qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,297,319 | <p>integration equation </p>
<p>$$\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$$</p>
<p>my work </p>
<p>$t= \sqrt{(1-4x^2)} $</p>
<p>$dt = -4x/\sqrt{(1-4x^2)} dx $</p>
<p>stuck here also </p>
| Joseph Martin | 203,227 | <p>You can use integration by substitution $ 2x=sin(u) $. Then $ \frac{d}{du}x=\frac{1}{2}\cos(u) $.</p>
<p>We can rearrange our substitution equation $ 2x=sin(u) $ into $ u = arcsin(2x) $. So we can find our limits with respect to u. When $ x = \frac{1}{8} \: \: u = arcsin(2\frac{1}{8}) = arcsin(\frac{1}{4}) $ and ... |
3,789,060 | <p>I was asked the following question:</p>
<blockquote>
<p>Determine if the following set is a vector space:<br />
<span class="math-container">$$W=\left\{\left[\begin{matrix}p\\q\\r\\s\\\end{matrix}\right]:\begin{matrix}-3p+2q=-s\\p=-s+3r\\\end{matrix}\right\}$$</span></p>
</blockquote>
<p>I know the answer is yes and... | Brian Moehring | 694,754 | <p>Write <span class="math-container">$\{x\} = x - \lfloor x\rfloor$</span>. Since <span class="math-container">$$\arctan(\cot(\pi x)) = \arctan(\cot(\pi\{x\})) = \arctan(\tan(\frac{\pi}{2}-\pi\{x\})) \\ -\frac{\pi}{2} < \frac{\pi}{2} - \pi\{x\} \leq \frac{\pi}{2}$$</span> it follows that <span class="math-containe... |
1,879,395 | <p>I am trying to learn generating functions so I am trying this recurrence:</p>
<p>$$F(n) = 1 + \frac{n-1}{n}F(n-1)$$</p>
<p>But I am struggling with it. Luckily the base case can be anything since $F(1)$ will multiply it by $0$ anyway, so let's say $F(0) = 0$. Then I tried this:</p>
<p>$$G(x) = \sum_{n=0}^{\infty}... | André Nicolas | 6,312 | <p>Hint: Life will be more pleasant if we let $kF(k)=W(k)$. Then we are looking at $W(n)=n+W(n-1)$. The generating function is straightforward, and then we can obtain the generating function of $F$.</p>
|
1,879,395 | <p>I am trying to learn generating functions so I am trying this recurrence:</p>
<p>$$F(n) = 1 + \frac{n-1}{n}F(n-1)$$</p>
<p>But I am struggling with it. Luckily the base case can be anything since $F(1)$ will multiply it by $0$ anyway, so let's say $F(0) = 0$. Then I tried this:</p>
<p>$$G(x) = \sum_{n=0}^{\infty}... | Jack D'Aurizio | 44,121 | <p>The usual GF-approach may go through the following lines. We have $F(0)=0$ and $n F(n) = n + (n-1) F(n-1)$. Assuming that
$$ G(x) = \sum_{n\geq 0}F(n) x^n = \sum_{n\geq 1}F(n) x^n,\tag{1} $$
we have:
$$ x\cdot G'(x) = \sum_{n\geq 1} n F(n)\,x^{n}=\sum_{n\geq 0} n F(n)\,x^{n}, \tag{2}$$
$$ x^2\cdot G'(x) = \sum_{n\ge... |
733,553 | <p>It's been a long time since high school, and I guess I forgot my rules of exponents. I did a web search for this rule but I could not find a rule that helps me explain this case:</p>
<p>$ 2^n + 2^n = 2^{n+1} $</p>
<p>Which rule of exponents is this?</p>
| Kaj Hansen | 138,538 | <p>$2^{N} + 2^{N} = 2^{N}(1+1) = 2^{N}(2) = 2^{N+1}.$</p>
<p>Just a little trickery with the distributive law.</p>
|
121,362 | <p>I have a set of sample time-series data below of monthly prices for two companies. </p>
<p>Q1. I want to calculate monthly and quarterly log returns.what is the most expedient way to do this? <code>TimeSeriesAggregate[]</code> only has the standard <code>Mean</code>, etc. </p>
<p>Q2. With the returns from Q1, wha... | wuyingddg | 18,981 | <p>Anothor way by <code>NestList</code></p>
<pre><code>randomTriPlot[n_] := Module[{next},
next[polys_] :=
Join[Map[# + {-1, -Sqrt[3]} &,
polys, {2}], {MapAt[# - 2 Sqrt[3] &,
polys[[-1]], {1, 2}], # + {1, -Sqrt[3]} & /@ polys[[-1]]}];
(*get coordinate of the next layer by translate thi... |
26,893 | <p>Does there exists a function $f \in C^2[0,\infty]$ (that is, $f$ is $C^2$ and has finite limits at $0$ and $\infty$) with $f''(0) = 1$, such that for any $g \in L^p(0,T)$ (where $T > 0$ and $1 \leq p < \infty$ may be chosen freely) we get
$$
\int_0^T \int_0^\infty \frac{u^2-s}{s^{5/2}} \exp{\left( -\f... | Zarrax | 3,035 | <p>As Joriki pointed out in his comment, this is equivalent to finding an $f(u)$ such that for all $0 \leq s \leq T$ one has
$$\int_0^{\infty}(u^2 - s)\exp{(-{u^2 \over 2s})}f(u) \,du = 0$$
Write $\int_0^{\infty}u^2\exp{-({u^2 \over 2s})}f(u)\,du$ as
$-\int_0^{\infty}-{u \over s}\exp{(-{u^2 \over 2s})}suf(u)\,du$ and... |
4,146,081 | <p>How can I demonstrate the Jacobi identity:</p>
<p><span class="math-container">\begin{equation}
[S_{i}, [S_{j},S_{k}]] + [S_{j}, [S_{k},S_{i}]] + [S_{k}, [S_{i},S_{j}]] = 0 ~,
\end{equation}</span></p>
<p>using the infinitesimal generators <span class="math-container">$S_{\kappa}$</span> for a continuous group, wher... | paul garrett | 12,291 | <p>Recapitulating @DietrichBurde's point, in different terms:</p>
<p>Again, emphatically, if we have a real or complex vector space <span class="math-container">$V$</span> with an anti-commutative binary (bilinear) operation <span class="math-container">$[,]$</span>, this does not imply that <span class="math-container... |
370,007 | <p>A river boat can travel a 20km per hour in still water. The boat travels 30km upstream against the current then turns around and travels the same distance back with the current. IF the total trip took 7.5 hours, what is the speed of the current? Solve this question algebraically as well as graphically..</p>
<p>I st... | Diego | 301,198 | <p>Suppose $(ab)^t=a^t b^t=1$. Then $t/mn$. Note that $t$ cannot be a multiple of $m$ and not of $n$, since then $a^tb^t=b^t \neq 1$. Now suppose $t$ is neither a multiple of $n$ nor a multiple of $m$. Then let $n=p_{1}^{\alpha_1}...p_k^{\alpha_k}$ and $m=q_1^{\beta_1}...q_n^{\alpha_n}$. Then $t$ takes a proper piece $... |
936,138 | <p>I need help approaching a proof which deals with inequalities:</p>
<p>If p and r are the precision and recall of a test, then the F1 measure of the test is
defined to be
$$F(p, r) = \frac{2pr}{p+r}$$</p>
<p>Prove that, for all positive reals p, r, and t, if t ≥ r then F(p, t) ≥ F(p, r)</p>
<p>What's the first ste... | Khosrotash | 104,171 | <p>$$\frac{1}{p}+\frac{1}{r}=\frac{p+r}{pr}\\so\\ f(p,r)=2 \frac{1}{\frac{1}{p}+\frac{1}{r}}\\\frac{1}{\frac{1}{p}+\frac{1}{r}}\\\\is -harmonic-mean$$</p>
|
3,416,895 | <p>here's the relevant question: <a href="https://math.stackexchange.com/q/193157/716946">If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$</a></p>
<p>In the accepted answer, <strong>doesn't the last inequality only work if <span class="math-container">$... | user284331 | 284,331 | <p><span class="math-container">\begin{align*}
&\limsup_{n}\left(\dfrac{1}{n}\sum_{j=1}^{k}s_{j}+\dfrac{n-k}{n}\sup_{l\geq k}s_{l}\right)\\
&\leq\limsup_{n}\dfrac{1}{n}\sum_{j=1}^{k}s_{j}+\limsup_{n}\dfrac{n-k}{n}\sup_{l\geq k}s_{l}\\
&=\lim_{n}\dfrac{1}{n}\sum_{j=1}^{k}s_{j}+\lim_{n}\dfrac{n-k}{n}\sup_{l\g... |
1,970,235 | <p>If I remember right, $f(x)$ is continuous at $x=a$ if</p>
<ol>
<li><p>$\lim_{x \to a} f(x)$ exists</p></li>
<li><p>$f(a)$ exists</p></li>
<li><p>$f(a) = \lim_{x \to a} f(x)$</p></li>
</ol>
<p>So $\lim_{x \to 0^{-}} \sqrt{x}$ exists? Thus $\lim_{x \to 0^{-}} \sin(\sqrt{x})$ <a href="https://math.stackexchange.com/q... | avs | 353,141 | <p>If we define $f(x) = \sqrt{x}$ only for $x \geq 0$, then the function is <em>continuous on the right</em>; i.e., there the right-sided limit
$$
\lim_{x \rightarrow 0+} f(x).
$$
As for the left-sided limit, it all depends on how $f(x)$ is defined for $x<0$.</p>
|
2,189,832 | <p>Take the matrix
$$
\begin{matrix}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{matrix}
$$</p>
<p>I tried to calculalte the eigenvalues of this matrix and got to a point where I found that the eigenva... | Ofek Gillon | 230,501 | <p>How did you calculate the eigenvalues? </p>
<p>It is easy to see that Wolfram is correct by multiplying $Av$ for each vector.</p>
<p>More over, you can see that the rank of the matrix is $1$, meaning there are $3$ linearly independent vectors that satisfy</p>
<p>$$Av = 0 = 0v$$</p>
<p>Meaning there needs to be 3... |
1,925,867 | <p>I can't find any. For saying $H$ is a subgroup of $G$ we have notation but it seems none exists for subrings.</p>
| quid | 85,306 | <p>This is correct. There is a notation for ideal, yet no notation for subring, as far as I know.
One just writes: "Let $S \subset R$ be a subring." </p>
<p>Anyway, also the notation for subgroup, ideal and alike is not used all that much. "Let $H\subset G$ be a subgroup." Is what I see all the time. </p>
|
1,925,867 | <p>I can't find any. For saying $H$ is a subgroup of $G$ we have notation but it seems none exists for subrings.</p>
| Keith Kearnes | 310,334 | <p>I have used and seen $S\leq R$ to say that $S$ is a substructure of $R$.</p>
|
1,119,027 | <p>I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this.</p>
<p>My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that I'll hit a straight flush?</p>
<p>My reasoning is like this:</p>
<p>$$\Pr(\text{straight flush}|H4H5) = (\Pr(... | Ross Millikan | 1,827 | <p>Hint: You still have three cards to draw out of $50$. How many combinations of three cards result in a straight flush? How many total draws are there?</p>
|
3,223,705 | <p>I have a task for school and we need to plot a polar function with MATLAB. The function is <span class="math-container">$r = 1-2\cos(6\theta)$</span>.</p>
<p>I did this and I'm getting exactly the same as on Wolfram Alpha: <a href="https://www.wolframalpha.com/input/?i=polar+plot+r%3D1-2" rel="nofollow noreferrer">... | Bernard | 202,857 | <p>As the function <span class="math-container">$r$</span> has period <span class="math-container">$\frac\pi3$</span>, the curve is invariant by rotations of angle <span class="math-container">$\frac\pi 3$</span>. Hence you have to draw a petal of the curve for <span class="math-container">$0\le\theta\le\frac\pi 3$</sp... |
1,186,825 | <p>Prove $$\lim_{n\to\infty}\int_0^1 \left(\cos{\frac{1}{x}} \right)^n\mathrm dx=0$$</p>
<p>I tried, but failed. Any help will be appreciated.</p>
<p>At most points $(\cos 1/x)^n\to 0$, but how can I prove that the integral tends to zero clearly and convincingly?</p>
| xpaul | 66,420 | <p>Note that if $x=\frac{1}{k\pi}$ ($k\in\mathbb{N}$), $\cos\frac{1}{x}=(-1)^k$. Fix $\varepsilon\in(0,1)$ such that $\frac1{\varepsilon \pi}$ is not an integer. Let $M=[\frac1{\varepsilon \pi}]$. Clearly if $k<M$, then $\frac{1}{k\pi}\in(\varepsilon,1]$
let
$$ I_k=(\frac{1}{k\pi}-\frac{\varepsilon}{2^k}, \frac{1... |
2,789,002 | <p>How can I calculate the height of the tree? I am with geometric proportionality.</p>
<p><img src="https://i.stack.imgur.com/m4zMD.png"></p>
| fleablood | 280,126 | <p><a href="https://i.stack.imgur.com/R4bUb.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R4bUb.jpg" alt="enter image description here"></a></p>
<p>$FG \approx DE$ as $AG \approx AE$.</p>
<p>And $FG \approx AG$ as $DE \approx AE$ and as $BC \approx AC$.</p>
<p>Or perhaps most sophisticatedly: If... |
3,029,208 | <p>Hi I have been trying to find a way to find a combinatorial proof for <span class="math-container">${kn \choose 2}= k{n \choose 2}+n^2{k \choose 2}$</span>. </p>
| user | 505,767 | <p>Consider a set of <span class="math-container">$k\cdot n$</span> elements allocated in a grid with <span class="math-container">$n$</span> rows and <span class="math-container">$k$</span> columns then </p>
<ul>
<li>on the LHS we have the ways to choose <span class="math-container">$2$</span> elements among all of t... |
1,704,410 | <p>If we have two groups <span class="math-container">$G,H$</span> the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product <span class="math-container">$G\times H$</span> into a group it is certainly by defining the multiplication</p>
<p><span class... | Alex Provost | 59,556 | <p>Forget about the actual construction of the semidirect product for now. I argue that the semidirect product is important because it arises naturally and beautifully in many areas of mathematics. I will list below many examples, and I urge you to find a few that interest you and look at them in detail.</p>
<p>Before... |
1,704,410 | <p>If we have two groups <span class="math-container">$G,H$</span> the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product <span class="math-container">$G\times H$</span> into a group it is certainly by defining the multiplication</p>
<p><span class... | Ari Royce Hidayat | 435,467 | <p>Many have given good answers here, so I just want to answer specifically for the intuition behind it.</p>
<p>Semi direct product came into light when we found out that if a group <span class="math-container">$H$</span> is a normal subgroup, and another group <span class="math-container">$K$</span> is also a subgrou... |
2,103,436 | <p>Suppose we have the vector space $V$ and the non-empty subspace $W$. I know there is a theorem that states that if $\bar{v}_1$ and $\bar{v}_2$ are vectors in a subspace $W$ then the vector $(\bar{v}_1 + \bar{v}_2)$ will also be in the subspace $W$. However is the converse true? Would having the vector $(\bar{v}_1 + ... | Siong Thye Goh | 306,553 | <p>This is not true. Consider the trivial subspace that consist of only the zero vector.</p>
<p>Pick any non-zero vector, $v$, it is not inside $W$. but $v-v=0$ </p>
|
2,316,042 | <p><strong>Problem:</strong> Consider the set of all those vectors in $\mathbb{C}^3$ each of whose coordinates is either $0$ or $1$; how many different bases does this set contain?</p>
<p>In general, if $B$ is the set of all bases vectors then,
$$B=\{(x_1,x_2,x_3),(y_1,y_2,y_3),(z_1,z_2,z_3)\}.$$</p>
<p>There are $8... | deinst | 943 | <p>Knowing that none of x,y,z can be (0,0,0) there are only 7 choices for each. Since they must be different you only have $\binom{7}{3}=35$ choices to make.</p>
<p>This is small enough to sort through by hand.</p>
|
4,176,646 | <p>I need to find the directional derivatives for all vectors <span class="math-container">$u=[u_1\ \ u_2]\in \mathbb R^2$</span> with <span class="math-container">$\|u\|=1$</span> at <span class="math-container">$P_0=(0,0)$</span>, and determine whether <span class="math-container">$f$</span> is differentiable at <spa... | Asher2211 | 742,113 | <p>The number of trialing zeros in binary notation is the highest power of <span class="math-container">$2$</span> that divides the number in decimal notation.</p>
<p><span class="math-container">$t(n)=\displaystyle \sum_{k=1}^\infty \left( 1-\left\lceil\left\{\frac{n}{2^k}\right\}\right\rceil\right)$</span> satisfies ... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | jmm | 57,731 | <pre><code>Table[
If[Mod[n, 3] != 0, data[[n]], Nothing], {n, 1, Length[data]}]
</code></pre>
|
1,617,269 | <p>Let X and Y be independent random variables with probability density functions
$$f_X(x) = e^{-x} , x>0$$
$$f_Y(y) = 2e^{-2y} , y>0$$</p>
<p>Derive the PDF of $Z_1 = X + Y$</p>
<p>other cases: $Z =min(X,Y)$ , $Z =1/Y^2 $ , $Z =e^{-2y} $ </p>
<p>Just considering the 1st part, I understand to go from the fac... | Graham Kemp | 135,106 | <p>You have obtained:$$\begin{align}
f_{X+Y}(z) & = \frac{\operatorname d \mathsf P(X+Y\leq z)}{\operatorname d z}
\\[1ex] & = \frac{\operatorname d }{\operatorname d z} \int_0^z \int_0^{z-x} f_X(x)\,f_Y(y)\operatorname d y\operatorname d x
\\[1ex] & = \int_0^zf_X(x)\,f_Y(z-x)\operatorname d x
\\[1ex] &... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Gadi A | 1,818 | <p>A nice example in Algebraic number theory is the solution of the $p=x^2+ny^2$ problem, described in details in <a href="http://rads.stackoverflow.com/amzn/click/0471190799" rel="nofollow">Cox's book</a>. Very much like Fermat's last theorem it has its roots in the 17th century (actually, it originated from Fermat...... |
39,684 | <p>In order to have a good view of the whole mathematical landscape one might want to know a deep theorem from the main subjects (I think my view is too narrow so I want to extend it).</p>
<p>For example in <em>natural number theory</em> it is good to know quadratic reciprocity and in <em>linear algebra</em> it's good... | Community | -1 | <ul>
<li>Spectral theorem in Functional Analysis.</li>
</ul>
|
2,530,458 | <p>Find Range of $$ y =\frac{x}{(x-2)(x+1)} $$</p>
<p>Why is the range all real numbers ? </p>
<p>the denominator cannot be $0$ Hence isn't range suppose to be $y$ not equals to $0$ ?</p>
| 5xum | 112,884 | <p>The range is $\mathbb R$ because for every $y\in\mathbb R$, there exists some $x$ such that $$\frac{x}{(x-2)(x+1)}=y.$$</p>
<p>For example, for $y=0$, you have $$\frac{0}{(0-2)(0+1)}=\frac{0}{(-2)\cdot 1} = -\frac{0}{2}=0.$$</p>
<hr>
<p>For a general $y$, you have to show that the equation above has a solution, w... |
2,799,123 | <p>Prove the following equation by counting the non-empty subsets of $\{1,2,\ldots,n\}$ in $2$ different ways:</p>
<p>$1+2+2^2+2^3\ldots+2^{n-1}=2^n-1$.</p>
<p>Let $A=\{1,2\ldots,n\}$. I know from theory that it has $2^n-1$ non-empty subsets, which is the right-hand side of the equation but, how do count the left one... | N. F. Taussig | 173,070 | <p>The right-hand side counts non-empty subsets of the set $\{1, 2, 3, \ldots, n\}$. </p>
<p>The left-hand side counts non-empty subsets of the set $\{1, 2, 3, \ldots, n\}$ whose largest element is $k$, $1 \leq k \leq n$. The number of such subsets is $2^{k - 1}$ since such a subset is determined by choosing which o... |
3,009,387 | <p>I'm asking the following: it is true that if <span class="math-container">$K$</span> is a normal subgroup of <span class="math-container">$G$</span> and <span class="math-container">$K\leq H\leq G$</span> then <span class="math-container">$K$</span> is normal in <span class="math-container">$H$</span>? I tried to pr... | Mefitico | 534,516 | <p><strong>Hint:</strong> Try factorization!</p>
<p><span class="math-container">$$
\frac{2x^2-50}{2x^2+3x-35}=\frac{2(x^2-25)}{(1/2)(4x^2+6x-70)}=\frac{4(x-5)(x+5)}{(2x+10)(2x-7)}
$$</span></p>
|
2,853,401 | <p>Assume $E\neq \emptyset $, $E \neq \mathbb{R}^n $. Then prove $E$ has at least one boundary point. (i.e $\partial E \neq \emptyset $).</p>
<p>================= </p>
<p>Here is what I tried.<br>
Consider $P_0=(x_1,x_2,\dots,x_n)\in E,P_1=(y_1,y_2,\dots,y_n)\notin E $.<br>
Denote $P_t=(ty_1+(1-t)x_1,ty_2+(1-t)x_2,\... | Joe | 524,659 | <p>Here's a proof that doesn't use connectedness at all. Suppose that $\emptyset \neq E \subsetneq \mathbb{R}^n$. Now take $x \in \mathbb{R}^n \setminus E$. If $x$ is a boundary point of $E$, we're done! Otherwise, take $\delta = \sup \{\epsilon : \epsilon > 0, B_{\epsilon}(x) \cap E = \emptyset \}$, where $B_\epsil... |
1,156,874 | <p>How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well?</p>
<p>I know how to show that it is an integral domain, because that follows very quickly.</p>
| Jyrki Lahtonen | 11,619 | <p>A small variation of Arthur's argument. I wanted to do this without using the fact that the complex norm works as a Euclidean domain norm as well.</p>
<hr>
<p>The claim is not true, if $I=\{0\}$, so let's assume that is not the case. So then there exists an element $z=a+bi\in I$ with either $a$ or $b$ non-zero. Be... |
1,156,874 | <p>How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well?</p>
<p>I know how to show that it is an integral domain, because that follows very quickly.</p>
| S. Venkataraman | 457,895 | <p>Let us count the number of elements in <span class="math-container">$\mathbb{Z}[i]/n\mathbb{Z}[i]$</span>. When do two elements <span class="math-container">$a+ib$</span>, <span class="math-container">$a_1+ib_1$</span> in <span class="math-container">$\mathbb{Z}[i]$</span> belong to the same coset of <span class="ma... |
1,123,777 | <p><strong><span class="math-container">$U$</span> here represents the upper Riemann Integral.</strong></p>
<p><img src="https://i.stack.imgur.com/GbNm2.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack.imgur.com/KtRI4.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack... | Brian M. Scott | 12,042 | <p>HINT: As an alternative to an inductive approach: for each possible choice of $k$ (the number of terms), there is exactly one choice of $a_1$ that allows the inequalities to be satisfied and the sum of the $a_i$’s to be $n$.</p>
|
1,123,777 | <p><strong><span class="math-container">$U$</span> here represents the upper Riemann Integral.</strong></p>
<p><img src="https://i.stack.imgur.com/GbNm2.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack.imgur.com/KtRI4.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack... | anomaly | 156,999 | <p>The given partitions $(a_1, \dots, a_k)$ are all given by
\begin{align*}
a_i &= \begin{cases}
a_1 & \text{if $i \leq p$}; \\
a_1 + 1 & \text{if $i > p$}
\end{cases}
\end{align*}<br>
for $p = 1, \dots, k$. Count the number of pairs $(a_1, p)$ for which the partition sums to $n$.</p>
|
130,564 | <p>Hi, everyone.</p>
<p>I am looking for some references for period matrix of abelian variety over arbitrary field, if you know, could you please tell me?</p>
<p>For period matrix of abelian varieties, I means that if $A$ is an abelian variety over complex number field, $A \cong V/\Gamma$. If we chose a basis of $V$,... | S. Carnahan | 121 | <p>You won't get $A$ as a quotient of a vector space in general without some kind of strange transcendentality. For example, if $V$ is defined over $\mathbb{F}_p$, then any $S$-valued point of $V$ is $p$-torsion for any test object $S$. In particular, you can't possibly obtain any of the prime-to-$p$ torsion in $A$ wi... |
749,926 | <p>I have a group of 10 players and I want to form two groups with them.Each group must have atleast one member.In how many ways can I do it?</p>
| vonbrand | 43,946 | <p>The "proper" way to solve this is that you want to know the ways to partition 10 elements into 2 parts; the ways to partition $n$ elements into $k$ groups is given by the <a href="http://en.wikipedia.org/wiki/Stirling_number" rel="nofollow">Stirling number</a> of the second kind $\genfrac{\{}{\}}{0pt}{}{n}{k}$, in y... |
1,840,352 | <blockquote>
<p>For every $A \subset \mathbb R^3$ we define $\mathcal{P}(A)\subset \mathbb R^2$ by
$$ \mathcal{P}(A) := \{ (x,y) \mid \exists_{z \in \mathbb R}:(x,y,z) \in A \} \,. $$
Prove or disprove that $A$ closed $\implies$ $\mathcal{P}(A)$ closed and $A$ compact $\implies$ $\mathcal{P}(A)$ compact.</p>
</bl... | John Gowers | 26,267 | <p><strong>Hints:</strong></p>
<ol>
<li><p>The fact that you are unable to prove this might suggest that the statement is actually untrue. Try and work out where your proof is falling apart and use that to construct a closed set $A\subset \mathbb R^3$ such that $P(A)$ is not closed.</p></li>
<li><p>This is not as dif... |
362,926 | <p>I have a problem that looks like this:</p>
<p>$$\frac{20x^5y^3}{5x^2y^{-4}}$$</p>
<p>Now they said that the "rule" is that when dividing exponents, you bring them on top as a negative like this:</p>
<p>$$4x^{5-2}*y^{3-(-4)}$$</p>
<p>That doesn't make too much sense though. A term like $y^{-4}$ is essentially say... | Javier | 2,757 | <p>What the other solution did is a straightforward application of the rule: $\frac1{y^n} = y^{-n}$. For $n = -4$, you get $\frac1{y^{-4}} = y^{-(-4)} = y^4$. Does that make it clear?</p>
|
1,339,649 | <p>Summation convention holds. If $\frac{\partial}{\partial t}g_{ij}=\frac{2}{n}rg_{ij}-2R_{ij}$, then ,I compute:
$$
\frac{1}{2}g^{ij}\frac{\partial}{\partial t}g_{ij}=\frac{1}{2}g^{ij}(\frac{2}{n}rg_{ij}-2R_{ij})=\frac{1}{n}r(\sum\limits_i\sum\limits_jg^{ij}g_{ij})-g^{ij}R_{ij}=nr-R
$$</p>
<p>But on the Hamilton's ... | Chappers | 221,811 | <p>$$ \sum_j g^{ij} g_{kj} = \delta^i_k $$
Now, $\delta^i_i = 1$, with no summation, so
$$ \sum_i \sum_j g^{ij} g_{ij} = \sum_i \delta_i^i = \sum_i 1 = n. $$
You may be confusing whether or not you are using summation convention.</p>
|
3,219,428 | <p>Sorry for the strange title, as I don't really know the proper terminology.</p>
<p>I need a formula that returns 1 if the supplied value is anything from 10 to 99, returns 10 if the value is anything from 100 to 999, returns 100 if the value is anything from 1000 to 9999, and so on.</p>
<p>I will be translating th... | P Vanchinathan | 28,915 | <p>Is a mathematical formula really needed ? Or a function (in the sense of a programming language) that does this job ok? As you say you are going to translate this into code it is simpler to directly translate your description to code, without having to find a mathematical formula</p>
<p>Here is it in Python: (you ... |
679,544 | <p>How to prove this for positive real numbers?
$$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$</p>
<p>I tried AM-GM, CS inequality but all failed.</p>
| Community | -1 | <p>Here other two answers used Cauchy-Scwartz Inequality. I am giving a simple $AM\ge GM$ inequality proof.</p>
<p>You asked, $$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}\ge a+b+c\\\implies a^4+b^4+c^4\ge a^2bc+b^2ca+c^2ab$$</p>
<p>Now, from, $AM\ge GM$, we have $$\frac {a^4+
a^4+b^4+c^4}4\ge \left(a^4\cdot a^4\cdo... |
679,544 | <p>How to prove this for positive real numbers?
$$a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$$</p>
<p>I tried AM-GM, CS inequality but all failed.</p>
| Michael Rozenberg | 190,319 | <p>By Holder
$$\sum_{cyc}\frac{a^3}{bc}\geq\frac{(a+b+c)^3}{3(ab+ac+bc)}=\frac{(a+b+c)\cdot(a+b+c)^2}{3(ab+ac+bc)}\geq a+b+c$$</p>
|
5,896 | <p>$\sum_{n=1}^{\infty} \frac{\varphi(n)}{n}$ where $\varphi(n)$ is 1 if the variable $\text n$ has the number $\text 7$ in its typical base-$\text10$ representation, and $\text0$ otherwise.</p>
<p>I am supposed to find out if this series converges or diverges. I think it diverges, and here is why.</p>
<p>We can see ... | Douglas S. Stones | 139 | <p>Your argument seems fine to me... so I'll give the argument that popped into my head when I read the question.</p>
<p>Sum phi(n)/n for n congruent to 7 (mod 10) and multiply by 10 (which does not affect divergence/convergence). Note that</p>
<pre><code>10/7 > 1/7 +1/8 +...+1/16
10/17 > 1/17+1/18+...+1/26
... |
5,896 | <p>$\sum_{n=1}^{\infty} \frac{\varphi(n)}{n}$ where $\varphi(n)$ is 1 if the variable $\text n$ has the number $\text 7$ in its typical base-$\text10$ representation, and $\text0$ otherwise.</p>
<p>I am supposed to find out if this series converges or diverges. I think it diverges, and here is why.</p>
<p>We can see ... | Aryabhata | 1,102 | <p>Yes your argument seems fine.</p>
<p>Another argument is:</p>
<p>Divide the integers into blocks of 10</p>
<p>$$[1 \dots 10] [11 \dots 20] [21 \dots 30] \dots$$</p>
<p>Each block will have an number with digit $7$ in it and so the series has sum at least</p>
<p>$$\frac{1}{10} + \frac{1}{20} + \dots + \frac{1}{1... |
4,183,263 | <p>If Tychonoff's theorem is true, why closed ball in <span class="math-container">$\mathbb{R}^n$</span> is not compact?</p>
<p>The theorem says that if <span class="math-container">$X_i$</span> is compact, for every <span class="math-container">$i\in I$</span>, so <span class="math-container">$\prod_{i\in I}X_i$</span... | Henno Brandsma | 4,280 | <p>An infinite product of copies of <span class="math-container">$[-1,1]$</span>, say, is indeed compact in the <em>product</em> topology, but a Banach space topology is <em>not</em> like a product topology; the "closest" we can get to that kind of a topology in an infinite-dimensional Banach space is the so-... |
1,634,325 | <blockquote>
<p><strong>Problem</strong>:
Is there sequence that sublimit are $\mathbb{N}$? If it's eqsitist prove this.</p>
</blockquote>
<p>I try to solve this problem by guessing what type of sequence need to be. <br>For example:
$a_n=(-1)^n$ has two sublimit $\{1,-1\}$.
<br>
$a_n=n
\times\sin(\frac{\pi}{2})$ h... | Jimmy R. | 128,037 | <p>Such a sequence exists and this is possible due to the (not so intuitive perhaps) fact, that $\mathbb N\times \mathbb N$ and $\mathbb N$ have the same cardinality (surprise!). But since they have the same cardinality, there is a bijection (actually many), $b:\mathbb N \times \mathbb N\to \mathbb N$. Now, for any $m ... |
1,556,298 | <p>If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.</p>
<p>So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.</p>
<p>Let $p$ be 'I insert a coin', and $q$ - 'I ge... | Ove Ahlman | 222,450 | <p>Well, yes, you can not say that any formula $\varphi$ (or more explicit $p\rightarrow q$) is allways true just based on one row of a truth table. You have to check all the rows. However If $p$ is evaluated to $True$ and $q$ is evaluated to $True$ then $p\to q$ is evaluated to $True$, thus $p\to q$ is true in this <s... |
2,364,742 | <p>$$\log_5\tan(36^\circ)+\log_5\tan(54^\circ)=\log_5(\tan(36^\circ)\tan(54^\circ)).$$ I cannot solve those 2 tangent functions above. Here calculator comes in handy to calculate it. Is there a method of evaluating this problem without a calculator?</p>
| user362325 | 362,325 | <p>Note that $\tan(36^\circ)=\tan(90^\circ-54^\circ)=\frac{1}{\tan54^\circ}$</p>
<p>$$\log_5(\tan(36^\circ)\tan(54^\circ))$$</p>
<p>$$=\log_5(\frac{1}{\tan54^\circ}\tan(54^\circ))$$</p>
<p>$$=\log_5(1)=0$$</p>
|
2,364,742 | <p>$$\log_5\tan(36^\circ)+\log_5\tan(54^\circ)=\log_5(\tan(36^\circ)\tan(54^\circ)).$$ I cannot solve those 2 tangent functions above. Here calculator comes in handy to calculate it. Is there a method of evaluating this problem without a calculator?</p>
| Khosrotash | 104,171 | <p>HInt: $36^\circ+54^\circ =90$ so
$$tan(54^\circ)=cot(36^\circ)\\$$
$$\log_5\tan(36^\circ)+\log_5\tan(54^\circ)=\log_5(\tan(36^\circ)\tan(54^\circ))=\\\log_5(\tan(36^\circ)\cot(36^\circ))=log_5(1)=0$$</p>
|
1,413,145 | <p>I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing.</p>
<p>$\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for $n\ge6$.)</p>
<p>I was led to this problem by trying to prove by induction that $\big(1+\frac{1}{n}\big)^n\le3-\f... | Jack D'Aurizio | 44,121 | <p>As suggested by Clement C., let:
$$ f(x)=\left(1+\frac{1}{x}\right)^{x}+\frac{1}{x}.\tag{1}$$
Then:
$$ f'(x) = \left(1+\frac{1}{x}\right)^{x}\left(\log\left(1+\frac{1}{x}\right)-\frac{1}{x+1}\right)-\frac{1}{x^2}\tag{2} $$
but, due to convexity:
$$\log\left(1+\frac{1}{x}\right)-\frac{1}{x+1}=-\frac{1}{x+1}+\int_{x}^... |
1,413,145 | <p>I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing.</p>
<p>$\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for $n\ge6$.)</p>
<p>I was led to this problem by trying to prove by induction that $\big(1+\frac{1}{n}\big)^n\le3-\f... | Arin Chaudhuri | 404 | <p>Here is another way to approach this problem.
The function $$f(z) = 1 - z/2 + z^2/3 + \ldots + (-1)^{k+1} z^k/(k+1) + \ldots $$ is analytic on the unit disc $ \{ z : |z| < 1\}$, which implies $ g(z) = \exp f(z)$ is also analytic on $ \{ z : |z| < 1\}$ and hence can be expanded as a power series $$g(z) = a_0 + ... |
622,552 | <p>In the context of (most of the times convex) optimization problems -</p>
<p>I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum of the primal problem from the one of the dual. Now I want to find the primal optimum point (i.e. the point in... | user3589786 | 168,570 | <p>If you solved the dual and have optimal values for the dual variables, then plug those optimal dual values back into the Lagrange equation. Now you have an equation with only x as unknown. Minimize, and you've recovered the value for x.</p>
|
622,552 | <p>In the context of (most of the times convex) optimization problems -</p>
<p>I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum of the primal problem from the one of the dual. Now I want to find the primal optimum point (i.e. the point in... | OtZman | 431,258 | <p>I know this question was asked some time ago, but I just stumbled upon it myself wondering the same thing, so I will post what I have found out in case it is helpful to someone else.</p>
<p>I think the answer you seek is in Boyd and Vandenberghe's Convex Optimization (freely available on Boyd's website: <a href="ht... |
96,080 | <p>The empty clause is a clause containing no literals and by definition is false.</p>
<p>c = {} = F</p>
<p>What then is the empty set, and why does it evaluate to true?</p>
<p>Thanks!</p>
| Ted | 15,012 | <p>Remember, we take the disjunction over the elements of a <em>clause</em>, then the conjunction over the entire <em>clause set</em>. So if the <em>clause set</em> is empty, then we have an empty <em>conjunction</em>. If the <em>clause itself</em> is empty, then we have an empty <em>disjunction</em>.</p>
<p>What do... |
8,695 | <p>I have a parametric plot showing a path of an object in x and y (position), where each is a function of t (time), on which I would like to put a time tick, every second let's say. This would be to indicate where the object is moving fast (widely spaced ticks) or slow (closely spaced ticks). Each tick would just be... | rm -rf | 5 | <p>To create the ticks perpendicular to the curve, I calculate the direction of the normal to the curve where the ticks are to be placed and then orient a line segment along that direction. The following code does that.</p>
<pre><code>ClearAll[ParametricTimePlot]
SetAttributes[ParametricTimePlot, HoldAll]
ParametricTi... |
974,656 | <p><img src="https://i.stack.imgur.com/LyqzL.jpg" alt="enter image description here"></p>
<p>One way to solve this and my book has done it is by : </p>
<p><img src="https://i.stack.imgur.com/2wYSn.jpg" alt="enter image description here"></p>
<hr>
<p>This is a well known way, but I have a different method, and it se... | A. Breust | 184,254 | <p><strong>Hints</strong></p>
<p>The polynomial is of even degree.
What does this means about the limits at $x\rightarrow-\infty$ and $+\infty$?</p>
<p>Now what is $P(0)$? Using the fact that $a_na_0<0$ means that $a_n$ and $a_0$ are of opposite signs, you should be able to finish.</p>
|
2,738,957 | <p>I did the following to derive the value of $\pi$, you might want to grab a pencil and a piece of paper:</p>
<p>Imagine a unit circle with center point $b$ and two points $a$ and $c$ on the circumference of the circle such that triangle $abc$ is an obtuse triangle. you can see that if $\theta$ denotes the angle $\an... | dxiv | 291,201 | <blockquote>
<p>$$\lim_{\theta \to 0} -\frac{\theta\pi}{90} = \lim_{\theta \to 0} \sin(2\theta) $$</p>
</blockquote>
<p>The above is correct, but it does not imply the following line:</p>
<blockquote>
<p>$$\pi = -\lim_{\theta \to 0} \frac{90\sin(2\theta)}{\theta}$$</p>
</blockquote>
<p>The fallacy here is expect... |
28,532 | <p><code>MapIndexed</code> is a very handy built-in function. Suppose that I have the following list, called <code>list</code>:</p>
<pre><code>list = {10, 20, 30, 40};
</code></pre>
<p>I can use <code>MapIndexed</code> to map an arbitrary function <code>f</code> across <code>list</code>:</p>
<pre><code>{f[10, {1}],... | tenure track job seeker | 6,251 | <p>You can also do this:</p>
<pre><code>list = {10, 20, 30, 40};
newlist =list/.{a_,b_,c_,d_}:>{a,f[b,2],f[c,3],d}
</code></pre>
<p>In this way, you give a lable to any element of your list, and using <code>:></code>, you can map some other function on some of elements of your list. In the above code for examp... |
3,425,415 | <p>I need to define a bijection <span class="math-container">$f:\mathbb Q\to\mathbb Q$</span> such that <span class="math-container">$f(0) = 0$</span> and <span class="math-container">$f(1) = 1$</span> while also preserving order (i.e. if <span class="math-container">$a < b$</span>, then <span class="math-container"... | Ross Millikan | 1,827 | <p>One approach is just to use two straight lines.
<span class="math-container">$$f(x)= \begin {cases} \frac x2 & x \le \frac 12\\
\frac 14+\frac 32(x-\frac 12)& \frac 12 \lt x\end{cases}$$</span>
[<img src="https://i.stack.imgur.com/zkUBn.png" alt="enter image description here"></p>
|
4,383,557 | <p>This question came up in an oral exam. During the course we studied a bit of the theory of lie algebras and some representation theory.</p>
<p>The question: show that the lie algebra <span class="math-container">$\mathfrak{g_2}$</span> has a dimension <span class="math-container">$14$</span> representation, where di... | Dietrich Burde | 83,966 | <p>Let <span class="math-container">$\mathfrak{g}$</span> be the simple Lie algebra of type <span class="math-container">$G_2$</span>, with positive roots <span class="math-container">$R^+=\{\alpha,\beta,\alpha+\beta,2\alpha+\beta,3\alpha+\beta,3\alpha+2\beta \}$</span>. With
<span class="math-container">$\lambda=m_1\... |
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Zarrax | 3,035 | <p>Substituting $y = x^{1 \over 3}$ you have
$$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1} = \lim_{y\to 1}\frac{y^{3/5}-1}{y-1}$$
Note the right hand side is the definition of ${d \over dy} y^{3/5}|_{y = 1}$, which gives you
${3 \over 5}(1)^{-{2 \over 5}} = {3 \over 5}$.</p>
|
915,054 | <p>I'm trying to find a closed form of this sum:
$$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$
<a href="http://www.wolframalpha.com/input/?i=Sum%5BGamma%28n%2B1%2F2%29%2F%28%282n%2B1%29%5E4+4%5En+n%21%29%2C+%7Bn%2C1%2CInfinity%7D%5D"><em>WolframAlpha</em></a> gives a large ex... | user153012 | 153,012 | <p>By now, I've found a closed-form by doing some integral evaluation, a lot of hypergeometric, polylogarithm and polygamma manipulation.
$$
S = \sqrt{\pi}\left(\frac{\pi}{12}\zeta(3)+\frac{1}{192\sqrt3}\psi^{(3)}\left(\tfrac13\right)-\frac{\pi^4}{72\sqrt3}-1\right).
$$</p>
|
915,054 | <p>I'm trying to find a closed form of this sum:
$$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$
<a href="http://www.wolframalpha.com/input/?i=Sum%5BGamma%28n%2B1%2F2%29%2F%28%282n%2B1%29%5E4+4%5En+n%21%29%2C+%7Bn%2C1%2CInfinity%7D%5D"><em>WolframAlpha</em></a> gives a large ex... | Tito Piezas III | 4,781 | <p>The OP gives the evaluation</p>
<p><span class="math-container">$$S=\frac{\pi^{3/2}}{3}-\sqrt{\pi}-\frac{\sqrt{\pi}}{324}\left[9\,_3F_2\left(\begin{array}{c}\tfrac{3}{2},\tfrac{3}{2},\tfrac{3}{2}\\\tfrac{5}{2},\tfrac{5}{2}\end{array}\middle|\tfrac{1}{4}\right)\\+3\,_4F_3\left(\begin{array}{c}\tfrac{3}{2},\tfrac{3}{... |
1,681,205 | <p>I would like a <strong>hint</strong> for the following, more specifically, what strategy or approach should I take to prove the following?</p>
<p><em>Problem</em>: Let $P \geq 2$ be an integer. Define the recurrence
$$p_n = p_{n-1} + \left\lfloor \frac{p_{n-4}}{2} \right\rfloor$$
with initial conditions:
$$p_0 = P ... | marty cohen | 13,079 | <p>I don't know if you
can show that
$\frac{p_n}{z^n}
= 1
$.
If the sequence
$\frac{p_n}{p_{n-1}}
$
approaches $z$ from the same side,
each term in the product
exceeds $z$,
so the product will
always exceed
$z^n$.</p>
<p>What you <em>can</em> show
is that
$\lim \frac{p_n^{1/n}}{z}
= 1
$.
I will now give the
standard,
... |
335,116 | <p>As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination be compatible and / or useful? Is it feasible?</p>
<p>I'd be grateful for an online resource to look into.</p>
<p>... | Mare | 61,949 | <p>The book "D-Modules, Perverse Sheaves, and Representation Theory " by Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki is the perfect source for this topic. The introduction gives a very nice (and elementary) explanation how representation theory of D-modules and symstems of partial differential equations ar... |
745,674 | <p>Let $E$ be a complex vector space of dimension 3. Let $f$ be a non zero endomorphism such that $f^2=0$. I want to show that there is a basis $B=\{b_1,b_2,b_3\}$ of $E$ such that
$$f(b_1)=0, f(b_2)=b_1,f(b_3)=0$$</p>
<p><strong>Edit</strong> Here is how i see the answer now: </p>
<p>$f$ being non zero there exists ... | user2566092 | 87,313 | <p>You can use Jordan-Normal form to solve this easily. In fact the proof would go along the lines of proving the general Jordan-Normal form theorem directly, so you may want to look at the proof for that if you don't want to use the theorem directly.</p>
|
1,246,705 | <p>I was doing some linear algebra exercises and came across the following tough problem :</p>
<blockquote>
<p>Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\phi:M_{n\times n}(\mathbf{R})\to M_{n\times n}(\mathbf{R})$ is a nonzero linear transform (i.e. t... | user1551 | 1,551 | <p>This kind of problems are known as <em>linear preserver problems</em> in the literature. The following is a sketch of proof that immediately comes to my mind. Certainly there are simpler ways to solve the problem (especially if one makes use of existing results on linear preserver problems), but anyway, let $\{e_1,\... |
4,598,275 | <p>Let <span class="math-container">$(X, \mathcal{F})$</span> be a measurable space, and <span class="math-container">$\mu_{n}, \mu$</span> probability measures on it. <span class="math-container">$\mu_{n}$</span> is said to converge weakly to <span class="math-container">$\mu$</span> if for any bounded continuous func... | Damian Pavlyshyn | 154,826 | <p>This is because all probability measures on <span class="math-container">$\mathbf{R}^d$</span> are inner-regular.</p>
<p>The relevant consequence of this fact is that for all <span class="math-container">$\epsilon > 0$</span>, there exists a compact <span class="math-container">$K \subseteq \mathbf{R}^d$</span> s... |
290,903 | <p>I am unable to understand the fundamental difference between a Gradient vector and a Tangent vector.
I need to understand the geometrical difference between the both. </p>
<p>By Gradient I mean a vector $\nabla F(X)$ , where $ X \in [X_1 X_2\cdots X_n]^T $</p>
<p>Note: I saw similar questions on "Difference betw... | Paul Orland | 42,566 | <p>Say you are standing on the side of a hill. Imagine somewhere beneath the hill, there is a flat $x,y$ plane that you can use to determine your position. Let's say $+x$ is east and $+y$ is north.</p>
<p>If the hill is smooth, then the height of the hill above this plane is some continuous function $f(x,y)$.</p>
<... |
136,086 | <p>I've been given the following problem as homework:</p>
<blockquote>
<p>Q: <strong>Compute the number of subgraphs of <span class="math-container">$K_{15}$</span> isomorphic to <span class="math-container">$C_{15}$</span></strong>.</p>
<p><span class="math-container">$K_{15}$</span> means complete graph with 15 verti... | Arturo Magidin | 742 | <p>The binomial coefficient $\binom{15}{n}$ selects the $n$ vertices that will be in the cycle.</p>
<p>I think you would be less confused if you rewrote the other factor: assuming $n\geq 3$,
$$\binom{n-1}{2}(n-3)! = \frac{(n-1)(n-2)}{2}(n-3)! = \frac{(n-1)!}{2}.$$
It's even more suggestive if we write it as
$$\frac{... |
1,407,683 | <p>I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. </p>
<p>Can we bypass this ugly object? Only intrinsic quantities should appear in a textbook. </p>
| Anthony Carapetis | 28,513 | <p>Connections are not tensors, but that does not mean they are not coordinate-independent objects! A linear connection is a map sending two vector fields $X,Y$ to another vector field $\nabla_X Y$ which satisfies the rules $\nabla_{fX+Z} Y = f \nabla_X Y + \nabla_Z Y$ and $\nabla_X (fY + Z) = f\nabla_X Y + (\nabla_X f... |
1,501,876 | <blockquote>
<p>I want to prove $A_n$ has no subgroups of index 2. </p>
</blockquote>
<p>I know that if there exists such a subgroup $H$ then $\vert H \vert = \frac{n!}{4}$ and that $\vert \frac{A_n}{H} \vert = 2$ but am stuck there. I have tried using the proof that $A_4$ has no subgroup of order 6 to get some idea... | Espen Nielsen | 45,874 | <p>Hint: What do we know about a subgroup of $G$ whose index is the smallest prime dividing $|G|$?</p>
|
2,869,898 | <p>I want to prove that <span class="math-container">$$
f(x,y)=
\begin{cases} \frac{xy^2}{x^2+y^2} &\text{ if }(x,y)\neq (0,0)\\
0 &\text{ if }(x,y)=(0,0)
\end{cases}
$$</span>
is not differentiable at <span class="math-container">$(0,0)$</span>.</p>
<p>I thought that I can prove that it is not continuous arou... | Kavi Rama Murthy | 142,385 | <p>At the origin the directional derivative in the direction of $(1,1)$ is $\frac 1 2$ whereas the derivative in the direction of $x-$ axis and $y-$axis are $0$. This implies that the derivative does not exist. </p>
|
2,869,898 | <p>I want to prove that <span class="math-container">$$
f(x,y)=
\begin{cases} \frac{xy^2}{x^2+y^2} &\text{ if }(x,y)\neq (0,0)\\
0 &\text{ if }(x,y)=(0,0)
\end{cases}
$$</span>
is not differentiable at <span class="math-container">$(0,0)$</span>.</p>
<p>I thought that I can prove that it is not continuous arou... | José Carlos Santos | 446,262 | <p>Since $f_x(0,0)=f_y(0,0)=0$, if $f$ was differentiable at $(0,0)$, $f'(0,0)$ would be the null function. Therefore$$\lim_{(x,y)\to(0,0)}\frac{\bigl|f(x,y)-f(0,0)\bigr|}{\sqrt{x^2+y^2}}=0,$$which means that$$\lim_{(x,y)\to(0,0)}\frac{|xy^2|}{(x^2+y^2)^{\frac32}}=0.$$However, this is false. See what happens when $x=y$... |
2,240,405 | <p>The question asks me to find the Laurent series of $$f(z) = {5z+2e^{3z}\over(z-i)^6}\,\,at\,\,z=i$$I know the following $$e^z=\sum_{n=0}^\infty {z^n\over n!}$$ What I want to know, is if I can do this: $$={1\over (z-i)^6}\sum_{n=0}^\infty ({2(3z)^n\over n!}+5z)$$ $$=\sum_{n=0}^{n=6}(z-i)^{n-6}({2(3z)^n\over n!}+5z)... | xpaul | 66,420 | <p>Noting
$$ 5z+2e^{3z}=5(z-i)+5i+2e^{3i}e^{3(z-i)}==5(z-i)+5i+2e^{3i}\sum_{k=0}^\infty\frac{1}{k!}(z-i)^k $$
one has
$$ f(z) = {5z+2e^{3z}\over(z-i)^6}={5i\over(z-i)^6}+{5\over(z-i)^5}+2e^{3i}\sum_{k=0}^\infty\frac{1}{k!}(z-i)^{k-6}.$$</p>
|
14,238 | <p>In question #7656, Peter Arndt asked <a href="https://mathoverflow.net/questions/7656/why-does-the-gamma-function-complete-the-riemann-zeta-function">why the Gamma function completes the Riemann zeta function</a> in the sense that it makes the functional equation easy to write down. Several of the answers were from... | JBorger | 1,114 | <p>Your questions are a part of what Deninger has been writing about for 20 years. He's proposed a point of view that sort of explains a lot of things about zeta functions. It's important to say that this explanation is more in a theoretical physics way than in a mathematical way, in that, as I understand it, he's pred... |
1,369,990 | <p>I came across a quesion -
<a href="https://www.hackerrank.com/contests/ode-to-code-finals/challenges/pingu-and-pinglings" rel="nofollow">https://www.hackerrank.com/contests/ode-to-code-finals/challenges/pingu-and-pinglings</a></p>
<p>The question basically asks to generate all combinations of size k and sum up the ... | Lucian | 93,448 | <p>Are you asking about a more efficient algorithm than merely summing $\displaystyle{n\choose2}=\dfrac{n(n-1)}2$ products ? If so, then you can sum only <em>n</em> products, namely $S_2=\dfrac12\cdot\displaystyle\sum_1^na_k(S_1-a_k),$ where $S_1=\displaystyle\sum_1^na_k.$</p>
|
1,353,498 | <p>The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. </p>
<p>I don't think such a group exists. Here is a brief outline of my proof:</p>
<p>Assume for a contradiction that there exists a noncyclic abelian group of order $51$.</p>
<p>We know that every element (except the iden... | mathcounterexamples.net | 187,663 | <p>You're right.</p>
<p>If you know the theorem that classifies finite abelian groups, then the only possible abelian groups of order $51$ are $\mathbb Z/51 \mathbb Z$ which is cyclic and $\mathbb Z/3 \mathbb Z \times \mathbb Z/17 \mathbb Z$ which is also cyclic because $3$ and $17$ are prime and $\gcd(3,17)=1$ so $\m... |
11,090 | <p>In <em>MMA</em> (8.0.0/Linux), I tried to to create an animation using the command</p>
<pre><code>Export["s4s5mov.mov", listOfFigures]
</code></pre>
<p>and got the output</p>
<p><img src="https://i.stack.imgur.com/bFWPP.png" alt="enter image description here"></p>
<p>Doing a little research, one can <a href="htt... | halirutan | 187 | <p>The function converting strings to integer is <code>FromDigits</code>. It is the counterpart of <code>IntegerString</code> and both functions can be used with whatever basis you like. Therefore, if you want to convert from base 16 you do</p>
<pre><code>FromDigits["6b", 16]
</code></pre>
|
1,912,570 | <p>There is a proof from stein for this assertion,</p>
<p><a href="https://i.stack.imgur.com/fqRKU.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fqRKU.jpg" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/fhZmE.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgu... | DanielWainfleet | 254,665 | <p>The proof looks unnecessarily complicated. Let $\mu^o$ denote outer measure. </p>
<p>First. Show that $\mu^o(A)+\mu^o(B)=\mu^o(A\cup B)$ when $A, B$ are non-empty open sets and $d(A,B)>0.$</p>
<p>Second. We have $\mu^o(E_1\cup E_2)\leq \mu^o(E_1)+\mu^o(E_2)$ for any $E_1,E_2.$ And if $\mu^o(E_1)=\infty$ or $\mu... |
3,118,462 | <p>cars arrives according to a Poisson process with rate=2 per hour and trucks arrives according to a Poisson process with rate=1 per hour. They are independent. </p>
<p>What is the probability that <strong>at least</strong> 3 cars arrive before a truck arrives? </p>
<p>My thoughts:
Interarrival of cars A ~ Exp(2 p... | Community | -1 | <p>A few quick notes:</p>
<ul>
<li>1 variable polynomials, can generalize, base 10 multiplication.</li>
<li>as each digit in base ten is between 0 and 9, That's the part of the table we need to know (arguably with tricks a lot less).</li>
<li>We group like terms,by the power of the variable (generalizing grouping by p... |
531,342 | <p>Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are pennies (1-cent coins), nickels (5-cent coins), and quarters (25-cent coins).</p>
<p>Does anyone know how to solve this?</p>
| Emily | 31,475 | <p>Since the each coin divides the face value of every larger coin, a single larger coin will always represent an integer multiple of smaller coins.</p>
|
3,345,329 | <p>In Bourbaki Lie Groups and Lie Algebras chapter 4-6 the term displacement is used a lot. For example groups generated by displacements. But I can not find a definition of the term displacement given anywhere. I also looked at Humphreys Reflection Groups and Coxeter groups book but I could not find it. Can someone pr... | Olivier | 381,016 | <p>For definiteness, consider odds <span class="math-container">$n$</span>, say <span class="math-container">$2m+1$</span>.</p>
<p>This is an additional comment more than an answer (the second comment by @LinAlg refers to a Theorem that gives the complete answer anyway), to grasp intuition about about the 1/n factor</p... |
2,961,971 | <blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty\frac{(2n)!}{2^{2n}(n!)^2}$$</span></p>
</blockquote>
<p>Can I have a hint for whether this series converges or diverges using the comparison tests (direct and limit) or the integral test or the ratio test?</p>
<p>I tried using the ratio test but it fa... | davidlowryduda | 9,754 | <p>For a more direct approach, you might directly expand the terms as follows:
<span class="math-container">$$\begin{align}
\frac{(2n)!}{4^n (n!)^2} &= \frac{1}{4^n}\frac{2n(2n-1)}{n^2}\frac{(2n-2)(2n-3)}{(n-1)^2}\cdots\frac{(4)(3)}{2^2} \frac{(2)(1)}{1^2} \\
&= \frac{2^n}{4^n}\frac{2n-1}{n} \frac{2n-3}{n-1}\cd... |
16,795 | <p>Consider a finite simple graph $G$ with $n$ vertices, presented in two different but equivalent ways:</p>
<ol>
<li>as a logical formula $\Phi= \bigwedge_{i,j\in[n]} \neg_{ij}\ Rx_ix_j$ with $\neg_{ij} = \neg$ or $ \neg\neg$ </li>
<li>as an (unordered) set $\Gamma = \lbrace [n],R \subseteq [n]^2\rbrace$ </li>
</ol>
... | Mariano Suárez-Álvarez | 1,409 | <p>The usual definition of graph isomorphism implies that in general a graph is not isomorphic to its complement, and it is generally agreed that that definition is sensible. So the claim in your second to last paragraph is false.</p>
|
3,383,687 | <p>I'm interested in ideas for improving and fixing the proof I wrote for the following theorem:</p>
<blockquote>
<p>Let <span class="math-container">$f \colon \mathbb{R}^n \to \mathbb{R} $</span> be differentiable, and <span class="math-container">$ \lim_{\| x \| \to \infty} f(x) = 0 $</span>. Then <span class="mat... | Lázaro Albuquerque | 85,896 | <p>I'll assume <span class="math-container">$f$</span> is bounded since you seem to get that part.</p>
<p>Let <span class="math-container">$\alpha = \inf_{x \in \mathbb{R}^n} f(x)$</span> and <span class="math-container">$\beta = \sup_{x \in \mathbb{R}^n} f(x)$</span>. </p>
<p>Then there are sequences <span class="ma... |
382,526 | <p>I can't calculate the Integral:</p>
<p>$$
\int_{0}^{1}\frac{\sqrt{x}}{\sqrt{1-x^{6}}}dx
$$</p>
<p>any help would be great!</p>
<p>p.s I know it converges, I want to calculate it.</p>
| xpaul | 66,420 | <p>Use substitution $u=x^6$ and $B$ function:
$$\int_0^1\frac{\sqrt{x}}{\sqrt{1-x^6}}dx=\frac{1}{6}\int_0^1u^{-\frac{3}{4}}(1-u)^{-\frac{1}{2}}du=\frac{1}{6}\int_0^1u^{\frac{1}{4}-}(1-u)^{\frac{1}{2}-1}du=\frac{1}{6}B(x,y)=\frac{1}{6}\frac{\Gamma(\frac{1}{4})\Gamma(\frac{1}{2})}{\Gamma(\frac{3}{4})}$$</p>
|
435,936 | <p>Does anyone know when
$x^2-dy^2=k$ is resoluble in $\mathbb{Z}_n$ with $(n,k)=1$ and $(n,d)=1$ ?
I'm interested in the case $n=p^t$</p>
| hot_queen | 72,316 | <p>Yes. For example, divide $\mathbb{N}$ into intervals $[n_k, n_{k+1})$ where $n_k$ is sufficiently fast growing and take the union of every other interval.</p>
|
2,032,241 | <p>In Euler's (number theory) theorem one line reads: since $d|ai$ and $d|n$ and $gcd(a,n)=1$ then $d|i$. I've been staring at this for over an hour and I am not convinced why this is true could anyone explain why? I have tried all sorts of lemma's I've seen before but I honestly just can't see it and I feel I'm going ... | Joffan | 206,402 | <p>Since $d \mid s_i$ and $d \mid n$, clearly $d \mid (s_i+An)$. And $s_i+An = ai$, so $d \mid ai$. (maybe you understood this, but it was highlighted).</p>
<p>So $d\mid ai$ and $d\mid n$. Now since $\gcd(a,n)=1$, $a$ and $n$ have no common factors, and also $a$ and $d$ have no common factors. That means that $d \nmid... |
972,281 | <p>I have to find the inverse laplace transform of:</p>
<p>$\mathcal{L}^{-1}(\frac{s}{-8+2s+s^2})$</p>
<p>I found it was </p>
<p>$\frac{2}{3}e^{-4t}+\frac{1}{3}e^{2t}$</p>
<p>But the question I'm asked is, determine $A,B,C,D$ such that $e^{At}(Bcosh(Ct)+Dsinh(Ct))$ is a solution of the inverse laplace transform.</p... | MPW | 113,214 | <p>Use the fact that
$$e^{At}(B\cosh Ct + D\sinh CT) = \tfrac{B+D}{2}e^{(A+C)t} + \tfrac{B-D}{2}e^{(A-C)t}
$$
Compare this to your expression. The coefficients give you
$$\left\{
\begin{array}{cc}\tfrac{B+D}{2}=\tfrac23 \\
\tfrac{B-D}{2}=\tfrac13
\end{array}\right.
$$
The exponents give you
$$\left\{
\begin{array}{cc}A... |
24,593 | <p>Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have until mid-December to finish). Luckily, I have great flexibility with what I can do to the syllabus, so for the first t... | Steven Gubkin | 117 | <p>I hope this does not come across as overly harsh: I do not think that thinking of teaching as "covering material" in a particular order is a useful framework.</p>
<p>If your students are solving equations like <span class="math-container">$3x+4 = 19$</span>, but are unable to evaluate <span class="math-co... |
3,933,069 | <p>Given
<span class="math-container">$f(x+1)+f(x-1)=x^2$</span></p>
<p>I have subtituted <span class="math-container">$(a=x+1)$</span> and <span class="math-container">$(a=x-1)$</span> and got
<span class="math-container">$$f(x)+f(x-2)=(x-1)^2 \text{ and } f(x+2)+f(x)=(x+1)^2$$</span>
Combining those equations, I got<... | Hagen von Eitzen | 39,174 | <p>Note that <span class="math-container">$f(x)=1$</span> leads to <span class="math-container">$f(x+1)+f(x-1)=2$</span>,
<span class="math-container">$f(x)=x$</span> leads to <span class="math-container">$f(x+1)+f(x-1)=2x$</span>,
<span class="math-container">$f(x)=x^2$</span> leads to <span class="math-container">$f(... |
3,933,069 | <p>Given
<span class="math-container">$f(x+1)+f(x-1)=x^2$</span></p>
<p>I have subtituted <span class="math-container">$(a=x+1)$</span> and <span class="math-container">$(a=x-1)$</span> and got
<span class="math-container">$$f(x)+f(x-2)=(x-1)^2 \text{ and } f(x+2)+f(x)=(x+1)^2$$</span>
Combining those equations, I got<... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>WLOG let <span class="math-container">$$f(x)=g(x)+\sum_{r=0}^na_rx_r$$</span></p>
<p><span class="math-container">$$x^2=g(x-1)+g(x+1)+2a_0+2a_1x+a_2((x+1)^2+(x-1)^2)+\cdots$$</span></p>
<p>Set <span class="math-container">$a_r=0$</span> for <span class="math-container">$r\ge3$</span></p>
<p><span class=... |
3,040,110 | <p>What is the Range of <span class="math-container">$5|\sin x|+12|\cos x|$</span> ?</p>
<p>I entered the value in desmos.com and getting the range as <span class="math-container">$[5,13]$</span>.</p>
<p>Using <span class="math-container">$\sqrt{5^2+12^2} =13$</span>, i am able to get maximum value but not able to fi... | Ross Millikan | 1,827 | <p>The four quadrants give the four combinations of signs of <span class="math-container">$\sin$</span> and <span class="math-container">$\cos$</span>. Let us work initially in the first quadrant, where both functions are positive. We can then remove the absolute value signs, take a derivative, and set to zero.
<span... |
646,183 | <p>I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation.</p>
<p>One can find the real and complex parabolic, elliptic, hyperbolic, subgroups, $\mathrm{SU}(2)$, $\mathrm{SU}(1,1)$ and $\mathrm... | Peter Crooks | 101,240 | <p>Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. There a bijective correspondence between the connected closed subgroups $H$ of $G$ and the Lie subalgebras $\mathfrak{h}$ of $\mathfrak{g}$. The correspondence associates to $H$ its Lie algebra, and to $\mathfrak{h}$ the closure of the image $\mathfrak{h}$ unde... |
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | Wojciech Karwacki | 242,866 | <p>$4n^{\frac{3}{2}}=8^{-\frac{1}{3}} \iff 4n^{\frac{3}{2}}=\frac{1}{2} \iff n^{\frac{3}{2}}=\frac{1}{8} \iff n^3=\frac{1}{64} \iff n= \frac{1}{4}$</p>
|
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | Taylor Ted | 225,132 | <p>We have $$(4n)^\frac{3}{2} = (8)^{-\frac{1}{3}}$$</p>
<p>So we write as $4^{3/2} n^{3/2} = (2^{3})^{-1/3}$</p>
<p>Now writing as</p>
<p>$(2^{2})^{3/2} n^{3/2}= 2^{-1}$</p>
<p>we get as</p>
<p>$8n^{3/2}=1/2$</p>
<p>so $n^{3/2}=1/16$
Now squaring both sides we get</p>
<p>$n^{3}= (1/16) (1/16)$</p>
|
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