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 |
|---|---|---|---|---|
839,043 | <p>I have been studying power functions, and started to think about imaginary powers. Take the function $x^i$. Because I don't know how to multiply a number $i$ times, I tried to simplify the equation</p>
<p>$x^i = x^{\sqrt{-1}} = x^{(-1)^{1/2}}$</p>
<p>Then, using the property of exponents that states an exponent to... | Graham Kemp | 135,106 | <blockquote>
<p>Then, using the property of exponents that states an exponent to an exponent is the two multiplied, </p>
</blockquote>
<p>This is where you go wrong. You are thinking of $\left(a^b\right)^c=a^{(bc)}$, but you have $a^{(b)^c}$. That's not the same thing at all.</p>
<p>Thus $x^{(-1)^{1/2}}\neq x^{-1... |
4,501,286 | <blockquote>
<p>If a, b, c are positive real numbers such that <span class="math-container">$a^2+ b^2+ c^2 = 1$</span>
<br>Show that:
<span class="math-container">$$\frac{1}{a} +\frac{1}{b} +\frac{1}{c}+a +b +c \geq 4\sqrt{3}.$$</span></p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>First , I used Holder's : <... | Mengchun Zhang | 420,459 | <p>By your argument, we already have
<span class="math-container">$$\frac1a+\frac1b+\frac1c+a+b+c\geq\frac9{a+b+c}+\frac{(a+b+c)^3}3=\frac9u+\frac{u^3}3\equiv f(u)$$</span>
where <span class="math-container">$u=a+b+c>0$</span>. Try to show that <span class="math-container">$f'(x)=0$</span> only has one root <span cl... |
1,013,346 | <p>Given a box which contains $3$ red balls and $7$ blue balls. A ball is drawn from the box and a ball of the other color is then put into the box. A second ball is drawn from the box, What is the probability that the second ball is blue? </p>
<p>could anyone provide me any hint? </p>
<p>Please, don't offer a comple... | Marco Cantarini | 171,547 | <p>You can also use the well known summation formula $$\sum_{n\in\mathbb{Z}}f\left(n\right)=-\sum\left\{ \textrm{residue of }\pi\cot\left(\pi z\right)f\left(z\right)\textrm{ at }f\left(z\right)\textrm{ poles}\right\}
$$ which is a consequence of the residue theorem. So it is sufficient to note that $$ \sum_{n\geq1}\f... |
81,267 | <p>I have the following problem: I have a (a lot)*3 table, meaning that I have 3 columns, say X, Y and Z, with real values. In this table some of the rows have the same (X,Y) values, but with different value of Z. For instance</p>
<pre><code>{{12.123, 4.123, 513.423}, {12.123, 4.123, 33.43}}
</code></pre>
<p>have th... | kglr | 125 | <pre><code>f1 = Composition[DeleteDuplicatesBy[Most], SortBy[Last]]
SeedRandom[1]
data = Append @@@ Transpose[{RandomInteger[2,{10,2}], RandomReal[1,{10}]}];
data2 = f1 @ data;
Row[MatrixForm /@ {data, data2}]
</code></pre>
<p><img src="https://i.stack.imgur.com/TVshL.png" alt="Mathematica graphics"></p>
<p>Alternat... |
3,609,906 | <p>I need to compute <span class="math-container">$$\lim_{n \to \infty}\sqrt{n}\int_{0}^{1}(1-x^2)^n dx.$$</span>
I proved that
for <span class="math-container">$n\ge1$</span>,
<span class="math-container">$$\int_{0}^{1}(1-x^2)^ndx={(2n)!!\over (2n+1)!!},$$</span>
but I don't know how to continue from here.</p>
<p>I a... | Ninad Munshi | 698,724 | <p>Let <span class="math-container">$y = \sqrt{n}x$</span>. Then we have that</p>
<p><span class="math-container">$$\lim_{n\to\infty} \int_0^{\sqrt{n}} \left(1-\frac{y^2}{n}\right)^n\:dy \longrightarrow \int_0^\infty e^{-y^2}\:dy = \frac{\sqrt{\pi}}{2}$$</span></p>
<p>by dominated convergence.</p>
<hr>
<p><span cla... |
255,217 | <p>Let a polynomial with integer coefficients be nice if</p>
<ol>
<li>this polynomial has distinct integer roots;</li>
<li>its derivative has also integer roots.</li>
</ol>
<p>For instance
<span class="math-container">$$p(x)=x(x-9)(x-24),\\
p'(x)=3(x-4)(x-18)$$</span>
is the smallest known nice cubic polynomial. Smalle... | Cesareo | 62,129 | <p>We can have an insight about the solutions behavior for each <code>m</code> by doing</p>
<pre><code>tmax = 1000;
solution = ParametricNDSolve[{-((m (1 + m) + 4/(9 (-2/3 + t) t)) y[t]) + 2 (-1/3 + t) y'[t] + (-2/3 + t) t y''[t] == (-4 (1 + c/2))/(9 (-2/3 + t) t), y[1] == 1, y'[1] == c}, y, {t, 1, tmax}, {m, c}]
m = ... |
255,217 | <p>Let a polynomial with integer coefficients be nice if</p>
<ol>
<li>this polynomial has distinct integer roots;</li>
<li>its derivative has also integer roots.</li>
</ol>
<p>For instance
<span class="math-container">$$p(x)=x(x-9)(x-24),\\
p'(x)=3(x-4)(x-18)$$</span>
is the smallest known nice cubic polynomial. Smalle... | user64494 | 7,152 | <p>Here is an answer for <code>m==2</code> in version 13.1 on Windows 10 (<code>C</code> is reserved in WL.)</p>
<pre><code>ClearAll[m, c]; $Assumptions = {m \[Element] Integers, c \[Element] Reals}; m = 2;
solution = AsymptoticDSolveValue[{-((m (m + 1) + 4/(9 (-2/3 + t) t)) y[t]) +
2 (-1/3 + t) y'[t] + (-2/3 + t) t... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Community | -1 | <p>Historically, quantificational logic came before propositional logic.</p>
<p>So maybe start with <strong>quantified</strong> sentences instead. It is reasonable to say both <em>"Every ten-foot-tall person plays basketball"</em> and <em>"No ten-foot-tall person plays basketball"</em>.</p>
<p>This leads to acceptin... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Benoît Kloeckner | 187 | <p>Let me add the tricks I used already to the interesting ones proposed in the other answers. They are partly, but not completely redundant.</p>
<p>First, implication is often best understood in quantified propositions; <a href="https://matheducators.stackexchange.com/a/368">this has already been pointed out by Matt ... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | G. Allen | 4,845 | <p>I use a specific example: If I'm in the rain then I'm wet. I also emphasize that we're talking about whether it's possible for the statement to be true. </p>
<p>Now think about the four cases:</p>
<p><strong>true/true:</strong> If you are in the rain then it's possible for you to be wet so this is a true resu... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | riceissa | 9,917 | <p>In my opinion, the truth-table definition of material implication is disturbing because "if ..., then ..." is used in mathematics in two distinct ways (and no similar distinction exists for the other connectives like NOT, AND, and OR):</p>
<ol>
<li>"If <span class="math-container">$p$</span>, then <sp... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | Alex Ortiz | 305,215 | <p>One discontinuous phenomenon in physics is the electric field above and below a surface with uniform positive charge density $\sigma$. The field above the sheet is pointing directly upwards with $E_\text{top} = +{\sigma \over 2\varepsilon_0} \hat k$ and that below being $E_\text{below} = -{\sigma \over 2\varepsilon_... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | Rob Arthan | 23,171 | <p>Navigation may provide interesting examples: you can fly west forever (if you don't start at one of the poles) but you can't fly north forever (from anywhere on the earth).</p>
|
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | Community | -1 | <p>One form of the Kronecker delta function (aka discrete delta function) is the one satisfying</p>
<p>$$ \delta(x) = \begin{cases} 1 & x = 0 \\ 0 & x \neq 0 \end{cases} $$</p>
<p>This usually crops up in discrete contexts, but it makes sense in this context too.</p>
<p>Another appearance of the same functio... |
3,283,606 | <p>Good Evening,</p>
<p>I know this is a basic question, but I haven't been able to find a clear explanation for how to simplify the follow equation:
<span class="math-container">$$n\log_2n=10^6$$</span>
Solving this equation is part of the solution for Problem 1-1 from the Intro. to Algorithms book by CLRS:
<a href="... | Claude Leibovici | 82,404 | <p>Starting from Parcly Taxel's answer, the problem is to compute <span class="math-container">$W(x)$</span> for a very large argument. If you do not have access to Lambert function, let me suggest the very fast algorithm presented in <a href="https://arxiv.org/pdf/1504.01964.pdf" rel="nofollow noreferrer">this paper</... |
3,283,606 | <p>Good Evening,</p>
<p>I know this is a basic question, but I haven't been able to find a clear explanation for how to simplify the follow equation:
<span class="math-container">$$n\log_2n=10^6$$</span>
Solving this equation is part of the solution for Problem 1-1 from the Intro. to Algorithms book by CLRS:
<a href="... | Arthur | 15,500 | <p>Note that <span class="math-container">$2^{20}=16\cdot 2^{16}=1\,048\,576$</span> is pretty darn close to one million, so <span class="math-container">$n=2^{16}$</span> is pretty close to the answer. Also note that decreasing <span class="math-container">$n$</span> by a relatively small amount won't make a big chang... |
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Claude Leibovici | 82,404 | <p>According to a CAS, $$I = \int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt =\,
_4F_3\left(\frac{1}{2},\frac{1}{2},1,1;\frac{3}{2},\frac{3}{2},\frac{3}{2};1\right
)+\frac{\pi ^3}{48}-\frac{1}{4} \pi \log ^2(2)$$</p>
<p>Enjoy !</p>
|
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Jack D'Aurizio | 44,121 | <p>Following Anastasiya-Romanova's approach, we have:</p>
<p>$$ I = \frac{1}{2}\sum_{n\geq 1}\frac{1}{n^2}\int_{0}^{\pi/2}\sin^n x\,dx =\frac{\pi}{16}\sum_{n\geq 1}\frac{\binom{2n}{n}}{n^2 4^n}+\frac{1}{4}\sum_{n\geq 1}\frac{4^n}{\binom{2n}{n}n(2n-1)^2}\tag{1}$$
where:
$$ S_1 = \sum_{n\geq 1}\frac{\binom{2n}{n}}{n^2 4... |
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Sangchul Lee | 9,340 | <p><strong>My attempt.</strong> This is by no means closer to the answer, but I want to address several equivalent forms that might be helpful for future calculations.</p>
<p>First, from <a href="http://en.wikipedia.org/wiki/Polylogarithm#Dilogarithm" rel="noreferrer">Landen's identity</a> of the following form</p>
<... |
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Jack D'Aurizio | 44,121 | <p>An alternative approach. As shown by <em>nospoon</em> <a href="https://math.stackexchange.com/questions/1438381">here</a>,
\begin{equation}\label{shalev} \int_{0}^{1}\frac{\log^2(x)}{\sqrt{x(1- x\sin^2\theta )}}\,dx = \frac{8}{\sin\theta}\left[\frac{\theta^3}{3}-\text{Im}\,\text{Li}_3\left(1-e^{2i\theta}\right)\righ... |
939,212 | <p>Apologies if this isn't at the level of questions expected here!</p>
<p>I've got two simultaneous equations to solve.</p>
<p>(Equation 1): $ x y = 4 $</p>
<p>(Equation 2): $ x + y = 2 $</p>
<p>They produce the following curves:</p>
<p><img src="https://i.stack.imgur.com/dMrNi.png" alt="enter image description h... | Kim Jong Un | 136,641 | <p>Suppose that the curves meet: there are $x,y$ such that $xy=4$ and $x+y=2$. Then, $x$ and $y$ are real roots to the polynomial
$$
P(z)=z^2-2z+4.
$$
But $P(z)=(z-1)^2+3$ clearly has no real roots, so contradiction. (Ref: <a href="http://en.wikipedia.org/wiki/Vieta%27s_formulas" rel="nofollow">Vieta's formulas</a>.)</... |
1,979,876 | <p>Currently I started studying about ray-casting when I came across this following problem based on ray-triangle intersection. The problem was:</p>
<p>You are provided with a triangle with vertices ,<strong>(x1,y1,z1)</strong>, <strong>(x2,y2,z2)</strong> and <strong>(x3,y3,z3)</strong>. A ray with origin <strong>(a1... | K. Miller | 264,375 | <p>The ray can be parameterized by the equation $\mathbf{y}(t) = (a_1,b_1,c_1) + t(a_2,b_2,c_2)$ for all $t\geq 0$. The boundaries of the triangle may be parameterized by</p>
<p>\begin{align}
\mathbf{s}_1(\tau) &= (x_1,y_1,z_1) + \tau(x_2-x_1,y_2-y_1,z_2-z_1)\\
\mathbf{s}_2(\tau) &= (x_2,y_2,z_2) + \tau(x_3 - ... |
3,404,673 | <p>Let <span class="math-container">$G$</span> be a unipotent connected linear algebraic group over a field <span class="math-container">$F$</span>. Then <span class="math-container">$G$</span> is called <em>split</em> if there is a series of closed subgroup schemes <span class="math-container">$1 = G_0 \subset G_1 \su... | WhatsUp | 256,378 | <p>It's not that complicated. I believe you have all the needed knowledge, but just have to unpack the definitions.</p>
<p>By fixing a basis of <span class="math-container">$E/F$</span>, we know that <span class="math-container">$E$</span> is isomorphic to <span class="math-container">$F^2$</span> as <span class="math... |
623,703 | <blockquote>
<p>Find the exact value of $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\right )$ without using a calculator. </p>
</blockquote>
<p>I started by finding $\sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )=\dfrac{\pi}{4}$</p>
<p>So, $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\r... | Zhoe | 99,231 | <p>$\hskip2in$ <img src="https://i.stack.imgur.com/fYbLE.gif" alt="enter image description here"></p>
<p>Using the triangle above...& the fact that $$\tan x = \frac{\text{opp}}{\text{adj}}, \space \tan \left(\frac{\pi}{4}\right)=...$$</p>
|
623,703 | <blockquote>
<p>Find the exact value of $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\right )$ without using a calculator. </p>
</blockquote>
<p>I started by finding $\sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )=\dfrac{\pi}{4}$</p>
<p>So, $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\r... | N74 | 288,459 | <p>You have</p>
<p><span class="math-container">$
\sin x=\dfrac{\sqrt{2}}{2}
$</span></p>
<p>so</p>
<p><span class="math-container">$
\cos x=\sqrt{1-(\sin x)^{2}}=\dfrac{\sqrt{2}}{2}
$</span></p>
<p>and</p>
<p><span class="math-container">$
\tan x =\dfrac{\sin x}{\cos x}\ = 1
$</span></p>
|
309,395 | <p>I am currently learning about <em>Direct Proofs</em>. I am struggling trying to find a starting point to prove the Statement: <em>For all real numbers $c$, if $c$ is a root of a polynomial with rational coefficients, then c is a root of a polynomial with integer coefficients</em>. Based on a definition given by the ... | half-integer fan | 54,125 | <p>Any polynomial with rational coefficients can be made into a polynomial with integer coefficients by multiplying through by the product of the denominators of the coefficients (or the LCM of those denominators). All that remains, then, is for you to show that if a number $c$ is the root of a polynomial, then it is ... |
1,816,807 | <blockquote>
<p>If $x,y,z\in \mathbb{R}$ and $x+y+z=4$ and $x^2+y^2+z^2=6\;,$ Then range of $xyz$</p>
</blockquote>
<p>$\bf{My\; Try::}$Using $$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$</p>
<p>So we get $$16=6+2(xy+yz+zx)\Rightarrow xy+yz+zx = -5$$ and given $x+y+z=4$</p>
<p>Now let $xyz=c\;,$ Now leyt $t=x,y,z$ be the ... | Ángel Mario Gallegos | 67,622 | <p>By CS inequality we get
$$|x^3+y^3+z^3|\le\sqrt{x^2+y^2+z^2}\sqrt{x^4+y^4+z^4}\tag{1}$$
Where
\begin{align*}
x^4+y^4+z^4&=(x^2+y^2+z^2)^2-2[(xy+yz+xz)^2-2xy^2z-2xyz^2-2x^2yz]\\
&=36-2[5^2-2xyz(4)]\\
x^4+y^4+z^4&=16xyz-14\tag{2}
\end{align*}
Now, from the identity
$$x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy... |
1,816,807 | <blockquote>
<p>If $x,y,z\in \mathbb{R}$ and $x+y+z=4$ and $x^2+y^2+z^2=6\;,$ Then range of $xyz$</p>
</blockquote>
<p>$\bf{My\; Try::}$Using $$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$</p>
<p>So we get $$16=6+2(xy+yz+zx)\Rightarrow xy+yz+zx = -5$$ and given $x+y+z=4$</p>
<p>Now let $xyz=c\;,$ Now leyt $t=x,y,z$ be the ... | Samrat Mukhopadhyay | 83,973 | <p>Note that $$c=t((t-2)^2+1),\ t\in \mathbb{R}$$, so according to the calculations, $c\in [a,b]$ where $$a=t((t-2)^2+1)|_{t=5/3}=\frac{50}{27},\ b=t((t-2)^2+1)|_{t=1}=2$$</p>
|
3,874,020 | <p>As it says in the title, any help would be greatly appreciated. I've tried some things but just can't get it to click.</p>
| Tryst with Freedom | 688,539 | <p><span class="math-container">$$ \vec{r(t) } = \langle1 , 0 ,-1\rangle +\langle 3 \cos t , 4 \cos t , 5 \sin t\rangle$$</span></p>
<p>And,</p>
<p><span class="math-container">$$ |\langle 3 \cos t , 4 \cos t + 5 \sin t\rangle | = \sqrt{ ( 3 \cos t)^2 + (4 \cos t)^2 + (5 \sin t)^2} = 5$$</span></p>
<p>So we have that ... |
3,874,020 | <p>As it says in the title, any help would be greatly appreciated. I've tried some things but just can't get it to click.</p>
| Rubén Colomina Citoler | 37,795 | <p><span class="math-container">$r(t)$</span> defines a curve in <span class="math-container">$3$</span>-dimensions as a functions of every real <span class="math-container">$t$</span> from <span class="math-container">$\mathbb R$</span> onto <span class="math-container">$\mathbb R^3$</span>. We will have to get rid of... |
712,697 | <p>What is the radius of convergence?</p>
<p>$$\sum_{n=0}^{\infty} n^3 (5x+10)^n$$</p>
| Makoto Kato | 28,422 | <p>Since $X$ is quasi-compact and locally noetherian, there exists a finite open cover $V_1,\ldots,V_n$ each of which is noetherian.
Let $\mathcal V$ be any non-empy family of open sets.
Write $\mathcal V_i=\{U\cap V_i\mid U\in\mathcal V\}$.</p>
<p>We define inductively an element $U_i \in \mathcal V_i$ and a non-empt... |
3,337,382 | <p>I want to find how many roots of the equation <span class="math-container">$z^4+z^3+1=0$</span> lies in the first quadrant.</p>
<p>Using Rouche's Theorem how to find ?</p>
| Jack D'Aurizio | 44,121 | <p>All the roots of the given polynomial lie inside <span class="math-container">$|z|\leq 2$</span>: if we assume <span class="math-container">$|z|>2$</span> and <span class="math-container">$z^4=-(z^3+1)$</span>, we get <span class="math-container">$2<|z|=|z^{-3}+1|<\frac{9}{8}$</span> which is a contradictio... |
3,337,382 | <p>I want to find how many roots of the equation <span class="math-container">$z^4+z^3+1=0$</span> lies in the first quadrant.</p>
<p>Using Rouche's Theorem how to find ?</p>
| Andrés Martínez | 1,103,687 | <p>For completeness, here I present a more standard solution to this problem:</p>
<p>Let <span class="math-container">$p(z):=z^4+z^3+1$</span>. Consider the following path</p>
<p><a href="https://i.stack.imgur.com/VDgt4m.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VDgt4m.png" alt="enter image des... |
635,301 | <p>I need some help with the following problem: </p>
<blockquote>
<p>Let $f:\Bbb C \to \Bbb C$ be continuous satisfying that $f(\Bbb C)$ is an open set and that $|f(z)| \to \infty$ as $z\to \infty$. Prove that $f(\Bbb C)=\Bbb C$. </p>
</blockquote>
<p>My idea on this one is to prove by contradiction and assume that... | Igor Rivin | 109,865 | <p>By your condition, the image of $f$ is closed (why? take a convergent sequence of points in the image, and their preimages. The preimages do not diverge to infinity, by assumption, so they have a limit point, which, by continuity, goes to the limit you want). So, $f(\mathbb{C})$ is open (by assumption) and closed. S... |
465,001 | <p>$\mathbf{h}_i\in\mathbb{C}^{M}$ are column vectors $\forall i=\{1, 2, \cdots, K\}$.</p>
<p>$q_i\in\mathbb{R}_+$ are scalars $\forall i=\{1, 2, \cdots, K\}$</p>
<p>$\lvert\bullet\rvert$ denotes determinant of a square matrix or Euclidean norm of a vector according to the context. </p>
<p>From Sylvester's theorem, ... | Ben Grossmann | 81,360 | <p>You are correct in saying that since $\cos\left(\frac1x\right)$ is an even function: if the limit as $x\to0$ from one direction exists, it must be equal to the limit from the other direction. However, neither $\lim_{x\to0^+}\cos\left(\frac1x\right)$ nor $\lim_{x\to0^-}\cos\left(\frac1x\right)$ exist. In fact, these... |
8,013 | <p>Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. Hence there are at least countably infinite many orders of infinity.</p>
<p><strong>But do there exist uncountably in... | morgan | 317 | <p>I can't believe this question is not closed.</p>
<p>As near as I can tell, Chloe solved the problem which has stymied Dr. Rush by successfully integrating a constant. Apparently Rush's expertise in ancient technology does not cover first year calculus. </p>
<p><img src="https://i.stack.imgur.com/kv6EZ.jpg" alt="Be... |
8,013 | <p>Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. Hence there are at least countably infinite many orders of infinity.</p>
<p><strong>But do there exist uncountably in... | SpaceMika | 55,118 | <p>Awww, if you want screenshots of my Crazy Hall, all you have to do is ask! Yes, it's all legit math, and all internally consistant throughout the seasons. The core science is usually from peer-reviewed publications, just blended together in ways that don't typically make sense in the Real World (when would you ever ... |
2,793,077 | <blockquote>
<p>Suppose $R$ is a Boolean ring. Prove that $a+a=0$ for all $a\in R$.
Also prove that $R$ is commutative. Give an example (with explanation)
of a Boolean ring.</p>
</blockquote>
<p>From what I know, a Boolean ring is a ring for which $a^2=a$ for all $a\in R$.</p>
<p>Under addition a ring is a comm... | hmakholm left over Monica | 14,366 | <p><strong>Hint.</strong> Compute $(a+1)^2$ in two ways -- once by the Boolean ring property, another time by using the distributive law that works in all rings to multiply out $(a+1)(a+1)$. Then apply $a^2=a$ once more and cancel terms that appear in both results.</p>
|
34,204 | <p>I have several contour lines and one point. How can I find a point in one of those contour lines which is nearest to the given point?</p>
<pre><code>(*Create the implicit curves*)
Data={{10,20,1},{10,40,2},{10,60,3},{10,80,4},{20,25,2},{20,45,3},{20,65,4},{30,30,3},{30,50,4},{40,35,4},{40,55,5},{50,20,4},{50,40,5},... | cormullion | 61 | <p>One way you could try is to examine the points of the plot. If <code>cp</code> is the plot:</p>
<pre><code>points = Cases[cp, {x_?NumericQ, y_?NumericQ} :> {x, y}, Infinity];
nearest = Last@Nearest[points, First[pt], 2]
</code></pre>
<blockquote>
<pre><code>{36.7666, 51.0523}
</code></pre>
</blockquote>
<p>Thi... |
1,878,806 | <p>I am a graduate school freshman.</p>
<p>I did not take a probability lecture.</p>
<p>So I don't have anything about Probability.</p>
<p>Could you suggest Probability book No matter What book level?</p>
| EHH | 133,303 | <p>This is a standard and good introduction to probability:
<a href="https://www.amazon.co.uk/Probability-Introduction-Oxford-Science-Publications/dp/0198532644" rel="nofollow">https://www.amazon.co.uk/Probability-Introduction-Oxford-Science-Publications/dp/0198532644</a></p>
<p>I would also recommend avoiding the app... |
848,415 | <p>If the limit of one sequence $\{a_n\}$ is zero and the limit of another sequence $\{b_n\}$ is also zero does that mean that $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?</p>
| Jika | 143,855 | <p>Of course not.</p>
<ul>
<li>$a_n=\dfrac{(-1)^n}{n}$,</li>
<li>$b_n=\dfrac{1}{n}$, </li>
<li>Hence, $r_n=\dfrac{a_n}{b_n}=(-1)^n$ and the limit of $r_n$ does not even exist.</li>
</ul>
|
848,415 | <p>If the limit of one sequence $\{a_n\}$ is zero and the limit of another sequence $\{b_n\}$ is also zero does that mean that $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?</p>
| Tom | 103,715 | <p>If we want an example such that the limit exists: Let $r \in \Bbb{R}$ be your favorite number, suppose that $b_n \to 0$, and let $a_n = r b_n$. Then $a_n \to 0$ and $a_n/b_n \to r$. </p>
<p>If we want an example such that the limit does not exist: Let $b_n \to 0$ and $\{c_n\}$ be your favorite <em>bounded</em> sequ... |
136,996 | <p>What is known about $f(k)=\sum_{n=0}^{k-1} \frac{k^n}{n!}$ for large $k$?</p>
<p>Obviously it is is a partial sum of the series for $e^k$ -- but this partial sum doesn't reach close to $e^k$ itself because we're cutting off the series right at the largest terms. In the full series, the $(k+i-1)$th term is always at... | Robert Israel | 8,508 | <p>$e^{-k} f(k) = P(X_k < k)$ where $X_k$ is a Poisson random variable with parameter $k$. Since the sum of $k$ independent Poisson random variables with parameter $1$ is Poisson with parameter $k$, the Central Limit Theorem says the probability distribution of $Z_k = (X_k - k)/\sqrt{k}$ approaches a standard normal... |
2,536,185 | <p><strong>Lemma 3.21</strong>. </p>
<p>Let $S$ be a subset of $R$. Then $\bar{S}$ is a closed set. $\bar{S}$ denotes the closure of $S$.</p>
<p>The following is a proof of Lemma 3.21:</p>
<p><em>Proof</em>. </p>
<blockquote>
<p>By Corollary 3.16 it is enough to show that $\bar{S^c}$
is open.</p>
</blockquote>
... | Mastrem | 253,433 | <p>Let $m,n\ge 3$ be given and suppose that $m\neq n$. Without loss of generality, we can say that $n>m$.</p>
<p>Assume there are primes between $m$ and $n$. Let $p$ be the largest prime number with $n>p\ge m$. It is clear that $p\not\in P_m$. If $p\not\in P_n$, we must have $p\mid n$, so $n\ge 2p$. However, Bet... |
4,085,635 | <p>What is the volume of an n-dimension cube? Consider the length of each side to be <span class="math-container">$a$</span>. How to solve this problem?</p>
| Jaap Scherphuis | 362,967 | <p>If there are 4 or more sides, you can always reduce the number of sides by joining together two adjacent ones into one long edge (essentially setting the vertex angle to 180). If that would result in a side longer than all the other sides put together, simply change your mind and choose an adjacent pair of those oth... |
2,681,451 | <p>For the following question I'm getting stuck on a proof. Below I've just written out all the things/steps I've tried (even if they might be wrong). Could someone steer me in the right direction?</p>
<blockquote>
<p>Suppose $f$ is a real-valued function $f:\mathbb{R}\to\mathbb{R}$,
which is continuous at $0$, wi... | Arian | 172,588 | <p>It is clear that any function of the form $f(x)=\lambda x$ satisfies $f(x+y)=f(x)+f(y)$ and $\lim_{x\to 0}f(x)=f(0)$. Now let $f:\mathbb{R}\to\mathbb{R}$ be such that these two conditions hold. It is not difficult to see that: $f(0+0)=f(0)+f(0)\Rightarrow f(0)=0$ and that for $x+y=0$ we have $f(x+y)=f(0)=f(x)+f(-x)\... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | PierreCarre | 639,238 | <p>Part b) can be obtained just by induction. The result is immediate for <span class="math-container">$n=1$</span> and
<span class="math-container">\begin{align*}
\int_0^1 (1-x^2)^{n+1}dx = & \int_0^1 (1-x^2)(1-x^2)^ndx = \int_0^1(1-x^2)^n dx-\int_0^1 x^2 (1-x^2)^n dx\\
= & \frac{2^{2n}(n!)^2}{(2n+1)!} + \fra... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | mathstackuser12 | 361,383 | <p>Here is an approach based on your attempt using a series</p>
<p><span class="math-container">$$F\left( n \right)=\int\limits_{0}^{1}{{{\left( 1-{{x}^{2}} \right)}^{n}}dx}=\sum\limits_{k=0}^{n}{\left( \begin{matrix}
n \\
k \\
\end{matrix} \right){{\left( -1 \right)}^{k}}}\int\limits_{0}^{1}{{{x}^{2k}}dx}=\sum... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | Quanto | 686,284 | <p>Integrate by parts</p>
<p><span class="math-container">$$I_n= \int_0^1 (1-x^2)^n dx=\int_0^1 \frac{(1-x^2)^n}{x^{2n}}d(\frac{x^{2n+1}}{2n+1})=\frac{2n}{2n+1}I_{n-1},\>\>\>I_0=1
$$</span>
which leads to <span class="math-container">$I_n= \frac{2^{2n}(n!)^2}{(2n + 1)!}$</span>.</p>
|
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | robjohn | 13,854 | <p><strong>Beta Integral</strong>
<span class="math-container">$$
\begin{align}
\int_0^1\left(1-t^2\right)^n\mathrm{d}t
&=\frac12\int_0^1\left(1-t\right)^nt^{-1/2}\mathrm{d}t\tag{1a}\\
&=\frac{\Gamma(n+1)\Gamma(1/2)}{2\Gamma(n+3/2)}\tag{1b}\\
&=2^{2n}\frac{\Gamma(n+1)\Gamma(n+1)}{\Gamma(2n+2)}\tag{1c}\\
&am... |
16,290 | <p>Hi I am new here and have a calculus question that came up at work.</p>
<p>Suppose you have a $4' \times 8'$ piece of plywood. You need 3 circular pieces all equal diameter. What is the maximum size of circles you can cut from this piece of material?
I would have expected I could write a function for the area of th... | Isaac | 72 | <p>I believe the optimal configuration is as shown below, with the circles tangent to the sides of the rectangle and to one another.</p>
<p><img src="https://i.stack.imgur.com/nT3kG.png" alt="diagram"></p>
<p>Let the rectangle have sides of length $Q$ and $2Q$, as you specified, and let the radius of all three circle... |
1,435,603 | <blockquote>
<p>Let $S$ be a subset of $\mathbb{R}$ with strictly positive Lebesgue measure. Prove that almost every (with respect to Lebesgue measure) real number can be written as the sum of an element of $S$ and an element of $\mathbb{Q}$. </p>
</blockquote>
<p>So, I remember proving a long time ago that $S-S$ c... | maibaita | 159,452 | <p>Let $R=\mathbb{R}-(S+\mathbb{Q})$. Assume its measure is strictly positive. Then there are $a,b\in\mathbb{R}$ with $a<b$ such that $(a,b)\subset{}R$. Since the measure of $S$ is strictly positive as well, the same is true for $S$, so $(c,d)\subset{}S$. That clearly contradicts $R=\mathbb{R}-(S+\mathbb{Q})$, since... |
724,045 | <p>I believe I'm not correctly understanding the concept of unique factorization and irreducibles.</p>
<p>Consider $R = \mathbb{F}_7$ and $h \in R[x]$ where $h = x^4 + 4x^3 + 3x^2 + 5x + 6$. Now $h$ has the following factorizations:</p>
<p>$h_1 = (3x^2 + 3x + 4)(5x^2 + x + 5)$</p>
<p>and</p>
<p>$h_2 = (x^2 + x + 6)... | Martin Argerami | 22,857 | <p>Consider the following sequence of elements in the space $V$ of finite sequences:
$$
u_1=(1,0,0,\ldots),\ \ u_2=(0,\frac12,0,\ldots),\ \ u_3=(0,0,\frac13,0,\ldots)
$$
Then
$$
\sum_{k=1}^nu_k=(1,\frac12,\frac13,\ldots,\frac1n,0,\ldots)
$$
Now consider these two norms on $u=(a_1,a_2,\ldots)$:
$$
\|u\|_1=\sum_{k=1}^\in... |
3,163,317 | <p>I am given the question "Find the equation of the line in standard form with slope <span class="math-container">$m = -3$</span> and passing through the point <span class="math-container">$(1, \frac{1}{3})$</span>"</p>
<p>The solution is provided as <span class="math-container">$x + 3y = 2$</span></p>
<p>I arrived ... | J. W. Tanner | 615,567 | <p>You made a mistake: <span class="math-container">$ 3(-\frac{1}{3}(x - 1))= -(x-1),$</span>
not
<span class="math-container">$ -3(x - 1).$</span></p>
|
575,232 | <p>Consider all strings whose letters belong to the set:</p>
<p>$A = \{ a, b, c, d, e\}$</p>
<p>How many strings of length $6$ are there that contain exactly one $a$?</p>
<p>Attempt:</p>
<p>Since we are only using $\frac{4}{5}$ letters for the rest of the string, </p>
<p>There are $1* 4^5$ strings that contain exa... | Daniel Robert-Nicoud | 60,713 | <p>Ok, a couple of months have passed since I posted this question, and I have begun this project on quantum groups. I am now in the condition of answering a couple of the questions I posed.</p>
<ol>
<li>There are many different definitions of quantum groups, all of which are related somehow. A possible definition is:... |
3,336,742 | <p>Can we evaluate <span class="math-container">$\displaystyle\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$</span> ?</p>
<p>where <span class="math-container">$H_n=\sum_{k=1}^n\frac1n$</span> is the harmonic number.</p>
<p>A related integral is <span class="math-container">$\displaystyle\int_0^1\frac{\ln^2(1-x)\operatorna... | Ali Shadhar | 432,085 | <p><strong>Different Solution with a big bonus:</strong></p>
<p>First lets define the following:</p>
<p><span class="math-container">$$M=\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$$</span></p>
<p><span class="math-container">$$N=\sum_{n=1}^\infty\frac{H_n^{(2)}}{n^32^n}$$</span></p>
<p>I am going to establish two relati... |
983,200 | <p>Find the derivative of the following: $$f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$$ Would I use the chain rule and product rule?
So far I have:</p>
<p>$$\begin{align}g(x)=x^3-4x+6
\\g'(x)=2x^2-4\end{align}$$</p>
<p>would $h(x)$ be $\ln(x^4-6x^2+9)$?
If so, how would I find $h'(x)$?</p>
| heropup | 118,193 | <p>Something to notice is that $$x^4 - 6x^2 + 9 = (x^2 - 3)^2$$ and consequently, $$f(x) = 2 (x^3 - 4x + 6) \log (x^2 - 3).$$ This makes differentiation slightly easier: $$f'(x) = 2\left( (3x^2 - 4)\log(x^2 - 3) + \frac{x^3-4x+6}{x^2-3} \cdot 2x \right).$$</p>
|
1,523,427 | <p>Is it possible to cover all of $\mathbb{R}^2$ using balls $\{ B(x_n,n^{-1/2})\}_{n=1}^\infty$ of decreasing radius $n^{-1/2}$? I know that if we chose e.g. radius $n^{-1}$ it could never work because $\sum \pi (n^{-1})^2 < \infty$. But in this case the balls cover an infinite amount of area, so it seems that it m... | David C. Ullrich | 248,223 | <p>It's true for any $r_n$ with $\sum r_n^2=\infty$.</p>
<p>Say $S(x,r)$ is the square with center $x$ and side length $r$.
It's enough to show that if $\sum r_n^2>4$ then there exist $x_n$ such
that $S(x_n,r_n)$ covers $[0,1]^2$.</p>
<p>For every $n$ there exists $j$ with $2^{-j}\le r_n<2^{-j+1}$. Replace $r_n... |
26,259 | <p>I've been reading <em>generationgfunctionology</em> by Herbert S. Wilf (you can find a copy of the second edition on the author's page <a href="http://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow">here</a>).</p>
<p>On page 33 he does something I find weird. He wants to shuffle the index forward and does so... | ypercubeᵀᴹ | 7,367 | <p>Are you allowed to write the following?</p>
<p>\begin{align*}
X = \frac{X^2}{X}\
\end{align*}</p>
|
26,259 | <p>I've been reading <em>generationgfunctionology</em> by Herbert S. Wilf (you can find a copy of the second edition on the author's page <a href="http://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow">here</a>).</p>
<p>On page 33 he does something I find weird. He wants to shuffle the index forward and does so... | Bill Dubuque | 242 | <p>The ring of formal power series over a domain is itself a domain, hence nonzero elements are cancellable, quotients are unique, etc. But you don't need these results for the whole ring - only for the element $\rm\:X\:,\: $ and this case can be proved quite simply. It is obvious that $\rm\:X\:$ is not a zero-divisor,... |
1,511,733 | <p>B = matrix given below. I is identity matrix.</p>
<pre><code> [1 2 3 4]
[3 2 4 3]
[1 3 2 4]
[5 4 3 7]
</code></pre>
<p>So What will be the relation between the matrices A and C if AB = I and BC = I?</p>
<p>I think that A = C because both AB and BC have B in common and both of their product is an identity matri... | Peter | 82,961 | <p>Hint : Multiply $AB=I$ from right with $B^{-1}$ and $BC=I$ from left with $B^{-1}$. What do you get ?</p>
|
4,274,085 | <p>A linear equation in one variable <span class="math-container">$x$</span>, <span class="math-container">$$ax+b=k$$</span> has only one non-negative integer solution. For example, <span class="math-container">$2x+3=85$</span> has a solution 41.</p>
<p>How to find the number of non-negative integer solutions of a line... | Robert Shore | 640,080 | <p>Assume <span class="math-container">$\gcd(a, b)=1$</span>. If there are any such solutions at all in non-negative integers, there will be a unique solution where <span class="math-container">$0 \leq x \lt b$</span>. Start there. Increment <span class="math-container">$x$</span> by <span class="math-container">$b$... |
4,274,085 | <p>A linear equation in one variable <span class="math-container">$x$</span>, <span class="math-container">$$ax+b=k$$</span> has only one non-negative integer solution. For example, <span class="math-container">$2x+3=85$</span> has a solution 41.</p>
<p>How to find the number of non-negative integer solutions of a line... | Thomas Andrews | 7,933 | <p>One way to solve this, when <span class="math-container">$(a,b)=1,$</span> is to use partial fractions:</p>
<p><span class="math-container">$$\frac1{(1-x^a)(1-x^b)}=\frac{\alpha}{(1-x)^2}+\frac\beta{1-x}+\frac{p(x)}{1-x^a}+\frac{q(x)}{1-x^b}$$</span> where <span class="math-container">$\deg p<a, \deg q<b,$</sp... |
380,530 | <p>It's easy to show that there's a function such that $\int_1^\infty f $ diverges, but $\int_1^\infty |f|$ converges, such as $f = 1/(-1+x)$. </p>
<p>But is there a function such that $\int_1^\infty f $ converges, but $\int_1^\infty |f| $ diverges?</p>
| Community | -1 | <p>First I recall that
$$\text {if}\quad \int_1^\infty |f|\quad\text{ converges then so does }\quad \int_1^\infty f$$</p>
<p>The integral
$$\int_1^\infty\frac{ \sin x}{x}dx$$
is convergent but
$$\int_1^\infty\frac{ |\sin x|}{x}dx$$
is divergent.</p>
|
4,016,921 | <p>Let <span class="math-container">$X, Y, Z$</span> be three random variables. Is the equality
<span class="math-container">$$
P(X=x, Y=y |Z=z) = P(X=x|Y=y,Z=z) P(Y=y|Z=z)
$$</span>
true?
Note that <span class="math-container">$P(X,Y)$</span> denotes the joint probability of random variables <span class="math-containe... | user | 293,846 | <p>First of all we can cancel one parameter:
<span class="math-container">$$\mbox{$_2$}F_1(a,b;a; x)=\mbox{$_1$}F_0(b;;x)$$</span></p>
<p>Further:
<span class="math-container">$$
\mbox{$_1$}F_0\left(-\frac12;;x\right)=\sum_{n\ge0}\frac{\left(-\frac12\right)^\overline{n}}{n!}x^n=\sum_{n\ge0}\binom{\frac12}n(-x)^n=\sqrt{... |
32,997 | <p>I have made a rather obvious yet peculiar observation while calculating with quadratic inequalities. Take a simple quadratic inequality like the one below</p>
<p>$\frac{x^2+1}{x}>1$</p>
<p>by multiplying both sides by $x$, then subtracting $x$ from both sides we get</p>
<p>$x^2-x+1>0$</p>
<p>Hence, both th... | Rasmus | 367 | <p>If you multiply an inequality with a negative number, you have to exchange the inequality signs $<$ and $>$.</p>
<p>For instance, multiplying $1<2$ with $(-1)$ you get
$$-1>-2.$$</p>
|
32,997 | <p>I have made a rather obvious yet peculiar observation while calculating with quadratic inequalities. Take a simple quadratic inequality like the one below</p>
<p>$\frac{x^2+1}{x}>1$</p>
<p>by multiplying both sides by $x$, then subtracting $x$ from both sides we get</p>
<p>$x^2-x+1>0$</p>
<p>Hence, both th... | Joseph Malkevitch | 1,369 | <p>Another way of thinking about of some of the previous answers and comments is to use a geometric point of view. Draw a graph of the function y = f(x) =((x^2+ 1)/x) - 1. Often this can be done without specialized algebraic knowledge. When the graph is above the line y = 0 you can read off the answer from what you hav... |
2,784,697 | <p>Find the solutions to:$\displaystyle\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^2$. </p>
<p>I got the following solutions:-</p>
<p>$\left(\frac{dy}{dx}\right)=0\Rightarrow y=c_1$ is a solution</p>
<p>$\left(\frac{dy}{dx}\right)=1\Rightarrow y=x+c_2$ is another solution </p>
<p>Are there any other solutions?</p>... | Bill O'Haran | 551,590 | <p><strong>Hint:</strong>
Let $p(x) = \frac{dy}{dx}$. Then:
$$
p'(x) = p(x)^2
$$
is a first order DE.</p>
|
1,610,616 | <blockquote>
<p>$6$ letters are to be posted in three letter boxes.The number of ways
of posting the letters when no letter box remains empty is?</p>
</blockquote>
<p>I solved the sum like dividing into possibilities $(4,1,1),(3,2,1)$ and $(2,2,2)$ and calculated the three cases separately getting $90,360$ and $90... | Empy2 | 81,790 | <p>Inclusion-Exclusion. </p>
<p>$3^6$ ways total.<br>
$2^6$ ways if you only mail to $1\&2,2\&3$ or $1\&3$, subtract them off (3 alike cases).<br>
$1^6$ way if you only mail to $1$, or to $2$, or to $3$, add them back in because they were subtracted twice (agai, 3 cases all alike).</p>
<p>$3^6 - 3 \cdot 2... |
245,921 | <p>I'm new to Mathematica (trying it out for an electromagnetism course) and struggling with plotting magnetic field lines using a <strong>line</strong> of current. Not a torturous, not a cylinder, not a circle. A line. Almost every example I see is <em>far</em> more complex and it's beyond what I'm trying to do... I k... | Daniel Huber | 46,318 | <p>in your course you will have learned that the magnetic field at r of a a current infinitesimal dI at position r0 is proportional to:</p>
<pre><code>Cross[dI,(r-r0)]/Norm[r-r0]^3
</code></pre>
<p>Now if we consider a current of 1 along the x ax at {x0,0,0}: dI= {1,0,0} dx0 and the field at {x,y,z};</p>
<pre><code>dFi... |
245,921 | <p>I'm new to Mathematica (trying it out for an electromagnetism course) and struggling with plotting magnetic field lines using a <strong>line</strong> of current. Not a torturous, not a cylinder, not a circle. A line. Almost every example I see is <em>far</em> more complex and it's beyond what I'm trying to do... I k... | Alex Trounev | 58,388 | <p>We can use 3D graphics as follows</p>
<pre><code>Graphics3D[{{Red, Arrowheads[0.025],
Arrow[Line[{{0, 0, -4}, {0, 0, 4}}]]},
Table[{Blue, Arrowheads[0.025],
Arrow[BSplineCurve@
Table[{r Cos[t], r Sin[t], z0}, {t, 0, 2 Pi, Pi/50}]]}, {r, {1,
2, 3}}, {z0, {-3, 0, 3}}]}, Boxed -> False]
</cod... |
388,561 | <p>I am trying to do this question in Bredon's <em>Topology and geometry</em> about using the transversality theorem to show that the intersection of two manifolds is a manifold.</p>
<p>Now it goes as follows:</p>
<p>Let $f(x,y,z)=(2-(x^2+y^2)^{1/2})+z^2$ on $\mathbb{R}^3 - (0,0,z)$. Then one can show that $M=f^{-1}(... | Ted Shifrin | 71,348 | <p>I couldn't fit my response in the character limit of the comment box.</p>
<p>No, your first function needs to be the function $f$ you started with :) You're correct that I was guiding you just to use the regular value set-up. Of course, this is a special case of the transversality theorem. If you try to study the f... |
1,170,602 | <p>How to evaluate the integral </p>
<p>$$\int \sqrt{\sec x} \, dx$$</p>
<p>I read that its not defined.<br>
But why is it so ? Does it contradict some basic rules ?
Please clarify it .</p>
| kryomaxim | 212,743 | <p>$$\int \sqrt{\sec x} \,dx = \int \frac{1}{\sqrt{\cos x}}\, dx$$; Now substituting $u = \cos x$ then it follows $$du = - \sin x \, dx = - \sqrt{1-u^2} dx.$$ Hence:</p>
<p>$$\int \sqrt{\sec x} \, dx = - \int \frac{1}{\sqrt{u(1-u^2)}}du$$ This integral is an <a href="http://mathworld.wolfram.com/EllipticIntegral.html"... |
646,010 | <p>So i kinda think i have figured this out, i'm not very good at math, and need a formula to figure out some stats for a game i'm playing.</p>
<p>I have a Weapon with a reload speed of X sec.. however, i also have a modifier attached, that will make the weapon reload faster by +Y%</p>
<p>i made this formula, mostly ... | T.J. Gaffney | 34,380 | <p>$\sin(x^2)>.5$ on $(\sqrt{\pi/6+2k\pi},\sqrt{5\pi/6+2k\pi}),k\in\mathbb{N}$. When $x>e$, $\log^2(x)>1$, so we can see that the positive part of the integral is greater than $.5\sum_{k=2}^K \left(\sqrt{5\pi/6+2k\pi}-\sqrt{\pi/6+2k\pi}\right)$ for any $K\in\mathbb{N}\setminus\{1\}$. Now see that </p>
<p>$$... |
2,869,031 | <p>Let $A$ a domain, i.e. $ab\in A\implies a=0$ or $b=0$. It's written that all domains are commutative. Is it by definition, or can we prove that domains are commutative? I mean, do we only consider domaind for commutative rings, or is a ring that is a domain then commutative?</p>
| José Carlos Santos | 446,262 | <p>That's a mistake: <a href="https://en.wikipedia.org/wiki/Domain_(ring_theory)" rel="noreferrer">a domain doesn't have to be commutative</a>. When it is, we call it an <em>integral domain</em>.</p>
|
2,869,031 | <p>Let $A$ a domain, i.e. $ab\in A\implies a=0$ or $b=0$. It's written that all domains are commutative. Is it by definition, or can we prove that domains are commutative? I mean, do we only consider domaind for commutative rings, or is a ring that is a domain then commutative?</p>
| rschwieb | 29,335 | <p>Some authors do allow "domain" to refer to a noncommutative ring. But nearly universally "integral domain" will imply commutativity.</p>
<p>Here are <a href="http://ringtheory.herokuapp.com/search/results/?H=8&L=1" rel="noreferrer">nine examples of noncommutative rings with no nonzero zero divisors</a>.</p>
<p... |
3,050,235 | <p>A polynomial is exactly divided by <span class="math-container">$x+1$</span>, and when it is divided by <span class="math-container">$3x-1$</span>, the remainder is <span class="math-container">$4$</span>. Given that the polynomial gives a remainder <span class="math-container">$hx+k$</span> when divided by <span cl... | Deepak | 151,732 | <p>Givens:</p>
<p><span class="math-container">$P(x) = Q_1(x)\cdot (x+1)$</span> [note here that <span class="math-container">$P(-1) = 0$</span>]</p>
<p>and</p>
<p><span class="math-container">$P(x) = Q_2(x)\cdot (3x-1) + 4$</span> [note here that <span class="math-container">$P(\frac 13) = 4$</span>]</p>
<p>Need t... |
3,050,235 | <p>A polynomial is exactly divided by <span class="math-container">$x+1$</span>, and when it is divided by <span class="math-container">$3x-1$</span>, the remainder is <span class="math-container">$4$</span>. Given that the polynomial gives a remainder <span class="math-container">$hx+k$</span> when divided by <span cl... | Bill Dubuque | 242 | <p><span class="math-container">$\ \overbrace{f\!-\!4\bmod (3x\!-\!1)(x\!+\!1)}^{\Large {\rm factor\ out}\,\ 3x\,-1\ \mid\ f-4\ \ \ \rightarrow\!\!\!\!\!\!\!} =\, (3x\!-\!1)\!\!\underbrace{\overbrace{\left[\dfrac{\color{#c00}f\!-\!4}{\color{#0a0}{3x}\!-\!1}\bmod x\!+\!1\right]}^{\Large\ \color{#c00}f\bmod x+1\ =\ \col... |
307,529 | <p>I am trying to prove that if $L/K$ is an algebraic extension and if $\alpha \in L$, then </p>
<ul>
<li><p>$\alpha$ is separable over $K$ if $\mathrm{char}(K)=0$. This is clear because $K$ is perfect which in turn implies that $L/K$ is seperable . </p></li>
<li><p>Now if $\mathrm{char}(K)=p$ is prime, then the state... | Barbara Osofsky | 59,437 | <p>In characteristic $p$, the Frobenius map $\Phi:K\rightarrow K$ with $\Phi(\kappa)=\kappa^p$ is additive (that is, $\Phi(\kappa +\lambda)=\Phi(\kappa)+\Phi(\lambda)$ because all the binomial coefficients that arise in $(\kappa+\lambda)^p$ are divisible by $p$. Taking $p$ powers is always multiplicative so $\Phi$ mus... |
3,278,797 | <p>I tried to solve it and I got answer '3'. But that is just my intuition.I don't have concrete method to prove my answer .I did like this, in order to maximize the fraction, we need to minimize the denominator .So if plug in '1' in expression, denominator becomes '1'.Now denominator is minimalized,the result of ex... | nonuser | 463,553 | <p>Hint: <span class="math-container">$$y= \frac{x^2+x+1}{x^2-x+1}=\frac{x+1+{1\over x}}{x-1+{1\over x}} = {t+2\over t} = 1+{2\over t}$$</span></p>
<p>where <span class="math-container">$t=x-1+{1\over x}$</span>. Since for positive <span class="math-container">$x$</span> have <span class="math-container">$$x+{1\over x... |
3,278,797 | <p>I tried to solve it and I got answer '3'. But that is just my intuition.I don't have concrete method to prove my answer .I did like this, in order to maximize the fraction, we need to minimize the denominator .So if plug in '1' in expression, denominator becomes '1'.Now denominator is minimalized,the result of ex... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: It is <span class="math-container">$$\frac{x^2+x+1}{x^2-x+1}\le 3$$</span> if <span class="math-container">$$x^2+x+1\le 3x^2-3x+3$$</span> and this is true since <span class="math-container">$$0\le x^2-2x+1$$</span> . This is <span class="math-container">$$(x-1)^2\geq 0$$</span> and the equal sign holds if <sp... |
130,806 | <p><strong>Qusestion:</strong> Let f be a continuous and differentiable function on $[0, \infty[$, with $f(0) = 0$ and such that $f'$ is an increasing function on $[0, \infty[$. Show that the function g, defined on $[0, \infty[$ by $$g(x) = \begin{cases} \frac{f(x)}{x}, x\gt0& \text{is an increasing function.... | noname1014 | 27,743 | <p>Base on @chemeng $g(x)$is increasing in$(0,\infty)$.when $x=0$,
$$\lim_{x \to 0^+} g(x)=\lim\frac{f(x)-f(0)}{x-0}=f'(0)=g(0)
$$,so $g(x)$ is contentious at $0$,last g(x) is an increasing function in
$[0,\infty)$</p>
|
3,336,506 | <p>Let <span class="math-container">$V=\mathbb R^3$</span> be an inner product space with the standard inner product (that means <span class="math-container">$\langle(x_1,y_1,z_1),(x_2,y_2,z_2)\rangle=x_1y_1+x_2y_2+x_3y_3$</span> ).<br>
<span class="math-container">$U=span\{(1,2,3),(1,2,1)\}\subseteq V$</span> </p>
<... | Ninad Munshi | 698,724 | <p>As you noted, unless <span class="math-container">$\phi(x) = 0$</span> this distribution isn't really a distribution since distributions need to be continuous linear functionals. But we can modify it by taking the action <span class="math-container">$$T_x(\phi) = \text{p.v.}\int_{\mathbb{R}^n} \frac{\phi(y)-\phi(x)}... |
2,097,557 | <blockquote>
<p>If $0< \alpha, \beta< \pi$ and $\cos\alpha + \cos\beta-\cos (\alpha + \beta) =3/2$ then prove $\alpha = \beta= \pi/3$</p>
</blockquote>
<p>How do I solve for $\alpha$ and $\beta$ when only one equation is given? By simplification I came up with something like
$$
\sin\frac{\alpha}{2} \sin\frac{... | egreg | 62,967 | <p>Set $p=\cos\beta$ and $q=\sin\beta$; then your equality can be written
$$
\cos\alpha+p-p\cos\alpha+q\sin\alpha=\frac{3}{2}
$$
or
$$
(1-p)\cos\alpha+q\sin\alpha=\frac{3}{2}-p
$$
Set $X=\cos\alpha$ and $Y=\sin\alpha$; then the equation becomes
$$
\begin{cases}
(1-p)X+qY=\frac{3}{2}-p\\[6px]
X^2+Y^2=1
\end{cases}
$$
an... |
2,485,276 | <p>Is the statement true? if it is, how to prove it?</p>
<p>If $\binom{p}{k} \mod p=0 $ for $k=1,2,..,p−1$ then $p$ is prime.</p>
| user19405892 | 282,223 | <p>Suppose that $N = ab$ is composite with positive integers $a,b > 1$ and $a$ is the smallest prime factor of $N$. Then $$\binom{N}{a} = \binom{ab}{a} = \dfrac{ab(ab-1) \cdots (ab-(a-1))}{a!} = \dfrac{b(ab-1) \cdots (ab-(a-1))}{(a-1)!}.$$ Then $(ab-1)(ab-2) \cdots (ab-(a-1)) \equiv \pm (a-1)!\pmod{ab}$. Thus, $\bin... |
1,102,668 | <p><a href="https://math.stackexchange.com/q/67994/198434">This question</a> shows how dividing both sides of an equation by some $f(x)$ may eliminate some solutions, namely $f(x)=0$. Naturally, all examples admit $f(x)=0$ as a solution to prove the point.</p>
<p>I tried to find a simple example of an equation that co... | Peter Woolfitt | 145,826 | <p>Well, assuming that we are looking for solutions over the reals, we can solve say $$(x^2+1)(x+1)=0$$ by dividing both sides by $x^2+1$ to find that the only solution is $x=-1$. However, we still have to check that $f(x)=x^2+1=0$ has no solutions in the reals (easy to see because $f(x)\ge1$ for every $x$). Is this th... |
2,210,531 | <blockquote>
<p>Prove that the limit is zero:
$$ F(x, y)= \frac{x^2y^3}{3x^2+ 2y^3}$$</p>
</blockquote>
<p>Definition1. Let $ U ⊂ R^n$ be an open set and letf : $U→R^m$ beafunction with domain U. Let x0 be a vector in U or on the boundary of U. Let b∈Rm. We say that the limit of f as x approaches x0 is b, written... | Barry Cipra | 86,747 | <p>The limit actually does not exist. Here's one way to see why:</p>
<p>Let $(x,y)\to(0,0)$ along the curve defined by $2y^3=kx^4-3x^2$. Then</p>
<p>$${x^2y^3\over3x^2+2y^3}={kx^6-3x^4\over2kx^4}={1\over2}x^2-{3\over2k}\to-{3\over2k}$$</p>
<p>Since we can use any value we like for $k$ (except $0$, of course), we c... |
280,156 | <p>I have a code as below:</p>
<pre><code>countpar = 10;
randomA = RandomReal[{1, 10}, {countpar, countpar}];
randomconst = RandomInteger[{0, 1}, {countpar, 1}];
For[i = 1, i < countpar + 1, i++,
If[randomconst[[i, 1]] != 0,
randomA[[All, i]] = 0.; randomA[[i, All]] = 0.;
randomA[[i, i]] = 1;
];
];
</code></pre... | bill s | 1,783 | <p>The formula is a recursion, and so can be written:</p>
<pre><code>Clear[y]; x = 2; n = 2; y[1] = 1;
y[k_] := y[k] = (y[k - 1]^(x - 1) + n)/(y[k - 1]^(x - 1) + y[k - 1]^(x - 2))
</code></pre>
<p>Then you can get individual values as</p>
<pre><code>{y[1], y[2], y[3]}
</code></pre>
<p>or get 10 at once:</p>
<pre><code>... |
178,342 | <p>This is an exercise from Kunen's book.</p>
<p>Write a formula expressing $z=\langle \langle x,y\rangle, \langle u,v\rangle \rangle$ using just $\in$ and $=$.</p>
<p>What I've tried: because $\langle x,y\rangle= \{\{x\},\{x,y\}\}$ and $\langle u,v\rangle= \{\{u\},\{u,v\}\}$, and hence $z=\{\{\langle x,y\rangle\},\... | Thanaton | 22,787 | <p>Although I am not an expert in this field (merely a beginning student), maybe I can help you out.</p>
<p>We will have to find a formula only using <span class="math-container">$\in$</span> and <span class="math-container">$=$</span> to describe <span class="math-container">$z = \langle \langle x, y \rangle, \langle ... |
1,279,328 | <p>Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable?
Thank you</p>
| Snehasish | 241,303 | <p>May be what you are looking for is, solving poission's equation using separation of variable method. Here is a link,<a href="http://farside.ph.utexas.edu/teaching/em/lectures/node66.html" rel="nofollow">http://farside.ph.utexas.edu/teaching/em/lectures/node66.html</a>, to explore more about it.</p>
|
3,916,130 | <p>Assume the weight of a person follows a normal distribution N(71,7). What is the probability of 4 people weighing more than 300kg?</p>
<p>I tried solving this by multiplying the values by 4, so it'd be N(284,28). I converted that into <span class="math-container">$x=284+28z$</span> which lead to <span class="math-c... | BruceET | 221,800 | <p>I assume heights are independently normally distributed with mean <span class="math-container">$\mu = 71$</span> and <em>standard deviation</em> <span class="math-container">$\sigma = 7.$</span> [Sometimes the second parameter is the variance and sometime the variance, you should always say which convention you are ... |
167,848 | <p>I am trying to solve numerically an equation and generate some results. I use the following code </p>
<pre><code>u[c_] := (c^(1 - σ) - 1)/(1 - σ)
f[s_] := g s (1 - s/sbar1)
h[s_] := (2 hbar)/(1 + Exp[η (s/sbar - 1)])
co[a_] := ϕ (a^2)/2
ψ[k_] := wbar (ω + (1 - ω) Exp[-γ k])
</code></pre>
<p>The equation I try to s... | Hugh | 12,558 | <p>I always find it a good idea to plot the functions you are investigating. Thus</p>
<pre><code>tmax1 = 10;
Column@Table[
Plot[Evaluate[adap[k, i] /. paramFinal2], {k, 0, 20}], {i, 1 tmax1}]
</code></pre>
<p><img src="https://i.stack.imgur.com/Thz8L.png" alt="Mathematica graphics"></p>
<p>They all look fine exce... |
3,399,586 | <p>Suppose that <span class="math-container">$f$$:$$\mathbb{R}\to\mathbb{R}$</span> is differentiable at every point and that </p>
<p><span class="math-container">$$f’(x) = x^2$$</span></p>
<p>for all <span class="math-container">$x$</span>. Prove that </p>
<p><span class="math-container">$$f(x) = \frac{x^3}{3} + C$... | wonderman | 715,632 | <p>Let <span class="math-container">$g(x) = f(x) - \frac{x^3}{3}$</span>. What can you say about the derivative of <span class="math-container">$g(x)$</span>?</p>
|
165,560 | <p>To find the volume of the following region:</p>
<pre><code>fn[x_, y_, z_]:= Abs[0.7*x*Exp[I*y] + 0.3*Sqrt[x^2 + 8*10^-5]
+ Sqrt[x^2 + 3*10^-3]*0.02*Exp[I*z]]
R = ImplicitRegion[fn[x, y, z]<=3*10^-3, {{x, 0, 0.015}, {y, 0, 2*Pi}, {z, 0, 2*Pi}}]
RegionPlot3D[
fn[x, y, z] <= 3*10^-3, {x, 0, 0.015}, {y, 0, 2... | José Antonio Díaz Navas | 1,309 | <p>I have tried this with MMA 11.2 (macOs 10.13.3) in my old Intel I3. First, I have simplified a bit your <code>fn</code> as it involves <code>Abs</code>, and by setting the definition of <code>ImplicitFucntion</code> in a standard form:</p>
<pre><code>fn[x_, y_, z_] := 8.4*10^-6 + 0.5804 x^2 + 0.00375659 x Sqrt[1 + ... |
3,346,676 | <blockquote>
<p><strong>Question.</strong> Find a divergent sequence <span class="math-container">$\{X_n\}$</span> in <span class="math-container">$\mathbb{R}$</span> such that for any <span class="math-container">$m\in\mathbb{N}$</span>,
<span class="math-container">$$\lim_{n\to\infty}|X_{n+m}-X_n|=0$$</span></p>
... | copper.hat | 27,978 | <p>Let <span class="math-container">$x_n= 1+ {1 \over 2} + \cdots + {1 \over n}$</span>, then <span class="math-container">$x_n \to \infty$</span>.</p>
<p>For any <span class="math-container">$m$</span>, we see that
<span class="math-container">$x_{n+m} - x_n = {1 \over n+1} + \cdots + {1 \over n+m} \le {m \over n}$</... |
767,304 | <p>Prove that there are no real numbers $x$ such that</p>
<p>$$\sum_{n\,=\,0}^\infty \frac {(-1)^{n + 1}} {n^x} = 0$$</p>
<p>Can I have a hint please?</p>
| Bruno Joyal | 12,507 | <p>This is an observation which is too long for a comment, but which someone will probably appreciate. Remark that</p>
<p>$$\sum_{n=1}^\infty \frac{(-1)^n}{n^s} = \zeta(s) - 2\times 2^{-s}\zeta(s) = (1-2^{1-s})\zeta(s).$$</p>
<p>The Riemann zeta function is equal to the Euler product $\prod_p (1-p^{-s})^{-1}$ in the ... |
2,978,605 | <p>Is there a sophisticate way to proof that:</p>
<p><span class="math-container">$$\frac{n}{N}\cdot\frac{n-1}{N-1}\cdot\frac{n-2}{N-2}\cdot\ldots\cdot \frac{1}{N-n+1} = \frac{1}{{N\choose n}}$$</span></p>
<p>where <span class="math-container">${N\choose n}$</span> denotes combinations.</p>
<p>When replacing <span c... | KM101 | 596,598 | <p>Simplify the numerator and denominator.</p>
<p><span class="math-container">$$\frac{n}{N}\cdot\frac{n-1}{N-1}\cdot\frac{n-2}{N-2}\cdot\ldots\cdot \frac{1}{N-n+1} = \frac{n!}{\frac{N!}{(N-n)!}}$$</span></p>
<p>Simplify by <span class="math-container">$n!$</span>.</p>
<p><span class="math-container">$$\frac{1}{\fra... |
2,558,988 | <blockquote>
<p>Let us consider a function $f(x,y)=4x^2-xy+4y^2+x^3y+xy^3-4$. Then find the maximum and minimum value of $f$.</p>
</blockquote>
<p>My attempt. $f_x=0$ implies $8x-y+3x^2y+y^3=0$ and $f_y=0$ implies $-x+8y+x^3+3xy²=0$ and $f_{xy}=3x^2+3y^2-1$. Now $f_x+f_y=0$ implies $(x+y)((x+y)^2+7)=0$ implies $x=-y... | Bart Michels | 43,288 | <p>Yes. It is the image of the connected space $X\times \mathbb R_{>0}$ under the continuous map $$(x, t)\mapsto (x, f(x)-t)$$</p>
|
843,634 | <p>I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, \mathfrak{L'}$, $\mathfrak{P}$. I don't really know how to approach this, because I was never taught thinking about such a prob... | lab bhattacharjee | 33,337 | <p>Do you know the difference between Congruence Equation and Algebraic equation which obeys <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra" rel="nofollow">Fundamental theorem of algebra</a> unlike the previous one?</p>
<p>For example $$x^2\equiv2\pmod3$$ is not solvable unlike $$x^2=2$$ </p>
<p... |
1,584,594 | <p>Find all $n$ for which $n^8 + n + 1$ is prime. I can do this by writing it as a linear product, but it took me a lot of time. Is there any other way to solve this? The answer is $n = 1$.</p>
| Paolo Leonetti | 45,736 | <p>Since $n^2+n+1$ divides $n^8+n+1$ and $1<n^2+n+1<n^8+n+1$ for $n>1$, then $n=1$ is the unique solution (which indeed gives a prime).</p>
|
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