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 |
|---|---|---|---|---|
395,678 | <p>Please, I need to know the proof that </p>
<p>$$\left(\sum_{k=0}^{\infty }\frac{n^{k+1}}{k+1}\frac{x^k}{k!}\right)\left(\sum_{\ell=0}^{\infty }B_\ell\frac{x^\ell}{\ell!}\right)=\sum_{k=0}^{\infty }\left(\sum_{i=0}^{k}\frac{1}{k+1-i}\binom{k}{i}B_in^{k+1-i}\right)\frac{x^k}{k!}$$</p>
<p>where $B_\ell$, $B_i$ are Be... | TCL | 3,249 | <p>The right-hand side is just the Cauchy product of the two series on the left-hand side. Generally, the Cauchy product of $\sum_{k=0}^\infty a_k$ and $\sum_{l}^\infty b_l$ is $$\sum_{j=0}^\infty\sum_{i=0}^j b_i a_{j-i}.$$
Now you should have no problem finishing the problem. (Change $j$ to $k$ at the end to match yo... |
1,037,316 | <p>I want to write this polynomial in factored form:</p>
<p>$$x^4+4$$</p>
<p>The reason I want to do this is to be able to make <strong>partial-fraction decomposition</strong> on it to make an integrand easier to integrate. <em>What's the general method?</em></p>
<p>In addition to this, I searched on how to figure o... | Sarah Bryant | 195,227 | <p>$x^4 + 4 = (x^2 + 2i)(x^2 - 2i)$</p>
<p>or without complex numbers :</p>
<p>$x^4 + 4 = (x^2 - 2x + 2)(x^2 + 2x + 2)$</p>
<p>Maybe that helps you ?</p>
|
1,037,316 | <p>I want to write this polynomial in factored form:</p>
<p>$$x^4+4$$</p>
<p>The reason I want to do this is to be able to make <strong>partial-fraction decomposition</strong> on it to make an integrand easier to integrate. <em>What's the general method?</em></p>
<p>In addition to this, I searched on how to figure o... | Count Iblis | 155,436 | <p>You equate the polynomial to zero and solve the equation. In this case, you find that the roots are</p>
<p>$$x_n = \sqrt{2}\exp(i \pi n/2)$$</p>
<p>for $n = 0,1,2,3$</p>
<p>The partial fraction expansion will then be of the form:</p>
<p>$$\sum_{n=0}^3 \frac{A_n}{x-x_n}$$</p>
<p>where $A_n$ is the residue of the... |
338,625 | <p>P-adic numbers are complete in one sense and incomplete in another sense. Is it so?</p>
<p>Firstly, does not complete mean connected? I read somewhere that there is not intermediate value theorem for p-adics because they are not connected. (if I am correct).</p>
<p>It seems I need elaboration of this "It can be sh... | Albertas | 126,197 | <p>Alternatively, one can verify that, e.g., the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $$x_n = \sum_{j = 0}^n{p^{j^2}}$$ is Cauchy with respect to the $p$-adic valuation, but does not have a limit in $\mathbb{Q}$.</p>
|
267,442 | <p>Suppose $a,b\in\Bbb N$ are odd coprime with $a,b>1$ then is it true that if all four of $$x_1a+x_2b,\mbox{ }x_2a-x_1b,\mbox{ }x_1\frac{(a+b)}2+x_2\frac{(a-b)}2,\mbox{ }x_2\frac{(a+b)}2-x_1\frac{(a-b)}2$$ are in $\Bbb Z$ for some $x_1,x_2\in\Bbb R$ then $x_1,x_2\in\Bbb Z$ should hold?</p>
| T. Amdeberhan | 66,131 | <p>Here is a more elementary approach.</p>
<p>Label the $4$ expressions in the question as (1), (2), (3), and (4), in the order they appear. Then subtracting and adding equations (3) and (4) from each other we get new integral expressions; namely, (3'): $x_1a-x_2b$ and (4'): $x_2a+x_1b$. Bring back (1): $x_1a+x_2b$ an... |
2,707,941 | <p>$\lim_{n\rightarrow \infty} n^\alpha (\sqrt[5]{n^2+n}-\sqrt[5]{n^2+2n+1})$, $\forall \alpha \in R$</p>
<p>I can change the form of this limit saying that
$n^\alpha (\sqrt[5]{n^2+n}-\sqrt[5]{n^2+2n+1})=n^\alpha (\sqrt[5]{n^2(1+\frac{1}{n})}-\sqrt[5]{n^2(1+\frac{2}{n}+\frac{1}{n^2})})=n^{\alpha+\frac{2}{5}} (\sqrt[5]... | egreg | 62,967 | <p>It's simpler if you consider the limit you get by (formally) substituting $n=1/x$:
$$
\lim_{x\to0^+}\frac{1}{x^{\alpha}}\frac{\sqrt[5]{1+x}-\sqrt[5]{1+2x+x^2}}{\sqrt[5]{x^2}}
$$
If this limit exists (finite or infinite), then it's the same as the limit of your sequence. Now use Taylor:
$$
\frac{1+\dfrac{1}{5}x-1-\df... |
35,014 | <p>I want to display a rational number in <em>Mathematica</em> in periodic style.
<code>PeriodicForm</code> isn't working anymore. It worked in <em>Mathematica</em> 5 and now I'm using <em>Mathematica</em> 9.</p>
<p>I want to display the number $3.13678989898989898989\ldots$, where the repeating $89$ part should be di... | rm -rf | 5 | <p>You can always write your own version of <code>PeriodicForm</code>:</p>
<pre><code>ClearAll@PeriodicForm
PeriodicForm[n_] := RealDigits[n] /. {{d___Integer, {i__Integer} ...}, l_Integer} :>
PeriodicForm[n, l, {d}, {i}]
Format[PeriodicForm[n_, l_, d_, i_]] ^:= Interpretation[Row[{
FromDigits[d ~Take~ l]... |
2,892,755 | <p>I know that the gradient descent flow of the Dirichlet energy $$\min_u E(u) = \int_{\Omega}|\nabla u|^2 dA$$
Is the diffusion/heat equation:</p>
<p>$$u_t = \Delta u$$</p>
<p>Is there a change in the Dirichlet energy such that it gives an anisotropic diffusion flow?</p>
| yousuf soliman | 55,547 | <p>Sure. Just make the obvious change to the Dirichlet energy. Let $A\in\mathcal{C}^1(\Omega;\mathbb{R}^{n\times n})$ be a matrix-valued function that is symmetric and positive definite for every $x\in\Omega$ (I think the regularity of $A$ can be relaxed to only having measurable coefficients, but I'm not too familiar ... |
3,824,959 | <p>In reading a recent paper, I came across the inequality: <span class="math-container">$e^x - 1 \le e x$</span> for <span class="math-container">$x \in [0, 1]$</span>.</p>
<p>I tried to prove this using (the reverse) Bernoulli's inequality i.e. <span class="math-container">$(1 + y)^r \le 1 + ry$</span>, for <span cla... | Barry Cipra | 86,747 | <p>For <span class="math-container">$0\le x\le1$</span> we have</p>
<p><span class="math-container">$$e^x-1=\int_0^xe^t\,dt\le\int_0^xe\,dt=ex$$</span></p>
<p>since <span class="math-container">$e^t$</span> is an increasing function of <span class="math-container">$t$</span> (and <span class="math-container">$x\le1$</s... |
996,626 | <p>The simple graphs upto <span class="math-container">$11$</span> vertices do not have <span class="math-container">$5,7,9,...$</span> automorphisms, in other words,
the only odd numbers appearing are <span class="math-container">$1$</span> and <span class="math-container">$3$</span>. Is this true for all graphs ?</p>... | Casteels | 92,730 | <p>An old paper of <a href="http://cms.math.ca/cjm/v1/p365" rel="nofollow noreferrer">Frucht</a> (referenced on the wikipedia page on graph automorphisms) contains the following results (Theorems 3.1 and 3.2):</p>
<blockquote>
<p>If <span class="math-container">$n>2$</span>, then there is a cubic graph <span class="... |
284,996 | <p>For every $f\in C[0,1]$ there is a sequence of even polynomials which converges uniformly on $[0,1]$ to f ? </p>
<p>What I have tried:</p>
<p>f is continuous on $D:=[0,1]$, let $(x_k)_{k\in \mathbb{N}} \in D$ converge to $y \in D$, then it must hold that (sequence definition of continuity): $$\lim _{k \rightarr... | Hagen von Eitzen | 39,174 | <p>Let $g(x)=f(\sqrt x)$, find a sequence of polynomials with $p_n\to g$. Then $p_n(x^2)\to g(x^2)=f(x)$.</p>
|
955,967 | <p>Prove that there are no positive integer solutions to $$x^2+1=y^3$$
This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this problem be solved using pre-university mathematics?</p>
<p>I am talking about elementary number theoretical solution... | Kieren MacMillan | 93,271 | <p><strong>Lemma.</strong> If <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are integers such that <span class="math-container">$3a^4+3a^2+1=b^2\!$</span>, then <span class="math-container">$a=0$</span> and <span class="math-container">$b=\pm 1$</span>.</p>
<p><em>Proof.</em> We m... |
955,967 | <p>Prove that there are no positive integer solutions to $$x^2+1=y^3$$
This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this problem be solved using pre-university mathematics?</p>
<p>I am talking about elementary number theoretical solution... | Community | -1 | <p><strong>An alternative proof and a generalisation</strong></p>
<p>As stated in the accepted answer, it is sufficient to prove that <span class="math-container">$3a^4+3a^2+1=b^2$</span> has no positive integer solution. With a change of notation we shall prove the following generalisation by using the same method us... |
1,566,163 | <p>I am writing a paper where I have more than one lemma (Lemma 1, Lemma 2, and Lemma 3) and when I cite them together I was wondering is it more appropriate to say, for example, </p>
<blockquote>
<p>because of Lemmas 1-3</p>
</blockquote>
<p>or </p>
<blockquote>
<p>because of Lemmata 1-3</p>
</blockquote>
<p>I... | Patrick Da Silva | 10,704 | <p>I am guessing we never write Lemmata in the literature because no one knows that the actual plural is Lemmata and not Lemmas. Since English is a very internationalized language and most English speakers are not native speakers, it becomes a reflex to write Lemmas for most people and sometimes this reflex can easily ... |
1,057,295 | <p>Sorry if this has already been asked before. Is there any formula for such questions?</p>
<p><img src="https://i.stack.imgur.com/rhWTj.png" alt="How many triangles in the picture?"></p>
<p>EDIT:</p>
<p>I have numbered the smallest triangles in the picture and marked the pentagon as x.
Then I listed all the possib... | Karthik S | 198,543 | <p>Thanks to help from <a href="https://math.stackexchange.com/users/3567/raskolnikov">Raskolnikov</a> and <a href="https://math.stackexchange.com/users/196858/wood">Wood</a> the answer is <strong>24</strong> triangles.</p>
|
1,620,526 | <p>In Mathematical Analysis by Apostol he mentions that the "Intersection of all open intervals of the form $(-\frac{1}{n}, \frac{1}{n})$ is $\{0\}$"</p>
<p>Obviously this is a super basic question but I thought that an open interval does not include the endpoints, so from the limit as $n\to\infty$ on both sides we ge... | seunglim | 303,409 | <p>$\cap_{n=1}(-\frac{1}{n}, \frac{1}{n}) = {0}$ is equivalent to the statement that for all $n \in \mathbb{N}, 0 \in (-\frac{1}{n}, \frac{1}{n})$, which is obviously true.</p>
<p>As $n \rightarrow\infty$, $(-\frac{1}{n}, \frac{1}{n})$ approaches the zero, but never reaches the zero</p>
|
485,043 | <p>If $x_n \to x$ in a Hilbert space $X$, is it true that $|x_n| \leq C|x|$ for all $n$ for some constant $C$?</p>
<p>It is true for $n$ big enough. But not sure about all $n$.</p>
| Jim | 56,747 | <p>This need not be true for any $n$, just take a sequence of nonzero $x_n$ such that $x_n \to 0$. But that's basically the only way it can fail, if $x \neq 0$ then such a $C$ exists. If you are already convinced that it exists for large enough $n$, i.e., for all $n \geq N$, then observe that there are only finitely ... |
3,951,593 | <p>Suppose we have 2 normal distributions <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> with mean <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span> and variance <span class="math-container">$\sigma_1^2$</span> and <span class="math-container">$... | grand_chat | 215,011 | <p>The density approach will work. In the simplest case, assume that <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are each <em>standard</em> normal, with correlation <span class="math-container">$\rho$</span>, so that <a href="https://en.wikipedia.org/wiki/Multivariate_normal_dist... |
301,176 | <p>What on Earth do Russian <a href="https://en.wikipedia.org/wiki/Matryoshka_doll" rel="noreferrer">Matryoshka dolls</a> have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how:</p>
<p>As illustrated in the pictures, a Matryoshka set is a self-replicating <em>container</em> i... | Pietro Majer | 6,101 | <p>I like Jochen Wengenroth's approach, and I think there is a point that it is worth to clarify. If we want to make a norm out of $A$, we need it to be a balanced set, so we'd like to pass to the bounded absolutely convex set $\overline{\operatorname{co}}\left(A\cup(-A)\right)$ or to $A-A$. Any family of linear opera... |
4,294,950 | <p>I am interested in knowing how often (in terms of percentage) you would expect the total of 3 fair rolled dice to exceed the total of 2 fair rolled dice.</p>
<p>Thanks</p>
| PGibson | 963,933 | <p>I agree with Lorenzo's answer having both simulated the experiment in Python and calculated the probabilities as follows.</p>
<p>If A = total score on 2 dice then the probability distribution for A is simply:</p>
<p><a href="https://i.stack.imgur.com/q9Ka0.png" rel="nofollow noreferrer"><img src="https://i.stack.img... |
1,302,738 | <p>I'm trying to evaluate this sum </p>
<p>$$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$</p>
<p>I have no idea how to deal with it. </p>
<p>With one sum I can, with partial-fraction decomposition, express it as a function of Digamma function and I stuck there.</p>
| ajotatxe | 132,456 | <p>Let $p(x)=x^2+x-1$.</p>
<p>Then
$$\sum_{n=1}^\infty\sum_{k=n}^\infty\frac1{p(n)p(k)}=\frac12\left[\left(\sum_{n=1}^\infty\frac1{p(n)}\right)^2+\sum_{n=1}^\infty\frac1{p(n)^2}\right]$$</p>
|
753,231 | <p>Solve $$z^2+2iz-1+2i$$</p>
<p>I tried:</p>
<p>$(z+i)^2-1-1+2i$</p>
<p>$(z+i)^2 = 2-2i$</p>
<p>Which gives me $a^2-b^2 = 2, 2ab = -2, a^2+b^2 = \sqrt(8)$ And this I cannot solve.</p>
| MJD | 25,554 | <p>We want to solve an equation $$z^2+2iz+(2i-1) = 0$$ which is in the correct form for the quadratic formula, so let's use that. The formula says that $$
\begin{align}
z & = \frac12\left(-2i\pm \sqrt{(2i)^2 - 4(2i-1)}\right)\\
& =\frac12\left(-2i\pm \sqrt{-8i}\right) \\
& = \frac12\left(-2i\pm\sqrt{2i}\sq... |
240,637 | <p>I'm trying to use an example to show that Fatou's lemma can not be strengthened to equality. I was given a hint, which I'm not quite sure how to use. I was told that if I look at the one-dimensional case, and let $f_k(x)=\begin{cases}
k, &\quad\text{if } - \frac{1}{k} \leq x \leq \frac{1}{k}\\
0,... | Julián Aguirre | 4,791 | <p>You cannot use an specific example to show that a general statement is true. For this you need a "proof".</p>
<p>But you can use an specific example to show that a general statement is not true.</p>
<p>For instance, the example $f(x)=|x|$ shows that the statement "All continuous functions are differentiable" is no... |
1,995,165 | <p>The Bauer–Fike theorem states that if $A = VDV^{-1}$ for $D = \operatorname{diag}(\lambda_{1},\ldots,\lambda_{n})$ is a diagonalizable matrix and $\tilde{A}=A+E$ is a perturbed matrix than the eigenvalues of $\tilde{A}$ lie in the union of the discs $D_{i}$ where every $D_{i}$ is centered at $\lambda_{i}$ and has ra... | DanDan | 53,212 | <p>If you expand the radius by a factor of roughly $2n$ than the answer is indeed that a disc cannot be empty.</p>
<p>The Bauer-Fike theorem tells us that the distance measure $d_1 = \max_{i}\min_{j}|\lambda_{i}-\mu_{j}|$ is small.
What I wanted to infer is that the distance measure $d_2 = \min_{\sigma \in S_{n}}\max_... |
4,401,567 | <p>The question is that find all numbers <span class="math-container">$x$</span> such that
<span class="math-container">$x+3^x <4$</span>.</p>
<p>This is a question from problem <span class="math-container">$4$</span> from Spivak Calculus book,</p>
<p>My attempt:</p>
<p>It easy to see that <span class="math-containe... | Peter Szilas | 408,605 | <p>Option:</p>
<p>1)<span class="math-container">$x\le 0$</span> satisfies the inequality.</p>
<p>2)<span class="math-container">$x>0$</span>;</p>
<p><span class="math-container">$f(x):=x+3^x$</span> is stricly increasing
(f is the sum of the two increasing functions <span class="math-container">$x$</span> and <span... |
640,648 | <p>What quadrilaterals in the real projective plane can be obtained by a projective transformation of the real projective plane from a square?</p>
| Ted Shifrin | 71,348 | <p>With a (unique) projective transformation of the projective plane you can take any four points, no three of which are collinear, to any other four points, no three of which are collinear. (You can arrange for the points all to be in your real plane inside the projective plane, but some points of the plane may concei... |
640,648 | <p>What quadrilaterals in the real projective plane can be obtained by a projective transformation of the real projective plane from a square?</p>
| John Sullivan | 66,948 | <p>I think a discussion of convexity needs to be added here. If the quadrilateral in nonconvex, then of course you can map its vertices to those of a square (by any of the methods from the other answers). But the edges won't map to the edges of the square – instead two of them will go through infinity. Note that in ... |
4,522,273 | <p>Let <span class="math-container">$F^{\bullet}$</span> and <span class="math-container">$ I^{\bullet }$</span> be two bounded below complexes of sheaves of <span class="math-container">$O_X$</span>-modules (on a scheme X) and let <span class="math-container">$F^{\bullet}\rightarrow I^{\bullet}$</span> be a quasi-isom... | Daniel | 411,639 | <p>To flesh out Zhen Lin's initial comment, if <span class="math-container">$F$</span> is a sheaf of modules, and <span class="math-container">$I^\bullet$</span> is an injective resolution of <span class="math-container">$F$</span>, then there is a quasi-isomorphism from the complex <span class="math-container">$F[0]=\... |
2,005,981 | <p>Let
$f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that:
$$f(x+y) = f(x) + f(y)$$
$$f(1) = 1$$ Show that $$f(x) = x$$ I have been having trouble approaching this problem. I have shown, through a system of equations, that $f(x+y) = x + y$, but that's about as far as I can get. Appreciate an... | Simply Beautiful Art | 272,831 | <p>See that</p>
<p>$$f(0)=f(0)+f(0)=2f(0)\implies f(0)=0$$</p>
<p>$$\begin{align}f(a+b+c+\dots+z) & =f(a)+f(b+c+\dots+z) \\&=f(a)+f(b)+
f(c+\dots+z)\\&=\dots\\&=f(a)+f(b)+f(c)+\dots+f(z)\end{align}$$</p>
<hr>
<p>For natural numbers $x$, we have</p>
<p>$$f(x)=f(\underbrace{1+1+1+\dots+1}_x)=\underbr... |
2,653,503 | <p>How to find a <strong>bijective</strong> function $f: 3\mathbb{N}+1\to 4\mathbb{N}+1$ such that $$f(xy)=f(x)f(y),\forall x,y\in 3N+1$$</p>
<p>If i let $x,y\in 3\mathbb{N}+1$ then there exists $n,m\in \mathbb{N}$ such that $x=3n+1,y=3m+1$ </p>
<p>but I have no idea how I can find a such $f$, Is there a method pleas... | hamam_Abdallah | 369,188 | <p>$$a=\sum_{i=0}^{2017}10^i $$
$$=\frac {10^{2018}-1}{10-1} $$
$$b=2\sum_{i=0}^{1008}10^i $$
$$=2\frac {10^{1009}-1}{10-1}$$</p>
<p>$$9 (a-b)=10^{2018}-2.10^{1009}+1$$
$$=(10^{1009}-1)^2$$</p>
<p>then</p>
<p>$$\sqrt {a-b}=\frac {10^{1009}-1}{3} $$</p>
<p>$=333...333.$ (1009 consecutive times).</p>
|
2,653,503 | <p>How to find a <strong>bijective</strong> function $f: 3\mathbb{N}+1\to 4\mathbb{N}+1$ such that $$f(xy)=f(x)f(y),\forall x,y\in 3N+1$$</p>
<p>If i let $x,y\in 3\mathbb{N}+1$ then there exists $n,m\in \mathbb{N}$ such that $x=3n+1,y=3m+1$ </p>
<p>but I have no idea how I can find a such $f$, Is there a method pleas... | Donald Splutterwit | 404,247 | <p>Let
\begin{eqnarray*}
N=\sum_{i=0}^{1008} 10^i =\underbrace{11\cdots 1}_{\text{1009 ones}} \\
M=10^{1009}+1
\end{eqnarray*}
note that $M-2=9N$ and<br>
\begin{eqnarray*}
a=NM \\
b=2N
\end{eqnarray*}
So $a-b=9N^2$ and
\begin{eqnarray*}
\sqrt{a-b} = 3N=\underbrace{\color{red}{33\cdots 3}}_{\text{1009 threes}}.
\end{... |
1,950,701 | <p>I was asked to prove or disprove the above question and "If there are functions g: A⟹B and f: B⟹C, if f◦g and f are one-to-one then g is one-to-one"</p>
<p>I think I have found a way to disprove it using the counterexample:
if f(x) = √x
and g(x) = x^2
then (f ◦ g)(x) = x, f and f◦g are injective, but g is not.</p>
... | drhab | 75,923 | <p>If $A$ is a singleton then $g:A\to B$ and $f\circ g:A\to B$ are automatically one-to-one. </p>
<p>Now let $B$ have more than one element, and let $f$ be constant. Then $f$ is not one-to-one.</p>
|
40,181 | <p>Currently, my math training includes Calc 1-3, linear algebra, and some introduction to set theory/discrete math.
What would you recommend that I study over summer in preparation for the Putnam? Real analysis, topology, abstract algebra (all of the above)? What would be the most pertinent? Thanks!</p>
| user11135 | 11,135 | <p>Putnam does not require any knowledge of analysis/algebra/topology, just do past exams like suggested above by user6312.</p>
|
2,460,155 | <p>I solved it by myself, but I don't know I did it right.
I hope you guys to check this out.</p>
<p>Suppose $(AB-BA)^{T}=BA-AB$</p>
<p>$(AB-BA)^{T}=B^{T}A^{T}-A^{T}B^{T}$</p>
<p>then,</p>
<p>$B^{T}=B$</p>
<p>$A^{T}=A$</p>
<p>therefore,</p>
<p>$B^{T}A^{T}-A^{T}B^{T}=BA-AB$</p>
<p>It's my first time to ask quest... | Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
(AB-BA)^T =(AB)^T-(BA)^T=B^TA^T-A^TB^T=BA-AB=-(AB-BA).
\end{eqnarray*}</p>
|
199,916 | <p>Let $\mathbb F_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces $W_1,W_2,\dots,W_k$ of $V$ of dimension $r$, there always exists a subspace $U$ of $V$ of dimension $n-r$ which satis... | Dave Witte Morris | 68,305 | <p>I think we have $k = q$ (for all $r$). </p>
<p>First, suppose $r = n - 1$. The vector space $V$ can be covered by $q + 1$ codimension-one hyperplanes $W_1,W_2, \ldots,W_{q+1}$ (but no fewer). (If we think of the projective space $[V]$, the most efficient covering is by taking the hyperplane at infinity, plus a set ... |
897,324 | <p>How many boundary values are needed to uniquely determine an automorphism of unit disk?</p>
<p>More precisely, find the $n \in \mathbb{N} $ such that for any two sets $ \{ x_1, x_2, ..., x_n \} $ and $ \{ y_1, y_2, ..., y_n \} $ of $n$ distinct points on $ \mathbb{S}^1 $, there exists a unique automorphism $ f: \ma... | Lee Mosher | 26,501 | <p>Your guess based on dimension is correct. </p>
<p>To come up with a proof, it helps to recognize that your formula for an automorphism of the disc generalizes to the concept of a fractional linear transformation
$$T(z) = \frac{az+b}{cz+d}
$$
where $a,b,c,d \in \mathbb{C}$ and $ad-bc \ne 0$. Furthermore, after a nor... |
3,462,835 | <p>I am supposed to evaluate the following limit:</p>
<p><span class="math-container">$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}$$</span> </p>
<p>My solution:</p>
<p><span class="math-container">$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{... | Tom Ryddle | 504,126 | <p>The answer is wrong, that limit doesn't exist.</p>
<p><span class="math-container">$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}=\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}+2xy-2xy}{x+y}=\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right ... |
132,936 | <p>In propositional logic, a <strong>theory</strong> <span class="math-container">$T$</span> consists of a set of logical symbols and statements which we call axioms. A logical statement <strong><span class="math-container">$A$</span> can be proven by <span class="math-container">$T$</span></strong> if there is a proof... | Levon Haykazyan | 11,753 | <p>I'll try to clarify your confusion. </p>
<p>Basically in logic there are formulas, which are sequences of symbols, and interpretations, that give <em>meaning</em> to formulas. This meaning usually is a truth value, i.e. a value in $\{0, 1\}$. Your case is more general, the meaning is a value in some boolean algebra... |
132,936 | <p>In propositional logic, a <strong>theory</strong> <span class="math-container">$T$</span> consists of a set of logical symbols and statements which we call axioms. A logical statement <strong><span class="math-container">$A$</span> can be proven by <span class="math-container">$T$</span></strong> if there is a proof... | FWE | 170,600 | <p>Intuitively you might want to think of a <strong>model $M$ of a theory $T$</strong> as an <strong>example</strong>. </p>
<p>Say $\mathcal L$ is a PL-1 language, $T$ is a set of $\mathcal L$ sentences then a $T$-model $M$ is defined as an $\mathcal L$-structure making the sentences in $T$ true when interpreted in $M... |
1,331,010 | <p>I am trying to solve $\dfrac{d\mathbf{x}}{dt} = \left[\begin{array}{cc}
-4 &1\cr
-6 &1
\end{array}\right] \mathbf{x}$ and I need to find the general solution of the system in the form $x=c_1x_1+c_2x_2$.</p>
<p>Finding the eigenvalues, I have $\det(P - \lambda I) = 0 \implies \lambda = \dfrac{-5 \pm \sqrt{21... | Ivo Terek | 118,056 | <p>I think you messed up the eigenvalues.. If $A \in \Bbb R^{2 \times 2}$, then $p_A(t) = t^2 - {\rm tr}(A)\,t + \det(A)$, so we have to solve: $$t^2 + 3t +2 = 0,$$which clearly has roots $-1$ and $-2$. Now you find eigenvectors $\bf v$ and $\bf w$, respectively. Your solution will be: $${\bf x} = e^{-t}{\bf v} + e^{-2... |
394,665 | <p>I'm wondering based on the definition of monotonicity:</p>
<blockquote>
<p>A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. </p>
</blockquote>
<p>So given that the sequence $a_n = 3$ is all the same numbers and is neither increasing or decreasing, is it monotonic? </p>
| André Nicolas | 6,312 | <p>Yes, every constant sequence is monotone, in fact simultaneously monotone non-decreasing and monotone non-increasing. </p>
|
588,042 | <p>Lets say that for every $n\in \mathbb{N}$$,a_n\ne0$.</p>
<p>If, $$\lim_{n\to 0}\frac{a_{n+1}}{a_n}=l$$ how do I prove that the radius of the convergence of the series:$$\sum_{n=0}^{\infty}a_nx^n$$
is $\frac{1}{|l|}$ if $l\ne 0$ and $\infty$ is $l=0$.</p>
<p>I have no idea how to prove it...</p>
<p>Thank you! </p... | Lutz Lehmann | 115,115 | <p>According to the paper, which is a much much better reference than the outdated version of the wikipedia article, you got to solve</p>
<p>$$\frac{x_{k}-x_{k-1}}6 a_{k-1}+\frac{x_{k+1}-x_{k-1}}3 a_k + \frac{x_{k+1}-x_k}6 a_{k+1}=\frac{y_{k}-y_{k-1}}{x_{k}-x_{k-1}}-\frac{y_{k+1}-y_k}{x_{k+1}-x_k}$$</p>
<p>for $k=1,.... |
588,042 | <p>Lets say that for every $n\in \mathbb{N}$$,a_n\ne0$.</p>
<p>If, $$\lim_{n\to 0}\frac{a_{n+1}}{a_n}=l$$ how do I prove that the radius of the convergence of the series:$$\sum_{n=0}^{\infty}a_nx^n$$
is $\frac{1}{|l|}$ if $l\ne 0$ and $\infty$ is $l=0$.</p>
<p>I have no idea how to prove it...</p>
<p>Thank you! </p... | Matt Timmermans | 308,022 | <p>There is a simple convolutional method for generating a natural cubic spline interpolation. See:</p>
<p><a href="http://nbviewer.jupyter.org/github/mtimmerm/IPythonNotebooks/blob/master/NaturalCubicSplines.ipynb" rel="nofollow">http://nbviewer.jupyter.org/github/mtimmerm/IPythonNotebooks/blob/master/NaturalCubicSp... |
3,770,506 | <p><span class="math-container">$\newcommand{\rank}{\operatorname{rank}}$</span>Suppose <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <span class="math-container">$n\times n$</span> matrices. Let <span class="math-container">$\rank(A)=s$</span> and <span class="math-container">... | markvs | 454,915 | <p>There is no general low bound. If <span class="math-container">$B=-A$</span>, then <span class="math-container">$A+B=0$</span> with rank <span class="math-container">$0$</span> independently of the rank of <span class="math-container">$A$</span>. For different <span class="math-container">$s,t$</span> you can probab... |
2,470,463 | <p>If $x+iy=\sqrt\frac{1+i}{1-i}$, where $x$ and $y$ are real, prove that $x^2+y^2=1$</p>
<p>I tried multiplying $\sqrt{(\frac{1+i}{1-i})(\frac{1+i}{1+i})}=\sqrt{i}$ but I'm not sure what to do after</p>
<p>thanks in advance :)))</p>
| Michael Rozenberg | 190,319 | <p>We need to prove that $$\left|\sqrt\frac{1+i}{1-i}\right|=1$$ or
$$\sqrt{\left|\frac{1+i}{1-i}\right|}=1$$ or
$$\sqrt1=1.$$
Done!</p>
|
4,628,654 | <blockquote>
<p>Prove the following statement:
<span class="math-container">$$(\forall a,b \in \Bbb R)(a<b\implies(\exists r\in\mathbb Q)(a^7<r-4<b^7))$$</span>
Hint: You may need to use the theorem: Given any two real numbers <span class="math-container">$a < b$</span>, there is some <span class="math-cont... | Mark Saving | 798,694 | <p>Since <span class="math-container">$a < b$</span> and <span class="math-container">$7$</span> is odd, it follows that <span class="math-container">$a^7 < b^7$</span>. Therefore, by the hint, take some <span class="math-container">$q \in \mathbb{Q}$</span> such that <span class="math-container">$a^7 < q <... |
1,790,370 | <p>If $\{x_n\}$ is Cauchy and has a convergent subsequence with limit $x$, show that $\{x_n\}$ is convergent with limit $x$.</p>
| Ege Erdil | 326,053 | <p>A result is that if $ f \in K[x] $ is irreducible and $ f(\alpha) = 0 $, then we have an isomorphism $ K[x]/(f) \cong K(\alpha) $. Therefore, it suffices to find an irreducible polynomial of which $ \alpha = \sqrt{1 + \sqrt{5}} $ is a root. We can see that $ (\alpha^2 - 1)^2 - 5 = 0 $, so $ \alpha $ is a root of $ X... |
1,790,370 | <p>If $\{x_n\}$ is Cauchy and has a convergent subsequence with limit $x$, show that $\{x_n\}$ is convergent with limit $x$.</p>
| Pipicito | 93,689 | <p>Observe that $\sqrt{5} \in \mathbb{Q}(\sqrt{1+\sqrt{5}})$ so $\mathbb{Q}(\sqrt{5})$ is an intermediate field of the extension $\mathbb{Q}(\sqrt{1+\sqrt{5}}) / \mathbb{Q}$. It's easy to see that $\sqrt{1+\sqrt{5}}$ is not in $\mathbb{Q}(\sqrt{5})$ and therefore $[\mathbb{Q}(\sqrt{1+\sqrt{5}}) : \mathbb{Q}(\sqrt{5})] ... |
21,318 | <p>Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's endomorphism object $\mathcal End(X):=[X,X]$ - it is actually a monoid in $\mathcal V$. Can the assignment
$$X\mapsto \... | Tilman | 4,183 | <p>It's hard to answer this question on this abstract level in any other way that by saying "no, it's not a functor". Of course it is a bifunctor $\mathcal{C}^{op} \times \mathcal{C} \to \mathcal{V}$, and the language of ends and coends deals with such functors (Mac Lane's <em>Categories for the working mathematician</... |
1,673,399 | <p>Let the ring of $p$-adic integers be the projective limit
$$
\mathbb{Z}_p = \varprojlim_{n\in\mathbb{Z}_{\geq 1}}(\mathbb{Z}/p^n\mathbb{Z}),
$$
and denote an element $x\in\mathbb{Z}_p$ as a sequence $x=(x_n)_{n\in\mathbb{Z}_{\geq 1}}$ with $x_n \in \mathbb{Z}/p^n\mathbb{Z}$.</p>
<p>I want to prove the following:</... | Marc | 132,141 | <p>Given that you have a finite sample size from a distribution/DGP, it usually is impossible to find the exact parameters that describe the DGP. In fact, there is a whole field of mathematics which studies ways to determine approximations to the true parameters, namely statistics. </p>
<p>A popular estimator for para... |
1,951,733 | <blockquote>
<p>Can any polynomial $P\in \mathbb C[X]$ be written as $P=Q+R$ where $Q,R\in \mathbb C[X]$ have all their roots on the unit circle (that is to say with magnitude exactly $1$) ? </p>
</blockquote>
<p>I don't think it's even trivial with degree-1 polynomials... In this supposedly simple case, with $P(X)=... | user374753 | 374,753 | <p>There are general rules for all bases.</p>
<p>One such rule that I've stumbled across is described on <a href="http://everything2.com/title/m_turner%2527s+postulate" rel="noreferrer">everything2</a>.</p>
<blockquote>
<p>For any integer base <strong>B</strong> greater than 2, the multiples of the number B-1, when... |
1,951,733 | <blockquote>
<p>Can any polynomial $P\in \mathbb C[X]$ be written as $P=Q+R$ where $Q,R\in \mathbb C[X]$ have all their roots on the unit circle (that is to say with magnitude exactly $1$) ? </p>
</blockquote>
<p>I don't think it's even trivial with degree-1 polynomials... In this supposedly simple case, with $P(X)=... | Richard Rast | 12,652 | <p>Lots of answers touch on the exact reason these rules don't (or occasionally, do) work in other bases. Let me try it from a slightly different tack though, which I think might be more generally helpful.</p>
<p>Basically: <strong>there is no reason they ought to hold in other bases</strong>. The reason is that you'r... |
1,520,118 | <p>I'm not sure what I have done here. I'm guessing it's something to do with the absolute values but I really have no idea what it is.</p>
<p><a href="https://i.stack.imgur.com/yBrKq.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yBrKq.jpg" alt="enter image description here"></a></p>
<blockquote>... | sav | 70,541 | <p>Check out this graph here:</p>
<p><a href="http://m.wolframalpha.com/input/?i=1%2Fx+%2C+x&x=10&y=14" rel="nofollow">http://m.wolframalpha.com/input/?i=1%2Fx+%2C+x&x=10&y=14</a></p>
<p>The values you are looking for are the ones where the hyperbola is below the straight line. </p>
<p>Values between... |
168,619 | <blockquote>
<p>Can $a^2+b^2+2ac$ be a perfect square if $c\neq \pm b$? </p>
</blockquote>
<p>$a,b,c \in \mathbb{Z}$.<br>
I have tried some manipulations but still came up with nothing. Please help. </p>
<p>Actual context of the question is:<br>
Let say I have an quadratic equation $x^2+2xf(y)+25$ that I have to m... | Integral | 33,688 | <p>You can always do $a^2=u^2+v$ for some $u,v\in\mathbb{Z}$, then $a^2+b^2+2ac=u^2+v+b^2+2ac=u^2+(2ac+v)+b^2$.</p>
<p>If $2ac+v = 2ub$, then its a perfect square.</p>
<p>The example given by Old John works cause we can write $6^2=4^2+20$, and $2\cdot 6\cdot 1 + 20= 2\cdot 4\cdot 4$. In this case, $u=4, v = 20$.</p>
|
533,315 | <p>What is the dimension of vector space of $R$ over $Z_p$ ? I think it is $p$.
Fruitful suggestion on how to look at it would be great. </p>
| DonAntonio | 31,254 | <p>You can't turn a field of characterstic zero into a vector space over a field of positive characteristic (keeping the usual structures in both), because then</p>
<p>$$1\in\Bbb R\implies \underbrace{1_{\Bbb R}+1_{\Bbb R}+\ldots+1_{\Bbb R}}_{p\;\text{times}}\neq 0$$</p>
<p><strong>But</strong> if we had a multiplica... |
270,360 | <p>The <em>Sunday New York Times magazine</em> has a puzzle, <strong>CRAZY EIGHTS</strong> such as this:</p>
<p><a href="https://i.stack.imgur.com/C5fo8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/C5fo8.png" alt="enter image description here" /></a></p>
<p>The goal is to fill in the <span class="... | Domen | 75,628 | <p>Set background color to <code>Row</code> inside <code>LegendLayout</code>:</p>
<pre><code>LegendLayout -> (Column[Row[##, Alignment -> {Center, Center},
Background -> LightGray] & /@ #, Spacings -> -1] &)
</code></pre>
<p><a href="https://i.stack.imgur.com/O5n16.png" rel="noreferrer"><img... |
425,611 | <p>If $$10^{20} +20^{10}$$ is divided with 4 then what would be its remainder?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>As $4$ divides $20 \implies 4$ divides $20^n$ for any integer $n\ge 1$</p>
<p>As $4$ divides $10^2\implies 4$ will divide $10^m$ for any integer $m\ge2$</p>
|
1,543,722 | <p>We are learning about inequalities. I originally assumed it would be the same as equations, except with a different sign. And so far, it has been - except for this.</p>
<p>Take the simple inequality:
$-5m>25$ To solve it, we divide by $-5$ on both sides, as expected.
$m>-5$.</p>
<p>But, I have been told that... | Martin Argerami | 22,857 | <p>It is just the fact that the product of a negative number by a positive is negative.</p>
<p>You have $25-5m>0$. If you multiply by the negative number $-1/5$, the result is negative:
$$
(-1/5)(25-5m)<0.
$$
After expanding,
$$
m-5 <0.
$$</p>
|
1,543,722 | <p>We are learning about inequalities. I originally assumed it would be the same as equations, except with a different sign. And so far, it has been - except for this.</p>
<p>Take the simple inequality:
$-5m>25$ To solve it, we divide by $-5$ on both sides, as expected.
$m>-5$.</p>
<p>But, I have been told that... | Steven Alexis Gregory | 75,410 | <p>One way to look at this problem, is to change the problem to one with a positive coefficient.</p>
<p>\begin{array}{lrcl}
\text{We start with : } & -5m &\gt &25\\
\text{Add $5m$ to both sides : } & 0 &\gt &5m + 25\\
\text{Add $-25$ to both sides : } ... |
298,259 | <p>This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?"
A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-bounce spectrum.<sup>1</sup> </p>
<p>Suppose there is a polygon $P$ hidden inside a disk $D$. All its edges are mirrors... | Adam P. Goucher | 39,521 | <p>For question 3, the answer is yes: take a solid disc and excavate half of the Penrose unilluminable room from it. Then, there are boundary arcs which can never be touched, and you can perturb them without altering the bounce behaviour.</p>
<p>It's possible to make this simple closed curve $C^{\infty}$ if you carefu... |
4,023,924 | <p>I'm doing the following induction proof and wanted to know if this was valid. I think it is, but I'm seeing more complicated solutions than what I did. What I did seems much easier.</p>
<p>Prove that <span class="math-container">$3^n+4^n<5^n$</span> for all <span class="math-container">$n>2$</span>.</p>
<p>Whe... | marty cohen | 13,079 | <p>Here's a method I like
for showing
<span class="math-container">$f(n) > g(n)$</span>
for <span class="math-container">$n \ge n_0$</span>.</p>
<ol>
<li><p>Show that
<span class="math-container">$f(n_0) > g(n_0)$</span>.</p>
</li>
<li><p>Show that,
if <span class="math-container">$n \ge n_0$</span>
and <span cla... |
825 | <p>What resources are available for any grade level K- 12 that are aligned with the Common Core Mathematics Standards and Mathematical Practices that have sets of problems or problem banks that can be used by teachers for instruction or homework?</p>
| Michael Pershan | 84 | <p>The CME Project! It's a full four-year high school curriculum.</p>
<p><a href="http://cmeproject.edc.org/" rel="nofollow">http://cmeproject.edc.org/</a></p>
|
825 | <p>What resources are available for any grade level K- 12 that are aligned with the Common Core Mathematics Standards and Mathematical Practices that have sets of problems or problem banks that can be used by teachers for instruction or homework?</p>
| benblumsmith | 140 | <p>David Wees already mentioned in but I'd like to highlight <a href="http://www.illustrativemathematics.org/" rel="nofollow">Illustrative Mathematics</a> - it's designed precisely for this purpose. (It's the brainchild of Bill McCallum, one of the lead authors of the Common Core math standards.)</p>
|
3,226,815 | <blockquote>
<p>Let <span class="math-container">$y(t)$</span> be a nontrivial solution for the second order differential equation</p>
<p><span class="math-container">$\ddot{x}+a(t)\dot{x}+b(t)x=0$</span></p>
<p>to determine a solution that is linearly independent from <span class="math-container">$y$</span> we set <sp... | the_candyman | 51,370 | <p>First of all, we say that <span class="math-container">$y$</span> and <span class="math-container">$z = yv$</span> are linearly independent iff:</p>
<p><span class="math-container">$$Ay + Bz = 0 ~\forall t \iff A=B=0.$$</span></p>
<p>Notice that:</p>
<p><span class="math-container">$$Ay + Byv = y(A+Bv).$$</span><... |
1,271,580 | <p>If $A \in {M_n}$ has exactly $k$ nonzero eigenvalue and $A$ is normal, why is $\operatorname {rank}(A)=k$?</p>
| copper.hat | 27,978 | <p>If $A$ is normal, it is unitarily diagonalisable.</p>
<p>If a matrix is diagonalisable, then we have
$\Lambda = V^{-1} A V$ for some $V$ with $\Lambda$ diagonal. It is easy to see that the rank of $\Lambda$ is $k$.</p>
|
1,290,444 | <p>It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the two, but I'm not really sure how to make that. I've been trying to think of functions between groups and rings and ways ... | Zev Chonoles | 264 | <p>The collection of <strong>all</strong> groups and the collection of <strong>all</strong> rings are, like the collection of <strong>all</strong> sets, <a href="http://en.wikipedia.org/wiki/Class_%28set_theory%29">proper classes</a> (colloquially, they are "too big to be a set").</p>
<p>For any set $S$ whatsoever, yo... |
600,063 | <p>If the function $f(x)$ is such that
$$f^2(x)=x+f(x+1),$$
find a closed-form expression for $f$.</p>
<p>I found
$$f(x)=\sqrt{x+\sqrt{x+1+\sqrt{x+2+\sqrt{x+3+\cdots}}}}$$
is such an $f$. Does anyone have other solutions? Thank you.</p>
| Lucian | 93,448 | <p>Notice that $f(0)$ is nothing else than the square root of the <a href="http://mathworld.wolfram.com/NestedRadicalConstant.html" rel="nofollow">Nested Radical Constant</a>, which is yet unknown to possess a closed form. Obviously, if <em>f</em> were to possess such a form, then so would this constant, meaning your q... |
1,131,475 | <p>If $f(x)$, is a twice differentiable function, and $f"(x)=0$ at $x=c$, then $f(x)$ has an inflection point at $x=c$. </p>
<p>Does the above statement always apply? It seems so to me, because if the second derivative is set equal to zero, and there is a solution, then there must be an inflection point. Can anyone pr... | Thomas | 128,832 | <p>No this need not be true. Consider a function which has $f^{\prime\prime}(x)=x^2$ as second derivative (e.g. by integrating). The second derivative is then nonnegative, hence the graph does not have an inflection point.</p>
|
1,349,656 | <blockquote>
<p>If
$f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$
find the density of their average $\frac{X+Y+Z}{3}$</p>
</blockquote>
<p>I'm a little lost on how to solve this exercise, $f_{X,Y,Z}(x,y,z)$ It looks like the product of three exponential random variables $X\sim... | Conrado Costa | 226,425 | <p>Consider $F(a) = \Bbb{P}(\frac{X+Y + Z}{3} \leq a) =\Bbb{P}(X+Y + Z \leq 3a)$.</p>
<p>Now calculate</p>
<p>$$F(a) = \int_0^{3a} \int_0^{3a-x} \int_0^{3a - x - y} e^{-z}e^{-y}e^{-x} \, dz\,dy\,dx = \\
= \int_0^{3a} \int_0^{3a-x} (1 - e^{-3a + x + y})e^{-y}e^{-x} \,dy\,dx \\
= \int_0^{3a} \int_0^{3a-x} (e^{-y} - e... |
656,791 | <p>For all $n\geq 1$, prove with mathematical induction </p>
<p>$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$</p>
<p>So far.. I have substituted 1 and saw that the statement is true and I have plugged in n+1 to show that the proof is true for all integers but I don't know how to g... | Lucian | 93,448 | <p><em>Hint:</em> $\dfrac1{n^2}<\dfrac1{n(n-1)}$ , whose sum is <a href="http://en.wikipedia.org/wiki/Telescoping_series" rel="nofollow">telescopic</a>.</p>
|
12,787 | <p>My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian</p>
| Aryabhata | 1,102 | <p>Guessing one root sometimes opens up the whole equation for you.</p>
<p>First notice that $\displaystyle x=1$ gives 0. So $\displaystyle x-1$ is a factor.</p>
<p>Next, rewrite as</p>
<p>$\displaystyle 2x^4 - 2x^3 + 5x^3 - 5x^2 -6x^2 + 6x - 15x + 15$
</p>
<p>This is to try and get $x-1$ as a factor.</p>
<p>This ... |
4,362,278 | <p>I am reading the proof of <a href="https://en.wikipedia.org/wiki/Subbase#Alexander_subbase_theorem" rel="nofollow noreferrer">Alexanders subbase theorem</a> on wikipedia. The proof begins</p>
<blockquote>
<p>Suppose for the sake of contradiction that the space <span class="math-container">$X$</span> is not compact (... | Robert Shore | 640,080 | <p>The set of all open covers of <span class="math-container">$X$</span> is a subset of <span class="math-container">$\mathscr P(X)$</span>, so it exists as a set, and since we’re assuming <span class="math-container">$X$</span> is not compact, at least one of those open covers must lack a finite subcover.</p>
|
4,362,278 | <p>I am reading the proof of <a href="https://en.wikipedia.org/wiki/Subbase#Alexander_subbase_theorem" rel="nofollow noreferrer">Alexanders subbase theorem</a> on wikipedia. The proof begins</p>
<blockquote>
<p>Suppose for the sake of contradiction that the space <span class="math-container">$X$</span> is not compact (... | Henno Brandsma | 4,280 | <p>For a given space <span class="math-container">$(X, \mathcal T)$</span></p>
<p><span class="math-container">$$\Bbb S= \left\{\mathcal U \in \mathscr{P}(\mathcal T)\mathrel{}\middle|\mathrel{} \bigcup \mathcal U = X \land \left( \forall \mathcal U’ \subseteq \mathcal U \text{ finite } : \bigcup \mathcal U’ \neq X\rig... |
629,550 | <p>Investigate the convergence of $\sum a_n$ where $a_n = \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx$.</p>
<p>We have thought about using the dominated convergence theorem to find $\lim a_n$, but that would result in something like $\lim a_n = \lim \displaystyle\int_0^1 \dfrac{x^n}{1-x}\sin(\pi x) \,dx = \... | Nick Peterson | 81,839 | <p>Remember that you can view sequences as functions $\mathbb{Z}_{\geq0}\rightarrow\mathbb{R}$; in this sense, for a sequence $a:\mathbb{Z}_{\geq0}\rightarrow\mathbb{R}$, the series $\sum_{n=0}^{\infty}a(n)$ is precisely $\int_{\mathbb{Z}_{\geq0}}a\,d\nu$, where $\nu$ is the counting measure on $\mathbb{Z}_{\geq0}$.</p... |
180,937 | <p>Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality
$$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?</p>
| Community | -1 | <p>Although the function $f(x)=\sqrt{x}/(1-x)$ is <strong>not</strong> convex on (0,1), its tangent at $x=1/3$ lower bounds the
function and passes through the origin.
That is, for $0\leq x\leq 1$, we have
$${\sqrt{x}\over 1-x}\geq {3\sqrt{3}\over2}\, x.$$</p>
<p>Plugging in $a,b,c$ and adding gives<br>
$${\sqrt{a}\... |
3,877,891 | <p>Suppose we consider two graphs to be the equivalent if they are isomorphic. The idea is that if we relabel the vertices of a graph, it is still the same graph. Using this definition of “being the same graph”, can you conclude that the set of trees over countably infinite vertices is countable?<br>
I know that for an... | Community | -1 | <p>There are as many isomorphism classes as the cardinality of <span class="math-container">$\Bbb R$</span>. To see a family of <span class="math-container">$\beth_1$</span> non-isomorphic trees on <span class="math-container">$\aleph_0$</span> vertices, start by considering <span class="math-container">$$0- 1- 2- 3-... |
3,877,891 | <p>Suppose we consider two graphs to be the equivalent if they are isomorphic. The idea is that if we relabel the vertices of a graph, it is still the same graph. Using this definition of “being the same graph”, can you conclude that the set of trees over countably infinite vertices is countable?<br>
I know that for an... | Especially Lime | 341,019 | <p>Probably the simplest way to do this is to just look at the set of degrees of the tree. This is an isomorphism invariant. It is a subset of the positive integers, and apart from <span class="math-container">$\varnothing$</span> and <span class="math-container">$\{1\}$</span>, any subset is possible.</p>
<p>To see th... |
3,877,891 | <p>Suppose we consider two graphs to be the equivalent if they are isomorphic. The idea is that if we relabel the vertices of a graph, it is still the same graph. Using this definition of “being the same graph”, can you conclude that the set of trees over countably infinite vertices is countable?<br>
I know that for an... | bof | 111,012 | <p>In fact, for any infinite cardinal <span class="math-container">$\kappa$</span>, there are <span class="math-container">$2^\kappa$</span> nonisomorphic <em>asymmetric</em> trees of order <span class="math-container">$\kappa$</span>. (A graph is asymmetric if it has no nontrivial automorphism,) For a proof see my ans... |
3,379,893 | <p><a href="https://i.stack.imgur.com/pfWuG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pfWuG.png" alt="enter image description here"></a></p>
<p>I am completely lost, I have no idea where to even start. I am sure someone here would be able to do this easily which is why I'm posting this here. B... | Blue | 409 | <p>More-generally, suppose that a vertex of the equilateral divides the short leg of the <span class="math-container">$30^\circ$</span>-<span class="math-container">$60^\circ$</span>-<span class="math-container">$90^\circ$</span> into segments of length <span class="math-container">$2p$</span> and <span class="math-con... |
2,580,232 | <p>Suppose that $A$ and $B$ are vector subspaces of $V$, and let $C$ and $D$ be bases for $A$ and $B$, respectively.</p>
<p>Then is it true that </p>
<ol>
<li>$C \cup D$ is a basis for $A+B$?</li>
<li>$\operatorname{dim}(A+B) \le |C| + |D|$ (where $|\cdot|$ denotes cardinality)?</li>
</ol>
<p>I am really bad at dime... | haakoek | 516,272 | <p>In order to be sure whether such a function exist you would need to solve the differential equations. b),c) and d) can be solved by separation of variables. a) is on the form $$y^\prime(x) + P(x)y(x) = Q(x), \ \ P(x) = -1, \ \ Q(x) = e^x$$
and can thus be solved by the method of integrating factor <a href="https://... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | pinaki | 1,508 | <p>A number of people I know and work with, including myself, are math PhDs currently working in industry after having tried to pursue a career in academia. Several of them, including myself, are trying to stay active in research. Some observations:</p>
<ol>
<li><p>It is possible, but very difficult. Primary obstructio... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Tom Wilde | 313,687 | <p>I hope I may add my personal, anecdotal experience - which I think mostly backs up the answers already given. I did a PhD in an area of mathematics not directly related to my favorite subject, character theory, and then saw no opportunity to remain in academia and trained as an accountant, later working in quantitat... |
422,196 | <p>After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.</p>
<p>I will therefore move towards a master's degree in engineering and probably work in industry. However, I'm still passionate abo... | Phil Harmsworth | 106,467 | <p>The "tl;dr" answers to your questions:</p>
<ol>
<li>Not really</li>
<li>Quite possibly</li>
<li>Probably not</li>
</ol>
<p>What you have at the moment is just an aspiration. You really love algebraic geometry and number theory. But you do not know for sure whether or not you're capable of undertaking resea... |
456,106 | <p>Was solving some exercise of Number theory, and used this theorem
$$m=[a,b]\Longleftrightarrow \left(\frac{m}{a},\frac{m}{b}\right)=1$$Remembered that the teacher showed it in class, but I do not remember how, and I think I may be in the evaluation, and I am also curious how to proof? $\;\;\;$ Please do not use modu... | benjamin_ee | 77,961 | <p>$$m=[a,b]\Longleftrightarrow(\frac{m}{a},\frac{m}{b})=1$$
$\Longrightarrow$$$[a,b]=\frac{ab}{(a,b)}\Rightarrow m=\frac{ab}{(a,b)}\Rightarrow m(a,b)=ab\Rightarrow(ma,mb)=ab\Rightarrow$$$$\frac{(ma,mb)}{ab}=1\Rightarrow(\frac{m}{b},\frac{m}{a})=1$$$\Longleftarrow$$$(\frac{m}{a},\frac{m}{b})=1\Rightarrow(mb,ma)=ab\Righ... |
4,519,350 | <p>Let
<span class="math-container">$$
I_k=\int_0^\infty (t+a)^k e^{-t}\exp\left(-\frac{(t-\mu)^2}{2\sigma^2}\right)\,\mathrm dt,
$$</span>
with <span class="math-container">$k\in\Bbb N_0$</span> and <span class="math-container">$a>0$</span>. Since <span class="math-container">$k$</span> is an integer we can expand ... | Tyma Gaidash | 905,886 | <p>A simpler integral is which is transformable into your is the following. The infinite radius of convergence makes the sum and integral switch possible: <span class="math-container">$$\int t^a e^{t^2+bt}dt=\int t^a\sum_{n=0}^\infty\frac{b^nt^n}{n!}\sum_{m=0}^\infty\frac{t^{2m}}{m!}dt= \sum_{n=0}^\infty\frac{b^n}{n!}\... |
363,377 | <p>Is there a world where circle is square? (like when triangle can have sum of degrees more than 180 on sphere)
What is the mathematical or at least common-sense proof?</p>
| user54358 | 54,358 | <p>If you define the norm to be the <a href="http://en.wikipedia.org/wiki/Maximum_norm">maximum norm</a>, circles in that normed space would look like a square.</p>
|
2,692,112 | <p>Let $A$ be $m\times n$ matrix with full column rank where $m > n$. Let $P = A(A^TA)^{-1}A^T$. How do we show that $P$ is SPD (symmetric positive definite)? Proving that it is symmetric is trivial, but how can I show it is positive definite?</p>
| Arian | 172,588 | <p>Symmetry:
$$(A(A^TA)^{-1}A^T)^T=(A^T)^T((A^TA)^{-1})^TA^T=A(((A^TA)^T)^{-1}A^T=A(A^TA)^{-1}A^T$$</p>
<p>Positive semidefinite:</p>
<p>$$\langle A(A^TA)^{-1}A^Tx,x\rangle=\langle(A^TA)^{-1}A^Tx,A^Tx\rangle=\langle(A^TA)^{-1}y,y\rangle$$
where $y:=A^Tx$, also $A^TA$ is invertible since $A$ has full column rank so $... |
1,521,745 | <p>I'm really confused about this math problem. I'm currently taking Calculus, but this problem seems to be like something basic in Algebra that I should understand.... Unfortunately, I don't remember what equivalent expression was used for this. </p>
<p>The problem is:</p>
<p><a href="https://i.stack.imgur.com/cjudA... | Brevan Ellefsen | 269,764 | <p>The transform equations to use for rotating any curve are
$$ x' = (x \cos θ - y \sin θ)$$
$$y' =(x \sin θ + y \cos θ)$$
(These basically rotate the axes, but when we view them with static axes the graph rotates) </p>
<hr>
<p>In this case, we get
$$x' = \frac{x+y}{\sqrt{2}}$$
$$y' = \frac{y-x}{\sqrt{2}}$$
$$$$$... |
1,521,745 | <p>I'm really confused about this math problem. I'm currently taking Calculus, but this problem seems to be like something basic in Algebra that I should understand.... Unfortunately, I don't remember what equivalent expression was used for this. </p>
<p>The problem is:</p>
<p><a href="https://i.stack.imgur.com/cjudA... | benleis | 148,578 | <p>I'm going to assume the OP wanted the vertex and focus of the original tilted parabola since they already had rotated it to a standard form where those values are easier to find.</p>
<p>Rather than rotating, its convenient to do everything in place since we're looking for the focus and by definition: for any point... |
3,530,608 | <p>Let <span class="math-container">$E_1,E_2$</span> be a smooth <strong>orthonormal</strong> frame of vector fields on a simply connected open domain <span class="math-container">$U \subseteq\mathbb{R}^2$</span>. </p>
<p>Let <span class="math-container">$E^i$</span> be the associated dual coframe of one-forms, and su... | amd | 265,466 | <p>Observe that the entries of the <span class="math-container">$2\times2$</span> block aren’t just any old values. They are the real and imaginary parts of the complex eigenvalues of <span class="math-container">$A$</span>. Here are some useful facts about complex eigenvectors and eigenvalues of a real matrix. </p>
... |
3,060,767 | <p>I have an expression I am to simplify:</p>
<p><span class="math-container">$$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$$</span></p>
<p>I arrived at <span class="math-container">$15\sqrt[4]{25}$</span>. My textbook tells me that the answer is in fact <span class="math-container">$15\sqrt{5}$</span>. Here is my thought pr... | Dr. Sonnhard Graubner | 175,066 | <p>Write <span class="math-container">$$\sqrt[4]{\frac{125}{5}}=\sqrt[4]{25}=\sqrt{5}$$</span></p>
|
3,060,767 | <p>I have an expression I am to simplify:</p>
<p><span class="math-container">$$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$$</span></p>
<p>I arrived at <span class="math-container">$15\sqrt[4]{25}$</span>. My textbook tells me that the answer is in fact <span class="math-container">$15\sqrt{5}$</span>. Here is my thought pr... | KM101 | 596,598 | <p><span class="math-container">$$\frac{\sqrt[4] {125}}{\sqrt[4] 5} = \frac{\sqrt[4] {5^3}}{\sqrt[4] 5} = \sqrt[4]{5^2} = 5^{\frac{2}{4}} = 5^{\frac{1}{2}} = \sqrt 5$$</span></p>
<p>Even quicker, <span class="math-container">$\sqrt[4]{25}$</span> means <span class="math-container">$\sqrt{\sqrt{25}}$</span>, which beco... |
649,169 | <p>I know this has to do with Euclidean division, I just can't prove the => direction.
Any tips would be appreciated!</p>
| Magdiragdag | 35,584 | <p>Every common divisor $d$ of $a$ and $n$ is also a common divisor of $n - a$ and $a$. Conversely, every common divisor $d$ of $n - a$ and $a$ is also a common divisor of $n - a + a = n$ and $a$. Therefore the <em>greatest</em> common divisor of $a$ and $n$ is the same as the <em>greatest</em> common divisor of $n - a... |
4,132,786 | <p>When describing the Kleene Star, is the following correct:</p>
<p>The Kleene Star applied to a word w allows for the word to be concatenated with itself 0 to n times, with n [element] N.</p>
<p>where N is the set of the natural numbers. Is this fine as is, or would a better way of writing this be to say "[...] ... | José Carlos Santos | 446,262 | <p>You have<span class="math-container">$$\lim_{x\to0}\frac{\frac{e^{\sqrt x}-1}x}{\frac1{\sqrt x}}=\lim_{x\to0}\frac{e^{\sqrt x}-1}{\sqrt x}=1.$$</span>So, since the integral <span class="math-container">$\int_0^1\frac{\mathrm dx}{\sqrt x}$</span> converges, then so does your integral.</p>
<p>Your approach is wrong, b... |
761,947 | <p>Suppose that we are given the following integral:</p>
<p>$$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx.$$</p>
<p>(<a href="https://i.stack.imgur.com/SLzxw.png" rel="nofollow noreferrer">Original screenshot</a>)</p>
<p>And the answer is one of these :- </p>
<blockquote>
<ol>
<li>$4\sqrt3-4-\frac\... | Community | -1 | <p><strong>Hint:</strong> $$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx=\int_0^\pi \sqrt{\left(1-2\sin\left(\frac{x}{2}\right)\right)^2}\;dx=\\
\int_0^{\pi/3} \left(1-2\sin\left(\frac{x}{2}\right)\right)\;dx+\int_{\pi/3}^\pi \left(1-2\sin\left(\frac{x}{2}\right)\right)\;dx\\
\text{This you can evaluate.}$$
... |
274,926 | <p>I'm learning real analysis.</p>
<blockquote>
<p>A subset $G$ of $X$ is called open if for each $x \in G$ there is a
neighborhood of $x$ that is contained in G</p>
</blockquote>
<p>My question is that is $\emptyset$ a open set in $X$? </p>
<p>The set $\emptyset$ has no elements, so there is no neighborhood o... | Community | -1 | <p>Like you said:</p>
<p>A subset G of X is called open if for each x∈G there is a neighborhood of x that is contained in G</p>
<p>If G is empty you can not choose a x that not satisfies this. So G is open.</p>
|
411,268 | <p>Please, somebody can help me with this problem?</p>
<p><hr>
Let $V$ and $W$ be two closed subspaces of a Hilbert $(H, \langle \cdot,\cdot\rangle)$, and let $P:H\rightarrow V$ and $Q:H\rightarrow W$ the orthogonal projectors respectively. Show that
$$\langle\ (Q-P)(x),\ x\ \rangle\ \geq\ 0,\ \forall\ x\in H\quad \mb... | Jared | 65,034 | <p>Wolfram|Alpha says:$$\sum_{k=3}^n\binom{n}{k}\binom{k-1}{2}=2^{n-3}n^2-5\cdot2^{n-3}n+2^n-1$$</p>
|
2,139,774 | <blockquote>
<p>Let $V$ be an $n$-dimensional vector space. If $f,g\in V^*$ are linearly independent, then find $\dim(\ker f\cap \ker g)$.</p>
</blockquote>
<ol>
<li><p>$\dim(\ker f)=\dim(\ker g)=n-1$ because $f,g\neq 0$, correct?</p></li>
<li><p>Hint for answer, please.</p></li>
</ol>
| kingW3 | 130,953 | <p>BIG HINT:
$$2a^2+2b^2+2c^2\geq 2ab+2bc+2ac\\a^2+b^2+b^2+c^2+a^2+c^2\geq 2ab+2bc+2ac$$</p>
|
2,139,774 | <blockquote>
<p>Let $V$ be an $n$-dimensional vector space. If $f,g\in V^*$ are linearly independent, then find $\dim(\ker f\cap \ker g)$.</p>
</blockquote>
<ol>
<li><p>$\dim(\ker f)=\dim(\ker g)=n-1$ because $f,g\neq 0$, correct?</p></li>
<li><p>Hint for answer, please.</p></li>
</ol>
| Nilabro Saha | 304,908 | <p>$
a^2 + b^2 + c^2 \ge ab + bc + ca\\
\Longleftrightarrow \frac12(2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca) \ge 0\\
\Longleftrightarrow (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ac + a^2) \ge 0 \\
\Longleftrightarrow (a-b)^2 +(b-c)^2 +(c-a)^2 \ge 0
$</p>
<p>which is obviously true. Giving a hint for this problem i... |
4,415,371 | <p>I'm currently reading <a href="https://www.ma.imperial.ac.uk/%7Ebuzzard/maths/research/notes/functoriality_of_satake_iso.pdf" rel="nofollow noreferrer">Buzzard's note</a> and trying to calculate the integral on page 5:
<span class="math-container">$$
S(f)(\mathrm{diag}(\varpi, 1)) = q^{-1/2} \int_N f \left( \begin{p... | Lukas Heger | 348,926 | <p>First off, I think that Buzzard means by <span class="math-container">$N$</span> the upper unitriangular matrices, i.e. with <span class="math-container">$1$</span> on the main diagonal. As a group, we have that <span class="math-container">$N$</span> is isomorphic to the additive group of <span class="math-containe... |
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