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 |
|---|---|---|---|---|
3,109,036 | <p>I must prove this tautology using logical equivalences but I can't quite figure it out. I know it has something to do with the fact that not p and p have opposite truth values at all times. Any help would be appreciated.</p>
| Bram28 | 256,001 | <p>Use the fact that <span class="math-container">$p \rightarrow q \Leftrightarrow \neg p \lor q$</span></p>
<p>Applied to your statement:</p>
<p><span class="math-container">$(q \land (p \rightarrow \neg )) \rightarrow \neg p \Leftrightarrow$</span></p>
<p><span class="math-container">$\neg (q \land (p \rightarrow ... |
2,523,000 | <p>Three players are playing a game and have a fair six-sided die. This is an arbitrary game conditioned on the following rule:</p>
<p>Player 1 rolls first, Player 2 roles until he has a different number to 1, Player 3 rolls until he has a different number to players 1 and 2. </p>
<p>$\underline{Question}$: How do I... | David K | 139,123 | <p>You might show that the distribution of numbers that the players receive
is the same as if they each (one at a time) chose a numbered ball from a bag without replacement, where the bag originally contained six balls numbered $1$ through $6.$</p>
<p>Then consider whether there are any two balls such that one is more... |
147,965 | <p>What is the method to count multiplicities of intersection? for example suppose we have the projective line $x=0$ in $\mathbb{P}^{2}$ and the curve $V(z^{2}y^{2}-x^{4}) \subseteq \mathbb{P}^{2}$.</p>
<p>Clearly they intersection consists of two points $p=[0:1:0]$ and $r=[0:0:1]$. So for example Bezout's theorem say... | guaraqe | 28,818 | <p><strong>For the first one</strong>:</p>
<p>We have a sum of the form $\sum_{t=X}^{100} t$. Now, lets write it as:
$$
\frac{\sum_{t=X}^{100} t + \sum_{t=X}^{100} t}{2}=\frac{\sum_{t=X}^{100} t + \sum_{t=X}^{100} (100+X-t)}{2}
$$
This second sum comes from the fact that we change the order of it, we sum up from the ... |
2,242,865 | <p>I have seen proofs using the delta-epsilon definition of continuity, and they make perfect sense, but I have not found one proof using the sequential definition of continuity. </p>
<p>For example, when given functions, $f$ and $g$ that are continuous on [$a,b$], prove that the function $h=f+g$ is also continuous on... | helloworld112358 | 300,021 | <p>You could certainly use the sequence definition of continuity, if you are already willing to accept the sum and product rule for limits of sequences. For instance,</p>
<p>Let $f$ and $g$ be continuous at $a$ and let $a_n\to a$. Then $$\lim_{n\to\infty}f(a_n)g(a_n)=\lim_{n\to\infty}f(a_n)\lim_{n\to\infty}g(a_n)=f(a)... |
261,461 | <p>I would like a good <strong>hint</strong> for the following problem that takes into account the position at which I am stuck. The problem is as follows</p>
<blockquote>
<p>Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Find a simple graph $G$ such that $\mathrm{Aut}(G) = \mathbb{Z}_n.$ </p>
</blockquote>
<... | hmakholm left over Monica | 14,366 | <p>Make a cycle of repeated units that are <em>not</em> individually symmetric, such as</p>
<pre><code> *
|
[--*---*--]^n
\ /
*
</code></pre>
|
114,278 | <p>If I wanted to show that a group of order $66$ has an element of order $33$, could I just say that it has an element of order $3$ (by Cauchy's theorem, since $3$ is a prime and $3 \mid 66$), and similarly that there must be an element of order $11$, and then multiply these to get an element of order $33$? I'm pretty... | Mariano Suárez-Álvarez | 274 | <p>Alternatively:</p>
<ul>
<li><p>show that the Sylow $11$-subgroup is normal.</p></li>
<li><p>show that a cyclic group of order $11$ has no automorphisms of order $3$.</p></li>
<li><p>pick an element of order $3$ in your group and conclude that it commutes with any element of order $11$.</p></li>
</ul>
|
114,278 | <p>If I wanted to show that a group of order $66$ has an element of order $33$, could I just say that it has an element of order $3$ (by Cauchy's theorem, since $3$ is a prime and $3 \mid 66$), and similarly that there must be an element of order $11$, and then multiply these to get an element of order $33$? I'm pretty... | Alex Becker | 8,173 | <p>To compliment the other answers, I will address why commutativity is necessary. Let $G$ be your group $o(x)$ denote the order of $x\in G$. You are making use of the statement that $o(a)o(b)=o(ab)$, but this is not necessarily true. It holds when $o(a),o(b)$ are coprime and $ab=ba$ (an interesting consequence of this... |
1,572,065 | <p>I came across this question in Probability & Statistics by Rohatgi Exercise 1.6.9.</p>
<p>Let $P(A|B) = P(A|B\cap C)P(C) + P(A|B\cap C^c)P(C^c)$ with $P(A|B\cap C) \neq P(A|B)$.</p>
<p>If all the three events have non-zero probability, prove that $B$ and $C$ are independent.</p>
<p>After several efforts, I c... | Varun Iyer | 118,690 | <p><strong>HINT</strong></p>
<p>Proof by contradiction: Assume that events $B$ and $C$ are dependent events.</p>
<p>Therefore we can say that:</p>
<p>$$P(B \cap C) = P(B) + P(C) - P(B \cup C)$$</p>
<p>Now substitute this in your formula above and if the equality doesn't hold, then the latter is true and $B$ and $C$... |
3,987,416 | <p>I need to prove that, for two linearly independent vectors <span class="math-container">$\mathbf{a},\mathbf{b}\in\mathbb{R}^3$</span>,</p>
<p><span class="math-container">$$(\mathbf{a} · \mathbf{a}) (\mathbf{b} · \mathbf{b}) - (\mathbf{a} · \mathbf{b})(\mathbf{a} · \mathbf{b}) = (\mathbf{a} \times \mathbf{b})·(\math... | bonsoon | 48,280 | <p>Observe that <span class="math-container">$(A\times B)\cdot (A\times B) = |A\times B|^2 = |A|^2|B|^2\sin^2\theta$</span>, where <span class="math-container">$\theta$</span> is the angle between them. Now we have <span class="math-container">$\sin^2\theta = 1-\cos^2\theta$</span>, and I will remind you that <span cla... |
3,987,416 | <p>I need to prove that, for two linearly independent vectors <span class="math-container">$\mathbf{a},\mathbf{b}\in\mathbb{R}^3$</span>,</p>
<p><span class="math-container">$$(\mathbf{a} · \mathbf{a}) (\mathbf{b} · \mathbf{b}) - (\mathbf{a} · \mathbf{b})(\mathbf{a} · \mathbf{b}) = (\mathbf{a} \times \mathbf{b})·(\math... | Christoph | 86,801 | <p>The left hand side is the <a href="https://en.wikipedia.org/wiki/Gramian_matrix#Gram_determinant" rel="nofollow noreferrer">determinant of the Gramian matrix</a> of <span class="math-container">$\mathbf a$</span> and <span class="math-container">$\mathbf b$</span>:
<span class="math-container">$$
\det\begin{pmatrix}... |
3,987,416 | <p>I need to prove that, for two linearly independent vectors <span class="math-container">$\mathbf{a},\mathbf{b}\in\mathbb{R}^3$</span>,</p>
<p><span class="math-container">$$(\mathbf{a} · \mathbf{a}) (\mathbf{b} · \mathbf{b}) - (\mathbf{a} · \mathbf{b})(\mathbf{a} · \mathbf{b}) = (\mathbf{a} \times \mathbf{b})·(\math... | robjohn | 13,854 | <p>For a unit vector <span class="math-container">$u$</span>, <span class="math-container">$|u\cdot v|$</span> is the length of the projection of <span class="math-container">$v$</span> onto <span class="math-container">$u$</span> (which is also the distance of <span class="math-container">$v$</span> from the plane per... |
341,728 | <p>Trivially, for any Lie Algebra (LA) g, g':=[g,g] is an ideal. What's wrong with the following argument?</p>
<p>Be g a simple LA, then it has to be g'=g by definition of simple LA. But [g,g]=g seems to be an alternative way of characterizing a semi simple LG. Furthermore, for sl(2) it doesn't seem to be true z=[x,y]... | yatima2975 | 360 | <p>The general method to compute the order of a permutation group involves is called the <a href="http://en.wikipedia.org/wiki/Schreier%E2%80%93Sims_algorithm" rel="nofollow">Schreier-Sims algorithm</a>, and involves computing a so-called <a href="http://en.wikipedia.org/wiki/Base_%28group_theory%29" rel="nofollow">Bas... |
635,893 | <p>I am trying to prove following statement:</p>
<blockquote>
<p>$[m,n]$ is a set of functions defined as $f \in [m,n] \leftrightarrow f: \{1,...,m\} \rightarrow \{1,...,n\}$. The size of $[m,n]$ is $n^m$ for $m,n \in \mathbb{N}_{\gt0}$.</p>
</blockquote>
<p>I have tried to prove it but I am not entirely sure about... | Alec Teal | 66,223 | <p>Use the fundamental theorem of calculus to show it is constant.</p>
<p>imagine the next line with a strike through:</p>
<p>Assume that $f=c$ and $c\ne 0$ and reach a contradiction. Thus proving f=0.</p>
<p><strong>Fleshing this out:</strong></p>
<p>$\frac{d}{ds}(\int^s_0f(t)dt)=f(s)$</p>
<p>You know that $f(s)=... |
635,893 | <p>I am trying to prove following statement:</p>
<blockquote>
<p>$[m,n]$ is a set of functions defined as $f \in [m,n] \leftrightarrow f: \{1,...,m\} \rightarrow \{1,...,n\}$. The size of $[m,n]$ is $n^m$ for $m,n \in \mathbb{N}_{\gt0}$.</p>
</blockquote>
<p>I have tried to prove it but I am not entirely sure about... | Stefan Smith | 55,689 | <p>Define $F(x) = \int_0^x f(s)\,ds$. By the Fundamental Theorem of Calculus, $F'=f$. But by your assumption, $F(x) = 0$ for all $x \in [0,1]$. So $f \equiv 0$.</p>
|
1,605,100 | <p>$$\lim_{n\to\infty}\sum_{k=1}^n \frac{n}{n^2+k^2}$$</p>
<p>I tried this using powerseries just putting as $x=1$ there ,even tried thinking subtracting $s_{n+1} - s_{n}$ would be of some help, also I thought of writing the denominator as product of two complex numbers and then doing the partial fractions but it did ... | Asinomás | 33,907 | <p>Every such subgroup $aHa^{-1}=aHHa^{-1}$ is of the form $bH Hc$ and there are only a finite number of options for $bH$ and a finite number of options for $Hc$ (in each case there are $[G: H]$ options). So the number of subgroups is bounded by $[G:H]^2$ (this bound is most likely very weak).</p>
|
656,742 | <p>One <a href="http://en.wikipedia.org/wiki/Nilpotent_matrix" rel="nofollow">property</a> of nilpotent matrices is that a matrix $N$ is nilpotent if and only if $\operatorname{tr}(N^k)=0$ for all $k>0$. How can this property be proved?</p>
| user39082 | 97,620 | <p>This follows from Engel's theorem: a nilpotent Matrix can ( by a suitable Base change) be brought in strictly triangular form (with 0's on the diagonal) from which the claim is immediate.</p>
|
1,536,632 | <p>I have the following system of equations </p>
<p>\begin{align}
\frac{dx}{d \tau} &= x \left(1-x-\frac{y}{x+b} \right) \\
\frac{dy}{d \tau} &= cy \left(-1+a\frac{x}{x+b} \right)
\end{align}</p>
<p>I am asked to show that if $a<1$, the only nonnegative equilibria are $(0,0), (1,0)$.</p>
<p>So first it is... | SchrodingersCat | 278,967 | <p>By Fermat's Little theorem, we have <br>
$25^{60} \equiv 1 \pmod {61}$<br>
$\left(25^{60}\right)^{20} \equiv 1^{20} \pmod {61}$<br>
$25^{1200} \equiv 1 \pmod {61}$<br>
$25^{1202} \equiv 25^2 \equiv 625 \equiv 15 \pmod {61}$<br>
$25^{1202}+3 \equiv 18 \pmod {61}$<br>
$\left(25^{1202}+3\right)^2 \equiv 18^2 \equiv 324... |
4,157,288 | <p>High school student here just wanting to be better at maths and to know how to approach and solve problems and also how to think like a mathematician. Please recommend a book to me it would help a lot.</p>
| Boshu | 257,404 | <p>Both Courant and Polya's books suggested above are excellent books you should read anyway. Another such book is Gowers' <em>The Princeton Companion to Mathematics</em> available <a href="https://rads.stackoverflow.com/amzn/click/com/0691118809" rel="nofollow noreferrer" rel="nofollow noreferrer">here</a>. However, ... |
1,264,353 | <blockquote>
<p>Evaluate the determinants given that $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-6.$</p>
</blockquote>
<ol>
<li>$\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}$ </li>
<li>$\begin{vmatrix} a & b &am... | Frank | 237,453 | <p>I see what you're doing now. You are using the properties of determinants. You look fine, but the only time you change signs is when you swap rows. I think there is a theorem that states that elementary row operations do not change the value of the det of a matrix. You were right to factor that the coefficient b... |
104,368 | <p>I can show that there infinitely many solutions to this equation. Is it possible that the set
of rational solutions is dense?</p>
| Thomas Riepe | 451 | <p>Concerning q.2, this looks like about it: <a href="https://www.math.psu.edu/calendars/meeting.php?id=24844" rel="nofollow">https://www.math.psu.edu/calendars/meeting.php?id=24844</a></p>
|
343,248 | <p>Let <span class="math-container">$\chi$</span> be a primitive Dirichlet character of modulus <span class="math-container">$q>1$</span>. Write, as is customary, <span class="math-container">$B(\chi)$</span> for the constant in the expression
<span class="math-container">$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} =... | Greg Martin | 5,091 | <p>When I was a graduate student at the University of Michigan (this would be the mid-1990s), I took an analytic number theory class from Montgomery, from notes that would eventually become his book with Vaughan. I remember learning directly from Montgomery in that class that the real part of <span class="math-containe... |
3,587,990 | <p>I am facing the following combination problem : </p>
<p>I do have <span class="math-container">$B$</span> colored boxes, each one containing <span class="math-container">$N_b$</span> balls of the given color.
The total number of balls is thus <span class="math-container">$\sum_{b=1}^{B} N_b = N$</span>.</p>
<p>I w... | Anubhab | 602,341 | <p>Well, it is easy to divide the balls such that <span class="math-container">$\displaystyle\sum_{p=1}^{P} c_p=B+P-1$</span>. You first give out as many balls as you can from box <span class="math-container">$1$</span> to player <span class="math-container">$1$</span>. Then if the box becomes empty, you give away from... |
1,438,026 | <p>Out of curiousity, are there functions who's domain is discrete but the range is continuous? Furthermore, is there also a real-world example of such a function, in physics for instance?</p>
| Jerry Guern | 223,953 | <p>Example: A probability distribution of the number of children women have. The domain is the integer number of children. The range is the probability of a woman having that many, which is a continuous variable.</p>
|
3,130,749 | <p>I am trying to differentiate <span class="math-container">$\frac{v^4-10v^2\sqrt{v}}{4v^2}$</span>.</p>
<p>I have tried splitting the fraction and doing the division before finding the differential, but I am still not getting the right answer.</p>
<p><span class="math-container">$$
\frac{v^4-10v^2\sqrt{v}}{4v^2}
=\... | TonyK | 1,508 | <p>You are forgetting to divide the first term by <span class="math-container">$4$</span>, and the second term by <span class="math-container">$2$</span>.</p>
|
143,173 | <p>I have a small question that I think is very basic but I am unsure how to tackle since my background in computing inequalities is embarrassingly weak - </p>
<p>I would like to show that, for a real number <span class="math-container">$p \geq 1$</span> and complex numbers <span class="math-container">$\alpha, \beta$... | MathArt | 319,307 | <p>To generalize this <span class="math-container">$L_p$</span> (p>1) norm for <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, we can introduce weights such that <span class="math-container">$q+r=1$</span>, so we have
<span class="math-container">\begin{align}
q\alpha+r\... |
2,634,408 | <p>Here is the question:</p>
<p>Define the set WFF as follows--</p>
<p>a) Every element in set $A = \{s_1, s_2, ...\}$ is in WFF. $A$ is countably infinite.</p>
<p>b) If $a$ is in WFF, so is $(\neg a)$. If $n$ and $m$ are in WFF, so is $(n \lor m)$.</p>
<p>c) No other elements are in WFF.</p>
<p>Show that WFF is c... | hmakholm left over Monica | 14,366 | <p>This style of definition can be confusing if you see it for the first time in a source that does not bother to explain very carefully how it should be understood.</p>
<p>What part (c) means is that $WFF$ is defined to be the <strong>smallest possible</strong> set that satisfies the conditions in parts (a) and (b).<... |
57,057 | <p>Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} \ge{\frac{{83}}{10}}\ \ ?$$</p>
| Michael Rozenberg | 190,319 | <p>We can use the Vasc's EV Method.</p>
<p>See here: <a href="https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf" rel="nofollow noreferrer">https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf</a></p>
<p>Indeed, let <span class="math-container">$a+b+c+d+e=constant.$</span></p>
<p>Thus, by c... |
840,445 | <p>Stirling number of the second kind is the number of ways to partition a set of n objects into k non-empty subsets - S(n,k). I want to restrict/constrain this partition so I can count the ways to partition a set of n objects into k non-empty subsets while at least p subsets out of k will have size r. When p=k the ans... | Adi Dani | 12,848 | <p>Formula that count the number of partitions of a n-set into k-parts of size 1 or 2 is<br>
$$\overline{p}_k(n,N_2) =\binom{k}{n-k} \frac{n!}{k!\cdot2^{n-k}}$$</p>
|
3,295,973 | <p>Let <span class="math-container">$(\Omega,\mathcal A,\mu)$</span> be a measure space, <span class="math-container">$p,q\ge1$</span> with <span class="math-container">$p^{-1}+q^{-1}=1$</span> and <span class="math-container">$f:\Omega\to\mathbb R$</span> be <span class="math-container">$\mathcal A$</span>-measurable ... | GSofer | 509,052 | <p>When you square the equation, you add a solution you didn't have before. Take for example the very simple equation - <span class="math-container">$$x=1$$</span> Squaring both sides we get: <span class="math-container">$$x^2=1\Rightarrow x=\pm 1$$</span>
Not all operations leave the information given in the original ... |
647,757 | <blockquote>
<p>How many integers $n$ are there such that $\sqrt{n}+\sqrt{n+7259}$ is an integer? </p>
</blockquote>
<p>No idea on this one.</p>
| Bart Michels | 43,288 | <p>Imranfat's answer is good, however you might be looking for a more rigorous argument to show that both $n$ and $7259+n$ have to be squares. Here it is:</p>
<p>If $\sqrt n+\sqrt{7259+n}$ is integer, then $$\sqrt n-\sqrt{7259+n}=\frac{-7259}{\sqrt n+\sqrt{7259+n}}$$ is rational.</p>
<p>Adding and substracting these ... |
3,887,526 | <p>I need to prove <span class="math-container">$\frac{|x+y+z|}{1+|x+y+z|} \le \frac{|x|}{1+|y|+|z|}+\frac{|y|}{1+|x|+|z|}+\frac{|z|}{1+|x|+|y|}$</span>. I've tried to use triangle inequality or to explore the form of <span class="math-container">$(a+b+c)^2$</span> but it won't get me anywhere. I would be grateful for ... | Noureddine Ouertani | 348,708 | <p>Let's check whether: <span class="math-container">$$S(N) = \frac{|\sum_{i=0}^{N} x_i|}{1+|\sum_{i=0}^{N} x_i|}\le \sum_{i=0}^{N} \frac{|x_i|}{1+ |\sum_{j=0;j \ne i}^{N} x_j|}$$</span> for all natural numbers <span class="math-container">$N$</span> and all natural numbers <span class="math-container">$i<N+1$</span... |
2,123,926 | <p>I have a triangle with the coordinates (0, 20), (15, -10), (-15, -10). So the centre of the triangle is 0,0. </p>
<p>I want to rotate this by one degree. Im using the formula:</p>
<pre><code>x' = cos(theta)*x - sin(theta)*y
y' = sin(theta)*x + cos(theta)*y
</code></pre>
<p>Yet when I work out the coordinate for a... | Julio Maldonado Henríquez | 406,412 | <p>For the homogeneous solution you can solve the characteristic polynomial as the ODE is lineal. So you replace $y^{i}$ by $\lambda^{i}$ when the first is derivate and the second a power. The idea is to find roots so: $$\lambda^2-2\lambda+1=0 \iff (\lambda-1)^2=0$$ for each root $r_i$ we associate a solution $A_i e^{r... |
69,373 | <p>Let $N$ be the number of rolls until the same number appears $k$ consecutive times. Show the expected value $E[N]=\dfrac{6^k-1}{5}$. I've tried conditioning this on the first occurrence of the expected number, but I'm having a hard time generalizing further than 2 consecutive times. I think I need to use the conditi... | MAK | 16,080 | <p>Well, for k=1, the value is trivially 1.</p>
<p>For k=2, you can find the probability of your event happenning in 2,3,4,.... throws.</p>
<p>The probability of N=2 , is the probability of the first two throws having the same
outcome. So the expected number of throws needed for N=2 is:</p>
<p>P(Outcome aa).2+ [P(O... |
2,122,539 | <p>A functor sends $X$ to $R^{(X)}$ what does this notation mean? Is this functor faithful and full?</p>
| egreg | 62,967 | <p>Usually, $R^{(X)}$ denotes the free $R$-module on the set $X$, that is, the set of “formal sums”
$$
\sum_{x\in X}r_xx
$$
where $\{x\in X:r_x\ne0\}$ is finite, with the obvious operations making it into an $R$-module. The elements of $X$ are those where just one coefficient is $1$ and the others are zero.</p>
<p>How... |
482,793 | <p>As title says, how does one show that a function is continuous over some interval (let us say over some interval of real numbers?)</p>
<p>Would(Can) this involve derivative?</p>
| Dan Brumleve | 1,284 | <p>Continuity has various equivalent definitions. The existence of a derivative is a stronger property than continuity, so if you already know the function is differentiable on an interval, then it is certainly continuous, but the <a href="http://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow">converse</a> ... |
2,819,950 | <p>I am a university graduate with a B.S. in mathematics who has been developing software for the past 8 years. I recently discussed a mutual interest in topology with a friend who is just about to complete his degree. Due to time constraints both of us missed our chance to take a topology course during our undergradua... | Mark Fischler | 150,362 | <p>I had great pleasure working through Munkries, doing all the exercises. I'm not sure what you want in terms of "foundational introduction" but his (extensive) first chapter will be an excellent ramp-up getting back into shape. In particular, before even getting into topology, his exercises leading to the proof tha... |
2,805,803 | <p>The question is as follows: </p>
<blockquote>
<p>A typical long-playing phonograph record (once known as an LP) plays for about $24$ minutes at $33 \frac{1}{3}$ revolutions per minute while a needle traces the long groove that spirals slowly in towards the center. The needle starts $5.7$ inches from the center an... | Break-in Master | 953,558 | <p>You can't assume that each revolution is exactly the same length! Each one will be slightly shorter than the one before. At best, you'd have to get the length of the outer-most groove and the inner most and figure out what the average of the two would be and then multiply by <span class="math-container">$800$</spa... |
3,929,952 | <p><strong>Question:</strong> How do I estimate the following integral? <span class="math-container">$$\int_{2}^{\infty} \frac{|\log \log t|}{t\, (\log t)^2} \, dt$$</span></p>
<p><strong>Attempt:</strong> Setting <span class="math-container">$u=\log t$</span>, we get</p>
<p><span class="math-container">$$\int_{2}^{\in... | Raffaele | 83,382 | <p>Integration by substitution <span class="math-container">$t=e^u;\;dt=e^u du;\;\log t = u$</span>
<span class="math-container">$$\int \frac{\log (\log t)}{t \log ^2 t} \, dt=\int\frac{ \log u}{u^2}\,du=-\frac{\log u+1}{u}+C$$</span></p>
<p><span class="math-container">$$\int \frac{\log (\log t)}{t \log ^2 t} \, dt=-\... |
2,066,765 | <p>Are there any examples of subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?</p>
| Batman | 127,428 | <p>Take the subspace of $\ell^2$ (or $\ell^\infty$) consisting of sequences with finitely many non-zero coordinates.</p>
<p>Clearly, this is a subspace (sum of sequences still has at most # of nonzeros of first seq + # of nonzeros in second seq, scaling of a sequence has same non-zeros). However, you can take any sequ... |
14,828 | <p>I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is... | yatima2975 | 360 | <p>In addition to the other answers, it's also noteworthy to know that there is a number system (more-or-less) integrating N, Z, Q and R (and <em>a lot</em> more, but not C!). </p>
<p>The <a href="http://en.wikipedia.org/wiki/Surreal_number">surreal numbers</a> take the basic idea from Dedekind cuts, assigning to each... |
192,537 | <p>Can you give me an example that there is a $f \in C_0^{\infty}(\mathbb C)$, such that the equation $\bar \partial u=f$ has no $C_0^{\infty}(\mathbb C)$ solution?</p>
| girianshiido | 39,741 | <p>If $f\in C^\infty_0(\mathbb{C})$, then $\displaystyle u:\zeta\mapsto \frac{1}{2i\pi}\int_{\mathbb{C}}\frac{f(z)}{z-\zeta}dz\wedge d\bar{z}$ is also in $C^\infty_0(\mathbb{C})$ and is a solution to $\bar{\partial}u=f$</p>
|
1,940,837 | <p>I don't understand the difference between $\bigoplus_{i=1}^n M_i$ and $\prod_{i=1}^n M_i$ where $\{M_i\}_{i=1,...,n}$ is a collection of $R-$module. When $I$ is not finite, $$\prod_{i\in I}M_i=\{(x_i)_{i\in I}\mid x_i\in I\}$$ and $$\bigoplus _{i\in I}M_i=\{(x_i)_{i\in I}\mid x_i=0\text{ except for finite number of ... | HarrySmit | 332,761 | <p>In the case that the $M_i$ are ideals, $\prod_{i \in I} M_i$ is sometimes used for the product of the ideals. Thus, to avoid confusion, $\bigoplus_{i \in I} M_i$ is used instead.</p>
|
2,568,544 | <p>The exercise is to construct a cube (inscribed in the unit spher) with one of the corners at:
$$ \vec{v} = \frac{1}{\sqrt{10}} (3,1,0) \in S^2 $$</p>
<p>I'm a bit stuck constructing the other seven vertices. My first guess was to use the eight reflections available to me:</p>
<p>$$ \square \stackrel{?}{=}\left... | user | 505,767 | <p><strong>HINT</strong></p>
<p>1) Find the opposite corner: $\vec{w}=-\vec{v} = -\frac{1}{\sqrt{10}} (3,1,0) \in S^2 $</p>
<p>2) Find the vertical axis $A$ of the cube, i.e. the line through the origin which form an angle $\alpha$ s.t. $\cos \alpha=\frac {1}{\sqrt 3}$ with the line $\vec v-\vec w$</p>
<p>3) Find t... |
2,571,629 | <blockquote>
<p>Let $\mathbb Z$ denote the set of integers and $\mathbb Z_{\ge 0}$
denote the set $\{0,1,2,3,...\}$. Consider the map $f:\mathbb Z_{\ge
0}\times \mathbb Z \to \mathbb Z$ given by $f(m,n)=2^m\cdot(2n+1)$. Then
the map $f$ is</p>
<p>(A)injective but not surjective.</p>
<p>(B)surjective b... | Rory Daulton | 161,807 | <p>Note that $f(m,n) = 2^m\cdot(2n+1)$ can never be zero, since neither factor can be zero. Therefore the mapping is not surjective.</p>
|
2,190,394 | <p><a href="https://i.stack.imgur.com/X1WFP.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/X1WFP.jpg" alt="enter image description here"></a></p>
<p>The required angle can be found out by using cosine rule and then we get it as be $90^{\circ}$.
Can someone provide a bit more geometric approach? Or ... | Futurologist | 357,211 | <p>This problem does not require any formulas, calculations or even exact lengths for $AB$ and $CD$. The condition $AB = CD = \sqrt{3}$ is absolutely irrelevant information.</p>
<p>1) Since $AB$ is orthogonal to the plane determined by the three points $B, C, D$, it is in fact orthogonal to any line in that plane. In ... |
2,793,144 | <p>Consider the following
Proposition: </p>
<ul>
<li>Let $A\subseteq B$. Also, let $B\subseteq C$. Thus, $A\subseteq C$.</li>
</ul>
<p>Proof: </p>
<ul>
<li>Let $A\subseteq B$. Also, let $B\subseteq C$.
<blockquote>
<p><strong>What goes here?</strong> Assume $x\in A$. As $x\in A$ and $A\subseteq B$, $x\in B$.
A... | Michael Hardy | 11,667 | <p>$$
\text{Is } S\left( \cdots + \frac{A^n}{n!} + \cdots \right) S^{-1} \text{ equal to } \cdots+\frac{(SAS^{-1})^n}{n!} +\cdots \text{ ?}
$$
$$
\text{Is } SA^n S^{-1} = (SAS^{-1})^n \text{ ?}
$$
\begin{align}
(SAS^{-1})^n & = (SAS^{-1}) (SAS^{-1}) \cdots\cdots (SAS^{-1}) (SAS^{-1}) \\[10pt]
& = SA(S^{-1}S) A(... |
184,241 | <p>I am having issues with the code:</p>
<pre><code>CC = p0^2/(p0 - L)
Dbuy = (1 - CC)*(L/p0) + (CC - p0)*Log[CC/p0]
Dwait = (1 - p1)*(L/(p1 - p0))
Dlock = (1 - p1)*(1 - L/p0 - L/(p1 - p0)) + (p1 - CC)*(1 - L/p0) - L*Log[(p1 - p0)/(CC - p0)]
RevLocking1 = Table[NMaximize[{x*Dbuy*p0 + x*Dwait*0.4*p1 + x*Dlock*(p0 + 0.4... | kglr | 125 | <pre><code>Merge[listData, Identity]
</code></pre>
<blockquote>
<p><|"Input" -> {1, 2, 3}, "double" -> {2, 4, 6}, "squared" -> {1, 4, 9}|></p>
</blockquote>
<pre><code>listFormatData = {<|"Input" -> input1, "OutputKey1" -> OutputKey1Input1Output|>,
<|"Input" -> input2, "OutputKey1" -> Ou... |
256,806 | <p>How can I prove or disprove that $\lim\limits_{n\to \infty} (n+1)^{1/3}−n^{1/3}=\infty$?</p>
<p>My guess is that it is false but I can't prove it.</p>
| Mario Carneiro | 50,776 | <p>Let $x^3=n$. Then $$\lim_{n\to\infty}(n+1)^{1/3}-n^{1/3}=\lim_{x\to\infty}(x^3+1)^{1/3}-x.$$
Now for $x\geq1/3$, $$x^3+1\leq x^3+3x\leq x^3+3x+\frac3x+\frac1{x^3}=\left(x+\frac1x\right)^3,$$ so $$\lim_{x\to\infty}(x^3+1)^{1/3}-x\leq\lim_{x\to\infty}\left(\left(x+\frac1x\right)^3\right)^{1/3}-x=\lim_{x\to\infty}\frac... |
256,806 | <p>How can I prove or disprove that $\lim\limits_{n\to \infty} (n+1)^{1/3}−n^{1/3}=\infty$?</p>
<p>My guess is that it is false but I can't prove it.</p>
| robjohn | 13,854 | <p>Since $x^3-y^3=(x-y)(x^2+xy+y^2)$, we have
$$
(n+1)^{1/3}-n^{1/3}=\frac{(n+1)-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}}
$$
Therefore,
$$
\frac1{3(n+1)^{2/3}}\le(n+1)^{1/3}-n^{1/3}\le\frac1{3n^{2/3}}
$$
The limit follows by the <a href="http://en.wikipedia.org/wiki/Squeeze_theorem" rel="nofollow">Squeeze Theorem</a>... |
1,026,135 | <p>Let a, b be elements of a group G and H a normal subgroup of G. Is it true that $aH = bH$, then $a^{-1}H = b^{-1}H$? How can I prove this?</p>
| egreg | 62,967 | <p>I assume that $H$ is a subgroup of the group $G$.</p>
<p>Saying $aH=bH$ is equivalent to $a^{-1}b\in H$; so $a^{-1}H=b^{-1}H$ is equivalent to $ab^{-1}\in H$ or $ba^{-1}\in H$ (because $(ab^{-1})^{-1}=ba^{-1})$.</p>
<p>Are you sure that $a^{-1}b\in H$ implies $ba^{-1}\in H$?</p>
<p>Check with $S_3$ and $H$ a two ... |
1,026,135 | <p>Let a, b be elements of a group G and H a normal subgroup of G. Is it true that $aH = bH$, then $a^{-1}H = b^{-1}H$? How can I prove this?</p>
| Ma Ming | 16,340 | <p>Untrue.</p>
<p>Hint:
$$
aH=bH\quad<=>\quad a^{-1}b\in H.
$$ </p>
<p>$$
a^{-1}H=b^{-1} H \quad<=>\quad ba^{-1}\in H.
$$</p>
<p>Pick up $bhb^{-1}\not\in H$, let $a=bh$.</p>
|
134,213 | <p>Is there a way to simplify the following expression</p>
<p>$$A = \frac{\left (a^6-b^6 \right )^2}{c^6-4a^3b^3}$$</p>
<p>assuming</p>
<p>\begin{eqnarray}
a = &\ (a-b)^2+b\ (a+1) \cr
b = &\ (b-c)^2+c\ (b+1) \cr
c = &\ (c-a)^2+a\ (c+1)
\end{eqnarray}</p>
<h3>Edit</h3>
<p>I would like to determi... | Bill | 18,890 | <p>I'm not certain if this is what you want</p>
<pre><code>Reduce[{A == (a^6 - b^6)^2/(c^6 - 4 a^3 b^3), a == (a - b)^2 + b (a + 1),
b == (b - c)^2 + c (b + 1), c == (c - a)^2 + a (c + 1)}, A]
</code></pre>
<p>tells you that <code>A == -3 (5 + 2 c^2 + c^4)</code> when <code>c== + or - 1</code> times the square root... |
2,416,848 | <p>Unfortunately,I've no idea of dealing with this problem.The term <em>isomorphism class</em> leads me to think of fundamental theorem of finitely generated abelian groups but i think i'm going in wrong direction.</p>
<p>Please give me some hint?suggestions?Anything?</p>
| Thomas Andrews | 7,933 | <p>Since $(2+i)(2-i)=5$ in $\mathbb Z[i]$, with the two factors relative prime there, and $\mathbb Z[i]$ is a PID, we can conclude that $$(\mathbb Z[i])/\langle 5\rangle = \mathbb Z[i]/\langle 2+i\rangle \times \mathbb Z[i]/\langle 2-i\rangle.$$</p>
<p>But, by conjugation, $\mathbb Z[i]/\langle2+i\rangle\cong \mathbb ... |
80,922 | <p>I am generally confused about integer valued polynomials, and how to count them. Trying to learn the subject I started by listing permutations of the rows in a lower triangular table:</p>
<p>$$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&... | Bill | 18,890 | <p>One way you can let Mathematica know what you are assuming here is</p>
<pre><code>Simplify[DSolve[{y'[x]==1/2 (1-y[x]) y[x] (a-y[x]+a y[x]), y[0]==y0}, y[x], x], 0<a<1/2]
</code></pre>
<p>One way of assigning the result and passing value of a, y0, etc is</p>
<pre><code>yf = y[x]/.DSolve[{y'[x]==1/2 (1-y[x])... |
1,341,231 | <p>Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is integrable and to what extent. It also gets really messy. Besides, I am not sure if I am to express the limit using $f$, ... | Shubham Avasthi | 240,149 | <p>I would recommend An Introduction to the Theory of Numbers By G.H. Hardy and E.M. Wright . </p>
|
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | TonyK | 1,508 | <p>Light both ends of rope $A$, and one end of rope $B$. When rope $A$ is burnt out, after 30 minutes, light the other end of rope $B$. Rope $B$ will be burnt out after a further 15 minutes.</p>
|
970,872 | <p>There is a common brain teaser that goes like this:</p>
<p>You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it d... | Eugene K | 273,801 | <p>Here's my proof why the answer is wrong as well, if you assume nothing about the rope properties. Essentially things like feedback might affect the rope burning. </p>
<p>Rope = [ 1 | 2 | 3 | 4 | 5 ]</p>
<p>Imagine the rope split into 5 parts.</p>
<p>The rope’s units are not guaranteed to burn at the same rate dep... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Jon Bannon | 6,269 | <p>I'm not really qualified to answer this...or perhaps I am, since I'm no Terry Tao! I think what we bring to mathematics is the unique perspective afforded by our own individual experience. The more of us who are trying to do mathematics, the better, because this increases the diversity of perspectives in approaching... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | drbobmeister | 8,472 | <p>I had the privilege of discussing similar concerns with regard to theoretical physics
with the late Richard Feynman. He told me the following, which has always served me in
good stead: "You keep on learning and learning, and pretty soon you learn something
no one has learned before." That was his "advice"; my adv... |
43,690 | <p>I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of question and since it is worrying me increasingly as of late I feel that I must ask it.</p>
<p>My question is: what ca... | Franz Lemmermeyer | 3,503 | <p>Let met add two quotations:</p>
<ol>
<li><p>Fermat's motto was "Multi pertransibunt et augebitur scientia" (many will pass
through and knowledge will be increased). At another occasion he wrote about
"passing the torch to the next generation", which I find particularly nice.</p></li>
<li><p>"When kings are build... |
4,038,151 | <p>In Bernt Oksendal's <em>Stochastic Differential Equations</em>, Chapter 4, one has the following stochastic differential equation (whose solution is geometric Brownian motion):
<span class="math-container">$$dN_t=rN_tdt+\alpha N_tdB_t\;\;\;\text{ ie } \;\;\; N_t-N_0=r\int_0^t N_sds+\alpha\int_0^tN_sdB_s,$$</span>
wh... | Jan Stuller | 503,335 | <p>I find notation such as <span class="math-container">$\frac{dN_t}{N_t}=rdt+\alpha dB_t$</span> incredibly unfortunate, it leads to the type of confusion you have encountered (and that I used to encounter myself).</p>
<p>That is why I advise against using short-hand notation (at least until one becomes very confident... |
57,223 | <p><strong>No error in 10.0.2</strong></p>
<hr>
<p>When using <code>Correlation[]</code> function I sometimes get the strange warning:</p>
<pre><code>CorrelationTest::nortst: "At least one of the p-values in {0.0527317}, resulting from a test for normality, is below 0.05`. The tests in \!\({\"PearsonCorrelation\"}\)... | Nasser | 70 | <p>No error nv 10.0.2. On windows 7, 64 bit</p>
<p><img src="https://i.stack.imgur.com/3Rsxu.png" alt="Mathematica graphics"></p>
<pre><code>SeedRandom[0];
x = RandomReal[{-5, 5}, 100];
y = 2 x + 1 + RandomReal[{-0.1, 0.1}, 100];
X = Transpose[{x, y}];
ListPlot[X]
Correlation[X] // MatrixForm
CorrelationTest[X, 99995... |
541,541 | <blockquote>
<p>Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product $f_1\otimes f_2\colon V_1\otimes V_2\to W_1\otimes W_2$ is defined, and it's obvious that $\ker f_1\otimes V_2+ V... | pre-kidney | 34,662 | <p>The isomorphism
$$
\ker (f_1\otimes f_2)=V_1\otimes \ker f_2 + \ker f_1 \otimes V_2
$$
is equivalent to the isomorphism
$$
\operatorname{coker} (f_1\otimes f_2)=\operatorname{coker} f_1\otimes \operatorname{coker} f_2,
$$
since for any ideals $I,J\subset R$ we have
$$
R/(I+J)=R/I\otimes R/J
$$</p>
|
904,547 | <p>I have the sum ( $M$ is any integer $> 1$ ):</p>
<p>$$
\sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor
-\left\lfloor\, 2M \over h\,\right\rfloor\,\right)
$$</p>
<p>and looking for a way to simplify it. In the sense of either finding a <strong>simple closed form or a good approximation for M... | Ara | 169,954 | <p>$$ 2\pi\ \int_0^{\sqrt{2}} \sqrt{4-x^{2}} $$ </p>
<p>$$ \pi \left(x \sqrt{4-x^2} + 4 * \arcsin \frac {x}{2} \right)$$ </p>
<p>from 0 to $\sqrt{2}$ which gives $\pi^{2}$ + 2$\pi$.</p>
|
2,326,551 | <p>Prove that $\varphi(n^2)=n\cdot\varphi(n)$ for $n\in \Bbb{N}$, where $\varphi$ is Euler's totient function.</p>
| DonAntonio | 31,254 | <p><strong>Hint</strong>: Use that for any $\;n=p_1^{a_1}\cdot\ldots\cdot p_k^{a_k}\in\Bbb N\;,\;\;p_i\;$ primes, $\;a_i\in\Bbb N\;$ , we have</p>
<p>$$\varphi(n)=n\prod_{i=1}^k\left(1-\frac1{p_i}\right)$$</p>
|
4,312,353 | <p>Let <span class="math-container">$f_n$</span> and <span class="math-container">$f$</span> functions of <span class="math-container">$L_1\mathbb R^n$</span> such that
<span class="math-container">$\lim_{n \rightarrow \infty} f_n = f$</span> almost always in <span class="math-container">$\mathbb R^n$</span> and <span ... | operatorerror | 210,391 | <p>Here is a hint to get you started: Apply Fatou's lemma to
<span class="math-container">$$
|f_n|+|f|-|f_n-f|\geq 0
$$</span>
to find <span class="math-container">$f_n\to f$</span> in <span class="math-container">$L^1$</span>.</p>
|
4,312,353 | <p>Let <span class="math-container">$f_n$</span> and <span class="math-container">$f$</span> functions of <span class="math-container">$L_1\mathbb R^n$</span> such that
<span class="math-container">$\lim_{n \rightarrow \infty} f_n = f$</span> almost always in <span class="math-container">$\mathbb R^n$</span> and <span ... | Mason | 752,243 | <p>This is an easy application of a "generalized" dominated convergence theorem. You have <span class="math-container">$|f_n\chi_{A}| \leq |f_n|$</span>, and <span class="math-container">$\int |f_n| \to \int |f|$</span>. Since <span class="math-container">$f_n\chi_{A} \to f\chi_{A}$</span> almost everywhere, ... |
3,310,599 | <p>Let <span class="math-container">$Y$</span> be a random variable on <span class="math-container">$[0,1]$</span> and let <span class="math-container">$X\sim U[0,1]$</span> be a uniformly distributed random variable that is independent of <span class="math-container">$Y$</span>.</p>
<p>Prove that <span class="math-co... | zoli | 203,663 | <p>Consider a probability space on which <span class="math-container">$X,Y$</span> are independent and (on <span class="math-container">$[0,1]$</span>) uniformly distributed random variables. By the definition of conditional expectation one has to prove that</p>
<p><span class="math-container">$$\int_A P(X<Y\mid Y... |
1,989,532 | <p>The equation I am considering is </p>
<p>$$x^{3}-63x=162$$</p>
<p>Here is how far I got:</p>
<p>$$3st=-63$$</p>
<p>$$s=\frac{-63}{3t}=-21t$$</p>
<p>$$s^{3}t^{3}=-162$$</p>
<p>$$(-21t)^{3}-t^{3}=-162$$</p>
<p>$$-9261t^{3}-t^{3}=-162$$</p>
<p>Where would I go from here?</p>
| Mark Fischler | 150,362 | <p>Cardano's method would be to write
$$
x = t - \left( \frac{-63}{3t} \right)= t+\frac{21}{t}
$$
which gives
$$
t^3 -162 +\frac{9261}{t^3} = 0
$$
and if you multiply thru by $t^3$ and write $t^3 = p$ this gives
$$p^2-162p+9261 = 0$$
where the solution is
$$p=81\pm30 i \sqrt{3}$$
and to recover $t$ one has to take t... |
2,090,452 | <p>I have a signature given with the real numbers as its universe and addition and multiplication as functions. I need to write the following expression in First Order Logic.</p>
<ul>
<li>$x$ is a rational number</li>
</ul>
<p>My idea: $\varphi(x) = \exists a \, \exists b \, x = \frac{a}{b}$.</p>
<p>The problem is ... | Stella Biderman | 123,230 | <p><strong>EDIT:</strong> I just noticed you said first order arithmetic. This answer uses second order arithmetic. It's a theorem that it is impossible to define $\mathbb{Z}$ in $(\mathbb{R},+,\cdot)$</p>
<p>As mentioned in the comments, division is easy: define $a/b$ to be the number $c$ such that $cb=a$. It's ident... |
1,312,783 | <p>I have come across the following formula:</p>
<p>$$u(n)=\sum_{m=-\infty}^{n}\delta(m)$$</p>
<p>where $u(n)$ is the Unit Step and $\delta(m)$ is the Delta Function:</p>
<p>What I can't understand is how this formula "works".</p>
<p>Expanding the formula we have:</p>
<p>$$u(n)=...+\delta(0)+\delta(1)+\delta(2)+..... | Teoc | 190,244 | <p>$$\frac{d}{dx}\sqrt{x+1}-\sqrt{x}=\frac1{2\sqrt{x+1}}-\frac{1}{2\sqrt x}=\frac{1}{2\sqrt{x+1}}\frac{2\sqrt x}{2\sqrt x}-\frac{1}{2\sqrt x}\frac{2\sqrt{x+1}}{2\sqrt{x+1}}$$
Can you take it from there?</p>
|
284,685 | <p>I was wondering if there is an explicit estimate on the probability that the lowest eigenvalue of a $n \times n$ GOE matrix is larger than some number $x \in \mathbb{R}$. I am aware of the fact that there is in principle an explicit formula for that, but if $n$ becomes large, this event is really difficult to comput... | Carlo Beenakker | 11,260 | <p>See <A HREF="https://arxiv.org/abs/0801.1730" rel="nofollow noreferrer">Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices</A> (2008), in particular the large-$n$ result:</p>
<p>$$\text{Prob}(E_{\rm smallest}\geq x)\rightarrow\exp\left[-n^2\Phi\left(\frac{x+\sqrt{2n}}{\sqrt{n}}\right)\right],\;\;-\... |
284,685 | <p>I was wondering if there is an explicit estimate on the probability that the lowest eigenvalue of a $n \times n$ GOE matrix is larger than some number $x \in \mathbb{R}$. I am aware of the fact that there is in principle an explicit formula for that, but if $n$ becomes large, this event is really difficult to comput... | Adrien Hardy | 15,517 | <p>By symmetry you're looking at the probability that the maximal eigenvalue is smaller than some number. Explicit inequalities for such events can be obtained by using the tridiagonal representation for the GOE, see the last 20 slides from Michel Ledoux : <a href="https://www.math.univ-toulouse.fr/~ledoux/Leipzig.pdf"... |
3,697,011 | <blockquote>
<p>For what value of <span class="math-container">$a$</span> would the following function have exactly one solution?
<span class="math-container">$$a^2x^2+3x-5\frac{1}{a}=0$$</span></p>
</blockquote>
<p>I know that it needs to become
<span class="math-container">$$\frac{3}{2}x^2+3x+\frac{3}{2}=0$$</... | Vishu | 751,311 | <p>For exactly one solution, we need <span class="math-container">$D=0$</span> i.e. <span class="math-container">$$3^2 -4\cdot a^2\cdot\left(-\frac 5a\right)=0$$</span></p>
|
3,296,638 | <p>Are two planes parallel if the magnitude of the cross product of their normal vectors is equal to <span class="math-container">$0$</span>?</p>
<p><span class="math-container">$|| \vec n_1 \times \vec n_2|| = 0$</span></p>
| José Carlos Santos | 446,262 | <p>Yes, because<span class="math-container">\begin{align}\left\lVert\vec n_1\times\vec n_2\right\rVert=0&\iff\vec n_1\times\vec n_2=0\\&\iff\vec n_1\text{ and }\vec n_2\text{ are parallel.}\end{align}</span></p>
|
1,012,652 | <p>I have a problem with the following question.</p>
<p>For which $n$ does the following equation have solutions in complex numbers</p>
<p>$$|z-(1+i)^n|=z $$</p>
<p>Progress so far.</p>
<ol>
<li><p>Let $z=a+bi$.</p></li>
<li><p>Since modulus represents a distance, the imaginary part of RHS has to be 0. This immedia... | inkievoyd | 93,500 | <p>$|z-(1+i)^n|=z$. LHS $\in \mathbb{R}$ so clearly $z \in \mathbb{R}$. I'll rewrite $z$ as $a$. $(1+i)^n =(\sqrt2)^ne^{\frac{in\pi}{4}}$. </p>
<p>Thus, RHS is the square root of $(a-(\sqrt2)^n\cos(\frac{n\pi}{4}))^2+((\sqrt2)^n\sin(\frac{n\pi}{4}))^2$ (it is getting cumbersome so I didn't include the square root).</p... |
2,029,279 | <blockquote>
<p>Let <span class="math-container">$T:\Bbb R^2\to \Bbb R^2$</span> be a linear transformation such that <span class="math-container">$T (1,1)=(9,2)$</span> and <span class="math-container">$T(2,-3)=(4,-1)$</span>.</p>
<p>A) Determine if the vectors <span class="math-container">$(1,1)$</span> and <span cla... | Community | -1 | <p>You can't scalair multiply $(1,1)$ to get $(2, -3)$, so the vectors are linear independent. So the span of these vectors are a basis for $\mathbb{R^2}$ (dimension is also ok).</p>
|
4,117,954 | <p>Let <span class="math-container">$l^\infty$</span> be the space of all real bounded sequences equipped with supremum norm. Let <span class="math-container">$S$</span> be the shift operator defined on <span class="math-container">$l^\infty$</span> by <span class="math-container">$(Sx)_n=x_{n+1}$</span>, <span class="... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$ L \in \widehat{(l^1)} $</span> then there exists <span class="math-container">$(a_n) \in l^{1}$</span> such that <span class="math-container">$\lim \inf x_n \leq L(x)=\sum a_nx_n$</span> for all <span class="math-container">$x=(x_n) \in l^{\infty}$</span>. Taking <span class="math-c... |
998,633 | <p>I'm having trouble to understand exactly <em>how we are using Fubini's theorem</em> in the following proof involving the distribution function, since it newer explicitly involves an integral with product measure.</p>
<p>The proof is given to show that we can calculate an integral over some measure space $X $ as an ... | Oliver Díaz | 121,671 | <p>The <span class="math-container">$\sigma$</span>-additivity assumption on <span class="math-container">$(X,\mathscr{F},\mu)$</span> can be relaxed. Here is a slightly more general result:</p>
<p><strong>Theorem:</strong> Suppose <span class="math-container">$\nu$</span> is a positive Radon measure on <span class="m... |
1,309,111 | <p>I am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space.</p>
<p>I am defining the a norm of $f(=(f_1,f_2)')$ as $\rho(f)=\int_0^{2\pi}(f^2_1(x)+f^2_2(x))^{\frac{1}{2}}dx$. </p>
<p>Could anyone please suggest me how ... | Lucia | 244,743 | <p>Well, I'm not sure if we could define it, but let's try to check if $L^2 $ is a vector space and $||.||_2$ is a norm for any $1\leq p \leq\infty$,
The cases $p=1$ and $p=\infty$ are clear, so let's check $1< p <\infty$ and let $f,g \in L^2$</p>
<p>we have $$|f(x)+g(x)| \leq (|f(x)|+|g(x)|)^p \leq 2^{p}(|f(x)|... |
546,083 | <p>Thanks to internet, I found and understand how to solve diophantine $x^2 - Dy^2 = 1$. Now I would like to solve the following diophantine equation :
$$x^2 - 2y^2 = x - 2y$$
but I don't know how to do it, even if I could read articles explaining how to solve the diophantine $x^2 - Dy^2 = c$.</p>
| newzad | 76,526 | <p>$i=e^{i(\pi/2+2k\pi)}$ therefore $i^i=(e^{i(\pi/2+2k\pi)})^i=e^{-\pi/2-2k\pi}$ </p>
<p>for $k=0$ you get $0.207879576350761908...$</p>
<p>for $k=1$ you get $0.000388203203926766...$</p>
<p>....</p>
<p>Or </p>
<p>$i=1\times i$, then $i^i=1^i\times i^i$: as tomasz commented, this post may be helpful: <a href="htt... |
1,139,885 | <p>If $a$ and $b$ are in a group $G$ and $ab=ba$, show that $xax^{-1}$ commutes with $xbx^{-1}$ for any $x \in G$.</p>
<p>So I wrote: </p>
<p>WWTS: $\bf{xax^{-1} \times xbx^{-1}=xbx^{-1}\times xax^{-1} }$</p>
<p>Now, the problem I have is I don't know where to start. Let's say if I start with what is given:</p>
<p>... | AlexR | 86,940 | <p>There's not much to it and you basically found it already:</p>
<p>$$\begin{align*}
(x a x^{-1}) (x b x^{-1}) & = x a \underbrace{(x^{-1} x)}_{=e}bx^{-1} \\
& = x (ab) x^{-1} \\
& = x(ba)x^{-1} \\
& = xb(x^{-1} x)a x^{-1}\\
& = (xbx^{-1})(xax^{-1})
\end{align*}$$</p>
<p>Note that the group opera... |
1,879,509 | <p>Please forgive the crudeness of this diagram.</p>
<p><a href="https://i.stack.imgur.com/AoS2Q.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AoS2Q.png" alt="enter image description here"></a></p>
<p>(I took an image from some psychobabble website and tried to delete the larger circle that's not relevant... | Doug M | 317,162 | <p>Draw two lines from the center of a circle, one to the center of the diagram, one to the center of the diagram, the other to a tip of the petal.</p>
<p>This sector has an angle of 90 degrees.
The area of the sector = $\frac 14$ the area of the cirlce or $\frac \pi4 r^2$
Now join the endpoints, creating an isosceles... |
1,879,509 | <p>Please forgive the crudeness of this diagram.</p>
<p><a href="https://i.stack.imgur.com/AoS2Q.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AoS2Q.png" alt="enter image description here"></a></p>
<p>(I took an image from some psychobabble website and tried to delete the larger circle that's not relevant... | Marcus Andrews | 97,648 | <p>Another approach:</p>
<p><a href="https://i.stack.imgur.com/cpNbg.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cpNbg.png" alt="enter image description here"></a> </p>
<p>Consider the square enclosed in red. It has side length $r$, and therefore area $r^2$. Notice that a quarter portion of two... |
710,374 | <p>Let $A$ be an $m\times n$ matrix. If the rank of $A$ is $m$, then prove there exists a matrix $B$, wich is $n \times m$, such that $AB=\text{I}_m$</p>
| mookid | 131,738 | <p><strong>Hint:</strong></p>
<p>try to prove that there are matrices $P,Q$ such as
$
A = P \bigl(\begin{smallmatrix}
I_m & 0
\end{smallmatrix} \bigr)Q^{-1}
$.</p>
|
3,588,068 | <p>I have the following system of 3 equations and 3 unknowns:
<span class="math-container">$$c_{0} = \frac{x_0}{x_0 + x_1},\ \ c_{1} = \frac{x_1}{x_1 + x_2},\ \ \ c_{2} = \frac{x_2}{x_2 + x_0},$$</span>
where all <span class="math-container">$c_i\!\in\!(0,1)$</span> are known and all <span class="math-container">$x_i &... | Community | -1 | <p>If <span class="math-container">$(x_0,x_1,x_2)$</span> is a solution, so is <span class="math-container">$\lambda(x_0,x_1,x_2)$</span>. So we may choose <span class="math-container">$x_0$</span> arbitrarily and draw</p>
<p><span class="math-container">$$x_1=\frac{1-c_0}{c_0}x_0,\\x_2=\frac{c_2}{1-c_2}x_0.$$</span><... |
200,063 | <p>I am looking to evaluate</p>
<p>$$\int_0^1 x \sinh (x) \ \mathrm{dx}$$</p>
| filmor | 7,247 | <p>You can also make use of the fact, that you can rewrite the integral:</p>
<p>$$\int_0^1 xf(x)\,\mathrm dx = \int_0^1 f(x) \int_0^x 1 \,\mathrm dy\,\mathrm dx.$$</p>
<p>This is just the integral over the right-angled triangle and using Fubini's theorem you arrive at
$$\int_0^1 1 \int_y^1 f(x) \,\mathrm dx \,\mathrm... |
216,055 | <p>Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra? </p>
| Carlo Beenakker | 11,260 | <p>Weyl algebra computations are implemented in
<A HREF="http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.8.2/share/doc/Macaulay2/Dmodules/html/" rel="noreferrer">Macaulay 2</A> (package D-modules). Here is a <A HREF="http://www.math.kobe-u.ac.jp/~taka/2007/knx/leykin-2002-m2.pdf" rel="noreferrer">manual.</A></p>
|
2,186,560 | <p>I was hoping someone can help me with the following.</p>
<p>I had to solve the following ODE and get the implicit form. </p>
<p>$$\frac{dy}{dx}=\frac{1+3y}{y} * x^2$$</p>
<p>this give by separating and integrating the following:
$1/9*(3y-ln(3y+1))=1/3*x^3 +c $
which can also be written as (c is an arbitrary cons... | Jan Eerland | 226,665 | <p>Well, we have that:</p>
<p>$$\text{y}'\left(x\right)=\frac{1+\text{n}\cdot\text{y}\left(x\right)}{\text{y}\left(x\right)}\cdot x^2\space\Longleftrightarrow\space\int\frac{\text{y}\left(x\right)\cdot \text{y}'\left(x\right)}{1+\text{n}\cdot\text{y}\left(x\right)}\space\text{d}x=\int x^2\space\text{d}x\tag1$$</p>
<p... |
14,858 | <p>Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, V)$.</p>
<ol>
<li>Why the degree of $G$ is 14? or in general, how to calculate the degree of a Plucker embedding?</li>
</ol>
<p>Let ${\... | Dmitri Panov | 943 | <p>To be able calculate the degree it is worth to read a bit of Griffiths-Harris about Grassmanians (chapter 1 section 5). To prove that $S$ is $K3$ one needs to caluclate the canonical bundle of $G$, use simple facts about Plucker embedding, use adjunction formula and finally the fact that a simply connected surface w... |
3,949,525 | <p>This question is from Ponnusamy and silvermann complex analysis and I am making some mistake in this question and unable to find it.</p>
<blockquote>
<p>Show that <span class="math-container">$\prod_{n=1}^{\infty} (1-z/n) e^{z/n +5z^2/n^2}$</span> is entire.</p>
</blockquote>
<p><span class="math-container">$1+ f_n ... | DanielWainfleet | 254,665 | <p>Hint.</p>
<p>Theorem: Suppose <span class="math-container">$C$</span> is an open subset of <span class="math-container">$\Bbb C$</span> and <span class="math-container">$(f_n)_n$</span> is a sequence of complex functions that are analytic on <span class="math-container">$C,$</span> such that <span class="math-contai... |
2,794,228 | <p>If we define a sequence $a_n$ where $n\in\mathbb{N}$ by $$a_n=\frac{1}{({2n\pi})^{\frac{1}{3}}}$$
Then how could one show the inequality $$a_n-a_{n+1}\leq Cn^{-\frac{4}{3}}$$</p>
<p>I have tried got to a point where $$a_n-a_{n+1}\leq(\frac{2\pi}{(2n\pi)(2(n+1)\pi)})^{1/3}$$</p>
| zhw. | 228,045 | <p>We can factor out the $(2\pi)^{-1/3}$ in each term. We're left with $n^{-1/3}-(n+1)^{-1/3}.$ By the MVT, this equals</p>
<p>$$(-1/3)c^{-4/3}\cdot (-1), \, \text { for some }c\in (n,n+1).$$</p>
<p>This is less than $(1/3)n^{-4/3}$ and we're done.</p>
|
691,927 | <p>Now before I begin, I know this question has been asked multiple times but all the answers but I had so many questions of my own that I figured I should make a new question as my thoughts are different than previous answers. </p>
<p>Now I will ask the question first then explain my thoughts and troubles :)</p>
<p>... | Jimmy R. | 128,037 | <p>Mathematically you can explain it as follows. You want to take $b$ as small as possible to maximize your derivative. Ok, but can you take $b$ smaller as the largest value you observed in the sample? No! That would be a contradiction to the fact that your sample comes from the (unkonwn) interval $[a,b]$ Could it be b... |
1,416,661 | <p>Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, measuring the resulting angle (divided by 2π) as a number from 0 to 1.</p>
<p>I am bit confused, when i look into the... | BruceET | 221,800 | <p>If you're using radians, then you have $U_1 \sim Unif(0, 2\pi).$
Then the density function must have height $1/2\pi$ over $(0, 2\pi).$</p>
<p>If you're using degrees, then you have $U_2 \sim Unif(0, 360),$
which has density $f_{U_2}(x) = 1/360,$ for $0 < x < 360\,$ (and $0$
elsewhere).</p>
<p>If you define $... |
549,014 | <p>If $y=\sin x$, then find the value of $$\frac{d^2(\cos^7 x)}{dy^2}$$ </p>
<p>I have no idea on how to proceed in this problem. Please help.</p>
| Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> notice that</p>
<p>$$ \cos x = \sqrt{1-y^2}$$</p>
|
2,484,027 | <p>I've been trying to solve this permutation problem. I know that it's been posted on this site before, but my question is about the specific approach I take to solving it.</p>
<p>Here's what I thought I'd do : I could first figure out the total number of permutation where AT LEAST $2$ girls are together, and then su... | N. F. Taussig | 173,070 | <p>The reason your answer is wrong is that your count of how many arrangements have at least two girls in consecutive seats is incorrect. When you subtracted those arrangements in which a pair of girls in consecutive seats, you subtracted those arrangements in which all the girls sit in consecutive seats twice, once w... |
427,596 | <p>Just a quick question. I'm trying to understand the answer to <a href="https://math.stackexchange.com/questions/424063/group-action-on-a-manifold-with-finitely-many-orbits">one of my previous questions</a>. The precise problem I want to show is as follows. </p>
<blockquote>
<p>Let $G$ be a group acting faithful... | Jim Belk | 1,726 | <p>This is not true. For example, $SO(3)$ acts faithfully on the $2$-sphere $S^2$, but $SO(3)$ has dimension $3$, and $S^2$ only has dimension $2$.</p>
|
3,244,365 | <blockquote>
<p>Prove that the product of any four consecutive integers is one less than a perfect square.</p>
</blockquote>
<p>My first idea is the let k be a member of the integers. Let <span class="math-container">$m$</span> which also belongs to the integers be equal to <span class="math-container">$k(k+1)(k+2)(... | Dr. Sonnhard Graubner | 175,066 | <p>Note that <span class="math-container">$$(x-2)(x-1)x(x+1)+1=(x^2-x-1)^2$$</span></p>
|
3,589,926 | <p>I wanted to come up with a proof using contrapositive, so <strong>I needed the negation of the following: "one of x and y is congruent to 1 modulo 6 while the other is congruent to 5 modulo 6."</strong></p>
<p>I interpreted this statement as being the same as "one of x and y is congruent to 1 modulo 6 and the other... | Albert | 331,549 | <p>I don't know Abbott's book. But I strongly recommend Apostol as a first book and Spivak as a second one. </p>
<p>Apostol's book begins from the very basics (the real numbers). He first presents integral calculus, even before the concepts of limits and continuous functions. But he takes a different (but equivalent) ... |
2,991,811 | <p>I think it converges to <span class="math-container">$\frac{\pi}{2}$</span>, but I am not sure how to prove it. I've tried using induction on <span class="math-container">$n$</span> but have had no luck. Any help would be appreciated thanks.</p>
| Doug M | 317,162 | <p>use that <span class="math-container">$|\sin x| < |x|$</span></p>
<p><span class="math-container">$\sin \frac{\pi}{2^{n+1}} < \frac{\pi}{2^{n+1}}$</span></p>
<p><span class="math-container">$0<2^{n}\sin \frac{\pi}{2^{n+1}} < \frac {\pi}{2}$</span></p>
<p>The sequence is bounded... can we show that it ... |
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