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 |
|---|---|---|---|---|
142,819 | <p>I am currently studying Serge Lang's book "Algebra", on page 25 it is proved that if $G$ is a cyclic group of order $n$, and if $d$ is a divisor of $n$, then there exists a unique subgroup $H$ of $G$ of order $d$.</p>
<p>I have trouble seeing why the proof (as explained below) settles the uniqueness part.</p>
<p>T... | Brian M. Scott | 12,042 | <p>What you're missing is that the homomorphism $f$ is fixed. <strong>Every</strong> subgroup $H$ of $G$ of order $d$ <strong>is</strong> the group $f[m\Bbb Z]$, so they're all the same subgroup of $G$.</p>
|
4,504,768 | <p>We've to prove that
<span class="math-container">$$
\lim_{(x,y)\to(0,0)} \frac{x^3+y^4}{x^2+y^2} =0
$$</span></p>
<p>Kindly check if my proof below is correct.</p>
<p><b>Proof</b></p>
<p>We need to show there exists <span class="math-container">$\delta>0$</span> for an <span class="math-container">$\varepsilon>... | David A. Craven | 804,921 | <p>(Here I prove that the intersection of two subgroups of a symmetric group that are generated by transpositions is itself generated by transpositions.)</p>
<p>Let <span class="math-container">$G$</span> be a subgroup of <span class="math-container">$S_n$</span> generated by transpositions. We claim that <span class="... |
637,897 | <p>I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. </p>
<p>My first thought was to take a free subgroup $A_k$ of rank $k$ of the isometries of the hyperbolic plane $H$, which acts fix point free and discontinuous... | Giulio Belletti | 99,588 | <p>That's not necessarily the case.</p>
<p>Take the sphere with 3 punctures: it has fundamental group free on two generators, right?</p>
<p>Now take the torus with one puncture (which has indeed a hyperbolic structure). You can easily see this has the same fundamental group.
So, to answer your question you probably n... |
276,310 | <p>I'm trying to eliminate variables in some fairly simple sets of equations. A typical example is: </p>
<p>$$ 9 x^2 + 18 xy + 9 y^2 - 32 = 256z$$
$$ 9 x^2 + 6 xy - 3 y^2 - 8 = 376z$$
$$ 9 x^2 - 6 xy + y^2 = 512z$$</p>
<p>I'd like to eliminate $x$ and $y$. Mathematica tells me that the answer is $ 161 z^2 -162z + 1 ... | Bojan Serafimov | 56,860 | <p>Factorize the left sides to separate x and y, and it's easy later.</p>
|
276,310 | <p>I'm trying to eliminate variables in some fairly simple sets of equations. A typical example is: </p>
<p>$$ 9 x^2 + 18 xy + 9 y^2 - 32 = 256z$$
$$ 9 x^2 + 6 xy - 3 y^2 - 8 = 376z$$
$$ 9 x^2 - 6 xy + y^2 = 512z$$</p>
<p>I'd like to eliminate $x$ and $y$. Mathematica tells me that the answer is $ 161 z^2 -162z + 1 ... | Mark Bennet | 2,906 | <p>You can use gaussian elimination (or other equivalent methods) to find expressions for $x^2$, $xy$ and $y^2$ in terms of $z$ - treating them as independent variables, and substitute these to get an equation for z. Then you still have to solve for $x$ and $y$. </p>
<p>There is an easier way in this case, but in a ge... |
276,310 | <p>I'm trying to eliminate variables in some fairly simple sets of equations. A typical example is: </p>
<p>$$ 9 x^2 + 18 xy + 9 y^2 - 32 = 256z$$
$$ 9 x^2 + 6 xy - 3 y^2 - 8 = 376z$$
$$ 9 x^2 - 6 xy + y^2 = 512z$$</p>
<p>I'd like to eliminate $x$ and $y$. Mathematica tells me that the answer is $ 161 z^2 -162z + 1 ... | DonAntonio | 31,254 | <p>Mark's idea with the unknowns $\,x^2\,,\,xy\,,\,y^2\,$ , using the augmented matrix:</p>
<p>$$\begin{pmatrix} 9&18&9&\;\;256z+32\\9&6&\!\!\!\!-3&\;\;376z+8\\9&\!\!\!\!-6&1&\;\;512z\end{pmatrix}\stackrel{\begin{cases}R_2-R_1\\R_3-R_1\end{cases}}\longrightarrow \begin{pmatrix} 9&am... |
16,831 | <p>As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presu... | Qiaochu Yuan | 232 | <p>The answer is no. Actually I believe the following is a theorem whose name totally escapes me at the moment: assume that $f$ is continuous and let $a_n$ be a sequence of increasing positive integers such that $\int_0^1 f(x) x^{a_n} \, dx = 0$ for $n \ge 1$. If $\sum \frac{1}{a_n}$ diverges, then $f$ is identically... |
140,294 | <p>Generative adversarial networks (GAN) is regarded as one of "the most interesting idea in the last ten years in machine learning" by Yann LeCun. It can be used to generate photo-realistic images that are almost indistinguishable from the real ones.</p>
<p>GAN trains two competing neural networks: a generator networ... | Pavel Perikov | 67,512 | <p>Please ignore the whole text below, just don't use MMA for nn learning NOW — you'll spend unbelievable amounts of time into studing MMA way of doing thins and step into many bugs on your ways.</p>
<p>If your goal is not to become an expert in Wolfram Mathematica NN framework — don't go this way. Invest your time in... |
2,583,454 | <p>Consider for instance the linear system:</p>
<p>$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\right)$$</p>
<p>This is over determined and thus has no solution. Y... | Robert Israel | 8,508 | <p>The solutions to $Ax=b$ are the same as those of $PAx=Pb$ if $P$ is one-to-one, i.e. $\ker(P) = \{0\}$. If $P$ is not one-to-one, so that $P y = 0$ for some $y \ne 0$, then any $x$ such that $Ax = b + y$ is a solution of $PAx = Pb$ but not a solution of $Ax = b$. </p>
|
30,292 | <p>One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a natural continuous analog of discrete self-avoiding walks?
I am particularly interested in self-avoiding polygons,
i.... | PeterR | 4,553 | <p>For SLE see <a href="http://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution" rel="nofollow">http://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution</a></p>
|
24,230 | <p>$f$ is continuous between $[0,1]$, and $f(0)=f(1)$.</p>
<p>I want to prove that there is an $a \in [0,0.5]$ such that $f(a+0.5)=f(a)$.</p>
<p>ok, so Rolle's theorem can be useful here, but I can't see the connection to the derivative,</p>
<p>(Weierstrass, Uniform continuity?) I'll be glad to instructions.</p>
<p... | Shai Covo | 2,810 | <p>HINT: Work with the function $g(a)=f(a+0.5)-f(a)$. Consider $g(0)$ and $g(0.5)$ (and their sum).</p>
|
97,062 | <p><strong>Bug introduced in 9.0 or earlier and fixed in 10.4.0</strong></p>
<hr>
<p>Why does this work?</p>
<pre><code>Solve[5 Tan[t] + 9 == 0 && 0 <= t < 2 Pi , t]
{{t -> π - ArcTan[9/5]}, {t -> 2 π - ArcTan[9/5]}}
</code></pre>
<p>But this doesn't.</p>
<pre><code>NSolve[5 Tan[t] + 9 == 0 &a... | m_goldberg | 3,066 | <p>As I noted in a comment to the question, I made a query about this issue to WRI tech support. I have now received a reply. I quote the relevant part.</p>
<blockquote>
<p>It does appear that NSolve is not behaving properly, and I have forwarded an incident report to our developers with the information you provided. I... |
2,474,355 | <p>I have this system of linear equations.</p>
<p><em>2x<sub>1</sub> - 4x<sub>2</sub> - x<sub>3</sub> = 1</em></p>
<p><em>x<sub>1</sub> - 3x<sub>2</sub> + x<sub>3</sub> = 1</em></p>
<p><em>3x<sub>1</sub> - 5x<sub>2</sub> - 3x<sub>3</sub> = 1</em></p>
<p>What is the best way or is there any special way to solve this... | Theoretical Economist | 388,944 | <p>Consider the set of natural numbers $\mathbb N$. Let $d$ be the discrete metric, so that $d(m,n)=1$ if $n \neq m$ and $d(n,n)=0$. Define an alternative metric </p>
<p>$$\rho(m,n)=\left\vert \frac{1}{m} - \frac{1}{n} \right\vert.$$</p>
<p>Both $d$ and $\rho$ induce the discrete topology on $\mathbb N$, and hence ar... |
926,168 | <p>In how many ways can 3 teachers and 4 pupils be arranged in a line if the pupils and teachers must alternate?
.
how to get the answer?
the ans :144</p>
| lab bhattacharjee | 33,337 | <p>Using <a href="http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html">Prosthaphaeresis Formulas</a>,</p>
<p>$$\sin2A+\sin2B=2\sin(A+B)\cos(A-B)$$</p>
<p>and $$\sin(2A+2B+2C)-\sin2C=2\sin(A+B)\cos(A+B+2C)$$</p>
<p>Finally,
$$\cos(A-B)-\cos(A+B+2C)=2\sin(A+C)\sin(B+C)$$</p>
|
157,587 | <p>I know the following is a well-known result.</p>
<p>Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is equality if and only if $f$ is linear.</p>
<p>I need some reference about the second part, i.e. there is equal... | Alexandre Eremenko | 25,510 | <p>Edit. </p>
<p>Let us first prove the inequality:
$$\sup_{z,w}|f(z)-f(w)|\geq\sup_z|f(z)-f(-z)|=\sup|g(z)|\geq|g'(0)|=2|f'(0)|,$$
where the Schwarz Lemma was applied to $g(z)=f(z)-f(-z)$. Equality in Schwarz lemma can
happen only if $g(z)=kz$, thus $f(z)=kz+\phi(z)$, where $\phi$ is even.</p>
<p>Now let us see when... |
1,000,705 | <p>I have been trying to solve this problem for hours. </p>
<p>$\dfrac{9e^{2x}}{8x+3}$</p>
<p>I know $u'(x)$ will be $18e^{2x}$
and $v'(x)$ will be $8$</p>
<p>Written out, it will be $\dfrac{(8x+3)(18e^{2x})-(9e^{2x})(8)}{(8x+3)^2}$</p>
<p>I get to the part above^^ and I'm not sure what to do. I know it's probably... | Aaron Maroja | 143,413 | <p>The determinant is given by</p>
<p>$$\det A =
\begin{vmatrix}
1 & 2 & -1 \\
\color{red}2 & \color{red}0 & \color{red}2\\
-1 & 2 & k
\end {vmatrix} =
-\color{red} 2\begin{vmatrix}2 & -1 \\ 2 & k \end {vmatrix}
+\color{red} 0\begin{vmatrix}1 & -1 \\ -1 & k \end {vmatrix}
-... |
2,916,685 | <p>What are some common, preferably uncomplicated functions $ f: [a,b] \rightarrow \mathbb{R} $ that are Riemann integrable on $ [c,b] $ for all $ c \in (a,b) $ but not integrable on $ [a,b] $. </p>
<p>I know $ f = 1/x $ is one such function for the interval $ [0,1] $. Are there any other examples? </p>
| Slepecky Mamut | 180,179 | <p>$ sin(1/x) $ is integrable on $ [0,1] $, integral converges to $sin(1)-cosintegral(1)=0.504067...$</p>
<p>Correct example is $ sin(1/x) /x $</p>
|
2,916,685 | <p>What are some common, preferably uncomplicated functions $ f: [a,b] \rightarrow \mathbb{R} $ that are Riemann integrable on $ [c,b] $ for all $ c \in (a,b) $ but not integrable on $ [a,b] $. </p>
<p>I know $ f = 1/x $ is one such function for the interval $ [0,1] $. Are there any other examples? </p>
| katosh | 494,716 | <p>You can construct many more functions by first selecting a differentiable $F\colon[a,b]\to\mathbb R$, that is not defined or infinity at $x=a$ and finite for $x\in(a,b]$. Then just take the derivative as your example $f(x) := \frac{d}{dx}F(x)$.</p>
<p>For example, with all differentiable functions $g\colon\mathbb R... |
1,371,580 | <p>i'm reading "A concise introduction ti pure mathematics" by Liebeck and in the exercises of the second chapter i found this question:</p>
<p>"Show that the decimal expression for $\sqrt 2 $ is not periodic"</p>
<p>If i write $\sqrt 2$ in its decimal form, i should obtain something like:</p>
<p>$\sqrt 2 = {a_o}.{... | barak manos | 131,263 | <p>Assume by contradiction a periodic representation of $\sqrt2$ on base $10$.</p>
<p>Let $A$ denote the digit-sequence in the integer part of $\sqrt2$.</p>
<p>Let $B$ denote the digit-sequence in the non-periodic prefix of the fractional part of $\sqrt2$.</p>
<p>Let $C$ denote the digit-sequence in the periodic rem... |
560,929 | <p>Consider a circle with two perpendicular chords, dividing the circle into four regions $X, Y, Z, W$(labeled):</p>
<p><img src="https://i.stack.imgur.com/2TDK5.png" alt="enter image description here"></p>
<p>What is the maximum and minimum possible value of </p>
<p>$$\frac{A(X) + A(Z)}{A(W) + A(Y)}$$</p>
<p>where... | MvG | 35,416 | <p>Assume that the line $BD$ between areas $X\cup Y$ and $W\cup Z$ is always horizontal, so the other chord will always be vertical. You may assume your circle to be the unit circle (since the radius will cancel out of the expression in any case). You can parametrize your whole setup by the coordinates of the intersect... |
47,603 | <p>Is it possible to express the functions $S(x)=x+1$ and $Pd(x)=x\dot{-}1$ in terms of the functions $f_1$, $f_2$, $f_3$ and $f_4$, where $f_1(x)=0$ if $x$ is even or $1$ if $x$ is odd, $f_2(x)=\mbox{quot}(x,2)$, $f_3(x)=2x$ and $f_4(x)=2x+1$? For example, $S(x)=f_4(f_2(x))$ if x is even. Is there a similar formula if... | drbobmeister | 8,472 | <p>As I recall from reading and classes in mathematical logic, the natural numbers are
"defined"--postulated might be a better word--more or less as follows:</p>
<p>A.) There is a set $N$;</p>
<p>B.) $N$ has a "preferred element" called $0$;</p>
<p>C.) There is a function $s:N \to N$, such that i.) $0$ is not in the... |
144,864 | <p>This is my homework question:
Calculate $\int_{0}^{1}x^2\ln(x) dx$ using Simpson's formula. Maximum error should be $1/2\times10^{-4}$</p>
<p>For solving the problem, I need to calculate fourth derivative of $x^2\ln(x)$. It is $-2/x^2$ and it's maximum value will be $\infty$ between $(0,1)$ and I can't calculate $h... | mrf | 19,440 | <p>As already noticed, $f(x)$ is not $C^4$ on the closed interval $[0,1]$, and a direct estimate on the error in Simpson's method is troublesome. One way to handle things is to remove the left end point as described by Jonas Meyer. Another way to handle weak singularities as these is to start with a change of variables... |
3,256,767 | <p>So I'm trying to understand a solution made by my teacher for a question that asks me to determine whether the following is true. I'm having trouble understanding where some values in the steps are coming from.</p>
<p>Like for the first part, I don't really get where n≥5 came from. My guess is getting 16n^2 + 25 to... | copper.hat | 27,978 | <p>Find two vectors <span class="math-container">$b_3,b_4$</span> such that with <span class="math-container">$b_1 = (1,1,1,1)^T, b_2 = (1,0,1,0)^T$</span>, the
vectors <span class="math-container">$b_1,...,b_4$</span> span <span class="math-container">$\mathbb{R}^4$</span>.</p>
<p>Note that <span class="math-containe... |
3,256,767 | <p>So I'm trying to understand a solution made by my teacher for a question that asks me to determine whether the following is true. I'm having trouble understanding where some values in the steps are coming from.</p>
<p>Like for the first part, I don't really get where n≥5 came from. My guess is getting 16n^2 + 25 to... | zipirovich | 127,842 | <p>I can offer you a hint for an ad hoc solution for this exercise specifically (rather than all such questions in general).</p>
<p>One way to construct the matrix of such a linear transformation is to know the images under <span class="math-container">$\varphi$</span> of the standard basis vectors <span class="math-c... |
2,673,465 | <p>Suppose that $n \in \mathbb{N}$ is composite and has a prime factor $q$. If $k \in \mathbb{Z}$ is the greatest number for which $q^k$ divides $n$, how can I show that $q^k$ does not divide ${{n}\choose{q}}$?</p>
<p>Clearly, since
$$
{{n}\choose{q}} = \frac{n!}{(n-q)!q!} = \frac{n(n-1)(n-2) \dots (n-q+1)}{q!}
$$
So ... | Francisco José Letterio | 482,896 | <p>We can also use Kummer's Theorem (<a href="https://en.m.wikipedia.org/wiki/Kummer%27s_theorem" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Kummer%27s_theorem</a>) since it was designed for this kind of problems</p>
<p>Is it possible that when adding n-q and q we get k carry-overs?</p>
<p>First we mus... |
1,950,809 | <p>I'm fairly certain that the probability of both dice returning an even number is $1/4$.</p>
<p>I got this by saying that since these are independent events, with each die returning an even number being $1/2$, then the probability of both being even is $1/2 \times 1/2 = 1/4$.</p>
<p>Further, there are 36 outcomes, ... | avs | 353,141 | <p>Instead of thinking of pairs of outcomes, incorporate the two dice into one, more complicated probability space. It is true that there are equally many odd and even score <em>for a single die</em>. But, with two dice scores being viewed as one outcome to a new, more complicated experiment, we have an exhaustive an... |
3,071,076 | <p>Let <span class="math-container">$ABC$</span> be an acute angled triangle whose inscribed circle touches <span class="math-container">$AB$</span> and <span class="math-container">$AC$</span> at <span class="math-container">$D$</span> and <span class="math-container">$E$</span> respectively. Let <span class="math-con... | Oldboy | 401,277 | <p><a href="https://i.stack.imgur.com/gCq7u.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gCq7u.png" alt="enter image description here"></a></p>
<p>Let us first show that <span class="math-container">$\angle BXC=\angle BYC=90^\circ$</span>.</p>
<p>Notice that triangle <span class="math-container"... |
3,066,530 | <p><span class="math-container">$$\lim_{x\to 0} \frac {(\sin(2x)-2\sin(x))^4}{(3+\cos(2x)-4\cos(x))^3}$$</span> </p>
<p>without L'Hôpital.</p>
<p>I've tried using equivalences with <span class="math-container">${(\sin(2x)-2\sin(x))^4}$</span> and arrived at <span class="math-container">$-x^{12}$</span> but I don't kn... | Mike_ | 632,850 | <p>One can evaluate all such lims using series expansions.
<span class="math-container">$$
\sin(x) = x - \frac{x^3}{6} + o(x^4)\\
\cos(x) = 1 - \frac{x^2}{2} + o(x^4)
$$</span>
<a href="https://en.wikipedia.org/wiki/Taylor_series#Trigonometric_functions" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Taylor_se... |
3,587,387 | <p>Assume draw that you draw a card from a standard deck.Find the probability of drawing a heart Given that your drew a face card (JQK) Using probability formulas how do I figure this out
Given in this equations mean what exactly??</p>
| Noah Schweber | 28,111 | <p>It looks like there's some confusion over the basic definitions here.</p>
<hr>
<blockquote>
<p>I am able to show that there are an infinite number of non-prime natural numbers, but I don't know how to show that the entire set of non-prime natural numbers is infinite.</p>
</blockquote>
<p>These are the same thin... |
3,587,387 | <p>Assume draw that you draw a card from a standard deck.Find the probability of drawing a heart Given that your drew a face card (JQK) Using probability formulas how do I figure this out
Given in this equations mean what exactly??</p>
| mathematics2x2life | 79,043 | <p>You have a few issues, first, you say that <span class="math-container">$T$</span> is infinite because it is contained in the set of all composite (non-prime) numbers. But this is the set you wanted to prove was infinite! What you really mean to say is that because there are infinitely many distinct prime numbers, t... |
2,724,744 | <p>I have a basic math question.</p>
<p>If I have the following inequality:
$$-a-b > -1$$
and I want to flip (or reverse) the sign. What is the correct way of the following? And why?</p>
<p>i) $a+b \le 1$<br>
ii) $a+b < 1$</p>
<p>Many thanks! (:</p>
| user | 505,767 | <p>The step is</p>
<p>$$-a-b > -1\iff (-a-b)(-1) \stackrel{reversed}{\color{red}<} (-1)(-1)\iff a+b<1$$</p>
<p>Let consider for a numerical example</p>
<p>$$1 > -1\iff 1(-1) < (-1)(-1)\iff -1<1$$</p>
<p>Note also that for $-a-b \ge -1$ the following holds</p>
<p>$$-a-b \ge -1\iff a+b\le1$$</p>
|
2,724,744 | <p>I have a basic math question.</p>
<p>If I have the following inequality:
$$-a-b > -1$$
and I want to flip (or reverse) the sign. What is the correct way of the following? And why?</p>
<p>i) $a+b \le 1$<br>
ii) $a+b < 1$</p>
<p>Many thanks! (:</p>
| Rócherz | 451,007 | <p>Start with $-a-b>-1$.</p>
<p>Add $1+a+b$ to both sides to get $1>a+b$.</p>
<p>Which is the same as $a+b<1$. So ii).</p>
<p>That's why multiplying by a negative number reverses the inequality sign.</p>
<p>Just a comment: $a+b<1$ implies $a+b \leq 1$, but $a+b \leq 1$ <strong>does not</strong> imply $a... |
1,301,509 | <p>I've the following integral, which should result in 1, as shown by the scetch, but in my calculation I get the result 0. What's my mistake?</p>
<p>Sorry the comments are in German and please note that a German 1 often looks like an English 7. Anything in the picture which looks like a 7 to you is in fact a 1.</p>
... | nullUser | 17,459 | <p>The issue is that you changed the bounds wrong. You correctly wrote
$$
\int_{z(-1)}^{z(0)}
$$
and then incorrectly changed it to
$$
\int_{0}^{1}
$$
when it should be
$$
\int_{1}^{0}
$$</p>
<p>One final note though. You should not use substitution to deal with a multiplicative constant. Just use linearity:
$$
\int_{... |
3,022,921 | <p>If 6 divides x and 8 divides x how do you deduce 24 divides x</p>
| Henry Lee | 541,220 | <p>what we know is:</p>
<p><span class="math-container">$\frac{x}{6}$</span> is an integer so : <span class="math-container">$\frac{x}{2}$</span> and <span class="math-container">$\frac{x}{3}$</span> are also integers. Also:</p>
<p><span class="math-container">$\frac x8$</span> is an integer so :<span class="math-con... |
4,609,001 | <p>The following definition of pmf is on page51 from Probability and statistical inference by Robert V. Hogg, etc.</p>
<p>The pmf <span class="math-container">$f(x)$</span> of a discrete random variable X is a function that satisfies the following properties:<br />
(a)<span class="math-container">$f(x)\gt 0, x\in S;$</... | Pavel Kocourek | 1,134,951 | <p>You want to show that for any fixed <span class="math-container">$x_1<x<x_2$</span> in <span class="math-container">$I$</span> the following statements are equivalent:</p>
<p>(a) <span class="math-container">$\quad$</span> <span class="math-container">$(x,f(x))$</span> is below the line joining <span class="ma... |
1,757,092 | <p>I want to find an explicit formula for $\sum_{n=0}^\infty n^3x^n$ for $|x|\le1$.Is the idea that first to show that this series is convergent and then we can find the number that it converges to? I tried to use ratio test, but it didn't work. Any suggestion? Thanks!</p>
| pancini | 252,495 | <p>By the root test
$$\limsup_{n\to\infty} \sqrt[n]{|n^3x^n|}=|x|\limsup_{n\to\infty}n^{3/n}=|x|$$
so we have convergence for $|x|<1$.</p>
<p>Now note that
$$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$
so
$$\sum_{n=0}^\infty nx^{n-1}=\frac{1}{(1-x)^2}$$
and
$$\sum_{n=0}^\infty nx^{n}=\frac{x}{(1-x)^2}.$$
You should be ab... |
4,022,415 | <p>My attempt:</p>
<p><span class="math-container">$\lbrace6,19,30\rbrace$</span> is sufficient to show that two sets are impossible.</p>
<p>Using a computer program with a brute force method I found that separating the numbers <span class="math-container">$1$</span> through <span class="math-container">$85$</span> int... | WhatsUp | 256,378 | <p>Without bruteforcing, I find the list <span class="math-container">$\{1058, 6338, 10823, 13826\}$</span> which shows that three sets is not enough.</p>
<p>My approach:</p>
<p>I start from the equations
<span class="math-container">\begin{eqnarray}
a + b &=& u^2\\
c + d &=& v^2\\
a + c &=& x^2... |
224,019 | <p>I am trying to compute the volume of intersection of the following two regions:</p>
<pre><code>a = 0.857597;
b = 1.653926;
hexagon = Polygon[{{0, (b - a)/2, 1/2}, {(b - a)/2, 0, 1/2},
{1/2, 0, (b - 1)/(2 a)}, {1/2, (b - 1)/2, 0}, {(b - 1)/2, 1/2, 0},
{0, 1/2, (b - 1)/(2 a)}}];
octahedron = ImplicitRegion[Ab... | user21 | 18,437 | <p>Here is an approach based on creating exact regions:</p>
<pre><code>a = Rationalize[0.857597, 10^-16];
b = Rationalize[1.653926, 10^-16];
hexagon =
Polygon[{{0, (b - a)/2, 1/2}, {(b - a)/2, 0, 1/2}, {1/2,
0, (b - 1)/(2 a)}, {1/2, (b - 1)/2, 0}, {(b - 1)/2, 1/2, 0}, {0,
1/2, (b - 1)/(2 a)}}] // Simpl... |
1,435,590 | <p>Suppose I have a statement like this:</p>
<p>(~p ^ ~q) V (p ^ q)</p>
<p>If I understand this correctly, I can apply the law to both sides separately while leaving the OR in the middle intact. Leaving this:</p>
<p>(p V q) V (~p V ~q)</p>
<p>Is this valid? (as opposed to taking the negation of the entire statement... | Brian Tung | 224,454 | <p>Depending on the level of familiarity that your assignments are assuming, you may be able to get some traction by trying some values. For instance, assuming you know the quadratic formula, you know that if you can find one factor by trial-and-error, you can use the quadratic formula to factor the remaining two (if ... |
1,435,590 | <p>Suppose I have a statement like this:</p>
<p>(~p ^ ~q) V (p ^ q)</p>
<p>If I understand this correctly, I can apply the law to both sides separately while leaving the OR in the middle intact. Leaving this:</p>
<p>(p V q) V (~p V ~q)</p>
<p>Is this valid? (as opposed to taking the negation of the entire statement... | E.H.E | 187,799 | <p>$$x^3-x^2-5x-3=x^3+2x^2-3x^2-6x+x-3$$
$$x^3+2x^2+x-3x^2-6x-3=0$$
$$x(x^2+2x+1)-3(x^2+2x+1)=0$$
$$(x^2+2x+1)(x-3)=0$$</p>
|
1,641,922 | <p>I've came accros this excersize:<br>
Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$
suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where $\bar z$ is the complex conjugate of $z$. Prove that $f$ cannot be analytic in $D$.<br>
I started with assuming th... | mvw | 86,776 | <blockquote>
<p>How to solve such questions?</p>
</blockquote>
<p>The approach is to model the given information as some variables and their relationship as equations or inequalities and then try one of the mathematical methods to find a solution, which is then translated back into the terms of the original problem.... |
1,641,922 | <p>I've came accros this excersize:<br>
Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$
suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where $\bar z$ is the complex conjugate of $z$. Prove that $f$ cannot be analytic in $D$.<br>
I started with assuming th... | paw88789 | 147,810 | <p>It can be helpful to organize the information into a 'now-and-then' chart. Let Jason's age now be $J$.</p>
<p>$$\begin{array} {c|c|c} &\mbox{Jason's age}&\mbox{Catherine's age} \\ \hline
\mbox{Now} &J&2J\\ \mbox{6 years ago} &J-6& 2J-6 \end{array}$$</p>
<p>Then $2J-6=5(J-6)$, which you c... |
134,001 | <p>what is the basic difference between the Discrete Fourier Transform and the Wavelet Transform ? and why does JPEG2000 preferred DWT over DCT or DFT ? </p>
| Emre | 9,901 | <p>Sinusoids and wavelets are the <a href="http://en.wikipedia.org/wiki/Basis_%28linear_algebra%29" rel="nofollow">bases</a> used in DCT (JPEG) and DWT (JPEG2000), respectively. Lossy compression works by finding a basis (think: alphabet) that represents the signal using as few elements (think: words or letters) as pos... |
2,391,624 | <p>This question pertains to Mosteller's classic book <em>Fifty Challenging Problems in Probability</em>. Specifically, this in regards to an algebraic operation Mosteller performs in the solution to the first question, entitled "The Sock Drawer."</p>
<p>Mosteller writes:</p>
<blockquote>
<p>Then we require the pro... | Ross Presser | 83,388 | <p>If $x>y$, then $x*x > xy$. But $xy = \frac12$ so $x^2 > \frac12$.</p>
<p>Similar for $\frac12 > y^2$.</p>
|
679,135 | <p><img src="https://i.stack.imgur.com/uJEMx.png" alt="enter image description here">The question is find the δ by The maximum likelihood estimation?
My answer is δ=0 but I am not sure whether it is correct and how tho show its biasness?</p>
| sas | 21,699 | <p>Geometric solution.</p>
<p>You have quadrilateral: two sides are your numbers $r_1$ and $r_2$, two diagonals are $r_1-r_2$ and $r_1+r_2$. </p>
<p>Diagonals are equal — so quadrilateral is rectangle.</p>
|
2,549,690 | <p>Is a direct sum of cyclic groups cyclic? I know every abelian group is a direct sum of cyclic groups of prime power orders, but I can't make use of this.</p>
| Mariah | 319,890 | <p>Especially in $\mathbb{R^n}$ you can picture homeomorphisms, which are a smooth deformation of a sort.</p>
<p>How to get from one ball with radius $r$ to another with radius $s$? Simply shrink or expand the balls by some proper constant..</p>
|
3,988,084 | <p>I have 3 points:</p>
<p><span class="math-container">$$A = (0,4) \\
B=(-5,0) \\
C=(5,0)$$</span></p>
<p>I need to find a polynomial that goes through B and C, and is tangent to <span class="math-container">$f(x) = (2/3)x+4$</span> at A.</p>
<p>I know that tangent means it must be equal to the derivative of f(x) at t... | Piquito | 219,998 | <p><span class="math-container">$P(x)=(x^2-25)(ax+b)$</span></p>
<p><span class="math-container">$P(0)=-25b=4\iff b=\dfrac{-4}{25}$</span></p>
<p><span class="math-container">$P'(x)=3ax^2-25a-\dfrac{-4x}{25}\Rightarrow P'(0)=-25a=\dfrac23\Rightarrow a=\dfrac{-2}{75}$</span></p>
|
332,603 | <p>I've passed by this article:
<a href="http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/" rel="noreferrer">http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/</a></p>
<p>and this paper:
<a href="http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf" rel="no... | Robert Israel | 8,508 | <p>It depends on what you mean by "formula". Certainly no formula is known such that using it to find a new very large prime would be "very simple".</p>
|
2,877,578 | <p>Yesterday, I asked the question: <a href="https://math.stackexchange.com/questions/2876740/prove-that-if-a-b-are-closed-then-exists-u-v-open-sets-such-that-u-cap?noredirect=1#comment5938458_2876740">Prove that if $A,B$ are closed then, $ \exists\;U,V$ open sets such that $U\cap V= \emptyset$</a>. </p>
<p>Here is th... | Lev Bahn | 523,306 | <p>If you our space is metric space, $X$, let's define</p>
<p>$f_S(x)=dist(x,S)=\inf_{y\in S}|x-y|$ and note that, for each closed set $S$, it is a continuous function on the space. </p>
<p>Let $U=\left\{x\in X: f_A(x) < f_B(x) \right\}=(f_A-f_B)^{-1}((-\infty,0))$</p>
<p>and $V=\left\{x\in X : f_B(x)<f_A(x) ... |
333,360 | <p>I know the series for $\cos(x)$ it is $\sum \limits_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}$ </p>
<p>which will result in $\sum \limits_{n=0}^\infty \dfrac{\left(-1\right)^ n x^{2n+1}}{\left(2n\right)!}$ </p>
<p>Which is great when you already know the series; however, my question is how does one find Maclaurin ... | Jared | 65,034 | <p>Are you trying to find the Maclaurin series for $x\cos(x)$? If so, what you've done is valid.</p>
<p>Practically, it is much easier to find the Maclaurin series representing a given function by relating it to some function with known Maclaurin series, as you've done here. In general, you must take care to note wh... |
1,665,533 | <p>Let $\mathcal{E}_1, ...,\mathcal{E}_n$ be collections of measurable sets on $(\Omega,\mathcal{F},P)$, each closed under intersection. Suppose
\begin{align*}
P(A_1\cap...\cap\ A_n)=P(A_1)\cdot ... \cdot P(A_n),
\end{align*}
for all $A_i \in \mathcal{E}_i$ for $1 \leq i \leq n$. </p>
<p>Now I want to show that the $... | John | 105,625 | <p>Fix $A_i,\forall i=2,\cdots,n$, define $G=\{B\in \sigma(\mathcal{E}_1)|P(B\cap\cdots\cap\ A_n)=P(B)\cdot \cdots \cdot P(A_n)\}$,</p>
<p>By assumption $\mathcal{E}_1\subset G$, it suffices to show $G$ is a $\lambda$-system by definition.</p>
<p>Then by $\pi-\lambda$ theorem, we have $\sigma(\mathcal{E}_1)\subset G$... |
886,626 | <p>I want to solve the following system of congruences:</p>
<p>$ x \equiv 1 \mod 2 $</p>
<p>$ x \equiv 2 \mod 3 $</p>
<p>$ x \equiv 3 \mod 4 $</p>
<p>$ x \equiv 4 \mod 5 $</p>
<p>$ x \equiv 5 \mod 6 $</p>
<p>$ x \equiv 0 \mod 7 $</p>
<p>I know, but do not understand why, that the first two congruences are redund... | Steven Alexis Gregory | 75,410 | <p>$x \equiv 1 \mod 2$</p>
<p>$x \equiv 2 \mod 3$</p>
<p>$x \equiv 3 \mod 4 \implies x \equiv 1 \pmod 2$</p>
<p>$x \equiv 4 \mod 5$</p>
<p>$x \equiv 5 \mod 6 \iff
\left.\begin{cases}
x \equiv 1 \mod 2 \\
x \equiv 2 \mod 3
\end{cases} \right\}$
By the CRT.</p>
<p>$x \equiv 0 \mod 7$</p>
<hr>
<p>So first we ... |
399,948 | <p>How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or $\csc(50)$, etc.. I tried looking for how to do it but neither my textbook or any other place has a tutorial for it. I w... | M. Strochyk | 40,362 | <p>Value of $\sin{x}$ with prescribed accuracy can be calculated from Taylor's representation
$$\sin{x}=\sum\limits_{n=0}^{\infty}{\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}}$$ or infinite product
$$\sin{x}=x\prod\limits_{n=1}^{\infty}{\left(1-\dfrac{x^2}{\pi^2 n^2} \right)}.$$
For some partial cases numerous <a href="http://en.... |
463,190 | <p>How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$?</p>
<p>I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?</p>
| Community | -1 | <p>Hint: Write $a^2+b^2=(a + b)(a − b)+2b^2$. </p>
<p>Now you can show that $\gcd(a+b, b^2)=1$ so that $\gcd(a + b, 2b^2) = 1\text{ or }2$.</p>
|
177,124 | <h1>Definitions and notations.</h1>
<p>Let $\mathcal{P}(X)$ the <strong>power set</strong> of $X$.</p>
<p>Let $\tau_X\subseteq\mathcal{P}(X)$ a <strong>topology</strong> on X.</p>
<p>We call $A$ <strong>irreducible</strong> if every time $A=B\cup C$ with $B,C$ closed set then $(B=A)\vee(C=A)$.</p>
<p>We call $X$ <s... | მამუკა ჯიბლაძე | 41,291 | <p>A self-contained proof is in the book "Continuous lattices and domains" by G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott.</p>
<p>The particular place you need is Lemma II-1.19 on page 146.</p>
|
177,124 | <h1>Definitions and notations.</h1>
<p>Let $\mathcal{P}(X)$ the <strong>power set</strong> of $X$.</p>
<p>Let $\tau_X\subseteq\mathcal{P}(X)$ a <strong>topology</strong> on X.</p>
<p>We call $A$ <strong>irreducible</strong> if every time $A=B\cup C$ with $B,C$ closed set then $(B=A)\vee(C=A)$.</p>
<p>We call $X$ <s... | Corrado | 54,610 | <p>We have (only) to prove that for every open set <span class="math-container">$A\supseteq P$</span>, where <span class="math-container">$P=\bigcap F$</span>, we have <span class="math-container">$A\in F$</span> (for the whole notation you can see the question).</p>
<h1>The missing step.</h1>
<p>(we refer to [3], than... |
3,170,871 | <p>Could anyone please give me a hint on how to compute the following integral?</p>
<p><span class="math-container">$$\int \sqrt{\frac{x-2}{x^7}} \, \mathrm d x$$</span></p>
<p>I'm not required to use hyperbolic/ inverse trigonometric functions.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>Write your integrand in the form <span class="math-container">$$\frac{\sqrt{x-2}}{x^{7/2}}$$</span> and then substitute <span class="math-container">$$u=\sqrt{x}$$</span> so you will get <span class="math-container">$$2\int\frac{\sqrt{u^2-2}}{u^6}du$$</span> after this substitute <span class="math-container">$$u=\sq... |
373,357 | <p>I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my <a href="https://math.stackexchange.com/questions/372747/what-are-the-uses-of-split-complex-numbers?noredirect=1">previous question</a>). I then found out using both together, we can have trouble on the pr... | Willie Wong | 1,543 | <p><a href="https://math.stackexchange.com/a/373468/1543">rschwieb</a> already gave you the high powered answer. Here let me give you the low-powered version of what he wrote. </p>
<p>Consider the collection of $2\times 2$ matrices with real entries. We can write each matrix as
$$ \begin{pmatrix} A & B \\ C & ... |
55,918 | <blockquote>
<p><strong>Zariski's Main Theorem</strong> (<a href="http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1966__28__5_0" rel="noreferrer">EGA IV</a>, Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as ... | Qing Liu | 3,485 | <p>I realized that I completely missed the second part of the question (the example). Note that ZMT implies that $f$ is a quasi-affine morphism. Then $X\to \mathit{Spec}(f_*\mathcal O_X)$ is always an open immersion (see <a href="http://www.math.columbia.edu/algebraic_geometry/stacks-git/morphisms.pdf" rel="nofollow no... |
272,173 | <p>I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature. </p>
<p>I'd appreciate multiple suggestions that explore the topic using different approaches too.</p>
<p>I'm par... | Shahrooz | 19,885 | <p>Actually, it is dangerous to answer this question, since there are a lot of good resources in this direction and there are many famous professors here that know this field much better than me. But I want to introduce a nice book to you which I believe it is a nice one in your direction:</p>
<p>"Spectral Theory in R... |
3,977,081 | <p>I’m just a high school student, so I may be somewhat logically flawed in understanding this.</p>
<p>According to wikipedia, the definition of function requires an input <span class="math-container">$x$</span> with its domain <span class="math-container">$X$</span> and an output <span class="math-container">$y$</span... | Michael Burr | 86,421 | <p>We talk about domains and codomains of a function, not of the variables of a function. You might not have come across the term codomain before, but I think that it's the best for what you're trying to describe.</p>
<p>So, the domain of <span class="math-container">$f$</span> is <span class="math-container">$X$</spa... |
3,977,081 | <p>I’m just a high school student, so I may be somewhat logically flawed in understanding this.</p>
<p>According to wikipedia, the definition of function requires an input <span class="math-container">$x$</span> with its domain <span class="math-container">$X$</span> and an output <span class="math-container">$y$</span... | CyclotomicField | 464,974 | <p>The notation <span class="math-container">$f(1)=0$</span> means that <span class="math-container">$f$</span> maps <span class="math-container">$1 \in X$</span> to <span class="math-container">$0 \in Y$</span>. This agrees with the familiar notation <span class="math-container">$y=f(x)$</span> form that most people e... |
4,064,209 | <p><a href="https://i.stack.imgur.com/Ux3cH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ux3cH.png" alt="enter image description here" /></a></p>
<p>Above is the exercise. Showing that <span class="math-container">$S$</span> is bounded is straightforward by <span class="math-container">$A$</span> ... | RRL | 148,510 | <p>The weakest condition under which this holds is with at least one of the series <span class="math-container">$\sum a_k$</span> and <span class="math-container">$\sum b_k$</span> converging absolutely.</p>
<p>Suppose that <span class="math-container">$\sum_{k=0}^\infty a_k$</span> is absolutely convergent and <span c... |
3,898,411 | <p>How can I prove that <span class="math-container">$ (a_n) = \frac{n^3 -1}{2n^3-n} $</span> converges?</p>
<p>I've calculated the limit and got a result of a 1/2.</p>
<p>Now I need to prove that this limit exists. So, I tried to use the definition and find an <span class="math-container">$M$</span> that <span class="... | Jack LeGrüß | 831,874 | <p>You can solve the problem in three cases:</p>
<p><strong>Case 1</strong>: <span class="math-container">$m\ge2$</span> and <span class="math-container">$2^m>3^n$</span>.</p>
<p>Here, looking modulos <span class="math-container">$3$</span> and <span class="math-container">$4$</span> demands that <span class="math-c... |
191,796 | <blockquote>
<p>I met with the following difficulty reading the paper <a href="http://www.cnki.com.cn/Article/CJFDTotal-ZZDZ198801008.htm" rel="nofollow">Li, Rong Xiu "The properties of a matrix order column" (1988)</a>:</p>
<p>Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
j+k\cdot ... | Włodzimierz Holsztyński | 8,385 | <p>A modest introductory step only. The following partial algebraization might be useful: the present matrix is given by:</p>
<ul>
<li>$\quad a_{kk}\ :=\ 2\cdot(k\ +\ i\cdot k)$</li>
<li>$\quad a_{km}\ :=\ \min(k\ m)\ +\ \imath\cdot\max(k\ m)$</li>
</ul>
<p>for $\,\ k\,\ m=1\ldots n\,\ $ and $\,\ k\ne m.\ $ However, ... |
2,820,696 | <p>We were just discussing with colleagues the number of combinations you could get with two "normal", $6$-sided dice. Almost all of my colleagues were saying $36$ ($6^2$), which I agree with as such, but you will get almost half of the possible combinations counted twice.
If I count, the number of different combinati... | drhab | 75,923 | <p>It agrees with the number of sums $a_1+\cdots+a_s=n$ where the $a_i$ are nonnegative integers.</p>
<p>Here $a_i$ stands for the number of dice that show face $i$.</p>
<p>Applying <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)" rel="nofollow noreferrer">stars and bars</a> we find $$\binom{n+s... |
1,299,127 | <p>Can someone please help me answer this question as I cannot seem to get to the answer.
Please note that the Cauchy integral formula must be used in order to solve it.</p>
<p>Many thanks in advance!
\begin{equation*}
\int_{|z|=3}\frac{e^{zt}}{z^2+4}=\pi i\sin(2t).
\end{equation*}</p>
<p>Also $|z| = 3$ is given the ... | Olivier Oloa | 118,798 | <p><strong>Hint.</strong> The denominator $\displaystyle z^2+4=(z-2i)(z+2i)$ of the function
$$
z \longmapsto \frac{e^{zt}}{z^2+4}
$$ gives two poles inside $|z|=3$ and by the Cauchy integral formula we have
$$
\int_{|z|=3}\frac{e^{zt}}{z^2+4}dz=2i\pi\left({\rm{Res}}_{z=-2i}f(z)+{\rm{Res}}_{z=2i}f(z)\right).
$$ You con... |
2,873,474 | <p>My staff room is having a debate about the construction of sample spaces.</p>
<blockquote>
<p>When you toss a coin twice, do you consider the sample space to be
$$\{H,H\}, \{H,T\}, \{T,T\}$$ or $$\{H,H\}, \{H,T\}, \{T,H\},\{T,T\}$$</p>
</blockquote>
<p>In my humble opinion, I feel there is no single correct an... | Community | -1 | <p>From my book: </p>
<blockquote>
<p>The set of all possible outcomes is the <em>sample space</em> corresponding to an
experiment</p>
</blockquote>
<p>The key word is an experiment. That is the sample corresponds to events that are possible for pertaining to experiment. I.e an example </p>
<blockquote>
<p>The... |
3,493,151 | <p>This is a calculus problem from a high school math contest in Greece,from 2012.</p>
<p>I wish to know some solutions for this. I attempted to solve it.</p>
<blockquote>
<p>Let <span class="math-container">$f:\Bbb{R} \to \Bbb{R}$</span> differentiable such that <span class="math-container">$\lim_{x \to +\infty}f(... | Community | -1 | <p>I think you can just use de l’Hopital inductively:</p>
<p>Given <span class="math-container">$n \in \Bbb{N}$</span>, we have (since the numerator and denominator diverge):
<span class="math-container">$\lim_{x \to \infty}\frac{f(x)}{x^n} = \lim_{x \to \infty} \frac{f’(x)}{nx^{n-1}} = \lim_{x \to \infty} \frac{f(x) ... |
118,029 | <p>It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their symmetries: </p>
<p>$$\sum_{i=1}^{n+1}x_i^{2n-3}=0.$$</p>
<p>Fermat hypersurfaces have a group of automorphisms of ord... | Sasha | 4,428 | <p>Assume for example that $n = 2k + 1$ is odd. Let $\xi^{2n-3} = -1$. Then for any $(y_0,y_1,\dots,y_k) \in \mathbb{CP}^k$ the point $(y_0,\xi y_0,y_1,\xi y_1,\dots,y_k, \xi y_k)$ is on the Fermat hypersurface. So, it contains $\mathbb{CP}^k$. In particular, if $k \ge 2$ (and so $n \ge 5$) the number of lines is infin... |
149,161 | <p>A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in each degree. My question is: does the converse hold? that is, is every simplicial set, having a finite number of simp... | Peter LeFanu Lumsdaine | 2,273 | <p>Take $X$ to be the “infinite-dimensional dunce’s cap”, with a unique non-degenerate simplex $x_n$ in each dimension, and with every face of $x_n$ equal to $x_{n-1}$.</p>
<p>Explicitly, $X_n = \coprod_{m \leq n} \mathrm{Surj}([n],[m])$. So it’s clear that this has finitely many simplices in each dimension, but infi... |
62,526 | <p>I want to show the following:</p>
<p>$X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ where $K$ is a cell complex of dimension $\leq n$ is homotopic to a constant map</p>
<p>For this I think I can use the following:
$X$ $n$-connected $\iff $ every continuous map $f: S^n \rightarrow X$ is homotopic ... | AlexE | 7,110 | <p>This is an application of the second theorem of chapter 10.3 in <em>May: A Concise Course in Algebraic Topology</em>, i.e. there you can find your proof.</p>
|
215,531 | <p>I must solve following inequation:</p>
<p>$\frac{x-3}{1-2x}<0$</p>
<p>Now the text says that I have to solve the inequation "direct" without solving the according equations.</p>
<p>What does that mean?</p>
<p>I would say that I have to multiply by $(1-2x)$ then I get</p>
<p>$x-3<0$ and </p>
<p>$L_1 = [x... | Salech Alhasov | 25,654 | <p>Since</p>
<p><span class="math-container">$$\frac{x-3}{1-2x}=\frac{(x-3)(1-2x)}{(1-2x)(1-2x)},\quad x\neq \frac{1}{2}$$</span></p>
<p>and <span class="math-container">$(1-2x)(1-2x)=(1-2x)^2>0$</span>, then</p>
<p><span class="math-container">$$\frac{x-3}{1-2x}=\frac{(x-3)(1-2x)}{(1-2x)^2}<0.$$</span></p>
<... |
2,617,621 | <p>$$z^4 =\lvert z \lvert , z \in \mathbb{C}$$</p>
<p>Applying the formula to calculate $ \sqrt[4]{z} $, I find that solutions have to have this form:</p>
<p>$$z=\sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \frac{\pi}{2}}=i \ \sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \fra... | nonuser | 463,553 | <p>From $|z|=z^4$ we get $|z|=|z|^4$ so $|z|=1$ or $|z|=0$ (so $z=0$.)</p>
<p>In the first case $z^4 =1$ so $$(z+i)(z-i)(z+1)(z-1)=0$$ </p>
<p>so <strong>yes you find all the solutions.</strong></p>
|
2,617,621 | <p>$$z^4 =\lvert z \lvert , z \in \mathbb{C}$$</p>
<p>Applying the formula to calculate $ \sqrt[4]{z} $, I find that solutions have to have this form:</p>
<p>$$z=\sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \frac{\pi}{2}}=i \ \sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \fra... | Martin Argerami | 22,857 | <p>Your first method is a bit unclear because of the use of the fourth root: it is not true, even for a real number, that $\sqrt {z^2}=z$. A good piece of advice is to avoid "taking roots" unless unavoidable. </p>
<p>Your second method works by chance due to the fact that in this particular case all solutions have eit... |
2,617,621 | <p>$$z^4 =\lvert z \lvert , z \in \mathbb{C}$$</p>
<p>Applying the formula to calculate $ \sqrt[4]{z} $, I find that solutions have to have this form:</p>
<p>$$z=\sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \frac{\pi}{2}}=i \ \sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \fra... | ArsenBerk | 505,611 | <p>If we define $z = re^{i\theta}$, we have $|z| = r$ and $z^4 = r^4e^{4i\theta}$. Notice that they are equal when $r^3e^{i(4\theta)} = 1$ or $r = 0$. From here, you can easily find $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ with $r = 1$ and corresponding $z$ values will be $1, i, -1, -i$ and for $r = 0$, we have... |
2,617,621 | <p>$$z^4 =\lvert z \lvert , z \in \mathbb{C}$$</p>
<p>Applying the formula to calculate $ \sqrt[4]{z} $, I find that solutions have to have this form:</p>
<p>$$z=\sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \frac{\pi}{2}}=i \ \sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \fra... | Bernard | 202,857 | <p>The result is correct, but it faster with the exponential form of complesx numbers: if $z=r\,\mathrm e^{i\theta}$, the equation becomes
$$r=r^4\,\mathrm e^{4i\theta}\iff\begin{cases}r=0\quad\text{or}\\
r^3=1\:\wedge\:\mathrm e^{4i\theta}=1\end{cases}\iff \begin{cases}z=0\quad\text{or}\\ r=1\:\wedge\: \theta\equiv 0\... |
2,617,621 | <p>$$z^4 =\lvert z \lvert , z \in \mathbb{C}$$</p>
<p>Applying the formula to calculate $ \sqrt[4]{z} $, I find that solutions have to have this form:</p>
<p>$$z=\sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \frac{\pi}{2}}=i \ \sqrt[4]{\lvert z \lvert}$$
$$z=\sqrt[4]{\lvert z \lvert} \ e^{i \fra... | user | 505,767 | <p>Yes you are correct, indeed note that</p>
<p>$$\begin{cases}\lvert z \lvert = z^4 \iff |z|=0 \quad \lor\quad |z|=1\\\\z^4 = \lvert z \lvert =\bar z^4 \iff \mathcal{Im}(z)=0\end{cases}$$</p>
<p>thus all the <strong>non trivial solutions</strong> $z\neq0$ are</p>
<p>$$z=e^{ik\frac{\pi}{2}} \quad \forall k \in \math... |
20,771 | <p><strong>Background:</strong></p>
<p>Let $G$ be a profinite group. If $M$ is a discrete $G$-module, then $M=\varinjlim_U M^U$, where the direct limit is taken with respect to inclusions over all open normal subgroups of $G$, and one naturally has $H^n(G,M)\simeq\varinjlim H^n(G/U,M^U)$, where the cohomology groups o... | Ahmed Matar | 10,766 | <p>Hi Keenan,</p>
<p>You're right that the projective limit of discrete $G$-modules is not necessarily discrete. To take the cohomology of such "topological $G$-modules" you can use continuous cochain cohomology and this continuous cochain cohomology commutes with inverse limits under certain conditions. See section 7... |
3,471,684 | <p>Is always correct statement that if natural numbers <span class="math-container">$a,b \in \Bbb N$</span> for which LCM<span class="math-container">$(a,b)=16\cdot(a,b)$</span>, then <span class="math-container">$a|b$</span> or <span class="math-container">$b|a$</span>?</p>
<p>I used formula that LCM<span class="math... | Calvin Lin | 54,563 | <p>You're nearly there!</p>
<p>Let <span class="math-container">$ a = k_a (a,b) , b = k_b (a,b)$</span> where <span class="math-container">$(k_a, k_b ) = 1$</span>. </p>
<p><strong>Hint:</strong> What is the value of <span class="math-container">$ k_a k_b$</span> according to your equation?</p>
<p><strong>Hint:</... |
3,990,195 | <p>I need some assistance solving what seems to be a very <a href="https://i.stack.imgur.com/arMiv.png" rel="nofollow noreferrer">intuitive problem</a>, but becomes tough when only using strict natural deduction and not assuming De Morgan laws.</p>
<p>Laws allowed: Implication, And, Or, MT, PBC, Copy Rule, Negation, Do... | Mauro ALLEGRANZA | 108,274 | <p><em>Hint</em></p>
<ol>
<li><p><span class="math-container">$\lnot (P \land \lnot Q)$</span> --- premise</p>
</li>
<li><p><span class="math-container">$(\lnot P \to S) \land \lnot Q$</span> --- premise</p>
</li>
<li><p><span class="math-container">$\lnot Q$</span> --- from 2) by <span class="math-container">$(\land \... |
4,108,926 | <p>I was reading Axler's Linear Algebra Done Right, and the following appears as exercise <span class="math-container">$3$</span> in chapter <span class="math-container">$5$</span>, section A:</p>
<blockquote>
<p>Suppose <span class="math-container">$T \in \mathcal{L}(V)$</span> and <span class="math-container">$T^2 = ... | tchappy ha | 384,082 | <p>I solved this exercise as follows:</p>
<blockquote>
<p>Since <span class="math-container">$-1$</span> is not an eigenvalue of <span class="math-container">$T$</span>, <span class="math-container">$T+I$</span> is invertible by 5.6 on p.134 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.<br />
S... |
529,053 | <p>I have already proved that if ${X_k}$ converges to a limit $L$, then any subsequence of it also converges to $L$. And now the question asks to show that if ${X_k}$ has two subsequence which converge to two different limits, then ${X_k}$ can not be convergent.</p>
| Luke Skywalker | 54,762 | <p>I believe you are getting a little confused in the logic of the question. If you think it through, when you say that you have already proved that if a sequence converges, then every subsequence converges to the same limit, you can readily answer the question using just that.</p>
<p>To see this more carefully, argue... |
1,195,175 | <p>In <a href="http://en.wikipedia.org/wiki/Solid_angle" rel="nofollow">wikipedia</a> the solid angle is defined as follows:</p>
<blockquote>
<p>In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point. </p>
</blockquote>
<p>Why solid angle is... | ASB | 111,607 | <p>$ \dfrac{f(x+\delta x)-f(x)}{(x+\delta x)-x}=\dfrac{\sqrt{x+\delta x}-\sqrt{x}}{\delta x}=\dfrac{(\sqrt{x+\delta x}-\sqrt{x})(\sqrt{x+\delta x}+\sqrt{x})}{\delta x (\sqrt{x+\delta x}+\sqrt{x})}=\dfrac{1}{\sqrt{x+\delta x}+\sqrt{x}} $</p>
|
497,015 | <p>So this is an excercise.. Does anyone have a hint? </p>
<p><strong>If A is both orthogonal and a orthogonal projector. What can you then conlcude about A?</strong></p>
<p>I know that an $n\times n$ matrix $P$ is an orthogonal projector if it is both idempotent ($P^2 = P$) and symmetric ($P = P^T$ ). Such a matrix ... | Owen Sizemore | 1,193 | <p>Hint: Assume $P$ is diagonal...</p>
|
652,446 | <p>I just ran into the next problem: The random variables $X$ and $Y$ are independent, where $X \sim Normal(1,1)$ and $Y \sim Gamma(\lambda,p)$ with $E(Y) = 1$ and $Var(Y) = 1/2$ How do we find $E(X+Y)^3$ ?? I've tried a convolution, which leads to a really ugly looking integral from which I then have to get the third ... | Dilip Sarwate | 15,941 | <p>$$E[(X+Y)^3] = E[X^3+3X^2Y+3XY^2+Y^3] = E[X^3]+3E[X^2]E[Y] +3E[X]E[Y^2]+E[Y^3]$$ when $X$ and $Y$ are independent.</p>
|
105,535 | <p>In a thread in <a href="https://math.stackexchange.com/questions/186292/derivatives-of-the-riemann-zeta-function-at-s-0">MSE</a> I proposed an older routine of mine for the efficient computation of coefficients; I use a very similar routine for the quick&dirty computation of the Stieltjes-constants. </p>
<p>... | Fredrik Johansson | 4,854 | <p>I have recently computed a large table of rigorous values of the Stieltjes constants. Thanks to some coding by Jon Bober, the table can be browsed using a web interface on LMFDB.org (the L-functions and modular forms database):</p>
<p><a href="http://beta.lmfdb.org/riemann/stieltjes/" rel="nofollow">http://beta.lmf... |
1,731,382 | <p>Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?</p>
| Plutoro | 108,709 | <p>Neglecting air resistance, and assuming constant gravity, the trajectory of anything will be a parabola. If there is air resistance, trajectories of roughly spherical objects become a lot more complicated, and are not described easily using nice geometric terms. However, the trajectories do involve the hyperbolic co... |
1,731,382 | <p>Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?</p>
| Student of physics | 682,067 | <p>The relativistic trajectory of an object under the influence of a constant force field perpendicular to its initial direction of motion is a catenary. The trajectory reduces to a parabola in the non-relativistic limit.</p>
<p>(Eg : motion of electron under constant electric field)</p>
|
83,607 | <p>While solving the heat equation in one spatial variable $u_t = u_{xx} $ (x goes from 0 to L) with the initial temperature distribution $T_0 \frac{x(L-x)}{L^2}$ , and with neumann boundary conditions $u_x(0,t) = u_x(L,t) = 0$, I got some really weird behaviour from NDSolve.</p>
<p>My code looks like this:</p>
<pre... | toadatrix | 120 | <p>I believe that all you need to do is select the following method option in NDSolve</p>
<p>Method->{"PDEDiscretization"->{"MethodOfLines",{"SpatialDiscretization"->"FiniteElement"}}}</p>
|
4,643,832 | <p><strong>Question</strong></p>
<p><a href="https://i.stack.imgur.com/XSeNR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XSeNR.png" alt="enter image description here" /></a></p>
<p>I am trying to prove this using balls (that is what we use in my school). The definition is that a subset <span clas... | TheBestMagician | 815,074 | <p>Taking the first two terms of the Binomial Theorem,
<span class="math-container">$$\left(1+\frac{1}{y}\right)^{x-y}\ge 1^{x-y}+\frac{x-y}y\cdot 1^{x-y-1}=\frac{x}{y}$$</span></p>
|
1,085,702 | <p>It's said that a computer program "prints" a set <span class="math-container">$A$</span> (<span class="math-container">$A \subseteq \mathbb N$</span>, positive integers.) if it prints every element of <span class="math-container">$A$</span> in ascending order (even if <span class="math-container">$A$</span... | Mark Bennet | 2,906 | <p>If the number of printable sequences can be taken to be countable, then count the sequences and then create a sequence as follows:</p>
<p>The first term is one greater than the first term of the first sequence
The second term is the smallest integer greater than both the first term already determined and the second... |
1,548,667 | <p>Consider the following steady state problem</p>
<p>$$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$</p>
<p>$$ T(0,y) = 300, \space \space T(4,y) = 600$$</p>
<p>$$ \frac{\partial T}{\partial y}(x,0) = 0, \space \space \frac{\partial T}{\partial ... | Evgeny | 87,697 | <p><strong>Sort of guide</strong></p>
<ol>
<li><p>Transform the equation such that it'll have homogeneous boundary conditions.<br>
I suggest to use $W(x, y) = 100 + 50x$ : this is the simplest function that has $W(0, y) = 100$, $W(2, y) = 200$ and by the way $\frac{\partial W}{\partial y} \equiv 0$. What will happen t... |
1,341,385 | <p>I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my own. My dad wants me to memorize about 30 trig identities (though some of them are very similar) since I'm missing tri... | Keith | 244,951 | <p>Let me first note that physicists may be better people to ask this question than mathematicians are.</p>
<p>I think it's worth remembering a few, and knowing how to rederive the others.</p>
<p>The important ones to remember initially are: </p>
<p>(a) The definitions of $\tan$, $\cot$, $\sec$, $\csc$, in terms of ... |
2,674,799 | <blockquote>
<p>Let $X_{2n}$ be the group whose presentation is$\langle x,y\,|\,x^n=y^2=1, xy=yx^2\rangle$. From $x=xy^2$, it is seen that $x^3=1$, hence $X_{2n}$ has at most $6$ elements. I have to show that if $n=3k$, then $X_{2n}$ has exactly $6$ elements. </p>
</blockquote>
<p>I can't see where I am having probl... | cansomeonehelpmeout | 413,677 | <p>You know that $y^2=1$, to show that $x^3=1$, notice that $xyxy=x(x^2)=x^3$, also $xyxy=yx^2yx^2=yx(xyx)x=yxyx^4=x^6$, so $x^3=1$. From here on, you can show that the 6 elements are: $$1,x,x^2,y,xy,yx$$ and that multiplying either of these yields no new element.</p>
|
65,002 | <p>I am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are often dropped. This strikes me as very sloppy notation if $1 \times 1$ matrices are not at least <em>functionally equivale... | charles.y.zheng | 7,862 | <p>No. To give a concrete example, you can multiply a 2x2 matrix by a scalar, but you can't multiply a 2x2 matrix by a 1x1 matrix.</p>
<p>It is sloppy notation.</p>
|
1,531,646 | <p>Find the following limit</p>
<p>$$
\lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac1{x^2}
$$</p>
<p>I have used natural logarithm to get</p>
<p>$$
\exp\lim_{x\to0}\frac1{x^2}\ln\left(\frac{1+x2^x}{1+x3^x}\right)
$$</p>
<p>After this, I have tried l'opital's rule but I was unable to get it to a simplified for... | Hosein Rahnama | 267,844 | <p><strong>Solution Procedure Using Taylor Series and L'Hoptial</strong></p>
<p>Try to understand or prove each of the following steps:</p>
<p>1) $\ln \left( {\frac{{1 + x{2^x}}}{{1 + x{3^x}}}} \right) = \ln \left( {1 + x\frac{{{2^x} - {3^x}}}{{1 + x{3^x}}}} \right)$ </p>
<p>2) $\ln (1 + x) = x + o({x^2})$</p>
<p>... |
15,033 | <p>I have noticed in <a href="https://meta.mathoverflow.net/questions/833/who-are-the-mathoverflow-moderators">this post</a> that at MO they have e-mail address <code>moderators@mathoverflow.net</code>, which can be used to contact moderators.</p>
<p>Is there a similar address for moderators of this site? If not, woul... | Community | -1 | <p>No, there is no such mail address for this site. Using email for moderator purposes has certain drawbacks, and SE tries to keep all moderator communication inside the SE platform for that reason. Keeping this all inside the SE software makes it much easier to see what happened afterwards, which is very useful if the... |
78,368 | <p>Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?</p>
| The Bridge | 2,642 | <p>Here what I found in my e-library the follwing articles (that you can find on arxiv or SSRN) :</p>
<p>Brahimi, Necir - A Semiparametric Estimation of Copula Models Based on the Method of Moments</p>
<p>Chicheportiche, Bouchaud - Goodness-of-Fit tests with Dependent Observations</p>
<p>Amblard, Girard - Estimation... |
2,965,193 | <p>Basically the question is asking us to prove that given any integers <span class="math-container">$$x_1,x_2,x_3,x_4,x_5$$</span> Prove that 3 of the integers from the set above, suppose <span class="math-container">$$x_a,x_b,x_c$$</span> satisfy this equation: <span class="math-container">$$x_a^2 + x_b^2 + x_c^2 = 3... | DeepSea | 101,504 | <p>Let's look at any <span class="math-container">$3$</span> of them,say <span class="math-container">$a,b,c$</span> among <span class="math-container">$a,b,c,d,e$</span>. You must have <span class="math-container">$2$</span> cases: <span class="math-container">$a^2 = 0, b^2=1, c^2=0$</span> or <span class="math-contai... |
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