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 |
|---|---|---|---|---|
1,672,666 | <p>I'm currently stuck on a proof that's giving me a hard time, not sure where to start:</p>
<p>Prove or disprove: If an integer $q \gt 1$ is composite, then $q$ will have some divisor $d \gt 1$ such that $d \lt \sqrt{q}$</p>
<p>Any ideas on how to approach these problems or any help at all would be greatly appreciat... | ForgotALot | 295,090 | <p>No, $\lim a_n=0$ is not sufficient for a series to converge. However, for large $n$ the series above is like $\sum 1/n^4$ and it converges because $\sum 1/n^A$ converges for all $A>1$.</p>
|
1,064,534 | <p>$$x^3-3y^2x=-1$$
$$3yx^2 -y^3=1$$</p>
<p>This was the real part and imaginary part on a previous question I asked, instead of the system it was easier to just use polar coordinates to solve, but if this was just a system unrelated to the problem, how would one solve it? I seem to have trouble when the terms are lik... | David | 119,775 | <p>Eliminating $y$, we have
$$y^2=\frac{x^3+1}{3x}\ ,\quad y(3x^2-y^2)=1$$
and so
$$\frac{x^3+1}{3x}\Bigl(3x^2-\frac{x^3+1}{3x}\Bigr)^2=1\ .$$
Multiplying both sides by $27x^3$, the equation becomes a cubic in $x^3$. The cubic happens to have one rational root, so all roots can easily be found.</p>
|
844,528 | <p>I cannot figure out an aesthetic way to do this.</p>
<p>Can someone give a beautiful solution to this ugly question?</p>
<hr>
<p>This is what I have tried yet. </p>
<p>I used the fact that $$x = \dfrac{1}{1-\left(1-\dfrac{1}{x}\right)}$$</p>
<p>Hence $$x = \sum_{n=0}^{\infty} \left(1-\dfrac{1}{x}\right)^n$$</p>... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
3,118,410 | <p>I've been trying to use the change of base property but I'm not having much luck. Can anyone give me any ideas on how I should approach this problem? The answer is 49/5</p>
<p>Thanks.</p>
| Community | -1 | <p><strong>Hint:</strong> Use properties of logs: <span class="math-container">$\log xy=\log x+\log y\\\log\frac xy=\log x-\log y\\\log a^x=x\log a\\b^{\log_bx}=x=\log_bb^x\\\log_ba=\frac{\log_xa}{\log_xb}$</span>.</p>
|
4,441,421 | <p>I am studying PDEs, in particular hyperbolic conservation laws.
In particular we are using the method of characteristic to solve some of the problems.</p>
<h1>Setting</h1>
<p>Given the quite general PDE
<span class="math-container">$$
U_t + \partial_x \big( a(x,t)U(x,t)\big) = 0\iff U_t + a\cdot U_x = - a_x\cdot U
$... | marty cohen | 13,079 | <p>Another proof that
this is only true for
a 45-45-90 right triangle.</p>
<p>If the angle bisector
of the right angle
is perpendicular
to the hypotenuse,
then the triangle
formed by the bisector
has a 45 angle
(bisects the right angle)
and a 90 angle
(perpendicular)
so the other angle has to be 45.</p>
<p>Therefore th... |
35,139 | <p>I've came upon a few methods of how to display "processing" information while <em>Mathematica</em> is running. But, from what I've understood, this only works in between lines of codes. Is it possible to display a "how far is something being processed"-information during a line of code?</p>
<p>For example, I wish t... | Ymareth | 880 | <p>Provided you can put your intermediate results or progress indicator into a common scope with a ScheduledTask you can monitor proceedings as follows...</p>
<pre><code>Module[{x = 0, monitor},
(* Create a monitoring ScheduledTask *)
StartScheduledTask[monitor = CreateScheduledTask[PrintTemporary[x], 1]];
(*SomeB... |
234,153 | <p>Prove: we always have at least one x>0 is a $x^3+bx^2+cx-d^2=0$ 's root (b, c, d are real numbers and $d≠0$)</p>
| lab bhattacharjee | 33,337 | <p>If $f(x)=x^3+bx^2+cx-d^2$,</p>
<p>using <a href="http://mathworld.wolfram.com/DescartesSignRule.html" rel="nofollow">this</a>, there is only one change of sign in $f(x),$ so $f(x)$ can have at most one positive real root and the number of positive real root will be exactly one as <a href="http://en.wikipedia.org/wi... |
781,994 | <p>Let $z= 1+i$.
The polar form of $z^{1/5}$ can be easily calculated: $$z = 2^{1 \over 10}\left\{\cos({\pi/4 + 2\pi k \over 5}) + i\cdot \sin({\pi/4 + 2\pi k \over 5})\right\}$$</p>
<p>the "principal root" as wolframalpha denotes is: $$e^{i\pi \over 20}$$
How can one "jump" easily to the next root, which is: $$e^{9i... | 5xum | 112,884 | <p>The set $$\{(x,0)| x\in\mathbb R\}$$
is a subset of $\mathbb R^2$ (obtained by "eliminating" vectors from $\mathbb R^2$ if you want to call it that) and is a vector space (it is a <em>subspace</em> of $\mathbb R^2$).</p>
|
781,994 | <p>Let $z= 1+i$.
The polar form of $z^{1/5}$ can be easily calculated: $$z = 2^{1 \over 10}\left\{\cos({\pi/4 + 2\pi k \over 5}) + i\cdot \sin({\pi/4 + 2\pi k \over 5})\right\}$$</p>
<p>the "principal root" as wolframalpha denotes is: $$e^{i\pi \over 20}$$
How can one "jump" easily to the next root, which is: $$e^{9i... | Ellya | 135,305 | <p>There are certain conditions that must remain, for instance, you cannot remove the additive identity element $0$, since without it we would no longer have a vector space.</p>
<p>But for instance on any general vector space if we remove all elements except $0$ we are okay.</p>
|
3,519,261 | <p>Here is the question.</p>
<blockquote>
<p>If <span class="math-container">$s_1=0$</span>, prove that <span class="math-container">$s_{n+1}=s_n+\frac{1}{2}(x-s_n^2)$</span> is monotone and converges if <span class="math-container">$0<x<1$</span>. </p>
</blockquote>
<p>I tried this question by induction, bu... | Martin Argerami | 22,857 | <p>You cannot use the formula with <span class="math-container">$2+n$</span>, because that means you are calculating <span class="math-container">$\sum_{j=1}^{2+n} j^2$</span>. What you want to do is
<span class="math-container">$$
\sum_{j=1}^m(2+j)^2=\sum_{j=3}^{m+2} j^2=\frac{(m+2)(m+3)(2(m+2)+1)}{6}-1^2-2^2.
$$</sp... |
954,658 | <p>How does
$$
\frac{1}{\log(1+\frac{d}{a})}
$$
simplify to $\frac{a}{d}$ in the limit that $d$ is tiny relative to $a$? I was wondering if a Taylor expansion would work though I am not sure how to work it.</p>
| Semiclassical | 137,524 | <p>Here's a non-Taylor series approach. Consider the denominator $\log(1+\frac{d}{a})=\log(a+d)-\log(a)$, which goes to zero as $d\to0$. This should remind you of the definition of the derivative: indeed,</p>
<p>$$\lim_{d\to 0}\dfrac{\log(a+d)-\log(a)}{d}=\dfrac{d}{dx}\log(x)|_{x=a}=\frac{1}{a}$$</p>
<p>Rearranging, ... |
582,804 | <p>I want to draw phase plane diagram of the following differential equation </p>
<p>$$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 10 y = 0.$$</p>
<p>Please check if my approach is correct. I have some doubts about it. They are in <strong>bold</strong>.</p>
<p>First of all I am changing it to a system of differential equat... | Ben Grossmann | 81,360 | <p><strong>Long story short:</strong> suppose that $\pmatrix{a&b\\-b&a} = T^{-1}AT$ where $a,b$ are real numbers. If $\det(T)$ and $b$ have the same sign, then the trajectory will spiral clockwise. If they have opposite signs, then the trajectory will spiral counterclockwise.</p>
<p>The system will spiral ou... |
328,021 | <p><span class="math-container">$f(x_1,x_2,\ldots x_n)$</span> is polynomial with integer coefficients.
<span class="math-container">$f$</span> is rather large to be computed explicitly, but an algorithm can
compute it efficiently at integers and complex number and "lazy" form using
parenthesis.
The degree of... | Max Alekseyev | 7,076 | <p>Since the degree of <span class="math-container">$f$</span> is <span class="math-container">$n$</span>, the monomial <span class="math-container">$x_1x_2\dots x_n$</span> is the only one containing every variable. Then by inclusion-exclusion, we have
<span class="math-container">$$C = \sum_{v\in\{0,1\}^n} (-1)^{n-s(... |
241,394 | <p>The convergence in the Central Limit Theorem is weak convergence, which is weaker than convergence in probability. I set it as an exercise to find an example that convergence in distribution does not imply convergence in probability:</p>
<blockquote>
<p>Let $(X_j)_{j\geq 1}$ be i.i.d. with $E[X_1]=0$ and $\sigma_... | Yury | 35,791 | <p>Hint. For convenience of notation, assume that <span class="math-container">$\sigma =1$</span>. Assume that <span class="math-container">$S_n/\sqrt{n} \to Z$</span> in probability. Choose a sufficiently large <span class="math-container">$n$</span>. Consider <span class="math-container">$A = S_n/\sqrt{n}$</span> and... |
1,604,226 | <p>Let $a,b,c$ be positive real numbers. Prove that </p>
<p>$\frac{bc}{a^2+bc}+\frac{ca}{b^2+ca}+\frac{ab}{c^2+ab}\le{\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}}$.</p>
<p>I have no idea how to solve this question.</p>
| nguyenhuyenag | 410,198 | <p>$$\sum \left(\frac{a}{b+c} - \frac{ab}{c^2+ab}\right) = (a^3+b^3+c^3+abc) \sum {\frac {ca \left( {b}^{2}+{c}^{2} \right) \left( a-b \right) ^{2}}{\displaystyle \prod (a+b)(a^2+bc)}} \geqslant 0.$$</p>
|
2,918,287 | <p>We have
\begin{align}
U_{0} &= 1 &&\text{and} & V_{0} &= 2 \\
U_{n+1} &= \frac{U_{n}+V_{n}}{2} &&\text{and} & V_{n+1} &= \sqrt{U_{n+1}V_n}
\end{align}
How to prove two sequence have a common limit?</p>
<p>I found $(U_n)$ is increasing and $(V_n)$ is decreasing but I don't kn... | Community | -1 | <p>If $U_n,V_n\in[1,2]$,</p>
<p>$$U_n+V_n\in[2,4]\implies U_{n+1}\in[1,2]$$</p>
<p>and</p>
<p>$$U_{n+1}V_n\in[1,4]\implies V_n\in[1,2].$$</p>
<p>Hence both sequences are bounded and both converge.</p>
<p>And if the limits exist, they are equal. Indeed,</p>
<p>$$U=\frac{U+V}2,V=\sqrt{UV}$$ imply $U=V$.</p>
|
824,255 | <p>Suppose I have a polynomial (of any order) and I'm not able to calculate the roots. Is there a way to get at least some information about the roots such as how many of them are complex, negative or positive? For example, I can safely identify the behavior (and therefore roots' character) of $f(x)=ax+b$ or even a qua... | Mr.Fry | 68,477 | <p>$$\text{Suppose the limit did exist then use the below to get a contradiction.}$$</p>
<p>$$\text{Limit Rule:} \ \lim x_n \to X \ \text{and} \ \lim y_n \to Y \Rightarrow \lim x_n y_n = XY$$</p>
|
273,412 | <p>I wrote the following table:</p>
<pre><code>k = 4;
T = Table[{a[n], 0, 1}, {n, 1, k}]
</code></pre>
<p>Which generates:</p>
<pre><code>{{a[1], 0, 1}, {a[2], 0, 1}, {a[3], 0, 1}, {a[4], 0, 1}}
</code></pre>
<p>Now in the I want to use this in the next table:</p>
<pre><code>Table[Table[a[n], {n, 1, 4}], x]
</code></pr... | userrandrand | 86,543 | <pre><code>x={{a[1], 0, 1}, {a[2], 0, 1}, {a[3], 0, 1}, {a[4], 0, 1}}
Table[Table[a[n], {n, 1, 4}], Evaluate[Sequence@@x]]==Table[Table[a[n], {n, 1, 4}], {a[1], 0, 1}, {a[2], 0, 1}, {a[3], 0, 1}, {a[4], 0, 1}]
True
</code></pre>
<p>In the documentation for Table, in section "properties & relations" ther... |
3,009,919 | <p>Give <span class="math-container">$n$</span>-th derivatives, can we find a function ?<br>
For example, if <span class="math-container">$f^{(i)}(x)=i,i=1,2,\cdots,n.$</span> , can we construct a function such that it satisfies the condition.<br>
It just occurs in my mind. I didn't know if it has a solution.</p>
| Seth | 610,132 | <p>If <span class="math-container">$f^{(i)}(x)=i$</span>, then <span class="math-container">$f^{(i-1)}(x)=ix+k=i-1$</span>, so this could work for specific values of <span class="math-container">$x$</span>, but it will not hold in general </p>
|
46,663 | <p>suppose $X$ is a topological space and $G$ and $H$ are groups acting on it.</p>
<p>1) if $G$ is isomorphic to $H$ do we have necessarely $X/G$ is homeomorphic to $X/H$</p>
<p>2) suppose $G$ and $H$ are two conjugate subgroups of a group $K$ acting on $X$, in this case do we have $X/G$ is homeomorphic to $X/H$</p>
... | Amitesh Datta | 10,467 | <p>The following are hints:</p>
<p>(1) Consider the case $G=H$ and consider one action of $G$ on $X$ to be defined by $g\cdot x=x$ for all $x\in X$. (Prove that this is indeed an action of $G$ on $X$ if you have not done so already.) The quotient space $X/G$ is naturally homeomorphic to $X$. Find an example of a non-t... |
623,873 | <p>I'm given with the following function:</p>
<p>$f(x) = x^n \arctan\left(\dfrac{1}{x}\right) , x\ne0 $ , and $f(0)=0$ . </p>
<p>I'm asked to:</p>
<ol>
<li>determine when is it continuous in $x=0$ </li>
<li>determine when is it differentiable in $x=0$</li>
<li>determine when is it continuously differentiable in $x=0... | Eric Naslund | 6,075 | <p><a href="https://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1/49434#49434">In this answer</a>, I use partial summation to show that</p>
<blockquote>
<p><strong>Asymptotic:</strong> For $k>-1$ we have
$$\sum_{p\leq x}p^{k}=\text{li}\left(x^{k+1}\right)+O\left(x^... |
1,942,342 | <p>This is not a homework, just something I am curious about. There are two independent variables in this problem: $x$, and $n$, the number of $x$'s. I am wondering if it is possible to express Graham's number with "reasonable" values for $x$ and $n$, say, $x \le$ a trillion, and $n\le 6$? If no, how much loser do the ... | Noah Schweber | 28,111 | <p>To move this off the unanswered list: <strong>no, you cannot</strong>.</p>
<p>The expression you're talking about is just $x\uparrow\uparrow n$, in the <a href="https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation" rel="nofollow noreferrer">up-arrow notation</a>. Now, Graham's number is defined using this nota... |
2,423,846 | <blockquote>
<p>a) Parametrize the line $L$ through $P = (2, 1, 2)$ that intersects the
line </p>
<p>$x = 1 + t$, $y = 1 − t$, $z = 2t$ </p>
<p>perpendicularly.</p>
<p>b) Parametrize the $z$ axis.</p>
<p>c) What is the distance from this line $L$ to the $z$-axis?</p>
</blockquote>
<p><strong>My... | Michael Rozenberg | 190,319 | <p>Let $B(1+t,1-t,2t)$ and $A(2,1,2)$.</p>
<p>Thus, $$\vec{AB}\perp\vec{(1,-1,2)}$$ or
$$\vec{(t-1,-t,2t-2)}\vec{(1,-1,2)}=0$$ or
$$t-1+t+4t-4=0$$ or
$$t=\frac{5}{6}$$ and since
$$\vec{\left(\frac{5}{6}-1,-\frac{5}{6},\frac{5}{3}-2\right)}=\vec{\left(-\frac{1}{6},-\frac{5}{6},-\frac{1}{3}\right)},$$
we obtain:
$$L:(2,... |
3,460,598 | <p>Suppose that <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> have the same distribution. Then, <span class="math-container">$$\phi_{t X_1} (1) = \phi_{X_1}(t) = \phi_{X_2}(t) = \phi_{t X_2} (1).$$</span></p>
<p>Why do <span class="math-container">$tX_1$</span> and <span class... | NCh | 413,376 | <p>For any <span class="math-container">$u\in\mathbb R$</span> <span class="math-container">$$\phi_{tX_1}(u)=\phi_{X_1}(tu)=\phi_{X_2}(tu)=\phi_{tX_2}(u),$$</span> so <span class="math-container">$tX_1$</span> and <span class="math-container">$tX_2$</span> have the same characteristic function and therefore the same di... |
485,402 | <p>I read earlier that if $A$ is a real $3\times 3$ matrix satsifying $A^3=I$, then $A$ is similar to a matrix of form
$$
\begin{bmatrix}
1 & 0 & 0\\
0 & \cos\theta & -\sin\theta\\
0 & \sin\theta & \cos\theta
\end{bmatrix}
$$</p>
<p>From the structure theorems for normal operators, I know that ... | wendy.krieger | 78,024 | <p>The matrix that governs the heptagonal isomorphs satisfies $A^3=I$.</p>
<pre><code> 1 1 1 1 0 1 1 0 0
0 0 1 -> 0 -1 -1 -> 0 1 0
0 -1 -1 0 1 0 0 0 1
</code></pre>
<p>When this matrix is applied to numbers (i,a,b), where these are the chords... |
1,928,142 | <p>I need help with this geometric problem. </p>
<p>$ \overline {AB}$ is a diameter in circle $M$ , $C \in$ $\overrightarrow{AB}$ lying outside the circle , $\overline{CD}$ is drawn tangent to the circle at point $D$ , then $\overrightarrow{DH} \bot \overrightarrow{AB} $ to intersect it at $H$.</p>
<p><strong>Prove t... | StackTD | 159,845 | <p>If you compare
$$\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i$$
with the form
$$r\left( \cos\theta + i \sin \theta \right) = r\cos\theta + i r\sin \theta$$
it is clear that you're looking for $r$ and $\theta$ such that:
$$r\cos\theta = \frac{1}{\sqrt{2}} \quad\mbox{ and } \quad r\sin \theta = \frac{1}{\sqrt{2}}$$
Squari... |
257,368 | <p>I have the following system of ODEs:</p>
<p>dx/dt = a/(1 + z) - Q*z</p>
<p>dy/dt = Qx - qy</p>
<p>dz/dt = qy - cz/(K + z).</p>
<p>Assuming K = 1, Q = q < c, and a = c*(Sqrt[c/Q] - 1), is there a way to use Mathematica to test whether a Hopf bifurcation exists?</p>
<p>Any help would be greatly appreciated!</p>
<p>... | Chris K | 6,358 | <p>We can use my <a href="https://github.com/cklausme/EcoEvo" rel="noreferrer">EcoEvo package</a>, despite the fact that this doesn't seem like an ecological model.</p>
<p>First, install the package (only needs to be done once):</p>
<pre><code>PacletInstall["EcoEvo", "Site" -> "http://raw.gi... |
257,368 | <p>I have the following system of ODEs:</p>
<p>dx/dt = a/(1 + z) - Q*z</p>
<p>dy/dt = Qx - qy</p>
<p>dz/dt = qy - cz/(K + z).</p>
<p>Assuming K = 1, Q = q < c, and a = c*(Sqrt[c/Q] - 1), is there a way to use Mathematica to test whether a Hopf bifurcation exists?</p>
<p>Any help would be greatly appreciated!</p>
<p>... | florin | 54,979 | <p>I will make a guess that this system comes from an application involving positive variables and parameters. This "essential nonnegativity" is only possible if the first equation is wrong, and Q z is in fact Q x. The fixed point becomes then unique. I will ignore the assumption on a, since Mathematica can h... |
3,924,376 | <p>Given <span class="math-container">$i,a_i,b_i\in\{1...n\},\space a_i\neq a_j,b_i\neq b_j,\forall i\neq j$</span> prove that
<span class="math-container">$$\sum_{i=1}^n|a_i-b_i|\le\big\lfloor \frac{n^2}{2}\big\rfloor$$</span>
This problem was proposed by the new contributor @user3458994 and it was closed by five use... | Calvin Lin | 54,563 | <p>Here's an almost immediate answer.</p>
<p>By expanding each term of <span class="math-container">$ |a_i - b_i|$</span> into the corresponding <span class="math-container">$\pm (a_i - b_i)$</span>, we ge that
<span class="math-container">$$ \sum |a_i - b_i | = \sum c_i i, $$</span> where <span class="math-container"... |
3,924,376 | <p>Given <span class="math-container">$i,a_i,b_i\in\{1...n\},\space a_i\neq a_j,b_i\neq b_j,\forall i\neq j$</span> prove that
<span class="math-container">$$\sum_{i=1}^n|a_i-b_i|\le\big\lfloor \frac{n^2}{2}\big\rfloor$$</span>
This problem was proposed by the new contributor @user3458994 and it was closed by five use... | John Omielan | 602,049 | <p>For any permutation, there's some <span class="math-container">$1 \le k \le n$</span> values of <span class="math-container">$i$</span> where</p>
<p><span class="math-container">$$a_i - b_i \lt 0 \tag{1}\label{eq1A}$$</span></p>
<p>Thus, the remaining <span class="math-container">$n - k$</span> values of <span class... |
61,172 | <p>Can anyone show me how to set all matrix rows to variables.
For example,
I have matrix <code>w={{1,2,3},{3,2,1},{0,0,0}}</code> and I need to set each matrix row to be separate variable that contains the row number of matrix <code>w</code> in its name.</p>
<p>For example,
<code>t1={1,2,3} t2={3,2,1} t3={{0,0,0}</c... | xyz | 9,627 | <p>The code as below can achieve the result that you need.</p>
<pre><code>Clear@t;
mat = RandomInteger[{1, 10}, {20, 4}];
Evaluate[t /@ Range[Length@mat]] = mat
</code></pre>
<p>Because of (=)<a href="http://reference.wolfram.com/language/ref/Set.html" rel="nofollow">Set</a> has the attribute <a href="http://referenc... |
168,089 | <p>Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim \text{exp}(\mu), L \sim \text{exp}(\lambda)$ are mutually independent, $\lambda \neq \mu$ are two positive integers,... | esg | 48,831 | <p>Call the left resp. right hand sum $R_m$ resp. $R_{m+1}$. As $L$ is $\exp(\lambda)$ and independent of $(R_m,R_{m+1})$ , taking expection with resp. to $L$ first gives
$$\mathbb{P}(R_m\leq L < R_{m+1})=\mathbb{E}(e^{-\lambda R_m}-e^{-\lambda R_{m+1}})= \left(\dfrac{1}{2}\dfrac{\mu}{\mu+\lambda}\right)^m\left(1-\... |
105,577 | <p>The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an intractable problem.</p>
<p>I'm posting this question in the hope that someone may have investigated this curve already ... | James Weigandt | 4,872 | <p>This particular curve, which I'll call $E$, may be quite challenging. The analytic rank is probably 2, but as far as I know, the only way to prove this is to show that the algebraic rank is not 0. Assuming the analytic rank is 2 and the full BSD formula, the product of the Regulator and the order of the Tate-Shafare... |
3,769,681 | <p>What's the name for sequences sitting between geometric and arithmetic?</p>
<p>E.g. let <span class="math-container">$x_{n+1}=ax_n+b$</span></p>
<p>I can't find a general name for these.</p>
<p>These sequences may also be a <a href="https://en.wikipedia.org/wiki/Lucas_sequence" rel="nofollow noreferrer">Lucas Sequen... | Prasanna Maddila | 627,540 | <p>I think the term you are looking for is <a href="https://brilliant.org/wiki/linear-recurrence-relations/#" rel="nofollow noreferrer">Linear recurrence Relations</a></p>
|
111,835 | <p>If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol.</p>
<p>Who first proved this? Is there a proof simple enough for an undergraduate number theory course? (We will get to quadratic reciprocity by th... | Samuel Hambleton | 16,825 | <p>The Pell conic $\mathcal{C} : x^2 - 5 y^2 = 4$ has $p - (5/p)$ points modulo $p$. The polynomials $$f_{-1} = x, f_0 = 2, f_n = x f_{n-1} - f_{n-2}$$ $$g_{-1} = -1, g_0 = 0, g_n = x g_{n-1} - g_{n-2}$$ perform multiplication by $n$ in the group $\mathcal{C}(\mathbb{F}_p)$, $n (x, y) = (f_n(x), y \cdot g_n(x))$. Let $... |
2,438,594 | <p>Let $\left({z_n}\right)$ be a sequence of distinct points in $D\left(0,1\right)$=$\{{z \in \mathbb{C} :\vert{z}\vert \lt{1}} \}$ with $\lim_{n\to\mathbb{\infty}} {z_n}=0$. Consider the following statements P and Q.</p>
<p>P: There exists a unique analytic function $f$ on $D\left(0,1\right)$ such that $f\left({z_n}\... | Steven Alexis Gregory | 75,410 | <p>If $a+b=4$, then $a^2 + ab = 4a$. So </p>
<p>$$a^2-4a+k=0. \tag 1$$</p>
<p>If $a+c=-k$, then $a^2+ac=-ka$. So </p>
<p>$$a^2 + ka -4 = 0. \tag 2$$</p>
<p>So $0 = (a^2 + ka -4)-(a^2-4a+k)=(k+4)a-(k+4)=(k+4)(a-1)$</p>
<p>So $k=-4$ or $a=1$</p>
<p>If $k=-4$, then both trinomials become $x^2-4k-4$ and so they have ... |
2,805,136 | <p>sadly, i don't know how to solve this question, so i cannot give my input on what i did, only my thoughts.</p>
<p>let $φ$ be a formula with a free variable $x$. for another variable, let's say $y$: $φ(y)=sub(φ,x,y)$ is $φ$ where each occurrence of $x$ is replaced with $y$.</p>
<p>1)prove that if $y$ is not in $φ$,... | user21820 | 21,820 | <p>It's basically a matter of the meaning of the quantifier. There is a related kind of substitution <span class="math-container">$φ[y/x]$</span> that denotes "<span class="math-container">$φ$</span> with every <strong>free</strong> occurrence of <span class="math-container">$x$</span> substituted by <span class="... |
3,654,326 | <p>I have tried to solve the following equation in rational(<span class="math-container">$x,y,z$</span> are rational numbers) <span class="math-container">$$\frac{x^3}{y^3+z^3}+\frac{y^3}{x^3+z^3}+\frac{z^3}{y^3+x^3}=1$$</span> , I can't find such pairs of rational numbers <span class="math-container">$(x,y,z)$</span> ... | Servaes | 30,382 | <p>Here's a roadmap to a solution; the details are too tedious for me to figure out.</p>
<p>First note that if <span class="math-container">$(x,y,z)\in\Bbb{Q}^3$</span> satisfies
<span class="math-container">$$\frac{x^3}{y^3+z^3}+\frac{y^3}{x^3+z^3}+\frac{z^3}{y^3+x^3}=1,\tag{1}$$</span>
then so does <span class="math-... |
2,973,278 | <p>In the book of <em>The theory of Functions</em> by Titchmarsh, at page 83, in the proof of the fact that analytic functions can be expressed as power series, it is stated that</p>
<p><span class="math-container">$$\frac{1}{w-z} = \frac{1}{w-a} + \frac{z-a}{(w-a)^2} + \dots \frac{(z-a)^n}{(w-a)^{n+1}} \dots ,$$</s... | Robert Z | 299,698 | <p>Hint. We have that
<span class="math-container">$$\frac{1}{w-z}=\frac{1}{w-a-(z-a)}=\frac{1}{w-a}\cdot \frac{1}{1-\frac{z-a}{w-a}}.$$</span>
Now consider the case when <span class="math-container">$\left|\frac{z-a}{w-a}\right|<1$</span> and recall the main properties of the <a href="https://en.wikipedia.org/wiki/... |
3,529,240 | <p>The equation I want to graph is: <span class="math-container">$$4x^2 -3y^2 + 8x +12y-4 =0$$</span></p>
<p>My textbook says that the above equation has no graph. </p>
<p>Yet, I rearranged it to the following: </p>
<p><span class="math-container">$$\frac{(y-2)^2}{(2/\sqrt3)^2} - \frac{(x+1)^2}{1^2} = 1^2$$</span></... | Community | -1 | <p>If the book indeed says "no graph", it is wrong.</p>
<p>Because when plugging <span class="math-container">$(10,0)$</span>, you get a positive value, and when plugging <span class="math-container">$(0,10)$</span>, a negative one. Hence there are two regions of opposite signs and there <em>must</em> be a continuous ... |
3,529,240 | <p>The equation I want to graph is: <span class="math-container">$$4x^2 -3y^2 + 8x +12y-4 =0$$</span></p>
<p>My textbook says that the above equation has no graph. </p>
<p>Yet, I rearranged it to the following: </p>
<p><span class="math-container">$$\frac{(y-2)^2}{(2/\sqrt3)^2} - \frac{(x+1)^2}{1^2} = 1^2$$</span></... | gandalf61 | 424,513 | <p><span class="math-container">$4(x+1)^2-3(y-2)^2=-4 \\ \displaystyle \Rightarrow \frac{3(y-2)^2}{4} - (x+1)^2=1$</span></p>
|
293,505 | <p>It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. I was wondering about the similar classification for a general compact four-manifolds possibly with boundaries or eve... | Daniele Zuddas | 23,193 | <p>As Paul Siegel pointed out, the fundamental group of a smooth closed orientable 4-manifold can be an arbitrary finitely presented group, and for this reason a general classification is not possible, unless, possibly, for some classes of fundamental groups, e.g. the trivial group, by Freedman's work. However, the kno... |
347,088 | <p>In a famous paper Rademacher used the circle theorem to give a formula for the fourier coefficients of the partition function <span class="math-container">$1/f(q)$</span> where <span class="math-container">$f(q) = \prod_{n=1}(1-q^n)$</span>, and in another paper he gave the coefficients of the j-invariant function ... | Henri Cohen | 81,776 | <p>Since <span class="math-container">$f(q)$</span> is lacunary and with coefficients only 0 and <span class="math-container">$\pm1$</span>, it should be
trivial to prove that the asymptotics of the Fourier coefficients of
<span class="math-container">$j(q)f(q)$</span> are the same as those of <span class="math-contain... |
247,755 | <p>We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X,\rho)\rightarrow (X,\rho)$.</p>
<p>Does there exist a topology $\tau$ that is similar to the euclidean topology on $\mathbb R$?</p>
<... | Gabriel C. Drummond-Cole | 3,075 | <p>The only topology similar to the Euclidean topology on $\mathbb{R}$ is the Euclidean topology. </p>
<hr>
<p>Suppose there is such a topology $\tau$. I'll use "open," "continuous," etc. to mean with respect to the Euclidean topology and "$\tau$-open" etc. for $\tau$.</p>
<p>Since $\tau$ is a refinement of the Eucl... |
585,044 | <blockquote>
<p>Let $R(n)$ be the number of ordered pairs of integers $(x,y)$ with $x^2+y^2 = n$. Using the identity $$(a+b)^2+(a-b)^2 =2(a^2 +b^2),$$
prove that for all integers $n$, $R(2n) = R(n)$.</p>
</blockquote>
<p>I'm not really sure how to start this question. Any help would be appreciated.</p>
| Hagen von Eitzen | 39,174 | <p>Let $$A(n)=\{\,(x,y)\in\mathbb Z^2\mid x^2+y^2=n\,\}$$
so that $R(n)=|A(n)|$.
Show that $(x,y)\mapsto(x+y,x-y)$ gives you a bijection $A(n)\to A(2n)$. The identity from the hint shows that this indeed maps elements of $A(n)$ to elements of $A(2n)$ as required. Why is it injective and surjective?</p>
|
3,148,803 | <p>I am reading a paper in which the following result is used, but I can’t see the proof of this.</p>
<blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative ring with only two maximal ideals, say <span class="math-container">$M_1$</span> and <span class="math-container">$M_2$</span>. Suppose <s... | Alex Mathers | 227,652 | <p>First notice that <span class="math-container">$1-m_1$</span> cannot be a unit, because this would imply <span class="math-container">$m_1$</span> is in the Jacobson radical of <span class="math-container">$R$</span>, and in particular we would have <span class="math-container">$m_1\in M_2$</span>.</p>
<p>Now it fo... |
679,343 | <p>I have been researching this for the past hour or so, but I have not really found a sufficient answer. What I have found is that vaguely, math education PhDs set policies for math education. However, universities almost uniformly have math professors teach mathematics courses. I believe that usually the math prof... | user952679 | 952,679 | <p>Actually the answer is very clear. Math professors deliver their mathematics knowledge. They are good at math, not necessarily good at pedagogy.
Math educators research about how can learners learn math more effectively. We observe, work with data, use research designs, do statistical analyses, etc. to develop/contr... |
1,117,081 | <p>In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's.
For instance to count the sum of the logarithms of the primes below some given integer.</p>
<p>(second Chebyshev Function)</p>
<p>Consider the floor function : <a href="http://mathworld.wolfram.com/FloorFunction.html" rel="n... | Peter Humphries | 2,689 | <p>The standard way in analytic number theory to deal with sums involving the floor function is to either (a) estimate it trivially or (b) use Fourier analysis. In both cases, the idea is that
\[\lfloor x \rfloor = x - \{x\},\]
where $\{x\}$ denotes the fractional part of $x$. Trivially, one has
\[0 \leq \{x\} < 1,\... |
343,549 | <p>For <span class="math-container">$\Omega$</span> a bounded open set of <span class="math-container">$\mathbf{R}^d$</span> and <span class="math-container">$f\in L^p(\Omega)$</span> the infimum
<span class="math-container">\begin{align*}
\inf_{C\in\mathbf{R}} \|f-C\|_p
\end{align*}</span>
is reached (by compactness)... | Lior Silberman | 327 | <p>The constant <span class="math-container">$C$</span> that you've constructed is called the <em>barycenter</em> in the metric geometry literature. Specifically, you're asking about the <span class="math-container">$p$</span>-barycenter of the measure <span class="math-container">$f_*m$</span> on <span class="math-co... |
1,605,864 | <p>Definition $1$: </p>
<p>A poset $P=(X, \prec)$ is called Interval order if there exists a function $f$ which maps each $x \in X$ to an interval $I_x$ of real line such that $x \prec y$ iff $I_x \prec I_y$ ($I_x$ lies totally to the left of $I_y$).</p>
<p>Definition $2$:</p>
<p>Interval order dimension of a poset ... | hmakholm left over Monica | 14,366 | <p>The notation $P_1 \cap \cdots\cap P_k$ is a little strange, but what makes best sense in context is that $P_1=(X, R_1)$, $P_2=(X,R_2)$ and so forth are all partial orders <em>on the same set $X$</em>, and $P_1 \cap \cdots\cap P_k$ then means $(X,R_1 \cap\cdots\cap R_k)$. It is easy to see that this is a poset.</p>
... |
1,135,416 | <p>I have two arrays with 16 unsigned integers. I compute the sum of the first array = x and the sum of the second array = y. What is the chance they will be the same?
Also, how many combinations out of the total combinations are the same?</p>
<p>An example:
Lets say an array1 contains these numbers (low values just f... | Damian Yerrick | 93,170 | <h2>Not anymore.</h2>
<p>Older versions of <em>Tetris</em> had a uniform stateless randomizer. This would allow a repeating SZ-sequence that eventually causes a loss. This is possible yet unfathomably unlikely in practice because a competent player can survive this sequence for 150 Tetriminos on a standard 10x20-cell m... |
3,491,239 | <p>Can somebody explain me this equation:
<span class="math-container">$$ \omega(X_H, Y) = \mathrm{d}H(Y), $$</span>
where <span class="math-container">$\omega$</span> is symplectic form and <span class="math-container">$Y$</span> is any vector field on manifold <span class="math-container">$\mathcal{M}$</span>. </p>
... | Tsemo Aristide | 280,301 | <p><span class="math-container">$\omega$</span> is non degenerated, this is equivalent to saying that for every <span class="math-container">$x\in M$</span>, the application <span class="math-container">$h:T_xM\rightarrow T^*_xM$</span> defined by <span class="math-container">$h(X)(Y)=\omega_x(X,Y)$</span> is an isomor... |
3,491,239 | <p>Can somebody explain me this equation:
<span class="math-container">$$ \omega(X_H, Y) = \mathrm{d}H(Y), $$</span>
where <span class="math-container">$\omega$</span> is symplectic form and <span class="math-container">$Y$</span> is any vector field on manifold <span class="math-container">$\mathcal{M}$</span>. </p>
... | K B Dave | 534,616 | <p>Suppose for the moment that <span class="math-container">$M$</span> is <span class="math-container">$\mathbb{R}^{n}$</span>, <span class="math-container">$n$</span> even, with coordinates and coordinates bases
<span class="math-container">$$(z^i,\mathbf{e}_i,\mathbf{e}^i)\in(\Gamma(M,\mathbb{R}),\Gamma(M,\mathrm{T}M... |
2,306,309 | <p>Induction on $n$: </p>
<p>If $n=1$, then $A_{n+2}=A_{3} \simeq \mathbb{Z}_3$ has subgroup $1 \simeq S_n = S_1$. Assume for some $n>1$, $S_n \simeq H$ for some $H \leq A_{n+2}$. I want to show that $S_{n+1} \simeq H'$ for some $H'\leq A_{n+3}$.</p>
<p>Note that $H \simeq S_n \leq S_{n+1}$, and by inductive hypot... | bof | 111,012 | <p>$A_{n+2}$ is the group of all even permutations of $\{1,2,\dots,n+2\}.$ Let $H$ be the subgroup of $A_{n+2}$ consisting of all permutations that leave the sets $\{1,2,\dots,n\}$ and $\{n+1,n+2\}$ fixed. Show that the obvious map $H\to S_n$ is an isomorphism.</p>
|
3,111,628 | <blockquote>
<p>Let <span class="math-container">$U \in M_{n,k}(\mathbb{R})$</span> such that : <span class="math-container">$^t UU = I_k$</span>. Then I would like to understand <strong>geometrically</strong> why <span class="math-container">$U ^t U$</span> is the orthogonal projection on <span class="math-container... | Daniel Mathias | 627,720 | <p>We have <span class="math-container">$6$</span> I's and <span class="math-container">$8$</span> other letters. Start with "_X_X_X_X_X_X_X_X_" where "_" represents an I or nothing and "X" represents one of the other <span class="math-container">$8$</span> letters. There are <span class="math-container">$\binom{9}{6}$... |
3,284,801 | <p>Let <span class="math-container">$D\subset \mathbb{R}^2$</span> be the closed unit disk, boundary of <span class="math-container">$D$</span> is <span class="math-container">$\mathbb{S}^1$</span>, let <span class="math-container">$a,b\in \mathbb{S}^1$</span> are two distinct points, <span class="math-container">$A,B\... | Matsmir | 685,805 | <p>At first let me introduce some simple notation. For a closed bounded set <span class="math-container">$A \subset \mathbb{R}^2$</span> I will denote by <span class="math-container">$\tilde{A}$</span> the union of <span class="math-container">$A$</span> and all bounded components of <span class="math-container">$\math... |
2,899,316 | <p>Let $(a_n)$ be a sequence of real numbers. Consider the two points,</p>
<ul>
<li>1.$\quad$ $(a_n)$ diverges,</li>
<li>2.$\quad$ $(a_n)$ <a href="https://proofwiki.org/wiki/Definition:Divergent_Sequence_to_Infinity" rel="nofollow noreferrer">tends to <strong>plus</strong> infinity</a>.</li>
</ul>
<p>Using the f... | Kusma | 514,933 | <p>If $(a_n)$ is (weakly) increasing, then by the monotone sequence theorem it either converges to a finite value or diverges to $+\infty$. However, there are sequences that tend to $+\infty$ that are not eventually increasing, for example</p>
<p>$(a_n)=(1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,...)$. </p>
|
3,595,653 | <p>Assume that I have a scalar variable which takes values on <span class="math-container">$\mathbb{R}$</span> and prescribes polar coordinates (in radians) on, say, a unit circle <span class="math-container">$r=1$</span>. Assume now that I have two such (possibly different) variables <span class="math-container">$a$</... | Community | -1 | <p><span class="math-container">$OAB$</span> is an isoceles triangle of angle at apex <span class="math-container">$|b-a|$</span> and sides <span class="math-container">$r$</span>. The base is</p>
<p><span class="math-container">$$2r\sin\frac{|b-a|}2.$$</span></p>
<hr>
<p>Note that for two points not on the same cir... |
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | user | 505,767 | <p>Note that when we choose $\binom{10}{5}$ we are counting twice the same teams, as for example</p>
<p>$$ \{ A,B,C,D,E \} , \{ G,H,I,L,M \} \equiv \{ G,H,I,L,M \},\{A,B,C,D,E\}$$</p>
<p>therefore the number of ways to obtain 2 teams is $\frac12 \binom{10}{5}$.</p>
|
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | Jaroslaw Matlak | 389,592 | <p>In the first case each choice is about selecting 5 out of ten, let's say from $\{A,B,C,D,E,F,G,H,I,J\}$. Note that selecting $\{A,B,C,D,E\}$ id different, than selecting $\{F,G,H,I,J\}$.</p>
<p>In the second case you select 5, but the other 5 are also forming a team. Thus your selection is for example $\{\{A,B,C,D,... |
3,458,467 | <p>Given a matrix <span class="math-container">$A \in M_{2}(\mathbb{Q}) = R$</span>. I don't believe that <span class="math-container">$A$</span> is similar to <span class="math-container">$-A$</span>, but I just want to check. Also, is the only similarity class of matrices in <span class="math-container">$R$</span> wi... | user1551 | 1,551 | <p>The field is mostly unimportant. <span class="math-container">$A\in M_2(\mathbb F)$</span> is similar to <span class="math-container">$-A$</span> over a field <span class="math-container">$\mathbb F$</span> of characteristic <span class="math-container">$\ne2$</span> if and only if <span class="math-container">$A$</... |
4,270,745 | <blockquote>
<p>I am trying to find this limit:
<span class="math-container">$$\lim_{n \to \infty}\left[\,\sqrt{\,2\,}\,\frac{\Gamma\left(n/2 + 1/2\right)}{\Gamma\left(n/2\right)} - \,\sqrt{\,n\,}\right]$$</span></p>
</blockquote>
<p>I know that
<span class="math-container">$$\frac{\sqrt{2}\Gamma(\frac{n+1}{2})}{\Gamma... | Gary | 83,800 | <p>Using the first few terms of <a href="http://dlmf.nist.gov/5.11.E13" rel="nofollow noreferrer">http://dlmf.nist.gov/5.11.E13</a> with <span class="math-container">$a=1/2$</span>, <span class="math-container">$b=0$</span> and <span class="math-container">$z=n/2$</span>, we find
<span class="math-container">$$
\frac{{... |
146,054 | <p>Let $X = (x_{ij}) \in \mathbb{R}^{n \times p}$ be a matrix with independent $N(0,1)$ entries. </p>
<p>We know that $\max_j x_{ij} < \sqrt{2\log(p/\delta)}$ with probability at least $1-\delta$. </p>
<p>I would like to obtain a lower bound for $\min_i (\max_j x_{ij})$ that holds with probability at least $1-\del... | passerby51 | 7,202 | <p>You might try Gordon's inequality, which is a sort of comparison inequality for Gaussian processes. Here is a lecture note:</p>
<p><a href="http://www-personal.umich.edu/~romanv/teaching/2006-07/280/lec11.pdf" rel="nofollow">http://www-personal.umich.edu/~romanv/teaching/2006-07/280/lec11.pdf</a></p>
<p>Not sure i... |
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | Joe Goldiamond | 516,719 | <p>I presume the integral is:
$$\pi = 2\int_{-1}^1{ \frac{1}{\sqrt{1-t^2}}}dt$$
Hint: use the standard integral:
$$2\int_{-1}^1{ \frac{1}{\sqrt{1-t^2}}}dt=2 \bigg[\arcsin t \bigg]_{-1}^{1}= 2\bigg[ \ arcsin (1) - \arcsin(-1)\ \bigg]>3$$</p>
|
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | Riemann | 27,899 | <p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}dt$$ is the area of the unit disk.We can consider the area of regular dodecagon inside the unit circle which is very easy to compute.</p>
|
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | mucciolo | 222,084 | <p>Let $f(t) = \sqrt{1-t^2}$. Since $f$ is even
$$ 2 \int_{-1}^{1} f(t)\,dt = 4 \int_{0}^{1} f(t)\,dt $$</p>
<p>Bound $f$ from below with easily integrable functions, like straight lines. For example, in this case, the concatenation of the line segments connecting in order the points $(0, f(0))$, $\left(\frac{1}{2}, f... |
3,106,223 | <blockquote>
<p><span class="math-container">$$A = \left( \begin{array} { c c c c c } { 1 } & { 4 } & { 15 } & { 1
} & { - 9 } \\ { 0 } & { 1 } & { 4 } & { 0 } & { - 3 } \\ { 2 } & { -
4 } & { - 18 } & { - 1 } & { 15 } \\ { 1 } & { 0 } & { - 1 } & { 0 } &a... | enedil | 126,823 | <p>The null space is the set of vectors <span class="math-container">$v$</span>, such that <span class="math-container">$f_A(v) = 0$</span>. This means that you need to solve <span class="math-container">$Av = [0, 0, 0, 0]^T$</span>. This amounts to Gaussian elimination. In this case, the echelon form of A is
<span cla... |
3,306,110 | <p>the expression <span class="math-container">$x(x-9)$</span> is resolved in the manner
<span class="math-container">$$x(x-9)=x^2-9x$$</span>
but why not like this?
<span class="math-container">$$x(x-9)=x^2-9$$</span>
where does the <span class="math-container">$9x$</span> come from?</p>
| N. Bar | 641,014 | <p><span class="math-container">$$x(x-9)$$</span>
By the Distributive Law
<span class="math-container">$$x^2-9x$$</span></p>
<p>Here's a short "proof" to show that you can't derive <span class="math-container">$x^2-9$</span> from <span class="math-container">$x(x-9)$</span>.</p>
<p><span class="math-container">$$\tex... |
43,326 | <p>Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (This would be a special case of the general result of O. Izhboldin a... | The Bridge | 2,642 | <p>Hi, </p>
<p>Maybe I haven't fully understood your question but I think there is a counter-example (in a simpler context) given in Protter's book on Stochastic Integration in the form of a theorem. It's theorem 71, chapter IV, section 7 (Local Times). </p>
<p>It's claimed there that for any $\alpha \in(0,1/2)$ and ... |
43,326 | <p>Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (This would be a special case of the general result of O. Izhboldin a... | has2 | 3,370 | <p>What follows is too long to fit in a comment.</p>
<p>Some thoughts on the second problem. Let us consider the problem for $t$ taking values in the compact interval $[0,T]$. The general case perhaps can be approximated by this case. If the $t$ variable were discrete and finite the problem would be: we have a sequenc... |
1,922,325 | <p>How do you prove that if $m\phi(m)=n\phi(n)$ then $m=n$?</p>
<p>Here, $\phi$ is the Euler Phi function.</p>
| Caleb Stanford | 68,107 | <p>We can prove this by strong induction on $(m,n)$.</p>
<p>First consider a base case where $m = 1$ (or $n = 1$). Then $n \phi(n) = 1$, so $n = 1$.</p>
<p>Now for the induction step, suppose that $m\phi(m) = n\phi(n)$, and that the result is true for all $m' < m$, $n' < n$.
Let $p$ be the <strong>largest prime... |
3,948,378 | <p>For <span class="math-container">$x\in(-\frac{\pi}{2}, -\frac{\pi}{2})\setminus \{0\}$</span> we can say that since the area of sector of circle is greater than the triangle (ABC) area I have <span class="math-container">$2|\sin{x}|<2|x|$</span>. But what's about x in the 2nd and 3rd quadrant?
Look at the figure ... | tomasz | 30,222 | <p><strong>Hint</strong>: The inequality is trivial when <span class="math-container">$\lvert x\rvert>1$</span>. (It is also not true when <span class="math-container">$x=0$</span>.)</p>
|
4,312,761 | <p>I'm trying to understand any of the <strong>LK sequent calculus (logic) inference rules for quantifiers</strong> (if I can understand one well enough, perhaps I can use that to inform my understanding of the other three). Per <a href="https://en.wikipedia.org/wiki/Sequent_calculus#Inference_rules" rel="nofollow nore... | Noah Schweber | 28,111 | <p>Because we <strong>don't</strong> just want to slap a "<span class="math-container">$\forall x$</span>" on the front of the formula <span class="math-container">$A$</span>. <em>(For simplicity, below I work with <span class="math-container">$\Gamma=\emptyset$</span> and <span class="math-container">$\Delta... |
4,312,761 | <p>I'm trying to understand any of the <strong>LK sequent calculus (logic) inference rules for quantifiers</strong> (if I can understand one well enough, perhaps I can use that to inform my understanding of the other three). Per <a href="https://en.wikipedia.org/wiki/Sequent_calculus#Inference_rules" rel="nofollow nore... | mohottnad | 955,538 | <p>You had a careful observation for the difference between arbitrary term <span class="math-container">$t$</span> and arbitrary variable <span class="math-container">$y$</span> used in the 4 quantifier rules in <span class="math-container">$LK$</span>. Indeed there's some purpose for this difference intentionally in m... |
361,143 | <p>Suppose I have a set containing $n$ elements and I consider the number of partitions of it in which all the constituent classes are even. I let $R_n$ denote this number ($R_0=1$) and proceed as follows to obtain the exponential generating function for $R_n$.</p>
<p>Since $$R_n=\sum_{k=1}^{\lfloor n/2\rfloor}(\text{... | vonbrand | 43,946 | <p>The problem is that the empty set can't be partitioned, i.e., $R_0 = 0$. Your boundary condition is off.</p>
|
575,892 | <p>I know I need to choose an open cover and then show it has no finite subcover.
If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?</p>
| math student | 60,174 | <p>Let A >0 arbitrary . Consider $n \in N$ such that $\sqrt{(n)^2 + (n^2 - 0 )^2} > A$.</p>
<p>We have $(n,n^2)$ an element of your set and the norm of this vector greater than A. Then your set is not bounded. And then is not compact .</p>
|
575,892 | <p>I know I need to choose an open cover and then show it has no finite subcover.
If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?</p>
| Brian M. Scott | 12,042 | <p>What do you mean by $(\langle -n,-n^2\rangle,\langle n,n^2\rangle)$? If you’re talking about a cover by open sets in $\Bbb R^2$, you’re using interval notation for what are supposed to be open sets in $\Bbb R^2$, which doesn’t have a linear structure that would allow interval notation to be meaningful. If you’re tal... |
1,177,111 | <p>Let $(a_n)$ be defined by $a_1=1$, $a_{n+1}=a_n+{1\over a_n}$. Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. </p>
<p>I tried many things but they wouldn't work. I would appreciate your help.</p>
| user2566092 | 87,313 | <p>As the second part of your question suggests, and confirmed by one of the comments, the limit must be $\infty$ because your sequence is strictly increasing and a limit $L$ would have to satisfy $L = L + 1/L$ which is impossible. </p>
<p>For the second part of the question, note that your sequence grows on the order... |
3,724,594 | <p>Let <span class="math-container">$g: \mathbb{R}^2 \to \mathbb{R}$</span>, <span class="math-container">$g(x,y) = |xy|$</span>. Find all the <span class="math-container">$(x,y) \in \mathbb{R}^2$</span> where <span class="math-container">$g$</span> is differentiable.</p>
<p>I tried to compute the partial derivatives:<... | badjohn | 332,763 | <p>You seem to have the definition backwards in your first sentence.<br />
<span class="math-container">$$\forall \epsilon > 0 \; \exists \delta > 0 \; ...$$</span></p>
<p>In English: for all <span class="math-container">$\epsilon > 0$</span> there exists <span class="math-container">$\delta > 0$</span> ...... |
2,230,507 | <p>I want to compute the smallest nonzero singular value of the matrix A, which is defined as follows. </p>
<p>Let $B = rand(500, 250)$, $A = B*B^t$, where $t$ denotes the transpose of the matrix. </p>
<p>I found the following matlab code to compute singular values of the matrix A which is based on the Singular val... | Community | -1 | <p>Since $A=BB^T$, it is symmetric and positive semidefinite, and its singular values are identical to its eigenvalues. Furthermore, its nonzero eigenvalues are identical to the nonzero eigenvalues of $B^TB$. To get the smallest eigenvalue of $B^TB$, use <code>eigs(B'*B,1,'sm')</code>.</p>
|
3,264,330 | <p>I am really struggling with a contour integration question, which I am revising for an exam. I want to show that the contour integral of </p>
<p><span class="math-container">$$\int_\Gamma e^{t(z+z^{-1})}z^{-2} dz=\sum_{m=0}^\infty b_mt^{2m+1}$$</span> where the <span class="math-container">$b_m$</span> are constant... | Anders Beta | 464,504 | <p><span class="math-container">$$e^{tz} = \sum_{n=0}^{+\infty}t^n\frac{z^n}{n!}\,,$$</span>
<span class="math-container">$$e^{t/z} = \sum_{m=0}^{+\infty}t^m\frac{z^{-m}}{m!}\,.$$</span></p>
<p>Therefore, the integrand is:</p>
<p><span class="math-container">$$e^{tz}e^{t/z}z^{-2}=\sum_{n=0}^{+\infty}\sum_{m=0}^{+\inf... |
190,914 | <p>In <a href="https://math.stackexchange.com/questions/38350/n-lines-cannot-divide-a-plane-region-into-x-regions-finding-x-for-n">this</a> post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. </p>
<p>What happens if we use circles instead of lin... | Willie Wong | 1,543 | <p>For the question stated in the title, the answer is yes, if more is interpreted as "more than or equal to".</p>
<p>Proof: let <span class="math-container">$\Lambda$</span> be a collection of lines, and let <span class="math-container">$P$</span> be the extended two plane (the Riemann sphere). Let <span cla... |
2,057,788 | <p><strong>Is the following true?</strong></p>
<p>If $k \geq 0$...</p>
<p>$x^2 > k \implies$</p>
<p>$|x| > \sqrt k \implies$</p>
<p>$x > \sqrt k$ or $x < -\sqrt k$</p>
<p>If $k < 0$...</p>
<p>$x^2 > k \implies$</p>
<p>$x^2 \geq 0$</p>
| Adam Hughes | 58,831 | <p>Use the definition of the derivative.</p>
<blockquote>
<p>$$0\le\lim_{x\to y} \left|{f(x) - f(y)\over x-y}\right|\le \lim_{x\to y}|x-y|=0$$</p>
</blockquote>
<p>So the function is differentiable with derivative $0$ at all points.</p>
|
1,897,814 | <blockquote>
<p>Find the co-efficient of $x^3y^3zw^2$ in the expression of $(x-y+2z-2w)^9$.</p>
</blockquote>
<p>I have clearly understood how to use the bin theorem for $(a+b)^n$, but am unable to extend the theorem to $4$ terms .</p>
| angryavian | 43,949 | <p>We have $9$ copies of $x-y+2z-2w$. We need to choose $3$ factors for the $x^3$, $3$ factors for the $(-y)^3$, and $1$ factor for the $(2z)$; the remaining two factors will be for the $(-2w)^2$. This produces a term $-8 x^3 y^3 z w^2$.</p>
<p>It remains to count how many ways to make the choices above. This is $\bin... |
2,031,779 | <p>Let $0<\beta <1$ and $a_0, a_1 \in \mathbb R$. I want to show that this sequence converges: </p>
<p>$$a_{n+1} = \beta a_n + (1-\beta) a_{n-1}$$</p>
<p>Furthermore, it is given that there you can explicitly formulate the sequence as $$c_1\lambda_1^n+c_2\lambda_2^n$$</p>
<p>I played it through with $a_5$ and... | Mohamed | 33,307 | <p>To solve the kind of séquences satisfying:$$(1) \quad u_{n+1}=au_n+bu_{n-1}$$ we solove the algebric equation:$$(E) \quad t^2-at-b=0$$ If $(E)$ has two solutions $q_1$ and $q_2$ the sequences realizing $(1)$ are :$$u_n=\lambda q_1^n+\mu q_2^n$$ where $\lambda$ and $\mu$ are two constants. </p>
<p>Using this, you ca... |
1,783,841 | <p>If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$.
And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.</p>
<p>I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?</p>
<p>I am referring to a case li... | πr8 | 302,863 | <p>If $f(x_0)=f'(x_0)$, this tells us that if we define $g(x)=f(x)\exp(-x)$, that $g$ has an extreme point at $x_0$ (excepting inflection points etc.). This is because:</p>
<p>$$\frac{dg}{dx}=\frac{d}{dx}f(x)\exp(-x)=f'(x)\exp(-x)-f(x)\exp(-x)$$</p>
<p>$$\implies\frac{dg}{dx}\vert_{x=x_0}=f'(x_0)\exp(-x_0)-f(x_0)\exp... |
2,174,766 | <p>If we omit the "0." part of diagonalization table in <a href="http://www.cs.virginia.edu/~lat7h/blog/posts/124.html" rel="nofollow noreferrer">http://www.cs.virginia.edu/~lat7h/blog/posts/124.html</a> we would get natural numbers. So does it mean that the diagonalization argument applies to natural numbers as well? ... | hmakholm left over Monica | 14,366 | <p>If you omit the "$0.$", the diagonal list you get out is just an infinite list of digits without any decimal point anywhere. Such a list does not represent any natural number -- or, indeed, any <em>real</em> number either.</p>
|
2,174,766 | <p>If we omit the "0." part of diagonalization table in <a href="http://www.cs.virginia.edu/~lat7h/blog/posts/124.html" rel="nofollow noreferrer">http://www.cs.virginia.edu/~lat7h/blog/posts/124.html</a> we would get natural numbers. So does it mean that the diagonalization argument applies to natural numbers as well? ... | Noah Schweber | 28,111 | <p>This is a common question, and the answer is: <strong>natural numbers only have finitely many digits</strong>.</p>
<p>The sequence "$1111111...$" does not correspond to a natural number. That is, your mistake is the claim that removing the "$0.$" yields a natural number. </p>
<hr>
<p>EDIT: The OP comments </p>
<... |
197,470 | <p>I'm new in Mathematica software, and I need help with a simple exercise. </p>
<p>Given the list</p>
<pre><code>list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
</code></pre>
<p>I need to make some rules to generate a new list with only three elements.</p>
<ul>
<li>first element is the product of its symbols </li>
<l... | Carl Woll | 45,431 | <p>This doesn't involve any rules, but you could use <a href="http://reference.wolfram.com/language/ref/Cases" rel="noreferrer"><code>Cases</code></a> to select the desired types, using <a href="http://reference.wolfram.com/language/ref/Through" rel="noreferrer"><code>Through</code></a> and <a href="http://reference.wo... |
197,470 | <p>I'm new in Mathematica software, and I need help with a simple exercise. </p>
<p>Given the list</p>
<pre><code>list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
</code></pre>
<p>I need to make some rules to generate a new list with only three elements.</p>
<ul>
<li>first element is the product of its symbols </li>
<l... | m_goldberg | 3,066 | <p>A way of doing it with <code>Reap</code> and <code>Sow</code> that would work in versions as old as V5, long before newfangled functions like <code>GroupBy</code> were introduced (V.10).</p>
<pre><code>list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
Reap[
Switch[#,
_Symbol, Sow[#, "s"],
_Integer, Sow[#, "i... |
197,470 | <p>I'm new in Mathematica software, and I need help with a simple exercise. </p>
<p>Given the list</p>
<pre><code>list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
</code></pre>
<p>I need to make some rules to generate a new list with only three elements.</p>
<ul>
<li>first element is the product of its symbols </li>
<l... | bill s | 1,783 | <p>One approach is to select the elements you want, and then mutliply them. </p>
<pre><code>Times @@@ {Select[list, Not[NumberQ[#]] &],
Select[list, IntegerQ[#] &],
Select[list, NumberQ[#] && Not[IntegerQ[#]] &]}
{x y z, 12, -67.7251}
</code></pre>
<p>The first one selects ... |
505,804 | <p>Let $X$ be a metric space and let $G$ be a group of homeomorphisms $X \to X$ acting on $X$. We say $G$'s action is properly discontinuous in case for every $x \in X$ and compact $K \subseteq X$, there are at most finitely many $g \in G$ such that $g(x) \in K$. Equivalently (and this is not hard to show), $G \cdot x$... | Yassine Guerboussa | 84,265 | <p>I taken a look in Svetlana Katok's "Fuchsian groups", and I'm really confused.</p>
<p>For instance, it is proved in that book that $G$ acts properly discontinuously on $X$ iff every point $x \in X$ has a neighborhood $U_x$ such that $U_x \cap gU_x$ is not empty for only finitely many $g \in G$. (I don't understand ... |
1,770,180 | <p>I was solving this limit from the Demidovich's book of exercises:
$$\lim_{n\to\infty} \frac{\sqrt[n]{\vphantom{\Large a}\, n!\,}}{n}$$</p>
<p>and I managed to get it to this state but then I got stuck:</p>
<p>$$e^{\frac{1}{n}(\log(n) + \log(n - 1) + ... + \log(1)) - \log(n)}$$</p>
<p>where $\log x$ is a natural l... | Mark Viola | 218,419 | <p>Here is a brute force approach.</p>
<p>$$\frac{\sqrt[n]{n!}}{n}=e^{\frac1n \log(n!)-\log(n)}$$</p>
<p>Now, note that the term $\log(n!)$ can be written as</p>
<p>$$\begin{align}
\log(n!)&=\sum_{k=1}^n \log(k)\\\\
&=n\left(\frac1n \sum_{k=1}^n\log(k/n)+\log(n)\right) \tag 1
\end{align}$$</p>
<p>Note that ... |
1,938,341 | <p>Prove, that $\forall n \in \mathbb{N}$ the following identity holds:
$$a^n - b^n = (a-b)\sum_{k=0}^{n-1} a^k b^{n-k-1}.$$</p>
<hr>
<p>For $n = 1$ we get $a^1 - b^1 = a - b$ on LHS, and on RHS we get $(a-b)\sum_{k=0}^{0}a^k b^{n-k-1} = a-b.$ Both sides are equal, so identity hold for $n = 1$.</p>
<hr>
<p>Assume t... | Piquito | 219,998 | <p>Using induction hypothesis for $n$, we have
$$a^{n+1}-b^{n+1}=a(a^n-b^n)+(a-b)b^n=(a-b)\left[a\sum_{i+j=n-1}a^ib^j\right]+(a-b)b^n$$ $$a^{n+1}-b^{n+1}=(a-b)\left[a^n+a^{n-1}b+.....+b^n\right]=(a-b)\sum_{i+j=n}a^ib^j$$</p>
|
3,313,655 | <p><span class="math-container">$$a^2 (a + 2 b) (a + b^2 + b)^{\frac{3}{2}}>b^3 (a^2 + a + b)^{\frac{3}{2}}$$</span></p>
<p>This is true with <span class="math-container">$a>b>0$</span>, <a href="https://www.wolframalpha.com/input/?i=a%5E2%20(a%20%2B%202%20b)%20(a%20%2B%20b%5E2%20%2B%20b)%5E(3%2F2)%3Eb%5E3%20... | Ross Millikan | 1,827 | <p>We prove the stronger result that comes from replacing <span class="math-container">$a+2b$</span> on the left by <span class="math-container">$a$</span></p>
<p>Note that for <span class="math-container">$0 \lt b \lt a, \frac 1b \gt \frac 1a$</span> and <span class="math-container">$\frac a{b^2} \gt \frac b{a^2}$</s... |
1,029,515 | <p>Show that $\gcd(x,x+2)$ is $1$ if $x$ is odd and $2$ if $x$ is even.</p>
<p>I am looking for a much simpler proof beside the one which I have posted.</p>
| albo | 22,610 | <p>Case 1:</p>
<p>If $x$ is odd, then $x$ can be written as $x=2n-1$ where $n\in \mathbb{Z}$. Now $(x,x+2)$ becomes $(2n-1,2n+1)$.</p>
<p>Let's find the $(2n-1,2n+1)$ by using Euclidean Algorithm.
$$ 2n+1=1\cdot(2n-1)+2$$
$$ 2n-1=2\cdot(n-1)+1$$
$$ n-1=1\cdot(n-1)+0$$
Clearly, $(2n-1,2n+1)=1.$</p>
<p>Case 2: (Trivia... |
434,543 | <p>I have an $m \ge5$ dimensional vector $x$. How i can construct a rotation matrix $A$ s.t: $x \cdot A$ would be a vector rotated by $\theta$ angle. This is trivial for 2 dimensions, but what for this case? For instance will it work if A is this:
$A = \begin{bmatrix}
cos \theta & -\sin\theta&0&0&0 \\
... | Lee Mosher | 26,501 | <p>A rotation in the plane $\mathbb{R}^2$ is determined by a fixed point and a rotation angle around that point. A rotation in 3-dimensional space $\mathbb{R}^3$ is determined by a fixed line and a rotation angle around that line. The common generalization of this is that a rotation in $m$-dimensional space is determin... |
60,486 | <p>I have noted several instances <a href="https://mathematica.stackexchange.com/a/51377/7167">here</a> and <a href="https://mathematica.stackexchange.com/a/55412/7167">here</a> where <code>ConvexHullMesh</code>, introduced in v10 has been used to greatly simplify some geometry problems. Determining the volume of a re... | Community | -1 | <pre><code>Area@RegionBoundary@ConvexHullMesh@pts
</code></pre>
<p>For example, consider a cube, which has volume 1 and surface area 6:</p>
<pre><code>pts = Tuples[{0, 1}, 3];
convexHull = ConvexHullMesh[pts];
convexHullSurface = RegionBoundary[convexHull];
RegionMeasure[convexHull, #] & /@ {0, 1, 2, 3}
(* {∞, ∞,... |
404,372 | <p>There are $7$ empty seats on a bus and four people get on. How many different ways can they be seated?</p>
<p>Would it be ${}^7C_{4}$ or ${}^7P_{4}$?</p>
| Trudie | 393,784 | <p>This is what I would do . . . especially to make it more visual:</p>
<p>Call the people A, B, C, D, and call the empty seats E, E, E.</p>
<p>See how many choices you have for each seat:</p>
<pre><code>choices: 7 6 5 4 3 2 1
</code></pre>
<p>Using the Fundamental Counting this gives 7! = 5040</p>
<p>But, s... |
127,645 | <p>In Lay's Linear Algebra and Its Applications textbook, he defines the matrix
$$A=\begin{bmatrix}
4 & 11 & 14\\8 & 7 & -2
\end{bmatrix}$$
and claims that the transformation $T(x)=A x$ maps the sphere to an ellipse.</p>
<p><a href="https://i.stack.imgur.com/grGkv.png" rel="nofollow noreferrer"><img sr... | Michael E2 | 4,999 | <p>Another way:</p>
<pre><code>mat = {{4, 11, 14}, {8, 7, -2}};
sph[{x_, y_, z_}] := x^2 + y^2 + z^2;
ContourPlot[sph[PseudoInverse[mat].{u, v}] == 1,
{u, -20, 20}, {v, -20, 20},
Epilog -> {Red, PointSize@Large, Point[{{18, 6}, {3, -9}}],
Line[{{{18, 6}, -{18, 6}}, {{3, -9}, -{3, -9}}}]},
PlotLabel -> S... |
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