) and degree(R) < degree(D)</p>
<p>This rearrangement is known as the division transformation.</p>
<p>In this particular case $R(x)=3x-3$ so deg... |
3,239,540 | <p><span class="math-container">$$S=\sum_{k=2}^{n}\frac{k^{2}-2}{k!}, n\geq 2$$</span></p>
<p>I got <span class="math-container">$S=\sum_{k=2}^{n}\frac{1}{(k-2)!}+\frac{1}{(k-1)!}-\frac{1}{k!}-\frac{1}{k!}$</span></p>
<p>I give k values but not all terms are vanishing.I remain with <span class="math-container">$\frac... | G Cab | 317,234 | <p><span class="math-container">$$
\eqalign{
& \sum\limits_{k = 2}^n {\left( {{1 \over {\left( {k - 2} \right)!}} + {1 \over {\left( {k - 1} \right)!}} - {2 \over {k!}}} \right)}
= \sum\limits_{k = 0}^{n - 2} {{1 \over {k!}} + \sum\limits_{k = 1}^{n - 1} {{1 \over {k!}}} - \sum\limits_{k = 2}^n {{2 \over {k!}}... |
2,627,131 | <blockquote>
<p>let <span class="math-container">$f(x)= 1+\sqrt{x+k+1}-\sqrt{x+k} \ \ k \in \mathbb{R}$</span>
Number of answers :</p>
<p><span class="math-container">$$f(x)=f^{-1}(x) \ \ \ \ :f^{-1}(f(x))=x$$</span></p>
</blockquote>
<p>MY Try :</p>
<p><span class="math-container">$$y=1+\sqrt{x+k+1}-\sqrt{x+k} \\( y... | prog_SAHIL | 307,383 | <p><strong>Hint:</strong></p>
<p>Point of intersection of $f(x)$ and $f^{-1}(x)$ while same as that of $f(x)$ and the line $y=x$.</p>
|
102,814 | <p>Is it possible to construct a nontrivial homomorphism from $C_6$ to $A_3$? I have tried to construct one but failed. Is there a good way to see when there will be a homomorphism?</p>
| Clive Newstead | 19,542 | <p>Obviously you can't construct an isomorphism $C_6 \to A_3$, since $C_6$ has order $6$ and $A_3$ has order $3$. So you'll need to construct a homomorphism $\theta : C_6 \to A_3$ which is not injective. </p>
<p>A useful fact to note is that $A_3 \cong C_3$, so write $C_6 = \{e, a, \dots, a^5 \}$ and $A_3 = \{ e, b, b... |
3,579,346 | <p>I've been learning some introductory analysis on manifolds and have had a small issue ever since the notion of tangent spaces at points on a differentiable manifold was introduced.</p>
<p>In our lectures, we began with the definition using equivalence classes of curves. But it is also possible to define tangent spa... | anomaly | 156,999 | <p>1) The tangent space <span class="math-container">$T_p M$</span> is a vector space and, as you pointed out, any two vector spaces of the same dimension are isomorphic. There are two issues that make the definitions in the post nontrivial. First, that isomorphism is not canonical; it depends on (e.g.) a choice of bas... |
1,960,911 | <p>I am trying to evaluate this limit for an assignment.
$$\lim_{x \to \infty} \sqrt{x^2-6x +1}-x$$</p>
<p>I have tried to rationalize the function:
$$=\lim_{x \to \infty} \frac{(\sqrt{x^2-6x +1}-x)(\sqrt{x^2-6x +1}+x)}{\sqrt{x^2-6x +1}+x}$$</p>
<p>$$=\lim_{x \to \infty} \frac{-6x+1}{\sqrt{x^2-6x +1}+x}$$</p>
<p>Th... | DonAntonio | 31,254 | <p>You should have gotten, after the last step:</p>
<p>$$\lim_{x \to \infty} \frac{-6+\frac1x}{\sqrt{1-\frac6x +\frac1{x^2}}+1}=\frac{-6}{2}=-3$$</p>
<p>so in fact you only had a minor, though pretty influential, arithmetical mistake.</p>
|
1,960,911 | <p>I am trying to evaluate this limit for an assignment.
$$\lim_{x \to \infty} \sqrt{x^2-6x +1}-x$$</p>
<p>I have tried to rationalize the function:
$$=\lim_{x \to \infty} \frac{(\sqrt{x^2-6x +1}-x)(\sqrt{x^2-6x +1}+x)}{\sqrt{x^2-6x +1}+x}$$</p>
<p>$$=\lim_{x \to \infty} \frac{-6x+1}{\sqrt{x^2-6x +1}+x}$$</p>
<p>Th... | haqnatural | 247,767 | <p>it should be $$\lim _{ x\to \infty } \frac { -6x+1 }{ \sqrt { x^{ 2 }-6x+1 } +x } =\lim _{ x\to \infty } \frac { x\left( -6+\frac { 1 }{ x } \right) }{ x\left( \sqrt { 1-\frac { 6 }{ x } +\frac { 1 }{ { x }^{ 2 } } } +1 \right) } =\frac { -6 }{ 2 } =-3$$</p>
|
1,966,122 | <p>$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k}$$</p>
<p>I am trying to prove this inductively, so I thought that I would expand the right side out of sigma form to get</p>
<p>$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k} = \frac{2}{2n(2n+1)} - \frac{1}{n}$$</p>
<p>which simplified to</p>
<p>$$\sum... | felasfa | 55,243 | <p>Good work so far. To complete the problem, the key is to understand the meaning of the matrix of a linear map. Consider a linear map $T:V \rightarrow W$. For simplicity, assume the basis for $V$ is $\{\alpha_{1},\alpha_{2},\alpha_{3}\}$ and the basis for $W$ is $\{\beta_{1},\beta_{2}\}$. The first step, as you did a... |
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| Hagen von Eitzen | 39,174 | <p>Don't trust Wolfram when also you have pen and paper available.</p>
<p>Of course, $x\mapsto \sin x$ and $x\mapsto \cos x$ are functions with period $2\pi$. Composing them with some other function (here the absolute value) gives us functions having $2\pi$ as a period as well. But since $\sin(x+\pi)=-\sin x$ (and sim... |
3,244,073 | <p>Let <span class="math-container">$A = \{1, 3, 5, 9, 11, 13\}$</span> and let <span class="math-container">$\odot$</span> define the binary operation of multiplication modulo <span class="math-container">$14$</span>.</p>
<p>Prove that <span class="math-container">$(A, \odot)$</span> is a group. </p>
<p>While comple... | Jack D'Aurizio | 44,121 | <p>I strongly agree with the comment by J.G.: the improper Riemann integral equals
<span class="math-container">$$\int_{-\infty}^{+\infty}\frac{z^2 e^z}{(e^z-1)^2}\,dz = \int_{-\infty}^{+\infty}\left(\frac{z}{2\sinh\frac{z}{2}}\right)^2\,dz\stackrel{sym}{=}4\int_{0}^{+\infty}\frac{u^2}{\sinh^2 u}\,du$$</span>
or, by in... |
2,906,917 | <p>I have this problem but I don't know how to continue.<br>
Here it is:
Compute $\int \sin(x) \left( \frac{1}{\cos(x) + \sin(x)} + \frac{1}{\cos(x) - \sin(x)} \right)\,dx.$<br>
So I can anti differentiate the sin x to be cos x but I am unsure on where to go off that for the fraction. I don't want to multiply the fract... | bjcolby15 | 122,251 | <p>An alternate (but longer) solution:</p>
<p>Going back to the beginning - if we have simplified the original to $$\int\sin x\left(\frac{2\cos x}{\cos^2x-\sin^2x}\right)dx,$$ we can convert the denominator to $2 \cos^2 x - 1$ (as it's equivalent to $\cos^2x -\sin^2 x$) and use the substitution $$u = \cos x, du = -\si... |
2,648,516 | <p>I am studying Fourier analysis from the text "Stein and Shakarchi" and there is this thing on Dirichlet Kernel. It's fine to define it as a trigonometric poylnomial of degree $n$ , but what is the mathematical intuition behind calling it a Kernel ? I have also thought of Kernel as being a set of zeroes of sum functi... | David C. Ullrich | 248,223 | <p>In general, if you have a linear operator $T$ on some space of functions, defined by an integral $$Tf(x)=\int f(t) K(x,t)\,dt,$$then $K$ is the "kernel". </p>
|
397,347 | <p>I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies J=-I'(1)$, but I couldn't figure out what $I(s)$ was. My other idea was contour integration, but I'm not sure how to deal... | Start wearing purple | 73,025 | <p>Mathematica says that the answer is
$$\pi^2\zeta(3)+12\zeta(5)$$
I will try to figure out how this can be proven.</p>
<hr>
<p><strong>Added</strong>: Let me compute the 2nd integral in <strong>Ron Gordon</strong>'s answer:
\begin{align}\int_{0}^{\infty}\frac{x^3 e^{-x}}{1-e^{-x}}\ln(1-e^{-x})\,dx
&=-\frac32\in... |
4,487,489 | <p>I'm currently working on completing their first unit on calculus ab and I've encountered this roadblock. That's probably an exaggeration but I honestly can't figure out what they mean by "for negative numbers". I did the math and got the right number (at least the right absolute value) but the missing nega... | Dark Malthorp | 532,432 | <p>The issue here has to due with non-integer exponents being weird. When we say <span class="math-container">$\sqrt{x}=y$</span>, what we mean is that <span class="math-container">$y$</span> is a number which, when squared, gives <span class="math-container">$x$</span>. However, there are two such numbers (unless <spa... |
1,984,843 | <p>if $\cup$ is finite, say $n$, I came up with formula</p>
<p>$f(x) = n x + i$, where $x \in [\frac{i}{n}, \frac{i+1}{n}]$, $n$ is non negative integer and $i$ differs between $0$ and $n-1$.<br><br></p>
<p>I'm not sure whether it's correct to assume the bijection holds if $n$ approaches infinity.</p>
| hmakholm left over Monica | 14,366 | <p>If you just want to know that a bijection <em>exists</em>, use the <a href="https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" rel="nofollow">Cantor-Schröder-Bernstein theorem</a>:</p>
<ul>
<li><p>The <em>identity function</em> is an injection from $[0,1]$ into $[0,1]\cup[2,3]\cup\cdots$.</p></l... |
1,908,844 | <p>The following example is taken from the book "Introduction to Probability Models" of Sheldon M. Ross (Chapter 5, example 5.4).</p>
<blockquote>
<p>The dollar amount of damage involved in an automobile accident is an
exponential random variable with mean 1000. Of this, the insurance
company only pays that amou... | Paolo Leonetti | 45,736 | <p>The result is $\binom{n+m}{k}$.</p>
<p>This is known as <a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity" rel="nofollow">Vandermonde's identity</a>.</p>
|
1,574,003 | <p>I know that if A and C are finite sets then |AxC|=|A||C|. This makes the problem quite simple but the sets may not be finite. </p>
<p>I am guessing that the concept of cardinally of infinite sets and ℵ <sub>0</sub> are part of the solution but those are concepts that my class did not go into much and I do not... | cr001 | 254,175 | <p>$|A|=|B|$ means there is a bijection $f:A\rightarrow B$ that $f(a)=b$ for $a\in A, b\in B$.</p>
<p>Similarly there is $g:C\rightarrow D$ that $g(c)=d$.</p>
<p>Now we can easily show that $h(a,c)=(f(a),g(c))$ is a bijection over $A\times C\rightarrow B\times D$</p>
|
4,220,972 | <p>I'm studying a for the GRE and a practice test problem is, "For all real numbers x and y, if x#y=x(x-y), then x#(x#y) =?</p>
<p>I do not know what the # sign means. This is apparently an algebra function but I cannot find any such in several searches. I'm an older student and haven't had basic algebra in over 4... | Shffl | 395,362 | <p>I used the form of Chebyshev's inequality found <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_inequality#Statement" rel="nofollow noreferrer">here</a>.</p>
<p>Because <span class="math-container">$\operatorname{Var}(X) = \frac{1}{4}$</span>, you have that <span class="math-container">$\sigma = 0.5$</span>. T... |
1,179,497 | <p>Let $(F,+,\cdot)$ be a field. </p>
<p>Then to prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.</p>
<p>I am facing difficulty in finding the map to bring a contradiction!!</p>
| Michael Hardy | 11,667 | <p>Quid's answer is in one sense the same as what I was about to post when I read it. But the way I phrased it may make it clear to some people first learning the subject, in a way that quid's might not.</p>
<p>Suppose an isomorphism $\varphi$ from the multiplicative group to the additive group exists. In a field in... |
2,111,402 | <p>Simple exercise 6.2 in Hammack's Book of Proof. "Use proof by contradiction to prove"</p>
<p>"Suppose $n$ is an integer. If $n^2$ is odd, then $n$ is odd"</p>
<p>So my approach was:</p>
<p>Suppose instead, IF $n^2$ is odd THEN $n$ is even</p>
<p>Alternatively, then you have the contrapositive, IF $n$ is not even... | Dylan | 409,257 | <p>Last night I read this as a perfectly acceptable claim, but, as has been pointed out, your negation was not simply a "harder case" but instead the converse. Apologies! Your proof happened to work here because the stronger relationship (i.e. $\iff$)</p>
<p>A couple of things to note though. For a proof by contradict... |
2,638,028 | <p><strong>Question:</strong></p>
<blockquote>
<p>If $p,q$ are positive integers, $f$ is a function defined for positive numbers and attains only positive values such that $f(xf(y))=x^py^q$, then prove that $p^2=q$.</p>
</blockquote>
<p><strong>My solution:</strong></p>
<p>Put $x=1$. So, $f(f(y))=y^q$, then eviden... | Community | -1 | <p>There is a major gap in your argument.</p>
<p>You've correctly argued that any $f$ satisfying the functional equation $f(x f(y)) = x^p y^q$ must also satisfy the functional equation $f(f(y)) = y^q$.</p>
<p>It is correct that $f(t) = t^\sqrt{q}$ is <em>one</em> solution to the functional equation $f(f(y)) = y^q$.</... |
26,083 | <p>I have a data set in form of:(this is just an example)</p>
<pre><code>1324501020
3241030205
4332020134
</code></pre>
<p>the data are stored in a text file (e.g. data.txt) but I need to convert them into a matrix format such that each number be place in a cell like this:</p>
<pre><code>1 3 2 4 5 0 1 0 2 0
3 2 4 1 ... | HyperGroups | 6,648 | <p><code>(IntegerDigits@ImportString["1324501020
3241030205
4332020134"]) // Flatten[#, 1] &</code>
list data no need spaces?</p>
<p>If you wanna string still add them
<code>Riffle[#, " "] & /@ Characters /@ StringSplit["1324501020
3241030205
4332020134"]</code></p>
<p>If data in txt file, impo... |
1,507,710 | <p>I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold?
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group.
I think I read somewhere... | C. Falcon | 285,416 | <p>$GL(n,\mathbb{Z})$ is a group for the multiplication law. One can show that : $$GL(n,\mathbb{Z})=\{A\in\mathcal{M}(n,\mathbb{Z})\textrm{ s.t. }|\det(A)|=1\}.$$
As far as $(A,+,\times)$ is a commutative ring with an identity element for $\times$, $GL(n,A)$ is a group.</p>
|
1,507,710 | <p>I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold?
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group.
I think I read somewhere... | mathcounterexamples.net | 187,663 | <p>In a general way, if you consider a matrix $A \in \mathcal{M}_n(\mathbb Z)$, you have the relation $$A.\mathbf{adj}(A)=\det(A)I_n \tag{1}$$ where $\mathbf{adj}(A)$ stands for the <a href="https://en.m.wikipedia.org/wiki/Adjugate_matrix" rel="nofollow noreferrer">adjugate matrix</a> of $A$. The adjugate matrix $\math... |
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