qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,673,452 | <p>Let $\{a_j\}_{j=1}^N$ be a finite set of positive real numbers. Suppose </p>
<p>$$\sum_{j=1}^{N} a_j = A,$$ prove</p>
<p>$$\sum_{j=1}^{N} \frac{1}{a_j} \geq \frac{N^2}{A}.$$ </p>
<p>Hints on how to proceed?</p>
| DanielWainfleet | 254,665 | <p>For positive $a,b$ we have $a/b+b/a= (\sqrt {a/b}-\sqrt {b/a})^2+2\geq 2 .$</p>
<p>Therefore $\sum_1^na_i \sum_1^n 1/a_i= \sum_1^n a_i(1/a_i)+\sum_{1\leq i<j\leq n}(a_i/a_j+a_j/a_i)=n+\sum_{1\leq i<j\leq n}(a_i/a_j+a_j/a_i)\geq n+\sum_{1\leq i<j\leq n}2=n+\binom {n}{2}2=n^2.$
The Cauchy-Schwarz Inequality... |
25,782 | <p>Hello I'm having trouble showing the following:</p>
<p>Let $u$ be a positive measure. If $\int_E f\, du= \int_E g\, du$ for all measurable $E$ then $f=g$ a.e.</p>
<p>I was trying to argue by contradiction: if $f\neq g$ a.e. then there must exist some set $E=\{x: f(x)\neq g(x)\}$ such that $u(E) \gt 0$. Then let $E... | Willie Wong | 1,543 | <p><strong>hint</strong>:</p>
<ol>
<li>The difference of two measurable functions is measurable</li>
<li>$(0,\infty)\subset\mathbb{R}$ is Borel, so for a measurable function $F$, the set on which it takes positive values is a measurable set. </li>
<li>The collection of measurable sets form a $\sigma$ algebra, and in p... |
2,590,068 | <p>$$\epsilon^\epsilon=?$$
Where $\epsilon^2=0$, $\epsilon\notin\mathbb R$.
There is a formula for exponentiation of dual numbers, namely:
$$(a+b\epsilon)^{c+d\epsilon}=a^c+\epsilon(bca^{c-1}+da^c\ln a)$$
However, this formula breaks down in multiple places for $\epsilon^\epsilon$, yielding many undefined expressions l... | zoli | 203,663 | <p><strong>To prove that $\epsilon^{\epsilon}$ does not exist</strong></p>
<p>If $\epsilon^{\epsilon}$ has a value in the set of dual numbers and if $$\epsilon^{\epsilon}\not=\epsilon$$ then $$\epsilon^{\epsilon}-\epsilon=a+b\epsilon$$ and either $a$ or $b$ is different from zero. From here $$\epsilon\left[\epsilon^{\... |
2,033,485 | <p>I have an equilateral triangle with each point being a known distance of N units from the center of the triangle.</p>
<p>What formula would I need to use to determine the length of any side of the triangle?</p>
| hamam_Abdallah | 369,188 | <p><strong>Hint</strong></p>
<p>let $L$ be the length of the triangle sides.</p>
<p>then</p>
<p>$$L^2=N^2+N^2-2N.N.\cos(\frac{2\pi}{3})$$</p>
<p>and</p>
<p>$$L=2N\sin(\frac{\pi}{3})=N\sqrt{3}$$</p>
|
2,033,485 | <p>I have an equilateral triangle with each point being a known distance of N units from the center of the triangle.</p>
<p>What formula would I need to use to determine the length of any side of the triangle?</p>
| Community | -1 | <p>If you label the triangle $ABC$ (from bottom left to right and top point $C$). Take the center to be $M$. Then consider the triangle formed by $AM$ and the middle of $AB:=Q$.
So now you have $AMQ$ with angle $AMQ = 90$ degrees, and angle $MAQ = 60$ degrees.
This means you have a $1:2:\sqrt{3}$ triangle with $|AM|=N$... |
2,674,853 | <blockquote>
<p>Suppose $E \subset \mathbb R^d$ has measure $0$ and $f: \mathbb R^d \longrightarrow \mathbb R$ is measurable. Does $f (E)$ necessarily have measure $0$?</p>
</blockquote>
<p>I tried to find a counter-example though I failed.It is clear that countable subset will not work for otherwise the image of it... | Angina Seng | 436,618 | <p>You can take the Cantor set in $\Bbb R$ (measure zero) and map it onto
the closed interval $[0,1]$ by a continuous increasing function. So, even
continuous functions on $\Bbb R$ don't preserve measure zero.</p>
|
34,521 | <p>I have been using the following result:</p>
<p>Given a polynomial $f(x,t)$ of degree $n$ in $\mathbb{Q}[x,t]$, if a rational specialization of $t$ results in a separable polynomial $g(x)$ of the same degree, then the Galois group of $g$ over $\mathbb{Q}$ is a subgroup of that of $f$ over $\mathbb{Q}(t)$.</p>
<p>Ho... | damiano | 4,344 | <p>You can find the first statement, for instance, in van der Waerden, Modern Algebra I, Section 61. In particular, under the inclusion of the Galois group of the reduction in the Galois group of the original polynomial, the cycle structure matches up.</p>
<p>For the statement when you drop the separability assertion,... |
34,521 | <p>I have been using the following result:</p>
<p>Given a polynomial $f(x,t)$ of degree $n$ in $\mathbb{Q}[x,t]$, if a rational specialization of $t$ results in a separable polynomial $g(x)$ of the same degree, then the Galois group of $g$ over $\mathbb{Q}$ is a subgroup of that of $f$ over $\mathbb{Q}(t)$.</p>
<p>Ho... | KConrad | 3,272 | <p>Here is a broader setup for your question. Let $A$ be a Dedekind domain with fraction field $F$, $E/F$ be a finite Galois extension, and $B$ be the integral closure of $A$ in $E$.
Pick a prime $\mathfrak p$ in $A$ and a prime $\mathfrak P$ in $B$ lying over $\mathfrak p$.
The decomposition group $D(\mathfrak P|\ma... |
3,582,585 | <p>Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the
die again and if any other number comes, toss a coin. Find the conditional probability of the event <strong>‘the coin shows a tail’</strong>, given that <em>‘at least one throw of die shows a 3’</em>.</p>
<p>I don't know how to deal w... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$$3^{15a}=5^{5b}=15^{3c}=k$$</span></p>
<p>What if <span class="math-container">$k=1$</span></p>
<p>Else</p>
<p><span class="math-container">$3=k^{1/15a}$</span> etc.</p>
<p>As <span class="math-container">$15=3\cdot5$</span></p>
<p><span class="math-container">$k^{1/3c}=k^{1/5b... |
3,582,585 | <p>Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the
die again and if any other number comes, toss a coin. Find the conditional probability of the event <strong>‘the coin shows a tail’</strong>, given that <em>‘at least one throw of die shows a 3’</em>.</p>
<p>I don't know how to deal w... | Z Ahmed | 671,540 | <p>Take <span class="math-container">$$3^{15a}=5^{5b}=15^{3c}=K \implies a=\frac{\log k}{15\log 3}, b=\frac{\log K}{5\log 5}, c=\frac{\log K}{3 \log 15} $$</span>
Then <span class="math-container">$$ F=5 ab-bc-3ac=abc(5/c-1/a-3/b)=anc\left(\frac {15 \log 15}{\log K}-\frac{15 \log 3}{\log K}-\frac{15 \log 5}{\log K}\rig... |
4,577,266 | <blockquote>
<p>Let <span class="math-container">$X_n$</span> be an infinite arithmetic sequence with positive integers term. The first term is divisible by the common difference of successive members. Suppose, the term <span class="math-container">$x_i$</span> has exactly <span class="math-container">$m>1$</span> ... | templatetypedef | 8,955 | <p>It might be easier to think of things this way. Start at <span class="math-container">$w$</span> and follow the edge from <span class="math-container">$w$</span> to <span class="math-container">$x$</span>. Now walk down path <span class="math-container">$P$</span> toward <span class="math-container">$y$</span>. Sinc... |
213,639 | <blockquote>
<p>Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom?</p>
</blockquote>
<p>This question was to some extent provoked by <a href="https://math.stackexchange.com/users/3515/dan-christensen">Dan Christensen</a>'s <a ... | Community | -1 | <p>Here's an (IMO) interesting example. Surely you're familiar with the set-theoretic interpretation of ordered pairs given by $(x,y) = \{ \{ x \}, \{ x, y \} \}$. You may wonder, why not use $(x,y) = \{ x, \{ x, y\}\}$?</p>
<p>We can... <em>if</em> we assume some amount of regularity. If there exists a set $S$ satisf... |
3,742,189 | <p>in a game I play there's a chance to get a good item with 1/1000.
After 3200 runs I only got 1.</p>
<p>So how can I calculate how likely that is and I remember there are graphs which have 1 sigma and 2 sigma as vertical lines and you can tell what you can expect with 90% and 95% sureness.</p>
<p>Sorry if that's aske... | Siong Thye Goh | 306,553 | <p>From <span class="math-container">$5+3x < 14$</span> and <span class="math-container">$-x<1$</span>, by adding them up, you have conclude those inequalities imply that <span class="math-container">$x<5$</span>. <span class="math-container">$x<5$</span> is necessarily true. But from <span class="math-con... |
3,726,772 | <p>For finite-dimensional vector space <span class="math-container">$V$</span>, there exist linear operators <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on <span class="math-container">$V$</span> such that <span class="math-container">$AB=BA$</span> commutative relation holds.</p... | Ben Grossmann | 81,360 | <p>Note that <span class="math-container">$\deg(A)$</span> is the dimension of the subspace consisting of all polynomials of <span class="math-container">$A$</span>.</p>
<p>Let <span class="math-container">$m = \deg(A), n = \deg(B)$</span>. Every polynomial of <span class="math-container">$p(A)$</span> can be written a... |
844,832 | <p>How to find the derivative of this function $$ 7\sinh(\ln t)?$$</p>
<p>I don't know from where to start, so i looked at it in wolfram alpha and it was saying that the $$ 7((-1 + t^2) / 2t) $$ I did not get that. How did they jump from $$ 7\sinh(\ln t) $$ to this step? Is there an equation for it that I am missing?<... | amWhy | 9,003 | <p>As Jean-Claude has shown you:</p>
<p>$$f(x) = \sinh (\ln t)=\frac{e^{\ln t}-e^{-\ln t}}{2}=\frac{t-\frac{1}{t}}{2}=\frac{t^2-1}{2t}$$</p>
<p>So $$7\sinh (\ln t)=\frac{e^{\ln t}-e^{-\ln t}}{2}=\frac{t-\frac{1}{t}}{2}=\frac{7(t^2-1)}{2t}$$</p>
<p>So Wolfram has not returned the value of the <em>derivative</em> of $... |
3,269,112 | <p>Theorem:</p>
<p><span class="math-container">$ x \lt y + \epsilon$</span> for all <span class="math-container">$\epsilon \gt 0$</span> if and only if <span class="math-container">$x \leq y$</span></p>
<p>Suppose to the contrary that <span class="math-container">$x \lt y + \epsilon$</span> but <span class="math-con... | Jose Brox | 146,587 | <p>We have that if <span class="math-container">$G/Z(G)$</span> is cyclic then <span class="math-container">$G$</span> is abelian. But since <span class="math-container">$G$</span> abelian means <span class="math-container">$G=Z(G)$</span>, this forces <span class="math-container">$|G/Z(G)|=1$</span>. Now, if <span cla... |
208,008 | <p>Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ in $C$. Also, let $\{C_{11},C_{12}\}$ and $\{C_{21},C_{22}\}$ be the sets of two arbitrary tangent circles with radiu... | Joseph O'Rourke | 6,094 | <p>This is <em>not</em> a answer, just an animation illustrating the challenge.
The blue circle $C$ contains $9$ points, and the red circles $C_1$ and $C_2$ each contain
$2$ points, except at four discrete times
(times corresponding to rotations that are multiples of $\pi/4$)
when they contain just $1$ each (e.g., at ... |
3,265,403 | <p>While trying to compute the line integral along a path K on a function, I need to parametrize my path K in terms of a single variable, let's say this single variable will be <span class="math-container">$t$</span>.
My path is defined by the following ensemble: <span class="math-container">$$K=\{(x,y)\in(0,\infty)\ti... | A.G. | 115,996 | <p>Notice that <span class="math-container">$x^2-y^2=(x+y)(x-y)=1600$</span>, therefore you are dealing with a hyperbola with asymptotes <span class="math-container">$x+y=0$</span> and <span class="math-container">$x-y=0$</span> as shown here:</p>
<p><a href="https://i.stack.imgur.com/B5Ebv.png" rel="nofollow noreferr... |
1,711,653 | <p>Let's define:</p>
<p>$f(t) = A_1 \cos(\omega_1t) + A_2 \cos(\omega_2t) $</p>
<p>I am interested in finding an expression for the peak of this function. It is not true in general that this peak will have the value:</p>
<p>$max{f(t)} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2}$</p>
<p>To find the value of max(f), I did the f... | Jean Marie | 305,862 | <p>If the ratio $r=\dfrac{\omega_1}{\omega_2}$ is not rational, function $f$ can reach values arbitrarily close to $|A_1|+|A_2|$.</p>
<p>A graphical example in the case $f(t)=2 \cos(1.5t)-3 \cos(\sqrt{2}t)$ with values arbitrarily close to $|2|+|-3|=5$.</p>
<p>A sketch of proof: As remarked by Yves Daoust, the case $... |
3,245,270 | <p>From Statistical Inference by Casella and Berger:</p>
<blockquote>
<p>Let <span class="math-container">$X_1, \dots X_n$</span> be a random sample from a discrete distribution
with <span class="math-container">$f_X(x_i) = p_i$</span>, where <span class="math-container">$x_1 \lt x_2 \lt \dots$</span> are the pos... | jgon | 90,543 | <p>The <a href="https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space" rel="nofollow noreferrer">Sierpinski space</a> is not pseudometrizable. The <a href="https://en.wikipedia.org/wiki/Kolmogorov_space#The_Kolmogorov_quotient" rel="nofollow noreferrer">Kolmogorov quotient</a> of a pseudometric space <a href="https://en.... |
473,874 | <p>Three people consider as A,B,C went for sight seeing.
<br/>
A,B and C each individually saw a bird that no other saw.(Eg: If A saw a bird the same is not seen by B and C)
<br/>
Each pair saw a yellow bird that the other pair dint see(Eg: If AB saw a bird the same is not seen by BC and CA)
<br/>
3 people together saw... | Hagen von Eitzen | 39,174 | <p>If I don't overlook any subtlety in the problem, in the minimal case, each bird determines a nonempty subset of $\{A,B,C\}$ (its obervers), hence there are at least $7$ birds, and at least $4$ of these are yellow.</p>
|
473,874 | <p>Three people consider as A,B,C went for sight seeing.
<br/>
A,B and C each individually saw a bird that no other saw.(Eg: If A saw a bird the same is not seen by B and C)
<br/>
Each pair saw a yellow bird that the other pair dint see(Eg: If AB saw a bird the same is not seen by BC and CA)
<br/>
3 people together saw... | Willemien | 88,985 | <p>I disagree I think only 6 birds are needed<br>
A sees bird A not seen by B & C<br>
B sees bird B not seen by A & C<br>
C sees bird C not seen by A & B<br>
A and B see yellow bird AB not seen by C<br>
A and C see yellow bird AC not seen by B<br>
B and C see yellow bird AC not seen by A</p>
<p>A, B and C... |
422,233 | <p>I was asked to find a minimal polynomial of $$\alpha = \frac{3\sqrt{5} - 2\sqrt{7} + \sqrt{35}}{1 - \sqrt{5} + \sqrt{7}}$$ over <strong>Q</strong>.</p>
<p>I'm not able to find it without the help of WolframAlpha, which says that the minimal polynomial of $\alpha$ is $$19x^4 - 156x^3 - 280x^2 + 2312x + 3596.$$ (True... | GregHL | 364,116 | <p>I know nothing about Galois Theory, and haven't really thought about this one, and am replying quickly (and am not in shape) so I might be saying stupid things, but I guess we could use the fact that $\alpha \in \mathbb{Q}(\sqrt{5},\sqrt{7})$, and then define $x \mapsto \bar{x}$ as sending -for example- $\sqrt{7}$ t... |
666,217 | <p>If $a^2+b^2 \le 2$ then show that $a+b \le2$</p>
<p>I tried to transform the first inequality to $(a+b)^2\le 2+2ab$ then $\frac{a+b}{2} \le \sqrt{1+ab}$ and I thought about applying $AM-GM$ here but without result</p>
| JLA | 30,952 | <p>Let $a=\sqrt{2}\cos\theta$, $y=\sqrt{2}\sin\theta$. Then $a^2+b^2=2$, and $a+b=\sqrt{2}(\cos\theta+\sin\theta)$, which is a maximum at $\theta=\frac{\pi}{4}$, at which case $a+b=2$. So $a+b\le 2$.</p>
|
666,217 | <p>If $a^2+b^2 \le 2$ then show that $a+b \le2$</p>
<p>I tried to transform the first inequality to $(a+b)^2\le 2+2ab$ then $\frac{a+b}{2} \le \sqrt{1+ab}$ and I thought about applying $AM-GM$ here but without result</p>
| Steven Stadnicki | 785 | <p>Yet another method, inspired by looking at the problem geometrically (try drawing the region $a^2+b^2\leq2$ and the line $a+b=2$): let $s=a+b$, $t=a-b$. Then $a=\frac12(s+t)$ and $b=\frac12(s-t)$, so $a^2+b^2=\frac14\bigl((s^2+2st+t^2)+(s^2-2st+t^2)\bigr) = \frac12(s^2+t^2)$ and $a+b=s$, so the problem becomes: </p... |
1,530,702 | <p>Can anybody help me out with getting an expression of the values of $\lambda$ for a matrix $A$ for which $det(A-\lambda I)$ equals the determinant of a matrix with on the main diagonal $-\lambda$, on the diagonal above the main diagonal $\dfrac{1}{2}$ and on the diagonal under the main diagonal $\frac{1}{2} \lambda$... | Bernard | 202,857 | <p>The determinant of such tridiagonal matrices of order <span class="math-container">$n$</span> are computed with the linear recurrence of order <span class="math-container">$2$</span>:
<span class="math-container">$$D_n=-\lambda D_{n-1}-\frac\lambda4 D_{n-2}$$</span>
and the initial conditions <span class="math-conta... |
1,787,806 | <p>I've recently had this problem in an exam and couldn't solve it.</p>
<p>Find the remainder of the following sum when dividing by 7 and determine if the quotient is even or odd:</p>
<p>$$\sum_{i=0}^{99} 2^{i^2}$$</p>
<p>I know the basic modular arithmetic properties but this escapes my capabilities. In our algebra... | Gareth McCaughan | 8,474 | <p>$2^{i^2}$ mod 7 is determined by $i^2$ mod 6, hence by $i$ mod 6. That should suffice to sort out the remainder. Now the parity of the quotient depends only on the remainder of the sum mod 14, which you can get from knowing the sum mod 7 and mod 2.</p>
|
159,761 | <p>I have two lists:</p>
<pre><code>list1 = {"a", "b"};
list2 = {{{1, 2}, {3, 4}}, {{1, 2}}};
</code></pre>
<p>My goal is to create a new list which would be:</p>
<pre><code>{"a u 1:2","a u 2:3","b u 1:2"}
</code></pre>
<p>In other words first element in <code>list1</code> would be distributed before each subelemen... | WReach | 142 | <p>The first <code>Map</code> attempt listed in the question gets us close -- a tweak involving <code>MapIndexed</code> can get us all the way:</p>
<pre><code>MapIndexed[StringRiffle[{list1[[#2[[1]]]], #}, " u ", ":"] &, list2, {2}] // Flatten
(* {"a u 1:2", "a u 3:4", "b u 1:2"} *)
</code></pre>
<p>Another way ... |
2,637,914 | <p>I would like to teach students about the pertinence of the Axiom of Infinity. Are there any high school-level theorems of arithmetic, algebra, or calculus, whose proof depends on the Axiom of Infinity? If there are no such examples, what would be the simplest theorem which demands the Axiom of Infinity?</p>
<p>It... | Tsemo Aristide | 280,301 | <p>Suppose that $\sum_{i=1}^{i=n}{i\over{i+1}}\leq {n^2\over{n+1}}$, </p>
<p>$\sum_{i=1}^{i=n+1}{i\over{i+1}}\leq {n^2\over{n+1}}+{{n+1}\over{n+2}}$.</p>
<p>${n^2\over{n+1}}+{{n+1}\over{n+2}}=$</p>
<p>${{n^2(n+2)+(n+1)^2}\over{{(n+1)(n+2)}}}$</p>
<p>$\leq {{n(n^2+2n+1)+(n+1)^2}\over{{(n+1)(n+2)}}}$</p>
<p>$={{n(n+... |
1,077,594 | <p>Let $C[a,b]$ be the space of continuous functions on $[a,b]$ with the norm
$$
\left\Vert{f}\right\Vert=\max_{a \leq t \leq b}\left| f(t)\right|
$$</p>
<p>Then $C[a,b]$ is a Banach space. </p>
<p>Let's view $C^1[a,b]$ as a subspace of it. My question is, is this $C^1[a,b]$ a Banach space?</p>
<p>I think it is, sin... | sabachir | 201,840 | <p>we use $$f\left( z \right) = \frac{{e^{iz} }}{{1 + z^2 }}$$
then take real parts of the resulting integral .using the same contour $C$ </p>
<h2><img src="https://i.stack.imgur.com/NMBBI.jpg" alt="enter image description here"></h2>
<p>\begin{array}{l}
i \in C; - i \notin C \\
{\mathop{\rm Re}\nolimits} s\left(... |
1,219,129 | <p>For any vector space $V$ over $\mathbb{C}$, let $X$ be a set whose cardinality is the dimension of $V$. Then $V \cong \bigoplus\limits_{i \in X} \mathbb{C}$ as vector spaces.</p>
<p>Is there a similar description of arbitrary Hilbert spaces? Is there something they all "look" like?</p>
| TZakrevskiy | 77,314 | <p>On the one hand, you can apply the same thing as you did with a generic vectors space and obtain that $$H \cong \bigoplus\limits_{i \in X} \mathbb{C}.$$
Notes that this relies on the notion of Hamel dimension of the space of a vector space.</p>
<p>However, you can say that a Hilbert space has its Hilbert dimension ... |
2,115,532 | <blockquote>
<p>Let $\mu$ be a $\sigma$-finite measure on $(A,\mathcal{A})$. Then
there are finite measures $(\mu_n)_{n \in \mathbb{N}}$ on
$(X,\mathcal{A})$ such that $$\mu = \sum_{n \in \mathbb{N}}\mu_n$$</p>
</blockquote>
<p>So if $\mu$ is $\sigma$-finite, we have that $$X = \bigcup_{n \in \mathbb{N}}X_n$$ fo... | C. Dubussy | 310,801 | <p>Since $\theta \in [0,2\pi]\mapsto e^{i\theta}$ is a parametrization of the circle, one has $$\int_{0}^{2\pi} e^{e^{i\theta}}d\theta = \int_{C(0,1)} \frac{e^z}{iz} dz = 2\pi$$ by Cauchy's theorem.</p>
|
1,553,354 | <p>Help me to find an example of a sequence of differentiable functions defined on $[0,1]$ that converge uniformly to a function $f$ on $[0,1]$ such that there exists $x \in (0,1)$ such that $f$ is not differentiable at $x$.</p>
| Prahlad Vaidyanathan | 89,789 | <p>Any continuous function is a uniform limit of polynomials, so pick your favourite non-differentiable continuous function, and that would work!</p>
<p>In fact, if $f$ is such a function, say $f(x) := |x- 0.5|$, then take the sequence to be the sequence of Bernstein polynomials
$$
B_n(f)(x) := \sum_{k=1}^n {n\choose ... |
117,608 | <p>We know that if $G$ is a simple group with $p+1$ Sylow $p$-subgroups, then $G$ is 2-transitive. Now let $G$ be almost simple group with $p+1$ Sylow $p$-subgroups. Is $G$ 2-transitive group?</p>
| Derek Holt | 35,840 | <p>I am sure that the answer is yes, but you might have to do a bit of work to write down a completely rigorous proof.</p>
<p>Let $S \unlhd G$ with $S$ simple and $G \le {\rm Aut}(S)$, and suppose that $G$ has $p+1$ Sylow $p$-subgroups. If $p$ divides $|S|$, then $S$ has at most $p+1$ and hence exactly $p+1$ Sylow $p$... |
187,974 | <p>If $ \cot a + \frac 1 {\cot a} = 1 $, then what is $ \cot^2 a + \frac 1{\cot^2 a}$? </p>
<p>the answer is given as $-1$ in my book, but how do you arrive at this conclusion?</p>
| Jyrki Lahtonen | 11,619 | <p>Hint:
$$
x^2+\frac1{x^2}=\left(x+\frac1x\right)^2-2.
$$</p>
|
187,974 | <p>If $ \cot a + \frac 1 {\cot a} = 1 $, then what is $ \cot^2 a + \frac 1{\cot^2 a}$? </p>
<p>the answer is given as $-1$ in my book, but how do you arrive at this conclusion?</p>
| Apratim Ran Chak | 299,618 | <p>cota+ (1/cota)=1</p>
<p>Therefore,
Squaring on both sides we get:</p>
<pre><code> cot^2a + (1/cot^2a)+ 2 = 1
Hence,
cot^2a + (1/cot^2a) = -1
</code></pre>
|
2,461,615 | <p>I am still at college. I need to solve this problem.</p>
<p>The total amount to receive in 1 year is 17500 CAD.
And the university pays its students each 2 weeks (26 payments per year). </p>
<p>How much does a student have to receive for 4 months?
I have calculated this in 2 ways (both seem ok) but results are di... | Guillemus Callelus | 361,108 | <p>If you consider that a month has $4$ weeks, you have a total of $48$ weeks, not $52$.</p>
|
2,216,601 | <p>Alright so I have this Transformation that I know isn't one to one transformation, but I'm not sure why. </p>
<p>A Transformation is defined as $f(x,y)=(x+y, 2x+2y)$.</p>
<p>Now my knowledge is that you need to fulfill the 2 conditions: Additivity and the scalar multiplication one. I tried both of them and somehow... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>$$f (1,-1)=(0,0) $$ and</p>
<p>$$f (0,0)=(0,0) $$</p>
|
1,943,328 | <p>I know about $S_n$, $D_n$ and $A_n$. And from my limited understanding there seem to be many more. I would like to know whether there is some kind of relation that links a small set of non Abelian groups to create the other ones. Something like with the Abelian groups and the Fundamental Theorem of Abelian Groups.</... | P Vanchinathan | 28,915 | <p>There is a construction called semi-direct product. It is somewhat like direct product with a twist. This creates a non-abelian group even if the two factors were abelian. And there is a generalisation called group extensions which creates more non-abelian groups.</p>
<p>It is difficult to classify them. Because d... |
1,983,129 | <p>In the tripple integral to calculate the volume of a sphere
why does setting the limits as follows not work?</p>
<p>$$ \int_{0}^{2\pi} \int_{0}^{\pi}
\int_{0}^{R} p^2 \sin{\phi}
\, dp\,d\theta\,d\phi $$</p>
| GEdgar | 442 | <p>Geometrically...</p>
<p>The center of gravity of the set of vertices of the polygon $\{1, \epsilon, \cdots, \epsilon^{n-1}\}$ is the center of that polygon. Proof: the polygon is invariant under rotation by $\epsilon$ about the center, so the center of gravity is also inveriant under that rotation.</p>
|
2,046,521 | <p>Of course, faster calculations help solve problems quickly. But does that also mean that faster calculations open more opportunities for a career in mathematics (like a researcher)? I like mathematics and can spend weeks trying to solve any problem or understanding any concept. But nowadays, there are many contests ... | SchrodingersCat | 278,967 | <p>This is a vague concept. True mathematics does not deal with numerical calculations, let alone faster calculations. Science has devised machines called calculators to perform this task of faster calculations. What real mathematics deals with are <em>concepts</em>, mathematical subtleties, reasoning, logic and new li... |
2,046,521 | <p>Of course, faster calculations help solve problems quickly. But does that also mean that faster calculations open more opportunities for a career in mathematics (like a researcher)? I like mathematics and can spend weeks trying to solve any problem or understanding any concept. But nowadays, there are many contests ... | Christopher.L | 347,503 | <p>In general I would say that problem solving skills are more important than being 'skilled' in arithmetic. The contests you mention, often test both problem solving and speed as well.</p>
<p>However, being fast at doing calculations often means that you understand theorems and proofs faster, and thus tend to get stu... |
557,543 | <p>Does there exists a positive decreasing sequence $\{a_i\}$ with $\sum_{i\in\mathbb{N}} a_i$ convergent, such that $\forall I\subset\mathbb{N},\sum_{i\in I}a_i$ is an irrational number?</p>
<p>Such an example would give rise to a <strong>closed perfect set containing no rationals</strong>. I can only do it for infin... | Andrés E. Caicedo | 462 | <p>Another easy-to-describe example of a perfect set of irrationals consists of the set of all $x\in(0,1)$ whose continued fraction has the form $$\cfrac1{a_0+\cfrac1{a_1+\cfrac1{a_2+\cfrac1{\dots}}}},$$ where each $a_i$ is either $1$ or $2$. In fact, the set of irrationals in $(0,1)$ is precisely the set of numbers wh... |
182,785 | <p>I haved plot a graph from two functions:</p>
<pre><code>η = 52;
h = 0.5682;
dpdx = -4.092*10^(-2);
Fg = dpdx;
Fl = dpdx/η;
Bl = ((Fg - Fl) h^2 - Fg)/(2 h - 2 η*h + 2 η);
Cg = -Fg/2 - η*Bl;
Bg = η*Bl;
Ut1[y_] := Fg*y^2/2 + Bg*y + Cg;
Ut2[y_] := Fl*y^2/2 + Bl*y;
Plot1 = Plot[Ut1[y]*1000, {y, h, 1}];
Plot2 = Plot[U... | kglr | 125 | <p>You can also post-process the <code>Show</code> output using <a href="https://reference.wolfram.com/language/ref/RotationTransform.html" rel="nofollow noreferrer"><code>RotationTransform</code></a> and <a href="https://reference.wolfram.com/language/ref/ReflectionTransform.html" rel="nofollow noreferrer"><code>Refle... |
2,801,433 | <p>I have made the following conjecture, and I do not know if this is true.</p>
<blockquote>
<blockquote>
<p><strong>Conjecture:</strong></p>
</blockquote>
<p><span class="math-container">\begin{equation*}\sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2\verb| such that we denote by | p_n\verb| ... | marty cohen | 13,079 | <p>If $a > 1$
Then,
since
$p_n \sim n \ln n$
and
$1/a^{1/n}
=e^{-\ln a/n}
\sim 1-\ln a/n
$,
$\sum_{k=1}^n 1/(a^{1/k}p_k)
\sim \sum_{k=1}^n 1/p_k-\sum_{k=1}^n\ln a/(kp_k)
\sim \ln. \ln n-c
$
for some $c$
since the second sum converges.</p>
<p>Therefore the sum diverges like
$\ln \ln n$.</p>
|
3,355,270 | <p>I know that</p>
<p><span class="math-container">$$\int \frac{1}{x}dx = \ln |x| + C$$</span></p>
<p>however I have seen differential equation notes and solutions claim that the integrating factor for <span class="math-container">$P(x)=-\frac{1}{x}$</span> is</p>
<p><span class="math-container">$$\mu(x)=e^{\int P(x... | Arthur | 15,500 | <p>The differential equation breaks down at <span class="math-container">$x=0$</span>, so what happens for negative <span class="math-container">$x$</span> is something we cannot tell from the given information. We only care about positive <span class="math-container">$x$</span> because we <em>can</em> only care about ... |
3,355,270 | <p>I know that</p>
<p><span class="math-container">$$\int \frac{1}{x}dx = \ln |x| + C$$</span></p>
<p>however I have seen differential equation notes and solutions claim that the integrating factor for <span class="math-container">$P(x)=-\frac{1}{x}$</span> is</p>
<p><span class="math-container">$$\mu(x)=e^{\int P(x... | Oscar Lanzi | 248,217 | <p>We can get the integrating factor without logarithms. Use the quotient rule for differentiation:</p>
<p><span class="math-container">$\dfrac{(u/v)}{dx}=\dfrac{v(du/dx)-u(dv/dx)}{v^2}$</span></p>
<p>Put <span class="math-container">$u=y, v=x$</span> to get</p>
<p><span class="math-container">$\dfrac{d(y/x)}{dx}=\... |
798,897 | <p>In our lecture we ran out of time, so our prof told us a few properties about measure: He said that a measure is $\sigma$-additive iff it has a right-side continuous function that it creates. And he was not only referring to probability measures.
After going through my lecture notes, I thought that this would imply... | Community | -1 | <p>There are many other measures. For example, the counting measure: $\mu(A)$ is the number of elements of $A$, with $\mu(A)=\infty$ if $A$ is infinite. This is not a Lebesgue-Stieltjes measure. Neither are the <a href="http://en.wikipedia.org/wiki/Hausdorff_measure" rel="nofollow noreferrer">Hausdorff measures</a> $\m... |
3,839,878 | <p>Am currently doing a question that asks about the relationship between a quadratic and its discriminant.</p>
<p>If we know that the quadratic <span class="math-container">$ax^2+bx+c$</span> is a perfect square, then can we say anything about the discriminant?</p>
<p>Specifically, can we be sure that the discriminant... | The Chaz 2.0 | 7,850 | <p>By hypothesis, the quadratic is a perfect square if it is <em>something</em> (linear) squared, say <span class="math-container">$$(ux + v)^2 = u^2x^2 + 2uvx + v^2$$</span></p>
<p>Then the discriminant <span class="math-container">$b^2 - 4ac = (2uv)^2 - 4(u^2)(v^2) = 0$</span></p>
|
3,839,878 | <p>Am currently doing a question that asks about the relationship between a quadratic and its discriminant.</p>
<p>If we know that the quadratic <span class="math-container">$ax^2+bx+c$</span> is a perfect square, then can we say anything about the discriminant?</p>
<p>Specifically, can we be sure that the discriminant... | tomi | 215,986 | <p>A perfect square takes the form <span class="math-container">$(px+q)^2$</span></p>
<p>By expanding the brackets this can be shown to be equal to <span class="math-container">$p^2x^2+2pqx+q^2$</span></p>
<p>You want <span class="math-container">$ax^2+bx+c \equiv p^2x^2+2pqx+q^2$</span></p>
<p>Comparing coefficients o... |
977,232 | <p>We have in and out degree of a directed graph G. if G does not includes loop (edge from one vertex to itself) and does not include multiple edge (from each vertex to another vertex at most one directed edge), we want to check for how many of the following we have a corresponding graph. the vertex number start from 1... | user140161 | 140,161 | <p>You are right. For $(t^2, -3t, (6-t)^{1/2}$ to be parallel to $(2,-3,1)$ it must be $p$ times $(2,-3,1)$, where $p$ is a scalar multiple.
Set up 2 equation by equating any two of the $i$ $j$ and $k$ components and solving them simultaneously for $p$ and $k$. When you have obtained these values, plug back into the or... |
1,136,278 | <p>Prove that $n(n-1)<3^n$ for all $n≥2$. By induction.
What I did: </p>
<p>Step 1- Base case:
Keep n=2</p>
<p>$2(2-1)<3^2$</p>
<p>$2<9$ Thus it holds.</p>
<p>Step 2- Hypothesis: </p>
<p>Assume: $k(k-1)<3^k$</p>
<p>Step 3- Induction:
We wish to prove that:</p>
<p>$(k+1)(k)$<$3^k.3^1$</p>
<p>We ... | rafforaffo | 91,488 | <p>I think that your solution is fine. However, I would phrase it slightly different.</p>
<p>Step-2. To be completely formal, I would say: Let $k>2$ and assume $k(k-1)<3^k$.</p>
<p>Step 3. We need to show $k(k+1)<3^{k+1}$. We have $$k(k+1)=k(k-1)+2k<3^k+2k<3^k+3^k+3^k=3^{k+1}$$
Where we have used the i... |
1,677,359 | <p>$\sum_{i=0}^n 2^i = 2^{n+1} - 1$</p>
<p>I can't seem to find the proof of this. I think it has something to do with combinations and Pascal's triangle. Could someone show me the proof? Thanks</p>
| GoodDeeds | 307,825 | <p>Let
$$\tag1S=1+2+2^2+\cdots+2^n$$
Multiplying both sides by $2$,
$$\tag22S=2+2^2+2^3+\cdots+2^{n+1}$$
Subtracting $(1)$ from $(2)$,
$$S=2^{n+1}-1$$</p>
<p>This is a specific example of the sum of a geometric series. In general,
$$a+ar+ar^2+\cdots+ar^n=a\left(\frac{r^{n+1}-1}{r-1}\right)$$</p>
|
1,677,359 | <p>$\sum_{i=0}^n 2^i = 2^{n+1} - 1$</p>
<p>I can't seem to find the proof of this. I think it has something to do with combinations and Pascal's triangle. Could someone show me the proof? Thanks</p>
| copper.hat | 27,978 | <p>\begin{eqnarray}
2^{n+1} &=& 2^n+2^n \\
&=&2^n + 2^{n-1} + 2^{n-1} \\
&\vdots& \\
&=& 2^n + 2^{n-1} +2^{n-1} + \cdots + 2 +2 \\
&=& 2^n + 2^{n-1} +2^{n-1} + \cdots + 2 +1 + 1
\end{eqnarray}</p>
|
1,677,359 | <p>$\sum_{i=0}^n 2^i = 2^{n+1} - 1$</p>
<p>I can't seem to find the proof of this. I think it has something to do with combinations and Pascal's triangle. Could someone show me the proof? Thanks</p>
| choco_addicted | 310,026 | <p>Mathematical induction will also help you.</p>
<ul>
<li>(Base step) When $n=0$, $\sum_{i=0}^0 2^i = 2^0 = 1= 2^{0+1}-1$.</li>
<li>(Induction step) Suppose that there exists $n$ such that $\sum_{i=0}^n 2^i = 2^{n+1}-1$. Then $\sum_{i=0}^{n+1}2^i=\sum_{i=0}^n 2^i + 2^{n+1}= (2^{n+1}-1)+2^{n+1}=2^{n+2}-1.$</li>
</ul>
... |
745,613 | <p>I've been pondering this since yesterday. I</p>
<blockquote>
<p>Is it true that given two irreducible polynomials <span class="math-container">$f(x)$</span> and <span class="math-container">$ g(x)$</span> will <span class="math-container">$f(g(x))$</span> or <span class="math-container">$g(f(x))$</span> be irreducib... | voldemort | 118,052 | <p>This need not be true. Note that in <span class="math-container">$\mathbb{R}[x]$</span> any irreducible polynomial has degree either <span class="math-container">$1$</span> or <span class="math-container">$2$</span>. So, you can take two irreducible polynomials of degree 2, and compose them to get a reducible polyno... |
745,613 | <p>I've been pondering this since yesterday. I</p>
<blockquote>
<p>Is it true that given two irreducible polynomials <span class="math-container">$f(x)$</span> and <span class="math-container">$ g(x)$</span> will <span class="math-container">$f(g(x))$</span> or <span class="math-container">$g(f(x))$</span> be irreducib... | Pipicito | 93,689 | <p>I don't know a good answer to this question. But here are a couple of simple ideas for easy cases.</p>
<p>1) If $\mathbb{K}$ is a field then for any two irreducible polynomials $f$ and $g$ in $\mathbb{K}[x]$ such that $g$ has degree one we have that $f \circ g$ is irreducible. To see this, write $g=ax+b$ with $a\ne... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Gerald Edgar | 127 | <p>Fatou's Lemma states: for nonnegative measurable functions $f_n$,
$$
\int_E \liminf_{n\to\infty} f_n\;d\mu
\le
\liminf_{n \to \infty}\int_E f_n\;d\mu
$$
The mnemonic is
$$
\text{ILLLLLI},
$$
meaning "the Integral of the Lower Limit is Less than the Lower Limit of the Integral".</p>
|
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | Nick C | 470 | <p>Recently, a student in my beginning algebra course offered the following to the class, regarding signed number multiplication:</p>
<p>Assuming positivity is like <em>love</em>, and negativity is like <em>hate</em>, then...</p>
<ul>
<li>"If you love love, that's love." $\Rightarrow$ <em>positive</em> $\times$ <em>p... |
14,007 | <p>A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). With an upper bound of 35 students a class, there is potential for a large amount of grading that needs to happen wi... | ZirJohn | 9,865 | <p>For the 4 quadrants of a Cartesian graph I say "All Students Take Calculus" counterclockwise (in order) to remember which trig fxns are positive in which quadrants.</p>
|
1,451,745 | <p>Can someone check my logic here. </p>
<p><strong>Question:</strong> How many ways are there to choose a an $k$ person committee from a group of $n$ people? </p>
<p><strong>Answer 1:</strong> there are ${n \choose k}$ ways. </p>
<p><strong>Answer 2:</strong> condition on eligibility. Assume the creator of the comm... | dcstup | 126,393 | <p>Usually the logic is:</p>
<p>If $A$ is in the committee, the problem reduces to choosing $k-1$ people from $n-1$ people.</p>
<p>If $A$ is not in the committee, the problem reduces to choosing $k$ people from $n-1$ people, which we keep dividing:</p>
<p>$\,$ If $B$ is in the committee, the problem reduces to choos... |
4,008,488 | <p>While looking for the answer for my question I came across <a href="https://math.stackexchange.com/questions/1840801/why-is-ata-invertible-if-a-has-independent-columns?rq=1">this</a> post. It may be a silly idea, but if <span class="math-container">$A^{t}$</span> has independent rows can I just transpose it and get ... | egreg | 62,967 | <p>Suppose <span class="math-container">$A$</span> is <span class="math-container">$m\times n$</span> and that <span class="math-container">$AA^T$</span> is not invertible. Then there exists <span class="math-container">$v\ne0$</span> (an <span class="math-container">$m\times1$</span> column vector) such that
<span cla... |
4,351,990 | <p>I have just finished my undergrad and while I haven't studied much in representation theory I find it a very fascinating subject. My current interest is in differential equations, and I am wondering is there any ongoing research that combines these two areas?</p>
| markvs | 454,915 | <p>There are strong connections. For example look at <a href="https://mathoverflow.net/questions/335116/is-there-a-connection-between-representation-theory-and-pdes">this question in</a> MO and its answers.</p>
<p>There are also older books and papers about connections between these subjects. For example, "Theory ... |
270,849 | <p>I am trying to show that </p>
<p>$P(E\mid E\bigcup F) \geq P(E \mid F)$.</p>
<p>This is intuitively clear. But when expanding I get $P(E)\ P(F)\geq P(E\bigcup F)\ P(E \bigcap F)$. How to continue?</p>
| mathemagician | 49,176 | <p>Let $a=P(E\cap F^c)$, $b=P(E\cap F)$ and $c=P(F\cap E^c)$. You have that $P(E)=a+b$, $P(F)=b+c$. Since $E\cup F=((E\cap F^c)\cup(E\cap F)\cup (F\cap E^c))$ and since the the union is disjoint you have that $P(E\cup F)=a+b+c$. Therefore, the problem you stated reduces to showing $(a+b)(b+c)\geq b(a+b+c)$ which follo... |
4,722 | <p>Is there a necessary and sufficient condition for the boundary of a planar region to be a finite union of Jordan curves?</p>
| some guy on the street | 1,631 | <p><em>Videtur</em> I can't post comments of my own? This is not a complete answer.</p>
<p>@buzzard, I'd say yours probably <em>isn't</em> a facetious comment, in that I can imagine a union of two Jordan curves --- that is, an intersection of two connected open planar sets --- looking particularly wild. For example,... |
4,722 | <p>Is there a necessary and sufficient condition for the boundary of a planar region to be a finite union of Jordan curves?</p>
| Greg Kuperberg | 1,450 | <p>I can think of an important necessary condition: The boundary has to be locally contractible; in particular, locally connected. The topologist's sine curve is not locally connected, while the Hawaiian earring is not locally simply connected, and both occur in boundaries of open sets.</p>
<p>If you throw in the co... |
2,368,827 | <p>I would like to know how a piecewise function and its derivative would look like under these circumstances. Suppose that the function is continuous (and also nice like poly, trig etc) but defined differently for points $\le a$ and point $\gt a $</p>
<p>1) The function is differentiable at $a$. Then the derivative w... | Mundron Schmidt | 448,151 | <p>You have to be very careful. Consider
$$
f(x)=\begin{cases} x^2\sin\left(\frac1x\right) & x>0\\0& x\leq 0\end{cases}
$$
This function is differentiable at $0$ since
$$
\lim_{h\to 0^+}\left|\frac{f(h)-f(0)}h\right|=\lim_{h\to 0^+}h\left|\sin\left(\frac1h\right)\right|=0\text{ and }\lim_{h\to 0^-}\frac{f(h)... |
1,656,136 | <p>I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)</p>
| RghtHndSd | 86,816 | <p>Edit: By "ring" I mean commutative ring with identity.</p>
<p>Every ring is a quotient of an infinite polynomial ring. Let $R$ be any ring, and let $\{r_i\}_{i \in I}$ be a set of generators (under the ring operations). For example, we could simply take every single element of the ring $R$. Then define a ring homom... |
1,656,136 | <p>I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)</p>
| Will Byrne | 214,346 | <p>By definition, such a ring is non-Noetherian. A good example of a non-Noetherian ring is $F[X_1, X_2, X_3,...]$, the ring of polynomials over a field F in countably infinite indeterminates. In this ring, we have the infinite chain of generated ideals $(X_1) \subsetneq (X_1, X_2) \subsetneq (X_1, X_2, X_3) \subsetneq... |
1,656,136 | <p>I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)</p>
| zyx | 14,120 | <p>For a bi-infinite chain of ideals (and bounded transcendence degree), start from $k[X]$ and adjoin $2^n$-th roots of $X$ for all $n$. </p>
<p>The inclusion ordering of the principal ideals $(X^{2^i})$ for $i \in \mathbb{Z}$ is equivalent to the reversed ordering of integers, via the correspondence $2^i \leftrighta... |
2,664,341 | <blockquote>
<p>Simplify $$\frac{1}{\sqrt[3]1+\sqrt[3]2+\sqrt[3]4}+\frac{1}{\sqrt[3]4+\sqrt[3]6+\sqrt[3]9}+\frac{1}{\sqrt[3]9+\sqrt[3]{12}+\sqrt[3]{16}}$$</p>
</blockquote>
<p>I have no idea how to do this. I tried using the idea of multiplying the conjugate to every term, but I seem to be getting no where. Is the... | I was suspended for talking | 474,690 | <p>Hint: Let $f_n := f\chi_{E_n}$, where $\chi$ is the characteristic function. Then $f \chi_E = \lim_{n\to\infty} f_n$. Hence you would like to move the limit inside the integral, which theorem would be useful for that?</p>
|
633,858 | <p>If G is cyclic group of 24 order, then how many element of 4 order in G?
I can't understand how to find it, step by step. </p>
| Mikasa | 8,581 | <p>If $G=\langle a\rangle =\{a^0,a^1,a^2,\cdots,a^{23}\}$ then $$|a|=24\\|a^2|=k\to 24\mid2k\to12|k\to k=12.~\text{(because 12 is the least positive integer in this case)}\\|a^3|=k\to 24|3k\to8|k\to k=8.~\text{(because 8 is the least positive integer in this case)}\\|a^4|=k\to 24|4k\to6|k\to k=6.~\text{(because 6 is th... |
2,993,979 | <p>I tried to determine if <span class="math-container">$n\cdot \arctan (\frac 1n)$</span> is divergent or convergent. </p>
<p>My solution is in the two pictures. I really have no clue as how to solve it, so I tried something, but it cannot be right. At least that's what I think.</p>
<p>I am sorry in advance for my ... | paf | 333,517 | <p><span class="math-container">$$n\arctan\left(\dfrac1n\right) = \dfrac{\arctan\left(\dfrac1n\right)}{\dfrac1n} = \dfrac{\arctan\left(t\right)}{t} =\dfrac{\arctan\left(t\right) - \arctan 0}{t-0} $$</span>
if <span class="math-container">$t:=1/n$</span>.
This last expression tends to <span class="math-container">$\arct... |
2,993,979 | <p>I tried to determine if <span class="math-container">$n\cdot \arctan (\frac 1n)$</span> is divergent or convergent. </p>
<p>My solution is in the two pictures. I really have no clue as how to solve it, so I tried something, but it cannot be right. At least that's what I think.</p>
<p>I am sorry in advance for my ... | saulspatz | 235,128 | <p>The <span class="math-container">$n$</span>th term doesn't go to zero, so the series diverges. </p>
|
2,993,979 | <p>I tried to determine if <span class="math-container">$n\cdot \arctan (\frac 1n)$</span> is divergent or convergent. </p>
<p>My solution is in the two pictures. I really have no clue as how to solve it, so I tried something, but it cannot be right. At least that's what I think.</p>
<p>I am sorry in advance for my ... | Community | -1 | <p>From </p>
<p><span class="math-container">$$1-x^2\le\frac1{1+x^2}\le1$$</span> you draw, by integration from <span class="math-container">$0$</span></p>
<p><span class="math-container">$$x-\frac{x^3}3\le\arctan x\le x$$</span> and for <span class="math-container">$x>0$</span></p>
<p><span class="math-container... |
2,706,141 | <p>I've been working on a math problem recently whose small subpart part is this. I don't want to post the whole problem and be spoon fed it, but I've been struggling with this sub part of it and since my math skills are still trivial the solution may require maths which I have to learn so,</p>
<p>Can the product $\ma... | MrYouMath | 262,304 | <p>Hint: Use partial fractions</p>
<p>$$\int\dfrac{dx}{(\sqrt{3}x-\sqrt{7})(\sqrt{3}x+\sqrt{7})}=a\int\dfrac{dx}{\sqrt{3}x-\sqrt{7}}+b\int\dfrac{dx}{\sqrt{3}x+\sqrt{7}}$$</p>
<p>In which the coefficients can be directly determined by Euler's trick:</p>
<p>$$a=\left.\dfrac{1}{\sqrt{3}x+\sqrt{7}}\right|_{x=\dfrac{\sqr... |
203,111 | <p>Assume $(A_{i})_{i\in\Bbb N}$
to be an infinite sequence of sets of natural numbers, satisfying</p>
<p>$$A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq A_{3}\cdots\subseteq\Bbb N\tag{*}$$</p>
<p>For each property $p_{i}$
shown below, state whether </p>
<p>• the hypothesis (*)
is sufficient to conclude that $p... | Brian M. Scott | 12,042 | <p>You really shouldn’t have any trouble with $p_1$ or $p_6$. I’ll do $p_3$ as an illustration.</p>
<p>First, $p_3$ is consistent with (*). For example, let $A$ be the set of odd positive integers, and for $n\in\Bbb N$ let $A_n=A\cup\{2k:k\le n\}$: then $2(n+1)\in A_{n+1}\setminus A_n$, and $\bigcup_{n\in\Bbb N}A_n=\B... |
380,177 | <p>In mathematics, I want to know what is indeed the difference between a <strong>ring</strong> and an <strong>algebra</strong>?</p>
| Community | -1 | <p>In the contexts I'm used to, rings are defined to have a constant $1$ and the corresponding axioms making it a multiplicative unit. Algebras, however, do not.</p>
<p>And while you can talk about (for some fixed ring $R$) "rings over $R$" just as you can "algebras over $R$", the latter phrase is far more common than... |
498,785 | <p>I'm trying to solve this problem, but I'm not even sure how to formulate it in a coherent mathematical manner, or even what branch of mathematics this might fall in to.</p>
<p>Basically I have a set of weights, where each weight individually must remain in the range $[0,1]$. I want to change the mean of the weight... | JoshS | 95,647 | <p>I'd use an exponential transform, eg raise each value to the same real power. I'm not sure it's possible to get a closed form for the power at which to raise to achieve a specific target mean, though. Iterative approximation should work for this, though, if performance is not too much of a concern.</p>
<p>see <a ... |
498,785 | <p>I'm trying to solve this problem, but I'm not even sure how to formulate it in a coherent mathematical manner, or even what branch of mathematics this might fall in to.</p>
<p>Basically I have a set of weights, where each weight individually must remain in the range $[0,1]$. I want to change the mean of the weight... | Alecos Papadopoulos | 87,400 | <p>What you describe is a minimization problem under constraints. I will provide a mathematical formalization, and work the problem up to a point.</p>
<p>We have $n$ weights $w_1,...,w_n$, with $w_i \in [0,1], \; i=1,...n$. We want to arrive at new weights $w_i^* = w_i+d_i$, with $ |d_i|\le 1-w_i$. This is a constrain... |
576,519 | <p>Assume that $x+\frac{1}{x} \in \mathbb{N}$. Prove by induction that $$x^2+\frac1{x^2}, x^3+\frac1{x^3}, \dots , x^n+\frac1{x^n}$$ is also a member of $\mathbb{N}$.</p>
<p>I have my <em>base</em>, it is indeed true for $n=1$..</p>
<p>I can assume it is true for $x^k+x^{-k}$ and then proove it is true for $x^{k+1}+x... | lhf | 589 | <p>Let $a_n = x^n + \frac{1}{x^n}$. Then $x^2 = a_1x - 1$ implies $a_{n+2} = a_1a_{n+1} -a_n$ for all $n$.</p>
<p>Since $a_0=2$ and $a_1 \in \mathbb Z$, we have $a_n \in \mathbb Z$ for all $n \in \mathbb N$ by induction.</p>
|
884,362 | <blockquote>
<p>Compute the integral
$$\int_{0}^{2\pi}\frac{x\cos(x)}{5+2\cos^2(x)}dx$$</p>
</blockquote>
<p>My Try: I substitute $$\cos(x)=u$$</p>
<p>but it did not help. Please help me to solve this.Thanks </p>
| lab bhattacharjee | 33,337 | <p>Using $\displaystyle\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx,$</p>
<p>$$I=\int_0^{2\pi}\frac{x\cos x}{5+2\cos^2x}dx=\int_0^{2\pi}\frac{(2\pi-x)\cos(2\pi-x)}{5+2\cos^2(2\pi-x)}\ dx=\int_0^{2\pi}\frac{(2\pi-x)\cos x}{5+2\cos^2 x}\ dx$$</p>
<p>$$2I=2\pi\int_0^{2\pi}\frac{\cos x}{5+2\cos^2x}dx$$</p>
<p>$$\implies I=\pi\... |
4,264,558 | <p>I calculated homogenous already, I'm just struggling a bit with the right side. Would <span class="math-container">$y_p$</span> be <span class="math-container">$= ++e^x$</span> or <span class="math-container">$= ++e^{2x}$</span>?</p>
<p>Would the power in front of the root be the roots found from the homogenous part... | MachineLearner | 647,466 | <p>Hint: <span class="math-container">$\cos(x) = 1-\dfrac{x^2}{2}+O(x^4)$</span> or <span class="math-container">$\cos(1/x) = 1 - \dfrac{1}{2x^2}+O\left(\dfrac{1}{x^4}\right)$</span></p>
|
1,677,868 | <p>The sequence is:</p>
<p>$$a_n = \frac {2^{2n} \cdot1\cdot3\cdot5\cdot...\cdot(2n+1)} {(2n!)\cdot2\cdot4\cdot6\cdot...\cdot(2n)} $$</p>
| Brian M. Scott | 12,042 | <p>Investigating the ratio of consecutive terms, as suggested by <strong>André Nicolas</strong>, is probably easiest. Alternatively, note that</p>
<p>$$2\cdot 4\cdot 6\cdot\ldots\cdot(2n)=2^nn!\;,$$</p>
<p>and</p>
<p>$$1\cdot3\cdot5\cdot\ldots\cdot(2n+1)\le 2\cdot4\cdot6\cdot\ldots\cdot 2n\cdot(2n+2)=2^{n+1}(n+1)!\;... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | JMJ | 295,405 | <p>One good motivating example I have is the <a href="https://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow noreferrer">Weierstrass Function</a>, which is continuous everywhere but differentiable nowhere. Throughout the 18th and 19th centuries (until this counter example was discovered) it was thought that ... |
2,208,943 | <p>I am about to finish my first year of studying mathematics at university and have completed the basic linear algebra/calculus sequence. I have started to look at some real analysis and have really enjoyed it so far.</p>
<p>One thing I feel I am lacking in is motivation. That is, the difference in rigour between the... | asv | 403,888 | <p>You can try to read this <a href="https://en.wikipedia.org/wiki/Fluxion" rel="noreferrer">https://en.wikipedia.org/wiki/Fluxion</a> to understand the motivations to introduce the definition of limit.</p>
<p>Important is the example in the indicated web page:</p>
<blockquote>
<p>If the fluent ${\displaystyle y}$ ... |
1,393,265 | <p>How to prove that$(n!)^{1/n}$ tends to infinity as limit tends to infinity?
I tried to do this by expanding $n!$ as $n\times (n-1)\times (n-2)\cdots 4\times3\times2\times 1$ and taking out n common from each factor so that I can have $n$ outside the radical sign, But then the last terms would be $(4/n)\times(3/n)\ti... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> for any $a > 1$, we have $n! > a^n$ for sufficiently large $n$. </p>
|
403,631 | <p>$a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1 $
<br/>
Hint $u_{2n}$ = $u_{n}^2$</p>
<p>I have totally no idea how to prove this, this looks obvious but i found out proof is really hard...
I am doing a real analysis course and there's a lot of proving and I stuck there.
Any advices? Pra... | Did | 6,179 | <p>Replacing $a$ by $|a|$, one can assume without loss of generality that $a$ is a nonnegative real number. If $a=0$, the result is direct. If $0\lt a\lt1$, the sequence defined by $u_n=a^n$ is decreasing and positive hence it converges to some finite nonnegative limit $\ell$. Since $u_{n+1}=au_n$, $\ell=a\ell$. Since ... |
92,020 | <p>Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$.<br>
Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that<br>
$$\limsup_{\omega,t \in\Omega\times[S,T]} \sum_i^m \sum_j^n | g_{ij}(\omega,t)|<\infty$$ and<br>
$$\int_S^T E|G(\omega,t)... | Fers | 21,345 | <p>I've found something...<br>
let's define
$$\eta(t) = \int_S^t G(\omega,t)dW_t$$ </p>
<p>if we apply the Ito's formula we obtain ( for the general case with $2n=6$ )
$$ E \left[\eta(T)^{2n} \right] = E \left[ \frac{2n(2n-1)}{2}\int_S^T\eta(s)^{2n-2} G^2(s,\omega)ds \right] \le$$
using the hint
$$ \le \frac{2n(2n-1)... |
177,519 | <p>Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1}]$. In a sequence of papers, Kazhdan and Lusztig constructed a braided monoidal structu... | BWW | 50,658 | <p>I don't have the references to hand but as no-one else has offered an explanation, here is how I understand it. The representation category of the Kac-Moody algebra is the fusion category of a rational conformal field theory. The category associated to the quantum group needs to be defined a bit more carefully than ... |
4,244,187 | <blockquote>
<p>Find the equation of the tangent line to <span class="math-container">$\sin^{-1}(x) + \sin^{-1}(y) = \frac{\pi}{6}$</span> at the point <span class="math-container">$(0,\frac{1}{2})$</span></p>
</blockquote>
<p>This is in the context of learning implicit differentiation.</p>
<p>First, I apply <span clas... | Elias Costa | 19,266 | <p>The trace <span class="math-container">$\mathop{\rm trace}(U^\ast V)$</span> of the product of two matrices <span class="math-container">$U,V\in \mathbb{C}^{n\times n}$</span> behaves as an inner product</p>
<p><span class="math-container">$$
\langle U, V \rangle= \mathop{\rm trace}(U^\ast V) =\sum_{i=1}^{n}\sum_{j=... |
1,512,171 | <p>I want to show that there exists a diffeomorphic $\phi$ such that the following diagram commutes:
$$
\require{AMScd}
\begin{CD}
TS^1 @>{\phi}>> S^1\times\mathbb{R}\\
@V{\pi}VV @V{\pi_1}VV \\
S^1 @>{id_{S^1}}>> S^1
\end{CD}$$
where $\pi$ is the associated projection of $TS^1$, and $\pi_1(x,y)=x$ is ... | Element118 | 274,478 | <p>If $f$ is not continuous, the Riemann integral may not exist. In this case, by splitting the range into intervals and picking numbers in the range, it is possible to pick only rational numbers, forcing the value of the integral to $0$. It is also perfectly valid to pick only irrational numbers, forcing the value of ... |
1,341,440 | <p>I came across a claim in a paper on branching processes which says that the following is an <em>immediate consequence</em> of the B-C lemmas:</p>
<blockquote>
<p>Let $X, X_1, X_2, \ldots$ be nonnegative iid random variables. Then $\limsup_{n \to \infty} X_n/n = 0$ if $EX<\infty$, and $\limsup_{n \to \infty} X_... | Adelafif | 229,367 | <p>If x^2 is not in the center then the subgroup does not intersect the center except in e. It follws that .Z=G and G is Abelian but then x^2 is in the center.</p>
|
173,112 | <blockquote>
<p>Solve for $x$. $12x^3+8x^2-x-1=0$ all solutions are rational and between $\pm 1$</p>
</blockquote>
<p>As mentioned in my previous answers, I'm guessing I have to use the Rational Root Theorem. But I've done my research and I do not understand what to plug in or anything about it at all. Can someone p... | Ross Millikan | 1,827 | <p>The rational root theorem tells you that any rational roots of the polynomial have numerators that divide the constant term and denominators that divide the coefficient of the highest power. So here the numerator must be $\pm1$ and the denominator can be any of $1,2,3,4,6,12$. So you have $12$ possibilities for ra... |
173,112 | <blockquote>
<p>Solve for $x$. $12x^3+8x^2-x-1=0$ all solutions are rational and between $\pm 1$</p>
</blockquote>
<p>As mentioned in my previous answers, I'm guessing I have to use the Rational Root Theorem. But I've done my research and I do not understand what to plug in or anything about it at all. Can someone p... | Arturo Magidin | 742 | <p>The Rational Root Theorem tells you that if the equation has any rational solutions (it need not have any), then when you write them as a reduced fraction $\frac{a}{b}$ (reduced means that $a$ and $b$ have no common factors), then $a$ must divide the constant term of the polynomial, and $b$ must divide the leading t... |
1,390,976 | <p>Similar to <a href="https://math.stackexchange.com/questions/54763/what-do-algebra-and-calculus-mean">What do Algebra and Calculus mean?</a>, what is the difference between a logic and a calculus?</p>
<p>I am learning about the different kinds of logics, and often when I look them up in a different resource, some p... | Lance | 5,266 | <p>The short answer is, after a few more months of letting this sit, is that in reality there is no difference between algebra, logic, and calculus. They are all just saying "a formal collection of mathematical rules". But each of these words have a history, and so when authors use them, they are mentally invoking that... |
1,390,976 | <p>Similar to <a href="https://math.stackexchange.com/questions/54763/what-do-algebra-and-calculus-mean">What do Algebra and Calculus mean?</a>, what is the difference between a logic and a calculus?</p>
<p>I am learning about the different kinds of logics, and often when I look them up in a different resource, some p... | Community | -1 | <p><strong>In proof theory there is a difference between logic and calculi</strong></p>
<p>There might be one semantic consequence relation $\vDash$, but many different syntactic consequences relations $\vdash_1$, $\vdash_2$, $\vdash_3$, ... . Thats the usual modus operandi of mathematical logic in the formal sciences... |
2,451,350 | <p>Currently I am reading into functional data analysis. A common assumption is that the expected value of some random function is $0$, i.e. $\mathbb{E}(x) = 0$ where $x \in L^2$, the space of all squared integrable functions with inner product $\langle x,y \rangle = \int x(t)y(t) \text{d}t$. </p>
<p>My question might... | José Carlos Santos | 446,262 | <ol>
<li>If $Ax=b$, then $A(u+v)=Au+Av=b+0=b$.</li>
<li>What I wrote above holds for every solution $v$ of the equation $Av=0$, not just to one or some of them.</li>
<li>If $Ax=b$ has a solution $u$, then, since the equation $Ax=0$ has, at least, one solution $v\neq0$, then every $\lambda v$ ($\lambda\in\mathbb R$) is ... |
139,021 | <p>Can you, please, recommend a good text about algebraic operads?</p>
<p>I know the main one, namely, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/LodayVallette.pdf" rel="nofollow noreferrer">Loday, Vallette "Algebraic operads"</a>. But it is very big and there is no way you can read it fast. Also there are no... | Dan Petersen | 1,310 | <p>The book of Markl, Stasheff and Shnider is also a standard reference. </p>
<p>Also, a good jumping-in point could be Ginzburg and Kapranov's "Koszul duality for operads".</p>
|
1,034,335 | <p>I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?" </p>
<p>I understand the following:</p>
<p>Let the sequence $a_n$ exist such that $a_n =\frac{1}{n^2}$ </p>
<p>Then $\lim_{n\to\infty} a_n=\lim_{n... | Juan123 | 527,550 | <p>You could say that all series are sequences, meaning that you could make a sequence taking its partial sums and checking whether or not it approaches a limit, but certainly not all sequences are series. </p>
|
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