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 |
|---|---|---|---|---|
3,397,548 | <p>For a sequence <span class="math-container">$\{x_n\}_{n=1}^{\infty}$</span>, define <span class="math-container">$$\Delta x_n:=x_{n+1}-x_n,~\Delta^2 x_n:=\Delta x_{n+1}-\Delta x_n,~(n=1,2,\ldots)$$</span> which are named <strong>1-order</strong> and <strong>2-order difference</strong>, respectively. </p>
<p>The pro... | Arthur | 15,500 | <p>Let <span class="math-container">$\{x_n\}$</span> be bounded by <span class="math-container">$X$</span> (i.e. <span class="math-container">$|x_n|< X$</span> for all <span class="math-container">$n$</span>) and <span class="math-container">$\lim_{n\to\infty} \Delta^2x_n = 0$</span>. Take an arbitrary <span class="... |
3,811,753 | <p>Show that the equation:</p>
<p><span class="math-container">$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$</span></p>
<p>Has an integration factor that depends on <span class="math-container">$x$</span> And solve it that way.</p>
<hr />
<p>Already we got to:</p>
<p><span class="math-container">$$
y’ + \frac{xy^3}{3x^2y^2} = \fra... | robjohn | 13,854 | <p><span class="math-container">$$
y’=\frac{2-xy^3}{3x^2y^2}\tag1
$$</span>
Multiply <span class="math-container">$(1)$</span> by <span class="math-container">$3y^2g$</span> and shuffle some terms:
<span class="math-container">$$
\left(y^3\right)'g+y^3\frac{g}x=\frac{2g}{x^2}\tag2
$$</span>
We want the integrating fact... |
230,504 | <p>Again, this question is related (**) to a <a href="https://mathoverflow.net/questions/101700/large-cardinals-without-the-ambient-set-theory?rq=1">previous one</a>:</p>
<p>in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: t... | Joel David Hamkins | 1,946 | <p>This is not exactly what you asked for, but there is an interesting axiomatization of Ordinals + Sets of Ordinals, which turns out to be precisely equiconsistent with ZFC. </p>
<ul>
<li>Peter Koepke, Martin Koerwien, <a href="http://arxiv.org/abs/math/0502265" rel="nofollow">The Theory of Sets of Ordinals</a>.</li>... |
230,504 | <p>Again, this question is related (**) to a <a href="https://mathoverflow.net/questions/101700/large-cardinals-without-the-ambient-set-theory?rq=1">previous one</a>:</p>
<p>in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: t... | Andreas Blass | 6,794 | <p>Are these papers of Takeuti the sort of thing you want?</p>
<p>MR0086751 (19,237e) 02.0X
Takeuti, Gaisi,
On the theory of ordinal numbers.
J. Math. Soc. Japan 9 (1957), 93–113.</p>
<p>MR0099918 (20 #6354) 02.00
Takeuti, Gaisi,
On the theory of ordinal numbers. II.
J. Math. Soc. Japan 10 1958 106–120</p>
<p>MR0197... |
231,187 | <p>I'm wondering if there's an efficient way of checking to see if two context free grammars are equivalent, besides working out "test cases" by hand (ie, just trying to see if both grammars can generate the same things, and only the same things, by trial and error).</p>
<p>Thanks!</p>
| Jurgen Vinju | 111,672 | <p>This paper provides an answer "Comparison of Context-Free Grammars Based on Parsing Generated Test Data", <a href="http://link.springer.com/chapter/10.1007%2F978-3-642-28830-2_18" rel="nofollow">http://link.springer.com/chapter/10.1007%2F978-3-642-28830-2_18</a>. The authors propose to generate sub-sentences from th... |
231,187 | <p>I'm wondering if there's an efficient way of checking to see if two context free grammars are equivalent, besides working out "test cases" by hand (ie, just trying to see if both grammars can generate the same things, and only the same things, by trial and error).</p>
<p>Thanks!</p>
| Anderson Green | 32,826 | <p>In some cases, it is relatively easy to prove that two formal grammars are equivalent. If two grammars generate a <em>finite language</em>, then you only need to compare the sets of strings generated by both grammars to prove that they are <a href="https://en.wikipedia.org/wiki/Equivalence_(formal_languages)" rel="n... |
1,560,411 | <p>If $B_1$ and $B_2$ are the bases of two integer lattices $L_1$ and $L_2$, i.e.</p>
<p>$L_1=\{B_1n:n\in\mathbb Z^d\}$ and $L_2=\{B_2n:n\in\mathbb Z^d\}$,</p>
<p>is there an easy way to determine a basis for $L_1\cap L_2$? Answers of the form "Plug the matrices into a computer and ask for Hermite Normal Form, etc" a... | Andrew Young | 491,550 | <p>L1 and L2 are column vector latices<br/>
A is a 2x2 block matrix<br/>
0 is an all zeros matrix<br/>
$$
\begin{align}
A=
\begin{bmatrix}
L1 & L2\\
L1 & 0
\end{bmatrix}
\end{align}
$$</p>
<p>hermite normal form<br/>
$$
\begin{align}
B=hnf(A)
\end{align}
$$</p>
<p>$$
\begin{align}
B=
\begin{bmatrix}
C ... |
43,956 | <p>There is this example at the Wikipedia article on Quotient spaces (QS):</p>
<blockquote>
<p>Consider the set $X = \mathbb{R}$ of all real numbers with the ordinary topology, and write $x \sim y$ if and only if $x−y$ is an integer. Then the quotient space $X/\sim$ is homeomorphic to the unit circle $S^1$ via the h... | Aaron | 9,863 | <p>Given a set $S$ and an equivalence relation $\sim$ on $S$, then we can quotient out the set by the equivalence relation to obtain the quotient $S/\sim$, which is the set of all equivalence classes of $S$ under $\sim$.</p>
<p>If $S$ isn't just a set but is actually a topological space, we wish to give a topology to ... |
4,015,741 | <p>I want to find the solutions of <span class="math-container">$(x+1)^{63}+(x+1)^{62}(x-1)+\cdots+(x-1)^{63}=0$</span>.</p>
<p>It is not hard to see <span class="math-container">$x=0$</span> is a root of the equation. but I don't know how to solve this equation in general. I can see terms of the equation looks very s... | Shubham Johri | 551,962 | <p>Note that <span class="math-container">$x=-1$</span> is not a solution. This is a GP with common ratio <span class="math-container">$r=(x-1)/(x+1)\ne1$</span> and the sum is<span class="math-container">$$(x+1)^{63}\left[\frac{r^{64}-1}{r-1}\right]=0\implies r=\pm1$$</span>Since <span class="math-container">$r\ne1$</... |
4,015,741 | <p>I want to find the solutions of <span class="math-container">$(x+1)^{63}+(x+1)^{62}(x-1)+\cdots+(x-1)^{63}=0$</span>.</p>
<p>It is not hard to see <span class="math-container">$x=0$</span> is a root of the equation. but I don't know how to solve this equation in general. I can see terms of the equation looks very s... | Shrivardhan | 884,246 | <p>You can solve it by using the formula of GP<br />
Here <span class="math-container">$a = (x+1)^{63}$</span> and <span class="math-container">$r = \dfrac{x-1}{x+1}$</span></p>
|
596,374 | <p>I solved this , but I am not sure if I did in the right way.</p>
<p>$$2^{2x + 1} - 2^{x + 2} + 8 = 0$$</p>
<p>$$2^{x + 2} - 2^{2x + 2} = 8$$</p>
<p>$$\log_22^{x + 2} - \log_22^{2x + 2} = \log_28$$</p>
<p>$$x + 2- 2x - 2 = 3$$</p>
<p>solving for $x$:</p>
<p>$$x = -2$$</p>
<p>any feedback would be appreciated.<... | Ahaan S. Rungta | 85,039 | <p>You can easily check the solution you get. Checking $x=-2$, you get $$ 2^{2x+1}-2^{x+2}+8=2^{-3}-2^{0}+8=\dfrac{1}{8}-1+8\ne0, $$which means you have an error. Specifically, you transition from the first line to the second line is incorrect. </p>
|
1,539,350 | <p>orthogonal matrixes conserve length of every vector and scalar products of them, I think only rotate transformation satisfy those property, but I don't know how to prove it, whether it's true or not.</p>
| Gautam Shenoy | 35,983 | <p>Ok. I got it. The Lemma is true. Given a sequence of probability measures $P_n$, we consider $P_n(x)$ for every $x \in \mathcal{X}$. Using Weierstrass Bolzano theorem, noting that $P_n(x)$ is a bounded sequence of real numbers, there exists a convergent subsequence. We do this iteratively:
WLOG, assume $\mathcal{X} ... |
1,539,350 | <p>orthogonal matrixes conserve length of every vector and scalar products of them, I think only rotate transformation satisfy those property, but I don't know how to prove it, whether it's true or not.</p>
| Ilya | 5,887 | <p>When $X$ is finite, you can treat $\mathcal P$ as a subset of $\Bbb R^X$. Total variation is the norm there, and all the norms over finite-dimensional spaces are equivalent, so the induced metric is equivalent to the Eucledian one. Embedded $\mathcal P$ is bounded and closed in Eucledian metric, hence compact.</p>
|
55,404 | <p>I have been searching for a version of the isoperimetric inequality which is something like:</p>
<p>$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ are the inner and outer radius of a given set. There are of course details which I am missing such as what kind... | Andrey Rekalo | 5,371 | <p>There is a sharpened version of the plane isoperimetric inequality due to Benson which involves the inner and outer radii. Let $$\Gamma=\{(r,\theta):\ r=r(s),\theta=\theta(s)\}$$ be a simple closed rectifiable curve on the plane, parametrized by the arc length $s$, and let
$$r_1=\sup\{r:\ (r,\theta)\in\Gamma\},\qqu... |
3,176,629 | <p>In the evening, pizza was ordered nine people sat around a round table, 50 slices of pizza were served to these nine people. Prove that there were two people sitting next to each other who ate at least 12 pizza slices.</p>
<p>I used the pigeon hole principle to determine 50/9 = 5.5 => 6</p>
<p>Therefore, at least ... | Mike Earnest | 177,399 | <p>Since on average people ate <span class="math-container">$50/9<6$</span> slices, there exists a person who ate <em>at most</em> <span class="math-container">$5$</span> slices. The other eight people together ate at least <span class="math-container">$45$</span> slices. Can you conclude?</p>
|
2,637,983 | <p>I was working on a program to carry out some computations, and ran into an issue of needing to compare some algebraic numbers, but not having enough precision to do it without exact arithmetic, and not knowing how to do it with exact arithmetic.</p>
<p>A little algebra shows that the statement
$$a+b\sqrt{n}>0$$
... | orangeskid | 168,051 | <p>Just an observation, you may already know it:</p>
<p>If all the conjugates of the real algebraic number <span class="math-container">$\alpha$</span> are complex ( that is, <span class="math-container">$\alpha$</span> is the only real root of its minimal polynomial), then there exists a unique ordering on <span class... |
95,819 | <p>I think I have solved a problem in <em>Topology</em> by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight what is troubling me.</p>
<p>We prove by contradiction: Assume $X$ is not Hausdorff. Then there exist points... | David Mitra | 18,986 | <p>A straightforward proof is a bit simpler:</p>
<p>If the diagonal is closed, then for $x\ne y$ in $X$, the point $(x,y)$ is not on the diagonal. So, there is an nhood of $(x,y)$ in $X\times X$ containing $(x,y)$ disjoint from the diagonal. But, then, this nhood is of the form $U\times V$ where $U$ and $V$ are open ... |
7,575 | <p>How could I display text that flashed red for a half second or so and then reverted to black? (Or was put in bold and reverted to normal, etc.)</p>
| István Zachar | 89 | <p>By specifying different values for <code>time</code> and <code>frequencyInverse</code>, the behavior of flashing can be finetuned.</p>
<pre><code>time = 100;
frequencyInverse = 4;
i = 0;
Dynamic@Style["TESTESTEST", Bold, RGBColor[color, 0, 0]]
RunScheduledTask[(i = i + 1; color = Rescale[Sin[i/frequencyInverse], {... |
3,573,334 | <blockquote>
<p>Given positives <span class="math-container">$a, b, c$</span> such that <span class="math-container">$a + b + c = 3$</span>, prove that <span class="math-container">$$\frac{1}{c^2 + 4a^2 + b^2} + \frac{1}{a^2 + 4b^2 + c^2} + \frac{1}{b^2 + 4c^2 + a^2} \le \frac{1}{2}$$</span></p>
</blockquote>
<p>We ... | LHF | 744,207 | <p>The idea is indeed that of using Cauchy-Schwarz. The problem is you went from a homogeneous expression (<span class="math-container">$a^2+4b^2+c^2$</span>) to non-homogeneous terms and that may be harder to prove than the original inequality.</p>
<p>Instead, I would apply Cauchy-Schwarz like this:</p>
<p><span cla... |
3,573,334 | <blockquote>
<p>Given positives <span class="math-container">$a, b, c$</span> such that <span class="math-container">$a + b + c = 3$</span>, prove that <span class="math-container">$$\frac{1}{c^2 + 4a^2 + b^2} + \frac{1}{a^2 + 4b^2 + c^2} + \frac{1}{b^2 + 4c^2 + a^2} \le \frac{1}{2}$$</span></p>
</blockquote>
<p>We ... | Michael Rozenberg | 190,319 | <p>We need to prove that
<span class="math-container">$$\sum_{cyc}\frac{1}{b^2+c^2+4a^2}\leq\frac{9}{2(a+b+c)^2}$$</span> or
<span class="math-container">$$\sum_{cyc}(2a^6-4a^5b-4a^5c+13a^4b^2+13a^4c^2-4a^4bc-12a^3b^3-12a^3b^2c-12a^3c^2b+20a^2b^2c^2)\geq0$$</span> or
<span class="math-container">$$\sum_{cyc}(a-b)^2(2c^... |
4,008,152 | <p>Question itself: Throw a coin one million times. What is the expected number of sequences of six tails, if we <strong>do not allow overlap</strong>?</p>
<p>I know when overlap is allowed, the answer is (1,000,000-5)/(2^6). Not sure if we can just do (1,000,000-5)/(2^6) divided by 6 if overlap is not allowed?</p>
<p>... | BruceET | 221,800 | <p><strong>Comment.</strong> I'm not sure I've got the rules exactly right, but I did some checking
with simulation in R. The <code>rle</code> procedure in R (for Run Length Encoding)
gives run values (<code>0</code>s for Tails, <code>1</code>s for Heads) and lengths.</p>
<p>For example:</p>
<pre><code>set.seed(2021)
x... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | guest | 11,700 | <p>For most courses, I feel like you learn the material better if it is no notes on the tests. You've really internalized the material. It sticks with you better years down the road (so even if you need to look at a book then, it comes back fast since you really learned it down pat before). And then in near term, it... |
743,227 | <p>I've this question:</p>
<blockquote>
<p>Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$.</p>
</blockquote>
<p>Is this second equation even a closed shape? If one were to plot points satisfying that equation, one gets things like $(2, \sqrt{-2})$, $(... | MarkisaB | 139,345 | <p>$$x^2+y^2-y=0$$</p>
<p>$$x^2+(y^2-2y\frac{1}{2}+\frac{1}{4}-\frac{1}{4})=0$$</p>
<p>$$x^2+(y-\frac{1}{2})=(\frac{1}{2})^2$$</p>
<p>This is an equation of cylinder, in the xy plane we have a circle moved from origin with $R=1/2$ and in + nad - z directions we have a constant - so this is cylinder. </p>
|
98,700 | <blockquote>
<p>Suppose you wanted to write the number 100000. If you type it in ASCII, this would take 6 characters (which is 6 bytes). However, if you represent it as unsigned binary, you can write it out using 4 bytes.</p>
</blockquote>
<p>(from <a href="http://www.cs.umd.edu/class/sum2003/cmsc311/Notes/BitOp/asc... | TonyK | 1,508 | <p>You can, in fact, write it out using three bytes. My current project uses 3-byte integers extensively, to save memory in an embedded system.</p>
|
3,313,697 | <p>To calculate the <span class="math-container">$n$</span>-period payment <span class="math-container">$A$</span> on a loan of size <span class="math-container">$P$</span> at an interest rate of <span class="math-container">$r$</span>, the formula is:</p>
<p><span class="math-container">$A=\dfrac{Pr(1+r)^n}{(1+r)^n-1... | Claude Leibovici | 82,404 | <p>Your approximation is not bad at all.</p>
<p>We can make it a bit better considering (as you wrote)
<span class="math-container">$$i=\bigg(\dfrac{nr(1+r)^n}{(1+r)^n-1}\bigg)^{\frac{1}{n}}-1$$</span> Expand the rhs as a Taylor series around <span class="math-container">$r=0$</span>. This would give
<span class="mat... |
37,252 | <p>Let $V$ be a vector space with a finite Dimension above $\mathbb{C}$ or $\mathbb{R}$.</p>
<p>How does one prove that if $\langle\cdot,\cdot\rangle_{1}$ and $\langle \cdot, \cdot \rangle_{2}$ are two Inner products</p>
<p>and for every $v\in V$ $\langle v,v\rangle_{1}$ = $\langle v,v\rangle_{2}$ so $\langle\cdot,\... | t.b. | 5,363 | <p><strong>Hint:</strong> Note that the associated norms satisfy $\|v\|_1 = \sqrt{\langle v,v\rangle_1} = \sqrt{\langle v,v\rangle_2} = \|v\|_2$ and then use the <a href="http://en.wikipedia.org/wiki/Polarization_identity" rel="nofollow">polarization identity</a> to recover the scalar products and see that they are equ... |
37,252 | <p>Let $V$ be a vector space with a finite Dimension above $\mathbb{C}$ or $\mathbb{R}$.</p>
<p>How does one prove that if $\langle\cdot,\cdot\rangle_{1}$ and $\langle \cdot, \cdot \rangle_{2}$ are two Inner products</p>
<p>and for every $v\in V$ $\langle v,v\rangle_{1}$ = $\langle v,v\rangle_{2}$ so $\langle\cdot,\... | yoyo | 6,925 | <p>(symmetric, bilinear, char $\neq2$)
$$\langle v+w,v+w\rangle_1=\langle v+w,v+w\rangle_2$$
$$\langle v,v\rangle_1+\langle w,w\rangle_1+2\langle v,w\rangle_1=\langle v,v\rangle_2+\langle w,w\rangle_2+2\langle v,w\rangle_2$$
$$\langle v,w\rangle_1=\langle v,w\rangle_2$$
if the product is sesquilinear over $\mathbb{C}$ ... |
334,701 | <p>The question is really simple, its just terminology.</p>
<p>For simplicity we work on smooth algebraic surfaces and we consider the intersection form on curves on the surface.</p>
<p>So let $S$ be a surface and $D \in \operatorname{Pic}(S)$ a divisor. Then $D$ is said to be nef if
$$
D.C \geq 0
$$
for all curves $... | Rhys | 47,565 | <p>Everything you have written is correct; the terminology <em>is</em> slightly confusing at first, because as you said, not all effective divisors are numerically effective.</p>
|
2,426,361 | <p>What would be the best mathematical tool/concept to measure how far a matrix is from being singular? Could it be the condition number?</p>
| Roger | 581,343 | <p>A matrix gets the rank it deserves. Technically only a square matrix can be nonsingular, but any m by n matrix, m>0 and n>0, can have a rank greater than zero if at least one entry is nonzero.</p>
<p>I interpret the question as, "How far a matrix is from losing rank?"</p>
<p>Gaussian elimination with complete pivo... |
3,443,137 | <p>Find the radius of the circle tangent to <span class="math-container">$3$</span> other circles <span class="math-container">$O_1$</span>, <span class="math-container">$O_2$</span> and <span class="math-container">$O_3$</span> have radius of <span class="math-container">$a$</span>, <span class="math-container">$b$</s... | MvG | 35,416 | <p>If the given circles are tangent to one another, use <a href="https://en.wikipedia.org/wiki/Descartes%27_theorem" rel="nofollow noreferrer">Descartes' theorem</a>.</p>
<p>Otherwise you will need more information, namely not just the radii but also the distances between the given circles, and following one of the me... |
3,501,879 | <p>I have been stuck at this problem for some time now. I'd really apprechiate your help. Thanks.</p>
<p><span class="math-container">$$2\sin^2(x)+6\cos^2(\frac x4)=5-2k$$</span></p>
| gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>If you are trying to solve for <span class="math-container">$k$</span> in terms of <span class="math-container">$x$</span>, why not use the fact that <span class="math-container">$\sin^2 a + \cos^2 a = 1$</span> for all real <span class="math-container">$a$</span>, and eliminate <span c... |
4,595,208 | <p>I'm having trouble calculating this limit directly :</p>
<p><span class="math-container">$$\lim_{n\to\infty}\frac{(2n＋1)(2n＋3)\cdots(4n＋1)}{(2n)(2n＋2)\cdots(4n)}$$</span></p>
<p>It can be calculated using the inventory method and the result is:<br />
<span class="math-container">$$\lim_{n\to\infty}\frac{(2n＋1)(2n＋3)... | Veliko | 172,553 | <p>You have to split the problem into three scenarios:</p>
<ol>
<li><span class="math-container">$\alpha>\pi/2$</span></li>
<li><span class="math-container">$\alpha=\pi/2$</span> (we have parallel lines, it does not work)</li>
<li><span class="math-container">$\alpha<\pi/2$</span></li>
</ol>
<ul>
<li><p>You cover... |
19,815 | <p>Problem:</p>
<blockquote>
<p>Prove that if gcd( a, b ) = 1, then gcd( a - b, a + b ) is either 1 or 2.</p>
</blockquote>
<p>From Bezout's Theorem, I see that am + bn = 1, and a, b are relative primes. However, I could not find a way to link this idea to a - b and a + b. I realized that in order to have gcd( a, b ) =... | NebulousReveal | 2,548 | <p>Note that $d|(a-b)$ and $d|(a+b)$ where $d = \gcd(a-b, a+b)$. So $d$ divides the sum and difference (i.e. $2a$ and $2b$).</p>
|
1,375,085 | <p>It is the first time I met such a question:</p>
<blockquote>
<p>Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?</p>
</blockquote>
<p>Intuitively I think $f(n)$ would gradually become larger as $n$ gets larger, but I find it hard to produce an argument.
Is there any trick to use for... | Eoin | 163,691 | <p>Take logs</p>
<p>$$\log\log\log(f)=\log\log\log(2^{2^{2^n}})=n\log(2\log(2\log(2)))$$
$$\log\log(g)=\log\log(100^{100^n})=n\log (100(\log (100)))$$</p>
<p>Which are both linear and grow at the same rate, so to say. Now exponentiate to retrieve $f,g$.</p>
|
2,699,621 | <p>To show $1 + \frac12 x - \frac18 x^2 < \sqrt{1+x}$ is it enough to tell that the taylor series expansion of $\sqrt{1+x}$ around $0$ has more positive terms?</p>
| Jack D'Aurizio | 44,121 | <p>The Maclaurin series of $\sqrt{1+x}$ is only convergent in a neighbourhood of the origin (due to the singularity at $x=-1$) hence it is not a good idea to use it for proving such inequality. Basic algebra performs much better: it is obvious that for $x>0$ we have $1+\frac{x}{2}>\sqrt{1+x}$ (it is enough to squ... |
79,337 | <p>Let $V$ be a vector space over $\mathbb C$ and $W$ a $\operatorname{End}(V)$-module. I'm having difficulty seeing why the map
$$
\operatorname{Hom}_{\operatorname{End}(V)}(V,W) \otimes V \to W
$$
$$
\phi \otimes v \mapsto \phi(v)
$$
is surjective. I'm having trouble because I can't construct elements of $\oper... | bradhd | 5,116 | <p>In the case that $V$ and $W$ are finite-dimensional, here is a "non-intrinsic" way to see this isomorphism. If $V$ is finite-dimensional, then every $\operatorname{End}(V)$-module is semisimple, and moreover every simple $\operatorname{End}(V)$-module is isomorphic to $V$:</p>
<p>Choose a basis $\{e_i\}$ for $V$ an... |
79,337 | <p>Let $V$ be a vector space over $\mathbb C$ and $W$ a $\operatorname{End}(V)$-module. I'm having difficulty seeing why the map
$$
\operatorname{Hom}_{\operatorname{End}(V)}(V,W) \otimes V \to W
$$
$$
\phi \otimes v \mapsto \phi(v)
$$
is surjective. I'm having trouble because I can't construct elements of $\oper... | Eric O. Korman | 9 | <p>I just thought of the following argument. Suppose first that $W$ is finite dimensional. Then the map I gave is a map of $\operatorname{End}(V)$ mods (with the trivial action on $Hom_{\operatorname{End}(V)}(V,W)$) and so must be an isomorphism by Schur's lemma. In the general case we'll have $W = \bigoplus_i W_i$ ... |
697,668 | <p>How many combinations are there to arrange the letters in MISSISSIPPI requiring that the 2 S's must be separated? </p>
<p>I found there are 34650 combinations to arrange without restriction. </p>
<p>How to approach this question?</p>
| Daniel Li | 294,291 | <p>Find all possible ways we can arrange MIIIPPI: 7!/(4!*2!). Then "insert" four S into the 8 space between each possible arrangement of MIIIPPI, e.g. [1]M[2]I[3]I[4]I[5]P[6]P[7]I[8]; This is really <span class="math-container">${8}\choose{4}$</span>. Multiplying the two!</p>
|
232,930 | <p>Let $f(n)$ denote the number of integer solutions of the equation $$3x^2+2xy+3y^2=n $$</p>
<p>How can one evaluate the limit $$\lim_{n\rightarrow\infty}\frac{f(1)+...f(n)}{n}$$</p>
<p>Thanks</p>
| Sangchul Lee | 9,340 | <p>Let us elaborate the solution by Thomas Andrews. Though the idea is simple and repeatedly used in other answers, here we want to give a meticulously detailed solution.</p>
<p>As we have observed, the number $f(1) + \cdots + f(n)$ is equal to the integer solution $(x, y)$, or equivalently the number of integer point... |
2,174,061 | <p>in $\Delta ABC$ if the $AD\perp BC$,$D\in BC$,and such $$|BC|=2|AD|$$
show that
$$\dfrac{|AB|}{|AC|}\le\sqrt{2}+1$$
<a href="https://i.stack.imgur.com/SXDvI.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SXDvI.png" alt="enter image description here"></a></p>
<p>since
$$\cot{B}+\cot{C}=\dfrac{BD}... | Michael Rozenberg | 190,319 | <p>In the standard notation <span class="math-container">$AD=\frac{a}{2}$</span> and let <span class="math-container">$\frac{c}{b}=x$</span>.</p>
<p>Thus, <span class="math-container">$$\frac{a^2}{4}=S_{\Delta ABC}$$</span> or
<span class="math-container">$$\frac{a^2}{4}=\frac{1}{4}\sqrt{2(a^2b^2+a^2c^2+b^2c^2)-a^4-b^... |
2,157,914 | <p>I am struggling with the next exercise of my HW:</p>
<p>How many conjugacy classes are in $GL_3(\mathbb{F}_p)$? And how many in $SL_2(\mathbb{F}_p)$?</p>
<p>It's on the topic of Frobenius normal form of finitely generated modules over $\mathbb{F}_p$.</p>
<p>I'd appreciate any idea.</p>
| Community | -1 | <p>Taking a cue from <a href="http://oeis.org/A000194" rel="noreferrer">sequence A000194 in the OEIS</a>, we have $$S = \sum_{k=1 }^{2016} \frac{1}{f (k)}$$ $$ = \frac {2}{1} + \frac {4}{2} + \frac {6}{3} + \cdots + \frac {2 \lfloor \sqrt {2016} \rfloor}{\lfloor \sqrt {2016} \rfloor} + \frac {36}{\lceil \sqrt {2016} \r... |
2,157,914 | <p>I am struggling with the next exercise of my HW:</p>
<p>How many conjugacy classes are in $GL_3(\mathbb{F}_p)$? And how many in $SL_2(\mathbb{F}_p)$?</p>
<p>It's on the topic of Frobenius normal form of finitely generated modules over $\mathbb{F}_p$.</p>
<p>I'd appreciate any idea.</p>
| Tig la Pomme | 419,035 | <p>If $j$ is a positive integer, the integers $n$ for which the closest integer to $\sqrt{n}$ is $j$ are those who verify $j-1/2<\sqrt{n}<j+1/2$, i.e. $j^2-j<n \leqslant j^2+j$. Then you can group terms:
$$\sum_{k=1}^{m^{2}+m}\frac{1}{f(k)}=\sum_{j=1}^{m}\sum_{k=j^2-j+1}^{j^2+j}\frac{1}{f(k)}=\sum_{j=1}^{m}\su... |
353,947 | <p>Problem. Let $f:[a,b]\to\mathbb{R}$ be a function such that $ f\in C^3([a,b])$ and $f(a)=f(b)$. Prove that $$ \left|\int\limits_{a}^{\frac{a+b}{2}}f(x)dx-\int\limits_{\frac{a+b}{2}}^{b}f(x)dx\right|\leq\frac{(b-a)^4}{192}\max_{x\in [a,b]}|f'''(x)|.$$
Any idea are welcome.</p>
| robjohn | 13,854 | <p>Let $f(x)=g(t)+h(t)$, where $g$ is odd, $h$ is even, and $x=\frac{b-a}{2}t+\frac{b+a}{2}$; that is $t=\frac{2x-b-a}{b-a}$.</p>
<p>Then, since $g$ is odd and $h$ is even,
$$
\max_{t\in[-1,1]}|g'''(t)|\le\left|\frac{b-a}{2}\right|^3\max_{x\in[a,b]}|f'''(x)|\tag{1}
$$
and
$$
\begin{align}
\int_{\frac{b+a}{2}}^bf(x)\,\... |
4,170,940 | <blockquote>
<p><a href="https://www.isical.ac.in/%7Eadmission/IsiAdmission2017/PreviousQuestion/BStat-BMath-UGA-2016.pdf" rel="nofollow noreferrer">Question 36</a>: Finding graph corresponding to <span class="math-container">$\int_0^{\sqrt{x} } e^{ -\frac{u^2}{x} } du$</span>
<a href="https://i.stack.imgur.com/KIVRA.p... | Jean Marie | 305,862 | <p>I don't know if this can be called a trick, but I have been guided by the "normal law" in probability :</p>
<p>Setting <span class="math-container">$\sigma=\sqrt{x}$</span> and <span class="math-container">$u=\tfrac{1}{\sqrt{2}}U$</span>, we get :</p>
<p><span class="math-container">$$ I=\frac{1}{\sqrt{2}}... |
2,261,927 | <p>How to get alternative form from equation 1)</p>
<p>$$ 1) -a^2 + a + b^2 -b $$</p>
<p>to equation 2)</p>
<p>$$ 2) (a-b)(a+b-1)$$</p>
| tomi | 215,986 | <p>First rearrange the equation slightly:</p>
<p>$$b^2-a^2+a-b$$</p>
<p>Note that we have a 'difference of two squares'</p>
<p>$$(b-a)(b+a)+a-b$$</p>
<p>Recognise that $a-b\equiv -(b-a)$</p>
<p>$$(b-a)(b+a)-(b-a)$$</p>
<p>Now we have $(b-a)$ as a common factor so factorise:</p>
<p>$$(b-a)\left ((b+a)-1 \right)$$... |
3,467,523 | <p><a href="https://i.stack.imgur.com/G47bX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/G47bX.png" alt="Attached is the picture of the problem."></a></p>
<p>I was doing some trig problems for leisure. This one particularly seems not trivial. So I thought someone may be interested to take a look.... | Eric Towers | 123,905 | <p>If we assume the claim is true and substitute <span class="math-container">$x = 0$</span>, we find <span class="math-container">$4-m=0$</span>, which is straightforward to solve for <span class="math-container">$m$</span>.</p>
<p>If we'd like to check another angle, try <span class="math-container">$x = \pi/4$</spa... |
2,250,638 | <p>I feel I am doing the problem correctly however my answers are not following the solution.</p>
<p>My attempt: </p>
<p>$y^{2}+2y+12x-23=0$</p>
<p>$(y^{2}+2y+1) +12x = 23-1$</p>
<p>$(y+1)^{2}+12x=22$ </p>
<p>$\dfrac{(y+1)^{2}}{22}+\dfrac{6x}{11}=1$</p>
<p>Note $a > b$</p>
<p>$a^{2}=22$</p>
<p>$b^{2}=11$</p>... | Blue | 409 | <p><a href="https://i.stack.imgur.com/C09ng.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C09ng.png" alt="enter image description here"></a></p>
<p>Let $A$ and $B$ be points on $x=1$ such that $|\overline{AB}| = 2$. With $R$ the point where $x=1$ meets the circle, define $\alpha := \angle ROA$ and... |
987,895 | <blockquote>
<p>Let $(G,\cdot)$ be a group, $g \in G$.</p>
<p>For $a,b \in G$ define $a * b := a \cdot g^{-1} b$. Show that $(G,*)$ is a group with the neutral element $g$ and $f : (G,*) \rightarrow (G,\cdot), a \mapsto a \cdot g^{-1}$ is a group isomorphism.</p>
</blockquote>
<p>In order to show that $(G,*)$ ... | mfl | 148,513 | <p>To get the inverse of an element $a:$</p>
<p>$$a*b=ag^{-1}b=g\implies g^{-1}b=a^{-1}g\implies b=ga^{-1}g,$$</p>
<p>that is, the inverse of $a$ with respect to $*$ is $ga^{-1}g.$</p>
<p>To show that $f$ is injective:</p>
<p>$$f(a)=f(b)\implies ag^{-1}=bg^{-1}\implies a=b$$ (just multiply by the right by $g$ in th... |
4,317,945 | <p>A function <span class="math-container">$h : A → \mathbb{R}$</span> is Lipschitz continuous if <span class="math-container">$\exists K$</span> s.t.</p>
<p><span class="math-container">$$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$</span></p>
<p>Suppose that <span class="math-container">$I = [a, b]$</span... | MachineLearner | 647,466 | <p>You can apply the <a href="https://en.wikipedia.org/wiki/Binomial_theorem" rel="nofollow noreferrer">binomial formula</a> <span class="math-container">$(1+x)^n=\sum_{k=0}^n\dfrac{n!}{(n-k)!k!}x^k1^{n-k}=\sum_{k=0}^n\dfrac{n!}{(n-k)!k!}x^k$</span></p>
<p><span class="math-container">$$g(x)=x^2(x + 1)^n = x^2\sum_{k=0... |
163,465 | <p>I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.</p>
<p>1) <strong>Does this quantity $f(X,t)$ have a name</strong>? As the title question suggests, it is a tail of the integral that is cut out when computing t... | Santiago | 10,203 | <p>The paper "Bounds on Conditional Moments of Weibull and Other Monotone Failure Rate Families" by Patel and Read (1975) gives some useful bounds for distributions with monotone failure rates (either increasing or decreasing), where the failure rate is given by $h(t) = p(t) / (1 - P(t) )$. </p>
<p>Let $H(t) = \int_0^... |
189,689 | <pre><code>CountryData[ "UnitedStates", {"Population", 2014} ]
</code></pre>
<blockquote>
<p>322 422 965 people</p>
</blockquote>
<pre><code>CountryData[ "UnitedStates", {"Population", 2015} ]
</code></pre>
<blockquote>
<p>Missing[ "NotAvailable" ]</p>
</blockquote>
<p>How can I update it?
I am new to Mathemati... | Jason B. | 9,490 | <p><code>CountryData</code> does appear to be out of date, and that is a shame. Many of the XXXData functions now act as wrappers to call <code>EntityValue</code>. Run <code>TracePrint[PlanetData["Venus", "AngularDiameterFromEarth"],_EntityValue]</code> to see that this is true. </p>
<p>But <code>CountryData</code> ... |
600,404 | <p>I'm trying to study line bundle over $S^2$. <a href="https://mathoverflow.net/questions/113924/line-bundle-on-s2">In this post</a> was outlined the method based on clutching functions. But now I'm interesting in another approach. </p>
<p>For the sphere there is two maps : upper hemisphere and lower hemisphere with ... | Jean-Claude Arbaut | 43,608 | <p>You can write</p>
<p>$$\frac{1^p+2^p+...+n^p}{n^{p+1}}=\frac{1}{n} \left( \left( \frac{1}{n}\right)^p + \left( \frac{2}{n}\right)^p + ... + \left( \frac{n}{n}\right)^p \right)= \frac{1}{n}\sum_{k=1}^n\left( \frac k n\right)^p$$</p>
<p>This is a <a href="https://en.wikipedia.org/wiki/Riemann_sum" rel="nofollow">Rie... |
194,472 | <p>It is well-known that there are continuous curves $f:I \to \mathbb R^2$ (where $I \subset \mathbb R$ is an interval) whose image have positive measure (e.g Peano curve).
I have read somewhere that if we require the curve to be differentiable evrywhere then this cannot happen; but if we require it to be almost everyw... | Sangchul Lee | 9,340 | <p>Let $\phi : [0, 1] \to [0, 1]$ be the <em><a href="http://en.wikipedia.org/wiki/Cantor_function" rel="noreferrer">Cantor-Lebesgue function</a></em>, and $\alpha : [0, 1] \to \Bbb{R}^n$ be a space-filling curve.</p>
<p>Since $\phi$ is stationary outside the Cantor set, it is locally constant almost everywhere. That ... |
194,472 | <p>It is well-known that there are continuous curves $f:I \to \mathbb R^2$ (where $I \subset \mathbb R$ is an interval) whose image have positive measure (e.g Peano curve).
I have read somewhere that if we require the curve to be differentiable evrywhere then this cannot happen; but if we require it to be almost everyw... | Will Jagy | 10,400 | <p>The other direction is called (mini)-Sard's Theorem. Page 205 in Appendix 1 of Guilleman and Pollack, <em>Differential Topology</em>. The mini version is just this: Let $U$ be an open set of $\mathbb R^n,$ and let $f:U \rightarrow \mathbb R^m$ be a smooth map. Then, if $m > n,$ we can conclude that $f(U)$ has mea... |
3,810,733 | <p><span class="math-container">$$''' + y' = 2^2 + 4\sin(x)$$</span></p>
<p>Find the general solution of the differential equation by using the Indefinite Coefficients Method.</p>
| Physor | 772,645 | <p><span class="math-container">$u$</span> is a unit vector means <span class="math-container">$||u|| = \sqrt{u_1^2 +u_2^2 + u_3^2 + u_4^2} = 1$</span>. Consider it as a constraint on the vector you seek <span class="math-container">$u$</span> in the form
<span class="math-container">$$
g(u_1,u_2,u_3,u_4) = 0\qquad {\r... |
108,253 | <p>I would like to assign 'x' individuals to 'y' groups, randomly. For example, I would like to divide 50 individuals into 100 groups randomly. Of course, with more groups than individuals many of the groups will have zero individuals, while some groups will have multiple individuals. That is fine. With random assignme... | David G. Stork | 9,735 | <pre><code>x = 50;
myList = Range[x];
maxgroupSize = 5;
numberGroups = 100;
Table[DeleteDuplicates@RandomChoice[myList, maxgroupSize], numberGroups]
</code></pre>
<p>or</p>
<pre><code>Table[RandomSample[myList, maxgroupSize], numberGroups]
</code></pre>
|
108,253 | <p>I would like to assign 'x' individuals to 'y' groups, randomly. For example, I would like to divide 50 individuals into 100 groups randomly. Of course, with more groups than individuals many of the groups will have zero individuals, while some groups will have multiple individuals. That is fine. With random assignme... | LLlAMnYP | 26,956 | <p>Randomly partitioning a list into <code>y</code> groups is as easy, as splitting a random permutation of it at <code>y-1</code> positions.</p>
<pre><code>set = RandomSample[Range[50]] (* works with any list, though *)
split = Partition[{1}~Join~RandomChoice[Range[Length[set] + 1], 99]
~Join~{Length[set] +... |
3,306,571 | <p>I know that the function <span class="math-container">$f(x) = \frac{x}{x}$</span> is not differentiable at <span class="math-container">$x = 0$</span>, but according to the definition of differentiable functions:</p>
<blockquote>
<p>A differentiable function of one real variable is a function whose derivative exi... | Theo Bendit | 248,286 | <p>The function <span class="math-container">$f(x) = \frac{x}{x}$</span> has a natural domain of <span class="math-container">$\Bbb{R} \setminus \{0\}$</span>; you can sensibly substitute in any value except <span class="math-container">$0$</span>. In order to be continuous or differentiable at a point, it is <strong>r... |
802,960 | <p>$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$
Find value of $n$ for which equation is satisfied. </p>
| enigne | 143,559 | <p>n=3.
By drawing this figure, you can easily know
<img src="https://i.stack.imgur.com/iA9Tw.png" alt="enter image description here"></p>
|
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | Zarrax | 3,035 | <p>No need for L'hopital: Just use the fact that by the definition of derivative,
$$\lim_{x \rightarrow 0} {2^x - 1 \over x} = {d \over dx} 2^x\bigg|_{x = 0}$$
$$ = \ln 2$$
So as in bjatr's answer, this means that
$$\lim_{n \rightarrow \infty} {2^{1 \over n} - 1 \over {1 \over n}} = \ln 2$$
So by the limit comparison... |
1,672,131 | <p>A card game is played with a deck whose cards can be one of 6 suits, one of the suits being hearts, and one of 11 ranks. A hand is a subset of 3 cards. What is the probability that a hand has exactly two hearts given that it has the 2 of hearts? Please explain.</p>
| Whiz_Geek | 230,983 | <p>There are exactly three solutions- 1,8,9. 10^n has exactly n+1 digits.</p>
|
804,882 | <p>If both $L:V\rightarrow W$ and $M:W\rightarrow U$ are linear transformations that are invertible, how can you prove that the composition $(M\circ L):V\rightarrow U$ is also invertible.</p>
| Galactus | 152,561 | <p>You can prove that a transformation is invertible $\Leftrightarrow $ its associated matrix is also invertible. Then use the matrix associated to the composition to prove that it is invertible.</p>
|
1,560,209 | <p>Prove that $f(x):\mathbb{R}\to\mathbb{R}$ , $x \mapsto x^3$ is injective.</p>
<hr>
<p>I want to prove this claim is true. </p>
<p>Here is my outline so far:</p>
<hr>
<p>We want to show that $f(a)=f(b)$ implies that $a=b$, for all $a,b \in \mathbb{R}$</p>
<p>We have $f(a)=a^3$, and $f(b)=b^3$</p>
<p>So, if $f... | Martin Argerami | 22,857 | <p>Think of it as a quadratic on $a$: if we complete the square,
$$
a^2+ab+b^2=\left(a+\frac b2\right)^2+b^2-\frac {b^2}4=\left(a+\frac b2\right)^2+\frac{3b^2}4.
$$
For this to be zero we need both summands to be zero (because both are nonnegative). Then we get first that $b=0$, and then that $a^2=0$, so $a=0$. </p>
<... |
1,560,209 | <p>Prove that $f(x):\mathbb{R}\to\mathbb{R}$ , $x \mapsto x^3$ is injective.</p>
<hr>
<p>I want to prove this claim is true. </p>
<p>Here is my outline so far:</p>
<hr>
<p>We want to show that $f(a)=f(b)$ implies that $a=b$, for all $a,b \in \mathbb{R}$</p>
<p>We have $f(a)=a^3$, and $f(b)=b^3$</p>
<p>So, if $f... | bartgol | 33,868 | <p>Your strategy works fine. You do not need to show that the function is increasing (although, that would be one way to do it).</p>
<p>You just need to show that $a^2+ab+b^2=0$ does not hold for any $a\neq b$. To this end, fix $b$ and try to use the quadratic formula to find whether there is a value of $a$ that satis... |
2,699,170 | <p>How to evaluate
$$ \int \frac{1}{ \ln x} \ \mathrm{d} x, $$
where $\ln x$ denotes the natural logarithm of $x$? </p>
<p>My effort: </p>
<blockquote>
<p>We note that
$$ \int \frac{1}{ \ln x} \ \mathrm{d} x = \int \frac{x}{x \ln x} \ \mathrm{d} x = \int x \frac{ \mathrm{d} }{ \mathrm{d} x } \left( \ln \ln x \r... | The Integrator | 538,397 | <p>I = $\large\int\frac{1}{ln(x)}dx$</p>
<p>let ln(x) = u</p>
<p>$\,e^u = x$</p>
<p>$\,dx = e^udu$</p>
<p>I = $\,\int \frac{e^u}{u}du$</p>
<p>expanding e$^u$,</p>
<p>I=$\,\int\frac{1+u+\frac{u^2}{2!}+\frac{u^3}{3!}+\frac{u^4}{4!}+\frac{u^5}{5!}..............}{u}du$</p>
<p>I = $\,\int\frac{1}{u}+1+\frac{u}{2!}+\... |
181,367 | <p>It is well known that compactness implies pseudocompactness; this follows from <a href="https://secure.wikimedia.org/wikipedia/en/wiki/Heine%E2%80%93Borel_theorem">the Heine–Borel theorem</a>. I know that the converse does not hold, but what is a counterexample?</p>
<p>(A <a href="https://secure.wikimedia.org/wikip... | GEdgar | 442 | <p>A favorite example (and counterexample) to may things is the first uncountable ordinal $\omega_1$ in its order topology: $[0,\omega_1)$. It is pseudo-compact but not compact.</p>
|
1,692,757 | <p>I was required to find the derivative of $2\sqrt{\cot(x^2)}$.</p>
<p><strong>My solution</strong></p>
<p><a href="https://i.stack.imgur.com/N98SM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/N98SM.jpg" alt="enter image description here"></a></p>
<p>I can't find any mistake in my solution but... | heropup | 118,193 | <p>Note that $$f(x) = 2 \sqrt{\cot x^2}$$ is real-valued if and only if $\cot x^2 \ge 0$, so for $f : \mathbb R \to \mathbb R$, we must have $$x^2 \in \bigcup_{k=-\infty}^\infty (\pi k, \pi(k+1/2)].$$ On this domain, we typically take the nonnegative square root. Thus $f \ge 0$ for all such $x$. We also note that be... |
3,086,024 | <p>I am trying to solve an exercise from the book "Theory of Numbers" by B.M.Stewart. The exercise is the following one:</p>
<blockquote>
<p>Let <span class="math-container">$T=2^ap_1^{a_1}p_2^{a_2} \dots p_n^{a_n}$</span>, where <span class="math-container">$a \ge0, n\ge0, 2<p_1<p_2<\dots p_n, p_j$</span> ... | poetasis | 546,655 | <p>I'm not sure this helps because I don't understand the function completely. The formula I've alway seen for generating triples is:
<span class="math-container">$$A=m^2-n^2\qquad B=2mn\qquad C=m^2+n^2$$</span>
and it is useful but it generates extraneous and trivial triples if <span class="math-container">$m\le n$</s... |
1,384,735 | <p>What is the ODE satisfied by $y=y(x)$ </p>
<p>given that $$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$</p>
<p>I understand that I need to get it in some form of $\int \cdots \;dy = \int \cdots \; dx$, but am not sure how to go about it.</p>
| Dmytro Chasovskyi | 215,170 | <p>If you have this equation:</p>
<p>$$\frac{dy}{dx} = \frac{-x-2y}{y-2x}$$</p>
<p>you may do:
$$ ( y - 2 \cdot x) \cdot dy = - ( x + 2 \cdot y ) dx $$
After that
$$ y dy + x dx = -2 y^2 \cdot \frac{ydx-xdy}{y^2}$$
Small tricks:
$$y~dy = \frac{y^2}{2} + C $$
$$ d\bigg(\frac{x}{y}\bigg) = \frac{ydx-xdy}{y^2} $$</p>
<... |
64,544 | <blockquote>
<p>Please let me know what is the standard notation for group action.</p>
</blockquote>
<p>I saw the following three notations for group action.
(All the images obtained as <code>G\acts X</code> for different deinitions of <code>\acts</code>.) </p>
<p>(1) <img src="https://lh5.googleusercontent.com/_7... | Dick Palais | 7,311 | <p>I guess I am somewhat of a minimalist when it comes to notation. I have spent a lot of my career writing about group actions, and what I usually have done is start out defining what a group action is, and then say something like
``If we have in mind a fixed action of G on X then we will say that X is a G-space... |
316,865 | <p>How do you find this limit?</p>
<p>$$\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x$$</p>
<p>I was given a clue to use L'Hospital's rule.</p>
<p>I did it this way:</p>
<p><strong>UPDATE 1:</strong>
$$
\begin{align*}
\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x
&= \lim_{x \rightarrow \infty} x\begin{p... | muzzlator | 60,855 | <p>Your working out is fine and you've shown all the steps now. $$\sqrt[5]{x^5 - x^4} < x$$ and so a negative limit is more likely than a positive limit :)</p>
|
2,083,127 | <p>How to show that $\lim_{n \rightarrow \infty} \frac{[a^{n+1}]}{[a^n]}=a$, where
$[a]$ = integer part of a?<br>
Here $a>1$. But I suspect it is true for all $a \ne 0$. </p>
| Community | -1 | <p>For $|a|>1$,</p>
<p>$$\frac{[a^{n+1}]}{[a^n]}=\frac{a^{n+1}-\{a^{n+1}\}}{a^n-\{a^n\}}=a\frac{1-\dfrac{\{a^{n+1}\}}{a^{n+1}}}{1-\dfrac{\{a^{n}\}}{a^n}}\to a.$$</p>
<p>As the fractional parts are bounded, the numerator and denominator both tend to $1$.</p>
<hr>
<p>This can be extended to $|a|\ge1$ as with $|a|=... |
1,594,722 | <p>The ODE is</p>
<p>($xy^{3} + x^{2}y^{7}) \frac{dy}{dx} = 1$</p>
<p>I have tried everything like integrating factor,it is not homogenous and not linear differential equation..What should be done now?</p>
| Ekaveera Gouribhatla | 31,458 | <p>HINT: The equation can be written as:</p>
<p>$$y^3\frac{dy}{dx}=\frac{1}{x(1+xy^4)}$$</p>
<p>Put $y^4=t$</p>
|
2,037,704 | <p>What symmetry property in complex space is related to the fact that the absolute value of numbers $|a+ib| = |b+ia|$ are equals?</p>
| israel.sincro | 394,550 | <p>reading the comment of fleablood and Widawensen about the numbers laid of a circle, I think that the correct answer is:</p>
<p>The wanted property is that the absolute value of a number must be invariant under any axis rotation. In this case all numbers $z_1 = a + ib$, $z_2 = -a + ib$, $z_3 = a - ib$, $z_4 = -a - ... |
517,282 | <p>Suppose $a,n \in \mathbb{Z}$, and $n>a>0$. How do I prove that $\nexists x \in \mathbb{Z}$ s.t. $nx = a$ ? I'm really not sure where to start on this one. I'd be happy if someone could give me a hint.</p>
<p>Edit: I've solved this by contradiction, but I will not be 'accepting' an answer from below because I ... | Henry Swanson | 55,540 | <p>Assume there is such an $x$. Since $nx = a$, then $0 < nx$ and $nx < n$. Can you now prove that $0 < n$ and $n < 1$? And can you prove that that is a contradiction?</p>
<p>Edit: changed $p < q < r$ statements to $p < q$ and $q < r$ statements. Because hypothetically, I forgot what $<$ mea... |
3,981,809 | <p>Imagine that we have two pairs of integers <span class="math-container">$(a_1,b_1)$</span> and <span class="math-container">$(a_2, b_2)$</span> where</p>
<p><span class="math-container">$$ a_1b_1\equiv 0,\,\ a_2b_2\equiv 0,\,\ a_1b_2+a_2b_1\equiv 0\pmod n$$</span></p>
<p>Does this imply that
<span class="math-contai... | Community | -1 | <p>The result is true.</p>
<p>Let <span class="math-container">$x=a_1b_2$</span> and <span class="math-container">$y=a_2b_1$</span> and let <span class="math-container">$p^m$</span> divide <span class="math-container">$n$</span> for some prime <span class="math-container">$p$</span>.</p>
<p>Then <span class="math-conta... |
149,558 | <p>I always use <code>InputForm</code> to check the result object,such as <code>Dataset</code> or <code>Graphics</code> or other objects.But if you are in the result of <code>InputForm</code>,you cannot use the Front-End function of balance the bracket. Note this gif</p>
<p><a href="https://i.stack.imgur.com/51OYd.gif"... | kglr | 125 | <p>If you have Version 11 you can use the function <code>PrettyForm</code> instead of processing <code>InputForm</code> output to get the desired result: </p>
<pre><code>Needs["GeneralUtilities`"]
Interpolation[{1, 2, 3, 4}] // PrettyForm
</code></pre>
<p><a href="https://i.stack.imgur.com/VE2qd.jpg" rel="noreferrer... |
6,931 | <p>One of the key steps in <a href="http://en.wikipedia.org/wiki/Merge_sort">merge sort</a> is the merging step. Given two sorted lists</p>
<pre><code>sorted1={2,6,10,13,16,17,19};
sorted2={1,3,4,5,7,8,9,11,12,14,15,18,20};
</code></pre>
<p>of integers, we want to produce a new list as follows:</p>
<ol>
<li>Start w... | Heike | 46 | <p>Here's another approach. </p>
<pre><code>mergeLists[lista_, listb_, crit_: LessEqual] :=
Module[{merge},
merge[list1_, list2_] /; crit[First[list1], First[list2]] :=
With[{part = TakeWhile[list1, crit[#, First[list2]] &]},
Sow[part];
If[Length[part] == Length[list1],
Sow[list2],
merge[l... |
6,931 | <p>One of the key steps in <a href="http://en.wikipedia.org/wiki/Merge_sort">merge sort</a> is the merging step. Given two sorted lists</p>
<pre><code>sorted1={2,6,10,13,16,17,19};
sorted2={1,3,4,5,7,8,9,11,12,14,15,18,20};
</code></pre>
<p>of integers, we want to produce a new list as follows:</p>
<ol>
<li>Start w... | Daniel Lichtblau | 51 | <p>Forget Leonid and Heike's recursive stuff (okay, actually I upvoted both as they are both good responses). But here is a simple, direct version. Note that it will not sort, so if the inputs are unsorted the result will be as well.</p>
<pre><code>mergeSortedLists[lista_, listb_, crit_: LessEqual] := Module[
{resul... |
248,710 | <p>The organizers of a cycling competition know that about 8% of the racers use steroids. They decided to employ a test that will help them identify steroid-users. The following is known about the test: When a person uses steroids, the person will test positive 96% of the time; on the other hand, when a person does not... | André Nicolas | 6,312 | <p>$\Pr(P|S)$ is the probability that the person tests positive, <strong>given</strong> that she uses steroids. We are told explicitly that this is $0.96$. </p>
<p>$\Pr(S|P)$ is the probability she is a steroid user, <strong>given</strong> that she tests positive. That is what we are asked to find. Informally, if we ... |
439,918 | <p>I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't find anything.<br>
Thank you for any help.</p>
| Brian M. Scott | 12,042 | <p>Let $Y$ be an uncountable set, let $p$ be a point not in $Y$, and let $X=\{p\}\cup Y$. Let</p>
<p>$$\mathscr{B}=\{\{x\}:x\in Y\}\cup\{X\setminus F:F\text{ is a finite subset of }Y\}\;;$$</p>
<p>then $\mathscr{B}$ is a base for a Hausdorff, locally compact, compact topology $\tau$ on $X$ that is not second countabl... |
237,031 | <p>The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the strongest large cardinal not known to be inconsistent with choice, as I understand)? This is implicit in the ordering of t... | Master | 141,402 | <p>Mohammed Golshani's link doesn't work, so I have reconstructed a sketch of a proof. The key fact is this (You can find a proof in most Set Theory textbooks):</p>
<p><strong>Theorem:</strong> If <span class="math-container">$DC_\omega$</span> holds and <span class="math-container">$D$</span> is an <span class="math-... |
1,135,005 | <p>Assume that the order of $a$ modulo $n$ is $h$ and the order of $b$ modulo $n$ is $k$.
Show that the order of $ab$ modulo $n$ divides $hk$; in particular, if $\gcd(h, k) = 1$, then $ab$ has order $hk$. </p>
<p>My attempt : </p>
<p>$$(ab)^{hk}\equiv 1\pmod{n} \implies \text{ord}_n(ab) | hk$$</p>
<p>But I feel st... | lab bhattacharjee | 33,337 | <p>If ord$_na=h,$ord$_nb=k$ with $(h,k)=1$</p>
<p>If ord$_n(ab)=d\implies (ab)^d\equiv1$</p>
<p>$\implies a^{hd}b^{hd}=\left[(ab)^d\right]^h\equiv1\implies b^{hd}\equiv1\implies k|hd$</p>
<p>As $(h,k)=1,k|d$</p>
<p>Similarly, $h|d\implies$lcm$[h,k]|d\implies hk|d\ \ \ \ (1)$ as $(h,k)=1$</p>
<p>Now $(ab)^{hk}=... |
1,135,005 | <p>Assume that the order of $a$ modulo $n$ is $h$ and the order of $b$ modulo $n$ is $k$.
Show that the order of $ab$ modulo $n$ divides $hk$; in particular, if $\gcd(h, k) = 1$, then $ab$ has order $hk$. </p>
<p>My attempt : </p>
<p>$$(ab)^{hk}\equiv 1\pmod{n} \implies \text{ord}_n(ab) | hk$$</p>
<p>But I feel st... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\,\ \color{#c00}{a^{\large h}}\equiv 1\equiv\color{#0a0}{b^{\large k}}\,\Rightarrow\, (ab)^{\large hk} \equiv (\color{#c00}{a^{\large h}})^{\large k}(\color{#0a0}{b^{\large k}})^{\large h}\equiv \color{#c00}1^{\large k}\color{#0a0}1^{\large h}\equiv 1\,\Rightarrow\, {\rm ord}(ab)\mid hk$</p>
|
3,663,054 | <p>In my introductory abstract algebra course, the quotient group <span class="math-container">$G/H$</span> was defined as
<span class="math-container">$$G/H=\{gH:g\in G\}$$</span>
which is a <strong>set of sets</strong>. In an exercise, I should show that for the group of invertible matrices <span class="math-contain... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$E|X_t|^{2}=\int_0^{t} e^{-2s} ds=\frac 1 2(1-e^{-t}) <1$</span> for all <span class="math-container">$t$</span> and this implies <span class="math-container">$E|X_t|$</span> is bounded. </p>
<p>The limiting distribution is <span class="math-container">$N(0,\int_0^{\infty} e^{-2s} ... |
470,739 | <p>Assume $S$ and $T$ are diagonalizable maps on $\mathbb{R}^n$ such that $S\circ T$=$T \circ S$. Then $S$ and $T$ have a common eigenvector.</p>
<p>I already have proof, but I just need validation in one part.
My proof:
Let $F$ be an eigenvector of $T$. This means $\exists \; \lambda \in R$ such that $T(v)=\lambda v$... | Zavosh | 28,494 | <p>You've shown that the eigenspaces of $T$ are <em>invariant</em> under $S$. If $E_\lambda$ is the $\lambda$-eigenspace of $T$ inside $\mathbb{R}^n$, then it makes sense to speak of $S'=S|_{E_\lambda}: E_\lambda \rightarrow E_\lambda$. Then the key fact is that the characteristic polynomial $p'(T)$ of $S'$ is a factor... |
2,412,454 | <p>I was obviously not clear enough in my first question, so I will reformulate. I have the following equation
$$
A=\frac{B\sin 2\theta}{C+D\cos 2\theta}
$$
where $A,B,C,D$ are variables.
I need to solve or rewrite the equation to easily obtain $\theta$ (or $2\theta$), given known values for $A, B, C, D$.
Thanks for a... | Acccumulation | 476,070 | <p>The precise definition of a relation is that it is a set of ordered pairs. We then say that x~y if (x,y) is in that set. However, that's a rather cumbersome definition; if one wishes to discuss the relation of one integer being larger than another, it would not be possible to list every ordered pair in which the fir... |
368,114 | <blockquote>
<p>Prove that this set is closed:</p>
<p><span class="math-container">$$ \left\{ \left( (x, y) \right) : \Re^2 : \sin(x^2 + 4xy) = x + \cos y \right\} \in (\Re^2, d_{\Re^2}) $$</span></p>
</blockquote>
<p>I've missed a few days in class, and have apparently missed some very important definitions if they ex... | Philippe Malot | 39,781 | <p>Your set is the preimage of the closed set $\{0\}$ by the continuous function $\Bbb R^2\ni(x,y)\mapsto\sin (x^2+4xy)- x\cos y$, hence it's closed. </p>
<p>Let $f:X\to Y$ a continuous function (between two metric spaces for example) and $F\subset Y$ a closed set, $G=f^{-1}(F)$. Here are two ways of proving that $G$ ... |
2,684,805 | <p>This question is asked by my 12 yr old cousin and I seem to be failing to give him a convincing explanation. Here is the summary of our discussion so far - </p>
<p>Case1 : $a>0, b>0$<br>
<a href="https://i.stack.imgur.com/fuoZS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fuoZS.png" alt=... | Parcly Taxel | 357,390 | <p>Without loss of generality, let $A$ be red. Then $B$ and $D$ can be independently coloured blue or green. If they are different (two ways), $C$ must be red. If they are the same (two ways), $C$ can be either red or the colour that $B$ and $D$ are not.</p>
<p>Similar reasoning applies if $A$ is blue or green. Thus t... |
3,775,749 | <p>So far I know that it’s possible to draw angles which are multiples of <strong>15°</strong> (ex. <em><strong>15°</strong></em>, <em><strong>30°</strong></em>, <em><strong>45°</strong></em> etc.).</p>
<p>Could anybody please tell me if it's possible to draw other angles which are not multiples of 15° using only a com... | loved.by.Jesus | 272,774 | <p>Concerning the nice answer of Ethan, I must say that not every angle constructed with rule and compass must form a regular n-gon. This is only a <em>sufficient</em> condition.</p>
<h2>Angles with rational trigonometric values</h2>
<p>I can give you another sufficient condition to create angles with compass and ruler... |
110,162 | <p>One can use <code>$Epilog</code> to do something when the Kernel is quit or put an <code>end.m</code> file next to the <code>init.m</code>.</p>
<blockquote>
<p>For Wolfram System sessions, <code>$Epilog</code> is conventionally defined to read in a file named end.m.</p>
</blockquote>
<p>But if <code>$Epilog</code> i... | Szabolcs | 12 | <p>In principle this can be done using LibraryLink. Just run an action on library unload. The library will be unloaded on kernel exit, if you don't unload it manually before.</p>
<p><strong>Warning:</strong> This is a heavyweight solution that just won't be practical in most cases. But it does work and it does not ... |
158,896 | <p>Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting properties of the complex plane and some researches, I realized that defining multiplication as repeated addition will lead m... | Community | -1 | <p>First you need to define $\mathbb{R}$ in your construction!</p>
<p>To define $\mathbb{R}$, one way is to go about defining $\mathbb{N}$, then defining $\mathbb{Z}$, then defining $\mathbb{Q}$ and then finally defining $\mathbb{R}$. Once you have these things set up, proving associativity, commutativity of addition ... |
2,948,045 | <p>In Eric Gourgoulhon's "Special Relativity in General Frames", it is claimed that the two dimensional sphere is not an affine space. Where an affine space of dimension <em>n</em> on <span class="math-container">$\mathbb R$</span> is defined to be a non-empty set E such that there exists a vector space V of dimension ... | Squid with Black Bean Sauce | 593,937 | <p>We could start from an easier problem---show that the <span class="math-container">$1$</span>-dimensional sphere, i.e. the circle <span class="math-container">$S^1$</span>, is not an affine space. </p>
<p>This one seems pretty easy---we can pick a point <span class="math-container">$A$</span> in <span class="math-c... |
3,267,883 | <p>I apologize in advanced as my literacy in this subject is not too great and this question may either be trivial or impossible as of yet. </p>
<p>I have seen many questions on stack exchange utilizing the Chinese Remainder Theorem to find solutions of <span class="math-container">$a^2\equiv 1\mod (p*q)$</span>, wher... | hmakholm left over Monica | 14,366 | <p>Obviously <span class="math-container">$a$</span> needs to be odd, and by going to <span class="math-container">$-a$</span> if necessary we can assume <span class="math-container">$a\equiv 1\pmod 4$</span>.</p>
<p>Therefore assume <span class="math-container">$a=2^nm+1$</span> with <span class="math-container">$n\g... |
3,267,883 | <p>I apologize in advanced as my literacy in this subject is not too great and this question may either be trivial or impossible as of yet. </p>
<p>I have seen many questions on stack exchange utilizing the Chinese Remainder Theorem to find solutions of <span class="math-container">$a^2\equiv 1\mod (p*q)$</span>, wher... | lab bhattacharjee | 33,337 | <p>As <span class="math-container">$a$</span> is odd, <span class="math-container">$a\pm1$</span> are even</p>
<p>If <span class="math-container">$2^k$</span> divides <span class="math-container">$(a+1)(a-1)$</span> for <span class="math-container">$k-2\ge1$</span></p>
<p><span class="math-container">$\implies2^{k-2}... |
1,061,311 | <p>Suppose $\sum_{n=0}^\infty a_n$ and $\sum_{m=0}^\infty b_m$ converge absolutely. I have to show that $$\left(\sum_{n=0}^\infty a_n\right) \cdot \left(\sum_{m = 0}^\infty b_m\right) = \sum_{m, n}^\infty a_nb_m.$$ But I do not understand what the sum on the right-hand side means (i.e. what limit this represents). Coul... | user149792 | 149,792 | <p>The notation $$\sum_{m, n = 0}^\infty a_nb_m$$ should be interpreted as the sum $$\sum_{k=1}^\infty a_{n_k}b_{m_k}$$ where for each ordered pair $(m, n)$ of nonnegative integers, there exists $k$ such that $(m_k, n_k) = (m, n)$. The sum is <strong>not well-defined</strong> in general. Since we are not given a way to... |
2,213,807 | <p>I was solving a problem to discover n and after I transformed the problem it gave me this equation:</p>
<p>\begin{equation*}
\left\lfloor{\frac{2}{3}\sqrt{10^{2n}-1}}\right\rfloor = \frac{2}{3}(10^{n}-1)
\end{equation*}</p>
<p>So I tried to simplify it by defining:
\begin{equation*}
k = 10^{n}-1
\end{e... | George Law | 141,584 | <p>Note that
$$-8(10^n-1)\ \le\ 0\ <\ 4\cdot10^n+5$$
is true for all $n\in\mathbb Z^+$. In other words
$$-8\cdot10^n+4\ \le\ -4\ <\ 4\cdot10^n+1$$
$$\iff\ 4(10^n-1)^2\ \le\ 4(10^{2n}-1)\ <\ (2\cdot10^n+1)^2$$
$$\iff\ 2(10^n-1)\ \le\ 2\sqrt{10^{2n}-1}\ <\ 2\cdot10^n+1$$
$$\iff\ \frac23(10^n-1)\ \le\ \frac23\... |
1,987,230 | <p>On Socratica, I saw a video demonstrating writing groups by writing the Cayley's table satisfying three conditions of the desired order. (1) Neutral element row and column are copies of the row and column headers. (2) Every row and column has neutral element once (3) All the elements of the set are present in each ... | dxiv | 291,201 | <p>Any integer $a \equiv 0,\pm1,\pm2 \pmod 5$. Since $2^2=4 \equiv -1 \pmod 5$ it follows that $a^2 \equiv 0, \pm 1 \pmod 5$. This proves both that $a^2 \not \equiv 3 \pmod 5$ which was the original question, and also that $a^2 \not \equiv 2 \pmod 5$ for a bonus conclusion.</p>
|
98,088 | <p>I can't understand a sentence in a textbook: if $x$ is a transitive set, then $\bigcup x^+=x$? Could someone help me to understand?</p>
<p><strong>added:</strong> $x^+=x\cup\{x\}$</p>
| Asaf Karagila | 622 | <p>Let us deconstruct this:</p>
<ol>
<li>$\bigcup A = \{z\mid\exists y\in A: z\in y\}$.</li>
<li>$x^+=x\cup\{x\}$</li>
</ol>
<p>From this we have: $\bigcup x^+ = \{z\mid\exists y\in x\cup\{x\}: z\in y\}$. We then follow the definition of a transitive set:</p>
<blockquote>
<p>The set $x$ is transitive if for every ... |
3,584,113 | <p>When we prove things like continuity in real analysis, why do we always aim for the result <span class="math-container">$<\epsilon$</span> when any positive multiple of <span class="math-container">$\epsilon$</span> proves the same result?</p>
| Ben W | 227,789 | <p>Well, first of all, that's not always what we do, because as you say, <span class="math-container">$<c\epsilon$</span> is enough, where <span class="math-container">$c\in(0,\infty)$</span>.</p>
<p>However when you're taking a real analysis class (especially for the first time!) we want to make sure that what's o... |
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