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 |
|---|---|---|---|---|
160,818 | <p>Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite groups are very complex and to understand them requires advanced theory on groups, rings, field and Galois Theory. Know a ... | Pete L. Clark | 299 | <p>Yes. Let $G = \prod_{i=1}^{\infty} \mathbb{Z}/2\mathbb{Z}$ be the direct product of (countably) infinitely many copies of the cyclic group of order $2$. This profinite group, sometimes called (well, by me at least) the <em>Bernoulli group</em>, occurs naturally in probability theory. As a topological space it is ... |
4,056,073 | <p>I need help with this task, if anyone had a similar problem it would help me !</p>
<p>The task is: Determine the type of interruption at the point x = 0 for the function</p>
<p><span class="math-container">$$f(x)=2^{-\frac{1}{x^{2}}}$$</span></p>
<p>I did:</p>
<p><span class="math-container">$$L=\lim_{x\to 0^{-}} 2^... | jjagmath | 571,433 | <p>What is happening here is just a consequence that an infinite set and a proper subset can be in bijective correspondence. That's an well known fact about infinite sets. And it is a paradox in the sense that it is anti-intuitive, but not in the sense that it leads to a contradiction.</p>
|
4,056,073 | <p>I need help with this task, if anyone had a similar problem it would help me !</p>
<p>The task is: Determine the type of interruption at the point x = 0 for the function</p>
<p><span class="math-container">$$f(x)=2^{-\frac{1}{x^{2}}}$$</span></p>
<p>I did:</p>
<p><span class="math-container">$$L=\lim_{x\to 0^{-}} 2^... | Paul Frost | 349,785 | <p>The core of your paradox seems to be that you claim that in some sense there are "more" (ordered) tuples <span class="math-container">$(c_{n-1},\dots,c_0)$</span> than (unordered) sets <span class="math-container">$\{r_{n-1},\dots,r_0\}$</span>. This is not true for infinite sets like <span class="math-con... |
698,743 | <blockquote>
<p>Let the real coefficient polynomials
$$f(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$$
$$g(x)=b_{m}x^m+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}$$
where $a_{n}b_{m}\neq 0,n\ge 1,m\ge 1$, and let
$$g_{t}(x)=b_{m}x^m+(b_{m-1}+t)x^{m-1}+\cdots+(b_{1}+t^{m-1})x+(b_{0}+t^m).$$
Show that</p>
<p>... | Calvin Lin | 54,563 | <p>Here is a series of steps, which seem mostly true to me.</p>
<p><strong>Fact:</strong> $f(x)$ has at most $n$ distinct roots.</p>
<p><strong>Claim:</strong> There exists a map $G: [0,1] \rightarrow (\alpha_1, \alpha_2, \ldots, \alpha_m)$ which is differentiable in each coordinate, and $ \alpha_i$ are roots of $ g... |
48,629 | <p>Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and Riemannian geometry. Therefore, I'm looking for the relations between the two areas. I should also mention, that I'm interes... | Daniel Loughran | 5,101 | <p>Any non-singular complex variety $V$ of dimension $n$ (in either affine space or projective space) can be endowed with the structure of a complex manifold of dimension $n$. Moreover as a submanifold of a Kahler manifold, it will also be Kahler.</p>
<p>The passage to a real manifold can be slightly subtle, as there ... |
88,199 | <p>Is there a function that would satisfy the following conditions?:</p>
<p>$\forall x \in X, x = f(f(x))$ and $x \not= f(x)$,</p>
<p>where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in \{0,1,\ldots,255\}$.</p>
<p>I would like to find a function that will have as an input RGB color values (tri... | Dilip Sarwate | 15,941 | <p>If you are doing these calculations on a computer and can get away from
insisting on interpreting $x_1$, $x_2$, $x_3$ as integers in the range
$[0,255]$, consider thinking of $x_1$, $x_2$, $x_3$ as eight-bit <em>bytes</em>
or vectors of length $8$ over $\mathbb F_2$ to be a bit more formal about it.
Then, for any t... |
1,643,201 | <p>The spectrum-functor
$$
\operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set}
$$
sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a prime ideal of R}\}$ and a morpshim $f:S\to R$ to the map $\operatorname{Spec}(R)\to \operatorname{Spec}(S)$ with $\m... | Travis Willse | 155,629 | <p>Yes, except that:</p>
<ol>
<li>I would replace $\frac{100}{7}$ with $\frac{99}{7}$ (to see why this is important consider the analogous question asking for how many multiples of $7$ there are in $\{700, \ldots, 999\}$), and</li>
<li>the quotients aren't quite right as written, but we can repair them with floor nota... |
1,386,682 | <p>How do you calculate $\lim_{z\to0} \frac{\bar{z}^2}{z}$?</p>
<p>I tried $$\lim_{z\to0} \frac{\bar{z}^2}{z}=\lim_{\overset{x\to0}{y\to0}}\frac{(x-iy)^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}=\lim_{\overset{x\to0}{y\to0}}\frac{x^2-2xyi-y^2}{x+iy}\cdot\frac{x-iy}{x-iy} \\ \\ =\lim_{\overset{x\to... | Empty | 174,970 | <p><strong>From your calculation :</strong></p>
<p>$$=\lim_{\overset{x\to0}{y\to0}}\frac{(x^2-2xyi-y^2)(x-iy)}{x^2+y^2}$$</p>
<p>$$=\lim_{(x,y)\to (0,0)}\frac{x^3-3xy^2}{x^2+y^2}-i\lim_{(x,y)\to (0,0)}\frac{3x^2y-y^3}{x^2+y^2}$$</p>
<p>From here, show that both the limits are <strong>zero</strong> by changing polar ... |
2,113,777 | <p>I have the following IVP (Initial value problem, Cauchy-Problem), and I do not know how to solve this.</p>
<p>$$y'=e^{-x}-\frac{y}{x} \qquad \qquad y(1)=2$$</p>
<p>I hope you can help me, cause I really do not know how to start.</p>
<p>Thank you! :)</p>
| Mercy King | 23,304 | <p>First, one need to solve the homogeneous differential equation associated to the original equation, that is:
$$
y'=-\dfrac{y}{x}
$$
We have
\begin{eqnarray}
\dfrac{y'}{y}&=&-\dfrac1x\\
\int\dfrac{dy}{y}&=&-\int\dfrac{dx}{x}\\
\ln|y|&=&\ln|C|-\ln|x|\\
y&=&\dfrac{A}{x}.
\end{eqnarray}
N... |
3,916,092 | <blockquote>
<p>A ball rotates at a rate <strong><span class="math-container">$r$</span></strong> rotations per second and simultaneously revolves around a stationary point <strong><span class="math-container">$O$</span></strong> at a rate <strong><span class="math-container">$R$</span></strong> revolutions per second... | Prakasan S.P | 1,142,365 | <p>Given:
Frequency of rotation = 'r' times/sec
Frequency of revolution = 'R' times/sec.<br />
(the rotation and the revolution are in the same sense), hence
Relative frequency of rotation = (r-R) times/sec.<br />
As per this relative frequency of rotation, the centre of the ball and the point of its surface come in ... |
3,357,841 | <p>In the diagram (which is not drawn to scale) the small triangles each have the area shown. Find the area of the shaded quadrilateral.</p>
<p><a href="https://i.stack.imgur.com/DK8sn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DK8sn.png" alt="enter image description here"></a></p>
| Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/RZvDU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RZvDU.png" alt="enter image description here"></a></p>
<p>It can be deduced that the area <span class="math-container">$[FED] = 7\cdot 4/14=2$</span>. </p>
<p>Furthermore, examine the ratios below,</p>
<p><... |
2,317,625 | <p>How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)</p>
<p>Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative ge... | NaOH | 398,864 | <p>There is still some hope in taking the first minus the second in this case:
$$6-2\sqrt{3} - (3\sqrt{2}-2) = 8 - (2\sqrt{3} + 3\sqrt{2})$$</p>
<p>So now the question boils down to if the expression with the square root exceeds $8$. We know that $8^{2} = 64$ and:
$$(2\sqrt{3}+3\sqrt{2})^{2}=4*3+2(2\sqrt{3})(3\sqrt{2}... |
192,125 | <p>Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<</p>
| Amzoti | 38,771 | <p>Questions:</p>
<ol>
<li><p>Is the problem written correctly.</p></li>
<li><p>Are there restrictions on x?</p></li>
</ol>
<p>Something does not seem right in the problem as posed.</p>
<p>Hint: Plot the left hand side and then plot the right hand side and see what it looks like.</p>
|
583,030 | <p>I have to show that the following series convergences:</p>
<p>$$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$</p>
<p>I have tried the following:</p>
<ul>
<li>The alternating series test cannot be applied, since $\frac{2+(-1)^n}{n+1}$ is not monotonically decreasing.</li>
<li>I tried splitting up the series in ... | 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{\ds}[1]{\displayst... |
111,183 | <p><img src="https://i.stack.imgur.com/1MOuo.jpg" alt="Problem">
<img src="https://i.stack.imgur.com/bdRXi.png" alt="New Solution"></p>
<p>I believe I have gotten all of the ways now - thanks for the hints below Yun, Andre Nicolas, and Gerry Myerson. If anyone could confirm my answer (I feel there should be more poss... | Lin | 348,610 | <p>2 + 4 + 4 + 4 + 4 + 8 also equals 26. I didn't see that on the list.</p>
|
351,846 | <p>The following problem was on a math competition that I participated in at my school about a month ago: </p>
<blockquote>
<p>Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.</p>
</blockquote>
<p>I will outline my proof below. I think it has some holes. My approach to the problem was to... | Christian Blatter | 1,303 | <p>The function
$$f(x):=\cos(\sin x)-\sin(\cos x)$$
is even and $2\pi$-periodic; therefore it suffices to consider $x\in[0,\pi]$. When $x=0$ or $x\in\bigl[{\pi\over2},\pi\bigr]$ then obviously $f(x)>0$. Finally, when $0<x<{\pi\over2}$ then $\cos x$ and $\sin x$ both lie in the interval $\ ]0,1[\ \subset\ ]0,{\... |
670,781 | <p>Given $y=x\sqrt{a+bx^2}$. the tangent to $y$ at point $x=\sqrt5$ is also passing at point </p>
<p>$(3\sqrt5,\sqrt5).$ the area between $y=x\sqrt{a+bx^2}$ and the $x$-axis is equal to $18$.</p>
<p>Need to find $a,b$.</p>
<p>I have tried to differentiate and eliminate but failed...</p>
| 5xum | 112,884 | <p>Hint: For a curve $y=f(x)$, the slope of the tangent at $x_0$ is given by $f'(x_0)$. This means that you have a tangent with the formula $y=f'(x_0)\cdot x + n$ for some real $n$. Plugging in the value $x=x_0$ and $y=f(x_0)$ can then help you calculate $n$ in terms of $a$ and $b$, giving you one of the equations for ... |
3,077,629 | <p>Assume <span class="math-container">$(f_i)_{i\in I}$</span> is an orthonormal/orthogonal system in an (complex) inner product space. Does <span class="math-container">$$\sum_{i\in I}\langle f_i,f\rangle f_i$$</span> always converges for any <span class="math-container">$f$</span> (may not to <span class="math-contai... | boojum | 882,145 | <p>The Lagrange equations are not all consistent, as we obtain
<span class="math-container">$$ \frac{-2x}{(x^2+y^2+z^2)^2} \ = \ 2x· \lambda \ \ \Rightarrow \ \ -2x \ · \left[ \ \lambda \ + \ \frac{1}{(x^2+y^2+z^2)^2} \ \right] \ = \ 0 \ \ , $$</span>
<span class="math-container">$$ \frac{-2y}{(x^2+y^2+z^2)^2} \ = \ -2... |
2,336,535 | <p>I have a limit:</p>
<p>$$\lim_{(x,y)\rightarrow(0,0)} \frac{x^3+y^3}{x^4+y^2}$$</p>
<p>I need to show that it doesn't equal 0.</p>
<p>Since the power of $x$ is 3 and 4 down it seems like that part could go to $0$ but the power of $y$ is 3 and 2 down so that seems like it's going to $\infty$.</p>
<p>I wonder if t... | Rom | 446,025 | <p><em>Graphically</em> <strong>f</strong> looks like this : <a href="https://i.stack.imgur.com/23eVc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/23eVc.png" alt="pic1"></a>
You can see that the value limit $(x,y) \rightarrow (0,0)$ <strong>depends</strong> on the path we take. </p>
<p>$\\
\\ ... |
3,561,664 | <p>I did part of this question but am stuck and don't know how to continue</p>
<p>I let <span class="math-container">$x= 2k +1$</span></p>
<p>Also noticed that <span class="math-container">$x^3+x = x(x^2+1)$</span></p>
<p>therefore
<span class="math-container">$4m+2 = 2k+1((2k+1)^2+1)$</span></p>
<p>I simplified th... | fleablood | 280,126 | <p>You didn't do your math right Let <span class="math-container">$x= 2k + 1$</span> so <span class="math-container">$x(x^2 + 1) = (2k+1)((2k+1)^2 + 1)=$</span></p>
<p><span class="math-container">$(2k+1)((2k+1)^2 + 1) = 8k^3 + 12k^2 + 8k +2$</span> so you had a typo.</p>
<p>But it's easier to do</p>
<p><span class... |
246,071 | <p>How do I solve the following equation?</p>
<p>$$x^2 + 10 = 15$$</p>
<p>Here's how I think this should be solved.
\begin{align*}
x^2 + 10 - 10 & = 15 - 10 \\
x^2 & = 15 - 10 \\
x^2 & = 5 \\
x & = \sqrt{5}
\end{align*}
I was thinking that the square root of 5 is iregular repeating 2.23606797749979 nu... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ \ $ If $\rm\ \ f(n) =\, \dfrac{5}6 - \dfrac{1}{n\!+\!1} - \dfrac{1}{n\!+\!2} - \,\cdots\, - \dfrac{1}{2n\!+\!1}\ $ then $$\rm f(n\!+\!1)-f(n) = \dfrac{1}{n\!+\!1}-\dfrac{1}{2n\!+\!2}-\dfrac{1}{2n\!+\!3} = \dfrac{1}{2(n\!+\!1)(2n\!+\!3)} > 0$$</p>
<p>thus $\rm\:f(n)\:$ is increasing, so ... |
3,703,981 | <p>If we consider an equation <span class="math-container">$x=2x^2,$</span> we find that the values of <span class="math-container">$x$</span> that solve this equation are <span class="math-container">$0$</span> and <span class="math-container">$1/2$</span>. Now, if we differentiate this equation on both sides with res... | Batominovski | 72,152 | <p>Alternatively, if <span class="math-container">$p$</span> is a prime natural number that divides <span class="math-container">$x^2+xy+y^2$</span>, then <span class="math-container">$$(2x+y)^2+3y^2=4(x^2+xy+y^2)\equiv 0\pmod{p}\,.$$</span> Thus, either <span class="math-container">$p$</span> divides both <span class... |
3,274,172 | <p>Let <span class="math-container">$X$</span> a compact set. Prove that if every connected component is open then the number of components is finite.</p>
<p>Ok, <span class="math-container">$X = \bigcup C(x)$</span> where <span class="math-container">$C(x)$</span> is the connected component of <span class="math-conta... | Paulo Mourão | 673,659 | <p><span class="math-container">$\textbf{Hint:}$</span> Argue by contradiction: assume <span class="math-container">$X$</span> has infinite <span class="math-container">$\textit{open}$</span> connected components and prove this implies that <span class="math-container">$X$</span> is not compact.</p>
<p>Recall that <sp... |
3,757,213 | <blockquote>
<p>Prove that the maximum area of a rectangle inscribed in an ellipse <span class="math-container">$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$</span> is <span class="math-container">$2ab$</span>.</p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Equation of ellipse: <span class="math-container">$\dfrac{x^... | heropup | 118,193 | <p>Consider the coordinate transformation <span class="math-container">$$(u,v) = (x/a, y/b),$$</span> which maps the ellipse <span class="math-container">$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$</span> to the unit circle <span class="math-container">$$u^2 + v^2 = 1.$$</span> Such a transformation preserves the ratios... |
1,040,932 | <p>I have a system of congruence equations</p>
<p>$$
\begin{cases}
x \equiv 17 \pmod{15} \\
x \equiv 14 \pmod{33}
\end{cases}
$$</p>
<p>I need to investigate the system and see if they've got any solutions.</p>
<p>I know that I should use the Chinese remainder theorem "in a reverse order" so I think I should split e... | Landon Carter | 136,523 | <p>$x\equiv2(\mod15)\implies x=15k+2$ for some integer $k$.
Similarly $x=33m+14$.</p>
<p>Thus $15k+2=33m+14\implies15k-33m=12\implies5k-11m=4$.</p>
<p>$\gcd(5,11)=1$ so the equation above has solutions.</p>
|
1,462,379 | <p>I have been given the task to compute $\int_{-1}^1 \sqrt{1-x^2} dx$ by means of calculus. We got the hint to substitute $x=\sin u$, but that only seems to make things more complicated:</p>
<p>$$\int_{-1}^1 \sqrt{1-x^2}dx = \int_{\arcsin-1}^{\arcsin1} \sqrt{1-\sin^2u} \frac{d\arcsin u}{du} du = \int_{\arcsin-1}^{\ar... | Hosein Rahnama | 267,844 | <p>This is how it works:</p>
<p>$$\left\{ \matrix{
x = \sin (u) \hfill \cr
x = 1\, \to \,\,\,\,\,\,\,u = {\pi \over 2} \hfill \cr
x = - 1 \to u = - {\pi \over 2} \hfill \cr} \right.$$</p>
<p>and also</p>
<p>$$\sqrt {1 - {x^2}} = \sqrt {1 - {{\sin }^2}(u)} = \sqrt {{{\cos }^2}(u)} = \left| {\cos (u)} ... |
22,839 | <p>Is it possible to have the text generated by <code>PlotLabel</code> (or any other function) aligned to the left side of the plot instead of in the center?</p>
| Mr.Wizard | 121 | <p>Not directly that I am aware of. You could of course fake it with spacing:</p>
<pre><code>Plot[Sinc[x], {x, 0, 9}, PlotLabel -> Row@{"Text", Spacer[300]}]
</code></pre>
<p>Simpler may be use <code>Column</code> but it is not part of the <code>Graphics</code> object itself:</p>
<pre><code>Column@{"Text", Plot[... |
22,839 | <p>Is it possible to have the text generated by <code>PlotLabel</code> (or any other function) aligned to the left side of the plot instead of in the center?</p>
| DavidC | 173 | <p>You can use Epilog to insert the plot label wherever you wish within the <code>PlotRange</code>. </p>
<p><em>Edit</em>: There is a drawback to this approach. Because the title will be within the graph region, there is a possibility that the title will be appear over part of the graph of the function.</p>
<pre><co... |
256,666 | <p>Let $X$ be a set. Suppose $\beta$ is a basis for the topology $\tau_\beta$ of $X$. Since each base element is open (with respect to $\tau_\beta$) we have that $$B\in \beta\Rightarrow B\in \tau_\beta.$$ Thus, $\beta\subset \tau_\beta$. </p>
<p>However, since $\beta$ is a union of base elements (I assume a set can al... | Tom Oldfield | 45,760 | <p>$\beta$ is a set of sets, and is not the union of it's elements .$\beta$ is not a union of basis elements, but a union of <em>sets containing</em> basis elements. Consider for example $A = \{[0,1], [1,2]\}$. It should be clear that $A \not= [0,2]$.</p>
<p>However, in general it is possible for a set to be both a s... |
3,438,653 | <p>I have this thing written on my notes: let <span class="math-container">${x}, {y}\in\mathbb R^n$</span> be two distinct points, then the set <span class="math-container">$$\{ \lambda x + (1-\lambda){y}\;\lvert\; \lambda \in [0,1] \}$$</span> contains all the points on the line segment that connects <span class="math... | Matematleta | 138,929 | <p>We use your (correct) definition of the line <span class="math-container">$\vec x(t)= \vec x_0 + t\vec{v}$</span>, where <span class="math-container">$\vec v$</span> marks the direction of <span class="math-container">$l$</span>. Note that <span class="math-container">$\textit{any}$</span> vector that has the same d... |
2,386,689 | <p>I know I can find solution for These equations using subtraction</p>
<p>$3x+2y=14.....(\text{i})$</p>
<p>$4x+3y=20.....(\text{ii})$</p>
<p>My question is can I divide $(\text{i})$ by $(\text{ii})$ to get values of $x$ and $y$ ?</p>
| Bill Dubuque | 242 | <p>Yes, eliminatiion by division always works. But you also need to separately consider the case that the eliminated variable $=0.\,$ Let's compare eliminating vs. isolating $x$ then dividing for the system</p>
<p>$$\begin{align} a x + b y = c\\
d x + e y = f \end{align}$$</p>
<p>Eliminating $x$ we obtain the equati... |
92,983 | <p><strong>Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a <em>topological</em> triangulation with complexity $O(n)$?</strong></p>
<p>Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ triangular facets, possibly with positive genus. A <em>topological</em> triangulation of $P$... | Misha | 21,684 | <p>Correction: My "answer" below has a fatal mistake, but the idea still could be useful, although, seems to be hard to implement. One could try to use the fact that complexity of a topological triangulation is bounded from below by hyperbolic volume and that for alternating knots/links hyperbolic volume is $O(t)$ wher... |
155,455 | <p>I want to find the maximum of a function (f) over a variable (t). The function is huge and it's not possible to maximize f(t) directly. So I want to create f inside a Table and then find the highest value over a small range of t. How can I add the steps to construct f into a Table? It seems "/." is not working. </p... | bbgodfrey | 1,063 | <p>The code contains numerous syntax errors. It could, for instance, be written as</p>
<pre><code>v = Table[(TR1 + TR2) /. Flatten@Solve[TR1 - t == 0, TR1] /. TR2 -> t + 1, {t, 1, 3, 1}]
tstar = Max[v]
(* {3, 5, 7} *)
(* 7 *)
</code></pre>
|
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| The Mathemagician | 3,546 | <p>Grubb's recent <em>Distributions And Operators</em> is supposed to be quite good. </p>
<p>There's also the recommended reference work, Strichartz, R. (1994), <em>A Guide to Distribution Theory and Fourier Transforms</em> </p>
<p>The comprehensive treatise on the subject-although quite old now-is Gel'fand, I.M.; Sh... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| abcd | 4,582 | <p>If you want a comparatively elementary approach to distribustion theory with applications to integral equations and difference equation no books come close to <strong>Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications</strong> by A H Zemanian. another plus is it i... |
20,314 | <p>Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.</p>
| wildildildlife | 10,729 | <p>I'd like to point out a recent (Birkhäuser Cornerstones) <a href="http://www.springer.com/birkhauser/mathematics/book/978-0-8176-4672-1?cm_mmc=Google-_-Book%2520Search-_-Springer-_-0" rel="noreferrer">textbook</a> on Distribution Theory by Duistermaat and Kolk. </p>
<blockquote>
<p>The present text has evolved fr... |
4,291,864 | <p>I have the following equation:</p>
<p><span class="math-container">$y=\frac{3x}{x^{2}+1}$</span></p>
<p>and I want to obtain x in terms of y, so far what I have done is the following:</p>
<p><span class="math-container">$3x=y(x^{2}+1)$</span></p>
<p><span class="math-container">$3x=x^{2}y+y$</span></p>
<p><span clas... | Mr.Gandalf Sauron | 683,801 | <p><span class="math-container">$(a,b)$</span> itself is infinite.</p>
<p>Any open set is of the form <span class="math-container">$\mathbb{R}\setminus F$</span> where <span class="math-container">$F$</span> is some finite set.</p>
<p>So you can see that any open set only excludes atmost finitely many elements of <span... |
842,266 | <p>I have a tiny little doubt related to one proof given in Ahlfors' textbook. I'll copy the statement and the first part of the proof, which is the part where my doubt lies on.</p>
<p><strong>Statement</strong>
The stereographic projection transforms every straight line in the $z$-plane into a circle on $S$ which pas... | Joshua P. Swanson | 86,777 | <p>Here's another way to say it. Suppose $n \cdot v = n \cdot v_0$ is the equation of a plane (so $v = (x, y, z)$ are the variables, $n$ is a non-zero vector, and $v_0$ is an arbitrary point on the plane). It turns out $n \cdot v_0$ is $|n|$ times the minimum distance from the origin to the plane (times $\pm 1$).</p>
... |
3,369,069 | <p>Let <span class="math-container">$l_1$</span> and <span class="math-container">$l_2$</span> be two distributions in disjoint variables <span class="math-container">$x_1, ..., x_n$</span> and <span class="math-container">$y_1, ..., y_m$</span>. Then it is said to be possible to define a product distribution.</p>
<p>... | reuns | 276,986 | <p>Concretely for <span class="math-container">$\phi \in C^\infty_c(\Bbb{R}^2)$</span> take <span class="math-container">$\varphi_n,\psi_n \in C^\infty_c(\Bbb{R})$</span> such that <span class="math-container">$$\sum_{n=1}^N \varphi_n(x)\psi_n(y)\to \phi$$</span> in test function topology.</p>
<p>Then for <span class=... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | Hagen von Eitzen | 39,174 | <p>Clearly, <span class="math-container">$x=y$</span> is impossible as <span class="math-container">$x\mid 2x^n=x^n+y^n=z^n$</span> leads to <span class="math-container">$x=z$</span>, which is absurd.
So wlog. <span class="math-container">$x<y<z$</span>.
Note that <span class="math-container">$y^n=z^n-x^n$</span... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | Barry Cipra | 86,747 | <p>The three primes cannot all be odd, so one of them must be <span class="math-container">$2$</span>. It cannot be <span class="math-container">$z$</span>, so let's let it be <span class="math-container">$x$</span>, in which case <span class="math-container">$y$</span> and <span class="math-container">$z$</span> are o... |
3,752,455 | <blockquote>
<p><strong>Problem.</strong> Show that for <span class="math-container">$n\ge 2$</span> there are no solution <span class="math-container">$$x^n+y^n=z^n$$</span> such that <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> are prime num... | Jyrki Lahtonen | 11,619 | <p>Reposting my answer from the deleted thread here by Martin's request.</p>
<p>Modulo two consideration shows that one of <span class="math-container">$a,b,c$</span> needs to be even. Given that there is a single even prime, we can conclude that <span class="math-container">$a$</span> or <span class="math-container">$... |
2,647,000 | <p>Consider a function $ϕ$ such that $$\lim_{h→0} ϕ(h) = L$$ and $$L − ϕ(h) ≈ ce^{−1/h}$$ for some constant $c$. By combining $ϕ(h)$, $ϕ(h/2)$, and $ϕ(h/3)$, find an accurate estimate of $L$.</p>
<p>Isn't $ϕ(h)=-ce^{−1/h}+L$? I think I am over-simplfying this...</p>
| Ri-Li | 152,715 | <p>Normally in my opinion in research there is nothing called number theory. We start with a subject called number theory and we develop tools to solve some problems. </p>
<p>Yes, it is true that to work in number theory we need to know a bit of stuffs but not a lot always. It depends on problems and possible projects... |
3,807,550 | <p>I am stuck in a true/false question. It is</p>
<p>In a finite commutative ring, every prime ideal is maximal.</p>
<p>The answer says it's false.</p>
<p>Well what I can say is (Supposing the answer is right)</p>
<p><span class="math-container">$(1)$</span> The ring can't be Integral domain since finite integral domai... | markvs | 454,915 | <p>The answer is false. <span class="math-container">$I$</span> is prime means <span class="math-container">$R/I$</span> is a domain. Which implies <span class="math-container">$R/I$</span> is a field which implies that <span class="math-container">$I$</span> is maximal.</p>
|
2,035,186 | <p>This is a probability question where I am asked to integrate a region that represents the probability of a scenario. X, Y, and U are random variables, where U = X-Y. I need to find the probability </p>
<p>$P(U \leq u) = P(X-Y \leq u)$, where the density function I'm integrating over is defined by f(x,y) = 1, for 0 ... | Community | -1 | <p>$\newcommand{\cm}{\mathrm{cm}}$Call $AOB$ the triangle, and label the tangency points $P,\,Q,\,R$ going from left to right.</p>
<p>You know $AO=\frac{120\cm^2}{15\cm}=8\cm$ and $AB=\sqrt{AO^2+BO^2}=\sqrt{289\cm^2}=17\cm$.</p>
<p>Moreover, you know that $PA=AQ$, that $QB=BR$, that $PA+AO=BR+BO$ and that $QB+PA=AB$.... |
1,724,554 | <p>Say, $A$ is an $ n\times n $ matrix over $\Bbb R$, with</p>
<p>$$ A_{ij} = \begin{cases} a \qquad \text{if } i=j\\
b \qquad \text{otherwise.}
\end{cases} $$</p>
<p>How do we compute the determinant of this symmetrix matrix $A$?</p>
| lhf | 589 | <p>We can write $A=B-(b-a)I$, where $B$ is the matrix with all entries equal to $b$.</p>
<p>Therefore, $\det A=(-1)^n\chi(b-a)$, where $\chi$ is the characteristic polynomial of $B$.</p>
<p>Since $B$ has rank $1$, we have $\chi(x)=x^n-tr(B)x^{n-1}=x^n-nbx^{n-1}$ (see <a href="https://math.stackexchange.com/a/458125">... |
161,024 | <p>I was recently having a discussion with someone, and we found that we could not agree on what an exponential function is, and thus we could not agree on what exponential growth is. </p>
<p>Wikipedia claims it is $e^x$, whereas I thought it was $k^x$, where k could be any unchanging number. For example, when I'm doi... | Qiaochu Yuan | 232 | <p>If $x$ has units (e.g. time), then there's no way to distinguish between these possibilities; they're all equivalent up to change of units. </p>
|
3,491,816 | <p>Find the min and max values of the function <span class="math-container">$$f(x,y)=10y^2-4x^2$$</span> with the constraint <span class="math-container">$$g(x,y)=x^4+y^4=1$$</span>
I have done the following working;
<span class="math-container">$$\frac{\partial f}{\partial x} = \lambda \frac{\partial g}{\partial x... | Siddhartha | 257,185 | <p>A different approach without using Lagrange multipliers:</p>
<p>Let <span class="math-container">$x^2=\cos\theta$</span> and <span class="math-container">$y^2=\sin\theta$</span> where <span class="math-container">$\displaystyle 0\le\theta\le\frac{\pi}{2}$</span>.</p>
<p><span class="math-container">$f(\theta)=10\s... |
3,676,284 | <p><a href="https://i.stack.imgur.com/9xfxz.png" rel="nofollow noreferrer">This is link to question</a>
[Here is my attempt, but the answer key is convergent. I dont think I count it wrong.<a href="https://i.stack.imgur.com/nAcEs.jpg" rel="nofollow noreferrer">][1]</a></p>
| J.G. | 56,861 | <p>The usual proof that well-ordering implies a choice function <span class="math-container">$f$</span> exists on <span class="math-container">$X\not\owns\emptyset$</span> notes that if <span class="math-container">$\le$</span> well-orders <span class="math-container">$\bigcup X$</span>, any <span class="math-container... |
2,969,203 | <p>Let <span class="math-container">$f$</span> be a <span class="math-container">$C''$</span> function on <span class="math-container">$(a, b)$</span> and suppose there is a point <span class="math-container">$c$</span> in (a, b) with <span class="math-container">$$f(c)= f'(c)=f''(c) = 0$$</span> Show that there is a c... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>For <span class="math-container">$x$</span> in <span class="math-container">$(a,b)$</span>,</p>
<p><span class="math-container">$$f(x)=f(c)+(x-c)f'(c)+\frac{(x-c)^2}{2}f''(c)+(x-c)^2\epsilon(x)$$</span></p>
|
4,086,485 | <blockquote>
<p>We can regard <span class="math-container">$\pi_1(X,x_0)$</span> as the set of basepoint-preserving homotopy classes of maps <span class="math-container">$(S^1,s_0)\rightarrow(X,x_0$</span>). Let <span class="math-container">$[S^1,X]$</span> be the set of homotopy classes of maps <span class="math-conta... | feynhat | 359,886 | <p>One can write an explicit expression for the homotopy but I am quite convinced by this picture, <img src="https://i.imgur.com/IU7h3A5.jpg" alt="this picture" /></p>
<p>where <span class="math-container">$x_1$</span> is a point on the pink loop <span class="math-container">$b$</span> (more generally suppose <span cla... |
508,791 | <p>I have an integer list that is <code>n</code> long and each value can be ranging from <code>1 .. n</code>.</p>
<p>I need a formula that tells me how many of all possible lists for a given n, that have one or more consecutive sequences of a length of exactly 2 of the same number and no other consecutive sequences th... | Marko Riedel | 44,883 | <p>Here is a different solution that may interest you. Introduce three sequences
$a_{n,k}$, $b_{n,k}$ and $c_{n,k}$ that count the number of strings over $\Sigma^k$ where $|\Sigma|=n,$ that end in a digit that is not repeated, a digit that is repeated twice and a digit that is repeated at least three times. In fact we ... |
2,073,410 | <p>If $$ a-(a \bmod x)<b$$ how do I prove that $$c-(c\bmod x)<b \;\forall c<a?$$ </p>
| Bart Michels | 43,288 | <p>We have to prove that
$$c-(c\bmod x)\leq a-(a\bmod x)$$
for all $a,c,x$, $c\leq a$.</p>
<p>Both are multiples of $x$, and their difference is
$$\begin{align*}a-(a\bmod x)-(c-(c\bmod x))&=(a-c)+(c\bmod x-a\bmod x)\\&\geq 0+c\bmod x-a\bmod x\\&>0+0-x\end{align*}$$
and it is a multiple of $x$, hence $\g... |
22,207 | <p>How to make a defined symbol stay in symbol form?</p>
<pre><code>w = 3; g = 4;
{w, g}[[2]]
</code></pre>
<blockquote>
<p><code>3</code></p>
</blockquote>
<p>I want the output to be <strong><code>g</code></strong> and not <code>3</code>. For example, if I want to save different definitions by <code>DumpSave</co... | Mr.Wizard | 121 | <p>I would do this:</p>
<pre><code>w = 3; g = 4;
HoldForm[w, g][[{2}]]
</code></pre>
<blockquote>
<pre><code> g (* wrapped in HoldForm *)
</code></pre>
</blockquote>
<p>The <code>{}</code> brackets are critical. See <a href="https://mathematica.stackexchange.com/a/2450/121">this answer</a> for an explanation.</p... |
372,198 | <blockquote>
<p>If $G$ is a group, $H$ and $K$ both subgroups of $G$, $K \subseteq H$, $\left[G:H\right]$ and $\left[H:K\right]$ both finite then $\left[G:K\right]=\left[G:H\right]\cdot\left[ H:K \right].$</p>
</blockquote>
<p>I am not sure if this is standard notation but $\left[ G : K \right]$ denotes the number o... | bfhaha | 128,942 | <p>Consider the splitting field $E$ for $f(x)$ over $F$.<br>
There are three possible factorizations of $f(x)$ in $E[x]$.<br>
(i) $f(x)=(x-r_1)(x-r_2)\cdots (x-r_p)$.<br>
(ii) $f(x)=(x-r_1)^s(x-r_2)^s\cdots (x-r_t)^s$, where $s\geq 2$, $t\geq 2$.<br>
(iii) $f(x)=(x-r_1)^p$.<br>
Since $\gcd{(f(x),f'(x))}\neq 1$,
the ca... |
2,797,717 | <p>Prove that exist function $\varphi :\left( {0,\varepsilon } \right) \to \mathbb{R}$ such that
$$\mathop {\lim }\limits_{x \to {0^ + }} \varphi \left( x \right) = 0,\mathop {\lim }\limits_{x \to {0^ + }} \varphi \left( x \right)\ln x = - \infty .$$
I think $\varphi \left( x \right) = \frac{1}{{\ln \left( {\ln \left(... | Ceeerson | 427,680 | <p>I guess you want something that shrinks to zero, but does so slower than $\ln$ grows. So what about $$\varphi(x) = |\ln(x)|^{-1/2}$$ I think that should work.</p>
|
3,800,521 | <p>Let <span class="math-container">$x=\tan y$</span>, then
<span class="math-container">$$
\begin{align*}\sin^{-1} (\sin 2y )+\tan^{-1} \tan 2y
&=4y\\
&=4\tan^{-1} (-10)\\\end{align*}$$</span></p>
<p>Given answer is <span class="math-container">$0$</span></p>
<p>What’s wrong here?</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p><span class="math-container">$$\sin^{-1}\dfrac{2(-10)}{1+(-10)^2}=\sin^{-1}\left(-\dfrac{20}{101}\right)$$</span></p>
<p><span class="math-container">$$\tan^{-1}\dfrac{2(-10)}{1-(-10)^2}=\tan^{-1}\dfrac{20}{99}=u(\text{say})$$</span></p>
<p><span class="math-container">$\implies\dfrac\pi2>u>0$</sp... |
1,240,212 | <blockquote>
<p>How to find the degree of an extension field ?</p>
</blockquote>
<p>Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ?</p>
<p>I've seen somewhere that $[\mathbb Q(r):\mathbb Q]\le n$ if $r$ is a root of an irre... | egreg | 62,967 | <p>The polynomial is irreducible over the rationals, because its possible rational roots are to be found among $\pm1$, $\pm2$, $\pm4$ and $\pm8$. A direct check shows these numbers are not roots.</p>
<p>Since the polynomial has degree $3$, reducibility over $\mathbb{Q}$ coincides with having a rational root.</p>
<p>S... |
1,178,361 | <p>The surface with equation $z = x^{3} + xy^{2} $ intersects the plane with equation $2x-2y = 1$ in a curve. What is the slope of that curve at $x=1$ and $ y = \frac{1}{2} $</p>
<p>So I put $ x^{3} + xy^{2} = 2x - 2y - 1 $</p>
<p>We have $ x^{3} + xy^{2} - 2x + 2y + 1 $</p>
<p>Do I then differentiate wrt x and y si... | drhab | 75,923 | <p>Let $c\in C$ and $c\notin D$. </p>
<p>Since $f$ is onto the set $f^{-1}(\{c\})\subset f^{-1}(C)$ is not empty and its elements do not belong to $f^{-1}(D)$ so that $f^{-1}(C)\neq f^{-1}(D)$.</p>
<p>Of course this also works if $d\notin D$ and $d\in D$ and proved is now $C\neq D\implies f^{-1}(C)\neq f^{-1}(D)$. So... |
4,058,884 | <p>I have an orthonormal basis <span class="math-container">${\bf{b}}_1$</span> and <span class="math-container">${\bf{b}}_2$</span> in <span class="math-container">$\mathbb{R}^2$</span>. I want to find out the angle of rotation. I added a little picture here. I essentially want to find <span class="math-container">$\t... | David K | 139,123 | <p>You have drawn the angle <span class="math-container">$\theta$</span> as an arc from <span class="math-container">${\bf e}_1$</span> to <span class="math-container">${\bf b}_1,$</span>
so apparently you already know intuitively that this is the angle you want,
implying that you want you use the angle
<span class="ma... |
2,646,363 | <p>Let $A_1, A_2, \ldots , A_{63}$ be the 63 nonempty subsets of $\{ 1,2,3,4,5,6 \}$. For each of these sets $A_i$, let $\pi(A_i)$ denote the product of all the elements in $A_i$. Then what is the value of $\pi(A_1)+\pi(A_2)+\cdots+\pi(A_{63})$?</p>
<p>Here is the solution </p>
<p>For size 1: sum of the elements, whi... | Community | -1 | <p>I have a way of doing it, but for some reason I don't get the same result that you've got.</p>
<p>Generally, if you have a finite set $A$ of numbers, and you want $\sum_{X\subseteq A}\prod_{x\in X}x$, the result will be $\prod_{x\in A}(x+1)$.</p>
<p>In your case it will be $(1+1)(2+1)(3+1)(4+1)(5+1)(6+1)=2\cdot 3\... |
3,101,098 | <p>From 11, 12 in the book Logic in Computer Science by M. Ryan and M. Huth:</p>
<p>**</p>
<blockquote>
<p>"What we are saying is: let’s make the assumption of ¬q. To do this,
we open a box and put ¬q at the top. Then we continue applying other
rules as normal, for example to obtain ¬p. But this still depends o... | Dan Christensen | 3,515 | <p>I don't find the "box" analogy to be very helpful. Just understand that, in classical logic:</p>
<p>(1) If you assume only that proposition P is true and you are subsequently able to prove that proposition Q is true, then you can conclude P implies Q. Having done so, you can no longer assume that P is true. That is... |
143,324 | <p>I want to know how to simplify the following expression by using the fact that $\sum_{i=0}^\infty \frac{X^i}{i!}=e^X$. The expression to be simplified is as follows:</p>
<p>$$\sum_{i=0}^{\infty} \sum_{j=0}^i \frac{X^{i-j}}{(i-j)!} \cdot \frac{Y^j}{j!}\;,$$ where $X$ and $Y$ are square matrices (not commutative). (... | Community | -1 | <p>Multiply and divide the innermost term by $\displaystyle i!$ and use binomial theorem. Move the mouse over the gray area to get the answer.</p>
<blockquote class="spoiler">
<p>This gives us $$\displaystyle \sum_{i=0}^{\infty} \frac1{i!} \sum_{j=0}^{i} \frac{x^{i-j} y^j i!}{(i-j)!j!} = \sum_{i=0}^{\infty} \frac1{i... |
1,222,064 | <p>Given is an ellipse with $x=a\cos(t),~~y=b\sin(t)$</p>
<p>I do this by using $S=|\int_c^d x(t)y'(t) dt|$, so calculating the area regarding the vertical axis.
Since $t$ runs from $0$ to $2\pi$ I figured I only had to calculate it from $c=\pi/2$ to $d=3\pi/2$ and then this times $2$. But when I integrate over those... | neslrac | 1,124,426 | <p>The "easy" one is the perspective divide in matrix form, which has to follow the "awful" one.</p>
<p>The "easy" matrix shows the right column starting with <span class="math-container">$x' = x_c$</span>. x, y and z all have identity; only w is z/D. But with contradicting indices (s, c, ... |
2,414,965 |
<p>I am following along and reading this notes: <a href="https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf" rel="nofollow noreferrer">https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf</a></p>
<p>I am having trouble understanding why we necessarily have $e_n=d_n+1$,
$d_{n+1}= d_{n+2} =···= 2$
and $e_{n+1} = e_{... | eyeballfrog | 395,748 | <p>All numbers have at least one ternary expansion. For example,
\begin{eqnarray}
\frac{1}{2} &=& 0.11111111... \\
\frac{2}{9} &=& 0.02000000... \\
\frac{\pi}{8} &=& 0.21001211... \\
\gamma &=& 0.12012021...
\end{eqnarray}
However, some numbers have two ternary expansions
\begin{eqnarray... |
2,414,965 |
<p>I am following along and reading this notes: <a href="https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf" rel="nofollow noreferrer">https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf</a></p>
<p>I am having trouble understanding why we necessarily have $e_n=d_n+1$,
$d_{n+1}= d_{n+2} =···= 2$
and $e_{n+1} = e_{... | Community | -1 | <p>As in the case of $\frac 13$ there are two ternary expressions. In this example $n $ as they have defined it $=1$. That is, the first place they don't agree is the first place after the decimal. ..</p>
<p>So, $0.e_1e_2\dots e_n=0.10\dots0$ and $0.d_1d_2\dots d_n=0.02\dots 2\dots $. Looking at this you should b... |
4,289,129 | <p>Let <span class="math-container">$H$</span> be a group with identity <span class="math-container">$1_H$</span> that is generated by 2 elements <span class="math-container">$a,b$</span> that commute (<span class="math-container">$ab=ba$</span>) and where each has at most order <span class="math-container">$3$</span>.... | Asinomás | 33,907 | <p>Every element of <span class="math-container">$H$</span> is a finite product of <span class="math-container">$a,b$</span> and <span class="math-container">$a^{-1}$</span> and <span class="math-container">$b^{-1}$</span>. Fortunately <span class="math-container">$a^{-1}$</span> is <span class="math-container">$a^2$</... |
1,920,994 | <p>My calculus teacher gave us this interesting problem: Calculate</p>
<p>$$ \int_{0}^{1}F(x)\,dx,\ $$ where $$F(x) = \int_{1}^{x}e^{-t^2}\,dt $$</p>
<p>The only thing I can think of is using the Taylor series for $e^{-t^2}$ and go from there, but since we've never talked about uniform convergence and term by term in... | grand_chat | 215,011 | <p>Hint: First (1) flip the limits on the inner integral (assuming those limits are correctly stated), then (2) switch the order of integration:
$$
\int_{x=0}^1\int_{t=1}^x e^{-t^2}dt\,dx
\stackrel{(1)}=-\int_{x=0}^1\int_{t=x}^1 e^{-t^2}dt\,dx
\stackrel{(2)}=-\int_{t=0}^1\int_{x=0}^t e^{-t^2}dx\,dt
=-\int_{t=0}^1t e^{-... |
2,404,176 | <p>From the days I started to learn Maths, I've have been taught that </p>
<blockquote>
<p>Adding Odd times Odd numbers the Answer always would be Odd; e.g.,
<span class="math-container">$$3 + 5 + 1 = 9$$</span></p>
</blockquote>
<p>OK, but look at this question </p>
<p><a href="https://i.stack.imgur.com/TmYsJ.... | dxiv | 291,201 | <blockquote>
<p>you can also repeat the numbers</p>
</blockquote>
<p>Wonder if that means $\,11,5+13,5+5=30\,$ (where the $\,,\,$ comma works as
decimal separator).</p>
|
2,854,671 | <p>Consider a $n\times n$ Hankel Matrix</p>
<p>$$
H = \begin{bmatrix}
x_{1} & x_{2} & \dots & x_{n} \\
x_{2} & x_{3} & \dots & x_{n+1} \\
\vdots \\
x_{n} & x_{n+1} & \dots & x_{2n}
\end{bmatrix}
$$
, where all $x_i \in \mathbb{Z}_p = \{ 0,\dots,p-1 \}$, where $p$ is ... | Community | -1 | <p>There is an algorithm called Levinson Recursion for Toeplitz matrices which is <span class="math-container">$\mathcal{O}(n^{2})$</span>. There is exists a similar algorithm for Hankel matrices called <a href="https://ac.els-cdn.com/S0024379510006294/1-s2.0-S0024379510006294-main.pdf?_tid=2586b0e6-a947-4789-8049-4a10... |
1,555,697 | <p>So, I'm working out one of my assignments and I'm a little bit stuck on this problem:</p>
<blockquote>
<p>A fish store is having a sale on guppies, tiger barbs, neons,
swordtails, angelfish, and siamese fighting fish (6 kinds). How many
ways are there to choose 24 fish with at least 1 guppy, at least 2
tige... | Community | -1 | <p>Here is an approach using generating functions. The generating function for the guppies
is $g+g^2+g^3+\cdots ={g\over 1-g}$. In the same way, the generating functions for the other
5 types of fish are ${t^2\over 1-t}$ (tiger barbs), ${n^3\over 1-n}$ (neons), $sw$ (swordtails),
${a^2\over 1-a}$ (angelfish), and $... |
539,027 | <p>How many ways are there to distribute $18$ different toys among $4$ children?</p>
<ol>
<li><p>without restrictions</p></li>
<li><p>if $2$ children get $7$ toys each and $2$ children get $2$ toys each.</p></li>
</ol>
<p>For $1$ since toys are different, then there are $4^{18}$ ways to distribute .</p>
<p>for $2$, ... | Community | -1 | <ol>
<li><p>$4^18$ works.</p></li>
<li><p>Choose 2 of them. Then choose 7 toys. There are 7^2 ways to distribute them. Then choose 2 toys and give 1 to 1 child and the other to the other. So we have ${4 \choose 2}{18 \choose 7}7^2{11 \choose 2}2^2$. </p></li>
</ol>
<p>I know that Andres' answer assumes that the child... |
1,105,971 | <p>I've got two independent bernoulli distributed random variables $X$ and $Y$ with parameter $\frac{1}{2}$. Based on those I define two new random variables </p>
<p>$X' = X + Y , E(X') = 1$</p>
<p>$Y' = |X - Y|, E(Y') = \sum_{x=0}^1\sum_{y=0}^1|x-y|*P(X=x)*P(Y=y) = \frac{1}{2}$ </p>
<p><strong>How can I calculate E... | tau_cetian | 206,595 | <p>Someone else can answer more authoritatively for the general case, but for a small experiment such as this one can we build up all possible values of $X' \cdot Y'$ from the four possible outcomes of $(X,Y)$?</p>
<p>$$
\begin{array}{l|l|l|l|l}
(X,Y) & X' & Y' & X' \cdot Y' & P(\ \ ) \\
\hline
(0,0) &... |
1,118,259 | <p>Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the remaining solid?</p>
<p>Can someone help me at least setting up the integral? I know that there is a similar problem... | Christian Blatter | 1,303 | <p>(<strong>Edit:</strong> This answer does not cover the case of large holes; see the comments by coffeemath.)</p>
<p>A hint:</p>
<p>If $b<{a\over\sqrt{2}}$ then the intersection $B$ of the two cylinders is completely in the interior of the sphere. In this case you can proceed as follows:</p>
<p>Do the problem ... |
405,205 | <p>Some friends and I have a family of polynomials (in one variable) with rational coefficients and we would very much like a formula for them. Grasping at straws, we computed many examples and wrote them in the basis of binomial coefficients. Specifically, I mean the basis <span class="math-container">$\left\{\binom... | Ira Gessel | 10,744 | <p>The basis <span class="math-container">$\binom{x}{0}$</span>, <span class="math-container">$\binom{x+1}{1}$</span>, <span class="math-container">$\binom{x+2}{2}$</span>, <span class="math-container">$\dots$</span> has this property. More generally, if <span class="math-container">$i$</span> is a nonnegative integer ... |
4,460,778 | <p>I have a more general question on the importance of fixed-point theorems. In mathematics youre being introduced to so many fixed-point theorems but i still could not figure out why they are so important. Why would be a simply looking statement as <span class="math-container">$f(x)=x$</span> be so important. I would ... | Michael Greinecker | 21,674 | <p>One important reason is that the existence of solutions to systems of equations are equivalent to fixed-points of appropriate functions. Suppose you want to show <span class="math-container">$f(x)=0$</span> for some <span class="math-container">$x$</span>. This is equivalent to <span class="math-container">$f(x)+x=x... |
4,460,778 | <p>I have a more general question on the importance of fixed-point theorems. In mathematics youre being introduced to so many fixed-point theorems but i still could not figure out why they are so important. Why would be a simply looking statement as <span class="math-container">$f(x)=x$</span> be so important. I would ... | ml0105 | 135,298 | <p>I second lisyarus' link. Nash's thesis on the existence of Nash equilibria in normal form games relied on the Brouwer Fixed Point Theorem. It was famously dismissed by von Neumann (who did work in both Game Theory and Functional Analysis) as being "just a fixed point theorem." More modern proofs rely on Ka... |
4,545,364 | <blockquote>
<p>Solve the quartic polynomial :
<span class="math-container">$$x^4+x^3-2x+1=0$$</span>
where <span class="math-container">$x\in\Bbb C$</span>.</p>
<p>Algebraic, trigonometric and all possible methods are allowed.</p>
</blockquote>
<hr />
<p>I am aware that, there exist a general quartic formula. (Ferrari... | Qiaochu Yuan | 232 | <p>We can look for a difference of squares factorization. Completing the square gives</p>
<p><span class="math-container">$$\left( x^2 + \frac{1}{2} x + c \right)^2 - \left( 2c + \frac{1}{4} \right) x^2 - (c + 2) x - (c^2 - 1)$$</span></p>
<p>and we want to find a value of <span class="math-container">$c$</span> such t... |
4,545,364 | <blockquote>
<p>Solve the quartic polynomial :
<span class="math-container">$$x^4+x^3-2x+1=0$$</span>
where <span class="math-container">$x\in\Bbb C$</span>.</p>
<p>Algebraic, trigonometric and all possible methods are allowed.</p>
</blockquote>
<hr />
<p>I am aware that, there exist a general quartic formula. (Ferrari... | Anders Kaseorg | 38,671 | <p>Since this quartic has no real roots, it has two pairs of complex conjugate roots, so it must factor into two conjugate quadratics:</p>
<p><span class="math-container">$$(x^2 + ax + b)(x^2 + \overline ax + \overline b) = x^4 + (a + \overline a)x^3 + (a\overline a + b + \overline b)x^2 + (a\overline b + \overline ab)... |
68,145 | <p>All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;</p>
<p>It is well-known that every regular ring is Gorenstein and every Gorenstein ring is Cohen-Macaulay. There are some examples to demonstrate that the converse of the above statements do not hold. For... | Hailong Dao | 2,083 | <p>I will argue that the examples you gave are "simplest" in some strong sense, so although they look unnatural, if Martians study commutative algebra they will have to come up with them at some point. </p>
<p>Let's look at the first one $A=k[[x,y,z]]/(x^2-y^2, y^2-z^2, xy,yz,zx)$. Suppose you want </p>
<blockquote>
... |
1,537,676 | <p>$f(x,y)=\begin{cases}
|\frac{y}{x^2}|exp(-|\frac{y}{x^2}|) & , x\ne0\\
0 & , x=0
\end{cases}$</p>
<p>I need to show that $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ the limit does not exist in $(0,0)^T$. I tried to prove it with the sequence criteria but I could not find a good sequence. Polar form doesn't get ... | MASL | 266,735 | <p>Choose two paths:</p>
<p>1) $y=e^x-1$. Then $\lim_{(x,y)\to 0^+}f(x,y)=\lim_{x\to0^+}\left|\frac{e^x-1}{x^2}\right|\exp(-\frac{e^x-1}{x^2})=\lim_{x\to 0^+}\left|\frac{1}{x}\right|\exp(-\frac{1}{x})=0$</p>
<p>This is a path for which $y$ reaches zero at the same pace as $x$. You can convince yourself that the limit... |
3,164,280 | <p>By accident, I find this summation when I pursue the particular value of <span class="math-container">$-\operatorname{Li_2}(\tfrac1{2})$</span>, which equals to integral <span class="math-container">$\int_{0}^{1} {\frac{\ln(1-x)}{1+x} \mathrm{d}x}$</span>.</p>
<p>Notice this observation</p>
<p><span class="math-co... | user90369 | 332,823 | <p>With this answer I show an indirect method to the wished result of the integral <span class="math-container">$~\int\limits_0^1\frac{\ln(1-x)}{1+x}dx~$</span>, </p>
<p>and <em>indirect</em> means here: It’s used <span class="math-container">$~\text{Li}_2\left(\frac{1}{2}\right)~$</span> <em>without</em> knowing it’... |
1,921,302 | <p>I can't believe I am asking such a silly question. So I have the function
$$\ln\tan^{-1}x$$
I am asked to find the range of this function. I know that the range of $\ln x$ is all real numbers and that the range of $\tan^{-1}(x)$ is $(-\frac\pi2$, $\frac\pi2)$. Wouldn't the range of $\ln\tan^{-1}x$ also be $(-\frac\p... | Ege Erdil | 326,053 | <p>You are given the inclusion $ \beta I \subset I $, multiplying both sides by the fractional ideal $ I^{-1} $ gives $ (\beta) \subset \mathcal O_K $, in particular, $ \beta \in \mathcal O_K $.</p>
|
1,921,302 | <p>I can't believe I am asking such a silly question. So I have the function
$$\ln\tan^{-1}x$$
I am asked to find the range of this function. I know that the range of $\ln x$ is all real numbers and that the range of $\tan^{-1}(x)$ is $(-\frac\pi2$, $\frac\pi2)$. Wouldn't the range of $\ln\tan^{-1}x$ also be $(-\frac\p... | tracing | 200,415 | <p>You don't need to argue with principal ideals if you don't want to. Instead, note that $I$ is finitely generated over $R$, say by $x_1,\ldots,x_n$.
Then $\alpha \cdot x_i = \sum_j r_{ij} x_j$ for some $r_{ij} \in R$
(by the assumption), and so the matrix $\bigl( \alpha\delta_{ij} - r_{ij}
\bigr)$ annihilates $I$. ... |
3,294,369 | <p>I am reading a paper which asserts that the value of the function <span class="math-container">$f(x) = x^{\frac{1}{x}}$</span> at <span class="math-container">$x=0$</span> is equal to <span class="math-container">$0$</span>. I can believe that this is so if I write <span class="math-container">$f(x) = e^{{\frac{1}{x... | hmakholm left over Monica | 14,366 | <p>The expression <span class="math-container">$x^{1/x}$</span> is not defined when <span class="math-container">$x=0$</span>, because <span class="math-container">$1/0$</span> has no value.</p>
<p>What you <em>can</em> ask is whether the function defined on <span class="math-container">$(0,\infty)$</span> can be <em>... |
510,130 | <p>Let $(r_i)_{i=1}^m$ be a sequence
of positive reals such that
$\sum_i r_i < 1$
and let $t$ be a positive real.
Consider the sequence $T(n)$
defined by $T(0) = t$,
$T(n) = \sum_i T(\lfloor r_i n \rfloor) $
for $n \ge 1$.</p>
<p>Show that
$T(n) = o(n)$,
that is,
$\lim_{n \to \infty} \dfrac{T(n)}{n}
= 0
$.</p>
<p>... | Marko Riedel | 44,883 | <p>Here is a proof for a special case that is not too involved yet does occur in actual settings. Suppose your $r_k$ are all inverse integer powers of some positive integer $p$, where $p\ge 2,$ so that $r_k = 1/p^{q_k}$ with $q_k\ge 1$ and the $q_k$ distinct.</p>
<p>Your recurrence now looks like this:
$$ T(n) = \sum_... |
1,811,528 | <p>The definition of the order of an element in a group is:</p>
<blockquote>
<p>The order of an element $x$ of a group $G$ is the smallest positive integer $n$ such that $x^{n}=e$.</p>
</blockquote>
<p>Doesn't this definition assume that the integers are somehow relevant to every group? </p>
<p>All of the other de... | Ben Grossmann | 81,360 | <p>Every group has an associative "multiplication" (binary operation), so there is no ambiguity in referring to the integer power of an element.</p>
<p>In particular, $x^n$ is just short hand for
$$
x^n = \overbrace{x x \cdots x}^n
$$
where we note that this expression is this same, no matter how the $x$s are "grouped... |
992,068 | <p>I am having a little trouble understanding this question.</p>
<p>For a DFA M = (Q, Σ, δ, q0, F), we say that a state q ∈ Q is reachable if there
exists some string w ∈ Σ∗ such that q = δ∗(q0, w).</p>
<p>Give an algorithm that, given as input a DFA expressed as a five-tuple M =
(Q, Σ, δ, q0, F), returns the set of ... | Brian M. Scott | 12,042 | <p>Someone hands you a DFA $M=\langle Q,\Sigma,\delta,q_0,F\rangle$, i.e., a set of states, an input alphabet, a transition function, an initial state, and a set of acceptor states. You’re to come up with an algorithm — a systematic procedure — that takes this information as input and produces as output a list of the r... |
4,324,493 | <p>According to Rick Miranda's Algebraic curves and Riemann surfaces, a hyperelliptic curve is defined as the Riemann surface obtained by gluing two algebraic curves, <span class="math-container">$y^2=h(x)$</span> and <span class="math-container">$w^2 = k(z)$</span> (where <span class="math-container">$h$</span> has di... | hm2020 | 858,083 | <p><strong>Question1:</strong> "Is there any geometric way to visualise the construction? Formally, is there any embedding/immersion of this construction into R3, that helps see, atleast topologically, atleast for specific examples, what parts of genus g surface are being glued together?"</p>
<p><strong>Answe... |
2,193,171 | <p>Question: Let $\{a_n\}$ and $\{b_n\}$ be convergent sequences with $a_n \Rightarrow L$ and $b_n \Rightarrow M$ as $n \Rightarrow \infty$. </p>
<p>Prove that $a_nb_n \Rightarrow LM$</p>
<p>Solution: (My Attempt). Instead of redoing it could someone just tell me what I'm doing wrong. Thx</p>
<p>WTS: </p>
<p>(1) $\... | Ethan Alwaise | 221,420 | <p>Pick $\epsilon > 0$. By taking the derivative one sees that $e^{-1/t}t^{-k}$ is increasing on some interval $(0,a)$. Choose $m \in \mathbb{N}$ such that $2^{-m} < a$ and
$$mk - 2^m < \log_2\epsilon.$$
Therefore if $0 < t < 2^{-m}$ we have
$$\frac{e^{-1/t}}{t^k} < \frac{2^{-1/t}}{t^k} < \frac{2^{... |
2,246,629 | <p>I'm trying to evaluate the following limit but I'm stuck.</p>
<p>$$
\lim_{x\to +\infty} {x^3\cos(1/x)\over \sin x}
$$ </p>
<p>I tried the squeeze theorem but I was led to a dead-end. Any help would be appreciated.</p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>prove that for any $N>0$ we can find $x_1>N$ such that the function is positive and $x_2>N$ such that the function is negative. So the limit cannot exists.</p>
|
2,246,629 | <p>I'm trying to evaluate the following limit but I'm stuck.</p>
<p>$$
\lim_{x\to +\infty} {x^3\cos(1/x)\over \sin x}
$$ </p>
<p>I tried the squeeze theorem but I was led to a dead-end. Any help would be appreciated.</p>
| Lakshya Gupta | 439,351 | <p>We can exclude cos(1/x) as when x approaches infinity cos(1/x) would approach 1.
Secondly.
We are left with x^3/sin(x).
If we apply L hospital rule thrice. We are left with -6/cos(x) and cos(x) is not defined when x approaches infinity. Thus the limit is not defined.</p>
|
1,237,450 | <p>I couldn't follow a step while reading this <a href="https://math.stackexchange.com/a/1237316/135088">answer</a>. Since I do not have enough reputation to post this as a comment, I'm asking a question instead. The answer uses "partial integration" to write this $$ \int \frac{dv}{(v^2 + 1)^\alpha} = \frac{v}{2(\alpha... | N. S. | 9,176 | <p>Try integration by parts
$$f= (v^2+1)^{-\alpha} ; g' =1 \\
f'=(-\alpha) (v^2+1)^{-\alpha-1}(2v) ; g=v$$</p>
<p><strong>P.S.</strong> After integration by Parts, you write the $v^2$ in the numerator as
$$v^2=(v^2+1)-1$$
and split the integral in two.</p>
|
122,546 | <p>There is a famous proof of the Sum of integers, supposedly put forward by Gauss.</p>
<p>$$S=\sum\limits_{i=1}^{n}i=1+2+3+\cdots+(n-2)+(n-1)+n$$</p>
<p>$$2S=(1+n)+(2+(n-2))+\cdots+(n+1)$$</p>
<p>$$S=\frac{n(1+n)}{2}$$</p>
<p>I was looking for a similar proof for when $S=\sum\limits_{i=1}^{n}i^2$</p>
<p>I've trie... | Thomas Andrews | 7,933 | <p>We use that <span class="math-container">$n^2=\sum_{k=1}^{n} (2k-1).$</span> Then:</p>
<p>Not quite Gaussian, but similar:</p>
<p><span class="math-container">$$\begin{align}S&=\sum_{n=1}^{m} n^2 \\&= \sum_{n=1}\sum_{k=1}^n (2k-1)\\
&=\sum_{k=1}^{m}\sum_{n=k}^{m}(2k-1)\\
&=\sum_{k=1}^m(2k-1)\sum_{n... |
1,747,696 | <p>First of all: beginner here, sorry if this is trivial.</p>
<p>We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .</p>
<p>My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual formula is that the first n... | C. Dubussy | 310,801 | <p>Assume you have a sequence $$x, x+a, x+2a, x+3a, ..., x+na.$$</p>
<p>Let us note $$S = \sum_{i=0}^n (x+ia).$$ Then by the trick you mentioned, we see that $$S+S = (2x+na)+(2x+na)+...+(2x+na) = (n+1)(2x+na).$$ Hence $$S = \frac{(n+1)(2x+na)}{2}.$$</p>
|
3,554,646 | <p>Let <span class="math-container">$f:(a,b)\to \mathbb{R}$</span> be injective and continuous. Prove that</p>
<ul>
<li><span class="math-container">$f$</span> is monotonic.</li>
<li>The image of <span class="math-container">$f$</span> is <span class="math-container">$(c,d)$</span> or maybe (<span class="math-containe... | Martund | 609,343 | <p>Without loss of generality, we may assume that <span class="math-container">$f$</span> is monotonic increasing function (Consider <span class="math-container">$-f$</span> otherwise). Let <span class="math-container">$$c:=\lim_{x\to a+}f(x)\in\mathbb R\cup\{-\infty\}$$</span>
<span class="math-container">$$d:=\lim_{x... |
1,932,961 | <p>Prove by mathematical induction that
$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$
holds $\forall n\in\mathbb{N}$.</p>
<hr>
<p>(1) Assume that $n=1$. Then left side is $1^2 =1$ and right side is $6/6 = 1$, so both sided are equal and expression holds for $n = 1$.</p>
<p>(2) Let $k \in \mathbb{N}$ is given. As... | Dr. Sonnhard Graubner | 175,066 | <p>it should be $$\frac{(xy+yz+zx-3)^2}{(x-1)^2(y-1)^2(z-1)^2}\geq 0$$
Hint: set $$x=a/b,y=b/c,z=c/a$$ in the given term
after the substituion we obtain
$${\frac {{a}^{2}}{{b}^{2}} \left( {\frac {a}{b}}-1 \right) ^{-2}}+{
\frac {{b}^{2}}{{c}^{2}} \left( {\frac {b}{c}}-1 \right) ^{-2}}+{
\frac {{c}^{2}}{{a}^{2}} \left( ... |
185,177 | <p>Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.</p>
<p>Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?</p>
<p>What if $X$ is an algebraic space (i.e., trivial stabilizers)?</p>
<p>Edit: I changed the old question to a different question which should... | Daniel Litt | 6,950 | <p>It seems to me that the answer is NO if $X$ is a DM stack. If I'm not mistaken, it suffices to give a smooth finite type separated connected Deligne-Mumford stack over $\mathbb{C}$ which is simply connected (since such a thing has no non-trivial finite etale covers, let alone finite etale covers by a scheme). But ... |
2,631,230 | <p>So, I'm studying mathematics on my own and I took a book about Proofs in Abstract Mathematics with the following exercise:</p>
<p>For each $k\in\Bbb{N}$ we have that $\Bbb{N}_k$ is finite</p>
<p>Just to give some context on what theorems and definitions we can use:</p>
<ol>
<li>Definition: $\Bbb{N}_k = \{1, 2, ..... | Jack D'Aurizio | 44,121 | <p>An alternative approach:
$$ \begin{eqnarray*}\iint_{(0,1)^2}4xy\sqrt{x^2+y^2}\,dx\,dy&=&\iint_{(0,1)^2}\sqrt{X+Y}\,dX\,dY\\&=&2\iint_{0\leq Y\leq X\leq 1}\sqrt{X+Y}\,dX\,dY\\&=&2\iint_{(0,1)^2}X\sqrt{X}\sqrt{K+1}\,dX\,dK\\&=&2\int_{0}^{1}X\sqrt{X}\,dX\int_{0}^{1}\sqrt{K+1}\,dK\end{eqn... |
424,514 | <p>Suppose one has a generating function <span class="math-container">$$F(z) = \sum_{k\ge 0} f(k) z^k$$</span>
for some <span class="math-container">$f:\mathbb{Z}\rightarrow \mathbb{Z}$</span>. Is there a way to express an iteration of <span class="math-container">$f$</span> in terms of <span class="math-container">$F(... | Joe | 171,026 | <p>You could explore this conjecture by the following method: Suppose <span class="math-container">$f(f(k))= h(f(k))$</span> for different specified <span class="math-container">$h$</span>, then look for <span class="math-container">$G(z) = H(F(z))$</span>. So eg. <span class="math-container">$h(k) = a k + b$</span> gi... |
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