qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,190,781 | <blockquote>
<p>Suppose the polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfies the following conditions:<br>
(A) If <span class="math-container">$P(x)$</span> is divided by <span class="math-container">$x^2 − 4x + 3$</span>, the remainder is <span class="math-container">$65x −... | Viki 183 | 664,768 | <p>Since <span class="math-container">$P(x)=g(x)(x^2-4x+3)+(65x-68)$</span>, for <span class="math-container">$x=1$</span> you get <span class="math-container">$P(1)=-3$</span>. This imply that
<span class="math-container">$P(1)=h(1)\cdot 0+(-5+a)$</span>, i.e. <span class="math-container">$a=2$</span>.</p>
|
2,588,225 | <p>To start off, I'm very new to the complex plane and complex numbers.</p>
<p>I'm trying to determine if the series $\sum_{n=1}^\infty\frac{1}{n^2 + i}$ is convergent or divergent.</p>
<p>However, I'm not exactly sure how to express the value of the function. Could I just treat the denominator of the expression as t... | mechanodroid | 144,766 | <p>When in doubt, you can always decompose it to the real and imaginary part:</p>
<p>$$\frac{1}{n^2 + i} = \frac{n^2-i}{n^4+ 1} = \frac{n^2}{n^4+1}-i\cdot\frac{1}{n^2}$$</p>
<p>Both the real and imaginary parts converge:</p>
<p>$$\sum_{n=1}^\infty\frac{n^2}{n^4+1} \le \sum_{n=1}^\infty\frac{n^2}{n^4} = \sum_{n=1}^\i... |
4,155,453 | <p>I am trying to evaluate <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^\infty \frac{n}{n^2+k^2}.$$</span> Now I am aware that clearly <span class="math-container">$$\lim_{n\to \infty} \sum_{k=1}^n \frac{n}{n^2+k^2} = \int_0^1 \frac{1}{1+x^2}dx = \tan^{-1}(1) = \frac{\pi}{4},$$</span> but I do not know ... | Claude Leibovici | 82,404 | <p><em>Another solution.</em></p>
<p><span class="math-container">$$\frac{n}{n^2+k^2}=\frac{n}{(k+in)(k-in)}$$</span> Using partial fraction decomposition and summing
<span class="math-container">$$S_m=\sum_{k=1}^m\frac{n}{n^2+k^2}=\frac{i}{2} (\psi(m+i n+1)-\psi (m-i n+1)+\psi (1-i n)-\psi (1+i n))$$</span>
<span cla... |
2,335,334 | <p>I'm trying to solve the following ODE:
$$3xy''+(3x+1)y'+y=0$$
So I started by assuming a solution(and its derivates) with the form:</p>
<p>$$
\tag{1}y = \sum_{n=0}^{\infty } a_{n} x^{n + r}$$</p>
<p>$$y' = \sum_{n=0}^{\infty } {(n+r)}a_{n} x^{n + r-1}$$</p>
<p>$$y'' = \sum_{n=0}^{\infty } {(n+r-1)(n+r)}a_{n} x^{n... | N. F. Taussig | 173,070 | <p>For each of the four questions, there are two possible outcomes. Thus, in total, there are $2^4 = 16$ ways to answer the four questions.<br>
\begin{array}{c c c c}
T & T & T & T\\
T & F & T & T\\
T & T & F & T\\
T & F & F & T\\
T & T & T & F\\
T & F &a... |
2,979,755 | <p>I wish to show that <span class="math-container">$2^{2^{2^x}}<100^{100^x}$</span> for <span class="math-container">$x$</span> sufficiently large. I have taken logs (base 10) of both sides to get <span class="math-container">$2^{(2^x-1)}\log_{10} 2$</span> and <span class="math-container">$100^x$</span>. It is not... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Since <span class="math-container">$\log$</span> function is strictly increasing we have</p>
<p><span class="math-container">$$2^{2^{2^x}}<100^{100^x}\iff \log_{10}\left(2^{2^{2^x}}\right)<\log_{10}\left(100^{100^x}\right) \iff 2^{2^x}\log_{10}2<2\cdot 100^x$$</span></p>
<p>t... |
2,979,755 | <p>I wish to show that <span class="math-container">$2^{2^{2^x}}<100^{100^x}$</span> for <span class="math-container">$x$</span> sufficiently large. I have taken logs (base 10) of both sides to get <span class="math-container">$2^{(2^x-1)}\log_{10} 2$</span> and <span class="math-container">$100^x$</span>. It is not... | fleablood | 280,126 | <p><span class="math-container">$100^{100^x}= 2^{[\log_2{100}]*100^x} $</span>. Let <span class="math-container">$k = \log_2{100}$</span>. (<span class="math-container">$6< k < 7$</span>)</p>
<p><span class="math-container">$100^{100^x} = 2^{k*2^{kx}}= 2^{2^{\log_2k}*2^{kx}}$</span>. Let <span class="math-cont... |
3,579,457 | <p>Last Christmas, a kid in my neighborhood dressed in a pirate suit and knocked my door for "trick or treat". He explained to me that Dec 25, which is 25 written in base 10, is equal to 31 in base 8, which is Oct 31. So ingeniously, Halloween coincides with Christmas. </p>
<p>I really wonder which other two dates wou... | emonHR | 565,609 | <p><strong>which other two dates would possibly coincide with each other</strong> </p>
<blockquote>
<p>There are too many to list out. </p>
</blockquote>
<p>Maybe you could find them using converting decimal number to octal number. <a href="https://www.tutorialspoint.com/how-to-convert-decimal-to-octal" rel="nofoll... |
1,860,243 | <p>Let $|G|$ be an abelian group and let $H = \{g \in G : |g| \text{ divides } 12 \}.$ Prove that $H$ is a subgroup of $G$.</p>
<p>I know that I have to show that $a,b \in H \Rightarrow ab^{-1} \in H$ or $(ab \in H \land a^{-1} \in H).$ But I can't figure out how $|a|$ and $|b|$ dividing $12$ relates to $|ab|$ or $|... | lhf | 589 | <p>The crucial observation is that $|g|$ divides $12$ if and only if $g^{12} = e$.
With this, it is simple to verify the requirements for subgroup:</p>
<ul>
<li><p>$e \in H$ because $e^{12}=e$.</p></li>
<li><p>$a \in H \implies a^{-1} \in H$ because $(a^{-1})^{12}=(a^{12})^{-1}=e^{-1}=e $ (or simply that $|a^{-1}|=|a... |
44,310 | <p>I wanted to draw (lattice-aligned) unit cubes inside a sphere with a given radius centered on the origin, so I wrote this program:</p>
<pre><code>Cube[x_, y_, z_] := Cuboid[{x - 1, y - 1, z - 1}, {x, y, z}];
Coords[r_] := {#1, #2, Floor[Sqrt[r^2 - #1^2 - #2^2]]} & @@ # & /@
With[{t = Sqrt[r^2 - 1]},
... | Kuba | 5,478 | <p><strong>New, not the best but can do</strong></p>
<pre><code>r = 50;
da = Pi/(4. r);
sa = da;
ea = Pi/2. - da;
Composition[
Graphics3D[{#,
GeometricTransformation[#, ReflectionTransform /@ {{0, 0, 1}, {0, 1, 1}}]
}, Axes -> True, PlotRange -> All, ImageSize -> 600] &,
C... |
1,016,585 | <p>I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$</p>
<p>How do I do this?</p>
<p>It looks like it is close to being an $\operatorname{arcsin}$?</p>
<hr>
<p>I would have thought I could just do:
$$\int \left(\sqrt{x^2 - c}\right)^{-\frac12}\, dx=\frac{2\sqrt{c+x^2}}{2x}\text{??... | Community | -1 | <p>Let us set $c=1$ for a moment.</p>
<p>From hyperbolic trigonometry, we know that $\cosh^2t-1=\sinh^2t$ (equivalent of $1-cos^2=\sin^2t$) and the substitution $x=\cosh t$ seems natural. Then</p>
<p>$$\int\frac{dx}{\sqrt{x^2-1}}=\int\frac{\sinh t\ dt}{\sinh t}=t\text{ (!)}$$</p>
<p>Similarly, set $c=-1$ and use $x=... |
1,016,585 | <p>I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$</p>
<p>How do I do this?</p>
<p>It looks like it is close to being an $\operatorname{arcsin}$?</p>
<hr>
<p>I would have thought I could just do:
$$\int \left(\sqrt{x^2 - c}\right)^{-\frac12}\, dx=\frac{2\sqrt{c+x^2}}{2x}\text{??... | Vishwa Iyer | 71,281 | <p>To answer this question. You need to use a technique called Trigonometric Substitution (of course there are other ways, but this way is the easiest in my opinion).</p>
<p>First draw a right triangle. Call the height $\sqrt{x^2-c}$, the base $\sqrt{c}$, and the hypotenuse $x$. Call the angle between the base and the... |
3,249,926 | <p>I’ve tried applying Vandermonde’s identity, but got stuck. Any help would be appreciated!</p>
| Julian Mejia | 452,658 | <p>What happens is that a square is always <span class="math-container">$\geq 0$</span>. What I would do is split in 2 cases (when <span class="math-container">$x$</span> is positive or negative)</p>
<p>If <span class="math-container">$0\leq x\leq 3$</span>, then <span class="math-container">$0\leq x^2\leq 9$</span>.<... |
3,949,872 | <p>I am to expand <span class="math-container">$\log_3\frac{7x^2+21x}{7x(x-1)(x-2)}$</span> using the quotient rule of logs. I arrived at:</p>
<p><span class="math-container">$$\log_3(7)+\log_3(4)+\log_3(x)-\log_3(x-1)-\log_3(x-2)$$</span></p>
<p>Whereas my textbook solution is:</p>
<p><span class="math-container">$$\l... | rishav | 842,762 | <p>"My train of thought was to divide 7<span class="math-container">$x^2$</span> in the numerator by the 7 in the denominator leaving me with 7+21=28 in the numerator and just (−1)(−2) in the denominator."</p>
<p>Once re-read your statement!!
when you divide 7<span class="math-container">$x^2$</span> with 7x ... |
403,977 | <blockquote>
<p>Let $u_n>u_{n+1}>0$ for all $n \in \Bbb N$, and suppose that $u_2+u_4+u_8+u_{16}+\dots$ diverges. Prove that $\sum_{n=1}^{\infty}\frac{u_n}{n}$diverges.</p>
</blockquote>
<p>Please provide me a hint or a full solution.</p>
| marty cohen | 13,079 | <p>$\sum_{n=2^m}^{2^{m+1}-1}\frac{u_n}{n}
>\sum_{n=2^m}^{2^{m+1}-1}\frac{u_{2^{m+1}}}{2^{m+1}}
= 2^m \frac{u_{2^{m+1}}}{2^{m+1}}
=u_{2^{m+1}}/2
$.</p>
<p>Therefore
$\sum_{n=1}^{2^m-1} \frac{u_n}{n}
> \frac1{2}\sum_{k=1}^m u_{2^k}
$
and, if the latter sum diverges, so does the former.</p>
<p>Note that an almost... |
2,228,913 | <p>For some geometric problem that I am trying to solve, I am interested
in characterizing the cross product between a Bezier curve and its second order derivative.</p>
<p>So, let's $\mathbf{c}(t)$ be a Bezier curve, and $\ddot{\mathbf{c}}(t)$ it's derivative.</p>
<p>Is $\mathbf{w}(t) = \mathbf{c}(t) \times \ddot{\... | hmakholm left over Monica | 14,366 | <p><em>Every</em> parametric curve where each coordinate is a polynomial in the parameter can be expressed as a Bézier curve.</p>
<p>Thus also the cross product of Bézier curves. The <em>order</em> of the resulting curve will in general (except in degenerate cases) be the sum of the orders of the two curves you multip... |
1,531,525 | <p>I would like to compute the following sum:</p>
<p>$$\sum_{n=0}^{\infty} \frac{\cos n\theta}{2^n}$$</p>
<p>I know that it involves using complex numbers, although I'm not sure how exactly I'm supposed to do so. I tried using the fact that $\cos \theta = {e^{i\theta} + e^{-i\theta}\over 2}$. I'm not sure how to proc... | QiLin Xue | 624,739 | <p>I know I'm late but there's a slightly different solution I want to present that doesn't involve any exponentiation.</p>
<p>We can write the summation as the real part of</p>
<p><span class="math-container">$$\sum_{n=0}^{\infty} \frac{\cos n\theta + i\sin m\theta}{2^n}$$</span></p>
<p>Using De Moivre's theorem:</... |
159,777 | <p>In his <a href="http://www.ams.org/mathscinet-getitem?mr=44124">short paper</a> (1951) and <a href="http://www.ams.org/mathscinet-getitem?mr=74819">longer monograph</a> (1955), Postnikov introduced what I believe are now called <a href="http://en.wikipedia.org/wiki/Postnikov_system"><em>Postnikov systems</em> or <em... | Ronnie Brown | 19,949 | <p>One question which has puzzled me is, just to take the <span class="math-container">$2$</span>-type, how does one specify an element of <span class="math-container">$H^3(\pi_1,\pi_2)$</span>?</p>
<p>In 1972, Philip Higgins and I discussed with Saunders Mac Lane the possibility of a <span class="math-container">$2$<... |
495,863 | <p>I am trying to show that this sequence $\{a_n\}$ = (2n+1)/(3n+5) does not converge to $42$ if it is not bounded above. I have already showed that it converges to $2/3$. For this I want to use a proof by contradiction,i.e, I assume that the sequence does converge to $42$, which will lead to a contradiction to the ini... | André Nicolas | 6,312 | <p>We will show that there is an $\epsilon\gt 0$ such that
$$\left|\frac{2n+1}{3n+5}-42\right|\gt \epsilon$$
for infinitely many $n$. </p>
<p>Pick $\epsilon=1$. It is obvious that for positive $n$, we have $\frac{2n+1}{3n+5}\lt 1$. It follows that for all positive $n$, we have $\left|\frac{2n+1}{3n+5}-42\right|\gt 41... |
3,926,580 | <p>I'm trying to prove with squeeze theorem that the limit of the following series equals 1:</p>
<p><span class="math-container">$$\frac{1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}}{n}$$</span></p>
<p>For the left side of the inequality I did:</p>
<p><span class="math-container">$$\frac{1+\sqrt{1}+\sqrt[3]{1}+...+\sqrt[n]{1... | Misha Lavrov | 383,078 | <p>You'll want to know some things about how big <span class="math-container">$\sqrt[n]{n}$</span> is. The key facts to prove are:</p>
<ul>
<li>For <span class="math-container">$n$</span> a positive real number, it's increasing when <span class="math-container">$n < e$</span> and decreasing when <span class="math-co... |
3,926,580 | <p>I'm trying to prove with squeeze theorem that the limit of the following series equals 1:</p>
<p><span class="math-container">$$\frac{1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}}{n}$$</span></p>
<p>For the left side of the inequality I did:</p>
<p><span class="math-container">$$\frac{1+\sqrt{1}+\sqrt[3]{1}+...+\sqrt[n]{1... | CHAMSI | 758,100 | <p>First of all we have, for any <span class="math-container">$ n\in\mathbb{N}^{*} $</span>, the following : <span class="math-container">$$ \sqrt[n]{n}=1+\frac{\ln{n}}{n}\int_{0}^{1}{n^{\frac{x}{n}}\,\mathrm{d}x} $$</span></p>
<p>Since : <span class="math-container">\begin{aligned}0\leq\int_{0}^{1}{n^{\frac{x}{n}}\,\m... |
1,221,206 | <p>In this computer, numbers are stored in <span class="math-container">$12$</span>-bits. We will also assume
that for a floating point (real) number, <span class="math-container">$6$</span> bits of these bits are reserved for
the mantissa (or significand) with <span class="math-container">$2^{k-1}-1$</span> as the exp... | Olivier Oloa | 118,798 | <p>One may observe that, by expanding and noticing the cancellations,
$$
(1-x)(1+x+x^2+...+x^n)=1-x^{n+1},
$$
giving $$
1+x+x^2+...+x^n=\frac{1-x^{n+1}}{1-x}, \quad x \neq 1, \tag1
$$ and then put $x=\dfrac14$ in $(1)$.</p>
|
15,875 | <p>It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. I have always seen the converse as: if $f$ is <em>continuously</em> differentiable as a function from $U \subset \mat... | NebulousReveal | 2,548 | <p>See <a href="http://www.jstor.org/stable/2321164" rel="noreferrer"><em>When is a Function that Satisfies the Cauchy-Riemann Equations Analytic?</em></a> J. D. Gray and S. A. Morris
The American Mathematical Monthly
Vol. 85, No. 4 (Apr., 1978), pp. 246-256. </p>
|
1,482,564 | <p>In school I'm learning about concavity and finding points of inflection using the second-derivative test. A typical question will look like</p>
<blockquote>
<p><strong>Determine the open intervals on which the graph is concave upward or concave downward.</strong></p>
</blockquote>
<p>Here is how I would solve a prob... | Paul Sinclair | 258,282 | <p>Concavity generally stays the same the same on intervals containing critical numbers. The exception is when the critical number is an inflection point instead of a maximum or minimum. Think about it: consider a typical maximum. The curve reaches a high point and then heads back down. Does it switch from concave down... |
4,551,543 | <p>You have ten coins, each with a bias for head drawn from a uniform distribution [0,1]. For example, a coin with a 0.7 bias means that the probability of flipping a head is 0.7. You are allowed 100 flips and you are free to choose any coins to flip. Each time you flip a head, you get $1. How much money would you pay ... | quasi | 400,434 | <p>Let <span class="math-container">$C$</span> be the number of coins, and let <span class="math-container">$N$</span> be the number of flips (e.g., <span class="math-container">$C=10,\,N=100$</span>).</p>
<p>
Here's a proposed strategy which I think is pretty good, although I'm fairly certain it's not optimal . . .
<p... |
2,491,943 | <p>In general, for two sets $A$ and $B$, $|A^B|$=the number of functions from $B$ into $A$. </p>
<p>Can someone please show that $|\{1,2\}^{\{3,4,5\}}|=8$ by showing that there are eight functions from $\{3,4,5\}$ to $\{1,2\}$?</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. By using polar coordinates one gets, for $(x,y)\ne (0,0)$,
$$
g(x,y)=\frac{x^3+xy^2+2x^2+2y^2}{x^2+y^2}=\frac{r^3\cos^3 \theta+r^3\cos\theta \sin^2\theta +2r^2}{r^2},\quad r\ne0,
$$ that is
$$
g(x,y)=r\cos^3 \theta+r\cos\theta \sin^2\theta +2,\quad r\ne0,
$$ then this tends to $2$ as $r \to 0^... |
2,491,943 | <p>In general, for two sets $A$ and $B$, $|A^B|$=the number of functions from $B$ into $A$. </p>
<p>Can someone please show that $|\{1,2\}^{\{3,4,5\}}|=8$ by showing that there are eight functions from $\{3,4,5\}$ to $\{1,2\}$?</p>
| Ahmed S. Attaalla | 229,023 | <p>For $(x,y) \neq (0,0)$, with some simplifications we may get:</p>
<p>$$g(x,y)=x+2$$</p>
<p>In order for the function to be continuous at the origin, $g(0,0)$ must exist and we must have that $\lim_{x \to (0,0)} g(x,y)=g(0,0)$.</p>
<p>When is that the case? </p>
<blockquote class="spoiler">
<p> When $c=g(0,0)=... |
800,802 | <p>Can someone give me a reference in which I can find the following result </p>
<p>Let $C$ be a curve, then $$g(C)=\frac{(n-1)(n-2)}{2}-s.$$</p>
<p>where $g=$ genus of $C,$ $n=$ degree of curve, $s=$ number of singular points.</p>
<p>Any help would be appreciated.</p>
| Hoot | 127,490 | <p>I know this is done in a series of exercises (this may be good news or not) at the end of Appendix A in Arbarello, Cornalba, Griffiths, and Harris, <em>Geometry of Algebraic Curves, Volume I</em>. They call it the "Gorenstein relation".</p>
|
978,202 | <p>When I use $\sin x \sim x$ , the answer is $1$ , is the answer correct?</p>
| John | 105,625 | <p>No. $|sinx^2| \leq 1$, then apply squeeze theorem, you get 0 </p>
<p>And you can only use $x$ to approximate $sinx$ when $x$ is close to 0. This is because $\lim_{x\to 0} \frac{sinx}{x}=1$. It's not true when $x$ is large.</p>
|
978,202 | <p>When I use $\sin x \sim x$ , the answer is $1$ , is the answer correct?</p>
| Hakim | 85,969 | <p><strong>Hint:</strong> $$-\dfrac{1}{x^2+x}\leqslant\dfrac{\sin x^2}{x^2+x}\leqslant\dfrac{1}{x^2+x},\tag{$x\gt0$}$$ then squeeze everything.</p>
|
4,426,699 | <p>I want to solve this functional</p>
<p><span class="math-container">\begin{equation}
L(y)=\int_{-1}^7\sqrt{1+y'^2} \ dx
\end{equation}</span></p>
<p>with IC: <span class="math-container">$y(0)=1,\ y(1)=2$</span></p>
<p>I start using the Euler Lagrange equation</p>
<p><span class="math-container">\begin{equation}
\f... | Dr. Sundar | 1,040,807 | <p>Method I:</p>
<p>Obviously, the functional
<span class="math-container">$$
L(y)=\int_{a}^b \ \sqrt{1+y'^2} \ dx
$$</span>
represents the arc-length of a curve, and the variational problem reduces to finding a curve of shortest arc-length connecting two points on the plane. This is one of the classical problems in th... |
2,193,306 | <p>I have to evaluate the integral of the $\sec x$ function and I do it as follows $$\int\sec xdx=\int\frac{dx}{\cos x}=\int\frac{\cos^2 x+\sin^2 x}{\cos x}dx=\sin x+\int\frac{\sin^2x}{\cos x}dx$$ Now we make a change of variables $\cos x=t$ so our integral becomes $$\sin x-\int\frac{\sqrt{1-t^2}}{t}dt$$ Now we make an... | Community | -1 | <p><strong>The fast way</strong>:</p>
<p>$$\int\frac{dx}{\cos x}=\int\frac{\cos x\,dx}{\cos^2x}=\int\frac{\sin'x\,dx}{1-\sin^2x}=\text{artanh}(\sin x).$$</p>
|
3,995,954 | <p>Today during a Calc I exam we(me and my classmates) have been ask to prove <span class="math-container">$$-a\cdot e\cdot \ln(x)\le x^{-a}$$</span>, <span class="math-container">$\forall x>0$</span> and <span class="math-container">$\forall a\in \mathbb{R}$</span>. But, noneone in the room has known how to do it. ... | Barry Cipra | 86,747 | <p>The inequality is obviously true if <span class="math-container">$a=0$</span>. If <span class="math-container">$a\not=0$</span>, let <span class="math-container">$x=e^{-u/a}$</span>. The inequality to prove becomes <span class="math-container">$eu\le e^u$</span> for all <span class="math-container">$u\in\mathbb{R}$<... |
3,984,652 | <p>Let <span class="math-container">$(X_n,d_n) $</span> a family of metric spaces and <span class="math-container">$(X=\prod_{n \in \mathbb N} X_n,\mathcal d ) $</span> a metric space with <span class="math-container">$\mathcal T_p$</span> the product topology is equal to the topology defined by the distance <span cla... | Siong Thye Goh | 306,553 | <p>I think you have an indexing problem, if you start indexing from <span class="math-container">$0$</span> and in total there are <span class="math-container">$n$</span> digits, the last index should be <span class="math-container">$n-1$</span>.</p>
<p><span class="math-container">$$N = \sum_{i=0}^{n-1} a_i \cdot 10^i... |
1,373,759 | <p>Given the natural number $n$,who is in the form $p^2 \cdot q^2$,with $p$,$q$ prime numbers.Also $φ(n)$ is given.Describe a fast algorithm(polynomial time) that calculates the $p$ and $q$.Apply your algorithm to calculate the $p$,$q$ when $n=34969$ and $φ(n)=29920$</p>
<p>I found this problem on a mathematical compe... | callculus42 | 144,421 | <p>I post the derivative w.r.t T. You can first seperate the variable T from the others:</p>
<p><span class="math-container">$n=\frac{1}{DL}\cdot \sqrt{\frac{g}{\pi \sigma}}\cdot \sqrt{T}$</span></p>
<p>Then you write <span class="math-container">$\sqrt T$</span> as <span class="math-container">$T^{1/2}$</span></p>
... |
4,143,759 | <p><strong>What I understand so far :</strong></p>
<p>Intuitively this question gives me a yuuuuuge hint, it tells me at t=0, t = -4. So that let's me know that I have a phase shift of -4 immediately.</p>
<p>I understand what a period is, I understand what a phase shift is, I understand what amplitude and vertical disp... | MPW | 113,214 | <p><strong>Hint:</strong> Think of <span class="math-container">$V$</span> and <span class="math-container">$r$</span> as being functions of <span class="math-container">$t$</span> (time), so that
<span class="math-container">$$V(t) = \frac43\pi \cdot r(t)^3$$</span>
and therefore by the chain rule
<span class="math-co... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Georges Elencwajg | 450 | <p>[In front of a blackboard, in an office at Real College] </p>
<p>Skeptic: And why should I care about holomorphic functions?</p>
<p>Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}$ ? Here $a$ is one of your cherished real numbers, but not an integer.</p>
<p>Ske... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Georges Elencwajg | 450 | <p>Let us call "division algebra over $\mathbb R$" a finite-dimensional vector space $A$ equipped with a bilinear map $A \times A \to A: (a,b) \mapsto a \bullet b$ , such that $a\bullet b=0$ implies $a=0$ or $b=0$. ( Associativity is not required).</p>
<p>Examples : the reals, the complexes, the real quaternions and ... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Mariano Suárez-Álvarez | 1,409 | <p>One problem that one can solve with Fourier analysis very easily is the isoperimetrical inequality and the corresponding characterization of the circle. Of course, this can also be done in many. many other ways, but using Fourier series it becomes particularly simple.</p>
|
3,808,111 | <p>Let <span class="math-container">$(X_n)_{n \geq 1}$</span> be a sequence of pairwise independent random variables such that :</p>
<p><span class="math-container">$$\sum_{n=1}^{\infty} n^{-1} P\left\{\max _{1 \leq m \leq n}\left|\sum_{k=1}^{m}\left(X_{k}-E X_{k}\right)\right|>\varepsilon n\right\}<\infty$$</spa... | user | 505,767 | <p>Yes it can be either a maximum/minimum or it could be an inflection point.</p>
<p>Let consider for example</p>
<p><span class="math-container">$$f(x)=x^3 \implies f'(x)=3x^2\quad f''(x)=6x$$</span></p>
<p>and in this case <span class="math-container">$x=0$</span> is an inflection point.</p>
<p>For the case</p>
<p><s... |
106,466 | <ol>
<li>I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers?</li>
<li>I know that the order type of rationals $\eta$ is a homogeneous order type (meaning that for any two its elements there is an automorp... | Michael Hardy | 11,667 | <p>Suppose $f$ is a strictly increasing bijection function from the linearly ordered set of all rational numbers to the linearly ordered set of all real algebraic numbers. Let $g:\mathbb R \to \mathbb R$ be defined as follows:</p>
<p>$$
g(x) = \sup \{ f(u) : u \in \mathbb Q \text{ and }u < x \}.
$$</p>
<p>If I'... |
106,466 | <ol>
<li>I know that rational numbers are order-isomorphic to real algebraic numbers. Does it imply that irrational numbers are order-isomorphic to real transcendental numbers?</li>
<li>I know that the order type of rationals $\eta$ is a homogeneous order type (meaning that for any two its elements there is an automorp... | Brian M. Scott | 12,042 | <p>Here’s an answer to part of (2).</p>
<p>The order type of $\mathbb{P}$, the irrationals, is homogeneous, because $\langle\mathbb{P},\le\rangle$ is order-isomorphic to $\langle\mathbb{Z}^\omega,\preceq\rangle$, where $\preceq$ is the lexicographic order, which is order-homogeneous.</p>
<p>To construct the order-iso... |
1,117,924 | <p>How can I show with linear approximation that $y \approx x$ for small x? I know the rule $$f(x) \approx f(a) + f^{\prime}(a) (x-a),$$ but I don't know how to put it to use in this case.</p>
| Null_Space | 798,900 | <p>You are looking at the first degree <strong>Maclaurin series</strong> of <span class="math-container">$\ln{(1+x)}$</span></p>
|
569,960 | <p>Suppose $\mathcal{A}$ and $\mathcal{B}$ are categories with products, and $T$ a functor between them. If $X$ and $Y$ are objects in $\mathcal{A}$, what does it mean when we say there is a <em>natural</em> morphism $f\colon T(X\times Y)\to T(X)\times T(Y)$?</p>
<p>In $\mathcal{A}$, we have the product $X\times Y$, w... | Sig TM | 49,336 | <p>Consider the categories $\mathcal B^2$ and $\mathcal C^2$ of pairs of objects in $\mathcal B$ and $\mathcal C$, respectively. The Cartesian product on $\mathcal B$ is a functor $\times_\mathcal B$ from $\mathcal B^2$ to $\mathcal B$ and similarly for $\mathcal C$. There is also the functor $T^2:\mathcal B^2\to\mat... |
569,960 | <p>Suppose $\mathcal{A}$ and $\mathcal{B}$ are categories with products, and $T$ a functor between them. If $X$ and $Y$ are objects in $\mathcal{A}$, what does it mean when we say there is a <em>natural</em> morphism $f\colon T(X\times Y)\to T(X)\times T(Y)$?</p>
<p>In $\mathcal{A}$, we have the product $X\times Y$, w... | Abolfazl Tarizadeh | 269,262 | <p>In (modern) mathematics those morphisms which arise from the definitions or the universal properties are called the $\mathbf{canonical}$ morphisms. The functorial morphisms are called the $\mathbf{natural}$ morphisms; for instance if $\phi:F\rightarrow G$ is a natural transformation between the two functors $F,G:\ma... |
1,258,414 | <p>I want to prove that $$c = \{ (x_n)_{n\geq 0} \mid x_n \in \Bbb C \text{ and the sequence converges} \}$$ with the norm $$ \left\|(x_n)_{n\geq 0}\right\|_{\infty} =\sup_{n\geq 0} |x_n|$$ is a Banach space. I have done some work, but I am having trouble concluding.</p>
<p>So far: let $(\xi_n)_{n\geq 0} = \left( (x_... | Tim.ev | 291,929 | <p>Here is an alternative proof, using as a starting point that $\ell^{\infty}$ is a Banach space. As $c$ is a subspace of $\ell^{\infty}$, it suffices to show that $c$ is closed in $\ell^{\infty}$.</p>
<p>To see this let $\mathbf{x}=(x_1,x_2,\dotsc)\in\ell^{\infty}$, and suppose that $\{\mathbf{x}^n\}_{n=1}^{\infty}$... |
4,415,254 | <p>When is <span class="math-container">$n!>x^n$</span>? assuming that x is a fixed positive. I know we can take the log of both sides and use the following formula:</p>
<p><span class="math-container">$$n\log(x) = \log(x^n) < \log(n!) = \sum_{i = 1}^n\log(i)$$</span>.
But this still gives me difficulty trying to... | Claude Leibovici | 82,404 | <p>There is an almost exact solution for the equation <span class="math-container">$$n!=x^n\tag 1$$</span> Have a look at @robjohn's answer to <a href="https://math.stackexchange.com/questions/1333449/could-this-approximation-be-made-simpler-solve-n-an-10k">this question</a> of mine.</p>
<p>Adapted to <span class="math... |
1,923,226 | <p>I have been trying to prove this with some continuity theorems but haven't put together a good proof yet. </p>
| celtschk | 34,930 | <p>Let's take the definition that a function is continuous if all pre-images of open sets are open. Now $\mathbb Z$ has the discrete topology, and therefore <em>all</em> its subsets are open, including the pre-images of open sets under $f$.</p>
|
1,515,817 | <p>I conjecture that in a consecutive sequence of $n$ natural numbers all greater than $n$, there exists at least one number which is not divisible by any prime number less than or equal to $n/2$.</p>
<p>Can any one prove or disprove this?</p>
| Konstantinos Gaitanas | 99,437 | <p>The conjecture is false.<br>
We can cover $O(n^2)$ natural numbers using the primes not exceeding $n$.<br>
Check the <a href="http://oeis.org/wiki/Jacobsthal_function." rel="nofollow">Jacobsthal function</a></p>
|
2,229,532 | <p>I have a circle which has a triangle inscribed in it.</p>
<p>The circle radius R = 4</p>
<p>The triangle ABC vertices divide circle into 3 arcs in 1:2:3 ratio</p>
<p>Find the perimeter and area of triangle.</p>
<p>Can you guys help me with this one? </p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>If the three arcs are in $1:2:3$ ratio ( and the sum is $2 \pi$), than the first arc goes form $0$ to $\pi/3$, the second from $\pi/3$ to $\pi$ and the third from $\pi$ to $2\pi$.</p>
<p>Can you do from this? ( see the figure)</p>
<p><a href="https://i.stack.imgur.com/lbERW.jpg" rel="nofollow norefe... |
1,401,661 | <blockquote>
<p>Dice are cubes with pips (small dots) on their sides, representing
numbers 1 through 6. Two dice are considered the same if they can be
rotated and placed in such a way that they present matching numbers on
the top, bottom, left, right, front, and back sides.</p>
<p>Below is an example of t... | mjqxxxx | 5,546 | <p>With no loss of generality, place the <span class="math-container">$1$</span> on top. Any of the <span class="math-container">$5$</span> remaining numbers could be on the bottom (opposite the <span class="math-container">$1$</span>). For each of these cases, there are <span class="math-container">$4$</span> number... |
553,103 | <p>consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$.
It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal convergence. I want to know whether there is uniform convergence towards $f$ or not. </p>
<p>I tried to disprove unifo... | imranfat | 64,546 | <p>You can also replace the quadratic trig terms with the double angle formula involving $cos2x$ Your denominator then will consists of a constant and a $cos2x$ term. Then sub away $2x=v$ and then use the Weierstrass substitution converting the integral into a rational function. This is hilarious of course, but what th... |
17,398 | <p>I am teaching my daughter, who is currently about <span class="math-container">$46$</span> months old, additions. She is very curious and asks a lot of <em>good</em> questions. For example, when I told her that <span class="math-container">$2+6=8$</span> and <span class="math-container">$4+4=8$</span>, she asked me ... | Sue VanHattum | 60 | <p>Blocks work well for thinking about addition. Have her count out 8 blocks, and then ask her about all the addition problems that have 8 blocks as the answer.</p>
<p>A lovely children's book which looks at all the sum pairs for 7 is <em>Quack and Count</em>, by Keith Baker. (You can <a href="https://www.abebooks.com... |
510,672 | <p>My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:</p>
<p>1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows</p>
<p>In addition to those, why isn't removing zero rows an elementary operation? It d... | egreg | 62,967 | <p>Doing an elementary row operation on the matrix $A$ can be seen as multiplying $A$ by a suitable invertible matrix, call it $E_1$. So the whole process leads to writing
$$
U=E_{k}\dots E_{2}E_{1}A
$$
where $E_j$ $(j=1,2,\dots,k)$ correspond to the elementary row operation we performed and $U$ is a row reduced matrix... |
4,014,144 | <p>Let <span class="math-container">$f(x) = \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$</span>.
where <span class="math-container">$x, c_1, c_2, \dots, c_k \in \mathbb R^n$</span>.
What fast iterative methods are available for finding the (approximate) min of <span class="math-container">$f$</span> with the constraint <span c... | V.S.e.H. | 443,030 | <p>EDIT: I have included all of the MATLAB code here: <a href="https://gitlab.com/TheRTGuy/point-cloud-support-function-minimization" rel="nofollow noreferrer">https://gitlab.com/TheRTGuy/point-cloud-support-function-minimization</a>. There I also have other evaluations.</p>
<p>Given the assumption that <span class="ma... |
3,423,138 | <p>A) Always positive </p>
<p>B) Always negative </p>
<p>C)Always non-negative </p>
<p>D) Always zero </p>
<p>Obviously, this isn’t a solving question, so I can’t really show any working here (I would have otherwise). I didn’t really understand the question, so help would be greatly appreciated </p>
<p>Thanks!</p>... | clathratus | 583,016 | <p>In case you're interested in a general formula, <a href="https://en.wikipedia.org/wiki/Polylogarithm" rel="nofollow noreferrer">Wikipedia</a> gives
<span class="math-container">$$\begin{align}
\mathrm{Li}_{-n}(x)&=\left(z\frac{\partial}{\partial z}\right)^n\frac{z}{1-z}\\
&=(-1)^{n+1}\sum_{k=0}^{n}\frac{\lef... |
1,794 | <p>I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose their meaning. (You know that feeling when you wake up after a lucid dream but as you think about it, it just evapora... | Benjamin Dickman | 262 | <p>I'm not exactly sure what's being asked here, since you say you want to to learn how "to do advanced math," but also that you "want to try algebra and trigonometry, to play with equations." I can't tell exactly what you mean by <em>do advanced math</em>, but you say:</p>
<blockquote>
<p>I'm pretty good at underst... |
1,794 | <p>I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose their meaning. (You know that feeling when you wake up after a lucid dream but as you think about it, it just evapora... | Jessica B | 4,746 | <p>I think the short answer to your question is that you are (like many people) confusing the arithmetic that you called 'mathematics' at school with the mathematics that mathematicians do. I very rarely add up numbers above 10, or write down ones with several digits. I think I possibly do own a calculator... somewhere... |
3,459,178 | <p>Is there a general way to factor <span class="math-container">$ax^2 + bx$</span> into <span class="math-container">$($</span>some expression<span class="math-container">$)^2$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are constants?</p>
<p>The reason I am askin... | Saketh Malyala | 250,220 | <p><span class="math-container">$\displaystyle ax^2+bx=a\left(x^2+\frac{b}{a}x\right)=a\left(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}\right)-\frac{b^2}{4a}=a\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a}$</span>.</p>
<p>Then, <span class="math-container">$$\displaystyle \int_{-\infty}^{\infty}e^{ax^2+bx}\,dx$$</span> <span cla... |
2,501,450 | <p>I was trying for a while to prove non-existence of theu following limit:</p>
<p>$$\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}$$</p>
<p>Unfortunately, with no results.</p>
<p>My hope was to show, that:</p>
<p>$$\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}\not=0$$</p>
<p>But showing that wa... | Community | -1 | <p>$$\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}=\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n}\frac1{\sqrt[n]{5}-1}$$</p>
<p>is of the form $\dfrac10$.</p>
|
235,425 | <p>Ok, here is what I have for the proof of this conjecture. Let me know if I'm on the right path? all input appreciated.</p>
<p>There exist integers $j$, $k$, and $m$, such that,
$b = aj $ and $ c = ajk.$ Then $c = ajk $ (substituting $aj$ for $b$)
let $m = jk$, then $c = ma, => a|c.$ </p>
| MarnixKlooster ReinstateMonica | 11,994 | <p><em>(In this answer all variables are integers, i.e., elements of $\mathbb{Z}$.)</em></p>
<p>By the definition of divisibility we are given that $\;n * a = b\;$ and $\;m * b = c\;$ for some $\;n\;$ and $\;m\;$. Now we are asked to find a $\;k\;$ which makes $\;k*a = c\;$:</p>
<p>\begin{align}
& k*a = c \\
\eq... |
223,521 | <p>Suppose that $a^2-b^2 =x$ where $a,b,x$ are natural numbers.</p>
<p>Suppose $x$ is fixed. If there is one $(a,b)$ found, can there be another $(a,b)$?</p>
<p>Also, would there be a way to know how many such $(a,b)$ exists?</p>
| Harish Kayarohanam | 30,423 | <p>Any number N can be represented as product of two no's .</p>
<p>$$ N= a*b =({a+b\over 2})^2 - ({a-b\over 2})^2 $$</p>
<p>$$ To\ be\ difference\ of\ perfect\ squares\ the\ condition\ is\ ({a+b\over 2}) and \ ({a-b\over 2}) \ should\ be\ integers\ .$$ </p>
<p>This happens only iff </p>
<p>1) a and b are odd</p>
... |
591,014 | <p>I need help with this proof:</p>
<p>$f: X\rightarrow Y$</p>
<p>$C,D\subseteq Y$</p>
<p>$f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$</p>
<p>Thanks.</p>
| amWhy | 9,003 | <p>To show set <em>equality</em>,$$f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$$ we typically proceed by showing that both the following inclusions hold:</p>
<p>$$f^{-1}(C \cap D) \quad \subseteq \quad f^{-1}(C) \cap f^{-1}(D)\tag{1}$$
$$f^{-1}(C) \cap f^{-1}(D) \quad \subseteq \quad f^{-1}(C \cap D)\tag{2}$$</p>
<p>... |
2,327,181 | <p>I'm a bit confused about the definition of finite sets/intervals. I know that a set S is called finite when it has a finite number of elements, or formally, when there exists a bijection $f:S\to\{1,...,n\}$ for some natural number n. </p>
<p>However, the interval $(1,2)$ is called finite. I don't understand why; $(... | Woett | 65,418 | <p>With 'finite' the length is meant. So a finite interval is an interval with finite length, where the length (or measure) of an interval $(a,b)$ is generally defined as $b-a$.</p>
|
2,327,181 | <p>I'm a bit confused about the definition of finite sets/intervals. I know that a set S is called finite when it has a finite number of elements, or formally, when there exists a bijection $f:S\to\{1,...,n\}$ for some natural number n. </p>
<p>However, the interval $(1,2)$ is called finite. I don't understand why; $(... | JanneK | 342,700 | <p>The limits of (1,2) are very clear. Hence not infinite.
The paradox of 'Achilles and the tortoise'can be applied here to clear up the difference between a limit and infinity.</p>
|
2,661,718 | <p>My basic understanding is that each time
I have 30% chance of winning the prize
so between 3 and 4 tries I should win it</p>
<p>cause .3 +.3 +.3 = 90% as I need to win only once with 3 try
.3+.3+.3+.3 = 120% with 4 try</p>
<p>I do remember a formula saying 1-(0.7)^3 = 65.7% chance but I dont remember what that ... | sku | 341,324 | <p>Let $y = x+1$</p>
<p>In general, $y^y = e^{y\ln y} \implies e^y <= y^y <= e^{y\ln y} <= e^{y^2}$ </p>
<p>Hence, $y^{y\ln y} <= e^{y(\ln y)^2} <= e^{y^2 \ln y}$ </p>
<p>we need to prove that $e^{y(\ln y)^2} <= e^{y^2}$ which can be trivially proven. </p>
|
2,752,421 | <p>a) Consider the equation $x_1 + x_2 + x_3 + x_4 = 35$. How many different solutions does this equation have
if all the variables must be positive integers? Enter the exact numeric answer.</p>
<p>b) Suppose that a license plate consists of three letters followed by three digits. How many different license plates st... | Tal-Botvinnik | 331,471 | <p>Hint for a)</p>
<p>Write
$$35=1+1+\dots+1$$
Now, in how many ways can you place three commas?
For instance $1,1+1,1+1+1,1+\dots+1$.</p>
<p>For b), you are not considering that the digits come after the letters.</p>
|
2,061,655 | <p>Could we compute the limits
$$\lim_{x\rightarrow 0}\frac{\sin (x)-x+x^3}{x^3} \\ \lim_{x\rightarrow 0}\frac{e^x-\sin (x)-1}{x^2}$$
without using the l'Hospital rule and the Taylor expansion?</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. If one recalls that, for any function $f$ differentiable near $0$,
$$
\lim_{x \to 0}\frac{f(x)-f(0)}{x-0}=f'(0) \tag1
$$ then one may write
$$
\begin{align}
l:=\lim_{x \to 0}\frac{e^x-\sin (x)-1}{x^2}=\lim_{x \to 0}\:\frac1x\left(\frac{e^x-1}{x} -\frac{\sin x}x\right)
\end{align}
$$ then one ... |
1,083,141 | <p>According to Zeidler, 1995, in his book "Applied Functional Analysis: Application to Mathematical Physics".</p>
<p>Dirichlet problem is a problem to minimize $$F(u)=\frac{1}{2}\int_G(u')^2\ dx-\int_G fu\ dx,\qquad u=g \textrm{ in } bd(G).$$
Then, He says</p>
<blockquote>
<p>If $G$ is open, bounded, nonempty subs... | Siminore | 29,672 | <p>Let me try to explain. If you have a solution $u \in C^2(G)$ of the equation $-u''=f$ in $G$, then for any $v \in C_c^\infty(G)$ an integration by parts shows that $$\int_G u' v' = \int_G f v.$$ Notice that there is no reference to the value of $u$ on $\partial G$, since $v$ vanishes on $\partial G$ and the boundary... |
1,647,399 | <p>Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is nonempty.</p>
<p><strong>What I know:</strong> Since $X$ is compact, we need to use the fact that every sequence in $X... | Sam | 189,840 | <p>For each $n$ choose some $x_n \in E_n$ to get a sequence $\{ x_n \}$. By passing to a subsequence if necessary we may assume that $x_n \to x$ for some $x \in X$. Now for any $n$ all but the first $n$ terms (at most) of this sequence lie in $E_n$. If we ignore these first few terms we still have a sequence in $E_n$ c... |
1,327,755 | <p>Prove that the antipodal mapping $A: S^n \to S^n$ given by $A(p)=-p$ is an isometry. </p>
<p>I know that in order to prove that a map $f$ is an isometry of a smooth manifold $M$ it must hold true that
$$\langle v,u\rangle_p = \langle d_f v, d_fu \rangle_{f(p)}$$</p>
<p>Now I found online <a href="http://image.scie... | Amitai Yuval | 166,201 | <p>As user 7530 suggests, the antipodal map on the sphere is the restriction to the sphere of the $-id$ map on $\mathbb{R}^{n+1}$. Note that $-id$ is linear, and hence, its derivative at a point is also equal to $-id$. More explicitly, for a point $p\in\mathbb{R}^{n+1}$, identify the tangent space $T_p\mathbb{R}^{n+1}$... |
1,098,028 | <p>Let $T: \mathbb{R}_{\leq 3}[X] \rightarrow \mathbb{R}_{\leq 2}[X]$ be a linear map defined as $T(f(x)) = f'(x)$, and let $\beta$ and $\gamma$ be the standard ordered bases for resp. $\mathbb{R}_{\leq 3}[X]$ and $\mathbb{R}_{\leq 2}[X]$. Then the matrixrepresentation of the linear map is given as \begin{align*} A = [... | Emily | 31,475 | <p>First condition (given): $f(-2) = 0$.</p>
<p>Second condition (from evenness, may not be useful!): $f(2) = 0$.</p>
<p>Third condition: $f'(x_0) = 6$.</p>
<p>Fourth condition: $f(x_0) = 6-6x_0$.</p>
<p>Fifth condition: At some $x_1$, $f'(x_1) = 0$ and $g(x_1) = f(x_1)$.</p>
<p>Sixth condition: $g'(0.5) = 0, g(0.... |
3,467,203 | <p><span class="math-container">$$\frac{(-1)^{k+1}k}{3^k}$$</span></p>
<p>Correct me if I am wrong but with <span class="math-container">$(-1)^{k+1}$</span> this makes it an alternating series test therefore i think I need to show that <span class="math-container">$\frac{k}{3^k}$</span> is decreasing and non-negative.... | Peter Szilas | 408,605 | <p>1)<span class="math-container">$a_k= \dfrac{k}{3^k} >0$</span> is decreasing.</p>
<p><span class="math-container">$\dfrac{a_{k+1}}{a_k}=\dfrac{(k+1)3^k}{k3^{k+1}}=$</span></p>
<p><span class="math-container">$(1/3)\dfrac{k+1}{k} =(1/3)(1+1/k) <1$</span> for <span class="math-container">$k=1,2,3....$</span></... |
3,415,211 | <p>I was solving my Rubik's cube when I though that if i toss my cube what is the probability of getting red colour twice in a row. So I calculate that it is 1/6 for first time, 1/6 possibility for second time getting a red and 1/6 * 1/6 = 1/36 is the answer.</p>
<p>But then I thought if the colour is unspecified i.e ... | user | 505,767 | <p>Note that more in general for any <span class="math-container">$p\le 1$</span> the integral</p>
<p><span class="math-container">$$\int_0^{1} \frac{1}{x (\ln x)^p} dx$$</span></p>
<p>diverges, indeed by <span class="math-container">$y=\ln x \implies dy = \frac1x dx$</span> we have</p>
<p><span class="math-containe... |
1,764,162 | <p>I have $$f(x) = x \left| x - 1 \right|$$</p>
<p>Here my given value for $x$ is 1</p>
<p>And I need to test if the function is "continuous" when $x$ is $1$ and also when reaching
$$ f(1)$$</p>
<p>$$ \lim\limits_{x \to 1} f(x)$$</p>
<p>So I plug in 1 to the equation to find out the value of $f(1)$</p>
<p>$$ f(1)... | A.Riesen | 329,330 | <p>The specific value matters for the function might be continuous at some values and not at others. For example $|x|/x=f(x)$ is continuous everywhere except $x=0$ for if you approach zero from the left you will get -1 but if you approach it from the right you will get 1. Also you can not just plug in x=0 for f(x) sinc... |
3,592,368 | <p>I managed to solve this question using trigonometry. But I wondered if there'd be anyway of doing it using only synthetic geometry. Here it is.</p>
<blockquote>
<p>Let <span class="math-container">$ABC$</span> be a right isosceles triangle of hypotenuse <span class="math-container">$AB$</span>. Let also <span class=... | Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/fa7mB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fa7mB.png" alt="enter image description here"></a></p>
<p>Draw the parallel line BX || AP and extend PC to meet it at X. Note that <span class="math-container">$\angle XCB = 90 - \beta = \gamma$</span>. Along... |
1,369,428 | <p>I am trying to solve the following equation;
$$\int_{-1}^{1}e^{i(x+a\cos x)} \, \mathrm{d}(\cos x)$$ or $$\int_{0}^{\pi}e^{i(x+a\cos x)} \sin x \, \mathrm{d}x$$</p>
<p>I tried this in <em>Wolfram Alpha</em>, but it says that answer cannot be obtained.</p>
| Claude Leibovici | 82,404 | <p>In the same spirit as Josh Broadhurst's answer, you should arrive to $$\int_{0}^{\pi}e^{i(x+a\cos(x))} \sin(x)\,dx=\frac{\pi a J_1(a)-2 a \cos (a)+ 2 \sin (a)}{a^2}\,i$$ I do not find any way to compute the antiderivative itself.</p>
|
2,073,794 | <p>Let's consider the following variant of Collatz (3n+1) : </p>
<p>if $n$ is odd then $n \to 3n+1$</p>
<p>if $n$ is even then you can choose : $n \to n/2$ or $n \to 3n+1$</p>
<p>With this definition, is it possible to construct a cycle other than the trivial one, i.e., $1\to 4 \to 2 \to 1$?</p>
<p>Best regards</p>... | sTertooy | 336,630 | <p>Yes! With the standard Collatz conjecture, every number must eventually end up at the cycle $4 \to 2 \to 1 \to 4 \cdots$ . This has been verified for all numbers up to $2^{60}$.</p>
<p>With your altered definition, you can start at $2$, apply $3n+1$ instead of $n/2$, and then continue like the standard Collatz agai... |
2,073,794 | <p>Let's consider the following variant of Collatz (3n+1) : </p>
<p>if $n$ is odd then $n \to 3n+1$</p>
<p>if $n$ is even then you can choose : $n \to n/2$ or $n \to 3n+1$</p>
<p>With this definition, is it possible to construct a cycle other than the trivial one, i.e., $1\to 4 \to 2 \to 1$?</p>
<p>Best regards</p>... | goodvibration | 402,209 | <p>$4\to13\to40\to20\to10\to5\to16\to8\to4$</p>
|
1,549,181 | <p><strong>Calculate Using Polar Coordinates</strong></p>
<p>This is a drawing I made to illustrate the problem. <a href="http://tube.geogebra.org/m/ZzvL0a38" rel="nofollow">http://tube.geogebra.org/m/ZzvL0a38</a></p>
<p>$$\int_{\frac 12}^{1} \int_{0}^{\sqrt{1-x^2}} 1 \quad dydx $$</p>
<p>What I am confused about in... | E.H.E | 187,799 | <p>$$\int_{\frac 12}^{1} \int_{0}^{\sqrt{1-x^2}} 1 \quad dydx=\int_{0}^{\frac{\pi}{3}} \int_{1-0.5\sec\theta}^{1} \quad rdrd\theta$$
<a href="https://i.stack.imgur.com/A07HN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A07HN.png" alt="enter image description here"></a>
some details:</p>
<hr>
<p... |
106,853 | <p>I have taken a lot of math in university, but chose to omit differential equations. Unfortunately, now I have to read computer science proofs that use them, mostly ODEs, and this is always a struggle. What textbook(s) should I read to take me from the basics to practical use of the theory?</p>
| Gerry Myerson | 8,269 | <p>Boyce and DiPrima, Elementary Differential Equations, Eighth Edition. <a href="http://bcs.wiley.com/he-bcs/Books?action=index&itemId=047143339X&itemTypeId=BKS&bcsId=2026" rel="nofollow">Website</a>. </p>
|
106,853 | <p>I have taken a lot of math in university, but chose to omit differential equations. Unfortunately, now I have to read computer science proofs that use them, mostly ODEs, and this is always a struggle. What textbook(s) should I read to take me from the basics to practical use of the theory?</p>
| Community | -1 | <p>The books which I referred are:</p>
<ul>
<li><p>Ordinary differential equations by E. Coddington.</p></li>
<li><p>Differential Equations With Applications and Historical Notes by George F. Simmons. This book is a classic and at the end of each chapter has historical notes of Mathematicians which is fun to read.</p>... |
828,668 | <p>Let $G$ be a finite group. If there exists an $a\in G$ not equal to the identity such that for all $x\in G$,$\phi(x) = axa^{-1}=x^{p+1} $ is an automorphism of $G$ then $G$ is a $p$-group.</p>
<p>This is what I have so for.</p>
<p>The order of $a$ is $p$ since $\phi(a) = a= a^pa\rightarrow a^p=e$ therefore the $o... | Dan Shved | 47,560 | <p>Here is a plan.</p>
<p>Let $N = \{x \in G \mid x^p = 1\}$.</p>
<ol>
<li><p>Note that $x \in N$ if and only if $x$ commutes with $a$. It follows that $N$ is a subgroup of $G$. It is easy to see that $N$ is characteristic and therefore normal.</p></li>
<li><p>Take an arbitrary $y \in G$. We have $a y a^{-1} = y^{p+1... |
3,828,982 | <blockquote>
<p><strong>Statement</strong></p>
<p>If <span class="math-container">$U$</span> is open and <span class="math-container">$\text{int}(S)\neq\emptyset$</span> then <span class="math-container">$\text{int}(U\cap S)\neq\emptyset$</span> too when <span class="math-container">$U\cap S\neq\emptyset$</span> and wh... | riemann_lebesgue | 539,416 | <p>Ok here's a counter example. Below, set A is the interior of the blue ball. Set B is the red line and the interior of the red ball. Clearly, the interior of B is <span class="math-container">$R^2$</span> is the interior of the red ball. And the interior of the blue ball clearly doesn't intersect with that. Making th... |
165,888 | <p>I want to return the final value from the Table.For example something like Table[n-1] value.</p>
| Henrik Schumacher | 38,178 | <p>Not faster, but maybe easier to implement</p>
<pre><code>makeKMat2[k1_, k2_] := Module[{A, m, n},
m = Length[k1];
n = Length[k2];
A = Map[SparseArray[Band[{1, 1}] -> #, {n, n}, 0.] &, k1, {2}];
With[{B = SparseArray@k2}, Do[A[[i, i]] += B, {i, 1, m}]];
ArrayFlatten[A]
]
</code></pre>
<p>Your imp... |
3,025,375 | <p>What is<span class="math-container">$$\lim_{n→∞}n^3(\sqrt{n^2+\sqrt{n^4+1}}-n\sqrt2)?$$</span>So it is<span class="math-container">$$\lim_{n→∞}\frac{n^3(\sqrt{n^2+\sqrt{n^4+1}})^2-(n\sqrt{2})^2}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}=\lim_{n→∞}\frac{n^3(n^2+\sqrt{n^4+1}-2n^2)}{\sqrt{n^2+\sqrt{n^4+1}}+n\sqrt{2}}.$$</span... | lab bhattacharjee | 33,337 | <p>Let <span class="math-container">$1/n=h$</span></p>
<p><span class="math-container">$$\lim_{h\to0^+}\dfrac{\sqrt{1+\sqrt{1+h^4}}-\sqrt2}{h^4}$$</span></p>
<p><span class="math-container">$$=\lim_{h\to0^+}\dfrac{1+\sqrt{1+h^4}-2}{h^4}\cdot\lim_{h\to0^+}\dfrac1{\sqrt{1+\sqrt{1+h^4}}+\sqrt2}$$</span></p>
<p><span cl... |
292,041 | <p>The Helly theorem in the Euclidean plane asserts that if $S_1, \dots, S_n$ are $n \ge 3$ convex subsets such that $S_i \cap S_j \cap S_k \ne \emptyset$ for all distinct triples $i,j,k$, then the total intersection $\bigcap_{i = 1}^n S_i$ is also nonempty. </p>
<p>I'm wondering if the same theorem is true in the hyp... | Igor Rivin | 11,142 | <p>I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative statement about convex sets in the Euclidean plane holds in the Hyperbolic plane and <em>vice versa</em>, since the model ... |
4,365,005 | <p>The set of all homomorphisms between two spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> is denoted as <span class="math-container">$\text{Hom}(X,Y)$</span>, the set of endomorphisms of a space <span class="math-container">$X$</span> is denoted as <span class="math-containe... | user829347 | 719,291 | <p>Common ways to denote the set of all functions from a set <span class="math-container">$X$</span> to a set <span class="math-container">$Y$</span> that I have encountered (some of which have already been mentioned) are <span class="math-container">$\mathcal{F}(X;Y)$</span>, <span class="math-container">$\text{Map}(... |
1,633,846 | <p>I'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof:</p>
<blockquote>
<p>Let <span class="math-container">$f(x)$</span> be a polynomial with real coefficients and a positive leading coefficient.</p>
<p>Let <span class="math-container">$a \geq 0$</span>. Then,... | Noble Mushtak | 307,483 | <p>Proof of Upper Bound Theorem:</p>
<p>Let <span class="math-container">$f(x)=q(x)*(x-a)+r$</span>. It is given that all of the coefficients of <span class="math-container">$q(x)$</span> and <span class="math-container">$r$</span> are non-negative. Factoring out a <span class="math-container">$x-a$</span> from the ri... |
1,234,320 | <p>It is given that $4 x^4 + 9 y^4 = 64$.</p>
<p>Then what will be the maximum value of $x^2 + y^2$?</p>
<p>I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .</p>
| Community | -1 | <p>If $64= (3y^2)^2 + 4x^4$, then $3y^2 =\sqrt{ 64 - 4x^4}$. Plugging that to $x^2+y^2$ we get:
$$x^2+\frac{\sqrt{64 - 4x^4}}{3}.$$</p>
<p>Now you have to maximalize a function of one variable. Remember that if $64 = 4x^4+9y^4$, then (setting $y = 0$) we obtain $16 \ge x^4$, therefore $|x| \le 2$.</p>
|
760,216 | <p>The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of the curve?</p>
<p>The same question can apply to the sewing pattern on a tennis ball.</p>
<p>Answers about why this s... | hmakholm left over Monica | 14,366 | <p>Actually I thought the seams on a basketball were two great circles and <em>two ovals</em> that each intersect the same great circle twice. And so apparently do many people who make basketball texture maps one can find on the web. (<a href="http://www.xkcd.com/1365/" rel="nofollow noreferrer">Randall Munroe too</a>)... |
4,183,734 | <p>I just started trigonometry, and I came across this problem. I know that this problem would be very simple with a calculator, but without one, I'm lost. How would you determine if <span class="math-container">$\sin 4$</span> is bigger than <span class="math-container">$\sin 3$</span> in negativity or positivity? How... | Ritam_Dasgupta | 925,091 | <p>Here's a possible method:
<span class="math-container">$$\sin 3+\sin 4=2\sin\left(\frac {3+4}{2}\right) \cos\left(\frac {3-4}{2}\right)=2\sin\left(\frac 72\right)\cos\left(\frac 12\right)$$</span>
Now, we know that <span class="math-container">$0<\frac 12<\frac {\pi}{2}$</span>, hence the cosine is positive. A... |
2,738,023 | <p>We know we can check if two vectors are 'orthogonal' by doing an inner product.</p>
<p>$a*b=0$</p>
<p>tells us that these two vectors are orthogonal</p>
<p>here comes the question:</p>
<p>if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction. </p>
| Anastassis Kapetanakis | 342,024 | <p>You can check the followings:</p>
<p>1) Find their slope if you have their coordinates. The slope for a vector $\vec v$ is $λ=\frac{y_v}{x_v}$. If the slope of $\vec a$ and $\vec b$ are equal, then they are parallel.</p>
<p>2) Find the if $\vec a= k\vec b$ where $k \in \mathbb{R}$. If there is a value that satisfi... |
4,240,892 | <p><strong>My attempt</strong>:
<span class="math-container">$f(x+y)=f(x)f(y)$</span></p>
<p>Differenting wrt <span class="math-container">$y$</span>,</p>
<p><span class="math-container">$$f'(x+y)=f(x)f'(y)$$</span></p>
<p>Putting <span class="math-container">$y=0$</span>,</p>
<p><span class="math-container">$$f'(x)=f(... | Infinity_hunter | 826,797 | <p>Actually, it does not matter as long as we are dealing with exponential functions.</p>
<p>You can still write the solution in terms of <span class="math-container">$a^{jx}$</span> where <span class="math-container">$a$</span> is a non-negative real number.</p>
<p>Take <span class="math-container">$j = \frac{k}{\log ... |
673,229 | <blockquote>
<p>The medians to the two legs of a right angled triangle are $10$ and $4\sqrt{10}$. Find the hypotenuse of the right angled triangle.</p>
</blockquote>
<p>I'm confused. Can somebody please explain to me how to do this step by step? Not just the answer I want to know what you did to get the answer and w... | Mick | 42,351 | <p>No one area in mathematics can be considered as fully developed. This is because some day, some one will come up with some brilliant ideas that will</p>
<p>extend our research to a new dimension; or even disprove what we have accepted as must-be-true.</p>
|
673,229 | <blockquote>
<p>The medians to the two legs of a right angled triangle are $10$ and $4\sqrt{10}$. Find the hypotenuse of the right angled triangle.</p>
</blockquote>
<p>I'm confused. Can somebody please explain to me how to do this step by step? Not just the answer I want to know what you did to get the answer and w... | Moishe Kohan | 84,907 | <p>Here are some "fully developed areas" (and not having PhDs awarded in), which once upon a time were active research areas:</p>
<ol>
<li><p>Squaring the circle and trisecting the angle using compass and ruler. </p></li>
<li><p>Proving the 5th postulate from the rest of Euclidean axioms. </p></li>
<li><p>Proving Ferm... |
3,245,838 | <p>We have <span class="math-container">$n$</span> voltages <span class="math-container">$V_1, V_2,\dots , V_n$</span> that are received in a condensator or sum, such that <span class="math-container">$V=\sum V_i$</span> is the sum of received voltages in that point. Every voltage <span class="math-container">$V_i$</sp... | José Carlos Santos | 446,262 | <p>Yes, it is fine, except that you forgot to multiply by <span class="math-container">$2\pi i$</span> at the last line. The answer is <span class="math-container">$\pi\bigl(\cos(4+i)-\cos(4-i)\bigr)$</span>.</p>
|
1,620,186 | <p>I have to find an example of a surface of revolution excluding a sphere and a cone. </p>
<p>Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example? </p>
<p>$$$$ </p>
<p>I also have to find an example of a surface the image of which is not the graph of a smooth function $z=f(x,y)$. </p>
<p>Is $\sigma(x,y)=(3\sqrt... | Rafael | 303,887 | <p><strong>1.</strong> There are $p,q\in\mathbb{Z}$, with $gcd(p,q)=1$ such that $\frac{x+a}{x+b}=\frac{p}{q}$. Then $$(x+a)q=(x+b)p\Leftrightarrow x(q-p)=bp-aq\Leftrightarrow x=\frac{bp-aq}{q-p}\in\mathbb{Q}\Leftrightarrow x=0.$$ This is absurd. Therefore $p=q$ and $x+a=x+b$, so $a=b$.</p>
<p><strong>2.</strong> Let ... |
621,438 | <p>Is there any Characterization for all measurable sets in $\mathbb{R}$?
Can I say that a set is measurable if an only if it has the property of Baire? (differs from an open set by a first category set). If the answer is no, Is there an example of a measurable set that does not have the property of Baire?</p>
<p>Than... | Listing | 3,123 | <p>To answer your second question, it is <a href="https://math.stackexchange.com/questions/62560/does-every-lebesgue-measurable-set-have-the-baire-property">not true</a> that a set is measurable iff it has the Baire property.</p>
<p>There is a way to see how measurable subsets $U\subset \mathbb{R}$ look like though, i... |
3,799,888 | <p>So I was attempting to solve a topology exercise when the following question came to me.</p>
<p>The objective was trying to fine a bijection <span class="math-container">$f$</span> between the following disks in <span class="math-container">$\Bbb R^2$</span>:</p>
<p><span class="math-container">$$D_1 :=\{(x,y)\in \B... | José Carlos Santos | 446,262 | <p>Your function is simply <span class="math-container">$(x,y)\mapsto2(x,y)$</span>, and therefore it is injective (indeed, bijective).</p>
<p>Concerning your final questions, all the pairs <span class="math-container">$(0,\varphi_1)$</span>, with <span class="math-container">$\varphi\in[0,2\pi]$</span>, describe the s... |
3,799,888 | <p>So I was attempting to solve a topology exercise when the following question came to me.</p>
<p>The objective was trying to fine a bijection <span class="math-container">$f$</span> between the following disks in <span class="math-container">$\Bbb R^2$</span>:</p>
<p><span class="math-container">$$D_1 :=\{(x,y)\in \B... | PeteBabe | 815,764 | <p>The correct way to define polar co-ordidinates is using the domain <span class="math-container">$r > 0$</span> , <span class="math-container">$\varphi$</span> in <span class="math-container">$(0, 2\pi)$</span> so that the transformation between cartesian and polar co-ordinates is a diffeomorphism (has a different... |
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