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 |
|---|---|---|---|---|
4,574,446 | <p>This is question 13 on page 294 of Vector Calculus by Marsden and Tromba.</p>
<blockquote>
<p>Find the volume of the region determined by <span class="math-container">$x^2 + y^2 + z^2 \leq 10, z \geq 2.$</span></p>
</blockquote>
<p>I have attempted it as follows.</p>
<p>The region can be described by using spherical... | Mike | 544,150 | <p>We first note the following:</p>
<p><strong>Claim 1:</strong> <em>The condition that <span class="math-container">$n$</span> is odd implies that <span class="math-container">$n \not = (a)(a+1)$</span> for all integer <span class="math-container">$a$</span>, because as always, one of <span class="math-container">$a,a... |
165,888 | <p>I want to return the final value from the Table.For example something like Table[n-1] value.</p>
| Henrik Schumacher | 38,178 | <p><em>This</em> is definitely faster. It took me some time to figure out the combinatorics and there might still be some potential for improvement within the compiled functions.</p>
<pre><code>getColumnIndices = Compile[{{m, _Integer}, {n, _Integer}, {i, _Integer}},
Transpose[Partition[
Partition[
Join[... |
2,786,610 | <p>I have to calculate $\int\operatorname{arccot}(\cot(x))\ dx.$ If I had to find the derivative it would be easy with the chain rule. How can I do this?</p>
| user | 505,767 | <p>From here we can solve as follow </p>
<ul>
<li><p>$Ab_1=b_1\implies (A-I)b_1=0$ and find $b_1$</p></li>
<li><p>$Ab_2=b_2\implies (A-I)b_2=0$ and find $b_2$</p></li>
<li><p>$Ab_3=b_2+b_3 \implies (A-I)b_3=b_2$ and find $b_3$</p></li>
<li><p>$Ab_4=b_3+b_4 \implies (A-I)b_4=b_3$ and find $b_4$</p></li>
</ul>
|
292,041 | <p>The Helly theorem in the Euclidean plane asserts that if $S_1, \dots, S_n$ are $n \ge 3$ convex subsets such that $S_i \cap S_j \cap S_k \ne \emptyset$ for all distinct triples $i,j,k$, then the total intersection $\bigcap_{i = 1}^n S_i$ is also nonempty. </p>
<p>I'm wondering if the same theorem is true in the hyp... | Andy Putman | 317 | <p>The original proof of Helly's theorem was topological and only uses basic homological properties of convex sets. It generalizes to all sorts of contexts, including the one you are interested in. Here is a general statement of what it can do. A <em>homology cell</em> is a topological space whose reduced singular h... |
1,633,846 | <p>I'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof:</p>
<blockquote>
<p>Let <span class="math-container">$f(x)$</span> be a polynomial with real coefficients and a positive leading coefficient.</p>
<p>Let <span class="math-container">$a \geq 0$</span>. Then,... | user574848 | 574,848 | <p>Here's what I think is a more intuitive proof. </p>
<p>For the first part:</p>
<p>Suppose that <span class="math-container">$f(x)/(x-a)$</span> leaves only non-negative coefficients. So <span class="math-container">$f(x)=(x-a)q(x)+r(x)$</span>. But by the definition of polynomial division, <span class="math-contai... |
3,699,640 | <p>How to show <span class="math-container">$x^2=20$</span> has no solution in <span class="math-container">$2$</span>-adic ring of integers <span class="math-container">$\mathbb{Z}_2$</span> ?</p>
<p>What is the general criterion fora solution of <span class="math-container">$x^2=a$</span> in <span class="math-contai... | nguyen quang do | 300,700 | <p>Here is an elementary solution, but which has the advantage to show "experimentally" why <span class="math-container">$2
$</span> is a "distinguished" prime. If <span class="math-container">$20$</span> is a square in <span class="math-container">$\mathbf Z_2$</span>, we can factor out <span class="math-container">$4... |
51,345 | <p>Do all infinite fields of char p contain a subfield isomorphic to $F_{p}(x)$?</p>
| Qiaochu Yuan | 232 | <p>Perhaps this is an opportunity to walk through a mechanical process from the statement of the problem to its solution. Suppose $F$ is a field of characteristic $p$ which does not contain a subfield isomorphic to $\mathbb{F}_p(x)$. In other words, no element of $F$ is transcendental over $\mathbb{F}_p$. It follows th... |
51,345 | <p>Do all infinite fields of char p contain a subfield isomorphic to $F_{p}(x)$?</p>
| Bill Dubuque | 242 | <p><strong>HINT</strong> $\ $ Consider the algebraic closure of $\rm\:\mathbb F_p\:,\:$ recalling that algebraically closed fields are infinite, since for any finite field $\rm\:\mathbb F\:$ with elements $\rm\:a_1,\cdots,a_n,\:$ a la Euclid, we can construct the polynomial $\rm\:1 + (x-a_1)\cdots(x-a_n)\:$ coprime to ... |
3,707,132 | <p>I believe my proof of this simple fact is fine, but after a few false starts, I was hoping that someone could look this over. In particular, I am interested in whether there is an alternate proof.</p>
<blockquote>
<p>For a real number <span class="math-container">$a$</span> and non-empty subset of reals <span cla... | fleablood | 280,126 | <p>Given <span class="math-container">$B$</span> is non-empty, <span class="math-container">$B$</span> is bounded above and <span class="math-container">$\sup B$</span> is the least upper bound of <span class="math-container">$B$</span> then</p>
<p>Claim 1: <span class="math-container">$a + B$</span> is non-empty.</p>... |
1,797,243 | <p>Find the relative extrema of the function by applying the first-derivative test:</p>
<p>$$f(x)=x^5-5x^3-20x-2$$</p>
<p>So I found the $f'(x)$</p>
<p>$$f'(x) = 5x^4-15x^2-20$$</p>
<p>Now, I'm trying to find the critical values, which $x=0$ or undefined, so I can apply the first-derivative test. However, I can't s... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. If one set $X=x^2$ then one has to solve a classic quadratic equation
$$
5X^2-15X-20=0.
$$</p>
|
1,004,837 | <p>The Warsaw circle is defined as a subset of $\mathbb{R}^2$：
$$\left\{\left(x,\sin\frac{1}{x}\right): x\in\left(0,\frac{1}{2\pi}\right]\right\}\cup\left\{(0,y):-1\leq y\leq1\right\}\cup C\;,$$
where $C$ is the image of a curve connecting the other two pieces. </p>
<p>A map from Warsaw circle to a single point space ... | Cheerful Parsnip | 2,941 | <p>Its Cech cohomology is equal to $\mathbb Z$ in degree $1$. You can either check this directly, or you could apply Alexander Duality (which relates the reduced Cech cohomology with the reduced homology of the complement.) In this case, since the complement has two path components, the Cech cohomology of the Warsaw ci... |
463,814 | <p>I am struggling to find the values of these integrals after trying many substitution it did't worked for me </p>
<p>1) $$ \int_{0}^{a}\frac{dx}{\sqrt{ax-x^2}} $$
2) $$ \int_{}^{}\frac{3x^5dx}{1+x^{12}} $$</p>
| Elias Costa | 19,266 | <p><strong>Hint (a):</strong> Use change of variables, then trigonometric substitutions.
\begin{align}
\int_0^a \frac{dx}{\sqrt{ax-x^2}}
=
&
\int_0^a \frac{dx}{\sqrt{\left(\frac{a}{2}\right)^2-\left(\frac{a}{2}\right)^2+2\cdot \left(\frac{a}{2}\right)x-x^2}}
\\
=
&
\int_0^a \frac{\left(x-\left(\frac{a}{2}\right... |
463,814 | <p>I am struggling to find the values of these integrals after trying many substitution it did't worked for me </p>
<p>1) $$ \int_{0}^{a}\frac{dx}{\sqrt{ax-x^2}} $$
2) $$ \int_{}^{}\frac{3x^5dx}{1+x^{12}} $$</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>For $$\int_0^a\frac{dx}{\sqrt{ax-x^2}}=\int_0^a\frac{dx}{\sqrt{x(a-x)}}=\int_0^a\frac{dx}{|a|\sqrt{\frac xa(1-\frac xa)}}$$</p>
<p>Putting $x=a\sin^2y,dx=2a\sin y\cos y dy$ and as $x=0\implies y=0;x=a\implies y=\frac\pi2$</p>
<p>$$\int_0^a\frac{dx}{|a|\sqrt{\frac xa(1-\frac xa)}}=\int_0^{\frac\pi2}\f... |
3,287,067 | <p>I'm reading a solution to the following exercise:</p>
<p>"Assume that <span class="math-container">$\lim_{x\to c}f\left(x\right)=L$</span>, where <span class="math-container">$L\ne0$</span>, and assume <span class="math-container">$\lim_{x\to c}g\left(x\right)=0.$</span> Show that <span class="math-container">$\li... | grand_chat | 215,011 | <p>How about the <em>even</em> function <span class="math-container">$f(x):=x^2+c$</span> for suitable choice of <span class="math-container">$c$</span>? For simplicity you can take <span class="math-container">$a=1$</span>. Then setting the definite integral to zero yields an equation that <span class="math-container"... |
3,245,838 | <p>We have <span class="math-container">$n$</span> voltages <span class="math-container">$V_1, V_2,\dots , V_n$</span> that are received in a condensator or sum, such that <span class="math-container">$V=\sum V_i$</span> is the sum of received voltages in that point. Every voltage <span class="math-container">$V_i$</sp... | trancelocation | 467,003 | <p>You can avoid partial fraction decomposition by noting that</p>
<p><span class="math-container">$$\oint_{\partial D_3(0)}\frac{\cos(z+4)}{z^2+1}dz = \oint_{\partial \color{blue}{D_1(i)}}\frac{\frac{\cos(z+4)}{z+i}}{z-i}dz + \oint_{\partial \color{blue}{D_1(-i)}}\frac{\frac{\cos(z+4)}{z-i}}{z+i}dz$$</span>
<span cla... |
1,620,186 | <p>I have to find an example of a surface of revolution excluding a sphere and a cone. </p>
<p>Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example? </p>
<p>$$$$ </p>
<p>I also have to find an example of a surface the image of which is not the graph of a smooth function $z=f(x,y)$. </p>
<p>Is $\sigma(x,y)=(3\sqrt... | Community | -1 | <p>For the first one, you can proceed directly: Notice that if the expression is equal to some rational $r$, then</p>
<p>$$x + b = r(x + a) \implies x(1 - r) = a - b$$</p>
<p>Now the right side is rational, but the left side is irrational unless.....</p>
<hr>
<p>For the second, I'd suggest proceeding similarly. Wri... |
1,426,264 | <p>Let $H$, $K$ be groups, and suppose that $H \cong K \times H$. Does it necessarily follow that $K$ is trivial?</p>
| Glen Whitney | 288,300 | <p>@joriki's answer demonstrates that there is no polyhedron of genus 0 satisfying the stated conditions, which leads one to wonder whether it's possible to satisfy the conditions with a higher-genus polyhedron. And in fact, it is. One can construct a toroidal polyhedron with 9 quadrilateral faces (four meeting at each... |
63,796 | <p>I'm looking for a neater way to achieve this, which to me looks awkward and suggests that I am missing something...</p>
<p>Given a Dataset...</p>
<pre><code>ds = Dataset[{
<|"a"->1,"b"->1,"c"->3|>,
<|"a"->1,"b"->2,"c"->4|>,
<|"a"->2,"b"->3,"c"->5|>}];
</code></pre>
<p>... | WReach | 142 | <p>Since the question states that we need not worry about the case where there are multiple maxima in a group, the following expression eliminates the need for <code>Flatten</code>:</p>
<pre><code>ds[GroupBy[#a &] /* Values, MaximalBy[#b &] /* First]
</code></pre>
<p><img src="https://i.stack.imgur.com/vVWIy.... |
199,374 | <p>Consider the two table set below</p>
<pre><code>t1= {0.44, 0.62, 0.77, 0.87, 0.93, 0.96, 0.98, 1}
t2= {0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25}
</code></pre>
<p>how can I create a plot which wil plot a rectangle with a heat map normalized to the values of the table?</p>
<p>The result should look like</p>
... | kglr | 125 | <pre><code>Column[Quiet @ DensityPlot[Interpolation[#][x], {x, 0, 1}, {y, 0, 1},
AspectRatio -> 1/5, Frame -> False, ColorFunction -> "Rainbow",
ColorFunctionScaling -> False, ImageSize -> Medium] & /@ {t1, t2}]
</code></pre>
<p><a href="https://i.stack.imgur.com/n3QBW.png" rel="nofollow norefer... |
51,732 | <p>A <em>Perron number</em> is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ such that some power of $M$ is strictly positive has a
unique positive eigenvector whose eigenvalue is a Pe... | Zili Huang | 60,542 | <p>There are many contexts in which one can establish that a family of polynomials, or even a single polynomial, has roots which are distributed according to some measure which approximates the uniform measure on the unit circle.</p>
<p>Let $f(x) = a_N x^N + \ldots + a_1 x + a_0$ be a polynomial with real coefficients... |
4,102,443 | <p>Let <span class="math-container">$a,b \in [0,\infty)$</span> with <span class="math-container">$a \leq b$</span>.</p>
<p><span class="math-container">$D(x,y)=\mid \frac{x}{2+x}-\frac{y}{2+y} \mid$</span>.</p>
<p>There exist constants <span class="math-container">$c_{1}, c_{2} \in [0,\infty)$</span> such that <span c... | Eric Towers | 123,905 | <p>A ring action on an abelian group is a module.</p>
<p>A linear group action on a vector space, <span class="math-container">$V$</span>, automatically produces a ring action (by the <a href="https://en.wikipedia.org/wiki/Group_ring" rel="nofollow noreferrer">group ring</a> <span class="math-container">$\Bbb{Z}G$</spa... |
4,102,443 | <p>Let <span class="math-container">$a,b \in [0,\infty)$</span> with <span class="math-container">$a \leq b$</span>.</p>
<p><span class="math-container">$D(x,y)=\mid \frac{x}{2+x}-\frac{y}{2+y} \mid$</span>.</p>
<p>There exist constants <span class="math-container">$c_{1}, c_{2} \in [0,\infty)$</span> such that <span c... | Aragogh | 665,469 | <p>I really like paul blart math cop's answer, and I'll give a slight modification with a more categorical flavor; note firstly that monoids and rings can be thought of as purely categorical gadgets, in the following sense; given a monoidal category <span class="math-container">$(\mathsf{C},\otimes, 1)$</span> (such as... |
1,626,721 | <blockquote>
<p>In the <span class="math-container">$n$</span>-cell below, we are to tile it completely with cells of the form <span class="math-container">$\boxdot$</span> and <span class="math-container">$\boxtimes \hspace{-0.45 mm}\boxtimes$</span>. How many tilings are possible for a <span class="math-container">$1... | Zubin Mukerjee | 111,946 | <p>It is possible to find a direct formula for $H_n$ in terms of $n$, then show that this is equivalent to the Fibonacci numbers, without using the recursion. </p>
<hr>
<p>If $n$ is even, the number of tiles of the second type ($\,\boxtimes \hspace{-0.45 mm}\boxtimes\,$) can be between $0$ and $n/2$, inclusive. If th... |
4,562,006 | <p>Prove that <span class="math-container">$\text{Ker}(A)=\text{Ker}(A^2)$</span> if <span class="math-container">$A$</span> is a normal operator.</p>
<p>I know that that <span class="math-container">$\text{Ker}(A)\subseteq \text{Ker}(A^2)$</span>, which I proved as follows:
<span class="math-container">$x\in \text{Ker... | Anne Bauval | 386,889 | <p>Any normal operator <span class="math-container">$A$</span> has the same kernel as its adjoint (since <span class="math-container">$\|Ax\|^2=(A^*Ax,x)=(AA^*x,x)=\|A^*x\|^2$</span>), hence</p>
<p><span class="math-container">$$A^2x=0\Rightarrow Ax\in\ker A=\ker A^*\Rightarrow0=(0,x)=(A^*Ax,x)=\|Ax\|^2.$$</span></p>
|
2,932,139 | <blockquote>
<p>How would I show that the line <span class="math-container">$A=[(x,y,z)=(0,t,t)\mid t\in\mathbb{R}]$</span> is parallel to the plane <span class="math-container">$5x-3y+3z=1$</span>?</p>
</blockquote>
<p>I know the normal vector would be <span class="math-container">$(5,-3,3)$</span>, but how would I... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Recall that the plane is parallel if its normal is orthogonal to the vector.</p>
|
4,390,127 | <p>I ma trying to prove that if <span class="math-container">$A$</span> is an <span class="math-container">$n\times n$</span> matrix, then <span class="math-container">$f:\mathbb{R}^n\to\mathbb{R},\ f(\vec{x})=A\vec{x}\cdot\vec{x}=\vec{x}^T A\vec{x}$</span> is differentiable and <span class="math-container">$Df(\vec{a}... | Kavi Rama Murthy | 142,385 | <p>For any neighborhood <span class="math-container">$V$</span> of <span class="math-container">$a$</span> there is a convex open set <span class="math-container">$U$</span> containing <span class="math-container">$a$</span> and contained in <span class="math-container">$V$</span>. The net <span class="math-container">... |
3,465,171 | <blockquote>
<p>Let <span class="math-container">$G=[0,\infty) \times [0,\infty)$</span>, <span class="math-container">$\alpha \in (0,1)$</span> and <span class="math-container">$$\phi (x,y)=x^{\alpha} y^{1-\alpha}$$</span> Then <span class="math-container">$\phi$</span> is concave; that is, <span class="math-contain... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$f$</span> has a pole of order <span class="math-container">$2$</span> at <span class="math-container">$z_0$</span> then the residue at <span class="math-container">$z_0$</span> is <span class="math-container">$\lim_{z \to z_0} \frac d {dz} (z-z_0)^{2}f(z)$</span>. In this case we g... |
4,014,768 | <blockquote>
<p>Find the sationary points of the curve and their nature for the equation <span class="math-container">$y=e^x\cos x$</span> for <span class="math-container">$0\le x\le\pi/2$</span>.</p>
</blockquote>
<p>I derived it and got <span class="math-container">$e^x(-\sin x+\cos x)=0$</span>.</p>
<p><span class="... | Gabrielek | 699,889 | <p>Since you are asking for <span class="math-container">$0\le x \le \pi/2$</span> it's quite trivial that <span class="math-container">$x = \pi/4$</span> is the only solution of <span class="math-container">$\sin x = \cos x$</span></p>
<p>Edit:</p>
<p>Reasoning in <span class="math-container">$0\le x \le \pi/2$</span>... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Humberto José Bortolossi | 8,828 | <p>I strongly recommend the film <a href="http://www.dimensions-math.org/Dim_E.htm" rel="noreferrer">Dimensions</a> by Jos Leys, Étienne Ghys and Aurélien Alvarez. It's free! The main tools used by the authors to explain the several dimensions are cross sections and steregraphic projections. The animation is very didac... |
18,136 | <p>I introduced the hypercube (to undergraduate students in the U.S.) in the
context of generalizations of the Platonic solids, explained its
structure, showed it rotating.
I mentioned <a href="https://matheducators.stackexchange.com/a/1824/511">Alicia Stott</a>, who discovered the <span class="math-container">$6$</spa... | Carlo Wood | 13,770 | <p>I invented higher dimensions at an early age and have scored "off the charts" in spatial insight on any test. Since I've made numerous "infinite dimensional" puzzles (because 3 dimensional ones were too boring).</p>
<p>This is how I grasp the concept of a hypercube:</p>
<p>Each hyper cube of N dimensions exists of... |
648,066 | <p>Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t \leq 2\pi$. Evaluate
$$\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz.$$</p>
<p>I know that
\begin{equation}
\int_{\gamma(0,1)}\frac{\sin(z)}{z^4}dz = \frac{1}{z^4}\left(z-\frac{z^3}{3!}+\frac{z^5}{5!}-\cdots\right)
= \frac{1}{z^3}-\frac{1}{6z}+\c... | Daniel Fischer | 83,702 | <p>Cauchy's integral formula is</p>
<p>$$f^{(n)}(z) = \frac{n!}{2\pi i} \int_\gamma \frac{f(\zeta)}{(\zeta-z)^{n+1}}\,d\zeta,$$</p>
<p>where $\gamma$ is a closed path winding once around $z$, and enclosing no singularity of $f$.</p>
<p>Thus in your example, $n = 3$, and you need the third derivative,</p>
<p>$$\int_... |
3,886,643 | <p>I have doubts about how to solve this issue about bases in numbering systems:</p>
<p>"Determine the base b in <span class="math-container">$(104)_b = 8285$</span>"</p>
<p>can anybody help me?
Thanks.</p>
| MPW | 113,214 | <p><strong>Hint:</strong> The number in base <span class="math-container">$10$</span> is <span class="math-container">$8285$</span>. Remember that this means the number can be expressed as
<span class="math-container">$$5\cdot 10^0 + 8\cdot 10^1 + 2\cdot 10^2 + 8\cdot 10^3 = 8285$$</span>
You are told that, in base <sp... |
2,767,977 | <p>I'm working through Williams' <em>Probability with Martingales</em> book and had a question. </p>
<p>Suppose we have an iid increment random walk on the integers. $S_n = \sum_1^n Y_i$ where $P(Y_i = 1) = p, P(Y_i = -1) = 1-p$.
On page 102, Williams proves that the random walk will almost surely hit $1$ in finite ... | Dr. Sonnhard Graubner | 175,066 | <p>We have $$f(x)=\sin(\frac{x}{2})$$ then we get by the chain rule: $$f'(x)=\cos(\frac{x}{2})\cdot \frac{1}{2}$$</p>
|
2,767,977 | <p>I'm working through Williams' <em>Probability with Martingales</em> book and had a question. </p>
<p>Suppose we have an iid increment random walk on the integers. $S_n = \sum_1^n Y_i$ where $P(Y_i = 1) = p, P(Y_i = -1) = 1-p$.
On page 102, Williams proves that the random walk will almost surely hit $1$ in finite ... | Community | -1 | <p>$$\sin'(x)=\cos(x)$$
$$\sin'(\frac{x}{2})=\sin'(u)=\cos(u) \cdot u'=
\cos(\frac{x}{2}) \cdot \frac{1}{2}$$</p>
|
2,254,025 | <p><strong>Please verify my answer to the following differential equation:</strong>
$$y''-xy'+y=0$$
Let $y = {\sum_{n=0}^\infty}C_nx^n$, then $y' = {\sum_{n=1}^\infty}nC_nx^{n-1}$ and $y''={\sum_{n=2}^\infty}n(n-1)C_nx^{n-2}$</p>
<p>Substituting this to the equation we get</p>
<p>$${\sum_{n=2}^\infty}n(n-1)C_nx^{n-2}... | levap | 32,262 | <p>Your equation is a second order linear equation so it should be a two-dimensional space of solutions (that is, solutions that depend on two free parameters) while your final answer depends only on one free parameter $C_0$ so this means you did something wrong.</p>
<p>Indeed, your equations don't determine what is $... |
3,742,707 | <p>How many different fractions can be made up of the numbers 3, 5, 7, 11, 13, 17 so that each fraction contains 2 different numbers? How many of them will be the proper fractions?</p>
| Triceratops | 574,334 | <p>Note that all the numbers are mutually <a href="https://en.wikipedia.org/wiki/Coprime_integers" rel="nofollow noreferrer">coprime integers</a> and therefore there cannot be cancellations in the fractions. Then you only need to count the number of possible pairs.</p>
|
2,585,828 | <p>I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known theorem in analysis, but I have never seen this proof. So could someone give a proof (preferably with a source) for ... | N. S. | 9,176 | <p>Let $\alpha$ be any real number. </p>
<p>Define
$$a_n = \frac{\lfloor n \alpha \rfloor}{n}$$
where $\lfloor , \rfloor $ denotes the floor function.</p>
<p>Then $a_n \in \mathbb Q$. Moreover we have
$$\frac{n \alpha -1}{n} \leq a_n \leq \alpha$$
and hence $a_n \to \alpha$.</p>
|
2,821,308 | <p>Consider below complex function
$$H(\omega) = \dfrac{1}{i\omega} \left(e^{i\omega} - e^{-i\omega}\right)$$</p>
<p>If I replace $i$ by $-i$ in $H(\omega)$, I get back the same $H(\omega)$.<br>
Easy to see that $H(\omega) = \dfrac{2}{\omega}\sin(\omega t)$. </p>
<p>Long back I heard somebody claim this, but I coul... | A.Γ. | 253,273 | <p>First of all, your KKT system has a wrong sign, it must be
\begin{align*}
x_1 (-1 + v) \color{red}{-} 2 x_2 &= 0 \tag{1'} ,\\ x_2 (-1 + v) \color{red}{-} 2 x_1 &= 0. \tag{2'}
\end{align*}
However, the conclusion $(x_1-x_2)(1+v)=0$ from (1') minus (2') is correct.</p>
<p>You cannot get all four solutions "on... |
237,960 | <p>How could I solve this problem?</p>
<blockquote>
<p>Find the first digit of $2^{4242}$ without using a calculator.</p>
</blockquote>
<p>I know how to find the last digit with modular arithmetic, but I can't use that here.</p>
| Ross Millikan | 1,827 | <p>If you have memorized that $\log_{10}2\approx 0.30103$ you can multiply by $4242$ and take the fractional part as $0.9692$ which looks like it should be greater than $\log_{10}9$ (and it is, but it is closer than I would have thought, it is $\approx 0.9542$). I don't know how to do it without log tables.</p>
|
1,626,840 | <blockquote>
<p>If I've played the lottery a certain $N$ number of times (and I didn't look into the results of each game), what this $N$ number must be to get me a $50\%$ chance of have already won at least one of my games? (imagine I 'll look up at all the results at once, after acheiving this chance) and the chanc... | lulu | 252,071 | <p>If $p$ is the probability of winning the lottery in a single try (so here $p=\frac 1{10^6}$) then the probability of not winning (in a single try) is $1-p$, and the probability of not winning in $n$ independent trials is $(1-p)^n$. It follows that the probability of winning at least once in $n$ independent trials ... |
444,865 | <p>Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.</p>
| Hagen von Eitzen | 39,174 | <p>If you consider the case $b=c+1$, you get $b^2-c^2=2c+1$, which can be any odd natural number $\ge3$.
Thus with $a=1$ you reach all even $n\ge4$ and with $a=2$ all odd $n\ge 7$.
This leaves only the cases $n\in\{1,2,3,5\}$ open. Can you find solutions for these $n$? (Hint: Try $b=c-1$).</p>
|
444,865 | <p>Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.</p>
| Oleg567 | 47,993 | <p>Consider $n\ge 6$.</p>
<p>If $n$ is odd, $n=2m+1$, then
$$
n = 2m+1 = 2^2 + (m-1)^2 - (m-2)^2;
$$
If $n$ is even, $n=2m$, then
$$
n = 2m = 1^2 + m^2 - (m-1)^2;
$$</p>
<p>Small $n$:<br>
$1=1^2+1^2-1^2$,<br>
$2=3^2+3^2-4^2$,<br>
$3=4^2+6^2-7^2$,<br>
$4=2^2+1^2-1^2$,<br>
$5=4^2+5^2-6^2$.</p>
|
444,865 | <p>Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.</p>
| Seirios | 36,434 | <p>First show the following lemma:</p>
<blockquote>
<p><strong>Lemma:</strong> Any odd postive integer can be written as the difference of two squares.</p>
</blockquote>
<p>For a proof without word, consider the area of the gray part with $a=n$ and $b=n-1$:</p>
<p><img src="https://i.stack.imgur.com/Iox9n.png" alt... |
2,709,878 | <p>How do I know that an equation will have an extraneous solution? </p>
<p>For example in this question:
2log9(x) = log9(2) + log9(x + 24)</p>
| Siong Thye Goh | 306,553 | <p>After you obtain the solution, substitute it in and check whether it is a valid solution. </p>
<p>For this particular question, a possible way to solve the problem is to bring the $2$ up and become the power and we might have included an extra solution. </p>
<p>$$x^2 = 2(x+24)$$</p>
<p>$$x^2-2x-48=0$$</p>
<p>$$(... |
324,557 | <p>Isbell gave, in <a href="https://eudml.org/doc/213746" rel="noreferrer"><em>Two set-theoretic theorems in categories</em> (1964)</a>, a necessary criterion for categories to be concretisable (i.e. to admit some faithful functor into sets). Freyd, in <a href="https://www.sciencedirect.com/science/article/pii/00224049... | Peter LeFanu Lumsdaine | 2,273 | <p>[Answer converted from a comment by Jiří Rosický on <a href="https://mathoverflow.net/a/324563/2273">another answer</a>.]</p>
<p>Isbell’s criterion is used directly in Libor Barto’s paper <em>Accessible set functors are universal</em> (<a href="http://www.karlin.mff.cuni.cz/~barto/Articles/accfununiv.pdf" rel="nofo... |
2,537,311 | <p>Question is, from finding a basis of e-vectors, determine an invertible matrix <strong>P</strong> such that: <img src="https://latex.codecogs.com/gif.latex?F=P%5E%7B-1%7DLP" title="F=L^{-1}AL" /> is diagonal. (write down the matrix F)</p>
<p><img src="https://latex.codecogs.com/gif.latex?L=%5Cbegin%7Bpmatrix%7D&am... | Rebellos | 335,894 | <p>$f$ is differentiable on $\mathbb R$ which means it's also continuous, thus we can apply the <em>Mean Value Theorem</em> on a domain $(0,x)$ :</p>
<p>$$f'(ξ) = \frac{f(x) - f(0)}{x-0}$$</p>
<p>By taking absolute values, we'd get : </p>
<p>$$|f'(ξ)| = \Bigg| \frac{f(x) - f(0)}{x-0} \Bigg| \Leftrightarrow |f'(ξ)||... |
4,474,095 | <p>There is one thing I can't grasp about the proof given in the Linear Algebra Done Right book by Sheldon Axler (attached below).</p>
<p>In the last part it says that <span class="math-container">$(T - \lambda_1I)...(T - \lambda_mI)v = 0$</span>, hence <span class="math-container">$T - \lambda_jI$</span> is not inject... | xbh | 514,490 | <ol>
<li>If <span class="math-container">$f, g$</span> are polynomials over <span class="math-container">$\Bbb C$</span>, then <span class="math-container">$f(T)g(T) = g(T)f(T)$</span> by merely definition.</li>
<li>You do not know <span class="math-container">$\lambda_j$</span> could be an eigenvalue beforehand. Then ... |
1,063,154 | <p>I Ran into this question and I can't find the right way to approach it.</p>
<p>We have $n$ different wine bootles numbered $i=1...n$. the first is 1 year old, the second is 2 years old ... the $n$'th bottle is $n$ years old.</p>
<p>Each bottle is still good at probability of $1/i$.</p>
<p>We pick out a random bot... | Ben S. | 199,026 | <p>You are asked to compute a conditional expectation (...'given that the bottle is good') for which you need a conditional probability distribution. Let $I$ be the age of the bottle you picked out, and recall that this variable take values over the range $i = 1...n$ By definition</p>
<p>$E[I\,|\, I\mbox{ is good}] = ... |
1,591,197 | <p>Let $ A \trianglelefteq G $ and $ B \trianglelefteq A $ a Sylow normal subgroup of $ A $. My textbook says then that $ B \trianglelefteq G $.</p>
<p>I don’t understand why that is.</p>
| Elle Najt | 54,092 | <p>A subgroup $H$ of a group $G$ is said to be a characteristic subgroup if it is fixed under all automorphisms of $G$ (not just conjugation). Precisely, if $\sigma \in Aut(G)$, then $\sigma H = H$ as a set. </p>
<p>Characteristic subgroups are of course normal, because conjugation is an automorphism.</p>
<p>The poin... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | user21820 | 50,073 | <p>It is easy to prove the following in Z+CC (Zermelo plus countable choice):</p>
<blockquote>
<p>Every uncountable closed set of reals is in bijection with the reals.</p>
</blockquote>
<p>I was informed by Asaf that it can be proven in ZF (no choice at all), but that proof appears to use replacement. I hence <a href="... |
416,407 | <blockquote>
<p>What examples are there of habitual but unnecessary uses of the axiom of
choice, in any area of mathematics except topology?</p>
</blockquote>
<p>I'm interested in standard proofs that use the axiom of choice, but where
choice can be eliminated via some judicious and maybe not quite obvious
rephrasing. ... | Timothy Chow | 3,106 | <p>The <a href="https://mathoverflow.net/a/22938/">highly-upvoted, accepted answer</a> (by Theo Johnson-Freyd) to another MO question, <a href="https://mathoverflow.net/q/22927/">Why worry about the axiom of choice?</a>, points out that the usual proof of the Poincaré–Birkhoff–Witt theorem assumes that every vector spa... |
836,753 | <p>$\{X_1, X2, \ldots, X_{121}\}$ are independent and identically distributed random variables such that $E(X_i)= 3$ and $\mathrm{Var}(X_i)= 25$. What is the standard deviation of their average? In other words, what is the standard deviation of $\bar X= {X_1+ X_2+ \cdots + X_{121} \over 121}$?</p>
| André Nicolas | 6,312 | <p>We first compute the variance of $W$, where
$$W=\frac{X_1+X_2+\cdots+X_{121}}{121}.$$
Here are the ingredients.</p>
<p>$1$) The variance of a sum of <strong>independent</strong> random variables is the sum of the variances. Each $X_i$ has variance $25$, so the variance of $X_1+X_2+\cdots+X_{121}$ is $(25)(121)$.</... |
2,306,122 | <p>Show $X=\{n \in \mathbb{N}: \text{n is odd and} \ n = k(k+1) \text{for some} \ k \in \mathbb{N}\}=\emptyset$</p>
<p>My proof is as follow, please point if I have made any mistake. </p>
<p><strong>proof:</strong></p>
<p>we have $\emptyset \subseteq X$
suppose $X≠\emptyset$ pick $n \in X$</p>
<p>Then there are 2 ... | mrnovice | 416,020 | <p>Your proof is correct, I would just be a bit pedantic about your format and ordering. So I'll follow on from your proof:</p>
<p>Suppose for contradiction that $X\neq \emptyset$ and pick $n\in X$</p>
<p>Then we know $n$ is odd and we can write $n=k(k+1)$ for some $k\in\mathbb{N}$</p>
<p>Then the two cases are:</p>... |
1,394,141 | <blockquote>
<p>Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$</p>
</blockquote>
<p><strong>My attempt:</strong>
I think I have computed the tangent space in the case that $M=SL(n)$. We can write $SL(n)=\det^{-1}(1)$ and I've proved in an earlier exe... | Thomas Rot | 5,882 | <p>Basically you write down the defining equation of your group. I'll do it for $O(n)$. Something like this:</p>
<p>$$M^TM=1$$</p>
<p>The tangent space can be defined as the equivalence class of directions of curves in your manifold. Let $M_t\in O(n)$ with $M_0=1$ and $\frac{d}{dt}\bigr|_{t=0}M_t=X$. Then</p>
<p>$$
... |
243,743 | <p>I have been studying this sequence (<a href="https://oeis.org/A266882" rel="nofollow">A266882</a> in the OEIS) and found the following pattern:</p>
<p>$13 + 17 + 19 + 23 + 29 = 101$ (101 is prime)</p>
<p>37 does not hold.</p>
<p>$223 + 227 + 229 + 233 + 239 = 1151$ (1151 is prime)</p>
<p>The same is true for 108... | assaferan | 74,819 | <p>No. A simple check shows that it is already false for the next few primes in the sequence, namely for 4783, 6079, 7331.
In fact, one could even ask whether these first numbers are the only ones for which this sum is prime, but it turns out that it later holds also for 11057 and 12269. </p>
|
729,441 | <p>This question comes up after going over Arzela-Ascoli theorems.</p>
<p>For a set of continuous functions $\mathbb F$ from $\mathbb R$ to $\mathbb R$ that is equicontinuous. How do I show that if sup{$|f(0)|:f \in F$} $< \infty$ , then $\mathbb F$ is pointwise bounded?</p>
<p>I know I need to show that since su... | hmakholm left over Monica | 14,366 | <p>A proof sketch could be:</p>
<p><em>1. Every (nonzero) vector is an eigenvector.</em> Let $v\ne 0$ and suppose $Tv$ is not a multiple of $v$. Then $v$ and $Tv$ are linearly independent; extend $\langle v,Tv\rangle$ to a basis $\langle v, Tv, v_3,v_4,\ldots,v_n\rangle$. By assumption $T$ has the same matrix represen... |
3,466,707 | <p>In that problem</p>
<p><span class="math-container">$$\lim\limits_{x \to \infty} \left(1-\frac{1}{x}\right)^\left(e^x\right)$$</span></p>
<p>I use <span class="math-container">$\ln$</span>, then it gave me <span class="math-container">$0\times\infty$</span> indeterminate, then I use L'Hospital Rule but I cannot re... | Focus | 254,076 | <p>Use <span class="math-container">$$0 \leq e^{e^x \ln(1-1/x)} \leq e^{-e^x /x}$$</span> and take the limit, which gives 0.</p>
|
506,286 | <p>This relates to a recent question which asked whether it was possible to approximate $\cos x$ with a series for an odd function (series with odd-powered terms). It occurred to me the answer maybe yes is we restricted to a finite positive interval and ask for approximation instead of absolute equality.</p>
<p>So her... | Ross Millikan | 1,827 | <p>The same sort of argument about positive and negative powers applies. The Legendre polynomials are orthonormal and complete on $[-1,1]$, which says that you can't approximate any one of them well in terms of the rest. Using linear algebra techniques, you can change the basis to the monomials in $x$. The (finite a... |
506,286 | <p>This relates to a recent question which asked whether it was possible to approximate $\cos x$ with a series for an odd function (series with odd-powered terms). It occurred to me the answer maybe yes is we restricted to a finite positive interval and ask for approximation instead of absolute equality.</p>
<p>So her... | Daniel Fischer | 83,702 | <p>If we have a sequence $0 \leqslant n_1 < n_2 < n_3 < \dotsc$ of admissible exponents (not necessarily integers), then the linear subspace of $C([a,b])$, for $0 < a < b$, generated by the functions $t\mapsto t^{n_k}$ is dense if and only if</p>
<p>$$\sum_{k=2}^\infty \frac{1}{n_k} = \infty.\tag{1}$$</... |
2,161,830 | <p>Hi i was reading a book called Symmetry and Pattern in Projective Geometry by Eric Lord, in his book the author give these axioms:</p>
<ol>
<li>Any two distinct points are contained in a unique line.</li>
<li>In any plane, any two distinct lines contain a unique common point.</li>
<li>Three points that do not lie o... | rschwieb | 29,335 | <p>As far as I can see, a line with two points satisfies this system of axioms.</p>
<p>None of these axioms postulate the existence of noncollinear points, but that is normally a feature of axioms for the projective plane and projective $3$-space.</p>
<p>Perhaps the author has given these axioms in addition to some o... |
2,161,830 | <p>Hi i was reading a book called Symmetry and Pattern in Projective Geometry by Eric Lord, in his book the author give these axioms:</p>
<ol>
<li>Any two distinct points are contained in a unique line.</li>
<li>In any plane, any two distinct lines contain a unique common point.</li>
<li>Three points that do not lie o... | mrnovice | 416,020 | <p>If we're allowed to use this definition for a line in $\mathbb{R}^{3}$:</p>
<p>$L = \vec{a} + \lambda \vec{u}: \lambda \in \mathbb{R}$, $\vec{a}, \vec{u} \in \mathbb{R}^{3}$</p>
<p>Where $\vec{a}$ and $\vec{u}$ are two distinct points contained by $L$</p>
<p>Then by changing the value of $\lambda$ we can show tha... |
1,360,658 | <p>prove: $$\frac{1-\cos 2\theta}{1-\cos\theta}=2\cos\theta-2$$</p>
<p>I'm thinking that there will be something to square in this? Because I notice that the $LHS$ looks like the half-angle identity....</p>
<p>Edit: I am so sorry guys, my grave mistake, the expression should have been equal to 2 instead like,</p>
<p... | Zain Patel | 161,779 | <p>We have, using $\cos 2\theta = 2\cos^2 \theta - 1$ $$\frac{1 - (2\cos^2 \theta -1)}{1-\cos \theta} = \frac{2(1-\cos^2 \theta)}{1-\cos \theta} = \frac{2(1-\cos \theta)(1+\cos \theta)}{1-\cos \theta} = 2(1 + \cos \theta)$$</p>
<p>Since $1 - \cos^2 \theta$ is a difference of two squares. The result you asked for in yo... |
2,619,344 | <p>I just struck with a doubt today</p>
<blockquote>
<p>Why do most of the standard inequalities require the variables to be positive.</p>
</blockquote>
<p>For example if we want to find minimum value of a certain expression say <span class="math-container">$a+b+c$</span> the very first thought that comes in our mind ... | Shashi | 349,501 | <p>Your series is absolutely convergent. There is no need for many calculations. Notice that the supremum norm of $f$ exists due to the continuity of $f$ on a compact set. You can see the absolutely convergence by using the following:
\begin{align}
\sum_{n\geq 1}\frac{1}{n}\bigg | \int^1_0 f(x) e^{-nx} dx\bigg |&... |
1,330,362 | <p>I need to define in ZFC the following things:</p>
<ul>
<li>image and domain of a binary relation ($\{ x \mid (x,y)\in f \}$ would be a definition of domain, but it is a class for which is for me is not quite clear why it is a set)</li>
<li>f[X] for a binary relation $f$ and a set $X$</li>
</ul>
<p>Please describe ... | Michael Hardy | 11,667 | <p>If $b>1$ then $b^x\to\infty$ as $x\to\infty$ and $b\to 0$ as $x\to-\infty$.</p>
<p>If $b>1$ then $b^x\to\infty$ as $x\to-\infty$ and $b\to\infty$ as $x\to-\infty$.</p>
<p>Either way, $b^x$ runs through all of the interval $(0,\infty)$.</p>
<p>Similarly for $c^x$.</p>
<p>If $b>1$ and $c>1$ then $b^x+c... |
1,330,362 | <p>I need to define in ZFC the following things:</p>
<ul>
<li>image and domain of a binary relation ($\{ x \mid (x,y)\in f \}$ would be a definition of domain, but it is a class for which is for me is not quite clear why it is a set)</li>
<li>f[X] for a binary relation $f$ and a set $X$</li>
</ul>
<p>Please describe ... | Claude Leibovici | 82,404 | <p>As Omnomnomnom answered, in general, there will not be explicit solutions and numerical methods, such as Newton, would be used.</p>
<p>Considering $$f(x)=b^x+c^x-a$$ we know that the function is such bracketted by $2 b^x-a$ and $2c^x-a$ which allows to define an approximate range for the solution.</p>
<p>But th... |
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | Rolf Hoyer | 228,612 | <p>I think you should try using a method other than Eisenstein's criterion. By Gauss' lemma it suffices to show that $f(x)$ does not factor over the integers, so you need only show the following:</p>
<ul>
<li>There is no integer root: ie $f(x) \ne 0 $ for $x$ an integer dividing $16$.</li>
<li>There is no quadratic f... |
1,260,722 | <blockquote>
<p>Prove that <span class="math-container">$f=x^4-4x^2+16\in\mathbb{Q}[x]$</span> is irreducible.</p>
</blockquote>
<p>I am trying to prove it with Eisenstein's criterion but without success: for <strong>p=2</strong>, it divides <strong>-4</strong> and the constant coefficient 16, don't divide the leading ... | Zarrax | 3,035 | <p>You've seen that $f(x)$ has no roots, so you want to exclude factorizations of the form
$$f(x) = (x^2 + ax + b)(x^2 + cx + d)$$
Since $f(x) = f(-x)$, the above implies
$$f(x) = (x^2 - ax + b)(x^2 - cx + d)$$
Here $a,b,c$, and $d$ are integers by Gauss's Lemma.</p>
<p>So a given root $r$ of $x^2 - ax + b$ is a root ... |
43,382 | <p>Let <span class="math-container">$C$</span> be a coalgebra and <span class="math-container">$\Delta: C \to C\otimes C$</span> a co-multiplication map. Then, due the co-associative property we can consider <span class="math-container">$\Delta^m$</span>. But how is defined <span class="math-container">$\Delta^{m}: C \... | Mariano Suárez-Álvarez | 1,409 | <p>We <em>usually</em> define $$\Delta^3=(\mathrm{id}_C\otimes\Delta)\circ\Delta,$$ and, more generally, $$\Delta^{k+1}=(\mathrm{id}_C^{k-1}\otimes\Delta)\circ\Delta^k.$$</p>
<p>(Your last formula does not make sense in a coalgebra: you seem to be multiplying elements of $C$; even in bialgebra, though, things like "$\... |
43,382 | <p>Let <span class="math-container">$C$</span> be a coalgebra and <span class="math-container">$\Delta: C \to C\otimes C$</span> a co-multiplication map. Then, due the co-associative property we can consider <span class="math-container">$\Delta^m$</span>. But how is defined <span class="math-container">$\Delta^{m}: C \... | user91132 | 6,827 | <p>To add to the answer above, I'd like to advertise the so-called <em>sumless Sweedler notation</em> here. </p>
<p>This notation works as follows: let $C$ be a coalgebra; then if $c \in C$ then we write $\Delta(c) = c_1 \otimes c_2$ as an abbreviation of the more precise $\Delta(c) = \sum_{i=1}^k c_{i1} \otimes c_{i2... |
6,090 | <p>I have been reading about Riemann Zeta function and have been thinking about it for some time.</p>
<p>Has anything been published regarding upper bound for the real part of zeta function zeros as the imaginary part of the zeros tend to infinity?</p>
<p>Thanks</p>
| Aryabhata | 1,102 | <p>I would suggest you read the book "Riemann's Zeta Function" by H.M.Edwards.</p>
<p>As to your question, this is one theorem (from the above book)</p>
<hr>
<p><strong>De La Vallee Poussin's Theorem (1899)</strong> : There exist constants $c > 0, K > 1$ such that</p>
<p>$$ \beta < 1 - \frac{c}{\log \gamma}$$</p... |
4,506,196 | <p>If <span class="math-container">$R$</span> is a commutative Hilbert ring then each of its prime ideals is an intersection of maximal ideals. Is there a similar class of commutative rings for which every ideal is an intersection of prime ideals (that is, in which all ideals are radical)? Does this class have a name? ... | Qiaochu Yuan | 232 | <p>This is an extremely rare condition. It implies that every ideal <span class="math-container">$I$</span> satisfies <span class="math-container">$I^2 = I$</span>; that is, every ideal is idempotent. <a href="https://math.stackexchange.com/questions/42727/finitely-generated-idempotent-ideals-are-principal-proof-withou... |
1,139,789 | <p>Is $f(z)=z^n$ holomorphic?</p>
<p>I have tested a number of other functions using the Cauchy Riemann equations $u_x=v_y$, $v_x=-u_y$. However in the case of $f(z)=z^n$ I cannot think of a way to find the functions $u(x,y)$ and $v(x,y)$ without using a binomial expansion of $(x+iy)^n$. </p>
<p>Any help or pointers ... | Adam Hughes | 58,831 | <p>The easiest way is just the definition</p>
<p>$$\lim_{z\to a}{z^n-a^n\over z-a}=\lim_{z\to a}(z^{n-1}+az^{n-2}+\ldots +a^{n-2}z+a^{n-1})=na^{n-1}$$</p>
<p>by definition, since $a$ was arbitrary, the derivative exists at all points, $a\in\Bbb C$.</p>
|
2,752,353 | <p>In general, if I have $|\Psi\rangle = (|\Psi_{1_1}\rangle \otimes |\Psi_{1_2}\rangle + |\Psi_{2_1}\rangle \otimes |\Psi_{2_2}\rangle)$, can I find $|\Psi_{3_1}\rangle$ and $|\Psi_{3_2}\rangle$, such that $|\Psi\rangle = |\Psi_{3_1}\rangle \otimes |\Psi_{3_2}\rangle$? Here $\otimes$ means tensor product and $|\Psi\r... | Arnaud Mortier | 480,423 | <p>In general the answer is no and you can clearly see why when you look at the dimensions: $\dim (V\otimes W)$ is much larger than $\dim (V\times W)$ whenever $V$ and $W$ are both of dimension greater than $1$. This tells you that in general a tensor is more than just a couple of vectors.</p>
<p>Tensors of the form $... |
2,752,353 | <p>In general, if I have $|\Psi\rangle = (|\Psi_{1_1}\rangle \otimes |\Psi_{1_2}\rangle + |\Psi_{2_1}\rangle \otimes |\Psi_{2_2}\rangle)$, can I find $|\Psi_{3_1}\rangle$ and $|\Psi_{3_2}\rangle$, such that $|\Psi\rangle = |\Psi_{3_1}\rangle \otimes |\Psi_{3_2}\rangle$? Here $\otimes$ means tensor product and $|\Psi\r... | chhro | 255,099 | <p>The matrix $A=\begin{pmatrix} 1& 0& 0& 0\\ 0&0& 0&0\\ 0&0&0&0\\0&0&0&1\end{pmatrix}$ is a sum of two simple tensors which is not a simple tensor itself. In particular, $A=E_{11}\otimes E_{11}+E_{22}\otimes E_{22}$ where $E_{11}=\begin{pmatrix} 1&0\\0&0\end{pmat... |
4,202,451 | <p>I have been reading about the Miller-Rabin primality test. So far I think I got it except the part where the accuracy is stated.<br />
E.g from <a href="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Accuracy" rel="nofollow noreferrer">wiki</a></p>
<blockquote>
<p>The error made by the primality t... | Ritam_Dasgupta | 925,091 | <p>The inductive hypothesis is:
<span class="math-container">$$\prod_{i=1}^{k} \frac {k+i}{2i-3}=2^k(1-2k)$$</span>
LHS can be written as:
<span class="math-container">$$\frac {k+1}{-1} \cdot \frac {k+2}{1} \cdot \frac {k+3}{3}...\cdot \frac {k+k}{2k-3}=\left(\frac {k+2}{-1} \cdot \frac {k+3}{1} \cdot \frac {k+4}{3}...... |
4,492,250 | <p>Say I have a function <span class="math-container">$f(x) = x^2 + 2$</span></p>
<p>This function never touches the x-axis, but it could be easily transformed to touch it by cancelling the constant as in <span class="math-container">$g(x) = (x^2 + 2) - 2$</span></p>
<p>Is there any way to generalize this, so that I ca... | trula | 697,983 | <p>if you mean touching in the sense that the x axes is a tangent o zur graph of f , than f(x9) must have at least one point x0 with f'(x0)=0 then you just take g(x)=f(x)-f(x0) and g touches the x- axes.</p>
|
253,208 | <p>Here's a problem I'm working on: </p>
<p>Find the matrix of T with respect to the standard bases $P_{3}$ and $\mathbb{R}^{3}$: </p>
<p>$T(p(x)) = \left( \begin{array}{cc} p'(0) \\ p(0) \\ p(0) - p'(0)\end{array} \right)$</p>
<p>So I'll list the steps that I've been taking and hopefully someone will be able to tel... | André Nicolas | 6,312 | <p>The following is a probably unreasonable interpretation of the question. It is motivated by the number-theoretic setting: we are maybe implicitly asked to assume that $T$ is an <em>integer</em>. </p>
<p>The man started watching, and watched <em>continuously</em> for several days, say $x$. Then the time spent wa... |
540,029 | <p>I know that for any matrix <span class="math-container">$A$</span>, <span class="math-container">$AA^{\ast}$</span> is positive semidefinite (where <span class="math-container">$A^\ast$</span> is <span class="math-container">$\overline{A}^T$</span>). Please help me show the following statement</p>
<blockquote>
<p... | Robert Lewis | 67,071 | <p>You cannot prove that any positive semidefinite matrix may be written $AA^*$ because it is not true. A complete explication for the real case is found in my answer to <a href="https://math.stackexchange.com/questions/482688/proof-of-a-matrix-is-positive-semidefinite-iff-it-can-be-written-in-the-form-x">this questio... |
401,967 | <p>This question is about logical complexity of sentences in third order arithmetic. See <a href="https://en.wikipedia.org/wiki/Arithmetical_hierarchy" rel="nofollow noreferrer">Wikipedia</a> for the basic concepts.</p>
<p>Recall that the Continuum Hypothesis is a <span class="math-container">$\Sigma^2_1$</span> sente... | Farmer S | 160,347 | <p>(As pointed out by @PaulBlainLevy, the following doesn't meet the requirement that it should have no known reduction even under large cardinal assumptions. But I think it's a natural <span class="math-container">$\Pi^2_2$</span> statement, so I'll leave it here.)</p>
<p>Consider the statement "For every set of ... |
3,502,507 | <p>This is very similar to the <a href="https://math.stackexchange.com/questions/3501693/given-that-you-started-with-one-chip-what-is-the-probability-that-you-will-win">question</a> I've just asked, except now the requirement is to gain <span class="math-container">$4$</span> chips to win (instead of <span class="math-... | Christian Blatter | 1,303 | <p>Let <span class="math-container">$p(n)$</span> be the probability that you win the game when you have <span class="math-container">$n$</span> chips in the pocket. Then <span class="math-container">$p(0)=0$</span>, <span class="math-container">$\>p(4)=1$</span>. Having <span class="math-container">$1\leq n\leq3$</... |
1,740,535 | <p>If <span class="math-container">$X\sim U(0,1)$</span> and <span class="math-container">$Y\sim U(0,X)$</span> what is the density (distribution) function <span class="math-container">$f_Y(y)$</span>?</p>
<p>I know the answer and I also found it on this site (link bellow). However, I just can't get the intuition why t... | Graham Kemp | 135,106 | <p>The support of the joint pdf is $0< Y<X <1$</p>
<p>Clearly to "integrate out" $X$ to obtain the marginal pdf of $Y=y$ requires integrating w.r.t. $x$ over the interval $y<x<1$</p>
<p>$$\begin{align}f_Y(y) ~=~& \int_\Bbb R \frac 1 x ~\mathbf 1_{y\in(0;x),x\in(0;1)}~\operatorname d x
\\[1ex] =~&a... |
1,879,149 | <p>Let $\mu$ be a Radon measure on $\mathbb{R}$. (<em>i.e.</em>, a locally finite Borel measure)</p>
<p>Let $I$ be an interval of $\mathbb{R}$.</p>
<p>Let $\gamma : I \mapsto \mathbb{R}$ be a <strong>monotone</strong> function that is Lebesgue-Stieltjes integrable with respect to $\mu$:
$$\int_I \gamma(x) \mathrm{d}\... | Luiz Cordeiro | 58,818 | <p>I don't think your version of Lusin's theorem can be used to give a short proof of the result, since we need some control over the continuous functions we get. Here's a solution, using a slightly weaker version of Lusin's theorem, and constructing the continuous functions more explicitly.</p>
<p>Let's assume $\gamm... |
132,957 | <p>Could you show me that how can I solve this as fast as possible, please? <br /></p>
<p>$$ ABCDEF \, \land \, DEFABC \; \large{ \in \, \mathbb{N^{+}} } $$ $$ ABCDEF \, = \, 6 \times (DEFABC) \\ $$ $$ (A+B+C+D+E+F)=\; ? $$</p>
<p>Thank you very much... :)</p>
| Dejan Govc | 19,588 | <p>Here's another solution: we immediately see that $D=1$, since otherwise $6\times DEFABC$ would have to have more than seven digits. Now write: $X=ABC$, $Y=DEF$. The equation becomes: $1000X+Y=6(1000Y+X)$ which can be rewritten as $994X=5999Y$. Now, since $994$ is divisible by the primes $2$ and $71$, and $5999$ isn'... |
4,404,751 | <p>I want to prove the following: Every symmetric matrix whose entries are calculated as <span class="math-container">$ 1/(n -1) $</span> with <span class="math-container">$n$</span> as the size of the matrix, except for the diagonal which is 0, has a characteristic polynomial with a root at <span class="math-container... | Diego Artacho | 761,744 | <p>The vector with all entries equal to <span class="math-container">$1$</span> is an eigenvector with eigenvalue <span class="math-container">$1$</span>.</p>
|
171,449 | <p><strong>Theorem</strong> Every sequence {$s_n$} has a monotonic subsequence whose limit is equal to $\limsup s_n$. I think to show that there exist a monotonic subsequence is kind of straight forward but I could show that there exist such subsequences whose limit is $\limsup s_n$.</p>
| copper.hat | 27,978 | <p>Plot in with a log scale on the y axis. Then the line will be straight and it will be easy to see where it is going.</p>
|
2,243,900 | <p>I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:</p>
<blockquote>
<p>What exactly is calculus? </p>
</blockquote>
<p>I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I ... | AspiringMathematician | 375,631 | <p>In a nutshell, Calculus (as seen in most basic undergraduate courses) is the study of change and behaviour of functions and sequences. The three main points are:</p>
<ul>
<li>Limits: How sequences and functions behave when getting closer and closer to a desired point (geometrically, what happens when you "zoom in" ... |
2,218,960 | <blockquote>
<p>Let $(W_t,\mathscr{F_t})$ be a Wiener process and let
$$M_t=M_0e^{W_t-t/2}\qquad t\ge0$$
where $M_0$ is deterministic.
Show that, for $\epsilon>0$, $\tau=\inf\{t\ge0:M_t\le\epsilon\}$ is a stopping time.</p>
</blockquote>
<p>I have been able to show that $M_t$ is a martingale but I'm a bit ... | drhab | 75,923 | <p>Yes if the convergence is absolute: if $\sum_{n=1}^{\infty}|a_n|<\infty$ then also $\sum_{n=1}^{\infty}|b_n|<\infty$.</p>
<p>No if the convergence is not absolute: see the counterexample given in the answer of Tsemo Aristide.</p>
|
3,033,202 | <p>I have been banging my head against this proof for a few days now, as I can visualize why it is true in my head, but don't know how to prove it in words:</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be nonempty subsets of <span class="math-container">$\mathbb{R}$</s... | KYHSGeekCode | 553,404 | <p>Problem: <span class="math-container">$$\int_0^s\frac 1{\sqrt{1-x^2}}dx$$</span></p>
<p>Let <span class="math-container">$$x=\sin \theta$$</span> then we get:
<span class="math-container">$$\frac{dx}{d\theta}=\cos \theta$$</span>
By derivating both side by <span class="math-container">$x$</span>.</p>
<p><span clas... |
4,240,598 | <blockquote>
<p>Simplify <span class="math-container">$\dfrac{18 - \dfrac 7 {3x}} {\dfrac 7 {18x} - 3}$</span>?</p>
</blockquote>
<p>I'm having a hard time simplifying this particular expression and am seeking any type of assistance in solving it.</p>
<p>In the expression
<span class="math-container">$$\frac{18 - \dfra... | Robert Israel | 8,508 | <p>The <span class="math-container">$1$</span> on the right is supposed to be the identity (operator or matrix, depending on what sort of thing <span class="math-container">$A$</span> is).</p>
|
3,986,785 | <p>On page 153 of <em>Linear Algebra Done Right</em> the second edition, it says:</p>
<blockquote>
<p>Define a linear map <span class="math-container">$S_1: \text{range}(\sqrt{T^*T} ) \to \text{range}(T)$</span> by:</p>
</blockquote>
<blockquote>
<p><strong>7.43:</strong> <span class="math-container">$S_1 (\sqrt{T^* ... | freakish | 340,986 | <p>What exactly a function <span class="math-container">$f:X\to Y$</span> is? Typically we define it as a subset of <span class="math-container">$X\times Y$</span> such that for any <span class="math-container">$x\in X$</span> there is precisely one <span class="math-container">$y\in Y$</span> such that <span class="ma... |
1,036,964 | <blockquote>
<p>Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis $\{e_i\otimes f_j\}$ of $\mathbb{R}^2\otimes\mathbb{R}^3$ where $x = (1, 1)$ and $y = (1, -2, 1)$.</p>
</blockquote>... | drhab | 75,923 | <p>Prescribing $g\left(x,y\right)=xy$ on $B=\left[0,4\right)\times\left[-2,2\right)$
gives $g\left[B\right]=f\left[A\right]$. </p>
<p>This actually is replacing $x^{2}$ with domain $\left(-2,1\right)$
by the 'less complex' $x$ with domain $\left[0,4\right)$.</p>
<p>For $t\in\left(-8,8\right)$ find some $r\in\left(0,4... |
311,617 | <p>If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. But whenever I actually think about the notation, I find it to be one of the most confusing conventions in algebra. I... | Community | -1 | <p><strong>Hint.</strong> Take a short exact sequence $0\to K\to F\to N\to 0$, where $F$ is projective (or free). Then use the long exact homology sequence for Tor and induction on $n$. </p>
|
3,932,883 | <p>Can someone suggest how to find the infinite series sum for</p>
<p><span class="math-container">$$\frac{k(k+1)3^k}{k!}$$</span> where k goes from <span class="math-container">$1$</span> to infinity.</p>
<p>I know that <span class="math-container">$\sum_0\frac{3^k}{k!}=e^3$</span> but I'm not sure if that helps here.... | Empy2 | 81,790 | <p><span class="math-container">$$\frac{k(k+1)3^k}{k!}=\frac{k(k-1)3^k}{k!}+\frac{2k3^k}{k!}=\frac{3^k}{(k-2)!}+2\frac{3^k}{(k-1)!}\\
=9\frac{3^{k-2}}{(k-2)!}+6\frac{3^{k-1}}{(k-1)!}$$</span>
Now change variables</p>
|
506,965 | <p>How can I prove that:
$E_π [ (dQ_X/dπ) S (T)| F_t ]= E_{Q_X} [S(T) | F_t]E_π [ dQ_X/dπ | F_t ]$.
Obviously $E_π [(dQ_X/dπ) S(T) ]= E_{Q_X} [S(T)]$ I know that much, but how to prove when it is conditioned on $F_t$.</p>
| Gordon | 169,372 | <p>Alternatively, note that, for ant $A \in \mathcal{F}_t$,
\begin{align*}
\int_A \frac{dQ_X}{d\pi}S(T) d\pi &=\int_A S(T) dQ_X\\
&=\int_A E_{Q_X}(S(T)\mid \mathcal{F}_t) dQ_X\\
&=\int_A E_{Q_X}(S(T)\mid \mathcal{F}_t) \frac{dQ_X}{d\pi} d\pi\\
&=\int_A E_{\pi}\left(E_{Q_X}(S(T)\mid \mathcal{F}_t) \frac... |
466,873 | <p>I'm trying to prove that if $K$ is a finite field, then every subset of $\mathbb A^n(K)$ is algebraic. I know that if $K$ is finite, then every element of $K$ is algebraic, i.e., for every $a\in K$ there is a polynomial $f\in K$ such that $f(a)=0$, but this didn't help me to solve the question. I almost sure that we... | M Turgeon | 19,379 | <p>A finite union of algebraic sets is an algebraic set. </p>
|
1,361,517 | <p>$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find
$LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.</p>
<p>Any suggestions as to how to solve this problem? I'm unsure what method to use. </p>
| DeepSea | 101,504 | <p>Use the law of cosine for each triangle $\triangle LOK_i, i = 1,\cdots, 11$, we have: $LK_i^2 = LO^2+OK_i^2 - 2LO\cdot OK_i\cos \theta_i, \theta_i = \angle LOK_i\Rightarrow LK_i^2 = 3^2+2^2-12\cos \theta_i=13-12\cos \theta_i\Rightarrow S = \displaystyle \sum_{i=1}^{11}LK_i^2=13\cdot 11-12\displaystyle \sum_{i=1}^{11... |
466,909 | <p>How many ways can you split the numbers 1 to 5 into two groups of varying size?
For example: '1 and 2,3,4,5' or '1,2 and 3,4,5' or '1,2,3 and 4,5'. How many combinations are there like this? What is the formula?</p>
| Sarthak Goyal | 528,258 | <p>This is a basic application of the famous "Stirling Numbers of the second kind".</p>
<p>In case of $n$-subset-$2$, the formula is given by $2^{n-1} - 1$.</p>
<p>So, here, we get the answer to be $2^{5-1} - 1$, i.e. $15$.</p>
|
3,912,722 | <blockquote>
<p>Circle of radius <span class="math-container">$r$</span> touches the parabola <span class="math-container">$y^2+12x=0$</span> at its vertex. Centre of circle lies left of the vertex and circle lies entirely within the parabola. What is the largest possible value of <span class="math-container">$r$</span... | Jake Mirra | 278,017 | <p>First of all, you have a typo, the circle is <span class="math-container">$ (x + r)^2 + y^2 = r^2 $</span>.</p>
<p>Clearly since <span class="math-container">$ x \leq 0 $</span> on the parabola, any intersections of the circle with the parabola will have <span class="math-container">$ x \leq 0 $</span>. <span class... |
2,611,450 | <p>I'm having trouble understanding the description of the event in italics (with P = 0.8) in the question below. </p>
<p>"The probability that an American industry will locate in Shanghai is P(S) = 0.7.<br>
The probability that it will locate in Beijing is P(B) = 0.4.<br>
And the <em>probability that it will locate i... | Graham Kemp | 135,106 | <blockquote>
<p>The problem is, I know how to check those properties for simple relations (i.e. checking that $xRy→yRx$), but I don't understand how to do it in this example, because I don't really understand the relation. </p>
</blockquote>
<p>$\def\R{\operatorname R}\R$ is symmetric if $\forall s\in S\;\forall t\i... |
4,531,998 | <p>As you increase the value of n, you will generate all pythagorean triples whose first square is even. Is there any visual proof of the following explicit formula and where does it come from or how to derive it?</p>
<p><span class="math-container">$(2n)^2 + (n^2 - 1)^2 = (n^2 + 1)^2$</span></p>
<div class="s-table-co... | Constantinos Pisimisis | 1,087,555 | <p>The formula that you are referring is a sub-case of the Euclid's formula<br />
Accroding to the Euclid's formula it is true that:
<span class="math-container">$$\text{Given an arbitrary pair of integers m and n with m$\gt$n and m,n$\gt$0}$$</span>
<span class="math-container">$$a=m^2-n^2 \text{ , } b=2mn \text{ , ... |
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