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 |
|---|---|---|---|---|
200,876 | <p>Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?</p>
<blockquote>
<blockquote>
<p>A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c_0) = x$ and $f(c_1) = y$.</p>
</blockquote>
</bloc... | Włodzimierz Holsztyński | 8,385 | <blockquote>
<p>I hope that my example below is as elegant as the continuous long line provided by Goldstern above, while my example is less expected. Also, while long line is simpler <strong>in itself</strong>, the <strong>proof</strong> is simpler in my case. Finally, perhaps logicians will find some advantages (I'... |
312,602 | <p>I am writing a paper right now, and part of the paper makes use of a (trivial) generalization of a number of really nice theorems and constructions from a paper that was never made public. The author has left pure mathematics and has no intention of publishing the paper, but I received a copy directly from him seve... | Alexandre Eremenko | 25,510 | <p>First of all, if you received a paper from the author privately, and it is unpublished, you need his/her permission to use the result, and permission to mention his/her work.</p>
<p>When you ask the permission, you may also ask whether the author is willing to post the paper on the arXiv, and you may propose him/he... |
312,602 | <p>I am writing a paper right now, and part of the paper makes use of a (trivial) generalization of a number of really nice theorems and constructions from a paper that was never made public. The author has left pure mathematics and has no intention of publishing the paper, but I received a copy directly from him seve... | T. Amdeberhan | 66,131 | <p>Why not you approach the author and ask to join forces to write as a co-author if the other person is willing to do so? That could be a fine resolution and fair to both parties.</p>
|
1,687,336 | <p>I've been searching through the internet and through SE to find something to help me understand generating functions, but I haven't found anything that would solve my problem with them.</p>
<p>I understand that </p>
<p>$$\frac1{1-x}=\sum_{n\ge 0}x^n\;,\tag{1}$$</p>
<p>gives the sequence $(1, 1, 1, 1,...) $ becaus... | anonymouse | 316,033 | <p>$$4x^0+0x^1+0x^2+4x^3+0x^4+0x^5+4x^6+0x^7+0x^8+4x^9\ldots$$</p>
<p>and</p>
<p>$$4+4x^3+4x^6+4x^9\ldots$$</p>
<p>are the same polynomial. Each has a $4$ term, each has a $4x^3$ term, et cetera.</p>
<p>So both can be represented by the sequence $(4, 0, 0, 4, 0, 0, 4, ...)$</p>
<p>The first term of the sequence is... |
2,130,776 | <p>Let $(L,R)$ be a pair of adjoint functor. </p>
<p>How to show that the commutativity of the left diagram induces the commutativity of the right one?</p>
<p><a href="https://i.stack.imgur.com/lOffl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lOffl.jpg" alt="diagram"></a></p>
| Geoff | 100,192 | <p>Let $(\eta,\epsilon):F \dashv G:\mathfrak{C} \to \mathfrak{D}$ be an adjoint pair of functors and assume you have the commuting diagram
$$
\array{
A & \xrightarrow{f} & GX \\
h \downarrow & & \downarrow G(k) \\
B & \xrightarrow{g} & GY
}
$$
in $\mathfrak{C}$. Then applying the functor $F$... |
3,835,514 | <p>For some c > <span class="math-container">$0$</span>, the cumulative distribution function of a continuous random variable X is given by:</p>
<p><span class="math-container">$$
F_X(x) = \begin{cases} 0 & \text{if } x \le0 \\ cx(x+1) & \text{if } 0 \lt x <1 \\ 1 & \text{if } x \ge 1\end{cases}
$$</s... | VIVID | 752,069 | <p>For a quadratic expression given as <span class="math-container">$$rx^2 + px + q$$</span> whose roots are, say, <span class="math-container">$m$</span> and <span class="math-container">$n$</span>, it can be factorised as <span class="math-container">$$rx^2 + px + q = r(x-m)(x-n)$$</span></p>
<p>So now, you should fi... |
2,166,897 | <blockquote>
<p>Let X be a complex Banach space. Let <span class="math-container">$T\in B(X)$</span> be a bounded linear operator on <span class="math-container">$X$</span>. Let <span class="math-container">$T^*\in B(X^*)$</span> be the adjoint of <span class="math-container">$T$</span>.</p>
<p>Prove: If <span class="m... | Vincent Boelens | 94,696 | <p>Let $x\in X$. By Hahn-Banach there is $f\in X^\ast$ with $\|f\|=1$ and
$|f(x)|=\|x\|$. Then we obtain
$$\begin{align}
\|x\|&=|f(x)| \\
&=|(T^\ast)^{-1}(T^\ast(f))(x)|\\
&\le \|(T^\ast)^{-1}\||(T^\ast(f))(x)|\\
&=\|(T^\ast)^{-1}\||(f\circ T)(x)|\\
&\le\|(T^\ast)^{-1}\|\|T(x)\|,
\end{align} $$
whic... |
281,504 | <p>Is there a trick for easily solving a matrix polynomial like
$$
p(A) = \left( 7\cdot A^4 - 4\cdot A^3 + 6\cdot A - 5\cdot E \right)
, A = \left(\begin{matrix}2 & -1 \\ 3 & 5\end{matrix}\right)
$$
or is it really just step-by-step calculation of
$$
7\cdot A\cdot A\cdot A\cdot A\cdot A\cdot A\cdot A-4\cdot A\c... | Felix | 463,163 | <p>Given a line <span class="math-container">$\overline r+t\cdot\overline{v}$</span> and a point to be reflected <span class="math-container">$\overline p$</span>, we must first find the closest point on the line to <span class="math-container">$\overline{p}$</span>. Let <span class="math-container">$t_0$</span> corres... |
2,491,818 | <p>My professor showed us that the Cauchy distribution does not have an expected value, that is, the integral $\int_{-\infty}^{\infty} x f(x) \text{d} x$ is not defined ($f(x)$ is the p.d.f. of the Cauchy distribution). I find that very counterintuitive. What does it actually mean, in the context of probability, to not... | Ken Wei | 243,183 | <p>The Law of Large Numbers fails for distributions without an expected value. So empirically, if you were to average many samples from the Cauchy distribution, you won't be able to say that the average converges to zero (as one might think).</p>
<p>A property of the Cauchy distribution is that if $X_1,...,X_n$ and $Z... |
2,617,235 | <p>Given a triangle $\Delta$ABC, how to draw any inscribed equilateral triangle whose vertices lie on different sides of $\Delta$ABC?</p>
| Christian Blatter | 1,303 | <p>Assume $A=(0,0)$, $B=(b,0)$, $C=(c,h)$ with $b>0$, $0<c<b$, and $h>0$. We shall construct an equilateral triangle with one side horizontal as follows: Draw a horizontal line $y=h'$, where $h'$ is determined by the condition
$$h'={\sqrt{3}\over2}\>{h-h'\over h}\>b\ .\tag{1}$$
This line intersects th... |
1,482,776 | <blockquote>
<p>Let $(X_t)$ be a continuous nonnegative supermartingale and $T = \inf\{t\geq 0 \colon X_t = 0 \}$ then $X_t = 0$ for every $t\geq T$.</p>
</blockquote>
<p>Idea of solution:</p>
<p>Since $T$ is stopping time, by Doob theorem:
$$E(X_{T+q} 1_{T < \infty} | F_T) \leq X_T 1_{T < \infty} =0 $$
for e... | Christian Blatter | 1,303 | <p>In order to simplify matters we are going to look at the reciprocal of the given expression. The denominator can be developed into a series as follows: From
$$\cos x=1-{1\over2}x^2+{1\over24}x^4+?x^6$$
and
$$\eqalign{\cos(\sin x)
&=1-{1\over2}x^2\left(1-{1\over 6}x^2+?x^4\right)^2+{1\over24}x^4(1+?x^2)^4+?x^6\cr... |
1,242,760 | <p>Now proving by induction is fairly simple. However, this is a multiple choice problem whose answers don't make any sense to me. The actual problem goes as follows:</p>
<p><em>To prove by induction that $n^2 - 7n - 2$ is divisible by $2$ is true for all positive integers $n$, we assume $k^2 - 7k - 2$ is divisible b... | ncmathsadist | 4,154 | <p>Use the fact that
$$ n^2 - 7n - 2 - ( (n-1)^2 - 7(n-1) - 2))
= 2n - 1 - 7 = 2(n-4).$$</p>
|
1,242,760 | <p>Now proving by induction is fairly simple. However, this is a multiple choice problem whose answers don't make any sense to me. The actual problem goes as follows:</p>
<p><em>To prove by induction that $n^2 - 7n - 2$ is divisible by $2$ is true for all positive integers $n$, we assume $k^2 - 7k - 2$ is divisible b... | John Joy | 140,156 | <p>In the problem statement <em>"we show that $k^2 - 7k - 2 + A$ is divisible by $2\dots$"</em>. I think that what is intended here is that $k^2 - 7k - 2 + A$ is the next case. In other words
$$k^2 - 7k - 2 + A = (k+1)^2 -7(k+1)-2$$
Do the math, and I'm sure that you'll come up with
$$A = 2(k-3)$$
To finish off the pro... |
345,589 | <p>I am interested in proving that the family of functions
$$\{f_{\omega}: \mathbb{C}^n\rightarrow\mathbb{C}, f_\omega(z) = \exp(i\langle \omega, z \rangle): \omega \in \mathbb{C}^n\},$$
where $\langle \cdot,\cdot\rangle$ is the usual hermitian dot product, is $\mathbb{C}$-linearly independent.</p>
<p>In the case $n=1... | Kofi | 17,439 | <p>Yes, you just need make make a slightly elaborate argument.</p>
<p>The function $E_\omega \in C(\mathbb{C}^n, \mathbb{C})$ defined by $z \mapsto \exp(i \langle \omega, z \rangle)$ is an eigenvector for each of the partial Derivative operators $\partial/\partial z_j$, $j = 1, \dots n$, and the corresponding eigenval... |
2,553,175 | <p>How can I verify that
$$1-2\sin^2x=2\cos^2x-1$$
Is true for all $x$?</p>
<p>It can be proved through a couple of messy steps using the fact that $\sin^2x+\cos^2x=1$, solving for one of the trigonemtric functions and then substituting, but the way I did it gets very messy very quickly and you end up with a bunch of ... | user284331 | 284,331 | <p>It should be very nice: $1-2\sin^{2}\theta=1-2(1-\cos^{2}\theta)=1-2+2\cos^{2}\theta$.</p>
|
35,987 | <p><em>cross post in <a href="https://stackoverflow.com/questions/3513660/multivariate-bisection-method">StackOverflow</a></em></p>
<p>I need an algorithm to perform a 2D bisection method for solving a $2$x$2$ non-linear problem. Example: two equations $f(x,y)=0$ and $g(x,y)=0$ which I want to solve simultaneously. I ... | KalEl | 8,602 | <ol>
<li><p>Check the pair of <em>opposite</em> corners to determine if zeroes lie within each of the four subdivided rectangles (zeroes can be there in more than one of them). Eg. if f(M)>0 and f(A)<0, then AEMG contains zeroes of f. Same is true also if f(G)>0 and f(E)<0.</p></li>
<li><p>Do this for all the fou... |
35,987 | <p><em>cross post in <a href="https://stackoverflow.com/questions/3513660/multivariate-bisection-method">StackOverflow</a></em></p>
<p>I need an algorithm to perform a 2D bisection method for solving a $2$x$2$ non-linear problem. Example: two equations $f(x,y)=0$ and $g(x,y)=0$ which I want to solve simultaneously. I ... | dranxo | 6,360 | <p>You might want to consider the vector field</p>
<p>$ \vec{F}(x,y) = (f(x,y), g(x,y)) $</p>
<p>and look for sources and sinks of $\vec{F}$. I think this could be done by recursively dividing up the plane into squares and calculating the winding number of each square. If it is nonzero then you have a critical point ... |
1,791,837 | <p>Recently I have been very fascinated by the claim and impact of Godel's incompleteness theorem. To understand the proof given by Godel, I felt the need to read an introductory book in logic to begin with. I have started reading the book titled "A Mathematical Introduction to Logic" by Herbert Enderton. As mentioned ... | R Mary | 309,615 | <p>I think the function you are looking for is $f(x)=\int_{\gamma_x} \alpha$ where $\gamma_x$ is a path from some chosen base point $x_0$ to $x$. But then for this to be well defined you have to say that all paths between $x$ and $x_0$ are homotopic, i.e. that your sphere is simply connected, so the whole argument gets... |
3,816,041 | <blockquote>
<p>How many ways <span class="math-container">$5$</span> identical green balls and <span class="math-container">$6$</span> identical red balls can be arranged into <span class="math-container">$3$</span> distinct boxes such that no box is empty?</p>
</blockquote>
<p>My attempt :</p>
<p>Finding coefficient... | Dietrich Burde | 83,966 | <p>The polynomial <span class="math-container">$f=x^4+x^3+x^2+x+1$</span> is irreducible over <span class="math-container">$\Bbb F_p$</span> if <span class="math-container">$p\not\equiv \pm 1\bmod 5$</span> with <span class="math-container">$p\neq 5$</span>. The proof goes as follows. Note that
<span class="math-contai... |
1,241,864 | <p>I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.</p>
<p>Thanks in advance.</p>
| Community | -1 | <p>For integer square roots, one should note that there are runs of equal values and increasing lengths</p>
<p>$$1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4\dots$$</p>
<p>For every integer $i$ there are $(i+1)^2-i^2=2i+1$ replicas, and by the Faulhaber formulas</p>
<p>$$\sum_{i=1}^m i(2i+1)=2\frac{2m^3+3m^2+m}6+... |
1,241,864 | <p>I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.</p>
<p>Thanks in advance.</p>
| Simply Beautiful Art | 272,831 | <p>For a better upper bound than Alex's answer,</p>
<p>$$\sum_{n=1}^xn^{1/2}\le\frac23\left(x+\frac12\right)^{3/2}$$</p>
<p>And if you want to improve upon that,</p>
<p>$$\sum_{n=1}^xn^{1/2}\approx\frac23\left(x+\frac12\right)^{3/2}\underbrace{-0.22474487139}_{\zeta(-1/2)}$$</p>
|
96,789 | <p>I've been working on understanding limits thoroughly, so I'm rewriting how I understand the chain rule. Please help me fill in my gaps in understanding.</p>
<p>$f$ is some function. Then</p>
<p>$f'(x) = \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$</p>
<p>Now I might want to evaluate something like </p>
<p>$$... | Michael Hardy | 11,667 | <p>This is correct except that it fails to address the fact that $k$ may be $0$ when $h$ is not. If there is some neighborhood of $0$ such that when $h$ is in that neighborhood and $h\ne0$, then $k\ne0$, then everything above works fine. If every open neighborhood of $0$ contains some value of $h\ne0$ for which $k=0$... |
96,789 | <p>I've been working on understanding limits thoroughly, so I'm rewriting how I understand the chain rule. Please help me fill in my gaps in understanding.</p>
<p>$f$ is some function. Then</p>
<p>$f'(x) = \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$</p>
<p>Now I might want to evaluate something like </p>
<p>$$... | Arturo Magidin | 742 | <p>The final idea is generally correct, as Michael Hardy points out, modulo some refinements to deal with tricky situations.</p>
<p>However, I wanted to point out that some of the arguments leading up to that final part are incorrect:</p>
<ul>
<li><p>You assume that
$$b\lim_{h\to 0}\frac{a}{h} = \lim_{h\to 0}\frac{ab... |
3,482,138 | <p><a href="https://i.stack.imgur.com/OUlV2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OUlV2.png" alt="enter image description here"></a></p>
<ol>
<li>I have found two planes trough the origin that meet the given plane
at right angles.</li>
</ol>
<p>I found three points in the plane, getting t... | bjorn93 | 570,684 | <p>It should be clear that <span class="math-container">$\triangle BEC$</span> is equilateral. Denote <span class="math-container">$\angle BFD=\alpha$</span> and <span class="math-container">$\angle AFD=\beta$</span>. We need to find <span class="math-container">$\alpha+\beta$</span>. You can show that <span class="mat... |
2,708,071 | <p>Question: Suppose $|x_n - x_k| \le n/k^2$ for all $n$ and $k$.Show that $\{x_n\}_{n=1}^{\infty}$ is cauchy. </p>
<p>Attempt : To prove this, I have to find $M \in N$ that for $\varepsilon >0$, $n/k^2 < \varepsilon$ for $n,k \ge M$. </p>
<p>Let $\varepsilon > 1/M$. </p>
<p>Then, $n/k^2 \le M/M^2$ (#) $= ... | Aweygan | 234,668 | <p>No, $(\#)$ is not necessarily true. $M=1$, $n=5$, $k=2$ provides a counterexample. </p>
<p>Note that since $|x_n-x_k|=|x_k-x_n|$, we have $|x_n-x_k|\leq\min\{n/k^2,k/n^2\}$. Given $\varepsilon>0$, choose $M\in\mathbb N$ such that $\varepsilon>1/M$. Suppose $n,k\geq M$ and $k\geq n$. Then we have
$$|x_n-x... |
196,303 | <p>All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings (circles) in a 11 dimensional manifold in order to model particles. Do they do this because there is no mathematical a... | Community | -1 | <p>This paper by Carlip <a href="http://arxiv.org/abs/gr-qc/0108040" rel="nofollow">http://arxiv.org/abs/gr-qc/0108040</a> is a good, relatively nontechnical explanation of why it's hard to reconcile quantum mechanics (QM) with general relativity (GR).</p>
<p>GR says that spacetime is a real manifold with a semi-Riema... |
2,871,655 | <p>I was trying some Cambridge past papers and it said to first separate into partial fractions and then find the sum of the sequence, however after splitting inot partial fractions I'm not getting the terms to cancel out like I normally do with these questions. Is there something I'm missing
.been trying to manipulate... | Simply Beautiful Art | 272,831 | <p>As you've noticed, we have</p>
<p>$$4^sf(s)=\sum_{n=0}^\infty\left(\frac1{(n+1/4)^s}-\frac1{(n+3/4)^s}\right)$$</p>
<p>which, for odd $s$, can be written as</p>
<blockquote>
<p>$$\begin{align}
&(s-1)!4^sf(s)\\
&=\lim_{x\to1/4}\frac{d^{s-1}}{dx^{s-1}}\sum_{n=0}^\infty\left(\frac1{n+x}+\frac1{x+1-n}\right... |
2,452,084 | <ul>
<li><em>I'm having trouble understanding why the arbitrariness of $\epsilon$ allows us to conclude that $d(p,p')<0$. It seems we could likely
conclude a value such as $\frac {\epsilon}{100}$ couldn't we?? The
other idea that would normally work is the limit (as $n$ approaches
$\infty$, $p$ approaches $p'$) but ... | Alekos Robotis | 252,284 | <p>The moral is that the only $x\ge 0$ satisfying $x< \epsilon$ for all $\epsilon>0$ is $0$. For, suppose that $x\ne 0$, then $x=\delta>0$. Choose $\epsilon = \frac{\delta}{2}$. Then $x\not< \epsilon$, contrary to assumption. </p>
<p>Because $\epsilon$ is arbitrary, the statement is true for all $\epsilon&... |
1,956,338 | <p>Let $f\in C_b(\mathbb{R})$ and $g$ any continuous function on $\mathbb{R}$ (or maybe there has to be a restiction on $g$?). Now let K be a compact subset of $\mathbb{R}$ and $U$ an open subset of $\mathbb{R}$. Then I want to show that the following set $$\Phi:=\{t\in\mathbb{R}_+:(e^{t\cdot g}f)(K)\subseteq U\}$$ is ... | user60589 | 60,589 | <p>For a compact set $C$ which is contained in an open set $U$ there is open ball around $1$ such that $B_{\varepsilon}(1) \cdot C \subset U$.</p>
<p>Then you have
$$ e^{B_{\delta}(t)\cdot g(K) }f(K) = e^{B_{\delta}(0)}e^{t\cdot g(K)}f(K)=B_{\varepsilon}(1) \cdot C \subset U $$
where $\varepsilon = \log \delta$ and $e... |
561,921 | <p>So far I have,
$$
\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{1^2+1}}}{x-1}=\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{2}}}{x-1}
$$</p>
<p>I have no idea how to keep going with this, every way I try I get stuck and can't do anything with it. </p>
| marty cohen | 13,079 | <p>More generally,
consider
$$\lim_{x\to a} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{a^2+1}}}{x-a}.$$</p>
<p>I will use just algebra.</p>
<p>If $x \ne a$,</p>
<p>$\begin{align}
\frac{\frac{x}{\sqrt{x^2+1}} - \frac{a}{\sqrt{a^2+1}}}{x-a}
&=\frac{x\sqrt{a^2+1}-a\sqrt{x^2+1}}{\sqrt{x^2+1}\sqrt{a^2+1}(x-a)}\\
&... |
561,921 | <p>So far I have,
$$
\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{1^2+1}}}{x-1}=\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{2}}}{x-1}
$$</p>
<p>I have no idea how to keep going with this, every way I try I get stuck and can't do anything with it. </p>
| Claude Leibovici | 82,404 | <p>Just as for aaa, L'Hopital's rule is, at least to me too, the simplest way to solve your problem. There is another one (which is usable if you know about Taylor series). Around x = 1, the first terms of the Taylor series of x / Sqrt[x^2 + 1] is 1 / Sqrt[2] + (x -1) / (2 Sqrt[2]) +... Then, replace the numerator by t... |
3,209,722 | <p>I saw in another post on the website a simple proof that <span class="math-container">$$\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n = \lim_{m\to\infty} \left( 1+\frac{1}{m} \right)^{mx}$$</span></p>
<p>which consists of substituting <span class="math-container">$n$</span> by <span class="math-container">$mx$</... | DINEDINE | 506,164 | <p>If <span class="math-container">$x<0$</span> then <span class="math-container">$-x>0$</span>. Now apply it for <span class="math-container">$-x$</span> and make the substitution </p>
|
1,969,625 | <p>Which of the following statements is/are true?</p>
<p>$A.$ $\sin(\cos x)=x$ has exactly one root in $[0,\pi/2]$</p>
<p>$B.$ $\cos(\sin x)=x$ has exactly one root in $[0,\pi/2]$</p>
<p>$C.$ Both $A$ and $B$ are true.</p>
<p>$D.$ Both $A$ and $B$ are false.</p>
<p>I tried as $\cos(\sin x):[0,\pi/2]\rightarrow [0,... | Mihir | 378,189 | <p>Use the property:
STEP1: Find the range of the function.
Suppose it comes out to be $[a,b]$
STEP2: Then find $f(a).f(b)$.
STEP3:. If it comes out to be negative(Think precisely). Then the function will have at least one solution in $[a,b]$</p>
<p>I'd go with options A,B &C.
I'll be glad if you'll do it yourself... |
1,969,625 | <p>Which of the following statements is/are true?</p>
<p>$A.$ $\sin(\cos x)=x$ has exactly one root in $[0,\pi/2]$</p>
<p>$B.$ $\cos(\sin x)=x$ has exactly one root in $[0,\pi/2]$</p>
<p>$C.$ Both $A$ and $B$ are true.</p>
<p>$D.$ Both $A$ and $B$ are false.</p>
<p>I tried as $\cos(\sin x):[0,\pi/2]\rightarrow [0,... | Vik78 | 304,290 | <p>By the intermediate value theorem there is at least one root (since 0 < sin(cos(0)) and 1 > sin(cos(1))). Since the derivative of sin(cos($x$)) is less than one everywhere we can use the fundamental theorem of calculus to conclude that this is the only root.</p>
|
1,793,318 | <p>[limit question][1]</p>
<p>Let $x_{k}$ be a sequence of strictly positive real numbers with $\lim \limits_{k \to \infty}\dfrac{x_{k}}{x_{k+ 1}} >1$. Prove that $x_{k}$ is convergent and calculate $\lim \limits_{k \to \infty} x_{k}$.</p>
<p><strong>Attempted answer attached as picture.</strong></p>
<p>I am not... | Zhanxiong | 192,408 | <p>You are on the right track. For clarity, let's denote the partial sum $\sum_{n = 1}^N \frac{x}{1 + n^2x^2}$ by $S_N(x)$. One way to show the result is to investigate $\sup_{x \in [0, 1]}|S_{2N}
(x) - S_{N}(x)|$:
\begin{align}
\sup_{x \in [0, 1]} |S_{2N}(x) - S_N(x)| & = \sup_{x \in [0, 1]}\sum_{n = N + 1}^{2N} ... |
1,397,576 | <p>To me there is a hierarchy where vectors $\subset$ sequences $\subset$ functions $\subset$ operators</p>
<ul>
<li><p>All vectors are sequences, but not all sequences are vectors because
sequences are infinite dimensional</p></li>
<li><p>All sequences are functions, but not all functions are sequences
because functi... | Race Bannon | 188,877 | <p>A vector is an object of a vector space that obeys the usual vector space properties. </p>
<p>A sequence is a function from the natural numbers to some set.</p>
<p>If $X$ is a set, and $Y$ is a set, a function is a subset of $X \times Y$ such that for $(x,y), (x,y') \in X \times Y$, we have $y = y'$.</p>
<p>An op... |
2,535,933 | <p>let assume i have a position function in 1 dimension with constant acceleration.</p>
<p>$$ x(t) = x_0 + v_0t + \frac{1}{2}at^2 $$</p>
<p>then it's first derivative is a velocity function:
$$ \frac{dx}{dt} = v(t) = v_0 + at $$</p>
<p>then it's second derivative is an acceleration function:</p>
<p>$$ \frac{dv}{dt}... | angryavian | 43,949 | <p>Suppose for sake of contradiction $P(X \ne E[X]) > 0$.
Then there exists some $\epsilon > 0$ such that $$P(|X - E[X]| > \epsilon) > 0.$$
But then
$$\text{Var}(X) = E[(X - E[X])^2] \ge E[(X - E[X])^2 \mathbf{1}_{|X - E[X]| > \epsilon}] > \epsilon^2 P(|X - E[X]| > \epsilon) > 0,$$
a contradicti... |
29,177 | <p>I had a quick look around here and google before, and didn't find any answer to this particular question, and it's beginning to really irritate me now, so I'm asking here:</p>
<p>How is one supposed to write l (little L), 1 (one) and | (pipe) so that they don't all look the same? One of my teachers draws them all a... | Kevin Seifert | 65,059 | <p>Another approach is to write the 1 serifs backwards (mirror image). That way, the top serif can be exaggerated, but doesn't look anything like a 2 or 7. Just make it pointy so it doesn't look like a C.</p>
|
997,634 | <p>Evaluate
$$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ </p>
<p>using polar coordinates.</p>
<p>My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct</p>
<p>And also, when I change from $dy\,dx$ to $dr \,d\theta$ ...how do I know if it should be $dr\,d\theta$ or $d\theta... | Gahawar | 129,839 | <blockquote>
<p>$$\int\limits_0^R \int\limits_0^{\sqrt{R^2-x^2}} e^{-x^2-y^2} \; dy \; dx = \int\limits_0^{\pi/2} \int\limits_0^{R} e^{-\rho^2} \rho \; d\rho \; d\theta = \frac{\pi}{4} \left(1 - e^{-R^2}\right).$$</p>
</blockquote>
<p>The domain of integration is a quarter circle of radius $R$, so when one converts ... |
44,226 | <p>Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed the disc $D$ real-analytically and isometrically into real Euclidean space ${\mathbb{R}}^n$ for some large $n$. (I th... | Robert Bryant | 13,972 | <p>The answer is 'no', there is no holomorphic curve in $\mathbb{C}^n$ (for any $n$) such that the induced metric has constant negative curvature. To my knowledge, this was first proved by E. Calabi many many years ago, essentially using the structure equations for holomorphic curves in $\mathbb{C}^n$. The proof is e... |
708,596 | <p>Suppose that $U$ and $V$ are vector spaces, and that $f:V \to W$ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in \ker f $ such that $v=u+w$.</p>
<p>I roughly understand what is kernel and its definition but I have no idea how to apply it to... | mookid | 131,738 | <p>We has necessarily:
$w - v-u$, let us check that it is in $\ker f$.</p>
<p>Using linearity:
$f(w) = f(v-u) = f(v) - f(u) = 0$
That's it.</p>
|
4,261,763 | <p>I was working with this problem:</p>
<p><a href="https://i.stack.imgur.com/AgVpu.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AgVpu.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$f(x) = 1, x \in(-\infty,0)\cup(0,\infty)\\
f(x) = -1, x = 0 $</span></p>
<p>Th... | L F | 221,357 | <p>What about
<span class="math-container">$$f(x)=\begin{cases}x & \text{if }x\in\mathbb{Q} \\ -x & \text{if }x\in\mathbb{R}-\mathbb{Q} \end{cases}$$</span>
Just separe points of domain on rationals and irrationals and give them opposite aditive functions, as this way you can have a lot of functions with the pr... |
976,617 | <p>Find supremum and infimum of the set:
$B={ \frac{x}{1+ \mid x \mid }} \ for \ x\in \mathbb{R}$
For me it is visible that it will be 1 and -1 respectively but how to prove it properly?</p>
| pitchounet | 61,409 | <p>Let :</p>
<p>$$ B = \left\{ f(x), \; x \in \mathbb{R} \right\}. $$</p>
<p>First, note that $B$ is bounded and not empty, which ensures that $\inf(B)$ and $\sup(B)$ exist. In order to prove that $\sup(B) = 1$, you need to prove that : $\forall y \in B, \, y \leq 1$ and either :</p>
<ul>
<li>$1 \in B$ and in this c... |
3,496,594 | <p>I have to factorize the polynomial <span class="math-container">$P(X)=X^5-1$</span> into irreducible factors in <span class="math-container">$\mathbb{C}$</span> and in <span class="math-container">$\mathbb{R}$</span>, this factorisation happens with the <span class="math-container">$5$</span>th roots of the unity. <... | Bernard | 202,857 | <p>A way to obtain an explicit factorisation using <em>Chebyshev polynomials</em>:</p>
<p>Using the recurrence relation:
<span class="math-container">$$P_{n+1}(t)=2tP_n(t)-P_{n-1}(t),\qquad P_0(t)=1,\;P_1(t)=t,$$</span>
we readily obtain
<span class="math-container">$$P_5(t)=2t\, P_4(t)-P_3(t)=16t^5-20t^3+5t,$$</span... |
1,707,675 | <p>How can I find the indefinite integral which is $$\int \frac{\ln(1-x)}{x}\text{d}x$$</p>
<p>I tried to use substitution by assigning $$\ln(1-x)\text{d}x = \text{d}v $$ and $$\frac{1}{x}=u$$ but, it is meaningless but true, the only thing I came up from integration by part is that $$\int \frac{\ln(1-x)}{x^2}\text{d... | Jack D'Aurizio | 44,121 | <p>The primitive is not an elementary function. Since in a neighbourhood of the origin we have:
$$ -\log(1-x) = \sum_{n\geq 1}\frac{x^n}{n} $$
it happens that:
$$ \int_{0}^{x}\frac{\log(1-z)}{z}\,dz = -\sum_{n\geq 1}\frac{x^n}{n^2} = \color{red}{-\text{Li}_2(x)}$$
for any $x\in(-1,1)$.</p>
|
654,239 | <p>Let $\phi_{\alpha}(z)=\frac{z-\alpha}{1-\bar{\alpha}z}$ for $0<|\alpha|<1$</p>
<p>Find all the line $L$ in the complex plane such that $\phi_{\alpha} (L)=L$</p>
<p>Can you help me?</p>
| user149088 | 149,088 | <p>$\newcommand{\CC}{\mathcal{C}}$
In this case $f$ is a 2-cycle (i.e. $f\circ f=Id$. ًWe take $f=-\Phi_\alpha$). Let $a$ be a fixed point of $f$ ($a^2=\frac \alpha{\bar \alpha}$). Let $\CC$ be the circle of center $\frac1{\bar \alpha}$ and passing through $a$, $L_1$ the straight line $(a,\alpha)$, $L_2$ the straight ... |
4,253,564 | <p>I am having trouble with the following integral</p>
<blockquote>
<p>Prove that <span class="math-container">$$ \int_0^1\frac{x\ln(x)}{1+x^2+x^4}dx=\frac{1}{36}\Big(\psi^{(1)}(2/3)-\psi^{(1)}(1/3)\Big)$$</span></p>
</blockquote>
<p><span class="math-container">$$I=\int_0^1\frac{x\ln(x)}{1+x^2+x^4}dx=\int_0^1\frac{\ln... | Quanto | 686,284 | <p>Alternatively</p>
<p><span class="math-container">\begin{align} \int_0^1\frac{x\ln x}{1+x^2+x^4}dx
= &\int_0^1\frac{x\ln x}{(e^{i\frac\pi3}+x^2)(e^{-i\frac\pi3}+x^2)}dx \\
=& \>\frac1{\sin\frac\pi3 }\>\Im \int_0^1 \frac{e^{i\frac\pi3}x\ln x}{1+ e^{i\frac\pi3}x^2 }dx
\overset{x^2\to x} =\frac{1}{2\sqrt3... |
401,898 | <p>the function $f_n(x)=x^n-x^{2n}$ converge to $f(x)=0$ in $(-1,1]$. Intuativly the function does not converge uniformally in (-1,1]. How can I prove it?
I tried using the definition $\lim \limits_{n\to\infty}\sup \limits_{ x\in (-1,1]}|f_n(x)-f(x)|$ function is continial fractional on $[-1,1]$ and $x=0,(\frac 1 2 )^{... | Community | -1 | <p>$f$ doesn't converge uniformly in $(-1,1]$ since
$$\lim_{x\to-1}\lim_{n\to\infty}f_n(x)=0\neq\lim_{n\to\infty}\lim_{x\to-1}f_n(x)=\lim_{n\to\infty}(-1)^n-1$$</p>
|
2,163,948 | <p><strong>Question:</strong></p>
<blockquote>
<p>Does there exist a Riemannian manifold, with a point $p \in M$, and <strong>infinitely many</strong> points $q \in M$ such that there is <strong>more than one</strong> minimizing geodesic from $p$ to $q$?</p>
</blockquote>
<p><strong>Edit:</strong></p>
<p>As demons... | HK Lee | 37,116 | <p>I will introduce two examples</p>
<p>(1) Consider a torus in $\mathbb{R}^3$</p>
<p>(2) Consider two dimensional regular triangle $T$ in
$\mathbb{R}^2\subset X=\mathbb{R}^3$ If $U$ is a <em>suitable</em> tubular
neighborhood of $T$ in $X$, then consider $\partial U$ which is
homeomorphic to $S^2$</p>
<p>There are ... |
3,163,580 | <p>I'm having troubles to show that if <span class="math-container">$0<|\alpha|<1$</span> then the elements <span class="math-container">$f_k=\lbrace 1, \alpha^k, \alpha^{2k}, \alpha^{3k}, \cdots \rbrace$</span> span <span class="math-container">$\ell^2$</span> for <span class="math-container">$k \geq 1$</span>. ... | Fnacool | 318,321 | <p>Recall that a set <span class="math-container">$S\subseteq \ell^2$</span> is dense if <span class="math-container">$({\bf c},s)=0$</span> for all <span class="math-container">$s\in S$</span> implies <span class="math-container">${\bf c}=0$</span>.</p>
<p>Fix <span class="math-container">$\alpha$</span> such that <... |
1,718,380 | <p>Simply: How do I solve this equation for a given $n \in \mathbb Z$?</p>
<p>$x^x = n$</p>
<p>I mean, of course $2^2=4$ and $3^3=27$ and so on. But I don't understand how to calculate the reverse of this, to get from a given $n$ to $x$. </p>
| marty cohen | 13,079 | <p>Simpler:</p>
<p>If
$x^x = n$,
then $x\ln(x) = \ln(n)
=y$.</p>
<p>Let
$f(x) = x\ln(x)-y
$.
$f'(x)
=\ln(x)+1
$.</p>
<p>Applying Newton's iteration,
starting with $x = \frac{y}{\ln y}$,
$x_{new}
=x-\frac{f(x)}{f'(x)}
=x-\frac{x\ln(x)-y}{\ln(x)+1}
=\frac{x\ln(x)+x-x\ln(x)+y}{\ln(x)+1}
=\frac{x+y}{\ln(x)+1}
$.</p>
<p... |
2,604,206 | <p>Can anyone provide links to a concrete proof? Intuitively, the two-dimensional real space is infinite. so there should be infinitely many subspaces. But how do I go about a proof?</p>
| Alekos Robotis | 252,284 | <p>Let $V$ be a $2-$dimensional vector space over $\mathbf{R}$. Fix a basis $v_1,v_2$ for $V$ so that $V$ can be identified with $\mathbf{R}^2$. Then we can see that the subspaces
$$X_\theta=\{\lambda(\cos\theta, \sin\theta):\lambda \in \mathbf{R}\}$$
are distinct $1-$dimensional vector spaces for each $0\le \theta<... |
1,811,109 | <p>How can we cause this relation to be true?</p>
<blockquote>
<p>$$x \sin\theta + y \cos\theta = \sqrt{ x^2 + y^2 } \tag{$\star$}$$</p>
</blockquote>
<p>I know the identity</p>
<p>$$x \sin\theta + y \cos\theta = \sqrt{x^2+y^2}\; \sin\left(\theta + \operatorname{atan}\frac{y}{x}\right)$$
What can make the sine pa... | Noam | 344,143 | <p>This identity is not correct. Take for example $x=2,\,y=3,\,\theta=0^\circ.$
Then $\sqrt{x^2+y^2} = \sqrt{4+9}=3.605\neq 3=x\sin \theta + y \cos \theta.$</p>
|
1,811,109 | <p>How can we cause this relation to be true?</p>
<blockquote>
<p>$$x \sin\theta + y \cos\theta = \sqrt{ x^2 + y^2 } \tag{$\star$}$$</p>
</blockquote>
<p>I know the identity</p>
<p>$$x \sin\theta + y \cos\theta = \sqrt{x^2+y^2}\; \sin\left(\theta + \operatorname{atan}\frac{y}{x}\right)$$
What can make the sine pa... | Tom-Tom | 116,182 | <p>In this answer, I will suppose that either $\theta$ is fixed and solve the equation for $(x,y)$ or that $x$ and/or $y$ are fixed and solve for $\theta$. But first,
let us write $r=\sqrt{x^2+y^2}$ and consider the angle $\alpha$ such that
$x=r\sin\alpha$ and $y=r\cos\alpha$. This is always possible. (Use for instanc... |
2,058,939 | <p>Let $f:\mathbb{R^3}$ $\rightarrow$ $\mathbb{R}$, defined as: </p>
<p>$$f(x,y,z)=\begin{cases} \left(x^2+y^2+z^2\right)^p \exp\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)& ,\,\text{if }\quad(x,y,z) \ne (0,0,0)\quad \\
0 &,\,\text{o.w}
\end{cases}$$</p>
<p>Where $\,p\in \mathbb{R}$. Is this function is contin... | Community | -1 | <p>$f$ is continuous in $\mathbb{R}^3\setminus\{(0,0,0)\}$, but not in the point $(0,0,0)$ since the limit of $f(x,y,z)$ when $(x,y,z)\to(0,0,0)$ does not exist: To see it recall that
$$
e^{t}=1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\frac{t^4}{4!}+\cdots,
$$</p>
<p>so for $t>0$ we have
$$
e^{t}\geq\frac{t^{2p+2}}{(2p+2)!... |
3,418,810 | <p>So my question is what does it mean to be <span class="math-container">$0$</span> in <span class="math-container">$S^{-1} M$</span>, where <span class="math-container">$S$</span> is a multi-closed subset of a ring <span class="math-container">$A$</span>, <span class="math-container">$M$</span>, lets assume to be a f... | Angina Seng | 436,618 | <p>In <span class="math-container">$S^{-1}M$</span> and element <span class="math-container">$m/s$</span> (with <span class="math-container">$s\in S$</span> and <span class="math-container">$m\in M$</span>) is zero iff
<span class="math-container">$tm=0$</span> for some <span class="math-container">$t\in S$</span>, tha... |
194,191 | <p>Test the convergence of $\int_{0}^{1}\frac{\sin(1/x)}{\sqrt{x}}dx$</p>
<p><strong>What I did</strong></p>
<ol>
<li>Expanded sin (1/x) as per Maclaurin Series</li>
<li>Divided by $\sqrt{x}$</li>
<li>Integrate</li>
<li>Putting the limits of 1 and h, where h tends to zero</li>
</ol>
<p>So after step 3, I get somethi... | Sasha | 11,069 | <p>Change variables $u = \frac{1}{x}$. Then:
$$
\int_0^1 \frac{\sin(1/x)}{\sqrt{x}} \mathrm{d}x= \int_1^\infty \sqrt{u} \sin(u) \frac{\mathrm{d}u}{u^2} =\int_1^\infty \frac{\sin(u)}{u^{3/2}}\mathrm{d}u
$$
The latter integral is absolutely convergent.</p>
|
987,054 | <p>Prove that the sequence
$$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$
Is decreasing.</p>
<p>I have calculated $b_n/b_{n-1}$ but it is obtain:
$$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$
But I can't go on.</p>
<p>Any suggestions please?</p>
| QmmmmLiu | 186,644 | <p>Why don't you change your goal to prove that $b_n-b_{n+1}>0$?</p>
<p>$b_n-b_{n+1}=(1+1/n)^n-(1+1/n)^n(1+1/n)=(1+1/n)^n(1-1-1/n)=(1+1/n)^n(-1/n)$</p>
<p>Since $n$ is a natural number, $-1/n < 0$. Since $1>0, 1/n>0$, we have $1+1/n>0$. It would be nice if you can use $(1+1/n)^n >0$ directly based o... |
1,425,519 | <p>I'm trying to solve <a href="http://poj.org/problem?id=2140" rel="nofollow">this problem</a> on POJ and I thought that I had it. Since I can't figure out what's wrong with my code, I'd like to test it against a huge list of correct answers. This will make my code much easier to debug.</p>
<p>If you don't want to go... | Dominik | 259,493 | <p>The other answers already answered how to calculate the number more efficiently, but since you've asked for a list of values I will provide one. If you don't count a "sum" of $1$ number as a sum, the wanted value is called <a href="https://en.wikipedia.org/wiki/Polite_number#Politeness" rel="nofollow">politeness of ... |
2,356,813 | <p>Let $f:[0,\infty)\to\mathbb R$ be a function in $C^2$ such that
$\lim_{x\to\infty} (f(x)+f'(x)+f''(x)) = a.$
Prove that $\lim_{x\to\infty} f(x)=a$</p>
| RRL | 148,510 | <p>Note that with $\alpha = e^{i \pi/3}$ and $\beta = e^{-i \pi/3}$ we have $\alpha \beta = 1$ and $\alpha + \beta = 1$ and , therefore,</p>
<p>$$\tag{1}f(x) + f'(x) + f''(x) = \alpha\beta f(x) + ( \alpha + \beta)f'(x) + f''(x) \\ =\alpha[ \, \beta f(x) + f'(x) + (\beta f(x) + f'(x))' \, ]$$</p>
<p>One can prove the ... |
70,582 | <p>For which n can $a^{2}+(a+n)^{2}=c^{2}$ be solved, where $a,b,c,n$ are positive integers?
I have found solutions for $n=1,7,17,23,31,41,47,79,89$ and for multiples of $7,17,23$...
Are there infinitely many prime $n$ for which it is solvable? </p>
| Thomas Andrews | 7,933 | <p>The general primitive solution to $x^2+y^2 = z^2$ is given by: $x=u^2-v^2$, $y=2uv$, $z=u^2+v^2$, with $u,v$ relatively prime and not both odd.</p>
<p>For $(a,a+n,z)$ to be a primitive triple, we'd have to have a $(u,v)$ such that: $|u^2 - v^2 - 2uv| = n$. We can rewrite that as: $(u-v)^2 - 2v^2 = \pm n$</p>
<p>S... |
1,860,615 | <p>I have a simple quadratic (with $x^2$) equation, x can Be complex too:</p>
<p>$$x^2+x+1=0$$</p>
<p>But it could be any equation, the equation above is just an example. I need to compute $x_1^{10}+x_2^{10}$, but it could have another exponents (ex: $x_1^{50}+x_2^{50}$).</p>
<p>I need to know, on a general case, ho... | Brian Tung | 224,454 | <p>One approach is to represent the roots in polar form, and take advantage of their symmetry. For the sake of ease of exposition, let the quadratic be monic; that is, $a = 1$. If it's not already in that form, it's trivial to convert it to a monic form.</p>
<p>If the roots are real—if $b^2-4c \geq 0$—I assume you k... |
1,860,615 | <p>I have a simple quadratic (with $x^2$) equation, x can Be complex too:</p>
<p>$$x^2+x+1=0$$</p>
<p>But it could be any equation, the equation above is just an example. I need to compute $x_1^{10}+x_2^{10}$, but it could have another exponents (ex: $x_1^{50}+x_2^{50}$).</p>
<p>I need to know, on a general case, ho... | Ng Chung Tak | 299,599 | <p>\begin{align*}
\alpha+\beta &= -\frac{b}{a} \\
\alpha \beta &= \frac{c}{a} \\
\alpha^n+\beta^n &=
(\alpha+\beta)^{n}+\sum_{k=1}^{\left \lfloor \frac{n}{2} \right \rfloor}
\binom{n-k}{k} \frac{n(-\alpha \beta)^{k} (\alpha+\beta)^{n-2k}}{n-k} \\
&=
(\alpha+\beta)^{n}-n\alpha \beta (\alp... |
1,774,294 | <p>If you have $$y^2=2x^2+C$$</p>
<p>why is this not equivalent to</p>
<p>$$y=\sqrt{2x^2}+C$$</p>
| lab bhattacharjee | 33,337 | <p>$$y=c+\sqrt{2x^2}\implies y-c=\sqrt2|x|$$</p>
<p>Squaring we get $$(y-c)^2=2x^2\iff2x^2=?$$</p>
<p>But by the first case $$2x^2=y^2-c$$</p>
|
3,270,944 | <p>Let <span class="math-container">$A$</span> be a bounded linear operator on a separable Hilbert space <span class="math-container">${\cal H}$</span>, and suppose that <span class="math-container">$A$</span> is distinct from its adjoint <span class="math-container">$A^*$</span>. </p>
<p><strong>Question:</strong> Ca... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$\frac{2(x+h)^2+1-(2x^2+1)}{(x + h) - (x)} =\frac{2(x^2+2xh+h^2)+1-(2x^2+1)}{(x + h) - (x)}=$$</span></p>
<p><span class="math-container">$$\frac{4xh+2h^2}{ h}=4x+2h$$</span></p>
|
751,138 | <p>Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$. </p>
<p>using proof of contradiction </p>
<p>can someone prove this using contradiction method please</p>
| user2345215 | 131,872 | <p>Prove the contrapositive:</p>
<blockquote>
<p>If $3\mid a+1$, then $3\nmid a^2+1$.</p>
</blockquote>
<p>Which is easy, because then $a=3k-1$ and $a^2+1=9k^2-6k+2$, which isn't divisible by $3$.</p>
|
386,799 | <blockquote>
<p>P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation.</p>
<p>P1087: If <span class="math-container">$S$</span> is a smooth orientable surface given in parametric form by a v... | Ted | 15,012 | <p>There's no fixed answer to this, because whichever one you choose, if you were to reverse $u$ and $v$, you would have to choose the other one. It depends on how the $u$ and $v$ coordinates are oriented on the surface.</p>
<p>However, "$\color{brown}{\text{if the positive $u$ and $v$ tangent vectors at a point are ... |
2,353,193 | <p>I've recently been learning some homological algebra, mainly out of Northcott and some other sources, and I'm having trouble with the notion of projective dimension. In particular, I have a question (not from Northcott) that says</p>
<blockquote>
<p>Let $R = k[x,y]$ for a field $k$ and $M$ a finitely generated $R... | Mariano Suárez-Álvarez | 274 | <p>A simpler way to see that the claim is false in one direction is to notice that the automorphism group of $R$ acts transitively on the set of modules of the form $R/(x-a,y-b)$, so that they all have the same projective dimension. As there are no nonzero maps between them, the claim is false.</p>
<hr>
<p>Let's do t... |
162,836 | <p>I would like to find the surface normal for a point on a 3D filled shape in Mathematica. </p>
<p>I know how to calculate the normal of a parametric surface using the cross product but this method will not work for a shape like <code>Cone[]</code> or <code>Ball[]</code>.</p>
<ol>
<li>Is there some sort of <code>Reg... | Bill Watts | 53,121 | <p>If you know the equation of your surface, you can do this. For instance a circle at the origin of radius 5.</p>
<pre><code>f[x_,y_,z_]=x^2+y^2+z^2-5^2;
grad[x_,y_,z_]=Grad[f[x,y,z],{x,y,z}]
</code></pre>
<p>The unit normal vector on that surface is</p>
<pre><code>normal[x_,y_,z_]=Simplify[grad[x,y,z]/Sqrt[grad[... |
162,836 | <p>I would like to find the surface normal for a point on a 3D filled shape in Mathematica. </p>
<p>I know how to calculate the normal of a parametric surface using the cross product but this method will not work for a shape like <code>Cone[]</code> or <code>Ball[]</code>.</p>
<ol>
<li>Is there some sort of <code>Reg... | Jesse Wilson | 83,195 | <p>I just came across the UnitNormal[] function that might be applicable:
<a href="https://resources.wolframcloud.com/FunctionRepository/resources/UnitNormal" rel="nofollow noreferrer">https://resources.wolframcloud.com/FunctionRepository/resources/UnitNormal</a></p>
<p>Also, SurfaceData[surface,"NormalVector"... |
2,698,098 | <p>The question:</p>
<blockquote>
<p>Suppose <span class="math-container">$0< \delta < \pi$</span>, <span class="math-container">$f(x) = 1$</span> if <span class="math-container">$|x| \leq \delta$</span>, <span class="math-container">$f(x) = 0$</span> if <span class="math-container">$\delta < |x| \leq \pi$</sp... | Kavi Rama Murthy | 142,385 | <p>There is a small mistake in the calculation of $\hat {f} (n)$. Also, your calculation does not hold for $n=0$. Calculate $\hat {f} (0)$ separately and use the fact that $\sum \hat {f} (n) =f(0)$. You will also have change $n$ to $-n$ when you sum over negative values of $n$.</p>
|
3,368,655 | <p>I came across a problem that asked if it is posible for a function to be Riemann integrable function in <span class="math-container">$[0,+\infty)$</span> but also <span class="math-container">$|f(x)|\geq 1$</span> for all <span class="math-container">$x\geq 0$</span>. </p>
<p>At first I thought it was imposible, bu... | Chris Culter | 87,023 | <p>The <a href="https://en.wikipedia.org/wiki/Fresnel_integral" rel="nofollow noreferrer">Fresnel integral</a>:
<span class="math-container">$$\int_0^\infty e^{ix^2}dx=\frac{(1+i)\sqrt{2\pi}}{4}$$</span>
(The problem statement doesn't say <span class="math-container">$f$</span> has to be real!)</p>
|
292,331 | <p>Suppose that $(n_k)_{k\in \mathbb{N}}$ is a given increasing sequence of positive integers. </p>
<p>Does there exist an (irrational) number $a$ such that
$\{an_k\}:=(a n_k)\text{mod }1 \rightarrow 1/2$ as $k \rightarrow \infty$? </p>
| Nick S | 11,552 | <p>The sequence $n_{2k}=k^2, n_{2k+1}=k^2+1$ is a counterexample.</p>
<p>Indeed, if $\{ n_k a \} \to \frac{1}{2}$ then $\{ n_{2k}a \} \to \frac{1}{2}$ and $\{ n_{2k+1} a\} \to \frac{1}{2}$.</p>
<p>This implies that $a= n_{2k+1}a-n_{2k} a= \lfloor n_{2k+1}a\rfloor+ \{ n_{2k+1}a \} - \lfloor n_{2k}a\rfloor- \{ n_{2k}a ... |
3,055,649 | <p>Can there be more than four different types of polygons meeting at a vertex? How?
(The polygons must be convex, regular and different)</p>
<p>There are two ways to fit 5 regular polygons around a vertex, what are they?
(The polygons must be regular, they may not be of different types)</p>
| user630708 | 630,708 | <p>This is easy using Residue Theory:</p>
<p>Note that by symmetry <span class="math-container">$\int_{0}^{\pi/2}...dx=1/4\int_{-\pi}^{\pi}…dx$</span> (use parity and a sub <span class="math-container">$y=\pi-x$</span> to Show that).</p>
<p>employing <span class="math-container">$z=e^{ix}$</span> we get</p>
<p><span... |
315,457 | <p>I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point.</p>
<p>Currently, I am thinking first write $x=\pi+x'$ where $x'=-0.14159265358979312$ and then use Taylor series $\cos(x)=\sum_{i=1}^n(-1)^n\... | user1551 | 1,551 | <p>A square matrix is reducible iff the associated directed graph has smaller <a href="http://en.wikipedia.org/wiki/Strongly_connected_component" rel="noreferrer">strongly connected components</a>. So you may use a <a href="http://en.wikipedia.org/wiki/Path-based_strong_component_algorithm" rel="noreferrer">strong comp... |
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| Sophie Clad | 190,787 | <p>Abstract algebra by John Fraleigh and JA Gallian Contemporary Abstract Algebra </p>
|
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| Gridley Quayle | 202,118 | <p>I am also in high school and two books I've used and found very accessible are:</p>
<p>1) 'Visual Group Theory' by Nathan Carter (the diagrams and illustrations are excellent, a bit pricey though) </p>
<p>2) 'Book of Abstract Algebra' by Charles C. Pinter (the 'Dover books on Mathematics' series of mathematics boo... |
3,882,261 | <p>I have the following question. It's basically my first day doing complex numbers, so I am absolutely lost here. <a href="https://i.stack.imgur.com/JTebK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JTebK.png" alt="enter image description here" /></a>
I have read that the modulus-arg form is
<sp... | Karan Elangovan | 497,101 | <p>I believe a valid parametrisation would be:</p>
<p><span class="math-container">$$ x = a * cosh(t)$$</span>
<span class="math-container">$$ y = b * sinh(t)$$</span>
<span class="math-container">$$ t \in \mathbb{R} $$</span></p>
|
3,882,261 | <p>I have the following question. It's basically my first day doing complex numbers, so I am absolutely lost here. <a href="https://i.stack.imgur.com/JTebK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JTebK.png" alt="enter image description here" /></a>
I have read that the modulus-arg form is
<sp... | Intelligenti pauca | 255,730 | <p>Your equation can be factored as
<span class="math-container">$$
\left({x\over a}+{y\over b}\right)\left({x\over a}-{y\over b}\right)=1.
$$</span>
Set then:
<span class="math-container">$$
{x\over a}+{y\over b}=t
\quad\text{and}\quad
{x\over a}-{y\over b}={1\over t}.
$$</span></p>
|
1,453,010 | <p>A certain biased coin is flipped until it shows heads for the first time. If the probability of getting heads on a given flip is $5/11$ and $X$ is a random variable corresponding to the number of flips it will take to get heads for the first time, the expected value of $X$ is:
$$E[x] = \sum_{x=1}^\infty{x\frac{5}{1... | Math1000 | 38,584 | <p>In general, if $X\sim\operatorname{Geo}(p)$, i.e. $\mathbb P(X=n)=p(1-p)^{n-1}$, $n=1,2,\ldots$, then
\begin{align}
\mathbb E[X] &= \sum_{n=1}^\infty np(1-p)^{n-1}\\
&= -p\sum_{n=1}^\infty \frac{\mathsf d}{\mathsf dp}\left[ (1-p)^n\right]\\
&= -p\frac{\mathsf d}{\mathsf dp}\left[\sum_{n=1}^\infty (1-p)^n... |
2,666,772 | <blockquote>
<p>$W$ = $\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix}$. Use W to build an 8x8 matrix encoding an orthonormal basis in $R^8$ by scaling A = $\begin{bmatrix} W & W \\ W & -W \end{bmatrix}$ in t... | José Carlos Santos | 446,262 | <p>No. First of all, $W$ is not orthonormal: its columns don't have norm $1$. Normalize it first. Call it $U$. Then consider $\left[\begin{smallmatrix}U&0\\0&U\end{smallmatrix}\right]$.</p>
|
2,090,790 | <p>We are given $g(x)=\frac{x \sin x}{x+1}$, and as I said we need to show it has no maxima in $(0,\infty)$.</p>
<p><strong>My attempt</strong>: assume there is some $x_0>0$ that yields a maxima. then for all $x$</p>
<p>$$-1+\frac{1}{x+1}\leq \frac{x \sin x}{x+1}\leq \frac{x_0 \sin x_0}{x_0+1}\leq 1-\frac{1}{x_0+1... | Dr. Sonnhard Graubner | 175,066 | <p>for the first derivative we get $$\frac{(x^2+x)\cos(x)+\sin(x)}{(x+1)^2}$$
but this derivative hase Solutions in $$0<x<\infty$$</p>
|
2,090,790 | <p>We are given $g(x)=\frac{x \sin x}{x+1}$, and as I said we need to show it has no maxima in $(0,\infty)$.</p>
<p><strong>My attempt</strong>: assume there is some $x_0>0$ that yields a maxima. then for all $x$</p>
<p>$$-1+\frac{1}{x+1}\leq \frac{x \sin x}{x+1}\leq \frac{x_0 \sin x_0}{x_0+1}\leq 1-\frac{1}{x_0+1... | Maverick | 171,392 | <p>Consider the interval $x\in [2n\pi,(2n+1)\pi]$</p>
<p>$f(2n\pi)=f((2n+1)\pi)=0$</p>
<p>Invoking Rolle's theorem it is clear that there exists atleast one $x=c$ for which $f'(c)=0$ where $c\in(2n\pi,(2n+1)\pi)$.</p>
<p>Also,on this interval $f(x)$ is clearly positive and since $f(x)$ is continuous and differentia... |
3,504,422 | <blockquote>
<p>Find: <span class="math-container">$$\displaystyle\lim_{x\to
\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln x}$$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$\displaystyle\lim_{x\to\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln x}=\lim_{... | bjorn93 | 570,684 | <p>Use <span class="math-container">$\lim_{t\to 1}\frac{\ln(t)}{t-1}=1$</span> twice: first for <span class="math-container">$t=\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}$</span> and then for <span class="math-container">$t=\frac{x^2+3x+4}{x^2+2x+3}$</span>. The rest is simplification.
<span class="math-container">$$\begin{al... |
979,144 | <p>I am searching for a formula of sum of binomial coefficients $^{n}C_{k}$ where $k$ is fixed but $n$ varies in a given range? Does any such formula exist?</p>
| Felix Marin | 85,343 | <p>$\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]{\displaystyle{#1}}
\newcomma... |
1,891,496 | <p>For example, can we say: $\infty=\lim\limits_{n\rightarrow\infty} n < \aleph_0$?</p>
<p>These are two different types of structures. The limit being like the length, extension, or just generic magnitude and the other being cardinality of a set. Can we compare magnitude to cardinality?</p>
<p>Intuitively, we can... | jdods | 212,426 | <p>To summarize information I've gotten from the existing answers and discussion in comments:</p>
<ul>
<li>Cardinals and real numbers are not comparable with the standard relations for real numbers nor those for cardinals (e.g. the usual $=$, $<$, etc., $<$ for real numbers is not the same $<$ as for cardinal... |
1,243,661 | <p>Let $\Theta$ be an unknown random variable with mean $1$ and variance $2$. Let $W$ be another unknown random variable with mean $3$ and variance $5$. $\Theta$ and $W$ are independent.</p>
<p>Let: $X_1=\Theta+W$ and $X_2=2\Theta+3W$. We pick measurement $X$ at random, each having probability $\frac{1}{2}$ of being c... | grand_chat | 215,011 | <p>The choice between $X_1$ and $X_2$ is a coin toss independent of everything else. Let $Y$ equal 1 if $X_1$ is chosen, and 0 if $X_2$ is chosen. Then
$$
X = X_1 Y + X_2(1-Y)\;.
$$
Calculate
$$E(X) = E(X_1Y)+ E[X_2(1-Y)] = 0.5 (EX_1+EX_2)
$$
and
$$E(X^2) = E(X_1^2Y^2) + 2E(X_1X_2Y(1-Y)) + E[X_2^2(1-Y)^2] = 0.5(EX_1^2... |
3,852,877 | <p>Is it possible to solve the eigenvalue problem of <span class="math-container">$$y''(x) - 2\gamma\, y'(x) + [\lambda^2 + \gamma^2 - (\frac{x^2}{2}+\alpha)^2 + x]\, y(x)=0$$</span> where <span class="math-container">$\lambda$</span> is the eigenvalue and <span class="math-container">$\alpha,\gamma$</span> are paramet... | doraemonpaul | 30,938 | <p><span class="math-container">$y''(x)-2\gamma y'(x)+\left(\lambda^2+\gamma^2-\left(\dfrac{x^2}{2}+\alpha\right)^2+x\right)y(x)=0$</span></p>
<p><span class="math-container">$y''(x)-2\gamma y'(x)-\left(\dfrac{x^4}{4}+\alpha x^2-x+\alpha^2-\lambda^2-\gamma^2\right)y(x)=0$</span></p>
<p>Let <span class="math-container">... |
3,852,877 | <p>Is it possible to solve the eigenvalue problem of <span class="math-container">$$y''(x) - 2\gamma\, y'(x) + [\lambda^2 + \gamma^2 - (\frac{x^2}{2}+\alpha)^2 + x]\, y(x)=0$$</span> where <span class="math-container">$\lambda$</span> is the eigenvalue and <span class="math-container">$\alpha,\gamma$</span> are paramet... | Disintegrating By Parts | 112,478 | <p>Starting with
<span class="math-container">$$
y''(x) - 2\gamma\, y'(x) + [\lambda^2 + \gamma^2 - (\frac{x^2}{2}+\alpha)^2 + x]\, y(x)=0,
$$</span>
let <span class="math-container">$y = e^{\gamma t} f$</span>. Then the above is reduced to potential form
<span class="math-container">$$
(e^{\gamma t}f''+... |
41,883 | <p>Let $G$ be an abelian group, $A$ a trivial $G$-module. We know that $\text{Ext}(G,A)$ classifies abelian extensions of $G$ by $A$, whereas $H^2(G,A)$ classifies central extensions of $G$ by $A$. So we have a canonical inclusion $\text{Ext}(G,A)\hookrightarrow H^2(G,A)$. Is there some naturally arising exact sequenc... | Torsten Ekedahl | 4,008 | <p>You get a description from the universal coefficient theorem which gives a (split) exact sequence
$$
0\to \mathrm{Ext}(H_1(G),A) \to H^2(G,A) \to \mathrm{Hom}(H_2(G),A) \to 0
$$
and the fact that $H_1(G)=G$. We have that $H_2(G)=\Lambda^2G$ and the map $H^2(G,A) \to \mathrm{Hom}(H_2(G),A)$ associates to an extension... |
3,963,479 | <p>In a quadrilateral <span class="math-container">$ABCD$</span>, there is an inscribed circle centered at <span class="math-container">$O$</span>. Let <span class="math-container">$F,N,E,M$</span> be the points on the circle that touch the quadrilateral, such that <span class="math-container">$F$</span> is on <span cl... | jlammy | 304,635 | <p>Here's a ridiculously high-powered solution, because why not :)</p>
<p>By Brianchon's theorem on the "hexagon" <span class="math-container">$ABCEDM$</span>, we get that <span class="math-container">$AE, BD, CM$</span> concur. So by Ceva's theorem,
<span class="math-container">$$\frac{AP}{PC}\cdot\frac{CE}{... |
362,854 | <blockquote>
<p>Show that every subgroup of $Q_8$ is normal.</p>
</blockquote>
<p>Is there any sophisticated way to do this ? I mean without needing to calculate everything out.</p>
| Cameron Buie | 28,900 | <p>Here, I consider $$Q_8=\langle -1,i,j,k\mid (-1)^2=1,i^2=j^2=k^2=ijk=-1\rangle.$$ Note that $-1$ commutes with everything, and that all other non-identity elements have order $4$, so their cyclic subgroups have index $2$, and are <a href="https://math.stackexchange.com/a/84663/28900">therefore</a> normal subgroups. ... |
747,789 | <p>I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite collection of polynomials $f_1, \dots, f_m$ in $n$ variables without any more restrictions. Then they define an algebraic ... | Georges Elencwajg | 3,217 | <p>Here are some personal reflections on the role of schemes in algebraic geometry, commenting on your very interesting remark: "Perhaps this is all a moot discussion since modern algebraic geometry is done with schemes".</p>
<p>Grothendieck introduced scheme theory in the late nineteen fifties and the high l... |
3,112,682 | <p>I was looking at</p>
<blockquote>
<p><em>Izzo, Alexander J.</em>, <a href="http://dx.doi.org/10.2307/2159282" rel="nofollow noreferrer"><strong>A functional analysis proof of the existence of Haar measure on locally compact Abelian groups</strong></a>, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). <a href="htt... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> <span class="math-container">$2\cos\theta=e^{i\theta}+e^{-i\theta}.$</span></p>
|
273,219 | <p>I have been studying for the AP BC Calculus exam (see this <a href="https://math.stackexchange.com/questions/272628/how-know-which-direction-a-particle-is-moving-on-a-polar-curve">previous question</a>) and most of the questions that deal with the first derivative in polar coordinates say that if ${dr\over d\theta}&... | Alexander Gruber | 12,952 | <p>It's pretty hard to answer this question when 'familiar' isn't defined further.</p>
<p>Every finite group (and thus every permutation group) has a composition series, which is <a href="http://mathworld.wolfram.com/Jordan-HoelderTheorem.html" rel="nofollow noreferrer">unique</a> in the sense that the length and comp... |
221,729 | <p>Till now, I have proved followings;</p>
<p>Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,</p>
<ol>
<li><p>$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous extension.</p></li>
<li><p>$Y$ is compact $\Rightarrow \exists$ a continuous extension.</p></li>... | Hagen von Eitzen | 39,174 | <p>Big-Oh is is not completely determined by derivatives. For example $\sin(x^2)\in O(1)$ but the derivative $2x\cos(x^2)$ is unbounded. </p>
<p>The claim that $f(n)\le g(n)$ implies $f(n)\in O(g(n))$ is false: Consider $g(n)=n$, $f(n)=-n^2$. But if you replace the condition with $|f(n)|\le g(n)$ then the claim is eas... |
221,729 | <p>Till now, I have proved followings;</p>
<p>Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,</p>
<ol>
<li><p>$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous extension.</p></li>
<li><p>$Y$ is compact $\Rightarrow \exists$ a continuous extension.</p></li>... | Alex | 38,873 | <p>One more easy way of looking at this type of problem is noticing that if $f(n) \leq g(n)$ then
$$
f(n)+g(n) \leq g(n) +g(n)=2g(n)= O(g(n))
$$
In fact it is easy to show that this sum is $\Theta(g(n))$</p>
|
4,120,827 | <p>Let's assume <span class="math-container">$P_1=(x_1, y_1)$</span> and <span class="math-container">$P_2=(x_2, y_2)$</span> and <span class="math-container">$P_3=(x_3, y_3)$</span>.</p>
<p>How to find the closest distance between <span class="math-container">$P_3$</span> and the line segment between <span class="math... | Cewein | 921,323 | <p>there is three area to take in count:</p>
<p><a href="https://i.stack.imgur.com/uutOo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uutOo.png" alt="enter image description here" /></a></p>
<p>For <span class="math-container">$P1$</span> the distance is <span class="math-container">$|P1-B|$</span... |
546,701 | <p>Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.</p>
<p>Can anyone give me any hints on how to solve this?</p>
| Christian Blatter | 1,303 | <p>Since $0$ has no weight we can consider all numbers as $7$-place decimals. It follows that we have to count the solutions of
$$\sum_{k=1}^7 d_k =42,\qquad 0\leq d_k\leq 9\quad(1\leq k\leq 7)\ .$$
Forgetting about the condition $d_k\leq9$ we have a standard stars-and-bars problem, which has ${42+7-1\choose 7-1}={48\c... |
1,229,467 | <p>I tried substituting the $4$ into the $3x-5$ equation, so my slope would be represented as $3(4)-5 = 7$. Then my equation for the line would be $y-3 = 7(x-4)$. That means the equation of the line would be $y = 7x - 25$. However, I'm trying to submit this answer online and it comes back as incorrect. Any help would b... | Rolf Hoyer | 228,612 | <p>Your equation only satisfies $y'(x) = 3x-5$ at the point $(4,3)$, rather than everywhere. You need to take the antiderivative to get the family of curves $y = (3/2)x^2 - 5x + C$ that satisfy the given differential equation everywhere. Solve for $C$ by plugging in $(4,3)$ to this family, instead.</p>
|
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