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 |
|---|---|---|---|---|
3,104,890 | <p>I'm trying to solve the following problem:</p>
<blockquote>
<p>Ten people are sitting around a round table. Three of them are chosen
at random to give a presentation. What is the probability that the
three chosen people were sitting in consecutive seats?</p>
</blockquote>
<p>I got the wrong answer but cannot... | Arthur | 15,500 | <p>You forgot the possibility that the second person can be chosen to sit <em>two</em> seats away from the first, and then the third person is chosen to be the one between the first and the second. This gives an additional <span class="math-container">$\frac29\cdot \frac18 = \frac1{36}$</span>, bringing the total up to... |
2,090,315 | <blockquote>
<ol>
<li>Let $\Omega \subset \mathbb{C}$ open. How do I prove that if $f(\Omega) \subseteq \text{ a line }$ then $f$ is constant?</li>
<li>How do I prove that if $f$ is holomorphic in $\mathbb{C}$ and there exists $r>0$ such that $f(\mathbb{C})\subset \mathbb{C}-B(0,r)$, then $f$ is constant?</li>... | Learnmore | 294,365 | <ol>
<li>Since $f$ is holomorphic it maps open sets to open sets,since $f(\Omega)\subset \text{line}\implies f \text{is constant}$ </li>
<li>Use Picards Theorem ,If the range of $f$ excludes more than two points of $\Bbb C$ then $f$ is constant.Here it excludes uncountably many points.</li>
</ol>
|
2,090,315 | <blockquote>
<ol>
<li>Let $\Omega \subset \mathbb{C}$ open. How do I prove that if $f(\Omega) \subseteq \text{ a line }$ then $f$ is constant?</li>
<li>How do I prove that if $f$ is holomorphic in $\mathbb{C}$ and there exists $r>0$ such that $f(\mathbb{C})\subset \mathbb{C}-B(0,r)$, then $f$ is constant?</li>... | Fred | 380,717 | <p>A proof of 2. without Picard:</p>
<p>We have $|f(z)| \ge r$ for all $z \in \mathbb C$. Let $g=1/f$. Then $g$ is a bounded entire function. By Liouville, $g$ is constant, hence $f$ is constant.</p>
|
1,115,117 | <p>Consider the initial value problem
$$y'=ty(4-y)/(1+t)$$ $$y(0)=y_{0}>0$$</p>
<p>(a)Determine how the solution behaves as $t$ tends to infinity.</p>
<p>(b)If $y_{0}=2$,find the time $T$ at which the solution first reaches the value of 3.99</p>
<p>(c)Find the range of initial values for which the solution lies... | Chinny84 | 92,628 | <p>$$
\cos\frac{1}{2}\theta + i\sin\frac{1}{2}\theta = \mathrm{e}^{i\frac{1}{2}\theta}
$$
using
$$
\frac{\mathrm{e}^{i\frac{1}{2}\theta} + \mathrm{e}^{-i\frac{1}{2}\theta}}{2} = \cos \frac{1}{2}\theta
$$
we get
$$
\left(\mathrm{e}^{i\frac{1}{2}\theta} + \mathrm{e}^{-i\frac{1}{2}\theta}\right)\mathrm{e}^{i\frac{1}{2}\th... |
4,590,677 | <p>This is perhaps a silly question related to calculating with surds. I was working out the area of a regular pentagon ABCDE of side length 1 today and I ended up with the following expression :</p>
<p><span class="math-container">$$\frac{\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}}}{4}$$</span></p>
<p>obtained by summing the... | Will Jagy | 10,400 | <p><span class="math-container">$10 + 2 \sqrt 5$</span> has norm <span class="math-container">$100 - 5 \cdot 4 = 80.$</span> <span class="math-container">$5 + 2 \sqrt 5$</span> has norm <span class="math-container">$25 - 5 \cdot 4 = 5.$</span> The ratio of the norms is <span class="math-container">$\frac{80}{5} ... |
3,443,137 | <p>Find the radius of the circle tangent to <span class="math-container">$3$</span> other circles <span class="math-container">$O_1$</span>, <span class="math-container">$O_2$</span> and <span class="math-container">$O_3$</span> have radius of <span class="math-container">$a$</span>, <span class="math-container">$b$</s... | Blue | 409 | <p>If you can endure the algebraic slog (a computer algebra system really helps), you can find some structure that simplifies the solution.</p>
<hr>
<p>Let the given circles have radii <span class="math-container">$r_A$</span>, <span class="math-container">$r_B$</span>, <span class="math-container">$r_C$</span>, and ... |
3,501,879 | <p>I have been stuck at this problem for some time now. I'd really apprechiate your help. Thanks.</p>
<p><span class="math-container">$$2\sin^2(x)+6\cos^2(\frac x4)=5-2k$$</span></p>
| lhf | 589 | <p><em>Hint:</em> The problem is about finding the minimum and maximum values of <span class="math-container">$
2\sin^2(x)+6\cos^2\left(\frac x4\right)$</span>.</p>
<p>The minimum value is easy: it's <span class="math-container">$0$</span>, taken at <span class="math-container">$x=10\pi$</span> for instance.</p>
<p>T... |
408,717 | <p>Let $n\in \mathbb N$ and $A_1,A_2,..,A_n$ be arbitrary sets. Now define $X=[x_{ij}]_{n \times n}$ where
$$x_{ij}=
\begin{cases}
1 , & \text{$A_i$$\subsetneq$}A_j \\
0 , & \text{otherwise} \\
\end{cases}.$$
How do you prove $X^n=0$?</p>
<p>Thanks in advance.</p>
| N. S. | 9,176 | <p>If you know some Graph Theory:</p>
<p>Define a digraph the following way: the vertices are $A_1,A_2,..., A_n$. The arcs:
Put an arc $A_i \rightarrow A_j$ if and only if $A_i\subsetneq A_j$.</p>
<p>Your matrix $X$ is exactly the incidence matrix of this digraph. Since the digraph contains no directed cycles, it con... |
4,595,208 | <p>I'm having trouble calculating this limit directly :</p>
<p><span class="math-container">$$\lim_{n\to\infty}\frac{(2n＋1)(2n＋3)\cdots(4n＋1)}{(2n)(2n＋2)\cdots(4n)}$$</span></p>
<p>It can be calculated using the inventory method and the result is:<br />
<span class="math-container">$$\lim_{n\to\infty}\frac{(2n＋1)(2n＋3)... | John Douma | 69,810 | <p>If the incident angle is <span class="math-container">$\beta$</span> then the final angle must be <span class="math-container">$\frac{\pi}{2} -\beta$</span>.</p>
<p>To see this, imagine a beam going to the right. Its perpendicular direction is straight up which is <span class="math-container">$\frac{\pi}{2}$</span> ... |
1,757,260 | <p>A little box contains $40$ smarties: $16$ yellow, $14$ red and $10$ orange.</p>
<p>You draw $3$ smarties at random (without replacement) from the box.</p>
<p>What is the probability (in percentage) that you get $2$ smarties of one color and another smarties of a different color?</p>
<p>Round your answer to the ne... | true blue anil | 22,388 | <p><strong>Whenever the sequence is unspecified</strong> in drawing w/o replacement,<br>
I much rather prefer using combinations to avoid getting into an unnecessary tangle by oversight.</p>
<p>$\dfrac{\left[\binom{16}{2}\binom{24}1 + \binom{14}{2}\binom{26}1 + \binom{10}{2}\binom{30}1\right]}{ \binom{40}{3}} = 67\%... |
3,043,846 | <p>I want to rewrite a question not so well written on this site and clarified by Mr. Lahtonen (thank you again).</p>
<p>So here the question:</p>
<blockquote>
<p>Let the extention <span class="math-container">$GF(p^m) \supset GF(p)$</span> that contains roots of
<span class="math-container">$p(x)=x^{p^{m}}-1$</s... | Dietrich Burde | 83,966 | <p>Besides your example, there is even an example with <em>finite</em> groups, as
<span class="math-container">$$
{\rm Aut}(S_3)\cong {\rm Aut}(C_2\times C_2)\cong S_3,
$$</span>
but <span class="math-container">$S_3$</span> is of course not isomorphic to <span class="math-container">$C_2\times C_2$</span>.</p>
|
697,668 | <p>How many combinations are there to arrange the letters in MISSISSIPPI requiring that the 2 S's must be separated? </p>
<p>I found there are 34650 combinations to arrange without restriction. </p>
<p>How to approach this question?</p>
| 2012ssohn | 103,274 | <p>We know that the string will take the form of</p>
<p>$$*S█S█S█S*$$</p>
<p>where $█$ MUST have at least one character and $*$ can be of any length (even 0). I would suggest the following steps:</p>
<ol>
<li>Find the number of ways you can put the $S$s (they can be in positions $(1,3,5,7)$, $(2,5,8,11)$, $(1,4,6,9)... |
254,253 | <blockquote>
<p>If the only contents of a container are 10 disks that are each numbered with a different positive integer from 1 through 10, inclusive. If 4 disks are to be selected one after the other, with each disk selected at random and without replacement, what is the probability that the range of the numbers on... | Gerry Myerson | 8,269 | <p>tofu, you have found the probability that you pick $1$, then $8$, then two others between $1$ and $8$. But you could also have picked $8$, then $1$, then the other two. Or one of the other two, then $1$, then $8$, then the other of the other two. So what you need to figure out is how many orderings there are of one-... |
4,131,747 | <p>I am having trouble with this problem in my Linear Algebra review:</p>
<blockquote>
<p>Find an equation for the plane parallel to <span class="math-container">$2x-y+2z=4 $</span> such that the
point <span class="math-container">$(3,2,-1) $</span> is equidistant from both planes.</p>
</blockquote>
<p>The answer is <s... | Toby Mak | 285,313 | <p>Since both planes are parallel, the <a href="https://math.stackexchange.com/questions/2472153/normal-vector-to-plane">normal vector</a> to both planes is <span class="math-container">$(2, -1, 2)$</span>. Thus the points <span class="math-container">$(2k + 3, -k + 2, 2k-1)$</span> and <span class="math-container">$(-... |
320,937 | <p>It is easy to think of $\mathbb{C}^2$ as an ordered pair. I just wonder if it is possible to put $\mathbb{C}^2$ into illustration, since $\mathbb{C}$ has taken the role of two dimensional Euclidean Space.</p>
| Brian M. Scott | 12,042 | <p>It’s basically the same argument. For $n\in\Bbb Z^+$ let $L_n$ be the $L$-shaped piece described in the question. Let $B=\{n\in\Bbb Z^+:L_n\text{ cannot be tiled}\}$. You want to show that $B=\varnothing$. Assume, to get a contradiction, that $B\ne\varnothing$. $\Bbb Z^+$ is well-ordered, so $B$ has a least element ... |
1,859,719 | <blockquote>
<p>Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$.</p>
</blockquote>
<p>I have tried to use the Lagrangian function to find the solution for the problem, with the equation</p>
<p>$$\nabla\mathscr{L}=\vec{0}$$</p>
<p>where... | Rodrigo de Azevedo | 339,790 | <p>We want to solve</p>
<p>$$\begin{array}{lc} \text{maximize} & x^\alpha y^\beta\\ \text{subject to} & p x + q y = r\end{array}$$</p>
<p>We define the Lagrangian</p>
<p>$$\mathcal{L} (x,y,\lambda) := x^\alpha y^\beta - \lambda (p x + q y - r)$$</p>
<p>Taking the partial derivatives and finding where they v... |
195,168 | <pre><code>f[2 x_] := f[x]
f[1] := 3
f[0] := 0
f[2 x_ + 1] := f[x] + f[x + 1]
a[x_] := f[x]/f[x + 1]
</code></pre>
<p>Will this work as an recursive function ?
I think there's something wrong with this because every integer will get an output of 3 </p>
<p>any help would be appreciated, thank you so much</p>
| Bill | 18,890 | <p>Perhaps try this</p>
<pre><code>f[0] := 0;
f[1] := 3;
f[x_/;EvenQ[x]] := f[x/2];
f[x_/;OddQ[x]] := f[(x-1)/2] + f[(x-1)/2 + 1];
a[x_] := f[x]/f[x + 1];
Table[{i,a[i]},{i,0,6}]
</code></pre>
<p>with output</p>
<pre><code>{{0, 0}, {1, 1}, {2, 1/2}, {3, 2}, {4, 1/3}, {5, 3/2}, {6, 2/3}}
</code></pre>
<p>Please chec... |
195,168 | <pre><code>f[2 x_] := f[x]
f[1] := 3
f[0] := 0
f[2 x_ + 1] := f[x] + f[x + 1]
a[x_] := f[x]/f[x + 1]
</code></pre>
<p>Will this work as an recursive function ?
I think there's something wrong with this because every integer will get an output of 3 </p>
<p>any help would be appreciated, thank you so much</p>
| Roman | 26,598 | <p>If you want to go to large values of <span class="math-container">$x$</span>, then some <a href="https://reference.wolfram.com/language/tutorial/FunctionsThatRememberValuesTheyHaveFound.html" rel="nofollow noreferrer">memoization</a> will speed up your recursion dramatically:</p>
<pre><code>f[0] = 0;
f[1] = 3;
f[x_... |
3,810,733 | <p><span class="math-container">$$''' + y' = 2^2 + 4\sin(x)$$</span></p>
<p>Find the general solution of the differential equation by using the Indefinite Coefficients Method.</p>
| Christian Blatter | 1,303 | <p>We are given the vector <span class="math-container">${\bf a}:=(3,-2,4,-1)$</span> and are looking for the unit vector <span class="math-container">${\bf u}\in{\mathbb R}^4$</span> for which the scalar product
<span class="math-container">$${\bf a}\cdot{\bf u}$$</span>
is maximal. By Schwarz' inequality we have <spa... |
108,253 | <p>I would like to assign 'x' individuals to 'y' groups, randomly. For example, I would like to divide 50 individuals into 100 groups randomly. Of course, with more groups than individuals many of the groups will have zero individuals, while some groups will have multiple individuals. That is fine. With random assignme... | Dr. belisarius | 193 | <p>it looks like a Poisson:)</p>
<pre><code>t = Tally[Flatten@RandomChoice[Range@100, {100000, 50}]][[All, 2]];
fdu = FindDistributionParameters[t, PoissonDistribution[u]];
Show[SmoothHistogram[t, PlotStyle -> Red],
Plot[PDF[PoissonDistribution[u /. fdu]][x], {x, Min@t, Max@t}, PlotStyle -> Blue]]
</code><... |
108,253 | <p>I would like to assign 'x' individuals to 'y' groups, randomly. For example, I would like to divide 50 individuals into 100 groups randomly. Of course, with more groups than individuals many of the groups will have zero individuals, while some groups will have multiple individuals. That is fine. With random assignme... | Eric Towers | 16,237 | <p>This function is "stable" in the sense that elements will appear in sublists in the same order they appear in the supplied list.</p>
<pre><code>Clear[randomSetPartition];
randomSetPartition[n_ /; IntegerQ[n], parts_] :=
randomSetPartition[Range[1, n], parts]
randomSetPartition[individuals_List, parts_] := Modul... |
3,306,571 | <p>I know that the function <span class="math-container">$f(x) = \frac{x}{x}$</span> is not differentiable at <span class="math-container">$x = 0$</span>, but according to the definition of differentiable functions:</p>
<blockquote>
<p>A differentiable function of one real variable is a function whose derivative exi... | Ross Millikan | 1,827 | <p><span class="math-container">$\frac xx$</span> is continuous at all points in its domain. It has derivative <span class="math-container">$0$</span> at all points of <span class="math-container">$\Bbb R$</span> except <span class="math-container">$0$</span>.</p>
|
336,196 | <p>Can anyone help me with the following SDE?</p>
<p>Solve the following stochastic differential equation:
$$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$
with $Y_0=0$.</p>
<p>Hint: Try a solution of the form $Z_tH_t$ where $Z_t = exp(cB_t+(a-\frac{1}{2}c^2t))$ and $dH_t=F(t)dt+G(t)dB_t$ for some adapted process F and G which need ... | Anders Muszta | 294,222 | <p>A linear SDE $$\text{d}Y_{t} = (\alpha(t)+\beta(t)Y_{t})\,\text{d}t+(\gamma(t)+\delta(t)Y_{t})\,\text{d}W_{t}$$ has an explicit (strong) solution which can be found on <a href="https://en.wikipedia.org/wiki/Stochastic_differential_equation#Linear_SDE:_general_case" rel="nofollow">Wikipedia</a> . Here $\alpha$, $\bet... |
510,814 | <p>I've seen in several places without further comment that if an equalizer is epic, it's an isomorphism. I've only proved one half of this:</p>
<p>Suppose $e:X \rightarrow A$ is an epimorphism and an equalizer for $f$ and $g$. Then $f \circ e = g \circ e \implies f = g$. Then any function $e': X' \rightarrow A$ trivi... | Community | -1 | <p>Suppose $B$ is the target of $f$ and $g$.</p>
<p>Since $e : X \to A$ is an equalizer of $f$ and $g$ and $X\xrightarrow{e} A \xrightarrow{f} B = X \xrightarrow{e} A \xrightarrow{g} B$ , hence there exists a unique $\ell : X \to X$ such that $X\xrightarrow{\ell} X \xrightarrow{e} A = X \xrightarrow{e} A$. </p>
<p>On... |
2,567,332 | <p>A Greek urn contains a red, blue, yellow, and orange ball. A ball is drawn from the urn at random and then replaced. If one does this $4$ times, what is the probability that all $4$ colors were selected?</p>
<p>I approached this questions by doing $(1/4)^4$ because there's always a $1/4$ chance of selected a specif... | Puffy | 513,387 | <p>The first ball drawn can be any colour. So, the probability is $\frac{4}{4}.$</p>
<p>Since the first ball is replaced, there is a $\frac{1}{4}$ chance that the same ball will be drawn. The chance for a different ball to be drawn is $\frac {3}{4}$.</p>
<p>There is $\frac{2}{4}$ chance that the two drawn balls will ... |
2,413,891 | <blockquote>
<p><strong>Question :</strong> Evaluate - $$\int_{0}^{1}2^{x^2+x}\mathrm dx$$</p>
</blockquote>
<p><strong>My Attempt :</strong> First I tried to evaluate the indefinite integral of $2^{x^2+x}$ in order to put the limits $0$ and $1$ later on, but couldn't integrate it. Then I checked on WA and came to k... | Satish Ramanathan | 99,745 | <p>Hint:</p>
<p>Put <span class="math-container">$y = 2^{x^2+x}$</span></p>
<p>Integral becomes <span class="math-container">$\int_{1}^{4} \frac{1}{\sqrt{1+ \frac{4}{\ln(2)}\ln(y)}}dy$</span></p>
<p>Again Put <span class="math-container">$\sqrt{1+ \frac{4}{\ln(2)}\ln(y)}= u$</span></p>
<p>Integral becomes <span class="... |
1,566,471 | <p>Hi can someone please help?</p>
<p>I need to evaluate this indefinite integral:</p>
<p>$$\int \frac{(\ln x)^5}x dx$$</p>
<p>I know I need to use substitution, so if I let <em>u= x</em> but I can't figure out the antiderivative for the top portion.</p>
<p>Thank you!</p>
| Peter | 82,961 | <p>The definition is correct in the case $P(A)=0$ (or $P(B)=0$), only if the event
$A$ (or $B$) is impossible.</p>
<p>As you have shown, the definition breaks down for events with $P(A)=0$, which can occur.</p>
|
2,110,286 | <p>Show that if $A$ and $B$ are subsets of a set $S$, then $\overline{A \cap B}=\overline{A}\cup \overline{B}$.</p>
<p>I tried to prove that $A \cap B=A \cup B$ because I didn't realize that the overline meant to prove it for the <em>closure</em> of the sets.</p>
<p>So, now I am confused about how to prove for closur... | Kanwaljit Singh | 401,635 | <p>Let M = (A ∩ B)' and N = A' U B'</p>
<p>Let x be an arbitrary element of M then x ∈ M ⇒ x ∈ (A ∩ B)'</p>
<p>⇒ x ∉ (A ∩ B)</p>
<p>⇒ x ∉ A or x ∉ B</p>
<p>⇒ x ∈ A' or x ∈ B'</p>
<p>⇒ x ∈ A' U B'</p>
<p>⇒ x ∈ N</p>
<p>Therefore, M ⊂ N …………….. (i)</p>
<p>Again, let y be an arbitrary element of N then y ∈ N ⇒ y ∈... |
874,300 | <p>I'm having trouble grasping how to set these types of problems. There are a lot of related questions but it's difficult to abstract a general procedure on finding constants that give the given function bounding constraints to make it big-theta(general function). </p>
<p>so $\frac{x^4 +7x^3+5}{4x+1}$ is $ \Theta ... | Mary Star | 80,708 | <p>Yes, you could take $c_1=\frac{1}{4}$, then you have the following:</p>
<p>$$ \frac{1}{4}x^3 \leq \frac{x^4 +7x^3+5}{4x+1} \Rightarrow \frac{4x^4+x^3}{4} \leq x^4+7x^3+5 \Rightarrow x^4+\frac{1}{4}x^3 \leq x^4+7x^3+5 \\ \Rightarrow \frac{27}{4}x^3 \geq -5 \Rightarrow x^3 \geq -\frac{20}{27} \Rightarrow x \geq - \sq... |
373,958 | <p>Is $\sum_{n=1}^\infty(2^{\frac1{n}}-1)$ convergent or divergent?
$$\lim_{n\to\infty}(2^{\frac1{n}}-1) = 0$$
I can't think of anything to compare it against. The integral looks too hard:
$$\int_1^\infty(2^{\frac1{n}}-1)dn = ?$$
Root test seems useless as $\left(2^{\frac1{n}}\right)^{\frac1{n}}$ is probably even harde... | John Dawkins | 68,151 | <p>Try the Comparison Test, using the elementary inequality
$$
2^{1/n} -1 > {\log 2\over n}
$$
for $n=1,2,\ldots$.</p>
|
2,282,818 | <p>I'm getting $f(x)=2x+f(0)$ and $f(x)=f(0)-2x$ by setting $y=0$, but I'd like to verify. Am I right?</p>
| Umberto P. | 67,536 | <p>The function $f$ is clearly continuous and one-to-one, since it satisfies the Lipschitz condition and $f(x) = f(y)$ implies $x=y$. Thus $f$ is monotone, and consequently either increasing or decreasing. </p>
<p>If $f$ is increasing, the condition $|f(x) - f(0)| = 2|x|$ leads to $$x > 0 \implies f(x) > f(0) \i... |
4,265,001 | <p>I'm a bit confused about expanding out the notation of product of matrices, in the context of quadratic forms.</p>
<p>If <span class="math-container">$x \in \mathbb{R}^n, \, \, A \in \mathbb{R}^{n \times n}$</span> then</p>
<p><span class="math-container">$$x^TAx = \sum_{i,j=1}^na_{ij}x_ix_j$$</span></p>
<p>But then... | Jean-Claude Arbaut | 43,608 | <p>It's easy to see why your attempt is wrong by considering a matrix <span class="math-container">$X$</span> of dimension <span class="math-container">$n\times p$</span>. Then <span class="math-container">$X^TAX$</span> has dimension <span class="math-container">$p\times p$</span>, while your formula yields a <span cl... |
3,086,024 | <p>I am trying to solve an exercise from the book "Theory of Numbers" by B.M.Stewart. The exercise is the following one:</p>
<blockquote>
<p>Let <span class="math-container">$T=2^ap_1^{a_1}p_2^{a_2} \dots p_n^{a_n}$</span>, where <span class="math-container">$a \ge0, n\ge0, 2<p_1<p_2<\dots p_n, p_j$</span> ... | saulspatz | 235,128 | <p>You're awfully close, if you accept my comment that only the <span class="math-container">$T=y$</span> case needs to be considered. It follows from <span class="math-container">$\gcd(u,v)=1, u \not\equiv v \pmod2$</span> that <span class="math-container">$\gcd(u+v,u-v)=1.$</span> There are <span class="math-contai... |
503,589 | <ol>
<li><p>Let $\epsilon>0$. Prove that the set of those $x\in [0,1]$ such that there exist infinitely many fractions $p/q$, with relatively prime integers $p$ and $q$ such that
$$\bigg |x-\frac{p}{q}\bigg|\leq \frac{1}{q^{2+\epsilon}}$$
is a set of measure zero.</p></li>
<li><p>Let $(a_n)$ be a sequence of real nu... | Etienne | 80,469 | <p>(1) First observe that if $x\in [0,1]$ and $\left\vert x-\frac{p}q\right\vert\leq \frac{1}{q^{2+\varepsilon}}$ for some $p,q$, then $\frac pq\leq x+ \frac{1}{q^{2+\varepsilon}}\leq 2$ and hence $p\leq 2q$. Consequently, if you put
$$A_q=\left\{ x\in [0,1];\; \exists p\leq 2q\;:\; \left\vert x-\frac{p}q\right\vert\... |
64,544 | <blockquote>
<p>Please let me know what is the standard notation for group action.</p>
</blockquote>
<p>I saw the following three notations for group action.
(All the images obtained as <code>G\acts X</code> for different deinitions of <code>\acts</code>.) </p>
<p>(1) <img src="https://lh5.googleusercontent.com/_7... | Rob Harron | 1,021 | <p>As this was the first hit on google for "latex action arrow", but didn't contain what I wanted, let me post what I figured out. But to address other people's issues with the original question: while I agree that in a sentence one should simply say "Let $G$ act on $X$...", I was interested in drawing a (what some peo... |
3,115,168 | <p>I've converted <span class="math-container">$\cos^3(x)$</span> into <span class="math-container">$\cos^2(x)\cos(x)$</span> but still have not gotten the answer. </p>
<p>The answer is <span class="math-container">$\dfrac{\sin(x)(3\cos^2x + 2\sin^2x)}{3}$</span></p>
<p>My answer was the same except I did not have a ... | Nikunj | 287,774 | <p>If you wish to do this using by parts,
use <span class="math-container">$\cos^2x$</span> as the first function (to differentiate) and integrate <span class="math-container">$\cos x$</span> to get:
<span class="math-container">$$\int \cos^2 x \cos x dx = \cos^2 x \sin x + 2\int \sin^2 x \cos {x}dx$$</span>
Now use <s... |
316,865 | <p>How do you find this limit?</p>
<p>$$\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x$$</p>
<p>I was given a clue to use L'Hospital's rule.</p>
<p>I did it this way:</p>
<p><strong>UPDATE 1:</strong>
$$
\begin{align*}
\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x
&= \lim_{x \rightarrow \infty} x\begin{p... | ferson2020 | 59,689 | <p>Yes, you are using L'Hopital's rule correctly, and the answer is $-\frac{1}{5}$. The steps are correct, and I double-checked the answer: <a href="http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427epe5q2eamff" rel="nofollow">http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf842... |
316,865 | <p>How do you find this limit?</p>
<p>$$\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x$$</p>
<p>I was given a clue to use L'Hospital's rule.</p>
<p>I did it this way:</p>
<p><strong>UPDATE 1:</strong>
$$
\begin{align*}
\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x
&= \lim_{x \rightarrow \infty} x\begin{p... | lab bhattacharjee | 33,337 | <p>If we are not compelled to use L'Hospital's Rule,</p>
<p>$$\lim_{x \rightarrow \infty} \sqrt[5]{x^5-x^4} -x$$
$$=\lim_{y\to0}\frac{(1-y)^\frac15-1}y$$</p>
<p>$$=\lim_{y\to0}\frac{(1-y)-1}{y\{(1-y)^\frac45+(1-y)^\frac35+(1-y)^\frac25+(1-y)^\frac15+1\}}$$ as $ a^n-1=(a-1)(a^{n-1}+a^{n-2}+\cdots+a+1)$</p>
<p>$$=\fra... |
2,889,835 | <p>If I have random lag times from <code>a=0.1</code> to <code>b=0.3</code> and a time to live (TTL) of <code>x=0.25</code>, what would be the packet loss in per cent?</p>
<p>Ok so basically I have packets that arrive in <code>Random [a,b]</code> time, if that random value is greater than <code>x</code> the packet get... | Sarvesh Ravichandran Iyer | 316,409 | <p>The issue is the following : $f$ continuous, means that if $x_n$ is a <em>convergent</em> sequence to $x$, then so is $f(x_n)$ , converging to $f(x)$.</p>
<p>Note that every convergent sequence is Cauchy. However, the converse is true(in $\mathbb R$), only in a closed set. Note that $(0,1)$ is not a closed set. The... |
4,640,732 | <p>Find roots of:
<span class="math-container">$$x^{6}\ -\ \left(x-1\right)^{6}=0 \tag {1}$$</span></p>
<p>I know this equation has <span class="math-container">$4$</span> complex roots and exactly one real roots of value <span class="math-container">$0.5$</span>.</p>
<p>However, my first instinct was to do this:
<span... | Henry | 6,460 | <p>If you had started with <span class="math-container">$x^2 - (x-1)^2=0$</span> then you can expand it to <span class="math-container">$x^2-x^2+2x-1=0$</span> and so <span class="math-container">$x=\frac12$</span>.</p>
<p>But let's take a version of your approach. You might say <span class="math-container">$a^2=b^2$</... |
3,599,008 | <p>I want to see if the set:</p>
<p><span class="math-container">$$X =\{f\in{}C[0,1]:|f(t)|<1\text{ for all } t\in[0,1]\}$$</span> is open or closed.</p>
<p>If it is open at each function <span class="math-container">$g\in{}X$</span> there is an epsilon sized open ball around it contained by <span class="math-cont... | Umberto P. | 67,536 | <p>Supposing the norm on <span class="math-container">$C[0,1]$</span> is <span class="math-container">$\|f\| = \max_{t \in [0,1]} |f(t)|$</span>, the corresponding metric on <span class="math-container">$C[0,1]$</span> is <span class="math-container">$d(f,g) = \|f-g\|$</span>.</p>
<p>Select an element <span class="ma... |
3,599,008 | <p>I want to see if the set:</p>
<p><span class="math-container">$$X =\{f\in{}C[0,1]:|f(t)|<1\text{ for all } t\in[0,1]\}$$</span> is open or closed.</p>
<p>If it is open at each function <span class="math-container">$g\in{}X$</span> there is an epsilon sized open ball around it contained by <span class="math-cont... | José Carlos Santos | 446,262 | <p>Assuming that you are working with the metric <span class="math-container">$(f,g)\mapsto\sup\lvert f-g\rvert$</span>, consider the map<span class="math-container">$$\begin{array}{rccc}\psi\colon&C[0,1]&\longrightarrow&\mathbb R\\&f&\mapsto&\sup\lvert f\rvert.\end{array}$$</span>Then <span cla... |
1,376,159 | <p>A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not achieved anything, so I decided to ask for your advice.
$$\int_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)... | Cleo | 97,378 | <p>\begin{align}
& \int_0^1\ln(2+x)\,\ln(1+x)\,\ln\left(1+x^{-1}\right)\ln x\,dx\\
& \quad=\frac{71}{36}\,\ln^42+2\ln^32\cdot\ln3+4\ln2\cdot\ln^33-7\ln^22\cdot\ln^23-\frac23\,\ln^32-\frac23\,\ln^33-\ln^22\cdot\ln3\\
& \quad \quad +6\ln^22+3\ln^23-12\ln2-\frac{\pi^4}{216}+\pi^2\!\left(\frac{49}{36}\,\ln^22-2... |
2,037,704 | <p>What symmetry property in complex space is related to the fact that the absolute value of numbers $|a+ib| = |b+ia|$ are equals?</p>
| levap | 32,262 | <p>If you identify the complex space $\mathbb{C}$ with $\mathbb{R}^2$ by sending $z = a + ib$ to $(a,b)$, the norm of a complex number $z$ is the same as (regular, Euclidean) the norm of the vector $(a,b) \in \mathbb{R}^2$. Consider the following two maps:</p>
<ol>
<li>The conjugation map $T \colon \mathbb{C} \rightar... |
1,969,169 | <p>We have to do the following integral.
$$\int_1^{\frac{1+\sqrt{5}}{2}}\frac{x^2+1}{x^4-x^2+1}\ln\left(1+x-\frac{1}{x}\right)dx$$
I tried it a lot.
I substitute $t=1+x-(1/x)$, $dt=1+(1/x^2)$</p>
<p>But then I stuck at
$$\int\limits_{1}^{2} \frac{\ln(t)}{(t-1)^{2} + 1} \mathrm{d}t$$</p>
<p>But now how to proceed.</p>... | David H | 55,051 | <hr>
<p>Let $I$ denote the integral</p>
<p>$$I:=\int_{1}^{\phi}\frac{x^{2}+1}{x^{4}-x^{2}+1}\ln{\left(1+x-\frac{1}{x}\right)}\,\mathrm{d}x,$$</p>
<p>with $\phi$ of course being the golden ratio, $\phi=\frac{1+\sqrt{5}}{2}$.</p>
<p>As a numerical approximation, we find</p>
<p>$$I\approx0.272198.$$</p>
<hr>
<p>Try... |
248,710 | <p>The organizers of a cycling competition know that about 8% of the racers use steroids. They decided to employ a test that will help them identify steroid-users. The following is known about the test: When a person uses steroids, the person will test positive 96% of the time; on the other hand, when a person does not... | broccoli | 50,577 | <p><a href="http://bayesianthink.blogspot.com/2012/08/understanding-bayesian-inference.html" rel="nofollow">Here</a> is a relatively plain English explanation for Bayesian statistics.</p>
<p>Here, you have an evidence which says a user tested positive and the hypothesis you want to test against is whether steroids wer... |
1,981,360 | <blockquote>
<p>Given function $f:\mathbb{R}_0^+ \to \mathbb{R},~f(x) = x^2 + 4x + 4$ prove that it is injective.</p>
</blockquote>
<p>Using definition of injectivity $(\forall x_1, x_2 \in \mathbb{R}_0^+)(x_1 \neq x_2 \implies f(x_1) \neq f(x_2))$ I'm doing the following:</p>
<p>$$x_1^2 + 4x_1 + 4 = x_2^2 + 4x_2 +... | Edouard L. | 378,837 | <p>Your final argument is not correct.<br>
Instead you have (from your calculation):<br>
Since $x_2 \geq 0$ that would imply $x_1 <0$ which is impossible (because out of the domain): therefore f is injective on the domain.<br>
It would also be better to give your assumptions: suppose there exists $x_1 \neq x_2$ such... |
1,981,360 | <blockquote>
<p>Given function $f:\mathbb{R}_0^+ \to \mathbb{R},~f(x) = x^2 + 4x + 4$ prove that it is injective.</p>
</blockquote>
<p>Using definition of injectivity $(\forall x_1, x_2 \in \mathbb{R}_0^+)(x_1 \neq x_2 \implies f(x_1) \neq f(x_2))$ I'm doing the following:</p>
<p>$$x_1^2 + 4x_1 + 4 = x_2^2 + 4x_2 +... | egreg | 62,967 | <p>Your argument is fine, but you can end a step earlier: $x_1>0$ and $x_2>0$ implies $x_1+x_2>0$, so $x_1+x_2=-4$ is a contradiction.</p>
<p>However you have better making evident where you're using the hypothesis $x_1\ne x_2$.</p>
<hr>
<p>Proofs of injectivity are often times simpler with the contrapositi... |
191,548 | <p>Say I have a list:</p>
<pre><code>{{Line[{{-Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, 1}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0,1}}]},
{Line[{{-Sqrt[5/8 + Sqrt[5]/8],1/4 (-1 + Sqrt[5])}, {Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}}],
Line[{{Sqrt[5/8 - Sqrt[5]/8], 1/4 (-1 - Sqrt[5])}, {0, ... | WReach | 142 | <p>To make things easier to read, let's use these lines (with the same nested structure as in the question):</p>
<pre><code>lines =
{ { Line[{{60, 50}, {60, 70}}]
, Line[{{40, 30}, {40, 50}}]
}
, { Line[{{20, 10}, {20, 30}}]
}
};
</code></pre>
<p>Furthermore, let's define the list <code>points</code... |
393,580 | <p>Show that $-Z$ is also a standard normal random variable; that is, show that $P[-Z < x] = P[Z < x] \,\forall x.$</p>
| response | 76,635 | <p>Hint: The standard normal is a symmetric distribution.</p>
|
507,467 | <p>How do you factor $x^3 + x - 2$?</p>
<p>Hint: Write it as $(x^3-x^2+x^2-x+2x-2)$ to get $(x-1)(x^2+x+2)$</p>
<p>Note the factored form <a href="http://www.wolframalpha.com/input/?i=x%5E3+%2B+x+-+2" rel="nofollow noreferrer">here</a>. Thanks!</p>
| Community | -1 | <p>By inspection, we see that $1$ is a root of $x^3 + x - 2$, so it is divisible by $x - 1$; alternatively, the rational roots theorem would suggest this too.</p>
<p>Now $x^2 + x + 2 = x^2 + x + \frac{1}{4} + \frac{7}{4} = (x + \frac{1}{2})^2 + \frac{7}{4}$ has no real roots, and is irreducible. If you're factoring ov... |
2,256,973 | <p>I'm writing a computer algorithm to do binomial expansion in C#. You can view the code <a href="https://gist.github.com/jamesqo/01015428601641347e436129c1ae0079#file-multinomial-cs-L29-L39" rel="nofollow noreferrer">here</a>; I am using the following identity to do the computation:</p>
<p>$$
\dbinom n k = \frac n k... | Robert Israel | 8,508 | <p>If division by $k$ would give you a non-integer, multiplication by $n$ would give you back an integer. So if you reduce $n/k$ to lowest terms, $n/k = a/b$ with $\gcd(a,b) = 1$, then you can safely divide by $b$ and then multiply by $a$.</p>
<p>EDIT: Another method:
Let $q$ be the result of integer division of $N... |
2,412,454 | <p>I was obviously not clear enough in my first question, so I will reformulate. I have the following equation
$$
A=\frac{B\sin 2\theta}{C+D\cos 2\theta}
$$
where $A,B,C,D$ are variables.
I need to solve or rewrite the equation to easily obtain $\theta$ (or $2\theta$), given known values for $A, B, C, D$.
Thanks for a... | Noah Schweber | 28,111 | <p>This is indeed somewhat confusing; let me break down what they're talking about.</p>
<p>The source of your confusion is I think the use of the word "in." In this context, it does <strong>not</strong> mean "$\in$." When Hrbacek/Jech say "Is $E$ a [someting] relation in $X$?," it might be better for them to use the w... |
2,781,017 | <p>I known that $\sum a_i b_i \leq \sum a_i \sum b_i$ for $a_i$, $b_i > 0$. It seems this inequality will also hold true when $a_i$, $b_i \in (0,1)$. However, I am unable to find out if</p>
<p>$\sum \frac{a_i}{b_i} \leq \frac{\sum a_i}{\sum b_i}$ </p>
<p>holds true for $a_i$, $b_i \in (0,1)$.</p>
| Fly by Night | 38,495 | <p>Is it true that $\frac{2}{1}+\frac{3}{1} \le \frac{2+3}{1+1}?$</p>
<p>If you want numbers between $0$ and $1$, then consider
$$\frac{0.2}{0.1}+\frac{0.3}{0.1} \ \ \le \ \ \frac{0.2+0.3}{0.1+0.1}$$</p>
|
1,182,523 | <p>I have stumbled across this question:
Let $a,b,c$ be integers, not all $0$ such that $\max(|a|,|b|,|c|)<10^6$. Prove that $|a+b \sqrt{2} + c \sqrt{3}| > 10^{-21}$. </p>
<p>Could anybody help by solving this? Elementary solution is preferred.</p>
| achille hui | 59,379 | <p>Let <span class="math-container">$0 < \epsilon \ll 1$</span> be any small and <span class="math-container">$M \gg 1$</span> be any large positive numbers.
Let <span class="math-container">$\lambda$</span> be a number of the form <span class="math-container">$a + b\sqrt{2} + c\sqrt{3}$</span> where <span class="ma... |
41,718 | <p>Currently I am working on creating the package on Mathematica version 9 on Windows 7. Here is my code as follows: </p>
<pre><code>BeginPackage["mypackage`"]
Begin["`Private`"]
getColumn[data_,branch_List]:=
Module[{pos},
pos = Position[data,#][[1,2]]&/@branch;
data[[All,pos]]
];
RemoveMissing[data_]:=Delete... | Szabolcs | 12 | <p>Notice how the active context changes as the package gets loaded line by line:</p>
<pre><code>BeginPackage["mypackage`"]
(* the active context is mypackage` *)
(* whatever gets mentioned for the first time within this section
becomes part of the mypackage` context *)
(* any symbol meant to be public, i.e. usabl... |
1,181,269 | <p>I have a function $f(x)=x+\sin x$ and I want to prove that it is strictly increasing. A natural thing to do would be examine $f(x+\epsilon)$ for $\epsilon > 0$, and it is equal to $(x+\epsilon)+\sin(x+\epsilon)=x+\epsilon+\sin x\cos \epsilon + \sin \epsilon \cos x$.</p>
<p>Now all I need to prove is that $x+\eps... | Jack D'Aurizio | 44,121 | <p>Let $f(x)=x+\sin x$. Then $f'(x)=1+\cos x\geq 0$ and:</p>
<p>$$ f(x+h)-f(x) = h\, f'(\xi),\quad \xi\in(x,x+h) $$
by Lagrange's theorem, hence $f(x+h)-f(x)\geq 0$. </p>
<p>In order to prove that the inequality is strict,
we can notice that:
$$ f(x+h)-f(x-h) = 2h + 2\cos x \sin h $$
can be zero only if $\cos x=-1$ a... |
95,126 | <p>Consider the finite sum</p>
<pre><code>rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
</code></pre>
<p>Is there a way to bring <em>Mathematica</em> to calculate the limit for <code>n -> ∞</code>?</p>
<p>I have tried <code>Limit[]</code> as well as <code>NLimit[]</code> without success.</p>
| Community | -1 | <p>It's not the solution you are looking for and surely you have tried the same.</p>
<pre><code>rs[x_, n_] := x/n Sum[n^2/(i + (n - i) x)^2, {i, 1, n}]
rs[1., \[Infinity]]
</code></pre>
<blockquote>
<p>Infinity::indet: Indeterminate expression 0 [Infinity] encountered.</p>
<p>Sum::div: Sum does not converge.</... |
2,302,966 | <p>Translate the following English statements into predicate logic formulae. The domain is the set of integers. Use the following predicate symbols, function symbols and constant symbols.</p>
<ul>
<li>Prime(x) iff x is prime</li>
<li>Greater(x, y) iff x > y</li>
<li>Even(x) iff x is even</li>
<li>Equals(x,y) iff x=y</... | Stefan Dawydiak | 98,106 | <p>Here is a whole family of examples: The matrix groups $\mathrm{SL}(n,\mathbb{C})$, $\mathrm{GL}(n,\mathbb{C})$, $SO(n,\mathbb{C})$, and $\mathrm{Sp}(n,\mathbb{C})$ are all real Lie groups. They are not compact (I think this is pretty easy to see). Each has a maximal compact subgroup though. In order, they are $\math... |
1,371,649 | <p>The question is:</p>
<blockquote>
<p>What does the following interation formula do?:
<span class="math-container">$$x_{k+1}=2x_k-cx_{k}^2.$$</span></p>
</blockquote>
<p>I tried to identify this with Newtons method. I.e. I tried to bring that into the form <span class="math-container">$x_{k+1}=x_k-\frac{f(x_0)}{f'(x_... | Lutz Lehmann | 115,115 | <p>You could also see the partial square of a binom on the right side, and complete it as
<span class="math-container">$$
1-cx_{k+1}=1-2cx_k+(cx_k)^2=(1-cx_k)^2.
$$</span>
The sequence <span class="math-container">$y_k=1-c_k$</span> is then a subsequence of the geometric sequence, and from <span class="math-container">... |
2,910,301 | <p>Most of the textbooks state that provided a nonzero field $F$, a nonzero polynomial $f\in F[x]$ of degree $n$ has at most $n$ <em>distinct</em> roots. I am wondering whether the word "distinct" can be removed? I guess the answer is yes, but I cannot come up with a nice proof. </p>
<p>Sorry for the confusion. The qu... | Mark Bennet | 2,906 | <p>The key thing you need is that these are no non-trivial zero divisors, so that if $ab=0$ then either $a=0$ or $b=0$. So the formula for maximum number of roots works if you are working within a domain (and not just a field - the integers will do too).</p>
<p>Then you can use the polynomial division algorithm with $... |
481,764 | <p>What type of symmetry does the function $y=\frac{1}{|x|}$ have?
Specify the intervals over which the function is increasing and the intervals where it is decreasing.</p>
| tylerc0816 | 53,243 | <p>Plotting the function is a good way to guess at where the function is going to be increasing or decreasing. Then you can think about why.</p>
<p>As you increase $x$, what is going to happen to $\frac{1}{|x|}$? What happens when $x$ approaches (but does not equal!) 0?</p>
|
229,703 | <p>Given the function
<span class="math-container">\begin{align*}
f \colon \mathbb{R}^n &\to \mathbb{R}^n\\
v&\mapsto \dfrac{v}{\|v\|},
\end{align*}</span>
I would like to compute the derivative of <span class="math-container">$f$</span>, that is <span class="math-container">$df(v)$</span>. It is possible t... | vsht | 20,240 | <p>A solution using <a href="https://feyncalc.github.io/" rel="noreferrer">FeynCalc</a> would be to write</p>
<pre><code>ex = CVD[v, i]/Sqrt[CSPD[v, v]]
</code></pre>
<p>which corresponds to
<span class="math-container">$
\frac{v^i}{\sqrt{v^2}}
$</span> (<code>CVD</code> denotes a <span class="math-container">$D-1$</sp... |
229,703 | <p>Given the function
<span class="math-container">\begin{align*}
f \colon \mathbb{R}^n &\to \mathbb{R}^n\\
v&\mapsto \dfrac{v}{\|v\|},
\end{align*}</span>
I would like to compute the derivative of <span class="math-container">$f$</span>, that is <span class="math-container">$df(v)$</span>. It is possible t... | Carl Woll | 45,431 | <p>Maybe you could use the following approach:</p>
<pre><code>Clear[VectorD]
VectorD[e_, v_] := ReplaceAll[
D[e, VectorD, NonConstants->{v}],
s_Dot:>TensorReduce[s,Assumptions->v ∈ Vectors[d]]
]
VectorD /: D[s_. v_,VectorD,NonConstants->{v_}] := s IdentityMatrix[d] +
TensorProduct[v, D[s, Vec... |
3,696,371 | <p>I was rolling stats for a set of characters (main + backup) with my DM, and he told me I could choose between 3 sets of two rolls. One rolled by him, one rolled by another player, and one rolled by me. Him and the other player use physical dice, rolling three dice, then rerolling the lowest value twice. Myself, I ro... | Xander Henderson | 468,350 | <p>You are both wrong:</p>
<ul>
<li><p>You are wrong because the two rolling procedures do not produce the same distributions of probability.</p></li>
<li><p>Your DM is wrong, because this has nothing to do with the Monty Hall (three doors) problem.</p></li>
</ul>
<p>Tackling these in reverse order: in the Monty Hal... |
2,669,524 | <p>I am reading <strong>Algebraic Geometry</strong>, Vol 1, <em>Kenji Ueno</em>. My problem is that
$$k\left[ x,y,t\right]/\left(xy-t\right)\otimes_{k\left[t\right]}k\left[t\right]/\left(t-a\right) \simeq k\left[x,y\right]/\left(xy-a\right) $$
where $k$ is a field and $a$ is an element in $k$. I don't understand how i... | user | 505,767 | <p>Note that $dV$ in the integral is just a symbol to identify the variable respect to we are integrating and $\Delta V$ in this case just indicate the difference $V(2)-V(1)$ according to the foundamental theorem of calculus</p>
|
976,910 | <p>i'm having a small issue with a certain question. </p>
<p>Given a parametric equation of a plane $x=5-2a-3b$, $y=3-4a+2b$, $z=7-6a-2b$, find a point $P$ on the plane so that the position vector of $P$ is perpendicular to the plane.</p>
<p>How would you go about this for a parametric equation? I think I could conve... | copper.hat | 27,978 | <p>Suppose $x \notin K$. Suppose $B(x,{1 \over n})$ intersects $K$ for all $n$, then $x$ is a limit point of $K$, which is a contradiction. Hence for some $n_x$ we have $K \cap B(x, {1 \over n_x}) = \emptyset$. Hence $K^c$ is open.</p>
|
4,004,978 | <blockquote>
<p>For all <span class="math-container">$a, b, c, d > 0$</span>, prove that
<span class="math-container">$$2\sqrt{a+b+c+d} ≥ \sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}$$</span></p>
</blockquote>
<p>The idea would be to use AM-GM, but <span class="math-container">$\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d}... | Neat Math | 843,178 | <p>How about Jensen's inequality? <span class="math-container">$\sqrt x$</span> is concave so</p>
<p><span class="math-container">$$\sqrt{\frac{a+b+c+d}{4}} \ge \frac 14 \left( \sqrt a +\sqrt b +\sqrt c +\sqrt d \right)$$</span></p>
|
1,929,445 | <blockquote>
<p>Is there a solution to the problem $$\left\{\begin{matrix}
y'=y+y^4\\
y(x_0)=y_0
\end{matrix}\right.$$
which is defined on $\mathbb{R}$? ($x_0,y_0$ might be any real numbers)</p>
</blockquote>
<p>It's easy to prove that for all $(x,y)\in\mathbb{R}^2$ there exists an open interval $I$ (with $x_0\in... | Futurologist | 357,211 | <p>Actually, there is a whole family of solutions of this equation that are defined on the whole real line. Look carefully at the equation itself
$$\frac{dy}{dx} = y + y^4$$ and write it in the form
$$\frac{dy}{dx} = y(1 + y^3)$$ As the equation is locally Lipschitz everywhere, because it is polynomial, locally everywh... |
2,429,164 | <p>I'd like to prove
$$\lim_{n\rightarrow+\infty} \int_0^\pi n\sqrt{n^2+x^2-2nx\cos\theta}-n^2 \mathrm{d}\theta = \frac{\pi}{4}x^2$$
for $x\in[0,\infty)$. I checked it numerically and derived it with Matlab Symbolic Toolbox, but cannot prove it by calculus.</p>
<p>I cannot use the Dominated Convergence Theorem since I... | zhw. | 228,045 | <p>The integrand equals</p>
<p>$$\tag 1 n^2(1+x^2/n^2 -(2x\cos \theta)/n)^{1/2} - n^2.$$</p>
<p>From Taylor we have $(1+u)^{1/2} = 1+u/2 -u^2/8 +O(u^3)$ as $u\to 0.$ Apply this with $u = x^2/n^2 -(2x\cos \theta)/n$ to see
$(1)$ equals</p>
<p>$$\frac{n^2}{2} (x^2/n^2 -(2x\cos \theta)/n) - \frac{n^2}{8}\frac{4x^2\cos^... |
3,063,053 | <p>I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped. <br /></p>
<p><span class="math-container">$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}$$</span> </p>
<p>You'll notice that using L'Hopital... | Noble Mushtak | 307,483 | <p>By your own reasoning, you have the following:
<span class="math-container">$$\lim\limits_{x\to \infty} \frac{x}{\sqrt{x^2 + 1}}=\lim\limits_{x\to \infty} \frac{\sqrt{x^2 + 1}}{x}$$</span></p>
<p>Now, the left side is clearly the reciprocal of the right side, so we have:
<span class="math-container">$$\lim\limits_{... |
621,409 | <p>I need some help with the following question:</p>
<p>We have $H$ acting by automorphisms on $N$, and let $\rho:H\to Aut(N)$ the associated representation by automorphisms.</p>
<p>Suppose that $G=H[N]_{\rho}$ is a semidirect product, and $K=\ker(\rho)$.</p>
<p>Prove that $K\unlhd G$ and that $G/K$ is also a semid... | user1337 | 62,839 | <p>Note that $$\frac{1-\cos(x^2+y^2)}{(x^2+y^2)^2}=\frac{2 \sin^2 \frac{x^2+y^2}{2}}{(x^2+y^2)^2}, $$ as well as that for small $\theta$ $$\sin \theta \sim \theta. $$</p>
|
748,325 | <p>In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen), argues as follows: Let $\sigma$ be the first singular value of A, and $v_{1}$ the corresponding singular vector. Let $w$ be another linearly independent vector such that $||Aw||=\sigma$, and construc... | Dubious | 32,119 | <p>This is not an answer but too long for a comment. Suppose that $||Av_2||=\sigma$, then:</p>
<p>$$\sigma^2=||A(w)||^2=\left<A(w),A(w)\right>=\left<aA(v_1)+bA(v_2),aA(v_1)+bA(v_2)\right>=$$
$$=a^2||A(v_1)||^2+b^2||A(v_2)||^2+2ab\left<A(v_1),A(v_2)\right>=\sigma^2+2ab\left<A(v_1),A(v_2)\right>$... |
2,900,294 | <p>I tried this and I only got $\sin( 53^\circ)= \sin( 127^\circ).$ How do I find the equal value in cosine or tangent? Please help me out. Thank you!</p>
| Jason Kim | 570,340 | <p>We can use the identity $\sin x^\circ = \cos {(90^\circ-x^\circ)}$ so $\sin{53^\circ}=\sin{127^\circ}=\cos{(127^\circ-90^\circ)}=\cos{37^\circ}.$</p>
<p>I can't think of an simple way to express it in tangent.</p>
|
864,237 | <p>Let's take a short exact sequence of groups
$$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$
I understand what it says: the image of each homomorphism is the kernel of the next one, so the one between $A$ and $B$ is injective and the one between $B$ and $C$ is surjective. I get it. But other than being a so... | jxnh | 132,834 | <p>As Jessica said in a previous answer, the set up initially tells us that in some sense <span class="math-container">$C = B/A$</span>. To see this, since <span class="math-container">$A \to B$</span> is injective, and has image precisely the kernel of some map, we may identify <span class="math-container">$A \unlhd B... |
3,682,661 | <p>I needed help with Part (A) without using L'Hopital's because its getting too lengthy.Can someone help me obtain solution with series without using L Hospitals rule</p>
<p>I'm trying something out with series </p>
<blockquote>
<p><a href="https://i.stack.imgur.com/FC6Xs.jpg" rel="nofollow noreferrer">Question Im... | Phicar | 78,870 | <p>Let <span class="math-container">$\begin{bmatrix}x_1 & x_2\\x_3 & x_4\end{bmatrix}\in SU(2),$</span> then <span class="math-container">$x_1\cdot x_4-x_3\cdot x_2=1$</span> because determinant is <span class="math-container">$1$</span>(this you did not mention) and <span class="math-container">$$\begin{bmatri... |
1,725,084 | <p>I am currently trying to practice the technique of transfinite induction with the following problem: </p>
<p>Suppose that $X$ is a non-empty subset of an ordinal $\alpha$, so that $X$ is well-ordered by $\in$. Show that $\text{type}(X; \in) \leq \alpha$. </p>
<hr>
<p>My approach thus far: </p>
<p>Let $\beta = \t... | Brian M. Scott | 12,042 | <p>It’s easier if you replace $f$ by its inverse and work with an order-isomorphism $f:\beta\to X$. Then $f$ is an order-isomorphism of $\beta$ into $\alpha$, and you want to show that $\xi\le f(\xi)$ for each $\xi\in\beta$. If not, let $\gamma\in\beta$ be minimal such that $f(\gamma)<\gamma$. Then by the minimality... |
1,676,505 | <p>Let $f:[0,1]\times[0,1]\to \mathbb R$,
$$f(x,y)=
\begin{cases}
\frac1q+\frac1n, & \text{if $(x,y)=(\frac mn,\frac pq) \in \Bbb Q\times\Bbb Q,$ $ (m,n)=1=(p,q)$ } \\
0, & \text{if $x$ or $y$ irrational$ $ or $0,1$}
\end{cases}
$$</p>
<p>Prove that f is integrable over $R=[0,1]\times[0,1]$ and find the va... | Julián Aguirre | 4,791 | <p>If <span class="math-container">$a,b\in[0,1]\setminus\mathbb{Q}$</span>, then <span class="math-container">$f$</span> is continuous in <span class="math-container">$(a,b)$</span>. Given <span class="math-container">$\epsilon>0$</span>, the number of rationals in <span class="math-container">$[0,1]$</span> with de... |
2,476,688 | <p>The Homework Exercise I am working on, is:</p>
<blockquote>
<p>Let $\overrightarrow{a}, \overrightarrow{b}$ be vectors. Show that
$\overrightarrow{a} \cdot
\overrightarrow{b}=\frac{1}{4}\left(\Vert{\overrightarrow{a}+\overrightarrow{b}\Vert^2}-\Vert{\overrightarrow{a}-\overrightarrow{b}\Vert^2}\right)$.</p>
<... | 5xum | 112,884 | <p>Use the fact that $$\vec a\cdot (\vec b + \vec c) = \vec a \cdot \vec b + \vec a\cdot \vec c$$</p>
<p>and write it all out. A lot of things should cancel out.</p>
|
2,476,688 | <p>The Homework Exercise I am working on, is:</p>
<blockquote>
<p>Let $\overrightarrow{a}, \overrightarrow{b}$ be vectors. Show that
$\overrightarrow{a} \cdot
\overrightarrow{b}=\frac{1}{4}\left(\Vert{\overrightarrow{a}+\overrightarrow{b}\Vert^2}-\Vert{\overrightarrow{a}-\overrightarrow{b}\Vert^2}\right)$.</p>
<... | Bernard | 202,857 | <p>Just expand the dot products by didtributivity in the right-hand side:
\begin{align}
(\vec a+\vec b)\cdot(\vec a+\vec b)-(\vec a-\vec b)\cdot(\vec a-\vec b)&=\begin{aligned}[t]\vec a\cdot\vec a&+\vec a\cdot\vec b+\vec b\cdot\vec a+\vec b\cdot\vec b\\
&-\vec a\cdot\vec a+\vec a\cdot\vec b+\vec b\cdot\vec ... |
2,267,935 | <p>There is a fibration $SO(n-1) \mapsto SO(n) \mapsto S^{n-1}$, from basically taking the first column of the matrix in $\mathbb{R}^n$. Is this fibration trivializable? </p>
| Tsemo Aristide | 280,301 | <p>If $n=3$, the Hopf fibration is the composition $S^3=Spin(3)\rightarrow SO(3)\rightarrow S^2$, so if $SO(3)\rightarrow S^2$ is trivial, so the hopf fibration will be flat and this is not true.</p>
<p><a href="https://en.wikipedia.org/wiki/Hopf_fibration#Geometric_interpretation_using_rotations" rel="nofollow noref... |
1,370,576 | <p>I am working on a trigonometry question at the moment and am very stuck. I have looked through all the tips to solving it and I cant seem to come up with the right answer. The problem is </p>
<blockquote>
<p>What is exact value of<br>
$$\cot \left(\frac{7\pi}{6}\right)? $$</p>
</blockquote>
| Zain Patel | 161,779 | <p>We have $$\cot \left(\frac{7\pi}{6}\right) = \frac{1}{\tan \left(\frac{7\pi}{6}\right)} = \frac{\cos \left(\frac{7\pi}{6}\right)}{\sin \left(\frac{7\pi}{6}\right)} \equiv \frac{-\cos \left(\frac{\pi}{6}\right)}{-\sin \left(\frac{\pi}{6}\right)} = \sqrt{3}$$</p>
<p>You can easily see this using a "CAST" diagram to r... |
149,830 | <p>As we know, the eigen vectors and eigen values of a real symmetric matrix always is are real numbers. I am trying to use the <em>Mathematica</em> to verify this theory. Suppose I have a matrix <code>A</code></p>
<pre><code>A={{6585, 7579, 6717}, {7579, 11002, 12324}, {6717, 12324, 17030}}
Eigenvalues[A]
</code></pr... | eldo | 14,254 | <p>In other situations you might have a look at <code>Map</code> (<code>/@</code>)</p>
<p>Vector of value pairs</p>
<pre><code>m1 = RandomInteger[{1, 10}, {10, 2}];
</code></pre>
<p>Matrix of value pairs</p>
<pre><code>m2 = Partition[m1, 5];
</code></pre>
<p>Some function</p>
<pre><code>f[{x_, y_}] := x*y
</code>... |
2,837,172 | <p>In complex analysis, sometimes we need to use some theorems which are results of measure theory. However, I know very very little about measure theory. So</p>
<blockquote>
<p>What are some very basic results of measure theory on complex functions/complex plane/complex calculus?</p>
</blockquote>
<p>I expect the ... | saulspatz | 235,128 | <p>Continuous functions are measurable. All the single-valued functions you'll see in complex variables are measurable. In particular, harmonic functions are measurable. </p>
<p>As for convergence theorems, I can't think of any but the dominated convergence theorem that are likely to apply. (Way back when I took co... |
3,454,682 | <p>I know that <span class="math-container">${\langle x, y \rangle}$</span> means the inner product but I've stumbled upon the notation <span class="math-container">${\langle x, y \rangle}_a$</span> with <span class="math-container">$a \in \mathbb{R}$</span> and I can't figure out what it means. Usually what's in the s... | J.G. | 56,861 | <p>The answer @Masacroso gave is probably right. I should mention subscripts on either side may instead be labels for the vectors themselves. However, this is usually used only in <a href="https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation" rel="nofollow noreferrer">bra–ket notation</a>, where we replace the comma ... |
156,376 | <p>I understand that when we are doing indefinite integrals on the real line, we have $\int f(x) dx = g(x) + C$, where $C$ is some constant of integration. </p>
<p>If I do an integral from $\int f(x) dx$ on $[0,x]$, then is this considered a definite integral? Can I just leave out the constant of integration now? I am... | robjohn | 13,854 | <p>$\int f(x)\,\mathrm{d}x$ is an <a href="http://en.wikipedia.org/wiki/Antiderivative">antiderivative</a>. It represents any function whose derivative with respect to $x$ is $f(x)$.</p>
<p>$\int_0^af(x)\,\mathrm{d}x$ is a <a href="http://en.wikipedia.org/wiki/Integral#Terminology_and_notation">definite integral</a>, ... |
1,714,278 | <p>Given the sequence $ y_{k}=2^k\tan(\frac{\pi}{2^k})$ for k=2,3,.. prove that $ y_{k} $ is recursively produced by the algorithm:
$$ y_{k+1}=2^{2k+1}\frac{\sqrt{1+(2^{-k}y_{k})^2}-1}{y_{k}} $$ for k=2,3,...</p>
<p>I used the identity $ {\tan^2({a})}=\frac{1-\cos{(2a)}}{1+\cos{(2a)}}$ but I couldn't get it right. An... | Noble Mushtak | 307,483 | <p>Let's try plugging in the explicit formula for $y_k$ into the recursive formula and seeing if we get the explicit formula for $y_{k+1}$ back.</p>
<p>$$y_{k+1}=2^{2k+1}\frac{\sqrt{1+\left(2^{-k}2^k\tan\left(\frac{\pi}{2^k}\right)\right)^2}-1}{2^k\tan\left(\frac{\pi}{2^k}\right)}$$</p>
<p>Cancel out the $2^{2k+1}$ o... |
4,004,827 | <p>I need to calculate:
<span class="math-container">$$\displaystyle \lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{1- e^{-2x}}}$$</span></p>
<p>I looks like I need to use common limit:
<span class="math-container">$$\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} = 1$$</span></p>
<p>So I take following steps:</p>
<p><span c... | Raffaele | 83,382 | <p><span class="math-container">$$\sqrt{1-e^{-2 x}}\sim \sqrt{2} \sqrt{x};\;\text{as }x\to 0$$</span></p>
<p><span class="math-container">$$\lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{2} \sqrt{x}}=\lim_{x \to 0^+}\left(\frac{3}{\sqrt 2}\sqrt x +\frac{1}{\sqrt 2} \right)=\frac{1}{\sqrt 2} $$</span></p>
|
1,073,628 | <p>I am trying to find generating functions which will give me a power logarithm. </p>
<p>I am trying to find generating sums in the form</p>
<p>$$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$</p>
<p>or </p>
<p>$$\sum_{n=1}^{\infty} a_n\,x^n = \frac{\log^2(x)}{x}.$$</p>
<p>Something, which will return ... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
789,407 | <p>If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. </p>
<p><strong>Edit:</strong> I forgot to mention in the question that $a$, $b$, and $c$ are natural numbers. Sorry for the inconvenience.<br>
<strong>Edit 2:</strong> As Hagen von Eitzen said abo... | MathGod | 101,387 | <p><strong>Given:</strong> Roots lie in $(0,1).$ </p>
<p>Let $f(x)=ax^2-bx+c$ and it's roots be $\alpha$ and $\beta$ </p>
<p>$\implies f(0) \times f(1) > 0$ (Can be easily verified from the parabolic graph of $f(x)$)</p>
<p>or $c(a-b+c)>0$ </p>
<p>$\implies \frac{c}{a}(1-\frac{b}{a}+\frac{c}{a})>0$ </... |
3,272,738 | <p>I've been trying to make sense of these two integrals, somehow the result seems intuitive, yet very hard to compute. We define</p>
<p><span class="math-container">$$
f(x)=\frac{1}{4\pi}\delta(|x|-R)$$</span>
and then note that
<span class="math-container">$$
-\frac{1}{2}\int\int\frac{f(x)f(y)}{|x-y|}=-\frac{1}{2R}$... | callculus42 | 144,421 | <p>Your calculation is right. <span class="math-container">$\mathbb E(X\cdot Y)=\mathbb E_X(X\cdot \mathbb E_{Y|X}[Y])=7.63$</span></p>
<p>You´re right as well that <span class="math-container">$\mathbb E(X)\cdot \mathbb E(Y)=7.63$</span>. That means that <span class="math-container">$Cov(X,Y)=0$</span>.</p>
<p>This ... |
4,515,517 | <p>Suppose that <span class="math-container">$E$</span> is a measruable set and <span class="math-container">$f: E \rightarrow [0, \infty]$</span> is a non-negative function with <span class="math-container">$\int_E f(x)^n dx = \int_E f(x) dx < \infty$</span> for all positive integers <span class="math-container">$n... | geetha290krm | 1,064,504 | <p><span class="math-container">$\lim t^{n}$</span> exists for every <span class="math-container">$t \geq 0$</span>. (It may be <span class="math-container">$+\infty$</span>, of course).</p>
<p>The hypothesis can be weakened to <span class="math-container">$\int_E f =\lim \int_E f^n$</span>.</p>
<p>Note that <span clas... |
4,033,831 | <p>Example:</p>
<p><img src="https://i.stack.imgur.com/nPzJb.png" alt="Notation" /></p>
<p>From this we can tell no negative real number can be the image of any element of the domain. Thus not surjective because the range is not equal to the codomain, which means the function associates a any real number to a positive ... | user10354138 | 592,552 | <p>It depends on how you define what a function is (i.e., whether you take the set of ordered-pairs alone a la Bourbaki, or use something like the source-target predicate).</p>
<p>If you use the Bourbaki definition (i.e., <span class="math-container">$f\colon A\to B$</span> is <span class="math-container">$\{(a,f(a))\m... |
849,433 | <blockquote>
<p>We have subspaces in $\mathbb R^4: $ </p>
<p>$w_1= \operatorname{sp} \left\{
\begin{pmatrix} 1\\ 1 \\ 0 \\1 \end{pmatrix} ,
\begin{pmatrix} 1\\ 0 \\ 2 \\0 \end{pmatrix},
\begin{pmatrix} 0\\ 2 \\ 1 \\1 \end{pmatrix} \right\}$,
$w_2= \operatorname{sp} \left\{
\begin{pmatrix} 1\\ 1 \\ 1 \\1 ... | James S. Cook | 36,530 | <p>If you consider $\text{rref}(B_1|B_2)$ then you can easily ascertain which vectors in $B_1$ fall in $w_2=\text{span}(B_2)$. Likewise, $\text{rref}(B_2|B_1)$ will tell you which vectors in $B_2$ fall in $w_1=\text{span}(B_1)$. Once you know both of these you ought to be able to put the answer together. </p>
<p>(<em>... |
750,751 | <p>if $V$ is a finite-dimensional vector space and $t \in \mathcal L (V,V) $is such that $t^2 = id_V$ prove that the sum of eigenvalues of t is an integer.</p>
<p>I started the prove as such:</p>
<p>Let $\lambda_1 ,...,\lambda_n $ be eigenvalues of $t$. </p>
<p>So $\lambda_1^2 ,... \lambda_n^2$ will be the eigenvalu... | Oliver | 84,451 | <p>The eigenvalues are 1 or -1, hence the sum is an integer.
The fact that the eigenvalues are $\pm1$ can be proved as follows: if $\lambda$ is an eigenvalue and $v$ an eigenvector corresponding to $\lambda$, then $v=t^2(v)=\lambda^2 v$, hence $\lambda=\pm1$.</p>
|
806,476 | <p>In Milnor's book <em>Topology from the Differentiable Viewpoint</em> there's the following problem:</p>
<p><strong>Problem $6$</strong> (Brouwer). Show that any map $S^n\to S^n$ with degree different from $(-1)^{n+1}$ must have a fixed point.</p>
<p><strong>My solution:</strong> Assume that the map $f:S^n\to S^n$ ... | Moishe Kohan | 84,907 | <p>I think, what you gave is (a sketch of) the simplest proof, unless you know about the <a href="http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem" rel="noreferrer">Lefshetz fixed point theorem</a>. Using the latter, then the argument goes like this: $H_k(S^n)$ is nonzero only in dimensions $0$ and $n$ and $H... |
315,551 | <p>So I'm going over my practice midterms (which all seem to have solutions like this one), </p>
<p><img src="https://i.stack.imgur.com/fC8Gu.png" alt="Image"></p>
<p>Can anyone help clarify this for me? I understand that you multiply by the reciprocal to get to line two. But after that I'm completely lost, I don't u... | Cousin | 54,755 | <p>Well, $x^2+1-[(x+h)^2+1]=x^2-(x+h)^2=(x-(x+h))(x+(x+h))$. The last equality comes from the difference of two squares. </p>
|
257,978 | <p>Is there any non-monoid ring which has no maximal ideal?</p>
<p>We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very easy Theorem there.</p>
<p>We say a ring $R$ is monoid if it has an multiplicative identity element, that if... | rschwieb | 29,335 | <p>If $D$ is a valuation domain with unique maximal ideal $M$, then there are some conditions where $M$ is an example of a commutative rng with no maximal ideals.</p>
<p>As I remember it, one can choose a domain with a value group within $\Bbb{R}$ such that the group has no least positive element. Then, one can argue ... |
159,529 | <p>The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as representation categories of vertex operator algebras?</p>
| Marcel Bischoff | 10,718 | <p>This answer is related to my answer here: <a href="https://mathoverflow.net/questions/153264/duality-between-orbifold-and-quasi-hopf-algebra-twisted-quantum-doubles/153270#153270">Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)</a> and the comment by Scott.</p>
<p>Every finite group $G$ ca... |
1,093,717 | <p>Let $\xi_1, \xi_2, \ldots \xi_n, \ldots$ - independent random variables having exponential distribution $p_{\xi_i} (x) = \lambda e^{- \lambda x}, \; x \ge 0$ and $p_{\xi_i} (x) = 0, \; x < 0$. Let $\nu = \min \{n \ge 1 : \xi_n > 1\}$. Need to find the distribution function of a random variable $g = \xi_1 + \xi... | wolfies | 74,360 | <p>Let:</p>
<ul>
<li><p><span class="math-container">$X \sim \text{Exponential}(\lambda)$</span> with pdf <span class="math-container">$p(x) = \lambda e^{-\lambda x}$</span>, for <span class="math-container">$x>0$</span>. </p></li>
<li><p><span class="math-container">$(X_1, X_2, \dots)$</span> denote successive ran... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.