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 |
|---|---|---|---|---|
1,242,570 | <p>I want to use the standard definition $x_n \rightarrow x$ if for all $\epsilon>0$ there exists $N$ such that if $n>N$ then $|x_N-x|<\epsilon$. </p>
<p>So my claim is $x_n\rightarrow 0$ If I set $N=\epsilon^2,$ then the following expression $|\sqrt{n^2+1}-n-0|<\epsilon$ will hold true. I solved for $N$... | André Nicolas | 6,312 | <p>Note that for positive $n$ we have
$$\left(n+\frac{1}{2n}\right)^2\gt n^2+1.$$
It follows that
$$n\lt \sqrt{n^2+1}\lt n+\frac{1}{2n},$$
and therefore
$$0\lt \sqrt{n^2+1}-n\lt \frac{1}{2n}.$$
We conclude by Squeezing that our limit is $0$.</p>
|
79,292 | <p>I recently realized that I don't know any non-linear diffeomorphisms of the plane (or $\mathbb{R}^n$ in general) except for linear ones, so I want to ask rather broad questions hoping to be pointed to the appropriate literature.</p>
<p><strike>1) Are there simple ways of constructing autodiffeomorphisms of $\mathbb... | Ryan Budney | 642 | <p>The answer to your question (2) is yes. </p>
<p>The proof goes like this. Let $f$ be a diffeomorphism. We find an isotopy from $f$ to a diffeomorphism $g$ with $g(0)=0$. The isotopy is</p>
<p>$$(x,t) \longmapsto f(x)-tf(0) $$</p>
<p>when $t=0$ this is $f$, when $t=1$ you get $f(x)-f(0)$. </p>
<p>Next, given a... |
2,717,007 | <p>the curves are $x^2 = 4y$ and $x^2=4y-4$
these are just the same parabolas but the other one is shifted up by one unit.</p>
<p>I have been thinking of 3 possibilities that might be the answer.</p>
<ol>
<li><p>The area is equal to infinite sq. units</p></li>
<li><p>The area is equal zero</p></li>
<li><p>The area is... | Mundron Schmidt | 448,151 | <p>While the other answers tell you why the first option is the correct answer, let me explain where your arguments and conclusions are correct or wrong.</p>
<p>While your first argument is a bit fishy, the second is true but the conclusion is wrong. And the third argument is correct to exclude the 3. option. Therefor... |
403,114 | <blockquote>
<p>Let <span class="math-container">$G$</span> be any group, and let <span class="math-container">$Z$</span> be its center.</p>
<p>(a) Show that <span class="math-container">$G/Z\cong \text{Inn}(G)$</span>.</p>
<p>(b) Conclude that <span class="math-container">$\text{Inn}(G)$</span> cannot be a nontrivial ... | DonAntonio | 31,254 | <p>Hints:</p>
<p>$$\forall x,y\in G\;\exists z_1,z_2\in Z(G)\;,\;n_1,n_2\in\Bbb N\;\;s.t.\;\; x=g^{n_1}z_1\;,\;y=g^{n_2}z_2\implies$$</p>
<p>$$xy=g^{n_1}z_1g^{n_2}z_2=g^{n_1}g^{n_2}z_1z_2=g^{n_2}g^{n_1}z_2z_1=\ldots$$</p>
<p>The above thus proves $\,G\,$ is abelian, but then $\,Z(G)=G\,$ , so$\;\ldots\;$</p>
|
3,095,310 | <p>There is a connection between type theory and logic, where types are propositions, and type checking performs the role of checking whether a proof of a proposition is correct (Curry-Howard isomorphism).</p>
<p>But I can imagine a different connection: There seems to be a similarity between type checking and checkin... | Nikolaj-K | 18,993 | <p>Yes, there are two notions of "fit into" here:</p>
<ul>
<li>Terms (what you call instances here) "fit into" types or don't, and that is checked when you do type checking</li>
<li>Structures (e.g. set theoretical constructions) "fit into" propositions or don't, and that's checked by when you verify whether the struc... |
3,095,310 | <p>There is a connection between type theory and logic, where types are propositions, and type checking performs the role of checking whether a proof of a proposition is correct (Curry-Howard isomorphism).</p>
<p>But I can imagine a different connection: There seems to be a similarity between type checking and checkin... | Alex Kruckman | 7,062 | <p>The key observation of the Curry-Howard correspondence is that the inductive structure of terms in type theory mirrors the inductive structure of proofs in logic.</p>
<p>For example, given two terms <span class="math-container">$t_1$</span> and <span class="math-container">$t_2$</span> of types <span class="math-co... |
2,749,624 | <p>Prove: $k^3 - k( b c + c a + a b ) + 2 a b c = 0$ always has a negative root with all positive parameters $a, b, c$</p>
<p>I tried: Write $f(x)=x^3-x(ab+ac+bc)+2abc$ then $f(-\infty)=-\infty,f(0)>0$. Now use the Intermediate Value Theorem. I can' t continue. Help me! Thanks!</p>
| Sonal_sqrt | 477,581 | <p>Since $a,b,c$ are positive the value of $f(0)=2abc>0$. And $f(-\infty)=-\infty$. So by intermediate value property, there exists $c\in(-\infty,0)$ such that $f(c)=0$</p>
<p><strong>Intermediate Value Theorem</strong> If there is a continuous function $f:[a,b]\rightarrow \mathbb{R}$. There is some $L$ such that $... |
2,749,624 | <p>Prove: $k^3 - k( b c + c a + a b ) + 2 a b c = 0$ always has a negative root with all positive parameters $a, b, c$</p>
<p>I tried: Write $f(x)=x^3-x(ab+ac+bc)+2abc$ then $f(-\infty)=-\infty,f(0)>0$. Now use the Intermediate Value Theorem. I can' t continue. Help me! Thanks!</p>
| dxiv | 291,201 | <p>Alt. hint (without IVT): by <a href="https://en.wikipedia.org/wiki/Vieta%27s_formulas" rel="nofollow noreferrer">Vieta's relations</a> the product of the three roots is $\,-2abc \lt 0\,$. Since the polynomial has real coefficients:</p>
<ul>
<li><p>either all three roots are real, in which case at least one m... |
138,079 | <p>I want to find an elegant method to rearrange these two sublists:</p>
<pre><code>SeedRandom[1]
list = {RandomInteger[10, {4, 2}], RandomInteger[{10, 30}, {4, 2}]}
</code></pre>
<blockquote>
<p>{{{1,4},{0,7},{0,0},{8,6}},{{11,20},{11,11},{25,17},{27,16}}}</p>
</blockquote>
<p>Make these two sublists’ elements ha... | Coolwater | 9,754 | <p>Borrowing WReach's idea of using <code>DeleteDuplicates</code>:</p>
<pre><code>Module[{L = Tuples[list]},
L = L[[Ordering[EuclideanDistance @@@ L, All, Less]]];
L = DeleteDuplicates[L, Or @@ MapThread[Equal, {##}] &];
{Reverse[L[[All, 1]]], L[[All, 2]]}]
</code></pre>
<hr>
<p>Edit by yode:</p>
<pre><co... |
324,557 | <p>Map the common part of the disks $|z|<1$ and $|z-1|<1$ on the inside of the unit circle. Choose the mapping sot hat the two symmetries are preserved.</p>
<p>I don't really know how to approach this??</p>
<p>Any suggestions on how to start constructing such a linear transformation??</p>
<p>Thanks in advance!... | Cameron Buie | 28,900 | <p>For a real-world example, suppose $A$ is the set of all humans who have ever lived and that $a\:R\:b$ means "$a$ is an ancestor of $b$." Well, if $S:=R\circ R$, then $a\:S\:b$ means that "$a$ is an ancestor of one of $b$'s parents." In particular, then, if $a$ is a parent of $b$, then $a\:R\:b$, but it is <strong>no... |
187,459 | <p>What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle?</p>
<p>Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one vertex in common, then can we characterize them in some way?</p>
<p>When is it possible to draw such a graph... | The Masked Avenger | 35,626 | <p>Consider two labeled squares, vertices on one labeled from the abcd alphabet, the other labeled
from 1234. We are going to identify one or more pairs of vertices while maintaining the constraint
that induced edges are not identified as well as not identifying vertices labeled from the same alphabet.</p>
<p>It is c... |
295,076 | <p>If a finite-dimensional vector space $V$ is a direct sum of two subspaces $W_1$ and $W_2$, prove that $V^* = W_1^0 \oplus W_2^0$.</p>
<p>Where $V^*$ is the dual space of $V$ and $W^0$ is the annihilator of $W$.</p>
| guest196883 | 43,798 | <p>We have $W_1^0\cap W_2^0 = \{0\}$ because a form annihilating both will certainly annihilate their sum. Therefore this sum can be considered direct. So if $\rho_U = \operatorname{id}$ on $U$ and 0 elsewhere for a subspace $U \subseteq V$, we have
$$\rho_{W_1}+\rho_{W_2}= \operatorname{id}$$
And so
$$f = f \circ(\rh... |
455,259 | <p>This person has been on all seven continents. But this same person has never been to Brazil.</p>
<p>Contrary/Consistent: I would say it's consistent because Brazil is not a continent.</p>
<p>am i right?</p>
| rurouniwallace | 35,878 | <p>I'll take a bit more of a rigorous approch to this:</p>
<p>Brazil is in South America. Therefore, someone having been to Brazil $\to$ he has been to South America. However, this is only a one way implication. That is to say, someone having been to South America $\not\to$ he has been to Brazil.</p>
<p>So even thoug... |
2,972,938 | <p>When is it possible to make a change of variables in the limit?</p>
<p>For example <span class="math-container">$\lim_{x \to \infty}(\ln x/x)$</span>, can I change <span class="math-container">$x=e^{y}$</span>?</p>
<p>Then <span class="math-container">$\lim_{x \to \infty}(\ln x/x)= \lim_{y \to \infty}(y/e^{y})$</s... | Community | -1 | <p>A solution without calculation.</p>
<p>from <span class="math-container">$tr(A)=a-b=1$</span>, we deduce that</p>
<p><span class="math-container">$A^2=I$</span> IFF <span class="math-container">$A$</span> is diagonalizable and <span class="math-container">$\{1,1\}\subset spectrum(A)$</span> </p>
<p>IFF <span clas... |
3,756,970 | <p>I know which step is wrong in the following argument, but would like to have contributors' explanations of <em>why</em> it is wrong.</p>
<p>We assume below that weather forecasts always predict whether or not it is going to rain, so <em>not forecast to rain</em> means the same as <em>forecast not to rain</em>. We sh... | J. W. Tanner | 615,567 | <p>Any set can have a metric, because the discrete metric can be applied to all sets.</p>
<p>See <a href="https://en.wikipedia.org/wiki/Discrete_space" rel="nofollow noreferrer">here</a> and <a href="https://en.wikipedia.org/wiki/Metric_space#Examples_of_metric_spaces" rel="nofollow noreferrer">here</a> for further det... |
45,973 | <p>Let $B,C,D \geq 1$ be positive integers and $(b_n)_{n\geq 0}$ be a sequence with $b_0 = 1, b_n = B b_{n-1} + C B^n + D$ for $n \geq 1$.</p>
<p>Prove that </p>
<p>(a) $\sum_{n\geq 0}^\infty b_n t^n$ ist a rational function</p>
<p>(b) identify a formula for $b_n$</p>
<hr>
<p>Hi!</p>
<p>(a)</p>
<p>As I know I ne... | Ross Millikan | 1,827 | <p>For part b), you can just find the first few terms by hand:
$b_0=1$</p>
<p>$b_1=B+CB+D=B(C+1)+D$</p>
<p>$b_2=B^2+CB^2+dB+CB^2+D=B^2(2C+1)+D(B+1)$</p>
<p>$b_3=B^3(3C+1)+D(B^2+B+1)$</p>
<p>Maybe you can see a pattern and prove it by induction.</p>
|
3,520,722 | <p>I have a few challenges setting up the bounds of integration for the region
<span class="math-container">$$U = \{(x,y) | -1 \leq x-y \leq 1 , \quad 1 \leq xy \leq 2 \}$$</span>
My ultimate goal is to solve <span class="math-container">$$\iint_U x^2y + xy^2 dxdy = \iint_U f(x,y) dxdy$$</span></p>
<p>Here is a plot ... | epi163sqrt | 132,007 | <p>We obtain
<span class="math-container">\begin{align*}
\sum_{k=0}^\infty\frac{k!}{(2k+3)!!}&=\sum_{k=0}^\infty\frac{k!(2k+2)!!}{(2k+3)!}\\
&=\sum_{k=0}^\infty\frac{k!2^{k+1}(k+1)!}{(2k+3)!}\\
&=\sum_{k=0}^\infty\binom{2k}{k}^{-1}\frac{2^k}{(2k+1)(2k+3)}\\
&=\sum_{k=0}^\infty\binom{2k}{k}^{-1}\frac{2^... |
3,520,722 | <p>I have a few challenges setting up the bounds of integration for the region
<span class="math-container">$$U = \{(x,y) | -1 \leq x-y \leq 1 , \quad 1 \leq xy \leq 2 \}$$</span>
My ultimate goal is to solve <span class="math-container">$$\iint_U x^2y + xy^2 dxdy = \iint_U f(x,y) dxdy$$</span></p>
<p>Here is a plot ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,450,176 | <p>I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$.</p>
<p>I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this </p>
<p>problem : ${1}^{-\infty }$ then i can't judge if this limit equal's $1$ , </p>
<p>because it is indeterminate case ,T... | DirkGently | 88,378 | <p>Let $y=1/x$ and $y\to 0^+$. We have
\begin{align*}
\left(\frac{1}{y}\sin y\right)^{1-1/y} & =\left(\frac{1}{y}\left(y+O\left(y^{3}\right)\right)\right)^{1-1/y}\\
& =\left(1+O\left(y^{2}\right)\right)^{1-1/y}\\
& =\exp\left(\ln\left(1+O\left(y^{2}\right)\right)\frac{y-1}{y}\right)\\
& =\exp\left(O\... |
4,652,260 | <p>I am self-studying Functional Analysis from Kreyszig's <em>Introductory Functional Analysis with Applications</em>. In section-1.3, he proves that the sequence space <span class="math-container">$\ell^\infty$</span> with the metric <span class="math-container">$d_\infty(x,y)=\sup_i\{|x_i-y_i|\}$</span> is not separa... | Robert Israel | 8,508 | <p>Other than misspelling "discrete" as "discreet" and writing the ambiguous "it is uncountable" rather than "<span class="math-container">$Y$</span> is uncountable", your proof is essentially correct.
You might note that in a separable metric space, any family of disjoint open s... |
3,270,245 | <p>I was provided this graph and asked if it passed the Extreme Value Theorem. I thought yes. I can see that this function is discontinuous...however, I was informed that this graph actually fails the Extreme Value Theorem due to the hole at x = 2. This caught me off guard, because I thought for certain there was a max... | Mohammad Riazi-Kermani | 514,496 | <p>The maximum and minimum of a set are elements of the set. In your case the maximum is not attained and as you have explained any number close to <span class="math-container">$2$</span> is not a maximum.</p>
<p>That is why we have the notion of supremum of a set which is the least upper bound of the set and it does... |
3,270,245 | <p>I was provided this graph and asked if it passed the Extreme Value Theorem. I thought yes. I can see that this function is discontinuous...however, I was informed that this graph actually fails the Extreme Value Theorem due to the hole at x = 2. This caught me off guard, because I thought for certain there was a max... | mlchristians | 681,917 | <p>All the hypotheses of the EVT are not satisfied; in this case, your function is not continuous over the closed interval <span class="math-container">$[1,3]$</span>.</p>
|
1,704,555 | <blockquote>
<p>If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering?</p>
</blockquote>
<p>Obviously it suffices to prove that $J \subseteq I$. I'm stuck with how to go forward ... | Matematleta | 138,929 | <p>An arbitrary vector in the plane is $(x,y,-x-5y)-(0,0,0)$, so we get the subspaces </p>
<p>$\left \{ x(1,0,-1) +y(0,1,-5)\right \}$.</p>
<p>A vector normal to these subspaces is $(1,2,5)$ so that if $\vec v\in \mathbb R^3$, then </p>
<p>$\vec v=x(1,0,-1) +y(0,1,-5)+t(1,2,5)$ and so </p>
<p>$T(\vec v)=x(1,0,-1) +... |
1,790,612 | <p>Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex semisimple Lie algebra ${\frak g}_{\Bbb C}={\frak g}\otimes\Bbb C$?</p>
<p>In particular:</p>
<blockquote>
<p>If $... | Andreas Cap | 202,204 | <p>The Killing form of $\mathfrak g_{\mathbb C}$ is complex bilinear by construction, so in any complex subspace of $\mathfrak g_{\mathbb C}$ you find non-zero vectors which are isotropic. </p>
|
1,904,354 | <p>Yeah the title says everything I will explain this quick if someone is so smart and nice than he has my ammiration! Here you are :: if we take an irrational number like π or e or whatever and we write this π+π-π… (ecc) at infinity of this series what could possibly come out?? I hope somebody can explain this thanks ... | Jack D'Aurizio | 44,121 | <p>If we set
$$ I_n = \int_{0}^{1}\frac{x^n}{x+5}\,dx \tag{1}$$
we clearly have $I_0=\log\frac{6}{5}$ and
$$ I_n+ 5I_{n-1} = \int_{0}^{1}\frac{x^n+5 x^{n-1}}{x+5}\,dx = \int_{0}^{1}x^{n-1}\,dx = \frac{1}{n}.\tag{2} $$</p>
|
587,275 | <p>I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) = 2^{x}$, but now I'm stuck.</p>
<p>Here's my final step:
<strong>$\displaystyle{{\rm f}'\left(x\right)
= \lim_{h \to 0}{2^{x}\left(2^{h} - 1\right... | xavierm02 | 10,385 | <p>$a>0$</p>
<p>$a^x := e^{x\ln a}$</p>
<p>$f:x\mapsto e^{x\ln a}$</p>
<p>$g:x\mapsto x\ln a$</p>
<p>$f=\exp \circ g$</p>
<p>$f'=g' \times(\exp '\circ g)=(x\mapsto \ln a)\times(\exp\circ g)=(\ln a) \times f$</p>
|
2,050,698 | <p>So I'm studying a few special families of square matrices, the diagonal matrices, upper triangular matrices, lower triangular matrices and symmetric matrices and I just had a few questions. </p>
<p>I know...</p>
<p>a diagonal matrix is if every nondiagonal entry is zero, $a_{ij}$=0 whenever $i$ doesn't equal $j
$.... | user361424 | 361,424 | <p>If you know Java, you can use <a href="https://docs.oracle.com/javase/7/docs/api/java/util/Random.html#nextGaussian()" rel="nofollow noreferrer">Random.nextGaussian()</a> to get a dataset with mean 0 and standard deviation $\sqrt{N}$. Multiplying each number in the dataset by the ratio of this to the desired standa... |
375,094 | <p>A metric space <span class="math-container">$(M,d)$</span> is <em>doubling</em> if there exists <span class="math-container">$n$</span> such that every ball of radius <span class="math-container">$r$</span> can be covered by <span class="math-container">$n$</span> balls of radius <span class="math-container">$r/2$</... | Ian Agol | 1,345 | <p>I think this follows from a standard ball-packing argument.</p>
<p>Suppose that <span class="math-container">$G$</span> with the metric <span class="math-container">$\rho$</span> induced from the Cayley graph has growth <span class="math-container">$V(R)=|B_R(1)| \sim R^d$</span>, i.e. <span class="math-container">$... |
85,351 | <p>It has been proven that:</p>
<p>1) if $s$ is a non trivial zero $\rho$ of $\zeta(s)$ then so is $1−s$.</p>
<p>2) $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$</p>
<p>3) $ 0 < \Re(\rho) <1$</p>
<p>From this it follows that when $s \to \rho$:</p>
<p>$\displaystyle \lim_{s \to \rho}... | Gerhard Paseman | 3,568 | <p>Here is an approach you might take. Consider the class of , oh, lets call them left-sections of f(x,y): fix x and define g_x(y) = f(x,y). Consider the equivalence classes induced by level sets of g_x. Associativity combined with the shape of the level sets of g_x, g_z, and g_w will determine whether it is feasible... |
2,526,695 | <p>I've got following sequence formula:
$ a_{n}=2a_{n-1}-a_{n-2}+2^{n}+4$</p>
<p>where $ a_{0}=a_{1}=0$</p>
<p>I know what to do when I deal with sequence in form like this:</p>
<p>$ a_{n}=2a_{n-1}-a_{n-2}$
- when there's no other terms but previous terms of the sequence.
Can You tell me how to deal with this typ... | John Doe | 399,334 | <p>Write:
$$\begin{align}a_n
&=2a_{n-1}-a_{n-2}+2^n+4\\
&=2(2a_{n-2}-a_{n-3}+2^{n-1}+4)-a_{n-2}+2^n+4\\
&=3a_{n-2}-2a_{n-3}+4\cdot2^{n-1}+4(1+2)\\
&=3(2a_{n-3}-a_{n-4}+2^{n-2}+4)-2a_{n-3}+8\cdot2^{n-2}+4(1+2)\\
&=4a_{n-3}-3a_{n-4}+11\cdot2^{n-2}+4(1+2+3)\end{align}$$</p>
<p>You can see a few patte... |
1,127,551 | <p>I have an equation, where I need to find <em>n</em>, that I need help solving.</p>
<p>I already cheated a little bit by using a CAS (<em>Maple</em>) to solve the equation, so i know what the result should be, but I need to know how to get the to result without using a CAS.</p>
<p>The equation:</p>
<p>$$\frac{2400... | Henrik supports the community | 193,386 | <p>You have already rewritten it to a quadratic, you just need to move the last term to have it in standard form allowing you to use the formula for the roots of a quadratic. </p>
|
2,681,621 | <p>I'm trying to calculate the following limit:</p>
<p>$$\lim_{x\to\pi} \dfrac{1}{x-\pi}\left(\sqrt{\dfrac{4\cos²x}{2+\cos x}}-2\right)$$</p>
<p>I thought of calculating this:</p>
<p>$$\lim_{t\to0} \dfrac{1}{t}\left(\sqrt{\dfrac{4\cos²(t+\pi)}{2+\cos(t+\pi)}}-2\right)$$</p>
<p>Which is the same as:</p>
<p>$$\lim_{... | xyzzyz | 23,439 | <p>Note that $a-b = \frac{a^2 -b^2}{a+b}$. Then:</p>
<p>$$
\frac{1}{t}\left(\sqrt{\frac{4\cos²t}{2-\cos t}}-2\right) =
\frac{1}{t}\left(\frac{\frac{4\cos²t}{2-\cos t}-4}{\sqrt{\frac{4\cos²t}{2-\cos t}}+2}\right) =
\frac{1}{t}\left(\frac{\frac{4\cos²t-8 + 4 \cos t}{2-\cos t}}{\sqrt{\frac{4\cos²t}{2-\cos t}}+2}\right) ... |
1,918,071 | <p>Sometimes you will see theorems of the form "Let $H_1, \dots, H_n$. If $A$, then $B$". Sometimes "suppose" or "if" is used instead of "let". Here's an example:</p>
<ol>
<li><p>Let $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>Suppose $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>If $... | haqnatural | 247,767 | <p>It should be $${ e }^{ 2x }+{ e }^{ x }-2=0\\ \left( { e }^{ x }+2 \right) \left( { e }^{ x }-1 \right) =0\\ { e }^{ x }+2=0,{ e }^{ x }-1=0\\ { e }^{ x }=-2,{ e }^{ x }=1$$
is only solution</p>
<blockquote>
<p>$$x=0$$</p>
</blockquote>
|
2,878,206 | <p>Let $a_n = \frac{9^n}{n + 5^n}$.</p>
<p>At large $n$ value, $a_n$ is expected to behave like $\frac{9^n}{5^n}$, therefore it diverges.</p>
<p>Using the direct comparison test, how can I find $b_n$ (has to be smaller than $a_n$ to prove that $a_n$ diverges)?</p>
| Kavi Rama Murthy | 142,385 | <p>By Binomial Theorem $5^{n}=(1+4)^{n}=1+4n+...+4^{n}>1+4n >n$ so $\frac {9^{n}} {n+5^{n}} > \frac {9^{n}} {2(5^{n})}$. Take $b_n=\frac {9^{n}} {2(5^{n})}$.</p>
|
104,195 | <p>Background: This year I'll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica's AbstractAlgebra package but (although visually appealing ) this is no longer sufficient (i.e. listing subgroups of <span class="math-container">$S_4$</span> takes ages). So, I want to learn more ... | Angel Blasco | 163,996 | <p>A book called <a href="http://www.springer.com/us/book/9783540654667" rel="noreferrer">"Computer Algebra Handbook" by Grabmeier, Kaltofen and Weispfenning (eds.)</a> (2003) includes some advanced topics in group theory and examples of code that you can use with GAP. </p>
<p>In particular, this book has chapters:</p... |
2,057,857 | <p>I am trying to complete my homework based on equivalence relation and I don't seem to understand it properly so I need help !</p>
<p>My question is that do all the elements in my set must satisfy all the three conditions then I can say there is an equivalence relation or I can say there is an equivalence relation o... | chelivery | 395,717 | <p>We can show that <span class="math-container">$\#S \ge {c}$</span> via the injection <span class="math-container">$f: \mathbb{R} \rightarrow S$</span>, <span class="math-container">$f(x) = \{x\}$</span>.</p>
<p>If you show <span class="math-container">$\#S \le c$</span> then you can conclude <span class="math-contai... |
4,037,295 | <p>The Cauchy Schwarz inequality says
<span class="math-container">$$
(ax+by+cz)^2 = (a^2+b^2+c^2)(x^2+y^2+z^2).
$$</span></p>
<p>I found that there is a kind of analogous inequality for <span class="math-container">$(ax+by+cz)^n$</span>
<span class="math-container">$$
(x+y+z)^n \leq 3^n(x^n+y^n +z^n).
$$</span>
if I r... | Falcon | 766,785 | <p>We have, for <span class="math-container">$x, y, z \ge 0$</span>,
<span class="math-container">$$(x + y + z)^n \le 3^n \max\{x, y, z\}^n \le 3^n(x^n + y^n + z^n).$$</span></p>
|
635,195 | <p>I'm trying to calculate the following limit: </p>
<p>$$\mathop {\lim }\limits_{x \to {0^ + }} {\left( {\frac{{\sin x}}{x}} \right)^{\frac{1}{x}}}$$</p>
<p>What I did is writing it as: </p>
<p>$${e^{\frac{1}{x}\ln \left( {\frac{{\sin x}}{x}} \right)}}$$</p>
<p>Therefore, we need to calculate: </p>
<p>$$\matho... | robjohn | 13,854 | <p>According to <a href="http://en.wikipedia.org/wiki/Bernoulli%27s_inequality" rel="nofollow noreferrer">Bernoulli's Inequality</a>, for $0\le x\le1$
$$
\left(\frac{\sin(x)}{x}\right)^{1/x}=\left(1-\frac{x-\sin(x)}{x}\right)^{1/x}\ge1-\frac1x\frac{x-\sin(x)}{x}\tag{1}
$$
<a href="https://math.stackexchange.com/a/32718... |
2,016,995 | <p>Is is possible to find a metric $d$ on $\mathbb{R}$ so that $(\mathbb{R}, d)$ would have a <strong>finite</strong> number of open sets?</p>
| miracle173 | 11,206 | <p>No.</p>
<p>If $(X,d)$ is a metric space and $X$ is not finite then choose $n$ arbitrary elements $x_1,\ldots,x_n$ of $X$ and let
$$d_0:=\frac{1}{3}\min\{d(x_i,x_j)|i \ne j\}$$
We have $d_0>0$ and the sets $$B(x_i,d_0)=\{x\in X|d(x_i,x)<d_0\}$$ are open and disjoint and therefore different.</p>
|
3,320,830 | <p>I was wondering if the inequality
<span class="math-container">$$\left|\int_0^T f(t,\omega )dW_t\right|\leq \int_0^T|f(t,\omega )|dW_t$$</span> holds for stochastic integral. In fact, I don't see such a property in any book, neither on Google, so I have some doubt. What do you think ?</p>
| Jyrki Lahtonen | 11,619 | <p>This is an old trick question.</p>
<p>Without loss of generality we can imagine that we are sitting on the train leaving from Nagpur at 7 a.m., before the train that left Raipur at 12.30 a.m. arrives, but after the arrival of 11.30 p.m. departure.
That train will arrive at Raipur at noon, after the 11.30 a.m. depar... |
356,574 | <blockquote>
<p>$$\aleph_2^{\aleph_0}=\aleph_2$$</p>
</blockquote>
<p>Appreciate your help</p>
| Brian M. Scott | 12,042 | <p>Yes, you can. Any function $f:\omega\to\omega_2$ is bounded, so ${}^\omega\omega_2=\bigcup_{\alpha<\omega_2}{}^\omega\alpha$, and therefore</p>
<p>$$\aleph_2\le\aleph_2^{\aleph_0}=\left|{}^\omega\omega_2\right|\le\sum_{\alpha<\omega_2}|\alpha|^\omega=\aleph_2\cdot\aleph_1^{\aleph_0}=\aleph_2\cdot2^{\aleph_0\c... |
1,692,346 | <p>I have heard of a statement like this:</p>
<blockquote>
<p>A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time.</p>
</blockquote>
<p>Is this possible (mathematics wise)?</p>
| Henricus V. | 239,207 | <p>Assume that the car engine is perfect and proportionally converts gas into distance. Then the answer is yes and no. Using the series
$$ \sum_{j=1}^\infty \frac{1}{2^j} = 1
$$
The car never runs out of gas since infinitely many terms are non-zero, but only travels a finite distance since the sum is finite.</p>
<p>Th... |
1,039,474 | <p>Solve the equation $x^4 - 14x^3 + 50x^2 -14x + 1 = 0$. <br/> I am not sure about how to best proceed, and would like a solution that does not involved the generalised quartic formula.</p>
| Community | -1 | <p>Not knowing the substitution trick, you can anyway infer that if $x$ is a solution, then $1/x$ as well, so that the polynomial can be factored in two polynomials of the second degree, and these will be palindromic too:</p>
<p>$$x^4 - 14x^3 + 50x^2 -14x + 1 =(x^2+Ax+1)(x^2+Bx+1).$$
Developing and identifying,
$$A+B=... |
3,370,076 | <p>The total mechanical energy is conserved when a ball is dropped from a height of 4.00 <span class="math-container">$\mathit{m}$</span>, and it makes a elastic collision with the ground. Assuming no non-conservative forces are acting find the period of the ball. g of course is 9.81.</p>
<p><span class="math-containe... | Marios Gretsas | 359,315 | <p>Let <span class="math-container">$a(x_1)=a(x_2)$</span> then <span class="math-container">$ax_1=ax_2$</span></p>
<p>So <span class="math-container">$x_1=a^{-1}ax_1=a^{-1}ax_2=x_2$</span></p>
<p>Now for <span class="math-container">$y \in G$</span> </p>
<p>Take <span class="math-container">$x=a^{-1}y$</span></p>
... |
114,733 | <p>Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?</p>
| davidlowryduda | 9,754 | <p>There is, but I'm not sure what you do or don't know. But we'll see what we can do.</p>
<p>So you know that it's a fractional linear transformation. It's continuous, bijective, open, and ultimately beautiful. It can be shown that it preserves circilinearity - i.e. it takes lines and circles to lines and circles (no... |
3,406,106 | <p>Want to prove rigorously (if possible, since I was not able to think of any counter-example) that <span class="math-container">$\lim_{x\to a} f(x)$</span> exists <span class="math-container">$\implies \lim_{x\to a} f(x^2)$</span> exists. (I also have a feeling that the limits equate.)</p>
<p>I started with the <s... | Randall | 464,495 | <p>This is false, as evidenced by the choices <span class="math-container">$a=2$</span> and <span class="math-container">$f(x) = \begin{cases} x, & x \leq 4 \\ x+1, & x>4.\end{cases}$</span></p>
|
764,947 | <p>I want to solve the following exercise:<br/>
<br/>
Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic.<br/>
$E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. <br/>
<br/>
I am trying to find a change of variables $(x,y)\mapsto(x',y')$ transforming the Weierstra... | Noam D. Elkies | 93,983 | <p>Typo: the equation of the first curve should be $y^2 = x^3 + x^2 - 2$. Hint: $y=y'$, so you need only find a transformation from $x$ to $x'$ that kills the quadratic term of the cubic on the right-hand side.</p>
|
1,748,547 | <p>Show that if the closed interval $[a,b]$ is covered by finitely many open intervals $(a_1,b_1), ...,(a_n,b_n)$, then $$b-a \le \sum^n_{i=1}(b_i-a_i)$$. </p>
<p>I know that $(a_1,b_1), ...,(a_n,b_n)$ form an open covering of $[a,b]$, and my thought is to show the inequality by mathematical induction, but not sure ho... | Lionel Ricci | 242,892 | <p>The base case is clear. Let $\{(a_i,b_i)\}_{i=1}^n$ be an open cover of $[a,b]$. Suppose the sets are not nested. Then there are two sets $(a_i,b_i)$ and $(a_j,b_j)$ with $a_i \leq a_j \leq b_i \leq b_j$. Without loss of generality we may assume $i=1, j=2$. Taking their union, i.e. forming $(a_1,b_2)$ we get a small... |
497,422 | <p>Which is bigger: $a$ or $a^2$ and what is the proof of that?</p>
<p>I'm kinda stuck and because there are cases where $a$ is bigger and other cases where $a^2$ is bigger.</p>
| Community | -1 | <p>$a>a^2:\ 0<a<1$</p>
<p>$a<a^2:\ \text{for all other }a,a\neq0,1$</p>
|
3,120,187 | <p>Suppose tall matrix <span class="math-container">$A$</span> is <span class="math-container">$n \times k$</span> and that its columns are orthogonal, i.e., <span class="math-container">$A' A = I_k$</span>. Suppose further that diagonal <span class="math-container">$M$</span> is <span class="math-container">$n \times ... | Mathiaspilot123 | 489,487 | <p>If I understood you correctly, the diagonal entries of <span class="math-container">$M$</span> is either <span class="math-container">$0$</span> or <span class="math-container">$1$</span>. If you let <span class="math-container">$M=diag(0\cdots 0)$</span> then <span class="math-container">$A'MA=$</span> zero matrix.... |
292,835 | <p>Does there exist a bijective function $f:{\mathbb R}\rightarrow{\mathbb R}$ that is nowhere-continuous, assuming that both domain and range have the "standard topology"? <sup>1</sup></p>
<p><sub><sup>1</sup> By this I mean the one generated by the open intervals $(a, b) \subset {\mathrm R}$. BTW, if this topo... | Asaf Karagila | 622 | <p>How about: $$f(x)=\begin{cases} x+1 & x\in\mathbb Q\\ x& x\notin\mathbb Q\end{cases}$$</p>
|
102,383 | <p>I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP. </p>
<p><strong>The Problem</strong><br>
Let $A$ be a $n\times x$ possibly complex matrix and $B$ a diagonal, real $n\times n$ matrix with m... | Sven E | 25,166 | <p>Thanks for this nice answer. For the stability-problem with only 1 constraint the formula is then really simple: $B_{11}=\mbox{diag}(1,\dots,1)=I_{n-1}$ and $A_{12},A_{21},A_{22}$ are only scalars. So that the GEVP is the solution of $(A_{11}-I_{n-1}(ac/b-\lambda))\cdot v=0$ with $a=A_{12},b=A_{22},c=A_{21}$ and $b\... |
70,146 | <p>I'm trying to use an image as a <code>ChartLabel</code> and I'm getting strange results.</p>
<p>Here is a bar chart, with labels, that looks ok:</p>
<p><img src="https://i.stack.imgur.com/YmRQp.png" alt="chart ok"></p>
<p>But when I try to replace the "A" label with an image, the output is confusing:</p>
<p><img... | MinHsuan Peng | 1,376 | <p>How about using <code>ChartElements</code> instead of <code>ChartLabels</code>.</p>
<pre><code>images = ExampleData[{"TestImage", #}] & /@ {"Lena", "Mandrill"};
BarChart[{{1, 2, 3}, {4, 5, 6}}, ChartElements -> {images, None}]
</code></pre>
<p><img src="https://i.stack.imgur.com/iZick.png" alt="enter image... |
2,564,217 | <p>For a project I'm doing, I'm wrapping an led strip light around a tube. The tube is 19mm in diameter and 915mm tall. I'm going to coil the led strip around the tube from top to bottom and the strip is 8mm wide, so the coils will be 8mm apart. How long does the led strip need to be to fully cover the tube?</p>
<p>Th... | karakfa | 14,900 | <p>Let's work on $X=3$ case, the other dice will take values $\{1..6\}$, where the corresponding $Y$ values will be $\{2,1,0,1,2,3\}$ Grouping the same values together will give (value, count) pairs as $\{(0,1), (1,2), (2,2), (3,1)\}$. Converting this the probability, you'll need to divide each count by $6*6$.</p>
<p... |
2,266,634 | <p>If $G$ is a finite abelian group then $G$ has a decomposition into two types:</p>
<p>(1) one is direct product of cyclic groups of some prime power order (may be with repitition)</p>
<p>(2) other is direct product of cyclic groups where order of one component divides order of next one (invariant factors).</p>
<p>... | Rajat | 177,357 | <p>Let say $T=\begin{bmatrix} 1 & 0\\ 1 & 0\end{bmatrix}$, $u=[0 \quad 1]^{\top}$ and $v=[1 \quad 0]^{\top}$. $Tu$ and $Tv$ are not linear independent.</p>
<p>Only, if $T$ is a one-to-one mapping, then we can say $Tu, Tv$ must be linearly independent.</p>
|
2,266,634 | <p>If $G$ is a finite abelian group then $G$ has a decomposition into two types:</p>
<p>(1) one is direct product of cyclic groups of some prime power order (may be with repitition)</p>
<p>(2) other is direct product of cyclic groups where order of one component divides order of next one (invariant factors).</p>
<p>... | Itay4 | 385,242 | <p>Consider the zero transformation:</p>
<p>$$Tv=0, \ \forall v \in V$$</p>
<p>It is a linear transformation (why?)</p>
<p>No matter what two linearly independent vectors $u$ and $v$ are, obviously $Tv$ and $Tu$ aren't. </p>
|
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | Kyle Miller | 8,981 | <p>It's worth noting that a real number is a point on the real number line (according to, for instance, Stewart's <em>Calculus</em>, Appendix A). The $y$-axis is a copy of the real number line. So, one could take the defensible position that a $y$-intercept is just as much a point $(0,b)$ as it is a point $b$ along t... |
2,974,747 | <p><strong>Q</strong>:Solve the equation <span class="math-container">$x^4+x^3-9x^2+11x-4=0$</span> which has multiple roots.<br><strong>My approach</strong>:Let <span class="math-container">$f(x)=x^4+x^3-9x^2+11x-4=0$</span>.And i knew that if the equation have multiple roots then there must exist H.C.F(Highest Common... | Mohammad Riazi-Kermani | 514,496 | <p><span class="math-container">$$x^4+x^3-9x^2+11x-4=0$$</span></p>
<p>By checking the divisors of <span class="math-container">$-4$</span> we see that <span class="math-container">$x=1$</span> satisfies our equation.</p>
<p>Upon synthetic division, we get <span class="math-container">$$x^4+x^3-9x^2+11x-4=(x-1)^3(... |
428,408 | <p>Consider a norm on <span class="math-container">$\mathbb C^2$</span> as <span class="math-container">$\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$</span></p>
<p><em>Question.</em> Is <span class="math-container">$(\mathbb C^2,\|.\|)$</span> linearly isometric to <span class="math-container">$(\m... | Gerald Edgar | 454 | <p><strong>comment</strong><br />
I think they are not isometric, having different structure for the set of extreme points.</p>
<p>The set of extreme points for the unit ball of <span class="math-container">$\|\cdot\|_\infty$</span> is a torus: <span class="math-container">$$T = \{(z_1,z_2) : |z_1| = |z_2| = 1\}.$$</sp... |
1,290,363 | <p>So I already proved Closure and Associativity, now I'm trying to find the identity element of this operation defined as:
$$
a * b = a + b - ab
$$</p>
<p>But my identity element gets cancelled...</p>
<p>(The set defined in this exercise is the real numbers.)</p>
<p><img src="https://i.stack.imgur.com/ZchjC.jpg" al... | 3x89g2 | 90,914 | <p><strong>Claim:</strong> The identity element in $(\mathbb{R},*)$ is the real number zero.</p>
<p><strong>Proof:</strong> For any $x\in \mathbb{R}$, $x*0=x+0-x\times 0=x$. Since the identity element in a group is unique, zero is the identity element.</p>
<p>Following your way, suppose the identity is $e$, it has to... |
672,736 | <p>Let $A = \begin{bmatrix}1&2&1\\0&1&0\\1&3&1\end{bmatrix}$. Find the eigenvalues of $A$.</p>
<p>I think I got a pretty steady ground on how I approached this, I just have some difficulty getting the right answer.</p>
<p>What I have done so far:</p>
<p>$P(\lambda) = det(A - \lambda I)$</p>
... | hjpotter92 | 27,741 | <p>You've made a little error in calculating your $ \det $ value.</p>
<p>$$ det\begin{bmatrix}1-\lambda&2&1\\0&1-\lambda&0\\1&3&1-\lambda\end{bmatrix} = 0 $$</p>
<p>will expand to give you:</p>
<p>$$ =(1-\lambda)(1-\lambda)^2 - 2(0) + 1 (\color{red}{-1 \times (1 - \lambda)}) = 0 $$</p>
<p>wh... |
672,736 | <p>Let $A = \begin{bmatrix}1&2&1\\0&1&0\\1&3&1\end{bmatrix}$. Find the eigenvalues of $A$.</p>
<p>I think I got a pretty steady ground on how I approached this, I just have some difficulty getting the right answer.</p>
<p>What I have done so far:</p>
<p>$P(\lambda) = det(A - \lambda I)$</p>
... | Brian Fitzpatrick | 56,960 | <p>You could expand your equation for the determinant about the second row. This gives
\begin{align*}
\det(A-\lambda\cdot I)
&=
\det\begin{bmatrix}1-\lambda&2&1\\0&1-\lambda&0\\1&3&1-\lambda\end{bmatrix} \\
&=
(1-\lambda)\cdot\det
\begin{bmatrix}
1-\lambda & 1 \\
1 & 1-\lambda
\e... |
272,846 | <p>Suppose I have a List of numbers:</p>
<pre><code>num = Range[5]
</code></pre>
<p>I want to combine the second and the third element into a sublist to get the result as {1,{2,3},4,5}.<br />
I tried using this:</p>
<pre><code>MapAt[List, num, {{2}, {3}}]
</code></pre>
<p>which is not giving me the desired result. What... | lericr | 84,894 | <p>If you want to do this at a position (i.e. you're not looking at individual elements or patterns to determine the nested list), then you'll probably need to break apart the list and reassemble it.</p>
<p>One way:</p>
<pre><code>FlattenAt[TakeList[num, {1, 2, All}], {{1}, {3}}]
(* you'd need to set the list argument ... |
3,014,453 | <p>If there is a number somewhere between 0 and 100 and you have to find it with the least attempts possible. Every attempt consists of you checking if the number is smaller (or bigger) than a number in the said interval (0 to 100). My guess would be you start with the half way point.</p>
<p>Is it smaller than 50?
yes... | gandalf61 | 424,513 | <p>Yes, this is the most efficient search method if you can only get a "yes" or a "no" answer on each attempt. Because there are two possible responses on each attempt, it is known as a binary search - see <a href="https://en.wikipedia.org/wiki/Binary_search_algorithm" rel="nofollow noreferrer">https://en.wikipedia.org... |
475,005 | <p>I want to check how many integral numbers in $\big[1,10^6\big]$ include the numbers $1,2,3,4,5$ and how many only them.<br>
how should I check it? this is a problem of inclusion-exclusion? <br>
I would like to get some advice!<br>
Thanks!</p>
| Henry | 6,460 | <p>Hints:</p>
<ul>
<li><p>You do not need use inclusion-exclusion, though you can if you want to. Another approach to the first question could be to look at those which do not include any of $1,2,3,4,5$</p></li>
<li><p>You might find it easier to treat $1000000$ and $0$ as special cases and instead look at the first ... |
2,698,960 | <p>Why this statement is false to $a \in \mathbb{C}$ </p>
<p>$(\sqrt[n]{a} * \sqrt[k]{a} ) - (a^{\frac{n+k}{nk}})= 0$</p>
<p>How you can prove it with high school maths ?</p>
| Clive Newstead | 19,542 | <p><strong>Hint:</strong> Left adjoints preserve colimits, so it suffices to prove that $U$ does not preserve colimits.</p>
<p>For an even bigger hint, hover over the box below.</p>
<blockquote class="spoiler">
<p> $U$ doesn't even preserve binary coproducts. The coproduct of two groups in $\mathbf{Ab}$ is their di... |
2,698,960 | <p>Why this statement is false to $a \in \mathbb{C}$ </p>
<p>$(\sqrt[n]{a} * \sqrt[k]{a} ) - (a^{\frac{n+k}{nk}})= 0$</p>
<p>How you can prove it with high school maths ?</p>
| Pece | 73,610 | <p>Clive gave a neat short way to answer your question. But let me try to explain what is really going on under the hood here.</p>
<p>Call a <em>Lawvere theory</em> any bijective-on-objects finite-coproducts-preserving functor $j : \aleph_0 \to \mathcal L$ where $\aleph_0$ is a skeleton of the category of finite sets.... |
309,380 | <p>Let me sum up my - hopefully correct - understanding of the <a href="https://en.wikipedia.org/wiki/Travelling_salesman_problem" rel="nofollow noreferrer">travelling salesman problem</a> and <a href="https://en.wikipedia.org/wiki/Complexity_class" rel="nofollow noreferrer">complexity classes</a>. It's about <a href="... | mhum | 9,840 | <p>I think that such a non-constructive method for solving an NP-complete problem can always be transformed into a polynomial-time constructive method by re-running the non-constructive method on a sequence of modified versions of the original problem. It's a little more straightforward in the case of <a href="https://... |
309,380 | <p>Let me sum up my - hopefully correct - understanding of the <a href="https://en.wikipedia.org/wiki/Travelling_salesman_problem" rel="nofollow noreferrer">travelling salesman problem</a> and <a href="https://en.wikipedia.org/wiki/Complexity_class" rel="nofollow noreferrer">complexity classes</a>. It's about <a href="... | Gerhard Paseman | 3,402 | <p>I suspect that one learns nothing about such proofs, even in the case that P equals NP. (If a specific polynomial time algorithm were given for 3-SAT, we could build one for any NP problem, of course, and we would then learn something. But you are not asking about that situation.) Let me illustrate with a particu... |
3,430,136 | <p>I would like to prove that the map <span class="math-container">$f: S^n \times S^m \to 2S^{m+n+1}: ((x_1,..,x_{n+1}), (y_1,...,y_{m+1})) \to (x_1,...,x_{n+1},y_1,...,y_{m+1})$</span>
is an imersion. Here <span class="math-container">$2S^{m+n+1}$</span> is the <span class="math-container">$m+n+1$</span> dimensional ... | Tsemo Aristide | 280,301 | <p>The map <span class="math-container">$h:\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+m+2}$</span>
defined by <span class="math-container">$h((x_1,..,x_{n+1}),(y_1,..,y_{m+1}))=(x_1,..,x_{n+1},y_1,..,y_{m+1})$</span> is an immersion. Its restriction to the submanifold <span class="math-container">$S... |
2,470,958 | <p>Let's say that I've got the following IVP:</p>
<p>$\frac{dy}{dx} = f(x,y)$</p>
<p>$y(x_0) = y_0$</p>
<p>And I want conditions that guarantee existence and uniqueness of its solution.</p>
<p>On the one hand I've got the Picard–Lindelöf theorem. It asks that there exists a rectangle $R = [a,b] \times [c,d]$, conta... | user247327 | 247,327 | <p>The first, that requires that the partial derivative of f with respect to y exist, is an "if and only if" theorem. The second, that requires "Lindelof" in y, says "if" but not "only if". Any function that is differentiable with respect to y is "Lindelof" but the converse is not true.</p>
|
2,144,520 | <p>I don't really understand Cantor's diagonal argument, so this proof is pretty hard for me. I know this question has been asked multiple times on here and i've gone through several of them and some of them don't use Cantor's diagonal argument and I don't really understand the ones that use it. I know i'm supposed to ... | Noah Schweber | 28,111 | <p>In the comments to your question, you indicate that your professor began by showing that $(0, 1)$ is uncountable. I actually think this is a bad way to start; it will be easier to understand the proof of the uncountability of set of <strong>infinite sequences of natural numbers</strong>, $\mathbb{N}^\mathbb{N}$. </... |
2,548,942 | <p>What would be the best approach to calculate the following limits </p>
<p>$$ \lim_{x \rightarrow 0} \left (1+\frac {1} {\arctan x} \right)^{\sin x}, \qquad \lim_{x \rightarrow 0} \frac {\tan ^7 x} {\ln (7x+1)} $$
in a basic way, using some special limits, without L'Hospital's rule? </p>
| Sebastiano | 705,338 | <p>For the first limit being <span class="math-container">$x\to 0$</span> we have <span class="math-container">$\arctan x\sim x$</span>, hence:</p>
<p><span class="math-container">$$\lim_{x \rightarrow 0} \left (1+\frac {1} {\arctan x} \right)^{\sin x}\sim \lim_{x \rightarrow 0} \Biggl[\left (1+\frac {1} {x} \right)^x... |
599,139 | <p>Let $A = B = N$, where N is the set of natural numbers.
Define $f:A \to B$ by $f(a)=2a$ and define $g:B\to A$ by $g(b)=3b$</p>
<p>Find $g^{-1}(\{2,4,6\})$.</p>
<p>Find $g^{-1} (\{2,4\})$</p>
<p>My trouble here is would $g^{-1}$ just be f?</p>
<p>Also an explanation of what the difference between $f(g(3))$ and $f... | Mauro ALLEGRANZA | 108,274 | <p>See <strong>Herbert Enderton</strong>, <em>Computability Theory An Introduction to Recursion Theory</em> (2011) [pag.4] :</p>
<blockquote>
<p>The concept of decidability can [...] be described in terms of functions: For a subset
$S$ of $\mathbb{N}$ , we can say that $S$ is <em>decidable</em> iff its characteris... |
599,139 | <p>Let $A = B = N$, where N is the set of natural numbers.
Define $f:A \to B$ by $f(a)=2a$ and define $g:B\to A$ by $g(b)=3b$</p>
<p>Find $g^{-1}(\{2,4,6\})$.</p>
<p>Find $g^{-1} (\{2,4\})$</p>
<p>My trouble here is would $g^{-1}$ just be f?</p>
<p>Also an explanation of what the difference between $f(g(3))$ and $f... | Mauro ALLEGRANZA | 108,274 | <p>About <em>semi-decidability</em>, we have that (again form Enderton's book, pag.5) :</p>
<blockquote>
<p>It is very natural to extend these concepts to the situation where we have half of
decidability: Say that S is semidecidable if its “semicharacteristic function”</p>
</blockquote>
<p>$c^S(x)$ = Yes if $x \i... |
1,986,402 | <blockquote>
<p>How can I simplify $\prod \limits_{l=1}^{a} \frac{1}{4^a} \cdot 16^l$?</p>
</blockquote>
<p>I've tried looking at the terms and finding something in there to conlcude what it might be and also took the $n^{th}$ term of $16^l$ into one fraction but that does rather the opposite of simplification.</p>
| Leucippus | 148,155 | <p>Following Jack's answer and using $\sum_{k=1}^{n} k = n(n+1)/2$ it can be seen that:
\begin{align}
\prod_{k=1}^{n} \left\{ \frac{x^{2k}}{y^{n}} \right\} &= \left( y^{-n} \right)^{n} \, \left( \prod_{k=1}^{n} x^{2k} \right) \\
&= y^{-n^{2}} \, x^{2 \, \sum_{k=1}^{n} k } = y^{-n^{2}} \, x^{n^{2} + n } \\
&... |
2,292,015 | <p>The question is:
the first three terms of an arithmetic series $c_{n}$ are
$$a(1+b), a(1+3b),a(1+5b)$$
I needed to find the common difference in terms of $a$ and $b$ and then find the expression for $c_{n}$.</p>
<p>The final part I struggled with where I have to find $a$ and $b$ and the information given is
$$c_{5... | Ahmed S. Attaalla | 229,023 | <p>If we write it as follows,</p>
<p>$$\vec r(t)=\langle t\cos t,t \sin t, t \rangle$$</p>
<p>Then we are able to see that $x^2+y^2=t^2$. As $t$ increases from $0$ we get higher an higher up the path, but the points stay circular in nature. The circles that a single point $(t\cos(t),t\sin(t),y\sin (t))$ are at get bi... |
4,247,637 | <p>There are several tea cups in the kitchen, some with handle and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly <span class="math-container">$1200$</span>. What is the maximum possible number of cups in the kitchen?<br>
Here's what I did:<br>
I... | Jack D'Aurizio | 44,121 | <p>I will prove that <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span> are equivalent definitions for <span class="math-container">$K_0$</span>:
<span class="math-container">$$ K_0(\xi)=\frac{1}{2}\int_{-\infty}^{+\infty}\frac{e^{i\xi x}}{\sqrt{x^2+1}}\,dx=\int_{0}^{+\infty}\frac{\... |
170,967 | <p>Cog $A$ is at position: $Ax$, $Ay$, rotation: $Ar$ and number of teeth: $At$</p>
<p>Cog $B$ is at position: $Bx$, $By$ and number of teeth $Bt$. What is Cog $B$'s rotation such that teeth between Cog $A$ and Cog $B$ line up. There will be the same number of answers as there are teeth, but a 'base angle' is desired.... | joriki | 6,622 | <p>You can calculate the angle $\alpha$ of the line from $A$ to $B$ as $\alpha=\arctan\frac{By-Ay}{Bx-Ax}$. You want the phases to be opposite at this angle, so $(\alpha-Ar)At=(\alpha+\pi-Br)Bt+\pi+ 2\pi n$, with $n$ an integer; you can solve this for $Br$ to determine $Br$ up to integer multiples of $2\pi/Bt$.</p>
|
2,746,222 | <p>Problem:</p>
<p>A boats speed is <strong>1,70 m/s</strong> in still water.<br />
It must cross a river with a width of <strong>260 m</strong>.<br />
The boats starting point is the <strong>origin on the xy-axsis</strong> (on the shore).<br />
It has to dock <strong>110 m</strong> to the right(in the positive x-direc... | Vasili | 469,083 | <p>Let $v_b$ be the speed of boat in still water, $v_r$ is the speed of river. The speed of boat in $x$ direction (in still water at 45 degree angle): $v_x=v_b/\sqrt{2}$, the speed of boat in $y$ direction is $v_y=v_b/\sqrt{2}$. The river speed has negative $x$ direction as you correctly concluded. Thus, we need to sub... |
2,704,394 | <p>Here is the formal statement:</p>
<blockquote>
<p>Let $\lambda_1, \lambda_2, \lambda_3$ be distinct eigenvalues of $n\times n$ matrix $A$. Let $S=\{v_1, v_2, v_3\}$, where $Av_i = \lambda_i v_i$ for $1\leq i\leq 3$. Prove $S$ is linearly independent. </p>
</blockquote>
<p>Many resources online state the general ... | RCT | 424,406 | <p>Here's one idea that comes to mind, although I don't promise there isn't a slicker way to do it. Suppose $c_1v_1 + c_2v_2 + c_3v_3 = 0.$ Applying $A$ gives $$\lambda_1c_1v_1 + \lambda_2c_2v_2 + \lambda_3c_3v_3 = 0.$$ On the other hand, multiplying the original equation by $\lambda_1$ gives $$\lambda_1c_1v_1 + \lambd... |
2,798,206 | <p>How do you prove this using the epsilon-delta definition? I'm unsure of using the min = { } function.</p>
<p>$\lim \limits_{x \to \infty}\frac{2x+1}{1-x}$</p>
<p>These are my steps: </p>
<p>$ |f(x) - L| < \epsilon => |\frac{2x+1}{1-x} +2|< \epsilon $</p>
<p>$ \qquad \qquad \; \; \; \; \; =>|\frac{3}{... | G.L. | 565,369 | <p>Thanks, Theo Bendit. I re-did my answer, not sure if it's correct.</p>
<p>$ |f(x) - L| = |\frac{2x+1}{1-x} +2|< \delta $</p>
<p>$ \qquad \quad \; \; \; =|\frac{3}{1-x} | < \delta $</p>
<p>$ \qquad \quad \; \; \; =|\frac{-3}{x-1} | < \delta $</p>
<p>$ \qquad \quad \; \; \; =|-3||\frac{1}{x-1} | < \del... |
1,083,277 | <p>$a,b,c \in \mathbb{R}$ and $a+b+c=0$.
Prove that: $8^{a}+8^{b}+8^{c}\geqslant 2^{a}+2^{b}+2^{c}$</p>
<p>I think that $2^{a}.2^{b}.2^{c}=1$, but i don't know what to do next</p>
| Blind | 207,277 | <p>We have
$$
8^a+1+1\geq3\sqrt[3]{8^a}=3\times 2^a,
$$
$$
8^b+1+1\geq3\sqrt[3]{8^b}=3\times 2^b,
$$
$$
8^c+1+1\geq3\sqrt[3]{8^c}=3\times 2^c,
$$
$$
2^a+2^b+2^c\geq 3\sqrt[3]{2^{a+b+c}}=3.
$$
It follows that
\begin{eqnarray}
8^a+8^b+8^c&\geq&3(2^a+2^b+2^c)-6\\
&=&(2^a+2^b+2^c)+2(2^a+2^b+2^c-3)\\
&\g... |
73,559 | <p>I have the following problem. I'd like to add a legend to <code>MatrixPlot</code>. Each colour should have a legend entry. I used <code>PlotLegends</code>, which in principle works. However, if I use more than five colours, this doesn't work anymore.</p>
<pre><code>a = RandomInteger[{1, 6}, {50}];
MatrixPlot[{a}, C... | Bob Hanlon | 9,362 | <pre><code>a = RandomInteger[{1, 6}, {50}];
colors = {1 -> Red, 2 -> Blue, 3 -> Green, 4 -> Gray, 5 -> Yellow,
6 -> Orange};
Column[{
MatrixPlot[{a},
ColorRules -> colors,
ImageSize -> 400],
SwatchLegend[
colors[[All, 2]],
colors[[All, 1]],
LegendLayout -> "Row"]},
... |
1,317,143 | <blockquote>
<p><em>Notation</em>: $\log:=\log_{10}$</p>
</blockquote>
<p>$\log x+\log_x 10$</p>
<p>$=\log x+ \frac{1}{\log x}$ </p>
<p>$=\log(x \cdot \frac{1}{x})$ </p>
<p>$=\log 1$ </p>
<p>$=0$ </p>
<p>Is the process correct? I doubt this is wrong. Please help.
Thanks.</p>
| Hagen von Eitzen | 39,174 | <p>Note that $\log_a b=\frac{\ln b}{\ln a}$. Hence you want ot minimize $y+\frac1y$ with $y=\frac{\ln x}{\ln 10}$. "Clearly", this minimum is $2$ and achieved when $y=1$, i.e., when $\ln x=\ln 10$ and finally $x=10$.</p>
<p>Why $y=\frac1y\ge 2$ for $y>0$? Well, we have $y-2+\frac1y=\left(\sqrt y-\frac1{\sqrt y}\rig... |
481,952 | <p>Why is a union of infinitely many bounded sets not necessarily bounded, please? In addition, what condition can we add to make this union bounded, please?</p>
| Community | -1 | <p>Since you're asking why it's not <em>necessarily</em> true that such a union is bounded, it suffices to consider a counterexample. Define $A_n = [n, n + 1]$; then </p>
<p>$$\bigcup_{n \in \Bbb{N}} A_n = [0, \infty)$$</p>
<p>is unbounded.</p>
<p>It is necessary and sufficient that there is a common bound on all th... |
3,403,255 | <p>I am trying to follow wikipedia's page about matrix rotation and having a hard time understanding where the formula comes from.</p>
<p><a href="https://en.wikipedia.org/wiki/Rotation_matrix" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Rotation_matrix</a> Wiki page about it.</p>
<p>what i have so far:</... | J.G. | 56,861 | <p>You've not shown how you got a different result, so I can't comment on your mistake. But it looks to me like you're examining the composition of rotations about the origin in <span class="math-container">$2$</span> dimensions. (@bounceback gave an answer that understood your aims differently, so I hope between us we... |
1,134,215 | <p>How can I determine whether {$\frac{z}{1+z^2}$; z $\in$ $\mathbb{C}$ \ {-i, i}} is bounded? My textbook is very poor at describing boundedness for complex functions. Thanks for the help!</p>
| ahulpke | 159,739 | <p>I don't think this is true without some further finiteness condition that limits the number of images of $K$:</p>
<p>Take $G$ the (two-sided) infinite sequences with entries in 0,1 and as operation component-wise addition (so its an infinite direct product of $C_2$ with itself), and as $K$ the kernel of the project... |
1,302,932 | <p>Given $$A=\begin{pmatrix}
2 & 0 & 0\\
a & 2& 0\\
a+3 & a &-1
\end{pmatrix}$$<br>
For which values of $a$ can $A$ be diagonal?<br>
I found that $p_A(x)=(x-2)^2(x+1)$ and tried to find the eigen subspace of 2, to see if the geomtric multiplicity of the eigenvalue $2$ is $2$.<br>
I got a set... | user84413 | 84,413 | <p>Since $A-2I=\begin{pmatrix}0&0&0\\a&0&0\\a+3&a&1\end{pmatrix}$, $\;\;\text{nullity}(A-2I)=2\iff \text{rank}(A-2I)=1 \iff a=0$,</p>
<p>so A is diagonalizable $\iff a=0$.</p>
|
3,350,021 | <blockquote>
<p>We have the following quadratic equation:</p>
<p><span class="math-container">$2x^2-\sqrt{3}x-1=0$</span> with roots <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>.</p>
<p>I have to find <span class="math-container">$x_1^2+x_2^2$</span> and <span clas... | Hussain-Alqatari | 609,371 | <p>Note that: if <span class="math-container">$a,b,c \in \mathbb{R}$</span> and <span class="math-container">$a\ne0$</span>, if <span class="math-container">$ax^2+bx+c=0$</span>, then <span class="math-container">$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$</span></p>
<p><span class="math-container">$2x^2-\sqrt{3}x-1=0$</... |
3,413,364 | <blockquote>
<p>Consider the set of points <span class="math-container">$$O = \{ x \in P \mid \alpha^* = C^T x \}$$</span> where <span class="math-container">$P \subseteq \mathbb R^n$</span> is a closed convex set, <span class="math-container">$C \in \mathbb R^n$</span> and <span class="math-container">$\alpha^* = \m... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$x_k \in O$</span> and <span class="math-container">$x_k \to x$</span> then <span class="math-container">$C^{T}x_k \to C^{T}x$</span> and <span class="math-container">$\alpha^{*}=C^{T}x_k$</span> for each <span class="math-container">$k$</span>. Hence <span class="math-container">$\a... |
2,959,686 | <p>I'm trying to see if I can find a bijection between two groups that are infinite of which one in the subset of the other. If I find the inverse <span class="math-container">$\phi^{-1}(x)=\frac{1}{5}x$</span> since it doesn't work for <span class="math-container">$x \in \mathbb{Z}$</span> (because I will have values ... | Dietrich Burde | 83,966 | <p>The subgroups of <span class="math-container">$\Bbb{Z}$</span> are given by <span class="math-container">$n\Bbb{Z}$</span>. For <span class="math-container">$n\neq 0$</span>, <span class="math-container">$n\Bbb{Z}$</span> is an infinite cyclic group with generator <span class="math-container">$n$</span>, and hence i... |
892,114 | <p>i have three number
1 2 3 which will always be in this order {123}, i want to find out number of cases can be made,
like {1},{2},{23},{13},{12},{123}{3},{}. but each number has two states like "a" "b", i.e, each one will become different entity,like 2a,2b,3a,3b,1a,
with only exception i.e. 1 will have only one stat... | Mufasa | 49,003 | <p>$$(x-2)^2=(x-2)(x-2)=x(x-2)-2(x-2)=x^2-2x-2x+4=x^2-4x+4$$</p>
|
187,975 | <p>Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states that for every $t \in [0,1]$, there exists a set $S \in \Sigma$ with $\mu(S) = t$.</p>
<p>I would like to use the foll... | user42397 | 42,397 | <p>It seems to be a special case of Lemma 2.5(chapter 2) of "interpolation of operators"(Bennett, Sharpley).</p>
|
43,956 | <p>There is this example at the Wikipedia article on Quotient spaces (QS):</p>
<blockquote>
<p>Consider the set $X = \mathbb{R}$ of all real numbers with the ordinary topology, and write $x \sim y$ if and only if $x−y$ is an integer. Then the quotient space $X/\sim$ is homeomorphic to the unit circle $S^1$ via the h... | Henno Brandsma | 4,280 | <p>The quotient space <span class="math-container">$Y = X / \sim$</span> as a set is just the set of equivalence classes of <span class="math-container">$X$</span> under <span class="math-container">$\sim$</span>, so the set <span class="math-container">$\{ [x]: x \in \mathbb{R} \} $</span> in your case. </p>
<p>The e... |
4,015,741 | <p>I want to find the solutions of <span class="math-container">$(x+1)^{63}+(x+1)^{62}(x-1)+\cdots+(x-1)^{63}=0$</span>.</p>
<p>It is not hard to see <span class="math-container">$x=0$</span> is a root of the equation. but I don't know how to solve this equation in general. I can see terms of the equation looks very s... | player3236 | 435,724 | <p><span class="math-container">$$\frac {a^n - b^n}{a-b} = a^{n-1} + a^{n-2} b + a^{n-3}b^2 + \dots + ab^{n-2} + b^{n-1}$$</span></p>
<p>Hence:</p>
<p><span class="math-container">$$(x+1)^{63} + (x+1)^{62}(x-1) + \dots+ (x-1)^{63} = \frac {(x+1)^{64}-(x-1)^{64}}{(x+1)-(x-1)} = \frac12((x+1)^{64} - (x-1)^{64})$$</span><... |
1,621,363 | <p>Integrate: $$\int \frac{\sin(x)}{9+16\sin(2x)}\,\text{d}x.$$</p>
<p>I tried the substitution method ($\sin(x) = t$) and ended up getting $\int \frac{t}{9+32t-32t^3}\,\text{d}t$. Don't know how to proceed further. </p>
<p>Also tried adding and substracting $\cos(x)$ in the numerator which led me to get $$\sin(2x) =... | Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$\int\frac{\sin(x)}{9+16\sin(2x)}\space\text{d}x=$$</p>
<hr>
<p>Use the double angle formula $\sin(2x)=2\sin(x)\cos(x)$:</p>
<hr>
<p>$$\int\frac{\sin(x)}{32\sin(x)\cos(x)+9}\space\text{d}x=$$</p>
<hr>
<p>Subsitute $u=\tan\left(\frac{x}{2}\right)$ and $\text{d}u=\frac{\sec^2\left(\frac{x}{2}\righ... |
2,887,440 | <p>We were asked in our Calculus class to prove that,</p>
<blockquote>
<p>$f(x+y) - f(x) = \frac {\sec^2(x) \tan(y)} {1 - \tan(x) \tan(y)}$ given that $f(x) = \tan(x)$</p>
</blockquote>
<p>I have gotten so far as:</p>
<p>$$f(x+y) - f(x)$$</p>
<p>$$\tan(x+y) - \tan(x)$$</p>
<p>$$\frac{\tan(x)+\tan(y)}{1-\tan(x)\t... | zahbaz | 176,922 | <p>You might be unnecessarily hung up on the $x$ in $f(x)=\tan(x)$. Remember that your function $f$ is simply a mapping between some input to some output. The definition of $f$ just so happens to use $x$ to stand for any real number in this context. </p>
<p>You can consider what happens to another real number $x'=x+y$... |
233,618 | <p>I want to be able to take a polynomial and take the 1st 5 derivatives, then add at least one root of each derivative to a list using a loop. However, each attempt I try only ends up outputting the roots of the 5th derivative, not the rest. So far I have:</p>
<pre><code>rootderivs[n_]:=(
p[x_]:= x^8-3x^5+x-1;
rootli... | cvgmt | 72,111 | <pre><code>Table[D[x^8 - 3 x^5 + x - 1, {x, n}], {n, 1, 8}]
(* {1 - 15 x^4 + 8 x^7, -60 x^3 + 56 x^6, -180 x^2 + 336 x^5, -360 x +
1680 x^4, -360 + 6720 x^3, 20160 x^2, 40320 x, 40320} *)
</code></pre>
<pre><code>Table[NSolve[D[x^8 - 3 x^5 + x - 1, {x, n}] == 0, x], {n, 1, 8}]
(* {{{x -> -0.628102 - 1.06836 I}, ... |
3,850,422 | <p>For a few days now I've been trying to find a closed form expression for the determinant of the following <span class="math-container">$n\times n$</span> tridiagonal matrix</p>
<p><span class="math-container">$$\begin{pmatrix}c_1+b_1+a_1 & b_1 & 0 & \ddots & 0 \\ c_2 & c_2+b_2+a_2 & b_2 &... | fewfew4 | 617,212 | <p>I believe I have an explicit solution!</p>
<p>Using the case that I had already figured out (when <span class="math-container">$a_k=0$</span>), we can Taylor expand around this solution. For finite <span class="math-container">$n$</span>, this will be a finite expansion.</p>
<p>First I define the quantity <span clas... |
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