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 |
|---|---|---|---|---|
1,070,008 | <p>Is being $T_1$ is a topological invariant?
Is being a first-countable space is a topological invariant?
I need a little hint as to whether or not these sets are topological invariants.</p>
| Sultan of Swing | 144,369 | <p>Let $X$ be $T_{1}$. Let $f:X\rightarrow Y$ be a homeomorphism. This tells us that $f$ is an open mapping (i.e., open sets map to open sets). Let $x,y ∈ X$, and let $B$ be an open set containing $x$ but not $y$. Since, $f$ is an open map, then $f(B)$ will be an open set in $Y$. Can we be certain that the open set $f... |
134,455 | <p>I have an expression which consists of terms with undefined function calls <code>a[n]</code>:</p>
<pre><code>example = 1 - c^2 + c a[1] a[2] + 1/2 c^2 a[1]^2 a[2]^2 + c a[1] a[3]
</code></pre>
<p>Now I want to transform each term with individual <code>a</code>s to a different function <code>v[m1,m2,m3]</code>, suc... | BoLe | 6,555 | <pre><code>v /: v[x__] v[y__] := Apply[v, {x} + {y}]
transform[expr_] :=
Module[{max, temp, free},
max = Max@Cases[expr, a[i_] :> i, Infinity];
temp = expr /. a[i_]^p_. :> (p + 1)*(v @@ UnitVector[max, i]);
free = Cases[temp, x_ /; FreeQ[x, v]];
With[{t = Total[free]},
temp - t + (v @@ ConstantArray[... |
85,126 | <p>Does anyone have an implementation for <code>AnglePath</code> (see <a href="http://reference.wolfram.com/language/ref/AnglePath.html" rel="nofollow"><code>AnglePath</code> Documentation</a> and <a href="http://blog.wolfram.com/2015/05/21/new-in-the-wolfram-language-anglepath/" rel="nofollow">example usage</a>) in <e... | KennyColnago | 3,246 | <p>For the first usage, with an input list <code>t</code> of angles, I used:</p>
<pre><code>anglePath[t_?VectorQ] :=
With[{a = Accumulate[t]},
Join[{{0., 0.}}, Accumulate[Transpose[{Cos[a], Sin[a]}]]]]
</code></pre>
<p>For the second usage, with an input matrix of <code>{r,t}</code> pairs, I used:</p>
<... |
2,011,754 | <p>Can somebody help me to solve this equation?</p>
<p>$$(\frac{iz}{2+i})^3=-8$$ ?
I'm translating this into</p>
<p>$(\frac{iz}{2+i})=-2$</p>
<p>But i recon it's wrong ...</p>
| Fred | 380,717 | <p>Determine the solutions $w_1,w_2,w_3$ of the equation $w^3=-8$.</p>
<p>Then solve $\frac{iz}{2+i}=w_j $ for j=1,2,3</p>
|
2,585,466 | <p>I have two growth curve data sets, A (Martians) and B (Venusians). Data point sets of age (0 (birth) - 250 months, X axis) against height (0 - 200 centimeters, Y axis). The first set (A) contains 67 X Y point pairs, the second set (B) contains 27 point pairs. I have fit both data sets to my favorite version of the L... | prog_SAHIL | 307,383 | <p>$$y^2=4ax$$ $$xy=c^2$$</p>
<p>As Arthur pointed, we need to find their point of intersection and calculate $dy\over{dx}$ of both curves at this point.</p>
<p>(Note that this is angle of tangents at that point, we cannot calculate the true angle due to mathematical limitations.)</p>
<p>Put $x$ from first equation ... |
815,661 | <p>Let $m$ be the product of first n primes (n > 1) , in the following expression :</p>
<p>$$m=2⋅3…p_n$$</p>
<p>I want to prove that $(m-1)$ is not a complete square.</p>
<p>I found two ways that might prove this . My problem is with the SECOND way . </p>
<p><strong>First solution (seems to be working) :</strong> <... | Mark Fischler | 150,362 | <p>The follow-on to the previous answer is that since $x^2 \equiv 2 \mod 3$ has no solution,
and $m-1 \equiv 2 \mod 3$, there cannot be a number $x$ such that $m-1 = x^2$. </p>
<p>It is worth pointing out that the first "solution" given is completely bogus because it contains a mistake in the first equation, which end... |
1,508,863 | <p>I have this homework problem assigned but I'm confused as to how to solve it:</p>
<p>For $n>2$ and $a\in\mathbb{Z}$ with $\gcd(a,n)=1$, show that $o_n(a)=m$ is odd $\implies o_n(-a)=2m$.</p>
<p>(where $o_n(a)=m$ means that $a$ has order $m$ modulo $n$).</p>
<p>We were also given this hint: Helpful to consider ... | Thomas Andrews | 7,933 | <p>Note that:</p>
<p>$$e^{-1}=\sum_{k=0}^\infty \frac{(-1)^k}{k!}$$</p>
<p>Then:</p>
<p>$$\frac{n!}e=n!e^{-1} = \left(\sum_{k=0}^{n} (-1)^k\frac{n!}{k!}\right) + \sum_{k=n+1}^{\infty} (-1)^{k}\frac{n!}{k!}$$</p>
<p>Show that if $a_n=\sum_{k=n+1}^{\infty} (-1)^{k}\frac{n!}{k!}$ then $0<|a_{n}|<1$ and $a_n>0... |
1,508,863 | <p>I have this homework problem assigned but I'm confused as to how to solve it:</p>
<p>For $n>2$ and $a\in\mathbb{Z}$ with $\gcd(a,n)=1$, show that $o_n(a)=m$ is odd $\implies o_n(-a)=2m$.</p>
<p>(where $o_n(a)=m$ means that $a$ has order $m$ modulo $n$).</p>
<p>We were also given this hint: Helpful to consider ... | Micah | 30,836 | <p>Following @Vladimir's comment, I can show that $a=3e$ has this property. I don't find the proof very enlightening, though...</p>
<p>We have</p>
<p>$$
\frac{n!}{3e} = \sum_{k=0}^n \frac{1}{3}\frac{n!}{k!}(-1)^k + E
$$
where $E$ is an error term that is less than $1$ in absolute value and also small by comparison wi... |
2,545,226 | <p>Suppose $a_n$ is a positive sequence but not necessarily monotonic. </p>
<p>For the series $\sum_{n=1}^\infty \frac{1}{a_n}$ and $\sum_{n=1}^\infty \frac{a_n}{n^2}$ I can find examples where both diverge: $a_n = n$, and where one converges and the other diverges: $a_n = n^2$.</p>
<p>Can we find example where both... | Reiner Martin | 248,912 | <p>No, by the Cauchy-Schartz inequality we have
$$
+\infty=\sum_{n=1}^\infty \frac{1}{n} = \sum_{n=1}^\infty \frac{1}{\sqrt{a_n}} \cdot \frac{\sqrt{a_n}}{n} \le \sqrt{\sum_{n=1}^\infty \frac{1}{a_n} \cdot \sum_{n=1}^\infty \frac{a_n}{n^2}}.
$$</p>
|
2,072,347 | <p>I was trying to solve this problem, but couldn't figure it out. The solution goes like this:</p>
<p><a href="https://i.stack.imgur.com/1KSWH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1KSWH.png" alt="http://www.tkiryl.com/Calculus/Problems/Section%201.4/Calculating%20Limits/Solutions/Calc_S_... | StephanCasey | 157,220 | <p>They were using the Squeeze theorem but I don't think it is necessary (<a href="https://www.khanacademy.org/math/differential-calculus/limits-from-equations-dc/squeeze-theorem-dc/v/proof-lim-sin-x-x" rel="nofollow noreferrer">https://www.khanacademy.org/math/differential-calculus/limits-from-equations-dc/squeeze-the... |
2,306,895 | <p>I want to find $Hom_{\mathtt{Grp}}(\mathbb{C}^\ast,\mathbb{Z})$, where $\mathbb{C}^\ast$ is the multiplicative group, and $\mathbb{Z}$ is additive.
$\mathbb{C}$ is the additive group of complex numbers. We have the following map: </p>
<p>$\large{\mathbb{C} \xrightarrow{exp} \mathbb{C}^\ast \xrightarrow{?} \mathbb{Z... | Joshua Ruiter | 399,014 | <p>This isn't a full answer, but I suspect that this Hom group may be the trivial group. Suppose $\phi:\mathbb{C}^* \to \mathbb{Z}$ is a group homomorphism. We know that $\phi(1) = 0$ since $1$ is the identity. Then
$$
\phi(-1)^2 = \phi(1) = 0 \implies 2 \phi(-1) = 0 \implies \phi(-1) = 0
$$
By a similar argument, any... |
4,077,917 | <p>If you have
<span class="math-container">$$
\int_0^2 \int_0^{\sqrt{4 - x^2}} e^{-(x^2 + y^2)} dy \, dx
$$</span>
and you convert to polar coordinates, you integrate from <span class="math-container">$0$</span> to <span class="math-container">$\pi/2$</span>) with respect to theta.</p>
<p>But, if you have
<span class=... | zkutch | 775,801 | <p>Let's construct formal proof. In first case we have set
<span class="math-container">$$\left\lbrace \begin{array}{l}0 \leqslant x \leqslant 2 \\
0 \leqslant y \leqslant \sqrt{4-x^2}
\end{array}\right\rbrace$$</span>
considering polar coordinates <span class="math-container">$x=r\cos \theta, y=r\sin \theta$</span> we... |
2,354,609 | <p>I have to approximate $\sqrt2$ using Taylor expansion with an error $<10^{-2}$.</p>
<p>I noticed that I can do MacLaurin expansion of $\sqrt{x+1}$ then put $x=1$</p>
<p>So: $$\sqrt{x+1}=1 + \dfrac{x}{2} - \dfrac{x^2}{8} + \dfrac{x^3}{16} + {{\frac1{2}}\choose{n+1}}x^{n+1}(1+\xi)^{-\frac1{2}-n}$$</p>
<p>I have ... | Christian Blatter | 1,303 | <p>The Taylor series of the function $f(x):=(1+x)^{1/2}$ just barely converges for $x:=1$. Evaluate the Taylor expansion of $g(x):=(1+x)^{-1/2}$ at $x:=-{1\over2}$ instead.</p>
|
3,389,542 | <blockquote>
<p><strong>Proposition.</strong> If <span class="math-container">$\text{Ran}(R) \subseteq \text{Dom}(S)$</span>, then <span class="math-container">$\text{Dom}(S \circ R) = \text{Dom}(R)$</span></p>
</blockquote>
<p>My attempt:</p>
<p>Suppose <span class="math-container">$\text{Ran}(R) \subseteq \text{D... | Trishan Mondal | 685,504 | <p><br>
Idk it is true or not ...
We can construct a complete bipartite graph which is balanced as early as possible .</p>
<p>Such construction is <em>per Tarun's construction</em>.so maximising x(G) can be done by finding maximal triangle free vertices.</p>
<p><em>Case1</em> if n(even) then we can construct bipari... |
4,065,797 | <p>Just to give a simple numerical example but in general the variables <span class="math-container">$x,y,z,u,v$</span> are not equal.</p>
<p><span class="math-container">$113= 2*4^2 + 2*4^2 +2*4^2 + 4^2 +1^2$</span></p>
<p>I am looking for a general method to solve this type of equation or a piece of software to do th... | Quanto | 686,284 | <p>Integrate by parts to obtain a recursive formula as follows</p>
<p><span class="math-container">\begin{align}
I_n&=\int_0^{\pi} \sin^{n}x \ln(\sin x) dx\\
&= -\int_0^{\pi} \sin^{n-1}x \ln(\sin x)\> d(\cos x)\>\\
& =\int_0^{\pi}((n-1) \sin^{n-2}x \cos^2x\ln(\sin x)+ \sin^{n-2}x\cos^2x)dx\\
&= (n... |
4,065,797 | <p>Just to give a simple numerical example but in general the variables <span class="math-container">$x,y,z,u,v$</span> are not equal.</p>
<p><span class="math-container">$113= 2*4^2 + 2*4^2 +2*4^2 + 4^2 +1^2$</span></p>
<p>I am looking for a general method to solve this type of equation or a piece of software to do th... | Igor Rivin | 109,865 | <p>Mathematica says:</p>
<p><span class="math-container">$$\fbox{$\frac{\sqrt{\pi } \left(H_{\frac{n-1}{2}}-H_{\frac{n}{2}}\right) \Gamma
\left(\frac{n+1}{2}\right)}{n \Gamma \left(\frac{n}{2}\right)}\text{ if }\Re(n)>-1$}.$$</span></p>
|
2,233,138 | <p>Let ${x_n}$ be defined by </p>
<p>$$x_n : = \begin{cases} \frac{n+1}{n}, &\text{if } n \text{ is odd}\\
0,&\text{if } n \text{ is even}.
\end{cases}$$</p>
<p>I am pretty sure about $\lim_{n\to\infty}\inf x_n = 0$ </p>
<p>because if ${x_1} = 2$, $x_2 = 0$, $x_3 = 4/3$, $x_4 = 0$ so $\lim_{n->\infty}\... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>Consider the vectors $\underbrace{(1,1,\dots,1)}_{n\;1\text{s}}$ and $\;(a_1, a_2,\dots,a_n)$.</p>
|
4,327,537 | <p>I have a question which states that
"In a group of 23 people what is the probability that there are two people with the same birthday? Assume there are 365 days in a year. Ignore leap years and such complications. Assume there is an equal probability of a person being born on each day of the year.". I solv... | Suzane | 901,114 | <p>A level curve of <span class="math-container">$f$</span> is a set of points <span class="math-container">$(x,y)$</span> satisfying <span class="math-container">$f(x,y)=c$</span> for some constant <span class="math-container">$c$</span>; here <span class="math-container">$c=0$</span>, but it could be any other value ... |
139,105 | <p>Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining disjoint (or at least never crossing) throughout?
(Imagine the arcs are made of rigid steel; but infinitely thin.)
The arc... | John Pardon | 35,353 | <p>I believe there is no such locked configuration. The proof is by induction, as you suggest.</p>
<p>Pick any arc and imagine moving it to infinity. Of course, to do this, it will have to pass through some other arcs, and thus this is not a valid motion. We can, however, by picking our motion "generically", ensure... |
3,712,256 | <p>I am trying to prove that: </p>
<blockquote>
<p>For nonempty subsets of the positive reals <span class="math-container">$A,B$</span>, both of which are bounded above, define
<span class="math-container">$$A \cdot B = \{ab \mid a \in A, \; b \in B\}.$$</span>
Prove that <span class="math-container">$\sup(A \cd... | Calum Gilhooley | 213,690 | <p><strong>Hint:</strong></p>
<p>Rather than <span class="math-container">$\sup A - \varepsilon$</span> and <span class="math-container">$\sup B - \varepsilon,$</span> subtract appropriate multiples of <span class="math-container">$\varepsilon$</span> from <span class="math-container">$\sup A, \sup B$</span> respectiv... |
2,117,054 | <p>Find all prime solutions of the equation $5x^2-7x+1=y^2.$</p>
<p>It is easy to see that
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$ or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$ </p>
<p>In the same way from $y^2+2x=1 \mod 5$ we have that $y^2=1 \mod 5$ and $x=0... | 1Emax | 324,326 | <p>Try working mod $3$ and mod $8$. Assuming $x, y>3$, we have $x,y = \pm 1$ mod $3$. Since $x, y$ are odd we have $x^2, y^2=1$ mod $8$, so
$$x^2, y^2 = 1 \text{ mod } 24.$$
Substituting in the equation gives $$x = 24k+11 $$ for some integer $k$.
Rearranging the original equation we get
$$x(5x-7)=(y-1)(y+1), \tag{1}... |
3,377,353 | <p>Given a purely real, rational integer <span class="math-container">$p$</span> that is prime in <span class="math-container">$\mathbb{Z}$</span>, we know very well that it ramifies in <span class="math-container">$\mathbb{Q}(\sqrt{pm})$</span> (where <span class="math-container">$m$</span> is a nonzero integer coprim... | Daniel Hast | 41,415 | <p>Here's the general situation for number fields: Let <span class="math-container">$L/K$</span> be a degree <span class="math-container">$n$</span> extension of number fields, let <span class="math-container">$\newcommand{\OO}{\mathcal{O}}\OO_K$</span> and <span class="math-container">$\OO_L$</span> be their rings of ... |
3,225,553 | <p>Show that <span class="math-container">$4x^2+6x+3$</span> is a unit in <span class="math-container">$\mathbb{Z}_8[x]$</span>.</p>
<p>Once you have found the inverse like <a href="https://math.stackexchange.com/questions/3172556/show-that-4x26x3-is-a-unit-in-mathbbz-8x">here</a>, the verification is trivial. But how... | Wuestenfux | 417,848 | <p>Hint: As in the hinted paper, a possible ansatz would be</p>
<p><span class="math-container">$(4x^2+6x+3) (ax+b) = 4ax^3+(4b+6a)x^2+ (6b+3a)x+3b=1$</span>.</p>
<p>This requires <span class="math-container">$4a\equiv 0\mod 8$</span> (so <span class="math-container">$a$</span> must be even), <span class="math-contai... |
1,473,318 | <blockquote>
<p>How many numbers can by formed by using the digits $1,2,3,4$ and $5$ without repetition which are divisible by $6$?</p>
</blockquote>
<p><strong>My Approach:</strong></p>
<p>$3$ digit numbers formed using $1,2,3,4,5$ divisible by $6$ </p>
<p>unit digit should be $2/4$ </p>
<p>No. can be $XY2$ &... | cr001 | 254,175 | <p>You have three sets {1,4}{2,5}{3}. For the exactly one from each set case, you already have the answer which is ${2\choose1}{1\choose1}2!+{2\choose1}{1\choose1}2!=8$.</p>
<p>For the one from first and one from second case, you have ${2\choose1}+{2\choose1}=4$</p>
<p>For the two from first and two from second case,... |
2,378,508 | <p>I am reading about Arithmetic mean and Harmonic mean. From <a href="https://en.wikipedia.org/wiki/Harmonic_mean#In_physics" rel="nofollow noreferrer">wikipedia</a>
I got this comparision about them:</p>
<blockquote>
<p>In certain situations, especially many situations involving rates and ratios, the harmonic mea... | Crazy | 449,016 | <p>I think you mixed up the degree of the differential equation and the degree of the polynomials.</p>
<p>Example,</p>
<p>$$\frac{d^3y}{dx^3}+\frac{dy}{dx}+y=0$$</p>
<p>is called a third order differential equation. The highest derivative inside the differential equation is $3$. So, it is a third order.</p>
<p>Cons... |
2,735,001 | <p>I was asked to find the corresponding series for the function $\ln(x^2+4)$</p>
<p>The obvious solution to me was to use the well known fact $$\ln(1+x)=\sum_{n=1}^\infty (-1)^{n-1}\frac{x^n}{n}$$
And substituting $x^2+3$ for $x$
$$\ln(1+(x^2+3))=\sum_{n=1}^\infty (-1)^{n-1}\frac{(x^2+3)^n}{n}$$
Using binomial theore... | Angina Seng | 436,618 | <p>The series
$$\ln(1+t)=\sum_{n=1}^\infty(-1)^{n-1}\frac{t^n}n$$
is only valid for $|t|<1$. You apply it for $t=x^2+3$. I don't
think $|x^2+3|<1$.</p>
|
1,557,015 | <p>This one looks simple, but apparently there is something more to it.
$$f{(x)=x^x}$$
I read somewhere that the domain is $\Bbb R_+$, a friend said that $x\lt-1, x\gt0$... </p>
<p>I'm really confused, because i don't understand why the domain isn't just all the real numbers.
According to any grapher online the domai... | Kamil Jarosz | 183,840 | <p>Split it into cases:</p>
<ol>
<li>When $x=p/q$ where $p\in \mathbb Z,q\in\mathbb N_{>1},p\ne0,\gcd(p,q)=1$, then:
$$x^x=\left(\frac{p}{q}\right)^\frac{p}{q}=\sqrt[q]{\left(\frac{p}{q}\right)^p}$$
<ul>
<li>when $p<0$ then
$$x^x=\sqrt[q]{\left(-\frac{q}{|p|}\right)^{|p|}}$$
if $p$ is even, then $\left(-\frac... |
270,985 | <p>A graph $G=(V,E)$ is said to be <em>vertex-critical</em> if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. <em>Edge-criticality</em> is defined in a similar manner. Moreover, $G$ is called <em>contraction-critical</em> if contracting any edge reduces the chromatic number.</p>
<p><em>Question... | Abdelmalek Abdesselam | 7,410 | <p>I think the graph studied in my article <a href="http://www.sciencedirect.com/science/article/pii/S0021869315005657" rel="nofollow noreferrer">"16,051 formulas for Ottaviani's invariant of cubic threefolds"</a> with Christian Ikenmeyer and Gordon Royle fits the bill. In the paper we considered a hypergraph on a set ... |
270,985 | <p>A graph $G=(V,E)$ is said to be <em>vertex-critical</em> if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. <em>Edge-criticality</em> is defined in a similar manner. Moreover, $G$ is called <em>contraction-critical</em> if contracting any edge reduces the chromatic number.</p>
<p><em>Question... | user1272680 | 90,417 | <p>Concerning your first question, every edge-critical graph without isolated vertices must be vertex-critical, but not vice versa. For instance, the complement of a $7$-cycle is vertex-critical but not edge-critical.</p>
<p>Concerning your second question, every vertex-critical graph must be contraction-critical as w... |
4,554,831 | <blockquote>
<p>Let <span class="math-container">$(X,d)$</span> be a metric space. Prove that if the point <span class="math-container">$x$</span> is on the boundary of the open ball <span class="math-container">$B(x_0,r)$</span> then <span class="math-container">$d(x_0,x)=r$</span>.</p>
</blockquote>
<p>I find this di... | Martin R | 42,969 | <p>Using the definition of a boundary point: For all <span class="math-container">$\epsilon > 0$</span> is <span class="math-container">$B(x,\epsilon)\cap B(x_0,r)\ne\emptyset$</span>, i.e. there is an <span class="math-container">$y \in X$</span> with
<span class="math-container">$$ \tag{$1$}
d(y, x) < \epsilon... |
3,080,124 | <p>Let <span class="math-container">$X$</span> be a topological space. Let <span class="math-container">$a\in X$</span>. Is it always true that <span class="math-container">$a$</span> is contained in a proper open set of <span class="math-container">$X$</span>? I don't know how to derive it directly by the axioms of a ... | Henno Brandsma | 4,280 | <p>No, this need not be the case: if <span class="math-container">$X$</span> is a set and <span class="math-container">$p \in X$</span> then the following defines a topology on <span class="math-container">$X$</span> (th excluded point topology w.r.t. <span class="math-container">$p$</span>):</p>
<p><span class="math-... |
2,005,555 | <p>When I was solving a DE problem I was able to reduce it to </p>
<p>$$e^x \sin(2x)=a\cdot e^{(1+2i)x}+b\cdot e^{(1−2i)x}.$$ </p>
<p>For complex $a,b$. Getting one solution is easy $(\frac{1}{2i},-\frac{1}{2i})$ but I was wondering what are all the values for complex $a,b$ that satisfy the equation. </p>
| snoram | 103,861 | <p>Going through it step by step without the use (of frankly very useful) shortcuts.</p>
<p>Start with:
$$e^x \sin(2x)=a\cdot e^{(1+2i)x}+b\cdot e^{(1−2i)x}$$</p>
<p>Clean up a bit and divide be $e^x$ on both sides:
$$ e^x \sin(2x)=a\cdot e^{x} e^{2xi}+b\cdot e^{x} e^{-2xi}$$
$$\sin(2x)=a\cdot e^{2xi}+b\cdot e^{-2... |
2,301,368 | <p>Below you see the <a href="https://en.wikipedia.org/wiki/Rhombicuboctahedron" rel="noreferrer">Rhombicuboctahedron</a>. If you put an additional point in the blue triangle, you make three blue triangles out of one. Now you connect a yellow square with two adjacent small blue triangle and you end up with a blue-yello... | zwim | 399,263 | <p>I think this has to be this since it has $12$ hexagons and $6$ squares as requested.</p>
<p><a href="https://en.wikipedia.org/wiki/Chamfer_(geometry)" rel="noreferrer">https://en.wikipedia.org/wiki/Chamfer_(geometry)</a></p>
<p><a href="https://i.stack.imgur.com/JGdGi.png" rel="noreferrer"><img src="https://i.stac... |
2,301,368 | <p>Below you see the <a href="https://en.wikipedia.org/wiki/Rhombicuboctahedron" rel="noreferrer">Rhombicuboctahedron</a>. If you put an additional point in the blue triangle, you make three blue triangles out of one. Now you connect a yellow square with two adjacent small blue triangle and you end up with a blue-yello... | lesath82 | 430,906 | <p>Here is your solid:</p>
<p><a href="https://i.stack.imgur.com/2wW66.png" rel="noreferrer"><img src="https://i.stack.imgur.com/2wW66.png" alt="enter image description here"></a></p>
<p>It's part of the family of the chamfered cubes, but I don't think it has a name on its own.</p>
|
2,881,914 | <p>Using a computer I found the double sum</p>
<p>$$S(n)= \sum_{j=1}^n\sum_{k=1}^n \frac{j^2 + jk + k^2}{j^2(j+k)^2k^2}$$
has values</p>
<p>$$S(10) \quad\quad= 1.881427206538142 \\ S(1000) \quad= 2.161366028875634 \\S(100000) = 2.164613524212465\\$$</p>
<p>As a guess I compared with fractions $\pi^p/q$ where $p,q$ ... | skbmoore | 321,120 | <p>$$S(\infty)=\sum_{j=1}^\infty\,\sum_{k=1}^\infty \frac{(j+k)^2 - jk}{j^2(j+k)^2k^2} = \underbrace{\Big(\sum_{k=1}^\infty \frac{1}{k^2}\Big)^2}_{=\zeta(2)^2} -
\underbrace{\sum_{j=1}^\infty\,\sum_{k=1}^\infty \frac{1}{j\,k}\int_0^\infty dt \,t \,e^{-t(j+k)}}_{:=U},$$
where the first step is algebra and the second is... |
2,881,914 | <p>Using a computer I found the double sum</p>
<p>$$S(n)= \sum_{j=1}^n\sum_{k=1}^n \frac{j^2 + jk + k^2}{j^2(j+k)^2k^2}$$
has values</p>
<p>$$S(10) \quad\quad= 1.881427206538142 \\ S(1000) \quad= 2.161366028875634 \\S(100000) = 2.164613524212465\\$$</p>
<p>As a guess I compared with fractions $\pi^p/q$ where $p,q$ ... | Jack D'Aurizio | 44,121 | <p>An alternative approach:</p>
<p>$$ S = \lim_{n\to +\infty}S(n) = \sum_{j,k\geq 1}\frac{1}{j^2 k^2}-\sum_{k,j\geq 1}\frac{1}{jk(j+k)^2}=\zeta(2)^2-\sum_{k,j\geq 1}\int_{0}^{+\infty}\frac{e^{-(j+k)x}}{jk}\,x\,dx $$
leads to
$$S = \zeta(2)^2-\int_{0}^{+\infty}x\log^2(1-e^{-x})\,dx=\frac{\pi^4}{36}+\int_{0}^{1}\frac{\l... |
2,303,163 | <blockquote>
<p>Let $T$ be a linear operator on the finite-dimensional space $V.$ Suppose there is a linear operator $U$ on $V$ such that $TU=I.$ Prove that $T$ is invertible and $U=T^{-1}.$</p>
</blockquote>
<p>Attempt: Let $\dim V=n$ and $\{\alpha_i\}_{i=1}^n$ a basis for $V$. We claim that $\{U(\alpha_i)_{i=1}^n\... | Ken Duna | 318,831 | <p>I suppose at this point you just need to show that $UT = I$ as well. You can use $TU = I$ and the fact that $T^{-1}$ exists for this:</p>
<p>\begin{align*}
TU &= I \\
TUT &= IT = T \\
T^{-1}TUT &= T^{-1}T \\
UT &= I
\end{align*}</p>
|
1,684,124 | <p>Here is my attempt:</p>
<p>$$ \frac{2x}{x^2 +2x+1}= \frac{2x}{(x+1)^2 } = \frac{2}{x+1}-\frac{2}{(x+1)^2 }$$</p>
<p>Then I tried to integrate it,I got $2\ln(x+1)+\frac{2}{x+1}+C$ as my answer. Am I right? please correct me if I'm wrong.</p>
| Travis Willse | 155,629 | <p><strong>Hint</strong> The factorization $(x + 1)^2$ of the denominator of the integrand suggests that we can rewrite the integral using the substitution $u := x + 1$, $du = dx$:
$$\int \frac{2 (u - 1)}{u^2} du = 2 \left(\int \frac{du}{u} - \int \frac{du}{u^2} \right) .$$</p>
|
511,304 | <p>Given the ODE: </p>
<p>$2(x+1)y' = y$</p>
<p>How can I solve that using Power Series? I started to think about it:</p>
<p>$
\\2(x+1)\sum_{n=1}^{\infty}{nc_nx^{n-1}}-\sum_{n=0}^{\infty}{c_nx^n}=0
\\2\sum_{n=1}^{\infty}{nc_nx^{n}}+2\sum_{n=1}^{\infty}{nc_nx^{n-1}}-\sum_{n=0}^{\infty}{c_nx^n}=0
\\\sum_{n=0}^{\infty}... | Sangchul Lee | 9,340 | <p>Note that</p>
<p>$$ \tanh x = 1 - \frac{2e^{-2x}}{1 + e^{-2x}} = 1 + O\left(e^{-2x}\right). $$</p>
<p>Thus </p>
<p>$$ \arctan\left(C^{-1} \tanh x \right) = \arctan\left( C^{-1} + O\left(e^{-2x}\right) \right) = \arctan(C^{-1}) + O\left(e^{-2x}\right). $$</p>
<p>This shows that</p>
<p>$$ \int_{0}^{\lambda} \arct... |
197,877 | <p>According to answer of Denis Serre to <a href="https://mathoverflow.net/questions/197773/a-geometric-property-of-singular-matrices">this question</a>, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$</p>
<p>So we define a (line b... | Alex Degtyarev | 44,953 | <p>No, this bundle is not trivial (starting from dimension $2$). Introduce a metric, consider projector to a hyperplane, and rotate this hyperplane through $\pi$ about an axis. You get an orientation reversing loop.</p>
|
90,112 | <p>When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am failing to prove it myself.</p>
<p>The result (slightly simplified) is</p>
<p>If $X$ is a uniformly convex space (i.e.... | Sergei Ivanov | 4,354 | <p>If the second $\delta(\varepsilon)$ is allowed to differ from the first one, then there is a simple implicit argument: Suppose the contrary, then there is a sequence $X_n$ of 2-dimensional normed spaces satisfying the definition with the same function $\delta(\varepsilon)$ and points $x_n,y_n\in X_n$ with $\|x_n\|\l... |
1,159,599 | <p>can someone give me a hint on how to calculate this integral?</p>
<p>$\int _0^{\frac{1}{3}} \frac{e^{-x^2}}{\sqrt{1-x^2}}dx$</p>
<p>Thanks so much!</p>
| Harry Peter | 83,346 | <p><span class="math-container">$\int_0^\frac{1}{3}\dfrac{e^{-x^2}}{\sqrt{1-x^2}}~dx$</span></p>
<p><span class="math-container">$=\int_0^{\sin^{-1}\frac{1}{3}}\dfrac{e^{-\sin^2x}}{\sqrt{1-\sin^2x}}~d(\sin x)$</span></p>
<p><span class="math-container">$=\int_0^{\sin^{-1}\frac{1}{3}}e^\frac{\cos2x-1}{2}~dx$</span></p... |
400,838 | <p>I need to find $$\lim_{x\to 1} \frac{2-\sqrt{3+x}}{x-1}$$</p>
<p>I tried and tried... friends of mine tried as well and we don't know how to get out of:</p>
<p>$$\lim_{x\to 1} \frac{x+1}{(x-1)(2+\sqrt{3+x})}$$</p>
<p>(this is what we get after multiplying by the conjugate of $2 + \sqrt{3+x}$)</p>
<p>How to proce... | Euler....IS_ALIVE | 38,265 | <p>Multiplying by the conjugate does indeed work. You just forgot to carry the negative sign throughout. After multiplying by the conjugate, the correct expression is $\frac{1-x}{(x-1)(2+\sqrt{3+x})}$</p>
|
1,775,649 | <p>True or false and explain why?: a matrix with characteristic polynomial $\lambda^3 -3\lambda^2+2\lambda$ must be diagonalizable.</p>
<p>First I found the lambda's that make this zero (eigenvalues) and got $0, 1, 2$ but I don't know if having $0$ as an eigenvalue means that the matrix is not diagonalizable? I know t... | Patrick Abraham | 337,503 | <p>Edit (better wording)</p>
<p>$0$ as an eigenvalue doesn't hinder the diagonalization, actually there is no eigenvalue that would hinder it.</p>
<p>That doesn't mean that every matrix is diagonalizable, but that the eigenvalues have no influence, at least in $\mathbb{C}$.</p>
<p>Let A be a complex $n x n$ matrix w... |
174,655 | <p>So I have 2 lists of 10000+ lists of 3 numbers, e.g.</p>
<pre><code>{{1,2,3},{4,5,6},{7,8,9},...}
{{2,1,3},{4,5,6},{41,2,0},...}
</code></pre>
<p>Wanting a result like </p>
<pre><code>{2,...}
</code></pre>
<p>Getting some sort of list of <code>True</code>/<code>False</code> is also probably enough, like this:</p... | kglr | 125 | <pre><code>a = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {1, 2, 3}};
b = {{2, 1, 3}, {4, 5, 6}, {41, 2, 0}, {1, 2, 3}};
Pick[Range@Length@a, Total /@ Unitize[Subtract[a, b]], 0]
</code></pre>
<blockquote>
<p>{2,4}</p>
</blockquote>
<p>Or using @Henrik's idea of using <code>Dot</code> in in the second argument of <code>Pic... |
174,655 | <p>So I have 2 lists of 10000+ lists of 3 numbers, e.g.</p>
<pre><code>{{1,2,3},{4,5,6},{7,8,9},...}
{{2,1,3},{4,5,6},{41,2,0},...}
</code></pre>
<p>Wanting a result like </p>
<pre><code>{2,...}
</code></pre>
<p>Getting some sort of list of <code>True</code>/<code>False</code> is also probably enough, like this:</p... | jkuczm | 14,303 | <p>If you need speed, you could use ugly compiled function:</p>
<pre><code>equalPosInt2 = Last@Compile[{{a, _Integer, 2}, {b, _Integer, 2}},
Module[{result, dimA, dimB, n, m, eq},
result = Internal`Bag@Most@{0};
dimA = Dimensions@a;
dimB = Dimensions@b;
n = Min[Compile`GetElement[dimA, 1], Compile`Ge... |
73,375 | <p>The example cells in the documentation each have a count of the cells inside their section:</p>
<pre><code> Cell[TextData[{"Basic Examples", " ", Cell["(4)", "ExampleCount"]}],
"ExampleSection", "ExampleSection"]
</code></pre>
<p>But this is static content, how exactly would this work dynamically? I'd like to ... | Kuba | 5,478 | <p>I think when it's done each time you save the notebook it should be nice enough :)</p>
<pre><code>SetOptions[
EvaluationNotebook[],
NotebookEventActions -> {
{"MenuCommand", "Save"} :> (Scan[
Module[{nr},
SelectionMove[#, All, CellGroup, AutoScroll -> False];
nr = Length @ Select... |
129,287 | <p>Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as</p>
<p>$q(x_1, x_2, \cdots, x_{n+1}) = u(x_1)p(x_2, x_3, \cdots, x_{n+1}) + u(x_2)p(x_1, x_3, \cdots, x_{n+1}) + \cdots \\ \phantom{q(x_1, x_2, \cdots, ... | Abdelmalek Abdesselam | 7,410 | <p>I don't know if the operation has a name in the context of the classical theory of symmetric functions. However, in mathematical physics this is essentially what is called a creation operator in a Boson Fock space.
See, e.g., Reed and Simon "Methods of Modern Mathemtatical Physics" vol 2, page 209, 1975 edition.</p>... |
3,210,295 | <p>I wondered if anybody knew how to calculate a percentage loss/gain of a process over time?</p>
<p>Suppose for example Factory A conducted activity over 6 periods.</p>
<p>In t-5, utilisation of resources was: 80%
t-4: 70%
t-3: 80%
t-2: 100%
t-1: 90%
t: 75%</p>
<p>Therefore, but for the exception of two periods ago... | Ross Millikan | 1,827 | <p>For each period, the loss is <span class="math-container">$100\%$</span> minus the utilization, so your losses are <span class="math-container">$20\%, 30\%, 20\%, 0\%, 10\%, 25\%$</span>. The total of these is <span class="math-container">$105\%$</span>, which means that in the six periods you have lost just over o... |
13,889 | <p><strong>Question:</strong> Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?</p>
<p><strong>Why do I care:</strong> For a number of math kids I know, doing algebraic topology is fi... | Paul VanKoughnett | 2,215 | <p>For simplicial/cellular cohomology, one way to think of it is in terms of dual cell structures: if $a_1,a_2,\dotsc$ are your $k$-simplices or $k$-cells, then they generate the $k$th chain group, and the $k$th cochain group is generated by their duals $\alpha_1,\alpha_2,\dotsc$, where $\alpha_i(a_j)=\delta_{ij}$. Yo... |
4,444,504 | <p>We have measure theory in this semester.I found the statement of Lusin's theorem on the internet to be:</p>
<blockquote>
<p>Let <span class="math-container">$f:\mathbb{R\to R}$</span> be a Lebesgue measurable function.Then for each <span class="math-container">$\epsilon>0$</span> there exists a closed set <span c... | Kishalay Sarkar | 691,776 | <p>Dave L. Renfro's comments and hints made me answer this question.First we prove the following:</p>
<blockquote>
<p>Let <span class="math-container">$f:\mathbb{R\to R}$</span> be measurable and <span class="math-container">$\epsilon>0$</span>,then there exists <span class="math-container">$E\subset \mathbb R$</spa... |
2,482,669 | <p>Find sum of the expression,
$$x^n+x^{n-1}y+x^{n-2}y^2+x^{n-3}y^3+\dots+xy^{n-1}+y^n$$
where $x,y$ are real numbers and $n$ is a natural number.</p>
| farruhota | 425,072 | <p>It is:
$$\frac{x^{n+1}-y^{n+1}}{x-y}.$$</p>
|
2,656,909 | <p>I recently begun to read Walter Rudin magnum upos "Principles of Mathematical Analysis" and i'm having a little trouble in understanding the proof of the the following statement:</p>
<p>2.41 Theorem: if a set $E$ in $R^k$ has one of the following three properties then it has the other two:</p>
<p>(a) $E$ is closed... | DonAntonio | 31,254 | <p>Answer to question: because either $\;\lim\limits_{n\to\infty} x_n\;$ doesn't exist or else it is $\;\pm\infty\;$ (one of the two). Either way, the limit is <strong>not</strong> in $\;\Bbb R^k\;$ .</p>
|
2,656,909 | <p>I recently begun to read Walter Rudin magnum upos "Principles of Mathematical Analysis" and i'm having a little trouble in understanding the proof of the the following statement:</p>
<p>2.41 Theorem: if a set $E$ in $R^k$ has one of the following three properties then it has the other two:</p>
<p>(a) $E$ is closed... | CopyPasteIt | 432,081 | <p>We have our set $S = \{x_n \;|\; n \ge 1\}$ with $|x_n| \gt n$.</p>
<p>Let $x$ be any point in $\Bbb R^k$ . Then $x$ is not a limit point of the set $S$.</p>
<p>To show this, first find an integer $N \ge 1$ so that $x$ is an interior point of the closed ball $B_N$ of radius $N$ about the origin (zero coordinates).... |
1,385,936 | <p><em>I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series.</em></p>
<p>The reason why I ask was because using this approximation, we can show that the sequence $(\cos(\pi{\sqrt{n^{2}-n}})_{n=1}^{\infty}$ converges to $0$. This done using a mean-value theorem or L... | Alex R. | 22,064 | <p>This requires nothing more than just the definition of a derivative. The function $f(x)=\sqrt{1+x}$ has derivative $f'(x)=\frac{1}{2\sqrt{1+x}}$ for $x\geq 0$. By the fundamental definition of derivatives:</p>
<p>$$f(x)-f(0)=f'(0)x+\epsilon(x),$$</p>
<p>where $\lim_{x\rightarrow 0^+}\epsilon(x)/x=0$ and $\lim_{x\... |
1,385,936 | <p><em>I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series.</em></p>
<p>The reason why I ask was because using this approximation, we can show that the sequence $(\cos(\pi{\sqrt{n^{2}-n}})_{n=1}^{\infty}$ converges to $0$. This done using a mean-value theorem or L... | Jack D'Aurizio | 44,121 | <p>For any $x>0$, $\sqrt{1+x}\leq 1+\frac{x}{2}$ is trivial by squaring. On the other hand:
$$ 1+\frac{x}{2}-\sqrt{1+x} = \frac{\frac{x^2}{4}}{1+\frac{x}{2}+\sqrt{1+x}}\leq\frac{x^2}{8+2x} $$
gives:</p>
<blockquote>
<p>$$ 1+\frac{x}{2}-\frac{x^2}{8+2x}\leq \sqrt{1+x} \leq 1+\frac{x}{2}-\frac{x^2}{8+4x}.$$</p>
</b... |
1,385,936 | <p><em>I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series.</em></p>
<p>The reason why I ask was because using this approximation, we can show that the sequence $(\cos(\pi{\sqrt{n^{2}-n}})_{n=1}^{\infty}$ converges to $0$. This done using a mean-value theorem or L... | Steven Alexis Gregory | 75,410 | <p>Try</p>
<p>$\sqrt{1+x} \approx 1 + \alpha x + \beta x^2 + O(x^3)$</p>
<p>$1 + x = 1 + 2 \alpha x + (\alpha^2 + 2 \beta)x^2 + O(x^3)$</p>
<p>$\alpha = \frac 12$, $\quad \beta = -\frac 18$</p>
|
208,883 | <p>Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.</p>
<p>A theorem of Tate tells us that we can always lift $\bar{\rho}$ to some $\rho: G_K \to GL_n(\mathbb{C})$. I am wondering if there is a lift $\rho$ whose kerne... | Jeremy Kahn | 8,252 | <p>I believe the problem is exactly this. A composition of $K$-quasiconformal maps is not necessarily $K$-quasiconformal, which makes them difficult to work with. And a locally quasiconformal map is not necessarily globally quasiconformal. Normally when you define a type of manifold in terms of a class of permitted ove... |
657,047 | <p>So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$?</p>
<p>Thanks in advance!</p>
| DryEraseMarker | 124,645 | <p>$$ a^n = b $$
$$ log_{a}b = n $$</p>
<p>Because the easily accessible <em>log</em> button on your calculator is probably <em>base 10</em> and <em>not base a</em>, you have to punch it in this way:</p>
<p>$$\frac {\log b} {\log a}$$</p>
<p>which will result in your answer, $n$.</p>
<p>If you have a TI-89 Titanium... |
289,923 | <p>As far as i know, both differential and gradient are vectors where their dot product with a unit vector give directional derivative with the direction of the unit vector. So what are the differences?</p>
| notmyname | 294,646 | <p>Essentially, and in an informal sense, it is the difference between the projection of the gradient onto the plane below the surface (this is the normal "gradient"), and a "risen" gradient which is embedded in the 3D surface.</p>
<p><strong>Note</strong>: Technically, the differential and gradient reduce to the same... |
868,943 | <p>Can you please tell me the sum of the seires</p>
<p>$ \frac {1}{10} + \frac {3}{100} + \frac {6}{1000} + \frac {10}{10000} + \frac {15}{100000} + \cdots $ </p>
<p>where the numerator is the series of triangular numbers?</p>
<p>Is there a simple way to find the sum?</p>
<p>Thank you.</p>
| Gerry Myerson | 8,269 | <p>$$S={1\over10}+{3\over100}+{6\over1000}+{10\over10000}+\cdots$$ $${S\over10}={1\over100}+{3\over1000}+{6\over10000}+\cdots$$ Subtracting, $${9S\over10}={1\over10}+{2\over100}+{3\over1000}+{4\over10000}+\cdots$$ Now do the same thing again, that is, divide by $10$ and subtract, to get $${81S\over100}={1\over10}+{1\ov... |
868,943 | <p>Can you please tell me the sum of the seires</p>
<p>$ \frac {1}{10} + \frac {3}{100} + \frac {6}{1000} + \frac {10}{10000} + \frac {15}{100000} + \cdots $ </p>
<p>where the numerator is the series of triangular numbers?</p>
<p>Is there a simple way to find the sum?</p>
<p>Thank you.</p>
| Mustafa Saad | 164,692 | <p>I thought I might add another derivation <em>(devised by me)</em>. This one is long and involves dissecting the sequence into its simplest terms.</p>
<blockquote>
<p>$1/10 + 3/100 + 6/1000 + \ldots$</p>
<p>$= 1/10 + (1+2)/100 + (1+2+3)/1000 + \ldots$ (from the definition of
triangular numbers.)</p>
<p... |
1,774,084 | <p>I think it is convergent to $1$ because as $n$ tends to $\infty$ , $1/\sqrt(n)$ tends to $0$. Is it true?</p>
<p>Thanks!</p>
| Will Jagy | 10,400 | <p>$$ n^{\left( \frac{1}{\log n} \right)} = e $$
$$ \lim_{n \rightarrow \infty} n^{\left( \frac{1}{ \log \log n} \right)} = \infty $$</p>
|
3,632,431 | <blockquote>
<p>Consider the function <span class="math-container">$f: \mathbb{N} \to \mathbb{N}$</span> defined by <span class="math-container">$f(x)=\frac{x(x+1)}{2}$</span>. Show that <span class="math-container">$f$</span> is injective but not surjective.</p>
</blockquote>
<p>So I started by assuming that <span ... | PrincessEev | 597,568 | <p>Suppose <span class="math-container">$a \ne b$</span> at that point. Then, without loss of generality, <span class="math-container">$b > a$</span>. But then <span class="math-container">$b^2 + b > a^2 + a$</span>. Why? Because squaring the inequality gives you <span class="math-container">$b^2 > a^2$</span... |
3,392,171 | <p>We have a partial fraction equation:
<span class="math-container">$$\frac{1}{x-5} +\frac{1}{x+5}=\frac{2x+1}{x^2-25}$$</span></p>
<p>I multiplied the equation by the common denominator <span class="math-container">$(x+5)(x-5)$</span> and got <span class="math-container">$0=1$</span>. Is this correct?</p>
| user | 505,767 | <p>Yes it is equivalent to</p>
<p><span class="math-container">$$\frac{2x}{x^2-25}=\frac{2x+1}{x^2-25}\iff 2x=2x+1$$</span></p>
<p>which indeed has not solutions for <span class="math-container">$x\in \mathbb R$</span>.</p>
|
223,955 | <p>How can we convert a list to an integer correctly? </p>
<p><strong>{5, 22, 4, 5} -> 52245?</strong></p>
<p>When I use the command <code>FromDigits</code> in Mathematica </p>
<pre><code>FromDigits[{5, 22, 4, 5}]
</code></pre>
<p>The result is incorrect, namely <strong>7245</strong></p>
| Mr.Wizard | 121 | <pre><code>FromDigits @ ToString @ Row @ {5, 22, 4, 5}
Head[%]
</code></pre>
<blockquote>
<pre><code>52245
Integer
</code></pre>
</blockquote>
|
1,043,090 | <p>A rectangle $ABCD$, which measure $9 ft$ by $12 ft$, is folded once perpendicular to diagonal AC so that the opposite vertices A and C coincide. Find the length of the fold. So I tried to fold a rectangular paper but there are spare edges. So the gray is my fold and I'm not sure if its in the middle of my diagonal ... | CiaPan | 152,299 | <p>Your <em>folding line</em> is perpendicular to the rectangle's diagonal, which is a hypotenuse of a right triangle with legs 9 and 12 feet, so the folding line itself is a hypotenuse of a right trianlge with legs $9\times\frac 9{12}$ and $12\times\frac 9{12}$ — so its length is $$\sqrt{\left(9\times\frac 9{12}\righ... |
1,324,062 | <p>Evaluate: </p>
<blockquote>
<p>$$\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h}$$</p>
</blockquote>
<p>Now one way would be using the Maclaurin expansion for $e^{2x}$</p>
<p>However, can we solve it using the definition of the derivative (perhaps considering $f(x)=e^x$)? Many thanks for your help! $$$$
EDIT: I forgo... | Ivo Terek | 118,056 | <p>$$\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h} = 2 \lim_{h \to 0}\frac{e^{2h}-1}{h} = 2\lim_{x \to 0}\frac{e^x-1}{x} = 2\cdot 1 = 2,$$ where I made the substituition $x = 2h$ just to make things easier for you to visualize. I used one of the fundamental limits along with the fact that $h \to 0 \iff x \to 0$.</p>
|
4,042,250 | <p>My idea is to use disjoint events and calculating the probability of getting at least two heads for each number rolled. For example, if I roll a 3, I would calculate the probability with the expression <span class="math-container">$(\frac{1}{6}) (\frac{1}{2})^3 \binom{3}{2} + (\frac{1}{6}) (\frac{1}{2})^3\binom{3}{3... | user | 293,846 | <p>You can compute this much simpler. The probability that you get not more than one head out of <span class="math-container">$n$</span> flips is <span class="math-container">$\frac{n+1}{2^n}$</span>. Therefore the probability in question is:
<span class="math-container">$$
\frac16\sum_{n=1}^6\left(1-\frac{n+1}{2^n}\ri... |
3,516,189 | <p>I've been struggling with the following exercise for quite some time already:</p>
<blockquote>
<p>Consider a linear space <span class="math-container">$\mathbb{V} = \mathcal{C}\left(\left[a, b\right]\right)$</span> and let <span class="math-container">$f_{1},\ldots, f_{n}$</span> be linearly independent functions... | Ben Grossmann | 81,360 | <p>Proceed by contrapositive. We suppose that for all <span class="math-container">$a\leq x_1 < \cdots < x_n \leq b$</span>,
<span class="math-container">$$
\det \begin{bmatrix}
f_{1}(x_{1}) & f_{1}(x_{2}) & \cdots & f_{1}(x_{n})\\
f_{2}(x_{1}) & f_{2}(x_{2}) & \cdots & f_{2}(x_{n}) \\
\... |
3,853,980 | <p>Let <span class="math-container">$A$</span> be an <span class="math-container">$n×n $</span> complex matrix such that the three matrices <span class="math-container">$A+I$</span> , <span class="math-container">$A^2+I $</span> , <span class="math-container">$ A^3+I$</span> are all unitary .Prove that<span class="mat... | StubbornAtom | 321,264 | <p>I am not sure of your logic for calculating <span class="math-container">$\operatorname E\left[X_1\max(X_1,X_2)\right]$</span>.</p>
<p>By definition, this is equal to</p>
<p><span class="math-container">\begin{align}
\operatorname E\left[X_1\max(X_1,X_2)\right]&=\iint x\max(x,y)f_{X_1,X_2}(x,y)\,\mathrm dx\,\mat... |
3,853,980 | <p>Let <span class="math-container">$A$</span> be an <span class="math-container">$n×n $</span> complex matrix such that the three matrices <span class="math-container">$A+I$</span> , <span class="math-container">$A^2+I $</span> , <span class="math-container">$ A^3+I$</span> are all unitary .Prove that<span class="mat... | G Cab | 317,234 | <p>A geometric approach (considering only the half square <span class="math-container">$0 \le X_1 \le X_2 \le 1$</span> because of symmetry)</p>
<p><a href="https://i.stack.imgur.com/Hof1j.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Hof1j.png" alt="Unif_max&sum_1" /></a></p>
<p>clearly shows ... |
56,103 | <p>A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?</p>
<p>Let $X$ be a nice space (eg a smooth manifold, or more generally a CW complex). The topological Picard group $Pic(X)$ is the set of i... | Andy Putman | 317 | <p>I noticed that someone voted this up today. Since this might indicate that someone else is interested in the answer, I thought I'd remark that Oscar Randal-Williams and I worked out a proof of this when I visited him earlier this year. A version of this proof can be found in Section 2.2 of my paper</p>
<p>The Pic... |
35,220 | <p>It is a basic result of group cohomology that the extensions with a given abelian normal subgroup <em>A</em> and a given quotient <em>G</em> acting on it via an action $\varphi$ are given by the second cohomology group $H^2_\varphi(G,A)$. In particular, when the action is trivial (so the extension is a central exten... | Torsten Ekedahl | 4,008 | <p>This looks like a (slightly) non-additive version of Grothendieck's theory of
"extensions panachées" (SGA 7/I, IX.9.3). There he considers objects (in some
abelian category) $X$ together with a filtation $0\subseteq X_1\subseteq
X_2\subseteq X_3=X$. In the first version he also fixes (just as one does for
extensions... |
4,348,455 | <p>Each digits of the decimal expansion of the integer <span class="math-container">$2022$</span> (this year) consists of <span class="math-container">$0$</span> or <span class="math-container">$2$</span> and also, each digits of the ternary expansion of the same integer <span class="math-container">$2022$</span> (whic... | Empy2 | 81,790 | <p>I think they will run out after a while. This is not a proof, just heuristics.<br />
There are <span class="math-container">$2^n$</span> of these numbers of length <span class="math-container">$n+1$</span>. One approach is to think of the base 3 version as a random set of digits. There will be about <span class="... |
3,852,362 | <p>Let <span class="math-container">$~X = \{(x,y)∈ℝ^2∶|x| ≤1,~|y|≤1\}$</span> and function <span class="math-container">$f : X →ℝ$</span> defined by <span class="math-container">$$f(x,y)=\dfrac{x\cos x + y \sin y}{x^2+y^2+\alpha}$$</span>where <span class="math-container">$\alpha\gt0$</span>, then the range of <span cl... | Kwin van der Veen | 76,466 | <p>The zero order hold discretization is easiest done in state space. The continuous state space model can be written as</p>
<p><span class="math-container">$$
\dot{x}(t) = A\,x(t) + B\,u(t-d), \tag{1}
$$</span></p>
<p>with <span class="math-container">$x$</span> the state, <span class="math-container">$u$</span> the i... |
283,360 | <p>Let $M$ be a simply connected topological 4-manifold with intersection form given by the E8 lattice. Does anyone know of examples of continuous self-maps of $M$ of degree 2 or 3? Or of degree any other prime for that matter?</p>
| Oscar Randal-Williams | 318 | <p>Such a map $f : M \to M$ of degree $d >0$ satisfies, with respect to the cup-product pairing $\langle -, - \rangle$,
$$\langle f^*(x), f^*(y) \rangle = d \langle x, y \rangle.$$</p>
<p>Conversely, I claim that any integer matrix $A$ satisfying
$$A^T E_8 A = d E_8$$
arises as $A = f^*$ for a map $f : M \to M$, ne... |
283,360 | <p>Let $M$ be a simply connected topological 4-manifold with intersection form given by the E8 lattice. Does anyone know of examples of continuous self-maps of $M$ of degree 2 or 3? Or of degree any other prime for that matter?</p>
| Will Sawin | 18,060 | <p>There are plenty of integer matrices $A$ with $A^T E_8 A = d E_8$, which give plenty of maps, as in Oscar's answer.</p>
<p>First, for $d=1$, these are the automorphisms of the $E_8$ lattice. There are <a href="https://en.wikipedia.org/wiki/E8_lattice#Symmetry_group" rel="nofollow noreferrer">696729600</a> of these.... |
623,709 | <p>I make the following conjecture: the function
$$
d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)}
$$
is a distance on $H$, where $H$ is a normed vector space or a Hilbert space, and $x, y \in H$ (the function $d$ is defined to be $0$ in the case $x=y=0$). Note that $d$ is scale invariant, i.e., $d(\lambda x, \lambda y)=... | Egor Maximenko | 118,806 | <p>The inequality purposed by BGA fails also in $(\mathbb{R}^2,\|\cdot\|_\infty)$,
but it seems to be true for the norms induced by inner products.</p>
<p>Hypothesis I: If $X$ is an inner product space and
$\|\cdot\|$ is the norm induced by the inner product,
then for every $x,y,z\in X$
$$\|x-y\|\,\|z\| \le \|x-z\|\,\... |
1,770,804 | <p>I am a high school student my maths teacher said that if $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d.\,$ Can someone give me a prove of this?</p>
| Community | -1 | <p>Let $x=0$, then $b=d$. So $ax+b=cx+b$. So $ax=cx$. Then let $x=1$ to get $a=c$</p>
|
2,342,051 | <p>I am totally new to statistics. I'm learning the basics.</p>
<p>I came upon this question while solving Erwin Kreyszig's exercise on statistics.
The problem is simple. It asks to calculate standard deviation after removing outliers from the dataset.</p>
<p>The dataset is as follows: 1, 2, 3, 4, 10.
What I did is, ... | Sahiba Arora | 266,110 | <p>$C$ is closed therefore $\mathbb R \setminus C$ is open. Let $x \in \mathbb R \setminus C$. Then there exists an open set $U$ such that $U \subseteq \mathbb R \setminus C$. Now, $f \equiv 0$ on U. Hence $f$ is differentiable at $x$. </p>
|
1,003,096 | <p>Let $G=(\mathbb{Q}-\{0\},*)$ and $H=\{\frac{a}{b}\mid a,b\text{ are odd integers}\}$.</p>
<ol>
<li>Show $H$ is a normal subgroup of $G$.</li>
<li>Show that $G/H \cong (\mathbb{Z},+)$</li>
</ol>
<p>I know that there are multiple definitions for normal subgroup and I am having a hard time to develop the proof for th... | Username Unknown | 62,874 | <p>$\textbf{Show that $G/H \cong (\mathbb{Z},+)$.}$</p>
<p>$\textbf{Proof:}$ We can construct a function $f:G/H \to (\mathbb{Z},+)$ to be able to show that $G/H \cong (\mathbb{Z},+)$. However we can notice that $f$ can be written as $f:2^kH \to k$. Why?</p>
<p>Note that $\mathbb{Q}=\{\frac{r}{s} | r,s \in \mathbb{Z},... |
3,106,574 | <p>Let <span class="math-container">$(a_n) _{n\ge 0}$</span> <span class="math-container">$a_{n+2}^3+a_{n+2}=a_{n+1}+a_n$</span>,<span class="math-container">$\forall n\ge 1$</span>, <span class="math-container">$a_0,a_1 \ge 1$</span>. Prove that <span class="math-container">$(a_n) _{n\ge 0}$</span> is convergent.<br>
... | Jean Marie | 305,862 | <p>Initial remarks : </p>
<p>a) In case of convergence to a limit <span class="math-container">$L$</span>, we would have <span class="math-container">$L^3+L=L+L$</span>, with solutions <span class="math-container">$L=-1,0,1$</span>. </p>
<p>b) We assume that, up to a switching operation, <span class="math-container">... |
1,295,453 | <p>In my assignment I have to calculate to following limit. I wanted to know if my solution is correct. Your help is appreciated:</p>
<p>$$\lim_{n \to \infty}n\cos\frac{\pi n} {n+1} $$</p>
<p>Here's my solution:</p>
<p>$$=\lim_{n \to \infty}n\cos \pi \frac{n} {n+1} $$</p>
<p>Since $\frac {n} {n+1}\to 1 $ and $\cos ... | Jan Eerland | 226,665 | <p>$$\lim_{n \to \infty}n\cos\left(\frac{\pi n} {n+1}\right) =$$</p>
<p>$$\lim_{n \to \infty}n\lim_{n \to \infty}\cos\left(\frac{\pi n} {n+1}\right) =$$
$$\lim_{n \to \infty}n\cos\left(\lim_{n \to \infty}\frac{\pi n} {n+1}\right)=$$
$$\lim_{n \to \infty}n\cos\left(\lim_{n \to \infty}\frac{\pi } {1+\frac{1}{n}}\right)=... |
73,277 | <p>Let $\boldsymbol{\theta}=(\theta_1,\ldots,\theta_m)$ be a vector of real numbers in $[-\pi,\pi]$. For $t\ge 0$, define
$$ f(t,\boldsymbol{\theta}) = \binom{m+t-1}{t}^{-1}
\sum_{j_1+\cdots+j_m=t} \exp(ij_1\theta_1+\cdots+ij_m\theta_m),$$
where the sum is over non-negative integers $j_1,\ldots,j_m$ with sum $t$.
Note... | Lucia | 38,624 | <p>We shall prove that
$$
f(t,{\theta}) \le \frac{m-1}{t+m-1} \min_{1\le j,k \le m} \frac{1}{|\sin((\theta_j-\theta_k)/2)|}.
$$
This shows that if the angles are not too close to each other, then the sum does get
small. </p>
<p>Suppose without loss of generality that the minimum in our bound occurs for $\theta_1$ a... |
3,910,345 | <p>Recently a lecturer used this notation, which I assume is a sort of twisted form of Leibniz notation:</p>
<p><span class="math-container">$$y\,\mathrm{d}x - x\,\mathrm{d}y \equiv -x^2\,\mathrm{d}\left(\frac{y}{x}\right)$$</span></p>
<p>The logic here was that this could be used as:</p>
<p><span class="math-container... | littleO | 40,119 | <p>Such arguments can always be rephrased to avoid treating <span class="math-container">$dx$</span> and <span class="math-container">$dy$</span>, etc, as individual "infinitesimal" quantities. (On the other hand, "infinitesimal intuition" is a powerful and intuitive way to derive calculus formulas,... |
2,280,203 | <p>How to transform the integral </p>
<p>$$\int _{0}^{\pi }\sin ^{2}\left( \psi \right) \sin \left( m\psi \right) d\psi $$</p>
<p>to </p>
<p>$$\int _{0}^{\pi }\left( \dfrac {1} {2}-\dfrac {1} {2}\cos 2\psi \right) \sin m\psi d\psi $$</p>
<p>What is the general method you need to solve trig questions like this. How ... | M.P | 441,344 | <p>Some other useful tools for basic trig. integrals: </p>
<p>$\sin(2x)=2 \sin (x)\cos (x)$<br>
$\cos(2x)=1-\cos^2(x)=1-2\sin^2(x)$<br>
and the most elementary one: $1=\sin^2(x)+\cos^2(x)$, </p>
<p>Trigonometric products to sums and identities for higher-exponent trigonometric functions are also handy, but harder t... |
3,995,986 | <p>Need help integrating:
<span class="math-container">$$\int _0^{\infty }\:\:\frac{6}{\theta}xe^{-\frac{2x}{\theta }}\left(1-e^{-\frac{x}{\theta }}\right)dx$$</span></p>
<p>I think I should multiply the <span class="math-container">$$xe^{-\frac{2x}{\theta }}$$</span> out and then use integration by parts but it is not... | Raffaele | 83,382 | <p><span class="math-container">$$I=\int_0^{\infty } \frac{6 x}{t} e^{-\frac{2 x}{t}} \left(1-e^{-\frac{x}{t}}\right) \, dx$$</span>
Set <span class="math-container">$$e^{-\frac{x}{t}}=u\to x=-t\log u;\;dx=-\frac{t}{u}$$</span>
<span class="math-container">$$I=\int_1^0-6 (1-u) u^2 \log u\left(-\frac{t}{u}\right)\,du=$$... |
351,030 | <p>for positive integer $n$, how can we show</p>
<p>$$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$</p>
<p>where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square free, and $0$ otherwise. Also, what is</p>
<p>$$ \sum_{d | n} \mu(d) \sigma (d) $$ where $\sigma(n)$ is t... | Norbert | 19,538 | <p>Since $L_\infty(\Omega)=L_1^*(\Omega)$ you need to show that for all $f\in L_1(\Omega)$
$$
\lim\limits_{n\to\infty}\langle A_n, f\rangle = \langle A, f\rangle\tag{1}
$$
where $A$ is the desired limit. In fact it is enough to check $(1)$ only for some functions $f\in S$, where $L_1(\Omega)=\overline{\mathrm{span}S}$.... |
133,418 | <p>Let $\langle R,0,1,+,\cdot,<\rangle$ be the standard model for R, and let S be a countable model of R (satisfying all true first-order statements in R). Is it true that the set 1,1+1,1+1+1,… is bounded in S? My intuition says "no", but I am yet to find a counter example. I read something about rational functions, ... | André Nicolas | 6,312 | <p>The (first-order) theory of real-closed fields is complete. So any real-closed field that has the desired properties (countable, non-Archimedean) will do. We can use devices from Model Theory. However, an <em>algebraically</em> natural approach is to start with the rational functions in $x$ with real algebraic coe... |
2,622,583 | <blockquote>
<p>Prove that if $f:\mathbb R \to \mathbb R$ is a measurable function and $f(x)=f(x+1)$ almost everywhere, then there exists a measurable function $g:\mathbb R \to \mathbb R$ with $f=g$ almost everywhere and $g(x)=g(x+1)$ for every $x \in \mathbb R$</p>
</blockquote>
<p>I'm trying to prove this by const... | ncmathsadist | 4,154 | <p>Let $E = \{x| f(x) \not= f(x+1)$; this has measure zero. Now let $Q$ be the union of all integer translates of $E$; this also has measure zero. Now define $g(x) = f(x)$ for $x\in Q$ and $0$ otherwise. </p>
|
306,461 | <p>Let $A = \{(x,y) \in\mathbb{R}^2: a \leq (x-c)^2+(y-d)^2 \leq b\}$ for given $a,b,c, d$ real numbers. I want to show that $A$ is path-connected.</p>
<p>How can I do that?</p>
<p>I know that every open subset of $\mathbb R^2$ that is connected is path connected. But this is obviously not open so I cannot use that. ... | Seirios | 36,434 | <p><strong>Hint:</strong> Translate $A$ so that you can suppose $c=d=0$ and use polar coordinates.</p>
|
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| XGS | 164,057 | <p>This should be pretty obvious. Take $\mathbb{R}$ (together with its equipped topology) for example. We have:
1. Since finite intersection of two open sets is open, it follows that $(1, 2) \cap (3, 4) = \emptyset$ must be open; The complement of $\emptyset$, which is $\mathbb{R}$, must be closed;<br>
2. Since any ... |
2,310,441 | <p>I consider the sequence of composite odd integers: 9, 15, 21, 25, 27, 33, 35, 41, ...</p>
<p>I observe that there are certain large gaps between the composite odd integers and this may contribute towards the solution.</p>
<p>So I start by considering some sums first:</p>
<p>9 + 9 + 9 = 27, 9 + 9 + 15 = 33. So thi... | Barry Cipra | 86,747 | <p>For every prime $p\gt3$, either $p-25$ or $p-35$ is divisible by $6$. (Both numbers are even, and $p-25\equiv p-1$ mod $3$ while $p-35\equiv p-2$ mod $3$.). If $p\ge53$, the difference ($p-25$ or $p-35$) is at least $18$, hence can be written in the form $6(a+b+1)=3(2a+1)+3(2b+1)$, with $a,b\ge1$. Thus every prime... |
480,727 | <p>If $$2^x=3^y=6^{-z}$$ and $x,y,z \neq 0 $ then prove that:$$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0$$</p>
<p>I have tried starting with taking logartithms, but that gives just some more equations.</p>
<p>Any specific way to solve these type of problems?</p>
<p>Any help will be appreciated.</p>
| Aroonalok | 142,139 | <p>My answer does not solve the problem at hand (because the existing solutions work fine) but it addresses an issue related to the condition $x,y,z \neq 0$.<br>
It may seem that $x = y = z = 0$ is the only solution to the above system of equations and by imposing the condition $x,y,z \neq 0$, we are driving even that ... |
1,029,650 | <p>In Four-dimensional space, the Levi-Civita symbol is defined as:</p>
<p>$$ \varepsilon_{ijkl } =$$
\begin{cases}
+1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\
-1 & \text{if }(i,j,k,l) \text{ is an odd permutation of } (1,2,3,4) \\
0 & \text{otherwise}
\end{cases}
</p>
<p>Let'... | Oscar Cunningham | 1,149 | <p>If you fix one of the indices of $\varepsilon_{ijkl}$ to be $4$ you get $\pm\varepsilon_{ijk}$ depending on wheather you fix an odd or an even positioned index. So $\varepsilon_{ijk4}=\varepsilon_{ijk}$ but $\varepsilon_{ij4k}=-\varepsilon_{ijk}$. To see why the signs come out this way, notice that when you substitu... |
386,649 | <p>If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that involved more than one digit?</p>
<p>For example, if we let $a_0$ represent $0$, and $a_n$ represent the number $n$ for any... | Foo Barrigno | 73,411 | <p>The <a href="http://en.wikipedia.org/wiki/Factorial_number_system" rel="nofollow">factorial number system</a> is one such system. Each place value has one more digit than the previous one. It also has the wonderful property that all rational numbers have a terminating factorial system representation.</p>
<p>In gene... |
565,046 | <blockquote>
<p>The center of $D_6$ is isomorphic to $\mathbb{Z}_2$.</p>
</blockquote>
<p>I have that
$$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$
$$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$
My method for trying to do this has been just checking elements that could be candid... | Ben West | 37,097 | <p>I wrote up a general classification for the centers of $D_n$, (the dihedral group of order $2n$, not $n$) just the other week. Perhaps it will be useful to read:</p>
<p>If $n=1,2$, then $D_n$ is of order $2$ or $4$, hence abelian, and $Z(D_n)=D_n$. Suppose $n\geq 3$. We have the presentation
$$
D_n=\langle x,y:x^2=... |
565,046 | <blockquote>
<p>The center of $D_6$ is isomorphic to $\mathbb{Z}_2$.</p>
</blockquote>
<p>I have that
$$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$
$$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$
My method for trying to do this has been just checking elements that could be candid... | Kevin Maguire | 336,007 | <p>It does not give proofs, but given the tone of the original question a good graphical tool for the OP would be Group Explorer. It does a lot of the donkey work, and can show you various visualisations, including the multiplication tables in helpful ways.</p>
<p><a href="http://groupexplorer.sourceforge.net/" rel="n... |
1,373,103 | <p>I was wondering if $|f(x)g(x)| = |f(x)| |(g(x)|$ is true all the time as in the case of real numbers.</p>
<p>I was not convinced enough that that was true.</p>
<p>But I can't think of any counterexample.</p>
<p>Thank you.</p>
| Bernard | 202,857 | <p>No, you have to solve $\;\lvert T_5(x)-\cos x\rvert\le 0.003406$.</p>
<p><em>Hint:</em> $\cos x$ is defined by an alternating series, so you have information on the error when you truncate the series at a given order.</p>
|
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