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 |
|---|---|---|---|---|
3,984,322 | <p>I have the 2nd order homogeneous ODE:
<span class="math-container">$$y(x)''+\frac{2}{x}y'(x)-By=0 $$</span>
Where <span class="math-container">$B$</span> is a constant. Since I am a bit rusty with analytical solutions, I plugged it in MatLab's symbolic solver which found the following solution.
<span class="math-con... | Gribouillis | 398,505 | <p><strong>Hint</strong> Try the substitution
<span class="math-container">\begin{equation}
y(x) = \frac{z(x)}{x}
\end{equation}</span></p>
|
1,291,657 | <p>Is Bezout's lemma enough to confirm the GCD of a number?</p>
<p>So suppose we have <span class="math-container">$$ax+by=z$$</span> does this mean <span class="math-container">$$\gcd(a,b)=z$$</span></p>
| davidlowryduda | 9,754 | <p>Consider the following:
$$ 1\cdot 2 + 1\cdot 3 = 5,$$
and so
$$\gcd (1,1) = 5.$$
You might try almost any combination of numbers and check for yourself the validity of your statement.</p>
|
1,291,657 | <p>Is Bezout's lemma enough to confirm the GCD of a number?</p>
<p>So suppose we have <span class="math-container">$$ax+by=z$$</span> does this mean <span class="math-container">$$\gcd(a,b)=z$$</span></p>
| Bill Dubuque | 242 | <p>A <em>common</em> divisor <span class="math-container">$\,c\mid a,b\,$</span> is <em>greatest</em> if it can be written in linear form <span class="math-container">$\,c = j a + k b,\,$</span> since then <span class="math-container">$\,d\mid a,b\,\Rightarrow\, d\mid ja+kb = c,\,$</span> so <span class="math-container... |
503,691 | <p>I'm a bit confused about a statement that I see often in the $\lambda$-calculus literature. Namely, what exactly does the following statement mean: "By induction on the length of $M\in\Lambda$." ?</p>
<p>In such literature, $\Lambda$ is defined as the smallest subset of $\mathcal{V}\cup\{(,),\lambda\}$ such that:</... | MJD | 25,554 | <p>As a practical matter the technique of structural induction and the technique of induction on length turn out to be the same, and you can take "induction on the length of the term" to be synonymous with "structural induction on the term".</p>
<p>In more detail, what is meant here is actually <a href="http://enwp.or... |
403,344 | <p>I'm trying to determine whether this statement is true or false:</p>
<p>Assume that $dim V = n$ and let $T \in L(V)$. Let $f(z) \in P_n(F)$ be a
monic polynomial of degree
$n$ such that $f(T) = 0$. Then $f(z)$ is the characteristic polynomial of $T$.</p>
<p>I know that because $f(T) = 0$, the minimal polynomial of... | Tom Oldfield | 45,760 | <p>This is not true in general. The minimum polynomial is the smallest degree polynomial that the linear map satisfies, and the characteristic polynomial is defined to be $\det(xI-T)$. If the minimum polynomial is degree $m$, we know that the characteristic polynomial will be degree $n$ so $m\leq n$. If $p(x)$ is the m... |
2,421,421 | <blockquote>
<p>Evaluate
$$ \lim_{x\to \pi/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}$$ </p>
</blockquote>
<p>I tried to solve this by L'Hospital's rule..but that doesn't give a solution..appreciate if you can give a clue.</p>
| Marios Gretsas | 359,315 | <p>$\cos{2x}=2\cos^2{x}-1$</p>
<p>Thus the nominator becomes:</p>
<p>$$\sqrt{1+\cos{2x}}=\sqrt{1+2\cos^2{x}-1}=\sqrt{2}|\cos{x}|$$</p>
<p>As $x \to \frac{\pi}{2}^-$ we have $|\cos{x}|=\cos{x}$</p>
<p>As $x \to \frac{\pi}{2}^+$ we have $|\cos{x}|=-\cos{x}$</p>
<p>Now you can use the L'Hospital rule to evaluate both... |
3,553,471 | <blockquote>
<p>For the triangle <span class="math-container">$\triangle ABC$</span>, prove the inquality,
<span class="math-container">$$\frac1{4+\cos A\cos(B-C)}+\frac1{4+\cos B\cos(C-A)}+\frac1{4+\cos C\cos(A-B)}\ge \frac23$$</span>
where <span class="math-container">$A$</span>, <span class="math-container">$B... | lab bhattacharjee | 33,337 | <p>Using <span class="math-container">$\cos(B+C)=\cos(\pi-A)=?$</span></p>
<p>and <a href="https://math.stackexchange.com/questions/345703/prove-that-cos-a-b-cos-a-b-cos-2a-sin-2b">Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$</a></p>
<p><span class="math-container">$$2(4+\cos A\cos(B-C))$$</span>
<... |
3,553,471 | <blockquote>
<p>For the triangle <span class="math-container">$\triangle ABC$</span>, prove the inquality,
<span class="math-container">$$\frac1{4+\cos A\cos(B-C)}+\frac1{4+\cos B\cos(C-A)}+\frac1{4+\cos C\cos(A-B)}\ge \frac23$$</span>
where <span class="math-container">$A$</span>, <span class="math-container">$B... | The 2nd | 751,538 | <p>First, we use the trig identity <span class="math-container">$ \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) $</span>: </p>
<p><span class="math-container">$$\frac1{4 + \cos A \cos(B-C)}+\frac1{4 +\cos B \cos(C-A)} + \frac1{4 + \cos C \cos(A-B)} \ge \frac23$$</span>
<span class="math-container">$$ \Leftright... |
954,281 | <p>Given a complex measure: $\mu:\Sigma\to\mathbb{C}$.</p>
<p><img src="https://i.stack.imgur.com/YS26y.jpg" alt="Variation Measure"></p>
<p>Consider its decomposition into positive measures:
$$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$</p>
<blockquote>
<p>Does it split i... | Alex R. | 22,064 | <p>Sure. Set $A$ to be the set on which $\mu$ is positive and $B$ the set on which its negative. </p>
|
2,280,323 | <p>I am given a linear transformation
$$T:\mathbb{R^3} \rightarrow \mathbb{R^2}$$
$$T((x,y,z)) = (x+y,-y+z)$$
The task if to find a basis in $\mathbb{R^3}$, let's call it $B=\{e_1, e_2, e_3\}$ and $\mathbb{R^2}$, let's call it $B'=\{f_1, f_2\}$ such that $A$ is the matrix of this transformation with respect to the foun... | egreg | 62,967 | <p>According to the definition of associated matrix with respect to given bases, you have to find $\{e_1,e_2,e_3\}$ and $\{f_1,f_2\}$ such that
\begin{align}
T(e_1)&=f_1\\
T(e_2)&=2f_2\\
T(e_3)&=0
\end{align}
Note that the problem is undetermined: you can find infinitely many bases with this property.</p>
... |
3,786,797 | <p>I have two variables <span class="math-container">$x$</span> and <span class="math-container">$y$</span> which are measured at two time steps <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Available to me is the percent change from the first timestep to the second, relative to t... | Fawkes4494d3 | 260,674 | <blockquote>
<p>First, just arrange only the <span class="math-container">$5$</span> men in a line. You can do that in <span class="math-container">$5!=120$</span> ways, number of permutations of <span class="math-container">$5$</span> distinct objects.</p>
</blockquote>
<blockquote>
<p>Take <strong>any such permutatio... |
2,592,581 | <p>Lets define the following series
$$S_n =\sum_{i=1}^{\infty} \frac 1 {i(i+1)(i+2)...(i+n)}.$$
I know that $S_0$ does not converge so let's suppose $n \in N$ and define $S_n$ for some $n$. We have $S_1=1$ , $S_2=\frac1 4$ , $S_3=\frac 1 {18}$ , $S_4=\frac 1{96}$ , $S_5=\frac 1{600}$ etc..<br/><br/>
the numerator in al... | Math Lover | 348,257 | <p>Hint: Telescopic sum and
$$\frac{n}{i(i+1)(i+2)\cdots(i+n)}=\frac{1}{i(i+1)\cdots(i+n-1)}-\frac{1}{(i+1)(i+2)\cdots(i+n)}.$$</p>
|
2,019,542 | <p>I read <a href="http://mathworld.wolfram.com/PolynomialOrder.html" rel="nofollow noreferrer">Polynomial order</a> defination, but i am still confuse how do we find polynomial order if we have a polynomial. What does Polynomial order represent?</p>
<p>How do we find polynomial order if we have polynomial forexample ... | iam_agf | 196,583 | <p>You want the minimum $n$ such that</p>
<p>$$\frac{x^n+1}{x^5+x^2+1}$$</p>
<p>In $\mathbb{F}_2$.</p>
<p>Doing the division without knowing the $n$ (only trying to get residue zero), you have that the polinomial $x^n+1$ will let you the quotient</p>
<p>$$x^{n-5}+x^{n-8}+ x^{n-10}+x^{n-11}+x^{n-14}+x^{n-15}+x^{n-16... |
2,019,542 | <p>I read <a href="http://mathworld.wolfram.com/PolynomialOrder.html" rel="nofollow noreferrer">Polynomial order</a> defination, but i am still confuse how do we find polynomial order if we have a polynomial. What does Polynomial order represent?</p>
<p>How do we find polynomial order if we have polynomial forexample ... | Gerry Myerson | 8,269 | <p>Let's write $F$ for the field of two elements, and $f(x)$ for $x^5+x^2+1$. You have to show $f$ is irreducible over $F$. Having done that, it follows that $K=F[x]/\langle\,f\,\rangle$ is a field of degree 5 as an extension of $F$. It follows that $K$ has $2^5=32$ elements. So there are 31 nonzero elements. The nonze... |
2,037,652 | <p>I'm taking a course on smooth manifolds and the following exercise was given to me:</p>
<blockquote>
<p>If $P\in\mathbb{R}[X_1, ..., X_{n+1}]$ is a homogeneous polynomial of degree $d$ such that $\left(\frac{\partial P}{\partial X_1}, ..., \frac{\partial P}{\partial X_{n+1}}\right)$ is never zero, prove that $Z(P... | Ted Shifrin | 71,348 | <p><strong>HINT</strong>: Consider, for example, the function
$$f_{n+1}(u_1,\dots,u_n) = P(u_1,\dots,u_n,1)$$
defined for $[u_1,\dots,u_n,1]\in U_{n+1} \cong \Bbb R^n$. Apply the regular value theorem (or constant rank theorem) to the functions $f_i$, $i=1,\dots,n+1$. Note that every $x\in Z(P)$ belongs to at least one... |
1,737,835 | <p>You have decided to buy candy for the trick-or-treaters and have estimated there will be 200 children coming to your door, and plan to give each children three pieces of candy. You have decided to offer Twix and 3 Musketeers. The cost of buying these two candies is<br>
$$C= 5T^2 + 2TM + 3M^2 + 800$$
Where T is... | Christian Blatter | 1,303 | <p><strong>Update:</strong> I had forgotten an essential point; see the comments to my answer.</p>
<p>Let $B\subset{\mathbb R}^n$ be any compact convex set which is symmetric to the origin: $-B=B$, and has nonempty interior. Then
$$\|x\|:=\inf\{\lambda\geq0\>|\>x\in\lambda B\}$$
is a norm on ${\mathbb R}^n$ havi... |
2,348,641 | <p><span class="math-container">$f$</span> is a periodic continuous function of period <span class="math-container">$T > 0 $</span>.</p>
<ol>
<li><p>Let <span class="math-container">$ a > 0$</span>, prove that: <span class="math-container">$ \int_{0}^{+ \infty} f(t)e^{-at}dt $</span> is absolutely convergent.</p>... | RRL | 148,510 | <p>Hint 2: </p>
<p>Using the substitution $t = u + (k-1)T$,</p>
<p>$$\int_0^\infty f(t)e^{-at} \, dt = \sum_{k=1}^\infty\int_{(k-1)T}^{kT}f(t)e^{-at} \, dt = \sum_{k=1}^\infty \int_{0}^{T}f(u+(k-1)T)e^{-au}e^{-a(k-1)T} \, du \\ = \sum_{k=1}^\infty e^{-a(k-1)T}\int_{0}^{T}f(u)e^{-au} \, du $$</p>
<p>Hint 3:</p>
<p... |
38,480 | <p>(Before reading, I apologize for my poor English ability.)</p>
<p>I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral below
$$ \int_0^{\frac{\pi}{2}} \arctan (1 - \sin^2 x \; \cos^2 x... | Mitch | 1,919 | <p>If you're talking about practical engineering applications, then really only numerical approximations are used (and studied in computer science as 'numerical analysis' or more recently 'scientific computing'). </p>
<p>As to an academic mathematical field nowadays that deals with symbolic integration, first some per... |
38,480 | <p>(Before reading, I apologize for my poor English ability.)</p>
<p>I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral below
$$ \int_0^{\frac{\pi}{2}} \arctan (1 - \sin^2 x \; \cos^2 x... | JWalter | 791,148 | <p>Symbolic integration becomes less popular indeed and most researchers prefer numerical integration. However, significance of the symbolic integration should not be underestimated. This can be shown by using the integration formula:
<span class="math-container">$$\int_0^1 f(x)dx = 2\sum_{m=1}^M{\sum_{n=0}^\infty{\fra... |
2,232,677 | <p>A parallellogram $ABCD$ is given, where $E$ is the midpoint of $BC$ and $F$ is a point on $AD$ so that $|FD| = 3|AF|$. $G$ is the point where $BF$ and $AE$ intersects.</p>
<p>Express the vector $AG$ in terms of vectors $AB$ and $AD$.</p>
<p>My solution to the problem is the following: Impose an <em>affine transfor... | Misha Lavrov | 383,078 | <p>This is straightforward once you show that $\overrightarrow{AG} = \frac13 \overrightarrow{AE}$. Then we have $$\overrightarrow{AG} = \frac13(\frac12\overrightarrow{AB} + \frac12\overrightarrow{AC}) = \frac16 \overrightarrow{AB} + \frac16 \overrightarrow{AC} = \frac16 \overrightarrow{AB} + \frac16 (\overrightarrow{AB... |
3,933,135 | <p>Let <span class="math-container">$m$</span> and <span class="math-container">$n$</span> are positive integer</p>
<blockquote>
<p>Can it be shown that, For every <span class="math-container">$m\ge 5$</span></p>
<p><span class="math-container">$$\sum_{i=1}^ni^m-m^i>0\iff n=2,3,...,m$$</span></p>
</blockquote>
<p>Ex... | Neat Math | 843,178 | <p><strong>Lemma 1:</strong> <span class="math-container">$2^m> m^2+m-1, \forall m>5$</span>.</p>
<p>When <span class="math-container">$m=5, 2^5=32>29=5^2+5-1$</span>. If <span class="math-container">$2^m>m^2+m-1$</span> then <span class="math-container">$$2^{m+1}-[(m+1)^2+(m+1)-1]\\>2(m^2+m-1)-(m+1)^2-(... |
1,575,920 | <p>The global error of $\int f(x) \mathrm{d}x$, between two
$x$-values by the trapezoidal rule is
$-(1/12)h^3f''(ξ)$</p>
<p>a) $f(x) = x^3, x := [0.2,0.5]$</p>
<p>Find the value for $ξ$. Not really sure where to start with this problem. We never did any examples in class.</p>
| abel | 9,252 | <p>you have $$\frac14\left(0.5^4 - 0.2^4\right)=\int_{0.2}^{0.5}x^3 \,dx = 0.3\left(\frac12 0.2^2 + \frac120.3^2\right) -\frac1{12}(0.3^3) 6\xi \tag 1 $$</p>
<p>equation $(1)$ determines what $\xi$ is in this case.</p>
|
309,841 | <p>Let $m$ be a positive integer. Prove that the dihedral group $D_{4m+2}$ of order $8m+4$ contains a subgroup of order 4.</p>
<p>Im fairly stumped as to where to start with this one, any input would be helpful.
Thanks</p>
| Haskell Curry | 39,362 | <p>The concept of tensor products of vector spaces is required for the description of <a href="http://en.wikipedia.org/wiki/General_relativity" rel="noreferrer">General Relativity</a>. For example, the metric tensor $ \mathbf{g} $ of a <a href="http://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold" rel="noreferrer">p... |
145,868 | <p>I have a data set like below:</p>
<pre><code>368 150 0.31895877
369 150 0.32141133
370 150 0.3190167
371 150 0.33890757
372 150 0.3649395
373 150 0.37585051
374 150 0.37916244
375 150 0.37619812
376 150 0.37287654
377 150 0.37249031
378 150 0.37102263
379 150 0.34839918
380 150 No Data
381 150 No Data
382 150 No Da... | JimB | 19,758 | <p>Because your distribution is (at least) bimodal and because the probability density is greater than zero at zero, you might consider using nonparametric density estimation rather than maybe a poor approximation with a mixture of a small number of distributions. (However, if it works with a mixture distribution, the... |
1,736,360 | <p>Using the binomial expansion, find the taylor series of the function $$f(z) = \frac{1}{(1+z)^3}$$ and find its radius of convergence.</p>
<hr>
<p>The solution is</p>
<p>$$f(z) = \sum_{n=0}^\infty (-1)^n\frac{(n+1)(n+2)}2z^n$$</p>
<p>How do we find this? I thought the binomial expansion was</p>
<p>$$(1+z)^n = \s... | Bernard | 202,857 | <p>The binomial formula is still valid for exponents $\alpha$ which are not natural numbers, but the binomial coefficients
$$\binom\alpha k=\frac{\alpha(\alpha -1)\dots(\alpha -k+1)}{1\cdot 2\dotsm k}$$
are never $0$, hence an infinite series.</p>
<p>Here, with your notations, you get:
$$\binom{-3}n=\frac{-3(-4)\dot... |
58,920 | <p>I have a csv that encodes the results of plugging certain parameters (say, <code>A</code> and <code>B</code>) into certain functions (say, <code>f</code> and <code>g</code>). Think matrix with (say, 4) columns and a million rows with the columns corresponding to <code>A, B, C = f(A, B)</code>, and <code>D = g(A, B)<... | kglr | 125 | <p>Perhaps you can use <a href="http://reference.wolfram.com/mathematica/ref/DiscretePlot3D.html" rel="nofollow noreferrer">DiscretePlot3D</a></p>
<pre><code>{alist, blist} = RandomReal[{-2 Pi, 2 Pi}, {2, 20}];
Manipulate[DiscretePlot3D[f[a + b], {a, alist}, {b, blist},
PlotStyle -> Hue[... |
1,160,741 | <p>Prove that continuous functions on $\mathbb{R}$ with period $1$ can be uniformly approximated by trigonometric polynomials, i.e. functions in the linear span of $\{e^{−2πinx}\}_{n \in \mathbb{Z}}$. Explain why $n \in \mathbb{N}$ is not enough.</p>
<p>I think I need to restrict the domain into a compact set and cons... | Berrick Caleb Fillmore | 85,964 | <p>Let us restrict the domain to the interval $ [0,1] $ and use some of the theory of inner-product spaces.</p>
<p>One reason why having $ \mathbb{N} $ as the index set is not enough is because $ e_{1}: x \mapsto e^{2 \pi i x} $ is orthogonal to the set $ \{ e_{- n}: x \mapsto e^{- 2 \pi i n x} \}_{n \in \mathbb{N}_{0... |
4,017,790 | <p>We consider the equation <span class="math-container">$y^2-4y+4+x^2y+\arctan(x)=0$</span>, the exercice of my book say that the normal vector <span class="math-container">$n$</span> of the cartesian curve implicitally defined by the previous equation is <span class="math-container">$n=l(1,0)$</span> for <span class=... | JMoravitz | 179,297 | <p>The difference between <span class="math-container">$\left(\frac{1}{2}\right)^8$</span> versus <span class="math-container">$\left(\frac{1}{2}\right)^7$</span> depends on how we interpret the problem.</p>
<p>Problem 1:</p>
<ul>
<li>We have two pizzas, each with eight pizza slices. Each time someone enters the room,... |
962,301 | <p>$$(\sqrt 2-\sqrt 1)+(\sqrt 3-\sqrt 2)+(\sqrt 4-\sqrt 3)+(\sqrt 5-\sqrt 4)…$$</p>
<p>I have found partial sums equal to natural numbers. The first 3 addends sum to 1. The first 8 sum to 2. The first 15 sum to 3. When the minuend in an addend is the square root of a perfect square, the partial sum is a natural number... | Stephen Quan | 170,354 | <p>@Jasper Joy is indeed right. It's a telescoping series whose partial sums eventually only have a fixed number of terms after cancellation:</p>
<p>$$(\sqrt{n+1} - \sqrt n) + ... + (\sqrt 4 - \sqrt 3) + (\sqrt 3 - \sqrt 2) + (\sqrt 2 - \sqrt 1)$$
$$= \sqrt{n+1} + (-\sqrt n + \sqrt n) + ... + (-\sqrt 3 + \sqrt 3) + (-... |
52,194 | <p>Assuming you have a set of nodes, how do you determine how many connections are needed to connect every node to every other node in the set?</p>
<p>Example input and output:</p>
<pre><code>In Out
<=1 0
2 1
3 3
4 6
5 10
6 15
</code></pre>
| Jake Petroules | 4,775 | <p>Figured it out. The formula is:</p>
<pre><code>x = n(n - 1) / 2
</code></pre>
|
2,679,525 | <p>Find the $\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$ over $\mathbb{Q}[x]$. Then find two polynomials $a(x),b(x) \in \mathbb{Q}[x]$ such that, $$a(x)(x^3-6x^2+14x-15) + b(x)(x^3-8x^2+21x-18)=\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$$ </p>
<p>I have managed to find,
$$x^3-6x^2+14x-15=(x-3)(x^2-3x+5)$$
$$x^3-8x^2+21x-18=... | Macavity | 58,320 | <p>You have the complete extended Euclid algorithm (which IMHO is the best way) given already in Will Jaggy's post. </p>
<p>Given that you already extracted $x-3$ as GCD, here is a perhaps quicker way (which may not generalise much). We simplify a bit by noting we seek <em>linear</em> $a(x), b(x)$, s.t.: </p>
<p>$a... |
2,866,389 | <p>Let $\gamma\colon\left[0,1\right]\to\mathbb{R}^{2}$ be a curve such
that $\gamma\in\mathcal{C}^{1}$ (Continuously Differentiable). I
need to show that $\gamma\left(\left[0,1\right]\right)$ has mesure
zero, that is to show there are generalized rectangles $R_{1},R_{2},\ldots$
such that $\gamma\left(\left[0,1\right]\r... | Angina Seng | 436,618 | <p>As $\gamma$ is $C^1$ on a compact set, it is Lipschitz. Let $K$ be a Lipschitz
constant for $\gamma$. Then, for each $N\in \Bbb N$, $\gamma([0,1])$
can be covered by $N$ squares, each of sidelength $2K/N$.</p>
|
47,655 | <p>Let $X_n$ be the set of all word of the length $3 n$ over the alphabet $\{A,B,C\}$ which contain each of the three letters <em>n</em> times.</p>
<p>The amount of elements of $X_n$ is $\frac{(3n)!}{(n!)^3}$, but why?</p>
<p>I tried to split the problem into three smaller ones by only looking at the distribution of ... | Arturo Magidin | 742 | <p>Two ways of doing it. The first follows your idea, but corrects the error:</p>
<ol>
<li><p>You are right that there are $\binom{3n}{n}$ possible ways to place one of the letters. However, <em>once</em> you place one of the letters (say, you place all the $A$s), and you try to place the $B$s, you'll find that you do... |
47,655 | <p>Let $X_n$ be the set of all word of the length $3 n$ over the alphabet $\{A,B,C\}$ which contain each of the three letters <em>n</em> times.</p>
<p>The amount of elements of $X_n$ is $\frac{(3n)!}{(n!)^3}$, but why?</p>
<p>I tried to split the problem into three smaller ones by only looking at the distribution of ... | trutheality | 3,638 | <p>If we number each occurence of a letter, e.g. $A_1, A_2,...$ there are $3n$ numbered letters rearrange, so $(3n)!$ permutations.</p>
<p>But permutations of the same letters ($A_1A_2$ vs $A_2A_1$) are indistinguishable,
so we have $n!$ permutations of the numbering of each kind of letter. Hence you divide by $n!$ th... |
3,107,755 | <p>When working with skew Schur functions, they can be defined as follows.</p>
<p>Let <span class="math-container">$C^{\lambda}_{\mu, \nu}$</span> be the integers such that </p>
<p><span class="math-container">$$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$</span></p>
<p>Then, we can define skew... | enedil | 126,823 | <p>You can count the number of ways you can divide <span class="math-container">$11$</span> elements into <span class="math-container">$8$</span> groups (a group can't be empty and order is important), then place an asterisk between every two groups.</p>
|
3,107,755 | <p>When working with skew Schur functions, they can be defined as follows.</p>
<p>Let <span class="math-container">$C^{\lambda}_{\mu, \nu}$</span> be the integers such that </p>
<p><span class="math-container">$$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$</span></p>
<p>Then, we can define skew... | saulspatz | 235,128 | <p>Put the <span class="math-container">$8$</span> *s in a row. We have to put <span class="math-container">$7$</span> #s between them. Now there are <span class="math-container">$9$</span> places to put the remaining <span class="math-container">$4$</span> asterisks. Use stars and bars. </p>
|
185,900 | <p>What is the <strong>fastest way</strong> to find the smallest positive root of the following transcendental equation:</p>
<p><span class="math-container">$$a + b\cdot e^{-0.045 t} = n \sin(t) - m \cos(t)$$</span></p>
<pre><code>eq = a + b E^(-0.045 t) == n Sin[t] - m Cos[t];
</code></pre>
<p>where
<span class="m... | John Doty | 27,989 | <p>Here's a method that requires no plotting or guessing of initial points. It uses <code>FindInstance</code> to find successively smaller solutions until no smaller solution can be found.</p>
<pre><code>try[tt_] := With[{r = FindInstance[eq && t < tt, t, Reals]},
If[r == {}, tt, t /. r[[1]]]]
FixedPoint... |
1,343,554 | <p>This is exercise 6.10 in Resnick's book "A Probability Path".</p>
<p>We're given a sequence of random variables $(X_n)$ and an increasing function
$f: [0, \infty) \rightarrow [0, \infty)$ such that </p>
<p>$$
\frac{f(x)}{x} \rightarrow \infty, \qquad x \rightarrow \infty,
$$</p>
<p>and</p>
<p>$$
\sup_{n \geq 1} ... | henderfen25 | 774,103 | <p>I just started using the math stack exchange community to help with my self study of probability theory, and I have a comment in general about this problem. It should really be a comment on the above post by David, but I don't have the reputation points for that, so I am posting it here. Also, at the end of this pos... |
2,181,633 | <p>Given a real polynomial $P(x)$, find another real polynomial $Q(x)$ such that the powers of $x$ in $P(x)Q(x)$ are multiples of an integer $n$. Is it always possible to find such a polynomial?</p>
<p>For e.g if $P(x) = 1+3x+2x^2$ I can find $Q(x)=(2 x - 1) (x - 1) (4 x^2 + 1) (x^2 + 1)$ such that $P(x)Q(x) = 16 x^... | A. Napster | 27,945 | <p>If $R(x)= x^{n} - a^{n} $ then by factorisation we can see that $ x^{n} - a^{n} = (x - a)(x^{n - 1} + ax^{n - 2}+ a^2x^{n - 3} + \dots + a^{n-1})$ or by simply observing that $R(a)=0$ implies $x-a$ is a factor of $x^{n} - a^{n}$. </p>
<p>In the same way we can deduce that $x+a$ is a factor of $x^n -a^n$ when $n$ i... |
1,382,003 | <p>I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem</p>
<blockquote>
<p>Compute
$$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$</p>
</blockquote>
<p>And here is my approach:</p>
<p>Since $A^\frac{1}{x}=\exp\left(\frac{\ln A}{x}\... | Leucippus | 148,155 | <p>For the integral in question:
$$\large\lim_{A\to\infty} \, \frac{1}{A} \, \int_{1}^{A} A^{\frac{1}{x}}\,dx$$
the following is considered:
\begin{align}
A^{\frac{1}{x}} = e^{\frac{1}{x} \, \ln A} = 1 + \frac{\ln A}{x} + \sum_{k=2}^{\infty} \frac{\ln^{k}A}{k! \, x^{k}}
\end{align}
for which
\begin{align}
\int_{1}^{A}... |
1,836,998 | <p>$$
\lim_{x\to 0^{+}}\left[\left(1+\frac{1}{x}\right)^x+\left(\frac{1}{x}\right)^x+\left(\tan(x)\right)^{\frac{1}{x}}\right]$$</p>
<p>Attempt: I used $\tan(x)\approx x$ also $(1+n)^{1/n}=e$ so I let $x=0+h$ and lim changes to lim h tending to $0$. But that gives me $e+\infty+h^{1/h}$. While the answer is an integer ... | Barry Cipra | 86,747 | <p>Note first of all that $(\tan x)^{1/x}\to0^\infty=0$ as $x\to0^+$, so the trickiest-looking part of the limit is actually the easiest. More formally, $0\le\tan x\le1$ for $0\le x\le\pi/4$ and $1/x\ge4/\pi\gt1$ for $0\lt x\le\pi/4$, so</p>
<p>$$0\le(\tan x)^{1/x}\le\tan x\quad\text{for }0\lt x\le{\pi\over4}$$</p>
... |
643,658 | <p>Let <strong><em>K</em></strong> be a finite field, $F=K(\alpha)$ a finite simple extension of degree $n$, and $ f \in K[x]$ the minimal polynomial of $\alpha$ over $K$. Let $\frac{f\left( x \right)}{x-\alpha }={{\beta }_{0}}+{{\beta }_{1}}x+\cdots +{{\beta }_{n-1}}{{x}^{n-1}}\in F[x]$ and $\gamma={f}'\left( \alpha ... | DonAntonio | 31,254 | <p>If $\;G=Gal(F/K)=\{\sigma_1:=Id,\sigma_2,...,\sigma_n\}\;$, then using your nice characterization </p>
<p>$$\;\gamma:=f'(\alpha)=\sum_{k=0}^{n-1}\beta_k\alpha^k\;$$</p>
<p>we get:</p>
<p>$$tr.(\alpha^i\beta_j\gamma^{-1})=\sum_{k=1}^m\sigma_k(\alpha^i\beta_j\gamma^{-1})=\sum_{k=1}^n\sigma_k(\alpha)^i\sigma_k(\beta... |
3,629,790 | <p>So I am a bit stuck on applying the limit definition for this problem. This is what I have so far: </p>
<p><span class="math-container">$\lvert x - c \rvert < \delta $</span> then <span class="math-container">$\lvert f(x) - l \rvert < \epsilon$$ $$\lvert x + 2 \rvert < \delta$</span> <span class="math-cont... | José Carlos Santos | 446,262 | <p>Note that<span class="math-container">\begin{align}x^2+x-5-(-3)&=x^2-4+x+2\\&=(x-2)(x+2)+x+2\\&=(x-1)(x+2).\end{align}</span>So, if <span class="math-container">$\lvert x+2\rvert<1$</span>, then <span class="math-container">$\lvert x-1\rvert\leqslant\lvert x+2\rvert+3<4$</span>. So, take <span clas... |
1,179,922 | <p>I would like to inquire whether there is a simple way to prove that a function is decreasing or not. For example how would I prove that the function</p>
<p>$$Y = (X^.5 - 1)/0.5$$ </p>
<p>is decreasing? I am not sure whether the negativity of the 2nd derivative solves this and would really like to understand the in... | user149418 | 149,418 | <p>I am posting this answer because you wanted to know the monotonicity in terms of the derivative, otherwise this answer does not make any sense as there is already a nice answer of your question.</p>
<p>So if a function is differentiable then the monotonicity can be described as follows.</p>
<blockquote>
<p>Let <span... |
1,898,839 | <p>Is the following inequality true?</p>
<p><span class="math-container">$$\mbox{Tr} \left( \mathrm P \, \mathrm M^T \mathrm M \right) \leq \lambda_{\max}(\mathrm P) \, \|\mathrm M\|_F^2$$</span></p>
<p>where <span class="math-container">$\mathrm P$</span> is a positive definite matrix with appropriate dimension. How a... | Branimir Ćaćić | 49,610 | <p>The heart of the matter is the following key fact:</p>
<blockquote>
<p>The assignment $(A,B) \mapsto \langle A,B \rangle := \operatorname{Tr}(A^T B)$ defines a Euclidean inner product on the real vector space $\mathbb{R}^{m \times n}$ of all $m \times n$ real matrices that induces the Frobenius norm $\|\cdot\|_F$... |
4,802 | <p>Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups a... | Steven Landsburg | 10,503 | <p>I believe Bloch's original insight was something like the following:</p>
<p>First, if $X$ is a regular scheme, you can filter $K_0$ by ``codimension of support''; that is, view $K_0(X)$ as the Grothendieck group of the category of all finitely generated modules and let $F^iK_0(X)$ be the part generated by modules w... |
2,898,616 | <p>My Math teacher told me that $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$. I asked for a proof and he gave me the result by using the chain rule. I don't understand why this is the case for every function because the function $y=x^2$ doesn't have an inverse function. If this is the case then why is $\frac{dx}{dy}$ no... | Christoph | 86,801 | <p>$\newcommand\R{\mathbb R}$Let's avoid Leibniz notation. In Lagrange notation, the <a href="https://en.wikipedia.org/wiki/Inverse_function_theorem" rel="nofollow noreferrer">inverse function theorem</a> for a differentiable function $f\colon\R\to\R$ states that when $f(a)=b$ then
$$
g'(b) = \frac 1{f'(a)},
$$
where $... |
2,007,123 | <p>Suppose $R=\mathbb{Q}[x]$ is a ring and define $T$ to be $T=\{f(x) \in R : f(0) \neq 0\}$ </p>
<p>Prove that $\frac{x}{1}$ is the only irreducible element in $T^{-1}R$ (disregarding associates).</p>
<hr>
<p>My approach: assume $\frac{x}{1}$ is reducible. So there exist non-unit elements $\frac{a}{b}, \frac{c}{d}$... | Han de Bruijn | 96,057 | <p>Not meant as a duplicate of someone else's answer - hope it's complementary,
but I want this section to be coherent and complete, for future reference as well.
Apologies if things have become somewhat overdone.<P>
The parent element that will be adopted for
our own purpose is <I>not quite the unit</I> square. Instea... |
3,401 | <p>Consider an idealized classical particle confined to a two-dimensional surface that is frictionless. The particle's initial position on the surface is randomly selected, a nonzero velocity vector is randomly assigned to it, and the direction of the particle's movement changes only at the surface boundaries where pe... | Joseph O'Rourke | 6,094 | <p>To supplement Andy's answer, there is a recent survey by Laura Demarco, "The conformal geometry of billiards," <a href="http://www.ams.org/journals/bull/2011-48-01/S0273-0979-2010-01322-7/home.html" rel="nofollow"><em>Bulletin AMS</em> 48(1), Jan 2011, pp. 33-52</a>.
She defines a billiard table as <em>ergodically o... |
4,253,485 | <p>I understand that if you roll <span class="math-container">$10$</span> <span class="math-container">$6$</span>-faced dice (D6) the different outcomes should be <span class="math-container">$60,466,176$</span> right? (meaning <span class="math-container">$1/60,466,176$</span> probability)</p>
<p>but if you are playi... | Oscar Lanzi | 248,217 | <p>When you roll a 60-face die, you are in effect rolling ten six-face dice but counting only one at a time, which is very different from counting the entire combination of ten independent six-face dice.</p>
<p>You can approximate a fair 60-face die with a red six-face die, a blue one and several white ones, call the n... |
2,828,023 | <p><strong>Four cards are dealt off the top of a well-shuffled deck.
Find the chance that:</strong></p>
<p><strong>You get a queen or a king.</strong> </p>
<p>The solution shows $ 1- \frac { \binom {44}{4}}{ \binom {52}{4}}$ </p>
<p>I get that in this case it's easier to use compliment rule, however I'm trying anot... | Boyku | 567,523 | <p>If the deal has no Queens and no Kings, the four cards are chosen among the 48 remaining and we have :</p>
<p>$ \binom {52-8}{4} $ posibilities among $ \binom {52}{4} $</p>
<p>If we get some queens or some kings, we get $1- \frac { \binom {44}{4}}{ \binom {52}{4}}$, the pointed solution.</p>
|
4,417,576 | <p>Say we divide 1858.54 cm by 5 -- where 5 is an exact number. We get the quotient 371.708</p>
<p>According to the rules of significant figures, we would answer write 371.708 cm, since "1858.54" is 6 significant figures, and 5, being an exact number doesn't count.</p>
<p>My question is, how do you make sense... | Ross Millikan | 1,827 | <p>Significant figures are a heuristic for keeping track of accuracy in a calculation. If you want to be careful, you need to assign an uncertainty to each number and keep track of how they build up. You also need to decide whether you want the error band on your result to be a worst case or some sort of "<span ... |
119,492 | <p>First I define a function, just a sum of a few sin waves at different angular frequencies:</p>
<pre><code>ubdat = 50;
ws = 10*{2, 5, 10, 20, 40}
fn = Table[Sum[Sin[w*x], {w, ws}], {x, 0, ubdat, .001}];
pts = Length@fn
ListPlot[fn, Joined -> True, PlotRange -> {{0, 1000}, All}]
{20, 50, 100, 200, 400}
</code>... | bill s | 1,783 | <p>Actually, LowpassFilter is pretty easy to use, if you understand the parameters it needs. For instance, replacing the OPs line with</p>
<pre><code>fnfilt = LowpassFilter[fn, 140, 200, SampleRate -> 1000];
</code></pre>
<p>gives exactly what you would expect.</p>
<pre><code>ListPlot[fnfilt, Joined -> True, P... |
396,363 | <p>I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to calculate both, the disitrbution functions $F_D(t), F_Z(t)$ and the the density functions $f_D(t),f_Z(t)$</p>
<p>To do... | Did | 6,179 | <blockquote>
<p>The area of a $\color{red}{\mathrm{right\ triangle}}$ being half the product of its legs is all that one needs, really...</p>
</blockquote>
<p>For example, $D\geqslant x$ with $0\leqslant x\leqslant\frac12$ means that $(X,Y)$ is either in the triangle $\color{red}{T_x}$ with vertices $(\frac12+x,\fra... |
2,845,033 | <p>Empirically this equality holds:</p>
<p>$\gcd(10,8) = 2 $ and $\gcd(\operatorname{mod}(10,8),8) = \gcd(2,8) = 2 $</p>
<p>$\gcd(18,9) = 9 $ and $\gcd( \operatorname{mod}(18,9),9) = \gcd(0,9) = 9 $</p>
<p>I am stuck on how to prove it,though , and do not understand why this holds true.</p>
| dan_fulea | 550,003 | <p>Let us take $\beta=-2$. Then:
$$
\begin{aligned}
&\frac{\partial}{\partial x}\Big(\ e^{-2x}\cdot y\ \Big)
+
\frac{\partial}{\partial y}\Big(\ e^{-2x}\cdot (-2x-y-3x^4+y^2)\ \Big)
\\
&\qquad = -2e^{-2x}\cdot y+ e^{-2x}\cdot (-1+2y)=-e^{-2x}
\\
&\qquad<0\ .
\end{aligned}
$$
This is enough to apply Dulac... |
3,107,194 | <p>Suppose <span class="math-container">$X$</span> is a compact metric space and <span class="math-container">$f: X \rightarrow \mathbb{R}$</span> is a function for which <span class="math-container">$f^{-1}([t,\infty))$</span> is closed for any real <span class="math-container">$t$</span>. Then <span class="math-conta... | Sangchul Lee | 9,340 | <p>Here is a slightly more general statement:</p>
<blockquote>
<p><strong>Claim.</strong> Let <span class="math-container">$K_1 \supseteq K_2 \supseteq \cdots $</span> be non-empty closed subsets of a metric space <span class="math-container">$X$</span>. If <span class="math-container">$x_n \in K_n$</span> for each ... |
3,451,782 | <blockquote>
<p>Let <span class="math-container">$M \in \mathcal{M}_n(\mathbb{R})$</span> invertible. Find a matrix <span class="math-container">$N$</span> such that <span class="math-container">$M + N$</span> is not invertible and <span class="math-container">$\|N\|_2$</span> is minimum.</p>
</blockquote>
<p>Is eas... | Fred | 380,717 | <p>No. We denote the unit in a <span class="math-container">$C^*$</span>- algebra by <span class="math-container">$e$</span>. Then take <span class="math-container">$a=ie$</span> and <span class="math-container">$b=e$</span> and <span class="math-container">$ \lambda=1-i.$</span> Then </p>
<p><span class="math-contain... |
3,451,782 | <blockquote>
<p>Let <span class="math-container">$M \in \mathcal{M}_n(\mathbb{R})$</span> invertible. Find a matrix <span class="math-container">$N$</span> such that <span class="math-container">$M + N$</span> is not invertible and <span class="math-container">$\|N\|_2$</span> is minimum.</p>
</blockquote>
<p>Is eas... | Sriram Balasubramanian | 273,433 | <p>It seems like one can conclude at the least, <span class="math-container">$a = a^*$</span> and <span class="math-container">$\lambda = \overline{\lambda}$</span> from just assuming that <span class="math-container">$a + \lambda b$</span> is self-adjoint, as follows. Have <span class="math-container">$a + \lambda b =... |
4,626,019 | <p>I recently encountered a quant interview question, which I think is not super hard, but still I am not very sure about my results.</p>
<p>Could you guys give me some hints on it?</p>
<p>The question goes as follows:</p>
<blockquote>
<p>An ant is put on a vertex of a simplex, and at each vertex, it has equal probabil... | Sarvesh Ravichandran Iyer | 316,409 | <h5>A brief note on simplification</h5>
<p>This question is related to the harmonic numbers because it can be formulated as a version of the coupon collector problem.</p>
<p>Indeed, this is how it would work. The "graph" of the <span class="math-container">$n$</span>-simplex, when you treat each of the corner... |
973,101 | <p>I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid.</p>
<p>If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables $\{x_1,x_2,x_3\}$, calculate the distance from the origin</p>
<p>$$d=\sqrt{x_1^2+x_2^2+x_3^2}$$</p>
<p>and calculat... | Stéphane Laurent | 38,217 | <ul>
<li><p>Simulate some points on the sphere (with arbitrary dimension) <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, <span class="math-container">$...$</span></p>
</li>
<li><p>Take the upper Cholesky factor <span class="math-container">$U$</span> of the shape matrix <span clas... |
913,239 | <p>Given a circle with known center $c$, known radius $r$ and perimeter point $x$:
$$
(x - c_x)^2 + (y - c_y)^2 = r^2
$$
with a tangent line that also goes through a point $p$ lying outside the circle. How do I find the point $x$ at which the line touches the circle?</p>
<p>Given that the tangent line is orthogonal to... | loldrup | 84,883 | <p>Inspired by the <a href="http://www.mathopenref.com/consttangents.html" rel="nofollow noreferrer">visual proof</a> supplied by <a href="https://math.stackexchange.com/users/43949/angryavian">angryavian</a> I have derived this proof:</p>
<p>Lets define the middle point between $c$ and $p$:
$$
m = ((c - p) / 2) + p
$... |
195,065 | <p>Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a nondecreasing function.</p>
<p>Let $a<f(a)$ and $f(b)<b$. Prove that there is a $a<c<b$ such that $f(c)=c$. </p>
<p>My attempt at a proof is as follows. Let $c:=\sup\{x:a\leq x\leq b\text{, }x\leq f(x)\}$.</p>
<p>This is where I'm stuck. Since I can't u... | Taylor | 22,255 | <p>Define another function $g(x) = f(x) - x$. Then try to find out if this function crosses $0$. </p>
|
4,361,063 | <blockquote>
<p>Suppose that <span class="math-container">$(X, \tau_1)$</span> is compact and <span class="math-container">$\tau_1 \subset \tau_2$</span>. Is the space <span class="math-container">$(X, \tau_2)$</span> compact? Does the converse hold i.e if <span class="math-container">$(X, \tau_1)$</span> is compact an... | User | 525,755 | <p>I think your confusion for the second one arises from the meaning of "finite subcover". A finite subcover is a finite cover consisting of elements of the initial cover. So, if you cover <span class="math-container">$X$</span> by elements of <span class="math-container">$\tau_2$</span>, these elements are a... |
4,361,063 | <blockquote>
<p>Suppose that <span class="math-container">$(X, \tau_1)$</span> is compact and <span class="math-container">$\tau_1 \subset \tau_2$</span>. Is the space <span class="math-container">$(X, \tau_2)$</span> compact? Does the converse hold i.e if <span class="math-container">$(X, \tau_1)$</span> is compact an... | MBolin | 251,330 | <p><span class="math-container">$X$</span> being <span class="math-container">$\tau_1$</span>-compact means that for every net there exists a <span class="math-container">$\tau_1$</span>-convergent subnet. If <span class="math-container">$\tau_2 \subseteq \tau_1$</span>, convergence in <span class="math-container">$\ta... |
1,401,116 | <p>An aeroplane is observed by two persons travelling at $60 \frac{km}{hr} $ in two vehicles moving in opposite directions on a straight road. To an observer in one vehicle the plane appears to cross the road track at right angles while to the observer in the other vehicle the angle appears to be $45°$. At what angle ... | Batominovski | 72,152 | <p>Although I didn't try to prove it, I think that all polynomial solutions to this functional equation (without the boundedness condition) are of the form $f(x,y)=a+bx+cy+dxy+e\left(x^2-y^2\right)$ for all $x,y\in\mathbb{Z}$, where $a,b,c,d,e$ are fixed real numbers. There are other solutions like $f(x,y)=\text{Re}\l... |
2,922,494 | <p>I came across the following problem today.</p>
<blockquote>
<p>Flip four coins. For every head, you get <span class="math-container">$\$1$</span>. You may reflip one coin after the four flips. Calculate the expected returns.</p>
</blockquote>
<p>I know that the expected value without the extra flip is <span class="m... | Thomas Andrews | 7,933 | <p>This particular problem can be solved simply by flipping five coins but winning a maximum of \$4.</p>
<p>That gives an expected value of: <span class="math-container">$$\frac{1\cdot 0 + 5\cdot 1 + 10\cdot 2 + 10\cdot 3 + (5+1)\cdot 4}{2^5}=\frac{79}{32}.$$</span></p>
|
392,441 | <p>Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional.</p>
<p>I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation $f\delta′=f(0)\delta′−f′(0)\delta$.</p>
| not all wrong | 37,268 | <p>I'm assuming you mean the delta function(al) $\delta(x)$.</p>
<p>If so, you should use formal integration by parts.
$$\int g\times (f\delta') =\int -(fg)'\delta = \int -(f'(0)g(x)+g'(x)f(0))\delta = -f'(0) \int g\delta-f(0)\int g'\delta$$</p>
<p>Integrating the last term by parts and stripping off the $\int g\time... |
2,996,476 | <p>So I was asked to solve the following sum using Fourier Series:</p>
<p><span class="math-container">$$\sum_{n\in\Bbb N}\frac{(-1)^n}{4n^2-1}^{(1)}$$</span></p>
<p>I decided to try splitting it up into two sums (for even and odd <span class="math-container">$n$</span>) in the hopes that it would telescope and be easi... | Andreas | 317,854 | <p>I would go about it in a similar way like you do, just shorter, as follows:</p>
<p>By partial fractions, </p>
<p><span class="math-container">$$
\frac{1}{4n^2-1} = \frac12 \left(\frac{1}{2n-1} - \frac{1}{2n+1} \right)
$$</span>
So
<span class="math-container">$$
\sum_{n\in\Bbb N}\frac{(-1)^n}{4n^2-1}^{(1)} = \fra... |
163,654 | <p>Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.</p>
| Włodzimierz Holsztyński | 8,385 | <p>For the sake of completeness, let's look at the following simple example of very <em>nice</em> continua, and of their inverse system of inclusions, for which there is no <strong>homological</strong> surjection.</p>
<p>All continua are subspaces of $\ \mathbb R^2$. Let $\ X:=(0\ 0)\ $ be a single-point space. Let $\... |
2,058,560 | <p>Can anyone please simplify this boolean expression ? My answer always reduces to a single variable i.e $x$ but my instructor reduced to three literals.
$$x'y'z+x'yz'+xy'z'+xyz$$ </p>
| barak manos | 131,263 | <p>You have $3$ variables, hence $2^3=8$ possible combinations of them.</p>
<p>Your expression is an OR on $4$ out of these $8$ combinations:</p>
<p>$x'y'z+x'yz'+xy'z'+xyz$.</p>
<p>So there isn't really any way to simplify it.</p>
<p>You can do an OR on the NOT of the other $4$ combinations:</p>
<p>$(x'yz)'+(xy'z)... |
3,954,010 | <p>The case <span class="math-container">$n=3$</span> is from <a href="https://usamts.org/Tests/Problems_31_1.pdf" rel="nofollow noreferrer">here</a>. It's straightforward to prove it's true:</p>
<p>First we notice that if any two of <span class="math-container">$x_1, x_2, x_3$</span> are equal then all must be equal.<... | Neat Math | 843,178 | <p>After reading <a href="https://math.stackexchange.com/a/3358130/843178">this answer</a> to another question, as well as <a href="https://math.stackexchange.com/a/3954365/843178">Andreas's answer</a> I decide to write up the following solution. I believe John Omielan had already figured out how to do it, though he ch... |
111,343 | <p>Hallo,</p>
<p>I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the leaves of $A$ are transvers to the leaves of $B$ at each point. Also the leaves of $A$ are $n$-dim complex manifol... | s-f | 27,789 | <p>We start by giving an answer that rephrases the Nijenhuis integrability condition
using the foliations, $A,B$, but
the main point here is
to present a more geometric<br>
criterion for integrability, related to complex structures of moduli spaces.
The main claim is that leafwise
holomorphicity of certain natural ... |
1,296,669 | <p>A tapas bar serves 15 dishes, of which 7 are vegetarian, 4 are fish and 4 are meat. A table of customers decides to order 8 dishes, possibly including repetitions.</p>
<p>a) Calculate the number of possible dish combinations. </p>
<p>b) The customers decide to order 3 vegetarian dishes, 3 fish and 2 meat. Calculat... | Diamantino | 133,038 | <p>a) The given that all dishes are different, then
$$
N_{orders}=15^8
$$</p>
<p>b) In this case, we can calculate how many different orders for each type and then combine them:
$$
N_{orders} = 7^3 \times 4^3 \times 4^2
$$</p>
|
2,925,450 | <p>I am trying to solve the following problem: <span class="math-container">$|4-x| \leq |x|-2$</span>. I am trying to do it algebraically, but I'm getting a solution to the problem that makes no sense. I fail to see the error in my reasoning though. I hope to get an explanation where I went wrong.</p>
<p><span class="... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>observe that there is no solution if the RHS is negative which gives $$|x|<2$$</p>
<p>so, solve in $[2,4], [4,+\infty)$ and $(-\infty,-2]$.</p>
|
2,692,124 | <blockquote>
<p>A group <span class="math-container">$G$</span> is finite iff all its elements have the same finite order?</p>
</blockquote>
<p>One direction (when <span class="math-container">$G$</span> is finite) is trivial following from Lagrange's theorem. But if a each element of <span class="math-container">$G$</... | Robert Z | 299,698 | <p>No, take for example the infinite group $G=\prod_{n=1}^\infty{\mathbb Z}_2$ where every non-identity element has order 2.</p>
|
549,438 | <p>Need to take the limit:
$$\lim_{x \to 1}\left(\frac{x}{x-1}-\frac{1}{\ln(x)}\right) = \lim_{x \to 1}\left(\frac{x\cdot \ln(x)-x+1}{(x-1)\cdot \ln(x)}\right)=(0/0)$$
Now I can use L'Hospital's rule:
$$\lim_{x \to 1}\left(\frac{1\cdot \ln(x)+x\cdot \frac{1}{x}-1}{1\cdot \ln(x)+(x-1)\cdot\frac{1}{x}}\right)= \lim_{x \t... | Ramanpreet Nara | 100,442 | <p>You're not guaranteed a solution with one application of L'Hopital's rule. Sometimes, you'll have to apply it multiple times to get the limit. </p>
|
1,808,183 | <p>Consider the Cauchy problem:
$$\left\{\begin{array}{lll}
x^2\partial_x u+y^2\partial_yu=u^2\\
u(x,2x)=1
\end{array}\right.$$
It is easy to show that the characteristic equations are given by:
$$\frac{dx}{x^2}=\frac{dy}{y^2}=\frac{dz}{px^2+qy^2}=\frac{dp}{2pu-2xp}=\frac{dq}{2qu-2yq}=dt$$
By the Lagrange-Charpit's met... | Gino | 780,023 | <p>As you well stated, the Brownian bridge is a GP. That means that given training outputs <span class="math-container">$f$</span> and test outputs <span class="math-container">$f_{\star}$</span>
the joint prior distribution is</p>
<p><span class="math-container">$$\begin{bmatrix}f\\f_{\star}\end{bmatrix} \sim N \Bigg... |
2,070,353 | <p>Let there be two arbitrary functions $f$ and $g$, and let $g \circ f$ be their composition.</p>
<ol>
<li><p>Suppose $g \circ f$ is one-one. Then prove that this implies $f$ is also one-one.</p></li>
<li><p>Suppose $g \circ f$ is onto. Then prove that this implies $g$ is also onto.</p></li>
</ol>
<p>I tried to do b... | Paolo Leonetti | 45,736 | <p>$$
1+\frac{1}{2}((2n+1)-7)=n-2.
$$</p>
|
4,610,640 | <p>I have this equation :</p>
<p><span class="math-container">$$\frac{d\alpha}{dz} = - \frac{dr}{dz} * \frac{\tan(\alpha)}r $$</span></p>
<p>I searched for some similar examples but non of these equations was like this one.
I'm confused about this one. As far as I know, I used RK4 for equations like this : <span class=... | Bob Dobbs | 221,315 | <p><span class="math-container">\begin{array}{rr|l}
&s+1 & s+2\\
&s+2 & \\
\hline &\text{(Remiander)}\color{lime}{-1}&\color{purple}{1} \text{(Quotient)}
\end{array}</span>
Hence,
<span class="math-container">$$\frac{s+1}{s+2}=\color{purple}{1}+\frac{\color{lime}{-1}}{s+2}=1-\frac{1}{s+2}.$$</... |
4,139,003 | <p>For this I used the method of making the determinant nonzero. First I create the array</p>
<p><span class="math-container">\begin{equation}
\begin{pmatrix}
a^2 & 0 & 1\\
0 & a & 2\\
1 & 0 & 1
\end{pmatrix}
\end{equation}</span></p>
<p>Then <span class="math-container">$\det(A)=a^3-a=a(a^2-1)$... | Jean Marie | 305,862 | <p><a href="https://i.stack.imgur.com/VdglK.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VdglK.jpg" alt="enter image description here" /></a></p>
<p><em>Fig. 1: (Some circles of) the pencil of circles <span class="math-container">$P_1$</span> in blue, and <span class="math-container">$P_2$</span> ... |
2,107,164 | <p>Let $G$ be gorup and $H$ be normal subgroup of $G$.</p>
<p>Suppose that $H \cong \mathbb{Z}$ ($\mathbb{Z}$:integer) and $G/H \cong \mathbb{Z}/ n \mathbb{Z} (\mathbb{Z} \ni n \geq 2 )$.</p>
<p>Then I'm stuck in next problems.</p>
<p>(1)If $n$ is odd number, G is abelian group.</p>
<p>(2)Classfy the group G up to... | Nicky Hekster | 9,605 | <p>Let me give you a headstart: in general, if $N \unlhd G$, then conjugation on $N$ induces an automorphism of $N$, hence $G/C_G(N) \hookrightarrow Aut(N)$. Here $C_G(N)=\{g \in G: g^{-1}ng=n, $ for all $n \in N\}$, the centralizer of $N$ in $G$. Now $Aut(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. Hence, if $n$ is odd... |
3,714,021 | <p>I've been working with the power series <span class="math-container">$\sum_{n=0}^{\infty}n(1-2^{-n})z^n$</span> for the last few hours and I'm starting to get a little lost in trying to figure out a few things for an exercise.</p>
<p>First of all, I've managed to find the radius of convergence ofr the series with r... | Fred | 380,717 | <p>If <span class="math-container">$b_n=a_n z^n$</span>, then we have for <span class="math-container">$|z|=1$</span> that</p>
<p><span class="math-container">$$|b_n|=|a_n|=n-n\frac{1}{2^n} \to \infty$$</span></p>
<p>as <span class="math-container">$n \to \infty$</span>.</p>
<p>Conclusion ?</p>
|
432,291 | <p>This question is related to <a href="https://mathoverflow.net/questions/421450/homotopy-type-theory-how-to-disprove-that-0-operatornamesucc0-in-the-ty">Homotopy type theory : how to disprove that <span class="math-container">$0=\mathrm{succ}(0)$</span>
in the type <span class="math-container">$\mathbb N$</span></a>.... | Daniel Gratzer | 76,636 | <p>It's a bit confused to view <span class="math-container">$\mathsf{code}$</span> as well-defined because 0 and its successor are not definitionally equal. Rather, it's well-defined because of the induction principle associated with the natural number type; the pattern-matching notation is a shorthand for invoking the... |
2,391,812 | <p>The question is to find the largest integer that divides all $p^4-1$, where p is a prime greater than 5. Being asked this question, I just assume this number exists. Set $p = 7$, then $p^4-1=2400$. I don't have any background in number theory and not sure what to do next. Thank you for your help! </p>
| Community | -1 | <p>Let $n$ be the number you are trying to find.</p>
<p>One of the main reasons to look at an example $p=7$ is to gain information about $n$. If $n$ divides all $p^4 - 1$, and $7^4 - 1 = 2400$, then $n$ divides $2400$.</p>
<p>Thus, you've gained information about $n$. Follow that up — what information does know... |
567,428 | <p>Let $A \ne V$ be a subspace of $V$ and $B$ a linearly independent subset of $V$. Prove that $B$ can be completed to a basis of $V$ with vectors from $V \setminus A$.</p>
<p>OK, I started with: </p>
<p>$K=\operatorname{lin}(v_1,v_2,v_3,\dotsc)$</p>
<p>and if any of $v_1,v_2,\dotsc$ belongs to $A$ we can replace th... | user99680 | 99,680 | <p>this should do it: Let {$B:=v_1,..,v_n$} and consider a basis $B_V$:={$v_1,..,v_m$} (i.e., let $DimV=m$ )for $V$; $n>m$ . Construct a matrix $M_{ij}$ with $n+m$ rows ,where the first $n$ rows are the vectors in $B$ and the remaining $m$ rows are the vectors of $B_V$. Now do Gaussian elimination on $M$ . The res... |
2,447,549 | <p>If so, could someone give me an example?</p>
| Ben Grossmann | 81,360 | <p>Yes. For example, take the transformation $T:\Bbb R \to \Bbb R^2$ given by
$$
T(x) = (x,0)
$$</p>
|
2,447,549 | <p>If so, could someone give me an example?</p>
| Fred | 380,717 | <p>Let $V$ be the vector space of all real sequences and let $T:V \to V$ be defined by</p>
<p>$T(x_1,x_2,x_3,....)=(0,x_1,x_2,x_3,...)$.</p>
<p>Then $T$ is injective but not surjective.</p>
|
4,214,892 | <p>In the spherical coordinate system</p>
<p><span class="math-container">$x=r \sin \theta \cos \phi$</span></p>
<p><span class="math-container">$y=r \sin \theta \sin \phi$</span></p>
<p><span class="math-container">$z=r \cos \theta$</span></p>
<p><span class="math-container">$\theta$</span> lies in <span class="math-c... | Mathman | 777,644 | <p>Imagine standing in the middle of your kitchen. You can turn around through a vertical axis(your phi) and with your arm by your side can raise or lower your hand(your theta). Then you can point to any object in your room without breaking your elbow (thats theta more than 180)</p>
<p>Also take care here, many sources... |
4,214,892 | <p>In the spherical coordinate system</p>
<p><span class="math-container">$x=r \sin \theta \cos \phi$</span></p>
<p><span class="math-container">$y=r \sin \theta \sin \phi$</span></p>
<p><span class="math-container">$z=r \cos \theta$</span></p>
<p><span class="math-container">$\theta$</span> lies in <span class="math-c... | miracle173 | 11,206 | <p>We have</p>
<p><span class="math-container">$$x=r \sin (-\theta) \cos \phi=r (-\sin \theta) \cos \phi=r \sin \theta (-\cos \phi)=r \sin \theta \cos(\phi+\pi)$$</span>
<span class="math-container">$$y=r \sin (-\theta) \sin \phi=r (-\sin \theta) \sin \phi=r \sin \theta (-\sin \phi)=r \sin \theta \sin (\phi+\pi)$$</spa... |
2,925,160 | <p>We know that <span class="math-container">$\mathbb{R}$</span> is a linear space over <span class="math-container">$\mathbb{Q}$</span>, denoted by <span class="math-container">$\mathbb{R_{\mathbb{Q}}}$</span>,where the vector addition is the real numbers and scalar multiplication is the number in <span class="math-co... | Magdiragdag | 35,584 | <p>The following may feel like cheating: pick your favorite transcendental element of <span class="math-container">${\mathbb R}$</span>, e.g., <span class="math-container">$\pi$</span>. Then <span class="math-container">$1, \pi, \pi^2, \dots, \pi^{n-1}$</span> are linearly independent over the field over algebraic real... |
3,490,024 | <p>This is part of a physics problem I was doing yesterday. I am supposed to find the maximum value of
<span class="math-container">$$\mu^{-1}\cos\theta+\sin\theta$$</span>
This is is supposed to produce the result of <span class="math-container">$\sqrt{1+\mu^{-2}}$</span>. However I am not getting that. By taking the... | rogerl | 27,542 | <p>You just need to simplify the result.</p>
<p>When you set <span class="math-container">$\theta = \tan^{-1}\mu$</span> you get
<span class="math-container">$$\cos\theta = \frac{1}{\sqrt{1+\mu^2}},\quad \sin\theta = \frac{\mu}{\sqrt{1+\mu^2}},$$</span>
so that
<span class="math-container">$$\mu^{-1}\cos\theta + \sin... |
1,836,236 | <p>For which $a$ and $b$ is this matrix diagonalizable?</p>
<p>$$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$</p>
<p>How to get those $a$ and $b$? I calculated eigenvalues and eigenvectors, but don't know what to do next?</p>
| Merkh | 141,708 | <p>Calculating the eigenvalues, $b, b+a, a-b,$ one can then easily calculate their respective eigenvectors. For eigenvalue $b,$
\begin{equation}
\begin{pmatrix} a-b&0&b\\0&0&0\\b&0&a-b \end{pmatrix}\begin{pmatrix} 0\\1\\0\end{pmatrix} = \begin{pmatrix} 0\\0\\0\end{pmatrix},
\end{equation}
There... |
3,453,626 | <p>I was playing around with the concepts of injectivity, surjectivity, and bijectivity recently. I used these three Wikipedia articles as references:</p>
<p><a href="https://en.wikipedia.org/wiki/Injective_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Injective_function</a></p>
<p><a href="https... | user | 293,846 | <p><span class="math-container">$$\begin{align}
m_a^4+m_b^4+m_c^4&=\frac1{16}\sum_\text{cyc.}(2b^2+2c^2-a^2)^2\\
&=\frac1{16}\sum_\text{cyc.}4b^4+4c^4+a^4+8b^2c^2-4a^2b^ 2-4c^2a^2\\
&=\frac1{16}(9a^4+9b^4+9c^4).
\end{align}$$</span></p>
|
1,180,293 | <p>This is from <em>Herstein</em>. </p>
<blockquote>
<p>$4.$ $\;a)$ Given a group $G$ and a subset $U$ denote by $\hat U$ the
smallest subgroup of $G$ which contains $U$ (the subgroup generated by
$U$). Prove there is a subgroup $\hat U$ of $G$. </p>
<p>$\;\;\;\;\;b)$ If $gug^{-1} \in U$ for all $g \in G$ ... | Frank Lu | 41,622 | <p><strong>Hint:</strong> Show that
$$V:=\{v\in \hat{U}: gvg^{-1}\in \hat{U},\forall g\in G\}$$
is a subgroup containing $U$. It is clear that $V\subseteq\hat{U}$. Also by minimality of $\hat{U}$, it follows that $V\supseteq\hat{U}$. Hence $V=\hat{U}$, this means for all $v\in\hat{U}$ we have $gvg^{-1}\in\hat{U},\fora... |
1,180,293 | <p>This is from <em>Herstein</em>. </p>
<blockquote>
<p>$4.$ $\;a)$ Given a group $G$ and a subset $U$ denote by $\hat U$ the
smallest subgroup of $G$ which contains $U$ (the subgroup generated by
$U$). Prove there is a subgroup $\hat U$ of $G$. </p>
<p>$\;\;\;\;\;b)$ If $gug^{-1} \in U$ for all $g \in G$ ... | Fan Zheng | 94,743 | <p>First show that for every $g\in G$ and $A\in \mathscr A$, $gAg^{=1}\in \mathscr A$.
Indeed,</p>
<p>$$A\in \mathscr A\Rightarrow U\subset A\Rightarrow U=gUg^{-1}\subset gAg^{-1}.$$</p>
<p>Now</p>
<p>$$\hat U=\bigcap_{A\in\mathscr A}A\subset gAg^{-1}$$</p>
<p>for every $A\in \mathscr A$. Hence</p>
<p>$$\hat U\sub... |
1,180,293 | <p>This is from <em>Herstein</em>. </p>
<blockquote>
<p>$4.$ $\;a)$ Given a group $G$ and a subset $U$ denote by $\hat U$ the
smallest subgroup of $G$ which contains $U$ (the subgroup generated by
$U$). Prove there is a subgroup $\hat U$ of $G$. </p>
<p>$\;\;\;\;\;b)$ If $gug^{-1} \in U$ for all $g \in G$ ... | athos | 26,632 | <p>First we prove that $\forall \hat u \in \hat U$, $\exists u, v \in U$, s.t. $\hat u = uv$.</p>
<p>To do so we define set $\hat W = \{ uv | uv \in \hat U, u, v \in U \}$, and prove $\hat W$ is exactly $\hat U$.</p>
<p>Consider $\forall w_1, w_2 \in \hat W$.</p>
<p>By definition of $\hat W$, we have $w_1, w_2 \in \... |
2,076,984 | <p>I have already asked <a href="https://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola">a similar question</a>. But the answer in that question is very difficult to understand. I am new to this concept so I am looking for an easier explanation.</p>
<blockquote>
<p>My main <strong... | Dan Uznanski | 167,895 | <p>$\int_a^b f(x)$ gives the area "under the curve" of $f(x)$ between $a$ and $b$: the area from the $x$-axis to the curve, across that interval.</p>
<p>In the cases given, you have two curves that you're dealing with instead; one (which I'll call $f(x)$) is higher and the other ($g(x)$) is lower. Finding the area be... |
2,076,984 | <p>I have already asked <a href="https://math.stackexchange.com/questions/2075949/how-to-draw-diagram-for-line-and-parabola">a similar question</a>. But the answer in that question is very difficult to understand. I am new to this concept so I am looking for an easier explanation.</p>
<blockquote>
<p>My main <strong... | Narasimham | 95,860 | <p>It is not complicated. One common area is falling on a similar area as overlap, some addition is involved and where from the minus sign? that seems to be your doubt ... which you expressed nicely with a graphic.I edited your picture to explain as follows.</p>
<p><a href="https://i.stack.imgur.com/ny5W9.png" rel="no... |
2,444,376 | <p>Let be $u:\mathbb{R}\rightarrow(-\infty,+\infty]$ a convex function and I suppose that $u$ admits a point of minimum.
I define:</p>
<p>$$(\varphi_\epsilon*u)(x)=\int_{\mathbb{R}^n}\varphi_\epsilon(y)u(x-y)dy, $$
where $\varphi_{\epsilon}$ is the standard mollifier.
Let's introduce the notation:
$$\tilde{u}_i=\varph... | H. H. Rugh | 355,946 | <p>Assuming that your mollifier has compact support, e.g. within $[-\epsilon,\epsilon]$ and is positive, the result follows from the fact that $u_\epsilon(x)=(\phi_\epsilon \star u)(x)$ lies in the convex hull of the values of $u$ within the interval $[x-\epsilon,x+\epsilon]$.</p>
<p>Suppose the minimum is $m=u(x_0)$ ... |
971,997 | <p>How do we find
$$\Re\left[\int_0^{\large\frac{\pi}{2}} e^{\Large e^{i\theta}}d\theta\right]$$</p>
<p>In the shortest and easiest possible manner?</p>
<p>I cannot think of anything good.</p>
| Arian | 172,588 | <p>$$e^{i\theta}=z\Rightarrow ie^{i\theta}d\theta=dz\Rightarrow d\theta=\frac{dz}{iz}$$
Consider the following integral
$$\oint_{\gamma}e^z\frac{dz}{iz}$$
where $\gamma$ is a contour of a quarter of a unit disk. Note that the integrand has a simple pole at $z=0$ so the chosen contour will go in the clockwise direction ... |
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