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 |
|---|---|---|---|---|
974,207 | <p>Given the sequence <span class="math-container">${b_n}$</span>, let <span class="math-container">$lim_{n \to \infty}\ b_n = b$</span>.</p>
<p>Suppose that the sequence <span class="math-container">${a_n}$</span> and the number <span class="math-container">$a$</span> have the property for which there exists <span c... | Community | -1 | <p>Let $\epsilon>0$. Since $\lim_{n\to\infty }b_n=b$ then there's $n_0\in\Bbb N$ such that </p>
<p>$$|b_n-b|\le\frac {\epsilon}M\quad\text{whenever}\; n\ge n_0$$
so for $n\ge n_0$ we have</p>
<p>$$|a_n-a|\le M|b_n-b|\le\epsilon$$
and the result follows.</p>
|
2,027,888 | <h2>Exercise</h2>
<p>If $H$ is the Heaviside function, prove, using the definition below, that $\lim \limits_{t \to 0}{H(t)}$ does not exist.</p>
<hr>
<h2>Definition</h2>
<blockquote>
<p>Let $f$ be a function defined on some open interval that contains the number $a$, except possible $a$ itself. Then we say that ... | John11 | 300,543 | <p>Your attempt expresses the right idea but doesn't directly use the definition. </p>
<p>Here is one way to do it following the hint:</p>
<p>Suppose $\lim_{t\rightarrow 0}H(t)=L$. Then for every $\epsilon >0$, there exists $\delta >0$ such that $\left | H(t)-L \right |< \epsilon $ if $\left | t-0 \right |=\... |
829,433 | <p><img src="https://i.stack.imgur.com/FgnTN.jpg" alt="enter image description here"></p>
<p>why is it when I get the average of the column "per hour", it's different from the total?</p>
| cnick | 133,048 | <p>Your total column is weighting the per hour numbers by the number of hours.</p>
<p>That is, if you straight average the per hour numbers,
$$
(20+10+11+3)/4 = 11,
$$
you are not getting the average number of 'per hours' because there is more time spent on 'audits' than on 'invoices'</p>
<p>The actual average would... |
163,585 | <p>Which (finite, undirected) graphs have this property?</p>
<p>Every vertex $v$ can be labeled with a positive integer $l(v)$.</p>
<p>Variant 1: For each vertex $v$, $l(v) \geq \Sigma_{[v,w] \in E, w \neq v} l(w)/2$.</p>
<p>Variant 2: For each vertex $v$, $l(v) > \Sigma_{[v,w] \in E, w \neq v} l(w)/2$.</p>
| Qiaochu Yuan | 232 | <p>The <a href="http://en.wikipedia.org/wiki/ADE_classification" rel="nofollow">simply laced Dynkin diagrams</a> $A_n, D_n, E_6, E_7, E_8$ are precisely the connected (simple, undirected) graphs with <a href="http://en.wikipedia.org/wiki/Spectral_radius" rel="nofollow">spectral radius</a> less than $2$. This is more or... |
3,270,382 | <p>I'm having trouble constructing, and understanding the construction, of a metric that induces the dictionary order topology on <span class="math-container">$\mathbb R \times \mathbb R$</span>. There are example metrics posted here on Math.SE, but that is not what I'm looking for. I'd like to understand how to <stron... | freakish | 340,986 | <p>I assume that by "dictionary" order you mean lexicographical order. Note that there is more than one way to define <a href="https://en.wikipedia.org/wiki/Product_order" rel="nofollow noreferrer">ordering on a product of posets</a>. In particular <span class="math-container">$(a,b)\leq (a',b')$</span> if and only if ... |
2,006,870 | <p>Is the following inequality true?</p>
<p>$\left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty \sum \limits_{k=1}^\infty \sum \limits_{l=1}^\infty a_{ij}\,a_{ik}\,a_{jl}\,a_{kl} \right) \leq \left( \sum \limits_{i=1}^\infty \sum \limits_{j=1}^\infty a_{ij}^2 \right)^{1/2}\left( \sum \limits_{i=1}^\infty \sum ... | RGS | 329,832 | <p>To see if your method works it is enough to see if</p>
<p>$$x\cdot y + x - y = x $$</p>
<p>or</p>
<p>$$x\cdot y + x - y = y $$</p>
<p>for any $x, y $.</p>
<p>Solving the first we get</p>
<p>$$ x\cdot y + x - y = x \iff x \cdot y - y = 0 \iff y(x-1) = 0 \iff x = 1 \vee y = 0$$</p>
<p>So your equation returns $... |
3,059,150 | <p>Does there exist any open linear (vector) subspace of a Hilbert space? I could not think of any example.</p>
<p>Actually, I was reading the book by Simmons, there almost in every theorem it assumed that "If M is a closed linear subspace".It seemed natural to me to think about subspaces which are not closed. I have ... | Robert Lewis | 67,071 | <p>No.</p>
<p>Let <span class="math-container">$N$</span> be a normed space and</p>
<p><span class="math-container">$M \subsetneq N \tag 1$</span></p>
<p>a <em>proper</em> subspace. Then <span class="math-container">$M$</span> contains no nonempty open set. For if</p>
<p><span class="math-container">$\emptyset \n... |
142,858 | <p>Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W_{-k} H_k(X) \to \mathrm{Gr}^W_{-k} H_k(Y) $$ is surjective.</p>
<p><strong>Question:</strong> Does anyone know a reference for this fact?</p>
<p>I have a proof, but it's not as simple ... | Geordie Williamson | 919 | <p>I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case). (This is an edited version of my first incorrect answer.)</p>
<p>Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ o... |
2,735,179 | <p>I am trying to sketch the curve given by the following two parametric equations.</p>
<p>$x=cos^3\theta$</p>
<p>$y=sin^3\theta$</p>
<p>Or the single cartesian equation:</p>
<p>$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$</p>
<p>So,</p>
<p>I put the graph in Desmos and got:</p>
<p><a href="https://i.stack.imgur.com/HtFN... | Henry | 6,460 | <p>If you want a Bayesian approach, you will want a <a href="https://en.wikipedia.org/wiki/Prior_probability" rel="nofollow noreferrer">prior distribution</a> for the probability $p$ of heads. With a Bernouilli or binomial random variable, the <a href="https://en.wikipedia.org/wiki/Conjugate_prior" rel="nofollow nore... |
3,243,655 | <p>Question:</p>
<blockquote>
<p>Prove the equation <span class="math-container">$2x - 6y = 3$</span> has no integer solution to <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
</blockquote>
<p>I need to verify my proof I think I did it correctly, but am not fully sure since... | Community | -1 | <p>Your proof, as it stands, is not correct. Some particular issues:</p>
<ul>
<li><p>You get to a point where <span class="math-container">$x = 3y + 3/2$</span> and <span class="math-container">$y = x/3 - 1/2$</span> and state that "since both <span class="math-container">$x,y$</span> are not integers..."; this assume... |
3,337,147 | <p>Let <span class="math-container">$R$</span> be a unique factorization domain (UFD). Given <span class="math-container">$a,b \in R$</span> not simultaneously equal to zero, an element <span class="math-container">$d \in R$</span> is by definition a greatest common divisor (GCD) of <span class="math-container">$a$</sp... | jvdhooft | 437,988 | <p>Your calculations on the third draw are off. Either you have selected two balls of the same color (probability <span class="math-container">$\frac{1}{3}$</span>) in which case you are certain to win, or you have selected two balls of different color (probability <span class="math-container">$\frac{2}{3}$</span>) in ... |
3,253,145 | <p>I am doing this question
<a href="https://i.stack.imgur.com/AEaIn.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AEaIn.jpg" alt="enter image description here" /></a>
As it can't be solved using separation of variables (my assumption according to what i did, after checking by substituting <span c... | Community | -1 | <p>Add <span class="math-container">$$2^{n+1}-1=\sum_{i=0}^n2^i$$</span> and you get</p>
<p><span class="math-container">$$\sum_{i=0}^n2^i+\sum_{i=0}^n\pm_i2^i=2\sum_{i=0}^nb_i2^i$$</span> where the <span class="math-container">$b_i$</span> are <span class="math-container">$0$</span> or <span class="math-container">$1... |
3,462,868 | <p>What is the minimal size of a set <span class="math-container">$\mathfrak S$</span> of <span class="math-container">$k$</span>-element subsets of <span class="math-container">$\{1,...,n\}$</span> such that for any <span class="math-container">$k$</span>-element subset <span class="math-container">$S$</span> of <span... | pico | 666,807 | <p>Its an integration rule:</p>
<p><span class="math-container">$\int f'(t)~dt = f(t) + C$</span></p>
<p>Or:</p>
<p><span class="math-container">$\int \frac{df(t)}{dt}~dt = f(t) + C$</span></p>
|
2,096,296 | <p>Recently I discovered that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{π^2}{6}$, and we know that sum of reciprocals of naturals to the first power diverges to infinity. So I was wondering just out of curiosity, whether there was a number between $1$ and $2$ where this sum of reciprocals raised to that power started dive... | T C | 368,626 | <p>This is what you need. <a href="https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)</a>, find 6.3 - p-series</p>
<p>To sum up what you ask</p>
<p>If $k>1$ then $\sum_{n=1}^\infty \frac{1}{n^k}$ converges</p>
<p>If $k... |
2,504,737 | <p>During one of my daily exercises, I was looking for properties of the elements of Cartan calculus. I stumbled on Wikipedia's page about interior products <a href="https://en.wikipedia.org/wiki/Interior_product" rel="nofollow noreferrer">(here)</a>, and I've noticed a property that sounds very useful:
$$
\iota_{[X,Y]... | Ted Shifrin | 71,348 | <p>Yes, you are off by a sign on the term you figured. Remember that (for a $2$-form $\phi$) $\iota_X\phi(\cdot) = \phi(X,\cdot)$, so $\iota_Y\big(\iota_X\phi\big)=\phi(X,Y)$.</p>
|
362,944 | <blockquote>
<p>Consider a Brownian bridge <span class="math-container">$B: [0,1]\to \mathbb{R}$</span> with <span class="math-container">$B(0)=B(1)=0$</span>. Let <span class="math-container">$M[0, 1/2]=\max_{x\in[0,1/2]}B(x)$</span>. How to prove that
<span class="math-container">$$\mathbb{P}(M[0, 1/2]\geq s)\le... | dohmatob | 78,539 | <p>You can prove it via standard computations. See (2.1), (2.7), and (2.8) of <a href="https://www.researchgate.net/publication/236984395_On_the_maximum_of_the_generalized_Brownian_bridge" rel="nofollow noreferrer">On the maximum of the generalized Brownian bridge</a>.</p>
|
3,212,488 | <p><span class="math-container">$$\int_{0}^{\infty} \ln{(1+a x)}{x^{-b-1}} dx$$</span><br>
I defined two branch cuts along the real axis: <span class="math-container">$[-\infty ,-\frac{1}{a}]$</span> & <span class="math-container">$[0,\infty]$</span> with the following contour: <a href="https://i.stack.imgur.com/4r... | Robert Israel | 8,508 | <p>Hint: for (finite) real numbers <span class="math-container">$x$</span>, <span class="math-container">$e^{-x} > 0$</span>.</p>
|
1,037,632 | <p>How to find the last 2 digits of $2014^{2001}$? What about the last 2 digits of $9^{(9^{16})}$?</p>
| Krishnamurari | 195,422 | <p>$9^{\phi(100)}=9^{40}\equiv 1\pmod{100}\implies9^{40k}\equiv 1\pmod{100}$</p>
<p>Now consider $9^{16}=40k+?$ , that is the same as finding $x\equiv 9^{16}\pmod{40}$</p>
<p>$9^{\phi(40)}=9^{16}\equiv 1\pmod{40}\implies 9^{16}=40k+1$</p>
<p>$9^{40k}\equiv 1\pmod{100}\implies 9^{40k+1}\equiv 9\pmod{100}$</p>
|
15,702 | <p>One way to prove that a field $K$ has no ideals except the entire field and the trivial ideal is to note the fact that every element $x$ has an inverse. By the definition of an ideal, if $x$ is in the ideal then $x^{-1}x$ is because $x^{-1} \in K$. But now we have that 1 is in the ideal, and so again by the definiti... | Matt E | 221 | <p>One way to think of an ideal as being the set of multiples of the (possibly non-existent) g.c.d of all the elements that contain it. (I am thinking here of the case of a commutative ring with $1$, so that distinctions between left, right, and two-sided don't matter.)</p>
<p>In the integers, for example, any set of... |
2,823,568 | <p>It is well known that the sum</p>
<p>$$
\sum _{{k=0}}^{\infty }{\frac {x^{k}}{k!}}
$$</p>
<p>converges to $e^{x}$. In particular, for $x=1$ we have $\sum _{{k=0}}^{\infty }{\frac {1}{k!}}=e$. But what about the sum over the reciprocals of primorials, i.e.,</p>
<p>$$
\sum _{{k=0}}^{\infty }{\frac {x^{k}}{k\#}},... | Pagode | 564,189 | <p>For the second one , D'Alembert criterion is also sufficient there.</p>
<p>Indeed ,
you have $$ \forall k \in \mathbb{N}^*, \ \frac{x^{k+1}}{p_{k+1}\#}\frac{p_{k}\#}{x^{k}}=\frac{x}{p_k} $$</p>
<p>which obviously converges to 0.</p>
<p><strong>Conclusion:</strong></p>
<p>The second radius is infinite.</p>
|
1,493,415 | <p>When I did exercises in probability theory I found this limits as follows and verified it with Mathematica 8.0, and also noticed when $p=\dfrac12$ it shows that $\displaystyle p^n\sum_{k=0}^{n-1}\binom{k+n-1}{k}q^k \equiv \frac12$, but how it works?</p>
<blockquote>
<p>$$
\lim_{n\to\infty}p^n\sum_{k=0}^{n-1}\... | Robert Israel | 8,508 | <p>Hint: there is a finite interval $E$ such that $\int_{E^c} |f(x)| \; dx < \epsilon$. Then approximate the restriction of $f$ to $E$ by simple functions...</p>
|
1,493,415 | <p>When I did exercises in probability theory I found this limits as follows and verified it with Mathematica 8.0, and also noticed when $p=\dfrac12$ it shows that $\displaystyle p^n\sum_{k=0}^{n-1}\binom{k+n-1}{k}q^k \equiv \frac12$, but how it works?</p>
<blockquote>
<p>$$
\lim_{n\to\infty}p^n\sum_{k=0}^{n-1}\... | John Dawkins | 189,130 | <p>I'd first prove the limit statement for $f\in L^1$ that is also a step function: $f(x) = \sum_{k=1}^n 1_{(a_k,b_k]}(x)\cdot c_k$, where $a_1<b_1\le a_2<b_2<\cdots\le a_n<b_n$, and the $c_k$ are real. Then show that a general $f\in L^1$ can be approximated in $L^1$ by such step functions. For this second... |
1,430,447 | <p>sketch the graph of the integrand function and use it to help evaluate the integral.</p>
<p>integration from(1 , -1) |x|-1 </p>
<p><a href="https://i.stack.imgur.com/NY3Ei.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NY3Ei.png" alt="enter image description here"></a></p>
<p>I think I can ev... | Mark Viola | 218,419 | <p>For $x\ge 0$, the integrand is $x-1$. The graph begins at $(0,-1)$ and ends at $(1,0)$.</p>
<p>For $x\le 0$, the integrand is $-x-1$. The graph begins at $(-1,0)$ and ends at $(0,-1)$.</p>
|
1,430,447 | <p>sketch the graph of the integrand function and use it to help evaluate the integral.</p>
<p>integration from(1 , -1) |x|-1 </p>
<p><a href="https://i.stack.imgur.com/NY3Ei.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NY3Ei.png" alt="enter image description here"></a></p>
<p>I think I can ev... | nathan.j.mcdougall | 181,447 | <p>The graph can be considered as a piecewise function of two lines either side of $x=0$. Specifically, these are the lines $$y=\pm x-1$$ If you graph these lines it should be clear that the integral can be found piecewise. The integral will be
$$\begin{align}\int\,(|x|-1)\;\mathrm{d}x=\begin{cases}\int\,(x-1)\;\mathrm... |
3,270,504 | <blockquote>
<p>Prove that <span class="math-container">$\log|e^z-z|\leq |z|+1$</span> where <span class="math-container">$z\in\mathbb{C}$</span> with <span class="math-container">$|z|\geq e$</span>.</p>
</blockquote>
<p><strong>Background:</strong></p>
<p>This is from a proof that <span class="math-container">$e^z... | A.Γ. | 253,273 | <p>Hint: estimate for <span class="math-container">$r=|z|\ge 0$</span>
<span class="math-container">$$
\log|e^z-z|\le\log(e^r+r)=\log(e^r[1+re^{-r}])=r+\log(1+re^{-r}).
$$</span>
Prove that the function <span class="math-container">$f(r)=re^{-r}$</span> attains maximum at <span class="math-container">$r=1$</span> and <... |
3,268,398 | <blockquote>
<p>General solution of the equation
<span class="math-container">$$x\left(\frac{dy}{dx}\right)^2+\left(y-x\right)\frac{dy}{dx}\:-y=0
$$</span>is</p>
</blockquote>
<p>Option are as follows:</p>
<p><span class="math-container">$a)\qquad (x-y+c)(xy-c)=0$</span></p>
<p><span class="math-container">$b)... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: <span class="math-container">$$\frac{dy}{dx}=1$$</span> or <span class="math-container">$$\frac{dy}{dx}=-\frac{y(x)}{x}$$</span></p>
|
3,268,398 | <blockquote>
<p>General solution of the equation
<span class="math-container">$$x\left(\frac{dy}{dx}\right)^2+\left(y-x\right)\frac{dy}{dx}\:-y=0
$$</span>is</p>
</blockquote>
<p>Option are as follows:</p>
<p><span class="math-container">$a)\qquad (x-y+c)(xy-c)=0$</span></p>
<p><span class="math-container">$b)... | nmasanta | 623,924 | <p><span class="math-container">$$x\left(\frac{dy}{dx}\right)^2+\left(y-x\right)\frac{dy}{dx}\:-y=0$$</span>
<span class="math-container">$$\implies xp^2+(y-x)p-y=0\qquad \text{where}\quad p\equiv \frac{dy}{dx}$$</span></p>
<p><span class="math-container">$$\implies xp(p-1)+y(p-1)=0$$</span>
<span class="math-containe... |
3,281,965 | <p>It is known that a way to check whether a number <span class="math-container">$n$</span> is prime, is to check for divisors of <span class="math-container">$n$</span> from <span class="math-container">$2$</span> to <span class="math-container">$\lfloor\sqrt{n}\rfloor$</span>. If we find any divisor, then <span class... | LAGRIDA | 634,307 | <p>The probability that <span class="math-container">$n$</span> is not divisible by <span class="math-container">$p$</span> is : <span class="math-container">$1-\frac{1}{p}$</span>, then</p>
<p>The probability that <span class="math-container">$n$</span> is prime is :</p>
<p><span class="math-container">$$\prod_{\sub... |
500,446 | <p>Let $p$ be a prime and $p \geq 5$, Consider the congruence $x^3 \equiv a$ (mod p) with $\gcd(a,p)=1$. Show that The congruence has either no solution or three incongruent solutions modulo $p$ if $p \equiv 1$ (mod 6) and has unique solution modulo $p$ if $p \equiv 5$ (mod 6).</p>
<p>My attempt: By Lagrange theorem... | N. S. | 9,176 | <p>As $p \neq 2$ you know that $2$ is invertible modulo $p$. Let $\alpha$ be its inverse.</p>
<p>By completing the square we get</p>
<p>$$(x^2+bx+b^2) \equiv 0 \Rightarrow x^2+bx+\alpha^2 b^2 \equiv \beta^2(\alpha^2-1) \equiv \beta^2 \alpha^2 (1-4) \pmod{p}$$</p>
<p>Now all you have to do is study if $-3$ is a qua... |
1,694,991 | <p>I tried doing $u$-substitution and got $-20e$ as my final answer, but I think the correct answer is just $20$. I'm not sure what I did wrong, but probably had to do with plugging in infinity... could someone explain the process of solving this integral?</p>
| User8128 | 307,205 | <p>I think integration by parts is the way to go here, not $u$-sub. Recall, we have $$\int^\infty_0 u(x) v'(x) dx = \left[u(x)v(x) \right]_0^\infty - \int^\infty_0 u'(x) v(x) dx.$$I'm abusing notation here; of course we shouldn't just "plug in" $\infty$, we should take a limit but of course this amounts to the same thi... |
3,299,863 | <p>Let <span class="math-container">$f_{1}(x) = e^{x^5}$</span> and <span class="math-container">$f_{2}(x) = e^{x^3}$</span>. Let <span class="math-container">$g(x) = f_{1}f_{2}$</span>. Find <span class="math-container">$g^{(18)}(0)$</span>.</p>
<p>By series expansion at <span class="math-container">$x = 0$</span>:</p... | Noah Schweber | 28,111 | <p>Just look at the usual <span class="math-container">$\epsilon$</span>-<span class="math-container">$\delta$</span> definition of convergence:</p>
<blockquote>
<p><span class="math-container">$\lim_{x\rightarrow a}f(x)=L$</span> iff for every <span class="math-container">$\epsilon>0$</span> there is a <span cla... |
2,921,390 | <blockquote>
<p>Aashna and Radhika see the integers $1$ to $211$ written on a blackboard. They alternate turns and in every step each of them wipes out any $11$ numbers until only $2$ numbers are left on the blackboard. If the difference of these $2$ numbers (by subtracting the smaller from the larger) is $\geq 111$,... | Batominovski | 72,152 | <p>There are $100$ pairs with difference $111$: $\{1,112\}$, $\{2,113\}$, $\ldots$, $\{100,211\}$. As you noticed, there are $11$ numbers that are unpaired are $101$, $102$, $\ldots$, $111$. Since the second player can eliminate at most $9\cdot 11$ numbers, the second player does not touch at least one pair of differ... |
184,575 | <p>I have a large jagged list, that is each sub-list has a different length. I would like to <code>Flatten</code> this list for <code>Histogram</code> purposes, but it seems to be taking an inordinate amount of time and memory</p>
<pre><code>jaggedList=Table[RandomReal[1,RandomSample[Range[400000,800000],1]],{n,100}]... | Carl Woll | 45,431 | <p>The difference between using <a href="http://reference.wolfram.com/language/ref/Flatten" rel="noreferrer"><code>Flatten</code></a> and using <a href="http://reference.wolfram.com/language/ref/Join" rel="noreferrer"><code>Join</code></a> as in @kglr's answer is that <a href="http://reference.wolfram.com/language/ref/... |
1,318,880 | <p>I'm trying to prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
I showed that both $n,m$ divides $nm/\gcd(n,m)$
but I can't prove that it is the smallest number.
Any help will be appreciated.</p>
| Atvin | 215,617 | <p>Hint: For any $a,b$ real numbers: $\min(a,b)+\max(a,b)=a+b$.</p>
<p>Now, if we have $a=a_1^{p_1} a_2^{p_2}\ldots$ and similarly with $b$, if you use the equation I just mentioned for all $p_i$, you will get, that $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=ab$.</p>
|
2,337,357 | <p><a href="https://i.stack.imgur.com/iWuCv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iWuCv.png" alt="enter image description here"></a></p>
<p>The derivative of $\frac1x$ is $\frac{-1}{x^2}$.</p>
<p>How do I find the $c$ if there is no zero in the derivative of the function?</p>
<p>I starte... | DreamConspiracy | 454,309 | <p>We will consider the probability that two of the same number do not appear. After the first role, this is of course $1$. Then, after the second roll, there is a $\frac {19}{20}$ chance that the die role is not repeated. On the third role, there is a $\frac {18}{20}$ chance that the role is repeated. Multiplying toge... |
15,093 | <p>For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the <a href="https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem" rel="noreferrer">Abel–Ruffini theorem</a> (there is no algebraic solution to general polynomial equations of degree five or higher wit... | paul garrett | 63 | <p>This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to "know" a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a... |
660,034 | <p>I wondered if all decimal expansions of $\frac{1}{n}$ could be thought of in such a way, but clearly for $n=6$,</p>
<p>$$.12+.0024+.000048+.00000096+.0000000192+...\neq.1\bar{6}$$</p>
<p>Why does it work for 7 but not 6? Is there only one such number per base, <em>i.e.</em> 7 in base 10? If so what is the general ... | Community | -1 | <p>Do you know about geometric series? Your series is really
$$7(0.02 + 0.0004 + \cdots) = 7 \frac{2/100}{1 - 2/100} = 7 \cdot \frac{1}{49}$$
When you replace $7$ by something else, say $n$, your series is similarly $n \frac{2/100}{1 - 2/100} = \frac{n}{49}$. </p>
<p>In particular, only when $n = 7$ would that be equa... |
1,656,963 | <p>If $(a_n) \to 0$, then applying the algebraic limit theorem, what is $ \lim_\limits{n\to\infty} \frac {1+2a_n}{1+3a_n - 4a^2_n}$.</p>
<p>Would just be able to do : $\lim_\limits{n\to\infty} \frac {1+2(0)}{1+3(0) - 4(0)}$. </p>
<p>$\lim \frac {1}{1} = 1$.</p>
| Roman83 | 309,360 | <p>If $(a_n)→a$ and $1+3a-4a^2 \not=0 (a\not=1; a \not = -\frac 14)$, then $ \lim_\limits{n\to\infty} \frac {1+2a_n}{1+3a_n - 4a^2_n}=\frac {1+2a}{1+3a - 4a^2}$</p>
|
199,842 | <p>I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:</p>
<p>The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the square all the way until the square length thus we have heig... | Will Orrick | 3,736 | <p>The main issue is that you don't form an area by placing <em>lines</em> next to each other--you need to place <em>strips</em> next to each other. As you say, to form an $a\times a$ square, you can place $n$ strips of dimension $a\times w$ next to each other, where $w=a/n$, giving total area $naw=a^2$.</p>
<p><img ... |
3,436,804 | <p>Does the following series converge? If yes, what is its value in simplest form?</p>
<p><span class="math-container">$$\left( \frac{1}{1} \right)^2+\left( \frac{1}{2}+\frac{1}{3} \right)^2+\left( \frac{1}{4}+\frac{1}{5}+\frac{1}{6} \right)^2+\left( \frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} \right)^2+\dots$$</... | Andronicus | 528,171 | <p>Notice, that:</p>
<p><span class="math-container">$$\left(\frac{1}{1}\right)^2+\left(\frac{1}{2} + \frac{1}{3}\right)^2 + \dots<\left(\frac{1}{1}\right)^2+\left(\frac{2}{2}\right)^2+\left(\frac{3}{4}\right)^2+\left(\frac{4}{7}\right)^2=2+\sum_{n=2}^{\infty}\left(\frac{n}{\frac{n(n-1)}{2}+1}\right)^2<2+\sum_{n=... |
3,436,804 | <p>Does the following series converge? If yes, what is its value in simplest form?</p>
<p><span class="math-container">$$\left( \frac{1}{1} \right)^2+\left( \frac{1}{2}+\frac{1}{3} \right)^2+\left( \frac{1}{4}+\frac{1}{5}+\frac{1}{6} \right)^2+\left( \frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10} \right)^2+\dots$$</... | robjohn | 13,854 | <p><strong>Approximation with Euler-Maclaurin Sum Formula</strong></p>
<p>The Euler-Maclaurin Sum Formula gives the fairly well-known asymptotic series for the Harmonic Numbers
<span class="math-container">$$
\begin{align}
H_n
&=\gamma+\log(n)+\frac1{2n}-\frac1{12n^2}+\frac1{120n^4}\\[3pt]
&-\frac1{252n^6}+\fra... |
1,105,787 | <p>I was given an excercise in my calculus class that i don't really understand, the problem says : Find the area limited by the curves $$ y = \frac{x+4}{x^2+1} ,\space x = -2 ,\space x = 3,\space y = 0 $$</p>
<p>I don't really know what approach to follow here, my guess would be to solve it using riemann sums or mayb... | mickep | 97,236 | <p>It is meant that you should calculate the integral
$$
\int_{-2}^3 \frac{x+4}{x^2+1}\,dx
$$
Also, see this picture where the curves are drawn.</p>
<p><img src="https://i.stack.imgur.com/Lw4JX.png" alt="picture"></p>
|
813,301 | <p>$f$ is strictly increasing and $g$ is decreasing. How to find whether $f\circ g$ and $g\circ f$ are increasing, decreasing, strictly increasing or strictly decreasing?</p>
<p>This is what I did,</p>
<p>$f \circ g=f(g(x))$</p>
<p>If we take $x_1 < x_2$,</p>
<p>$f(x_1) < f(x_2) $ and $\ g(x_1) \ge g(x_2) $<... | Christian Blatter | 1,303 | <p>It is not necessary to produce a contradiction.</p>
<p>When $x<y$ then $g(x)\geq g(y)$ and therefore $f\bigl(g(x)\bigr)\geq f\bigl(g(y)\bigr)$. This already proves that $f\circ g$ is decreasing. You cannot hope for more: It could be that $g(x)=g(y)$, so that we obtain an instance of $x<y$ and $f\circ g(x)=f\c... |
483,173 | <p>A non-zero matrix $A$ is said to be nilpotent for some positive integer $k\geq2$. If $A$ is nilpotent then is $I+A$ invertible?? Where $I$ is the identity matrix.</p>
| Ben Grossmann | 81,360 | <p>Short answer: yes.</p>
<p>Long answer:</p>
<blockquote class="spoiler">
<p>If $A^k=0$, consider the product $$A^{2n+1}+I=(A+I)(A^{2n}-A^{2n-1}+\cdots+I)$$ For a sufficiently large integer $n$.</p>
</blockquote>
|
492,031 | <p>I am currently reading through Hatcher's Algebraic Topology book. I am having some trouble understanding the difference between a deformation retraction and just a retraction. Hatcher defines them as follows:</p>
<p>A <strong>deformation retraction</strong> of a space $X$ onto a subspace $A$ is a family of maps $f_... | Giorgio Mossa | 11,888 | <p>As you noted the two notion are different, deformation retraction being a continuous family of continuous function, i.e. an homotopy, while a retraction being just a continuous function.</p>
<p>A retraction is just a map that sends all the point of $X$ in $A$ fixing the points of $A$.</p>
<p>A deformation retracti... |
3,220,273 | <p>It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a mathematical/geometric construct) contain greater than two pairs of parallel sides? In a hexagon for example, all six sides a... | Ethan Bolker | 72,858 | <p>Interesting question. Parallelograms are quadrilaterals for historical reasons. They could have been defined to include your examples, but weren't. Now the meaning is so common that it can't be changed. </p>
<p>I don't think there is a name for your class of polygons. The reason is in this:</p>
<blockquote>
<p>I... |
4,563,135 | <p>Denote the linear space of linear operators on the linear space <span class="math-container">$V$</span> with field <span class="math-container">$\mathbb{F}$</span> by <span class="math-container">$L(V)$</span> and the linear space of <span class="math-container">$n \times n$</span> matrices with entries in <span cla... | Hosein Rahnama | 267,844 | <p>For any matrix <span class="math-container">$A$</span> with entries in <span class="math-container">$\mathbb{R}$</span> we have</p>
<p><span class="math-container">$$(\bar \lambda I - A)^{(m)} = \Big((\overline{\lambda \bar I - \bar A)}\Big)^{(m)} =
\Big((\overline{\lambda I - A)}\Big)^{(m)} = \overline{(\lambda I -... |
4,024,871 | <p>I'm looking to build a function <span class="math-container">$f:S^2 \to \mathbb R^2$</span> such that <span class="math-container">$f(x)\neq f(−x)$</span> for all <span class="math-container">$x\in S^2$</span>.</p>
<p>By Borsuk-Ulam Theorem, this function must be discontinuous. I was trying to build a not too compli... | DanielWainfleet | 254,665 | <p><span class="math-container">$S^2=\{(\cos u,\sin u \cos v,\sin u \sin v):u\in [0,2\pi)\land |v|\le \pi /2\}.$</span></p>
<p>Consider the equator <span class="math-container">$E=\{(\cos u, \sin u,0):u\in [0,2\pi)\}.$</span></p>
<p>For <span class="math-container">$u\in [0,2\pi)$</span> let <span class="math-container... |
2,288,358 | <p>Is there a straightforward way to prove the following inequality:
$$|1 + k\big(\exp(it)-1\big)|\leq 1 $$
where $k\in(0,1)$ and $t \in \mathbb{R}$ (correction, see dxiv answer) with $|t| \leq 1$, other than writing the quantity into its real and imaginary parts and checking that they satisfy the required inequalities... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>Put $$y=\frac {xt}{\sqrt {3} }$$</p>
<p>the integral becomes</p>
<p>$$\frac {1}{\sqrt {3}}\int_0^1 x^3(\int_0^\sqrt{3}\sqrt {1+t^2}dt)dx $$</p>
<p>with $t=\sinh(u) $, you can finish.</p>
|
1,745,180 | <p>I want to prove that the polynomial </p>
<p>$$
f_p(x) = x^{2p+2} - cx^{2p} - dx^p - 1
$$</p>
<p>,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove that the polynomial $f_p(x)$ has distinct roots for any $c$,$d$ and $p$.</p>
<p>PS: There is a simila... | Community | -1 | <p>Here is a <strong><em>partial</em></strong> solution that shows there can be no multiple REAL roots. The proof doesn't work for complex numbers though (and I am not sure the result is even true in complex numbers).</p>
<p>$f' = (2p+2) x^{2p+1} - 2pcx^{2p-1} - pdx^{p-1}$. If $x$ is a multiple root of $f$, then both ... |
372,401 | <p>Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth.</p>
<p>Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in $H_0^1$, the norm is given via $\|v\|^2=|v|^2+|\nabla v|^2$. </p>
<p>In general, one cannot bound the $H_0^1$-no... | xyzzyz | 23,439 | <p>Let <span class="math-container">$C$</span> be that sine curve part, and <span class="math-container">$S$</span> be the vertical segment. Suppose there exists a path <span class="math-container">$\omega: [0, 1] \to A$</span> that connects a point <span class="math-container">$(0, 1) \in S$</span> with <span class="m... |
3,443,672 | <p>The equation is
<span class="math-container">$$\tan\frac{5\pi}{6} \cos x=1-\sin x$$</span>
<span class="math-container">$$\sin\frac{5\pi}{6} \cos x=\cos\frac{5\pi}{6}-\cos\frac{5\pi}{6} \sin x$$</span>
<span class="math-container">$$\sin\left(\frac{5\pi}{6}+x\right)=\cos \frac{5\pi}{6}$$</span> which looks weird to... | lab bhattacharjee | 33,337 | <p>Another way and probably a better one,</p>
<p><span class="math-container">$$2+\sqrt3=\csc30+\cot30=\dfrac{1+\cos30}{\sin30}=\cot(?)=\tan(?)$$</span></p>
<p>Use <a href="https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution#The_substitution" rel="nofollow noreferrer">https://en.m.wikipedia.org/wiki/Tang... |
3,588,605 | <p>We have an isosceles <span class="math-container">$\triangle ABC, AC=BC, \measuredangle ACB=40^\circ$</span> and a point <span class="math-container">$M$</span> such that <span class="math-container">$\measuredangle MAB=30^\circ$</span>, <span class="math-container">$\measuredangle MBA=50^\circ$</span>.
Find <span c... | bjorn93 | 570,684 | <p>Here's a trigonometric approach. Let <span class="math-container">$\angle BCM=\varphi\Rightarrow \angle ACM=40^{\circ}-\varphi$</span>. Apply the law of sines in <span class="math-container">$\triangle AMC$</span> and <span class="math-container">$\triangle BMC$</span>:
<span class="math-container">$$\frac{AC}{CM}=\... |
3,588,605 | <p>We have an isosceles <span class="math-container">$\triangle ABC, AC=BC, \measuredangle ACB=40^\circ$</span> and a point <span class="math-container">$M$</span> such that <span class="math-container">$\measuredangle MAB=30^\circ$</span>, <span class="math-container">$\measuredangle MBA=50^\circ$</span>.
Find <span c... | Quanto | 686,284 | <p><a href="https://i.stack.imgur.com/6THq1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6THq1.png" alt="enter image description here"></a></p>
<p>Construct the equilateral triangle <span class="math-container">$AHB$</span>. Given that <span class="math-container">$AC = BC, AH = BH$</span> and th... |
471,710 | <p>Why do small angle approximations only hold in radians? All the books I have say this is so but don't explain why.</p>
| Mostafa | 62,686 | <p>The reason why we use radians for $\sin x$ (or other trigonometric functions) <strong>in calculus</strong> is explained <a href="http://en.wikipedia.org/w/index.php?title=Trigonometric_functions&oldid=563025229#The_significance_of_radians">here</a> and <a href="http://en.wikipedia.org/w/index.php?title=Radian&am... |
4,524,393 | <p>I don't know if this question is best suited to this stack exchange. If it isn't, feel free to migrate it or close it. This question was inspired by a mistake I saw in a math class. I corrected the professor, and he acknowledged it. I then said, "Some students think professors never make mistakes". And he ... | John Wayland Bales | 246,513 | <p>Aristotle famously taught that the only regular solids that tiled space were the cube and the tetrahedron when, in fact, only the cube does so.</p>
<p>It took 1800 years for the mistake to be corrected.</p>
<p><a href="http://www.ams.org/notices/201211/rtx121101540p.pdf" rel="noreferrer">Mysteries in Packing
Regular... |
4,524,393 | <p>I don't know if this question is best suited to this stack exchange. If it isn't, feel free to migrate it or close it. This question was inspired by a mistake I saw in a math class. I corrected the professor, and he acknowledged it. I then said, "Some students think professors never make mistakes". And he ... | usr0192 | 275,654 | <p>Wiles initial proof of Fermat's Last Theorem contained an error, and in some documentary he said he was about to give up on it, but tried to understand one last time what the exact problem was, and then was able to fix it (he also had the help of Richard Taylor his previous graduate student)</p>
|
689,433 | <p>In class, my professor said that given a Markov chain $\{X_k\}$ it intuitively should be true that </p>
<p>$P(X_{k+1} = x_{k+1} \, \mid \, X_0 = a_0, \dots, X_{k-1}= a_{k-1}) = P(X_{k+1} = x_{k+1}\, \mid \,X_{k-1}= a_{k-1})$</p>
<p>and asked us to prove it as an exercise. Note that the indices in the conditional o... | Bjørn Kjos-Hanssen | 61,578 | <p>The idea is that $X_k$, whatever it is, has to be something, so you sum over it:
$$
P(X_{k+1} = x_{k+1} \mid X_0 = a_0, \dots, X_{k-1} = a_{k-1})
$$
$$
= \sum_{x_k} P(X_{k+1} = x_{k+1}, X_k=x_k \mid X_0 = a_0, \dots, X_{k-1} = a_{k-1} )
$$
$$
= \sum_{x_k} P(X_{k+1} = x_{k+1}\mid X_k=x_k, X_0 = a_0, \dots, X_{k-1} = ... |
689,433 | <p>In class, my professor said that given a Markov chain $\{X_k\}$ it intuitively should be true that </p>
<p>$P(X_{k+1} = x_{k+1} \, \mid \, X_0 = a_0, \dots, X_{k-1}= a_{k-1}) = P(X_{k+1} = x_{k+1}\, \mid \,X_{k-1}= a_{k-1})$</p>
<p>and asked us to prove it as an exercise. Note that the indices in the conditional o... | Did | 6,179 | <blockquote>
<p>The "intuitive" statement is actually true for any Markov chain. More generally, for every $0\leqslant i\lt j\leqslant k$,
$$
P(X_{j:k} = x_{j:k} \mid X_{0:i} = x_{0:i})=P(X_{j:k} = x_{j:k} \mid X_{i} = x_{i}).
$$</p>
</blockquote>
<p>Let $(\ast)=P(X_{k+1} = x_{k+1} \mid X_{0:k-1} = x_{0:k-1})$.
Th... |
2,710,656 | <p>I now there is a continuous surjective map from $\Bbb{R}\to\Bbb{R}^2$ thanks to Peano curve.</p>
<p>My question is simple: does there exist a $C^1$ surjective map from $\Bbb{R}\to\Bbb{R}^2$ ? I think that the answer is no and I have seen this long time ago but I was too young to understand the proof. Unfortunately ... | DanielWainfleet | 254,665 | <p>Notation: <span class="math-container">$f''S=\{f(x):x\in S\}$</span> when <span class="math-container">$S$</span> is a subset of the domain of <span class="math-container">$f.$</span></p>
<p>Let <span class="math-container">$f:\Bbb R\to \Bbb R^2$</span> be a continuous surjection. For <span class="math-container">... |
440,528 | <p>My question is about group theory:</p>
<blockquote>
<p>How many subgroups does a non-cyclic group contain whose order is 25?</p>
</blockquote>
<p>How can i answer that question?</p>
<p>Can you generalize the answer?</p>
<p>Thanks for your help.</p>
| amWhy | 9,003 | <p>There are three possibilities for the order of any subgroup $H$ of a group $G$ of order $25$:</p>
<ul>
<li>$|H| = 1 \iff H = e$</li>
<li>$|H| = 5,$ since $5\mid 25$.</li>
<li>$|H| = 25$, if $H = G$.</li>
</ul>
<p>We're given that $G$ is non-cyclic, so the order of any element $x \neq e$ must be $5$, <strong>else</... |
35,375 | <p>Good morning,
today I have read that "number theory is nothing but the study of $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$", here <a href="http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf" rel="nofollow">http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf</a>
can anyone give a very naiv... | Gerry Myerson | 8,269 | <p>You might want to let us know where you read that quote, the context could be helpful in determining the author's intentions. I'm guessing that it wasn't meant to be taken too seriously, and I'd recommend you take it to mean only that there are a few things in Number Theory that you can understand better if you know... |
3,480,857 | <p>For <span class="math-container">$x \in \mathbb{R}^n$</span> we define <span class="math-container">$\Vert x \Vert _\infty := \sup_{k = 1,..,n} |x_k|$</span> (meaning that <span class="math-container">$\Vert x \Vert _\infty $</span> is the biggest component of <span class="math-container">$x$</span> according to amo... | N. S. | 9,176 | <p><span class="math-container">$$\| x \|_\infty = \sup_{k = 1,..,n} |x_k| =\sqrt{ \sup_{k = 1,..,n} |x_k|^2 } \leq \sqrt{ \sum_{k = 1}^n |x_k|^2 }$$</span></p>
<p>For the second
<span class="math-container">$$\sqrt{ \sum_{j = 1}^n |x_j|^2 } \leq \sqrt{ \sum_{j = 1}^n \sup_{k = 1,..,n} |x_k|^2 }$$</span></p>
|
1,955,505 | <blockquote>
<p>$\sum_{n=0}^\infty \frac{n^2+3n+2}{4^n} = \frac{128}{27}$ Given hint: $(n^2+3n+2) = (n+2)(n+1)$</p>
</blockquote>
<p><strong>I've tried</strong> converting the series to a geometric one but failed with that approach and don't know other methods for normal series that help determine the actual conver... | dxiv | 291,201 | <p>Hint: for $|x| \lt 1$</p>
<p>$$f(x) = \sum_{n=0}^{\infty} x^{n+2} = \frac{x^2}{1-x} = -1 -x + \frac{1}{1-x}$$</p>
<p>Now consider $f''(\frac{1}{4})$.</p>
|
3,113,850 | <blockquote>
<p><span class="math-container">$$f(x)=\sum_{n=1}^{\infty}{\frac{x^{n-1}}{n}},$$</span></p>
<p>Prove that <span class="math-container">$f(x)+f(1-x)+\log(x)\log(1-x)=\frac{{\pi}^2}{6}$</span></p>
</blockquote>
<p>In my mind though,I think that this is related to Basel problem<span class="math-container">$\l... | Invisible | 721,644 | <p>I'm not sure if the proof is concise enough, but here is my attempt. Every remark is welcome!
<span class="math-container">$\text{LaPlace formulae}$</span></p>
<p>Let <span class="math-container">$A\in M_n(\mathbb N).$</span></p>
<blockquote>
<p><span class="math-container">$$A_{ij}:=\sum_{p\in S_n\\p(i)=j}^n(-1... |
4,627,133 | <p>We have <span class="math-container">$A$</span> <span class="math-container">$(3×3)$</span> matrix with real entries. We know that A is orthogonal and <span class="math-container">$\operatorname{trace}(A)>1$</span>. Show that matrix <span class="math-container">$A+I_{3}$</span> is invertible.</p>
<p>We can see th... | user1551 | 1,551 | <p>Without using eigenvalues: for any unit vector <span class="math-container">$u$</span>, complete it to an orthonormal basis <span class="math-container">$\{u,v,w\}$</span> of <span class="math-container">$\mathbb R^3$</span>. Then
<span class="math-container">$$
\begin{aligned}
1&<\operatorname{tr}(A)\\
&... |
65,392 | <p>Hello,</p>
<p>I was wondering if anyone is aware of an elementary proof of the claim in the title, assuming the existence of nilpotent injectors in soluble groups. By elementary I mean a proof that does not involve any recourse to Fitting classes etc.</p>
<p>Thanks in advance. </p>
| Geoff Robinson | 14,450 | <p>I take a nilpotent injector of a finite solvable group $G$ to be a nilpotent subgroup $M$ of $G$
such that $M \cap N$ is a maximal nilpotent normal subgroup of $N$ whenever $N$ is subnormal in $G$.
Assuming existence of $M$ , I think uniqueness up to conjugacy follows inductively. Notice that
$M \cap H$ is a nilpote... |
65,392 | <p>Hello,</p>
<p>I was wondering if anyone is aware of an elementary proof of the claim in the title, assuming the existence of nilpotent injectors in soluble groups. By elementary I mean a proof that does not involve any recourse to Fitting classes etc.</p>
<p>Thanks in advance. </p>
| Tom Morris | 15,601 | <p>Another good reference here is "Injectors and Normal Subgroups of Finite Groups" by Avinoam Mann. Israel Journal of Mathematics, Vol 9.</p>
|
507,867 | <p>I saw an inequality for $n\times n$ matrices. I was wondering if the inequality is true or not?</p>
<p>Does $\det(A)>0$ imply $\det(I+A)>0$?</p>
| mrf | 19,440 | <p>Let $$ A = \begin{bmatrix} -3 & 0 \\ 0 & -1/2 \end{bmatrix}.$$</p>
|
2,459,651 | <p>What additional properties must an operation have besides commutativity so that commutativity along with other properties implies associativity?</p>
<p>Where can I read about such structures?</p>
| Cameron Buie | 28,900 | <p>Suppose we have an operation $\star$ on a set $S$ such that for all $x,y,z\in S,$ we have $$x\star(y\star z)=(x\star z)\star y.$$ If $\star$ is also commutative, then $\star$ is associative.</p>
<p>The above is borrowed from Axiom 4 of <a href="https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals" r... |
87,319 | <p>How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?</p>
| Dirk | 3,148 | <p>There is no topology on $L^1([0,1])$ which describes the notion of "convergence almost everywhere".</p>
<p>Well, I just noticed that this has been answered a few seconds ago. Anyway, I'd like to point out the nice note "Convergence Almost Everywhere is Not Topological" which can be read <a href="http://ordman.net/M... |
87,319 | <p>How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?</p>
| David Mitra | 18,986 | <p>Just to add to the other answers (since it was not explicitly stated): there is a sequence in $L_1[0,1]$ that converges in measure but not pointwise a.e. </p>
<p>For example: </p>
<p>$f_1(x)=1$, </p>
<p>$f_2(x)= \chi_{[0,1/2]}$</p>
<p>$f_3(x)=\chi_{[1/2,1]}$</p>
<p>$f_4(x)=\chi_{[0,1/4]} $</p>
<p>$f_5(x)=\chi... |
1,818,281 | <p>Suppose I have a pair of 2 non-linear differential equations of the form:
$$\begin{matrix}
\frac{dy}{dt}=f(x,y)\\
\frac{dx}{dt}=g(x,y)
\end{matrix}$$
Equilibrium points are where the trajectory ends up on, when plotted on the $x-y$ plane.</p>
<p>What are the qualitative differences between a 'stable node' and a 's... | okrzysik | 246,433 | <p>Yes both stable nodes and stable spirals are stable equilibria (as indicated by their names). And they are qualitatively very similar to one another. </p>
<p>The only real difference between the two is that solution trajectories in the $x-y$ phase plane for a stable spiral tend to spiral around the equilibrium befo... |
1,031,304 | <p>Let $\varphi$ be a bounded, differentiable function on $\mathbb{R}$ such that $\varphi'$ is bounded and uniformly continuous on $\mathbb{R}$.</p>
<p>We want to prove that $\displaystyle\frac{\varphi(x+h)-\varphi(x)}{h}\to\varphi'(x)$ uniformly as $h\to 0$</p>
<p>I can prove $\varphi$ is uniformly continuous, but I... | copper.hat | 27,978 | <p>You have $\phi(x+h)-\phi(x)-\phi'(x)h = \int_0^1 (\phi'(x+th)-\phi'(t))h dt$.</p>
<p>Let $\epsilon>0$ and choose $\delta$ such that if $\|\xi\| < \delta$ then $\|\phi'(x+\xi)-\phi'(x) \| < \epsilon$.</p>
<p>Then
$\|\phi(x+h)-\phi(x)-\phi'(x)h \| \le \epsilon \|h\|$.</p>
<p>(There is no need to assume tha... |
283,473 | <p>I'd like to be able to construct polynomials $p$ whose graphs look like this:</p>
<p><img src="https://i.stack.imgur.com/8Rk3a.jpg" alt="enter image description here"></p>
<p>We can assume that the interval of interest is $[-1, 1]$. The requirements on $p$ are:</p>
<p>(1) Equi-oscillation (or roughly equal, anywa... | fedja | 12,992 | <p>Here the Newton version of the Asymptote code that does the job. Now it runs really fast in the range requested. The polynomial is encoded by its roots (stored in the x array). Thanks to the kind soul who had the patience to type 4 spaces in the beginning of each line :). </p>
<p><strong>Edit.</strong> Since bubba ... |
2,391,769 | <p>I'm concerned with the total number of ones, and the total number of runs, but not with the size of any of the runs.</p>
<p>For example, $N=8$, $R=3$, $C=5$ includes 11101010, 01101011 among the 24 total possible strings.</p>
<p>I can compute these for small $N$ easily enough, but I am specifically interested in t... | G Cab | 317,234 | <p><a href="https://i.stack.imgur.com/52D4H.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/52D4H.png" alt="runs_ones_1"></a></p>
<p>Consider a binary string and let's put an additional (dummy) fixed $0$ at the start of the it.
We individuate as a <em>run</em> a $0$ followed by contiguous $1$'s.</... |
2,019,711 | <p>Correction: For what values of the real number $a$, can
$$
a(x_1^2+x_2^2+x_3^2)+2(x_1x_2+x_1x_3+x_2x_3)
$$
be expressed as sum of the form
$$
\alpha x^2+\beta y^2+\gamma z^2
$$
where $\alpha,\beta,\gamma$ are real numbers? </p>
| user361424 | 361,424 | <p>The proof that it won't work in that case is trivial - you get an impossible fraction of $\frac{f(x)}{0}$. But if what you're looking for is "why" it won't work, beyond that obvious reason, consider it geometrically - what Newton's method is is following the tangent line to the x-intercept. If the derivative is ze... |
199,235 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/9505/xy-yx-for-integers-x-and-y">$x^y = y^x$ for integers $x$ and $y$</a> </p>
</blockquote>
<p>Determine the number of solutions of the equation
$n^m = m^n$
where both m and n are integers.</p>
| Cameron Buie | 28,900 | <p>Well, there are countably infinitely many, yes? There are (after all) only that many integer pairs $(m,n)$, so certainly no more than that will be solutions to the equation. On the other hand, any pair $(m,m)$ will be a solution, and there are countably infinitely many of those.</p>
|
38,597 | <p>Or equivalently, if $G$ is a group, do the projective and injective dimension of $Z$ (viewed as a $ZG$-module) agree?</p>
<p>Thanks! </p>
| Dan Ramras | 4,042 | <p>I like to think of group (co)homology topologically, so I would say the integral (co)homological dimension of G is the integral (co)homological dimension of BG. Thinking this way there are lots of geometric examples in which the integral homological dimension is less than the integral cohomological dimension: for e... |
1,226,162 | <p>\begin{align}
x' &= -x^3 + x^5 + (x^4)(y^5)\\[.7em]
y' &= -8y^3 + y^5 - 10(y^4)(x^5)
\end{align}
$(0,0)$ is obviously a critical point of the system, and we are given that it is asymptotically stable, but have to show it. </p>
<p>I have tried to make a Lyapunov function $V(x,y) = ax^2 + cy^2$, with ... | RTJ | 223,807 | <p>For
$$V(x,y)=\frac{1}{2}(x^2+y^2)$$
we have
$$\dot{V}=-x^4-8y^4+x^6+y^6-9y^5x^5\\
\leq -(1-x^2-|xy|)x^4-8(1-(1/8)y^2-|xy|)y^4$$
Thus if we define the region (neighborhood of the origin)
$$\Omega:=\left\{(x,y)|(x^2+|xy|<1)\textrm{ and }\frac{1}{8}y^2+|xy|<1\right\}$$
we have that $$\dot{V}(x,y)<0\qquad \for... |
77,485 | <p>Find the smallest integer $n$ such that $$\left(1-\frac{n}{365}\right)^n < \frac{1}{2}.$$</p>
<p>I cannot use a calculator, and I do not know where to begin.</p>
| Jens | 505 | <p>You could get an initial guess, by </p>
<p>$$
\begin{aligned}
&&\left(1-\frac{n}{365}\right)^n &< \frac{1}{2}.\\
&\Leftrightarrow & n\ln\left(1-\frac{n}{365}\right) &< -\ln 2\\
\end{aligned}
$$</p>
<p>and using $\ln(1 + x)\approx x$ for small $x$ to arrive at $n\approx \sqr... |
4,587,608 | <p>The following is the definition of a change-of-coordinate matrix from a textbook I'm using:</p>
<p>Let <span class="math-container">$\beta$</span> and <span class="math-container">$\beta'$</span> be two ordered bases for a finite-dimensional vector space V, and let <span class="math-container">$Q=[I_v]_{\beta}^{\bet... | Marian G. | 527,214 | <p>You want to so solve the given integral in a simple way. I suggest the following approach, where no substitution is needed.</p>
<p>First, modify the integral as follows:</p>
<p><span class="math-container">\begin{align*}
\int\frac{-t^2}{\sqrt{1-t^2}}\,\mathrm dt
&=\int\left(\frac{1-t^2}{\sqrt{1-t^2}}-\frac{1}{\... |
276,725 | <p>Is there a positive number $n$ of distinct odd integers $z_1,z_2, \ldots, z_n \geq 3$ such that $\frac{1}{z_1} + \frac{1}{z_2} + \cdots + \frac{1}{z_n} = 1$?</p>
| user6043040 | 321,541 | <p>Another solution: 1=1/3+1/5+1/7+1/9+1/11+1/13+1/23+1/721+1/979007+1/661211444787+1/622321538786143185105739+1/511768271877666618502328764212401495966764795565+1/209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815</p>
<p>You may have to verify it via symbolic softwares.</... |
4,320,437 | <p>I was thinking about linear tramsformations and i came up with this example:
<span class="math-container">$$f:\mathbb{R}^n \to \mathbb{C}^n\\
f(x)=ix$$</span>
for this example, domain and co-domain are not defined over the same field and all linear transformations that i encountered by now had domain and co-domain ... | Elliot Yu | 165,060 | <p>Linear transformations are always defined with a single underlying field <span class="math-container">$K$</span>, so that both the domain and the codomain are <span class="math-container">$K$</span>-vector spaces. This is because for linearity to make sense, we need to have the same notion of scalar multiplication i... |
1,166,127 | <p>I want to select an even rows of matrix. How can I show it using mathematical notations.</p>
<p>Something like this:</p>
<p>$X_{i:}$ where <code>i</code> is even.</p>
| Mathophile | 219,073 | <p>Let $A_{m\times n}$ be the original matrix, and $B$ be the matrix which consists of even rows of $A$. Then $B$ can be defined as follows:</p>
<p>$B_{i,j}=A_{2i,j}$ $\ \ (1\leq i\leq \lfloor m/2\rfloor) $</p>
|
839,516 | <p>What is the number of binary sequences of length $n$, with no two consecutive zeros, and if starts with $0$ has to end with $1$.</p>
<p>Would appreciate suggestions and help.</p>
<p>I tried counting the total sequences and than substractung the ones containing 2 consecutive zeros, and then substracting the ones st... | Pavan Sangha | 154,686 | <p>In the case of starting with a $0$</p>
<p>Let $A(n)$ be the number of such strings of length $n$ and suppose we know $A(k)$ for $k<n$. Now each string of length $n$ can end in a $01$ or $11$. If it ends in $01$ the number immediately before the $0$ must be a $1$ and so it must actually end in $101$ the first $n-... |
1,344,892 | <p>Let $K$ be a field, and let $A$ be a $K$-algebra such that $\alpha \in A$. Then the natural homomorphism $$ \phi: K[x] \to K[\alpha], \hspace{3mm} (x \mapsto \alpha )$$ has a kernel which is a principal ideal $ \langle f \rangle$ and so $$ K[x] / \langle f \rangle \cong K[\alpha]$$</p>
<p>Notice that $K[\alpha]$ is... | Noah Schweber | 28,111 | <p>Towards showing it spans, let's prove the easier proposition that $\alpha^n$ is in the span of $\{1, \alpha, . . ., \alpha^{n-1}\}$. (It's easy to get the rest of the way from this.) Write $f=c_nx^n+ . . . + c_0x^0$, with $c_n\not=0$. Then consider the element $$z=-{c_{n-1}x^{n-1}+c_{n-2}x^{x-2}+ . . . +c_0x^0\over... |
293,371 | <p>This is part of a homework assignment for a real analysis course taught out of "Baby Rudin." Just looking for a push in the right direction, not a full-blown solution. We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero. The first part of this question let me assum... | KarlG | 60,744 | <p>Since $f(t)=f(t+0)=f(t)f(0)$ we can conclude that $f(0)=1$. If there exists a real number $β$ such that $f(β)=0$, then for any real number $x$ we have </p>
<p>$f(x)=f(β+(x-β))=f(β)f(x-β)=0$</p>
<p>which implies that $f$ is identically zero. By the continuity of $f$ we can conclude that $f(x)>0$ for all real n... |
586,724 | <p>Please how to integrate this $$\frac{1}{(1-e^{2x})^{1/2}}$$
I have tried $u= e^x$
But I think that is wrong
So can anyone help me ?</p>
| Claude Leibovici | 82,404 | <p><strong>HINT</strong>. Would you try to define $u$ such that $u^2 = 1 - e^{2x} $ ? It will become quite simple.</p>
|
514,352 | <p>Can anybody give me an example of a multiplicative function $f$ such that
$$\prod_p \sum_{k=0}^\infty f(p^k)$$
converges absolutely and such that
$$\sum_{n=1}^\infty f(n)$$
diverges?</p>
| coffeemath | 30,316 | <p>The product $\Pi_p(1+a_p)$ converges absolutely iff the sum $\sum_p a_p$ does. </p>
<p>A multiplicative function may be defined as $f(1)=1$ and then for each prime $p$ and power $k \ge 1$ the values of $f(p^k)$ may be assigned arbitrarily, and then $f$ itself may be defined by extending from prime powers using the ... |
889,628 | <p>The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, and not 0 to 1? And also the same thing for y2. Thank you
<img src="https://i.stack.imgur.com/ECQQ7.png" alt="Questio... | JRN | 18,398 | <p>I teach infinity by using the number line. Tell the child that each point on the line represents a number, where the numbers are arranged such that the larger numbers are on the right. Then tell them that <em>infinity</em> is the number (the point) that is the largest (the farthest to the right). If they're smart... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Henno Brandsma | 4,280 | <p>In a locally connected space $X$, all connected components of open sets are open. This is in fact equivalent to being locally connected.</p>
<p>Proof: (one direction) let $O$ be an open subset of a locally connected space $X$. Let $C$ be a component of $O$ (as a (sub)space in its own right). Let $x \in C$. Then let... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | peter | 92,783 | <p>I hope that this is right as this is a lemma i've thought of and i plan to use in a project due in several days and it somewhat generalizes the question asked:</p>
<p>Suppose that $U$ is a set of intervals in $\mathbb{R}$ (closed, open, semi-closed, etc.). Then there exists a set of disjoint intervals $V$ in $\math... |
318,299 | <blockquote>
<p>Let <span class="math-container">$U$</span> be an open set in <span class="math-container">$\mathbb R$</span>. Then <span class="math-container">$U$</span> is a countable union of disjoint intervals. </p>
</blockquote>
<p>This question has probably been asked. However, I am not interested in just get... | Christian Bueno | 86,451 | <p>The following is certainly not the quickest approach to a proof, but when this question was first posed to me in class, my first intuition was to use some elementary graph theory:</p>
<hr>
<p>Let $U$ be an open set of $\mathbb{R}$. As we know, $\mathbb{R}$ has a countable basis $\mathcal{B}$ comprised of connected... |
183,768 | <p>Prove convergence\divergence of the series:
$$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$</p>
<p>Here is what I have at the moment:</p>
<p><strong>Method I</strong></p>
<p>My first way uses a result that is related to <strong><a href="http://en.wikipedia.org/wiki/Wallis_product" rel=... | Rijul Saini | 27,729 | <p>As robjohn notes, $$
\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}=\frac{(2n)!}{2^{2n}n!^2} = \frac 1{4^n} \binom{2n}{n}
$$
Noting that $$(2n+1) \binom{2n}{n} > \sum_{i=0}^{2n} \binom{2n}{i} = 4^n$$
As $\binom{2n}{n}$ is the largest binomial coefficient.</p>
<p>Therefore, $$\frac 1{4^n} \binom{2n}{n}... |
2,259,243 | <p>How to solve T(n) = T(n-2) + n using iterative substitution</p>
<pre><code>Base case:
T(0) = 1
T(1) = 1
Solve:
T(n) = T(n-2) + n
</code></pre>
<p>Currently I have:</p>
<pre><code>T(n) = T(n-2) + n
= T(n-4) + n - 2 + n = T(n-4) + 2n - 2
= T(n-6) + n - 4 + n - 2 + n = T(n-6) + 3n - 6
= T(n-8) + n - ... | Anindya Sengupta | 664,444 | <p>In your proof, you are assuming that a permutation cannot be both even and odd (for otherwise, you could express B as a product of an even number of transpositions, and B^(-1) as a product of an odd number of transpositions, and then your proof falls flat). To prove that a permutation cannot be BOTH even and odd, yo... |
1,553,530 | <p>I have a small question about how to finish the proof in the title. The main idea seems to be make an assumption of ∀x∀y (Px→(Py→x=y)) and to derive a contradiction between Raa and ¬Raa from that, which then proves the conclusion:</p>
<p>So</p>
<p>1.∀x∀y (Px→(Py→x=y))</p>
<ol start="2">
<li>∀y (Pa→(Py→a=y))</li>
</o... | Ollie | 293,142 | <p>For those who like a nice Gentzen-style proof: </p>
<p>Have ¬∀x∀y(Px→(Py→(x=y)) as your root. </p>
<p>Line1: ∃Elim. Have ∃y(Rcy∧Py) and ¬∀x∀y(Px→(Py→(x=y)) on the same line. </p>
<p>To prove ∃y(Rcy∧Py) it's just a simple ∀Elim from ∀x∃y(Rxy∧Py)</p>
<p>Line2: ¬Intro. Have Pa and ¬Pa on the same line. </p>
<p>To ... |
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | Petrus | 11,860 | <p>"Lectures on Chevalley Groups" - by Robert Steinberg</p>
|
64,905 | <p>Let's see if we could use MO to put some pressure on certain publishers...</p>
<p>Although it is wonderful that it has been put
<a href="http://www.jmilne.org/math/Books/DMOS.pdf" rel="nofollow">online</a>, I think it would make an even greater read if "Hodge Cycles, Motives and Shimura Varieties" by Deligne, Miln... | John Stillwell | 1,587 | <p>Paul Cohen - <em>Set Theory and the Continuum Hypothesis</em>.</p>
|
2,031,842 | <p>If 3 is the remainder when dividing $P(x)$ with $(x-3)$, and $5$ is the remainder when dividing $P(x)$ with $(x-4)$, what is the remainder when dividing $P(x)$ with $(x-3)(x-4)$?</p>
<p>I'm completely puzzled by this, I'm not sure where to start...</p>
<p>Any hint would be much appreciated. </p>
| Mark Bennet | 2,906 | <p>If you carry out the division you should be able to show that $$P(x)=Q(x)(x-3)(x-4)+R(x)$$</p>
<p>Two questions: What is the maximum degree of the remainder $R(x)$? Can you see how to use the remainder theorem for the cases $x=3,4$ that you know already so that $Q(x)$ becomes irrelevant?</p>
|
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