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 |
|---|---|---|---|---|
403,165 | <p>Suppose that $H\triangleleft G, K\le G\ $ and $K\nsubseteq H$. How we can prove that $HK=G?$ </p>
<p>Also $(G:H)=p$ where p is prime.</p>
| Kabalan Gaspard | 79,563 | <p>$K\nsubseteq H$, so there is a non-trivial element $k$ of $K$ such that $kH$
non-trivial in $G/H$. Since $(G:H)=p$, $G/H$ is cyclic, so this $kH$
generates $G/H$. So any element in $G$ can be written in the form $%
hk^{\alpha }$ (since $H$ normal in $G$) where $0\leq {\alpha }\leq p-1$ and $h\in H$.</p>
|
157,413 | <p>Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important). </p>
<p>Consider a family of curves $X \subset S \times T$ parametrised by a variety $T$ of dimension 2 (the fibres $X_t$ are disti... | Community | -1 | <p>For a $C^\ast$-algebraic approach to noncommutative two dimensional torus you may want to look at the Marc Rieffel paper <a href="http://msp.org/pjm/1981/93-2/pjm-v93-n2-p12-s.pdf">$C^\ast$-algebras associated with irrational rotations</a>. For more detailed study of noncommutative torus including higher dimensions ... |
1,248,667 | <p>I was wondering how do you get x from the triangle below: <img src="https://i.stack.imgur.com/Ws9Hv.jpg" alt="question"></p>
| Michael Hardy | 11,667 | <p><b> First method:</b>
$$
\frac n {\sqrt 3} = \frac{\text{opposite}}{\text{adjacent}} = \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\sqrt{1-\cos^2\theta}}{\cos\theta} = \frac{\sqrt{1-(1/3)^2}}{1/3}
$$</p>
<p><b>Second method:</b>
$$
\frac 1 3 = \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\s... |
2,285,276 | <p>A wheel is spun with the numbers $1, 2$, and $3$ appearing with equal probability of $1\over 3$ each. If the number $1$ appears, the player gets a score of $1.0$; if the number $2$ appears, the player gets a score of $2.0$, if the number $3$ appears, the player gets a score of $X$, where $X$ is a normal random varia... | GiantTortoise1729 | 219,849 | <p>$\mathcal{P}(\mathbb{R})$ has basis $\{1,x,x^2,\dots,x^n,\dots\}$. Since any linear functional in $\mathcal{P}(\mathbb{R})'$ is determined completely by its action on the basis, we may construct an injective linear map from $\mathcal{P}(\mathbb{R})'$ to $\mathbb{R}^\infty$ via the assignment $$\ell \mapsto (\ell(x^n... |
1,756,593 | <p>I have to prove that the series $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over n^2 }$ converges. The ratio test is inconclusive, so I should use the comparison test, but which series should I compare it with? I tried ${1\over n}$, ${1 \over n^2 }$, but I need a bigger series which converges to prove that this one c... | Alex | 38,873 | <p>Stolz-Cesaro theorem for $a_n = \log n, b_n = n^2$:
$$
\sum_{k=1}^{\infty} \frac{\log (k+1) -\log k}{(k+1)^2 - k^2} \sim \sum_{k=1}^{\infty} \frac{1}{k(2k+1)} \sim \frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{k^2}
$$</p>
<p>The second step is due to Maclaurin series expansion. The last is a famous $\zeta(2)$.</p>
|
1,640,110 | <p>Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix
with real entries such that its determinant is zero, but if one changes any single
entry one gets a matrix with non-zero determinant.</p>
<p>I think we may be able to construct such matrices.</p>
| 2'5 9'2 | 11,123 | <p>Take a look at $$M=\begin{bmatrix}1&0&0&\cdots&\cdots&0&(-1)^n\\1&1&0&\cdots&\cdots&0&0\\0&1&1&\cdots&\cdots&0&0\\0&0&1&\ddots&\cdots&0&0\\\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\0&0&... |
4,508,353 | <blockquote>
<p>Solve the differential equation and find its solution <span class="math-container">$$\frac{d^3y}{dx^3}=0$$</span></p>
</blockquote>
<p>I haven't learnt yet how to solve differential equations of order <span class="math-container">$3$</span>. Moreover, I don't think this can be solved as there is no vari... | Lorago | 883,088 | <p>What do you mean by there being no variable you can integrate? You can integrate the zero function just as well as any other function by just noticing that if <span class="math-container">$\alpha\in\mathbb{R}$</span>, then</p>
<p><span class="math-container">$$\frac{\mathrm{d}}{\mathrm{d}x}\alpha=0,$$</span></p>
<p>... |
2,241,326 | <p>I am not being able to understand the graphical method of solving this, any simple explanation will be appreciated.</p>
<p>A non-graphical calculation will be very helpful too.</p>
<p>Thank you so much in advance!</p>
| D Wiggles | 103,836 | <p>Let's use our Calculus skills. Define a function $f$ by
$$f(a)=\int_{a-1}^{a+1}\frac{1}{1+t^8}dt$$
You want to minimize $f$? Well, we should take a derivative and set it equal to zero...can you see the fundamental theorem of calculus in your future?</p>
<p>$$0=f'(a)=\frac{d}{da}\left[\int_{a-1}^0\frac{1}{1+t^8}dt... |
1,197,547 | <p>If $G=A * B$ is the free product of two groups $A$ and $B$ and $g \in G-A$, then prove that $gAg^{-1} \cap A=1$.</p>
<p>We know $A \cap B=1$, so if we write $g=a_1b_1a_2b_2 \ldots a_nb_n$, does not give me sufficient road to go? How should I approach it?</p>
| Robert Soupe | 149,436 | <p>The first thing that popped into my mind was bitwise rotation, like <code>ROR</code>, <code>ROL</code>, <code>RCR</code>, etc. But obviously a power of 2 in an integer data type rotated in whatever direction whatever number of bits simply gives another power of 2.</p>
<p>Your failed answer of 1024 and 2401 suggeste... |
1,948,880 | <p>Suppose I have some $n$-dimensional vector space $V$ and a finite collection of $m$ distinct points $v_1,\dotsc, v_m\in V$. Is there a basis of $V$ such that the first coordinate of each $v_i$ is distinct?</p>
<p>This obviously fails when the base field is finite, but my intuition over $\mathbb{R}^n$ has convinced ... | Asinomás | 33,907 | <p>Let $V$ be a vector space of dimension $n$ over a finite field.</p>
<p>Lemma: Given a finite set of vectors $F$ there exists $W\leq V$ of dimension $n-1$ with $W\cap F=\varnothing$.</p>
<p>We prove every maximal subspace with $W\cap F=\varnothing$ must have dimension $n-1$.</p>
<p>Suppose not: Let $W$ be a maxima... |
278,694 | <p>How to plot a function line with Markers:</p>
<pre><code>f[x_, m_] = m Sin[x]
g[x_, m_] = 2 m Cos[x]
Manipulate[
Plot[{f[x,m], g[x,m]}, {x, 0, 10},
PlotLegends -> Automatic], {m, 0, 5}]
</code></pre>
<p><a href="https://i.stack.imgur.com/wjVEs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Nasser | 70 | <p>I am not sure if this is what you meant or not. (I assumed you meant your Manipulate was not working). Will delete if not.</p>
<p><a href="https://i.stack.imgur.com/q73LH.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/q73LH.gif" alt="enter image description here" /></a></p>
<pre><code>f[x_, m_] :... |
3,782,170 | <p>This is probably just a minor notational issue, but I am unsure whether I should write <span class="math-container">$z=a+bi$</span> or <span class="math-container">$z=a+ib$</span> when denoting complex numbers. Though the former notation seems more common, Euler's identity tends to be written as
<span class="math-co... | Moko19 | 618,171 | <p>In my experience, <span class="math-container">$z=a+bi$</span> is used when b is a constant (for example <span class="math-container">$4+3i$</span>) and <span class="math-container">$z=a+ib$</span> when b is a variable (for example <span class="math-container">$x+iy$</span>). When b has both variable and constant c... |
1,548,076 | <p>Assume that $P(X_i = 1) =1/2, P(X_i =-1)= 1/4,\text{ and }P(X_i = 0)=1/4$.<br>
Consider the random walk starting at 1 given by
$$S_n = 1 + X_1 + X_2 + \cdots + X_n$$ where $X_1,X_2, ...$ are i.i.d.<br>
What is the probability that the random walk ever reaches $0$?</p>
<p>I have tried to solve this using Binomial... | David K | 139,123 | <p>There is a well-known algorithm for extracting the square root of a number.
Treating your desired string of digits as an integer, add $1$,
then use the standard square-root algorithm to compute the square root to
a sufficient number of digits.
The result (an approximate square root)
is a decimal number whose square ... |
194,220 | <p>Im trying to animate multiple points around multiple parametric plots which depicts orbital motion of body around a planet. Initial conditons are;</p>
<pre><code>μ = 3.986004418*10^14
a = {7.92597218162462`*^6, 7.359004757830201`*^6,
6.970300551929753`*^6}
r = {7.388961739897817`*^6, 7.352270990873303`*^6,
6.... | m_goldberg | 3,066 | <p>Your expression for assigning values to <code>x[t]</code> and <code>y[t]</code> look suspect.</p>
<pre><code>Table[
{x[t] = a[[i]] (1 - Ecc[[i]]^2)/(1 + Ecc[[i]]*Cos[t])*Cos[t],
{y[t] = a[[i]] (1 - Ecc[[i]]^2)/(1 + Ecc[[i]]*Cos[t])*Sin[t]}},
{i, 1, 3}]
</code></pre>
<p>generates</p>
<p><a href="https://i.st... |
3,473,147 | <p>Using the lema:
<span class="math-container">$F(u)=F(c\cdot u)=F(c+u)$</span> for <span class="math-container">$c\in F$</span></p>
<p>I could prove that:
If <span class="math-container">$r=t^2\cdot s$</span> for some <span class="math-container">$t\in \mathbb{Q}$</span>, then <span class="math-container">$\mathbb{Q... | Kenta S | 404,616 | <p>Let <span class="math-container">$\csc(x)=\frac1{\sin x}.$</span> Then, the integral becomes <span class="math-container">$\int \csc^4(x)dx.$</span> Then,</p>
<p><span class="math-container">$$\int \csc^4(x)dx=\int \csc^2(x)\cdot(1+\cot^2(x)) dx=\int \csc^2(x)dx+\int\cot^2(x)\csc^2(x)dx.$$</span></p>
<p>Now, letti... |
4,445,869 | <p>Having been introduced to the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Feynman technique of integration</a>, it seemed natural to wonder if it could be done the other way:</p>
<ol>
<li><p>Introduce a new parameter <span class="math-container">$a$</span></p>
</li>
<li><p... | Quanto | 686,284 | <p>It is a legitimate method. Here is an extension of it that is more convenient than integration by parts for evaluating the integral below</p>
<p><span class="math-container">$$\int_{0}^{\infty }x^ne^{-x}dx= (-1)^n \frac{d^n}{da^n}
\bigg( \int_{0}^{\infty }e^{-ax}dx\bigg)_{a=1}
= (-1)^n \frac{d^n}{da^n} \frac1a\big... |
3,809,546 | <blockquote>
<p>Using the change of variable evaluate <span class="math-container">$\iint_{R} x y\ dx\ dy,$</span> when the region <span class="math-container">$R$</span> is bounded by the curves <span class="math-container">$x y=1, x y=3, y=3 x, y=5 x$</span> in the <span class="math-container">$1^{\text {st }}$</span... | Christian Blatter | 1,303 | <p>You are told to integrate over a certain region <span class="math-container">$R$</span> in the <span class="math-container">$(x,y)$</span>-plane. This region <span class="math-container">$R$</span> is bounded by the hyperbolic arcs <span class="math-container">$xy=1$</span> and <span class="math-container">$xy=3$</s... |
1,367,091 | <p>In order to resolve a limit, I need to rationalize $\frac {x-8}{\sqrt[3]{x}-2}$. I tried multiplying it by $\sqrt[3]{x^3}$ or $\sqrt[3]{x^2}$ but with no much success. It seems that I can't use "Difference of Squares" identity too. The limit in question is:</p>
<p>$$
\textstyle \lim_{x \to 8}\frac {x-8}{\sqrt[3]{x}... | Harish Chandra Rajpoot | 210,295 | <p>Notice, $(a-b)(a^2+b^2+ab)=a^3-b^3$, now we have $$\frac{x-8}{x^{1/3}-2}$$ $$=\frac{(x-8)(x^{2/3}+4+2x^{1/3})}{(x^{1/3}-2)(x^{2/3}+4+2x^{1/3})}$$ $$=\frac{(x-8)(x^{2/3}+4+2x^{1/3})}{((x^{1/3})^3-(2)^3)}$$
$$=\frac{(x-8)(x^{2/3}+4+2x^{1/3})}{(x-8)}$$ $$=\color{blue}{x^{2/3}+2x^{1/3}+4}$$</p>
|
327,201 | <p>I'm reading Behnke's <em>Fundamentals of mathematics</em>:</p>
<blockquote>
<p>If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.</p>
</blockquote>
<p>I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible ex... | Metin Y. | 49,793 | <p>Just to make the matter maybe a bit more clear giving another example: </p>
<p>Similar to what André Nicolas stated, $ACF_0$ (the theory of algebraically closed fields of charateristic $0$) is axiomatized by $ACF$(the theory of algebraically closed fields) + $\{\neg\phi_p$ : $p$ is prime$\}$, where $\phi_p$ says ... |
999,389 | <p>In the "Make Money Game," the winning number is four digits, each selected at random from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, e.g. 0-3-9-6, 0-0-6-0, 9-4-7-9. A player may place any of the following types of bets. In each case tell the odds of the player winning.</p>
<p>I got, </p>
<pre><co... | Chris | 508,572 | <p>If you want an easier way to do it you should try to create a python codelike this:</p>
<p>from random import randint
print(randint(1, 49))</p>
<p>I found a really cool <a href="http://www.lotterypros.com/lucky-number-generator/" rel="nofollow noreferrer">lucky numbers generator</a> on this site. </p>
|
2,903,920 | <p>Let $X$ be a set and $M$ be a sigma algebra of $X$. If $U \subset X$, is it true that the set $M_U = \{E \cap U|E \in M\}$ is a sigma algebra? I'm getting the countable union part, but not compliments:</p>
<p>$$(E\cap U)^C = E^C \cup U^C$$</p>
<p>I'm not sure what to do with $U^C$</p>
| LucaMac | 586,942 | <p>Well, you want to show that $\mathcal{M}_U$ is an $U$ $\sigma-$ algebra, thus you have to show that $$U \cap (E \cap U)^c \in \mathcal{M}_U$$
But $$U \cap (E \cap U)^c = U \cap (E^c \cup U^c) = U \cap E^c \in \mathcal{M}_U$$</p>
|
3,057,924 | <p>I was reading <a href="http://advancedintegrals.com/wp-content/uploads/2016/12/advanced-integration-techniques.pdf" rel="nofollow noreferrer">Advanced Integration Techniques</a>, and found that<span class="math-container">$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$</span></p>
<p>The book provides ... | Frank W | 552,735 | <p>J.G. has the elementary method down. If you see a <span class="math-container">$1-x^2$</span> term inside your integrand, it might be wise to give a trig substitution a try. In this case, letting <span class="math-container">$x=\sin^2\theta$</span> works out beautifully.</p>
<p>There’s another less elementary way b... |
3,057,924 | <p>I was reading <a href="http://advancedintegrals.com/wp-content/uploads/2016/12/advanced-integration-techniques.pdf" rel="nofollow noreferrer">Advanced Integration Techniques</a>, and found that<span class="math-container">$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$</span></p>
<p>The book provides ... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$\sqrt{x-x^2}=y$</span>.</p>
<p>Thus, <span class="math-container">$y\geq0$</span> and <span class="math-container">$$x^2-x+y^2=0$$</span> or
<span class="math-container">$$\left(x-\frac{1}{2}\right)^2+y^2=\left(\frac{1}{2}\right)^2,$$</span> which is a semicircle with radius <span ... |
3,057,924 | <p>I was reading <a href="http://advancedintegrals.com/wp-content/uploads/2016/12/advanced-integration-techniques.pdf" rel="nofollow noreferrer">Advanced Integration Techniques</a>, and found that<span class="math-container">$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$</span></p>
<p>The book provides ... | Jack D'Aurizio | 44,121 | <p>Another technique just for fun (and in the meanwhile, <em>happy new year!</em>). We have
<span class="math-container">$$ \frac{1}{\sqrt{1-x}}\stackrel{L^2(0,1)}{=}2\sum_{n\geq 1}P_n(2x-1) $$</span>
hence by Bonnet's recursion formulas and symmetry
<span class="math-container">$$ \sqrt{1-x}=2\sum_{n\geq 0}\frac{1}{(1... |
1,636,807 | <p>Ok, here is what I think. Please correct me if I am wrong.
$$\sqrt{9} \neq 3$$
and also
$$\sqrt{9} \neq -3$$</p>
<p>Now let's assume, that above statements are false, then we have $-3 = \sqrt{9} = 3$ and since $3 \neq -3$ the assumption must be wrong.
Ok, square root must be equal to 3 and -3 at the same time. As o... | sinbadh | 277,566 | <p>You are partially correct:</p>
<blockquote>
<p>Now let's assume, that above statements are false, then we have $−3=\sqrt{9}=3$ and since $3≠−3$ the assumtion must be wrong.</p>
</blockquote>
<p>Yes. It is all correct. The key is that the prhase "above statements are false" means $\sqrt{9}\neq3$ AND $\sqrt{9}\neq... |
574,267 | <p>We know that the column-rank of an arbitrary matrix is equal to it's row-rank.
But what are possible interpretations of this equation or equality ?
How can we visualize this equation intuitively?</p>
| AnyAD | 107,693 | <p>For every leading entry we have a non-zero row in reduced form of the matrix. Every leading entry corresponds to a column that has been transformed to an elementary vector. The two numbers coincide.</p>
|
3,590,250 | <p>For <span class="math-container">$r>0$</span>, let be
<span class="math-container">$$I(r)=\int_{\gamma_r}\frac{e^{iz}}{z}dz$$</span>
where <span class="math-container">$\gamma_r:[0,\pi]\to\mathbb{C}, \gamma_r(t)=re^{it}$</span>. Prove that <span class="math-container">$\lim_{r\to\infty}I(r)=0$</span>.</p>
<p>I'v... | Kavi Rama Murthy | 142,385 | <p>Hint: When <span class="math-container">$z=re^{it}$</span> we have <span class="math-container">$|e^{iz}|=e^{\Re (ire^{it})}=e^{-r\sin t}$</span> and <span class="math-container">$\int_0^{\pi} e^{-r\sin t} dt \to 0$</span> as <span class="math-container">$r \to \infty$</span> by DCT. </p>
|
4,084,576 | <p><img src="https://i.stack.imgur.com/x3oOV.png" alt="image" /></p>
<p>I tried using x+2 as the longer side of the large unshaded rectangle, and subtracted the right triangles to get 192. My friend tells me this is incorrect, and I was wondering how to get the correct answer.</p>
| Some Guy | 730,299 | <p><a href="https://i.stack.imgur.com/Fhg7p.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Fhg7p.png" alt="enter image description here" /></a></p>
<p>By the Pythagorean theorem, we can clearly see that <span class="math-container">$EF = 2\sqrt{10}$</span>. Also, note that <span class="math-containe... |
75,844 | <p>I have the following question:</p>
<p>If I hold a Jack and a nine, what is the chance of making a straight (five cards in a row) when the next three cards have been dealt?</p>
<p>My attempt at an answer:</p>
<p>There are three possible 5 card hands that make a straight:</p>
<p><strong>7 8</strong> 9 <strong>T</s... | karmic_mishap | 17,529 | <p>There are 64 possible combinations for each straight, but the straight events can't all be independent because there is some overlap between them. For instance, the first two straights can't be independent because they both involve drawing an 8 and a T. Hopefully this illustrates how the second and third straight an... |
44,661 | <p>Assume that $(P,\le)$ is a notion of forcing. There are several ways to define what it means for $P$ being proper and I would like to know: What is the complexity (in terms of the Levy-Hierarchy) of the statement 'P is proper'?</p>
| Stefan Hoffelner | 4,753 | <p>I think that I might have found a solution to this rather dispensable question. I will sketch it:</p>
<p>Consider the following characterization of properness:</p>
<p>$P$ is proper iff for all $\lambda > 2^{|P|}$ there is a club $C$ of elementary submodels $M \prec (H_{\lambda},...)$ such that $\forall p \in P... |
44,661 | <p>Assume that $(P,\le)$ is a notion of forcing. There are several ways to define what it means for $P$ being proper and I would like to know: What is the complexity (in terms of the Levy-Hierarchy) of the statement 'P is proper'?</p>
| Joel David Hamkins | 1,946 | <p>Properness is observable in any sufficiently large $V_\alpha$, and therefore has complexity $\Sigma_2$. In oktan's answer, it suffices to consider sufficiently large $\lambda$, rather than all $\lambda$. I think this is proved in some of the standard accounts of proper forcing.</p>
|
1,498,801 | <p>Let $f(x)=cos(5x)+Acos(4x)+Bcos(3x)+Ccos(2x)+Dcos(x)+E$ and $T=f(0)-f(\pi/5)+f(2\pi/5)-f(3\pi/5)+..-f(9\pi/5)$.Then out of A,B,C,D which does T depend on?</p>
<p>Hints please!
P.S:KVPY 2011 question</p>
| Matt Dickau | 273,941 | <p>Over on the Wikipedia page on <a href="https://en.wikipedia.org/wiki/Conic_section" rel="noreferrer">conic sections</a> you can see that we can write a general conic section as:
$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
And this can be rewritten as an equation:
$$X^TMX + c=0$$
Where
$$X = \left[ \begin{matrix}x\\y\\\end{matrix} \... |
1,498,801 | <p>Let $f(x)=cos(5x)+Acos(4x)+Bcos(3x)+Ccos(2x)+Dcos(x)+E$ and $T=f(0)-f(\pi/5)+f(2\pi/5)-f(3\pi/5)+..-f(9\pi/5)$.Then out of A,B,C,D which does T depend on?</p>
<p>Hints please!
P.S:KVPY 2011 question</p>
| John Hughes | 114,036 | <p>Yes; affine transformations map conic sections to conic sections. </p>
<p>The correct statement that your text should have made is </p>
<p>"An axis-aligned ellipse, centered at the origin, has the equation .... When $A = B$, the ellipse is also called a <em>circle</em>. </p>
|
1,438,839 | <p>If we have 8 white balls and 5 black balls and one of them is picked at random, the probability of getting a white ball is 8/13. Suppose now we put 3 white balls and 2 black balls in one closed bag and the remaining balls in another identical bag. And now the experiment is that first a bag is chosen at random and th... | DanielWainfleet | 254,665 | <p>We are going to deke around the MVT in the case where $(a_n)_{n \in N}$ does not converge.......(1)For positive integers $ m,n$ with $m<n$ we have $$m^{-j} \binom {m} {j} \le n^{-j} \binom {n} {j} $$ for $ 0\le j$, from the def'n of the binomial co-efficient, so $$\binom {m}{j}\le (m/n)^j \binom {n}{j}. $$.From... |
419,091 | <blockquote>
<p><span class="math-container">$G$</span> is an infinite group.</p>
<ol>
<li><p>Is it necessary true that there exists a subgroup <span class="math-container">$H$</span> of <span class="math-container">$G$</span> and <span class="math-container">$H$</span> is maximal ?</p>
</li>
<li><p>Is it possible that... | Little Endian | 67,892 | <p>I like this example for its simplicity:</p>
<p>Let $A$ be any group with a proper subgroup $B$. Let $G = \prod_{i = 1}^{\infty}A$ and $H_n = \prod_{i = 1}^{n}A \times \prod_{i = n + 1}^{\infty}B$, then $H_1 < H_2 < \cdots < H_n < \cdots < G$.</p>
|
3,506,533 | <blockquote>
<p>Let <span class="math-container">$X,Y$</span> be Banach space and <span class="math-container">$T:X\to Y$</span> bounded . If <span class="math-container">$T(X)$</span> has finite dimensional, then <span class="math-container">$T$</span> is compact. </p>
</blockquote>
<p>Assume <span class="math-con... | Severin Schraven | 331,816 | <p>You want to show that the Theorem of Heine-Borel holds in any finite-dimensional Banach space (ie a set is compact iff it is bounded and closed).</p>
<p>Now consider the linear isomorphism <span class="math-container">$$F: T(X)\rightarrow \mathbb{R}^n, \sum_{j=1}^n a_j y_j \mapsto (a_1, \dots, a_n).$$</span>
We kn... |
907,336 | <p>show that:
$$\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}-\left(\dfrac{1}{a+b+c+1}+\dfrac{1}{b+c+d+1}+\dfrac{1}{c+d+a+1}+\dfrac{1}{d+a+b+1}\right)\ge 0$$
where $abcd=1,a,b,c,d>0$</p>
<p>I have show three variable inequality</p>
<p>Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.... | RE60K | 67,609 | <p>Partial Proof:
For general case of n variables, the inequality converts to:</p>
<blockquote>
<p>$$\sum_i^n \frac1{1+a_1+a_2+\cdots+a_n-a_i}\le \sum_i^n\frac1{n-1+a_i}$$</p>
</blockquote>
<p>Similiar to the given proof we can convert $\frac {a_1}{a_1+(n-1)}$ like this:
$$\frac{a_1}{a_1+(n-1)}=\frac{a_1}{a_1+(n-1)... |
3,238,780 | <p>I am trying to work through a few problems, and one asks to sum over the Fibonacci numbers which are even-valued (it is the Euler Project problem #2). I realized that (if we index like <span class="math-container">$\langle 1, 2, 3, 5, 8, \ldots \rangle$</span>) that <span class="math-container">$F_n$</span> is even ... | Minus One-Twelfth | 643,882 | <p>You have your quotient rule reversed for the numerator. So you ended up with the negative of the answer. The rule is <span class="math-container">$$\left(\frac{u}{v}\right)' =\frac{u'v-uv'}{v^2},$$</span>
but if you look closely, you'll see you did <span class="math-container">$\color{red}{uv'-u'v}$</span> in the nu... |
20,714 | <p>The first three expressions evaluate as expected and the polynomial is displayed in what I would call "textbook" form. The last expression, however, switches the order of terms. Mathematica employs this change for two-term polynomials if it results in getting rid of the leading negative sign (at least that is the be... | Jens | 245 | <p>Since the two other answers don't seem to do exactly what's needed, I'll try my luck:</p>
<pre><code>order[poly_] :=
Replace[Reverse@Sort[List @@ poly], List[x__] :> HoldForm[Plus[x]]]
order /@ {x^2 + x + 5, -x^2 + x + 5, x^2 + x, -x^2 - x}
</code></pre>
<blockquote>
<p>$\left\{x^2+x+5,-x^2+x+5,x^2+x,-x^2-... |
66,474 | <p>I am trying to solve a 1st order non-linear ODE</p>
<pre><code>W[y]*W'[y] + W[y]*v + Fnum == 0 /. v -> 10
Fnum = 0.05 - 1.66667 y + (270.27 y)/(1 + 270.27 y) - (660. y)/(1 + 1666.67 y)
</code></pre>
<p>The function Fnum has three zeros. two of them are at y=StartY and y=EndY.</p>
<pre><code>In[1]:= StartY = 0... | Acus | 18,792 | <p>The problem with NDSolve methods is that (as far as I know) it always check boundary points. And this a problem, as you noted. I would recommend you to look at this <a href="http://blog.wolfram.com/2013/07/09/using-mathematica-to-simulate-and-visualize-fluid-flow-in-a-box/" rel="noreferrer">wonderful approach</a>. I... |
1,744,160 | <p>I want to evaluate </p>
<p>$$\lim _{x\to \pi }\left(\frac{\sin x}{\pi ^2-x^2}\right)$$</p>
<p>without using L'Hopital's rule or Taylor series. My thinking process was something like this: in order to get rid of the undefined state, I need to go from $\sin x$ to $\cos x$. I tried this substitution: $t = \frac{\pi}{... | Mark Viola | 218,419 | <p>Note that from elementary geometry, the sine function is bounded as </p>
<p>$$|\theta \cos(\theta)|\le |\sin(\theta)|\le |\theta| \tag 1$$</p>
<p>for $0\le |\theta|\le \pi/2$. Letting $\theta =x-\pi$ in $(1)$ yields</p>
<p>$$|(x-\pi)\cos(x-\pi)|\le |\sin(x-\pi)|\le |x-\pi| \tag 2$$</p>
<p>for $0\le |x-\pi|\le \... |
246,445 | <p>I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know that there is a sequence of INDEPENDENT random variables of the same distribution on $\Omega$?</p>
| Michael Hardy | 11,667 | <p>Since they're i.i.d., you can just use a product measure on a product space $\Omega\times\Omega\times\Omega\times\cdots$.</p>
<p>And for many purposes, you can take $\Omega=\mathbb R$ and let the measurable subsets of $\Omega$ be the Borel sets.</p>
|
2,509,382 | <p>Two spheres of radii $r_1$ and $r_2$ intersect each other orthogonally. Prove that the circle formed by the intersection of the two spheres has a radius
$$\frac{r_1 r_2}{\sqrt{r_1^{2} + r_2^{2}}}.$$</p>
| Intelligenti pauca | 255,730 | <p>HINT.</p>
<p>See below a section of the spheres, passing through their centers $A$ and $B$. They intersect each other orthogonally if radii $AC$ and $BC$ are perpendicular. </p>
<p>It follows that $ABC$ is a right triangle with legs $r_1$ and $r_2$. And its altitude $CH$ is the radius of the intersection circle.</... |
2,894,954 | <p>What is $[\cos(\pi/12)+i\sin(\pi/12)]^{16}+[\cos(\pi/12)-i\sin(\pi/12)]^{16}$?</p>
<p>I can use De Moivre's formula for the left part:</p>
<p>$[\cos(\pi/12)+i\sin(\pi/12)]^{16} = \cos(4\pi/3) + i\sin(4\pi/3) = -\dfrac{\sqrt3}{2} + \dfrac{i}{2}$</p>
<p>but I'm stuck at the right part. Thanks in advance.</p>
| Bernard | 202,857 | <p>It's much simpler with the exponential notation:
\begin{align}
(\cos\pi/12+i\sin\pi/12) ^{16}&+(\cos\pi/12-i\sin\pi/12 )^{16}\\&=\mathrm e^{\tfrac{4i\pi}3}+\mathrm e^{\tfrac{-4i\pi}3}=2\cos\frac{4\pi}3=2\cdot\frac12
\end{align}</p>
|
766,629 | <p>Suppose I am given two finite groups $G$ and $H$ (not too large: let's say their orders are around $10000$ and $100$ respectively, and the order of $H$ divides the order of $G$). These may be represented as groups of permutations with known, fairly small, sets of generators. I would like to find, if possible, a su... | Derek Holt | 2,820 | <p>There are several ways you could attempt this, and finding the most efficient method for the types of groups you are interested in might require some experimentation. One way would be to start by finding all (conjugacy classes of) subgroups of $G$ of order dividing $|H|$ and then test them in decreasing order of siz... |
172,432 | <p>Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, <a href="https://math.stackexchange.com/questions/209512/binary-quadratic-forms-over-z-and-class-numbers-of-quadratic-%EF%AC%81elds/209543#comment1727526_209543">https://math.stackexchange.com/questions... | Pete L. Clark | 1,149 | <p>I have always been fuzzier on the theory of indefinite binary forms than the definite theory. This may come from the fact that I got to learn the definite theory by teaching a course out of Cox's book, and then I led some student research projects on the definite case. (Actually, just in the last week I spoke with... |
2,679,173 | <p>The question is:</p>
<p>A sports club has 3 departments, tennis, squash and badminton. We get
given the following information.</p>
<p>• 90 people are members of the tennis department.</p>
<p>• 60 people are members of the squash department.</p>
<p>• 70 people are members of the badminton department.</p>
<p>• 25... | quasi | 400,434 | <p>Just set up a Venn diagram with $3$ circles, and place a variable in each of the $7$ bounded regions.
<p>
Note that you have $7$ pieces of given information, each of which yields a linear equation.
<p>
Thus, you'll have $7$ linear equations in $7$ unknowns.
<p>
Of course, if some piece of information just gives an i... |
2,679,173 | <p>The question is:</p>
<p>A sports club has 3 departments, tennis, squash and badminton. We get
given the following information.</p>
<p>• 90 people are members of the tennis department.</p>
<p>• 60 people are members of the squash department.</p>
<p>• 70 people are members of the badminton department.</p>
<p>• 25... | N. F. Taussig | 173,070 | <p>Let $B$ denote the number of people who play badminton; let $S$ denote the number of people who play squash; let $T$ denote the number of people who play tennis. The number of people in the club is the number of people who play one of these sports, which by the <a href="https://en.wikipedia.org/wiki/Inclusion%E2%80... |
2,253,752 | <p>The ultrafilter lemma says: <em>Every filter $F$ is contained in a ultrafilter.</em></p>
<p>Question: Is this ultrafilter unique? Or can one find a filter $F$ such that there are several ultrafilters that contain $F$?</p>
| Stella Biderman | 123,230 | <p>This ultra-filter is not necessarily unique. Consider the <em>cofinite filter</em> on $\mathbb{N}$, where $S\in F\iff |\mathbb{N}\setminus S|<\infty$. Let $F_e$ be the filter generated by $F$ and the set of even numbers, and $F_o$ be the filter generated by $F$ and the set of odd numbers. These can both be extend... |
2,253,752 | <p>The ultrafilter lemma says: <em>Every filter $F$ is contained in a ultrafilter.</em></p>
<p>Question: Is this ultrafilter unique? Or can one find a filter $F$ such that there are several ultrafilters that contain $F$?</p>
| Reese Johnston | 351,805 | <p>No, it isn't even vaguely unique. For any set $X$, the set $\{X\}$ is a filter; this filter is not just contained in several ultrafilters, it's contained in <em>all</em> of them. So, for example, let $X$ be $\mathbb{N}$, and let $F$ be the filter $\{\mathbb{N}\}$. Let $U_1$ be the ultrafilter consisting of exactly t... |
202,493 | <p>I've wondered about the following question :</p>
<p>Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such that $x_n$ converges to $x$ with respect to $\|\cdot\|_1$, $x_n$ converges to $y$ with respect to $\|\cdot\|_2... | Blitzer | 39,112 | <p>Not sure but I would guess no. Maybe see if $(X, ||.||_1 + ||.||_2)$ is also a Banach space then you'll probably get a contradiction if you assume x is not y.</p>
|
202,493 | <p>I've wondered about the following question :</p>
<p>Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such that $x_n$ converges to $x$ with respect to $\|\cdot\|_1$, $x_n$ converges to $y$ with respect to $\|\cdot\|_2... | Davide Giraudo | 9,849 | <p><a href="https://mathoverflow.net/questions/53971/example-of-sequences-with-different-limits-for-two-norms">Bill Johnson's example</a> in MathOverflow seems answer the question.</p>
|
251,316 | <p>why the second image just has the positive part? Is the problem in the domain of definition? 0.0</p>
<p>The code is here:</p>
<pre><code>{Plot[1-Power[t^2, (3)^-1],{t,-1,1}],Plot[1-t^(2/3),{t,-1,1}]}
</code></pre>
<p><a href="https://i.stack.imgur.com/IT5Uc.png" rel="noreferrer"><img src="https://i.stack.imgur.com/I... | cvgmt | 72,111 | <pre><code>{Plot[1 - Power[t^2, (3)^-1], {t, -1, 1}],
Plot[1 - Surd[t, 3]^2, {t, -1, 1}]}
</code></pre>
<p>Or</p>
<pre><code>{Plot[1 - Power[t^2, (3)^-1], {t, -1, 1}],
Plot[1 - CubeRoot[t]^2, {t, -1, 1}]}
</code></pre>
<p><a href="https://i.stack.imgur.com/eNsnc.png" rel="noreferrer"><img src="https://i.stack.imgur... |
2,017,133 | <p>$$\sum_{i = 1}^n (2i+3) = n(n+4)$$
for all n >= 1.</p>
<p>Was a homework problem that was given no solution. Was told last lines weren't correctly written.
My attempt:
Let P(n) = n(n+4) for all n >= 1
Basis Step: P(2) = 2(6) = 12 >= 1
Inductive Step:
$$\sum_{(i=1}^{k+1} (k+1)(k+5)$$</p>
<p>= k(k+4) + (k+1)
= $$k... | mfl | 148,513 | <p>The inductive step is $$\sum_{i=1}^n(2i+3)=n(n+4)$$ and you need to show that $$\sum_{i=1}^{n+1}(2i+3)=(n+1)(n+5).$$ We have</p>
<p>$$\sum_{i=1}^{n+1}(2i+3)=\color{red}{\sum_{i=1}^{n}(2i+3)}+2n+5.$$ Use the induction step to susbtitute the value of the sum in red and you are done.</p>
|
2,036,533 | <p>I am trying to understand how $\mathrm W$ is an equivalence relation.</p>
<p>Let $A = \{1,2,3,4,5,6,7\}$ and $B = \{1,2,3,4\}$. </p>
<p>Let $\mathrm W$ be the relation on $P(A)$ defined by: \begin{equation}
\forall X, Y \in P(A), X \mathrm R Y \Leftrightarrow |X \cap B| = |Y \cap B|
\end{equation}</p>
| Graham Kemp | 135,106 | <p>$$\forall X\in \mathcal P(A),\forall Y\in\mathcal P(A): \Big[X\operatorname R Y \iff \lvert X\cap B\rvert = \lvert Y\cap B\rvert\Big]$$</p>
<p>In English: "Any two subsets of $A$, are said to be $\operatorname R$ related if their intersections with $B$ have the same cardinality."</p>
<p>The relation $\operatorname... |
2,871,490 | <p>Let $F$ be a vector field such that $$\vec{F}=\langle x^2,y^2,z\rangle$$ Integration over the line segments which form the triangle with vertices $(0,0,0)$,$(0,2,0)$,$(0,0,2)$ can be achieved by parametrizing each segment and then evaluating $$\sum_{i=1}^3\int_{C_i}\vec{F}(r_i^{(x)}(t),r_i^{(y)}(t),r_i^{(z)}(t))\... | José Carlos Santos | 446,262 | <p>Yes, those formulas exist. The reflection of a point $(p,q)$ on the line $ax+by+c=0$ is$$\left(\frac{p(a^{2}-b^{2})-2b(aq+c)}{a^{2}+b^{2}},\frac{q(b^{2}-a^{2})-2a(bp+c)}{a^{2}+b^{2}}\right).$$The rotation of $(p,q)$ with angle $\theta$ around $(a,b)$ is$$\bigl(a+(p-a)\cos(\theta )-(q-b)\sin(\theta),b+(p-a) \sin (\th... |
1,665,714 | <p>I am looking for the following sets for all $z \in \mathbb{C}$</p>
<p>$$\{z: \cos(z)=0\} \text{ and } \{z: \sin(z)=0\}$$</p>
<p>I believe the best way to do this is consider the exponential form so</p>
<p>$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2} \text{ and } \sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$</p>
<p>So for $\cos(z)=0... | user90369 | 332,823 | <p>It's senseful to use the Euler product formula for the Riemann zeta function.</p>
<p>With $\displaystyle \zeta(s)=\prod\limits_{p\, prime}\frac{1}{1-p^{-s}}$ and $\displaystyle -\ln (1-x)=\sum\limits_{k=1}^\infty\frac{x^k}{k}$ with $|x|<1$ one gets
$$\prod\limits_{k=1}^\infty\zeta(2nk)=\exp\sum\limits_{p\, prim... |
44,036 | <p>For any given topological group $G$ we have Segal's construction/definition of $BG$. I'm recalling it in case the details turn out to be relevant. </p>
<blockquote>
<p>Form the disjoint union of $G^n\times\Delta_n$ for $n\geq 0$ and identify points via $(d_i\cdot,\cdot)\sim (\cdot,\partial_i \cdot)$ where $\parti... | David Carchedi | 4,528 | <p>They're the same. You can construct the geometric realization of a simplicial space $X:\Delta^{op} \to Top$ by taking its co-end with the functor $\Delta \to Top$ which sends the $n$ to the "standard n-simplex" $\Delta^n$. Segal's construction is an explicit description of this co-end in the particular case that $X$... |
44,036 | <p>For any given topological group $G$ we have Segal's construction/definition of $BG$. I'm recalling it in case the details turn out to be relevant. </p>
<blockquote>
<p>Form the disjoint union of $G^n\times\Delta_n$ for $n\geq 0$ and identify points via $(d_i\cdot,\cdot)\sim (\cdot,\partial_i \cdot)$ where $\parti... | Tom Bachmann | 5,181 | <p>I came here looking for essentially an answer to the original question, but none of the replies so far actually provide a reference (HTT 4.2.4 does not contain the result, as far as I can tell.)</p>
<p>In [1] there is a proof that the ordinary and homotopy coherent nerve of a "fibrant groupoid" are homotopy equival... |
776,739 | <p>prove the inequality if you can: $\frac{1}{2}\cdot\frac{2}{3}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}$</p>
<p>Thanks.</p>
| the_candyman | 51,370 | <p>I assume that $n$ is a integer positive number.
Then:</p>
<p>$$\frac{1}{2}\cdot\frac{2}{3}\cdots\frac{2n-1}{2n} = \frac{(2n-1)!}{(2n)!} = \frac{1}{2n} < \frac{1}{\sqrt{2n+1}}$$</p>
<p>$$\sqrt{2n+1}<2n$$
$$2n+1 < 4n^2$$
$$4n^2-2n-1>0$$</p>
<p>Real solutions of $4n^2-2n-1=0$ are $n=\frac{1 \pm \sqrt{5}}... |
914,890 | <p>Let $X$ and $Y$ be independent random variables, both of exponential distibution, such that $EX=\frac{1}{2},EY=1$. Calculate the density of conditional distribution $f_{X+Y|X}(z|x)$. In fact I am interested in $s=3$.</p>
<p>My idea: I know that I should apply the formula $f_{Z|X}(z,x)=\frac{f_{Z,X}(z,x)}{f_X(x)}$. ... | Dilip Sarwate | 15,941 | <p>Hint: Given that $X = x$, the conditional distribution of $Y$ is the same as the unconditional distribution of $Y$ because $X$ and $Y$ are given to be independent
random variables, and so knowing the value of $X$ tells us nothing that we did
not already know about $Y$. Thus, the conditional distribution of $X+Y$, ... |
914,890 | <p>Let $X$ and $Y$ be independent random variables, both of exponential distibution, such that $EX=\frac{1}{2},EY=1$. Calculate the density of conditional distribution $f_{X+Y|X}(z|x)$. In fact I am interested in $s=3$.</p>
<p>My idea: I know that I should apply the formula $f_{Z|X}(z,x)=\frac{f_{Z,X}(z,x)}{f_X(x)}$. ... | Sasha | 11,069 | <p>Let $Z = X+Y \mid X=x$. The complementary cumulative distribution function of $Z$ is easy to work out:
$$
S_Z(z) = \Pr(Z > z) = \Pr\left(X+Y > z \mid X=x\right) =
\Pr\left(Y > z-x \mid X=x\right) = \Pr\left(Y > z-x \right)
$$
Hence
$$
S_Z(z) = \begin{cases} \mathrm{e}^{-(z-x)} & z \geqslant x ... |
2,469,690 | <blockquote>
<p>If $a+b+c=0$, for $a,b,c \in\mathbb R$, prove</p>
<p>$$ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $$</p>
</blockquote>
<p>I made this question as a more difficult (higher degree) version of <a href="https://math.stackexchange.com/questions/2469296/if-abc-0-prove-t... | Angina Seng | 436,618 | <p>Let's define
$$S_n=a^n+b^n+c^n$$
and consider the generating function
$$F(t)=\sum_{n=0}^\infty S_nt^n=\frac1{1-at}+\frac1{1-bt}+\frac1{1-ct}.$$
Using $a+b+c=0$ gives
$$(1-at)(1-bt)(1-ct)=1+pt^2+qt^3$$
for some $p$ and $q$, and
$$F(t)=\frac{3+pt^2}{1+pt^2+qt^3}.$$
From this we expand as a power series
$$F(t)=(3+pt^2)... |
3,455,552 | <p>In the <span class="math-container">$n$</span>-dimensional Euclidean space, given two vectors <span class="math-container">$\vec{OA}$</span> and <span class="math-container">$\vec{OB}$</span> (not collinear), their angle AOB is <span class="math-container">$\theta$</span>. Now there is a vector <span class="math-con... | gandalf61 | 424,513 | <p><em>(Note that I am assuming that <span class="math-container">$M^{49}$</span> in the question refers to the <span class="math-container">$49$</span>th Mersenne number i.e. <span class="math-container">$M^{49}=2^{49}-1$</span>).</em></p>
<p>You can use FLT, but here is an alternative approach:</p>
<p>We want to sh... |
3,455,552 | <p>In the <span class="math-container">$n$</span>-dimensional Euclidean space, given two vectors <span class="math-container">$\vec{OA}$</span> and <span class="math-container">$\vec{OB}$</span> (not collinear), their angle AOB is <span class="math-container">$\theta$</span>. Now there is a vector <span class="math-con... | B. Goddard | 362,009 | <p>Solve the Diophantine equation <span class="math-container">$18x+49y = 1$</span> and take the smallest positive solution for <span class="math-container">$y$</span>. This turns out to be <span class="math-container">$y = 7$</span>. Raise both sides of your congruence to the <span class="math-container">$7$</span> ... |
638,529 | <p>Find, using the power series: $$y(x)=\sum_{k=0}^\infty a_{k}x^k$$ a solution for the following differential equation: $$y'(x) = -x^2y(x),\,\, y(0)=1$$
What's the convergence radius of the constructed power series? Also give a closed formula.</p>
<p>So far I've come up with $\sum_{k=0}^\infty a_{k+1}(k+1)x^{k} = \su... | Ukhrir | 103,919 | <p>Since $y$ has a power series representation, it is equal to its taylor series within its radius of convergence. Therefore, taking the first terms of the taylor series we see that $f(0)=1$, $f'(0)=0$, $f''(0)=(-x^2y(x))'=(-2xy(x)-x^2y'(x))(0)=0, $ and finally $f'''(0)=(-2xy(x)-x^2y'(x))(0)=-2$.
We then merely have t... |
41,725 | <p>Asked this question in a different formulation in cstheory, got some pointers, but no definitive answer ... maybe someone here knows.</p>
<p>Suppose I need to compute the factorization of a block of consecutive numbers N, N+1, ... N+n. </p>
<p>As far as I understand, there are two extreme cases. On one hand, if n ... | Gerhard Paseman | 3,568 | <p>Short answer: I doubt you can do much better than trial factorization. However, for many
composite numbers, the run time is often up to the size of the second largest prime factor,
so you may actually end up with a better run time than pi(sqrt(N)).</p>
<p>Long answer: there are lots of things to try. First is to ... |
120,254 | <p>I want to use <code>Solve[]</code> inside a <code>Module</code>. If I make the variables solved for local
to the module, they are treated differently than if I leave them global.
For example,</p>
<pre><code>SolveIt[a_, b_] := Module[{x, soln},
soln = Solve[a x + b == 0, {x}];
Return[soln]
];
SolveIt[3, 4]
... | mikado | 36,788 | <p>As explained in Mathematica help, "<code>Module</code> creates a symbol with name <code>xxx\$nnn</code> to represent a local variable with name <code>xxx</code>. The number <code>nnn</code> is the current value of <code>$ModuleNumber</code>." This variable is not renamed after the module has completed.</p>
<p>If i... |
122,986 | <p>I've read the axioms of a field. To understand the generality of the axioms, could you give me an example of a field which is not (isomorphic to) a subset of complex number (with or without modulus operations).</p>
| Arturo Magidin | 742 | <ol>
<li><p>Any field of positive characteristic. For example, $\mathbb{F}_2 = \{0,1\}$, or more generally $\mathbb{F}_p$ for prime $p$ (integers modulo $p$ with modular addition and multiplication); $\mathbb{F}_{p^n}$, the Galois Field with $p^n$ elements (the splitting field over $\mathbb{F}_p$ of the polynomial $x^{... |
122,986 | <p>I've read the axioms of a field. To understand the generality of the axioms, could you give me an example of a field which is not (isomorphic to) a subset of complex number (with or without modulus operations).</p>
| André Nicolas | 6,312 | <p>Let $I$ be a set of cardinality greater than the cardinality of the reals (or equivalently the complex numbers). For every $i$ in $I$, invent a symbol $x_i$. Let $F$ consist of all ratios of polynomials in the $x_i$. By polynomial we mean say polynomial with real coefficients. Any polynomial will "mention" only fi... |
3,187,756 | <p>I looked for answers on how to do this on this on this site and couldn't find anything answering this question. Is this what a line integral is used for or is that only to find area under a function f(x,y) along a curve C (on xy-plane for example)?</p>
| marty cohen | 13,079 | <p>If <span class="math-container">$f(x) > 0$</span>
and
<span class="math-container">$\lim_{x \to \infty} f(x) = 0$</span>
then
<span class="math-container">$0
\lt \int_n^{n+1} f(x) dx
\le \max(f(x))|_{x=n}^{n+1}
\to 0
$</span>
as <span class="math-container">$n \to \infty$</span>.</p>
<p>If, in addition,
<span cl... |
808,507 | <p>Let $a,b \in S_n$ and $ab=ba$ and $b$ moves some points that not moved by $a$. Is it true that $a$ and $b$ should be disjoint permutations? </p>
<p>EDIT:
We can consider $b=a^kc$ where $a$ and $c$ are disjoint, to construct a counterexample class. Do you know other constructions? </p>
| David | 119,775 | <p><strong>Hint</strong> for the $4\times4$ version.</p>
<ul>
<li>Explain why expanding the determinant gives an equation of the form
$$Ax^2+Ay^2+Bx+Cy+D=0\ ,$$
where $A,B,C,D$ are constants. </li>
<li>Explain why $A\ne0$. </li>
<li>Explain why the three given points satisfy the equation.</li>
</ul>
<p>Good luck!</... |
48,227 | <p>I am stuck with this problem,</p>
<blockquote>
<p>A function $f(x)$ is defined as $f(x)
= \sinh(x)$. Another function $g(x)$ is such that $f(g(x)) = x$.</p>
<p>Find the value of $\large
g(\frac{e^{2012}-1}{2e^{1006}})$</p>
</blockquote>
<p>I tried representing $f(x) =\large \frac{e^{2x}-1}{2e^{x}}$,and th... | J126 | 2,838 | <p>If $f(g(x))=x$, then $f$ and $g$ are inverses. Thus $g(f(x))=x$ as well. The answer is then $g(f(1006))=1006$.</p>
|
2,452,466 | <p>This is the problem: </p>
<p>$$\int \frac{x+2}{x^2+x}$$</p>
<p>I am supposed to write $(y+2)$ as $\frac{1}{2}(2x+1)+\frac{3}{2}$</p>
<p>$$\frac{1}{2}\int \frac{2x+1}{x^2+x}+\frac{3}{2}\int \frac{3}{x^2+x}$$</p>
<p>Fine, is there supposed to be some way for me to know which fractions to use here or am I supposed ... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>As $x^2+x=x(x+1),$</p>
<p>Set $x+2=Ax+B(x+1)=x(A+B)+B$</p>
<p>$\implies B=2$</p>
|
1,620,900 | <p>Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$.</p>
<p>Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some $m\ge 1$?</p>
| Crostul | 160,300 | <p>The answer is no. For example, you can consider the fraction field of $\Bbb{F}_p[x,t]/(x^2-t)$: this is a quadratic extension of $\Bbb{F}_p(t)$, where the element $x= \sqrt{t}$ is added.</p>
|
359,903 | <p>I've a doubt on how do we prove this kind of stuff involving infinite intersections. My point is: the book I'm working with gives the following example to prove that the intersection of infinite sets may not be open: let $a \in \mathbb{R}^n$, given the familly of balls $B(a; 1/k)=\left\{x \in \mathbb{R}^n \mid d(x,a... | Ian Coley | 60,524 | <p>Your intuition is sound. Pick any other point $x\in\mathbb R^n$. Then we have $d(a,x)=\varepsilon>0$. We may find $N\in\mathbb N$ such that $\varepsilon>1/N>0$. Thus $x\notin B(a;1/N)$, and so it cannot be in the infinite intersection either. Does this help?</p>
|
359,903 | <p>I've a doubt on how do we prove this kind of stuff involving infinite intersections. My point is: the book I'm working with gives the following example to prove that the intersection of infinite sets may not be open: let $a \in \mathbb{R}^n$, given the familly of balls $B(a; 1/k)=\left\{x \in \mathbb{R}^n \mid d(x,a... | mathemagician | 49,176 | <p>Your intuition is perfectly fine. Formally, I will call the intersection $I$ and suppose that there were some element $b\in I$ s.t. $b\neq a$. Since $d$ is a metric function $a\neq b$ implies $d(a,b)>0$ Let this distance be $\epsilon$. Since the sequence $x_k=\frac{1}{k}$ converges to 0 for large $k$, there will ... |
3,062,765 | <p>i’d like to calculate Fourier coefficients of <span class="math-container">$\cos 2 \pi f_0 t$</span>.
This is what I did : </p>
<p><span class="math-container">$$ c_k = \frac{1}{T_0}\int_{0}^{T} \cos 2 \pi f_0 t \cdot e^{-2i\pi f_0 t}. $$</span></p>
<p>From Euler formulas:</p>
<p><span class="math-container">... | Noble Mushtak | 307,483 | <p>First off, you didn't set up the initial integral correctly. Assuming <span class="math-container">$T_0=\frac{1}{f_0}$</span> is the period of <span class="math-container">$\cos(2\pi f_0t)$</span>, it should be:</p>
<p><span class="math-container">$$c_k=\frac 1 {T_0}\int_0^{T_0}\cos(2\pi f_0t)e^{-2i\pi kf_0t}dt$$</... |
808,964 | <blockquote>
<p>Let $a_n$ be a positive sequence such that $S_n = \sum\limits_{k=1}^n a_k$ diverges.
I'm trying to prove $\sum\limits_{k=1}^n \frac{a_n}{S_n}$ diverges.</p>
</blockquote>
<p>I tried summation by parts, limit comparison and Stolz theorem in many different combinations and am still stuck with nothing... | Krokop | 153,019 | <p>HINT : If $\lim_{n\rightarrow +\infty} \frac{a_n}{S_n}=0$
$$
\frac{a_n}{S_n}\sim -\ln(1-\frac{a_n}{S_n})=\ln(\frac{S_n}{S_{n-1}})
$$</p>
<p>Details : </p>
<p>$$\sum_{p=1}^{n}\ln(\frac{S_p}{S_{p-1}})=\sum_{p=1}^n \ln(S_p)-\sum_{p=0}^{n-1}\ln(S_p)=\ln(S_n)-\ln{S_0}$$</p>
<p>As the series $\sum a_n$ diverges then th... |
176,569 | <p>How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.</p>
<p><strong>Example with $|A|=3$:</strong> Out of the set $A := \{1,7,11\}$ follow 19 sums 1,2,3,7,8,9,11,12,13,14,15,18,19,21,22,23,25,29,33 which are al... | GH from MO | 11,919 | <p>There is an algorithm: look at all $n$-element subsets of $S:=\{1,2,3,\dots,4^{n-1}\}$. There is at least one $B_3[1]$ subset among them, namely $\{1,4,4^2,\dots,4^{n-1}\}$. So list all $n$-element $B_3[1]$ subsets of $S$, and pick the one with the smallest maximum.</p>
|
2,077,831 | <p>This is not a homework problem. It is meant as a challenge for people who really enjoy math and have time to spare.</p>
<p><strong>Background Info</strong></p>
<p>Suppose you have a 2D Cartesian coordinate system. There are three shapes: <em>R</em>, <em>C</em>, and <em>P</em>.</p>
<p><em>R</em> is a large rectang... | Joseph O'Rourke | 237 | <p>Without some geometric condition on what it means for $R$ to be "large" and $C$ to be "small," there may not be a solution for $n=3$, i.e., $P$ a triangle:
<hr />
<a href="https://i.stack.imgur.com/T7uSG.jpg" rel="nofollow noreferrer"><img... |
3,854,740 | <blockquote>
<p>How do I show that <span class="math-container">$[\sqrt{n}] - [\sqrt{n-1}] = 1$</span> when <span class="math-container">$n $</span> is a
perfect square and <span class="math-container">$0$</span> otherwise for <span class="math-container">$n\in\mathbb{N}$</span>. Here <span class="math-container">$[.] ... | dan_fulea | 550,003 | <p><strong>Part I</strong></p>
<p>This is a long answer, beyond the allowed maximal size = 30K characters.
So i had to split it.</p>
<p>It's because of the question, but also because details for the performed steps are given, hoping that the text should be accesible, up to some complex analysis issues, to a wider circl... |
959,742 | <p>For a function $f(x) = x\sqrt{2+x^2}$ find out if it's bijective and if so, find its inverse.</p>
<p>The function is surjective because $x^2 > 0:\forall x\in\mathbb{R}$. I'm having difficulties proving that the function is injective.</p>
<p>I tried the following: $$f(x) = f(y) \iff x\sqrt{2+x^2} = y\sqrt{2+y^2}... | N. S. | 9,176 | <p><strong>Hint</strong> Let $a=x^2 \geq 0$ and $b=y^2 \geq 0$. The equation reduces to $$2a+a^2=2b+b^2$$ </p>
|
959,742 | <p>For a function $f(x) = x\sqrt{2+x^2}$ find out if it's bijective and if so, find its inverse.</p>
<p>The function is surjective because $x^2 > 0:\forall x\in\mathbb{R}$. I'm having difficulties proving that the function is injective.</p>
<p>I tried the following: $$f(x) = f(y) \iff x\sqrt{2+x^2} = y\sqrt{2+y^2}... | Winther | 147,873 | <p>Hint:</p>
<p>To find the inverse let $y= f^{-1}(x)$ then since $f(f^{-1}(x)) = x$ we have</p>
<p>$$f(f^{-1}(x)) = x = y\sqrt{2+y^2}$$</p>
<p>which gives</p>
<p>$$x^2 = y^2(2+y^2)\to y^4 + 2y^2 - x^2 = 0$$</p>
<p>This is a quadratic equation in $y^2$: $(y^2)^2 + 2y^2 - x^2 = 0$ for which you can use the standard... |
1,198,754 | <p>A rectangular prism has <strong>integer</strong> edge lengths. Find all dimensions such that its surface area equals its volume.</p>
<p>My Attempt at a Solution:</p>
<p>Let the edge lengths be represented by the variables $l, w, h$.</p>
<p>Then $$lwh = 2\,(lw +lh + wh) \implies lwh = 2lwh\left(\frac{1}{l} + \frac... | Christian Blatter | 1,303 | <p>Since you have already determined all solutions where two or three of the variables are equal it remains to find all integer triples $(a,b,c)$ with
$$a<b<c\quad{\rm and} \qquad{1\over a}+{1\over b}+{1\over c}={1\over2}\ .\tag{1}$$
According to your edit you suspect that there could be an infinity of solutions.... |
3,974,159 | <p>How many sub choices if there are 3 types of bread, 6 different types of meat, 8 different veggies, 4 different kinds of cheese</p>
<p>You must choose 1 bread.
You can choose any Meats, including none.
You can choose any veggies, including none.<br />
You must choose 1 cheese.</p>
<p>I am confident in finding the nu... | reuns | 276,986 | <p>Split <span class="math-container">$[0,1]=\bigcup_j [a_j,b_j]$</span> where <span class="math-container">$p$</span> is injective on <span class="math-container">$(a_j,b_j)$</span> (ie. <span class="math-container">$p'$</span> doesn't vanish) then <span class="math-container">$$\int_{a_j}^{b_j} e^{2i\pi np(x)}dx=\int... |
1,917,484 | <p>Let $f:X\rightarrow \mathbb{C}$ be an integrable function and $g_n:X\rightarrow \mathbb{C}$ be a sequence of integrable functions so that $\|g_n\|_1\rightarrow 0$ and $|g_n(x)|\leq 1$ for every $n, x$. Show that $\|fg_n\|_1\rightarrow 0$.</p>
<p>I think the $|g_n(x)|\leq 1$ part is only so we can say $g_n \geq g_n^... | kobe | 190,421 | <p>Let $(g_{n_k})_{k=1}^\infty$ be a subsequence of $(g_n)_{n=1}^\infty$ Since $\|g_n\|_1\to 0$, $(g_{n_k})$ has a further subsequence $(g_{n_{k_j}})_{j=1}^\infty$ such that $g_{n_{k_j}} \to 0$ pointwise a.e. on $X$. Use dominated convergence to show that $\|fg_{n_{k_j}}\|_1 \to 0$. Since $g_{n_k}$ was an arbitrary sub... |
3,848,934 | <p>Let <span class="math-container">$T$</span> be some operator on an inner product space <span class="math-container">$(V, \langle\cdot,\cdot\rangle)$</span>, and <span class="math-container">$T^\dagger$</span> be its adjoint. I found too many questions about the proof of
<span class="math-container">$$Im(T^\dagger) =... | alepopoulo110 | 351,240 | <p>What is true in any case is that the <em>closure</em> of <span class="math-container">$\text{im}(T^*)$</span> is equal to <span class="math-container">$\ker(T)^\bot$</span>. This is superfluous in the finite dimensional case, since all subspaces are automatically closed. In the general case though this is necessary.... |
124,522 | <p><a href="https://i.stack.imgur.com/CxrDw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CxrDw.png" alt="enter image description here"></a></p>
<p>Greetings,</p>
<p>I have created 4 sample data and classified them to 4 classes as shown. However, when I want to test the ClassifierFunction, the ou... | Sascha | 4,597 | <p>I assume what you want to do is classify individual values (such as for instance <em>12.232</em>) into one of four classes and that your <code>data1</code> to <code>data4</code> are training examples for each individual class. In this case the syntax for <code>Classify</code> is slightly different and this is how yo... |
124,522 | <p><a href="https://i.stack.imgur.com/CxrDw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CxrDw.png" alt="enter image description here"></a></p>
<p>Greetings,</p>
<p>I have created 4 sample data and classified them to 4 classes as shown. However, when I want to test the ClassifierFunction, the ou... | ubpdqn | 1,997 | <p>This is, perhaps, an extended comment. Mathematica comes with a lot of built-in classifiers (e.g. language, face detection etc). I think it is unreasonable to expect presenting any data form to classify. Mathematica is certainly'smart' in trying to match input with classifier. However, data pre-processing happens wh... |
1,052,512 | <p>Basically, the question started with a little argument I had with my friend. My friend said he thinks it's possible to draw only 2 lines on the letter "W" and make 6 triangles, and I played around with it, but I couldn't really do it, so I told him I don't think it's possible, and we need at least 3 lines. </p>
<p>... | WaveX | 323,744 | <p>No one said the W couldn't be "cursive"...</p>
<p><a href="https://i.stack.imgur.com/UkwmZ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UkwmZ.jpg" alt="enter image description here"></a>
Perhaps not what the OP was looking for (and I'm aware this post was brought back from YEARS AGO, but I jus... |
3,948,869 | <p>I used the definition.</p>
<p><span class="math-container">$1$</span>; <span class="math-container">$\cos(x)$</span>; <span class="math-container">$\cos^2(\frac{x}{2})$</span></p>
<p><span class="math-container">$c_1\cdot1+c_2\cdot\cos(x)+c_3\cdot\cos^2(\frac{x}{2}) = 0$</span></p>
<p>I tried converting <span class=... | heropup | 118,193 | <p>Once you know that <span class="math-container">$$\cos^2 \frac{x}{2} = \frac{1 + \cos x}{2},$$</span> you already know that <span class="math-container">$$(1/2, 1/2, -1) \cdot (1, \cos x, \cos^2 \tfrac{x}{2}) = 0.$$</span></p>
|
253,413 | <p>If we know that $\sin(n^\circ)$ is constructible where $n$ is some integer,then is $\sin((an)^\circ)$ also constructible for any integer $a$ ?</p>
<p>I am thinking it should be but not sure how to show it? Maybe using some recurrence relation for sin, to express it entirely in terms of powers of $\sin(n^\circ)$? </... | Jonas Meyer | 1,424 | <ol>
<li><p>You can use the fact that $f$ is bounded and $f_n\to f$ uniformly to show that there is a uniform bound $M>0$ such that $|f_n(x)|<M$ and $|f(x)|<M$ for all $n$ and all $x$.</p></li>
<li><p>You can use the identity $a^2-b^2 = (a+b)(a-b)$ to bound the difference of squares in terms of $M$ and the dif... |
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Yiftach Barnea | 5,034 | <p>I am not an expert, but I believe lots of work was done on the case of groups of Lie type by Seitz, Liebeck and others. I think the basic paper is "The maximal subgroups of classical algebraic groups" by Seitz. These results are of great importance in the development of the many of the random generation results that... |
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Igor Rivin | 11,142 | <p>I think the basic paper is actually this one:</p>
<p>Aschbacher, Michael. "On the maximal subgroups of the finite classical groups." Inventiones mathematicae 76.3 (1984): 469-514.</p>
|
4,627,869 | <p>Assume I have two convex objects in 3D, <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, and the two are in contact. How can I show that there always exists a vector <span class="math-container">$v$</span>, such that if I move <span class="math-container">$A$</span> along <span cl... | JMoravitz | 179,297 | <p>As for proof of the equivalence of the two specific conditions you cite without referring to other conditions... consider doing so by contrapositive. Suppose that there exists some <span class="math-container">$b$</span> such that <span class="math-container">$Ax=b$</span> has more than one solution, in other words... |
4,627,869 | <p>Assume I have two convex objects in 3D, <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, and the two are in contact. How can I show that there always exists a vector <span class="math-container">$v$</span>, such that if I move <span class="math-container">$A$</span> along <span cl... | Samuele Biscaro | 1,044,136 | <p>It is true, let <span class="math-container">$v_1$</span> and <span class="math-container">$v_2$</span> be two solutions for the system <span class="math-container">$Ax=b$</span>. If we calculate <span class="math-container">$A(v_1-v_2)$</span> we get: <span class="math-container">$$A(v_1-v_2) = Av_1-Av_2=b-b=0$$</s... |
3,318,909 | <p>I haven't learnt natural deduction yet so I'm completely stuck on how to proceed. One tip I was given was to use the properties of negation but again, that's not really helping.</p>
| Kavi Rama Murthy | 142,385 | <p><span class="math-container">$|\int_s^{t} f(x)dx | \leq M(t-s) \to 0$</span> as <span class="math-container">$0<s<t \to 0$</span> where <span class="math-container">$M$</span> is a bound for <span class="math-container">$|f|$</span>. This implies that <span class="math-container">$\lim \int_{\delta} ^{1} f(x)d... |
1,405,889 | <blockquote>
<p>Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$ for positive real $n$?</p>
</blockquote>
<p>I've come to this answer by inverse Fourier transform. But since there is n, I am quite confused that I didn't get n in the answer. Is this answer incorrect?
Thank you</p>
| Vincenzo Oliva | 170,489 | <p>Alternatively, rewrite the hypothesis as $$\lim_{n\to\infty}x_n\left(\frac{x_{n+1}}{x_n}-\frac{1}{2}\right)=0.$$ Even if we can't split the limit, as we don't know <em>a priori</em> that $x_n$ converges, we <em>do</em> know we must have either $x_n\to0$ or $x_{n+1}/ x_n \to 1/2 $, which still implies $x_n\to0$.</p>
|
2,923,346 | <p>Let $f_1,..., f_n: \mathbb{R}\rightarrow\mathbb{R}$ be measurable functions. And $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous. </p>
<p>For $g:\mathbb{R}\rightarrow\mathbb{R}$, $g(x):=F(f_1(x),f_2(x),...,f_n(x))$. Is $g$ measurable?</p>
<p>I think I was be able to do that if F is measurable, but I don't know... | Awoo | 590,336 | <p>I think I was confusing myself. All real continuous function should be Lebesgue measurable, which gives $F$ should be also measurable, which lead to $h$ is also measurable.</p>
|
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