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 |
|---|---|---|---|---|
4,131,916 | <p>Let <span class="math-container">$(a_n)_{n \in \mathbb{Z}}$</span> be some given sequence of positive numbers, such that <span class="math-container">$\lim_{n \to -\infty} a_n=0,\lim_{n \to \infty} a_n=+\infty$</span>.</p>
<p>Let <span class="math-container">$\Omega \subseteq \mathbb{R}^2$</span> be a bounded, con... | ibnAbu | 334,224 | <p>I think thus might be helpful:</p>
<p>Is it possible to calculate volume of a cube (with volume <span class="math-container">$L^3$</span>) by filling it with small balls each with a radius <span class="math-container">$r_N$</span> and the balls are disjoint ? The answer is 'No'.</p>
<p>Let the number of balls be <sp... |
95,176 | <p>Does the octic,</p>
<p>$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$</p>
<p>for any constant <em>n</em> have Galois group of order 1344? Its discriminant <em>D</em> is a perfect square,</p>
<p>$D = (1728n^4-341901n^3-11560361n^2+3383044089n+28121497213)^2$</p>
<p>Surely (1) is not an isolated result. ... | Gene Ward Smith | 26,327 | <p>In arxiv.org/abs/1209.5300 I give the following polynomial with the same Galois group over Q(t): $(y+1)(y^7-y^6-11y^5+y^4+41y^3+25y^2-34y-29)-t(2y+3)^2$. This has $(6912t^4-3456t^3-95472t^2+23976t-1417)^2$ for a discriminant. This doesn't quite answer the second question, but substituting $y \mapsto (x-3)/2$, $t \ma... |
30,728 | <p>Is this graph in the list among the so-called "standard" structures used in <code>GraphData</code>? However, I have not found yet anything like "Carpet" or "Sponge" in the list of the objects that can be built. Maybe, this graph has a different name? </p>
<p>For me, using <code>GraphData</code> helps to save time f... | Michael E2 | 4,999 | <p>I don't think there is a carpet graph built-in, but it's hard to be sure that something is <em>not</em> there. Still it's not hard to construct a <code>Graph</code> -- not quite the same thing as drawing it (I wasn't sure what you meant).</p>
<p>There are probably more efficient ways, but adapting <a href="https:/... |
2,071,226 | <p>Assuming $f : \mathbb R \to \mathbb R$ is differentiable, how can one prove that
$$
f'(x) = \lim_{h,k \to 0^+} \frac{f(x+h)-f(x-k)}{h+k},
$$
an alternate expression to the usual limit definition of the derivative $f'(x)=\lim_{h \to 0^+} \frac{f(x+h)-f(x)}h$?</p>
<p>I figured the problem out for the special case of ... | Ted Shifrin | 71,348 | <p><strong>HINT</strong>: You were on the right track to subtract and add $f(x)$ in the numerator. So then you'll have the sum of two fractions. You now need to do a similar trick with <em>multiplication</em> with each of those factors in order to make the usual difference quotients
$$\frac{f(x+h)-f(x)}h \qquad\text{an... |
2,071,226 | <p>Assuming $f : \mathbb R \to \mathbb R$ is differentiable, how can one prove that
$$
f'(x) = \lim_{h,k \to 0^+} \frac{f(x+h)-f(x-k)}{h+k},
$$
an alternate expression to the usual limit definition of the derivative $f'(x)=\lim_{h \to 0^+} \frac{f(x+h)-f(x)}h$?</p>
<p>I figured the problem out for the special case of ... | Zhanxiong | 192,408 | <p>It's better to use an $\varepsilon$-$\delta$ argument to present a rigorous proof: that is, evaluating the difference between the operand and the objective $f'(x)$.</p>
<hr>
<p>Details:
Given $\varepsilon > 0$, by the definition of $f'(x)$, there exists $\delta > 0$ such that $0 < h < \delta$ and $0 &l... |
3,631,212 | <p>I am trying to show that the functors <span class="math-container">$h^n(X)=\text{Hom}(H_n(X),\Bbb Z)$</span> do not define a cohomology theory on CW complexes. If a contravariant functor <span class="math-container">$h^n(X)$</span> is a cohomology theory, by definition it must satisfy the followings:</p>
<p>(1) If ... | Connor Malin | 574,354 | <p>A fun answer using more machinery: Any cohomology theory that has the cohomology of a point the same as singular cohomology is isomorphic to singular cohomology. If <span class="math-container">$\operatorname{Hom}(H_n(-); \mathbb{Z})$</span> were a cohomology theory, it must then be isomorphic to singular cohomology... |
252,957 | <p>I used to think naively the construction is straightforward, which is, if we add one layer innermost each time, then we could have one that corresponds to $\omega$ in Neumann's representation, which is. of course, constructible in <strong>ZF</strong> excluding axiom of foundation .</p>
<p>But I was told I can't def... | Asaf Karagila | 622 | <p>(I read this question as asking about constructing such sets in ZF-Foundation alone)</p>
<p>Essentially you are asking for a set $x=\{x\}$ in ZF without foundation.</p>
<p>The problem is that just without foundation you don't get enough. Every model of ZF is a model of ZF without foundation.</p>
<p>There are meth... |
2,799,257 | <p>I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-valued continuous function) when $t=\frac{1}{\pi}$.</p>
<p>When $t=\frac{1}{\pi}$, then we get that $u=\cos{(\pi x-u)... | EditPiAf | 418,542 | <p>Based on the proof in <a href="https://math.stackexchange.com/q/2445518/418542">this post</a>, we have the expression for the <em>breaking time</em>
$$
t_b = \frac{-1}{\min \partial_x u(x,0)} = \frac{1}{\pi} \, .
$$
At $t=t_b$, characteristics intersect and a shock wave occurs. This is illustrated in the figure belo... |
902,015 | <p>Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$.<br/><br/>
<strong>EDIT:</strong><br/>
Thsi is my attempt:<br/>
Let $x^2-ax=b$ and $x^3-ax=q$ for some $b,q\in\mathbb{Q}$. Then I tried to write $x^2$ and $x^3$ in linear terms. I got $x^2=ax+b$ a... | Kelenner | 159,886 | <p>Put $x^2-ax=b$ and $x^3-ax=c$ with $b,c\in \mathbb{Q}$. As $x\in \mathbb{R}$ is a real solution of $y^2-ay-b=0$, we have $D=a^2+4b\geq 0$. Now:</p>
<p>$$x^3=xx^2=x(ax+b)=ax^2+bx=a^2x+ab+bx=(a^2+b)x+ab$$
This is also $ax+c$. If we suppose that $x\not \in \mathbb{Q}$, we get that $a=a^2+b$. Hence $D=a^2+4b=4a-3a^2<... |
4,491,844 | <p>I want to show that every number in <span class="math-container">$[\frac{1}{2},1)$</span> is in a unique interval <span class="math-container">$[\frac{n}{n+1},\frac{n+1}{n+2}]$</span>, where <span class="math-container">$n$</span> is a positive integer. Intuitively, I think this is correct, but I do not know how to ... | nonuser | 463,553 | <p>You probably mean <span class="math-container">$[\frac{n}{n+1},\frac{n+1}{n+2})$</span>?</p>
<p>So you want to prove there is unique <span class="math-container">$n$</span> such that <span class="math-container">$${n\over n+1}\leq x<{n+1\over n+2}$$</span> which is equivalent to <span class="math-container">$${2x... |
3,441,225 | <p>Let <span class="math-container">$S=1-1/3+1/5-1/7+\cdots$</span>. As each term in the series is decreasing and tends to <span class="math-container">$0$</span>, it is known that their sum exists and is finite by alternating series test. And by considering <span class="math-container">$\int_0^11/(1+x^2)dx$</span>, it... | Patrick | 317,074 | <p>Assuming that you start from <span class="math-container">$n=0$</span>. Let <span class="math-container">$b_n = a_{2n} - a_{2n+1}$</span>. Then the infinite sum of <span class="math-container">$a_n$</span> is equal to the infinite sum of <span class="math-container">$b_n$</span>. But <span class="math-container">$b_... |
2,969,710 | <blockquote>
<p>Use mathematical induction to prove that
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}
$$</span></p>
</blockquote>
<p>I am unsure about the prove n+1 step!
I let
<span class="math-container">$$
\frac12 + \frac16 + \ldots + \frac{1}{n(n+1)} = 1 -... | PrincessEev | 597,568 | <p><strong>Overview of Induction:</strong></p>
<p>Keep in mind that an induction proof consists of three parts:</p>
<ul>
<li><em>Proving</em> a base case. </li>
<li><em>Assuming</em> that the hypothesis you're proving holds for some <span class="math-container">$n$</span>, where <span class="math-container">$n$</span... |
900,326 | <p>Let $G=(V,E)$ be a connected graph which is not a tree. Prove that if for every cycle $C$ of the graph G and for any $v \in V(G)- V(C)$ there are more than $\frac{|C|}{2}$ edges from $v$ to $V(C)$ then G is Hamiltonian.</p>
<p>My proof:</p>
<p>I will show that every vertex $w \in V$ is part of some cycle.
Proof b... | Smylic | 100,361 | <p>You are close to prove Hamiltonicity of such graphs. Just use for example induction to show that if there exists cycle $C$ of length $|C|$ in graph $G$ then there exists a simple cycle $C_k$ of length $k$ for any $k$ between $|C|$ and $|G|$, inclusive.</p>
|
35,014 | <p>I want to display a rational number in <em>Mathematica</em> in periodic style.
<code>PeriodicForm</code> isn't working anymore. It worked in <em>Mathematica</em> 5 and now I'm using <em>Mathematica</em> 9.</p>
<p>I want to display the number $3.13678989898989898989\ldots$, where the repeating $89$ part should be di... | bobthechemist | 7,167 | <p>To get <code>PeriodicForm</code> working in Mathematica 9 (and probably other versions after 6) you need to first download the obsolete package from the <a href="http://library.wolfram.com/infocenter/MathSource/6773" rel="noreferrer">Wolfram Library Archive</a>. Run the package, ignore the errors and have fun:</p>
... |
3,650,114 | <p>Why is there Bx+c term when we try to split partial fraction with irreducible quadratic?</p>
<p>Eg:</p>
<p><span class="math-container">$$\frac{1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$$</span></p>
<p>I think that splitting partial fraction is intuition when we directly put it as <span class="math-container">... | Jack D'Aurizio | 44,121 | <p><span class="math-container">$$f(x)=\frac{1}{x(x^2+1)}=x\cdot\frac{1}{x^2(x^2+1)}=x\left(\frac{1}{x^2}-\frac{1}{x^2+1}\right)=\frac{1}{x}-\frac{x}{x^2+1}.$$</span></p>
<p>Also
<span class="math-container">$$ A = \lim_{x\to 0} x f(x) = \lim_{x\to 0}\frac{1}{x^2+1} = 1$$</span>
so <span class="math-container">$Bx+C$<... |
2,011,261 | <p>A pizza restaurant has 3 crust options, 2 cheese options and 10 choices of toppings. On Saturday nights, the restaurant offers a special deal on 2-toppings pizzas including pizzas with double portions of one toppings. How many distinct special deal pizzas are possible.</p>
<p>My approach:
I assumed (not too sure if... | sTertooy | 336,630 | <p>$$\lim_{n \to +\infty} \left(1-\frac{1}{n}\right)^n = \lim_{n \to +\infty} \left(\frac{n-1}{n}\right)^n = \lim_{n \to +\infty}\frac{1}{\left(\frac{n}{n-1}\right)^n} = \lim_{n \to +\infty}\frac{1}{\left(\frac{n-1+1}{n-1}\right)^n} = \lim_{n \to +\infty}\frac{1}{\left(1+\frac{1}{n-1}\right)^n} = \left(\lim_{n \to +\... |
2,892,755 | <p>I know that the gradient descent flow of the Dirichlet energy $$\min_u E(u) = \int_{\Omega}|\nabla u|^2 dA$$
Is the diffusion/heat equation:</p>
<p>$$u_t = \Delta u$$</p>
<p>Is there a change in the Dirichlet energy such that it gives an anisotropic diffusion flow?</p>
| mathreadler | 213,607 | <p>Yes, multiply the gradient with some matrix. For example an outer product tensor $${\bf T} = \sum_{\forall n} \lambda_n {\bf \hat e}_n {{\bf \hat e}_n}^T$$
and then modify like this:
$$\min_u E(u) = \int_{\Omega}|{\bf T}\nabla u|^2 dA$$</p>
<p>This will make <em>cost of flow</em> in directions to depend on the eige... |
169,253 | <p>I know the formula for the area of a sector of an arc made by central angle is
$$\text{Area}_\text{Sector}= \frac{\text{Arc Angle} \times \text{Area of Circle} }{360}$$
Now my question is , Is this formula also applicable for Arcs formed by inscribed angles rather than Central Angles ? (I know that angle of an inte... | i. m. soloveichik | 32,940 | <p>No. Consider the case of a right angle. For the sector we get a quarter circle. For an inscribed angle, consider the base of the angle as a diameter and now move the other side of the angle so that it contains the quarter circle. The area can be made almost twice the size of the quarter sector.</p>
|
1,088,166 | <p>I'm so confused about cardinalities of some sets. What is the countable infinite product of a two points set $\{0,1\}$? Does it have the same cardinality as the real number $\mathbb R$? Or is the infinite product just countable?</p>
<p>Could anyone give me the answer?</p>
| egreg | 62,967 | <p>Let $X$ be a set and consider the set $\{0,1\}^X$ of all maps $X\to\{0,1\}$. Then there is a bijection
$$
f\colon \mathcal{P}(X)\to\{0,1\}^X
$$
($\mathcal{P}(X)$ is the power set of $X$) defined by sending each subset $A$ of $X$ to its characteristic function
$$
\chi_A(x)=\begin{cases}
1 & \text{if $x\in A$}\\
... |
1,566,163 | <p>I am writing a paper where I have more than one lemma (Lemma 1, Lemma 2, and Lemma 3) and when I cite them together I was wondering is it more appropriate to say, for example, </p>
<blockquote>
<p>because of Lemmas 1-3</p>
</blockquote>
<p>or </p>
<blockquote>
<p>because of Lemmata 1-3</p>
</blockquote>
<p>I... | Piquito | 219,998 | <p>At the time that "lemma" is admitted as a word in a particular language, it should be applied accordingly to pluralize in that language no matter what the ethymological Greek (or Latin) plural whatever it is (in French the plural of "lemme" is "lemmes" and so is in Spanish, with an S at the end of the word).</p>
<p... |
82,558 | <p>Given a function $f:\mathbb{R^n}\to \mathbb{R}$ that can be expressed as sum of roots of polynomials, i.e. $f = \sum_{i=0}^k (p_i)^{1/n_i}$ for some polynomials $p_i$ and integers $n_i$. Can one find a polynomial $p:\mathbb{R}^n \to \mathbb{R}$ such that in the domain where both function are defined, we have $p(x_1,... | Carl | 18,702 | <p>Take a sheet of paper, and curl the far end over to make a cylinder. The place where the edges of the paper meet is one of your one cells. Now fold the ends of the cylinder over to touch each other and make a torus. The circle where these ends meet is another one cell. The place where your circles intersect is a... |
2,047,748 | <p>Question:
suppose $\mathrm{log}_9 X + \mathrm{log}_{27} X = P$. write the value of $\mathrm{log}_3 X + \mathrm{log}_{81} X$ in terms of $P$.</p>
<p>I changed $\mathrm{log}_9 X + \mathrm{log}_{27} X = P$ into $\frac{1}{2} \mathrm{log}_3 X + \frac{1}{3}\mathrm{log}_3 X = P$.</p>
<p>I can't expand $\mathrm{log}_3 X +... | Community | -1 | <p>You changed $P = \log_9(x) +\log_{27}(x) \Rightarrow (\frac{1}{2} +\frac{1}{3})\log_3(x)$. Thus we have $$\log_3(x) =\frac{6}{5}P$$. Then we have $$\log_3(x)+\log_{81}(x)= \log_3(x) +\frac{1}{4}\log_3(x) = \frac{5}{4}\log_3(x) = \frac{5}{4}\times \frac{6}{5}P = \frac{3P}{2}$$. Hope it helps.</p>
|
2,848,177 | <p>Random variables $X$ and $Y$ have joint p.d.f</p>
<p>$$f_{x, y} (x, y) =\begin{cases}
c(x^3 + 2y^3) & 0 \leq x \leq 3, 0 \leq y \leq 4\\
0 & \text{otherwise }\\
\end{cases}
$$</p>
<p>Find the value of c that makes $f_{x, y}$ a joint density function.</p>
<hr>
<p>attempt:</p>
<p>$$1 = \int_{0}^{4}\int_... | spaceisdarkgreen | 397,125 | <p>Your work is right.</p>
<p>If we have $0\le x \le y\le 4$ then the integral can be set up as $$ \int_0^4\int_0^y c(x^3+2y^3)dx\; dy$$ or $$ \int_0^4\int_x^4 c(x^3+2y^3)dy\; dx$$</p>
|
69,900 | <p>Hello!</p>
<p>Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq \ldots \leq p_k$ are odd prime numbers. I have edited the answer and gave three attempts I tried to use in order t... | Jernej | 1,737 | <p>The question has been answered on math.stackexchange, the answer here is just for the sake of completeness.</p>
<p>From <a href="https://math.stackexchange.com/questions/52737/estimating-an-integral">https://math.stackexchange.com/questions/52737/estimating-an-integral</a> we see that an asymptotically equivalent e... |
599,394 | <p>A pack contains $n$ card numbered from $1$ to $n$. Two consecutive numbered cards are removed from</p>
<p>the pack and sum of the numbers on the remaining cards is $1224$. If the smaller of the numbers on</p>
<p>the removing cards is $k$, Then $k$ is.</p>
<p>$\bf{My\; Try}::$ Let two consecutive cards be $k$ and ... | codeapplied | 1,092,372 | <p>Sum of 1st n natural nos. <span class="math-container">$=\frac{\mathrm{n}(\mathrm{n}+1)}{2}$</span></p>
<p>Given: <span class="math-container">$\frac{\mathrm{n}(\mathrm{n}+1)}{2}-(\mathrm{k}+\mathrm{k}+1)=1224$</span></p>
<p><span class="math-container">$$
\Rightarrow \frac{\mathrm{n}(\mathrm{n}+1)}{2}-2 \mathrm{k}=... |
3,226,067 | <p>I have a triangle ABC and I know that <span class="math-container">$\tan\left(\frac{A}{2}\right)=\frac{a}{b+c}$</span>, where <span class="math-container">$a,b,c$</span> are the sides opposite of the angles <span class="math-container">$A,B,C$</span>. Then this triangle is:</p>
<p>a. Equilateral</p>
<p>b. Right tr... | Julian Mejia | 452,658 | <p>One way to see that c) is the correct answer is as follows. Draw yor triangle <span class="math-container">$ABC$</span>. Construct the angle <span class="math-container">$A/2$</span> by extending <span class="math-container">$BA$</span> until a point <span class="math-container">$M$</span> such that <span class="mat... |
730,130 | <p>I have data which are visualized in this chart:
<img src="https://i.stack.imgur.com/6ekLT.png" alt="enter image description here"></p>
<p>I need to compute slope of increasing / decreasing parts of the curve. I can't use any 2 points because of noise in data. Maybe numerical derivative can help but I don't know how... | Calvin Khor | 80,734 | <p>I believe its because you replaced $g'$ with $g$ by accident: you let $ε = \frac{|g'(x_0)|}{2}$ but then use $ε = \frac{|g(x_0)|}{2}$ in the next line. What you want is</p>
<p>$$\left|\frac{g(x) - g(x_0)}{x-x_0} - g'(x_0)\right| \leq \frac{|g'(x_0)|}{2}$$</p>
<p>And from this you can prove your inequality using a... |
5,877 | <p>I've recently run across a series of problems that didn't reflect reality. </p>
<p>For example - </p>
<ul>
<li>An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH. </li>
<li>A quadratic equation, "The football follows a path of....." but the equation didn't refl... | Ilmari Karonen | 526 | <h3>Absolutely!</h3>
<p>In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is <em>checking whether the answers make sense</em>. This is an absolutely invaluable part of making <em>any</em> practical use of mathematics, as opposed to just blindly... |
5,877 | <p>I've recently run across a series of problems that didn't reflect reality. </p>
<p>For example - </p>
<ul>
<li>An algebra problem with two teens on bicycles. The resulting times showed the bike was moving at 120MPH. </li>
<li>A quadratic equation, "The football follows a path of....." but the equation didn't refl... | Benoît Kloeckner | 187 | <p>I think trying to make "real-world" word problems is often posing more problem than it solves. One expects using maths in real-world problem would make them more appealing and show their usefulness, but honestly in most cases the problems are either ridiculous, or artificial, or boring to students, or all of these. ... |
2,782,492 | <p>Suppose we have the series $\sum a_n$. Define,</p>
<p>$$
L=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}
$$</p>
<p>Then,</p>
<ul>
<li>if $L<1$ the series is absolutely convergent (and hence convergent).</li>
<li>if $L>1$ the series is divergent.</li>
<li>if $L=1$ the series may be divergent, conditionally convergen... | Sam Spedding | 202,717 | <p>Clearly if $a_n=0$ for every $n\in\mathbb{N}$, the limit is undefined but the series still converges.</p>
|
1,369,669 | <p>We are given the following:</p>
<p>$$\int \sin(xy)dy$$</p>
<p>We start by assigning anything algebraic into our first variable, $u$. Recalling LIATE (Logarithmic, Inverse-Trig., Algebra, Trig., Exponential) we start with algebra.</p>
<p>If I assign $$u=xy$$ then</p>
<p>$$\int \sin(u)\frac{du}{x}dy$$</p>
<p>Henc... | silentkiller | 253,392 | <p>I guess you can take x as a constant to get $-\cos(xy)/x$.</p>
|
3,020,024 | <blockquote>
<p>Let <span class="math-container">$f: \Bbb R \to \Bbb R$</span> be a function such that <span class="math-container">$f'(x)$</span> exists and is continuous over <span class="math-container">$\Bbb R$</span>. Moreover, let there be a <span class="math-container">$T > 0$</span> such that <span class="... | Mike | 194,842 | <p>Say a period is $[a,a+T]$. Since $f$ is continuous, it attains a minimum on that interval, say at $c$. (If you have $c = a$, then change the period to $[a - T/2,a + T/2]$ so that $c$ becomes an interior point.)</p>
<p>We must have $f'(c) = 0$ there, so $f(c) \geq 0$. So the minimum value of $f$ is nonnegative.</p>
|
3,020,024 | <blockquote>
<p>Let <span class="math-container">$f: \Bbb R \to \Bbb R$</span> be a function such that <span class="math-container">$f'(x)$</span> exists and is continuous over <span class="math-container">$\Bbb R$</span>. Moreover, let there be a <span class="math-container">$T > 0$</span> such that <span class="... | David Holden | 79,543 | <p>$ f \ne 0$</p>
<p>$f(x)=0 \Rightarrow f'(x) \ge 0$, hence $f$ can cross the X-axis at most once. periodicity means that it cannot cross at all.</p>
<p>if $f$ is non-positive then $f' \ge 0$ so $f$ is monotone increasing. again this contradicts periodicity</p>
|
3,020,024 | <blockquote>
<p>Let <span class="math-container">$f: \Bbb R \to \Bbb R$</span> be a function such that <span class="math-container">$f'(x)$</span> exists and is continuous over <span class="math-container">$\Bbb R$</span>. Moreover, let there be a <span class="math-container">$T > 0$</span> such that <span class="... | Did | 6,179 | <p>Let $g(x)=\mathrm e^xf(x)$, then $g'(x)=\mathrm e^x(f'(x)+f(x))\geqslant0$ hence: $$(1)\ \textit{The function $g$ is nondecreasing.}$$
Since $f$ is continuous and periodic, $f$ is bounded, say $|f(x)|\leqslant C$ for every $x$, hence $|g(x)|\leqslant C\mathrm e^x\to0$ when $x\to-\infty$, that is: $$(2)\ \textit{The ... |
2,118,611 | <p>$$\lim\limits_{x \to 0}\left(\frac{e^2}{(1+ 4x )^{\frac1{2x}}}\right)^{\frac1{3x}}=e^{\frac43}$$</p>
<p>I need help with solving this limit. I don't know how to get to the solution. Thanks.</p>
| Claude Leibovici | 82,404 | <p>You can also avoid using L'Hospital.
$$A=\left(\frac{e^2}{(1+ 4x )^\frac{1}{2x}}\right)^\frac{1}{3x}\implies \log (A)=\frac{1}{3x}\left(2-\frac1 {2x}\log (1+4x)\right)$$ Now, using Taylor series around $x=0$ $$\log(1+4x)=4 x-8 x^2+\frac{64 x^3}{3}+O\left(x^4\right)$$ $$2-\frac1 {2x}\log (1+4x)=4 x-\frac{32 x^2}{3}+O... |
240,637 | <p>I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional case, and let $f_k(x)=\begin{cases}
k, &\quad\text{if } - \frac{1}{k} \leq x \leq \frac{1}{k}\\
0,... | Romeo | 28,746 | <p>There is a nice exercise on Rudin, <em>Real and Complex Analysis</em> (Chapter 1) about this. </p>
<p>Consider a measure space $(X,\mathcal A, \mu)$ and pick a measurable subset $E$. Then consider the sequence $(f_n)_{n\in\mathbb N}$ of real valued functions defined as $f_n=\chi_E$ if $n$ is even, $f_n=\chi_{X\setm... |
2,093,152 | <p>$A$ is a $2\times3$ matrix, $B$ is a $3\times2$ matrix, $\text{rank}(A)=\text{rank}(B)=2$</p>
<p>Does always $\text{rank}(AB)$ equal to $2$?</p>
| Yiorgos S. Smyrlis | 57,021 | <p>Hint. Assume that a polynomial of the form $y(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ satisfies the ODE, and pug it in the ODE. Then you shall find that $n=2$. Next, you shall find that $y=x^2-1/3$. Observe that if $y$ is a solution, then so is $cy$, for every $c\in \mathbb R$. Satisfaction of the condition $y(1)=2$, imp... |
640,648 | <p>What quadrilaterals in the real projective plane can be obtained by a projective transformation of the real projective plane from a square?</p>
| Phira | 9,325 | <p>One way to prove that all quadrilaterals $ABCD$ without three collinear vertices can be mapped by a projective transformation to a square is as follows:</p>
<p>Call the intersection points of opposite sides $X$ and $Y$. Map the line $XY$ to the line at infinity. (This is easy to do, just project to a plane that is ... |
4,522,273 | <p>Let <span class="math-container">$F^{\bullet}$</span> and <span class="math-container">$ I^{\bullet }$</span> be two bounded below complexes of sheaves of <span class="math-container">$O_X$</span>-modules (on a scheme X) and let <span class="math-container">$F^{\bullet}\rightarrow I^{\bullet}$</span> be a quasi-isom... | locally trivial | 581,923 | <p>It seems to me like the OP may not have meant to take global sections before taking cohomology: @Yuan Yang, did you really mean to take global sections <em>before</em> taking cohomology, or did you mean to take the right derived functor of global sections <em>as</em> cohomology? It seems to me like your original int... |
1,878,519 | <blockquote>
<p>$$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx$$</p>
</blockquote>
<p>$$f(x) = (\tan \ x)^{\frac{2}{3}}, \ f'(x) = \frac{2}{3} \cdot (\tan \ x)^{-\frac{1}{3}} \cdot \sec^2x$$</p>
<p>$$\therefore \int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx = \fra... | Zau | 307,565 | <blockquote>
<p>$$
\frac{3}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{2}{3}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx =
\frac{3}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{f'(x)}{f(x)}dx$$</p>
</blockquote>
<p>This step is wrong since</p>
<p>$$f(x) = (\tan \ x)^{\frac{2}{3}}, \ f'(x) = \frac{2}{3} \cdot (\tan \ x)^{-... |
1,236,603 | <p>Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to show that the order is $0$.</p>
<p>Clearly though not all entire functions that grow at roughly this rate are polyn... | Matthew Levy | 182,607 | <p>Nevermind, I think I figured it out. It just follows from the Weierstrass Factorization Theorem for entire functions of finite order that it will have the normal form for an entire function just that the polynomial over the $e$ will be degree 0, or in other words a constant and will have all the roots multiplied by ... |
9,840 | <p>The formula for finding the roots of a polynomial is as follows</p>
<p>$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4\cdot3\cdot24 } $$
$$... | Alex Basson | 506 | <p>The other answers are nice, so I won't reiterate them. It's worth pointing out, though, that when you say "I know there are other methods, i.e. factorisation and completing the square, but does this mean that this formula can only be used in specialised cases", you seem to be implying that these other met... |
318,754 | <p>This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.</p>
<p>It looks to me, that complex-analytic geometry has lost its relative positions since 50's, especially if we compare it to scheme theory. <em>Are there internal mathematical reasons for why tha... | Libli | 37,214 | <p>Though I am not an expert on this I think that the shift toward algebraic geometry is not entirely sociological. Consider the following statement which is true in both the category of schemes and analytic spaces :</p>
<p><em>The push-foward of a coherent sheaf by a proper map is coherent.</em></p>
<p>In algebraic ... |
1,237,425 | <p>I have to prove the following:</p>
<p>If $a, b \in \mathbb{C}$ and are both algebraic over $\mathbb{Z}$, then:</p>
<ol>
<li><p>$a + b$ is algebraic over $\mathbb{Z}$ </p></li>
<li><p>$a - b$ is algebraic over $\mathbb{Z}$</p></li>
<li><p>$ab$ is algebraic over $\mathbb{Z}$</p></li>
</ol>
<p>I tried this for the f... | MadMonty | 145,364 | <p>Probably the easiest way to tackle this is by saying:</p>
<p>$a$ is algebraic implies that the field extension $\mathbb{Q}(a):\mathbb{Q}$ is algebraic, and thus $[\mathbb{Q}(a):\mathbb{Q}] < \infty$.</p>
<p>Similarly, we have $[\mathbb{Q}(b):\mathbb{Q}] < \infty$.</p>
<p>So by the tower law, $[\mathbb{Q}(a,... |
3,163,123 | <p>I am trying to prove that the determinant of a magic square, where all rows, columns and diagonal add to the same amount, is divisible by 3. </p>
<p>I proved it for magic squares which have entries <span class="math-container">$1,\ldots, 9$</span>, but it turns out I need to show it for magic squares which can have... | Oscar Lanzi | 248,217 | <p>Render the elements <span class="math-container">$\bmod3$</span>. Then the sum of every row and every column is three times the central element, but this is <span class="math-container">$\equiv0\bmod3$</span>. To make all the column sums zero you then need the last row to be <span class="math-container">$\equiv$</sp... |
546,809 | <p>The question is :Find the derivative of $f(x)=e^c + c^x$. Assume that c is a constant.</p>
<p>Wouldn't $f'(x)= ce^{c-1} + xc^{x-1}$. It keeps saying this answer is incorrect, What am i doing wrong?</p>
| xekyu | 70,356 | <p>There are distinct derivation rules for two cases:</p>
<ul>
<li>A variable raised to a constant</li>
<li>A constant raised to a variable.</li>
</ul>
<p>You tried to apply the first case to the second term, $c^x$, where you really want the second case. Consider the general rule for constants raised to variables, $$... |
391,364 | <p>Let <span class="math-container">$\Sigma_{g,n}$</span> denote an <span class="math-container">$n$</span>-punctured surface of genus <span class="math-container">$g$</span>, with <span class="math-container">$2g+n-2 > 0$</span>. Let <span class="math-container">$\Pi_{g,n}$</span> be its fundamental group (for some... | Andy Putman | 317 | <p>It only has finite index in very low-complexity degenerate cases.</p>
<p>Here's a proof that it always has infinite index for <span class="math-container">$\Sigma_{g,1}$</span> with <span class="math-container">$g \geq 2$</span>. This proof generalizes in an obvious way to deal with all the other cases too, but the... |
1,025,881 | <p>Let $G$ be a finite group and $p$ be the smallest prime divisor of $|G|$ , let $x \in G$ be such that $o(x)=p$ , and suppose for some $h\in G $ , $hxh^{-1}=x^{10}$ , then is it true that $p=3$ ? </p>
| k170 | 161,538 | <p>There many different ways to show this.
$$ (cf)'=\frac{d}{dx}[cf(x)] =\lim_{h\to 0}\frac{cf(x+h)-cf(x)}{h}=c \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=c \frac{d}{dx}[f(x)] =cf'$$</p>
|
114,645 | <p>How can I make a moving point on the circle with control?</p>
<pre><code>Manipulate[
ParametricPlot[Sqrt[50]{Cos[x],Sin[x]},{x,0,10Pi},
Epilog->{PointSize[Large], Point[Table[{2,0}]]},
PlotRange->{{0,10},{0,10}}],{{Sqrt[50],2,"Play"}, 1, 10}
]
</code></pre>
| e.doroskevic | 18,696 | <p><strong>Example:</strong></p>
<pre><code>Manipulate[
Graphics[{
Circle[],
{Red, PointSize @ .05, Point@{Cos[x], Sin[x]}}
}],
{x, 0, 2 Pi}
]
</code></pre>
<p>Alternatively, if you would like to animate it: </p>
<pre><code>Animate[
Graphics[{
Circle[],
{Red, PointSize @ .05, Point@{Cos[x], Sin[x]... |
3,886,173 | <p>Please help me to prove this inequality</p>
<p><span class="math-container">$$\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} < \ln n < 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n - 1}$$</span></p>
| Milo Moses | 630,231 | <p>To solve (a), you are correct to use the Perron's formula. Namely, we get that</p>
<p><span class="math-container">\begin{align*}
\frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty}\log(\zeta(s))\frac{x^s}{s}ds&=\frac{1}{2 \pi i}\int_{c-i\infty}^{c+i\infty}\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^s\log(n)}\frac{x^s}{s}d... |
3,956,963 | <p>What is the difference between countable infinity and uncountable infinity? Are there any examples? How can I imagine it?
Can you offer some assistance? please.</p>
| nima | 852,926 | <p>For a finite set <span class="math-container">$A$</span> the size of <span class="math-container">$A$</span> or <span class="math-container">$card(A)$</span> is the number of elements in <span class="math-container">$A$</span>. Now, what's the size of infinite sets? Or first: what's the definition of infinite set?<b... |
3,336,886 | <p>How do I factorize this thing?<br>
<span class="math-container">$x^4+x^2+1$</span><br>
I tried to solve the integral <span class="math-container">$\int{\frac{1}{x^4+x^2+1}}$</span> and after trying some substitutions that did not work, I plugged the integral into an integral calculator and it turns out that <span cl... | battletwink69 | 682,764 | <p>You are right! But look at it this way:</p>
<p><span class="math-container">$a = x^2 + 1$</span></p>
<p><span class="math-container">$b = x$</span></p>
|
394,665 | <p>I'm wondering based on the definition of monotonicity:</p>
<blockquote>
<p>A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. </p>
</blockquote>
<p>So given that the sequence $a_n = 3$ is all the same numbers and is neither increasing or decreasing, is it monotonic? </p>
| dereje tigabu | 145,736 | <p>yes, because constant sequence is both increasing and decreasing sequence. so that it is monotonic.</p>
|
185,295 | <p>I would like to solve the following equation <span class="math-container">$y^2=x^2+ax^2y^2+by^2x^3+cy^3x^2$</span> where <span class="math-container">$a,b,c$</span> are small, so <span class="math-container">$y\approx x+O(x^3)$</span>. I would like to have a series approximation of the solution rather than an exact ... | AccidentalFourierTransform | 34,893 | <p>Quick-and-dirty solution:</p>
<pre><code>y^2 - (x^2 + a x^2 y^2 + b y^2 x^3 + c y^3 x^2) /.
y -> x + c1 x^2 + c2 x^3 + c3 x^4 + O[x]^5 // FullSimplify
Solve[% == 0, {c1, c2, c3}]
y -> x + c1 x^2 + c2 x^3 + c3 x^4 + O[x]^5 /. % // TeXForm
</code></pre>
<blockquote>
<p><span class="math-container">$$... |
185,295 | <p>I would like to solve the following equation <span class="math-container">$y^2=x^2+ax^2y^2+by^2x^3+cy^3x^2$</span> where <span class="math-container">$a,b,c$</span> are small, so <span class="math-container">$y\approx x+O(x^3)$</span>. I would like to have a series approximation of the solution rather than an exact ... | Michael E2 | 4,999 | <p>Substitute <span class="math-container">$y=xu$</span> to handle the node at the origin, differentiate and use <a href="https://reference.wolfram.com/language/ref/AsymptoticDSolveValue.html" rel="nofollow noreferrer"><code>AsymptoticDSolveValue</code></a>:</p>
<pre><code>x * AsymptoticDSolveValue[{
D[y^2 == (x^2... |
2,470,463 | <p>If $x+iy=\sqrt\frac{1+i}{1-i}$, where $x$ and $y$ are real, prove that $x^2+y^2=1$</p>
<p>I tried multiplying $\sqrt{(\frac{1+i}{1-i})(\frac{1+i}{1+i})}=\sqrt{i}$ but I'm not sure what to do after</p>
<p>thanks in advance :)))</p>
| Bernard | 202,857 | <p>Just observe that numerator and denominator have the same modulus since $1+i$ and $1-i$ are conjugate, and the modulus is a multiplicative function.</p>
|
18,772 | <p>Suppose $G$ is a semisimple group, and $V_{\lambda}$ is an irreducible finite-dimensional representation of highest weight $\lambda$. Suppose $H \subset G$ is a semisimple subgroup. What is the multiplicity of the trivial representation in $V_{\lambda}|_{H}$? Is there a simple way to read this off from $\lambda$ ... | Bruce Westbury | 3,992 | <p>I would like to add to the answers by Ben and Allen. First if we extend the question to include all multiplicities and not just the multiplicity of the trivial representation then there are a number of special cases that are of interest:</p>
<p>1.Take $H$ to be the trivial group then the question asks for the dimen... |
1,118,269 | <p>I'm reading: <em>Mathematical thought from ancient to modern times by Kline</em>. My question is about this pasasge:</p>
<blockquote>
<p>Beyond its achievements in subject matter, the nineteenth century
reintroduced rigorous proof. No matter what individual mathematicians may
have thought about the soundness ... | Nodt Greenish | 614,664 | <p>If you read proofs by Euler or Gauss for example they don't use induction as we do today but talk about infinite descending - what is true for small numbers has to be true for big numbers. Maybe that gives you an impression what is meant by empirical proofs. </p>
<p>Modern set theory and Zermelo-Fraenkel came up ar... |
1,951,733 | <blockquote>
<p>Can any polynomial $P\in \mathbb C[X]$ be written as $P=Q+R$ where $Q,R\in \mathbb C[X]$ have all their roots on the unit circle (that is to say with magnitude exactly $1$) ? </p>
</blockquote>
<p>I don't think it's even trivial with degree-1 polynomials... In this supposedly simple case, with $P(X)=... | Benson Lin | 371,844 | <p>The reason why you can simply sum the digits of a number to test for divisibility by 3 is because for all integers $n \ge 0$:</p>
<p>$$10^n \equiv 1 \pmod 3$$</p>
<p>To see why this is true, we know that $10^1 \equiv 1 \pmod 3$</p>
<p>Thus: </p>
<p>$$
10^n = 10*10*10 \cdots\\
\hspace{2.1cm} \equiv 1*1*1\cdots \... |
1,520,118 | <p>I'm not sure what I have done here. I'm guessing it's something to do with the absolute values but I really have no idea what it is.</p>
<p><a href="https://i.stack.imgur.com/yBrKq.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yBrKq.jpg" alt="enter image description here"></a></p>
<blockquote>... | HDE 226868 | 170,257 | <p>The error is on the left.</p>
<p>You went from
$$|y|>1$$
to
$$\left|\frac{1}{y}\right|>1$$
This is incorrect. You should have divided both sides by $|y|$, leaving
$$1>\frac{1}{|y|}$$
This agrees with the right-hand result.</p>
<p>However, something happened with the absolute value. Try the case where $y=-... |
3,917,129 | <p>I have some problems trying to solve this exercise: <br />
Consider the following Cauchy problem: <span class="math-container">$\left\{\begin{matrix}x'=t+\frac{y}{1+x^2}
\\ y'=txe^{-ty^2}
\\ x(0)=y(0)=1
\end{matrix}\right.$</span><br />
Discuss the existence of a solution and its uniqeness. Then I'm asked to prove o... | xpaul | 66,420 | <p>Lutz Lehmann gives a good solution. Here is another one which uses argument by contradiction. Suppose there is <span class="math-container">$t_0\in(0,\infty)$</span> such that
<span class="math-container">$$ \lim_{t\to t_0^-}\|(x(t),y(t))\|=\infty. $$</span>
Let <span class="math-container">$u=\|(x,y)\|^2$</span> an... |
2,451,327 | <p>I have the following propositions and statement</p>
<p>\begin{align}
1. & &q \to \neg p\\
2. & &p\vee s\\
3. && \neg q \to \neg r\\
4. && r\\
\therefore &&s
\end{align}</p>
<p>I need to demonstrate that is true.</p>
| Robert Z | 299,698 | <p>Recall the definition of <a href="https://en.wikipedia.org/wiki/Dense_set" rel="nofollow noreferrer">dense set</a>: we have to show that FOR ALL $a,b \in \mathbb{R}$ there EXISTS $x\in A$ such that $a<x<b$. The density of $\mathbb{Q}$ is too weak here because $A$ is a proper subset of $\mathbb{Q}$.</p>
<p>Hin... |
168,619 | <blockquote>
<p>Can $a^2+b^2+2ac$ be a perfect square if $c\neq \pm b$? </p>
</blockquote>
<p>$a,b,c \in \mathbb{Z}$.<br>
I have tried some manipulations but still came up with nothing. Please help. </p>
<p>Actual context of the question is:<br>
Let say I have an quadratic equation $x^2+2xf(y)+25$ that I have to m... | individ | 128,505 | <p>the equation: $y^2=a^2+b^2+2ac$</p>
<p>Has a solution:</p>
<p>$a=p^2-2(q+t)ps+q(q+2t)s^2$</p>
<p>$b=p^2+2(t-q)ps+(q^2-2t^2)s^2$</p>
<p>$c=p^2+2(q-t)ps-q^2s^2$</p>
<p>$y=2p^2-2(q+t)ps+2t^2s^2$</p>
<p>Has a solution:</p>
<p>$a=-p^2-2(q+t)ps+(8t^2-2qt-q^2)s^2$</p>
<p>$b=-p^2+2(5t-q)ps+(2t^2-8qt-q^2)s^2$</p>
<p... |
425,611 | <p>If $$10^{20} +20^{10}$$ is divided with 4 then what would be its remainder?</p>
| Américo Tavares | 752 | <p>Since
$$
\begin{eqnarray*}
10^{20}+20^{10} &=&\left( 10^{10}\right) ^{2}+2^{10}10^{10} \\
&=&10^{10}\left( 10^{10}+2^{10}\right) \\
&=&2^{10}5^{10}\left( 2^{10}5^{10}+2^{10}\right) \\
&=&2^{10}2^{10}5^{10}\left( 5^{10}+1\right) \\
&=&4^{10}5^{10}\left( 5^{10}+1\right) \\
&... |
2,385,599 | <p>ABC is a triangle. D is the center of BC . AC is perpendicular to AD. prove that $$\cos(A)\cdot \cos(C)=\frac{2(c^2-a^2)}{3ac}$$
problem and my attempts are shown in images. I cannot find the exact way to the answer.</p>
<p><a href="https://i.stack.imgur.com/PUcka.jpg" rel="nofollow noreferrer"><img src="https://i... | Dr. Sonnhard Graubner | 175,066 | <p>HINT: it is $$\left(\left(1+\frac{1}{n}\right)^n\right)^{n+1}$$</p>
|
298,259 | <p>This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?"
A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-bounce spectrum.<sup>1</sup> </p>
<p>Suppose there is a polygon $P$ hidden inside a disk $D$. All its edges are mirrors... | Newton fan 01 | 123,488 | <p>my answer for Q1:</p>
<p><em>"If the ray hits a vertex, it dies".</em>
I take this property as the starting point of my solution</p>
<p>Let two neighboring disks D1 and D2 contain two such incongruent equireflective polygons.
Let us take a vertex V1 of polygon P1; all rays hitting V1 will die; same behavior for... |
2,248,754 | <p>Take $b>a>1$ By considering $x^{-y}$ over $(1,\infty)\times (a,b)$, show that $$\int_{1}^{\infty}\frac{x^{-a}-x^{-b}}{\log(x)}dx$$ exists and find its value</p>
<p>I've assumed they want me to write the intagral as $$\int_{1}^{\infty}\int_{a}^{b}\frac{yx^{-y-1}}{\log(x)}dxdy$$ and use Tonelli's Theorem to jus... | Simply Beautiful Art | 272,831 | <p>Let $x=e^u$ to get</p>
<p>$$I=\int_0^\infty\frac{e^{-(a-1)u}-e^{-(b-1)u}}u\ du$$</p>
<p>This happens to be <a href="http://mathworld.wolfram.com/FrullanisIntegral.html" rel="nofollow noreferrer">Frullani's integral</a>, and one easily finds that</p>
<p>$$I=\ln\left(\frac{b-1}{a-1}\right)$$</p>
|
2,248,754 | <p>Take $b>a>1$ By considering $x^{-y}$ over $(1,\infty)\times (a,b)$, show that $$\int_{1}^{\infty}\frac{x^{-a}-x^{-b}}{\log(x)}dx$$ exists and find its value</p>
<p>I've assumed they want me to write the intagral as $$\int_{1}^{\infty}\int_{a}^{b}\frac{yx^{-y-1}}{\log(x)}dxdy$$ and use Tonelli's Theorem to jus... | preferred_anon | 27,150 | <p>How about instead,
$$\int_{1}^{\infty}\frac{x^{-a}-x^{-b}}{\log(x)}dx=\int_{1}^{\infty}\int_{a}^{b}x^{-y}dydx$$
EDIT: $\int x^{-y} dy = \int e^{-y\log(x)}= \frac{-1}{\log(x)}e^{-y\log(x)}= \frac{-1}{\log(x)}x^{-y}$.</p>
|
2,681,107 | <p>Given that $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfies </p>
<p>$2f^3(x)-3=2x-3f(x)$ , $x\in \mathbb{R}$, show that $f$ is continuous on $\mathbb{R}$.</p>
<p>How can we handle this problem?</p>
| Peter Szilas | 408,605 | <p>Hint:</p>
<p>MVT:</p>
<p>Suppose $f$ continuous on $[a,b$] , $f$ differentiable differentiable in $(a,b)$, then there is a point $t \in (a,b)$ with</p>
<p>$\dfrac{f(b)-f(a)}{b-a}=f'(t) \ge 1.$</p>
<p>1) Choose $x=b, a=0$, for example, and cosider $x \rightarrow \infty.$</p>
<p>2) Similarly : $x \rightarrow -\i... |
22,078 | <p>It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but assume smoothness for what follows.) The proof makes essential use of $k$ being a field, insofar as it uses freen... | Ben Wieland | 4,639 | <p>Let's assume unipotent and characteristic zero (and eventually commutative), so that we can pass to Lie algebras. After using smoothness to choose coordinates, a deformation of $G_0$ over the dual numbers is controlled by a cocycle $\mathfrak g \otimes \mathfrak g\to \mathfrak g$, where $\mathfrak g$ is the Lie alge... |
825 | <p>What resources are available for any grade level K- 12 that are aligned with the Common Core Mathematics Standards and Mathematical Practices that have sets of problems or problem banks that can be used by teachers for instruction or homework?</p>
| David Wees | 254 | <p>Can we include rich math tasks in our answers? If so, look at:</p>
<ul>
<li><a href="http://visualpatterns.org">http://visualpatterns.org</a></li>
<li><a href="http://graphingstories.com">http://graphingstories.com</a></li>
<li><a href="http://map.mathshell.org">http://map.mathshell.org</a></li>
<li><a href="http:/... |
168,040 | <p>Is there an algorithm which on input "$(a,p)$" (where $0\leq a<p$ are integers) takes time polynomial in $\log p$ and outputs "NOT PRIME" if $p$ is not prime and otherwise outputs the Legendre symbol $(a/p)$?</p>
<p>By the <a href="http://en.wikipedia.org/wiki/AKS_primality_test" rel="nofollow noreferrer">AKS pr... | Henry Cohn | 4,720 | <p>For quadratic reciprocity, you need the <a href="http://en.wikipedia.org/wiki/Jacobi_symbol">Jacobi symbol</a>. It's an extension of the Legendre symbol to composite $p$ that still satisfies quadratic reciprocity, so you can just apply quadratic reciprocity to your heart's content without worrying about primality. ... |
4,128,110 | <p>Find the <span class="math-container">$\max$</span> and the <span class="math-container">$\min$</span> with Lagrange multipliers, given <span class="math-container">$$f(x,y,z)=xyz^2,$$</span> <span class="math-container">$$g(x,y,z)=x^2+y^2+z^2-1=0.$$</span></p>
<p><a href="https://i.stack.imgur.com/Nr8G8.jpg" rel="... | JMoravitz | 179,297 | <p>Letting <span class="math-container">$A$</span> be the event the bus is late and <span class="math-container">$B$</span> the event the train is late. Martin being late corresponds (<em>presumably</em>) to the event <span class="math-container">$A\cup B$</span>, the bus or train being late. Assuming what he has exp... |
3,290,047 | <p>I understand the solution of <span class="math-container">$m^{2}+1=0$</span> is <span class="math-container">$\iota$</span>. However for sure this solution (<span class="math-container">$(m^{2}+1)^2=0$</span>) should contain four roots. The answer reads <span class="math-container">$\pm \iota$</span> and <span class... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong> <span class="math-container">$(m^2+1)^2=(m^2+1)(m^2+1)=(m+i)(m-i)(m+i)(m-i)$</span></p>
|
87,463 | <p>Hi all,</p>
<p>I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+1}}$.</p>
<p>I tried the way of setting: $x_n=f(n)$, and use the 1st order taylor expansion of $f(n+1)$ and $... | Per Alexandersson | 1,056 | <p>I decided to compute the ratio $x_{30}/x_{29}$ for various start values $x_0 = x_1 = s$
For $s>0.42$, the computations overflows for me, so I could not compute that part.</p>
<p>The image shows the ratio on the y axis, and start value on the x axis.
The images are essentially identical for $x_{31}/x_{30}$, so it... |
87,463 | <p>Hi all,</p>
<p>I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+1}}$.</p>
<p>I tried the way of setting: $x_n=f(n)$, and use the 1st order taylor expansion of $f(n+1)$ and $... | Barry Cipra | 15,837 | <p>Not really the answer you're looking for, but possibly helpful: The fact that the sequence $0,0,1/4,0,0,1/4,0,0,1/4,...$ satisfies the recursion equation offers a glimpse into what's making things hard here. At the very least, there are starting pairs $(x_1,x_2)$ close to $(0,0)$ that stay close to this three-peat... |
585,924 | <p>Given a function $$F(x) = \int_0^x \frac{t + 8}{t^3 - 9}dt,$$ is $F'(x)$ different when $x<0$, when $x=0$ and when $x>0$? </p>
<p>When $x<0$, is $$F'(x) = - \frac{x + 8}{x^3 - 9}$$ ... since you can't evaluate an integral going from a smaller number to a bigger number? That's what I initially thought, but ... | egreg | 62,967 | <p>No, we have
$$F'(x)=\frac{x+8}{x^3-9}$$
for all $x<\sqrt[3]{9}$. The limitation is due to the fact that the integral is meaningful only when the interval doesn't contain $\sqrt[3]{9}$ and so we must consider only the interval $(-\infty,\sqrt[3]{9})$ that contains $0$.</p>
<p>If $b<a$, one sets, <em>by definit... |
585,924 | <p>Given a function $$F(x) = \int_0^x \frac{t + 8}{t^3 - 9}dt,$$ is $F'(x)$ different when $x<0$, when $x=0$ and when $x>0$? </p>
<p>When $x<0$, is $$F'(x) = - \frac{x + 8}{x^3 - 9}$$ ... since you can't evaluate an integral going from a smaller number to a bigger number? That's what I initially thought, but ... | Christian Blatter | 1,303 | <p>Given any interval $I\subset{\mathbb R}$ and a continuous function $f:\>I\to{\mathbb R}$ the <em>indefinite integral</em> of $f$ over $I$ is the set of all functions $F:\>I\to{\mathbb R}$ (called <em>primitives</em> of $f$) that satisfy
$$F'(x)=f(x)\qquad\forall x\in I\ .$$
This set depends only on ($I$ and) $... |
3,013,384 | <p>Let <span class="math-container">$f(x) = \frac{1}{2}\langle Ax,x\rangle - \langle b,x \rangle + c$</span> with <span class="math-container">$A\in \mathbb{R}^{n\times n}$</span> and <span class="math-container">$b\in \mathbb{R}^n$</span>, <span class="math-container">$c\in \mathbb{R}$</span>. Assume that <span class=... | Calvin Khor | 80,734 | <p>Its easier to complete the square, but you can compute derivatives if you liked.
You need to know the following identities (using that <span class="math-container">$A$</span> is symmetric positive definite):
<span class="math-container">$$\nabla_x \langle b,x\rangle = b\\
\nabla_x \frac12\langle x,Ax \rangle\cdot ... |
1,349,656 | <blockquote>
<p>If
$f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$
find the density of their average $\frac{X+Y+Z}{3}$</p>
</blockquote>
<p>I'm a little lost on how to solve this exercise, $f_{X,Y,Z}(x,y,z)$ It looks like the product of three exponential random variables $X\sim... | Community | -1 | <ol>
<li>$X,Y,Z\sim_{iid}\text{Exp}(1)$ because $F_{X,Y,Z}(x,y,z)=F_X(x)F_Y(y)F_Z(z)$ for all $(x,y,z)\in\mathbb{R}^3$.</li>
<li>Denote $S_n=\sum_{i=1}^nX_i$ where $X_i\sim_{iid}\text{Exp}(\lambda)$. Distribution of $S_n/n$ follows $\text{Gamma}(n,n\lambda)$ because</li>
</ol>
<p>$$\varphi_{S_n/n}(t)=\left[\varphi_{X_... |
2,426,659 | <p>I have a very basic question on wether or not we use the axiom of choice when we prove the very simple fact that the union of open sets of $\mathbb{R}$ (defined as unions of open intervals) is an open set of $\mathbb{R}$. </p>
<p>Say that $(U_i)_{i\in I}$ is a family of open sets of $\mathbb{R}$. So each $U_i$ is o... | Rob Arthan | 23,171 | <p>So your definition of an open set in $\Bbb{R}$ is a set that can be written as a union of open intervals. This generalises: given a set $X$ and a system $\cal B$ of subsets of $X$ such that if $A, B \in \cal B$ then $A \cap B \in \cal B$, there is a topology $\cal T$ on $X$ whose open sets are the sets of the form ... |
629,550 | <p>Investigate the convergence of $\sum a_n$ where $a_n = \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx$.</p>
<p>We have thought about using the dominated convergence theorem to find $\lim a_n$, but that would result in something like $\lim a_n = \lim \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx = \... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
2,417,736 | <p>Someone told me that the following formula holds for $f$ differentiable and decreasing, with $\lim_{x\rightarrow +\infty}{f(x)}=0$.</p>
<blockquote>
<p>$$\sum_{k=n}^{\infty}{f(k)} = \int_{n}^{\infty}{f(t)dt} +\frac{f(n)}{2}+\mathcal{O}(f'(n))$$</p>
</blockquote>
<p>But I managed to prove only if the function is ... | An aedonist | 143,679 | <p>I believe a proof can go as follows. Let us for the moment use $M$ as upper limit for the series and the integral. We are looking for a bound on the difference between the series and the integral.
$$\begin{align}\sum_{k=n}^{M} f(k) - \int_{n}^{M} f(x)\mathrm{d}x &= \frac{1}{2}[f(M) + f(n)] + \sum_{j=n}^{M-1} \l... |
180,937 | <p>Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality
$$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?</p>
| Community | -1 | <p>This is as far as I got... </p>
<p>$\frac{(1-b)\sqrt{a}}{(1-b)(1-a)}$ + $\frac{(1-a)\sqrt{b}}{(1-a)(1-b)}$ + $\frac{\sqrt{c}}{(1-c)}$ $\geq$ $\frac{3\sqrt{3}}{2} $</p>
<p>$\Leftrightarrow$ </p>
<p>$\frac{(1-b)\sqrt{a} + (1-a)\sqrt{b}}{(1-b)(1-a)}$ + $\frac{\sqrt{c}}{(1-c)}$ $\geq$ $\frac{3\sqrt{3}}{2} $</p>
<p>... |
180,937 | <p>Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality
$$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?</p>
| Xeing | 48,358 | <p>$\sqrt{a} = x, b=y^2, c=z^2 => x^2+y^2+z^2=1$
We have to prove $$\frac{x}{y^{2}+z^{2}}+\frac{y}{x^{2}+z^{2}}+\frac{z}{x^{2}+y^{2}}\geq \frac{3\sqrt{3}}{2}$$:
$$\frac{2\sqrt{3}}{3}x\left ( y^{2}+z^{2} \right )\leq \left ( x^{2}+\frac{1}{3} \right )\left ( y^{2}+z^{2} \right )\leq \frac{\left ( x^{2}+y^{2}+z^{2}+\f... |
127,643 | <p>Hello everybody,</p>
<p>I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative algebra one needs to push himself some deeper than the elementary subjects. This can be seen just counting th... | Dmitry Vaintrob | 7,108 | <p>I want to offer a possibly heretical opinion based on conversations I've had with people who do algebraic geometry, especially Joe Harris. I think that it is not necessary to know very much commutative algebra in order to study and understand algebraic geometry. The fundamental objects algebraic geometry studies are... |
2,580,232 | <p>Suppose that $A$ and $B$ are vector subspaces of $V$, and let $C$ and $D$ be bases for $A$ and $B$, respectively.</p>
<p>Then is it true that </p>
<ol>
<li>$C \cup D$ is a basis for $A+B$?</li>
<li>$\operatorname{dim}(A+B) \le |C| + |D|$ (where $|\cdot|$ denotes cardinality)?</li>
</ol>
<p>I am really bad at dime... | Guy Fsone | 385,707 | <p>$$f′(x)=e^x+f(x)\Longleftrightarrow f'-f =e^x\Longleftrightarrow(fe^{-x})'=1\Longleftrightarrow fe^{-x}=x-c\Longleftrightarrow f(x)= e^x(x+c)$$</p>
|
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Roland Bacher | 4,556 | <p>There where a few famous 'amateurs' in the past: Fermat worked as lawyer (but at his time there were of course (almost) no permanent posititions).</p>
<p>Ramanujan had no formal instruction.</p>
<p>(The list is certainly much longer.)</p>
<p>I think the following tipps (with some overlap over previous answers) are u... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Hollis Williams | 119,114 | <p>Even most low-grade office jobs outside of academia these days are surprisingly high pressure and stressful. Your performance is constantly monitored and analysed. I would be surprised if jobs in industry were less high pressure.</p>
<p>If you want to continue mathematics on the side and are certain you can get resu... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Marco | 481,829 | <p>Although I studied Economics (about 15 years ago), never attending any math lecture at the university, I have finally published some original research papers in peer-review journals (about number theory and graph theory, mainly) after some hard work as an autodidact (studying online without any external help), focus... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Zinklestoff | 482,917 | <p>I have this image of science as an expanding sphere, with ever more unsolved problems, but at an ever larger distance from the center. In reality it is probably some kind of curved high dimensional manifold. Maybe there are some nooks and crannies you can find as an amateur, I couldn't really say. But if you enjoy l... |
4,519,350 | <p>Let
<span class="math-container">$$
I_k=\int_0^\infty (t+a)^k e^{-t}\exp\left(-\frac{(t-\mu)^2}{2\sigma^2}\right)\,\mathrm dt,
$$</span>
with <span class="math-container">$k\in\Bbb N_0$</span> and <span class="math-container">$a>0$</span>. Since <span class="math-container">$k$</span> is an integer we can expand ... | Jan Eerland | 226,665 | <blockquote>
<p>Still working on it. Too big for a comment.</p>
</blockquote>
<p>Well, we are trying to solve the following integral:</p>
<p><span class="math-container">$$\mathcal{I}_\text{k}\left(\alpha,\mu,\sigma\right):=\int\limits_{0}^{\infty}\left(x+\alpha\right)^\text{k}\exp\left(-x\right)\exp\left(-\frac{1}{2}\... |
384,501 | <p>$A= \left[ \begin{array}{ccc}
3 & -1 & 2 \\
-6 & 2 & 4 \\
-3 & 1 & 2 \end{array} \right]$</p>
<p>Applying, $R_{3}-\frac{1}{2}R_{2}$</p>
<p>~ $A= \left[ \begin{array}{ccc}
3 & -1 & 2 \\
-6 & 2 & 4 \\
0 & 0 & 0 \end{array} \right]$</p>
<p>Applying, $R_{2}+2R_{1}$</p>
... | amWhy | 9,003 | <p>Since you have a row of zeros, the rank is at most $2$: two non-zero rows. If the matrix can be reduced further, you may have another row of zeros, in which case one non-zero row remains: rank 1.</p>
<p>ADDED: Given your work/edit, how many NON-ZERO rows remain? That gives you the <strong>rank</strong> of the origi... |
2,713,293 | <p>Let</p>
<p>$$f(x_1, x_2, x_3) := \sum m(2, 3, 4, 5, 6, 7)$$</p>
<p>With the normal SOP expression for this function, it must be, with the use of minterm:</p>
<p>$$f = m_2 + m_3 + m_4 + m _6 + m_7 = x_1'x_2x_3'+x_1'x_2x_3+x_1x_2'x_3'+x_1x_2x_3'+x_1x_2x_3$$</p>
<p>How can this function be simplified?</p>
<p>(I co... | Bram28 | 256,001 | <p>First, you're missing $m_5=x_1x_2'x_3$</p>
<p>To simplify, either use a Karnaugh Map, or us some handy-dandy equivalences principles like:</p>
<p><strong>Adjacency</strong></p>
<p>$PQ + PQ' = P$</p>
<p>For example, $x_1'x_2x_3'+x_1'x_2x_3=x_1'x_2$, $x_1x_2'x_31+x_1x_21x_3=x_1x_2$, and $x_1x_2x_3'+x_1x_2x_3=x_1x_... |
2,432,128 | <p>If there are $5,000,000$ couples in a city, and the probability that a couple matches a specific description is $1\over 12,000,000$, what are the chances that there are two couples that match the specific description given that there is at least one couple that matches the description?</p>
<p>I guess I'm supposed t... | hmakholm left over Monica | 14,366 | <p>Let the random variable $X$ be the number of matching couples. You're then looking for (I think, it is not quite clear from the problem statement)
$$ P(X \ge 2 \mid P\ge 1) $$
which clearly equals
$$ 1 - P(X=1 \mid X\ge 1) = 1 - \frac{P(X=1)}{P(X\ge 1)} $$</p>
<p>Using the binomial distribution, we get
$$ P(X=1) = ... |
1,268,103 | <p>Definition: Let $X$ be a topological space and let $\sim_C$ be the equivalence relation on $X$ defined by $x \sim_C y$ if $x$ and $y$ lie in a connected subset of $X$. The components of $X$ are the equivalence classes of the equivalence relation $\sim_C$. </p>
<p>Question: Prove that each component of $X$ is a clos... | Matt Samuel | 187,867 | <p>Let $C$ be a connected component. Let $x\in X$ be outside $C$; then $B=C\cup \{x\}$ is not connected. Let $U,V$ form a separation of $B$. Suppose $x\in U$; I claim that $U\cap C=\emptyset$. If not, then $U\cap C,V\cap C$ would form a separation of $C$, which is not possible since $C$ is connected. Thus $U\cap C=\emp... |
4,247,968 | <p>I'm currently studying Euler's formula and the representation of</p>
<p><span class="math-container">$\frac{\sin(x)}x$</span> as <span class="math-container">$\frac{1}{2ix}(e^{ix}-e^{-ix})$</span></p>
<p>and my understanding would be taking the limit as <span class="math-container">$x$</span> approaches <span class=... | Alessio K | 702,692 | <p><a href="https://en.wikipedia.org/wiki/Euler%27s_formula" rel="nofollow noreferrer">Euler's formula</a> states that <span class="math-container">$e^{ix}=\cos(x)+ i\sin(x)$</span> for any real number <span class="math-container">$x$</span>, where <span class="math-container">$ i=\sqrt{-1}$</span> is the imaginary uni... |
4,247,968 | <p>I'm currently studying Euler's formula and the representation of</p>
<p><span class="math-container">$\frac{\sin(x)}x$</span> as <span class="math-container">$\frac{1}{2ix}(e^{ix}-e^{-ix})$</span></p>
<p>and my understanding would be taking the limit as <span class="math-container">$x$</span> approaches <span class=... | user | 505,767 | <p>We have that by standard limit <span class="math-container">$\frac{e^x-1}x \to 0$</span> as <span class="math-container">$x \to 0$</span> we obtain</p>
<p><span class="math-container">$$\frac{\sin x}x=\frac{e^{ix}-e^{-ix}}{2xi}=\frac1{e^{ix}}\frac{e^{2ix}-1}{2ix}\to 1\cdot 1=0$$</span></p>
|
2,512,212 | <p>When we work on a metric space, the concepts of open set is defined as</p>
<blockquote>
<p>A subset $A$ of the metric space (X,d) is called open iff $Int A = A$</p>
</blockquote>
<p>And in topology books (see Munkress), a open set is defined as </p>
<blockquote>
<p>We say that a subset $U$ of $X$ is an open s... | hmakholm left over Monica | 14,366 | <p>Yes, these define the same sets -- but for a decidedly boring reason: When $X$ is a metric space, the $\tau$ that appears in the second definiton is <em>defined to</em> consist of exactly those subsets of $X$ that satisfy the <em>first</em> definition.</p>
<p>Of course we could use all sorts of <em>other</em> $\tau... |
3,614,178 | <p>I was going through this article on geometric series on Wikipedia and found this diagrammatic representation of an infinite geometric series with a said common factor of (1/2) but shouldn't the common factor be (1/4) based on the diagram?
link to the diagram:
<a href="https://commons.wikimedia.org/wiki/File:Geometr... | Calum Gilhooley | 213,690 | <p>As there are now several explicit answers (there were none before), I'll expand my earlier comment:
<span class="math-container">$$
\log_5{17} > \log_5{15} = 1 + \log_53 > 1 + \frac23 = \frac53 > \log_23,
$$</span>
because
<span class="math-container">$$
3^3 = 27 > 25 = 5^2
$$</span>
and
<span class="mat... |
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