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 |
|---|---|---|---|---|
139,487 | <p>I have a not-so-complicated piecewise cubic function, shown below as the yellow curve on the right. It's derivative is on the left; the blue lines are references. Please see the code below where I call it <strong>myF</strong> (and its derivative myf).</p>
<p>To my surprise, the integration with a parameter $u$ and ... | Himalaya Senapati | 57,938 | <p>Inside If, separate statements with a semicolon, not &&, the code does something. But, it is wrong, and I don't yet know why it is giving wrong output. </p>
<pre><code> IC = {x[0] == 1, x'[0] == 0, a[0] == 10, b[0] == 1/10};
sol = NDSolve[{x''[t] + b[t] x'[t] + a[t] x[t] == 0,
WhenEvent[x[t] == 0,
... |
238,627 | <p>Kindly consider it soft question as I am a software engineer.and I know only software but I have a doubt in my mind that there would be something like null in mathmatics as well. </p>
<p>If Mathematics <code>NULL</code> IS Equivalent to <code>ZERO</code>?</p>
| Dan Brumleve | 1,284 | <p>The distinction between the empty set $\emptyset$ and the number $0$ is similar to that between <code>NULL</code> and <code>ZERO</code>. For example, the set of real solutions (or informally "the solution") to $x^2=-1$ is $\emptyset$, but the solution to $x^2=0$ is $0$.</p>
|
4,617,031 | <p>How would I order <span class="math-container">$x = \sqrt{3}-1, y = \sqrt{5}-\sqrt{2}, z = 1+\sqrt{2} \ $</span> without approximating the irrational numbers? In fact, I would be interested in knowing a general way to solve such questions if there is one.</p>
<p>What I tried to so far, because they are all positive ... | boojum | 882,145 | <p>A visual demonstration of sorts:</p>
<p>Starting from <span class="math-container">$ \ x \ = \ \sqrt3 - 1 \ \ , \ $</span> we have<br />
<span class="math-container">$$ x \ + \ 1 \ \ = \ \ \sqrt3 \ \ \Rightarrow \ \ x^2 \ + \ 2x \ + \ 1 \ \ = \ \ 3 \ \ . \ $$</span> So <span class="math-container">$ \ \sqrt3 - 1 \... |
113,843 | <p>Here comes some sample data</p>
<pre><code>data = {{0, 0.7, 0.4}, {1, 0.831177, 0.51854}, {2, 1.11106, 0.463533},
{3, 1.84226, -0.642571}, {4, 0.677049, -0.327877},
{5, 0.77886, -0.451322}, {6, 0.965874, -0.508772},
{7, 1.34397, -0.202473}, {8, 1.01761, -0.717013},
{9, -0.0507992... | Jason B. | 9,490 | <p>Use a <code>Graph</code> with directed edges</p>
<pre><code>labels = Thread[
Range[12] -> (Placed[#, Above] & /@
Join[{Subscript[x, 0]}, Range[10], {Subscript[x, f]}])];
Graph[# \[DirectedEdge] # + 1 & /@ Range[11], VertexCoordinates -> d0,
VertexLabels -> labels, VertexStyle -> {1 -... |
3,130,195 | <p>I'm interested of finding a closed formula for the fundamental matrix to the system
<span class="math-container">$$\eqalign{
& y'(t) = a(t)z(t) \cr
& z'(t) = \delta a(t)y(t) \cr} $$</span>
<span class="math-container">$$(y(0),z(0)) = ({y_0},{z_0})$$</span>
where <span class="math-container">$\delta$</s... | SchrodingersCat | 278,967 | <p>The given equation <span class="math-container">$\displaystyle \sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=1$</span>
can be simplified as
<span class="math-container">$$\displaystyle \sqrt{\frac{x}{a}} = 1 - \sqrt{\frac{y}{b}}$$</span>
Squarring both sides,
<span class="math-container">$$\displaystyle \frac{x}{a}= 1 - 2\... |
4,073,821 | <p>If <span class="math-container">$\sum a_n$</span> converges, then does <span class="math-container">$\sum |a_n|$</span> converge as well? This is the same as "absolute convergence", where if <span class="math-container">$\sum a_n$</span> converges, then <span class="math-container">$\sum|a_n|$</span> might... | S.Farr | 683,130 | <p>Yes, you are correct. If <span class="math-container">$\sum a_n$</span> converges, <span class="math-container">$\sum |a_n|$</span> does not have to converge as well.</p>
|
19,373 | <p>I posted this question earlier today on the Mathematics site (<a href="https://math.stackexchange.com/q/3988907/96384">https://math.stackexchange.com/q/3988907/96384</a>), but was advised it would be better here.</p>
<p>I had a heated argument with someone online who claimed to be a school mathematics teacher of man... | leftaroundabout | 6,902 | <p>Just to play the <del>devil's</del> teacher's advocate here: one <em>can</em> make a point that rounding should be generally avoided but measurement uncertainty instead be expressed explicitly. Specifically, rounding errors should always be <em>much smaller</em> than measurement errors. Now, if you have a figure of ... |
1,561,716 | <p>I have spent the last couple weeks in my Fourier Analysis course to solve PDEs with the method of separations of variables. However, I have come up with something that annoys me and I can't really explain it. Let me show an example.</p>
<p>I have this problem here</p>
<p><span class="math-container">$$u_{xx}+u_{yy}=... | Xoque55 | 88,545 | <p>The awesome thing about hyberbolic (trig) functions is how they can be represented as sums of exponentials (and vice versa). Recall that <span class="math-container">$\cosh(u) = \dfrac{e^u + e^{-u}}{2}$</span> and <span class="math-container">$\sinh(u) = \dfrac{e^u - e^{-u}}{2}$</span>. It's just a matter of how you... |
1,561,716 | <p>I have spent the last couple weeks in my Fourier Analysis course to solve PDEs with the method of separations of variables. However, I have come up with something that annoys me and I can't really explain it. Let me show an example.</p>
<p>I have this problem here</p>
<p><span class="math-container">$$u_{xx}+u_{yy}=... | marty cohen | 13,079 | <p>I think that
the reason that
hyperbolic functions are used
is that they satisfy
many identities similar to the ones
satisfied by the trig functions.</p>
<p>An example is
$\cosh^2(x)-\sinh^2(x)
= 1
$.</p>
|
102,721 | <p>This is probably a very simple question, but I couldn't find a duplicate.</p>
<p>As everybody knows, <code>{x, y} + v</code> gives <code>{x + v, y + v}</code>. But if I intend <code>v</code> to represent a vector, for example if I am going to substitute <code>v -> {vx, vy}</code> in the future, then the result <... | eldo | 14,254 | <p>You can use <code>Inactivate</code></p>
<pre><code>pl = Inactivate[{x, y} + v, Plus]
</code></pre>
<p><a href="https://i.stack.imgur.com/b7qQz.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/b7qQz.jpg" alt="enter image description here"></a></p>
<p>As an indication to not being "active" the plus sign is... |
3,703,495 | <p><a href="https://en.m.wikipedia.org/wiki/Doomsday_argument" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Doomsday_argument</a></p>
<p>Suppose each new human born has the knowledge of the total number of humans born so far. So in their life, each human multiplies that number by 20 to get the upper bound... | Ryder Rude | 445,404 | <p>I think the problem is that 60B (current population) is not a randomly selected number out of the set {1,2,3,4.....N}, where N is the total number of humans ever going to exist.</p>
<p>Assuming we know nothing about <span class="math-container">$N$</span>, and each natural number value of <span class="math-containe... |
2,072,059 | <blockquote>
<p>Let $x$ be an element of the group $G$. Prove that $x^2 = 1$ <em>iff</em> $|x|$ is either $1$ or $2$</p>
</blockquote>
<p>Now I know that $G$ is implicitly a group under multiplication, therefore $G = (A, \ \cdot \ )$, for some underlying set $A$.</p>
<p>The problem I'm facing is that $A$ could be a... | q.Then | 222,237 | <p>You need not attempt to deduce properties of this specific group or set. This is true because of the group properties in general, not of $G = (A, \cdot )$. You may also want to read your textbook further, by conventional context in group theory, $|a|$ is by definition the order, which is the lowest positive integer ... |
951 | <p>I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them?</p>
| Michael Lugo | 173 | <p>One way that functions from C to C can be represented is to show the image of a grid. That is, plot the images of the lines x = constant and y = constant under your function, where z = x + yi. </p>
<p>Another is what Wikipedia calls <a href="http://en.wikipedia.org/wiki/Domain_coloring">domain coloring</a> (see <... |
951 | <p>I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them?</p>
| Steve Bosman | 167 | <p>Open two browser windows side-by-side and use <a href="http://www.wolframalpha.com/" rel="nofollow">wolfram alpha</a>. Recast your function f(z) as f(x+iy) and plot the <a href="http://bit.ly/bEZgsA" rel="nofollow">real part</a> in one window and the <a href="http://bit.ly/bfeLyS" rel="nofollow">imaginary part</a> i... |
65,059 | <p>I have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to transform any point $P$ ssfrom one to the other.</p>
<p>There is no third point, but there <em>is</em> an extra constra... | joriki | 6,622 | <p>The problem is to find a rotation matrix $R$ and a translation vector $\vec t$ such that</p>
<p>$$R\vec p_{1B}+\vec t=\vec p_{1A}\;,\tag1$$
$$R\vec p_{2B}+\vec t=\vec p_{2A}\;.\tag2$$</p>
<p>Subtracting these yields</p>
<p>$$R\left(\vec p_{1B}-\vec p_{2B}\right)=\vec p_{1A}-\vec p_{2A}\;.\tag3$$</p>
<p>Since the... |
3,340,093 | <p>Is the following statement true?</p>
<blockquote>
<p>Two real numbers a and b are equal iff for every ε > 0, |a − b| < ε.</p>
</blockquote>
<p>I got that if a and b are equal then |a-b|=0 which is less than ε.
But I'm not sure if the converse also holds.</p>
| drhab | 75,923 | <p>The statement is correct. Here is a proof of the converse.</p>
<p>Suppose that <span class="math-container">$a\neq b$</span>. </p>
<p>Then <span class="math-container">$\epsilon:=|a-b|>0$</span> but we do not have <span class="math-container">$|a-b|<\epsilon$</span>.</p>
|
1,757,556 | <p>Solve the recursion $p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$</p>
<p>$p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$</p>
<p>$= p + (1-2p)p_{n-1}$ I can see that this step simply rearranges the expression, but what's the point of it? What are we trying to accomplish here? Is it to combine the $p_{n-1}$ terms?</p>
<p... | Christopher Carl Heckman | 261,187 | <p>$$p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}= p + (1-2p)p_{n-1}$$
<em>I can see that this step simply rearranges the expression, but what's the point of it? What are we trying to accomplish here? Is it to combine the $p_{n-1}$ terms?</em></p>
<p>Yes.</p>
<p>$$p_n - \frac{1}{2} = (1-2p)\left(p_{n-1} - \frac{1}{2}\ri... |
1,199,746 | <p>Let $f : (−\infty,0) → \mathbb{R}$ be the function given by $f(x) = \frac{x}{|x|}$. Use the $\epsilon -\delta$ definition of a $\lim\limits_{x \to 0^-} f(x) = -1.$</p>
<p>Workings:</p>
<p>Informal Thinking:
We want $|f(x) - L| < \epsilon$</p>
<p>$\left|\frac{x}{|x|} - -1\right| < \epsilon$</p>
<p>$\left|\f... | abcdef | 223,597 | <p>Take some $\epsilon >0$. We need to schow there exists a $\delta >0$ such that $\forall x: 0>x>-\delta \Rightarrow |f(x)-(-1)| <\epsilon$. <br/>
Hint: If $x<0$ what does it say about $f(x)$ what is $|f(x)+1|$ in this case?</p>
|
3,700,938 | <p>I know that the equations are equivalent by doing the math with the same value for x, but I don't understand the rules for changing orders or operations.<br>
When it is not the first addition or subtraction happening in the equation, parentheses make the addition subtraction and vice versa? Are there any other rules... | Community | -1 | <p>The sign before the parenthesis distributes to the content:</p>
<p><span class="math-container">$$\begin{align}+(a+b)&=a+b\\-(a+b)&=-a-b\\+(a-b)&=a-b\\-(a-b)&=-a+b.\end{align}$$</span></p>
|
2,561,968 | <p>Is there a sequence of integers such that for ∀ k it contains an arithmetic subsequence of length k but it does have no infinitely long arithmetic subsequence?</p>
| Misha Lavrov | 383,078 | <p>There's two approaches to constructing a sequence like this.</p>
<p>First, we might try to space out blocks of the sequence sufficiently far. For example,
$$
1, \qquad 3,4, \qquad 9,10,11, \qquad 23,24,25,26, \qquad 53,54,55,56,57, \dots
$$
where, whenever a block ends at $x$, the next block begins at $2x+1$.</... |
337,655 | <p>I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.</p>
| 40 votes | 85,506 | <p>spitespike gave a good explanation already, but here is a another point of view.
Suppose $\|\cdot\|$ is a translation-invariant norm on some space of functions on $\mathbb R$. Translation-invariant means $\|f\|=\|f(\cdot-h)\|$ for all $h\in\mathbb R$. Consider the convolution $f*\varphi$ with some <a href="http://en... |
243,336 | <p>This question was asked to me in an interview, I still cannot think of its solution. Can anyone help? Following is the question:</p>
<blockquote>
<p>Given an infinite number of ropes of length $R$, you have to calculate the
minimum number of cuts required to make $N$ ropes of length $C$. You can
append ropes ... | P.. | 39,722 | <ul>
<li><p>Suppose there are $k,m\in \mathbb{N}$ such that $k\cdot R=m \cdot C$
and the minimum $m$ is $m_0$.</p>
<ol>
<li>For $N=r\cdot m_0, \ \ r \in
\mathbb{N}$ the minimum number of cuts is $r\cdot (m_0-1)$ (Make
$r$ ropes of length $m_0\cdot C$ and with $m_0-1$ cuts in each rope
you can produce $m_0$ ropes ... |
1,802,020 | <p>Let $S$ be a set such that if $A,B\in S$ then $A\cap B,A\triangle B\in S,$ where $\triangle$ denotes the symmetric difference operator. I would like to show that if $S$ contains $A$ and $B$, then it also contains $A\cup B, A\setminus B$.</p>
<p>The difference was easy to find, but I am not succeeding with the union... | joriki | 6,622 | <p>$$
\left(A\triangle B\right)\triangle\left(A\cap B\right)=A\cup B\;.
$$</p>
|
1,750,592 | <p>Can anyone help with this? I got a wrong answer. </p>
<p>Problem: Joe's French poodle, FooFoo, is tied to the corner of the barn which measures 40 x 30. FooFoo's rope is 50 long. In terms of π, over how many square feet can FooFoo wander?</p>
<p><a href="https://i.stack.imgur.com/Nmrc4.jpg" rel="nofollow noreferr... | user19405892 | 282,223 | <p><a href="https://i.stack.imgur.com/dc2ps.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dc2ps.png" alt="enter image description here"></a></p>
<p>The picture above shows the limits that FooFoo can turn. It is then easy to compute our answer </p>
<p>$$\dfrac{3}{4} 50^2 \pi+\dfrac{1}{4}10^2 \pi ... |
300,900 | <p>I am learning random matrix theory. Unfortunately I do not like combinatorics, and have never really been good at it. But I found that random matrix theory has heavily relied on combinatorics, particularly in finding the limiting spectral distribution, at least for symmetric matrices, where the moment method is the ... | Bazin | 21,907 | <p>Using Fourier transformation, your question is about the existence of a subspace $X$ of $L^2(\mathbb R^d)$ such that
$$
\lim_{t\rightarrow 0_+}\left\{\sup_{v\in X, \Vert v\Vert=1}\int(1-e^{-t\vert \xi\vert^2})\vert v(\xi)\vert^2 d\xi\right\}=0.
$$
Of course, as you noted, taking $X=L^2(\mathbb R^d)$ does not work si... |
4,623,590 | <blockquote>
<p>Find all positive integers <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <span class="math-container">$z$</span> satisfying <span class="math-container">$(x+y+z)(xy+yz+zx)=12xyz$</span>.</p>
</blockquote>
<p>Firstly, I think of <span class="math-container">$\textsf... | achille hui | 59,379 | <p>There are no positive integer solutions for the equation</p>
<p><span class="math-container">$$(x+y+z)(xy + yz + zx) = 12xyz$$</span></p>
<p>The way I do it might be an overkill. I transform the problem to one finding rational points on an elliptic curve. It turns out the elliptic curve has rank zero, so it has fini... |
4,623,590 | <blockquote>
<p>Find all positive integers <span class="math-container">$x$</span>, <span class="math-container">$y$</span> and <span class="math-container">$z$</span> satisfying <span class="math-container">$(x+y+z)(xy+yz+zx)=12xyz$</span>.</p>
</blockquote>
<p>Firstly, I think of <span class="math-container">$\textsf... | Steffen Jaeschke | 629,541 | <p>The given equation can be rewritten as:</p>
<p><span class="math-container">$x^2 y+x y^2+x^2 z-9 x y z+y^2 z+x z^2+y z^2 = 0$</span></p>
<p>The biggest problem of the task is that this is a volume and volumes of that kind are not visibly nice to humans. That has impact an the solutions methodologies. Most of them at... |
3,105,546 | <p>Suppose we have two functions <span class="math-container">$g:\mathbb{R}\rightarrow [0,1]$</span> and <span class="math-container">$f:\mathbb{R}\rightarrow [0,1]$</span> such that
<span class="math-container">$$\hspace{1cm}\lim\limits_{y\rightarrow -\infty}g(y)=\lim\limits_{y\rightarrow -\infty}f(y)=0 \tag{1}$$</spa... | feynhat | 359,886 | <p>Your assertion is true.</p>
<p><span class="math-container">$\lim_{y\to-\infty} \frac{f(y)}{g(y)} = 0$</span> means that for any <span class="math-container">$\epsilon > 0$</span>, <span class="math-container">$\exists M > 0$</span> such that <span class="math-container">$\left|\frac{f(y)}{g(y)}\right| < \... |
1,255,376 | <blockquote>
<p>For all $q \in \Bbb Q$ there exists $n \in \Bbb Z$ so that $q + n = 271$.</p>
</blockquote>
<p>This is true? Because both $q$ and $n$ are rational numbers and $271$ is an integer thus it's a rational number? </p>
<p>Also,</p>
<blockquote>
<p>For all $n \in \Bbb Z$ there exists $q \in \Bbb Q$ so t... | Jonathan Hebert | 120,932 | <p>No, it's not the same thing.</p>
<p>Note that $\frac{1}{2}$, a rational number, cannot be added to any integer to get 271. The first one is false.</p>
<p>The second is true. For any integer, there is some integer I can add to it to yield 271, and all integers are also rational numbers.</p>
|
1,507,076 | <p>I'm supposed to give a constructive proof of the following claim, but to get some hints on what to do I tried to prove the inverse and see where the proof breaks down. I have the following.</p>
<p>Claim: Define $X * Y = \{ xy \mid x \in X, y \in Y\}$. There is a ring $R$ with ideal $U$ such that $R / U$ is not clos... | TJ Combs | 285,776 | <p>Let $R=M_2(\mathbb{F}_2)$, the ring of $2 \times 2$ matrices with entries in $\mathbb{F}_2$.</p>
<p>Let $ U = \{ \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} , \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \}$, the set of upper triangular matrices of $R$.</p>
<p>I claim that $R / U$ is not closed un... |
1,842,365 | <p>I recently got acquainted with a theorem:</p>
<p>If $f(x)$ is a periodic function with period $P$, then $f(ax+b)$ is periodic with period $\dfrac{P}{a}$ , $a>0$.</p>
<p>I am having a difficulty in understanding this theorem. Does this theorem mean that $f(ax+b)=f(ax+b+ \dfrac{P}{a})$? </p>
<p>If the above mean... | Jochen | 38,982 | <p>In addition to the identical distribution independence is used to obtain that $(Z_1,Z_1+Z_2)$ and $(Z_2,Z_1+Z_2)$ are identically distributed which is needed to conclude $\mathbb E(Z_1 \mid Z_1+Z_2) = \mathbb E(Z_2 \mid Z_1+Z_2)$. </p>
<p>Edit. The distribution of $(Z_1,Z_1+Z_2)$ is the image (push forward) of
$\ma... |
24,550 | <p>Let <span class="math-container">$H$</span> and <span class="math-container">$K$</span> be affine group schemes over a field <span class="math-container">$k$</span> of characteristic zero. Let <span class="math-container">$\varphi:H\to Aut(K)$</span> be a group action. Then we can form the semi-direct product <span ... | Victor Ostrik | 4,158 | <p>This is just a reiteration of David's answer: in this situation the group $H$ acts on the category
$Rep(K)$ (an element $h\in H$ sends a representation of $K$ into its twist by $\phi(h)$; I should add that the notion of an action of affine group on abelian category is a bit subtle..). In this language, the category ... |
2,178,428 | <p><strong>Goal</strong>: I need help to intuitively understand division by a fraction. </p>
<p><strong>Background</strong>: I've read <a href="https://math.stackexchange.com/questions/248385/what-is-the-physical-explanation-of-a-division-by-a-fraction">this</a> and <a href="http://www.mathsisfun.com/division.html" re... | mlchristians | 681,917 | <p>First of all, a <em>fraction</em> is any real number of the form <span class="math-container">$\frac{a}{b}$</span>, where <span class="math-container">$a, b$</span> are integers and <span class="math-container">$b \neq 0$</span>. So to answer your second question in the title, <em>yes</em>---you can only divide by n... |
549,159 | <p>How to simplify this:</p>
<p>$$(5-\sqrt{3}) \sqrt{7+\frac{5\sqrt{3}}{2}}$$</p>
<p>Dont know how to minimize to 11.</p>
<p>Thanks in advance!</p>
| njguliyev | 90,209 | <p><em>Hint:</em> $7+\dfrac{5\sqrt{3}}{2} = \dfrac{1}{2^2}\left(5^2+\left(\sqrt{3}\right)^2+2\cdot5\cdot\sqrt{3}\right)$.</p>
|
549,159 | <p>How to simplify this:</p>
<p>$$(5-\sqrt{3}) \sqrt{7+\frac{5\sqrt{3}}{2}}$$</p>
<p>Dont know how to minimize to 11.</p>
<p>Thanks in advance!</p>
| André Nicolas | 6,312 | <p>It helps to know the answer! (One might suspect it, for example, from a numerical calculation. )</p>
<p>Equivalently, we need to ask whether
$$\frac{11}{5-\sqrt{3}}\overset{??}{=}\sqrt{7+\frac{5\sqrt{3}}{2}}.\tag{1}$$</p>
<p>To simplify the left side of (1), rationalize the denominator by multiplying top and bott... |
416,253 | <p>I am playing around a bit with <span class="math-container">$W^*$</span>-algebras, and I'm trying to come up with a definition for the <span class="math-container">$W^*$</span>-algebraic tensor product. Here is my first attempt:</p>
<p><a href="https://i.stack.imgur.com/WE797.png" rel="noreferrer"><img src="https://... | Nik Weaver | 23,141 | <p>You can see this is false by taking <span class="math-container">$N = \mathbb{C}$</span>. Then, given a von Neumann algebra <span class="math-container">$M$</span>, you are asking for a von Neumann algebra <span class="math-container">$\widetilde{M}$</span> and a weak* dense embedding <span class="math-container">$\... |
416,253 | <p>I am playing around a bit with <span class="math-container">$W^*$</span>-algebras, and I'm trying to come up with a definition for the <span class="math-container">$W^*$</span>-algebraic tensor product. Here is my first attempt:</p>
<p><a href="https://i.stack.imgur.com/WE797.png" rel="noreferrer"><img src="https://... | bathalf15320 | 159,073 | <p>Let me add that the basic flaw in your attempt is that you are working with a l.c. topology which is not complete. This is a problem common to the array of standard topologies which are used in von Neumann algebras--weak, strong, strong <span class="math-container">$\ast$</span>, ultraweak, ultrastrong (not, of cou... |
4,380,748 | <p>I suppose by contradiction that <span class="math-container">$x+a$</span> is a factor of <span class="math-container">$x^n-a^n$</span> for all odd <span class="math-container">$n$</span>. In particular for <span class="math-container">$n=1$</span>, we have that <span class="math-container">$x+a$</span> is a factor o... | Misha Lavrov | 383,078 | <p>Your analysis is good. Here is another argument.</p>
<p>The three statements contradict each other, so at least two of them are false, and there are at least two werewolves.</p>
<p>If there are three werewolves, all three statements must be false - but one of the statements <em>says</em> that there are three werewol... |
2,859,463 | <blockquote>
<p>Prove or disprove. All four vertices of every regular tetrahedron in $ \mathbb{R}^3$ have at least two irrational coordinates.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/hYrWv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hYrWv.png" alt="enter image description here"... | S. Piérard | 317,824 | <p>False.</p>
<p>Proof.
Here is one possible solution:</p>
<ul>
<li>vertex 1: (1 0 0)</li>
<li>vertex 2: (0 0 1)</li>
<li>vertex 3: (0 1 0)</li>
<li>vertex 4: (1 1 1)</li>
</ul>
<p>You can check that:</p>
<ul>
<li>all coordinates of all vertices are rational;</li>
<li>and that all distances from one vertex to another a... |
1,184,327 | <p>Find $\lim_{x\rightarrow 0}\exp(-1/(x^2+y^2))$.</p>
<p>I've tried multiplying the power out by the conjugate, which gives me a power of $y^2-x^2/x^4-y^4$, but I still can't figure out how to whittle this down. The most confusing thing to me is that the answer is supposedly $0$. How can $e^\text{anything}$ limit to ... | Julián Aguirre | 4,791 | <p>The equation is in separated variables:
$$
\frac{dP}{k\,P \left (1- \dfrac P {P_\infty} \right)-H}=dt.
$$</p>
|
1,184,327 | <p>Find $\lim_{x\rightarrow 0}\exp(-1/(x^2+y^2))$.</p>
<p>I've tried multiplying the power out by the conjugate, which gives me a power of $y^2-x^2/x^4-y^4$, but I still can't figure out how to whittle this down. The most confusing thing to me is that the answer is supposedly $0$. How can $e^\text{anything}$ limit to ... | boojum | 882,145 | <p>There are in fact three cases to consider for the logistic model with harvesting. Multiplying out the expression on the right side of the differential equation produces <span class="math-container">$$ \frac{dP}{dt} \ \ = \ \ -\frac{k}{M} \ P^2 \ + \ k P \ - \ H \ \ , $$</span></p>
<p>where I am replacing <span cla... |
265,635 | <p>I'm trying to create a user-defined function that computes the equivalent resistance of <span class="math-container">$n$</span> resistors in parallel.</p>
<p>As we know, such formula is:</p>
<p><span class="math-container">$R_\text{eq.p} = \dfrac{1}{\displaystyle\sum_{k=1}^{n} \dfrac{1}{R_k}} = \left( \displaystyle\... | bmf | 85,558 | <pre><code>Quit[]
Rp[list_] := 1/Sum[1/list[[k]], {k, Length[list]}];
test = {1, 2, 3}
</code></pre>
<p>and then either</p>
<pre><code>Rp[test]
</code></pre>
<p>or my preference</p>
<pre><code>Rp@test
</code></pre>
<p>to get</p>
<blockquote>
<p><code>6/11</code></p>
</blockquote>
<p>which is the right result; see</p>... |
265,635 | <p>I'm trying to create a user-defined function that computes the equivalent resistance of <span class="math-container">$n$</span> resistors in parallel.</p>
<p>As we know, such formula is:</p>
<p><span class="math-container">$R_\text{eq.p} = \dfrac{1}{\displaystyle\sum_{k=1}^{n} \dfrac{1}{R_k}} = \left( \displaystyle\... | John Doty | 27,989 | <p><code>Sum</code> is really for symbolic sums. It's clumsy here. I suggest:</p>
<pre><code>Rp[r_List] := 1/Total[1/r]
</code></pre>
|
1,691,605 | <p>I've been given the following definition:</p>
<blockquote>
<p>For a THMC with one step transition matrix $\mathbf{P}$, the row vector $\mathbf{\pi}$ with elements $(\pi_{i})_{i \in S}$ (where $S$ is the state space) is a stationary distribution iff $\mathbf{\pi \; P} = \mathbf{\pi}$</p>
</blockquote>
<p>However,... | Paramanand Singh | 72,031 | <p>The expression $$\frac{a^{a^x} - a^{x^a}} {a^x-x^a} $$ can be written as $$\frac{\exp(a^x\log a) - \exp(x^{a} \log a) }{a^x-x^a}=\exp(x^a\log a) \cdot\frac{\exp((a^x-x^a) \log a) - 1} {(a^x-x^a)\log a} \cdot\log a$$ and the above clearly tends to $\exp(a^a\log a) \log a=a^{a^a} \log a$. </p>
|
3,567,563 | <p>We have the polynomial <span class="math-container">$ f= X^4+X^3+X^2+X+2$</span> with <span class="math-container">$f\in \Bbb C[X] $</span>, it asks to determine the quotient of the division of the polynomial <span class="math-container">$f$</span> by the polynomial <span class="math-container">$g$</span>, <span cla... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>As a complement, you can use <em>synthetic division</em> by <span class="math-container">$X-\cos\alpha+i\sin\alpha=X-\mathrm e^{-i\alpha}$</span>. </p>
<p>You obtain for the quotient:
<span class="math-container">$$q(X)=X^3+(1+\mathrm e^{-i\alpha})X^2+(1+\mathrm e^{-i\alpha}+\mathrm ... |
2,417,542 | <p>$$\sum_{i,j}{n\brack i+j}\binom{i+j}i$$
Does this have a combinatorial interpretation? I don't see how to use Stirling numbers of the first kind in interpretations. I know that the answer is $(n+1)!$ , but the original question didn't provide it.</p>
| Dan Robertson | 243,692 | <p>The question is asking for the angle through which a vector (perpendicular to the axis of rotation) is rotated by the rotation matrix. Consider the effect of the matrix on a vector and work out how much it is rotated by. You should find the answer to be $\frac{3\pi}2$ anti-clockwise.</p>
<p>For more general matrice... |
1,218,354 | <p>I've read in some textbooks that $\vdash$ and $\vDash$ are symbols present only in metalanguage. From this, I infer that their use in object language is unacceptable.</p>
<p>I would like to know why. Can't we define them as relation symbols in a structure? Or introduce them in statements for the sake of formal proo... | tomasz | 30,222 | <p>You can use whatever symbols you want to denote whatever you want. For example, you can use ${\mathfrak w}_\$^7$ as a binary function symbol for addition of natural numbers. The utility of that is questionable, though.</p>
<p>More seriously, you certainly can introduce them if the structure in question is rich enou... |
4,035,300 | <p>If someone could just do a very basic walkthrough on how you would go about answering this question it would be greatly appreciated as I'm practising for an exam!</p>
<p>'''</p>
<p>When a company bids for contracts it estimates the probability of winning each contract is <span class="math-container">$0.18$</span>, i... | Luke Collins | 301,095 | <p>Let <span class="math-container">$\alpha$</span> denote the real root of <span class="math-container">$X^3+X^2-1$</span>. Then <span class="math-container">$\alpha^3 = 1-\alpha^2$</span>. If we plug in <span class="math-container">$\alpha^2$</span> into <span class="math-container">$X^3-X^2+2X-1$</span>, we obtain
<... |
2,373,175 | <p>It is probably an easy solution to this problem but I am either too overwhelmed already or not smart enough. Please help me out with this problem from the last year test</p>
<p><a href="https://i.stack.imgur.com/rzQWv.jpg" rel="nofollow noreferrer">Here is the problem</a></p>
| Adarsh Sharma | 471,162 | <p>First, let's have a look at how we can find a pattern to get the number of perfect squares in the range $R = 1, 2, 3 ... n$:</p>
<p>Let $k$ be the number of perfect squares in $R$. We need to find $k$.</p>
<p>Difference between consecutive perfect squares is $1 + 2i$, where $i = 1, 2, 3 ... k-1$. Upon subtle obser... |
3,229,789 | <p>For proving <span class="math-container">$$\frac {16}{\cos (4x)+7} =\frac{1}{\sin^4x +\cos^2x} +\frac{1}{\sin^2x +\cos^4x} $$</span></p>
<p>I tried to use that:</p>
<p><span class="math-container">\begin{align}
\sin^4 x +\cos^4 x&=\sin^4 x +2\sin^2x\cos^2 x+\cos^4 x - 2\sin^2x\cos^2 x\\
&=(\sin^2x+\cos^2 x... | lab bhattacharjee | 33,337 | <p><span class="math-container">$c^4-s^4=(c^2+s^2)(c^2-s^2)=c^2-s^2$</span></p>
<p><span class="math-container">$\iff c^4+s^2=c^2+s^4=P$</span>(say) where <span class="math-container">$c=\cos x,s=\sin x$</span></p>
<p><span class="math-container">$P+P=1+c^4+s^4=1+(c^2+s^2)^2-2c^2s^2=2-\dfrac{\sin^22x}2=2-\dfrac{1-\co... |
27 | <p>Burt Totaro has a result that for a certain class of algebraic stacks, having affine diagonal is equivalent to the stabilizers at closed points begin affine. Is there an example of this equivalence failing in general?</p>
| David Rydh | 40 | <p>The answer is yes since a fiberwise condition (such as affine stabilizers) does not imply a global condition (such as affine diagonal) without extra hypotheses (such as having the resolution property). Think of quasi-finite+proper <=> finite.</p>
<p>There are (non-separated) <em>schemes</em> with non-affine diag... |
355,489 | <p>What are suggestions for reducing the transmission rate of the current epidemics?</p>
<p>In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:</p>
<p><em>If members o... | fedja | 1,131 | <p>Here is another model from Gil Kalai last idea:</p>
<p>We have the moving compressible fluid in which the particles can teleportate, diffuse, and organize the motion, while contamination can only diffuse. Each particle can be contaminated or not. You see contamination levels but not individual contamination. How to... |
355,489 | <p>What are suggestions for reducing the transmission rate of the current epidemics?</p>
<p>In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:</p>
<p><em>If members o... | Martin Rubey | 3,032 | <p>I have one rather practical answer, which does not answer the question in the body, but possibly in the title.</p>
<p>I was very surprised to learn that a (small) hospital in my home town was doing their scheduling ("rostering" might be the correct term, but I don't know) by hand.</p>
<p>They were clever enough to... |
2,467,228 | <p>I attempted to start with the $L_p$ norm and raise it to the power of $p$ but got stuck because I realized that I have no idea how to eliminate the integrand.</p>
<hr>
<p><strong>$L_p$ norm:</strong>
$||f||_p = ||f||_{L_p[a,b]} = (\int_{a}^{b}~|f(x)|^p~~dx)^{\frac{1}{p}}$</p>
<p>$\\$</p>
<hr>
<p><strong>C-norm:... | Learnmore | 294,365 | <p>Note that $(\int _a^b|f_n-f|^p)\le \max_{x\in [a,b]} |f_n(x)-f(x)|^p\times (b-a)^p$</p>
<p>Choose $\epsilon<1$. </p>
<p>Now since $f_n\to f$ in $C-$ Norm </p>
<p>$\implies ||f_n-f||=\max_{x\in [a,b]} |f_n(x)-f(x)|<\epsilon(<1)\implies |f_n(x)-f(x)|<\epsilon$</p>
<p>$\implies |f_n(x)-f(x)|^p<\epsi... |
2,477,107 | <p>Okay so I have to prove this. I can write that if 2 divides n and 7 divides n, then there must be integers k and m such that
$2*k=n$
and
$7*m=n$</p>
<p>So $14*k*m=n^2$</p>
<p>But what to do after that?</p>
<p>If I say that then 14 divides $n^2$, I get bit of a circular argument, but if I write that n divides $14*... | nonuser | 463,553 | <p>You can say much more: <strong>Say $a$ and $b$ are relatively prime. If $a|n$ and $b|n$ then $ab|n$.</strong> </p>
<p>Proof: Since $a|n$ we can write $n=ak$. Now since $b|ak$ we have, by Euclid lemma $b|k$, so $k=bl$. Thus $n=abl$ and so $ab|n$. </p>
|
2,477,107 | <p>Okay so I have to prove this. I can write that if 2 divides n and 7 divides n, then there must be integers k and m such that
$2*k=n$
and
$7*m=n$</p>
<p>So $14*k*m=n^2$</p>
<p>But what to do after that?</p>
<p>If I say that then 14 divides $n^2$, I get bit of a circular argument, but if I write that n divides $14*... | orangeskid | 168,051 | <p>Write $n = 7 m$. If $m$ were odd, $7m$ would also be odd, contradiction. So $m$ is even, $m=2k$, and $n = 14k$.</p>
|
71,822 | <p><em>I have moved this question here from MSE, because I did not receive any answers as of yet over there.</em></p>
<p>I know that there are statements that are neither provable nor disprovable within some set of axioms, and I also know that such statements are called undecidable. Please allow me to call these state... | Ron Maimon | 14,689 | <p>In questions like these, I think it is important to say right off the bat: Theorems are only undecidable in a given axiom system. For all we know, all meaningful mathematical questions, including all questions about the halting/nonhalting of any computer program, are settled from a strong enough axiom of higher infi... |
347,214 | <p>We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog starts running toward the woman from (0,0) they are both running at constant speed, the dogs path is curved and we w... | johannesvalks | 155,865 | <p>You ask:</p>
<p><em>At this point a dog starts running toward the woman from (0,0) they are both running at constant speed, the dogs path is curved and we wish to find the length of the curve until the dog reaches the woman.</em></p>
<hr>
<p>Path length is given by</p>
<p>$$
s = \int_0^T ds = \int_0^T \frac{ds}{... |
2,875 | <p>I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group <code>G</code> being <code>SL(2, R)</code>, can be completely described and that there is a discrete and continuous part of the spectrum of <code>L^2(G)</code>.</p>
<ol>
<li>How are those ... | Rob Harron | 1,021 | <p>I strongly recommend you read the article "Representations of semisimple Lie groups" by Knapp and Trapa in the park city/ias proceedings "Representation theory of Lie groups". It's a very nice introduction to the problem of describing the "unitary dual" (which is what you are asking about) which focusses on SL(2,R).... |
2,138,241 | <p>I tried to prove $$\lim_{x\to \infty}\frac 1x = 0$$
I started as thus
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2}$$
Applying <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hospital's Rule</a>
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2... | Simply Beautiful Art | 272,831 | <p>I too tried the same thing:</p>
<p>$$\lim_{x\to\infty}x=\lim_{x\to\infty}\frac{x^2}x\stackrel{L'H}=2\lim_{x\to\infty}x$$</p>
<p>Thus,</p>
<p>$$\lim_{x\to\infty}x=2\lim_{x\to\infty}x$$</p>
<p>And as you have said,</p>
<p>$$\lim_{x\to\infty}x=0$$</p>
<p>QED (?)</p>
|
2,138,241 | <p>I tried to prove $$\lim_{x\to \infty}\frac 1x = 0$$
I started as thus
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2}$$
Applying <a href="https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule" rel="nofollow noreferrer">L'Hospital's Rule</a>
$$\lim_{x\to \infty}\frac 1x=\lim_{x\to \infty}\frac x{x^2... | Barry Cipra | 86,747 | <p>What you have (very cleverly!) shown is that <em>if the limit $\lim_{x\to\infty}{1\over x}$ exists</em>, then, by L'Hopital, it can only equal $0$. Simply Beautiful Art's answer establishes the same result for $\lim_{x\to\infty}x$. The difference is, in your case the limit actually does exist, while in SBA's case ... |
228,224 | <p>I am faced with the following expression</p>
<p><span class="math-container">$$
-\frac{(1 - a x^{2})^{b/2}}{b} {{}_2F_1} (1, \frac{b}{2}; \frac{c}{2}; 1 - a x^{2}) = - p t
$$</span></p>
<p>where <span class="math-container">$ a, b, c, p $</span> are constant values. Also, <span class="math-container">$ {{}_2F_1} $</... | Αλέξανδρος Ζεγγ | 12,924 | <p>I think <a href="https://reference.wolfram.com/language/ref/ContourPlot.html" rel="nofollow noreferrer"><code>ContourPlot</code></a> works for your case because it can find the curves represented by equations. I do not know what the values of your those constants so I make wild choices in the code below:</p>
<pre><c... |
3,890,064 | <p>When Max is 8 m from a lamp post which is 6 m high his shadow is 2 m long. When Max is 3 m from the lamp post, what is the length of his shadow?</p>
| poetasis | 546,655 | <p>When he is <span class="math-container">$8m$</span> from the lamp post, the end of the shadow is <span class="math-container">$10m$</span> from the lamp post meaning we have two legs of a triangle with <span class="math-container">$6$</span> and <span class="math-container">$10$</span> respectively.</p>
<p>By propor... |
3,890,064 | <p>When Max is 8 m from a lamp post which is 6 m high his shadow is 2 m long. When Max is 3 m from the lamp post, what is the length of his shadow?</p>
| Toby Mak | 285,313 | <p>Alternately, from the gradient of <span class="math-container">$AH'$</span> being <span class="math-container">$\frac{h-6}{3}$</span>, we can work out <span class="math-container">$h$</span> using similarity: <span class="math-container">$\text{small $\Delta$ ratio} = \text{big $\Delta$ ratio} \Rightarrow \frac{h}{2... |
1,465,167 | <p>Given 2 points $p_1=(x_1^1, x_2^1, ..., x_n^1)$ and $p_2=(x_1^2, x_2^2, ..., x_n^2)$ in $n$-dimensional Euclidean space, how would you define the straight-line from $p_1$ to $p_2$ with these 2 points being the endpoints of the line.</p>
<p>There are a few of things I've been completely unable to figure out. One is ... | Arpit Kansal | 175,006 | <p>Note that for $a\gt 0$,
$$a^b = e^{b\ln a}.$$
So
$$\lim_{n\to\infty}x^{1/n} = \lim_{n\to\infty}e^{\frac{1}{n}\ln x}.$$
Since the exponential is continuous, we have
$$\lim_{n\to\infty}e^{\frac{1}{n}\ln x} = e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln x}.$$</p>
<p>Can you conclude now?</p>
<p><strong>OR</strong> Use ... |
1,465,167 | <p>Given 2 points $p_1=(x_1^1, x_2^1, ..., x_n^1)$ and $p_2=(x_1^2, x_2^2, ..., x_n^2)$ in $n$-dimensional Euclidean space, how would you define the straight-line from $p_1$ to $p_2$ with these 2 points being the endpoints of the line.</p>
<p>There are a few of things I've been completely unable to figure out. One is ... | Claudius | 218,931 | <p>First assume $x\ge 1$. Write $x_n = \sqrt[n]{x} - 1$. Notice that $x_n\ge0$, because $\sqrt[n]x\ge1$ if $x\ge1$. Using Bernoulli inequality we get
$$
x = (1+x_n)^n \ge 1 + n\cdot x_n
$$
for all $n$. This yields $0\le x_n\le \frac{x-1}n$ for all $n$. This shows $\lim_{n\to\infty} x_n = 0$ or $\lim_{n\to\infty} \sqrt... |
2,155,740 | <p>This is a homework problem for which I think I've missed the point or have incorrectly done the proof (or both). There are two parts to the problem: Let $(a_n)$ be a sequence with $a_n \ge 0,$ for all $n$.</p>
<p>Part 1:<br>
Suppose that $a_n \rightarrow 0$. Show that $\sqrt{a_n}\rightarrow 0$. I recognized that... | Squirtle | 29,507 | <p>For the first one it is NOT true that $\sqrt{x_n} < x_n$ but the limit is still zero all the same. Are you allowed to use the fact that you can "pass a limit through a continuous function"? If so... then
$$\lim_{n\to \infty} g(x_n) = g(\lim_{n\to \infty} x_n) = g(0)$$
where $g(x)=\sqrt{x}$ and we know that $g(... |
2,570,008 | <p>For what value of a will this equation have only one real root: </p>
<p>$$(2a−5)x^2−2(a−1)x+3=0$$</p>
<p>Note: $x$ is a variable</p>
<p>If found that $a=4$ works, but there seems to be another solution. Any help?</p>
| Tanuj | 438,329 | <p>A quadratic equation has equal roots or repeated roots , when it fulfills the condition that </p>
<p>$D=0$ </p>
<p>Here $D$ is the discriminant of the quadratic given by $D=b^2-4a\cdot c$</p>
<p>Here , $a$ is the coefficient of $x^2$ , $b$ is the coefficient of $x$ and $c$ is the constant term .</p>
<p>Can you t... |
4,157,472 | <p>Let <span class="math-container">$V$</span> be a vector space, and <span class="math-container">$T:V→V$</span> a linear transformation such that:</p>
<p><span class="math-container">$T(2v_1 + 3v_2) = -5v_1 - 4v_2$</span> and <span class="math-container">$T(3v_1 + 5v_2) = 3v_1 -2v_2$</span></p>
<p>Then:</p>
<p>T(v<su... | herb steinberg | 501,262 | <p>Since <span class="math-container">$||v||=\alpha$</span>, there is an <span class="math-container">$n$</span> such that <span class="math-container">$|v_n|=\alpha$</span> Let <span class="math-container">$u_n=-\alpha$</span> and <span class="math-container">$u_k=0$</span> for <span class="math-container">$k\ne n$</... |
279,064 | <p>I find this question interesting, but need to get it out of my system: is the space of connections (modulo gauge) on a compact four-manifold paracompact, in the Sobolev topology?</p>
<p>If so, I believe it would admit partitions of unity, which would surely make life easier in gauge theory. But I haven't seen the e... | Tobias Diez | 17,047 | <p>Yes, the space of gauge orbits of connections is paracompact (even when you the use Fréchet topology).</p>
<p>First, the space of all connections is paracompact since it is an affine space modelled on a nuclear Fréchet space (and/or it is metrisable). Narasimhan & Ramadas (Geometry of SU(2) Gauge Fields) showed... |
2,889,929 | <p>I have following function:</p>
<p>$$f(x)=x^2\cdot\left({\sin{\frac 1 x}}\right)^2$$</p>
<p>I want to find the limit of the function for $x\rightarrow0^\pm$. First I analyze $\frac 1 x$:</p>
<ul>
<li>$\frac {1}{x}\rightarrow +\infty$ for $x\rightarrow0^+$</li>
</ul>
<p>but the $\sin$ of infinity does not exist. T... | prog_SAHIL | 307,383 | <p><strong>HINT:</strong></p>
<p>$\text{P(Required)= P(A and B hits, C misses)+P(A and C hits, B misses) +}$</p>
<p>$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{P(C and B hits, A misses)+P(A,B,C all hit)}$</p>
<p>$\text{P(A)=A hits
, P(A')=A misses= 1- P(A)}$</p>
<p>$\text{P(B)=B hits
, P(B')=B misses= 1- P(B... |
1,925,182 | <p>I am trying to understand the representation theory of $GL_2(\mathbb{R})$, and I have a few questions. I know that it is generally inadvisable to put a few questions in the same topic, but since they are quite inter-related I will give it a shot like this.</p>
<p>I can understand the idea of classifying irreducible... | Kimball | 11,323 | <p>Let me just give a very basic answer to 1 (which may be too simple for you, I don't know). In general, the basic way to study representations of a group $G$ is to try to reduce the problem to the case of simpler subgroups $H$. For $G=GL(n)$, one of the simplest kinds of subgroups you can consider are the (proper) ... |
1,925,182 | <p>I am trying to understand the representation theory of $GL_2(\mathbb{R})$, and I have a few questions. I know that it is generally inadvisable to put a few questions in the same topic, but since they are quite inter-related I will give it a shot like this.</p>
<p>I can understand the idea of classifying irreducible... | tracing | 200,415 | <p>Regarding your question 1.,
there is an important underlying fact at work here, which you may not know:</p>
<p>It is a theorem of Casselman (the <em>subrepresentation theorem</em>) that for a real reductive Lie group $G$, any irreducible admissible $(\mathfrak g, K)$-module can be embedded as a subrepresentation of... |
1,925,182 | <p>I am trying to understand the representation theory of $GL_2(\mathbb{R})$, and I have a few questions. I know that it is generally inadvisable to put a few questions in the same topic, but since they are quite inter-related I will give it a shot like this.</p>
<p>I can understand the idea of classifying irreducible... | Devadatta Hegde | 628,020 | <p>For almost all that I am going to say below the reference is V.S. Varadarajan, An introduction to Harmonic analysis on semisimple Lie groups. </p>
<p>The concept of parabolic induction is due to Gelfand and Naimark from their monograph "Unitary representations of classical groups" (~1950). They used the principal s... |
2,936,608 | <blockquote>
<p>Let <span class="math-container">$a_n$</span> where <span class="math-container">$n \in \mathbb {N}$</span> be a sequence of rational numbers converging to <span class="math-container">$a$</span>. Suppose <span class="math-container">$a \neq 0$</span>, for <span class="math-container">$k = 1, 2, ...$<... | 1123581321 | 482,390 | <p><strong>Hint</strong></p>
<p>Because <span class="math-container">$a\not=0$</span> (let's say <span class="math-container">$a>0$</span>), show that <span class="math-container">$\exists n_0\in\mathbb{N}: \forall n\geq n_0\quad a_n>0$</span> </p>
|
3,855,736 | <p>Mathematics is not my primary discipline, but I know enough about both it and academics in general to know that many to most mathematical researchers do what they do because they enjoy doing it. This would seem to make "recreational mathematics" a rhetorical tautology, yet the term is used as if it were a ... | Robert Israel | 8,508 | <p>Recreational math tends to comprise topics that are fun, and easily accessible to people without a lot of specialized training, but, for one reason or another, are not usually topics of serious mathematical research. Factors in that lack of serious mathematical research may include lack of applications of the topic... |
4,380,274 | <p>Indefinite integral is pretty easy to solve, I did it by substitution and I'm pretty sure it can be done relatively easy via integration by parts.
The problem are boundaries.</p>
<p>After substitution <span class="math-container">$arcsin x=t$</span> we get</p>
<p><span class="math-container">$$\int_0^\frac{\pi}{2} \... | Svyatoslav | 869,237 | <p>We can also try to evaluate the integral simply integrating by part.
<span class="math-container">$$I=\int_0^1 \frac{x-\arcsin x}{x^3}dx=-\frac{1}{2}\int_0^1 (x-\arcsin x)d\Big(\frac{1}{x^2}\Big)$$</span>
<span class="math-container">$$=-\frac{x-\arcsin x}{2x^2}\Big|_0^1+\frac{1}{2}\int_0^1\frac{dx}{x^2}\Big(1-\frac... |
4,146,858 | <blockquote>
<p>Q) For every twice differentiable function <span class="math-container">$f:\mathbb{R}\longrightarrow [-2,2] $</span> with <span class="math-container">$[f(0)]^2+[f'(0)]^2=85$</span> , which of the following statement(s) is(are) TRUE?</p>
</blockquote>
<blockquote>
<p>(A) There exists <span class="math-c... | Shinrin-Yoku | 789,929 | <p>You can derive the quotient rule by considering <span class="math-container">$\dfrac{f(x)}{g(x)}$</span> as <span class="math-container">$f(x)\cdot(g(x))^{-1}$</span> and then using the product and chain rule.<br><br>The quotient rule gives a <em>formula</em> (under the right conditions) for evaluating the derivat... |
4,146,858 | <blockquote>
<p>Q) For every twice differentiable function <span class="math-container">$f:\mathbb{R}\longrightarrow [-2,2] $</span> with <span class="math-container">$[f(0)]^2+[f'(0)]^2=85$</span> , which of the following statement(s) is(are) TRUE?</p>
</blockquote>
<blockquote>
<p>(A) There exists <span class="math-c... | marty cohen | 13,079 | <p>I like using
<span class="math-container">$(\ln(f))'
=\dfrac{f'}{f}
$</span>
for general
products and quotients.</p>
<p>If
<span class="math-container">$f = \dfrac{\prod u_k}{\prod v_k}
$</span>
then
<span class="math-container">$\ln(f)
= \sum \ln(u_k)-\sum \ln(v_k)
$</span>
so
<span class="math-container">$(\ln(f)... |
2,984,918 | <p>How can I prove this? </p>
<blockquote>
<p>Prove that for any two positive integers <span class="math-container">$a,b$</span> there are two positive integers <span class="math-container">$x,y$</span> satisfying the following equation:
<span class="math-container">$$\binom{x+y}{2}=ax+by$$</span></p>
</blockquote... | Wolfram | 397,123 | <p>Case <span class="math-container">$a=b$</span> is trivial after dividing both sides by <span class="math-container">$x+y$</span>, <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are symmetric in the problem, so in the following text WLOG I assume <span class="math-container">$a &l... |
90,812 | <p>How do I use the Edmonds-Karp algorithm to calculate the maximum flow? I don't understand this algorithm $100\%$. What I need to know is about flow with minus arrow. Here is my graph: </p>
<p><img src="https://i.stack.imgur.com/nep5F.jpg" alt="the graph">. </p>
<p>Our $1-6-11-12$, the flow is $4$. On the next iter... | Gerry Myerson | 8,269 | <p>You are correct that the next step is to use the path $P$ given by 1-3-5-11-6-8-10-12, but since the flow from 6 to 11 is only 1, you can only send 1 unit of flow along $P$ (which is the solution rattle found). The flow in each edge must be non-negative at all times. </p>
|
4,498,296 | <p>Is there any subtle way to compute the following integral?</p>
<p><span class="math-container">$$\int \frac{\sqrt{u^2+1}}{u^2-1}~ \mathrm{d}u$$</span></p>
<p>The solution i had in mind was substituting <span class="math-container">$u=\tan (\theta)$</span>,then after a few calculations the integral became <span class... | Accelerator | 1,065,595 | <p>Here are a couple of ideas.</p>
<p>By using <span class="math-container">$\cos^{2}\left(\theta\right)\ =\ 1-\sin^{2}\left(\theta\right)$</span> and <span class="math-container">$\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$</span>, we can rewrite one of your integrals as</p>
<p><span class="math-contain... |
3,861,319 | <p>I have 3 features <code>age</code>, <code>income</code> and <code>rating</code>.</p>
<p>In case of age I have 3 buckets.</p>
<p>for income I have 4 buckets.</p>
<p>and for rating I have 2 buckets.</p>
<p>If one could filter data where a person could select 1 or more than 1 bucket from each feature what would be the ... | Parcly Taxel | 357,390 | <p>The outermost quantifier is <span class="math-container">$\exists$</span>, so you're supposed to come up with one <span class="math-container">$x$</span> that satisfies <span class="math-container">$\forall y(2\mid5x+4y)$</span>. We can take <span class="math-container">$x=0$</span>. I will let you continue from her... |
1,566,215 | <p>Can someone explain to me the difference between joint probability distribution and conditional probability distribution?</p>
| BLAZE | 144,533 | <p>I will answer this question with a typical example as it might be easier to understand the difference, but first for some definitions:</p>
<p><code>Definition 1:</code></p>
<p>Given two independent events $A$ and $B$ $$P(A \space\mathrm{and}\space B)=P(A)\times P(B)=\space$$The probability of event $A$ <em>and</em... |
1,179,195 | <p>Good day everyone. </p>
<p>I need to know automata theory. Can you advice me the best way to study math?
What themes will I need to know to understand automata theory. What a sequence of study? What level will I need to study intermediate themes? Maybe can you say something yet, what can help me quickly learn autom... | tomasz | 30,222 | <p>A contraction mapping can never have more than one fixed point: if $a,b$ are both fixed points, then $d(a,b)=d(f(a),f(b))\leq \lambda d(a,b)$. This is only possible if $d(a,b)=0$.</p>
<p>On the other hand, by Banach fixed-point theorem, any contraction mapping of a complete metric space into itself has a fixed poin... |
123,587 | <p>I always wonder how many inbuilt functions does Mathematica have (of course you can google for it) and how they are connected with each other! So I tried this (v10.1).</p>
<pre><code>SetDirectory[$InstallationDirectory<>"/Documentation/English/System/ReferencePages/Symbols"]
comms = FileNames[];
ncomms = Leng... | Wjx | 6,084 | <p>There is a way for you to find out the relations without internet, and I think it fit your need more as they are actually extracted directly from those "See Also" links.</p>
<p>I'm currently using v11, 5000+ Functions, it takes about 7 min before it finishes the main evaluation, and another 10 min or so for plottin... |
1,349,654 | <p>Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$</p>
<p>Note :By mathematica,the result is :
$\frac{Gamma\left(\frac1 4\right)Gamma\left(\frac5 4\right)}{\sqrt{\pi}}-\sqrt{2} Hypergeometric2F1\left(\frac1 4,\frac3 4,\frac5 4,\frac1 4\right).$
and ... | robjohn | 13,854 | <p>$$
\begin{align}
\int_{\pi/4}^{\pi/2}\frac{\sqrt{\tan(x)}}{\sin(x)}\,\mathrm{d}x
&=\int_{\pi/4}^{\pi/2}\frac{\sqrt{\tan(x)}}{\sin(x)}\frac{\mathrm{d}\tan(x)}{\sec^2(x)}\tag{1}\\
&=\int_1^\infty\frac{\sqrt{u}}{\frac{u}{\sqrt{1+u^2}}}\frac{\mathrm{d}u}{1+u^2}\tag{2}\\
&=\frac12\int_0^\infty\frac{\mathrm{d}... |
2,065,639 | <p>$\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$</p>
<p>$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$ </p>
<p>Instead of expanding the integrand, or doing integration by part, is there any faster way to compute this kind of integral?</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may use the fact that
$$
4uv=(u+v)^2-(u-v)^2 \tag1
$$ giving
$$
\begin{align}
4\int_a^b (x-a)(x-b)\:dx=4\int_a^b \left(x-\frac{a+b}2\right)^2\:dx-\int_a^b (b-a)^2\:dx \tag2
\end{align}
$$ which is <em>easier</em> to evaluate.</p>
<p>Then, one has $$
\begin{align}
&4\int_a^b (x-a)(x-b)... |
2,065,639 | <p>$\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$</p>
<p>$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$ </p>
<p>Instead of expanding the integrand, or doing integration by part, is there any faster way to compute this kind of integral?</p>
| Community | -1 | <p>If you're looking for elementary methods there's not going to be a faster one than expanding the integral or using integration by parts. For another solution you could remember the formula, and I think the following <strong>heuristic</strong> is a good mnemonic for that:</p>
<p>We know that $\int_a^b(x-a)(x-b)\,dx$... |
185,867 | <p>I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).</p>
<p>So, I'm just curious why don't expert mathematicians use ZF+GCH
instead of ZF+AC(ZFC).</p>
| GH from MO | 11,919 | <p>Warning: naive answer follows. I think AC is more intuitively true than GCH. Also, I believe, AC is more indispensable for mathematics than GCH. For example, there are many equivalent forms of AC within ZF that come up naturally, such as the Zorn lemma or the well-ordering principle.</p>
|
387,202 | <p>What is the smallest 3-regular graph to have a unique perfect matching?</p>
<p>With a large enough number of nodes, it is possible for a 3-regular graph to have no perfect matching (example can be seen in this question <a href="https://mathoverflow.net/questions/98385/cubic-graphs-without-a-perfect-matching-and-a-ve... | PickupSticks | 176,479 | <p>There is no such graph.</p>
<p>I have some reading to do as my intuition is off, but the details and a related question are available here:
<a href="https://mathoverflow.net/questions/349421/does-there-exist-an-r-regular-graph-r%E2%89%A52-with-a-unique-maximum-matching">Does there exist an r-regular graph (r≥2) with... |
947,626 | <p>What are the conditions under which the center of a group will have a cyclic subgroup? (with proof, of course)</p>
| Timbuc | 118,527 | <p>Always, certainly: if $\;Z(G)=1\;$ there's nothing to prove, and otherwise take any $\;z\in Z(G)\;$ and then $\;\langle z\rangle\;$ is a cyclic subgroup of the center.</p>
<p>In fact, we don't even need the first case $\;Z(G)=1\;$ .</p>
|
1,476,982 | <p>I'm trying to understand why the volume of a parallelepiped whos sides are $s,u,w$ is $ V = s \cdot(u \times w)$.</p>
<p>Even the units of measurement don't add up. The length of the vectors $s,u,w$ is measured in centimeters, the volume is measured in cubic cm.</p>
<p>$u\times w$ is a vector. It is a vector that ... | Narasimham | 95,860 | <p>The cross product has dimensions of area $ |A|\cdot|B| \sin \theta $. Just because it is a vector, it need not be a straight line always. We say an <em>area</em> is directed quantity. An area or even a volume can be a vector. </p>
<p>How do we accept angles, angular velocities, angular accelerations as vectors? For... |
4,173 | <p>I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead -- </p>
<p>Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?</p>
<p>$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$</p>
<p>Where $\mathbf{v}$ is a $d\times ... | ronaf | 1,547 | <p>a slight retouching of unkz's idea is the following orthogonal basis for $v^\perp$:</p>
<p>$$u_1 = (1,-1,0,\dots,0),\
u_2 = (1,1,-2,0,\dots,0),\
\dots, \
u_{d-1} = (1,\dots,1,-d+1)$$</p>
<p>in this form it is pretty easy to see that the $\{u_i\}$ are in $v^\perp$ and are mutually orthogonal.</p>
<p>i don't qu... |
2,000,013 | <p>On an NFA, how can the empty set ∅ and {ϵ} be considered regular languages? Does it make sense that a machine that accepts no symbols or a machine that takes the empty symbol exist? I could think of a machine (laptop) that is in off mode, where no entries (symbols) are accepted, but, is that of interest? or could it... | Brian M. Scott | 12,042 | <p>Let $M$ be a machine with any alphabet, one state, $q_0$, that is the initial state, and no transitions. If $q_0$ is not an acceptor state, $M$ accepts $\varnothing$, and if $q_0$ is an acceptor state, $M$ accepts $\Sigma^*$, where $\Sigma$ is the alphabet.</p>
<p>Now let $M$ have any alphabet and two states, the i... |
108,594 | <p>I would like to know if one can weaken conditions of Proposition 2.8 in
<a href="http://www.jmilne.org/math/xnotes/CA.pdf" rel="nofollow">http://www.jmilne.org/math/xnotes/CA.pdf</a></p>
<p>The proposition says that if an ideal $a$ in a ring $A$ is contained in the union of ideals $p_1,...,p_r$ with $p_2,...,p_r$ ... | Mohan | 9,502 | <p>For an infinite base field $k$, you are right. Assume $a\subset\cup_{i=1}^r a_i$ and choose $r$ the least such. Then we can pick $f_i\in a$ not in the union of the the $a_j$'s with $j\neq i$. Consider the $k$-vector space $V$ generated by the $f_i$'s and let $V_i=a_i\cap V$. Then $V\subset \cup V_i$, but $V_i$s are ... |
4,380,124 | <p>I understand that the double integral is
<a href="https://i.stack.imgur.com/29e8B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/29e8B.png" alt="enter image description here" /></a>
However what confuses me is when I try to visualize why this formula only accounts for the region inside the bounds... | CatsOnAir | 974,003 | <p><a href="https://i.stack.imgur.com/PkLBr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PkLBr.png" alt="enter image description here" /></a>
The equation works for non rectangular regions because dy or dx can be a function of x and y itself meaning the height and width of your rectangles dydx cha... |
4,380,124 | <p>I understand that the double integral is
<a href="https://i.stack.imgur.com/29e8B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/29e8B.png" alt="enter image description here" /></a>
However what confuses me is when I try to visualize why this formula only accounts for the region inside the bounds... | Andrew D. Hwang | 86,418 | <p>To illustrate Hans' (+1) answer:</p>
<p><a href="https://i.stack.imgur.com/E8J4X.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/E8J4X.png" alt="Riemann sums for a non-rectangular plane region" /></a></p>
<p>Extending the integrand by <span class="math-container">$0$</span> outside the triangle of... |
1,528,501 | <p>Let $P(n)$ be a property for all $n \geq 1$. For the phrase "there is some $N \geq 1$ such that $P(n)$ holds for all $n \geq N$" there are some suggestive, convenient abbreviations such as "$P(n)$ holds for large $n$" or "$P(n)$ holds eventually" and so on.</p>
<p>I wonder if there is in literature a like abbreviat... | BrianO | 277,043 | <p>In some contexts (set theory, order theory, point set topology, though probably never in probability) you can say <em>cofinally</em>, or <em>cofinally many</em>, <em>cofinally often</em>. Given a preorder $(A,\preceq)$, a subset $X\subseteq A$ is <em>cofinal</em> in $A \Leftrightarrow$ for every $a\in A$ there is $x... |
2,679,153 | <p>Let $A\in\mathbb{R}^{n\times n}$ be a generic <em>lower triangular</em> matrix and let $P\in\mathbb{R}^{n\times n}$ be a symmetric <em>positive definite</em> matrix.</p>
<blockquote>
<p><strong>True or false.</strong> Does $AP + PA^\top=0$ imply $AP=0$? </p>
</blockquote>
| Doug M | 317,162 | <p>The ratio of sides is 1:1.6</p>
<p>or lengths $x$ and $1.6x$ </p>
<p>Area $= \frac 12 bh\\ 0.8 x^2 = 51.2\\x^2=64\\x = 8$</p>
<p>Two of the sides then are $8$ and $12.8$</p>
<p>The 3rd side you can find using the Pythagorean theorem.</p>
|
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