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 |
|---|---|---|---|---|
2,856,121 | <p>Before I start explaining what I will say regarding the question, read the question before going any further.</p>
<p>Question: <em>Mariam and Leena both live on the bank of the same river, a distance apart. When Mariam visits Leena, she swims downstream (with the current), but then must swim upstream (against the c... | Ross Millikan | 1,827 | <p>The point is that current changes the velocity of swimming, not the time spent. Let the distance be $d$, the still water swimming speed be $v$ and the current speed be $c$. Swimming with the current is at speed $v+c$ and against the current is at speed $v-c$. If the dam is closed the round trip swimming time is $... |
1,844,846 | <p>I had an exam, and there was a question - <code>is cosh(z) injective?</code><br>
I presented it as cos(iz) (or cos(-iz), doesn't matter because it's an even function), but it didn't help because I don't know if <em>this</em> function is injective either.</p>
| shai horowitz | 339,356 | <p>If $\cos(iz)=\cos(-iz)$ then by definition its not injective from $C$ to $C$.</p>
|
1,609,845 | <p><strong>Problem</strong>: Let $\alpha, \beta$ be non-negative numbers.
Suppose the number of strictly increasing sequences $a_0, a_1, a_2 \cdots a_{2014}$ satisfying $0 \leq 3m$ is $2^{\alpha}(2\beta+1)$.
Find $\alpha$.</p>
<p>My attempt:</p>
<p>I tried to find an explicit form for the $nth$ of the sequence using... | Random Variable | 16,033 | <p>To supplement Dave's awesome answer, I'm adding a proof of the integral <span class="math-container">$$\zeta(s)-\frac{1}{s-1} = \frac{1}{\Gamma(s)} \int_0^{\infty} \left(\frac{1}{e^x-1}+1-\frac{1}{x} \right) x^{s-1} e^{-x} \, \mathrm dx \ , \quad \Re (s)>0. $$</span></p>
<p>All we need are the well-known integral... |
3,873,630 | <p>Hi Guys can some one help with this question?</p>
<p>Let <span class="math-container">$\mathcal{H}$</span> be a Hilbert space and F a linear bounded functional in <span class="math-container">$\mathcal{H}^∗$</span>
such that <span class="math-container">$F \neq 0$</span>. Prove that dim(kerF)<span class="math-contai... | Clement Yung | 620,517 | <p><span class="math-container">$
\renewcommand{\c}[1]{\langle #1 \rangle}
\renewcommand{\bb}[1]{\left( #1 \right)}
\renewcommand{\In}{\mathrm{In}}
$</span>
This proof is motivated by @M477's answer, and serves to fill in the details. Let <span class="math-container">$F,G_1,\dots,G_k$</span> be functions. We say that <... |
1,787,072 | <p>Boolean algebras aren't algebras (to the best of my understanding).</p>
<p>So why are they called algebras?</p>
<p>Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like that?</p>
| Noah Schweber | 28,111 | <p>They are algebras, in the sense of <em>universal algebra</em> (where "algebra" is basically synonymous with "first-order structure", except that it requires the language to have no relation symbols).</p>
<p>In fact, I think this notion of algebra = algebraic structure long preceded the definition of algebra as (rou... |
167,049 | <p>$p(z),t(z)$ are two mutually defined functional equations, while $\widehat{G}(z)$ is the exponential generation function of <a href="https://oeis.org/A182173" rel="nofollow noreferrer">A182173</a> (maybe, I am not sure...lol):</p>
<p>$$\begin{cases}
p(z)=e^{t(z)}-t(z)+2 z-1\\
t(z)=2\left(e^{p(z)}-e^{p(z)/2}+z\right... | Akku14 | 34,287 | <p>Find <code>Ge[z]</code>:</p>
<pre><code>r1 = p[z] -> 2 z + E^t[z] - 1 - t[z]
r2 = t[z] -> 2*(z + E^p[z] - E^(p[z]/2)) - p[z]
Ge[z_] = p[z] + t[z] - 2 z /. r1 /. r2 // FullSimplify
(* -1 + E^(2 (-E^((p[z]/2)) + E^p[z] + z) - p[z]) *)
</code></pre>
<p><code>SeriesCoefficient</code> produces derivatives ... |
176,386 | <p>I stumbled across this "dictionary for noncommutative topology" <a href="http://planetmath.org/noncommutativetopology" rel="nofollow">http://planetmath.org/noncommutativetopology</a>
and I would be very interested in learning more on the subject, particularly I'd like to see why these results are true. </p>
<p>Wha... | Michael | 33,290 | <p>K-Theory and C*-Algebras: A Friendly Approach (Oxford Science Publications)
by N.E. Wegge-Olsen. </p>
|
3,489,898 | <p>In order to show that a metric space <span class="math-container">$(X, d)$</span> is not complete one may apply the definition and look for a Cauchy sequence <span class="math-container">$\{x_n\}\subset X$</span> which does not converge with respect to the metric <span class="math-container">$d$</span>. Now I have o... | Marios Gretsas | 359,315 | <p>For the second method you mentioned note that <span class="math-container">$(1+1/n)^n$</span> is a Cauchy sequence on the rationals with respect to the usual metric restricted on <span class="math-container">$\Bbb{Q}$</span></p>
<p>Even if the sequence <span class="math-container">$x_n$</span> converged to a poi... |
648,946 | <p>I want to find
$$ \lim_{x\to0} \frac{(e^{-x^2}-1)\sin x }{x\ln(1+x^2)}$$
using a Maclaurin series and not using the l'Hôpital's rule.</p>
<p>However I can't seem to get it right.</p>
<p>Thanks for any possible answers.</p>
| Mr.Fry | 68,477 | <p>$(1)$ $z=(1+i\sqrt{3})^{-1+i} \iff z=e^{\ (-1+i) \log(1+i\sqrt{3})}$</p>
<p>$(2)$ Use $Log(z):= \ln |z| + i \arg(z)$</p>
<p>$(3)$ Here $|z|=2$ and $\arg(z)=\frac{\pi}{3}+2\pi n$ for any integer $n$</p>
<p>$(4)$ So, $z= e^{ (1+i)(2+i(\frac{\pi}{3}+2\pi n))}$</p>
<p>We can simplify this further, but I think this a... |
244,718 | <p>How can I plot the function</p>
<p><code>F[x_,a_]:=Integrate[Abs[x - z]^(-1-2a), {z, -Infinity, -1}] + Integrate[Abs[x - z]^(-1-2a), {z, 1, Infinity}]</code></p>
<p>for x in (-1,1) and a in (0,1)?
I have been using Wolfram Cloud to do
<code>Plot[F[x,1], {x,-1,1}]</code>,
but I get that the computational time is exc... | Andreas | 69,887 | <p>You can do numerical integration and a 3D plot like:</p>
<pre><code>Plot3D[NIntegrate[Abs[x - z]^(-1 - 2 a), {z, -Infinity, -1}] +
NIntegrate[Abs[x - z]^(-1 - 2 a), {z, 1, Infinity}], {x, -1, 1}, {a,0,1}]
</code></pre>
<p><a href="https://i.stack.imgur.com/HfFCf.png" rel="nofollow noreferrer"><img src="https://i... |
244,718 | <p>How can I plot the function</p>
<p><code>F[x_,a_]:=Integrate[Abs[x - z]^(-1-2a), {z, -Infinity, -1}] + Integrate[Abs[x - z]^(-1-2a), {z, 1, Infinity}]</code></p>
<p>for x in (-1,1) and a in (0,1)?
I have been using Wolfram Cloud to do
<code>Plot[F[x,1], {x,-1,1}]</code>,
but I get that the computational time is exc... | Roman | 26,598 | <p>Using some <a href="https://reference.wolfram.com/language/ref/Assuming.html" rel="nofollow noreferrer">assumptions</a> and an <a href="https://reference.wolfram.com/language/ref/Set.html" rel="nofollow noreferrer">immediate assignment</a> gives an analytic integral, which is easy to plot:</p>
<pre><code>F[x_, a_] =... |
1,752,468 | <p>I've tried some values of $m$, and found that the equality $C^{m}_{2m}=\Sigma_{i=1}^{m}C^{i}_{m+1}C^{i-1}_{m-1}$ holds. But I can't give it a proof. Can anybody give some suggestions?</p>
| Patrick Sole | 266,100 | <p>There is an obvious graphical picture of VanderMonde equality. Imagine a 2m set split between an <span class="math-container">$m+1$</span>--set A and an <span class="math-container">$m-1$</span>--set B. An <span class="math-container">$m$</span>--set E is then split itself into <span class="math-container">$E \cap A... |
4,468,883 | <p>In the book "<em>Primes of the form <span class="math-container">$x^2+ny^2$</span></em>", David Cox had shown that:
<span class="math-container">$$p=x^2+14y^2 \Longleftrightarrow (-14/p)=1 \;\text{and}\; (x^2+1)^2=8 \mod p \; \text{has an integer solution.} $$</span>
Is this imply that there are infinitely... | franz lemmermeyer | 23,365 | <p>According to a theorem due to Dirichlet (proved later by various mathematicians such as A. Meyer and H. Weber), every primitive binary quadratic form represents infinitely many prime numbers.</p>
<p>In the case of <span class="math-container">$x^2 + 14y^2$</span>, this follows from the observation above in addition ... |
1,048,260 | <p>I'm trying to read algebraic geometry on my own by doing homeworks on course hold in 2003. One of the problem is the following:</p>
<p>Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, $\mathbb{P}=\mathbb{P}_k^r=\operatorname{Proj}(S)$ and $\mathcal O$ the structure sheaf of $\mathbb{P}$. Let $r=3$ and let $f\in S_d,g\in ... | jflipp | 187,123 | <p>$S_d \subseteq S$ is the space of homogeneous polynomials of degree $d,$ i.e. the $k$-span of all monomials $T_0^{a_0}\cdots T_r^{a_r}$ with $a_0,\ldots,a_r \in \mathbb N_0$ and $a_0 +\cdots+a_r =d.$</p>
|
1,048,260 | <p>I'm trying to read algebraic geometry on my own by doing homeworks on course hold in 2003. One of the problem is the following:</p>
<p>Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, $\mathbb{P}=\mathbb{P}_k^r=\operatorname{Proj}(S)$ and $\mathcal O$ the structure sheaf of $\mathbb{P}$. Let $r=3$ and let $f\in S_d,g\in ... | Georges Elencwajg | 3,217 | <p>Now that the the meaning of the notations is clarified, here is a solution to the actual problem.<br>
We have a short exact sequence of coherent sheaves on $P=\mathbb P^3_k$: </p>
<p>$$0\to\mathcal O_P (-d-e)\to \mathcal O_P (-d)\oplus\mathcal O_P (-e) \to \mathcal O_P\to\mathcal O_Z\to 0$$ The first non trivial ... |
3,645,742 | <p>The number <span class="math-container">$2012 \cdot 2013 \cdot 2014 + 2013$</span> is the cube of </p>
<p>a) <span class="math-container">$2012$</span><br>
b) <span class="math-container">$2013$</span><br>
c) <span class="math-container">$2014$</span><br>
d) <span class="math-container">$2112$</span><br>
e) <span c... | Stackman | 595,519 | <p>As suggested by Andrew Chin in the comments to the question, let <span class="math-container">$n=2013,$</span> so that <span class="math-container">$2012=n-1$</span> and <span class="math-container">$2014=n+1$</span>. Then writing the given quantity in terms of <span class="math-container">$n$</span> we see that
<sp... |
3,827,422 | <p>Given a finite group <span class="math-container">$G$</span>, how do we know that there exists a map <span class="math-container">$\rho: G \rightarrow GL(V)$</span> such that <span class="math-container">$\rho(g_1\circ g_2) = \rho(g_1).\rho(g_2)$</span> for any <span class="math-container">$g_1, g_2\in G$</span>?</p... | Alekos Robotis | 252,284 | <p>Let <span class="math-container">$G$</span> be any finite group. We can form a free vector space on the group <span class="math-container">$G$</span> by <span class="math-container">$k[G]$</span>. That is, we define a basis <span class="math-container">$\{e_g\}_{g\in G}$</span>. Now, we can define a linear action of... |
3,306,209 | <p>I was given this problem and told to find it's derivative: <span class="math-container">$$f(x)=\int^{x^2}_1 t^2+t+1\ dt$$</span>I thought the derivative was simply the inside function in terms of x - <span class="math-container">$x^2+x+1$</span>. But, this is not correct. Where am I going wrong? What piece am I miss... | gt6989b | 16,192 | <p>If you think of
<span class="math-container">$$
F(t) = \int \left(t^2+t+1\right)dt
$$</span>
then
<span class="math-container">$$
f(x) = F\left(x^2\right) - F(1),
$$</span>
therefore
<span class="math-container">$$
f'(x)
= \frac{d}{dx} F\left(x^2\right)
= f\left(x^2\right) \frac{d\left[x^2\right]}{dx}
$$</span></... |
57,508 | <p>While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question: </p>
<p>Is it possible to prove that $\mathbb A^2-point$ is not an affine variety, if you don't know that the polynomia... | Guillermo Mantilla | 2,089 | <p>I'd say yes. I'll work over $\mathbb{C}$ to make my life easier. First it is enough to show that the ring of regular functions on $Y:=\mathbb{A}^2(\mathbb{C})−0$ is isomorphic $\mathbb{C}[x,y]$. If Y were affine, the identity map from $\mathbb{C}[x,y]$ to itself would induce an isomorphism between $Y$ and $X:=\mathb... |
1,106,320 | <p>I know the chance of hitting either red or black on the first roll is about 48%
1/37 chance on European 0
But shouldn't the chance of getting it right on either the first or second roll be high</p>
<p>like 66% or 75%</p>
<p>so to bet 25 1st roll and 50 second roll should have about 75% chance of a win?</p>
| 123 | 172,170 | <p>Things to consider: </p>
<ul>
<li><p>Each spin of the roulette wheel is an independent even. It is a common fallacy for a gambler to reason in the following manner: Because the ball has landed black three times consecutively, I should now bet red since the odds of the ball landing on black a fourth time are very sm... |
2,439,069 | <p>$$\int_0^{\infty}\frac{dx}{2+\cosh (x)}$$</p>
<p>I'm not sure how to approach this problem. I tried substituting $\cosh (x) = \frac{e^x + e^{-x}}{2}$, and following the substitution $y = e^x$, I ended up with the integral </p>
<p>$$2 \cdot \int_1^{\infty} \frac{1}{(y+2)^2-3} dy$$</p>
<p>Which I wasn't able to ev... | Angina Seng | 436,618 | <p>Using your substitution I get
$$2\int_1^\infty\frac{dy}{y^2+4y+1}=
2\int_3^\infty\frac{du}{u^2-3}.$$
This can be done by partial fractions.</p>
|
2,795,230 | <blockquote>
<p>Let $V$ be a vector space of continuous functions on $[-1,1]$ over $\Bbb R$. Let $u_1,u_2,u_3,u_4\in V$ defined as $u_1(x)=x,u_2(x)=|x|,u_3(x)=x^2,u_4(x)=x|x|$ then </p>
<ol>
<li>$\{u_1,u_2\}$ is linearly dependent.</li>
<li>$\{u_1,u_3,u_4\}$ is linearly dependent.</li>
<li>$\{u_1,u_2,u_4\}... | Adriano | 76,987 | <p>Correct. More generally, the four functions form a linearly independent set, because if:
$$
ax + b|x| + cx^2 + dx|x| = 0
$$</p>
<p>then it can be shown that $a = b = c = d = 0$. Just evaluate the equation at $x = \pm 1, \pm \frac{1}{2}$ and solve the resulting system of four equations in four unknowns.</p>
|
2,795,230 | <blockquote>
<p>Let $V$ be a vector space of continuous functions on $[-1,1]$ over $\Bbb R$. Let $u_1,u_2,u_3,u_4\in V$ defined as $u_1(x)=x,u_2(x)=|x|,u_3(x)=x^2,u_4(x)=x|x|$ then </p>
<ol>
<li>$\{u_1,u_2\}$ is linearly dependent.</li>
<li>$\{u_1,u_3,u_4\}$ is linearly dependent.</li>
<li>$\{u_1,u_2,u_4\}... | copper.hat | 27,978 | <p>Suppose $f(x) = \sum_k \alpha_k u_k(x) = 0$.</p>
<p>Note that $f$ is smooth for $x \neq 0$.</p>
<p>Note that $f''(x) = 2 \alpha_3 u_3''(x) + 2 \alpha_4 u_4''(x)$. Evaluating at $x=\pm 1$
shows that $\alpha_3 = \alpha_4 = 0$.</p>
<p>Then $f'(x) = \alpha_1 u_1'(x) + \alpha_2 u_2'(x)$, evaluating at $x= \pm 1$ shows... |
2,061,695 | <p>I'm examining function slope and determine relative extrema of function (local minimum and local maximum)</p>
<p>The example is as following:
$
y = x^8 e^{-5x}
$</p>
<p>To do this I have to determine derivative of this function, which is:
$$
y' = 8x^7e^{-5x}+x^8(-5e^{-5x}) = -e^{-5x}x^7(5x-8)
$$</p>
<p>Then accor... | Ross Millikan | 1,827 | <p>When you have a product that equals zero, at least one of the factors has to equal zero. $e^{-5x}$ is never zero, so you can divide by it. You are left with $x^7(5x-8)=0$ Now take each factor and set it to zero, which you should be able to solve. The solution is the union of these.</p>
|
2,061,695 | <p>I'm examining function slope and determine relative extrema of function (local minimum and local maximum)</p>
<p>The example is as following:
$
y = x^8 e^{-5x}
$</p>
<p>To do this I have to determine derivative of this function, which is:
$$
y' = 8x^7e^{-5x}+x^8(-5e^{-5x}) = -e^{-5x}x^7(5x-8)
$$</p>
<p>Then accor... | Simply Beautiful Art | 272,831 | <p>Well, we have three cases</p>
<p>$$\begin{cases}0=-e^{-5x}\\0=x^7\\0=5x-8\end{cases}$$</p>
<p>Clearly, the first case is not possible. For the second case, the solution is $x=0$. For the last case, the solution is $x=\frac85$.</p>
|
2,737,830 | <p>For some fixed b, r > 0, place b blue balls and r red balls in a box. Select balls from the box, one after the other, without replacement. Prove that the probability that the kth ball taken from the box is blue is the same for 1 ≤ k ≤ b.</p>
<p>Does this just come from the fact that if you do not know what you are ... | Hisoka Moroh | 459,561 | <p>We have $b$ blue balls and $r$ red balls, making for $b+r$ total balls in the box. Let $B_k$ be the event the $k^{th}$ ball drawn is blue.</p>
<p>Clearly, $P(B_1) = \frac{b}{b+r}$, because of the total number of balls in the box, we have $b$ blue balls to chose from. </p>
<p>What is $P(B_2)$? This requires a littl... |
176,600 | <p>The question is about factoring extremely large integers but you can have a look at <a href="https://stackoverflow.com/questions/11699464/mathematically-navigating-a-large-2d-numeric-grid-in-c-sharp">this question</a> to see the context if it helps. Please note that I am not very familiar with mathematical notation ... | MJD | 25,554 | <p>If the number in question is known to be a power of 2, you are just trying to find $n$ such that $N=2^n$. If the input number is in decimal notation, just count the digits, subtract 1, multiply by 3.3219, and round up. Then add 1 if the initial digit is 2 or 3, 2 if the initial digit is 4, 5, 6, or 7, and 3 if the... |
3,184,887 | <p>I want to divide a prize, say 1000, among an unknown number of contestants, as determined by their rank in a marathon race. I want the smallest portion to be above zero (so it is worthwhile to compete). So the higher the rank: the higher the prize.</p>
<p>PS, feel free to explain in detail as I am a math noob. Than... | MediaMaker | 663,415 | <p>Following my understanding of Chris Moorhead's comments, I'll try to answer this myself:</p>
<p>for 6 contestants</p>
<p>1+2+3+4+5+6</p>
<p>= 21</p>
<p>1/21 = .0476<br>
2/21 = .0952<br>
3/21 = .1428<br>
4/21 = .1905<br>
5/21 = .2381<br>
6/21 = .2857<br>
6/21+5/21+4/21+3/21+2/21+1/21 = 1
= sweet!</p>
<p>Then jus... |
1,321,704 | <p>For example: $3^\sqrt5$ versus $5^\sqrt3$</p>
<p>I tried to write numbers as this:</p>
<p>$3^{5^{\frac{1}{2}}}$ and then as
$3^{\frac{1}{2}^5}$</p>
<p>But this method gives the wrong answer because $a^{(b^c)} \ne a^{bc}$</p>
| Fermat | 83,272 | <p>Hint: Consider the function $$f(x)={\ln x\over \sqrt x}$$ and see where it is increasing or decreasing.</p>
|
1,321,704 | <p>For example: $3^\sqrt5$ versus $5^\sqrt3$</p>
<p>I tried to write numbers as this:</p>
<p>$3^{5^{\frac{1}{2}}}$ and then as
$3^{\frac{1}{2}^5}$</p>
<p>But this method gives the wrong answer because $a^{(b^c)} \ne a^{bc}$</p>
| YawarRaza7349 | 156,463 | <p>user134824's approach, but with the $\sqrt{3}$ instead, is a bit nicer:</p>
<p>$$
\begin{align}
3^\sqrt{5} &\text{ vs. } 5^\sqrt{3}\\
(3^\sqrt{5})^\sqrt{3} &\text{ vs. } (5^\sqrt{3})^\sqrt{3}\\
3^{\sqrt{5}\sqrt{3}} &\text{ vs. } 5^{\sqrt{3}\sqrt{3}}\\
3^\sqrt{15} &\text{ vs. } 5^3\\
\end{align}
$$</... |
540 | <p>A user has been using multiple unregistered accounts (with the same name and the same avatar). For reference, the accounts I've been able to find are 8280, 8281, 8284, 8289, 8294, 8308, 8309, 8310, 8311, 8312, 8313, 8314, 8315, 8316, 8317, 8318, and 8319, but I expect more to be added soon.</p>
<p>If these were re... | Chris Cunningham | 11 | <p>I helped [guest] combine two non-answers into an answer on the question here: <a href="https://matheducators.stackexchange.com/questions/12924/how-can-we-focus-students-on-the-various-data-types-in-multivariable-calculus/12941#12941">How can we focus students on the various data types in multivariable calculus?</a> ... |
746,751 | <p>I'm studying the theory of computation, and I know there are pumping lemmas for regular and context-free languages, but why not for recursively enumerable languages? Is there something about a Turing machine that would make a pumping lemma impossible? My guess is that no matter how you pump a string, a Turing machin... | ml0105 | 135,298 | <p>Suppose I have $L = \{ <M>,w :$ M is a TM that halts on w $\}$. This is clearly recursively enumerable. It's not regular though, nor is it context free. The pumping lemmas for regular and context free languages tell you that if you have a string in the language, building the string but maintaining the pattern... |
2,088,564 | <blockquote>
<p>Sum of binomial product $\displaystyle \sum^{n}_{r=0}\binom{p}{r}\binom{q}{r}\binom{n+r}{p+q}$</p>
</blockquote>
<p>Simplifying $\displaystyle \frac{p!}{r!\cdot (p-r)!} \cdot \frac{q!}{r!\cdot (q-r)!}\cdot \frac{(n+r)!}{(p+q)! \cdot (n+r-p-q)!}$.</p>
<p>Could some help me with this, thanks</p>
| G Cab | 317,234 | <p>The proposed identity can be derived (putting $r=s$) from this more general one
$$
\sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,n} \right)} {\left( \matrix{
m - \left( {r - s} \right) \cr
k \cr} \right)\left( \matrix{
n + \left( {r - s} \right) \cr
n - k \cr} \right)\left( \matrix{
r + k \... |
1,550,736 | <p>Consider Z a Normal (Gaussian) random variable with mean 0 and variance 1.<br>
It has density
$$f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} \text{for all x real numbers}$$
We consider $X=2Z+1$. Write the CDF of X in terms of the one of Z and take the derivative to get that the density of X is
$$f_X(x)=\frac{1}{2... | Ian | 83,396 | <p>Hint: let $x \in \mathbb{R}^q$ and write it as a linear combination of right singular vectors of $A$. Notice that $Ax$ is a linear combination of the corresponding <em>left</em> singular vectors of $A$. But these are orthogonal, so you have the Pythagorean theorem, which makes it straightforward to solve the minimiz... |
159,495 | <p>I have a text file where each line is a data point in the form:</p>
<pre><code>[ -495.01172, -158.35966, 2705.0 ]
[ -489.15576, -127.229675, 2673.0 ]
[ -487.6918, -97.679855, 2665.0 ]
[ -487.32578, -68.4594, 2663.0 ]
[ -485.86182, -39.19415, 2655.0 ]
[ -485.3128, -10.12311, 2652.0 ]
[ -484.03183, 18.853745, 2645.0 ... | Jack LaVigne | 10,917 | <p>I copied the 10 lines in your question and pasted them into a text file called <strong><em>out.txt</em></strong>.</p>
<pre><code>data = Import["D:\\at_work\\mathematica\\stack_exchange\out.txt",
"text"]
</code></pre>
<p>The basic problem with the way you were working is that what appears to be numbers are strin... |
50,406 | <p>This is a question about support of modules under extension of scalars.</p>
<p>Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module.
Recall that the <em>support</em> of $M$ is the set of prime ideals $\mathfrak{p}$ of $A$ such that the localizat... | S.Hamid Hassanzadeh | 47,945 | <p>The question is amount to the statement that if <span class="math-container">$f:(A,\mathfrak{m})\to (B,\mathfrak{n})$</span> is a local homomorphism and <span class="math-container">$M$</span> is a non-zero <span class="math-container">$A$</span>-module then <span class="math-container">$M\otimes _AB$</span> is also... |
85,651 | <p>Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.</p>
<p>How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $N \rightarrow \infty$?</p>
<p>Any suggestions?</p>
| S. Carnahan | 121 | <p>I don't have an answer, but here is a suggestion. From the environs of exercise 20 in section 3.7 of Terras's <em>Harmonic analysis on symmetric spaces I</em>, we find that a primitive geodesic whose beginning and end are joined by the hyperbolic transformation $\gamma$ has length $\log N(\gamma)$, where $N(\gamma)... |
85,651 | <p>Let $\Gamma$ be one of the classical congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$ of $SL(2, \mathbb{Z})$.</p>
<p>How does the lower bound for the length of primitive geodesics on $\Gamma \backslash \mathbb{H}$ depending on $N \rightarrow \infty$?</p>
<p>Any suggestions?</p>
| Igor Rivin | 11,142 | <p>For more general results and extensive references, see <a href="http://u.math.biu.ac.il/~vishne/publications/JDG-76-3-399-422.pdf" rel="nofollow">Katz, Schaps, Vishne, Logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups (JDG, 2007)</a></p>
|
2,475,054 | <p>I'm wondering if someone could give me some hints as to how to approach this question, or some theory to understand what it's actually asking:</p>
<p><a href="https://i.stack.imgur.com/FXvsz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FXvsz.png" alt="enter image description here"></a></p>
<p... | Robert Z | 299,698 | <p>The tangent plane at $(x_0,y_0,z_0)$ to a surface given in an implicit equation $F(x,y,z)=0$ is given by
$$F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0.$$
Can you take it from here? How do you recognize an horizontal plane?</p>
|
575,069 | <p>Let $\mathbb R_n[x]$ the vector space of polynomials with degree less or equal $n$ and we consider the linear transformation $f$ defined by
$$\forall P\in \mathbb R_n[x]\quad f(P)=(x^2-1)P''+2xP'$$
I proved that $f$ has the spectrum
$$\mathrm{sp}(f)=\{k(k+1),\ k=0,\ldots,n\}$$
I'm stuck in this question: Prove that... | Zhoe | 99,231 | <p>We know $$n^2 > (n - 1)^2$$ </p>
<p>Inverting the fractions the sign changes and we get </p>
<p>$$n^2 > (n - 1)^2 \implies \frac{1}{n^2} < \frac{1}{(n - 1)^2}$$</p>
<p>Take $b_n = \frac{1}{(n - 1)^2}$</p>
<p>Hence $a_n < b_n$ is satisfied.</p>
<p>P-series are of the form $$f(x) = \frac{1}{x^p}$$</p>... |
58,710 | <p>I've got a bunch of readings at various dates and times. I'd like to be able to interpolate and then integrate over, say days. I can do this by converting my date/time to <code>AbosluteTime</code> and get the interpolating function that way. Then by integrating and plugging in seconds as my limits of integration ... | kale | 1,560 | <p>There's a couple ways to do this:</p>
<pre><code>data = {{{2014, 8, 4, 10, 36, 0.}, 257.}, {{2014, 8, 4, 16, 28, 0.}, 385.},
{{2014, 8, 4, 22, 53, 0.}, 176.}, {{2014, 8, 5, 6, 52, 0.}, 148.},
{{2014, 8, 5, 11, 19, 0.}, 192.}};
</code></pre>
<p>1) Convert dates to absolute times:</p>
<pre><code>data[[All, 1]] = ... |
1,872,369 | <p>Prove that the equation $z^{3}e^{z}=1$ has infinitely many complex solutions.How many of them are real?</p>
<p>Use the argument principle,I choose a disk centered at $0$ with radius $R$ and get $\int_{\partial D}\dfrac {3z^{2}e^{z}+z^{3}e^{z}}{z^{3}e^{z}-1}dz$,and I don't how to do with this integral.Additionally,I... | Zach Blumenstein | 199,290 | <p>Set $f(z)=z^3e^z-1$, and let $g(z)$ be the meromorphic function defined by $g(z)=f(1/z)$. It is easy to check, by looking at a power series expansion, that $g$ has an essential singularity at $0$. So by Picard’s Theorem, for every neighborhood $U$ of $0$, there exists $w \in \mathbb{C}$ so that $g(U \setminus \{0\})... |
505,067 | <p>I am writing a computer program which generate 5 digit alpha-numeric codes. Each character can contain a-z,A-Z and 0-9 (62 different possibility per character).</p>
<p>So if there are 5 character how many possibility are there?</p>
| Shobhit | 79,894 | <p>$(\frac{p}q)^n$ cannot be an integer for $p,q$ and $n$ integers and $n\geq 1$ and $p$ is not divisible by $q$, as in your case. For if $p$ is not divisible by $q$, then so is the case for $p^n$ and $q^n$, hence it cannot be an integer for if you want it to be an integer the denominator must be $\pm 1$, or $q$ must d... |
11,172 | <p>As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$</span> because <span class="math-container">$\in... | Math Misery | 6,898 | <p>Wanted to write this as a comment, but I'm not strong enough [reputation]. This may be a bit heavy handed or you're already doing it, but what about challenging the very notion of arithmetic when dealing with infinities before going towards counterexamples? That's usually what the fuss tends to be about.</p>
<p>As ... |
11,172 | <p>As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times.</p>
<blockquote>
<ol>
<li><p><span class="math-container">$\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\sqrt{x}=0$</span> because <span class="math-container">$\in... | Henry Towsner | 62 | <p>This is a difficult problem, because the students you're dealing with aren't really prepared to think about logical reasoning and the role of counterexamples. (For example, I find students at that level often don't distinguish between examples and arguments.)</p>
<p>I'm of the view that you can't actually expect s... |
2,895,923 | <p>I am trying to solve this simple geometry problem but I am always tangled in so many equations it makes my head spin. I tried solving it via similar triangles but i cant seem to eliminate all the unwanted variables. Please help.</p>
<p>I have to prove $ r_1\times r_3=(r_2)^2$</p>
<p><a href="https://i.stack.imgur... | Community | -1 | <p>Due to the way they are constructed, the triangles $ABO,ACP,ADQ$ are similar, the figures $ABCPO$, $ACDQP$ are homothetic and $\dfrac{r_2}{r_1}=\dfrac{r_3}{r_2}.$</p>
|
3,497,554 | <blockquote>
<p>Show that <span class="math-container">$\frac{(2n-1)!}{(n)!(n-1)!}$</span> is odd or even according as <span class="math-container">$n$</span> is or is not a power of <span class="math-container">$2$</span>.</p>
</blockquote>
<p>I know that the index of the highest power of <span class="math-containe... | lulu | 252,071 | <p>This follows from <a href="https://en.wikipedia.org/wiki/Lucas%27s_theorem" rel="nofollow noreferrer">Lucas' Theorem</a>.</p>
<p>that theorem tells us that <span class="math-container">$\binom ab$</span> is divisible by a prime <span class="math-container">$p$</span> if and only if one of the base <span class="math... |
3,497,554 | <blockquote>
<p>Show that <span class="math-container">$\frac{(2n-1)!}{(n)!(n-1)!}$</span> is odd or even according as <span class="math-container">$n$</span> is or is not a power of <span class="math-container">$2$</span>.</p>
</blockquote>
<p>I know that the index of the highest power of <span class="math-containe... | Nεo Pλατo | 719,444 | <p>So my breakthrough discovery is for when n is a power of 2.</p>
<p>Let's do some rewriting.</p>
<p><span class="math-container">$\displaystyle\dfrac{(2n-1)!}{n!(n-1)!}=\dfrac{\dfrac{(2n)!}{2n}}{n! \cdot \dfrac{n!}{n}}=\dfrac{(2n)!}{2 \cdot (n!)^2}!$</span></p>
<p>And I'll replace <span class="math-container">$n=2... |
3,274,461 | <p>Suppose I have a recurrence relation like : </p>
<p><span class="math-container">\begin{equation}
f(n)=\begin{cases}
2*f(n-1) , & \text{n%2 = 1}.\\
f(n-1) + 1, & \text{n%2 = 0}.
\end{cases}
\end{equation}</span></p>
<p><span class="math-container">\begin{equation}
f(0) = f(1) = 1
\end{equation... | resound | 575,412 | <p>Refer this link : <a href="https://codeforces.com/blog/entry/53202" rel="nofollow noreferrer">Codeforces</a> and check Yeputons' second answer(the part where he breaks the solution). Has a nice way of breaking the function into two different functions. Maybe you can proceed with it.</p>
|
3,810,944 | <p>Consider a positive integer <span class="math-container">$n$</span> and the function
<span class="math-container">$f:\mathbb{N}\to \mathbb{N}$</span> (<span class="math-container">$\mathbb N$</span> includes <span class="math-container">$0$</span>) by</p>
<p><span class="math-container">$$f(x) = \begin{cases} \fra... | Tortar | 704,856 | <p>Stating <span class="math-container">$f(x) < x,\forall x > 2^n-1$</span> can be seen as the base step for an induction argument.</p>
<p>We can finish by induction proving :</p>
<p>if <span class="math-container">$\underbrace{\left( f\circ f\circ ....\circ f \right)}_{k\ f\text{'s}}\left( x \right) < x,\fo... |
3,810,944 | <p>Consider a positive integer <span class="math-container">$n$</span> and the function
<span class="math-container">$f:\mathbb{N}\to \mathbb{N}$</span> (<span class="math-container">$\mathbb N$</span> includes <span class="math-container">$0$</span>) by</p>
<p><span class="math-container">$$f(x) = \begin{cases} \fra... | Sebastian Spindler | 793,374 | <p>Note first that the inequality <span class="math-container">$f(x) < x$</span> only holds for <span class="math-container">$x \geq 2^n$</span> since <span class="math-container">$f(2^n - 1) = 2^n -1$</span>.</p>
<p>If there is an element <span class="math-container">$x \in A$</span> with <span class="math-containe... |
2,710,900 | <blockquote>
<p>Let $S=\{1,2t,-2+4t^2,-12t+8t^3\}$ be a set of polynomials in $P_3$.
Show that these polynomials make up a basis for $P_3$ and determine
the coordinates for $p=7-12t-8t^2+12t^3$ in this basis.</p>
</blockquote>
<p>The first part of the problem was easy, showing that they make up a basis by showin... | Arthur | 15,500 | <p>If you take a calculator and press <code>2^3^2</code>, whether you get $2^9$ or $2^6$ (or an error) will be entirely up to the manufacturer. There is no convention for this, and to human readers it is entirely ambiguous.</p>
<p>When writing, you should specify either $2^{3^2}$ or $(2^3)^2$, and when using a calcula... |
2,710,900 | <blockquote>
<p>Let $S=\{1,2t,-2+4t^2,-12t+8t^3\}$ be a set of polynomials in $P_3$.
Show that these polynomials make up a basis for $P_3$ and determine
the coordinates for $p=7-12t-8t^2+12t^3$ in this basis.</p>
</blockquote>
<p>The first part of the problem was easy, showing that they make up a basis by showin... | JMoravitz | 179,297 | <p>Be careful. Writing <code>2^2^3</code> can be thought of as ambiguous. It is standard to read power-towers from top down. That is to say, without parentheses, <code>a^b^c</code> should be interpreted as $a^{(b^c)}$, not as $(a^b)^c$</p>
<p>The property you think you are remembering is that $(a^b)^c=a^{b\times c}... |
2,710,900 | <blockquote>
<p>Let $S=\{1,2t,-2+4t^2,-12t+8t^3\}$ be a set of polynomials in $P_3$.
Show that these polynomials make up a basis for $P_3$ and determine
the coordinates for $p=7-12t-8t^2+12t^3$ in this basis.</p>
</blockquote>
<p>The first part of the problem was easy, showing that they make up a basis by showin... | Rob Arthan | 23,171 | <p>Your calculator is correctly implementing the (hotly disputed) correct answers to <a href="https://math.stackexchange.com/questions/1633790/what-is-the-order-when-doing-xyz-and-why/1633800#1633800">What is the order when doing $x^{y^z}$ and why?</a></p>
|
2,764,447 | <p>In the following question:</p>
<p><a href="https://i.stack.imgur.com/Rgzjr.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Rgzjr.jpg" alt="enter image description here" /></a>
I'm getting the answer as LHL=-infinity while RHL= infinity</p>
<p>The answer given to this question is D.</p>
<p>However ... | Hagen von Eitzen | 39,174 | <p>Speaking of sequences instead of functions (for which the limit is ultimately defined via sequences), we say that $a$ is the limit of $\{a_n\}$, if for every $\epsilon>0$, at most finitely many $a_k$ are outside the $\epsilon$-neighbourhood of $a$.</p>
<p>$$\tag1 a_n\to a:\iff \forall \epsilon>0\;\exists N\in... |
882,540 | <p>Does anyone know of a non-trivial (i.e. cardinality $\geq 2)$ algebraic structure $(X,+,-)$ satisfying the following identities?</p>
<ol>
<li><p>$(x+a)-a=x$</p></li>
<li><p>$(x-a)+a=x$</p></li>
<li><p>$(x+y)+a = (x+a)+(y+a)$</p></li>
<li><p>$(x-y)+a = (x+a)-(y+a)$</p></li>
</ol>
<p><em>Remark.</em> The Abelian gro... | Clinton Curry | 166,978 | <p>Below I will demonstrate that, if $+$ is associative, then $x+a = x$ and $x-a = x$ for all $x$ and $a$. This is enough, I think, to qualify as a "trivial" algebraic structure, even though the underlying set can be as large as you like.</p>
<p>Beginning from (3):</p>
<p>\begin{align*}
(x+y)+a &= (x+a) + (y+a)\... |
882,540 | <p>Does anyone know of a non-trivial (i.e. cardinality $\geq 2)$ algebraic structure $(X,+,-)$ satisfying the following identities?</p>
<ol>
<li><p>$(x+a)-a=x$</p></li>
<li><p>$(x-a)+a=x$</p></li>
<li><p>$(x+y)+a = (x+a)+(y+a)$</p></li>
<li><p>$(x-y)+a = (x+a)-(y+a)$</p></li>
</ol>
<p><em>Remark.</em> The Abelian gro... | James | 751 | <p>A structure with these properties is called an involuntary <a href="http://en.wikipedia.org/wiki/Racks_and_quandles" rel="nofollow">quandle</a>, or kei. They arise as important algebraic invariants of knots. See <a href="http://www.esotericka.org/cmc/quandles.html" rel="nofollow">here</a>.</p>
|
3,238,306 | <p>Hello I have the differential equation <span class="math-container">$$x'=\frac{8t+10x}{17t+x}$$</span>
Brought it in a eqation where I can substitute <span class="math-container">$$ u=\frac{x}{t}$$</span>
and after some transformations I got the equation
<span class="math-container">$$ \frac{17+u}{8-7u-u^2}du=\frac... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$ ab\ge 2a$$</span></p>
<p><span class="math-container">$$ab\ge 2b$$</span></p>
<p>Add both sides</p>
<p><span class="math-container">$$2ab \ge 2a+2b$$</span>
<span class="math-container">$$ ab\ge a+b $$</span></p>
|
290,173 | <p>Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\widetilde{d}(x,y)\to\infty$ if and only if $d(x,y)\to K$? One way would be to find some function $f$ such that $\widetil... | pteromys | 116,623 | <p>Not unless you're willing to lose the triangle inequality.</p>
<p>Nik Weaver's counterexample in the comments doesn't require $\tilde{d}$ to be of the form $f \circ d$: If $(X,d) = [0,1]$ with the usual metric, then you'd be requiring $\tilde{d}(0,1) = \infty$ while $\tilde{d}(0,0.5)$ and $\tilde{d}(0.5,1)$ are fin... |
290,173 | <p>Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\widetilde{d}(x,y)\to\infty$ if and only if $d(x,y)\to K$? One way would be to find some function $f$ such that $\widetil... | Nik Weaver | 23,141 | <p>I don't think there's any standard procedure, but here is one way to do this if there is a "base point" $e\in X$ satisfying $d(e,p) < K/2$ for all $p\in X$. I am not sure whether such a point can always be added. Wlog take $K=2$ and fix $e$. For any $p,q \in X$ find $p', q'$ in the open unit disc equipped with th... |
2,481,463 | <p>I am self-learning elliptic functions and modular form but struggling with the very basics.
I have programmed the Eisenstein series
$$E_6 = \sum_{m,n \in \mathbb{Z}} \frac{1}{(m +n \tau)^6} $$
and produced the following image (color denotes the argument)</p>
<p><a href="https://i.stack.imgur.com/NZeZb.png" rel="no... | Kenny Wong | 301,805 | <p>To answer your question, and to understand the picture of the "fundamental domains in the upper half plane" (your third picture), it's probably best if we forget about $E_6$ and $G_{2k}$ for the time being. They are not directly relevant to your question, although we will return to them.</p>
<p>Instead, I would rec... |
103,084 | <p><strong>Bug introduced in 8.0 or earlier and fixed in 11.0.0</strong></p>
<p>Case number: 305932.</p>
<hr>
<p>In the front-end notebook (Mathematica 9.0 and 10.2):</p>
<pre><code>g[x: Optional[_,default]]:=x;
</code></pre>
<p>has no problems, and works correctly (<code>g[]</code> outputs <code>default</code>, a... | jkuczm | 14,303 | <p>String <code>"x : Optional[_, default]"</code> in <code>StandardForm</code> is interpreted as:</p>
<pre><code>ToExpression["x : Optional[_,default]", StandardForm, HoldComplete] // FullForm
(* HoldComplete[Optional[Pattern[x, Blank[]], default]] *)
</code></pre>
<p>and in <code>InputForm</code> as:</p>
<pre><code... |
1,403,228 | <blockquote>
<p><strong>Question:</strong></p>
<p>Let <span class="math-container">$m, n, q, r \in \mathbb Z$</span>. If <span class="math-container">$m = qn + r$</span>, show that <span class="math-container">$\gcd(m, n) = \gcd(n, r)$</span>. Hence justify the Euclidean Algorithm.</p>
</blockquote>
<p>I found this que... | Manish Kumar Singh | 234,705 | <p><strong>To Prove $\gcd(m, n) = \gcd (n, r)$ if $m = qn + r$</strong></p>
<p>Let $\gcd(m, n) = d$.</p>
<p>So $d \mid m$ and $d \mid n$ implies $d \mid r$ (read $d$ divides...)</p>
<p>Similarly if $n = q_1r + r_1$ and $d \mid n$ and $d \mid r$ implies $d \mid r_1$.</p>
<p>Note $r_i$ are reducing by each successive... |
1,050,141 | <p>Find the roots of the equation $3^{x+2}$+$3^{-x}$=10 . By inspection the roots are $x=0$ and $x=-2$. But how can I solve this equation otherwise? </p>
| MPW | 113,214 | <p><strong>Hint:</strong> Multiply both sides by the positive quantity $3^x$ to clear the fraction implied by the negative exponent, and rearrange a bit:
$$3^{x+2} + 3^{-x}=10$$
$$3^x\cdot3^{x+2} + 3^x\cdot 3^{-x} = 3^x\cdot 10$$
$$3^{2x+2} + 1 = 3^x\cdot 10$$
$$3^{2x}\cdot 3^2 + 1 = 3^x\cdot 10$$
$$(3^x)^2\cdot 9 + 1 ... |
189,618 | <p>I'm trying to plot 2D vectors in Mathematica. Built in functions don't really work for me because I want to plot vectors of matrices from the origin to the their coordinates with an arrow on their tips. I made a function </p>
<pre><code>plotMatrixVectors[mat_List] :=
Graphics[Table[Arrow[{origin, i}], {i, mat}]... | Rainb | 59,208 | <p>I'd like to show this code, which is essentially the same as from m_goldberg, but it uses Standard colors. And adds label, the reason I did this, is because I needed this on my own plots and wanted to share.</p>
<pre class="lang-mathematica prettyprint-override"><code>plotVector[vectors : {{_?NumberQ, _?NumberQ} ..}... |
478,620 | <p>Let $S$ be a set that satisfies the following property:</p>
<blockquote>
<p>For every countable family $\{U_{i}\}_{i\in \mathbb{N}}$ of subsets of $S$ for which $$S=\bigcup_{i\in \mathbb{N}}U_{i}$$there is a finite set $I\subset \mathbb{N}$ for which $$S=\bigcup_{i \in I}U_{i}$$</p>
</blockquote>
<p>Does it foll... | A. H. | 160,657 | <p>To answer your question specifically, your idea is not correct because after you eliminate the integers that contain 5, you no longer have a 1:1 ratio of even:odd integers, so you can't simply divide by 2 to get your "number of odd integers that do not contain the digit 5."</p>
|
759,683 | <p>Given an undirected connected simple graph $G=(V,E)$ there are $2^{|E|}$ orientations. How many of these orientations are acyclic? </p>
| ml0105 | 135,298 | <p>I came across a research paper from 1972 which addresses this question. Let $\chi(G, \lambda)$ be the chromatic polynomial. Then $(-1)^{V} \chi(G, -1)$ is the number of acyclic orientations of $G$, your graph. I let $V$ be the vertex count of $G$. This is Theorem 1.3 of the paper.</p>
<p><a href="http://www-math.mi... |
380,696 | <p>Show that for any odd $n$ it follows that $n^2 \equiv 1 \mod 8$ and for uneven primes $p\neq 3$ we have $p^2 \equiv 1\mod 24$.</p>
<p>My workings so far: I proceeded by induction. Obviously $1^2 \equiv 1 \mod 8$. Then assume that for a certain uneven number $k$ we have $k^2 \equiv 1 \mod 8$. Then the the next uneve... | Martin Kochanski | 340,970 | <p>Never use induction at all. It is far too heavy a weapon for this kind of question.</p>
<ol>
<li>An odd number is a number of the form $4n±1$. Square it, and you get $16n^2±8n+1$, which clearly equals 1 modulo 8.</li>
<li>Primes are irrelevant: this is true for all odd numbers which are not multiples of 3. Such a n... |
2,478,026 | <p><a href="https://i.stack.imgur.com/TyhHX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TyhHX.png" alt="enter image description here"></a></p>
<p>It doesn't make sense to me that |E[x] - m| = |E[X-m]|</p>
| heropup | 118,193 | <p>I think the underlying issue is an apparent confusion regarding the meaning of "median" in this question. In the present context, the median refers to a property of the distribution of the random variable $X$, rather than a statistic of a sample drawn from such a distribution. Both are called "median" (for brevity... |
2,916,168 | <p>I am a beginner in proofs and, unfortunately, I cannot wrap my mind around how to prove the simplest things, so I need a bit of help getting started. This is the proof that I am dealing with:</p>
<p>$\text{If }x< y< 0\text{, then }x^{2}> y^{2}\text{.}$</p>
<p>Thank you in advance.</p>
| abc... | 470,090 | <p>$x^2>y^2\Rightarrow|x|>|y|\Rightarrow y>x$ for $x,y<0.$</p>
|
3,129,192 | <p>I've been trying to integrate
<span class="math-container">$$\int\frac{dx}{(x-a)(x-b)}$$</span></p>
<p>By using the substitution
<span class="math-container">$$x=a \cos^2 \theta + b \sin^2 \theta$$</span></p>
<p>The only problem here is I arrived at the result
<span class="math-container">$$\frac{2}{a-b} \ln |\cs... | user555203 | 447,605 | <p>Note: throughout, if <span class="math-container">$p\in I^2$</span> or <span class="math-container">$p\in A$</span>, <span class="math-container">$[p]$</span> will denote the equivalence class of <span class="math-container">$p$</span> in the respective quotient space.</p>
<p>You can first map <span class="math-con... |
3,129,192 | <p>I've been trying to integrate
<span class="math-container">$$\int\frac{dx}{(x-a)(x-b)}$$</span></p>
<p>By using the substitution
<span class="math-container">$$x=a \cos^2 \theta + b \sin^2 \theta$$</span></p>
<p>The only problem here is I arrived at the result
<span class="math-container">$$\frac{2}{a-b} \ln |\cs... | mechanodroid | 144,766 | <p>Let's show that the map <span class="math-container">$g : I^2/_\sim \to A/_{\approx}$</span> from @csprun's answer is indeed a homeomorphism:
<span class="math-container">$$g([(x,y)]) = [f(x,y)] = [((y+1)\cos(2\pi x), (y+1)\sin(2\pi x))].$$</span></p>
<p>First note that <span class="math-container">$g$</span> is we... |
1,326,525 | <p>Suppose I have some functions $f,g$ such that
$$f:\Bbb{R} \mapsto \Bbb{R}^2$$
$$g:\Bbb{R}^2 \mapsto \Bbb{R}^n$$</p>
<p><strong>My Question:</strong></p>
<p>For some $c \in \Bbb{R}$, is $g(f(c))$ a function of one variable? If so, why is the fact $g$ requires two arguments irrelevant? </p>
<p>EDIT: </p>
<p>Speci... | RowanS | 246,774 | <p>P( not cat | dog) = P( dog and not cat) /P (dog)</p>
|
1,212,177 | <p>Does $i\arg(e^{2z})=2iy?$ If it does I have solved my problem, and hence it seems like it must be the case, but I don't see it.</p>
<p>$$i\arg(e^{2z})=i\arg(e^{2x+2iy})=i\arg(e^{2x}e^{2iy})\implies \theta=2y(?)$$ Why does the $2x$ get 'ignored'?</p>
| Trajan | 119,537 | <p>$$\begin{align}i\arg(e^{2z})&=i\arg(e^{2x+2iy}) \\
&=i\arg(e^{2x}e^{2iy}) \\
&=i\arg(e^{2x}(\cos2y+i\sin2y)) \\
&=i\arg((\cos2y+i\sin2y)) \ \ \text{ as $e^{2x} \in \mathbb{R}$} \\
&=i(2y) \ \ \ \ \ \ \ \text{ by definition} \\
\end{align}$$</p>
|
61,174 | <p>I was just thinking about this recently, and I thought of a possible bijection between the natural numbers and the real numbers. First, take the numbers between zero and one, exclusive. The following sequence of real numbers is suggested so that we have bijection. </p>
<p>0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, ... | Vhailor | 887 | <p>If you know Cantor's diagonalization argument, then you should be able to find a counterexample to your problem using it. Say the method we use to make the $k$-th digit of our new real number different from the $k$-th digit of the $k$-th number in your list is adding 1 to it (and make it 0 if it's 9). Then Cantor's ... |
660,629 | <p>According to Wikipedia, factorial only is defined for non-negative integers.
How come Spotlight, the Windows calculator and the Google search engine come up with $\sqrt\pi$ if you try to solve $(-0.5)!$ ?</p>
| David H | 55,051 | <p>The area of the quadrilateral $BEDC$ is what's leftover if you subtract the triangle $ADE$ from triangle $ABC$. Since both triangles are right triangles, </p>
<p>$$area(BEDC)=area(ABC)-area(ADE)\\
=\frac12\|AB\|\cdot\|BC\|-\frac12\|AD\|\cdot\|DE\|\\
=\frac{24\cdot10}{2}-\frac{13}{2}\|DE\|.$$</p>
<p>To find $\|DE\|... |
660,629 | <p>According to Wikipedia, factorial only is defined for non-negative integers.
How come Spotlight, the Windows calculator and the Google search engine come up with $\sqrt\pi$ if you try to solve $(-0.5)!$ ?</p>
| Sawarnik | 93,616 | <p><img src="https://i.stack.imgur.com/1nQyf.png" alt="enter image description here"></p>
<p>Let us look at $\triangle AED$ and $\triangle CED$:</p>
<ul>
<li>$\angle EDA = \angle EDC = 90\,^{\circ}$</li>
<li>$DC= DA = 13$ cm</li>
<li>$ED$ is common</li>
</ul>
<p>So, by $SAS$ congruence, those two triangles are congr... |
2,939,605 | <p>The given task goes as follows:</p>
<blockquote>
<p>Show that <span class="math-container">$ f: \mathbb{R} \longrightarrow \mathbb{R}$</span> defined by <span class="math-container">$f(x) = \sqrt{1 + x^2} $</span> is not a polynomial function.</p>
</blockquote>
<p>I tried this approach - if <span class="math-con... | Angel Moreno | 327,493 | <p>Another demonstration:</p>
<p>A polynomial of degree one or higher when squared, should result in a polynomial that has all its roots double or more than doubles.</p>
<p><span class="math-container">$f ^ 2 = x ^ 2 + 1 = (x + i) (x-i)$</span> </p>
<p>That has two roots but they are not equal. And by the fundamenta... |
2,846,700 | <p>Consider the functions $f(x)$, $g(x)$, $h(x)$, where $f(x)$ is neither odd nor even, $g(x)$ is even and $h(x)$ is odd. Is it possible for $f(x) + g(x)$ to be </p>
<ol>
<li>even;</li>
<li>odd?</li>
</ol>
<p>For the second case I can imagine for example $f(x) = x - 1$ and $g(x) = 1$. Then $f$ is neither even nor od... | Hagen von Eitzen | 39,174 | <p>If $f+g$ is even, then $(f+g)-g$ is the difference of even functions, hence even</p>
|
873,224 | <p>I'm not mathematically inclined, so please be patient with my question.</p>
<p>Given </p>
<ul>
<li><p>$(x_0, y_0)$ and $(x_1, y_1)$ as the endpoints of a cubic Bezier curve.</p></li>
<li><p>$(c_x, c_y)$ and r as the centerpoint and the radius of a circle.</p></li>
<li><p>$(x_0, y_0)$ and $(x_1, y_1)$ are on the ci... | me22 | 875,771 | <p>I got inspired to attempt yet another way to interpret "best fit": most-constrained <em>curvature</em>.</p>
<p>Let's look at the first quarter-circle, using the four control points <span class="math-container">$<1,0>$</span>, <span class="math-container">$<1,α>$</span>, <span class="math-conta... |
104,328 | <p>This question is related to <a href="https://mathematica.stackexchange.com/questions/104310/export-bug-in-combination-with-paralleltable">Export bug in combination with ParallelTable?</a>.</p>
<p>The following code shows an even stranger bug than in the upper question.</p>
<p><strong>The y-axis should always be lo... | Ymareth | 880 | <p>Confirmation rather than answer. I'm not sure the original example is entirely functional, at least it was not for me. Mine is...</p>
<pre><code>SetDirectory["F:\\Temp"]; (* Adjust to suit your environment *)
hist = Histogram[RandomVariate[NormalDistribution[0, 1], 10000], ScalingFunctions -> "Log", ImageSize ... |
2,655,178 | <p>I am asked to find the equation of a cubic function that passes through the origin. It also passes through the points $(1, 3), (2, 6),$ and $(-1, 10)$. </p>
<p>I have walked through many answers for similar questions that suggest to use a substitution method by subbing in all the points and writing in terms of vari... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<p>Let the equation be $y=ax^3+bx^2+cx+d$, since it passes through $(1,3)$, we have</p>
<p>$$a(1)^3+b(1)^2+c(1)+d=3$$</p>
<p>Do the same thing for the other $3$ points. </p>
<p>Hence you will obtain $4$ linear equation with $4$ variables.</p>
<p>You can then solve it using elementary row operations t... |
2,329,003 | <p>Parametrise the circle centered at $ \ (1,1,-1) \ $ with radius equal to $ 3 $ in the plane $ x+y+z=1 $ with positive orientation . $$ $$ I have thought the parametriation: </p>
<p>\begin{align} x(t)=1+ 3 \cos (t) \hat j +3 \sin (t) \hat k \\ y(t)=1+3 \cos (t) \hat i+3 \sin (t) \hat k \\ z(t)=-1+3 \cos (t) \hat i... | Doug M | 317,162 | <p>$\mathbf u = (\frac {\sqrt 2}{2} \mathbf i - \frac {\sqrt 2}{2} \mathbf j)\\
\mathbf v = (\frac {\sqrt 6}{6} \mathbf i + \frac {\sqrt 6}{6} \mathbf j -\frac {\sqrt 6}{3} \mathbf k)$</p>
<p>$(x,y,z) = (1,1,-1) + 3\mathbf u \cos t + 3\mathbf v \sin t\\
x = 1 + 3\frac {\sqrt 2}{2} \cos t + \frac{\sqrt {6}}{2} \sin t\... |
212,466 | <p>Consider the list <code>list={1,2,3,4,5,6,7,8}</code> and imagine I wanted to change it to <code>{{1,2,3},{4,5,6},{7,8}}</code>. How can I do this?</p>
<p>Using <code>ArrayReshape</code> in the usual manner</p>
<pre><code>list = {1, 2, 3, 4, 5, 6, 7, 8};
ArrayReshape[list, {3, 3}]
</code></pre>
<p>yields <code>{{... | kglr | 125 | <p>It seems that using <code>ArrayReshape</code> and modifying, if needed, the last entry to remove trailing zeros is faster than using <code>Partition</code> or using <code>Inactive @ Nothing</code> padding:</p>
<pre><code>ClearAll[f1, f2, f3, f4]
f1 = Partition[#, UpTo[#2]] &; (* from Okkes's answer *)
f2 = Part... |
212,466 | <p>Consider the list <code>list={1,2,3,4,5,6,7,8}</code> and imagine I wanted to change it to <code>{{1,2,3},{4,5,6},{7,8}}</code>. How can I do this?</p>
<p>Using <code>ArrayReshape</code> in the usual manner</p>
<pre><code>list = {1, 2, 3, 4, 5, 6, 7, 8};
ArrayReshape[list, {3, 3}]
</code></pre>
<p>yields <code>{{... | Mr.Wizard | 121 | <p><a href="https://reference.wolfram.com/language/ref/ArrayReshape.html" rel="nofollow noreferrer"><code>ArrayReshape</code></a> is implemented using <code>Internal`Deflatten</code> (at least in version 10.1). This function requires <span class="math-container">$n \times m$</span> elements, or at least behaves strang... |
2,054,338 | <p>I had this question on my exam and I thought I could solve it using the row echelon method. Well, I couldn't, and I still don't know how to solve it. We were asked to "determine $a$ such the equation system has a unique solution" and to determine $a$ itself. The system had three equations: </p>
<p>$2x - y + az = 3$... | Mitchell Spector | 350,214 | <p>No calculus needed:</p>
<p>\begin{align}
\sum_{k=1}^\infty k(1-a)^{k-1}a &= \sum_{k=1}^\infty \sum_{j=1}^k (1-a)^{k-1}a
\\&= \sum_{j=1}^\infty \sum_{k=j}^\infty (1-a)^{k-1}a
\\&= \sum_{j=1}^\infty \frac{(1-a)^{j-1}}{1-(1-a)}a
\\&= \sum_{j=1}^\infty (1-a)^{j-1}
\\&=\frac1{1-(1-a))}
\\&=\frac1... |
2,044,362 | <p>(<em>This summarizes scattered results from <a href="https://math.stackexchange.com/questions/879089/prove-2f-1-left-frac13-frac13-frac56-27-right-stackrel-color808080">here</a>, <a href="https://math.stackexchange.com/questions/879854/prove-large-int-11-fracdx-sqrt394-sqrt5-x-left1-x2-right">here</a>, <a href="http... | Lucian | 93,448 | <p><strong>Too long for a comment :</strong> In general, for strictly positive values of <em>n</em> we have </p>
<blockquote>
<p>$$\begin{align}
\sqrt[n]2\int_0^\infty\frac{dx}{\sqrt[n]{\cosh2t~+~\cosh x}}
~&=~
\int_0^1\frac{dx}{\sqrt{1-x}\cdot\sqrt[n]{x^{n-1}~+~x^n\cdot\sinh^2t}}
\\\\
~&=~
\int_{-1}^1\frac{... |
2,411,811 | <p>Determine the limit:</p>
<p>$$\lim_{h\to 0}\frac{1-\cos(2h)}{h}$$</p>
<p>I know that the answer is $0$, I am just unsure of how to solve this. Thanks.</p>
| farruhota | 425,072 | <p>The third method is to multiply and divide by conjugate of numerator:
$$\lim_{h\to 0}\frac{1-\cos(2h)}{h}=\lim_{h\to 0}\frac{1-\cos(2h)}{h} \cdot \frac{1+\cos(2h)}{1+\cos(2h)}=\lim_{h\to 0}\frac{\sin^2(2h)}{h(1+\cos(2h))}=$$
$$\lim_{h\to 0}\frac{\sin^2(2h)}{2h}=\lim_{h\to 0}\frac{\sin(2h)}{2h}\cdot \lim_{h\to 0} \si... |
17,819 | <p>$\mathfrak{sl}_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = -2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for "Cartan.") My guess, based on similarities between these commutator relations and ones I have seen mentioned when pe... | LSpice | 2,383 | <p>I always thought that $H$ <em>did</em> stand for Cartan—at least, for ‘Henri’. However, I seem to recall that I had this discussion with <a href="https://mathoverflow.net/users/3927/brian-conrad">Brian Conrad</a>, and that he said it was actually Élie, not Henri, after whom the subgroups we... |
1,612,859 | <p>Take $i(G)$ to be the independence number of $G$, i.e. the maximum
number of
pairwise nonadjacent vertices in $G.$ I want to show that if $G$ has
$n$ vertices and $\frac{nk}{2}$ edges where $k \geq 1,$
$i(G) \geq \frac{n}{k+1}.$ I'm having difficulties making the
combinatorial argument on this graph. I get that if t... | Brian M. Scott | 12,042 | <p>I'm assuming that the word <em>then</em> is missing after $k\ge 1$. Suppose that $G$ is a counterexample: $G$ has $n$ vertices and $\frac{nk}2$ edges, and $i(G)<\frac{n}{k+1}$. Let $V_1$ be a maximal independent set of vertices in $G$, $V_2$ a maximal independent set of vertices in $G-V_1$, and so on through $V_m... |
1,836,785 | <p>Following are the two theorems that Hardy and Wright state in their book</p>
<blockquote>
<p><strong>Theorem A:</strong> The number of primes not exceeding <span class="math-container">$x$</span> is given by <span class="math-container">$\pi(x) \sim \frac{x}{\log{x}}$</span>.</p>
<p><strong>Theorem B</strong>: The ... | Community | -1 | <p>They are slightly different. The first gives a limit asymptotic constant equal to $1$, but no bracketing. The second doesn't give a single asymptotic constant, but a bracketing (for $x$ large enough). Obviously, $A'\le1\le A$.</p>
|
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | domotorp | 88,597 | <p>While there are many excellent suggestions, I would like to add that it is crucial to go abroad/other cities to get acquainted with many people, from whom you can learn a lot in many ways. This is something that's not good to delay, as later you might have no chance for family reasons!</p>
|
1,041,623 | <p>I have been suggested to read the <a href="http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf"><em>Advice to a Young Mathematician</em> section </a> of the <em>Princeton Companion to Mathematics</em>, the short paper <a href="http://alumni.media.mit.edu/~cahn/life/gian-carlo-rota-10-lessons.html"><em>Ten L... | fosco | 685 | <p>I don't know how many of these advices are already present in the pdfs, but I found these really valuable pieces of advice.</p>
<ol start="0">
<li>Choose a subject, an area of mathematics, which is "your favourite one". Live there as it was your home.</li>
<li>Relentlessly go back to the very basic fundame... |
2,377,139 | <p>My question:</p>
<p>Characterize those power series $\sum_{k=0}^\infty a_{k}(x-c)^{k}$ that converge uniformly on ($-\infty, \,\infty$).
What does it mean to characterize a power series?</p>
<p>Let {$a_{k}$} be a sequence of coefficients for a power series. By definition, the radius of convergence $R = \infty$ if ... | lhf | 589 | <p>A <em>characterization</em> in this case is a condition on the series (that is, on its coefficients) that is equivalent to uniform convergence in the whole real line.</p>
|
979,947 | <p>If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer?</p>
<p>UPDATE: I've done some more work and have come up with the following: Based on the equations I combined to arr... | Mathink | 166,778 | <p>You shouldn't guess for two reasons:
1-If you're wrong, you're giving proof that you haven't learned your lessons, rules, theorems...
2-You want to know your real level through SAT. If by guessing, you get some correct answers, then the score you'll have won't reflect your real value and skills.</p>
|
4,380,862 | <p>How can I solve the tangent point and <span class="math-container">$a$</span> when <span class="math-container">$f(x)=\left(\log_{a}{x}\right)^2$</span> is tangent to <span class="math-container">$g(x)=-ax+2$</span>?</p>
<p>Although this can be solved by substituting <span class="math-container">$a=e^2$</span> and <... | IV_ | 292,527 | <p>for real <span class="math-container">$a,x$</span></p>
<p><span class="math-container">$$\frac{d}{dx}\left(\log_a(x)^2\right)=\frac{d}{dx}\left(-ax+2\right)$$</span>
<span class="math-container">$$\frac{2\ln(x)}{\ln(a)^2x}=-a$$</span>
<span class="math-container">$a\to e^t$</span>:
<span class="math-container">$$t^2... |
278,637 | <p>After spending more than 1 hr on this, and looking at many questions, I give up as I am not able to figure a solution.</p>
<p>I have my current package in the standard location given by <code>FileNameJoin[{$UserBaseDirectory, "Applications"}]</code> which on windows is</p>
<pre><code>C:\Users\Owner\AppData... | Vitaliy Kaurov | 13 | <p>Perhaps something like that:</p>
<pre><code>Scalarize[x_List]:=If[Length[Flatten[x]]==1,Identity@@Flatten[x],x]
</code></pre>
|
631,348 | <p>Let $\sum a_n$ be a series of non-negative terms and let $$L = \lim_{n\to\infty}n\left(1-\frac{a_{n+1}}{a_n}\right)$$
Prove that the series converges (resp. diverges) if $L > 1$ (resp. $L<1$). I've tried, for example, that when $L<1$, $$n\left(1-\frac{a_{n+1}}{a_n}-L\right)= \frac{n(a_n-a_{n+1}-La_n)}{a_n}\... | QA Ngô | 122,043 | <p>The following is a new argument for the convergence part. Assume that <span class="math-container">$L>1$</span>. Choose <span class="math-container">$\varepsilon > 0$</span> small enough such that
<span class="math-container">$$L-\varepsilon > 1.$$</span>
There exists some <span class="math-container">$1 \l... |
2,226,386 | <p>Solve $$\frac{dx}{y+z}=\frac{dy}{z+x}=\frac{dz}{x+y}.$$ Solve the similtaneous equation by using method of multipliers. How can we choose these multipliers? Is there any specific method to know these multipliers?</p>
| Cye Waldman | 424,641 | <p>I think the difficulty you are having is the misconception that a complex number, say <em>z</em>, is a point in the complex plane. Rather, think of it as a vector (hence all those arrows). And they add and subtract just like vectors and they have scalar and vector products as well. For example, given two complex num... |
3,370,061 | <p>Could anyone clarify why this Boolean expression AB'+AB'AC' = AB'? I did not understand what happened to the C'</p>
| Andrei | 331,661 | <p>If <span class="math-container">$AB'$</span> is true, the <span class="math-container">$+$</span> will not change that. </p>
<p>If <span class="math-container">$AB'$</span> is false, then <span class="math-container">$(AB')(AC')$</span> is also false, no matter what <span class="math-container">$C'$</span> is. Then... |
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