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 |
|---|---|---|---|---|
2,917,848 | <p>Given a real square matrix <span class="math-container">$A$</span> and a vector <span class="math-container">$v$</span>, Krylov subspaces are given by:
<span class="math-container">$$\mathcal K_n(A,v) := \text{span}(v, Av, \cdots A^{n-1} v)$$</span>
These spaces are known to help solve numerical linear algebra probl... | user | 505,767 | <p><strong>HINT</strong></p>
<p>Let use <a href="https://en.wikipedia.org/wiki/Spherical_coordinate_system" rel="nofollow noreferrer"><strong>spherical coordinates</strong></a> with</p>
<ul>
<li><p>$h=r\sin \phi \cos \theta$</p></li>
<li><p>$k=r\sin \phi \sin \theta$</p></li>
<li><p>$t=r\cos \phi$</p></li>
</ul>
<p>... |
213,665 | <p><strong>I've tried 3 methods but all failed to do that.</strong></p>
<p>1st Method</p>
<pre><code>Apply[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>2nd Method</p>
<pre><code>Map[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>3rd Method</p>
<pre><code>Flatten[{1, {2, {3, 4}, 5}, 6}, {2}]
</co... | user1066 | 106 | <pre><code>lst[[2]]=Flatten@lst[[2]];lst
</code></pre>
<blockquote>
<p>{1, {2, 3, 4, 5}, 6}</p>
</blockquote>
|
2,507,247 | <p>Given a bimatrix game of
<span class="math-container">$$\left(\begin{matrix}(0,-1) & (0,0)\\(-90,-6)&(10, -10)\end{matrix}\right)$$</span>
<a href="https://i.stack.imgur.com/uY44c.jpg" rel="nofollow noreferrer">Source</a></p>
<p>How to find the nash equilibrium strategy for both players?</p>
| Gerhard S. | 474,939 | <p>I assume that the entries $(a,b)$ in the payoff-matrix are interpreted as: $a$ is the row player's payoff and $b$ is the column player's payoff. To find a mixed strategy Nash equilibrium you use the fact that for a mixed strategy to be optimal for a player, the player must be indifferent between the pure strategies ... |
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | KCd | 619 | <p>There is no such thing as an ideal method. Abandon the dream that there should be some kind of technique that "always works" unless you just want to find a general algorithm that works all the time, like the <a href="https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Zassenhaus_algorithm" rel="nofollow norefe... |
2,803,398 | <p>We know that in category of $\mathbb{Set}$ the inverse limit is the direct product. But I am looking for specific category in which inverse limit does not exist. Any comments would be highly appreciated.</p>
| hmakholm left over Monica | 14,366 | <p>You have an <em>injection</em> $(0,1)^2\to(0,1)$ -- but it is not <em>surjective</em>, because there is nothing that maps to, for example,
$$ \frac{1}{99} = 0.0101010101010\ldots $$</p>
<p>De-interlating the digits of this would produce $\langle 0,\frac19\rangle$, but that is not in $(0,1)^2$.</p>
<hr>
<p>However... |
2,637 | <p>Trying to round-trip expressions through JSON, I'm getting unexpected errors for held expressions, and would be grateful for advice or clues. Consider, first, something that works well</p>
<pre><code>Export[Environment["USERPROFILE"] <> "\\AppData\\Local\\test.json", {1, 2, 3},"JSON"]
</code></pre>
<p>and re... | celtschk | 129 | <p>The reason is that the JSON format is quite limited. It doesn't support arbitrary expressions. You can e.g. see that by trying the folllowing:</p>
<pre><code>Export["test.json",someSymbol,"JSON"]
</code></pre>
<p>You'll get the same error.</p>
<p>If your goal is just to pass the expressions around (i.e. on the ot... |
3,623,432 | <p>Say I have two independent normal distributions (both with <span class="math-container">$\mu=0$</span>, <span class="math-container">$\sigma=\sigma$</span>) one for only positive values and one for only negatives so their pdfs look like:</p>
<p><span class="math-container">$p(x, \sigma) = \frac{\sqrt{2}}{\sqrt{\pi}... | J.G. | 56,861 | <p>Your approach is workable (although the <span class="math-container">$\tfrac{1}{2\pi\sigma^2}$</span> should be <span class="math-container">$\tfrac{2}{\pi\sigma^2}$</span>), but there's a much easier way @callculus pointed out. Since <span class="math-container">$X,\,Y$</span> are independent, <span class="math-con... |
2,767,070 | <p>The intuition for $E[g(Y)|Y=y]$ would be that $g(Y)$ would play the role of a constant once $Y$ is fixed to a certain $y$ value. But how to show this more formally ? I can't seem to expand the equation below.</p>
<p>$E[g(Y)|Y=y]=\sum_{y} g(y)P[g(y)=y'|Y=y]$</p>
| B. Mehta | 418,148 | <p>It looks like you've got your variables confused. On the left, $y$ is a fixed value but on the right you're summing over it. I believe you should have $$\sum_{y'} g(y')P[Y=y'|Y=y].$$ From there, think about what the probability means in the cases $y = y'$ and $y \neq y'$, and the answer should become clear.</p>
|
163,672 | <p>Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$,
so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there are? </p>
| Aaron Meyerowitz | 8,008 | <p>Here are some minor comments and a few counts <strong>LATER</strong> and a (rather weak) lower bound of $$4\binom{2^{n-1}}{2^{n-2}} \approx 2^{2^{n-1}-n/2}$$</p>
<p>Another way to say this is label the $2^n$ vertices of an $n$-cube so that in each of the $n$ directions exactly half (i.e. $2^{n-2}$ ) of the edges ar... |
162,293 | <p>Consider a "curve" defined by a list of points in finite dimension (here, four):</p>
<pre><code> pts = Table[{Cos[t], 0, Sin[2 t], Sin[t]}, {t, Subdivide[0, 1, 99]}]
</code></pre>
<p>I used known functions to generate <code>pts</code> but of course I am not supposed to know the parametric equation of the curve the... | Ulrich Neumann | 53,677 | <p>The Square of curvature is <a href="https://i.stack.imgur.com/gdrRV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gdrRV.png" alt="enter image description here"></a></p>
<p>To solve your problem you only need good local approximations of the two derivatives, which you could get form Interpolatio... |
244,333 | <p>Consider this equation : </p>
<p><span class="math-container">$$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$</span></p>
<p>where <span class="math-container">$t$</span> varies from <span class="math-container">$0$</span> to <span class="math-container">$T$</span> , and <span class="math-container... | Egor Skriptunoff | 50,643 | <p>Let $\large t$ = time, $\large\phi$ = angle between velocities and $\large z$ = distance.<br>
The system would be as follows:<br>
$$\large\frac{dz}{dt}=u\cdot\cos\phi-v$$
$$\large\frac{d\phi}{dt}=-\frac{u\cdot\sin\phi}{z}$$
It can be easily proved that the following expression is an invar... |
3,327,435 | <p>I have no clue for the following problem: </p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a finite group, <span class="math-container">$p$</span> a prime number, <span class="math-container">$S$</span> a Sylow <span class="math-container">$p$</span> subgroup of <span class="math-container">$G$... | Arturo Magidin | 742 | <p>Because <span class="math-container">$X$</span> is contained in <span class="math-container">$Z(P)$</span>, it follows that <span class="math-container">$N_G(X)$</span> contains <span class="math-container">$P$</span>. That means that <span class="math-container">$N_G(Y) = N_G({}^gX)$</span> must contain <span class... |
2,156,357 | <p>if $H$ and $K$ are nonabelian simple groups prove that :</p>
<blockquote>
<p>$H$ $\times$ $K$ has exactly four distinct normal subgroups. </p>
</blockquote>
<p>Please help me prove this.</p>
| Nicky Hekster | 9,605 | <p>Hint: put $M=H \times \{1\}$, and $N=\{1\} \times K$. Then $M,N \unlhd G$, $G=MN$, and $M \cap N=\{(1,1)\}$. Now if $T \unlhd G$, what can you say about $T \cap M$ and $T \cap N$?</p>
|
787,894 | <p>Find the values of $x,y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$.</p>
<p>Tried Cauchy-Schwarz and AM - GM , unable to do.</p>
| sirfoga | 83,083 | <p><strong>Hint</strong>: take a look at the picture below, and all the problem will vanish...</p>
<p><img src="https://i.stack.imgur.com/tvQBv.png" alt="enter image description here"></p>
<p>In fact the picture shows the circle of equation $(x+5)^2 +(y-12)^2 =14$, and the line passing trough its centre and the origi... |
3,844,235 | <p>Suppose a matrix <span class="math-container">$A \in \text{Mat}_{2\times 2}(\mathbb{F}_5)$</span> has characteristic polynomial <span class="math-container">$x^2 - x +1$</span>. Is <span class="math-container">$A$</span> diagonalizable over <span class="math-container">$\mathbb{F}_5$</span>?</p>
<p>Normally, I would... | markvs | 454,915 | <p>The char polynomial has no roots in <span class="math-container">$\Bbb{F}_5$</span>, so the matrix is not diagonalizable over that field. It is diagonalizable over <span class="math-container">$\Bbb{F}_{25}$</span> because the char polynomial has two different roots in that field.</p>
|
954,130 | <p>I have to prove that the function $f(n)=3n^2-n+4$ is $O(n^2)$. So I use the definition of big oh:</p>
<blockquote>
<p>$f(n)$ is big oh $g(n)$ if there exist an integer $n_0$ and a constant $c>0$ such that for all integers $n\geq n_0$, $f(n)\leq cg(n)$.</p>
</blockquote>
<p>And it doesn't matter what those con... | Bridgeburners | 166,757 | <p>No proof is needed, it's true simply by definition of units. When dealing with units in any physics calculation, we generally respect the following unofficial axioms that define units:</p>
<p>1) A unit is treated as a variable in any calculation</p>
<p>2) Any number that is linear in a specific unit (with no inter... |
630,912 | <p>In <a href="https://math.stackexchange.com/questions/626256/choose-the-branch-for-1-zeta21-2-that-makes-it-holomorphic-in-the-upper">this</a> question I brought up a passage from Stein/Shakarchi's <em>Complex Analysis</em> page 232: </p>
<blockquote>
<p>...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z
\frac... | Disintegrating By Parts | 112,478 | <p>The defining property of $(1-\zeta)^{-1/2}$ is that its square is $1/(1-\zeta^{2})$. Both of the forms you gave check out that way because $i^{2}=-1$. This is why using the traditional view of branch cuts can be confusing. If you consider $f(z)=\sqrt{z}$ to be the branch cut where $z= x+i0$ for $-\infty \le x \le 0$... |
630,912 | <p>In <a href="https://math.stackexchange.com/questions/626256/choose-the-branch-for-1-zeta21-2-that-makes-it-holomorphic-in-the-upper">this</a> question I brought up a passage from Stein/Shakarchi's <em>Complex Analysis</em> page 232: </p>
<blockquote>
<p>...We consider for $z\in \mathbb{H}$, $$f(z)=\int_0^z
\frac... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
3,224,455 | <p>I derived the volume of a cone using two approaches and compared the results.</p>
<p>First I integrated a circle of radius <span class="math-container">$r$</span> over the height <span class="math-container">$h$</span> to get the expression: <span class="math-container">$$V_1=\frac{1}{3}\pi r^2 h$$</span></p>
<p>T... | Matthew Leingang | 2,785 | <blockquote>
<p>Equating <span class="math-container">$V_1$</span> and <span class="math-container">$V_2$</span> implies that: <span class="math-container">$$ \lim_{n \to \infty} \left(n\tan{\frac{180}{n}}\right) = \pi $$</span></p>
</blockquote>
<p>That is correct. 180 degrees is <span class="math-container">$\pi$... |
3,252,765 | <p>We are trying to codify in terms of modern algorithm the works of the ancient Indian mathematician <em>Udayadivakara</em> (CE 1073). In his work <em>Sundari</em>, he quotes one <em>Acarya Jayadeva</em> who has given methods to solve Pell's equations. In these methods, one can find the the cyclic <em>Chakravala</em> ... | Dmitry Ezhov | 602,207 | <p>This not answer, only illustration how work <strong>pari/gp</strong>.</p>
<p>gp-code:</p>
<pre><code>gpell(D,C)=
{
print("\nRoot solutions of Pell equation x^2-",D,"*y^2=",C,"\n");
Q= iferr(bnfinit('X^2-D), E, 0);
if(Q,
U= iferr(Q.fu, E, 0);
if(U, for(j=1, #U, u= U[j]; print("Q.fu: ",u,"\n");
N= ifer... |
2,879,035 | <p>$f(x) = \int_{1}^{\infty} \frac{2}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} dx$</p>
<p>find $P(X > 1)$</p>
<p>This is $X$ ~ $Norm(0, 1)$.</p>
<p>$P(X > 1) = 1 - P(X \leq 1) = 1 - 2 \phi(1) = 1-2(1-\phi(-1)) = 1 - 2(1-0.1587) = -0.6826$. </p>
<p>Yikes. Negative number. What am I doing wrong? </p>
| Aaron Montgomery | 485,314 | <p>I assume that by $\phi(t)$ you mean the area in the left tail of a normal distribution. If so, then the two below is incorrect and should be removed:
$$P(X > 1) = 1 - P(X \leq 1) = 1 - \color{red}{2}\phi(1) = \dots$$</p>
|
2,322,646 | <p>Let $f$ and $\varphi$ be continuous real valued functions on $\mathbb{R}$. Suppose $\varphi(x)=0$ for $|x|>5$ and that $\int_{\mathbb{R}}\varphi(x)\mathbb{d}x=1$. Show that
$$\lim_{h\to 0}\left[\frac{1}{h}\int_{\mathbb{R}}f(x-y)\varphi\left(\frac{y}{h}\right)\mathbb{d}y\right]=f(x).$$
I don't know how to proceed... | Robert Z | 299,698 | <p>Hint. The given limit is equivalent to
$$\lim_{h\to 0^+}\frac{1}{h}\int_{-5h}^{5h}(f(x-y)-f(x))\varphi\left(\frac{y}{h}\right)dy=\lim_{h\to 0^+}\int_{-5}^{5}(f(x-th)-f(x))\varphi\left(t\right)dt=0$$
Then use the fact that $\varphi$ is bounded in $[-5,5]$ and $f$ is uniformly continuous in $[x-6,x+6]$.</p>
|
4,498,263 | <p>I know that a group action is transitive when there is one orbit. Say that <span class="math-container">$G$</span> is a group acting on the set <span class="math-container">$A$</span>. The identity element of <span class="math-container">$G$</span> will clearly create <span class="math-container">$|A|$</span>-many o... | Oiler | 270,500 | <p>Elements of <span class="math-container">$G$</span> do <em>not</em> have orbits. Elements of <span class="math-container">$A$</span> have orbits (and orbits are subsets of <span class="math-container">$A$</span>). If a group <span class="math-container">$G$</span> is acting on a set <span class="math-container">$A$<... |
29,766 | <p>I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?</p>
| sotiris | 7,158 | <p>There is also a list about group theory open problems here : <a href="http://www.grouptheory.info/" rel="nofollow">http://www.grouptheory.info/</a></p>
|
29,766 | <p>I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?</p>
| Bogdan Grechuk | 89,064 | <p>Here <a href="https://theorems.home.blog/theorems-list/" rel="nofollow noreferrer">https://theorems.home.blog/theorems-list/</a> is the website you are asking for. </p>
<p>It covers all recently solved mathematical problems, which are important (for example, published in a top journal) but at the same time can be u... |
174,165 | <p>I have Maths test tomorrow and was just doing my revision when I came across these two questions. Would anyone please give me a nudge in the right direction?</p>
<p>$1)$ If $x$ is real and $$y=\frac{x^2+4x-17}{2(x-3)},$$ show that $|y-5|\geq2$ </p>
<p>$2)$ If $a>0$, $b>0$, prove that $$\left(a+\frac1b\right)... | Asaf Karagila | 622 | <p>There are two cases:</p>
<ol>
<li>$n$ is even, write it as $2k$ and then you have $\cos(k\pi)$ and $\sin(k\pi)$ which you already know.</li>
<li>$n$ is odd, write it as $2k+1$ and then you have $\cos(k\pi+\frac\pi2)$ and $\sin(k\pi+\frac\pi2)$. Recall that $\sin(x)=\cos(x+\frac\pi2)$, and deduce from the previous c... |
1,149,561 | <p>I've tried using mods but nothing is working on this one: solve in positive integers $x,y$ the diophantine equation $7^x=3^y-2$.</p>
| hjhjhj57 | 150,361 | <p>If you see the <a href="http://en.wikipedia.org/wiki/Euler_characteristic#Surfaces" rel="nofollow">wikipedia article</a> on the subject you'll see there are some examples. For instance, if $T$ is the 2-torus, and $K$ the Klein bottle we have that,
$$
\chi(T) = \chi(K) =\chi(S^1) = 0.
$$
On the other hand,
\begin{al... |
873,434 | <p>Let's say I want to find the product of $1,2,3, \dots, 10$. Do I need to do $1 \cdot 2 \cdot 3 \cdot \dots \cdot 10$ manually or is there an easier way to do it?</p>
<p>Something like the sumation of $1$ to $n$ which gives $\frac{n(n+1)}{2}$.</p>
<p>I tried to search but couldn't find a way to do it directly. </... | johannesvalks | 155,865 | <p>It is known as factorial and denoted as $n!$.</p>
<p>The case $10!$ can be reduced:</p>
<p>\begin{eqnarray}
1 \cdot 2 \cdot 3 \cdots \cdot 8 \cdot 9 \cdot 10 &=& 10 \Big(5-4\Big) \Big(5-3\Big) \cdots \Big(5+4\Big)\\
&=& 50 \Big(25-16\Big) \Big(25-9\Big) \Big(25-4\Big) \Big(25-1\Big)\\
&=& 5... |
3,491,867 | <p>I'm working on an integral used to illustrate <span class="math-container">$\pi > \frac{22}{7}$</span> and I'm stuck on finding the name of a theorem for the following:</p>
<p>Let <span class="math-container">$f(x)$</span> be a continuous Real Valued function on the interval <span class="math-container">$[a,b]$<... | Hans Lundmark | 1,242 | <p>I don't know if this helps (or if you really <em>need</em> a name for that theorem), but the property that <span class="math-container">$f \ge g$</span> on <span class="math-container">$[a,b]$</span> (where <span class="math-container">$a \le b$</span>) implies <span class="math-container">$\int_a^b f(x) \, dx \ge \... |
542,391 | <p>I understand the processes of putting a matrix into Jordan normal form and forming the transformation matrix associated to "diagonalizing" the matrix. So here's my question:</p>
<p>Why is it that when you have an eigenvalue x=0 with algebraic multiplicity greater than 1, that you don't put a 1 in the superdiagonal ... | Тимофей Ломоносов | 54,117 | <p>I can make some counter-example about $1$.</p>
<p>Let's see two matrices: $A=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $B=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.</p>
<p>In both cases eigenvalues are $1$ with multiplicity $2$, but $A$ has two eigenvectors such that $Ae=e (\{1,0\},\{0,... |
4,002,458 | <p>I'm a geometry student. Recently we were doing all kinds of crazy circle stuff, and it occurred to me that I don't know why <span class="math-container">$\pi r^2$</span> is the area of a circle. I mean, how do I <em>really</em> know that's true, aside from just taking my teachers + books at their word?</p>
<p>So I t... | Joffan | 206,402 | <p>Typically this kind of issue is handled by seeing a stable value emerge, with tolerance under some desired value, from successively finer approximation.</p>
<hr />
<p>In this case if you double the number of rectangles, you approximately halve the error - this is what you would expect from treating each sub-section ... |
1,682,341 | <p>While looking at another question on this site about constructable numbers I started wondering. If you can take a countable number of steps (possibly infinite) can you draw an interval of a length corresponding to a computable number?</p>
<p>More strictly if I have a unit interval, a straight edge, a compass, a fi... | user21820 | 21,820 | <p>If you can take an infinite number of steps you construct a non-constructible length?</p>
<p><strong>No.</strong> If you take infinitely many steps, you never finish drawing and hence you cannot draw what you want.</p>
<p>However, you might want to ask whether you can draw successive approximations that get arbitr... |
925,140 | <p>$$f(x)=\frac { x }{ x+4 } $$</p>
<p>I am not sure how to go about solving this but here is what I have done so far:</p>
<p>$$y=\frac { x }{ x+4 } $$</p>
<p>$$(x+4)y=\frac { x }{ x+4 } (x+4)$$</p>
<p>$$yx+4y=x$$</p>
<p>I feel stuck now. Where do I go from here?</p>
| Adriano | 76,987 | <p>Bring the terms containing $x$ together, factor out the $x$, then divide through:
\begin{align*}
yx - x &= -4y \\
x(y - 1) &= -4y \\
x &= \frac{-4y}{y - 1}
\end{align*}</p>
|
3,578,191 | <p>Without tables or a calculator, find the value of <span class="math-container">$\displaystyle\frac{(\sqrt5 +2)^6 - (\sqrt5 - 2)^6}{8\sqrt5}$</span>.</p>
<p>I do not understand how the positive/negative signs are obtained as shown in the book; is there a formula for expanding these kind of things (what kind of expre... | David Diaz | 431,789 | <p>The book is correct. Notice the signs in the identities:
<span class="math-container">$$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$</span>
<span class="math-container">$$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$</span>
Let <span class="math-container">$a = (\sqrt{5}+2)^2$</span> and <span class="math-container">$b = (\sqrt{5}-2)^... |
3,691,692 | <p>Find all real values of a such that <span class="math-container">$x^2+(a+i)x-5i=0$</span> has at least one real solution. </p>
<p><span class="math-container">$$x^2+(a+i)x-5i=0$$</span></p>
<p>I have tried two ways of solving this and cannot seem to find a real solution.</p>
<p>First if I just solve for <span cla... | Michael Rozenberg | 190,319 | <p>Now, <span class="math-container">$$\frac{(ad+bc)^2}{abcd}-4=\frac{a^2d^2-2abcd+b^2c^2}{abcd}=\frac{(ad-bc)^2}{abcd}\geq0.$$</span>Also, by C-S
<span class="math-container">$$\left(\dfrac{b}{a}+\dfrac{d}{c}\right)\cdot\left(\dfrac{a}{b}+\dfrac{c}{d}\right)\geq\left(\sqrt{\frac{b}{a}\cdot\frac{a}{b}}+\sqrt{\frac{... |
3,691,692 | <p>Find all real values of a such that <span class="math-container">$x^2+(a+i)x-5i=0$</span> has at least one real solution. </p>
<p><span class="math-container">$$x^2+(a+i)x-5i=0$$</span></p>
<p>I have tried two ways of solving this and cannot seem to find a real solution.</p>
<p>First if I just solve for <span cla... | Michael Hoppe | 93,935 | <p>Let <span class="math-container">$x=a/b$</span>, <span class="math-container">$y=c/d$</span>, you'll get <span class="math-container">$2+x/y+y/x$</span>. Now use that for any positive number the sum of that number and its reciprocal is at least <span class="math-container">$2$</span>.</p>
|
2,079,822 | <p>I am asked to find the maximum velocity of a mass. </p>
<p>I know that the equation for maximum acceleration is </p>
<p>$$a = w^2A$$</p>
<p>However I do not know how to find the maximum velocity. Is velocity just the same as acceleration? </p>
| eyeballfrog | 395,748 | <p>Velocity is the rate of change of position. It's pretty much just the speed of the object, with a little extra structure to keep track of the direction it's moving.</p>
<p>Acceleration is the rate of change of the velocity. It incorporates both the object speeding up and slowing down, and the object turning to move... |
1,893,540 | <p>I've been asked to prove the following,
if $x - ε ≤ y$ for all $ε>0$ then $x ≤ y$.
I tried proof by contrapositive, but I keep having trouble choosing the right $ε$. Can you guys help me out? </p>
| DeepSea | 101,504 | <p>Suppose $x > y \implies \epsilon = x-y > 0 \implies x = y + \epsilon > y + \dfrac{\epsilon}{2}$, contradiction.</p>
|
3,597,301 | <p>We know that formula of finding mode of grouped data is</p>
<p>Mode = <span class="math-container">$l+\frac{(f_1-f_0)}{(2f_1-f_0-f_2)}\cdot h$</span></p>
<p>Where, <span class="math-container">$f_0$</span> is frequency of the class preceding the modal class and <span class="math-container">$f_2$</span> is frequenc... | BruceET | 221,800 | <p>Here is an elementary example of the use of a density estimator in R.</p>
<p>First we generate a thousand observations from the gamma distribution <span class="math-container">$\mathsf{Gamma}(\mathsf{shape}=\alpha=2, \mathsf{rate} = \lambda = 1/3)$</span>
and plot their histogram in such a way that the 'modal bin' ... |
1,902,455 | <p>$x=e^t$ $y=te^(-t)$</p>
<p>$\frac{dy}{dx}= \frac{e^(-t)(1-t)}{e^(t)}$</p>
<p>$\frac{d^2y}{dx^2}= \frac{\frac{dy}{dx}}{\frac{dx}{dt}}= \frac{e^(-t)(1-t)}{e^t}$</p>
<p>any t's without proper enclosement are meant to be to the power...I don't know why its giving me this trouble. I entered these answers into my homew... | Community | -1 | <p>A substitution of 1+y^2 or such is tempting but the derivative produces a y term which needs removal by trig substitution. Thus a substitution in both y^2 and y is needed. The only way I can see is this substitution:</p>
<p>$$ u= \frac {y}{1+y^2} \Rightarrow \frac {du}{dx}= \frac {1-y^2}{(1+y^2)^2} \\ \therefore I... |
1,902,455 | <p>$x=e^t$ $y=te^(-t)$</p>
<p>$\frac{dy}{dx}= \frac{e^(-t)(1-t)}{e^(t)}$</p>
<p>$\frac{d^2y}{dx^2}= \frac{\frac{dy}{dx}}{\frac{dx}{dt}}= \frac{e^(-t)(1-t)}{e^t}$</p>
<p>any t's without proper enclosement are meant to be to the power...I don't know why its giving me this trouble. I entered these answers into my homew... | Jack Tiger Lam | 186,030 | <p>Divide throughout the fraction by $y^2$ to obtain:</p>
<p>$$\displaystyle \int \frac{\frac{1}{y^2} -1}{\left(y+\frac{1}{y}\right)^2} \text{d}y = \int \frac{-\text{d}\left(y+\frac{1}{y}\right)}{\left(y+\frac{1}{y}\right)^2}$$</p>
<p>Which yields immediately to the reverse chain rule.</p>
|
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | C. E. | 731 | <blockquote>
<h1>Status Completed</h1>
</blockquote>
<p>It would be neat if IntelliJ could detect this precedence issue:</p>
<pre><code>Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> Hue[#3] &]
</code></pre>
<p>it should be</p>
<pre><code>Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}, ColorFunction -> ... |
86,755 | <p>I need to solve the following integral equations for a problem I'm working on - </p>
<p>$\displaystyle \frac{-i}{2 \pi}$ $\int_{-a}^{a} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$ and $\displaystyle \frac{-i}{2 \pi}$ $\int_{a}^{\infty} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$</p>
<p>where $\tau, k \in \mat... | glS | 27,539 | <p>This is due to the sum of two very large numbers (coming from <code>CosIntegral</code> and <code>SinhIntegral</code>) being carried out without sufficient machine precision used to represent them.
You can fix it giving an appropriate value of <code>WorkingPrecision</code> as an option to plot.</p>
<p>You can see qu... |
957,400 | <p>S: Every employee who is honest and persistent is successful or bored.</p>
<p>Would this statement be the negations, converse, or contrapositive of S?</p>
<p>-> All employees who are dishonest or not persistent must be unsuccessful and not bored.</p>
| Platehead | 29,459 | <p>Sorry if my notation is unfamiliar.</p>
<p>Write
\begin{align*}
Hx &= \text{$x$ is honest}\\
Px &= \text{$x$ is persistent}\\
Sx &= \text{$x$ is successful}\\
Bx &= \text{$x$ is bored}
\end{align*}</p>
<p>Then $S$ can be written $\forall x (Hx \land Px) \to (Sx \lor Bx)$.</p>
<p>The next statement... |
4,515,488 | <p>I am making a computer program to play cards, for this algorithm to work I need to deal cards out randomly.
However, I know that some people cannot have some cards due to the rules of the card game.</p>
<p>To elaborate on this, imagine we have 3 players: <em>a</em>, <em>b</em> and <em>c</em>. Also, there are 4 cards... | kodlu | 66,512 | <p>Since each card <em>must</em> be dealt, this can be done recursively.</p>
<p>Let <span class="math-container">$P_i=\{x:$</span> x is a player that can be dealt card <span class="math-container">$i\},$</span> and <span class="math-container">$C_a=\{i:$</span> x is a card that can go to player <span class="math-cont... |
1,719,568 | <p>Can we say that $k$ grows faster than $\sqrt{k}$ when term is large? But what is the formal way write it ?</p>
| Henricus V. | 239,207 | <p>We can restrict $k > 4$, then it suffices to show that
$$ \lim_{k \to \infty} (k - 2\sqrt{k}) = \infty
$$
Using the fact that $2\sqrt{k} \in o(k)$,
$$ \lim_{k \to \infty} (k - 2\sqrt{k}) = \lim_{k \to \infty} k = \infty
$$</p>
|
1,719,568 | <p>Can we say that $k$ grows faster than $\sqrt{k}$ when term is large? But what is the formal way write it ?</p>
| Pedro | 70,305 | <p>Note that, for all positive integer $k$,
$$\frac{1}{k-2\sqrt{k}}=\frac{\frac{1}{k}}{1-2\frac{\sqrt{k}}{k}}=\frac{\frac{1}{k}}{1-2\frac{1}{\sqrt{k}}}$$</p>
<p>Thus, if you know that
$$\lim_{k\to\infty}\frac{1}{k}=0\quad\text{and}\quad\lim_{k\to\infty}\frac{1}{\sqrt{k}}=0,$$
then you can conclude that
$$\lim_{k\to\in... |
91,739 | <p>I have 2 groups: </p>
<ul>
<li>general linear $ k \times k $ with $\cdot$</li>
<li>top-triangle matrix $ n \times n $ with 1 on main diagonal. Operation is $\cdot$ too</li>
</ul>
<p>Is there isomorphism for any any non-trivial $n,k$ i.e $n \neq 2 \ or \ k \neq 1$ over $\mathbb{R}$ or $\mathbb{Q}$?</p>
<p>If no... | Bill Cook | 16,423 | <p>Upper-triangular matrices form solvable groups, general linear groups are not solvable (for $k>1$). Thus they cannot be isomorphic.</p>
|
1,559,485 | <p>Suppose $\sup_{x \in \mathbb{R}} f'(x) \le M$.</p>
<p>I am trying to show that this is true if and only if $$\frac{f(x) - f(y)}{x - y} \le M$$</p>
<p>for all $x, y \in \mathbb{R}$.</p>
<p><strong>Proof</strong></p>
<p>$\text{sup}_{x \in \mathbb{R}} f'(x) \le M$</p>
<p>$f'(x) \le M$ for all $x \in \mathbb{R}$</p... | Justpassingby | 293,332 | <p>The 'if' part follows from the definition of the derivative as a limit. If some expression is always less than or equal to $M,$ and the limit exists, then the limit also satisfies that inequality. That, in its turn, follows from the epsilon-delta definition of a limit.</p>
<p>The 'only if' part is the really intere... |
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | Hanul Jeon | 53,976 | <p>Substitute $t=1/(1+x^4)$ then we get
$$\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =\frac{1}{16}\int_0^1 \ln\left(\frac{1-t}{t}\right)(1-t)^{-3/4}t^{-1/4}dt.$$
And $\mathrm{B}(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$, we get
$$\frac{\partial}{\partial x}\mathrm{B}(x,y)=\mathrm{B}(x,y)[\psi(x)-\psi(x+y)]$$
where $\psi$ is... |
680,205 | <p>Milnor lemma 2 pg 34
"Any orientation preserving diffeomorphism f on $R^m$ is smoothly homotopic to the identity"</p>
<p>So he proves that $f\simeq df_0$ ,which he says is clearly homotopic to the identity.
Can you explain me why?</p>
<p>Here I found two explanations I don't understand:
1) $Gl^{+}(m,\mathbb{R})$ i... | Ted Shifrin | 71,348 | <p>First, $f$ is orientation-preserving. Second, $GL(n)^+$ is path-connected (e.g., use the $QR$ decomposition).</p>
|
1,417,286 | <p>So I'm trying to learn about RSA and have come across various subtopics, including the discrete logarithm problem. </p>
<p>This mentions primitive roots, which I do not understand.</p>
<p>Essentially all I want is an answer in simple terms of what a primitive root actually is.</p>
<p>Thanks</p>
| lulu | 252,071 | <p>I sometimes find it helpful to think of primitive roots as akin to logarithms...that is, a way to change multiplication into addition. For example, let's consider powers of $3$ mod$(17)$. They are: $$\{3,9,10,13,5,15,11,16,14,8,7,4,12,2,6,1\}$$</p>
<p>We note there are $16$ distinct values, so $3$ is indeed a pri... |
815,195 | <p>I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you.</p>
<p>Let $f$ be a polynomial such that
$|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$
for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$</p>
| Community | -1 | <p>This is a strange question to put on a complex analysis qual, since numerical estimates overshadow the complex analysis material (the maximal principle, as in the answer by Umberto P.). </p>
<p>We need a value of $|z|$ such that $1-|z|^2+|z|^{1000}$ can be bounded by $1/5$. I'll take $$|z| = \sqrt{\frac{10}{11}}$$... |
357,557 | <p>I have a function: $f(x)=-\frac{4x^{3}+4x^{2}+ax-18}{2x+3}$ which has only one point of intersection with the $x$-axis.</p>
<p>How can i find the value of $a$?</p>
<p>I tried polynomial division and discriminant, but it didn't help me.</p>
| Asaf Karagila | 622 | <p>If $x=2^k$ then every prime number which divides $x$ has to be $2$. Of course that saying "every prime number" is an unbounded assertion, but luckily no prime number which is larger than $x$ can divide $x$, so we can instead write it as follows:</p>
<blockquote>
<p>$x=2^k$ if and only if $x\neq 0$ and for every $... |
2,904,912 | <p>$$24a(n)=26a(n-1)-9a(n-2)+a(n-3)$$
$$a(0)=46, a(1)=8, a(2)=1$$
$$\sum\limits_{k=3}^{\infty}a(k)=2^{-55}$$
How can I prove it?</p>
| Deepesh Meena | 470,829 | <p>$$24r^3=26r^2-9r+1$$</p>
<p>solutions
$$r=\frac{1}{2},\frac{1}{3},\frac{1}{4}$$</p>
<p>$$a_n=x\left(\frac{1}{2}\right)^n+y\left(\frac{1}{3}\right)^n+z\left(\frac{1}{4}\right)^n$$
Determine $x,y,z$ using base conditions</p>
<p>$$a_0=46=x+y+z$$
$$a_1=8=x/2+y/3+z/4$$
$$a_2=1=x/4+y/9+z/16$$</p>
<p>$$\fbox{
x=... |
1,392,858 | <p>Is is known that the space of symmetric matrices $\mathbb{R}_{sym}^{n \times n}$ has $\binom{n}{2}$ dimensions.</p>
<p>And according to the spectral theorem every symmetric matrix $A \in \mathbb{R}_{sym}^{n \times n}$ has a spectral decomposition in terms of 1-rank matrices.</p>
<p>A = $\sum_{i=1}^n \lambda_i v_i... | Eugene Zhang | 215,082 | <p>The linear subspace of symmetric matrices is actually of dimension $n(n+1)/2$. The linear subspace of diagoanl matrices is of dimension $n$. The similar transformation (spectral decomposition) maps $n(n+1)/2$ space to $n$ space. You have confused 2 different spaces with a single space. </p>
|
48,077 | <p>First, I'm a beginner.</p>
<p>I can compute the sum of roots with the follwing:</p>
<pre><code>Roots[x^7 + 5 x^6 + x^5 + x + 1 == 0, x]
Plus @@ (x /. {ToRules[%]}) // Simplify
</code></pre>
<p>Of course I get, except the sign, the coefficient of x^6.</p>
<p>Now, is there a way to compute more elaborate symmetric... | Daniel Lichtblau | 51 | <p>You might compute the defining polynomial for the root quotients directly. Since you are interested in root quotients x/y, call the result z and we have x-y*z as a new polynomial relation. Now use iterated resultants to eliminate x and y.</p>
<pre><code>res =
Resultant[Resultant[x^7 + 5 x^6 + x^5 + x + 1, x - y*z... |
3,156,643 | <blockquote>
<p>Prove that <span class="math-container">$\sin(x) < x$</span> when <span class="math-container">$0<x<2\pi.$</span></p>
</blockquote>
<p>I have been struggling on this problem for quite some time and I do not understand some parts of the problem. I am supposed to use rolles theorem and Mean v... | Theo Bendit | 248,286 | <p>Using <span class="math-container">$f : [0, \infty) : x \mapsto x - \sin(x)$</span> and the mean value theorem, we can solve this problem. Note that
<span class="math-container">$$f'(x) = 1 - \cos(x) \ge 0.$$</span>
The mean value theorem tells us that <span class="math-container">$f$</span> is therefore non-decreas... |
1,211,287 | <p>Given that the angles between the consecutive lateral edges AB, AC & AD meeting at the vertex A of a tetrahedron ABCD are $ α, β, γ$ (as shown in the diagram below). Is there any set-formula to find out the solid angle subtended by the tetrahedron at the same vertex? </p>
<p>Note: A tetrahedron is a solid havin... | Harish Chandra Rajpoot | 210,295 | <p>The <strong>solid angle <span class="math-container">$\omega$</span> subtended at a vertex by any tetrahedron having (vertex) angles <span class="math-container">$\alpha, \beta$</span> & <span class="math-container">$\gamma$</span> between consecutive lateral edges meeting at the same vertex</strong>, is given b... |
2,623,735 | <p><a href="https://i.stack.imgur.com/5QfOQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5QfOQ.png" alt="enter image description here"></a></p>
<p>I have proven that $S$ is also a basis. But I am not sure about the second one. Is it just identity matrix $3\times 3$ as we don't change anything?</p... | lab bhattacharjee | 33,337 | <p>As $\arctan(-x)=-\arctan x,$</p>
<p>Set $x=-y$ with $x\ge0,$ in $$\arccos\dfrac{1-x^2}{1+x^2}=2\arctan x$$</p>
<p>to find $$\arccos\dfrac{1-(-y)^2}{1+(-y)^2}=2\arctan(-y)=?$$</p>
<p>In $$\arcsin\dfrac{2x}{1+x^2}=2\arctan x$$ with $x\le1$</p>
<p>set $x=\dfrac1y$ to find $$\arcsin\dfrac{2/y}{1+(/y)^2}=2\arctan(1/y... |
422,118 | <p>I'm a CS major working on social network analysis and its friends.</p>
<p>In page 15 of <a href="http://open.umich.edu/sites/default/files/1446/SI508-F08-Week3.pdf" rel="nofollow">this lecture note</a>, two very interesting questions have been asked. Given a social network graph, in which cases would we find nodes ... | hcp | 12,963 | <p><strong>No, it isn't.</strong></p>
<p>The basic idea of the proof is this: We can define an alternative interpretation for the logic such that the logic is correct with respect to said interpretation, but $\triangledown(\top, \top)$ isn't valid in the interpretation, therefore implying that $\triangledown(\top, \to... |
677,785 | <p>I have to evaluate this integral:</p>
<p>$$
\int_0^4 \int_\sqrt{y}^2 y^2 {e}^{x^7} \operatorname d\!x \operatorname d\!y\,
$$</p>
<p>I have no idea what to do with $\;{e}^{x^7}$.</p>
<p>I have even <a href="http://www.wolframalpha.com/input/?i=int+e%5Ex%5E7+dx" rel="nofollow">tried $\int{e}^{x^7} dx$ with Wolfram... | Community | -1 | <p>Change the order of integration; this leads to</p>
<p>$$\int_0^2 \int_0^{x^2} y^2 e^{x^7} dy dx = \frac 1 3\int_0^2 x^6 e^{x^7} dx$$</p>
<p>which is an easy integral.</p>
|
120,067 | <p>The <em>theta function</em> is the analytic function $\theta:U\to\mathbb{C}$ defined on the (open) right half-plane $U\subset\mathbb{C}$ by $\theta(\tau)=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 \tau}$. It has the following important transformation property.</p>
<blockquote>
<p><strong>Theta reciprocity</strong>: $\theta... | Stopple | 6,756 | <p>Going in the direction of more generality:</p>
<p>With $\theta(\tau)=\sum_n\exp(\pi i n^2 \tau)$, theta reciprocity describes how the function behaves under the linear fractional transformation $[\begin{smallmatrix} 0&1 \\ -1&0\end{smallmatrix}]$. From this one can show it's an automorphic form (of half in... |
684,892 | <p>My progress:</p>
<p>Let's take $a \in \mathbb{Z}\left[\frac{-1 + \sqrt{-3}}2\right]$ such that $a \mid 2$, and function $l(x) = x \bar x$.</p>
<p>$a \mid 2$ $\Rightarrow$ $2 = ab$ $\Rightarrow$ $l(ab) = l(a)l(b) = 4 = l(2)$</p>
<p>If $z \in \mathbb{Z}[\frac{-1 + \sqrt{-3}}2]$, then $z = x + y\frac{-1 + \sqrt{-3}}... | Astro Nauft | 122,372 | <p>Another simple proof is that your equation is reduced bilinear form, and the smallest integer that reduced bilinear form represents is x's coefficient.</p>
|
2,616,847 | <p>By definition, a function $f:\Bbb R^n \to \Bbb R^m$ is linear if</p>
<ol>
<li>$f(x+y)=f(x)+f(y) \forall x,y\in \Bbb R^n$</li>
<li>$f(ax)=af(x) \forall x\in \Bbb R^n$</li>
</ol>
<p>I want to prove that $f$ is linear iff $f(x)=Ax,A\in\Bbb R^{m\times n}$ and A is unique for any x. </p>
<p>I try to prove it by showin... | user | 505,767 | <p>We need to prove two implications</p>
<ol>
<li>if $f$ is linear $\implies$ $f(x)=Ax$</li>
<li>if $f(x)=Ax$ $\implies$ $f$ is linear</li>
</ol>
<p>To <strong>prove "1"</strong> you need to show that every $x$ can be expressed by a linear combination of a basis $x=\sum a_i\cdot v_i$ and that $f(x)$ for linearity is ... |
2,101,750 | <p>The WP article on general topology has a section titled "<a href="https://en.wikipedia.org/wiki/General_topology#Defining_topologies_via_continuous_functions">Defining topologies via continuous functions</a>," which says,</p>
<blockquote>
<p>given a set S, specifying the set of continuous functions $S \rightarrow... | Noah Schweber | 28,111 | <p>Your first question is easily addressed. Yes, on the face of it, there are a proper class of such maps. However, we can restrict attention to topological spaces $X$ which have cardinality no greater than $S$, since we can leave off the part of a space not in the image of $f$. Up to homeomorphism, there are only set-... |
1,841,958 | <p>This is a claim on Wikipedia <a href="https://en.wikipedia.org/wiki/Partially_ordered_set">https://en.wikipedia.org/wiki/Partially_ordered_set</a></p>
<p>I am not sure how to make sense of the claim</p>
<p>What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? </p>
<p>Can someone provide a small ... | user247327 | 247,327 | <p>"Ordered by inclusion" means "$A\le B$ if only if A is a subset of B". For example, the set, U, of all vectors of the form (a, b, 3a+ 2b) is a subspace of $R^3$ so is a subset so "$U\le R^3$". And the set, V, of all vectors of the form (a, 3a, 9a) is a subspace of U: $V\le U$.</p>
|
434,061 | <p>I am given a matrix $A\in M(n\times n, \mathbb{C})$ normal (in matrix form $AA^*=A^*A$) and $A^2=A$. The task is to prove that the matrix is Hermitian.</p>
<p>But when I try something like $A^*=\,\,...$ , then I can't reach $A$, because I can't "get rid of star" in expression. Also it is not enough to show $BA=BA^*... | tomasz | 30,222 | <p><strong>Hint</strong>: by spectral theorem, a normal matrix is hermitian if and only if all its eigenvalues are real. What complex numbers have the property that they are equal to their squares?</p>
|
1,039,141 | <blockquote>
<p>Let <span class="math-container">$X = \mathbb{R}$</span> and <span class="math-container">$Y = \{x \in \mathbb{R} :x ≥ 1\}$</span>, and define <span class="math-container">$G : X → Y$</span> by <span class="math-container">$$G(x) = e^{x^2}.$$</span>
Prove that <span class="math-container">$G$</span> is ... | mathcounterexamples.net | 187,663 | <p>You should write $G$ as a composite function of onto functions. Then as an exercise, prove again that a composite function of onto functions is onto.</p>
|
3,201,797 | <p>I have three points <span class="math-container">$(x_1, y_1),~ (x_2, y_2),~ (x_3, y_3)$</span> that are on the same line. How to efficiently find which is the point in between.</p>
<p><a href="https://i.stack.imgur.com/e2wHq.png" rel="nofollow noreferrer">Example</a></p>
<p>Also, is there any efficient way to chec... | Community | -1 | <p>Sort the <span class="math-container">$x$</span> and sort the <span class="math-container">$y$</span> (sorting three items takes three comparisons). Consider the axis on which the difference between the extremes is the largest, and return the point with the intermediate coordinate.</p>
<p><a href="https://i.stack.i... |
3,364,316 | <p>While I'm reading E. Landau's <em>Grundlagen der Analysis</em> (tr. <em>Foundations of Analysis</em>, 1966), I couldn't understand the proof of <em>Theorem 3</em> at the segment of <em>Natural Numbers</em> which I've quoted below.</p>
<blockquote>
<p><strong>Theorem 3:</strong> <em>If</em><br>
<span class="math... | Ben Grossmann | 81,360 | <p>In fact, your lim inf is a lim. First, observe that
<span class="math-container">$$
(n+1)\ln(n+1) = n \ln(n+1) + \ln(n+1) = \\
n[\ln(n+1) - \ln(n) + \ln(n)] + \ln(n+1) = \\
n \ln(n) + n \ln\left(\frac{n+1}{n}\right) + \ln(n+1)
$$</span>
Thus (by the prime number theorem), we have
<span class="math-container">$$
\l... |
3,364,316 | <p>While I'm reading E. Landau's <em>Grundlagen der Analysis</em> (tr. <em>Foundations of Analysis</em>, 1966), I couldn't understand the proof of <em>Theorem 3</em> at the segment of <em>Natural Numbers</em> which I've quoted below.</p>
<blockquote>
<p><strong>Theorem 3:</strong> <em>If</em><br>
<span class="math... | hunter | 108,129 | <p>If you're willing to use recent huge theorems in your proof, it follows from Zhang's prime gaps theorem (there is some <span class="math-container">$N$</span> and infinitely many <span class="math-container">$n$</span> with <span class="math-container">$p_{n+1} - p_n < N$</span>).</p>
|
4,274,314 | <blockquote>
<p>Find all <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span> such that
<span class="math-container">$$f\bigl(xf(y)+y\bigr)+f\bigl(-f(x)\bigr)=f\bigl(yf(x)-y\bigr)+y $$</span>
for all <span class="math-container">$x,y \in \mathbb{R}$</span>.</p>
</blockquote>
<p>Help me solving this. My ex... | Knas | 634,505 | <p>Throughout my answer, the expression <span class="math-container">$(n)(x_1, x_2, \ldots, x_n)$</span> will mean the substitution in the equation with the number <span class="math-container">$n$</span>.</p>
<p>We want to find all functions <span class="math-container">$f:\mathbb{R} \rightarrow \mathbb{R}$</span> such... |
2,354,467 | <p>I am trying to evaluate the following
\begin{equation}
I(a,b) = \int_{a}^{\frac{a+b}{2}} (x-a)^{\alpha-1} \, x^n \, dx + \int_{\frac{a+b}{2}}^{b} (b-x)^{\alpha-1} \, x^n \, dx,
\end{equation}
where $0<\alpha<1$. Wolfram alpha gives no solution. I tried integration by parts without success. My problem is that ... | JJacquelin | 108,514 | <p>Since the wording of the question was modified, my first anser is no longer valid. So, I post a new answer to the new wording :
\begin{equation}
I(a,b) = \int_{a}^{\frac{a+b}{2}} (x-a)^{p-1} \, x^n \, dx + \int_{\frac{a+b}{2}}^{b} (b-x)^{p-1} \, x^n \, dx,
\end{equation}</p>
<p>About the convergence of the first i... |
2,130,397 | <p>If I want to find the power series representation of the following function:</p>
<p>$$ \ln \frac{1+x}{1-x} $$</p>
<p>I understand that it can be written as </p>
<p>$$ \ln (1+x) - \ln(1-x) $$</p>
<p>And I understand that if I now write in the power series representations for $ln(1+x)$ and $ln(1-x)$:</p>
<p>$$\su... | Community | -1 | <p>We have $$I = \int_{0}^{\frac {\pi}{2}} \frac {1}{1+\cos^2 x} \mathrm{d}x = \int_{0}^{\frac {\pi}{2}} \frac {1}{\tan^2 x +2} \sec^2 x \mathrm {d}x = \int_{0}^{\infty} \frac {1}{u^2+2} \mathrm {d}u $$ by substituting $u = \tan x $. Hope you can take it from here. If you want to check, the answer is $\boxed {\frac {\p... |
2,130,397 | <p>If I want to find the power series representation of the following function:</p>
<p>$$ \ln \frac{1+x}{1-x} $$</p>
<p>I understand that it can be written as </p>
<p>$$ \ln (1+x) - \ln(1-x) $$</p>
<p>And I understand that if I now write in the power series representations for $ln(1+x)$ and $ln(1-x)$:</p>
<p>$$\su... | MoNtiDeaD MoonDogs | 408,248 | <p>Note $\displaystyle \cos x=\frac{1}{\sec x}$.</p>
<p>So, $$ \begin{align}I&=\int\frac{1}{\cos^2x+1}dx\\ &=\int\frac{\sec^2x}{\sec^2x+1}dx \\&=\int\frac{\sec^2x}{\tan^2x+2}dx \tag{1}\end{align}$$</p>
<p>Now Let $u=\tan x\rightarrow du=\sec^2xdx$ and substituting in $(1)$, we get</p>
<p>$$\begin{align}I... |
1,525,660 | <p><a href="https://i.stack.imgur.com/w2y9k.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w2y9k.png" alt="image"></a></p>
<p>Problem above. (Sorry I can't embed yet and the link seems to be removed when hyperlinked)</p>
<p>Hello,</p>
<p>Fairly simple (I imagine) question that I am stuck on.
The ... | Nicholas | 282,542 | <p>Perhaps it could be easier expand $(1-x^2)^{-4}$ using the binomial series.</p>
<p>$\begin{align}
(1-x^2)^{-4} &= 1+4x^2 + \frac{(-4)(-4-1)}{2}(-x^2)^2+\frac{(-4)(-4-1)(-4-2)}{3!}(-x^2)^3+ \dots\\
&=1+4x^2+10x^4+20x^6+35x^8+\dots\\
&=\sum_{n=1}^{\infty} \binom{n+2}{3}x^{2n-2}\\
\end{align}$</p>
|
1,300,853 | <p>Could somebody tell me the exact value of this series?
$$
\sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k}
$$
where
$$
H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n}
$$</p>
<p>Thanks!</p>
| Olivier Oloa | 118,798 | <p><strong>Hint.</strong> You may write
$$
\sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k}=\sum_{k=1}^{\infty} (-1)^k\frac{H_{k-1}^{(5)}}{k}+\sum_{k=1}^{\infty} \frac{(-1)^k}{k^6}=\zeta(\bar{1},5)-\frac{31 \pi ^6}{30240}.
$$
I am not sure the <a href="http://en.wikipedia.org/wiki/Multiple_zeta_function" rel="nofollow">M... |
24,873 | <p>It is very elementary to show that $\mathbb{R}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>1$: subtract a point and use the fact that connectedness is a homeomorphism invariant.</p>
<p>Along similar lines, you can show that $\mathbb{R^2}$ isn't homeomorphic to $\mathbb{R}^m$ for $m>2$ by subtracting a point ... | IV_ | 292,527 | <p><a href="https://en.wikipedia.org/wiki/Emanuel_Sperner" rel="nofollow noreferrer">Sperner</a> showed in his doctoral thesis [Sperner 1928] that invariance of open sets, invariance of domain and invariance of dimension can be proved already with elementary combinatorial methods alone.</p>
<p>see also: <a href="https:... |
1,299,474 | <p><img src="https://i.stack.imgur.com/EyYdm.jpg" alt="enter image description here"></p>
<p>Here is an attempt at a solution:</p>
<p><img src="https://i.stack.imgur.com/IWSah.jpg" alt="enter image description here"></p>
<p>Since $f(x)>0$, $f(x)>\delta$ for all x between $1$ and $2$ </p>
<p>Is this correct? <... | Paul | 17,980 | <p>It seems having no mistakes. I think it is okay.</p>
|
1,858,297 | <p>Suppose the diameter of a nonempty set $A$ is defined as </p>
<p>$$\sigma(A) := \sup_{x,y \in A} d(x,y)$$</p>
<p>where $d(x,y)$ is a metric.</p>
<p>Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable additivity for this particular case?</p>
| drhab | 75,923 | <p>Observe that the diameter of singletons is $0$ and the diameter of set $\{x,y\}$ is $d(x,y)>0$ if $x\neq y$. So there is no additivity.</p>
|
806,532 | <p>This question takes place in a general metric space $X$. </p>
<p>Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. </p>
<p>This is like the normal definition of "interior point", except it uses "deleted neighborhood" instead of "neighborhood",... | Lee Mosher | 26,501 | <p>A good test of a new definition is its power of expressiveness. What I mean by this is, when introducing the definition into mathematical discourse, does it help you express mathematical ideas or concepts in a way that enhances understanding, aids discovery, quickens comprehension of proofs, and so on? </p>
<p>As o... |
1,070,870 | <p>"Write down (say, as a power series) a holomorphic function $f(z)$ on $D(1, 1)$ which satisfies $f(z)^5 = z$ and $f(1) = 1$. What is the result of analytically continuing $f$ along a path which travels once counterclockwise around the origin, returning to the point $1$? What about if you go $N$ times counterclockwis... | megas | 191,170 | <p>As pointed out in the comments by Franco, you need $m \ge n$.
Under this assumption, for real and complex matrices, you could argue based on the (truncated) singular value decomposition.</p>
<p>Let $\mathbf{A} = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^{H}$ be the singular value decomposition of $\mathbf{A}$: $\mathbf{U... |
3,450,598 | <blockquote>
<p>Prove that <span class="math-container">$\sum_{i = m}^n a_i + \sum_{i = n + 1}^p a_i = \sum_{i = m}^p a_i$</span>, where <span class="math-container">$m ≤ n<p$</span> are integers, and <span class="math-container">$a_i$</span> is a real number assigned to each integer <span class="math-container">$... | SARTHAK GUPTA | 293,005 | <p>It should be pretty straightforward. </p>
<p>The left-hand side is <span class="math-container">$\sum_{i = m}^n a_i + \sum_{i = n + 1}^p a_i$</span> which after expansion (since <span class="math-container">$m,p,n$</span> are finite, we can always expand explicitly) gives us <span class="math-container">$(a_m+a_{m+... |
280,393 | <p>I want to show that $(3, \sqrt 15)$ is not a principal ideal in the ring $ R = \mathbb{Z}[\sqrt{15}]$ with norm $N(a + b \sqrt 15) = a^2 - 15b^2$.</p>
<p>My attempt:</p>
<p>Suppose $(3, \sqrt 15) = (x) $</p>
<p>Then $3 = x * r1$ and $\sqrt 15 = x * r2$ , $r1,r2 \in R$.</p>
<p>$N(3) = 9 = N(x) N(r1)$ and $N(\sqrt... | Gerry Myerson | 8,269 | <p>If the norm of $x$ is $\pm1$ then $x$ is a unit. </p>
<p>If the norm of $x$ is $\pm3$, you get a contradiction from working modulo $5$. </p>
|
4,124,324 | <p>I am trying to find the complex function <span class="math-container">$f(z)$</span> who's derivative equals the complex conjugate of its reciprocal</p>
<p><span class="math-container">$$\dfrac{\mathrm{d} f(z)}{\mathrm{d} z} = \dfrac{1}{f(z)^*}$$</span></p>
<p>which is equivalent to</p>
<p><span class="math-container... | José Carlos Santos | 446,262 | <p>There is no such function, at least if its domain is a non-empty open subset <span class="math-container">$D$</span> of <span class="math-container">$\Bbb C$</span>. Suppose otherwise. Take <span class="math-container">$w\in D$</span> and take <span class="math-container">$r>0$</span> such that <span class="math-... |
178,319 | <p>I asked this initially in <a href="https://math.stackexchange.com/questions/894399/identities-that-connect-antipode-with-multiplication-and-comultiplication">math.stackexchange</a>:</p>
<p>The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities:
$$
S\otimes S\circ \Delta=\sigm... | Zahlendreher | 33,854 | <p>This also holds in an arbitrary braided monoidal category and is not hard to see. See for example Majid: Foundations of quantum group theory. He gives a graphical calculus proof of this in Figure 9.14.
Strictly speaking, you need to turn this proof upside down (i.e. dualize) to get the corresponding identity for the... |
139,954 | <p>I'm having trouble understanding this question.</p>
<p>We have a path <span class="math-container">$h$</span> in <span class="math-container">$X$</span> from <span class="math-container">$x_0$</span> to <span class="math-container">$x_1$</span> and <span class="math-container">$\overline{h}$</span> its inverse path.... | Michael Albanese | 39,599 | <p><em>This is effectively an extended comment.</em></p>
<p>The question is for every path $h$, not $f$. As $[f] \in \pi_1(X, x_1)$, $f$ is a loop in $X$ based at $x_1$, not a path from $x_0$ to $x_1$.</p>
<p>Instead of writing $h\circ f\circ\bar{h}$ you should write $h\cdot f\cdot\bar{h}$ because you are not composi... |
139,954 | <p>I'm having trouble understanding this question.</p>
<p>We have a path <span class="math-container">$h$</span> in <span class="math-container">$X$</span> from <span class="math-container">$x_0$</span> to <span class="math-container">$x_1$</span> and <span class="math-container">$\overline{h}$</span> its inverse path.... | Dasheng Wang | 616,946 | <p>Actually <span class="math-container">$\beta_h$</span> does depend on the homotopy class of <span class="math-container">$h$</span>. Note that <span class="math-container">$\beta_h$</span> is an isomorphism between <span class="math-container">$\pi_1(X,x_0)$</span> and <span class="math-container">$\pi_1(X,x_1)$</sp... |
3,673,014 | <p>If you take an <span class="math-container">$2r\times 2r\times 2r$</span> cube, and divide it to 27 equal cubes, and then remove all the "axis" cubes (all the cubes which are straight left, straight right, straight up, etc. from the middle cube) then divide each cube into 125 equal cubes and remove all the axis cube... | Dr. Richard Klitzing | 518,676 | <ul>
<li><span class="math-container">$v_1, v_2, v_3$</span> belong to <span class="math-container">$x_3=0$</span></li>
<li><span class="math-container">$v_1, v_2, v_4$</span> belong to <span class="math-container">$x_2=x_3$</span></li>
<li><span class="math-container">$v_1, v_3, v_4$</span> belong to <span class="math... |
185,766 | <p>After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course? </p>
<p>And would an advanced linear algebra course be taught in graduate schools?</p>
| Gerry Myerson | 8,269 | <p>Different universities will teach different things under the heading "advanced linear algebra", and at different levels. I would suggest you go to a few university websites and see what they have on offer and what the contents are. </p>
<p>At my university, we teach a 3rd-year undergarduate course which is half Gal... |
2,317,391 | <p>This question popped up somewhere on the internet and I thought it was interesting. I attempted to solve it but I don't know if it is correct.</p>
<blockquote>
<p>Find the derivative of $$F(x)=\int_{\cos{x^3}}^{\int_{1}^{x} {1/(1+t^2)dt}} {\sin{w} dw}$$</p>
</blockquote>
<p>$$\begin{align}
\implies F(x) & =-... | Community | -1 | <p>Yes, it's that simple. Though, perhaps, you were meant to use the formula $$\frac{d}{dx}\left[\int_{a(x)}^{b(x)}f(t)\,dt\right]=b'(x)f(b(x))-a'(x)f(a(x))$$ which holds for continuous $f$ and differentiable $a$ and $b$.</p>
<p>Another possible simplification is using the identities $\cos\arctan x=\frac{1}{\sqrt{1+x^... |
1,230,112 | <p>Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by $\mathbf{x}^T\mathbf{A}^T\mathbf{A}\mathbf{x}=r^2$.</p>
<p>I tried to do this with tensor algebra but I'm pretty new ... | Nico | 101,332 | <p>The trick I used to memorize them actually stemmed from formal logic (which you may or may not have had any exposure to):</p>
<p>The symbol $\land$ is a way to symbolize the binary connective "and". Notice it looks like a "pointy" $\cap$. Similarly $\lor$ (or) looks similar to $\cup$.</p>
<p>Now,
$$x\in A\cap B$$
... |
1,618,042 | <p>Is there any operation that makes a set of primes i.e. {2,3,5,7.... .} a group with identity 2?</p>
| Sean English | 220,739 | <p>There are two ways to interpret your question.</p>
<ol>
<li><p>If you want the group to be a subgroup of $\mathbb{Z}$ with the usual addition then no. To see this, all we need to see is that $3$ has no inverse.</p></li>
<li><p>If you just want the group to have as its underlying set the set of all primes, then yes.... |
3,425,373 | <p>Consider:</p>
<p><span class="math-container">$$ 1+1/2^2+2/3^2+1/4^2+2/5^2+1/6^2+...$$</span></p>
<p>Does this sum have a closed form?</p>
<p>If all the numerators are <span class="math-container">$1$</span> then it does have a closed form. </p>
| Community | -1 | <p>You have </p>
<p><span class="math-container">$$1+\dfrac{2-1}{2^2}+\dfrac2{3^2}+\dfrac{2-1}{4^2}+\dfrac2{5^2}+\cdots$$</span></p>
<p>or</p>
<p><span class="math-container">$$2\left(1+\dfrac1{2^2}+\dfrac1{3^2}+\dfrac1{4^2}+\dfrac1{5^2}+\cdots\right)-1-\left(\dfrac1{2^2}+\dfrac1{4^2}+\dfrac1{6^2}+\dfrac1{8^2}+\cdot... |
3,231,387 | <p>I have been given the following quadratic equation and is asked to find the range of its roots <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, where <span class="math-container">$\alpha>\beta$</span>
<span class="math-container">$$(k+1)x^2 - (20k+14)x + 91k +40 =0,$$<... | Shantam Srivastava | 675,373 | <p>We know from the question that <span class="math-container">$ \alpha > \beta $</span> ,using the Quadratic formula, the root with the positive sign will be <span class="math-container">$ \alpha $</span> and
the one with the negative sign will be <span class="math-container">$ \beta $</span>, which you have reve... |
3,655,545 | <p>What is the asymptotic behaviour of the difference
<span class="math-container">$$
c_j - c_{j+1}
$$</span>
for <span class="math-container">$j\rightarrow \infty$</span> if <span class="math-container">$(c_j)_{j\in\mathbb{N}}$</span> is a null sequence?</p>
| Obriareos | 210,676 | <p>Oh..., it was obvious - now I have it.</p>
<p>The asymptotic behaviour is of course as <span class="math-container">$c_j$</span> since <span class="math-container">$(c_j - c_{j+1})/c_j = 1-c_{j+1}/c_j\rightarrow 1.$</span></p>
|
3,950,098 | <p>I can evaluate the limit with L'Hospital's rule:</p>
<p><span class="math-container">$\lim_{n\to\infty}n(\sqrt[n]{4}-1)=\lim_{n\to\infty}\cfrac{(4^{\frac1n}-1)}{\dfrac1n}=\lim_{n\to\infty}\cfrac{\dfrac{-1}{n^2}\times 4^{\frac1n}\times\ln4}{\dfrac{-1}{n^2}}=\ln4$</span></p>
<p>But is there any way to do it without us... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$f(x)=4^x$</span>, then <span class="math-container">$f'(x)=\log(4)4^x$</span> and, in particular, <span class="math-container">$f'(0)=\log(4)$</span>. In other words,<span class="math-container">$$\lim_{h\to0}\frac{4^h-1}h=\log(4)$$</span>and therefore<span class="math-container">$$\... |
2,995,643 | <p>Here is a thought experiment I have. </p>
<p>Say we flip a unique coin where we have a 99.99999999999% chance of it landing on heads, and a .000000000001% chance of it landing on tails (the two possibilities equal to 100%).</p>
<p>And say we have an <em>infinite</em> number of coins flipped all at once (and only o... | Michael Hoppe | 93,935 | <p>It will happen infinitely often.</p>
|
3,182,532 | <p>I am confused about converting a <strong>Probability Density Function</strong> from <strong>Polar coordinates</strong> to <strong>Cartesian coordinates</strong>. </p>
<p>Here is an example:</p>
<p>In Polar coordinates, we can have a <strong>Gaussian probability function</strong>:</p>
<p><strong><span class="math-... | Sharat V Chandrasekhar | 400,967 | <p>Your transformation takes a PDF in <span class="math-container">$r$</span> and converts it into a Joint Density Function in <span class="math-container">${x,y}$</span>. So what you really need to do is preserve the normalising property i.e., </p>
<p><span class="math-container">$$\int_{x=-\infty}^{x=\infty} \int_{y... |
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