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 |
|---|---|---|---|---|
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | David G. Stork | 9,735 | <pre><code>MemberQ[myeigens = Normalize/@Eigenvectors[h], Normalize[y]]|| MemberQ[myeigens, -Normalize[y]]
</code></pre>
<p>(* False *)</p>
|
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | AccidentalFourierTransform | 34,893 | <p>You already have several good answers. An alternative is to use a <a href="https://en.wikipedia.org/wiki/Rayleigh_quotient" rel="noreferrer">Rayleigh quotient</a>,</p>
<pre><code>r = First[y.h.ConjugateTranspose[{y}]/Norm[y]];
</code></pre>
<p>The vector <code>y</code> is an eigenvector of <code>h</code> if and on... |
2,080,042 | <blockquote>
<p>I am interested of finding examples of non-zero homomorphisms $f:R\to S$ of rings with unity such that $f(1_R)\neq 1_S$.</p>
</blockquote>
<p>I will provide one example and I will be glad if others can also give examples. </p>
| Juniven Acapulco | 44,376 | <p>Let $R=\mathbb{Z}$ and $S=\mathbb{Z}\times \mathbb{Z}$. Define $f:R\to S$ by $f(n)=(n,0)$ for all $n\in\mathbb{Z}$. Then $f$ is a nonzero homomorphism but
$$f(1_R)=f(1)=(1,0)\neq (1,1)=1_S.$$</p>
|
2,354,036 | <p>$\require{AMScd}\def\colim{\text{colim}}$I need this result in less generality, but I'd be happy to know this stronger version holds.</p>
<p>Let $\{\alpha_c : Fc \to Gc\}$ be arrows in a category $D$, indexed by the objects of a category $C$, for two functors $F,G: C\to D$. </p>
<p>Let $A\subseteq C$ be a dense su... | Younesse Kaddar | 581,427 | <p>$$
\newcommand{\colim}{\mathop{\rm colim}\nolimits}
\newcommand{\Nat}{\mathop{\rm Nat}\nolimits}
\newcommand{\cancom}[1]{\big((i/#1)\stackrel{\mathrm{pr}_}{→} \stackrel{i}{→} ℂ\big)}
$$</p>
<p>First, I thank <a href="http://www.cs.ox.ac.uk/people/ohad.kammar/main.html" rel="nofollow noreferrer">Ohad Kammar</a>, <a... |
438,166 | <p>Let $X$ be a bounded connected open subset of the $n$-dimensional real Euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support in $X$. </p>
<p>Does the closure of this operator generate a strongly continuous semigroup on $C_0(X)$ endowed with t... | Community | -1 | <p>The closure of the Laplacian on $C^\infty_0(X)$
cannot generate a strongly continuous semigroup on $C(X)$,
since there are two distinct (closed!) generators that extend $(\Delta, C^\infty_0(X))$; namely
the Dirichlet Laplacian and the Neumann Laplacian. Probabilistically,
these correspond to absorbing and reflecti... |
148,185 | <p>Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on which to use van Kampen. </p>
<p>Any help would be appreciated. Th... | Simon Markett | 30,357 | <p>I am not sure whether there is a nicer choice but this is how I think about it. Intuitively the fundamental group should be $\mathbb Z$ - a path may jump through the hoop a couple of times or not. I choose the open sets to model this somewhat. One open set is the interiour of a filled torus with the circle lying on ... |
23,471 | <p>I'm trying to find an explanation for the different sizes I'm seeing for fonts added to graphics in different ways, and haven't yet located an easy to understand explanation. Here's a minimal example:</p>
<pre><code>Graphics[
{LightGray,
Rectangle[{0, 0}, {72, 72}],
Red,
Style[
Text["Hig", {0, 0... | Alexey Popkov | 280 | <p>As it is pointed out in my old answer <a href="https://stackoverflow.com/a/6124065">here</a>, graphics and text are displayed inside of the FrontEnd in the style environment defined by the <code>ScreenStyleEvironment</code> option while are <code>Export</code>ed into <code>"PDF"</code> in the style environment defin... |
3,760,253 | <blockquote>
<p>The diagram shows the line <span class="math-container">$y=\frac{3x}{5\pi}$</span> and the curve <span class="math-container">$y=\sin$</span>
<span class="math-container">$x$</span> for <span class="math-container">$0\le x\le \pi$</span>.</p>
<p>Find (as an exact value) the enclosed area shown shaded in... | Henry | 6,460 | <p>If <span class="math-container">$L=1-x+{x}^2-\cdots$</span> converges for <span class="math-container">$|x|<1$</span></p>
<p>then <span class="math-container">$xL= x-x^2+{x}^3-\cdots$</span> also converges for <span class="math-container">$|x|<1$</span></p>
<p>and <span class="math-container">$L+xL = 1$</span>... |
2,604,093 | <p>I would like to study the convergence of the series:</p>
<p>$$\sum_{n=1}^\infty \frac{\log n}{n^2}$$</p>
<p>I could compare the generic element $\frac{\log n}{n^2}$ with $\frac{1}{n^2}$ and say that
$$\frac{1}{n^2}<\frac{\log n}{n^2}$$ and $\frac{1}{n^2}$ converges but nothing more about.</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>Let use <a href="https://en.wikipedia.org/wiki/Convergence_tests#Limit_comparison_test" rel="nofollow noreferrer">Limit comparison test</a> test with $$\frac{1}{n^{\frac32}}$$</p>
<p>Related OP with a more general discussion <a href="https://math.stackexchange.com/questions/2586107/det... |
3,968,905 | <p>I am trying to prove this:</p>
<p><span class="math-container">$\bullet$</span> Prove that <span class="math-container">$\Delta(\varrho_\epsilon \star u) = \varrho_\epsilon \star f $</span> in the sense of distributions, if <span class="math-container">$\Delta u = f$</span> in the sense of distributions, <span class... | dan_fulea | 550,003 | <p>We write the given equation equivalently:
<span class="math-container">$$
\begin{aligned}
0 &= x^2y^2 - 2x^2y + 2xy^2 + x^2 - 4xy + y^2 + 2x - 2y - 3\ ,\\
0 &= x^2(y-1)^2 + 2x(y-1)^2 + (y-1)^2-4\ ,\\
4 &= (x+1)^2(y-1)^2\ .
\end{aligned}
$$</span>
Now consider all possible ways to write <span class="math-... |
76,163 | <p>Represent the position of a unit-length, oriented segment $s$ in the plane
by the location $a$ of its <em>basepoint</em> and
an orientation $\theta$: $s = (a,\theta)$. So $s$ can be
viewed as a point in $\mathbb{R^2} \times \mathbb{S^1}$.
Now I'll define a metric on this space.
Define the distance $d(s_1,s_2)$ betw... | Anton Petrunin | 1,441 | <p>Let us start with the metric on $\mathbb R^4=\mathbb R^2\times \mathbb R^2$
defined by the norm $\|{*}\|$ defined by
$$\|(x,y)\|=\int_0^1|t\cdot x+(1-t)\cdot y|\,dt,$$
where $|{ * }|$ denotes the Euclidean norm on $\mathbb R^2$.
This norm is not strongly convex, so you should expect many geodesics between close poi... |
76,163 | <p>Represent the position of a unit-length, oriented segment $s$ in the plane
by the location $a$ of its <em>basepoint</em> and
an orientation $\theta$: $s = (a,\theta)$. So $s$ can be
viewed as a point in $\mathbb{R^2} \times \mathbb{S^1}$.
Now I'll define a metric on this space.
Define the distance $d(s_1,s_2)$ betw... | Sergei Ivanov | 4,354 | <p>In addition to Anton Petrunin's answer, here is a trick to simplify (and in some sense solve) the geodesic equation.</p>
<p>Since the metric has three-dimensional group of isometries (generated by rigid motions of the plane), the corresponding <a href="http://en.wikipedia.org/wiki/Noether%2527s_theorem" rel="norefe... |
813,825 | <p>In strong Induction for the induction hypothesis you assume for all K, p(k) for k
<p>If for example I am working with trees and not natural numbers can I still use this style of proof?</p>
<p>For example if I want my induction hypothesis to be that p(k) for k < n where n is a node in the tree and everything sma... | vadim123 | 73,324 | <p>Induction, and strong induction, are used to prove statements that are indexed by the natural numbers $\mathbb{N}=\{1,2,3,\ldots\}$. $k$ and $n$ need to be natural numbers, with a minimum value ($0$ or $1$ are the usual choices). The <em>statements</em> need not be natural numbers, for example, the first statemen... |
1,177,988 | <p>Comparing the equation
$$x^4+3x+20=0$$<br>
With the equation
$$(x^2+\lambda)^2-(mx+n)^2=0$$
we get </p>
<p>$m^2=2\lambda,$</p>
<p>$-2mn=3,$<br>
$n^2=\lambda^2-20$ </p>
<p>Now, $4m^2n^2=9\Rightarrow 4(2\lambda)(\lambda^2-20)=9\Rightarrow 8\lambda^3-160\lambda-9=0$. </p>
<p>How can I find easily the value... | Luigi D. | 164,401 | <p>The cubic equation is in depressed form (i.e., its quadratic coefficient is 0). To find the first root, you can use Cardano's formula $$\lambda_1=\sqrt[3]{-{q\over 2}+ \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}} +\sqrt[3]{-{q\over 2}- \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}$$ where $p = -20$ and $q = -\frac{9}{8}$.</p>
|
1,177,988 | <p>Comparing the equation
$$x^4+3x+20=0$$<br>
With the equation
$$(x^2+\lambda)^2-(mx+n)^2=0$$
we get </p>
<p>$m^2=2\lambda,$</p>
<p>$-2mn=3,$<br>
$n^2=\lambda^2-20$ </p>
<p>Now, $4m^2n^2=9\Rightarrow 4(2\lambda)(\lambda^2-20)=9\Rightarrow 8\lambda^3-160\lambda-9=0$. </p>
<p>How can I find easily the value... | Bernard | 202,857 | <p>You can't find them easily, as the discriminant of the equation is standard form:
$$x^3-20x+\frac 98=0$$
is $ 4\cdot (-20)^3+ 27\cdot\Bigl(\dfrac 98\Bigr)<0$.</p>
<p>In such a case, we know the equation has $3$ real toots, but Cardano's formula requires using complex numbers since the square root is that of a ... |
183,243 | <p>Gödel's incompleteness theorem states that: "<em>if a system is consistent, it is not complete.</em>" And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc.</p>
<p>However, why does this mean that ZF is consistent? What does "<em>relatively consistent</em>" actually mean?</p>
| user642796 | 8,348 | <p>We have to hedge quite a bit in what you said. First, there are many complete theories: the theory of all groups of order 7 is complete, as an example. However, Gödel showed that any "nice enough" (read: "recursively enumerable") consistent first-order theory capable of encoding basic arithmetic is incomplete. (T... |
481,173 | <p>The most common way to find inverse matrix is $M^{-1}=\frac1{\det(M)}\mathrm{adj}(M)$. However it is very trouble to find when the matrix is large.</p>
<p>I found a very interesting way to get inverse matrix and I want to know why it can be done like this. For example if you want to find the inverse of $$M=\begin{b... | Agustí Roig | 664 | <p>Because when you are performing Gauss reduction and elimination to</p>
<p>$$
(A \ \vert \ I) \ ,
$$</p>
<p>what your are doing in fact is finding the solutions of all these linear systems:</p>
<p>$$
AX_1 = e_1 , \dots , AX_n = e_n \ .
$$</p>
<p>Aren't you?</p>
<p>So, the matrix formed by the columns $X_1, \dot... |
488,258 | <p>What are the last two digits of $11^{25}$ to be solved by binomial theorem like $(1+10)^{25}$?
If there is any other way to solve this it would help if that is shown too.</p>
| Mher | 80,548 | <p>$$11^{25} = (1+10)^{25} = \dbinom{25}{0} + 10\cdot\dbinom{25}{1} + \sum_{k=2}^{25}\dbinom{25}{k}\cdot10^k =$$
$$ 1 + 5\cdot10 + 2\cdot 10^2 + \sum_{k=2}^{25}\dbinom{25}{k}\cdot10^k =$$
$$\underline{1} + \underline{5}\cdot10 + 10^2\cdot \left(2+\sum_{k=2}^{25}\dbinom{25}{k}\cdot10^{k-2}\right), $$
hence the last digi... |
640,554 | <p>For the system
$$
\left\{
\begin{array}{rcrcrcr}
x &+ &3y &- &z &= &-4 \\
4x &- &y &+ &2z &= &3 \\
2x &- &y &- &3z &= &1
\end{array}
\right.
$$
what is the condition to determine if there is no solution or unique solution or infinite solut... | shariva | 431,031 | <p>There is an easier way to determine whether a system of equations has unique, infinite or no solution. It is as follows: calculate determinant <span class="math-container">$D$</span> of the coefficients of the three variables in three equations, then calculate <span class="math-container">$Dx$</span>, where the x co... |
244,492 | <p>Find $m \in \mathbb R$ for which the equation $|x-1|+|x+1|=mx+1$ has only one unique solution. When does a absolute value equation have only 1 solution?</p>
<p>I solved for $x$ in all 4 cases and got $x=\frac{1}{-m-2},x=\frac{1}{2-m},x=\frac{1}{m},x=-\frac{3}{m}$</p>
| Martin Argerami | 22,857 | <p>You have
<span class="math-container">$$
|x-1|+|x+1|=\begin{cases} 2x,&\text{ if }x\geq1 \\2,&\text{ if }x\in(-1,1)\\ -2x,&\text{ if }x<-1\end{cases}
$$</span>
so
<span class="math-container">$$
g(x)=|x-1|+|x+1|-mx-1=\begin{cases} 2x-mx-1,&\text{ if }x\geq1 \\2-mx-1,&\text{ if }x\in(-1,1)\\ -2... |
90,263 | <p>Let $\mathcal{E} = \lbrace v^1 ,v^2, \dotsm, v^m \rbrace$ be the set of right
eigenvectors of $P$ and let $\mathcal{E^*} = \lbrace \omega^1 ,\omega^2,
\dotsm, \omega^m \rbrace$ be the set of left eigenvectors of $P.$ Given any two
vectors $v \in \mathcal{E}$ and $ \omega \in \mathcal{E^*}$ which correspond to
t... | deinst | 943 | <p>Mathematics may be universal, but learning styles are not. The relationship between teacher and student vary vastly between (and even within) cultures. Your job is to get knowledge from your brain to theirs. If this were a one size fits all problem, you would be replaced by a You-tube video.</p>
|
129,295 | <p>$$\int{\sqrt{x^2 - 2x}}$$</p>
<p>I think I should be doing trig substitution, but which? I completed the square giving </p>
<p>$$\int{\sqrt{(x-1)^2 -1}}$$</p>
<p>But the closest I found is for</p>
<p>$$\frac{1}{\sqrt{a^2 - (x+b)^2}}$$ </p>
<p>So I must add a $-$, but how? </p>
| Mike | 17,976 | <p>The main obstacle here is the square root. It is likely that eliminating it will allow us to proceed. So we want something squared minus 1 is the square of something. That leaves secant (or cosecant) as the best option.</p>
<p>As N3buchadnezzar states, it is not the only option. Consider the formulas</p>
<p>$(... |
84,711 | <p>This is a homework question I was asked to do</p>
<p>Of a twice differentiable function $ f : \mathbb{R} \to \mathbb{R} $ it is given that $f(2) = 3, f'(2) = 1$ and $f''(x) = \frac{e^{-x}}{x^2+1}$ . Now I have to prove that $$ \frac{7}{2} \leq f\left(\frac{5}{2}\right) \leq \frac{7}{2} + \frac{e^{-2}}{... | Arturo Magidin | 742 | <p>Using a local linear approximation (that is, a degree 1 Taylor polynomial approximation), we have that
$$f(x) \approx f(2) + f'(2)(x-2) = x+1.$$</p>
<p>Using the Lagrange Error Bound (with $n=1$) we have that
$$\left| f(x) - (x+1)\right| \leq \frac{M}{2!}|x-2|^2$$
where $\max|f''(x)|\leq M$ on the inter... |
1,422,990 | <p>How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$?</p>
<p>If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.</p>
| Spenser | 39,285 | <p>If $(2^n-1)^{1/n}=a/b$ for $a,b\in\Bbb Z$ and $n\ge 3$, then
$$a^n+b^n=(2b)^n,$$
contradicting <a href="https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem" rel="nofollow">Fermat's Last Theorem</a>.</p>
|
1,892 | <p>Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.</p>
<p>Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conv... | T.. | 467 | <p>If $P = NP$, computational revolution (once a specific algorithm is identified for an NP-hard problem, with explicit asymptotic runtime bounds).</p>
<p>If $P < NP$ <em>and one can prove it</em>, secure (classical) cryptography provably exists, and a huge missing piece in our understanding of computation is fille... |
4,277,616 | <blockquote>
<p>In how many different ways can we arrange <span class="math-container">$120$</span> students into <span class="math-container">$6$</span> groups for <span class="math-container">$6$</span> different classes so that the largest group has at most <span class="math-container">$2$</span> members more than t... | Asinomás | 33,907 | <p>Let <span class="math-container">$l$</span> be the size of the smallest group, we get <span class="math-container">$6l \leq 120 \leq 6l+5\times 2$</span>. Hence we must have <span class="math-container">$6l = 120$</span> or <span class="math-container">$6l= 114$</span>.</p>
<p>If <span class="math-container">$6l = ... |
4,277,616 | <blockquote>
<p>In how many different ways can we arrange <span class="math-container">$120$</span> students into <span class="math-container">$6$</span> groups for <span class="math-container">$6$</span> different classes so that the largest group has at most <span class="math-container">$2$</span> members more than t... | Robert Shore | 640,080 | <p>You can do this a smidgen more simply (at least, it's simpler to me).</p>
<p>First, the average size of a group is <span class="math-container">$20$</span>. If any group is smaller than average, then some other group must be larger than average, which means that there must be a group of size at least <span class="m... |
2,995,408 | <blockquote>
<p><span class="math-container">$$
\lim_{x\to 2^-}\frac{x(x-2)}{|(x+1)(x-2)|}=
\lim_{x\to 2^-}\left(\frac{x}{|x+1|}\cdot \frac{x-2}{|x-2|}\right)
$$</span></p>
</blockquote>
<p>So as the title says, is it okay to separate function under absolute value like this (i.e In form of Products) as shown in the... | Ross Millikan | 1,827 | <p>Yes, in general <span class="math-container">$|ab|=|a||b|$</span>. You can check all the sign combinations to justify it.</p>
|
1,281,627 | <p>Today I completed the chapter of '**Limits **' in my school, and I found this chapter very fascinating. But the only problem I have with limits and Derivatives is that I don't know How can I use it in my daily life. (Any Book Recommendation?)</p>
| drawnonward | 51,530 | <p>A great way to see applications of the derivative is to consider real life functions, and look at the units you get when you apply Newton's Quotient.</p>
<p>As an example, lets say you have a velocity function based on time, then if you apply Newton's Quotient, you will see you are left with $m/s^2$(acceleration), ... |
1,281,627 | <p>Today I completed the chapter of '**Limits **' in my school, and I found this chapter very fascinating. But the only problem I have with limits and Derivatives is that I don't know How can I use it in my daily life. (Any Book Recommendation?)</p>
| Jesse P Francis | 45,937 | <p>You might have calculated $\lim\limits_{n\to\infty}\frac{1}{n}$ during your course work ($=0$)- what does it mean? Does it mean $\frac{1}{n}=0$ for some n?</p>
<p>In other words, $\lim\limits_{n\to\infty}\frac{1}{n}=0$ is a neat and accurate way of saying, as the value of n gets bigger, $\frac{1}{n}$ is almost near... |
110,722 | <p>1) Many Mathematics departments ask to send a "list of publications" while applying for research postdoctoral jobs. My question is: how important is it to post my papers in arXiv. I know, posting on arXiv is always good, because people might search for the arXiv -ed papers, but how much difference is publication on ... | Per Alexandersson | 1,056 | <p>You should put the preprints on arxiv. This allows for other researchers to find your work, and to cite it.
In extreme cases, it can take <strong>2-3 years to get a a referee report</strong> after submission, and then perhaps another year before finally getting published.
I am not kidding - I am speaking of personal... |
2,178,714 | <p>Let $ f: (-1,1) \rightarrow \mathbb{R}$ be a bounded and continuous function . Prove that the function $ g(x)=(x^{2}-1)f(x) $ is uniformly continuous on $ (-1,1)$ . $$ $$ My little approach is, Since $f$ is bounded on $(-1,1)$ , there is positive $M \in \mathbb{R}$ such that </p>
<p>$$\forall x \in (-1,1)\,|f(x)|... | Adren | 405,819 | <p>Given $\epsilon>0$, there exists $a\in(-1,0)$ and $b\in(0,1)$ such that :</p>
<p>$$\forall x\in(-1,a],\,|g(x)|\le\epsilon$$</p>
<p>and similarly :</p>
<p>$$\forall x\in[b,1),\,|g(x)|\le\epsilon$$</p>
<p>(this is because $\lim_{x\to\pm 1}g(x)=0$)</p>
<p>By Heine theorem, $g$ is uniformly continuous on $[a,b]$... |
173,131 | <p>Let's suppose that for the following expression:</p>
<p>$\qquad \alpha\,\beta +\alpha+\beta$</p>
<p>I know that $\alpha$ and $\beta$ are of small magnitude (e.g., 0 < $\alpha$ < 0.02 and 0 < $\beta$ < 0.02). Therefore, the magnitude of $\alpha\,\beta$ is negligible, i.e., the original expression can be... | Henrik Schumacher | 38,178 | <p>You could use first order Taylor expansion, e.g. with</p>
<pre><code>f = α β + α + β;
(f /. {α -> 0, β -> 0}) + (D[f, {{α, β}, 1}] /. {α -> 0, β ->0}).{α, β}
</code></pre>
<blockquote>
<p>$\alpha +\beta$</p>
</blockquote>
<p>For your second example (with typos fixed), I obtain</p>
<pre><code>numera... |
3,765,398 | <p>The range of <span class="math-container">$\alpha$</span> for which all the points of local extrema of the function <span class="math-container">$f\left( x \right) = {x^3} - 3\alpha {x^2} + 3\left( {{\alpha ^2} - 1} \right)x + 1$</span> lie in the interval (–2, 4), is</p>
<p>(A) (–1, 3)</p>
<p>(B) (3, 4)</p>
<p>(C)... | Orenio | 783,418 | <p>Solve the quadratic equation with the parameter <span class="math-container">$\alpha$</span>:</p>
<p><span class="math-container">$x_{1,2}=\frac{6\alpha\pm\sqrt{({36\alpha^2-36\alpha^2+36})}}{6}$</span></p>
<p>Meaning <span class="math-container">$x_1=\alpha+1, x_2=\alpha-1$</span></p>
<p>Can you finish solving from... |
3,765,398 | <p>The range of <span class="math-container">$\alpha$</span> for which all the points of local extrema of the function <span class="math-container">$f\left( x \right) = {x^3} - 3\alpha {x^2} + 3\left( {{\alpha ^2} - 1} \right)x + 1$</span> lie in the interval (–2, 4), is</p>
<p>(A) (–1, 3)</p>
<p>(B) (3, 4)</p>
<p>(C)... | Quanto | 686,284 | <p>Note that</p>
<p><span class="math-container">\begin{align}
f'( x ) & = 3{x^2} - 6\alpha x + 3\left( {{\alpha ^2} - 1} \right)\\
&= 3[x^2 -2ax + (a+1)(a-1)] \\
& = 3[x-(a+1)][x-(a-1)]=0
\end{align}</span></p>
<p>which leads to the roots <span class="math-container">$x=a\pm 1$</span>. Then, solve <span c... |
2,214,236 | <p>The question:</p>
<blockquote>
<p>An object is dropped from a cliff. How far does the object fall in the 3rd second?"</p>
</blockquote>
<p>I calculated that a ball dropped from rest from a cliff will fall $45\text{ m}$ in $3 \text{ s}$, assuming $g$ is $10\text{ m/s}^2$.</p>
<p>$$s = (0 \times 3) + \frac{1}{2}\... | Narasimham | 95,860 | <p>Your teacher is correct. Question asks how much distance is covered between $t= 2$ and $ t= 3.$ Time lapse is 1 second, that is, in the <em>third second of duration</em>. In meters, distance travelled =</p>
<p>$$ s = \frac12 \cdot 10\cdot (3^2-2^2) = 25, $$ </p>
<p>and, if you draw the parabola graph, $s_2-s_1 = a... |
236,933 | <p>When I use ListPlot, I want to show the labels, such as</p>
<pre><code>ListPlot[Callout[#, #, Above] & /@ Range[10], Joined -> True, Mesh -> All]
</code></pre>
<p>Now all positions are <strong>Above</strong>, but sometimes the labels will over other text, so I want to set some of them <strong>Below</strong... | cvgmt | 72,111 | <p>One way is use <code>MapAt</code></p>
<pre><code>data = Range[10];
mapBelow[n_] := MapAt[Callout[#, #, Below] &, n];
mapAbove[n_] := MapAt[Callout[#, #, Above] &, n];
below = Table[mapBelow[n], {n, {2, 3, 8}}];
above = Table[mapAbove[n], {n, Complement[Range[10], {2, 3, 8}]}];
ListPlot[Composition[Sequence @... |
30,918 | <p>Judging by some of the posts on meta<sup>1</sup> and comments posted there it seems that there are users who try to improve the posts by correcting spelling mistakes. Of course, there are other ways to improve the posts via editing, some of them probably more important than grammar and spelling.<sup>2</sup> Still th... | Martin Sleziak | 8,297 | <p>Here are some words which have been misspelled in some posts on the site, also some SEDE queries are included.<sup>1</sup></p>
<ul>
<li>alegbra; <a href="//math.stackexchange.com/search?tab=active&q=alegbra">search</a></li>
<li>analisis; <a href="//math.stackexchange.com/search?tab=active&q=analisis">search... |
3,960,282 | <p>I am trying to find <span class="math-container">$z$</span> such that
<span class="math-container">$$\dot{z} = -1 + e^{-iz^*},$$</span>
where <span class="math-container">$*$</span> denotes complex conjugate and the dots represent derivatives with respect to time. The time dependence of the dependent variables is su... | JJacquelin | 108,514 | <p><span class="math-container">$$\begin{cases}
\frac{dx}{dt}=-1-e^{-y}\cos(x) \\
\frac{dy}{dt}=e^{-y}\sin(x)
\end{cases}\quad\implies\quad
\frac{dy}{dx}=\frac{e^{-y}\sin(x)}{-1-e^{-y}\cos(x)}$$</span>
<span class="math-container">$$\left(1+e^{-y}\cos(x) \right)dy+e^{-y}\sin(x)dx=0$$</span>
<span class="math-container"... |
1,940,446 | <p>I'm working through Bona's "A Walk Through Combinatorics" and I came across this problem:</p>
<blockquote>
<p>A company has $20$ employees, $12$ male and $8$ female. How many ways are
there to form a $5$ person committee that contains at least one male and
at least one female?</p>
</blockquote>
<p>I realise ... | barak manos | 131,263 | <p>The error in the other method is double-counting.</p>
<p>For example, the following combinations are essentially identical:</p>
<ul>
<li>$M_1+W_1,M_2+M_3+M_4$</li>
<li>$M_2+W_1,M_1+M_3+M_4$</li>
<li>$M_3+W_1,M_1+M_2+M_4$</li>
<li>$M_4+W_1,M_1+M_2+M_3$</li>
</ul>
|
2,530,820 | <p>Let $1\leq p<\infty$ and $q$ be the conjugate exponent of $p$. Suppose that $\lbrace a^n \rbrace_{n=1}^{\infty} \subset \ell^q$ in a sequence in $\ell^q$ such that $f_{a^n}(x) \mapsto 0~( n \mapsto \infty)$ for all $x \in \ell^p$ where $f_{a^n}(x)=\sum_{i=1}^{\infty} a_i^{(n)}x_i$. Show that the sequence $\lbrace... | Martin Argerami | 22,857 | <p>You have a sequence that converges weakly to zero, and you want to conclude that it is bounded. What you need is to use the <a href="https://en.wikipedia.org/wiki/Uniform_boundedness_principle" rel="nofollow noreferrer">Uniform Boundedness Principle</a>. </p>
|
1,232,420 | <p>Consider the ideal $I = (13x+16y, 11x+13y)$ in the ring R = $\mathbb{Z}[x,y].$</p>
<p>Prove that $I=(x-2y, 3x+y)$ by using mutual inclusion.</p>
<p>I'm confused on how to start...do I begin by multiplying the elements in the ideal by a general element of R?</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Use basis transformations to triangularize the basis, i.e. to the form $\,(x+cy, dy)$</p>
<p>$(a,b) = (x\!-\!2y,3x\!+\!y) = (x\!-\!2y,\overbrace{7y}^{b-3a})$</p>
<p>$(13x\!+\!16y,11x\!+\!13y) = (\underbrace{2x\!+\!3y}_{a-b},11x\!+\!13y) = (2x\!+\!3y,\underbrace{x\!-\!2y}_{b-5a}) = (\und... |
15,480 | <p>Say I have two lists,</p>
<pre><code>list1 = {a, b, c}
list2 = {x, y, z}
</code></pre>
<p>and I want to map a function f over them to produce</p>
<pre><code>{f[a,x], f[a,y], f[a,z], f[b,x], f[b,y], f[b,z], f[c,x], f[c,y], f[c,d]}
</code></pre>
<p>I would assume I map the function over the first list to produce a... | Sascha | 4,597 | <p>Or you could use <code>Tuples</code>, which appears a bit more natural to me.</p>
<pre><code>Tuples[{{a, b, c}, {x, y, z}}]
</code></pre>
<p>creates</p>
<pre><code>{{a, x}, {a, y}, {a, z}, {b, x}, {b, y}, {b, z}, {c, x}, {c, y}, {c, z}}
</code></pre>
<p>Afterwards <code>Apply</code> can be used to apply your fun... |
2,223,039 | <p>Consider an urn containing 90 balls numbered from 1 to 90, plus 3 balls attached with 3 distinct (known) numbers still from 1 to 90, say 1,2,3. I'm trying to find the probability that at least two equal numbers are extracted after 5 extractions without replacement.
I've tried two approaches which give different answ... | Stella Biderman | 123,230 | <p>You should approach this using the following formula, which is valid for all complex numbers $z$</p>
<p>$$\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$$</p>
<p>We also have the cooresponding formula for sine</p>
<p>$$\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$$</p>
<p>These can both be proven by using $e^{iz}=\cos(z)+i\sin(x)$ and s... |
2,223,039 | <p>Consider an urn containing 90 balls numbered from 1 to 90, plus 3 balls attached with 3 distinct (known) numbers still from 1 to 90, say 1,2,3. I'm trying to find the probability that at least two equal numbers are extracted after 5 extractions without replacement.
I've tried two approaches which give different answ... | The_Sympathizer | 11,172 | <p>To explain the deeper answer to the "why" of why it will happen is first off to point out that one <em>cannot</em> and <em>should not</em> expect that the behavior of at least analytic complex functions should mirror that on the real number line.</p>
<p>In particular, one very important theorem of complex analysis ... |
2,462,297 | <p>Let $(x_1,...,x_n)$ be real numbers and M be an $n \times n$ matrix whose its column is given by the entries $x_i,x_i^2, x_i^3,...x_i^n$. Compute the determinant of M.</p>
<p>I computed the formula for the determinant of M in terms of $x_1...x_n$ but I wonder if I can find a specific value.</p>
| Tsemo Aristide | 280,301 | <p>Hint: use this determinant formula</p>
<p><a href="https://en.wikipedia.org/wiki/Vandermonde_matrix" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Vandermonde_matrix</a></p>
<p>take $x_{n+1}=0$ and develop the Vandermonde $(n+1)\times (n+1)$-determinant relatively to the last line.</p>
|
899,249 | <p>I got this problem from my friend. I have been doing it for hours.</p>
<p>$a_1 = 2$</p>
<p>$a_{n+1} = 2a^2_n+1$</p>
<p>$a_n = ?$</p>
<p>Could you please tell me how to solve this? Thanks!</p>
<p>BTW: I failed to solve it by using Mathematica <code>RSolve[{a[1] == 2, a[n + 1] == 2 a[n]^2 + 1}, a[n], n]</code></p... | Claude Leibovici | 82,404 | <p>Assuming, as Will Jagy suspected, that the problem is $${ a_1 = 2, \; \; a_{n+1} = 2 a_n^2 - 1}$$ using a CAS I obtained, after some manipulations and simplifications, the surprizing form $$a_n=\cosh\Big(2^{n-1}\log(2+\sqrt 3)\Big)$$ or $$a_n=\frac{1}{2} \left(2+\sqrt{3}\right)^{-2^{n-1}}
\left(1+\left(2+\sqrt{3}... |
529,861 | <p>If $m,n$ are coprime positive integers and $m-n$ is odd, then $(m-n),(m+n),m,n,$ are coprime each other?</p>
<p>How do I prove it?</p>
<p>Especially how do I prove $(m-n), (m+n)$ are coprime?</p>
| Prahlad Vaidyanathan | 89,789 | <p>Hints :</p>
<ol>
<li><p>Show that it is a decreasing sequence bounded below.</p></li>
<li><p>Notice that $a_0 = x$, and $a_k = f(a_{k-1})$ where $f(y) = xy$. By (1), we know that $L = \lim a_k$ exists. Now
$$
a_k \to L \Rightarrow f(a_k) \to f(L)
$$
since $f$ is continuous. However, $f(a_k) = a_{k+1}$, and so the t... |
1,606,978 | <p>What is the value for $\lim \limits _{x\to\infty} \frac{\sin x} x$? </p>
<p>I solved it by expanding $\sin x$ as</p>
<p>$$\sin x = x - \frac {x^3} {3!} \dotsc$$</p>
<p>So $\lim \limits _{x\to\infty} \frac {\sin x} x = 1 -\infty = - \infty$,</p>
<p>but the answer is $0$. Why? What I am doing wrong?</p>
| Community | -1 | <p>Yes , the answer is $0$ . </p>
<p>One way to see this is by using the inequality :</p>
<p>$$\left |\frac{\sin x}{x}\right | \leq \frac{1}{x}$$ when $x>0$ (this happens because $|\sin x\ | \leq 1$ )</p>
<p>When $x \to \infty $ we have $\frac{1}{x} \to 0$ so the limit must be $0$ .</p>
|
1,606,978 | <p>What is the value for $\lim \limits _{x\to\infty} \frac{\sin x} x$? </p>
<p>I solved it by expanding $\sin x$ as</p>
<p>$$\sin x = x - \frac {x^3} {3!} \dotsc$$</p>
<p>So $\lim \limits _{x\to\infty} \frac {\sin x} x = 1 -\infty = - \infty$,</p>
<p>but the answer is $0$. Why? What I am doing wrong?</p>
| johnnyb | 298,360 | <p>The range of $\sin(x)$ will always be a value between -1 and 1, no matter what the input. However, there is no such restriction on the denominator. Therefore, if your numerator is restricted to a finite value, and your denominator is not, as the denominator goes to infinity the value of the whole expression will g... |
2,986,515 | <p>Can anyone help me with this problem? </p>
<p>Prove that for any real number <span class="math-container">$x > 0$</span> and for any <span class="math-container">$M > 0$</span> there is <span class="math-container">$N ∈ \mathbb N$</span> so that if <span class="math-container">$n > N$</span> then <span cla... | Seth | 610,132 | <p>If <span class="math-container">$\cfrac{3n^2-9n+6}{n^3+5n^2+8n+4}=0$</span>, then <span class="math-container">$3n^2-9n+6=0$</span>, or <span class="math-container">$3(n-2)(n-1)=0$</span>, so since the denominator is always positive and the poynomial is positive for <span class="math-container">$x \notin (1,2)$</spa... |
1,087,015 | <p>I'm looking for a polynomial $P(x)$ with the following properties:</p>
<ol>
<li>$P(0) = 0$.</li>
<li>$P\left(\frac13\right) = 1$</li>
<li>$P\left(\frac23\right) = 0$</li>
<li>$P'\left(\frac13\right) = 0$</li>
<li>$P'\left(\frac23\right) = 0$</li>
</ol>
<p>From 1 and 3 we know that $P(x) = x\left(x - \frac23\right)... | Lucian | 93,448 | <p><strong>Hint:</strong> $P'(a)=0$ implies $\Big(a,P(a)\Big)$ is either among the extrema, or an inflection point. My advice for you would be to draw a simple graphic. It is clear that we area dealing with something that is at least a cubic. Also, the addition of any other root or maxima or inflection points does not ... |
1,087,015 | <p>I'm looking for a polynomial $P(x)$ with the following properties:</p>
<ol>
<li>$P(0) = 0$.</li>
<li>$P\left(\frac13\right) = 1$</li>
<li>$P\left(\frac23\right) = 0$</li>
<li>$P'\left(\frac13\right) = 0$</li>
<li>$P'\left(\frac23\right) = 0$</li>
</ol>
<p>From 1 and 3 we know that $P(x) = x\left(x - \frac23\right)... | Brady Gilg | 188,927 | <p>This is the same solution as Ross's, but from a more linear algebra focused perspective.</p>
<p>You have five linearly independent conditions, so a polynomial with five parameters should work.</p>
<p>$P(x) = ax^4 + bx^3 + cx^2 + dx + e$</p>
<p>Now your five conditions can be written as such:</p>
<ol>
<li>$e = 0$... |
2,900,454 | <p>There are so many different methods I've found on SE and through Matlab, and they're all giving me different results.</p>
<p>Specifically, I have {v1} = (1,2,1) and {v2} = (2,1,0) in set S. What is the method to find {v3} vectors that are orthogonal to both v1 and v2?</p>
<p>I'm preparing for a final and I'm tryin... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<p>Your answer should be a non-zero scalar multiple of the $v_3$ that you provided since $v_1$ and $v_2$ are not parallel to each other.</p>
<p>Method $1$:</p>
<ul>
<li>Compute the cross product of $v_1$ and $v_2$, that will give you a valid solution.</li>
</ul>
<p>Method $2$:</p>
<ul>
<li>Solve the ... |
744,034 | <p>How do I show that for all integers $n$, $n^3+(n+1)^3+(n+2)^3$ is a multiple of $9$?
Do I use induction for showing this? If not what do I use and how? And is this question asking me to prove it or show it? How do I show it? </p>
| DeepSea | 101,504 | <p>Use the identity: $a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) + 3abc$,</p>
<p>and substituting $a = n$, $b = n + 1$, and $c = n + 2$ into the equation to get:</p>
<p>$n^3 + (n+1)^3 + (n+2)^3 = (3n + 3)(n^2 + (n+1)^2 + (n+2)^2 - n(n+1) - (n+1)(n+2) - n(n+2)) + 3n(n+1)(n+2) = 3(n+1)(n^2 + n^2 + 2n... |
3,623,368 | <p>for example this equation: 505x-673y=1 . x=4 and y = 3. but how can I find them with mathematics. What would be the approach here?</p>
| Jean Marie | 305,862 | <p>This was a comment, but I think deserves to be an answer.</p>
<p>As your question is asking for a method, here it is.</p>
<ul>
<li><p>1) First, divide by the common factor <span class="math-container">$d=GCD(a,b,c)$</span>.</p></li>
<li><p>2) Then apply "extended Euclidean algorithm" which will give you coefficie... |
2,901,783 | <p>I am having trouble solving a multi part question.</p>
<p>Express $ \frac x{x^2-3x + 2} $ in the partial fraction form.</p>
<p>The answer I got was $\frac2{x-2}-\frac1{x-1}$ .</p>
<p>The problem comes when they asked:</p>
<p>Show that, if $x$ is so small that $x^3$ and higher powers of $x$ can be neglected, then... | R zu | 587,462 | <p>Just cross out terms with $x^3$, $x^4$, ... in the approximation.</p>
<p>The question assumes these terms are too small, and we don't care.</p>
<p>For example, if I know x is between 0 and 0.1. Then $x^3$ is at most 0.001. $x^4$ is even smaller. </p>
<p>I think the sum of the terms $x^3$, $x^4$, ... are at least ... |
574,041 | <p>Consider a set of linear equations described by $A\vec{X}=\vec{B}$ is given, where $A$ is an $n\times n$ matrix and $\vec{X}$ and $\vec{B}$ are n-row vectors. Also suppose that this system of equations have a unique solution and this solution is given.</p>
<p>Imagine a new set of linear equations $A'\vec{X}=\vec{B}... | Community | -1 | <p>Changing a single element corresponds to a special rank one update of the form
$$(a'(i,j)-a(i,j))e_i e_j^T$$where $e_k$ is a vector with zero except at location $k$, which is one. You can use <a href="http://en.wikipedia.org/wiki/Woodbury_matrix_identity" rel="nofollow">Sherman-Morrison-Woodbury</a> to construct the... |
3,878,174 | <p>a) Prove that = {<span class="math-container">$_{,}$</span> | , ∈ ℝ, ≠ 0} is a group, where the operation is composition.<br />
Let 1 = {<span class="math-container">$\big(\begin{smallmatrix}
a & 0\\
b & 1
\end{smallmatrix}\big)$</span> ∶ , ∈ ℝ, ≠ 0}, where the operation is matrix multiplication, ... | User203940 | 333,294 | <p>We need to show that <span class="math-container">$G$</span> is closed under composition. Let <span class="math-container">$f_{a,b}, f_{c,d} \in G$</span>. Show that <span class="math-container">$f_{a,b} \circ f_{c,d} \in G$</span>.</p>
<p><em>Hint:</em> We see <span class="math-container">$f_{a,b} \circ f_{c,d}(x) ... |
3,878,174 | <p>a) Prove that = {<span class="math-container">$_{,}$</span> | , ∈ ℝ, ≠ 0} is a group, where the operation is composition.<br />
Let 1 = {<span class="math-container">$\big(\begin{smallmatrix}
a & 0\\
b & 1
\end{smallmatrix}\big)$</span> ∶ , ∈ ℝ, ≠ 0}, where the operation is matrix multiplication, ... | Mateus Figueiredo | 674,120 | <p>Let <span class="math-container">$a,b,c,d\in\mathbb{R}$</span> be arbitrary elements, <span class="math-container">$a,d\neq 0$</span>. Observe that for all <span class="math-container">$x\in \mathbb{R}$</span>, <span class="math-container">$$f_{a,b}\circ f_{c,d}(x)=f_{a,b}(f_{c,d}(x))=f_{a,b}(cx+d)=a(cx+d)+b=(ac)x+a... |
2,945,367 | <p><a href="https://i.stack.imgur.com/MGzHc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/MGzHc.png" alt="enter image description here"></a></p>
<p>We were given a couple formulas, but the one that immediately stood out to me was the Vfinal = Vinitial + at</p>
<p>so we know the patrol will consta... | Paramanand Singh | 72,031 | <p>You just need to consider the Riemann sum of <span class="math-container">$f$</span> over partition <span class="math-container">$P''=\{x_i, y, x_{i+1}\} $</span> of <span class="math-container">$[x_i, x_{i+1}]$</span>. By <span class="math-container">$(a) $</span> we have <span class="math-container">$$U(f, P'')\le... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Hong Liu | 13,033 | <p>I think fibre bundles should be introduced to give a modern viewpoint of tensor analysis.</p>
|
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Deane Yang | 613 | <p>I'm not sure if the original question is about a one semester or year course.</p>
<p>If this is the first course the students have ever had in differential geometry, then I still agree with Anton that at least the first semester should be about only 2-dimensional manifolds embedded in $R^3$ and Gauss-Bonnet. The po... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Spencer | 4,281 | <p>This is in agreement with Igor's comment on Anton's answer, but became too long.</p>
<p>I'd say whatever approach you ultimately take, for a first-year grad course it surely has to be done 'properly', i.e. starting from intrinsic definition of a smooth manifold and using the 'modern' language and general definition... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | John Klein | 8,032 | <p>I do have one addition to make to the above. At our university we usually use a combination of Guillemin and Pollack and Milnor. There is another approach at a first course which some have found useful: Bott and Tu's book, </p>
<pre><code> Differential forms in algebraic topology
</code></pre>
<p... |
195,832 | <p>I want to download the content of the website(contains text) and only a few lines from the content (from a specific number of the line up to the last line minus specific offset). Unfortunately, I do not know how to get the number of line in the content. For example, I want to replace 59 with the length of the conten... | amator2357 | 61,985 | <pre><code>length = ToExpression[ToString[StringCount[#,"\n"] & /@ FullForm[data1]]]+1
Snippet[data1,138;;length]
</code></pre>
<blockquote>
<p>This page was last edited on 27 January 2019, at 05:26 (UTC) . Text \
is available under the Creative Commons Attribution-ShareAlike \
License ; additional terms may... |
294,519 | <p>The problem I am working on is:</p>
<p>Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.</p>
<p>a) $∀x(C(x)→F(x))$ </p>
<p>b)$∀x(C(x)∧F(x))$</p>
<p>c) $∃x(C(x)→F(x))$ </p>
<p>d)$∃x(C(x)∧F(x))$</p>
<h2>-----------------------... | Damien L | 59,825 | <p>This is because the mathematical language is more accurate than the usual language. But your answers are right and they are the same as the book.</p>
|
857,801 | <p>"If in the obvious equalities $(k+1)^3−k^3=3k^2+3k+1$, for the different values $k=1,2,…,n−1$, we add the left and the right sides separately, we obtain the equation $n^3−1=3σ_n+\frac{3(n−1)n}{2}+n−1$, where $σ_n=1^2+2^2+…+(n−1)^2$."</p>
<p>I'm stuck trying to understand what the author has done in the paragraph be... | Sebastian Garrido | 62,734 | <p>He performed a summation over the given value of k on each side. You may want to look into series to get a clearer idea.</p>
|
857,801 | <p>"If in the obvious equalities $(k+1)^3−k^3=3k^2+3k+1$, for the different values $k=1,2,…,n−1$, we add the left and the right sides separately, we obtain the equation $n^3−1=3σ_n+\frac{3(n−1)n}{2}+n−1$, where $σ_n=1^2+2^2+…+(n−1)^2$."</p>
<p>I'm stuck trying to understand what the author has done in the paragraph be... | Mathmo123 | 154,802 | <p>It's a concise way of explaining the following:</p>
<p>We wish to calculate $\displaystyle \sum_{i=1}^ni^2$ by considering $S_n =\displaystyle \sum_{i=1}^n i^3$, and we know (by evaluating) that $(k+1)^3 - k^3 = 3k^2 + 3k+ 1$</p>
<p>So $$S_n - S_{n-1}= 1+\sum_{i=1}^{n-1} (i+1)^3 - i^3 = \sum_{i=1}^{n-1} 3i^2+ 3i +... |
947,358 | <p>Okay $g(x)= \sqrt{x^2-9}$</p>
<p>thus, $x^2 -9 \ge 0$</p>
<p>equals $x \ge +3$ and $x \ge -3$</p>
<p>thus the domains should be $[3,+\infty) \cup [-3,\infty)$ how come the answer key in my book is stating $(−\infty, −3] \cup[3,\infty)$. </p>
| Timbuc | 118,527 | <p>You can try the following argument:</p>
<p>$$x^2-9=(x-3)(x+3)\ge 0\iff x\le -3\;\;\text{or}\;\;x\ge 3$$</p>
<p>You can see the above easily and geometrically: the function $\;f(x)=x^2-9\;$ is a parabolla opening upwards, and if you draw it it is non-negative exactly when $\;x\le -3\;$ or $\;x\ge 3\;$ .</p>
|
3,366,569 | <p>I am trying to solve the following problem;</p>
<p>Write all elements of the following set: <span class="math-container">$ A=\left \{ x\in\mathbb{R}; \sqrt{8-t+\sqrt{2-t}}\in\mathbb{R}, t\in\mathbb{R} \right \}$</span> .</p>
<p>My assumption is that the solution is <span class="math-container">$\mathbb{R}$</span> ... | Oscar Lanzi | 248,217 | <p>Render <span class="math-container">$y=u/x$</span> where <span class="math-container">$1/x$</span> is the homogeneous solution with the lower power of <span class="math-container">$x$</span> (using the other solution could complicate the integral you eventually get by forcing unneeded fractions in the integrand). T... |
1,261,825 | <p>How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail. </p>
| Renato Faraone | 217,700 | <p>To find the inverse of the function $ y=x^x$ (where it is well defined) you first have to switch $y$ with $x$ and viceversa and now you proceed this way:</p>
<p>$x=y^y$</p>
<p>$\ln x=y\ln y$</p>
<p>$\ln x=e^{\ln y}\ln y$</p>
<p>Now we apply Lambert's W function defined as:</p>
<p>$$W(z)e^{W(z)}=z$$</p>
<p>$\ln... |
21,201 | <p>Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a good book where I can read about some basics? Thank you very much.</p>
| Bugs Bunny | 5,301 | <p>You may be able to salvage something by looking at the Bursnide ring of your group. The isomorphism class of the permutation representation of a set $X$ is governed by products $[X]e$ with some of the idempotents and you need to include missing idempotents (of the Burnside ring) to have a version of Brauer's permuta... |
1,445,702 | <p>i'm a little confused. </p>
<p>1)Which axis is which in 3 Dimensional system?</p>
<p>2)Does it matter if I switch the x-axis to y-axis?</p>
| uniquesolution | 265,735 | <p>1) Traditionally, the Cartesian coordinate system in $\mathbb{R}^3$ uses the notation $(x,y,z)$ to denote a general point in three-space. The reason is that these letters appear in this order in the alphabet. Now the $x$ axis is obtained by putting zeros for the $z$ and $y$, so that you are left with points of the f... |
2,585,265 | <p>I understand that $\cos(\theta) = \sin(\pi/2 - \theta)$ holds true. But, </p>
<blockquote>
<p>Does $\cos(\theta) = \sin(\pi/2 +\theta)$ always hold true?</p>
</blockquote>
<p>I am asking this question because I encountered the following question in my workbook.</p>
<p>If $h(x) = \cos x$, $g(x) = \sin x$, and $h... | lab bhattacharjee | 33,337 | <p>Let $$\sqrt{\dfrac{x+36}{x-36}}=n\implies x=\dfrac{36(n^2-1+2)}{n^2-1}=36+\dfrac{72}{n^2-1}$$</p>
<p>So $n^2-1(\ge-1)$ must divide $72$ and</p>
<p>$$n^2-1\le72\implies2\le n\le8$$</p>
|
366,401 | <p>Let <span class="math-container">$\nu$</span> be the uniform measure on the unit circle <span class="math-container">$\mathbb{S}^1 \subset \mathbb{R}^2$</span>, normalised so that <span class="math-container">$\nu(\mathbb{S}^1) = 1$</span>. Suppose <span class="math-container">$\mu$</span> is a Borel probability mea... | Piotr Hajlasz | 121,665 | <p>In general it is not true. Let <span class="math-container">$\{f_n\}_{n\geq 1}=\{1,z,\overline{z},z^2,\overline{z^2},\ldots\}$</span>, then as the OP pointed out <span class="math-container">$a_n=o(n^{-k})$</span>.
However, with a suitable permutation <span class="math-container">$\sigma$</span> of the basis <span... |
812,778 | <p>Prove that $(4/5)^{\frac{4}{5}}$ is irrational.</p>
<p><strong>My proof so far:</strong></p>
<p>Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational.</p>
<p>Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are integers.</p>
<p>Then $\dfrac{4^4}{5^4}=\dfrac{p^5}{q^5}$</p>
<p>$\therefore$ $4^4... | Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ By unique factorization, comparing powers of $\,2\,$ as below yields a contradiction $\,5\mid 8$ </p>
<p>$$\begin{align}2^{\large\color{#c00}8}\, Q^{\large\color{#0a0}5}\! &=\, P^{\large\color{#f70}5}\, 5^{\large 4}\\
\Rightarrow\ \color{#c00}8\! +\! \color{#0a0}5q &=\, \color{#f... |
1,304,529 | <p>I have come across these while studying the limsup & liminf of sequence of subset of a set. In order to understand that, I have to understand what least upper bound & greatest lower bound of a sequence of subset mean. I would be grateful if anyone helps me comprehend this concept intuitively as I am new &... | paw88789 | 147,810 | <p>$x$ is in the lim sup of a sequence of sets if and only if it is in infinitely many of the sets. </p>
<p>And $x$ is in the lim inf of a sequence of sets if and only if it is in all but finitely many of the sets (or equivalently if it is in all the sets from some point on).</p>
|
88,145 | <p>A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{Z}$ (implicitly as a group scheme) especially when $n$ is a prime power: <a href="https://mathoverflow.net/question... | Marty | 3,545 | <p>The places to look are: </p>
<ol>
<li>Steinberg, "Endomorphisms of linear algebraic groups." Memoir AMS 80, (1968), and </li>
<li>Gross, "The motive of a reductive group" Invent. math. 130, 287 ± 313 (1997).</li>
</ol>
<p>(I learned about the former from the latter).</p>
<p>To any (quasi-split) group $G$ with ma... |
88,145 | <p>A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{Z}$ (implicitly as a group scheme) especially when $n$ is a prime power: <a href="https://mathoverflow.net/question... | George McNinch | 4,653 | <p>I think the assumptions needed on a group scheme $G$ over $\mathbf{Z}$ are that $G$
should be smooth, affine, and of finite type over $\mathbf{Z}$. Actually,
I'm going to take the point of view that $p$ is fixed and so I'll suppose that $G$ is smooth, affine, and of finite type over the local ring $\mathbf{Z}_{(p)}$... |
746 | <p>There have been a number of questions in the Close part of Review lately which were basically asking for help creating an algorithm to do some mundane task (see <a href="https://mathoverflow.net/questions/140585/how-to-perform-divide-step-of-in-place-quicksort#comment362909_140585">here</a>, <a href="https://mathove... | Scott Morrison | 3 | <p>I don't really like our setup at the moment. Anna said we need to establish precedents of migrations before the open migration paths, but for now only moderators can actually do these migrations. Because we're actually a long established site where moderators area relatively rarely involved in closing questions, we'... |
1,177,493 | <p>If $p$ is a prime and $p \equiv 1 \bmod 4$, how many ways are there to write $p$ as a sum of two squares? Is there an explicit formulation for this?</p>
<p>There's a theorem that says that $p = 1 \bmod 4$ if and only if $p$ is a sum of two squares so this number must be at least 1. There's also the Sum of Two Squar... | user2566092 | 87,313 | <p>Here is a reference that gives an explicit formula for your problem, even extending to all integers (instead of just primes) <a href="http://mathworld.wolfram.com/SumofSquaresFunction.html" rel="nofollow">http://mathworld.wolfram.com/SumofSquaresFunction.html</a></p>
|
1,738,153 | <p>I know the definition is given as follows:</p>
<p>A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means</p>
<p>Can someone help me gain some intuition for this - perhaps a basic example?</p>
<p>Thanks</p>
| snulty | 128,967 | <p>If you know a little bit of group theory your familiar with the idea of homomorphism. </p>
<p>Now the idea of a linear representation is to take a group and find a homomorphism to a group of linear operators on a vector space. If you've fixed a basis then you're finding a homomorphism to a matrix group. If the vect... |
1,961,727 | <p>As far as I understood <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram–Schmidt orthogonalization</a> starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.</p>
<p>I have no problem understand... | user376902 | 376,902 | <p>Your choice of $w_1 = (1,0)$ and $w_2 = (0,1)$ fails one of the basic purposes of the Gram-Schmidt process: the result of the algorithm would not only have $\mathrm{Span}(v_1,v_2) = \mathrm{Span}(w_1,w_2)$ but also $\mathrm{Span}(v_1) = \mathrm{Span}(w_1).$</p>
<p>There are a few things I should mention:</p>
<p>(1... |
1,961,727 | <p>As far as I understood <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram–Schmidt orthogonalization</a> starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.</p>
<p>I have no problem understand... | Community | -1 | <p>If you want to represent a subspace, the standard basis will no do.</p>
<p>For example, assume a plane defined by two arbitrary vectors in 3D: G-S will define an orthonormal base that spans the plane.</p>
|
1,913,689 | <blockquote>
<p>Let $f: X \rightarrow Y$ be a function. $A \subset X$ and $B \subset Y$.
Prove $A \subset f^{-1}(f(A))$.</p>
</blockquote>
<p>Here is my approach. </p>
<p>Let $x \in A$. Then there exists some $y \in f(A)$ such that $y = f(x)$. By the definition of inverse function, $f^{-1}(f(x)) = \{ x \in X$ suc... | Caleb Stanford | 68,107 | <p>Your proof looks pretty good. The only thing to point out is when you said:</p>
<blockquote>
<p>By the definition of inverse function, $f^{-1}(f(x)) = \{ x \in X$ such that $y = f(x) \}$. Thus $ x \in f^{-1}(f(A)).$</p>
</blockquote>
<p>Two comments on this:</p>
<ol>
<li><p>This isn't usually called the <em>inv... |
2,007,373 | <p>At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. </p>
<p>For me the first instance that comes to memory was in 7th grade in a inner city USA school district. </p>
<p>Getting to the point, my geometry teacher taught,</p>
<p>"a p... | Ben Grossmann | 81,360 | <p>I don't know if your post has much to do with "life advice", but the question of whether there should be an "infinitely small but non-zero width" is something that bears answering.</p>
<p>The way math is done (with the standard set of axioms), it is indeed taken as fact that a point has <strong>exactly zero</strong... |
2,007,373 | <p>At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. </p>
<p>For me the first instance that comes to memory was in 7th grade in a inner city USA school district. </p>
<p>Getting to the point, my geometry teacher taught,</p>
<p>"a p... | Mikhail Katz | 72,694 | <p>After Abraham Robinson died in 1974 there was a bit of a pogrom against both the framework with infinitesimals that he developed, and against his students who had great trouble finding jobs. Most of the critics were less than well-informed, as richly illustrated in the current literature. </p>
<p>Thus, Alain Conne... |
1,017,707 | <p>Are there any proofs of this equality online? I'm just looking for something very simply that I can self-verify. My textbook uses the result without a proof, and I want to see what a proof would look like here.</p>
| Andreas H. | 44,523 | <p>I am not sure which proof you are looking for.</p>
<p>Your polynomial $Q_n$ is called the Chebyshev polynomial of degree $N$. </p>
<p>Probably the wikipedia page (<a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials" rel="nofollow">http://en.wikipedia.org/wiki/Chebyshev_polynomials</a>) will help you out in... |
1,976,382 | <p>Hölder's inequality for finite sums is given by
$$\sum_{k=0}^n|a_kb_k|\leq\left(\sum_{k=0}^n|a_k|^p\right)^{1/p}\left(\sum_{k=0}^n|b_k|^q\right)^{1/q},$$
where $1/p+1/q=1$, $p,q\in(1,\infty)$.</p>
<p>Is there a "similar" inequality which gives a lower bound for the left hand sum? I have searched, but found nothing ... | Jacky Chong | 369,395 | <p>Just consider Cauchy-Schwarz. We see that
\begin{align}
|a_1b_1 + a_2b_2| \leq \sqrt{a_1^2+a_2^2}\sqrt{b_1^2+b_2^2}.
\end{align}
If you are hoping for estimates of the nature
\begin{align}
f\left(\sqrt{a_1^2+a_2^2}, \sqrt{b_1^2+b_2^2}\right) \leq |a_1b_1+a_2b_2|
\end{align}
for some function $f(x, y)$ which is nonne... |
3,982,103 | <p>Let <span class="math-container">$(X,\tau)$</span> be a topological space. Prove that <span class="math-container">$\tau$</span> is the finite-closed topology on <span class="math-container">$X$</span> if and only if (i)<span class="math-container">$(X,\tau)$</span> is a <span class="math-container">$T_1$</span>-spa... | Kavi Rama Murthy | 142,385 | <p>Let <span class="math-container">$U$</span> be a non-empty open set. If <span class="math-container">$X \setminus U$</span> has an infinite number of points then the infinite set <span class="math-container">$X \setminus U$</span> cannot be dense. This is because it does not intersect <span class="math-container">... |
3,982,103 | <p>Let <span class="math-container">$(X,\tau)$</span> be a topological space. Prove that <span class="math-container">$\tau$</span> is the finite-closed topology on <span class="math-container">$X$</span> if and only if (i)<span class="math-container">$(X,\tau)$</span> is a <span class="math-container">$T_1$</span>-spa... | user126154 | 126,154 | <p>You have to prove that, under hypothesis <span class="math-container">$(i),(ii)$</span>, a non-trivial set (i.e. non-empnty and not <span class="math-container">$X$</span>) is closed if and only if it is finite.</p>
<p>Any closed set different from <span class="math-container">$X$</span> is not dense (because it coi... |
9,934 | <p>I have been given some code with the following line</p>
<pre><code>PeriodicExtension[g_, x_] := If[Abs[x] < Pi, g[x], PeriodicExtension[g, x - 2 Sign[x] Pi]]
</code></pre>
<p>I do not understand the syntax. I would appreciate if someone can explain what this
code does for different values of <code>x</code>.</p>... | stevenvh | 1,450 | <p>I agree with J.M.: don't use recursion if you don't need it. I was just running a benchmark when he posted his answer.</p>
<pre><code>PeriodicExtension[g_, x_] := If[Abs[x] < Pi, g[x], PeriodicExtension[g, x - 2 Sign[x] Pi]];
g[x_] := x^3
Timing[Plot[PeriodicExtension[g, x], {x, 0, 1000 Pi}]]
</code></pre>
<blo... |
3,133,695 | <p>A spotlight on the ground shines on a wal 12 m away. If a man 2m tall walks from the spotlight toward the building at a speed of 1.6m/s, how fast is the length of his shadow on the building decreasing when he is 4m from the building.</p>
<p>How do you solve this word problem. I have drawn a picture to figure out th... | Anders Kaseorg | 38,671 | <p>Yes. There are two degrees of freedom in the solution space, so we can add two extra constraints to make the computation easier. Here’s a solution with a 90° angle and a 45° angle:</p>
<p><a href="https://i.stack.imgur.com/uS9nN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/uS9nN.png" alt="dia... |
2,305,656 | <p>I solved this problem on my own, months ago, but the solution seems to me completely forgotten, a little help on it would be appreciated:</p>
<p>Suppose $\alpha= \alpha(t)$ on an interval $I$ is a smooth (of class C$^1$) parametric representation of the curve $C$, and for any $t \in I$ we have $\space\space\frac{d}... | Chandramauli Chakraborty | 375,555 | <p>We know that $r=4R\sin{(\dfrac{A}{2})} \sin{(\dfrac{B}{2})} \sin{(\dfrac{C}{2})}.$ where A, B and C are the angles of the triangles.
And $\dfrac{a}{c}=\tan{A}$, where A is the equal angles or base angles which can be proofed using sine rule.</p>
<p>Now $\frac{r}{R}=k=8\sin{(\dfrac{A}{2})}\cos{(\dfrac{A}{2})}$,
$\t... |
846,797 | <p>I encountered this calculation in a problem $\dfrac{\sin 150^o\times\sin 20^o}{\sin 80^o\times\sin 10^o}$ and calculated that it equals 1.</p>
<p>Is it just a coincidence or is there any identity that says $\sin 150^o\times\sin 20^o=\sin 80^o\times\sin 10^o$?</p>
<p>I am trying to use the addition formulae and<br... | puru | 151,916 | <p>$\sin 150^0=\sin 30^0=1/2$ </p>
<p>Hence LHS=$\frac{\sin 20^0}{2}$ </p>
<p>$(\sin 2\theta)/2= \sin \theta \cos \theta\dots(1)$</p>
<p>$\sin 80^0=\cos 10^0 \implies$ RHS= $\sin 10^0 \cos 10^0$ = LHS from $(1)$</p>
<p>Hope this helps! </p>
|
846,797 | <p>I encountered this calculation in a problem $\dfrac{\sin 150^o\times\sin 20^o}{\sin 80^o\times\sin 10^o}$ and calculated that it equals 1.</p>
<p>Is it just a coincidence or is there any identity that says $\sin 150^o\times\sin 20^o=\sin 80^o\times\sin 10^o$?</p>
<p>I am trying to use the addition formulae and<br... | Anonymous Computer | 128,641 | <p>You should know that $\sin 150^\circ=\sin 30^\circ=\frac 12$.</p>
<p>Now, $\sin 80^\circ=\cos 10^\circ$, because of the property $\sin x =\cos(90-x)$. </p>
<p>Now we have:
$$\frac 12 \sin 20^\circ=\cos 10^\circ \times \sin 10^\circ$$
Recall the double angle identity $\sin(2x)=2\sin x\cos x$. This means that $\frac... |
4,594 | <p>I would like to open an Excel file and manipulate it as a COM object. While I'm able to open an instance of excel with</p>
<pre><code>Needs["NETLink`"]
InstallNET[]
excel = CreateCOMObject["Excel.Application"]
</code></pre>
<p>This doesn't work for me:</p>
<pre><code> wb = excel@Workbooks@Open["D:\\prices.csv"]
<... | Chris Degnen | 363 | <p>You don't need the initial <code>InstallNET[]</code>. That should come after <code>Needs["NETLink"]</code>.</p>
<p>I made a post on this topic a while back, here: <a href="http://forums.wolfram.com/mathgroup/archive/2011/Oct/msg00386.html" rel="noreferrer">http://forums.wolfram.com/mathgroup/archive/2011/Oct/msg00... |
3,356,544 | <p>A lot of calculators actually agree with me saying that it is defined and the result equals 1, which makes sense to me because:</p>
<p><span class="math-container">$$ (-1)^{2.16} = (-1)^2 \cdot (-1)^{0.16} = (-1)^2\cdot\sqrt[100]{(-1)^{16}}\\
= (-1)^2 \cdot \sqrt[100]{1} = (-1)^2 \cdot 1 = 1$$</span></p>
<p>Howev... | Brandon O. Salazar | 480,071 | <p>I don't think you are allowed to split negative numbers up like that. Since <span class="math-container">$-1=e^{i\pi}$</span> then <span class="math-container">$$(-1)^{2.16}=e^{i\pi\cdot 2.16}$$</span> which is not a real number.</p>
|
2,281,932 | <p>If Peano axioms uniquely determine the natural numbers, doesn't this mean that Peano axioms are categorical and hence complete?</p>
<p>If above is true, how is it explained by Goedel's incompleteness theorem?</p>
| Noah Schweber | 28,111 | <p>It depends what you mean by "Peano axioms".</p>
<p>M.Winter's answer assumes you mean the <em>first-order</em> Peano axioms. By the compactness theorem, no infinite structure can <em>ever</em> be captured up to isomorphism by a first-order theory, and no countable rigid structure (like $\mathbb{N}$) can ever be cap... |
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