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 |
|---|---|---|---|---|
180,122 | <p>Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth?</p>
<p>Zero section is a map $\xi:M\rightarrow E$ defined by $$\xi(p)=0\qquad\forall p\in M.$$</p>
| Mariano Suárez-Álvarez | 274 | <p>Smoothness can be checked locally on $M$ and locally $E$ is trivial. </p>
<p>Can you use these two facts to conclude what you want?</p>
|
1,113,934 | <p>How to prove that $\mathrm{Hom}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$ ?</p>
<p>(We treat it as field homomorphisms. )</p>
<p>I know that $\mathrm{Aut}(\mathbb{R})=\{\mathrm{id}\}$ and $\mathrm{Mon}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$.</p>
| Krish | 177,430 | <p><em>Hint:</em> Let $\phi: \mathbb R \to \mathbb R$ is a ring homomorphism. Prove the followings: (1). $\phi(r) = r, \forall r \in \mathbb Q.$ (2). $\phi(x) > 0, \forall x >0.$ (3). $|x-y|<\frac{1}{m} \Rightarrow |\phi(x) - \phi(y)|<\frac{1}{m}.$</p>
|
195,432 | <p>Mathematica 12 has a new function <code>FunctionCompileExportLibrary</code>, which can export a pure function as a <code>.dll</code> file like this:</p>
<pre><code>FunctionCompileExportLibrary["function.dll", Function[
Typed[arg, "MachineInteger"], arg + 1]]
</code></pre>
<blockquote>
<p><code>"...\\function.... | ilian | 145 | <p>This is a quick example of calling the exported compiled function from a standalone executable on Windows.</p>
<p>A few things to note which are currently undocumented and that can definitely be subject to change (and improvement) in the future: </p>
<ul>
<li><p>the function is named <code>Main</code> by default a... |
2,161,913 | <p>Consider the angle $\theta$ with $$0<\theta<\frac{\pi}{2}$$ Suppose that $$cos\theta=\frac{l}{n} \space and \space \space sin\theta=\frac{m}{n}$$
are rational numbers, with postive integers $l,m$ and $n$. Show that $l$ and $m$ cannot both odd integers.<br>
MY ATTEMPT: Assume that $l$ and $m$ are both odd int... | Mark Viola | 218,419 | <p>Let $\epsilon>0$ be given. Take $N$ so large that $|\log(\cos(1/i))|<\epsilon/2$ whenever $i>N$. Then, we can write for $n>N+1$</p>
<p>$$\begin{align}
\left|\log\left(\sqrt[n]{\prod_{i=1}^n\sec(1/i)}\right)\right|&=\frac1n\left|\sum_{i=1}^n\log(\cos(1/i))\right|\\\\
&\le\frac1n\sum_{i=1}^N|\lo... |
4,037,206 | <p><a href="https://i.stack.imgur.com/VfORl.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VfORl.png" alt="enter image description here" /></a></p>
<p>I have done the following for the median <br/>
<span class="math-container">$2(x-1) = .50$</span> <br/>
results in <span class="math-container">$x = ... | heropup | 118,193 | <p>The median is the number <span class="math-container">$m$</span> for which <span class="math-container">$$\int_{x=-\infty}^m f(x) \, dx = \frac{1}{2}.$$</span> So you need to solve <span class="math-container">$$\frac{1}{2} = \int_{x=1}^m 2(x-1) \, dx = \left[x^2 - 2x\right]_{x=1}^m = m^2 - 2m - (1-2) = (m-1)^2.$$<... |
733,124 | <p>I understand that reals are defined as "completing" the rationals, which (at least in ZFC) are in turn derived from the natural numbers.</p>
<p>So, if ordinal numbers are viewed as an extension of the natural numbers, are there constructs of ordinals corresponding in the same way as rationals and reals are derived ... | Asaf Karagila | 622 | <p>These are called <strong>Surreal Numbers</strong>. It is a proper-class sized order field which is complete, in the sense that every subset has a least upper bound and greatest lower bound.</p>
<p>It embeds all the ordered fields in the universe [of set theory], and all the ordinals too. It does not, however, exten... |
733,124 | <p>I understand that reals are defined as "completing" the rationals, which (at least in ZFC) are in turn derived from the natural numbers.</p>
<p>So, if ordinal numbers are viewed as an extension of the natural numbers, are there constructs of ordinals corresponding in the same way as rationals and reals are derived ... | Stella Biderman | 123,230 | <p>The complex numbers are the completion of the reals in the algebraic sense, by which I mean the set of solutions to polynomials in $\mathbb{R}[x]$ is $\mathbb{C}$.</p>
|
1,609,446 | <p>I read the definition that $f$ is in $O(x^n)$ if $|f(x)|<C|x^n|$ for some $C$.</p>
<p>I'm struggling to understand how to check this. For example, supposedly $f(x) = 5x+3x^2$ is in $O(x)$ but not $O(x^2)$?</p>
<p>If I plot $f(x) = 5x + 3x^2$ and $g(x)=x$ I see that the first goes to infinity much quicker.</p>
... | Dr_Be | 242,338 | <p>The definition is not complete since the inequality is supposed to hold for all $x>k$ where $k$ is a constant. The idea behind the $\mathcal O$-notation is to provide an estimate (comparison) for large $x$. This makes life much easier for your example since $x < x^2 \ \forall\, x>1$, so that </p>
<blockquo... |
1,335,734 | <blockquote>
<p>A library leases its photocopier. One monthly bill was 750 dollars for
12,000 copies. Another month, the bill was $862.50 for 16,500 copies.
How much does the library pay for each copy?</p>
</blockquote>
<p>How should I start set up the equation for this problem?
How exactly do I find the linear... | meiji163 | 201,018 | <p>Use the chain rule.</p>
<p>$$3\left({d\over{dx}}(x^2+y^2)^2\right)=3\cdot2{d\over{dx}}(x^2+y^2)(x^2+y^2)=6\left(2x+2{d\over{dx}}(y)y\right)(x^2+y^2)=6\left(2x+2y{dy\over{dx}}\right)(x^2+y^2)=12(x^2+y^2)\left(y{dy\over{dx}}+x\right)$$</p>
|
2,111,579 | <p>A question:</p>
<p>If $\displaystyle{m}^{2}={1}-{m},{\quad\text{and}\quad}{n}^{2}={1}-{n},{\quad\text{and}\quad}{n}\ne{m};$</p>
<p>Proof that $\displaystyle{m}^{7}+{n}^{7}+{30}={1}$</p>
<p>Without finding the roots of equation $\displaystyle{x}^{2}+{x}-{1}={0}$.</p>
<p>Is there such a shortcut solution?</p>
| Algebraic Pavel | 90,996 | <p>You can also use the induction along the dimension $n$ (leading to a simple but not very constructive proof though). </p>
<p>Let $A_n$ denote the matrix $A$ of dimension $n$ and $B_n$ its inverse. The statement is obviously true for $n=1$ and reduces to showing that $1=1$. Let the statement be true for $n-1$, that ... |
1,557,206 | <p>Given the series $\sum\limits_{n=1}^\infty a_n^2$ and $\sum\limits_{n=1}^\infty b_n^2$ converge. Show that the series $\sum\limits_{n=1}^\infty a_n b_n$ converges absolutely.</p>
<p>My idea so far:</p>
<ul>
<li>It's quite quite obvious that both given series converge absolutely</li>
<li>So the Cauchy-Produc tells ... | Mirko | 188,367 | <p>$\sum |a_n b_n|\le \sum b_n^2 + \sum a_n^2$ </p>
<p>spoiler: </p>
<blockquote class="spoiler">
<p> if $|a_n|\le|b_n|$ then $|a_n\cdot b_n|\le b_n^2$ else $|a_n\cdot b_n|\le a_n^2$, so $\sum |a_n b_n|\le \sum b_n^2 + \sum a_n^2$ </p>
</blockquote>
<p>Edit: Second spoiler (complete solution with all details): </... |
1,557,206 | <p>Given the series $\sum\limits_{n=1}^\infty a_n^2$ and $\sum\limits_{n=1}^\infty b_n^2$ converge. Show that the series $\sum\limits_{n=1}^\infty a_n b_n$ converges absolutely.</p>
<p>My idea so far:</p>
<ul>
<li>It's quite quite obvious that both given series converge absolutely</li>
<li>So the Cauchy-Produc tells ... | Mark Viola | 218,419 | <p>HINT:</p>
<p>From Cauchy-Schwarz we have </p>
<p>$$\sum_{n=1}^{N}\left|a_n\, b_n\right|\le\sqrt{\left(\sum_{n=1}^{N}a_n^2\right)\,\left(\sum_{n=1}^{N}b_n^2\right)}$$</p>
|
2,278,436 | <p>Does there exist integers $a, n > 1$ such that $1 + \frac{1}{1 + a} + \frac{1}{1 + 2a} + ... + \frac{1}{1 + na}$ is an integer? I have no clue how to begin. I've tried to simplify this somehow, but with no effect.</p>
| Dr. Wolfgang Hintze | 198,592 | <p><strong>Extended proof</strong></p>
<p>Here I extend the proof of my original post for $a = 2$ to arbitary integers $a > 2$.</p>
<p>Notice that I consider the proof incompete as it makes an assumption which I could not prove (but which was used by myself and others before).</p>
<p>Let</p>
<p>$$c(k)=1 + a \;k$... |
3,133,152 | <blockquote>
<p><span class="math-container">$g(x)=|\sin{x}-1|+|3-\cos{x}-\sin{x}|+2\sin{x}$</span></p>
<p>Answer: Above equality is simplified to <span class="math-container">$$1-\sin{x}+3-\cos{x}-\sin{x}+2\sin{x}=4-\cos{x}$$</span></p>
</blockquote>
<p><span class="math-container">$$-1 \le\sin{x}\le1$$</span></p>
<p>... | Carl Schildkraut | 253,966 | <p>Usually, we hear that <span class="math-container">$x=|x|$</span> is <span class="math-container">$x$</span> is "positive" and <span class="math-container">$x=-|x|$</span> if <span class="math-container">$x$</span> is "negative," and often the case of <span class="math-container">$x=0$</span> isn't really discussed.... |
2,487,709 | <p>I have to proof that if for any $z$:
$$\cos(z+\omega) = \cos(z)$$</p>
<p>then $\omega = 2\pi k \quad (k = 0, \pm 1, \pm 2 \ldots)$</p>
<p>I have no idea how to prrof it strictly</p>
| Anders Beta | 464,504 | <p>This integral is in general not well defined in the theory of distributions. Note that $f(x)$ is not a test function, and it has a singularity at $x=0$. If we try to give a meaning to it one runs into difficulties. Using the step function $H(x)$ one could try to give meaning to $I(x)$ as follows:</p>
<p>$$\begin{al... |
3,264,655 | <p>I've found myself where I've got a bidirectional graph of roughly 5500 nodes, 14000 edges.</p>
<p>I'd like to visit ~85 of these nodes, each <em>at least</em> once, starting and ending at the same node. I'd like to get a short route (don't need explicitly shortest, though that would be nice!). What graph processing... | sebastian | 604,625 | <p>What you could do for instance is the following: </p>
<p>For every one of your <span class="math-container">$85$</span> nodes use a shortest path algorithm (like Dijkstra) to find shortest paths to all other <span class="math-container">$84$</span> nodes. Use this information to build a new complete graph with <spa... |
3,772 | <p>In <a href="https://math.stackexchange.com/questions/102764/non-polynomial-functions-on-fields-of-finite-characteristic">this</a> question, there are two answers, both of which I up-voted, but I didn't "accept" either since there didn't appear to be a reason why one stands out as clearly preferable to the other. Ap... | Bill Dubuque | 242 | <p>My opinion is that it is perfectly fine not to accept any answer if one has good reasons to do so. </p>
<p>For example, during the month I was testing MathOverflow, I posted a <a href="https://mathoverflow.net/questions/31595/how-would-you-solve-this-tantalizing-halmos-problem">question</a> on a tantalizing proble... |
44,310 | <p>I wanted to draw (lattice-aligned) unit cubes inside a sphere with a given radius centered on the origin, so I wrote this program:</p>
<pre><code>Cube[x_, y_, z_] := Cuboid[{x - 1, y - 1, z - 1}, {x, y, z}];
Coords[r_] := {#1, #2, Floor[Sqrt[r^2 - #1^2 - #2^2]]} & @@ # & /@
With[{t = Sqrt[r^2 - 1]},
... | Mr.Wizard | 121 | <p>Focusing only on syntax I think your code looks pretty good, but there are a few things I note that I think could be improved.</p>
<ul>
<li><p>The method <code>f @@ # & /@ expr</code> can be replaced with <code>f @@@ expr</code>, which is shorthand for <code>Apply[f, expr, {1}]</code>. See <a href="http://refe... |
330,677 | <p>I was thinking last days about the following problem - the sum of Darboux is a Darboux function?</p>
<p>Do You know a proof or counter example?</p>
| Dominic Michaelis | 62,278 | <p>No this is not true, as darboux functions are really a general class of functions, you can proof that you can write any (really any) real valued function as the sum of two Darboux functions, look <a href="http://en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29#Darboux_function" rel="nofollow">here</a></p>
<... |
77,466 | <p>There is a standard syllabus for a first graduate course in algebra. One teaches groups,
rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of
category theory, perhaps a little bit of representation theory, all this a little bit superficially, to give an idea of the fundamental
algebraic st... | David White | 11,540 | <p>I'm posting a separate answer because I realized my first might be too much algebraic topology. Another great second year course idea is to follow <em><a href="http://books.google.com/books/about/Lectures_on_modules_and_rings.html?id=r9VoYbk-8c4C" rel="nofollow">Lectures on Modules and Rings</a></em> by T.Y. Lam. Th... |
1,016,585 | <p>I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$</p>
<p>How do I do this?</p>
<p>It looks like it is close to being an $\operatorname{arcsin}$?</p>
<hr>
<p>I would have thought I could just do:
$$\int \left(\sqrt{x^2 - c}\right)^{-\frac12}\, dx=\frac{2\sqrt{c+x^2}}{2x}\text{??... | Community | -1 | <p>If you try this derivative (which is not helpful in itself),
$$(\sqrt{x^2-1})'=\frac x{\sqrt{x^2-1}},$$
you can notice that by adding an $x$ term,
$$(x+\sqrt{x^2-1})'=1+\frac x{\sqrt{x^2-1}}=\frac{x+\sqrt{x^2-1}}{\sqrt{x^2-1}},$$
and
$$\frac{(x+\sqrt{x^2-1})'}{x+\sqrt{x^2-1}}=\frac1{\sqrt{x^2-1}}.$$</p>
<p>From thi... |
3,249,926 | <p>I’ve tried applying Vandermonde’s identity, but got stuck. Any help would be appreciated!</p>
| nonuser | 463,553 | <p>Remember that <span class="math-container">$a\leq b \iff a^2\leq b^2$</span> only if <span class="math-container">$a,b\geq 0$</span>.</p>
<p>If <span class="math-container">$x\geq 0$</span> then from <span class="math-container">$x\leq 3$</span> after squaring, we get <span class="math-container">$x^2\leq 9$</span>... |
1,782,796 | <p>Is there a particular name given to a matrix of <strong><em>m</em></strong> rows and <strong><em>n</em></strong> columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance:</p>
<pre><code>0 1 0 0 0 0 0
0 0 0 0 1 0 0
0 1 0 0 0 0 0
0 0 0 0 0 0 1
0 0 1 0 0 0 0
</code></pre>
<p>Note: I... | Ben Grossmann | 81,360 | <p>I'm not aware of a common name for this type of matrix. However, a nice way to describe it is that the rows of your matrix are rows of the identity matrix with possible repetition.</p>
|
1,782,796 | <p>Is there a particular name given to a matrix of <strong><em>m</em></strong> rows and <strong><em>n</em></strong> columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance:</p>
<pre><code>0 1 0 0 0 0 0
0 0 0 0 1 0 0
0 1 0 0 0 0 0
0 0 0 0 0 0 1
0 0 1 0 0 0 0
</code></pre>
<p>Note: I... | Anon | 245,264 | <p>I haven't heard of a name for the class of such matrices. If I had to suggest a name for such a matrix, I would call it a "row selection matrix", since if $A \in R^{m\times n}$ is such a matrix and $B \in R^{n\times k}$ an arbitrary matrix, then every row of $AB$ is also a row of $B$.</p>
|
403,977 | <blockquote>
<p>Let $u_n>u_{n+1}>0$ for all $n \in \Bbb N$, and suppose that $u_2+u_4+u_8+u_{16}+\dots$ diverges. Prove that $\sum_{n=1}^{\infty}\frac{u_n}{n}$diverges.</p>
</blockquote>
<p>Please provide me a hint or a full solution.</p>
| Jonas Meyer | 1,424 | <p>I suggest that you read about the <a href="http://en.wikipedia.org/wiki/Cauchy_condensation_test">Cauchy condensation test</a>.</p>
|
2,228,913 | <p>For some geometric problem that I am trying to solve, I am interested
in characterizing the cross product between a Bezier curve and its second order derivative.</p>
<p>So, let's $\mathbf{c}(t)$ be a Bezier curve, and $\ddot{\mathbf{c}}(t)$ it's derivative.</p>
<p>Is $\mathbf{w}(t) = \mathbf{c}(t) \times \ddot{\... | Steve T. | 435,649 | <p>Thanks for your answers. I was in fact trying to compute the coefficients of this Bezier curve, while proving that the product is indeed a Bezier curve.</p>
<p>A friend helped me to obtain this:</p>
<p>We write:</p>
<ul>
<li>$\mathbf{c}(t) = \sum_{i=0}^n \mathbf{P}_i B^n_i(t)$</li>
<li>$\mathbf{\ddot{c}}(t) = \su... |
1,392,857 | <p>Evaluate the integral from: </p>
<p>$$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx$$</p>
<p>The way I approach this problem is </p>
<p>$$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx = \frac{1}{2}\int_{-\infty}^{\infty} \frac{z \cdot e^{i2z}}{(z - i\sqrt{3})(i+i\sqrt{3})}dz$$
and
$$ \text{Res}_{i\sqrt3}(f... | GeorgSaliba | 142,772 | <p>First of all, keep in mind that
$$\sin(2z)=\Im (e^{2iz})$$
Second, the residue is wrong, you've forgotten the $z$ in the numerator:
$$\text{Res}_{i\sqrt{3}}(f(z))=\frac{P(z).e^{i2z}}{Q'(z)}$$</p>
<p>$$\text{Res}_{i\sqrt{3}}(f(z))=\frac{z.e^{i2z}}{2z}$$
$$\text{Res}_{i\sqrt{3}}(f(z))=\frac{e^{i2z}}{2}$$</p>
<p>So ... |
1,392,857 | <p>Evaluate the integral from: </p>
<p>$$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx$$</p>
<p>The way I approach this problem is </p>
<p>$$\int_0^{\infty} \frac{x \cdot \sin(2x)}{x^2+3}dx = \frac{1}{2}\int_{-\infty}^{\infty} \frac{z \cdot e^{i2z}}{(z - i\sqrt{3})(i+i\sqrt{3})}dz$$
and
$$ \text{Res}_{i\sqrt3}(f... | Math-fun | 195,344 | <p>Here is another approach:</p>
<p>$$f(a)=\int_0^{\infty} \frac{x \cdot \sin(ax)}{x^2+3}dx$$ take a Laplace transform with respect to $a$ to obtain
\begin{align}
\mathcal{L}(f(a))&=\int_0^{\infty} \frac{x^2}{(x^2+3)(x^2+s^2)}dx\\
&=\frac{\pi}{2\sqrt3+2s}
\end{align}
now take an inverse Laplace to obtain
$$f(a... |
159,777 | <p>In his <a href="http://www.ams.org/mathscinet-getitem?mr=44124">short paper</a> (1951) and <a href="http://www.ams.org/mathscinet-getitem?mr=74819">longer monograph</a> (1955), Postnikov introduced what I believe are now called <a href="http://en.wikipedia.org/wiki/Postnikov_system"><em>Postnikov systems</em> or <em... | Neil Strickland | 10,366 | <p>First, you say:</p>
<blockquote>
<p>It is my understanding that Postnikov systems have since then been
widely adopted, as a way of totally encoding the homotopy type of a
topological space.</p>
</blockquote>
<p>I don't think that this is really true. There are very few cases where one can actually describe ... |
224,474 | <p>I want to generate the following matrix for any n:</p>
<pre><code> Table[A[i, j], {i, 0, n}, {j, 0, n}];
</code></pre>
<p>Where,</p>
<pre><code> A ={{0,1,0},{0,0,2},{0,0,0}} when n=2;
A={{0,1,0,0},{0,0,2,0},{0,0,0,3},{0,0,0,0}} when n=3;
A={{0,1,0,0,0},{0,0,2,0,0},{0,0,0,3,0},{0,0,0,0,4},{0,... | Alan | 19,530 | <p>The following approach is easily adapted to other cases. (But Bob Hanlon's answer is the right one for the original question.)</p>
<pre><code>makeMat[n_] := With[{
zeros = ConstantArray[0, {n, n}],
rules = (Rule[{#, 1 + #}, #] &) /@ Range[n - 1]
},
ReplacePart[zeros, rules]]
</code></pre>
|
3,926,580 | <p>I'm trying to prove with squeeze theorem that the limit of the following series equals 1:</p>
<p><span class="math-container">$$\frac{1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}}{n}$$</span></p>
<p>For the left side of the inequality I did:</p>
<p><span class="math-container">$$\frac{1+\sqrt{1}+\sqrt[3]{1}+...+\sqrt[n]{1... | AlanD | 356,933 | <p>This is not a full answer to the question, but many answers are implying that the function <span class="math-container">$n\mapsto n^{1/n}$</span> is strictly increasing. This is not the case. To see this:</p>
<p>Let <span class="math-container">$y=x^{1/x}$</span>. Then <span class="math-container">$\ln y=\frac 1x \l... |
3,898,017 | <p><span class="math-container">$$\frac{1}{2z-1} = f(z)$$</span></p>
<p>So I don't think CR theoreum applies. So if the denominator is 0, then it's not defined. If <code>z = 1/2</code>, then it is not defined at 0. So <code>f(z)</code> it is not analytic even within C right? So there does not exist a simply connected d... | Mark Viola | 218,419 | <p><strong>METHODOLOGY <span class="math-container">$1$</span>: CONTOUR DEFORMATION</strong></p>
<p>We can deform the contour <span class="math-container">$|z|=1$</span> and apply Cauchy's Integral Theorem to assert that for <span class="math-container">$0<r<1/2$</span></p>
<p><span class="math-container">$$\beg... |
3,898,017 | <p><span class="math-container">$$\frac{1}{2z-1} = f(z)$$</span></p>
<p>So I don't think CR theoreum applies. So if the denominator is 0, then it's not defined. If <code>z = 1/2</code>, then it is not defined at 0. So <code>f(z)</code> it is not analytic even within C right? So there does not exist a simply connected d... | Raffaele | 83,382 | <p>Using <a href="https://en.wikipedia.org/wiki/Contour_integration#Evaluation_with_residue_theorem" rel="nofollow noreferrer">residue theorem</a> we easily get
<span class="math-container">$$\oint\limits_C {\frac{{dz}}{{2z - 1}} = 2\pi i}\;\text{Res}\left(\frac{1}{2 z-1},\left\{z,\frac{1}{2}\right\}\right) =\pi i$$</s... |
388,875 | <p>For $V$ a vector space.</p>
<p>Let $T\colon V\rightarrow V$ be a linear transformation of a finite dimensional inner product space.</p>
<p>How can I prove that $\ker(T^* T + TT^*) = $\ker(T)$\cap$$\ker(T^*)$?</p>
<p>And what is the meaning of $\ker(T^*T + TT^*)$ ?</p>
<p>Is $\ker(T^*T + TT^*)=\ker(T^*T)+\ker(TT... | wantToLearn | 74,557 | <p>any suggestion ?</p>
<p>I know how to prove that for every vector that is in the kernel of T and T*</p>
<p>is also at the kernel of TT*+T*T (very trivial).</p>
<p>but how to prove the other side ?</p>
|
497,609 | <p>There are 30 balls in a bag. There are 6 balls each of 5 different colors. What is the minimum number of times I have to pick a ball to pick 2 balls of each color?</p>
<ol>
<li>24</li>
<li>25</li>
<li>26</li>
<li>27</li>
<li>28</li>
</ol>
<p>Is the question correct? Someone asks me this and I'm confused.</p>
| André Nicolas | 6,312 | <p>If we want to have <strong>certainty</strong> of picking at least two balls of each colour. The worst case is if we have picked up all the balls of every colour but yellow, and only one yellow. So $25$ is not enough. </p>
<p>But $26$ is enough. For if we pick $26$, there are only $4$ balls left, so we have at least... |
3,826,152 | <p>How can I verify this inequality by using induction?</p>
<p><span class="math-container">$$\frac1{2n}\le \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$$</span></p>
<p>I understand the principle of induction, but I'm struggling with how my teacher is solving this problem. I Highlighted the area I'm stuck ... | Bernard | 202,857 | <p>Your teacher simply uses transitivity of inequalities:</p>
<p>To simplify the notations, I'll use double factorials.</p>
<p>The inductive hypothesis is that <span class="math-container">$\;\dfrac1{2(k-1)}\le \dfrac{(2k-3)!!}{(2k-2)!!}$</span>, and he/she wants to deduce that
<span class="math-container">$$\dfrac1{2k... |
1,826,094 | <p>$i\cdot\bar{z} = 2+2i$</p>
<p>I know that $\bar{z} = a-bi$ so then i get $i(a-bi)=2+2i$
Then $ai+b=2+2i$ (because $i^2=-1$)</p>
<p>When 2 complex numbers are equal you usually can equal their parts
Ex: $2+2i=a+bi$ so $a=2$ and $b=2$</p>
<p>But in this case I aint got $a+bi$ I have $ai+b$</p>
<p>The answer is $z=... | egreg | 62,967 | <p>Suppose $z=a+bi$; then $\bar{z}=a-bi$ and $i\bar{z}=ai+b=b+ai$. Thus the real part of $i\bar{z}$ is $b$ and the imaginary part is $a$.</p>
<p>Hence, equating them gives $b=2$ and $a=2$.</p>
<p>However, you can do it more easily: since $i^{-1}=-i$, multiplying both sides by $-i$ gives
$$
\bar{z}=-i(2+2i)=2-2i
$$
an... |
2,057,014 | <p>$a_n, b_n \in \mathbb R; n \in \mathbb N; x \in \mathbb R \land x > 1$ are defined as:</p>
<p>$$a_1 = 1 \land b_1 = x$$
$$a_{n+1} = \frac {2a_nb_n} {a_n+b_n} \land b_{n+1} = \frac {a_n+b_n} 2$$</p>
<p>How can I prove that $a_nb_n < a_n+b_n$ ?</p>
| ajotatxe | 132,456 | <blockquote>
<p>Your claim is true if and only if $x\le 4$.</p>
</blockquote>
<p>We will use induction to show some facts about both sequences. Namely,</p>
<p>$$a_{n+1}b_{n+1}=\frac{2a_nb_n}{a_n+b_n}\cdot\frac{a_n+b_n}2=a_nb_n=x$$</p>
<p>$$b_{n+1}-a_{n+1}=\frac{a_n+b_n}2-\frac{2a_nb_n}{a_n+b_n}=\frac{(b_n-a_n)^2}{... |
2,057,014 | <p>$a_n, b_n \in \mathbb R; n \in \mathbb N; x \in \mathbb R \land x > 1$ are defined as:</p>
<p>$$a_1 = 1 \land b_1 = x$$
$$a_{n+1} = \frac {2a_nb_n} {a_n+b_n} \land b_{n+1} = \frac {a_n+b_n} 2$$</p>
<p>How can I prove that $a_nb_n < a_n+b_n$ ?</p>
| robjohn | 13,854 | <p>$\frac{2ab}{a+b}$ is the harmonic mean of $a$ and $b$ while $\frac{a+b}2$ is the arithmetic mean. When $a,b\gt0$, the harmonic mean is less than or equal to the geometric mean $\left(\sqrt{ab}\right)$, which is less than or equal to the arithmetic mean.</p>
<p>It is useful to note that the harmonic mean of $x$ and ... |
47,177 | <p>Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do with physical chemistry: cond-mat, fluids, QM of molecules, and analysis of spectra. I'm more interested in learning ... | pharmine | 15,371 | <p><strong>Organic chemistry</strong></p>
<p>S. Fujita's "<a href="http://rads.stackoverflow.com/amzn/click/0387541268">Symmetry and combinatorial enumeration in chemistry</a>" (Springer-Verlag, 1991) is one such endeavor. It mainly focuses on stereochemistry.</p>
<p><strong>Molecular biology and biochemistry</strong... |
3,312,959 | <p>If the distance from the point <span class="math-container">$(K, 4)$</span> to <span class="math-container">$(2, 4)$</span> and y axis are same find k. </p>
<p>I can't understand what to calculate. According to my calculation the answer is 1, but the given answer is 0 and 4.</p>
| Fareed Abi Farraj | 584,389 | <p>I think the question is not very clear. You understood the question as "the distance from <span class="math-container">$(2,4)$</span> to <span class="math-container">$(K,4)$</span> is equal to the distance from <span class="math-container">$(K,4)$</span> to the y-axis". But actaully the question is stating that "the... |
2,193,306 | <p>I have to evaluate the integral of the $\sec x$ function and I do it as follows $$\int\sec xdx=\int\frac{dx}{\cos x}=\int\frac{\cos^2 x+\sin^2 x}{\cos x}dx=\sin x+\int\frac{\sin^2x}{\cos x}dx$$ Now we make a change of variables $\cos x=t$ so our integral becomes $$\sin x-\int\frac{\sqrt{1-t^2}}{t}dt$$ Now we make an... | imranfat | 64,546 | <p>Just another approach: <span class="math-container">$\frac{1}{\cos x}=\frac{\cos x}{\cos^ {2x}}=\frac{\cos x}{1-\sin^{2x}}$</span>.And performing a substitution <span class="math-container">$cosx=t$</span> in <span class="math-container">$\int\frac{{\cos}xdx}{1-\sin^{2x}}$</span> results in a doable integral. </p>
|
114,831 | <p>$$A_n=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}$$
Try to prove $$\lim_{n \to \infty}n(\ln 2-A_n) = \frac{1}{4}$$</p>
<p>I try to decompose $\ln 2$ as $$\ln(2n)-\ln(n)=\ln\left(1+\frac{1}{2n-1}\right)+\dots+\ln\left(1+\frac{1}{n}\right)\;,$$ but I can't continue, is that right?</p>
| TMM | 11,176 | <p>Let's cheat and use one of Euler's many results:</p>
<p>$$\sum_{i=1}^{n} \frac{1}{i} = \ln n + \gamma + \frac{1}{2n} + O\left(\frac{1}{n^2}\right)$$</p>
<p>Note that:</p>
<p>$$A_n + \sum_{i=1}^{n} \frac{1}{i} = \sum_{i=1}^{2n} \frac{1}{i}$$</p>
<p>Substituting Euler's result for both summations, we get:</p>
<p>... |
114,831 | <p>$$A_n=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}$$
Try to prove $$\lim_{n \to \infty}n(\ln 2-A_n) = \frac{1}{4}$$</p>
<p>I try to decompose $\ln 2$ as $$\ln(2n)-\ln(n)=\ln\left(1+\frac{1}{2n-1}\right)+\dots+\ln\left(1+\frac{1}{n}\right)\;,$$ but I can't continue, is that right?</p>
| Ragib Zaman | 14,657 | <p>Trying to follow the idea of the OP, write $$ \log 2 = \log (2n+2) - \log (n+1) = \log \left( 1+ \frac{1}{2n+1} \right) + \log \left( 1+ \frac{1}{2n} \right) + \cdots + \log \left( 1+ \frac{1}{n+1} \right) $$</p>
<p>so then $$ n( \log 2 - A_n) = n\log \left(1+ \frac{1}{2n+1} \right) + n\sum_{k=1}^n \left( \log \lef... |
812,143 | <p>City a is separated by a 2km wide river and are located as shown in Figure 1 (not drawn to scale). A road is to be built between city A to B that crosses a bridge straight across the river. </p>
<p>Use the Sign Test to determine the type of stationary point </p>
<p>Test to see if you have a minimum</p>
| Claude Leibovici | 82,404 | <p>If you want to do with algebra, $A$ being defined as the origin, let us define two points $C(x,9)$ and $D(x,11)$. So, the distance between $A$ and $C$ is given by $$d_{AC}=\sqrt {(x-0)^2+(9-0)^2}=\sqrt {x^2+81}$$ The distance between $D$ and $B$ is given by $$d_{DB}=\sqrt {(17-x)^2+(14-11)^2}=\sqrt {x^2-34x+298}$$ S... |
3,005,591 | <p>I wish to find the integers of <span class="math-container">$a,b,c$</span> and <span class="math-container">$d$</span> such that:
<span class="math-container">$$225a + 360b +432c +480d = 3$$</span>
which is equal to:
<span class="math-container">$$75a + 120b +144c+ 160d =1$$</span></p>
<p>I know I have to use the... | user21820 | 21,820 | <p>You can <strong>systematically</strong> solve any such equation (or prove that there are no solutions) by the following:</p>
<p>Take any integers <span class="math-container">$a,b,c,d$</span>. Then the following correspond:</p>
<ul>
<li>Solutions of <span class="math-container">$75a+120b+144c+160d = 1$</span></li>... |
2,640,143 | <h2>Problem</h2>
<p>Compute following:</p>
<p>$$ \int_{1}^{2} \int_{0}^{z} \int_{0}^{y+z}\frac{1}{(z+y+x)^3}dxdydz $$</p>
<h2> Attempt to solve </h2>
<p>Now i could probably solving this by opening $(z+y+x)^3$ parenthesis and then solving 3 separate integrals involving variables $x,y,z$. Opening these parenthesis w... | Mohammad Riazi-Kermani | 514,496 | <p>While the logarithmic function is obviously the inverse of the exponential function it's main purpose is to convert large numbers to smaller numbers.</p>
<p>For example for $$x=100,000,000,$$</p>
<p>$$ log(x)= 8.$$ </p>
<p>From $$log(xy)=log (x) + log (y)$$ we get $$log(3600) = 2+log(36)=2+2(log2 +log3)$$
Applica... |
4,143,759 | <p><strong>What I understand so far :</strong></p>
<p>Intuitively this question gives me a yuuuuuge hint, it tells me at t=0, t = -4. So that let's me know that I have a phase shift of -4 immediately.</p>
<p>I understand what a period is, I understand what a phase shift is, I understand what amplitude and vertical disp... | DMcMor | 155,622 | <p><strong>Hint:</strong> You know that <span class="math-container">$V = \frac{4}{3}\pi r^{3}$</span>. Use implicit differentiation to take the derivative of both sides with respect to <span class="math-container">$t$</span>. You should have <span class="math-container">$\frac{dV}{dt}$</span> on the left, and someth... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Jonas Meyer | 1,119 | <p>The spectral theory of commutative Banach algebras led to an elegant proof, due to Gelfand, of the following (previously difficult) theorem of Wiener:
If $f$ is a nowhere vanishing complex valued function on the unit circle whose Fourier coefficients are absolutely summable, then the Fourier coefficients of $1/f$ ar... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Jonas Meyer | 1,119 | <p>Complex analysis and the theory of spaces of analytic functions on open subsets of $\mathbb{C}$ are used to prove the Riemann Mapping Theorem, which has the nice topological consequence that there is a unique homeomorphism class of (nonempty) simply connected open subsets of $\mathbb{C}$.</p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | José Figueroa-O'Farrill | 394 | <p>There are a number of algebraic theorems which are easier to prove using topology. The best known is probably <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra">the fundamental theorem of algebra</a>, but there are others. For example, an $n\times n$ matrix with positive entries always has an ei... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Ilya Grigoriev | 2,467 | <p>This is not an answer as much as a request for answers. Here are some topics
for which I don't know an example of this sort, but would really like to see
one (and nobody seems to have done them yet). Feel free to edit this post if
good examples appear for some of these topics, or if you have a topic to add to
the l... |
942,738 | <p>First of all I must state that I am not a mathematician, so please correct me if I use wrong terminology.</p>
<p>I am building a web application which needs to calculate the rating for each entity based on both the quantity and score of the reviews for that entity.</p>
<p>In other words, I don't want to just calcu... | Ross Millikan | 1,827 | <p>You can do all the things you speculate about. You need to decide how to weight them. You say you think 100 reviews of 9 are better than one review of 9.5, but how about 10 reviews of 9? Five? How about 90 reviews of 9 and 10 of 2-how does that compare to one review of 9.5? </p>
<p>One simple thing to do is gi... |
4,009,117 | <p>Suppose <span class="math-container">$X$</span>, <span class="math-container">$Y$</span>, and <span class="math-container">$Z$</span> are exhaustive events with <span class="math-container">$Y$</span> and <span class="math-container">$Z$</span> being mutually exclusive. Can <span class="math-container">$Y$</span> an... | Dylan C. Beck | 390,180 | <p>By the familiar definition (i.e., the "Wikipedia" definition), for a real-valued function <span class="math-container">$f(x),$</span> we have that <span class="math-container">$\lim_{x \to \infty} f(x)$</span> is a real number <span class="math-container">$L$</span> (if it exists) such that for every real ... |
3,634,490 | <p>Let <span class="math-container">$N \geq 1$</span> be an integer.</p>
<p>Let <span class="math-container">$X$</span> be a standard <span class="math-container">$\mathbb{R}^N$</span> Gaussian vector (all components are <span class="math-container">$\mathcal{N}(0, 1)$</span> and i. i. d.).</p>
<p>Let <span class="ma... | Leander Tilsted Kristensen | 631,468 | <p>I doubt, that there is an exact formula to do this (that is if we want exactly 95% confidence and not just at least 95% confidence). For instance consider the two matrices
<span class="math-container">$$A=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad , \quad B=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end... |
2,683,731 | <p>If a relation is not symmetric shall we say that it is anti symmetric?
In a quiz show's preliminary round i had this question looks easy but i couldn't answer.can anyone answer?</p>
| Levent | 261,831 | <p>Symmetric and anti-symmetric relations are not complements of each other. Consider $\{(a,b),(b,a),(a,c)\}$ over $\{a,b,c\}$. It is neither symmetric nor anti-symmetric.</p>
|
22,662 | <p>Suppose $V_1$ and $V_2$ are two $(g,K)$ modules of some reductive group $G$ with maximal compact $K$. Let $P$ be the minimal parabolic of $G$, $U$ its unipotent part, and $u$ its Lie algebra. Suppose the quotients $V_1/uV_1$ and $V_2/uV_2$ are isomorphic as modules for the Levi component of $P$, then what else do we... | Kevin Buzzard | 1,384 | <p>EDIT: I assumed the OP was asking about reductive groups over non-arch local fields. Emerton raises the possibility that the question is actually about groups over R or C, and he's probably right. So the answer below is probably irrelevant.</p>
<p>Do correct me if I'm wrong; I'm not an expert. But I thought that if... |
1,580,796 | <p>In a game players take turns saying up to 3 numbers (starting at 1 and working their way up) and who every says 21 is eliminated. So we may have a situation like the following for 3 players:</p>
<blockquote>
<p>Player 1: 1,2</p>
<p>Player 2: 3,4,5</p>
<p>Player 3: 6</p>
<p>Player 1: 7,8,9</p>
<p>Player 2:10,11</p>
<... | Antony | 517,240 | <p>You can use a strategy but only by the means on where you stand in the order of people, there is complex maths involved in probabilities but if numbers chosen ($1$-$3$) by each player are randomised then certain positions have a much lower chance of being forced into saying “$20+1$”</p>
<p>This was figured out usin... |
1,258,414 | <p>I want to prove that $$c = \{ (x_n)_{n\geq 0} \mid x_n \in \Bbb C \text{ and the sequence converges} \}$$ with the norm $$ \left\|(x_n)_{n\geq 0}\right\|_{\infty} =\sup_{n\geq 0} |x_n|$$ is a Banach space. I have done some work, but I am having trouble concluding.</p>
<p>So far: let $(\xi_n)_{n\geq 0} = \left( (x_... | shalop | 224,467 | <p>Here is a sketch. I will use all of the same notations you are using (You should definitely check it for mistakes)</p>
<p>For each $n$, then sequence $\xi_n=(x^{(n)}_k)_{k\geq 0}$ is a convergent sequence (because it is in $c$), so let us say that $\xi_n=(x^{(n)}_k)_{k\geq 0}$ converges to $u_n \in \mathbb{C}$.</p>... |
215,077 | <p>This question is partly connected with the following <a href="https://mathoverflow.net/questions/214931/connection-between-stallings-end-theorem-and-seifert-van-kampen-theorem">Connection between Stalling's end theorem and Seifert-van Kampen Theorem</a>. </p>
<p>By Stalling's Theorem a group with more than one ... | Lee Mosher | 20,787 | <p>While there's no single answer to this question, there are a lot of partial results.</p>
<p>To start with, there is the still unresolved <a href="http://eprints.soton.ac.uk/46386/1/KrophollerConjecture.pdf" rel="nofollow noreferrer">Kropholler-Roller conjecture</a>, although various special cases are known, see for... |
2,229,532 | <p>I have a circle which has a triangle inscribed in it.</p>
<p>The circle radius R = 4</p>
<p>The triangle ABC vertices divide circle into 3 arcs in 1:2:3 ratio</p>
<p>Find the perimeter and area of triangle.</p>
<p>Can you guys help me with this one? </p>
| Ángel Mario Gallegos | 67,622 | <p><strong>Hint:</strong></p>
<p>The ratio between the angles of the triangle is $1:2:3$, so the angles are $30^{\circ}$, $60^{\circ}$ and $90^{\circ}$. Now, the hypotenuse is also a diameter of the circle (why?) then the sides of the triangle are $4$, $4\cos 30^{\circ}$ and $4\sin 30^{\circ}$.</p>
|
2,229,532 | <p>I have a circle which has a triangle inscribed in it.</p>
<p>The circle radius R = 4</p>
<p>The triangle ABC vertices divide circle into 3 arcs in 1:2:3 ratio</p>
<p>Find the perimeter and area of triangle.</p>
<p>Can you guys help me with this one? </p>
| fz1795 | 435,765 | <p>Yes Angelo Mario. if you draw the triangle with these angles you could find the answer.(The longest side of the triangle goes through the center of the circle.)
the answer will be 13.856 </p>
|
2,229,532 | <p>I have a circle which has a triangle inscribed in it.</p>
<p>The circle radius R = 4</p>
<p>The triangle ABC vertices divide circle into 3 arcs in 1:2:3 ratio</p>
<p>Find the perimeter and area of triangle.</p>
<p>Can you guys help me with this one? </p>
| Alex | 293,781 | <p>The condition on the three arcs' $1:2:3$ ratio, together with the fact that the arcs add up to $360^\circ$, means that the arcs are $60^\circ$, $120^\circ$, and $180^\circ$. </p>
<p>The angles in triangle $ABC$ are inscribed angles, each being exactly half of the corresponding arc; so the angles are $30^\circ$, $60... |
1,401,661 | <blockquote>
<p>Dice are cubes with pips (small dots) on their sides, representing
numbers 1 through 6. Two dice are considered the same if they can be
rotated and placed in such a way that they present matching numbers on
the top, bottom, left, right, front, and back sides.</p>
<p>Below is an example of t... | Maverick Meerkat | 342,736 | <p>In the case there are <strong>no limitation</strong> on the way we sort the different values (i.e. it's not a normal dice which opposite sides sum up to 7), and there's also <strong>no rotation</strong> allowed (the cube is fixed) - then the answer would be 6! = <strong>720</strong> options.</p>
<p>(n-Factorial is ... |
1,401,661 | <blockquote>
<p>Dice are cubes with pips (small dots) on their sides, representing
numbers 1 through 6. Two dice are considered the same if they can be
rotated and placed in such a way that they present matching numbers on
the top, bottom, left, right, front, and back sides.</p>
<p>Below is an example of t... | Eric Padmore | 784,379 | <p>Assuming that the question refers to "normal" die, where the opposite sides must total seven, (why would you make any other kind?) I believe this is a spatial problem rather than a mathematical one.</p>
<p>Lets begin with the 4 side facing up, it is symmetrical around both the horizontal and vertical axis so it doe... |
200,414 | <p>Some people may carelessly say that you need calculus to find such a thing as a local maximum of $f(x) = x^3 - 20x^2 + 96x$. Certainly calculus is <em>sufficient</em>, but whether it's <em>necessary</em> is another question.</p>
<p>There's a global maximum if you restrict the domain to $[0,8]$, and $f$ is $0$ at t... | Community | -1 | <p>There is a kind of algebraic structure called a <em>real closed field</em>. It's a structure with constants $0$, $1$, and has operations $+$ and $\cdot$ and a relation $<$.</p>
<p>The theory of real closed fields is interesting, because any (first-order) logical statement you can make using just $0,1,+,\cdot,<... |
17,398 | <p>I am teaching my daughter, who is currently about <span class="math-container">$46$</span> months old, additions. She is very curious and asks a lot of <em>good</em> questions. For example, when I told her that <span class="math-container">$2+6=8$</span> and <span class="math-container">$4+4=8$</span>, she asked me ... | Jamie Watts | 13,034 | <p>Start with two equations that are the same in every way, like this:</p>
<p>2 + 2 = 4 and 2 + 2 = 4</p>
<p>Now, add "4" more to each of the two equations, one at a time, so you'll get 8 in each equation. But do it different ways in each of the two equations.</p>
<p>1.) Add all of the "4" more to the second "2" in ... |
933,515 | <p>Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ and the residual spectrum $\sigma_r(T)$ is defined to be $$\sigma_r(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has e... | Jonas Dahlbæk | 161,825 | <p>I think the property of having closed range is simply more useful in certain applications, so you want to single out the 'worst case scenario' by considering those $\lambda$ for which $(T-\lambda)$ fails to have closed range. Note for instance that in the Hilbert space setting, you have the very useful direct sum de... |
510,672 | <p>My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:</p>
<p>1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows</p>
<p>In addition to those, why isn't removing zero rows an elementary operation? It d... | Trevor Wilson | 39,378 | <p>Because we want the dimensions of the matrix to remain fixed. I can think of two reasons for this.</p>
<ol>
<li><p>So that performing an elementary row operation will correspond to multiplication on the left by an elementary matrix.</p></li>
<li><p>Changing the dimensions of the matrix by adding or removing zero r... |
510,672 | <p>My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:</p>
<p>1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows</p>
<p>In addition to those, why isn't removing zero rows an elementary operation? It d... | Marc van Leeuwen | 18,880 | <p>Whether you want addition or removal of rows with all entries zero to be allowed or not depends on the precise application one has for using row operations. If row operations are intended to model the action of the general linear group by left multiplication, then addition or removal of rows should be forbidden beca... |
510,672 | <p>My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:</p>
<p>1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows</p>
<p>In addition to those, why isn't removing zero rows an elementary operation? It d... | Community | -1 | <p>Here is a slightly more long-range answer: a matrix corresponds to a linear operator $T:V\to W$ where $V$ and $W$ are vector spaces with some chosen bases. The elementary row operations (or elementary column operations) then correspond to changing the basis of $W$ or of $V$ to give an equivalent matrix: one which re... |
13,553 | <p>(This issue has been discussed before, specifically <a href="https://math.meta.stackexchange.com/questions/8485/how-civil-should-we-be/8493">here</a>. A different kind of censorship has been discussed <a href="https://math.meta.stackexchange.com/questions/8306/defacement-censorship-of-peoples-answers">here</a>.)</p... | user642796 | 8,348 | <p>Answering just to provide some of my thoughts (as the ♦-mod in question):</p>
<ol>
<li><p>The answer was flagged. I don't go out of my way (and I don't think anyone goes out of their way) to find the odd swear word on the site. They're easy enough to find, however, for anyone who wants to.</p></li>
<li><p>I did fe... |
4,014,144 | <p>Let <span class="math-container">$f(x) = \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$</span>.
where <span class="math-container">$x, c_1, c_2, \dots, c_k \in \mathbb R^n$</span>.
What fast iterative methods are available for finding the (approximate) min of <span class="math-container">$f$</span> with the constraint <span c... | River Li | 584,414 | <p>We need to solve the following optimization problem (P<span class="math-container">$0$</span>):
<span class="math-container">\begin{align}
(\mathrm{P0})\qquad &\min_{x} \max\{c_1^\mathsf{T}x, \cdots, c_k^\mathsf{T}x\}\\
&\mathrm{s.t.}\quad \|x\|_2 = 1.
\end{align}</span></p>
<p>Assume that <span class="math-... |
1,794 | <p>I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose their meaning. (You know that feeling when you wake up after a lucid dream but as you think about it, it just evapora... | Joseph O'Rourke | 511 | <p>I look forward to the responses to this interesting question.
Instead of answering directly,
permit me to make an analogy to art.</p>
<p><a href="http://en.wikipedia.org/wiki/Frank_Stella" rel="nofollow noreferrer">Frank Stella</a> is a renowned artist whose influence and compositional talent cannot be questioned.
... |
235,425 | <p>Ok, here is what I have for the proof of this conjecture. Let me know if I'm on the right path? all input appreciated.</p>
<p>There exist integers $j$, $k$, and $m$, such that,
$b = aj $ and $ c = ajk.$ Then $c = ajk $ (substituting $aj$ for $b$)
let $m = jk$, then $c = ma, => a|c.$ </p>
| The Substitute | 17,955 | <p>Yes, $b=aj$ and $c=bk=ajk=(jk)a$. Since $jk \in \mathbb{Z}$, we have $a|c$.</p>
|
223,521 | <p>Suppose that $a^2-b^2 =x$ where $a,b,x$ are natural numbers.</p>
<p>Suppose $x$ is fixed. If there is one $(a,b)$ found, can there be another $(a,b)$?</p>
<p>Also, would there be a way to know how many such $(a,b)$ exists?</p>
| Mark Bennet | 2,906 | <p>Note that $a^2-b^2=(a+b)(a-b)$ and that $(a+b)=c$ and $(a-b)=d$ are either both odd or both even.</p>
<p>If $x=cd$ where $c>d$ and $c$ and $d$ have the same parity then $a=\frac {c+d}2, b=\frac {c-d}2$ are natural numbers which work for $x$.</p>
<p>It is always possible to express an odd number $x$ in this way,... |
223,521 | <p>Suppose that $a^2-b^2 =x$ where $a,b,x$ are natural numbers.</p>
<p>Suppose $x$ is fixed. If there is one $(a,b)$ found, can there be another $(a,b)$?</p>
<p>Also, would there be a way to know how many such $(a,b)$ exists?</p>
| ShreevatsaR | 205 | <p>You want $x = a^2 - b^2 = (a-b)(a+b)$. Let $m = a-b$ and $n = a+b$, then note that $a = (m+n)/2$ and $b = (n-m)/2$. For these to be natural numbers, you want both $m$ and $n$ to be of the same parity (i.e., both odd or both even), and $m \le n$. For any factorization $x = mn$ satisfying these properties, $a = (m+n)/... |
223,521 | <p>Suppose that $a^2-b^2 =x$ where $a,b,x$ are natural numbers.</p>
<p>Suppose $x$ is fixed. If there is one $(a,b)$ found, can there be another $(a,b)$?</p>
<p>Also, would there be a way to know how many such $(a,b)$ exists?</p>
| Adrián Barquero | 900 | <p>You need to exploit the fact that the right hand side of your equation can be factored. For example for part (1) of the exercise, if $x$ is odd, say $x = 2n+1$ for some integer $n$, then</p>
<p>$$
x = 2n + 1 = y^2 - z^2 = (y - z)(y + z)
$$ </p>
<p>Now try to consider a trivial factorization of $2n+1$ like $2n+1 = ... |
2,797,039 | <p><strong>1)</strong> Let $\{f_n\}$ be a sequence of nonnegative measurable functions of $\mathbb R$ that converges pointwise on $\mathbb R$ to $f$ integrable. Show that</p>
<p>$$\int_{\mathbb R} f = \lim_{n\to \infty}\int_{\mathbb R}f_n \Rightarrow \int_{E} f = \lim_{n\to \infty}\int_{E}f_n $$</p>
<p>for any measu... | Green.H | 456,457 | <p>Suppose the die shows $n$ (where $n \geq 2$). Then, the probability that among $n$ randomly pocked cards, the number of Jacks is equal to $2$ is </p>
<p>$$\frac{\binom{4}{2}\binom{48}{n-2}}{\binom{52}{n}}.$$</p>
<p>Namely, there are $\binom{4}{2}$ events that we choose $2$ out of $4$ Jacks and $\binom{48}{n-2}$ ev... |
2,365,120 | <p>I'm doing some revision on indices and surds. How do you simplify </p>
<blockquote>
<p>$(xy^2)^p\sqrt{x^q} $</p>
</blockquote>
<p>Bit confused because my textbook says the answer is </p>
<blockquote>
<p>$x^{p+q/2}y^{2p}$</p>
</blockquote>
<p>I can understand simplifying but only when it's the same base. I'm ... | Surb | 154,545 | <p>$$(x\cdot y^2)^p\cdot\sqrt{x^q}=x^p\cdot(y^2)^p \cdot(x^q)^{1/2}=x^p\cdot y^{2p}\cdot x^{q/2}\\=x^p\cdot x^{q/2}\cdot y^{2p}=(x^p\cdot x^{q/2})\cdot y^{2p}=x^{p+q/2}\cdot y^{2p}$$</p>
|
2,866,399 | <p>I was studying Ring and modules and now I am studying Group theory. I feel very confuse regarding generating set .
In Ring Theory if we have an Ideal $ \ I = < x,y > $
Then x doesn't contained in that Ideal, however in Group theory if we have $ \ G = < x, y >$ then x is in the group.
Can any one please... | Rob Arthan | 23,171 | <p>The right way to think of the subgroup $\langle X \rangle$ of a group $G$ generated by a subset $X$ of $G$ is that it is the smallest subgroup of $G$ that contains $X$. Or more formally, it is the intersection of all subgroups of $G$ that contain $X$. From that point of view, it is clear that $X \subseteq \langle X ... |
903,831 | <p>One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in parallel to ring of integers, I think converting a polynomial ring to a numerical (i.e. integer) ring is useful.</p>
... | Bill Dubuque | 242 | <p>If $\,R\,$ is a ring (or any algebraic structure) then one can <a href="https://math.stackexchange.com/a/190326/242">transport the structure</a> of $\,R\,$ to <em>any</em> set $\,S\,$ of the same cardinality, by push/pulling the algebra operations along any bijection of their underlying sets. When $\,S\,$ = $\,\Bbb ... |
2,873,309 | <p>Let's say each month a utility bill comes and it needs to be split among a variable number of roommates. Let's say each person occupied the house for a certain percentage of the month: person 1 100%, person 2 100%, person 3, 75%, person 4, 50%. How can I calculate how much each person has to pay? (The number of peop... | Micah | 30,836 | <p>If the goal is to get exactly the same result as the solution you've presented, there's pretty much no hope except doing it day-by-day. Just knowing the number of days each roommate occupied isn't enough information, because it's more expensive to live in the house when there are fewer other people living there.</p>... |
2,661,718 | <p>My basic understanding is that each time
I have 30% chance of winning the prize
so between 3 and 4 tries I should win it</p>
<p>cause .3 +.3 +.3 = 90% as I need to win only once with 3 try
.3+.3+.3+.3 = 120% with 4 try</p>
<p>I do remember a formula saying 1-(0.7)^3 = 65.7% chance but I dont remember what that ... | Dr. Sonnhard Graubner | 175,066 | <p>HINT: write $$\left(\frac{x}{\sqrt{x+1}}\right)^2\geq \left(\log(x+1)\right)^2$$
and split into two cases
$$\left(\frac{x}{\sqrt{x+1}}-\log(x+1)\right)\left(\frac{x}{\sqrt{x+1}}+\log(x+1)\right)\geq 0$$</p>
|
294,383 | <p>Evaluate :
$$\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$$</p>
| Mhenni Benghorbal | 35,472 | <p><a href="https://math.stackexchange.com/questions/513413/show-that-int-0-infty-frac-cosat-cosbtt-ln-fracba">Related technique</a>. Here is a closed form solution of the integral</p>
<p>$$\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x} = -\frac... |
1,219,851 | <p>If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ? Trivially $n$ cannot be even , so this leaves us only with the possibilities $n \equiv1,3,5( \mod 6) $ , but I cannot reduce the cases . Please help , Thanks in advance . </p>
| lab bhattacharjee | 33,337 | <p>$$2^{6a+1}+(6a+1)^2=64^a\cdot2+(6a+1)^2\equiv0\pmod3$$ as $64\equiv1\pmod3\implies64^a\equiv1^a$</p>
<p>Similarly,
$$2^{6a+5}+(6a+5)^2=64^a\cdot32+36a^2+60a+25\equiv0\pmod3$$ </p>
<hr>
<p>Alternatively, if $(n,3)=1,n\equiv\pm1\pmod3\implies n^2\equiv1$</p>
<p>$2^n+n^2\equiv2^n+1\pmod3\iff(-1)^n+1$</p>
<p>Now $... |
2,148,979 | <p>I want to find $A$ such that $$A\sum \limits_{n=m}^{\infty}1/n^{3}=1$$</p>
<p>for any natural value of $m$.</p>
| Robert Israel | 8,508 | <p>If you're interested in approximations,</p>
<p>$$A = 2m^2 - 2m + 1 - \frac{1}{6 m^2} - \frac{1}{6 m^3} + \frac{1}{12 m^4} + \frac{1}{3m^5} + O\left(\frac{1}{m^6}\right) \ \text{as}\ m \to \infty $$</p>
|
2,061,655 | <p>Could we compute the limits
$$\lim_{x\rightarrow 0}\frac{\sin (x)-x+x^3}{x^3} \\ \lim_{x\rightarrow 0}\frac{e^x-\sin (x)-1}{x^2}$$
without using the l'Hospital rule and the Taylor expansion?</p>
| Robert Israel | 8,508 | <p>Sure: use <a href="http://www.wolframalpha.com/input/?i=limit%20of%20(exp(x)-sin(x)-1)%2Fx%5E2%20as%20x%20-%3E%200" rel="nofollow noreferrer">Wolfram Alpha</a> or Maple or ...</p>
|
1,647,399 | <p>Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is nonempty.</p>
<p><strong>What I know:</strong> Since $X$ is compact, we need to use the fact that every sequence in $X... | Martín-Blas Pérez Pinilla | 98,199 | <p>Alternative idea:
$$\bigcap_{n=1}^\infty E_n = \emptyset\implies \bigcup_{n=1}^\infty(X\setminus E_n) = X$$</p>
|
1,327,755 | <p>Prove that the antipodal mapping $A: S^n \to S^n$ given by $A(p)=-p$ is an isometry. </p>
<p>I know that in order to prove that a map $f$ is an isometry of a smooth manifold $M$ it must hold true that
$$\langle v,u\rangle_p = \langle d_f v, d_fu \rangle_{f(p)}$$</p>
<p>Now I found online <a href="http://image.scie... | Lee Mosher | 26,501 | <p>To address the remaining question, the proof of the equation of tangent planes $T_p S^2 = T_{A(p)}S^2$ is obtained by tracing through the definitions and doing some multivariable calculus computations.</p>
<p>First, $S^2$ is the solution set of the equation
$$x^2 + y^2 + z^2-1=0
$$
Second, the gradient of the left... |
3,208,003 | <p>How do you calculate the sum of this series?</p>
<p><span class="math-container">$$ \sum_{n=1}^\infty 0.5n \left(\frac{2}{3}\right)^n $$</span></p>
| denklo | 562,687 | <p>There is a simple trick, for <span class="math-container">$a < 1$</span>:
<span class="math-container">$$\sum\limits_{n=0}^\infty n a^n = a\partial_a\sum\limits_{n=0}^\infty a^n = a\partial_a \frac{1}{1-a} = \frac{a}{(1-a)^2}$$</span></p>
|
3,208,003 | <p>How do you calculate the sum of this series?</p>
<p><span class="math-container">$$ \sum_{n=1}^\infty 0.5n \left(\frac{2}{3}\right)^n $$</span></p>
| Community | -1 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$\color{green}{\sum_{n=1}^\infty na^n}=\sum_{n=0}^\infty(n+1)a^{n+1}=a\color{green}{\sum_{n=1}^\infty na^n}+\sum_{n=0}^\infty a^{n+1}.$$</span></p>
<blockquote class="spoiler">
<p> <span class="math-container">$$|a|<1\implies(1-a)S=\frac a{1-a}.$$</s... |
3,208,003 | <p>How do you calculate the sum of this series?</p>
<p><span class="math-container">$$ \sum_{n=1}^\infty 0.5n \left(\frac{2}{3}\right)^n $$</span></p>
| PrincessEev | 597,568 | <p>Note that we can pull out the <span class="math-container">$0.5$</span> from the summation, and focus on </p>
<p><span class="math-container">$$\sum_{n=1}^\infty n \left( \frac 2 3 \right)^n$$</span></p>
<p>This bears resemblance to a geometric series. We know that, for a geometric series with ratio <span class="m... |
4,537,390 | <p>Let <span class="math-container">$C$</span> be a circle centred at <span class="math-container">$O$</span> and <span class="math-container">$P$</span> be a point inside the circle.</p>
<p>The line <span class="math-container">$OP$</span> intersects the circle <span class="math-container">$C$</span> at <span class="m... | Eugen Ionascu | 83,478 | <p>Let's extend PQ and have the intersection with the circle denoted by R. It is known that <span class="math-container">$$PQ\cdot QR=PQ_1\cdot PQ_2,$$</span> (similarity of triangles) so if we denote by <span class="math-container">$s=PQ+QR=PR$</span> and <span class="math-container">$p=PQ_1\cdot PQ_2$</span> we see t... |
1,764,162 | <p>I have $$f(x) = x \left| x - 1 \right|$$</p>
<p>Here my given value for $x$ is 1</p>
<p>And I need to test if the function is "continuous" when $x$ is $1$ and also when reaching
$$ f(1)$$</p>
<p>$$ \lim\limits_{x \to 1} f(x)$$</p>
<p>So I plug in 1 to the equation to find out the value of $f(1)$</p>
<p>$$ f(1)... | Siddharth Bhat | 261,373 | <p>I think you are confused between the definition of any old function and a continuous function.</p>
<p>As you said, it is <em>only</em> for continuous functions that the fact </p>
<p>$$\lim_{x \to a} f(x) = f(a)$$</p>
<p>Any discontinuous function <strong>will not</strong> exhibit this behaviour.</p>
<p>As an exa... |
26,736 | <p>I am still a newbie to <em>Mathematica</em>, so I am still getting used to the syntax and logic of the program - but I've been trying to do the following 3D plot:</p>
<pre><code>g[x_, t_] := y /. FindRoot[y == Tanh[(2*y + x)/t], {y, x, t}];
p1 = ContourPlot3D[{z ==
Sech[(2 g[x, t] + x)/t]^2/(t - 2 Sech[(2 g[x... | Michael E2 | 4,999 | <h2>Edit: Introduction</h2>
<p>In my original answer, I had noticed a few things about this problem but hadn't pursued them. Then every few days something would occur to me and I would poke around little, be done, and then another thing would occur to me. This problem has some features that I think make an interesti... |
3,592,368 | <p>I managed to solve this question using trigonometry. But I wondered if there'd be anyway of doing it using only synthetic geometry. Here it is.</p>
<blockquote>
<p>Let <span class="math-container">$ABC$</span> be a right isosceles triangle of hypotenuse <span class="math-container">$AB$</span>. Let also <span class=... | Michael Rozenberg | 190,319 | <p>Let <span class="math-container">$PD$</span> be a perpendicular from <span class="math-container">$P$</span> to the line <span class="math-container">$BC$</span>.</p>
<p>Thus, since a line <span class="math-container">$BC$</span> is a tangent to the circle and <span class="math-container">$\measuredangle DCP=\measu... |
701,677 | <ol>
<li><p>If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb C)$? E.g. branching rules</p></li>
<li><p>Similarly for the groups $\mathrm{SO}_n(\mathbb C) \subset \mathrm O_... | goblin GONE | 42,339 | <p>Assume for a contradiction that $k \in A \cap B$ and that $H$ is surjective. Then $H(K) = \langle \{k\}_A,\emptyset_B\rangle$ for some $K : A \cup B \rightarrow 2.$ Fix any such $K$.</p>
<p>Case 1. Assume $k \in K$. Then $k \in [H(K)]_0$ and $k \in [H(K)]_1$, thus $H(K) \neq \langle \{k\}_A,\emptyset_B\rangle,$ a c... |
1,218,912 | <p>My problem is following:
$$\binom{n}{r} + \binom{n+1}{r+1} + \binom{n+2}{r+2} + \dots + \binom{n+M}{r+M}$$</p>
<p>how can we reduce it to a more short solution</p>
<p>Here $\dbinom{n}{r} = \dfrac{n!}{r! (n-r)!}$ and thus same as regular.Please help me in solving the above expression</p>
| Brian M. Scott | 12,042 | <p>$$\begin{align*}
\sum_{k=0}^m\binom{n+k}{r+k}&=\sum_{k=0}^m\binom{n+k}{n-r}\\
&=\sum_{\ell=r}^{m+r}\binom{n-r+\ell}{n-r}\\
&=\sum_{\ell=0}^{m+r}\binom{n-r+\ell}{n-r}-\sum_{\ell=0}^{r-1}\binom{n-r+\ell}{n-r}\\
&=\binom{n+m+1}{n-r+1}-\binom{n}{n-r+1}\\
\end{align*}$$</p>
<p>by a standard identity.</p>... |
78,461 | <p>I'm trying to solve the following problem:</p>
<p>For a vector $v$ of length $c$, $\min \frac{\sum v[i]^4}{(\sum v[i]^2)^2}$ subject to $\sum v[i] = N$.</p>
<p>I can solve this numerically for a given $c$ using the following command: </p>
<pre><code>Minimize[{Total[z^4]/Total[z^2]^2, Total[z] == N}, z]
</code></... | David G. Stork | 9,735 | <p>Isn't this is a problem of mathematics more than programming? Under the stated conditions, by permutation symmetry of the indexes of $v$ all entries of $v$ should be the same (up to absolute value, because in the problem statement all elements are squared). Thus one answer is $v$ is an array of $c$ entries each of... |
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