qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
4,116,252 | <p>I'm trying to prove (or disprove) the following:</p>
<p><span class="math-container">$$ \sum_{i=1}^{N} \sum_{j=1}^{N} c_i c_j K_{ij} \geq 0$$</span>
where <span class="math-container">$c \in \mathbb{R}^N$</span>, and <span class="math-container">$K_{ij}$</span> is referring to a <a href="https://en.wikipedia.org/w... | g g | 249,524 | <p><strong>Warning: Only (very) partial answer!</strong></p>
<p>For <span class="math-container">$N=1$</span> and <span class="math-container">$u,v>0$</span> the function <span class="math-container">$K$</span> is indeed positive definite in the sense normally used for a Kernel function (see <a href="https://en.wiki... |
279,064 | <p>I find this question interesting, but need to get it out of my system: is the space of connections (modulo gauge) on a compact four-manifold paracompact, in the Sobolev topology?</p>
<p>If so, I believe it would admit partitions of unity, which would surely make life easier in gauge theory. But I haven't seen the e... | Vít Tuček | 6,818 | <p>I assume that by Sobolev topology you mean the topology induced by the Sobolev norm. Since all normed spaces are metric spaces the affirmative answer to your question follows from the fact that all metric spaces are paracompact. See e.g.
<em>A new proof that metric spaces are paracompact</em> by Mary Ellen Rudin (<a... |
3,748,392 | <blockquote>
<p>Let <span class="math-container">$G=\mathbb{Z}_{7}\rtimes_{\rho}\mathbb{Z}_{6}$</span> with <span class="math-container">$|\ker\rho| = 2$</span>. How many <span class="math-container">$3$</span>-Sylow subgroups are there in <span class="math-container">$G$</span>?</p>
</blockquote>
<p>I know that the nu... | dan_fulea | 550,003 | <p>The question is compactly answered and accepted already, as this message is written, so consider it as a comment. We can in fact construct explicitly the group <span class="math-container">$G$</span> as a group of matrices over the field <span class="math-container">$F=\Bbb F_7$</span> with seven elements. Note firs... |
3,003,672 | <p>Say I have an infinte 2D grid (ex. a procedurally generated world) and I want to get a unique number for each integer coordinate pair. How would I accomplish this?</p>
<p>My idea is to use a square spiral, but I cant find a way to make a formula for the unique number other than an algorythm that just goes in a squa... | Trebor | 584,396 | <p>There are also tools from number theory. We can first map all integers to non-negative ones, which is easy, just take <span class="math-container">$$f(n)=\left\{\begin{align}&2n&n\ge0\\&-2n-1&n<0\end{align}\right.$$</span> as Ross pointed out. Now us take the pair <span class="math-container">$(m,... |
3,003,672 | <p>Say I have an infinte 2D grid (ex. a procedurally generated world) and I want to get a unique number for each integer coordinate pair. How would I accomplish this?</p>
<p>My idea is to use a square spiral, but I cant find a way to make a formula for the unique number other than an algorythm that just goes in a squa... | Jungkwuen An | 788,678 | <p>If you want to convert <span class="math-container">$(a,b)$</span> into <span class="math-container">$c$</span>.
(<span class="math-container">$a$</span>,<span class="math-container">$b$</span>,<span class="math-container">$c$</span> are all positive integer)
<span class="math-container">$$c = 2^{a-1} * (2*b-1)$$</s... |
2,482,341 | <p>I have tried to solve $\frac{\mathrm{d}}{\mathrm{dx}}\int_{0}^{x^2}e^{x+t}\mathrm{dt}$ by two different ways and I'm getting two answers. Please let me know the mistake: </p>
<p><strong>Method One</strong><br>
Let $F(t)$ be the antiderivative of $e^{x+t}$.<br>
Thus $F^{'}(t)=e^{x+t}$ </p>
<p>So </p>
<p>\begi... | Zhuoran He | 485,692 | <p>Since $AB$ and $BC$ have lower upper bounds, they should be set to maximum. Then if $AC$ is also set to maximum, angle $B$ becomes blunt ($2^2+3^2<4^2$), which is not good. So we set angle $B$ to $90^\circ$ and the maximum area should be $\frac{1}{2}\times 2\times 3=3$. Once the maximum is found (guessed), it's n... |
656,423 | <p>This is a really simple problem but I am unsure if I have proved it properly.</p>
<p>By contradiction:</p>
<p>Suppose that $x \geq 1$ and $x< \sqrt{x}$. Then $x\cdot x \geq x \cdot 1$ and $x^2 < x$ (squaring both sides), which is a contradiction.</p>
| user76568 | 74,917 | <p>As a contra-positive, assuming $x$ is not negative:
$$\sqrt{x} > x \implies x>x^2 \land x \neq0 \implies 1>x$$
(1st implication is by squaring (Which is obviously an increasing function here), 2nd implication is by dividing by $x$) </p>
<p>So, equivalently:
$$x \geq 1 \implies x \geq\sqrt{x}$$</p>
<p>Al... |
816,249 | <p>I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential equations.</p>
<p>The example I know is the differential operator defined on the hilbert space $H=L^2(\mathbb{R})$ by
$$A... | Disintegrating By Parts | 112,478 | <p>If you have any densely-defined selfadjoint linear operator $A$ on a complex Hilbert space $X$, then $e^{itA}$ is a unitary semigroup, which is really $e^{tB}$ where $B^{\star}=-B$.</p>
<p>More generally, suppose that $U : [0,\infty)\rightarrow \mathscr{L}(X)$ is an isometric $C^{0}$ semigroup, meaning that
$$
\beg... |
4,146,858 | <blockquote>
<p>Q) For every twice differentiable function <span class="math-container">$f:\mathbb{R}\longrightarrow [-2,2] $</span> with <span class="math-container">$[f(0)]^2+[f'(0)]^2=85$</span> , which of the following statement(s) is(are) TRUE?</p>
</blockquote>
<blockquote>
<p>(A) There exists <span class="math-c... | KCd | 619 | <p>Logically the quotient rule in calculus is not needed, since it can be derived from the product rule, the power rule, and the chain rule every time, e.g., <span class="math-container">$(1/g)' = (g^{-1})' = -g^{-2}g' = -g'/g^2$</span>. But most students learn the quotient rule and don't have trouble after practicing ... |
2,984,918 | <p>How can I prove this? </p>
<blockquote>
<p>Prove that for any two positive integers <span class="math-container">$a,b$</span> there are two positive integers <span class="math-container">$x,y$</span> satisfying the following equation:
<span class="math-container">$$\binom{x+y}{2}=ax+by$$</span></p>
</blockquote... | Oldboy | 401,277 | <p><strong>Case 1:</strong> <span class="math-container">$a=b$</span>. </p>
<p><span class="math-container">$${x+y \choose 2}=ax+ay$$</span></p>
<p><span class="math-container">$$\frac{(x+y)(x+y-1)}{2}=a(x+y)$$</span></p>
<p><span class="math-container">$$x+y-1=2a$$</span></p>
<p><span class="math-container">$$x+y=... |
2,065,639 | <p>$\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$</p>
<p>$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$ </p>
<p>Instead of expanding the integrand, or doing integration by part, is there any faster way to compute this kind of integral?</p>
| Matthew Leingang | 2,785 | <p>You can do this with substitution and <a href="https://en.wikipedia.org/wiki/Cavalieri%27s_quadrature_formula" rel="nofollow noreferrer">Cavalieri's formula</a>:
$$
\int_0^1 u^n \,du = \frac{1}{n+1}
$$</p>
<p>For the first one, let $u= \frac{x-a}{b-a}$. Then $x = (b-a)u +a$, which means $x-a = (b-a)u$ and $x-b... |
93,099 | <p>Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?</p>
| Community | -1 | <p><strong>Update:</strong> By a result of Buchta (Zufallspolygone in konvexen Vielecken, Crelle, 1984; available on digizeitschriften.de) there is a general formula for this expected value, it is
$$1 -\frac{8}{3(n+1)} \bigl( \sum_{k=1}^{n+1} \frac{1}{k} (1 - \frac{1}{2^k}) - \frac{1}{(n+1)2^{n+1}} \bigr) $$
yielding... |
2,546,487 | <p>When we say $f(x)=x\, \mbox{sgn}( x)$ is continuous at $x=0$ when we say $f(x)=0$ but it is not differentiable at $x=0$. Furthermore, we say $g(x)=x^2\mbox{sgn}(x)$ is continuous and differentiable at $x=0$. But why or how? How can $f’(0)=0$ when $f(0)=0$?</p>
| Green | 357,732 | <p>It's because all the 1's are indistinguishable. For example, if you have 1001 and the 1's and 0's were distinguishable, you have $4!$ options, but since they aren't you have $4!$ divided by $2! * 2!$ (which is equivalent to 4 choose 2). </p>
<p>As a result, to find out the total number of ways to have r 1's in a st... |
2,546,487 | <p>When we say $f(x)=x\, \mbox{sgn}( x)$ is continuous at $x=0$ when we say $f(x)=0$ but it is not differentiable at $x=0$. Furthermore, we say $g(x)=x^2\mbox{sgn}(x)$ is continuous and differentiable at $x=0$. But why or how? How can $f’(0)=0$ when $f(0)=0$?</p>
| vrugtehagel | 304,329 | <p>Yes the order matters, but we still use $c(n,r)$ (more commonly denoted ${n\choose r}$) because we essentially want to find the number of ways to pick $r$ zeroes out of a string with length $n$ and change those to $1$'s, resulting in the amount of strings with exactly $r$ ones.</p>
|
1,953,251 | <blockquote>
<p>For what x does the exponential series $P_c(x) = \sum^\infty_{n=0} (-1)^{n+1}\cdot n\cdot x^n$ converge?</p>
</blockquote>
<p><strong>What I got so far:</strong></p>
<p>$\sum^\infty_{n=0} (-1)^{n+1}\cdot n\cdot x^n = (-1)\sum^\infty_{n=0} (-1)^{n}\cdot (\sqrt[n]n)^n\cdot x^n = (-1)\sum^\infty_{n=0}... | DonAntonio | 31,254 | <p>Cauchy-Hadamard:</p>
<p>$$\lim_{n\to\infty}\sqrt[n]{\left|(-1)^{n+1}n\right|}=\lim_{n\to\infty}\sqrt[n]n=1$$</p>
<p>so the series converges for $\;|x|<1\;$ .</p>
|
4,380,124 | <p>I understand that the double integral is
<a href="https://i.stack.imgur.com/29e8B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/29e8B.png" alt="enter image description here" /></a>
However what confuses me is when I try to visualize why this formula only accounts for the region inside the bounds... | Hans Lundmark | 1,242 | <p>The formula in your first picture seems to refer to double integrals over rectangles only. But if you know how to integrate over a rectangle, you can define the double integral over a non-rectangular bounded set <span class="math-container">$M$</span> by taking a rectangle <span class="math-container">$D$</span> whi... |
3,639,192 | <p>In their article on the <a href="https://en.wikipedia.org/wiki/Brauer_group#Galois_cohomology" rel="nofollow noreferrer">Brauer group</a> Wikipedia writes:</p>
<blockquote>
<p>Since all central simple algebras over a field <span class="math-container">$K$</span> become isomorphic to the matrix algebra over a sepa... | Quanto | 686,284 | <p>Let <span class="math-container">$y=(\frac35)^x$</span>. Then, divide the equation by <span class="math-container">$15^x$</span> to get</p>
<p><span class="math-container">$$y-\frac 1y+1=0$$</span></p>
<p>Solve to get</p>
<p><span class="math-container">$$y = \frac{-1\pm\sqrt5}2$$</span></p>
<p>Use <span class="... |
1,176,098 | <p>Here are some of my ideas:</p>
<p><strong>1. Addition Formula:</strong> <span class="math-container">$\sin{x}$</span> and <span class="math-container">$\cos{x}$</span> are the unique functions satisfying:</p>
<ul>
<li><p><span class="math-container">$\sin(x + y) = \sin x \cos y + \cos x \sin y $</span></p>
</li>
<li... | Rory Daulton | 161,807 | <p>The book <em>Principles of Mathematical Analysis</em> (also called <em>Baby Rudin</em>) by Walter Rudin, Second edition, pages 167-169, briefly develops the theory of trigonometric functions. This is after developing the theory of series of complex numbers as well as the theory of exponential and logarithmic functio... |
4,116,134 | <p>Find angle between <span class="math-container">$y=\sin x$</span> and <span class="math-container">$y=\cos x$</span> at their intersection point.</p>
<p>Intersection points are <span class="math-container">$\frac{\pi}{4}+\pi k$</span> and to find angle between them we need to compute derivatives at intersection poin... | Steven Alexis Gregory | 75,410 | <p>The slope of the tangent line to <span class="math-container">$y=\sin x$</span> at <span class="math-container">$x = \frac{\pi}{4}+\pi k$</span> is
<span class="math-container">$$m_s = \cos\left(\frac{\pi}{4}+\pi k \right)
= \frac{\cos \pi k - \sin \pi k}{\sqrt 2} = \frac{(-1)^k}{\sqrt 2} $$</span></p>
<p>The ... |
47,246 | <p>I got some text scraps with this <em>structure</em></p>
<pre><code>"Focal Plane: 198' Active Aid to Navigation: Yes *Latitude: 35.250 N *Longitude: -75.529 W"
</code></pre>
<p>But some of them lack of parts like this</p>
<pre><code>"Focal Plane: 198' Active Aid to Navigation: Yes *Longitude: -75.529 W"
</cod... | Kuba | 5,478 | <pre><code>str1 = "Focal Plane: 198' Active Aid to Navigation: Yes *Latitude: \
35.250 N *Longitude: -75.529 W";
str2 = "Focal Plane: 198.12' Active Aid to Navigation: Yes \
*Longitude: -75.529 W"
</code></pre>
<p>With string patterns because I do not use regex :) </p>
<pre><code>record[string_] := Map[
StringCas... |
47,246 | <p>I got some text scraps with this <em>structure</em></p>
<pre><code>"Focal Plane: 198' Active Aid to Navigation: Yes *Latitude: 35.250 N *Longitude: -75.529 W"
</code></pre>
<p>But some of them lack of parts like this</p>
<pre><code>"Focal Plane: 198' Active Aid to Navigation: Yes *Longitude: -75.529 W"
</cod... | Mr.Wizard | 121 | <p>A variation of Kuba's method, using a single <code>StringCases</code> pass with post processing:</p>
<pre><code>str2 = "Focal Plane: 198.12' Active Aid to Navigation: Yes *Longitude: -75.529 W"
fields = {"Focal Plane: ", "Latitude: ", "Longitude: "};
StringCases[str2, a : fields ~~ x : NumberString :> (a ->... |
3,155,229 | <p>Stumbled across this weird phenomenon using the equation <span class="math-container">$y = \frac{1}{x} $</span>.</p>
<p><strong>Surface Area:</strong>
When you calculate the surface area under the curve from 1 to <span class="math-container">$\infty$</span></p>
<p><span class="math-container">$$\int_1^\infty \fra... | Daan Seuntjens | 655,854 | <p>As @Minus One-Twelfth pointed out in the comments: this phenomenon is called Gabriel's horn.</p>
<p>Gabriel's horn is a geometric figure which has infinite surface area but finite volume.</p>
<p><a href="https://i.stack.imgur.com/zHAvX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zHAvX.png" alt... |
4,269,434 | <p>I want to compute the gradient of the vector function <span class="math-container">$f(\vec{x}) = \|\vec{x} - \vec{a}\|$</span>, I have a try, but the result is kind of strange to me.</p>
<p>so here is my steps
<span class="math-container">\begin{align*}
\nabla\|\vec{x} - \vec{a}\| & = \nabla\sqrt{\sum_{i = 1}^... | Steph | 993,428 | <p>The differential approach is much simpler
with almost no computation.</p>
<p>Denote your scalar function
<span class="math-container">$f = \| \mathbf{z} \|$</span>
where
<span class="math-container">$\mathbf{z=x-a}$</span>.</p>
<p>From the relation
<span class="math-container">$f^2 = \mathbf{z} : \mathbf{z}$</span>,... |
1,794,855 | <p>I need to prove that for every three integers $(a,b,c)$, the $\gcd(a-b,b-c) = \gcd(a-b,a-c)$. Assuming that a $a \ne b$.</p>
<p>Having:</p>
<p>$d_1 = \gcd(a-b,b-c)$</p>
<p>$d_2 = \gcd(a-b,a-c)$</p>
<p>How do i prove $d_1 = d_2$?</p>
| clark | 33,325 | <p>Let $X_n=B(n,p)$ be a binomially distributed random variable. Also notice that $X_n=Y_1+Y_2+\cdots+ Y_n$ where $Y_i$ are i.i.d. Bernoulli with parameter $p$.</p>
<p>Now observe that
\begin{align}
\sum_{k=0}^n k\frac{n!}{k!\,(n-k)!}p^k(1-p)^{n-k}&= \operatorname{E}(X_n)\\
&= \operatorname{E}( Y_1+Y_2+\cdots... |
3,065,572 | <p>Function <span class="math-container">$f: (0, \infty) \to \mathbb{R}$</span> is continuous. For every positive <span class="math-container">$x$</span> we have <span class="math-container">$\lim\limits_{n\to\infty}f\left(\frac{x}{n}\right)=0$</span>. Prove that <span class="math-container">$\lim\limits_{x \to 0}f(x)=... | EuxhenH | 281,807 | <p>Proof is wrong. You need <span class="math-container">$A\subseteq B$</span> and <span class="math-container">$B\subseteq A$</span> to show that <span class="math-container">$A=B$</span>. </p>
<p>Take an element <span class="math-container">$x\in A$</span> and see what that implies.</p>
|
267,051 | <p>Games appear in pure mathematics, for example, <a href="https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game" rel="noreferrer">Ehrenfeucht–Fraïssé game</a> (in mathematical logic) and <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game" rel="noreferrer">Banach–Mazur game</a> (in topo... | Igor Rivin | 11,142 | <p>Well, here is <a href="https://arxiv.org/abs/1408.1790" rel="noreferrer">one.</a> (A game-theoretic proof of Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for fair-coin tossing, 2014).</p>
<p>And this:</p>
<p><em>Tanaka, Kazuyuki</em>, <a href="http://dx.doi.org/10.1002/malq.19920380127" rel="noref... |
52,802 | <p>This is partly a programming and partly a combinatorics question.</p>
<p>I'm working in a language that unfortunately doesn't support array structures. I've run into a problem where I need to sort my variables in increasing order.</p>
<p>Since the language has functions for the minimum and maximum of two inputs (b... | Martin Sleziak | 8,297 | <p>I thought it might be helpful to add a completely different viewpoint. (Feel free to ignore this answer, if other answers are clearer for you or if this post contains some things that you have not studied yet.)</p>
<p>We work with subsets of some given set $M$. Such subsets can be identified with function from $M$ ... |
1,560,539 | <p>I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$</p>
<p>I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$</p>
<p>so that this number is pretty close to what I want.</p>
<p>Now I look at the difference between sum and integral of log:</p>
<p>$$\sum_{j=1}^n \log(j) - \int... | marty cohen | 13,079 | <p>You have
$\log(1+1/j)
= 1/j+O(1/j^2)$,
so
$\sum_{j=2}^n \frac1{\log(1+1/j)}
\approx \sum_{j=2}^n (-1/j+O(1/j^2))
\approx -\log(n)+O(1)
$
and
(I think you mean $n$
in this expression
instead of $j$)
$n\log(1+1/n)
\approx 1$
,
which,
I think,
gives you what you want.</p>
|
1,315,199 | <p>From the wikipedia article on sine waves:</p>
<blockquote>
<p>The sine wave is important in physics because it retains its wave
shape when added to another sine wave of the same frequency and
arbitrary phase and magnitude. It is the only periodic waveform that
has this property. This property leads to its i... | nullUser | 17,459 | <p>The statement means that $$\alpha_1 \sin(\omega x + \delta_1) + \alpha_2\alpha_3\sin(\omega x + \delta_2)$$ can be expressed as another sine wave of the form $\alpha_3\sin(\omega x + \delta_3)$, e.g. in the case $\alpha_1=\alpha_2 = 1$
$$
2\cos(\frac{\delta_2}{2}-\frac{\delta_1}{2})\sin(\omega x + \frac{\delta_1}{... |
1,315,199 | <p>From the wikipedia article on sine waves:</p>
<blockquote>
<p>The sine wave is important in physics because it retains its wave
shape when added to another sine wave of the same frequency and
arbitrary phase and magnitude. It is the only periodic waveform that
has this property. This property leads to its i... | Zach Effman | 135,903 | <p>This is referring to the sum identity for sines: $\sin(\alpha) + \sin(\beta) = 2\cos(\frac{\alpha-\beta}{2})\sin(\frac{\alpha+\beta}{2})$. </p>
<p>From this we see that we can add sine waves of the same frequency but different phases and still get a sine wave of the original frequency. Specifically, applying the ab... |
97,318 | <p>I use this code but it doesn't work.</p>
<pre><code>ZZ = {
{1, 2, 3, 4},
{5, 6, 7, 8},
{9, 10, 11, 12},
{13, 14, 15, 16}
} ;
ZZ // MatrixForm
K = ConstantArray[0, {4, 4, 4}];
K // MatrixForm
For[ i = 1, i = 4, i++,
For [j = 1, j = 4, j++,
K[[i, j, 1]] = ZZ[[i, 1]] ;
K[[i... | e.doroskevic | 18,696 | <p>Try something like: </p>
<pre><code>Table[10 i + j, {i, 4}, {j, 3}] // MatrixForm
</code></pre>
<p>Please refer to:
<a href="https://reference.wolfram.com/language/ref/Table.html" rel="nofollow">https://reference.wolfram.com/language/ref/Table.html</a></p>
|
3,829,377 | <p>We know trace of a matrix is the sum of the eigenvalues of the given matrix. Suppose we know the characteristics polynomial of the matrix, is there any result which gives us the sum of the positive eigenvalues of the matrix?</p>
<p>Note that I need the sum of only the positive eigenvalues...not all eigenvalues.</p>
| Misha Lavrov | 383,078 | <p>This is essentially just as hard as finding the individual eigenvalues...</p>
<p>...in particular, because if it were easy, you could <em>use</em> it to find the individual eigenvalues.</p>
<p>From the characteristic polynomial of a matrix <span class="math-container">$A$</span>, it is easy to get the characteristic... |
4,419,565 | <p>Consider the function <span class="math-container">$f(x)$</span> and let <span class="math-container">$g(x)=f(cx)$</span>.</p>
<p>By the definition of derivative</p>
<p><span class="math-container">$$f'(x)=\frac{df(x)}{dx}=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}\tag{1}$$</span></p>
<p>and so the definition of <s... | Carl Schildkraut | 253,966 | <p>This is definitely potentially confusing, but not an ambiguity.</p>
<p>The object written <span class="math-container">$f$</span> is a function, which takes an input and gives an output. The object written <span class="math-container">$f'$</span> is also a function, which is defined using the function <span class="m... |
4,419,565 | <p>Consider the function <span class="math-container">$f(x)$</span> and let <span class="math-container">$g(x)=f(cx)$</span>.</p>
<p>By the definition of derivative</p>
<p><span class="math-container">$$f'(x)=\frac{df(x)}{dx}=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}\tag{1}$$</span></p>
<p>and so the definition of <s... | Jackozee Hakkiuz | 497,717 | <p>I just want to emphasize one thing. You don't take the derivative of a function "relative to something". The derivative of a function <span class="math-container">$f:\mathbb R \to \mathbb R$</span> is always taken "with respect to" one thing and one thing only: its entry.</p>
<p>If you want to se... |
1,683,414 | <p>$$\begin{cases}
x^2 = yz + 1 \\
y^2 = xz + 2 \\
z^2 = xy + 4
\end{cases}
$$</p>
<p>How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got $(x - y)^2 + (y - z)^2 + (x - z)^2 = 14$, but it leads to nowhere.</p>
| Robert Israel | 8,508 | <p>Maple indicates that a Groebner basis for the corresponding polynomials is
$[z^2 - 4, y, z + 2 x]$. Thus there are two solutions: $x = \pm 1, y = 0, z = \mp 2$.</p>
|
2,482,868 | <p>I am trying to find</p>
<p>$$ \lim\limits_{x\to \infty }[( 1+x^{p+1})^{\frac1{p+1}}-(1+x^p)^{\frac1p}],$$</p>
<p>where $p>0$. I have tried to factor out as</p>
<p>$$(1+x^{p+1})^{\frac1{p+1}}- \left( 1+x^{p}\right)^{\frac{1}{p}} =x\left(1+\frac{1}{x^{p+1}}\right)^{\frac{1}{p+1}}- x\left(1+\frac{1}{x^{p}}\right... | Adayah | 149,178 | <p>We can use the expansion</p>
<p>$$(1+y)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} \cdot y^n.$$</p>
<p>Taking $y = \frac{1}{x^p}, \ \alpha = \frac{1}{p}$, we get </p>
<p>$$\left( 1 + \frac{1}{x^p} \right)^{\frac{1}{p}} = 1 + \frac{1}{p x^p} + o \left( \frac{1}{x^{p}} \right).$$</p>
<p>Similarly</p>
<p>$$\... |
685,642 | <p>I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries outside the diagonal, as they are not relevant.</p>
<p>I don't understand though how to generalize for the entries on th... | sunspots | 110,953 | <p>Consider the subspace <span class="math-container">$W = \{ A \in M_{n \times n}(F) : tr(A) = 0\} = \{ A \in M_{n \times n}(F) : \sum_{i=1}^{n} A_{ii} = 0\}.$</span> Now, we apply the standard representation of <span class="math-container">$M_{n \times n}(F)$</span> with respect to the matrix units <span class="math-... |
3,489,642 | <blockquote>
<p>Let <span class="math-container">$D_{2\cdot 8}$</span> be given by the group presentation <span class="math-container">$\langle x,y\mid xy = yx^{-1} , y^2 = e, x^8 = e\rangle$</span>. Let <span class="math-container">$G = F_{\{x,y\}}$</span> be the free group on two generators and <span class="math-cont... | Con | 682,304 | <p>Hint: Define <span class="math-container">$\varphi \colon F_{\lbrace x,y \rbrace} \rightarrow D_{2 \cdot 8}$</span>, by sending <span class="math-container">$x$</span> and <span class="math-container">$y$</span> to the two generators of the dihedral group that you also called <span class="math-container">$x$</span> ... |
1,412,862 | <p>The author of my textbook has an unsatisfactory proof when it is describing the properties of the closure of a set.
I'm using <span class="math-container">$E^*$</span> for E closure. Also, <span class="math-container">$E'$</span> indicates the set of limit points of <span class="math-container">$E$</span>.</p>
<bloc... | Gabriel Sanfins | 247,708 | <p>The main implication we will use when proving this theorem is that:</p>
<p>If $E\subset F $ then $ E^* \subset F^*$.</p>
<p><em>Proof:</em> take $x \in E^*$, then for every neigborhood $V$ of $x$, there is a point $y \in E$ such that $y \in V$ as well. But we have that $E \subset F$, so $y \in F$ and $y \in V$. Fr... |
4,154,205 | <p>I study maths as a hobby. I am stuck on this question:</p>
<p>Find the values of c for which the line <span class="math-container">$2x-3y = c$</span> is a tangent to the curve <span class="math-container">$x^2+2y^2=2$</span> and find the equation of the line joining the points of contact.</p>
<p>I have established t... | Community | -1 | <p>I think there is a subtle distinction between your case and one where I would use WLOG in my writings.</p>
<p>For me, WLOG is for when you have a number of cases that are dealt with in a nearly identical manner, so that you'd really like to treat them as a single case. For instance, if I am dealing with a rectangul... |
6,219 | <p>First we craft a function to return the quadrant boundary based on <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture" rel="noreferrer">Oppermann's Conjecture</a></p>
<pre><code>a[n_] := (Mod[n, 2] + n^2 + 2 n)/4
</code></pre>
<p>Then we create a few lists</p>
<pre><code>r = 10;
q = 1;
q1 = Table[a[q ... | Heike | 46 | <p>Starting with <code>Graph[u]</code> you can extract the coordinates of the vertices as follows</p>
<pre><code>gr = Graph[u];
crds = AbsoluteOptions[gr, VertexCoordinates];
</code></pre>
<p>The graph including the diagonals can then be drawn according to</p>
<pre><code>Graph[VertexList[gr], Union[u, q1, q2, q3, q4... |
6,219 | <p>First we craft a function to return the quadrant boundary based on <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture" rel="noreferrer">Oppermann's Conjecture</a></p>
<pre><code>a[n_] := (Mod[n, 2] + n^2 + 2 n)/4
</code></pre>
<p>Then we create a few lists</p>
<pre><code>r = 10;
q = 1;
q1 = Table[a[q ... | kglr | 125 | <p>You can also use <code>GraphEmbedding</code> to get the vertex coordinates:</p>
<pre><code>Graph[VertexList[g = Graph[u]], Union[u, q1, q2, q3, q4],
VertexCoordinates -> GraphEmbedding[g]]
</code></pre>
<p><a href="https://i.stack.imgur.com/BjC1R.jpg" rel="nofollow noreferrer"><img src="https://i.stack.im... |
1,992,789 | <blockquote>
<p>Let $\{x_n\}$ be a sequence that does not converge and let L be a real
number. Prove that there exist $\epsilon >0$ and a sub-sequence
$\{x_{p_n}\}$ of $\{x_n\}$ such that $|x_{p_n}-L|>\epsilon$ for all n.</p>
</blockquote>
<p>I don't have any idea on how to prove this. Any advice and sugge... | Robert Z | 299,698 | <p>Use <a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow">Inclusion-exclusion principle</a>. </p>
<p>There are $\frac{8!}{2^4}$ 8-digits numbers that can be formed using two 1s, two 2s, two 3s, and two 4s.</p>
<p>i) How many of these 8-digits numbers have two adjacent 1s?
Th... |
1,027,235 | <p>Let N = 12345678910111213141516171819. How can I use modular arithmetic to show that N is (or isn't) divisible by 11? In general, how can I apply modular arithmetic to determine the divisibility of an integer by a smaller integer? I am finding modular arithmetic very confusing and unintuitive. I can understand "simp... | Jessica B | 81,247 | <p>You want to find what that number is mod 11. The good thing is, you don't have to get the answer in one go. You just keep applying the rule 'any multiple of 11 goes away'. Notice that at the start of the number you have some power of 10 times 12. That it therefore the same as that multiple of 10 times 1. So replace ... |
1,027,235 | <p>Let N = 12345678910111213141516171819. How can I use modular arithmetic to show that N is (or isn't) divisible by 11? In general, how can I apply modular arithmetic to determine the divisibility of an integer by a smaller integer? I am finding modular arithmetic very confusing and unintuitive. I can understand "simp... | Community | -1 | <p><strong>Hint</strong>:</p>
<p>$f(x)=\displaystyle\sum_{i=0}^mc_ix^i\ \ (c_i \in \mathbb{Z} \ \forall i)\land a\equiv b \pmod n \implies f(a)\equiv f(b) \pmod n$</p>
|
3,305,677 | <p>I am learning Asymptotic complexity of functions from CLRS. I know that exponentiation functions like <span class="math-container">$a^n$</span>,<span class="math-container">$(a>0)$</span> are faster than <span class="math-container">$n!$</span> But what about <span class="math-container">$a^{a^n}$</span> vs <span... | logdev | 566,485 | <p>How do they compare?: <span class="math-container">$n!=\mathcal{O}(a^{a^n})$</span></p>
<p>Proof:
If we bring down both <span class="math-container">$n!$</span> and <span class="math-container">$a^{a^n}$</span> to <span class="math-container">$log$</span> scale then <span class="math-container">$n!$</span> in <span... |
3,933,011 | <p>I know how to solve for the basic about this. But in this problem</p>
<p><span class="math-container">$\displaystyle{\frac{1}{2\pi i} \oint_{|z|=1} \frac{\cos \left(e^{-z}\right)}{z^2} dz}$</span>,</p>
<p>I don't know how to start. Can somebody help me or guide me about this? Or just give me a hint. Thanks. I wo... | Thusle Gadelankz | 840,795 | <p>You could also do this using the residue theorem directly, as the headline asks for. I wish to emphazise that I am aware that the calulation is completely analogous, and that I simply show this as an alternative since it appears from the comments and the headline that the asker might be more comfortable with the res... |
454,426 | <blockquote>
<p>In set theory and combinatorics, the cardinal number $n^m$ is the size of the set of functions from a set of size m into a set of size $n$.</p>
</blockquote>
<p>I read this from this <a href="http://en.wikipedia.org/wiki/Empty_product#0_raised_to_the_0th_power" rel="nofollow noreferrer">Wikipedia pag... | Ben Grossmann | 81,360 | <p>Here's an attempt to explain it understandably through induction:</p>
<p>Suppose we want to make a function from $\{a,b,c\}$ to $\{1,2,3,4,5\}$. How many choices do we have?</p>
<p>Well, we have to start by choosing some $f(a)$. We have $5$ choices for that. We also have $5$ choices for $f(b)$. However, we can... |
63,348 | <p>This question arises from a discussion with my friends on a commonly encountered IQ test questions: "What's the next number in this series 2,6,12,20,...". Here a "number" usually means an integer. I was wondering whether there is a systematical way to solve such problems.Let us call a point on a plane integer point ... | Aaron Meyerowitz | 8,008 | <p>Just to make a few comments:</p>
<p>1) As noted, If we have a list of values $a_0,a_1,\cdots, a_n$ of integers then there is a (unique) polynomial $f(t)$ of degree no more than $n$ with integer coefficients which maps integers to integers and such that $f(k)=k$ for $0 \le k \le n$.</p>
<p>2) There is a method invo... |
1,577,174 | <p>If $a, b \in \mathbb{C}$, then we have the standard triangle inequality for the difference:</p>
<p>$$||a| - |b|| \le |a - b|.$$</p>
<p>I am wondering if this inequality generalizes to exponents greater that one.</p>
<blockquote>
<p>My question is, for $1 < p < \infty$, does there exists a constant $C_p$ s... | zhw. | 228,045 | <p>Hint: Is $(x+1)^2 -x^2$ bounded on $(0,\infty)?$</p>
|
743,819 | <p>I'm studying for my exam and I came across the following draw without replacement problem :</p>
<blockquote>
<p><span class="math-container">$N$</span> boxes filled with red and green balls.
The box <span class="math-container">$r$</span> contains <span class="math-container">$r-1$</span> red balls and <span class=... | drhab | 75,923 | <p>1) The number of green balls in total equals the number of red balls.
Picking out a box at random, taking out $2$ balls and then looking
at the second is actually 'the same' as picking out one ball out of
'big' box that contains all balls. The procedure followed has no
influence at all on the chances of a ball to be... |
1,521,649 | <p>I need to show that at most finitely many terms of this sequence are greater than or equal to $c$. </p>
<p>I don't know if it is the wording of the problem but I don't know what this is asking me to do. Help on this would be amazing! And thank you in advance.</p>
| Brevan Ellefsen | 269,764 | <p>$$x = \log_b(a^n)$$
$$b^x=a^n$$
$$(b^x)^{1/n}=(a^n)^{1/n}$$
$$b^{x/n}=a$$
$$x/n = \log_b(a)$$
$$x = n \log_b(a)$$
$$\log_b(a^n) = n\log_b(a)$$</p>
|
1,521,649 | <p>I need to show that at most finitely many terms of this sequence are greater than or equal to $c$. </p>
<p>I don't know if it is the wording of the problem but I don't know what this is asking me to do. Help on this would be amazing! And thank you in advance.</p>
| randomgirl | 209,647 | <p>$\ln(x)=\int_1^x \frac{1}{t} dt \\ \text{ so } \ln(x^r)=\int_1^{x^r} \frac{1}{t} dt \\ \text{ and then differentiating both sides gives} \\ [\ln(x^r)]'=(x^r)' \cdot \frac{1}{x^r}=\frac{r}{x} \\ \text{ now integrating both sides ... see if you can finish from here } \ln(x^r)=... \\ $</p>
|
1,570,754 | <p>Let $I$ be an interval and $f\colon I \to \mathbb{R}$ a differentiable function. Suppose the following definitions:</p>
<p>For $x_0 \in I$ the point $(x_0,f(x_0))$ is called <em>saddle point</em> if $f'(x_0) = 0$ but $x_0$ is not a local extremum of $f$.</p>
<p>For $x_W \in I$ the point $(x_W,f(x_W))$ is called <... | zhw. | 228,045 | <p>Your example does indeed show that a saddle point need not be an inflection point. (The function $x^2\sin(1/x)$ also works, but your example has the virtue of being continuously differentiable.)</p>
<p>In the other direction, if $(a,f(a))$ is a point of inflection and $f'(a) = 0,$ then $(a,f(a))$ is a saddle point.... |
1,914,752 | <p>dividing by a whole number i can describe by simply saying split this "cookie" into two pieces, then you now have half a cookie. </p>
<p>does anyone have an easy way to describe dividing by a fraction? 1/2 divided by 1/2 is 1</p>
| John | 362,662 | <p>1/2 is half of a cookie. 1/2 divided by 1/2 is simply seeing how many times half a cookie fits, or corresponds to half a cookie, which is one time. Or how many halves of a cookie you need to get half of a cookie, which is one (One half of a cookie) again.</p>
|
115,385 | <p>I heard that computation results can be very sensitive to choice of random number generator. </p>
<ol>
<li><p>I wonder whether it is relevant to program own Mersenne-Twister or other pseudo-random routines to get a good number generator. Also, I don't see why I should not trust native or library generators as rando... | hmakholm left over Monica | 14,366 | <p>The answers to those questions depend <em>completely</em> on what you need the pseudorandom numbers <em>for</em>.</p>
<p>In some applications, such as cryptography, one needs to use very, very random numbers, and it is essential that nobody can <em>predict</em> which number your generator produced, and also that no... |
28,456 | <p>I built a <code>Graph</code> based on the permutations of city's connections from :</p>
<pre><code>largUSCities =
Select[CityData[{All, "USA"}], CityData[#, "Population"] > 600000 &];
uScityCoords = CityData[#, "Coordinates"] & /@ largUSCities;
Graph[#[[1]] -> #[[2]] & /@ Permutations[largUSCitie... | E3labs | 8,351 | <p>I asked about something similar, but it's with data for municipalities in Brazil. Since I couldn't find the names and locations for them on WolframAlpha (well at least not elegantly) then I resorted to the sledgehammer approach. Where I the image of the map, added a layer to it and then put dots where I wanted the n... |
1,723,718 | <p>Knowing $f(x,y) = 2x^2 +3y^2 -7x +15y$, one simply proves $$|f(x,y)|\leq 5(x^2+y^2)+22 \sqrt{x^2 + y^2}$$ How can I use this info to compute
$$ \lim_{(x,y)\to(0,0)} \frac{f(x,y) - 2(x^2+y^2)^{1/4}}{(x^2+y^2)^{1/4}}\;\;\; ?$$</p>
<p>Thanks!</p>
| Klint Qinami | 318,882 | <p>This can be done quite easily if you convert to polar coordinates.</p>
<p>We convert </p>
<p>$$lim_{x, y \to (0, 0)} \frac{2x^2 + 3y^2 - 7x + 15y - 2(x^2 + y^2)^{\frac{1}{4}}}{(x^2 + y^2)^{\frac{1}{4}}}$$</p>
<p>turns into </p>
<p>$$lim_{r \to 0} \>\> \frac{2r^2\cos^2 \theta + 3r^2 \sin^2 \theta - 7r \cos ... |
839,431 | <p>Can someone be kind enough to show me the steps to integrate this, I'm sure it's by parts but how do I split up the sin part?
$$x\sin(1+2x)$$</p>
| Aapeli | 77,427 | <p>Let $u=x$ and $v'=\sin(1+2x)$, then $u'=2$, $v=-\frac{\cos(1+2x)}{2}$:
\begin{align}\int{uv'\ dx}&=uv-\int{u'v\ dx}\\
\int{x\sin(1+2x)\ dx}&=-\frac{x\cos(1+2x)}{2}-\int{-\frac{2\cos(1+2x)}{2}\ dx}+C\\
&=-\frac{x\cos(1+2x)}{2}+\frac{\sin(1+2x)}{4}+C\\
\therefore\int{x\sin(1+2x)\ dx}&=\frac{1}{4}\sin(1... |
13,989 | <p>Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.</p>
<p>But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E_d$. Now th... | Felipe Voloch | 2,290 | <p>The answer is a bit more complicated if $j=0,1728$ because the corresponding elliptic curves have a bigger automorphism group, so I'll leave those out and let you (or others) deal with this case. If $j \ne 0,1728$, then the automorphism group of $E$ is of order $2$ and all other elliptic curves isomorphic to $E$ ove... |
13,989 | <p>Suppose $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$.
Now we know that both curves are isomorphic over $\mathbb{C}$ iff
they have the same $j$-invariant.</p>
<p>But $E_1$ and $E_2$ could also be isomorphic over a subfield of $\mathbb{C}$.
As is the case for $E$ and its quadratic twist $E_d$. Now th... | stankewicz | 3,384 | <p>Question 1: Putting both curves in say, Legendre Normal Form (or else appealing the lefschetz principle) shows that if the two curves are isomorphic over <span class="math-container">$\mathbf{C}$</span> then they are isomorphic over <span class="math-container">$\overline{\mathbf{Q}}$</span>. Now we could say that f... |
2,759,407 | <p>Suppose there's a exam with 5 questions.
If the probability that Student $1$ correctly answers question $i$ is $P_{1.i}$, then</p>
<p>$P_{1.1} = 0.3$ , $P_{1.2} = 0.4$ , $P_{1.3} = 0.9$, $P_{1.4} = 0.7$ , $P_{1.5} = 0.1$</p>
<p>For Student $2$, </p>
<p>$P_{2.1} = 0.4$ , $P_{2.2} = 0.5$ , $P_{2.3} = 0.2$, $P_{2.4}... | BruceET | 221,800 | <p><strong>Comment:</strong> I wish you success with @HennoBrandsma's approach (+1). It seems there will be some bookkeeping in considering
all the possibilties. In case it is of any use (e.g., for checking intermediate results), here are simulated
distributions for Student 1, Student 2, and Difference scores. </p>
<p... |
3,540,045 | <p>The definite integral, <span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\:dt$$</span></p>
<p><strong>My question</strong>: for the trigonometric integral above the answer is <span class="math-container">$\frac{3\pi}{16}$</span>. What I want to know is how can I compute these integrals easily. Is there more... | Community | -1 | <p>Use <span class="math-container">$$\frac{1+\cos\left(ax\right)}{2}=\cos^{2}\left(\frac{ax}{2}\right)$$</span></p>
<p>Then we have:</p>
<p><span class="math-container">$$\int_{0}^{\pi}3\cos^{4}\left(t\right)dt-\int_{0}^{\pi}3\cos^{6}\left(t\right)dt=\frac{3}{4}\int_{0}^{\pi}\left(1+\cos\left(2t\right)\right)^{2}dt-... |
3,540,045 | <p>The definite integral, <span class="math-container">$$\int_0^{\pi}3\sin^2t\cos^4t\:dt$$</span></p>
<p><strong>My question</strong>: for the trigonometric integral above the answer is <span class="math-container">$\frac{3\pi}{16}$</span>. What I want to know is how can I compute these integrals easily. Is there more... | emonHR | 565,609 | <p><strong>Approach <span class="math-container">$(1)$</span></strong> <br>
Let
<span class="math-container">\begin{align}
y&=\cos t+i\sin t\implies y^n&=\cos t+i\sin nt\\
\frac{1}{y}&=\cos t-i\sin t\implies \frac{1}{y^n}&=\cos t-i\sin nt\\
\end{align}</span>
Then
<span class="math-container">\begin{al... |
175,971 | <p>Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals.</p>
<p>Thanks a lot!</p>
| Rudy the Reindeer | 5,798 | <p>As you say $\mathrm{Spec}(R)$ is defined to be the set of all prime ideals of $R$. If $R$ is a field, the only proper ideal is $0$ hence you get $\mathrm{Spec}(F) = \{0\}$. </p>
<p>It gets more interesting if your space is a ring that is not a field, like for example $R = \mathbb Z$. Then you can endow it with the ... |
3,669,700 | <blockquote>
<p>Let <span class="math-container">$f(x)$</span> be a monic, cubic polynomial with <span class="math-container">$f(0)=-2$</span> and <span class="math-container">$f(1)=−5$</span>. If the sum of all solutions to <span class="math-container">$f(x+1)=0$</span> and to <span class="math-container">$f\big(\fr... | AryanSonwatikar | 571,692 | <p>When <span class="math-container">$f(x+1)=0$</span>, we have,
<span class="math-container">$$(x+1)^3+a(x+1)^2+b(x+1)-2=0$$</span>
Whose sum of roots can be obtained by "negative of coefficient of <span class="math-container">$x^2$</span> upon coefficient of <span class="math-container">$x^3$</span>", which comes out... |
81,588 | <p>A certain function contains points $(-3,5)$ and $(5,2)$. We are asked to find this function,of course this will be simplest if we consider slope form equation </p>
<p>$$y-y_1=m(x-x_1)$$</p>
<p>but could we find for general form of equation? for example quadratic? cubic?</p>
| Greg Martin | 16,078 | <p>I think the following idea works. Let $f(x) = x^p-x+a$. They key observation is that $f(x+1)=f(x)$ in the field of $p$ elements. Now factor $f(x) = g_1(x) \cdots g_k(x)$ as a product of irreducibles. Sending $x$ to $x+1$ must therefore permute the factors $\{ g_1(x), \dots, g_k(x) \}$. But sending $x$ to $x+1$ $p$ t... |
2,506,182 | <p>The question speaks for itself, since the question comes from a contest environment, one where the use of calculators is obviously not allowed, can anyone perhaps supply me with an easy way to calculate the first of last digit of such situations?</p>
<p>My intuition said that I can look at the cases of $3$ with an ... | Cornman | 439,383 | <p>We observe $3^{2011}\mod 10$ to get the last digit.</p>
<p>It is $3^{2011}=3\cdot 3^{2010}=3\cdot 9^{1005}\equiv 3\cdot (-1)^{1005}\equiv -3\equiv 7\mod 10$</p>
|
2,506,182 | <p>The question speaks for itself, since the question comes from a contest environment, one where the use of calculators is obviously not allowed, can anyone perhaps supply me with an easy way to calculate the first of last digit of such situations?</p>
<p>My intuition said that I can look at the cases of $3$ with an ... | Community | -1 | <p>Actually 3^1=3 , 3^2=9 ,3^3=27 ,3^4=81 . These are the only four numbers that come at units place to powers of 3. So to find any last digit of 3^2011 divide 2011 by 4 which comes to have 3 as remainder . Hence the number in units place is same as digit in units place of number 3^3. Hence answer is 7.</p>
|
9,010 | <p>I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ambient space?), but it's unclear to me whether the only non-varietable manifolds should be those that don't admit smo... | Matt E | 221 | <p>In English (as opposed to French, in which language variety and manifold are synonyms) the word <em>variety</em> is short for <em>algebraic variety</em>. The main differences, then, between (algebraic) varieties and (smooth) manifolds are that:</p>
<p>(i) Varieties are cut out in their ambient (affine or projectiv... |
2,949,224 | <p>How do I show that if <span class="math-container">$\sqrt{n}(X_n - \theta)$</span> converges in distribution, then <span class="math-container">$X_n$</span> converges in probability to <span class="math-container">$\theta$</span>? </p>
<p>Setting <span class="math-container">$Y_n = \sqrt{n}(X_n - \theta)$</span> , ... | Kavi Rama Murthy | 142,385 | <p>You have almost finished the proof. Let <span class="math-container">$\delta >0$</span>. <span class="math-container">$P\{|X_n-\theta| >\delta\} \leq P\{|Y_n| >M\}$</span> for all <span class="math-container">$n$</span> such that <span class="math-container">$\delta \sqrt n >M$</span>, hence for all suff... |
3,657,428 | <p>My textbook says that</p>
<blockquote>
<p>If <span class="math-container">$f(x)$</span> is piecewise continuous on <span class="math-container">$(a,b)$</span> and satisfies <span class="math-container">$f(x) = \frac{1}{2} [f(x_{-})+f(x_{+})]$</span> for all <span class="math-container">$x\in(a,b)$</span>, and if ... | Community | -1 | <p>Observe that <span class="math-container">$\lvert f(x)\rvert>0$</span> and <span class="math-container">$f(x)\ne0$</span> are the same thing.</p>
<p>If <span class="math-container">$f$</span> is continuous at <span class="math-container">$x_0$</span>, then by definition of continuity <span class="math-container"... |
1,725,945 | <p>I'm reading the proof of "Fundamental Theorem of Finite Abelian Groups" in Herstein Abstract Algebra, and I've found this statement in the proof that I don't see very clear.</p>
<p>Let $A$ be a normal subgroup of $G$. And suppose $b\in G$ and the order of $b$ is prime $p$, and $b$ is not in $A$. Then $A \cap (b)=(e... | egreg | 62,967 | <p>$A\cap(b)$ is a subgroup of $(b)$. Since $b$ has prime order, the only subgroups of $(b)$ are $(e)$ and $(b)$ (by Lagrange's theorem).</p>
<p>If $A\cap(b)=(b)$ we have $(b)\subseteq A$, so $b\in A$. Since, by assumption, $b\notin A$, we must have $A\cap(b)=(e)$.</p>
|
736,749 | <p>Let us consider the Fourier transform of <span class="math-container">$\mathrm{sinc}$</span> function. As I know it is equal to a rectangular function in frequency domain and I want to get it myself, I know there is a lot of material about this, but I want to learn it by myself. We have <span class="math-container">... | CyTex | 30,854 | <p>We know that the Fourier transform of Sinc(z) is,</p>
<p>\begin{equation*} \int_{-\infty}^{+\infty} {{\sin(z)}\over{z}} e^{-i \omega z} dz\end{equation*} </p>
<p>and</p>
<p>\begin{equation*} \int_{-\infty}^{+\infty} {{\sin(z)}\over{z}} e^{-i \omega z} dz = \int_{-\infty}^{+\infty} {{e^{iz}-e^{-iz}}\over{2iz}} e... |
736,749 | <p>Let us consider the Fourier transform of <span class="math-container">$\mathrm{sinc}$</span> function. As I know it is equal to a rectangular function in frequency domain and I want to get it myself, I know there is a lot of material about this, but I want to learn it by myself. We have <span class="math-container">... | LL 3.14 | 731,946 | <p>This simple method is missing. Since for <span class="math-container">$\omega_0\geq 0$</span>,
<span class="math-container">$$
\int_{\Bbb R} \mathbf{1}_{[-\omega_0,\omega_0]}(\omega) \,e^{i\,\omega\,t}\,\mathrm d \omega = \int_{-\omega_0}^{\omega_0} \,e^{i\,\omega\,t}\,\mathrm d \omega = \frac{e^{i\, \omega_0\,t} - ... |
2,494,232 | <p>I'm trying to find the out if $\sum_{n=1}^\infty {{1\over \sqrt{n}}-{1\over{\sqrt{n+1}}}}$ is divergent or convergent.</p>
<p>Here are some rules my book gives that I will try to follow:</p>
<p><a href="https://i.stack.imgur.com/TNV6x.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TNV6x.png" al... | Mark Viola | 218,419 | <p><strong>Without recognizing the telescoping nature of the series</strong>, we have </p>
<p>$$\sqrt{n+1}-\sqrt n=\frac{1}{\sqrt{n+1}+\sqrt n}$$</p>
<p>Hence, we have </p>
<p>$$\frac{1}{\sqrt{n}}-\frac1{\sqrt{n+1}}=\frac{1}{\sqrt{n}\sqrt{n+1}\left(\sqrt{n+1}+\sqrt n\right)}\le \frac{1}{2n^{3/2}}$$</p>
<p>Inasmuch ... |
2,181,540 | <p>What approach we have to solve the following differential equation?</p>
<p>$$y''(x)- \frac{y'(x)^2}{y} + \frac{y'(x)}{x}=0$$</p>
<p>The known solution is $y(x) = c_2 * x^{c_1}$</p>
| Mike | 17,976 | <p>It dawned on me that the first couple terms look very close to a quotient rule. It seems this equation falls apart when you divide both sides by $y$.</p>
<p>$$\frac{yy''-y'^2}{y^2}+\frac{y'}{xy}=0$$</p>
<p>$$\left(\frac{y'}{y}\right)'+\frac{y'}{xy}=0$$</p>
<p>$$x\left(\frac{y'}{y}\right)'+\frac{y'}y=\left(\frac{... |
2,485,425 | <p>If $A$ is a non empty subset of the reals and $f$ is a bounded function from $A$ to the reals, how can we show that:</p>
<blockquote>
<p>$\sup|f(x)| - \inf|f(x)| \le \sup(f(x)) - \inf(f(x))$?</p>
</blockquote>
<p>I started by stating that since $f$ is bounded, $\inf(f(x)) \le f(x) \le \sup(f(x))$. And then $|f(x... | gen-ℤ ready to perish | 347,062 | <p>I assert that the equator is <em>not</em> a line.</p>
<p>From <a href="http://mathworld.wolfram.com/Line.html" rel="nofollow noreferrer"><em>Wolfram MathWorld</em></a>:</p>
<blockquote>
<p>A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions.</p>
</blockquot... |
1,593,282 | <p>Say we have the function $f:A \rightarrow B$ which is pictured below.<a href="https://i.stack.imgur.com/WfCxs.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WfCxs.jpg" alt="enter image description here"></a></p>
<p>This function is not bijective, so the inverse function $f^{-1}: B \rightarrow A$... | Gregory Grant | 217,398 | <p>Yes that's perfectly reasonable and it's done with regularity. BUT you have to interpret $f^{-1}(x)$ as a <em>set</em>, a subset of the domain, and not as an element of the domain. That's the case even if the set has one element. Just make sure you don't treat $f^{-1}$ as a real function and the notation is fine.... |
3,098,245 | <p><span class="math-container">$$ \frac{d}{dx}e^x =\frac{d}{dx} \sum_{n=0}^{ \infty} \frac{x^n}{n!}$$</span>
<span class="math-container">$$ \sum_{n=0}^{ \infty} \frac{nx^{n-1}}{n!}$$</span>
<span class="math-container">$$ \sum_{n=0}^{ \infty} \frac{x^{n-1}}{(n-1)!}$$</span>
This isn't as straightforward as I thought ... | David C. Ullrich | 248,223 | <p>Seems to me people are giving correct versions of the calculation, hence showing that your conclusion is wrong, but without pinning down exactly where the error in your version is. The error is here: The "identity" <span class="math-container">$$\frac k{k!}=\frac1{(k-1)!}$$</span>is not true if <span class="math-c... |
3,323,170 | <p>This question was originally posted <a href="https://crypto.stackexchange.com/q/72456/62225">here</a> on Crypto StackExchange. As suggested by an answer I am posting it here to help get a better perspective on the math side.</p>
<blockquote>
<p>Public-key cryptography was not invented until the 1970's. Apart from... | Henno Brandsma | 4,280 | <p>I think that a system like RSA would have been impractical in pre-computer days. How to generate large enough primes? There were tables for the smaller primes, but your opponent has the same tables, so then trial division would be a threat... </p>
<p>Also, modular exponentiation is no party to do by hand either. An... |
2,230,897 | <blockquote>
<p>A line passing through $P=(\sqrt3,0)$ and making an angle of $\pi/3$ with the positive direction of x axis cuts the parabola $y^2=x+2$ at A and B, then:<br>
(a)$PA+PB=2/3$<br>
(b)$|PA-PB|=2/3$<br>
(c)$(PA)(PB)=\frac{4(2+\sqrt3)}{3}$<br>
(d)$\frac{1}{PA}+\frac{1}{PB}=\frac{2-\sqrt3}{2}$ </p>... | Mick | 42,351 | <p>Let the feet of A and B on the x-axis be A’ and B’ respectively.</p>
<p><a href="https://i.stack.imgur.com/rV4mQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rV4mQ.png" alt="enter image description here"></a></p>
<p>Compute $A’P$. It is … $\dfrac {1}{6} + \dfrac {\sqrt(\delta)}{6}$.</p>
<p>S... |
3,600,528 | <p>Is there a general formula for determining multiplicity of <span class="math-container">$2$</span> in <span class="math-container">$n!\;?$</span>
I was working on a Sequence containing subsequences of 0,1. 0 is meant for even quotient, 1 for odd quotient.
Start with k=3, k should be odd at start, if odd find (k-1)... | Bernard | 202,857 | <p>You have <em>Legendre's formula</em>: for any prime <span class="math-container">$p$</span>, the multiplicity of <span class="math-container">$p$</span> in <span class="math-container">$n!$</span> is
<span class="math-container">$$v_p(n!)=\biggl\lfloor\frac{n}{p}\biggr\rfloor+\biggl\lfloor\frac{n}{p^2}\biggr\rfloor+... |
182,101 | <p>With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in </p>
<blockquote>
<p>$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$</p>
</blockquote>
<p>which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's <a href="http... | Belgi | 21,335 | <p>$x:=y$ means $x$ is defined to be $y$.</p>
<p>The notation $\equiv$ is also (sometimes) used to mean that, but it also have other uses such as $4\equiv0$ (mod 2).</p>
<p>I encountered $:=$ a lot more than $\equiv$ , and it is my personal favourite.</p>
<p>There is also the notation $\overset{\Delta}{=}$ to mean "... |
165,328 | <p>What is the difference between $\cap$ and $\setminus$ symbols for operations on sets?</p>
| Zev Chonoles | 264 | <p>Here is the <a href="http://en.wikipedia.org/wiki/Union_%28set_theory%29" rel="nofollow">Wikipedia article on $\cup$</a>, the <a href="http://en.wikipedia.org/wiki/Intersection_%28set_theory%29" rel="nofollow">Wikipedia article on $\cap$</a>, and the <a href="http://en.wikipedia.org/wiki/Complement_%28set_theory%29#... |
165,328 | <p>What is the difference between $\cap$ and $\setminus$ symbols for operations on sets?</p>
| Aru Ray | 13,129 | <p>(Answer to the edited question):</p>
<p>$\cap$ is for set intersection and $\backslash$ is for set difference. I'm sure you can look up the wikipedia entries for them. Here is a more descriptive example: </p>
<p>Suppose $A$ is the set of families with pet cats, and $B$ is the set of families with pet dogs. $A \cap... |
3,247,176 | <p>I have this statement:</p>
<blockquote>
<p>It can be assured that | p | ≤ 2.4, if it is known that:</p>
<p>(1) -2.7 ≤ p <2.3</p>
<p>(2) -2.2 < p ≤ 2.6</p>
</blockquote>
<p>My development was:</p>
<p>First, <span class="math-container">$ -2.4 \leq p \leq 2.4$</span></p>
<p>With <span class="math... | fleablood | 280,126 | <p>You getting confused by a <em>strong</em> premise implying a <em>weak</em> conclusion. </p>
<p>1 and 2 together say precisely: <span class="math-container">$p\in (-2.2,2.3) $</span></p>
<p>And you are asking to conclude <span class="math-container">$p\in [-2.4,2.4] $</span>.</p>
<p>That's simply a matter of noti... |
1,994,021 | <p>In one of the research article it is written that the following limit is equal to zero $$\lim_{x \to 0 }\frac{d}{2^{b+c/x}-1}\left[a2^{b+c/x}-a-a\frac{c\ln{(2)}2^{b+c/x}}{2x}-\frac{c\ln{(2)}}{2x^2}\frac{2^{b+c/x}}{\sqrt{2^{b+c/x}-1}}\right]\left(e^{-ax\sqrt{2^{b+c/x}-1}}\right)=0$$ where $a,b,c,d$ are all positive c... | Sil | 290,240 | <p>First notice that the function under limit is not defined in the left neighborhood of $0$, because assuming $x<0$, the expression under square root has to be non negative, i.e. $2^{b+c/x}-1 \geq 0$, and you can put these together to show that $x \leq -c/b$ then. So I'll assume you want to compute limit for $x \to... |
413,882 | <p>Let $\mathbb F$ be a field and $\mathbb F[x]$ the ring of polynomials with coefficients in $\mathbb F$. Let $p(x)$ be an irreducible polynomial in $\mathbb F[x]$. Let $k$ be a positive integer and consider the vector space $V$, over the field $\frac{\mathbb F[x]}{(p(x))}$with basis </p>
<p>$$1, p(x), p(x)^2, \ldot... | Community | -1 | <p>If I understand well $V=L^k$, where $L=\mathbb F[X]/(p)$ is a field. On $V$ you define a multiplication by $$(a_0,a_1,\dots,a_{k-1})(b_0,b_1,\dots,b_{k-1})=(a_0b_0,a_0b_1+a_1b_0,\dots,a_0b_{k-1}+\cdots+a_{k-1}b_0).$$</p>
<p>It's easy to see that $V\simeq L[Y]/(Y^k)$ and thus your problem reduces to the following: <... |
413,882 | <p>Let $\mathbb F$ be a field and $\mathbb F[x]$ the ring of polynomials with coefficients in $\mathbb F$. Let $p(x)$ be an irreducible polynomial in $\mathbb F[x]$. Let $k$ be a positive integer and consider the vector space $V$, over the field $\frac{\mathbb F[x]}{(p(x))}$with basis </p>
<p>$$1, p(x), p(x)^2, \ldot... | Jim | 56,747 | <p>No, the algebra you've defined is isomorphic to $\mathbb F[x]/x^k$, where $x$ is represented by $p(x) \in V$. It has dimension $k$ whereas $\mathbb F[x]/p(x)^k$ has dimension $k\cdot\deg p$.</p>
|
223,008 | <p>Ok so my teacher said we can use this sentence:
<strong>If $a$ is not a multiple of $5$, then $a^2$ is not a multiple of $5$ neither.</strong></p>
<p>to prove this sentence:
<strong>If $a^2$ is a multiple of $5$, then $a$ itself is a multiple of $5$</strong></p>
<p>I don't understand the logic behind it, I mean wh... | Brian M. Scott | 12,042 | <p>This is an example of an implication and its contrapositive. The contrapositive of an implication $\varphi\to\psi$ is the implication $\lnot\psi\to\lnot\varphi$; in words, the contrapositive of $$\text{if }\varphi\text{ is true},\text{ then }\psi\text{ is true}\tag{1}$$ is $$\text{if }\psi\text{ is not true},\text{ ... |
10,622 | <p>Given an open set $U \subset \mathbb R ^n $, there exists an exhaustion by compact sets, i.e. a sequence of compact sets $K_i$, s.t.</p>
<p>$\cup _{i=0}^{\infty} K_i = U$ and $\forall i \in \mathbb N : K_i \subset K_{i+1} ^{\circ}$
</p>
<p>We can imagine that different exhaustions by compact sets 'propagate' at di... | shuhalo | 3,557 | <p>Ok, one version on my own, more technical.</p>
<p>Let $V$ be any compact set included in $U$. It remains to show $\exists i \in \mathbb N : V \subset K_i$.</p>
<p>Note that $\forall i : V \cap K_i$ bounded again.</p>
<p>Suppose, there is no index $i$ s.t. $V \subset K_i$.</p>
<p>Then there is a nested sequence $... |
3,535,316 | <p><span class="math-container">$$\int_{0}^{\pi}e^{x}\cos^{3}(x)dx$$</span></p>
<p>I tried to solve it by parts.I took <span class="math-container">$f(x)=\cos^{3}(x)$</span> so <span class="math-container">$f'(x)=-3\cos^{2}x\sin x$</span> and <span class="math-container">$g'(x)=e^{x}$</span> and I got"</p>
<p><spa... | user577215664 | 475,762 | <p>Use the definition of the exponential:
<span class="math-container">$$
\begin{align}
E=&\sum_{n=0}^\infty \frac {n+1}{n!} \\
E=&\sum_{n=0}^\infty \frac {1}{n!}+\sum_{n=1}^\infty \frac {n}{n!} \\
E=&e+\sum_{n=1}^\infty \frac {1}{(n-1)!} \\
E=&e+\sum_{n=0}^\infty \frac {1}{n!} \\
\implies E=&2e
\en... |
3,166,419 | <blockquote>
<p>A fair coin is tossed three times in succession. If at least one of
the tosses has resulted in Heads, what is the probability that at
least one of the tosses resulted in Tails?</p>
</blockquote>
<p>My argument and answer: The coin was flipped thrice, and one of them was heads. So we have two unkn... | Jacob | 644,834 | <p>The answer <span class="math-container">$\frac{6}{7}$</span> is correct. Whatever the issue in your reasoning, I feel it must lie in the statement "there is no useful information" in saying one of the 3 flips was heads. To me it seems there is a difference between reading off the results of coins already flipped and... |
1,761,527 | <p>Let $f,g,h:X\to\mathbb{R}$ such that $f(x)\leq g(x)\leq h(x)$ for all $x\in X$. If $f$ and $h$ are differentiable in $a$ and $h(a)=f(a)$. Then $g$ is differentiable and $g'(a)=f'(a).$</p>
<p>How can I can prove that $g$ is differentiable in $a$.? Thanks</p>
| hmakholm left over Monica | 14,366 | <p>This is not even close to being true -- for example a counterexample could be
$$ f(x) = -4 \qquad g(x)=\arctan(x) \qquad h(x)=4 $$
(where $g'(a)$ exists for all $a$ but never equals $f'(a)$) or
$$ f(x) = 0 \qquad g(x)=\begin{cases}1 & x\in\mathbb Q \\ 0 & x\notin\mathbb Q \end{cases} \qquad h(x)=1 $$
(where ... |
794,842 | <p>Let the statement $?PQR$ be determined by the following truth-table.</p>
<pre><code>P Q R ?PQR
T T T T
T T F F
T F T F
T F F T
F T T T
F T F T
F F T F
F F F T
</code></pre>
<ol>
<li>After ‘Answer:’ below, give a logically equivalent sentence of ?PQR in FOL. Bu... | Asaf Karagila | 622 | <p><strong>HINT:</strong></p>
<p>For every line whose value is $\tt T$, write a sentence describe the assignment to the variables, e.g. $P\land Q\land R$ describes the first line of the table. </p>
<p>Then take the disjunction of these sentences, and show that the result has exactly this truth table.</p>
<p>You migh... |
4,623,022 | <p>I have a question that I have been curious about for years.</p>
<p>In differential geometry, since the exterior derivative satisfies property <span class="math-container">$d^2=0$</span>, we can make a de Rham cohomology from it.</p>
<p>Then if we write <span class="math-container">$\iota_X:\Omega^n\rightarrow\Omega^... | Quaere Verum | 484,350 | <p>This is more of a comment that is far too long for a comment. But I have also wondered about this, so I thought I would get started with the simplest possible examples. If anyone has a good interpretation for what these cohomology groups are, I would also love to know it.</p>
<p>For the simplest example, let's take ... |
1,557,039 | <h2>Background</h2>
<p>I am a software engineer and I have been picking up combinatorics as I go along. I am going through a combinatorics book for self study and this chapter is absolutely destroying me. Sadly, I confess it makes little sense to me. I don't care if I look stupid, I want to understand how to solve the... | Domenico Vuono | 227,073 | <p>For $n=1$ $3^2\equiv 1 \pmod 4$
Now suppose that $$3^{2n}\equiv 1\pmod 4$$ and you have to demonstrate that $3^{2n+2}\equiv 1\pmod 4$ Indeed $3^{2n}\cdot 3^2\equiv 1\pmod 4$</p>
|
307,458 | <p>Let <span class="math-container">$\mathbf{x}_1, \dots, \mathbf{x}_n \in \mathbb{R}^d$</span> be <span class="math-container">$n$</span> given vectors. Define the function</p>
<p><span class="math-container">$$ \mathcal{K}(\mathbf{x},\mathbf{y}) := \alpha\exp\left(-\frac{\|\mathbf{x}-\mathbf{y}\|^2}{2\sigma^2}\right)... | dchatter | 3,600 | <p>I don't see any connection between your problem and MPC.</p>
<p>Take a look at optimization via the technique known as stochastic approximation; it is extremely popular today for several reasons. Check out "Optimization Methods for Large-Scale Machine Learning" by Bottou, Curtis, and Nocedal on arXiv: <a href="http... |
1,811,081 | <blockquote>
<p>Let $1,x_{1},x_{2},x_{3},\ldots,x_{n-1}$ be the $\bf{n^{th}}$ roots of unity. Find: $$\frac{1}{1-x_{1}}+\frac{1}{1-x_{2}}+......+\frac{1}{1-x_{n-1}}$$</p>
</blockquote>
<p>$\bf{My\; Try::}$ Given $x=(1)^{\frac{1}{n}}\Rightarrow x^n=1\Rightarrow x^n-1=0$</p>
<p>Now Put $\displaystyle y = \frac{1}{1-x... | pmichel31415 | 326,840 | <p>Otherwise you could write $x_i=e^{j\frac {i2\pi} {n}}$ ($j^2=-1$)</p>
<p>Then
$$y_i=\frac {e^{-j\frac {i\pi} {n}}} {e^{-j\frac {i\pi} {n}}-e^{j\frac {i\pi} {n}}}=\frac {e^{-j\frac {i\pi} {n}}}{-2j\sin(\frac {i\pi} {n})}=2j\cot(\frac {i\pi} {n})+\frac 1 2$$</p>
<p>Now consider the fact that, if $x_i$ is an $n^{th}... |
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