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 |
|---|---|---|---|---|
3,260,776 | <blockquote>
<p>Suppose we are given any arbitrary collection of sets.</p>
<p>How do get a largest topology from the above arbitrary collection?</p>
</blockquote>
<p>How to construct a Topology from this collection and in which condition ?</p>
<p>I don't have any answer to me.</p>
<p>Any help or idea is appreciated.</p... | Kavi Rama Murthy | 142,385 | <p>First take all possible finite intersections of those sets together with the empty set and then take arbitrary unions of the new sets you have obtained. You get a topology and this topology is the smallest topology containing the given sets. You can read about base and sub-base in any book on topology for more inf... |
2,332,741 | <p>I have the following problem:</p>
<blockquote>
<p>Let $\Omega \subset \mathbb{R}^3$ be an open bounded set with a smooth boundary $\partial \Omega$ and the unit normal $v$. Calculate for the vector field $a(x,y,z)=(0,0,-pz)$ with $p>0$ the value of -$\int_{\partial\Omega}\langle a,v\rangle d\mu_{\partial\Omega... | Tony Blair's Witch Project | 445,451 | <blockquote>
<p>$\text{}$1. If $C$ is an irreducible nonsingular curve on a nonsingular projective surface $X$, is there always a very ample divisor $E\in|C|$?</p>
</blockquote>
<p>No. For example, a line on a nonsingular quadric surface has $L^2 = 0$. Your question is not grammatical, since very ample is a property... |
350,910 | <p>Is there any condition based on the coefficients of terms that guarantees all real solutions to a general cubic polynomial? e.g. $$ax^3+bx^2+cx+d=0\, ?$$</p>
<p>If not, are there methods rather than explicit formula to determine it?</p>
<p>Thank You.</p>
| Arthur | 15,500 | <p><a href="http://en.wikipedia.org/wiki/Cubic_polynomial#The_nature_of_the_roots">Wikipedia</a> says it's the following expression:
$$
18abcd-4b^3d + b^2c^2 -4ac^3 -27a^2d^2
$$
If it's positive there are three real roots. If it's negative there's one real and two (conjugate) imaginary roots. If it's equal to zero, the... |
3,278,761 | <p>Suppose a student says : "if 17 is even, then 2 is not a divisor of 17". </p>
<p>Surely his teacher would tell him he is wrong, saying that when a number is even, this number has 2 as divisor. The teacher would correct with " if 17 were even, then 2 would be a divisor of 17". In other words, the student's claim con... | David K | 139,123 | <p>You have stated a general rule, which is mathematically true:</p>
<blockquote>
<p>For every number <span class="math-container">$x,$</span> if <span class="math-container">$x$</span> is even, then <span class="math-container">$x$</span> has <span class="math-container">$2$</span> as divisor.</p>
</blockquote>
<p... |
134,796 | <p>Example list below. All elements are in the form {1 or 0, 1 or 0, 1 or 0}, with a least one of the numbers 0 and 1 in the element (so excluding {1,1,1} and {0,0,0}) </p>
<pre><code>ListA = {{1, 1, 0}, {1, 1, 0}, {1, 1, 0}, **{0, 1, 1}**, {1, 0, 1}, {1, 0, 1}, {1,
0, 1}}
</code></pre>
<p>I want a command to rep... | Mr.Wizard | 121 | <p>Written for clarity over efficiency:</p>
<pre><code>ListA =
{{1, 1, 0}, {1, 1, 0}, {1, 1, 0}, {0, 1, 1}, {1, 0, 1}, {1, 0, 1}, {1, 0, 1}};
Split[ListA]
% //. {a___List, lone : {{__}}, b__List} :> {a, {b}[[1, {1}]], b}
Join @@ %
</code></pre>
<blockquote>
<pre><code>{{{1, 1, 0}, {1, 1, 0}, {1, 1, 0}}, {{0, ... |
3,214,331 | <p>If i take the complex number <span class="math-container">$e^{i(3+2i)}$</span>, it's conjugate is <span class="math-container">$e^{i(-3+2i)}$</span>.</p>
<p>However, the conjugate of the function f, defined as <span class="math-container">$f(x+iy)=e^{i(x+iy)}$</span>, is, according to my book: <span class="math-con... | Peter Foreman | 631,494 | <p>I think the book meant that
<span class="math-container">$$f(\overline{x+iy})=f(x-iy)=e^{i(x-iy)}$$</span>
but otherwise the book is in fact wrong as you say.</p>
|
1,028,695 | <p>While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: </p>
<p>I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of three variables to functions... | Mike F | 6,608 | <p>It's not true. There is only one <a href="http://en.wikipedia.org/wiki/Hilbert_space#Separable_spaces" rel="nofollow">separable Hilbert space</a>, up to isomorphism. Since $L^2(\mathbb{R}^3)$ and $L^2(\mathbb{R}^4)$ are both separable, they are isomorphic as Hilbert spaces. That is, there is an isometric linear mapp... |
3,082,635 | <p>Prove that for a given prime <span class="math-container">$p$</span> and each <span class="math-container">$0 < r < p-1$</span>, there exists a <span class="math-container">$q$</span> such that </p>
<p><span class="math-container">$$rq \equiv 1 \bmod p$$</span></p>
<p>I've only taken one intro number theory ... | David | 119,775 | <p>There are lots of proofs of this, and which is best for you will depend very much on what results you know already. Here is one which uses only the following fact:</p>
<ul>
<li>if <span class="math-container">$p$</span> is prime and <span class="math-container">$x,y$</span> are integers and <span class="math-conta... |
24,704 | <p>It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least <span class="math-container">$n_p(p^n-1)$</span> elements, where <span class="math-container">$n_p$</span> is the number of Sylow <span class="math-container">$p$</spa... | wildildildlife | 6,490 | <p>Suppose $P$ and $Q$ are Sylow p-subgroups of prime order p (so not just any power of p; as others remarked, then it is not true in general). Note that $P\cap Q$ is a subgroup of $P$ (and of $Q$). So by Lagrange, the order $|P\cap Q|$ divides p. As p is prime, it is 1 or p. But it cannot be p, as $P$ and $Q$ are dist... |
4,244,966 | <p>I have the ODE <span class="math-container">$y^2(1+y'^2)=4$</span> to solve this I used the substitution <span class="math-container">$y'=p$</span>
<span class="math-container">$$y^2(1+p^2)=4$$</span>
<span class="math-container">$$2y(1+p^2)dy+2py^2dp=0$$</span>
<span class="math-container">$$(p^2+1)dy+py\;dp=0$$</s... | Ishan Tiwari | 771,724 | <p>I think you've only circulated in the differential equation. You have written the series of steps only to arrive at the restatement of the original differential equation.</p>
<p>From your solution,</p>
<p><span class="math-container">$$y\sqrt{p^2-1} = c$$</span>
or <span class="math-container">$$ y^2 (p^2-1) = c^2$... |
624,002 | <p>Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.</p>
<p>pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \righ... | Boris Novikov | 62,565 | <p>No, it is not correct. Suppose $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ is an isomorophism. Then you have only $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}/\mathbb{Z}\times \mathbb{Z}\times \left \{ 0 \right \}\cong \mathbb{Z}\times \mathbb{Z}/f^{-1}(\mathbb{Z}\times \mathbb{Z}... |
624,002 | <p>Determine whether $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ are isomorphic groups or not.</p>
<p>pf) Suppose that these are isomorphic. Note that $\mathbb{Z}\times \mathbb{Z}$ is a subgroup of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}\times\left \{ 0 \righ... | bof | 111,012 | <p>The class of Abelian groups is an equational class. $\mathbb Z\times\mathbb Z$ is a free Abelian group with $2$ generators. $\mathbb Z\times\mathbb Z\times\mathbb Z$ is a free Abelian group with $3$ generators. If the free Abelian group with $2$ generators were isomorphic to the free Abelian group with $3$ generator... |
2,829,990 | <p>I want to calcurate</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n $$</span></p>
<p>I met this in studying Lebesgue integral. But, I don't know how to do at all. I would really appreciate if you could help me!</p>
<p>[Add]</p>
<p>Thanks to e... | JanG | 266,041 | <p>This answer will be based om Fubini's theorem and DCT.
Put
\begin{equation*}
I_n = \int_{V_n}\dfrac{x_1^p+x_2^p+\dots +x_n^p}{x_1^q+x_2^q+\dots +x_n^q}\, dx_1dx_2\dots dx_n
\end{equation*}
where $ V_n=(0,1)^n. $ We will try to prove that</p>
<p>\begin{equation*}
\lim_{n\to \infty}I_n = \dfrac{q+1}{p+1} \tag{1}
\end... |
172,617 | <p>I need to plot two datasets on the same plot. The datasets have the same x-range. However, I want to show only parts of the plot. </p>
<p>A minimal example would be</p>
<pre><code> h = π/100.;
i1 = ListLinePlot[Table[{i*h, Sin[i*h]}, {i, 0, 100}], PlotStyle -> Red];
i2 = ListLinePlot[Table[{i*h, Cos[... | Henrik Schumacher | 38,178 | <p>Maybe</p>
<pre><code>h = π/100.;
a = Table[{i*h, Sin[i*h]}, {i, 0, 100}];
b = Table[{i*h, Cos[i*h]}, {i, 0, 100}];
α = FirstPosition[a[[All, 1]] - π/2, _?Positive][[1]]
i1 = ListLinePlot[a[[α ;;]], PlotStyle -> Red];
i2 = ListLinePlot[b[[;; α]], PlotStyle -> Blue];
l1 = Graphics[{Black, Dashed, Line[{{π/2, -1... |
3,728,963 | <p>(Exercise 21 Chapter 2, Baby Rudin) I am trying to prove</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be separated subsets of some <span class="math-container">$\mathbb{R}^k$</span>, suppose <span class="math-container">$\textbf{a} \in A$</span>, <span c... | William Elliot | 426,203 | <p>Since <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are seperated, there exist open disjoint <span class="math-container">$U,V$</span> with <span class="math-container">$A \subset U, B \subset V$</span>.
<span class="math-container">$A_0 \subset K = p^{-1}(U), B_0 \subset L = p^... |
136,240 | <p>Why does the subgroup have to be able divide the group? For example why isn't the group:<br/>
$S= \{-4,-3,-1,0,1,2,3,4\}$ <br/>
a subgroup of $G= \{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6 \}$? Aren't both additive, have a neutral number, each member has a reciprocal and every member in group $S$ is present in group $G$?</p>... | Henry T. Horton | 24,934 | <p>A group must be closed under the group operation. Since you said "both are additive," I am going to assume that the group operation on $G$ (and hence $S$) is addition. Then $G$ is not closed under addition ($6 + 1 = 7 \notin G$, for example), unless you are considering $G$ as $\mathbb{Z}/13$. The same issue happens ... |
136,240 | <p>Why does the subgroup have to be able divide the group? For example why isn't the group:<br/>
$S= \{-4,-3,-1,0,1,2,3,4\}$ <br/>
a subgroup of $G= \{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6 \}$? Aren't both additive, have a neutral number, each member has a reciprocal and every member in group $S$ is present in group $G$?</p>... | Mikko Korhonen | 17,384 | <p>If $G$ is a finite group, then according to <a href="http://mathworld.wolfram.com/LagrangesGroupTheorem.html" rel="nofollow">Lagrange's theorem</a> the order (number of elements) of a subgroup $H$ divides the order of $G$. A proof of this fact can be found in any introductory text on abstract algebra.</p>
<p>The ba... |
253,921 | <p>I am trying to come up with a measurable function on $[0,1]^2$ which is not integrable, but such that the iterated integrals are defined and unequal.</p>
<p>Any help would be appreciated.</p>
| Michael Hardy | 11,667 | <p>$$
\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dy\,dx \ne \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dx\,dy
$$</p>
<p>Obviously either of these is $-1$ times the other and if this function were absolutely integrable, then they would be equal, so their value would be $0$. But one is $\pi/2$ and the other is ... |
3,528,370 | <p>Maybe this is too obvious, but I what to be sure... Let <span class="math-container">$Y$</span> be a <span class="math-container">$p\times p$</span> symmetric random matrix (i.e. you can think about <span class="math-container">$Y$</span> as a matrix with random entries). Define <span class="math-container">$E[Y]$</... | gt6989b | 16,192 | <p>To add to the existing Ian's answer, the mistake in your proof is that <span class="math-container">$tr(Y) < 0$</span> <strong>does not imply</strong> that <span class="math-container">$\mathbb{E}[tr(Y)] < 0$</span>.</p>
<p>The reason for this is that <span class="math-container">$Y$</span> is some <em>realiz... |
1,438,999 | <p>If $(x-a)^2=(x+a)^2$ for all values of $x$, then what is the value of $a$?</p>
<p>At the end when you get $4ax=0$, can I divide by $4x$ to cancel out $4$ and $x$?</p>
| Peter | 111,826 | <p>Since we know that $$(x-a)^2 = (x+a)^2$$ holds for all $x$,
we particularly know that it holds for $x=a$ as well. Plugging that in the relation yields $$0=4a^2$$ from which we conclude that $a=0$.</p>
|
102,304 | <p>I have here a complex equation:</p>
<p>$$z^2 - (7+j)z + 24 +j7 = 0$$</p>
<p>How do we get the roots of this equation? I started using the quadratic formula $-b \pm \sqrt{ b^2-4ac}\over 2$, but it yielded too much complexity on it. Is there any way to directly attack this? Thanks.</p>
| Zarrax | 3,035 | <p>Note that the constant term $(24 + 7j)$ is one-half of the square of the $7 + j$ coefficient. This suggests writing $z = (7 + j)w$, and the equation becomes
$$(48 + 14j) w^2 - (48 + 14j)w + (24 + 7j) = 0$$
Divide through by $(24 + 7j)$ and you get
$$2w^2 - 2w + 1 = 0$$
By the quadratic formula this has roots ${1 \ov... |
235,661 | <p>Is this sufficient? Also, any good books/other suggestions regarding the subject will be very helpful.</p>
<p>Find min, max, inf, sup (if they exist):</p>
<p>$$B=\left\{\frac{m}{m+n}:m,n\in\mathbb{N}\right\}$$</p>
<p>Showing B has an upper bound:
Let $M=1$, we need to find $m,n$ fulfilling:$$\frac{m}{m+n}>1$$
... | Martin Argerami | 22,857 | <p>Your proof that $1$ is an upper bound is unnecessarily complicated: as $m,n>0$, we have $m<m+n$, and then $m/(m+n)<1$. </p>
<p>Also, as was mentioned, $0$ is a lower bound (since everything is positive). And it is the infimum, as $1/(1+n)\to 0$. </p>
|
1,514,628 | <p>I've been looking over some old assignments in my analysis course to get ready for my upcoming exam - I've just run into something that I have no idea how to solve, though, mainly because it looks nothing like anything I've done before. The assignment is as follows:</p>
<p>"Let $H$ be a Hilbert space, and let $(e_n... | levap | 32,262 | <p>For part one, Gram-Schmidt is indeed the way to go. Let $f_1 = e_1 + e_2$, $f_2 = e_3 + e_4$, $f_3 = e_2 + e_3$. Then, since $f_1$ and $f_2$ are already orthogonal to each other, you need only to normalize them. Since $e_1$ and $e_2$ are orthogonal, you have $||f_1||^2 = ||e_1||^2 + ||e_2||^2 = 2$ so $||f_1|| = \sqr... |
3,500,405 | <p>I have a similar question to what was asked already <a href="https://math.stackexchange.com/questions/2511111/prove-that-the-following-map-has-at-least-k-2-fixed-points">here</a></p>
<p>But I do not really understand the answer there.</p>
<p>The problem is: Let <span class="math-container">$x_0 \in S^1$</span> and... | Milo Brandt | 174,927 | <p>The general idea here is that <em>loops</em> with a fixed class in the fundamental group are uniquely associated to <em>paths</em> between a fixed pair of lifts in the universal cover. For this answer, let's have <span class="math-container">$S^1$</span> be the unit circle in the complex plane and the universal cove... |
2,275,951 | <p>The parabola y=x² is parameterized by x(t) = t and y(t) = t². At the point <strong>A</strong> (t,t²) a line segment <strong>AP</strong> 1 unit long is drawn normal to the parabola extending inward. Find the parametric equations of the curve traced by the point <strong>P</strong> as <strong>A</strong> moves along the... | A.Γ. | 253,273 | <p>Define
$$
F(\alpha)=\int_0^{\pi/2}e^{-\alpha\tan x}\,dx,\quad\alpha>0.
$$
Differentiating $F$ two times we get
\begin{align}
F''(\alpha)&=\int_0^{\pi/2}e^{-\alpha\tan x}\tan^2x\,dx=-F(\alpha)+\int_0^{\pi/2}e^{-\alpha\tan x}(\underbrace{1+\tan^2x}_{\tan'(x)})\,dx=\\
&=-F(\alpha)+\int_0^{+\infty}e^{-\alpha ... |
119,810 | <p>My question today is about the minimization of an error function with two parameters. It is a function that measures the error of a set of points. The two parameters are the weights of a regressor. </p>
<p>$$\frac{1}{N}\sum_{t=1}^{N}[r^t-(w_1x^t+w_0)]^2$$ </p>
<p>The minimum should be calculated by taking partia... | Chris Taylor | 4,873 | <p>I concur with the results in Henry's answer. In case you want to try it out for yourself, here's some Matlab code implementing the two solutions (note that due to Matlab's indexing rules, $w_0$ is <code>w(1)</code> and $w_1$ is <code>w(2)</code> etc.</p>
<pre><code>% Create a fake dataset
x = linspace(2,8,30)';
r ... |
267,355 | <p>Let $H_i = (V_i, E_i)$ be <a href="https://en.wikipedia.org/wiki/Hypergraph" rel="nofollow noreferrer">hypergraphs</a> for $i=1,2$. Then we say that $H_1\cong H_2$ if there is a bijection $\varphi:V_1\to V_2$ such that $A\in E_1$ implies $\varphi(A) \in E_2$ and $B\in E_2$ implies $\varphi^{-1}(B)\in E_1$.</p>
<p>I... | Gro-Tsen | 17,064 | <p>Others have already answered, but I think the following counting argument is worth pointing out:</p>
<ul>
<li><p>there are $2^{2^{\aleph_0}}$ hypergraphs on $\omega$ (since a hypergraph on $\omega$ is just a collection of nonempty subsets of $\omega$),</p></li>
<li><p>each isomorphism class contains at most $2^{\al... |
2,264,791 | <p>I have a problem that I'm having trouble figuring out the distribution with given condition.</p>
<p>It is given that 1/(<span class="math-container">$X$</span>+1), where <span class="math-container">$X$</span> is an exponentially distributed random variable with parameter 1.</p>
<blockquote>
<p><strong>Original Prob... | heropup | 118,193 | <p>You've misunderstood the difference between a random variable $X$ and its probability density function $f_X(x)$. These are not the same thing. $X$ represents the value of the random outcome. $f_X(x)$ represents a likelihood of observing a particular outcome.</p>
<p>With this in mind, given that $X \sim \operator... |
936,525 | <p>I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension $n\times 1$, $$\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $I$ is an $n\times n$ matrix, and $\|.||$ is a matrix norm.... | Batman | 127,428 | <p>$P=I - \frac{v v^T}{v^T v}$ is the orthogonal projection onto $v^\perp$. </p>
<p>Proof: Clearly, $P$ is symmetric. </p>
<p>$P^2 = (I - \frac{v v^T}{v^T v}) (I - \frac{v v^T}{v^T v}) = I - 2 \frac{v v^T}{v^T v} + \frac{v v^T}{v^T v} \frac{v v^T}{v^T v} = I - 2 \frac{v v^T}{v^T v} + \frac{v (v^T v) v^T}{(v^T v)^2} ... |
200,658 | <p>What is the value of :</p>
<p>$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$</p>
| user 1591719 | 32,016 | <p><strong>HINT</strong>: you may use $e^{kx}$. Then solve the geometrical progression, derive once/twice its both sides and then plug in $x=-\ln(3)$. </p>
|
2,661,443 | <p>For the equation $2^x = 7$</p>
<p>The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. </p>
<p>I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ to isolate the $x$. I understand this part.</p>
<p>I can alternatively solve it in an easier way by s... | Mohammad Riazi-Kermani | 514,496 | <p>Logarithmic functions enjoy many properties.</p>
<p>One of the very interesting properties of logarithms is the formula called change of base formula.</p>
<p>$$ \log_a x = \frac {\log_b x}{\log_b a} $$</p>
<p>For example $$ \log_2 7 = \frac {\log_{10} 7}{\log_{10} 2} $$</p>
<p>This formula makes finding logarith... |
77,379 | <p>It is to show for an $a\in \mathbb{C}^{\ast}$ that $aB_{1}(1)= B_{|a|}(a)$ </p>
<p>where B denotes a disc </p>
<p>Okay, maybe this is correct: </p>
<p>$aB_{1}(1) = a(e^{i\phi}) = ae^{i\phi} = |a|e^{i\phi} = B_{|a|}(a)$</p>
<p>But this seems very wrong! </p>
<p>V</p>
| VVV | 18,298 | <p>let a be a complex number of the form: $a:= u+vi$ and $z:= x+yi$ </p>
<p>$B_{1}(1)$ means that $ |z-1| < 1 $ and so $a|z-1| = (u+vi)|z-1| = (u+vi)(\sqrt{(x-1)^{2}+y^{2}} $ so we can write it as $|z-1|u+|z-1|vi< a|1| = |a| = \sqrt{u^{2}+v^{2}}$</p>
<p>This seems to be the wrong route also. ??</p>
<p>V</p>
|
77,379 | <p>It is to show for an $a\in \mathbb{C}^{\ast}$ that $aB_{1}(1)= B_{|a|}(a)$ </p>
<p>where B denotes a disc </p>
<p>Okay, maybe this is correct: </p>
<p>$aB_{1}(1) = a(e^{i\phi}) = ae^{i\phi} = |a|e^{i\phi} = B_{|a|}(a)$</p>
<p>But this seems very wrong! </p>
<p>V</p>
| t.b. | 5,363 | <p>First let me answer your specific question.</p>
<ol>
<li><p>Let $z \in B_{1}(1)$, that is to say, $|z - 1| \lt 1$. We want to show that $az \in B_{|a|}(a)$, that is, we want to show that $|az-a| \lt |a|$. But
$$
|az - a| = |a| \cdot \underbrace{|z-1|}_{\lt 1} \lt |a|,
$$
as we wanted, therefore $aB_1(1) \subset B... |
224,226 | <p>I am trying to count the number of distinct colours in a <span class="math-container">$5\times5$</span> box, (a radius 2 filter) at all points over a quantized image. I cannot seem to get anything out of the following code except for a black square:</p>
<pre><code>img = ColorQuantize[ExampleData[{"TestImage&quo... | MarcoB | 27,951 | <p>In your 1D <code>MovingMap</code> you are asking for a window of size 1, so the inputs to <code>CountDistinct</code> are lists with only one element, so the output is always 1 (you can see what happens by replacing <code>CountDistinct</code> with an undefined <code>f</code>). You would get closer with something like... |
727,752 | <blockquote>
<p>If S is a compact subset of R and T is a closed subset of S,then T is compact.</p>
<p>(a) Prove this using definition of compactness.</p>
<p>(b) Prove this using the Heine-Borel theorem.</p>
</blockquote>
<p>My solution: firstly I should suppose a open cover of T, and I still need to think of the
set S-... | John Hughes | 114,036 | <p>Here's how to get started on part a. </p>
<p>Start with an open cover of $T$. You need to show it has a finite subcover. If $U$ is in the open cover, then it's open in $T$, which means that there's an open set $U'$ in $S$ such that $U = T \cap U'$. For every $U$ in your cover, find a corresponding $U'$; now you <em... |
727,752 | <blockquote>
<p>If S is a compact subset of R and T is a closed subset of S,then T is compact.</p>
<p>(a) Prove this using definition of compactness.</p>
<p>(b) Prove this using the Heine-Borel theorem.</p>
</blockquote>
<p>My solution: firstly I should suppose a open cover of T, and I still need to think of the
set S-... | Simon Rose | 87,590 | <p>I will address part (b), since the others address (a) well.</p>
<p>The Heine-Borel theorem says that a subset $V \subset \mathbb{R}$ is compact if and only if it is both closed and bounded.</p>
<p>So suppose that $T \subset S \subset \mathbb{R}$. Since $S$ is compact, it is closed and bounded. What can you now say... |
4,242,561 | <p>Let <span class="math-container">$T: R^3 \rightarrow R^3$</span> be a linear transformation such that <span class="math-container">$T(x,y,z) = (x,0,0)$</span>. Which implies that the matrix that represents the transformation is <span class="math-container">\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\e... | Bible Bot | 963,424 | <p>Like what the other poster said, it depends on the context and the chosen convention of the authors.
I’ve seen matrices written with and without brackets around the letter so it’s up to you/the teacher.</p>
<p>There rarely is one “true” convention for any particular math concept.</p>
|
230,971 | <p>At the moment I use <code>Length[ DeleteDuplicates[ array ] ] == 1</code> to check whether an array is constant, but I'm not sure whether this is optimal.</p>
<p>What would be the quickest way to test whether an array consists of equal elements?</p>
<p>What if the elements would be integers?</p>
<p>What if they are ... | Sjoerd Smit | 43,522 | <p>Here are two methods that are quite fast for flat lists (you can flatten arrays to test at deeper levels):</p>
<pre><code>const = ConstantArray[1, 100000];
nonconst = Append[const, 2];
</code></pre>
<p>Using <code>CountDistinct</code> (or <code>CountDistinctBy</code>):</p>
<pre><code>CountDistinct[const] === 1
Count... |
2,086,006 | <p>You have $7$ boxes in front of you and $140$ kittens are sitting side-by-side inside the
boxes, $20$ in each box. You want to take some kittens as your pets. However the
kittens are very cowardly. Each time you chose a kitten from a box, the kittens that
are in that box to the left of it go to the box in the left, t... | MoebiusCorzer | 283,812 | <p>The function $$A:[1,+\infty)\to\Bbb R: x\mapsto xa^{\frac{1}{x}}(e^{\frac{1}{x}}-1)$$ is continuous as a product of composite functions of continuous functions.</p>
<p>Hence, if the limit when $x\to+\infty$ exists, it is the same along any sequence $(x_{n})_{n}$ such that $x_{n}\to+\infty$ as $n\to+\infty$. We shal... |
2,520,768 | <p>How would I approach this problem? </p>
<p>Let $(a, b, c) \in \mathbb{Z^3}$ with $a^2 + b^2 = c^2$. Show that:
$$
60 \,\mid\, abc
$$</p>
| David | 119,775 | <p>There are <a href="https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple" rel="nofollow noreferrer">formulas</a> for the integer solutions of $a^2+b^2=c^2$: you can use them to give a proof.</p>
<p>If you don't know these formulas, here is an alternative method using modular arithmetic.</p>
<p>First... |
158,916 | <p>Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out.</p>
<blockquote>
<p>For every $\varepsilon>0$ there is a $\delta>0$ such that every smooth 2-sphere in $M$ of area smaller than $\delta$ bounds a b... | Sam Nead | 1,650 | <p>In their paper "The classical Plateau problem and the topology of 3-manifolds", Meeks and Yau claim that for any fixed closed Riemannian 3-manifold $M$ there is a lower bound for the area of non-trivial two-spheres. Furthermore, the least area such is embedded (or double covers an essential $RP^2$.)</p>
<p>There a... |
3,190,828 | <p>Let <span class="math-container">$A\in M_n(\mathbb{C})$</span> be a matrix such that <span class="math-container">$A^n=aA$</span>,where <span class="math-container">$a\in \mathbb{R}-\{0,1\}$</span>.<br>
I wanted to find <span class="math-container">$A$</span>'s eigenvalues and I thought that they are the roots of th... | Arthur | 15,500 | <p>If <span class="math-container">$A$</span> has eigenvalue <span class="math-container">$b$</span>, and <span class="math-container">$v$</span> is a corresponding (non-zero) eigenvector, then <span class="math-container">$$0 = (A^n-aA)v = A^nv - aAv = b^nv-abv = (b^n-ab)v$$</span>This means <span class="math-containe... |
2,080,716 | <p>I have the quadratic form
$$Q(x)=x_1^2+2x_1x_4+x_2^2 +2x_2x_3+2x_3^2+2x_3x_4+2x_4^2$$</p>
<p>I want to diagonalize the matrix of Q. I know I need to find the matrix of the associated bilinear form but I am unsure on how to do this.</p>
| Harsh Kumar | 395,886 | <p>Sum of numbers divisible by $3=1683$</p>
<p>Sum of numbers divisible by $7=735$</p>
<p>Sum of numbers divisible by $21=210$</p>
<p><strong>We need the sum of divisor of $21$ since if we subtract only by the Sum of divisor of $3$ and Sum of divisor of $7$ then the numbers those are divisible by both $3$ & $7$ ... |
2,080,716 | <p>I have the quadratic form
$$Q(x)=x_1^2+2x_1x_4+x_2^2 +2x_2x_3+2x_3^2+2x_3x_4+2x_4^2$$</p>
<p>I want to diagonalize the matrix of Q. I know I need to find the matrix of the associated bilinear form but I am unsure on how to do this.</p>
| Nilabro Saha | 304,908 | <p>Number of multiples of $3$ between $1$ & $100$ = $[\frac{100}{3}]$ = $33$,</p>
<p>Number of multiples of $7$ between $1$ & $100 =$ $[\frac{100}{7}]$ = $14$,</p>
<p>Number of multiples of $21$ between $1$ & $100$ = $[\frac{100}{21}]$ = $4$,</p>
<p>Where $[x]$ is the box function.</p>
<p><strong>Sum of... |
2,595,247 | <p>What is equation of circle when two lines y=x and y=x-4 are tangent to a circle at (2,2) and (4,0) respectively.</p>
| QED | 91,884 | <p>Suppose $\{\alpha_n\}$ is a sequence of functions whose total variations are uniformly bounded on $[a,b]$. If $\{\alpha_n\}$ is pointwise convergent to $\alpha$ on $[a,b]$, then for every $f$ continuous on $[a,b]$,
$$\lim_{n\to\infty}\int_{a}^bf(x)d\alpha_n=\int_a^bf(x)d\alpha$$
Clearly,
$$\lim_{n\to\infty}x^{n+1}=\... |
1,515,478 | <p>Given a quadratic equation with one and only one root (for example $6-\sqrt{2}$ ). Does there exist integers $a,b$ and $c$ where $ax^2 + bx + c = 0$ for that root?</p>
| Jack Smith | 841,375 | <p>When there is only one root, the discriminant is zero and the root is <span class="math-container">$\frac{-b}{2a}$</span> because <span class="math-container">$\frac{-b+0}{2a}$</span> and <span class="math-container">$\frac{-b-0}{2a}$</span> are both equal to <span class="math-container">$\frac{-b}{2a}$</span></p>
<... |
2,863,995 | <p><em>Problem:</em></p>
<p>Let $f_n: [0,1] \to \mathbb{R}$ be a sequence of measurable functions. </p>
<p>Suppose that $\int_{0}^{1}|f_n(x)|^2 ~ dx \le 1$ for $n \in \mathbb{N}$ and $f_n$ converges to $0$ a.e. </p>
<p>Show that $\lim_{n \to \infty} \int_{0}^{1} f_n(x) ~ dx = 0$.</p>
<p><em>Question:</em> <strong>I... | Wraith1995 | 462,363 | <p>The proof is false. DCT requires that the function be dominated in the sense that $|f_{n}| \leq g$. </p>
<p>Also, the inequalities that you use don't apply in a sigma finite measure space. They apply in a finite measure space! They are false over the entire real line!</p>
|
1,986,172 | <p>I am asked to simplify $(\sqrt{t^3}) \times (\sqrt{t^5})$.</p>
<p>I get up to $\sqrt[3]{t^3}\times \sqrt{t^5}$ but I am not sure how to simplify this further as now roots are involved and not just powers.</p>
<p>When I checked the solutions the final answer should be $t^4$ but I'm not sure how this is achieved.</p... | Ross Millikan | 1,827 | <p>One way is to note that $\left( \sqrt t \right)^3=t^{\frac 32}$ and similarly for the other one. Then when you multiply terms you add exponents</p>
|
125,116 | <p>Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation?</p>
<p>In 2D, a simple angle satisfies this, as it can have additional multiples of $2\pi$. For example, rotating by a turn and a half would be $3\pi$.</p>
<p>Is there someth... | S. Carnahan | 121 | <p>Since you haven't described what you plan to do with this representation, I'm not sure what method would work well.</p>
<p>One of the problems with representing "turns" in more than two dimensions is that you don't have much in the way of discrete invariants. This is because the fundamental group of $SO(n)$ only h... |
203,464 | <p>I would like to exclude the point <code>{x=0,y=0}</code> in the function definition</p>
<pre><code>f = Function[{x, y}, {x/(x^2 + y^2), -(y/(x^2 + y^2))}]
</code></pre>
<p>So far I tried <code>ConditionalExpression</code>and <code>/;</code> without success.</p>
<p>Thanks!</p>
| Ulrich Neumann | 53,677 | <p>Try using <code>Distribute</code>:</p>
<pre><code>Distribute[{{a, b, c}, {d, e, f}}, List]
(*{{a, d}, {a, e}, {a, f}, {b, d}, {b, e}, {b, f}, {c, d}, {c, e}, {c,f}}*)
</code></pre>
|
12,949 | <p>Let $\kappa$ be an infinite cardinal. Then there exists at least one <a href="http://en.wikipedia.org/wiki/Real-closed_field">real-closed field</a> of cardinality $\kappa$ (e.g. <a href="http://en.wikipedia.org/wiki/Lowenheim-Skolem">Lowenheim-Skolem</a>; or, start with a function field over $\mathbb{Q}$ in $\kappa... | Marty | 3,545 | <p>Hi Pete!</p>
<p>There's been a lot of study of this and similar problems. I believe that Shelah's theorem, from his 1971 paper "The number of non-isomorphic models of an unstable first-order theory" (Israel J. of Math) answers your question about real closed fields in the positive.</p>
<p>The best big result on s... |
12,949 | <p>Let $\kappa$ be an infinite cardinal. Then there exists at least one <a href="http://en.wikipedia.org/wiki/Real-closed_field">real-closed field</a> of cardinality $\kappa$ (e.g. <a href="http://en.wikipedia.org/wiki/Lowenheim-Skolem">Lowenheim-Skolem</a>; or, start with a function field over $\mathbb{Q}$ in $\kappa... | Dave Marker | 5,849 | <p>For real closed fields this is fairly easy.</p>
<p>First show that for any infinite cardinal k there are 2^k nonisomorphic linear orders
of cardinality k</p>
<p>For example if X is a subset of k let A_x be Q+2+Q if x is in k and Q+3+Q if x is not in X. Let L_X be the sum of the A_x for x in k. It is easy to see ... |
2,301,198 | <p>Solve the initial value problem for the sequence $\left \{ u_{n}| n \in \mathbb{N} \right \}$ satisfying the recurrence relation:
$u_n − 5u_{n-1} + 6u_{n−2} = 0 $ with $u_0 = 1$ and $u_1 = 1$.</p>
<p>Ive gotten the general solution to be $u_n = A(2)^n + B(3)^n$. </p>
<p>Once I sub the initial values: </p>
<p>$u_0... | mvw | 86,776 | <p>$$
u_0 = A 2^0 + B 3^0 = A + B \\
u_1 = A 2^1 + B 3^1 = 2A + 3B
$$
So you got the first equation wrong.</p>
|
1,120,013 | <p>Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true:</p>
<p>E [ X | E [ X | Y ] ] = E [ X | Y]</p>
<p>Note that E [ X | Y ] is a random variable in it's own right. Also note that equality here is point-wise, for every point in the s... | Henry | 6,460 | <p>Essentially the <a href="http://en.wikipedia.org/wiki/Law_of_total_expectation" rel="nofollow">law of iterated expectation</a>, perhaps more commonly written like $$\operatorname{E_X} [X] = \operatorname{E}_Y [ \operatorname{E}_{X \mid Y} [ X \mid Y]].$$</p>
<p>For a discrete case, the essence of the proof is
$$\o... |
308,520 | <p>The DE is $y' = -y + ty^{\frac{1}{2}}$. </p>
<p>$2 \le t \le 3$</p>
<p>$y(2) = 2$</p>
<p>I tried to see if it was in the <a href="http://www.sosmath.com/diffeq/first/lineareq/lineareq.html" rel="nofollow">linear form</a>. I got:</p>
<p>$$\frac{dy}{dt} + y = ty^{\frac{1}{2}}$$</p>
<p>The RHS was not a function o... | Maesumi | 29,038 | <p>Set $y=z^2$ and simplify. You get $y'=2zz'$ and your equation is $2zz'+z^2=tz$ or $2z'+z=t$ which is linear and you can apply integrating factor to it.</p>
<p>(We assumed $y>0$, which it is near initial point.)</p>
|
308,520 | <p>The DE is $y' = -y + ty^{\frac{1}{2}}$. </p>
<p>$2 \le t \le 3$</p>
<p>$y(2) = 2$</p>
<p>I tried to see if it was in the <a href="http://www.sosmath.com/diffeq/first/lineareq/lineareq.html" rel="nofollow">linear form</a>. I got:</p>
<p>$$\frac{dy}{dt} + y = ty^{\frac{1}{2}}$$</p>
<p>The RHS was not a function o... | Mikasa | 8,581 | <p>Hint: </p>
<p>Another approach applying here is to see that your OD is a [Bernoulli] (<a href="http://en.wikipedia.org/wiki/Bernoulli_differential_equation" rel="nofollow">http://en.wikipedia.org/wiki/Bernoulli_differential_equation</a>) equation. Just to see that $n=1/2$. The substitution $w=y^{1-1/2}=\sqrt{y}$ wo... |
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$x^3-4x^2-4x+16=x(x^2-4x+4)-8x+16=x(x-2)^2-8(x-2)=$$</span></p>
<p><span class="math-container">$$=(x-2)(x^2-2x-8)=(x-2)(x+2)(x-4)=0$$</span></p>
|
3,858,362 | <p>Solve <span class="math-container">$$\dfrac{x^3-4x^2-4x+16}{\sqrt{x^2-5x+4}}=0.$$</span>
We have <span class="math-container">$D_x:\begin{cases}x^2-5x+4\ge0\\x^2-5x+4\ne0\end{cases}\iff x^2-5x+4>0\iff x\in(-\infty;1)\cup(4;+\infty).$</span> Now I am trying to solve the equation <span class="math-container">$x^3-4... | PierreCarre | 639,238 | <p>The other answers pretty much covered all the aspects, I'm just adding an heuristic way of obtaining integer solutions of polynomial equations that can come handy sometimes. The equation <span class="math-container">$x^3-4x^2-4x+16=0$</span> can be rewritten as
<span class="math-container">$$
x(x^2-4x-4)= -16
$$</sp... |
1,862,524 | <p>In the textbook I'm using to prepare the logic exam says that first order logic may be used to <strike>implement</strike> axiomatize data structures. There is an example of that:</p>
<p>"Stack": uses a language that contains the <strike>predicates</strike> functions <em>top</em>, <em>pop</em> and <em>push</em>, and... | jugglingmike | 341,620 | <p>Slight error in your final line. The original x limits were 0 and 2 which changed to 0 and 4 for u. However you seemed to have doubled them again when substituting into $4e^u$.</p>
|
4,280,426 | <blockquote>
<p>We have a bag with <span class="math-container">$3$</span> black balls and <span class="math-container">$5$</span> white balls. What is the probability of picking out two white balls if at least one of them is white?</p>
</blockquote>
<p>If <span class="math-container">$A$</span> is the event of first b... | José Carlos Santos | 446,262 | <p>Yes, <span class="math-container">$\overline A=\Bbb R$</span>, since <span class="math-container">$A$</span> is not closed, from which it follows that the only closed subset of <span class="math-container">$\Bbb R$</span> which contains <span class="math-container">$A$</span> is <span class="math-container">$\Bbb R$... |
1,364,554 | <p>Is it possible to select real values $a_{n, k}$ so that</p>
<p>$$f(x) =\lim_{n \to \infty}\sum_{k = 0}^{n - 1} a_{n, k} x^k = \frac{1}{1 - x} $$ for all $x \in \mathbb{R} \setminus \{1\}$ ?</p>
<p>Failing examples:</p>
<ol>
<li>$a_{n, k} = 1$ for all $n, k \in \mathbb{N}_0$ and $\lvert x \rvert < 1$ </li>
</ol... | Jack D'Aurizio | 44,121 | <p>In order that our power sums match the Taylor series of $\frac{1}{1-x}$ in a neighbourhood of the origin we must have $\lim_{n\to +\infty}a_{n,k}=1$ for every $k$, so we cannot have convergence outside the unit disk.</p>
|
1,364,554 | <p>Is it possible to select real values $a_{n, k}$ so that</p>
<p>$$f(x) =\lim_{n \to \infty}\sum_{k = 0}^{n - 1} a_{n, k} x^k = \frac{1}{1 - x} $$ for all $x \in \mathbb{R} \setminus \{1\}$ ?</p>
<p>Failing examples:</p>
<ol>
<li>$a_{n, k} = 1$ for all $n, k \in \mathbb{N}_0$ and $\lvert x \rvert < 1$ </li>
</ol... | Daniel Fischer | 83,702 | <p>By the <a href="http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_approximation_theorem" rel="nofollow">Stone-Weierstraß theorem</a>, we can uniformly approximate every continuous function on a compact subset of $\mathbb{R}$ by a sequence of polynomials.</p>
<p>Thus if we exhaust $\mathbb{R}\setminus \{1\}$ by... |
688,430 | <blockquote>
<p>How can I show the following
$$a^n|b^n \Rightarrow a|b$$ with $a,b$ integers.</p>
</blockquote>
<p>$$a^n|b^n \Rightarrow b^n=m \cdot a^n \Rightarrow b^n=(m\cdot a^{n-1}) \cdot a\qquad(1)$$
How can I continue? Do I maybe have to suppose the opposite and arrive at contradiction?
$$\text{So } a \nmid ... | Bill Dubuque | 242 | <p><strong>Hint</strong> $ $ Clear if $\,a=0.$ Else $\,(\color{#c00}{b/a})^n \! = k\in\Bbb Z\,\Rightarrow\, b/a\in \Bbb Z,\,$ because the $ $ <a href="http://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">Rational Root Test</a> $ $ implies that $ $ any $ $ rational $ $ root $ $ of $\ \ \color{#c00}x^n - k\... |
436,225 | <p><a href="http://en.wikipedia.org/wiki/Incidence_matrix">The incidence matrix</a> of a graph is a way to represent the graph. Why go through the trouble of creating this representation of a graph? In other words what are the applications of the incidence matrix or some interesting properties it reveals about its grap... | Jair Taylor | 28,545 | <p>Another example is the beautiful Matrix Tree Theorem, which says that the number of spanning trees of a graph is equal to a minor of the Laplacian of the graph, which is a matrix closely related to the incidence matrix.</p>
|
4,160 | <p>I am a guest here, having responded to a general invitation extended to the <a href="https://stats.stackexchange.com/questions">Cross Validated</a> community, to possibly contribute answers whenever some question related to Statistics comes up in this site.
I do not teach Mathematics, but I do occasionally teach Sta... | Community | -1 | <p>I've encountered something similar in <strong>logic</strong>, where symbols can be interpreted either syntactically ($\vdash$) or semantically ($\models$). I've had the same feeling that logic students were only playing along about appreciating the difference between the two.</p>
<p>The <strong>statistical results... |
4,160 | <p>I am a guest here, having responded to a general invitation extended to the <a href="https://stats.stackexchange.com/questions">Cross Validated</a> community, to possibly contribute answers whenever some question related to Statistics comes up in this site.
I do not teach Mathematics, but I do occasionally teach Sta... | Joseph Malkevitch | 1,865 | <p>You might find this article in the Journal of Mathematical Behavior of interest: Conceptual issues in understanding the inner logic of statistical inference: Insights from two teaching experiments, by Luis A. Saldanha, Patrick W. Thompson, pages 1-30, Volume 35, September, 2014.</p>
|
1,255,311 | <p><img src="https://i.stack.imgur.com/5V9e0.png" alt="enter image description here"></p>
<p>I understand inner product space with vectors, but the conversion to functions is throwing me off. Also why do they use an integral here, I've always seen summations. I think I'm missing something with notation here. Any help/... | Chappers | 221,811 | <p>Take real and imaginary parts, with <span class="math-container">$A=a+bi$</span>, <span class="math-container">$C=c+di$</span>. Then the real and imaginary parts are
<span class="math-container">$$ \binom{(a+c)\cos{t}+(d-b)\sin{t}}{(b+d)\cos{t}+(a-c)\sin{t}} = \begin{pmatrix} a+c & d-b \\ b+d & a-c \end{pmat... |
1,662,398 | <p>I am currently studying for my upcoming midterm and I am stumped on this example provided in the slides. Basically here is the question:</p>
<blockquote>
<p>Given 35 computers, what is the probability that more than 10 computers are in use(active)? We are told that each computer is only active 10% of the time. Th... | Ross Millikan | 1,827 | <p>The expression you have subtracted from $1$ is the correct computation of the probability that exactly $10$ computers are active. The question asks for the probability that more than $10$ are active, so you should sum the chances that $11, 12, 13, \dots$ are active. As you only expect $3.5$ to be active, these will ... |
1,662,398 | <p>I am currently studying for my upcoming midterm and I am stumped on this example provided in the slides. Basically here is the question:</p>
<blockquote>
<p>Given 35 computers, what is the probability that more than 10 computers are in use(active)? We are told that each computer is only active 10% of the time. Th... | rogerl | 27,542 | <p>As you have correctly noted, the probability of exactly 10 being active at any one time is
$${35 \choose 10} \cdot (0.10)^{10} \cdot (0.90)^{25}.$$
More generally, the probability of exactly $k$ being active at any time is
$${35 \choose k} \cdot (0.10)^{k} \cdot (0.90)^{35-k}.$$
Since you want the probability of mor... |
387,268 | <p>Let <span class="math-container">$A$</span> be an <span class="math-container">$N\times N$</span> nonnegative matrix with all diagonal entries equal to zero and such that there is <span class="math-container">$n_0$</span> such that all entries of <span class="math-container">$A^{n_0}$</span> are strictly positive. L... | Noam D. Elkies | 14,830 | <p>(1) No. Counterexample: the symmetric <span class="math-container">$3 \times 3$</span> matrix
<span class="math-container">$$
M(a,b) = \left[
\begin{array}{ccc} 0 & a & b \cr a & 0 & b \cr b & b & 0 \end{array}
\right]
$$</span>
with <span class="math-container">$0 < b < a$</span> has... |
830,977 | <p>I'm having some real trouble with lebesgue integration this evening and help is very much appreciated.</p>
<p>I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over $(0,\infty)$.</p>
<p>My first thought was to write the integral as $f(x) = \frac{\cosh(x)}{\cosh(2x)}$ and then no... | UserB1234 | 82,877 | <p>Hint: You should be able to show that $e^{-x}$ and $e^{-3x}$ are integrable without too much trouble (this should follow from the "standard" trick of writing $\int_{0}^{\infty}g=\int_{0}^{\infty}\lim_{n}g\chi_{(0,n)}$ and then switching limits using the monotone convergence theorem). Then you can do a comparison bas... |
2,613,484 | <p>Give an example of a vector space which has 125 elements.
I don't know how proceed!!!
Is there any technique about the field?? </p>
| cansomeonehelpmeout | 413,677 | <p>$\frac{\mathbb{Z}_5[x]}{(x^3)}$ as a $\mathbb{Z}_5$-module, which is $\{a+bx+cx^2\mid a,b,c\in\mathbb{Z}_5\}$.</p>
<p>This is the same as $$V=\left < [a,b,c]\mid a,b,c\in\mathbb{Z}_5\right>$$</p>
<p>which have $5^3=125$ elements.</p>
|
2,553,284 | <p>I know that
$$\ln e^2=2$$
But what about this?
$$(\ln e)^2$$
A calculator gave 1. I'm really confused.</p>
| Gustavo Mezzovilla | 396,214 | <p>The natural log are the log with base $e$ (<em>euler's number</em> our <em>napier constant</em>). Therefore
$$ \ln (x) = \log_e(x)$$
When you put $x=e$, we have $\ln(e)$, but that is simply $1$. Therefore $\big(\ln(e)\big)^2=1$.</p>
|
157,497 | <p>Let's suppose I have created a 3d image of gray scale Images with:</p>
<pre><code>image3d = Image3D[Table[readImage[i], {i, numberOfImages}]];
</code></pre>
<p>and </p>
<pre><code>image3dSlices = Image3DSlices[image3d]
</code></pre>
<p>To show the 3d image I can use:</p>
<pre><code>image3d
</code></pre>
<p>or... | Michael E2 | 4,999 | <p>The easiest way I've found is to apply the Fundamental Theorem of Calculus:</p>
<pre><code>Integrate[Evaluate@D[Sum[Sin[n x]/n, {n, ∞}], x], {x, Pi, x}]
</code></pre>
<p>Another approach is to apply a trig. function, simplify, apply the inverse function, and simplify again. The problem here is that <a href="http:... |
157,497 | <p>Let's suppose I have created a 3d image of gray scale Images with:</p>
<pre><code>image3d = Image3D[Table[readImage[i], {i, numberOfImages}]];
</code></pre>
<p>and </p>
<pre><code>image3dSlices = Image3DSlices[image3d]
</code></pre>
<p>To show the 3d image I can use:</p>
<pre><code>image3d
</code></pre>
<p>or... | Alexei Boulbitch | 788 | <p>You may like to proceed as follows. Here is your expression:</p>
<pre><code>expr1 = Simplify[Sum[Sin[n x]/n, {n, 1, \[Infinity]}], 0 < x < 2 \[Pi]]
(* 1/2 I (-Log[1 - E^(-I x)] + Log[1 - E^(I x)]) *)
</code></pre>
<p>Mma does not collect logarithms by itself. I use for this purpose the function entitled co... |
244,214 | <p>One major approach to the theory of forcing is to assume that ZFC has a countable <em>transitive</em> model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, uses the fact that $M$ is <em>countable</em> to prove that there exists a generic set $G \in V$, then defin... | Mohammad Golshani | 11,115 | <p>You may look at the paper ``<a href="https://ojs.victoria.ac.nz/ajl/article/view/1784">Forcing with non-wellfounded models</a>''. Australas. J. Log. 5 (2007), 20–57, by Paul Corazza:</p>
<p>Here is the abstract of the paper:</p>
<blockquote>
<p>We develop the machinery for performing forcing over an arbitrary (p... |
288,001 | <p>Points A and B are given in Poincare disc model. Construct equilateral triangle ABC.
Any kind of help is welcome.</p>
| MvG | 35,416 | <p>This very much depends on the geometric primitives you have at hand. If you know how to construct a circle given its midpoint and a point on the circle, then you can simply intersect the circle around $A$ through $B$ with the circle around $B$ through $A$. A circle in the Poincaré disc is an Euclidean circle, but th... |
3,830,204 | <p>Working through <em>Spivak's Calculus</em> and using old assignments from the course offered at my school I'm working on the following problem, asking me to find the integral <span class="math-container">$$\int \frac{1}{x^{2}+x+1} dx$$</span></p>
<p>Looking through Spivak and previous exercises I worked on, I though... | jacopoburelli | 530,398 | <p>Hint : <span class="math-container">$x^2+x+1 = (x+\frac{1}{2})^{2}+\frac{3}{4}$</span></p>
|
3,371,339 | <p>How to show <span class="math-container">$Pr(X>2E(X))\le 1/2$</span> given that <span class="math-container">$X$</span> is a continuous random variable and <span class="math-container">$P(X\le 0)=0$</span>? <span class="math-container">$E(X)$</span> here is the mean of <span class="math-container">$X$</span>.</p>... | Community | -1 | <p>Suppose <span class="math-container">$f_n: X\to \mathbb{\bar{R}}$</span> for all <span class="math-container">$n\in \mathbb{N}$</span>. If we can write
<span class="math-container">$$ \{x\in X:\lim_{n\to\infty}f_n(x)\;\mathrm{exists}\}=E\cup E_{\infty}\cup E_{-\infty} $$</span>
where <span class="math-container">$E_... |
60,071 | <p>Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean:</p>
<ul>
<li>Set inclusion) For all $a\in A$, $a$ is a $P$.</li>
<li>Set equality) For all $a$, $a\in A$ iff $a$ is a $P$.</li>
</ul>
... | Ilmari Karonen | 9,602 | <p>I'd say "$A$ is <em>a</em> set of $P$s" for the first, and "$A$ is <em>the</em> set of (all) $P$s" for the second.</p>
|
60,071 | <p>Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean:</p>
<ul>
<li>Set inclusion) For all $a\in A$, $a$ is a $P$.</li>
<li>Set equality) For all $a$, $a\in A$ iff $a$ is a $P$.</li>
</ul>
... | Steven Alexis Gregory | 75,410 | <p>If you want set inclusion, you should say A is a set of P's.</p>
<p>If you want set equality, you should say A is the set of all P's.</p>
<p>The words "a" and "the" (most often) have very specific meanings in mathematics.</p>
|
9,335 | <p>How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!</p>
| Arturo Magidin | 742 | <p>Use the definition, and double inclusion; that is, show that every element of $\limsup(A\cup B)$ must be either an element of $\limsup(A)$ or of $\limsup(B)$; then show that every element of $\limsup(A)$ must be in $\limsup(A\cup B)$ and that every element of $\limsup(B)$ must be in $\limsup(A\cup B)$.</p>
<p>Of co... |
267,971 | <p>I want to keep inside of a integral evaluated after some replacement inside it, but at the same time the integral itself unevaluated.</p>
<p>I start with:</p>
<pre><code>int=HoldForm[Integrate[x^n/(x + 1)^(n + 1), {x, 0, 1}]]
</code></pre>
<p>Output as desired:
<span class="math-container">$$\int_0^1 \frac{x^n}{(x+1... | Lukas Lang | 36,508 | <p>This is exactly what <a href="https://reference.wolfram.com/language/ref/Inactivate.html" rel="nofollow noreferrer"><code>Inactivate</code></a> was designed for:</p>
<pre><code>int = Inactivate[Integrate[x^n/(x + 1)^(n + 1), {x, 0, 1}], Integrate]
</code></pre>
<p><a href="https://i.stack.imgur.com/0nWLX.png" rel="n... |
1,993,693 | <blockquote>
<p>$$\lim_{x \rightarrow +\infty} \frac{2^x}{x}$$ $$\lim_{x \rightarrow
\infty} \frac{x^{50}}{e^x}$$</p>
</blockquote>
<p>I don't really know how to solve this.</p>
<p>As for the first one, I know that $\lim_{x \rightarrow \infty} a^x=0$ , I supposed that helps...?</p>
<p>How do I solve these (prefer... | Eff | 112,061 | <p>You say that $\lim_{x\to\infty} a^x = 0$, however this is only true if $|a| < 1$, and hence not true for $2^x$ or $e^x$. </p>
<p>With that said, you can solve both of these limits if you know that exponentials eventually dominate any polynomial (if the base of the exponential is larger than 1). Let $p$ be a poly... |
404,472 | <p>Let $F$ and $F′$ be two finite fields with nine and four elements respectively.
How many field homomorphisms are there from $F$ to $F′$?</p>
| pankaj kumar | 160,044 | <p>Recall that any finite field has a finite characteristic--- a smallest positive integer $n$ with the property that for all $x$ in the field, $x + x + ... + x \;(n \text{ times}) = 0$. One can show from the field axioms that such an $n$ must be prime, and a divisor of the number of elements in the field.</p>
<p>So h... |
4,338,285 | <p>I have been thinking about the problem of finding the sum of the first squares for a long time and now I have an idea how to do it.
However, the second step of this technique looks suspicious.</p>
<ol>
<li><p><span class="math-container">$$\sum_{i=1}^n i = \frac{n^2+n}{2}$$</span></p>
</li>
<li><p><span class="math-... | epi163sqrt | 132,007 | <p>Here we look at steps (1) and (2) and we will see that the left-hand side of (2) needs to be revised somewhat. We start with the identity
<span class="math-container">\begin{align*}
\sum_{i=1}^n i=\frac{n^2+n}{2}\tag{0}
\end{align*}</span></p>
<p><strong>Step 1.:</strong></p>
<blockquote>
<p>We consider the left-han... |
4,338,285 | <p>I have been thinking about the problem of finding the sum of the first squares for a long time and now I have an idea how to do it.
However, the second step of this technique looks suspicious.</p>
<ol>
<li><p><span class="math-container">$$\sum_{i=1}^n i = \frac{n^2+n}{2}$$</span></p>
</li>
<li><p><span class="math-... | bonsoon | 48,280 | <p>Just to add a possibly well-known way (thought not necessarily as generalizable as above) of showing the sum of consecutive squares, in the spirit of apocryphal story of Gauss:</p>
<p>Note that
<span class="math-container">$$1^2 = 1\\ 2^2= 2+2 \\ 3^2 = 3+3+3 \\ 4^2 = 4+4+4+4 \\\vdots$$</span> etc.</p>
<p>So we have ... |
159,446 | <p>The ordinary Thom isomorphism says $H^{*+n}(E,E_{0}) \simeq H^{*}(X)$, where $E$ is a vector bundle over $X$ and $E_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section for the vector bundle $E$. In each fiber $E_{x}$ we remove two points $0_{x}$ and $S(x)$. Then we put $E_{0,1}$for the... | Liviu Nicolaescu | 20,302 | <p>You can see from the special case when $X$ is a point that what you proposed cannot work.</p>
|
3,696,776 | <p>I was given this problem:</p>
<p><a href="https://i.stack.imgur.com/8ACor.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8ACor.png" alt="Problem"></a></p>
<p>These are my calculations and I'm asking for verification:</p>
<p>Pointwise limit:</p>
<p><span class="math-container">$\lim_{n \to \in... | Aryaman Maithani | 427,810 | <p>You have written "<span class="math-container">$|f_n(x) - f(x)| = f_n(x) - 1$</span>". That is not correct. The RHS is negative.</p>
<hr>
<p>To do it, an easy way is to note that for every <span class="math-container">$n \in \Bbb N$</span>, the function <span class="math-container">$f_n$</span> is increasing on <s... |
118,275 | <p>In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with <em>split</em> Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objec... | Stephan Müller | 28,011 | <p>These groups are mentioned in [Swan '68 - Algebraic K-Theory, p.69]. He constructs $K_0(\mathcal{A}, S)$ for a class $S$ of exact sequences in $\mathcal{A}$. Take the free abelian group mod the relations from sequences in $S$. For example the class of all exact sequences for the Grothendieck-group $K_0(\mathcal{A})$... |
919,562 | <p>I need to prove that:</p>
<p>$$\inf\{\frac{1}{3}+\frac{3n+1}{6n^2} \Big| n\in\mathbb N\}=\frac{1}{3}$$</p>
<p>I get stuck with my proof, I'll write it down.</p>
<p>$$n\geq1$$
$$3n\geq3$$
$$3n+1\geq4$$
$$\frac{1}{3}+3n+1\geq4+\frac{1}{3}$$</p>
<p>Now, I'm having a problem with $6n^2$ if I multiply by $6n^2$, I'll... | copper.hat | 27,978 | <p>First note that ${1 \over 3} + { 3n+1 \over 6n^2} \ge {1 \over 3}$, so ${1 \over 3} $ is a lower bound.</p>
<p>Now let $\epsilon >0$ and choose $n$ large enough so that ${ 3n+1 \over 6n^2} < \epsilon$. (One easy way is to choose $n$ large enough so that ${1 \over n } < \epsilon$, then $\epsilon > { 1\ov... |
919,562 | <p>I need to prove that:</p>
<p>$$\inf\{\frac{1}{3}+\frac{3n+1}{6n^2} \Big| n\in\mathbb N\}=\frac{1}{3}$$</p>
<p>I get stuck with my proof, I'll write it down.</p>
<p>$$n\geq1$$
$$3n\geq3$$
$$3n+1\geq4$$
$$\frac{1}{3}+3n+1\geq4+\frac{1}{3}$$</p>
<p>Now, I'm having a problem with $6n^2$ if I multiply by $6n^2$, I'll... | BigM | 90,395 | <p>Obviously $\frac{1}{3}+\frac{3n+1}{n^2}\geq\frac{1}{3}.$ The sequence $a_n=\frac{1}{3}+\frac{3n+1}{n^2}$ converges to $\frac{1}{3}$ since $\frac{3n+1}{n^2}$ behaves like $\frac{1}{n}.$ Thus $a_n\rightarrow \frac{1}{3}.$ whcih is to say $\inf=\frac{1}{3}$</p>
|
1,728,920 | <p>I am a software engineer trying to wrap his head around <strong>Fast Fourier Transform (FFT)</strong>. Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of the algorithm/operations, and in fact will likely just use an open source math library to do most o... | Wouter | 89,671 | <p>Differential equations such as <a href="https://en.wikipedia.org/wiki/Thermal_conduction#Fourier.27s_law" rel="nofollow">the one that describes heat flow</a> are often easily solvable when they have a sinusoidal (or complex exponential) source term. So, physicists solve it in the easy case and then make use of the F... |
200,920 | <p>I would like to create an array f containing n indices. The label of those indices is stored in a liste of length n, let's call it "list".</p>
<p>So I would like to have something like :</p>
<blockquote>
<p>{f[list[[0]]], f[list[[1]],...}</p>
</blockquote>
<p>The point is to affect the f[liste[[i]]] to some val... | Carl Woll | 45,431 | <p>One idea I like to use for these kinds of integrals is to add an auxiliary variable and a Dirac delta function, convert the Dirac delta function to it's integral formulation, and then do a bunch of simple 1D integrals. For your case, this would proceed as follows, starting from:</p>
<p><span class="math-container">... |
3,295,021 | <p>The Hopf fibration is a mapping <span class="math-container">$h:\mathbb{S^3} \mapsto\mathbb{S}^2$</span> defined by <span class="math-container">$r\mapsto ri\bar{r}$</span> where <span class="math-container">$r$</span> is a unit quaternion in the form <span class="math-container">$r=a+bi+cj+dk $</span> where <span c... | Tom Chen | 117,529 | <p>For (i), define <span class="math-container">$\{x\} = x - \lfloor x \rfloor$</span>. Then
<span class="math-container">\begin{align*}
\lfloor 2x \rfloor + \lfloor 2y \rfloor = 2\lfloor x\rfloor + 2\lfloor y \rfloor + \lfloor 2\{x\} \rfloor + \lfloor 2\{y\} \rfloor
\end{align*}</span>
and
<span class="math-container... |
1,946,881 | <p>Looking around I have found lots of material on continuous time Markov processes on finite or countable state spaces, say $\{0,1,\ldots,J\}$ for some $J\in\mathbb{N}$ or just $\mathbb{N}$. Similarly I have earlier worked with (discrete time) Markov chains on general state spaces, following the modern classic by Meyn... | Mikasa | 8,581 | <p>First one states that for all $x$ there is an elemnt $e$ satisfing that condition but the Second one is saying any $x$ has its own $e$ with that property. These two ones can be found out through definitions of Continuity and Uniformy Continiuty as well: $$\forall x \in I \, \exists \delta > 0\, \forall y \in I \,... |
1,440,470 | <p>Given two real valued independent random variables $X$ and $Y$, write their ratio as $R = \frac{X}{Y}$</p>
<p>I know various other ways of finding a formula for the distribution of $R$, but I'm specifically interested in understanding why the following derivation does not yield the correct result.</p>
<p>$$
P(R = ... | zhoraster | 262,269 | <hr>
<p>Sol'n, written by original asker</p>
<p>1) $U([0, \frac{1}{2}])$, density is 2 for values on it's support</p>
<p>2) $P(R \le r) = \int_{\mathbb{R}}P(R \le r | Y = y)f_Y(y)dy = \int_{\mathbb{R}}P(X\le ry)f_Y(y)dy$, if $X$ and $Y$ are independent.</p>
<p>3) $f_{R|Y = y} = \frac{d}{dr}P(R \le r | Y = y) = \fra... |
1,463,419 | <p>A letter has come from exclusively LONDON or CLIFTON, but on the postmark only $2$ consecutive letters ''ON'' are found to be visible. What is the probability that the letter came from LONDON?</p>
<hr>
<p>This is a question of conditional probability. Let $A$ be the event that the letter has come from LONDON. Let... | Soudipta Dutta | 952,126 | <p>E1: Letter (ON) came from CLIFTON</p>
<p>E2 : Letter (ON) came from LONDON</p>
<p>E : Event that 2 consecutive times the letter (ON) is visible.</p>
<p>P(E1) = P(E2) = 1/2</p>
<p>CLIFTON can be broken down to : CL,LI,IF,FT,TO,ON = 6 pairs in which one pair contains letter ON .
Therefore,
<span class="math-containe... |
1,554,603 | <p>Let $\theta \in \mathbb R$, and let $T\in\mathcal L(\mathbb C^2)$ have canoncial matrix</p>
<p>$M(T)$ = $$
\left(
\begin{matrix}
1 & e^{i\theta} \\
e^{-i\theta} & -1 \\
\end{matrix}
\right)
$$
(a) Find the eigenvalues of $T$... | Kushal Bhuyan | 259,670 | <p>$\tau(60)=12$ so there are $12$ ideals</p>
<p>Note: The ideals in $Z_n$ are precisely the sets of the form $<d>$ where $d$ divides $n$, so number of ideals are same as the number of divisors of $n$</p>
|
386,018 | <p>I've been working with Spivak's Differential Geometry exercises and I found myself confused with this one: "Let $C\subset \mathbb{R} \subset \mathbb{R}^2$ be the Cantor set. Show that $\mathbb{R}^2 - C$ is homeomorphic to the surface shown at the top of the next page."</p>
<p><img src="https://i.stack.imgur.com/rAR... | 75064 | 75,064 | <p>The surface consists of infinitely many <a href="http://en.wikipedia.org/wiki/Pair_of_pants_%28mathematics%29" rel="noreferrer">pairs of pants</a> sewn together in the way shown: waist line to leg opening. A pair of pants is homeomorphic to the following domain in the plane: </p>
<p><img src="https://i.stack.imgur.... |
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