qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,131,158 | <p>I'm having an exam from linear algebra in a few days and I'm trying to understand some definitions from my workbook. Here's a definition I can't wrap my head around:</p>
<p>Let's assume we have two bases in vector space $X$: <strong>a</strong>$=(a_1,...,a_n)$ and <strong>b</strong>$=(b_1,...,b_n)$. Let $f$ be an e... | P Vanchinathan | 28,915 | <p>Working through an illustrative example in a more concrete setting will help:
Take your vector space $X$ to be the collection of polynomials in a variable $t$ of degree less than or equal to, say 2. This is a vector space of dimension 3, with one obvious basis <strong>a</strong>=$\{1,t,t^2\}$. Can check easily that... |
812,379 | <p>Is the above true? (I think it is!) if so, please can somebody explain why? I don't see it!</p>
| drhab | 75,923 | <p>It can be proved by induction. </p>
<p>$a.a^{n-1}=a^n=0\in P$ implies that $a\in P\vee a^{n-1}\in P$ (characteristic for prime ideals)</p>
<p>hence (inductionstep) $a\in P$.</p>
<p>Note that the stronger $a^n\in P\Rightarrow a\in P$ is also true. It can be proved exactly the same way.</p>
|
3,502,551 | <blockquote>
<p>Does L' Hospital's rule pay-off at all in calculating:
<span class="math-container">$$\displaystyle\lim_{x\to 0}\frac{\sqrt{\cos2x}\cdot
e^{2x^2}-1}{\ln{(1+2x)\cdot\ln{(1+2\arcsin{x})}}}$$</span></p>
</blockquote>
<p>I posted <a href="https://qr.ae/TJIZ8z" rel="nofollow noreferrer">this question o... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$\lim_{x\to0}\dfrac{\sqrt{\cos2x}\cdot e^{2x^2}-1}{\ln(1+2x)\cdot\ln(1+2\arcsin x)}$$</span></p>
<p><span class="math-container">$$=\dfrac14\cdot\lim_{x\to0}\dfrac{\cos2x\cdot e^{4x^2}-1}{x^2}\cdot\dfrac1{\lim_{x\to0}\dfrac{\ln(1+2x)}{2x}}\cdot\dfrac1{\lim_{x\to0}\dfrac{\ln(1+2\arcsin ... |
452,889 | <p>My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with eigenvalues $\lambda_{1},...,\lambda_{n}$, $X=(x_{1},...,x_{n})$ where $x_{1},...,x_{n}$ are a set of orthonormal eigenv... | Mariano Suárez-Álvarez | 274 | <p>What about the matrix $(-1){}{}{}{}{}{}{}{}{}{}{}$?</p>
|
3,246,533 | <p>ABC is a triangle such that AB = AC. Let D be the foot of the perpendicular from C to AB and E the foot of the perpendicular from B to AC. Then</p>
<p>(a) <span class="math-container">$BC^3 < BD^3 + BE^3$</span></p>
<p>(b) <span class="math-container">$BC^3 = BD^3 + BE^3$</span></p>
<p>(c) <span class="math-co... | Dr. Sonnhard Graubner | 175,066 | <p>Let <span class="math-container">$$AB=AC=b,BC=a$$</span> and <span class="math-container">$$\angle{ABC}=\angle{ACB}=\beta$$</span> so we get
<span class="math-container">$$BD=\frac{a^2}{2b}$$</span> and <span class="math-container">$$BE=\frac{a}{b}\sqrt{b^2+\frac{a^2}{4}}$$</span>
Now we show that
<span class="math-... |
103,948 | <p>I have asked this sort of <a href="https://math.stackexchange.com/questions/69178/what-are-the-odds-of-rolling-a-3-number-straight-throwing-6d6">question before</a>, and I have a new similar question <a href="https://boardgames.stackexchange.com/questions/6375/is-there-a-best-character-in-button-men-is-there-a-worst... | David Mitra | 18,986 | <p>This is rather unimaginative, but you could do it in the straightforward way:</p>
<p>If you want "at least three rolls less than or equal to 5 and at least two less than or equal to 4":</p>
<p>First, split this up into the disjoint events </p>
<p>$\ \ A$: exactly three rolls less than or equal to 5 and at least... |
3,446,691 | <p>[Please excuse ignorance, I'm not a mathematician, though I do data analysis.]</p>
<p>The question is strictly for <span class="math-container">$2\times2$</span> real non-symmetric matrix <span class="math-container">$\bf A$</span>. Let us assume, having real eigenvalues.</p>
<p>I've learned how to compute the eig... | Mnifldz | 210,719 | <p>If your matrix is not symmetric you're not guaranteed they are orthogonal. There are some matrices where if you find their eigenvectors and orthogonalize them, then they will cease to be eigenvectors.</p>
|
647 | <p>For example, I gave an exam earlier today with a problem that ended in the sentence</p>
<blockquote>
<p>Use the chain rule to find $(f\circ g)'(3)$.</p>
</blockquote>
<p>During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition ... | vonbrand | 123 | <p>In a course I taught a while back I made it a rule: "I ask <em>you</em> 4 questions, each of you gets to ask <em>me</em> at most 4 questions." Did wonders to cut down on dumb questions. And obviously, the only questions that get answered (but all count as asked) are questions about what is being asked. "Can this be ... |
647 | <p>For example, I gave an exam earlier today with a problem that ended in the sentence</p>
<blockquote>
<p>Use the chain rule to find $(f\circ g)'(3)$.</p>
</blockquote>
<p>During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition ... | Eddie | 307 | <p>If you are worried about fairness, I'd offer the answer to any students' questions to the whole class, whether announced to the class verbally, or written down on the board.</p>
|
3,000,522 | <p>I'm working on a conjecture and I've come up to a point where I need to express every non-zero positive integers not divisble by 4 with only one parameter (as you would express every even number as 2p). Here the parameter p has to cross the whole natural integers domain <span class="math-container">$N*$</span>.</p>
... | eyeballfrog | 395,748 | <p><span class="math-container">$$
u(i) = 4\left\lfloor\frac{i}{3}\right\rfloor +\left(i\;\mathrm{mod}\;3\right) + 1
$$</span>
should do the trick for enumerating all natural numbers not divisible by 4.</p>
|
82,070 | <p>I reading on Sums and I am reading about the difference between using a generalized Sigma notation and the delimited form. Ok, I understand that the generalized form is more expressive. </p>
<p>But I found the following example confusing to me:<br>
It says: </p>
<blockquote>
<p>In the following we can change ... | vharavy | 19,161 | <p>The notation</p>
<p>$$ \sum_{k=1}^{n} a_k$$</p>
<p>is short form of the following</p>
<p>$$ \sum_{k=1}^{k=n} a_k$$</p>
<p>So, by changing index variable from $k$ to $k + 1$ we will get</p>
<p>$$ \sum_{k+1=1}^{k+1=n} a_{k+1} = \sum_{k=0}^{k=n-1} a_{k+1} = \sum_{k=0}^{n-1} a_{k+1} $$</p>
<p><strong>Note:</strong... |
196,173 | <p>Recently I have stumbled upon links which are closures of braids, of the form $\sigma = \tau^{n}$. Such links generalize torus links. Are there any papers studying such links? In particular I am interested in questions like which links appear in this way, what can we say about polynomial invariants of such links, ar... | anonmoose | 100,917 | <p><a href="https://arxiv.org/abs/1610.04582" rel="nofollow noreferrer">This</a> recent paper seems to answer your question partially. It shows that the limit of the Khovanov homology (and so in particular the Jones polynomial) behaves like that of the Jones-Wenzel projector as $n \to \infty$. This generalizes a resul... |
960,880 | <p>Could you help me to explain how to find the solution of this equation
$$y ′ (t)=−y(t)-\frac1{2}*(1+e^{-2t})+1$$
Given $y(0)=0$
Thank all
This is my answer
$$y ′ (t)=−y(t)-\frac1{2}e^{-2t}+\frac1{2}$$
$$e^{2t}y ′ (t)=e^{2t}(−y(t)-\frac1{2}e^{-2t}+\frac1{2})$$
where
$$(e^{2t}y(t))′=e^{2t}y(t)′+2(e^{2t}y(t))=e^{2t}... | Paul Magnussen | 168,511 | <p>Work it out assuming y = u*v, and y' = u'<em>v + u</em>v'.</p>
<p>Rearrange to y' + y = 1/2 - e<sup>-2t</sup>/2</p>
<p>Solve homogeneous equation</p>
<p>u' + u = 0</p>
<p>So u = C * e<sup>-t</sup>.</p>
<p>Now substitute for y in full equation</p>
<p>u'<em>v + u</em>v' + uv = 1/2 - e<sup>-2t</sup>/2</p>
<p>v*(... |
2,854,419 | <p>Are these first-order formulas equivalent?
$$(\forall x)[(Ax \to Bx)\to(Cx \to Dx)]\tag{1}$$
$$(\forall x)(Ax \to Bx)\to (\forall x)(Cx \to Dx)\tag{2}$$
$$(\forall x)(Ax \to Bx)\to (\forall y)(Cy \to Dy)\tag{3}$$
I think (2) and (3) are equivalent, but I am not sure about (1).</p>
| hmakholm left over Monica | 14,366 | <p>Here's a counterexample that shows (1) is not equivalent to (2): Let the universe be $\{1,2,3,4\}$, and let $Ax$ mean $x=1$, $Bx$ mean $x=2$, $Cx$ mean $x=3$, and $Dx$ mean $x=4$.</p>
<p>I will let you compute the truth values of the formulas in this interpretation. This is fairly quick to do with truth tables beca... |
2,455,306 | <p>I am trying to prove the following:</p>
<p>(Monotonicity) If <span class="math-container">$A \subset B$</span> , then
<span class="math-container">$m(A) \le m(B)$</span>.</p>
<p>Now, I've drawn some pictures and defined several things, including several different identities for a measurable/lebesgue set. However, I ... | AlvinL | 229,673 | <p>Let $(X,\Sigma,m)$ be a measure space. A measure is countably additive, therefore finitely additive i.e for every two disjoint sets $A,B\in\Sigma\quad$ $m(A\cup B) = m(A)+m(B)$.</p>
<p>Assume $A\subseteq B$, then $A\cup(B\setminus A) = B$. By additivity
$$m(B) = m(A)+m(B\setminus A)$$
Since a measure of a set is n... |
2,455,306 | <p>I am trying to prove the following:</p>
<p>(Monotonicity) If <span class="math-container">$A \subset B$</span> , then
<span class="math-container">$m(A) \le m(B)$</span>.</p>
<p>Now, I've drawn some pictures and defined several things, including several different identities for a measurable/lebesgue set. However, I ... | copper.hat | 27,978 | <p>One definition of $m$ is in terms of an outer measure $m^*$ and then
one defines the measurable sets and for these we have $m=m^*$.</p>
<p>Usually we have something like $m^* A = \inf \{ \sum_k l(I_k) \, | \, A \subset \cup_k I_k\}$, where the $I_k$ are some core sets (such
as intervals) and the union may be at mos... |
2,096,827 | <p><strong>Question</strong>: A sample size of 10 is taken with replacement from an urn that contains 100 balls, which are numbered 1, 2, ..., 100. (At each draw, each ball has the same probability of being selected).</p>
<p>There are 3 parts to the question, and I've included my work below. However, I'm not sure if i... | Frank | 332,250 | <p>By definition,$$\begin{align*} & (f+g)(x)=f(x)+g(x)\tag1\\ & (f-g)(x)=f(x)-g(x)\tag2\end{align*}$$
And since the problem states $(f+g)(x)=10-3x$ and $(f-g)(x)=5x-14$, we have$$\begin{align*} & f(x)+g(x)=-3x+10\tag3\\ & f(x)-g(x)=5x-14\tag4\end{align*}$$
Which can be easily solved by adding $(3)$ and ... |
23,564 | <p>In the <a href="http://www.ems-ph.org/journals/newsletter/pdf/2009-12-74.pdf" rel="noreferrer">December 2009 issue</a> of the <a href="http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X" rel="noreferrer">newsletter of the European Mathematical Society</a> there is a very interesting interview with Pierre C... | François G. Dorais | 2,000 | <p>Here is Pierre Cartier's abstract from the Oberwolfach meeting <em>Category Theory and Related Fields: History and Prospects</em> (February 2009).</p>
<blockquote>
<p><strong>Living in a contradictory world: categories vs. sets?</strong></p>
<p>In the present time, the ambition to offer global foundations fo... |
1,066,061 | <p>I was thinking about the following problem:</p>
<blockquote>
<p>Suppose <em>R</em> is a ring s.t. every left ideal is also right. Is <em>R</em> commutative?</p>
</blockquote>
<p>This actually continues the easier question:</p>
<blockquote>
<p>Suppose <em>G</em> is a group whose all subgroups are normal. Is <e... | k.stm | 42,242 | <p>Not sure if this is satisfactory, but division rings only have trivial left and right ideals.</p>
<p>Therefore finite products of noncommutative division rings are counterexamples.</p>
|
1,874,299 | <p>The question is: Consider the family $F$ of circles in the $xy$ plane, $(x-c)^2+y^2=c^2$ tangent to the $y$ axis at the origin. Find a differential equation that is satisfied by the family of curves orthogonal to $F$. </p>
<p>My thinking: Since the implicit equation represents the level sets of the function $$
f(x,... | H. H. Rugh | 355,946 | <p>The differential of the equation is $2(x-c)\;dx+2y \;dy=0$ and describes the tangents of a curve at a point $(x,y)$ and with $c=c(x,y)$ verifying $2cx =x^2+y^2$.
By exchanging coefficients and changing sign on one (so as to get the orthogonal vector) we get $-2y \; dx + 2(x-c) \; dy=0$ or
$0= -2yx \; dx + (2x^2-2xc... |
1,874,299 | <p>The question is: Consider the family $F$ of circles in the $xy$ plane, $(x-c)^2+y^2=c^2$ tangent to the $y$ axis at the origin. Find a differential equation that is satisfied by the family of curves orthogonal to $F$. </p>
<p>My thinking: Since the implicit equation represents the level sets of the function $$
f(x,... | Jean Marie | 305,862 | <p>@H. H. Rugh an equivalent way to yours that one could find in an old textbook of mine is:</p>
<p>"if a family (F) of curves is characterized as the set of solutions of differential equation $y'=f(x,y)$, the family of orthogonal curves to all curves of (F) is solution of the differential equation </p>
<p>$$-\dfrac{... |
1,874,299 | <p>The question is: Consider the family $F$ of circles in the $xy$ plane, $(x-c)^2+y^2=c^2$ tangent to the $y$ axis at the origin. Find a differential equation that is satisfied by the family of curves orthogonal to $F$. </p>
<p>My thinking: Since the implicit equation represents the level sets of the function $$
f(x,... | JJacquelin | 108,514 | <p>$$(x-c)^2+y^2=c^2$$
or :
$$y^2+x^2-2cx=0 \quad\to\quad 2c=\frac{y^2+x^2}{x}=\frac{y^2}{x}+x$$
The differential equation of this family of circles is obtained by differentiation :
$$dc=0=2\frac{y}{x}dy-\frac{y^2}{x^2}dx+dx$$
$$2\frac{y}{x}dy=\left(\frac{y^2-x^2}{x^2}\right)dx$$
$$\frac{dy}{dx}=\frac{y^2-x^2}{2xy}$$
T... |
310,363 | <p>Given $N$ boxes with the same capacity $C$, I toss coins into the boxes uniformly, one by one. When any one of the boxes is full, the sum of the coins in all boxes is denoted $S$. How to compute the probability density function of $S(N,C)$? </p>
| esg | 48,831 | <p>(I change notation from <span class="math-container">$N,C$</span> to <span class="math-container">$n,c$</span> since I use capitals throughout to denote rvs).</p>
<p>Let <span class="math-container">$X_i$</span> be the random variable "number of the box the <span class="math-container">$i$</span>-th coin", then <sp... |
1,182,644 | <p>The tittle says it all. I think it's true, and I tried to prove it by showing that the derivative of this function: $-2Bxe^{-Bx^2}$ is bounded from above with a bound less than 1, in order to do that, I tried to use Taylor series of $e^{-Bx^2}$, but it seems that leads nowhere. Any suggestion?</p>
<p>Here $B>0$ ... | randomgirl | 209,647 | <p>I think we can do this for $x<0$:
\begin{align}
& \text{ if } x<0 \text{ then } |x|=-x \text{ so } -|x|=x \\
&\text{ or } - (x^2)^\frac{1}{2} =x \\
&\int x \, d(x^2)=\int - (x^2)^\frac{1}{2} \, d(x^2) \\
&\text{ someone correct me If I'm wrong }\end{align}</p>
|
2,740,808 | <p>One excercise asked me to <strong>"Prove that the determinant of an inversible matrix can't be 0"</strong>. I couldn't remember the proof the teacher gave and I didn't want to "cheat" because I'm practising for an exam, so after some thinking I came up with this.</p>
<p>I'd like to know <strong>if someone has a sim... | alex | 520,974 | <p>Recall<br>
$$ det(AB) = det(A)det(B) $$
$$ AA^{-1}=I \iff \text{ $A$ is invertible.} $$
Suppose $A$ is an invertible matrix. Note that if $A$ is invertible, then it follows that $A \neq 0$. So<br>
$$ det(A)det(A^{-1}) = 1. $$
Suppose $ det(A) = 0 $. Then
$$ 0 = 1 $$
Which is a contradiction. Then $det(A) \neq 0$.</p... |
922,339 | <p>Let A={1,2,3,...,8,9} B={2,4,6,8} C={1,3,5,7,9} D={3,4,5} E={3,5}
Which of these sets can equal a set X under each of the following conditions?</p>
<p>a. X and B are disjoint</p>
<p>b. X is a subset of D but X is not a subset of B</p>
<p>c. X is a subset of A but X is not a subset of C</p>
<p>d. X is a subset o... | sranthrop | 95,054 | <p>Using the Laplace-operator $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$, your equation writes
$$
\Delta u = Ax + By + C.
$$
Now, if both $u$ and $v$ are solutions of this equation, we obtain by considering the difference $\Delta(u-v)=0$. Functions $h\in C^2$ with $\Delta h=0$ are called h... |
945,920 | <p>Let $X$ be a nonempty set and $d: X\times X\to R$ be a function such that for all $x,y\in X$ and all distinct $u, v\in X$ each of which is different from $x$ and $y$</p>
<p>(1) $ d(x,y)\geq 0$ ;</p>
<p>(2) $d(x,y)=0$ if and only if $x=y$;</p>
<p>(3) $d(x,y)=d(y,x)$;</p>
<p>(4) $d(x,y)\leq d(x,u)+d(u,v)+d(v,y)$.<... | mathlove | 78,967 | <p>Note that $$3^{2n+1}+5^{2n}\equiv (\color{red}{-1})^{2n+1}+1^{2n}=-1+1=0\pmod 4$$
and that
$$3^{2n+1}+5^{2n}=3\cdot 9^n+25^n\equiv 3\cdot 1^n+1^n\equiv 3+1\equiv 4\not\equiv 0\pmod8$$</p>
|
1,231,082 | <p>Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$.</p>
<ul>
<li>I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a <em>formal</em> proof?). However, I am confused about how I should calculate $\text{dim}(\text{Im(tr)})$ and $\text{dim}(\text... | Domenico Vuono | 227,073 | <p>You can write $2^{105} = 2^{6 \cdot 17}\cdot 8$ and $3^{105} = 3^{6 \cdot 17} \cdot 3^3$ and you can use Fermat's little theorem.</p>
<p>$2^{6 \cdot 17} \equiv 1 \pmod 7$, $8 \equiv 1 \pmod 7$ therefore $2^{2015} \equiv 1 \pmod 7$ while $3^{6 \cdot 17} \equiv 1 \pmod 7$ and $3^3 \equiv 6 \pmod 7$ therefore $3^{105}... |
966,482 | <p>Over algebraically closed fields $K$, the <a href="http://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem" rel="nofollow noreferrer">Ax–Grothendieck theorem</a> (see also <a href="https://math.stackexchange.com/questions/662293/polynomial-map-is-surjective-if-it-is-injective?rq=1">this thread</a>) states that ... | Hagen von Eitzen | 39,174 | <p>A polynomial function $f:\mathbb R\to\mathbb R$ is either of odd degree and surjective, or of even degree and not injective.</p>
|
2,768,249 | <p>In the middle of a proof, I have had to analyze the asymptototic behavior of
$$
\mathbb{E}\left[\frac{1}{(1+X)^2}\right] = \frac{1}{2^n}\sum_{k=0}^n \binom{n}{k}\frac{1}{(1+k)^2}\tag{1}
$$
where $X$ is Binomially distributed with parameters $n$ and $1/2$. (I also had to handle $\mathbb{E}\left[\frac{1}{(1+X)^4}\righ... | Clement C. | 75,808 | <p>You have, writing a tautology and using a crude bound,
$$
|X| = |X|\mathbb{1}_{\{|X|\leq M\}} + |X|\mathbb{1}_{\{|X|> M\}} \leq
M\mathbb{1}_{\{|X|\leq M\}} + |X|\mathbb{1}_{\{|X|> M\}} \tag{1}
$$
so that, by linearity of expectation,
$$
\mathbb{E}[|X|] \leq \mathbb{E}[M\mathbb{1}_{\{|X|\leq M\}}]
+ \mathbb{E... |
901,161 | <p>Let's consider the manifold $S^1$</p>
<p>It is well known that we need two charts to cover this manifold.</p>
<p>Nonetheless, we can cover the full space using a single coordinate $\theta$ which is just the angle from the center.</p>
<p>Now, is this a general feature? I mean, is it always possible to have in ever... | Sina | 37,230 | <p>By definition a coordinate system is a collection of charts $\{U_i,\phi_i\}_i$ where $U_i$ are open sets in your manifold that cover your whole manifold and $\phi_i:U_i \rightarrow \mathbb{R}^n$ are homeomorphisms and the transition functions $\phi^i \circ \phi^{-j}$ are smooth where ever defined.</p>
<p>In this ca... |
1,571,706 | <p>I have found the solutions by a little calculation $(2,3,5,7)$ and $(2,3,4,5)$. But I don't know if there's any other solutions or not?</p>
| Mirko | 188,367 | <p>Assume $1\le a\le b\le c$. Then $2^n=a!(1+b\cdots(a+1)+c\cdots(a+1)$.<br>
It follows that either $a=2$ or $a=1$. </p>
<p>Consider the case $a=2$. Then one of the numbers $b\cdots(a+1)$ and $c\cdots(a+1)$ must be even, and the other odd, and since $b\cdots(a+1)$ divides $c\cdots(a+1)$ it follows that $b\cdots(a+1)$ ... |
3,364,263 | <p>How many 8 digit (its digits from left to right are labeled a through g) satisfy the following constraints:</p>
<ol>
<li><span class="math-container">$a_1 < a_2 < a_3 < a_4$</span></li>
<li><span class="math-container">$a_4 > a_5$</span></li>
</ol>
| lab bhattacharjee | 33,337 | <p><span class="math-container">$$y=(x-2)^3\implies y^2-19y=216$$</span></p>
<p>Either use <a href="https://www.aplustopper.com/factorization-by-splitting-middle-term/" rel="nofollow noreferrer">Middle term factor</a> <span class="math-container">$$\implies(y-27)(y+8)=0$$</span></p>
<p>Or use Sridhara'fs <a href="htt... |
221,130 | <p>Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be different, of course. But is this information enough to reconstruct number of faces of $P$ of all dimensions? If yes, what i... | David Eppstein | 440 | <p>Consider a simplex $S$ in five dimensions, and one more face normal $\pi$ in general position with respect to $S$ (no hyperplane perpendicular to $\pi$ passes through more than one vertex of $S$). Intersect $S$ with a halfspace $H$ perpendicular to $\pi$, and translate $H$ continuously towards $S$, starting from a p... |
620,340 | <p>I am stuck by this question from Liu's algebraic geometry textbook on quasi-coherent modules.</p>
<p>Let X be an affine scheme $\mbox {Spec} A $. Let $\mathcal {F} $ be a quasi-coherent $\mathcal{O}_{X} $ module. Show that for any affine open subset U of X we have a canonical isomorphism</p>
<p>$\mathcal {F}(X)\ot... | Martin Brandenburg | 1,650 | <p>When $U$ is basic-open, it holds by definition. In general, there is a canonical homomorphism $F(X) \otimes_A \mathcal{O}_X(U) \to F(U)$ adjoint to the restriction map $F(X) \to F(U)$. If $U$ is quasi-compact, it is a finite union of basic-open subsets $U_i$ and their intersections $U_{ij} := U_i \cap U_j$ are also ... |
2,225,932 | <p>If given a convex function $f: \mathbb{R} \to \mathbb{R}$, then the conjugate function $f^*$ is defined as $$f^*(s) = \sup_{t \in \mathbb{R}} (st-f(t))$$</p>
<p>Now i want to understand what is the physical interpretation of this conjugate function? What is its exposition? Please help me. </p>
| littleO | 40,119 | <p>The basic idea behind duality in convex analysis is to view a (closed) convex set <span class="math-container">$C$</span> as an intersection of half spaces. Applying this idea to the epigraph of a convex function <span class="math-container">$f$</span> suggests that we should view <span class="math-container">$f$</... |
1,992,009 | <p>It is known that $$\lim_{x \to 0}\frac{f(x)}{x} = -\frac12$$ </p>
<p>Solve
$$\lim_{x \to 1}\frac{f(x^3-1)}{x-1}.$$</p>
<p>Beforehand, I know that I should aim to get rid of the denominator $(x-1)$ and as such I factor the numerator to get:</p>
<p>$$\lim_{x \to 1}{f(x^2+x+1)}{}.$$</p>
<p>Now that I factored the d... | mfl | 148,513 | <p>First of all, note that $$\lim_{x \to 1}\frac{f(x^3-1)}{x-1}=\lim_{x \to 1}\left(\dfrac{f(x^3-1)}{x^3-1}\dfrac{x^3-1}{x-1}\right)=\lim_{x \to 1}\dfrac{f(x^3-1)}{x^3-1}\lim_{x \to 1}\dfrac{x^3-1}{x-1}$$ where the last equality holds if both limits exist. Now, use that,</p>
<p>$$\lim_{x \to 1}\dfrac{f(x^3-1)}{x^3-1}=... |
1,992,009 | <p>It is known that $$\lim_{x \to 0}\frac{f(x)}{x} = -\frac12$$ </p>
<p>Solve
$$\lim_{x \to 1}\frac{f(x^3-1)}{x-1}.$$</p>
<p>Beforehand, I know that I should aim to get rid of the denominator $(x-1)$ and as such I factor the numerator to get:</p>
<p>$$\lim_{x \to 1}{f(x^2+x+1)}{}.$$</p>
<p>Now that I factored the d... | Community | -1 | <p>$$\begin{align}\lim_{x \to 1}\frac{f(x^3-1)}{x-1} &= \lim_{x \to 1}\frac{f(x^3-1)}{x-1}\cdot\frac{x^2+x+1}{x^2+x+1} \\ &= \lim_{x\to 1}\frac{f(x^3-1)}{x^3-1}(x^2+x+1) \\ &= \lim_{x\to 1}\frac{f(x^3-1)}{x^3-1}\lim_{x\to 1}(x^2+x+1) \\ &= \lim_{u\to 0}\frac{f(u)}{u}(3) \\ &= -\frac 32\end{align}$$<... |
4,370,574 | <p>A fixed point iteration formula for <span class="math-container">$2x^3-4x^2+x+1=0$</span> can be derived:
<span class="math-container">$$x_{r+1}=4x_r^2-2x_r^3-1$$</span></p>
<p>Starting with the initial value of 2:</p>
<p><span class="math-container">$$x_0=2$$</span>
<span class="math-container">$$x_1=-1$$</span>
<s... | pfuhlert | 346,938 | <p>I decided to write something like this:</p>
<p>[...] where <span class="math-container">$\lvert \lvert M \rvert \rvert_2$</span> represents the L2 matrix norm that is equivalent to the number of non-zero entries for <span class="math-container">$M \in \{0, 1\}^{n \times m}$</span>.</p>
<p>Thanks for the help.</p>
|
109,292 | <p>Let $\kappa$ be a regular, uncountable cardinal. Let $A$ be an unbounded set, i.e. $\operatorname{sup}A=\kappa$. Let $C$ denote the set of limit points $< \kappa$ of $A$, i.e. the non-zero limit ordinals $\alpha < \kappa$ such that $\operatorname{sup}(X \cap \alpha) = \alpha$. How can I show that $C$ is unboun... | Asaf Karagila | 622 | <p>Since $\kappa$ is regular this means that the order type of $A$ is $\kappa$, and for every $\delta<\kappa$ we have that the cofinality of $\delta<\kappa$ as well.</p>
<p>Now suppose that $A$ is bounded in all limit points above $\beta$, without loss of generality $A\cap\beta=\varnothing$. Define a regressive ... |
737,152 | <p>Give an algebraic proof that $\binom{n+1}{m+1} = \sum_{k=m}^{n} \binom{k}{m}$.</p>
<p>I've tried using Pascal's rule and looking for a telescopic sum, but I can't find one.</p>
<p>Any help is appreciated.</p>
| Einar Rødland | 37,974 | <p>As <em>did</em> has already pointed out, $T(n)\sim2\sqrt{n}$. Any easy way to arrive at this conclusion is to pretend $T(n)$ is defined for all real numbers and continuous, and approximate
$$
1=T(n)-T(n-\lceil\sqrt{n}\rceil)\approx\sqrt{n}T'(n)
$$
for large $n$, which makes $T'(n)\approx1/\sqrt{n}$ and in turn, by i... |
1,791,815 | <p>I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$.</p>
<p>This is part of an attempt to find the de Rham cohomology of the n sphere generally. I have shown that $H^k(S^n) = H^{k-1}(S^{n-1})$ (for $k>1$) but to find the de Rham cohomol... | Pedro | 23,350 | <p>You can imitate the argument that shows $\pi_1(S^n)=0$ if $n>1$ by removing a point! Try it!</p>
|
992,125 | <p>If I rolled $3$ dice how many combinations are there that result in sum of dots appeared on those dice be $13$?</p>
| John Rawls | 352,368 | <p>At the lower division math level we can do the following easily given the low number of combinations:</p>
<p>1)list the number of potential combinations
116</p>
<p>265</p>
<p>355</p>
<p>364</p>
<p>454</p>
<p>2) Now we find out the numbers of way that we can arrange the listed numbers in which it is: </p>
<p>3... |
2,918,497 | <p>From <em>Linear Algebra</em> by Friedberg, Insel, and Spence:</p>
<blockquote>
<p>Given <span class="math-container">$M_1=\begin{pmatrix} 1&0\\0 &1\end{pmatrix}$</span>, <span class="math-container">$M_2=\begin{pmatrix} 0&0\\0 &1\end{pmatrix}$</span> and <span class="math-container">$M_3=\begin{pmatr... | mechanodroid | 144,766 | <p>Clearly $M_1, M_2$ and $M_3$ are symmetric.</p>
<p>Conversely, for an arbitrary symmetric matrix $\begin{bmatrix} a & b \\ b & c\end{bmatrix}$ we have
$$\begin{bmatrix} a & b \\ b & c\end{bmatrix} = a \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} + (c-a)\begin{bmatrix} 0 & 0 \\ 0 & 1\e... |
4,001,192 | <p>Suppose <span class="math-container">$a, b$</span> in the field <span class="math-container">$K$</span> are algebraic over the subfield <span class="math-container">$F \subseteq K$</span>. Suppose <span class="math-container">$a$</span> is algebraic of degree <span class="math-container">$m$</span> over <span class=... | Saúl RM | 807,670 | <p>Yes. Call <span class="math-container">$f$</span> the minimal polynomial of <span class="math-container">$b$</span> in <span class="math-container">$F$</span>. Then, <span class="math-container">$f(b)=0$</span> and <span class="math-container">$f$</span> is in <span class="math-container">$F(a)[X]$</span>. So the mi... |
4,001,192 | <p>Suppose <span class="math-container">$a, b$</span> in the field <span class="math-container">$K$</span> are algebraic over the subfield <span class="math-container">$F \subseteq K$</span>. Suppose <span class="math-container">$a$</span> is algebraic of degree <span class="math-container">$m$</span> over <span class=... | nullUser | 17,459 | <p>If <span class="math-container">$b$</span> is algebraic of degree <span class="math-container">$n$</span> over <span class="math-container">$F$</span>, then there is a polynomial <span class="math-container">$P$</span> of degree <span class="math-container">$n$</span> with coefficients in <span class="math-container... |
4,634,263 | <p>I have the following problem that asks to find the shock curve for the following IVP
<span class="math-container">$$u_t + (u^2)_x = 0, \quad u(x,0) = \frac{1}{2}e^{-x},$$</span>
to then obtain a weak solution from this curve using Rankine Hugoniot condition. I'm pretty lost on how to raise the problem. In Evans ther... | Vercassivelaunos | 803,179 | <p>Here is another perspective: In most cases, we don't really care wether events are disjoint or not. More specifically, we don't care about the underlying measurable space <span class="math-container">$(\Omega,\Sigma)$</span> from which events are taken that much, we care about the probability distribution itself. An... |
2,832,025 | <p>$$\sum_{k=1}^{\infty}\ln\left[ \frac{(4k+1)^{1/(4k+1)^{n}}}{(4k-1)^{1/(4k-1)^{n}}} \right] = -\beta'(n)$$.
Where $\beta$ is the Dirichlet Beta Function and $n$ is a positive integer. </p>
<p>I cannot find this cited anywhere nor values of the beta function derivative apart from at $-1,0,1$. How can I go about find... | Nikos Bagis | 223,191 | <p>It is known that (uniformly and absolutely)
$$
\beta(n)=\sum^{\infty}_{k=1}\frac{\chi_4(k)}{k^n}\textrm{, }Re(n)>1.
$$
Hence writing $1/k^n=e^{-n\log(k)}$, we have easily
$$
-\beta'(n)=\sum^{\infty}_{k=2}\frac{\chi_4(k)\log(k)}{k^n}.
$$
But
$$
\chi_4(k)=\left\{
\begin{array}{cc}
0\textrm{ if }k\equiv 0 (mod)4... |
3,081,076 | <p>In do Carmo, one exercise gives a plane in <span class="math-container">$\mathbb R^3$</span>, <span class="math-container">$ax +by +cz+d = 0$</span>, and tells us to show that <span class="math-container">$|d|/\sqrt{a^2 + b^2 + c^2}$</span> measures the distance from the plane to the origin.</p>
<p>However, this se... | Jose | 19,799 | <p>I am not a sage expert. But I know you can call pari within sage and here you can compute the Gram/height matrix from the points and determine its determinant. Suppose you <code>e</code> is your elliptic curve (in pari this created by <code>ellinit</code>) and suppose <code>P1,P2,P3</code> are your points then you j... |
3,081,076 | <p>In do Carmo, one exercise gives a plane in <span class="math-container">$\mathbb R^3$</span>, <span class="math-container">$ax +by +cz+d = 0$</span>, and tells us to show that <span class="math-container">$|d|/\sqrt{a^2 + b^2 + c^2}$</span> measures the distance from the plane to the origin.</p>
<p>However, this se... | Viktor Vaughn | 22,912 | <p>In SageMath this can be accomplished using the command <code>height_pairing_matrix</code>. For example:</p>
<pre><code>sage: E = EllipticCurve([1, 0, 0, 0, 1])
sage: E
Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Rational Field
sage: pts = E.gens()
sage: pts
[(-1 : 1 : 1), (0 : -1 : 1)]
sage: M = E.height_pai... |
1,072,924 | <p><strong>Setting</strong></p>
<p>You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$.</p>
<p>I got up to here</p>
<p>$$\Pr[R > t] = \Pr[X_o/X_1 > t] = \Pr[X_o > X_1 t] = ... | Robert Israel | 8,508 | <p>One way to do this is to rescale: let $Y = t X_1$, so $X_0/X_1 > t$ is equivalent to $X_0 > Y$. Now $X_0$ and $Y$ are independent exponential random variables with rate parameters $\alpha$ and $\alpha/t$ respectively. Think of two independent Poisson processes with these rates. One way to realize this is to... |
2,290,295 | <p>I do not know if the following is true: </p>
<p>Question: If $\Omega\subseteq\mathbb{R}^n$ is open and connected, then there exists a covering by balls $\{B_j\}_{j=1}^{\infty}$ with the property $B_j\subseteq\Omega$ and $B_j\cap B_{j+1}\neq\emptyset$, for all $j\geq1$.</p>
<p>Motivation: I have a function $u$ that... | Mundron Schmidt | 448,151 | <p>It is just a short step. If $u(x)=c_j$ for all $x\in B_j$ then we get $c_j=u(y)=c_{j+1}$ for $y\in B_{j+1}\cap B_j\neq\emptyset$ for $j\geq 1$. Then all $c_j$ are the same and $u$ is constant on $\Omega$.</p>
|
1,340,888 | <p>I came across the picture below through random means.</p>
<p><img src="https://i.stack.imgur.com/7POpd.gif" alt="ellipse magic"></p>
<p>What is being demonstrated? All I could think of is <em>maybe</em> the center of the triangle is moving back and forth between the focii of the ellipse, but even if that's true (w... | Piquito | 219,998 | <p>We need to prove the following: <strong>when two vertices of a fixed triangle slide along two arms of a fixed angle, the locus described by the third vertex is an ellipse.</strong></p>
<p>In the figure all the letters excepting $t$ and $P(x,y)$ are data of the problem; the angle $t$ determines the position of the ... |
1,880,150 | <p>I work in a warehouse where we take components and put them together to make a finished good product.</p>
<p>We have these values for each component:</p>
<ol>
<li><p>Quantity Required</p></li>
<li><p>Quantity Used</p></li>
<li><p>Variance (Quantity Used- Quantity Required)</p></li>
<li><p>Overage ((Variance/Quanti... | Ethan Bolker | 72,858 | <p>You should definitely not ignore the zero rows.</p>
<p>The simplest thing to do would be just to sum the Quantity Required and the Quantity Used , then average by dividing by the number of rows in the report.</p>
<p>You can sum the Variance the same way. Summing the Overage makes no sense - mark it N/A.</p>
<p>Yo... |
3,408,839 | <p><span class="math-container">$$\frac{1}{\log_{2x-1}{(x)}} + \frac {1}{\log_{x+6}{(x)}}=1+\frac{1}{\log_{x+10}{(x)}}$$</span>
What should i do for the first step ?</p>
<p>Is it like <span class="math-container">$\frac{1}{A}+\frac{1}{B}$</span> then i simplify into <span class="math-container">$\frac{A+B}{AB}$</span>... | nonuser | 463,553 | <p>Hint: <span class="math-container">$$\log _ax = {\log x\over \log a}$$</span></p>
<p>So you have <span class="math-container">$${\log (2x-1)\over \log x}+{\log (x+6)\over \log x} = 1+{\log (x+10)\over \log x} $$</span></p>
|
258,622 | <p>How do you swap the values and keys of an Association in a manner similar to taking the transpose of a two-dimensional matrix or nested list?
I am trying to solve the Wolfram Challenge <a href="https://challenges.wolfram.com/challenge/capital-cities-near-a-latitude" rel="nofollow noreferrer">Capital Cities Near a La... | Rohit Namjoshi | 58,370 | <pre><code>TakeSmallest[data, 1]
(* <|Entity["City", {"Athens", "Attiki", "Greece"}] -> 0.02|> *)
</code></pre>
|
481,527 | <p>The following question came up at a conference and a solution took a while to find.</p>
<blockquote>
<p><strong>Puzzle.</strong> Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza has no crust on it.</p>
</blockquote>
<p>We can make this more concrete,</p>
<blockq... | robjohn | 13,854 | <p>Here is another with 12 pieces, but all pieces have the same orientation:</p>
<p>$\hspace{32mm}$<img src="https://i.stack.imgur.com/w03ah.png" alt="enter image description here"></p>
<p>Using this idea, the pizza can be divided into $6n$ equal pieces with the same orientation for any $n$. However, to have some pie... |
1,976,001 | <p>The statement goes as following: if $3 \mid 2a$, then $3 \mid a$ and $a$ is an integer. In my approach, I used prime factorization, but is this actually valid? This was my approach:</p>
<p>$$3 \mid 2a \implies 2a = 2 \cdot 3 \cdot k, k \in \mathbb{N}$$</p>
<p>$$\frac{2a}{2} = \frac{2\cdot 3 \cdot k}{2}$$</p>
<p>$... | Bill Dubuque | 242 | <p>Yes: $\,\ 3\mid 2a,3a\,\Rightarrow\, 3\mid 3a\!-\!2a = a.\ $ <strong>QED</strong> $\ $ We used Bezout $3 - 2 = 1.\,$ This generalizes:</p>
<hr>
<p>Generally $\,c\mid ab\,\Rightarrow\, c\mid a\ $ if $\ b,c\,$ are (Bezout) coprime $\ jb+kc = 1\,$</p>
<p><strong>Proof</strong> $\ \ c\mid abj,akc\,\Rightarrow\, c\... |
966,420 | <p>How can one write $x$ is a factor of $y$ (as a constraint)? I am also not sure what else to add to meet the question quality requirements. </p>
| David P | 49,975 | <p>$x$ is a factor of $y$ is the same thing as $y$ is a multiple of $x$. You can write </p>
<p>$$y=kx \ \text{ for some integer $k$}$$</p>
|
311,950 | <p>I'm trying to show that any subset bounded of $\Bbb{R}^k$ is totally bounded.</p>
<p>Here is what I did: </p>
<p>(1)A subset of a totally bounded Set is bounded:</p>
<p>Proof: Let $X$ be a totally bounded subset and $Y\subset X$ then there exists an $\epsilon /2$-net $\{x_1,x_2,..,x_n\}$ and $X\subset \displaysty... | Pedro | 23,350 | <p>I'll be systematic here, I think it can help.</p>
<p><strong>D</strong> Let $S$ be any subset of $\Bbb R^n$. Given $\epsilon >0$, we say that $N$ is an $\epsilon$-net for $S$ if the set of open balls</p>
<p>$$B_\epsilon(N)=\{B(x,\epsilon):x\in N\}$$</p>
<p>covers $S$. That is, the set of open balls of radius $... |
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Jeff Harvey | 10,475 | <p>Perhaps it would not be out of place to quote Miles Reid's Bourbaki seminar on the
McKay correspondence here:</p>
<p>"The physicists want to do path integrals, that is, they want to integrate
some "Action Man functional" over the space of all paths or loops
$ \gamma : [0; 1] \rightarrow Y $. This
impossibly large ... |
1,197,056 | <p>I tried to evaluate the following limits but I just couldn't succeed, basically I can't use L'Hopital to solve this... </p>
<p>for the second limit I tried to transform it into $e^{\frac{2n\sqrt{n+3}ln(\frac{3n-1}{2n+3})}{(n+4)\sqrt{n+1}}}$ but still with no success...</p>
<p>$$\lim_{n \to \infty } \frac{2n^2-3}{... | user37238 | 87,392 | <p>Hints :
$$ \frac{2n^2-3}{-n^2+7} = \frac{2 - \frac{3}{n^2}}{-1+\frac{7}{n^2}},$$
and
$$ \frac{3^n-2^{n-1}}{3^{n+2}+2^n} = \frac{1-\frac{1}{2}\left( \frac{2}{3} \right)^n}{3^2+\left( \frac{2}{3} \right)^n}.$$</p>
|
3,795,655 | <p>Let <span class="math-container">$H(\mu|\nu)$</span> be the relative entropy (or Kullback-Leibler convergence) defined in the usual way. I am looking for a proof or reference to the following fact: <span class="math-container">$\mu,\nu$</span> two-dimensional probability measures with marginals <span class="math-con... | stochasticboy321 | 269,063 | <p>Let the laws be over random variables <span class="math-container">$X,Y$</span>. By the chain rule of KL divergences, <span class="math-container">$$ D(P_{XY}\|Q_{XY}) = D(P_X\|Q_X) + D(P_{Y|X} \|Q_{Y|X}|P_X).$$</span></p>
<p>Thus the question boils down to asking if <span class="math-container">$$ D(P_{Y|X} \|Q_{Y|... |
918,510 | <p>I am trying to solve the following problem :
Find all the positive integers $n$ and $k$ such that it is possible to write integers in an $n \times n$ grid so that the sum of all elements in the grid is negative but the sum of elements of each $k \times k$ grid contained in it is positive. I am only looking for a sm... | user133281 | 133,281 | <p><strong>Hint:</strong> As explained in the comments, you cannot do so when $k \mid n$. So let's show it is possible when $k \nmid n$. Try to put positive numbers in only a few squares, such that every $k \times k$-grid contains exactly one positive number.</p>
|
3,975,162 | <p>Let <span class="math-container">$(X, A, µ)$</span> be a positive metric space. If <span class="math-container">$\mu(X) < \infty$</span> and <span class="math-container">$(A_n)_{(n \in N^*)},A \in X$</span> <br />
show that if <span class="math-container">$\mu(A\bigtriangleup A_n)\rightarrow 0$</span> then <span ... | nonuser | 463,553 | <p>Numbers <span class="math-container">$y$</span> and <span class="math-container">$y+1$</span> are consecutive and thus relatively prime and thus <span class="math-container">$y\ne \pm1$</span> can not divide <span class="math-container">$y+1$</span> and so it can not divide <span class="math-container">$(y+1)^2$</sp... |
137,006 | <p><code>FunctionDomain[(x^2-x-2)/(x^2+x-6),x]</code> </p>
<p>gives </p>
<blockquote>
<p><code>x < -3 || -3 < x < 2 || x > 2</code>.</p>
</blockquote>
<p>However, when I factor the numerator and denominator the result is different:</p>
<p><code>FunctionDomain[((x - 2) (x + 1))/((x - 2) (x + 3)),x]</co... | jkuczm | 14,303 | <p>In <em>Mathematica</em> trivial removable singularities of like <code>x/x</code> (in full form <code>Times[x, Power[x, -1]]</code>) are replaced by <code>1</code> during ordinary evaluation of <code>Times</code>, when appropriate pair of positive and negative <code>Power</code> of same expression is encountered. Sim... |
4,256,719 | <p>I have to teach a 1 hr class about the stereographic projection in the complex plane and i am looking for sources or some interesting fact about this.
The best I have found is in the Alhfors of Complex Analysis.</p>
<p>It would help me a lot to read your suggestions</p>
| Alessio K | 702,692 | <p>Note that by the <a href="https://en.wikipedia.org/wiki/Chain_rule" rel="nofollow noreferrer">chain rule</a> we have</p>
<p><span class="math-container">$$\frac{d}{dn}(2^{n})=\frac{d}{dn}\left(e^{n\ln(2)}\right)=\frac{d}{dn}\left(n\ln(2)\right)e^{n\ln(2)}=2^{n}\ln(2)$$</span></p>
<p>Then by the change of base formul... |
395,850 | <p>Is Dirac delta a function? What is its contribution to analysis? </p>
<p>What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.</p>
| Tomás | 42,394 | <p>To have a better understanding of what is the Delta Dirac "function", it is good to know what is a distribution (but this is not necessary). Let $\Omega\subset\mathbb{R}^N$ be a open set and $\mathcal{D}(\Omega)=C_0^\infty(\Omega)$. We define in $\mathcal{D}(\Omega)$ the following notion of convergence: we say that ... |
3,247,841 | <p>I'm working on an algorithm to colour a map drawn in an editor using 4 colours, as a visual demonstration of the four colour theorem. However, my (imperfect) algorithm was able to colour all maps except this one, which after giving it a go myself I struggled to do. I was also unable to collapse it into an 'untangled... | Radost | 641,644 | <p><a href="https://i.stack.imgur.com/TJJbY.png" rel="noreferrer"><img src="https://i.stack.imgur.com/TJJbY.png" alt="4 colored map"></a></p>
<p>This is a possible coloring. Done by hand, either really lucky or not that difficult. Perhaps it'll help with debug of the algorithm and shows that this map is definitely leg... |
1,577,098 | <p>I've been banging my head against the wall trying to handle these proofs for two hours now, it seems very simple but I guess I need a hand starting out. I hope I at least know what to show:</p>
<p>Show that the set $S_1=\{(x_1,x_2), x_2 \geq x_1^2\}$ is convex. Intuitively it makes sense since this is the area abov... | YYF | 16,595 | <p>\begin{align*}
&\lambda x_2+(1-\lambda)x_4-[\lambda x_1+(1-\lambda)x_3]^2\\
&=\lambda
x_2+(1-\lambda)x_4-(\lambda^2x_1^2+2\lambda(1-\lambda)x_1x_3+(1-\lambda)^2x_3^2)\\
&=\lambda[x_2-\lambda
x_1^2]+(1-\lambda)[x_4-(1-\lambda)x_3^2]-2\lambda(1-\lambda)x_1x_3\\
&\geq\lambda(1-\lambda)x_1^2+... |
1,577,098 | <p>I've been banging my head against the wall trying to handle these proofs for two hours now, it seems very simple but I guess I need a hand starting out. I hope I at least know what to show:</p>
<p>Show that the set $S_1=\{(x_1,x_2), x_2 \geq x_1^2\}$ is convex. Intuitively it makes sense since this is the area abov... | Kelenner | 159,886 | <p>An idea (but perhaps not the simpler...): Your set $S$ is the set of points that are above all the tangents to the parabola. If you write the equation of such a tangent, at the point $(t,t^2)$, then it is $x_2=2tx_1-t^2$. So $(x_1,x_2)\in S$, iff you have $x_2\geq 2tx_1-t^2$ for all $t\in \mathbb{R}$. (It is easy t... |
370,192 | <p>I have a basic question about the Heegaard diagrams involved in providing a framework
for calculation of Floer-Homology of three-manifolds.</p>
<p>Typically such diagrams look like Figure 1 and Figure 2 <a href="http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere" rel="nofollow noreferrer">here</a> or ... | Oğuz Şavk | 131,172 | <p>You are probably familiar with definitions and theorems. But I prefer to write those for completeness. And also excuse for a paint-like drawing. I hope that they will be useful.</p>
<p>A <em>handlebody of genus <span class="math-container">$g$</span></em> is a <span class="math-container">$3$</span>-manifold constru... |
1,856,171 | <p>Functional</p>
<p>$F(x)=(x^2-x+1)Q(x)+x-1$</p>
<p>$G(x)=(x^2-x+1)T(x)+x+1$</p>
<p>$F(x).G(x)=(x^2-x+1)H(x)+ax+b$</p>
<p>find a and b</p>
| Branimir Ćaćić | 49,610 | <p>Let $L$ be the line spanned by some non-zero vector $\mathbf{v} \in \mathbb{R}^3$, so that $L = \{a\mathbf{v} \mid a \in \mathbb{R}\}$ is the space of all scalar multiples of $\mathbf{v}$. Then the orthogonal projection of a vector $\mathbf{x} \in \mathbb{R}^3$ onto the line $L$ can be computed as
$$
\operatorname{... |
54,393 | <p>I used to think that in any Vector space the space spanned by a set of orthogonal
basis vectors contains the basis vectors themselves. But when I consider the vector space $\mathcal{L}^2(\mathbb{R})$ and the Fourier basis which spans this vector space, the same is not true ! I'd like to get clarified on possible mi... | ashutosh simha | 24,399 | <p>I want you guys to understand few things clearly:</p>
<p>1> The dirchlet conditions are "sufficient" not necessary for F' transform'bility</p>
<p>2> The sinusoid is DEFINITELY in the span of {exp(j2PIft)} this follows from linear algebra!
However It does not mean that the fourier TRANSFORM exists ! .. however in... |
707,667 | <p>From the wiki of <a href="http://en.wikipedia.org/wiki/Precision_and_recall" rel="nofollow">Precision and recall</a>:</p>
<blockquote>
<p>recall (also known as sensitivity) is the fraction of relevant
instances that are retrieved.</p>
</blockquote>
<p>I can understand the literal meaning of "sensitivity", but ... | Jochen | 38,982 | <p>One more proof: $f:\ell_\infty \to \mathbb R$, $x\mapsto \lim\sup |x_n|$ is continuous so that $c_0= f^{-1}(\lbrace 0\rbrace)$ is closed. </p>
|
3,651,173 | <p>I am trying to use generating function to find closed form formula for this expression:
<span class="math-container">$$\sum_{n \geq 0} \frac{n^2+4n+5}{n!}$$</span>
but I don't how to start this. any suggestion or hints. thank you</p>
| grand_chat | 215,011 | <p>We know
<span class="math-container">$\displaystyle
\sum_{n=0}^\infty \frac1{n!} = e.
$</span>
By mangling the numerator <span class="math-container">$n^2+4n+5$</span> of the given expression we can break off terms that are proportional to <span class="math-container">$n(n-1)$</span> and <span class="math-container"... |
924,177 | <p>I'm trying to draw quadratic bezier curve (as line).
I approximate quadratic bezier curve as parabola ($y=x^2$), according to this document <a href="http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html" rel="noreferrer">http://http.developer.nvidia.com/GPUGems3/gpugems3_ch25.html</a></p>
<p>There, in secti... | G.Kós | 141,614 | <p>The formula gives only an approximation. If $f(\mathbf x)$ is a continuously differentiable function and the point $\mathbf a$ is ,,close'' to the implicit curve/surface $f(\mathbf x)=0$ and the closest point of the curve is $\mathbf a_0$ then
$$
f(\mathbf a) = f(\mathbf a)-f(\mathbf a_0) \approx (\nabla f)\cdot (\m... |
177,915 | <p>I have two inequalities that I can't prove:</p>
<ol>
<li>$\displaystyle{n\choose i+k}\le {n\choose i}{n-i\choose k}$</li>
<li>$\displaystyle{n\choose k} \le \frac{n^n}{k^k(n-k)^{n-k}}$</li>
</ol>
<p>What is the best way to prove them? Induction (it associates with simple problems, but sometimes I find it difficult... | Karolis Juodelė | 30,701 | <p>The first, after writing down the factorials and some canceling becomes $$(i+k)! \geq i!k!$$ </p>
<p>The second works with induction. Show that the inequality is true with $n = k$ and then show that $$\binom{n+1}{k} = \binom{n}{k} * \frac{n+1}{n+1-k} \leq \frac{n^n}{k^k(n-k)^{n-k}} * \frac{n+1}{n+1-k} \leq \frac{(... |
1,828,669 | <p>Let $N\gg 1$ be a large parameter, which I ultimately want to let tend to infinity. I am reading an old <a href="http://math.mit.edu/classes/18.158/bourgain-restriction.pdf" rel="nofollow">paper</a> of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118)</p>
<p>$$\sum_{q=1}^{N^{1/2}-1}\sum_{{1\leq a &... | Junpei Iori | 348,176 | <p>I will give a hint. As always the entire ball game is picking the proper flow graph. As is the typical flavor, your flow graph will be bipartite with one side connected to the source, and the other connected to the sink. Now you want to attach weights to the edges to ensure that:</p>
<p>1) The flow "sees" the si... |
3,180,628 | <p>The following is a quotation from Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(6.7 The Affine n-Space) .</p>
<blockquote>
<p>For any <span class="math-container">$R$</span>-algebra <span class="math-container">$R'$</span> let <span class="math-container">$\mathbb{A}^n_S(R')$</span> be the set of al... | Kitamado | 547,107 | <p>From Siegfried Bosch「Algebraic Geometry and Commutative Algebra」(7.2 Fiber Products)</p>
<blockquote>
<p>First note that the universal property of fiber products just says that these products respect the formation of <span class="math-container">$T$</span>-valued points, namely,
<span class="math-container">$$... |
13,937 | <p>Prove that $$\sum_{\substack{x,y \in \mathbb N \\ ax-by \ne 0 }}\frac1{|ax-by|xy}$$ converges, where $a, b \in \mathbb N$
</p>
<p>This 2-D problem can be proved by an integral test. I'm looking for some other proofs that can be easily generalized to higher dimensional cases like below,</p>
<p>Prove that $$\sum_{\s... | Timothy Wagner | 3,431 | <p>Here is a partial answer (although it uses the ring structure on the set of endomorphisms). A torsion abelian group is cyclic if and only if any two elements in $End(A)$ commute with respect to composition (in other words, $End(A)$ is a commutative ring).</p>
<p><a href="http://www.springerlink.com/content/m7222244... |
2,419,391 | <p>Suppose there exists a Lebesgue Space, <span class="math-container">$L_1$</span> and functions functions <span class="math-container">$\phi$</span>, <span class="math-container">$\phi'$</span>, <span class="math-container">$f$</span>, and <span class="math-container">$f'$</span> functions where
<span class="math-con... | Jeremy | 472,252 | <p>By the definition of Lebesgue spaces,</p>
<p>If $\phi \in L_1$ then $\int_{-\infty}^{\infty} \phi(x)\, dx < \infty$</p>
<p>If $\lim_{x \to \infty} \phi(x) > 0$ or $\lim_{x \to \infty} \phi(x) < 0$ then $\int_{-\infty}^\infty |\phi(x)| dx = \infty$</p>
<p>The contrapositive of the preceding statement is $... |
275,937 | <p>Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. </p>
<p>Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\ldots,r_n)/\mathbb{Q})$$</p>
<p>What is the probability, as $n\to\infty,$ that $G$ is solvable? (I assume 0.) Who... | Alexander Gruber | 12,952 | <p>$G\cong S_n$ with probability $1$ as $n\rightarrow \infty$. This was proven first by </p>
<p>B. L. van der Waerden, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002276607">Die Seltenheit der Gleichungen mit Affekt</a>, <em>Mathematische Annalen</em> <strong>109</strong>:1 (1934), pp. 13–... |
42,411 | <p>I have some products that I want to increase in value such that a 20% discount gives their current value. It's been ~25 years since college algebra and so I'm a bit rusty on setting up the equation.</p>
<p>I've been trying to figure out how to solve for X being the percentage increase needed in order that 20% off ... | André Nicolas | 6,312 | <p>Let's work first with your particular numbers. Suppose that an item is originally priced at 100 (dollars). Imagine that you inrease the price by $x$ percent. Then the new price is $100+x$.</p>
<p>You want to make sure that if you apply a 20 percent discount to this new price $100+x$, you end up with a price of ... |
2,984,534 | <p>I've gone ahead and split up <span class="math-container">$4000$</span> into <span class="math-container">$2^{5} 5^{3}$</span> and solved each solution separately - as in applied Hensel's lemma for mod 2 and mod 5 solutions separately, I just don't understand how I would combine these solutions.</p>
<p>For mod <sp... | Alex Kruckman | 7,062 | <p>Let <span class="math-container">$c$</span> be the value of the constant function <span class="math-container">$f|_A$</span>. Since points are closed in <span class="math-container">$Y = \mathbb{R}$</span>, <span class="math-container">$\{c\}$</span> is closed. Since <span class="math-container">$f$</span> is contin... |
3,315,622 | <p>Given <span class="math-container">$\textbf{a}$</span> , <span class="math-container">$\textbf{h}$</span> <span class="math-container">$\in$</span> <span class="math-container">$\mathbb{R}^n$</span> and a function from some subset of <span class="math-container">$\mathbb{R}^n$</span> to <span class="math-container">... | Kavi Rama Murthy | 142,385 | <p>This need not be true if <span class="math-container">$v=0$</span>. Assume <span class="math-container">$v \neq 0$</span> and let <span class="math-container">$\epsilon >0$</span>. Choose <span class="math-container">$\delta >0$</span> such that <span class="math-container">$|f(a+h)| <\epsilon$</span> whene... |
105,413 | <p>I know: I'm going to make a poor showing, but really I can't understand this:</p>
<p><code>a</code> is an expressione whose FullForm is</p>
<pre><code>Power[Plus[Subscript[u,x],Times[Complex[0,-1],Subscript[u,y]],Times[Complex[0,1],Subscript[v,x]],Subscript[v,y]],2]
</code></pre>
<p>Why does the following code re... | thedude | 27,670 | <pre><code>a = Power[
Plus[Subscript[u, x], Times[Complex[0, -1], Subscript[u, y]],
Times[Complex[0, 1], Subscript[v, x]], Subscript[v, y]], 2]
b = ComplexExpand@a /. I -> H
</code></pre>
|
147,965 | <p>What is the method to count multiplicities of intersection? for example suppose we have the projective line $x=0$ in $\mathbb{P}^{2}$ and the curve $V(z^{2}y^{2}-x^{4}) \subseteq \mathbb{P}^{2}$.</p>
<p>Clearly they intersection consists of two points $p=[0:1:0]$ and $r=[0:0:1]$. So for example Bezout's theorem say... | Brian M. Scott | 12,042 | <p>In the first problem, $\sum_{t=0}^{100-X}X=(100-X+1)X$: there are $100-X+1$ terms because of the $t=0$ term. Fix this, and you’ll get Yuval’s answer.</p>
<p>In the second, notice first that $|i-j|=0$ when $i=j$, so we can remove those terms from the sum. Next, $|i-j|=|j-i|$, so we need only sum the terms in which $... |
147,965 | <p>What is the method to count multiplicities of intersection? for example suppose we have the projective line $x=0$ in $\mathbb{P}^{2}$ and the curve $V(z^{2}y^{2}-x^{4}) \subseteq \mathbb{P}^{2}$.</p>
<p>Clearly they intersection consists of two points $p=[0:1:0]$ and $r=[0:0:1]$. So for example Bezout's theorem say... | Community | -1 | <p>Too long for a comment.</p>
<p>As Brian has already given the answer, I will just add a slightly different round about way to go about summing the second sum.</p>
<p>$$\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\lvert i - j \rvert}{n^2}$$
Split the inner sum over $j$ from $1$ to $i$ and $i+1$ to $n$.</p>
<p>$$\sum_{i=1}... |
25,784 | <p>As many Americans know, the “traditional” high school sequence is:</p>
<p>Algebra 1</p>
<p>Geometry</p>
<p>Algebra 2</p>
<p>PreCalculus</p>
<p>Calculus</p>
<p>For those who take developmental education at the community college level, it consists of something like:</p>
<p>Developmental Algebra</p>
<p>Intermediate Alg... | Kevin Arlin | 5,233 | <p>"Geometry," the American high school course, is generally pseudo-axiomatic Euclidean geometry. I don't know whether your claim about the CC curriculum is broadly true, but assuming it is, it's probably because Euclidean geometry is simply not a prerequisite for calculus. Students need to know the Pythagore... |
108,209 | <p>A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global information, or alternatively, lack a complete, instantaneous understanding of the structure of the problem? </p>
<p... | Suvrit | 8,430 | <p>The question really makes no clear sense, and should be revised to ask something more precise. In any case, let me try to partially disabuse the OP of a few things.</p>
<p>Not all of non-convex optimisation is difficult; large parts of it are. A key difficulty comes not just from having a large number of local mini... |
1,109,671 | <p>Some things I know:</p>
<ul>
<li>$S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$</li>
<li>$(2,4) \in N_G(S)$</li>
<li>Number of conjugates = $[G: N_G(S)]$</li>
</ul>
<p>This seems like such a easy question but it made me realised that I do not know how to go about thinking about (and finding) the right cosets of $N... | Joe | 107,639 | <p>Consider the conjugates of the element $(1,2,3,4)$: they are all and only the elements of $S_4$ with the same cyclic structure. Hence they are
$$
(1,2,3,4),\;\;(1,4,3,2)\\
(1,2,4,3),\;\;(1,3,4,2)\\
(1,4,2,3),\;\;(1,3,2,4)\\
$$</p>
<p>Now consider the corrispondent cyclic groups: the first couple of element I wrote ... |
2,671,173 | <p>I need to find isometry between two spaces of continuous functions <span class="math-container">$C[a,b]$</span> and <span class="math-container">$C[0,1]$</span>. That means to find function <span class="math-container">$ \phi\colon C[a,b] \longrightarrow C[0,1] $</span> which is bijection and <span class="math-conta... | Aweygan | 234,668 | <p>Define $\psi:[0,1]\to[a,b]$ by $\psi(x)=(b-a)x+a$, and define $\phi:C([a,b])\to C([0,1])$ by $\phi f=f\circ\psi$. That $\phi$ is a bijection follows from the fact that $\psi$ is a bijection: Define $\rho:C([0,1])\to C([a,b])$ by $\rho f=f\circ \psi^{-1}$. Then $\rho\phi f=f\circ(\psi\circ\psi^{-1})=f$ and $\phi\... |
341,728 | <p>Trivially, for any Lie Algebra (LA) g, g':=[g,g] is an ideal. What's wrong with the following argument?</p>
<p>Be g a simple LA, then it has to be g'=g by definition of simple LA. But [g,g]=g seems to be an alternative way of characterizing a semi simple LG. Furthermore, for sl(2) it doesn't seem to be true z=[x,y]... | DonAntonio | 31,254 | <p>I don't think there is a general method for all cases but work each one separatedly. Let us put</p>
<p>$$a:=(123)(465)\;,\;\;b:=(257)(386)$$</p>
<p>Then we clearly have that $\,G:=\langle\,a,b\,\rangle\le A_8\,$ and the following relations hold:</p>
<p>$$a^3=b^3=1\;,\;\;ab=(1246)(3857)\neq(1543)(2867)=ba$$</p>
<... |
3,237,234 | <p>Find the maximum and minimum
value of <span class="math-container">$$\cos 2x + 3\sin x.$$</span></p>
| nmasanta | 623,924 | <blockquote>
<p><strong>Leibniz Integral Rule (Differentiation under the integral sign):</strong></p>
<p>Let <span class="math-container">$f(x, t)$</span> be a function of <span class="math-container">$x$</span> and <span class="math-container">$t$</span> such that both <span class="math-container">$f(x, t)$</span> and... |
222,653 | <p>We have encountered the following problem that we think that should be true. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity.</p>
<p>The question is the following: can we assure that the sequence does <em>NOT</em> converge in distribution to a Poisson? <... | Robert Israel | 13,650 | <p>As Nate Eldridge noted, the answer is no. For a positive result, you need some extra condition. Suppose e.g. the variances $\text{Var}(X_n) < c (E X_n)^2$ for $n$ sufficiently large, with some constant $c \in (0,1)$.
Then $X_n$ can't converge in distribution.</p>
|
33,361 | <p>I have a function like this</p>
<p>$f(x,y)=c\,y\,(y-x),\ \text{for}\ 0<x<2,\;-x<y<x$</p>
<p>and I need to find the value of $c$ such that $f(x,y)$ is a PDF. </p>
<p>How can I do that? </p>
<p>I know that the condition is double integral $f(x,y)=1$, but how can I impose that condition with <em>Mathe... | Dr. belisarius | 193 | <pre><code>s = Solve[Integrate[c y (y - x), {x, 0, 2}, {y, -x, x}] == 1]
Plot3D[c y (y - x) /. s[[1]], {x, 0, 2}, {y, -x, x}, PlotRange -> All, Mesh -> None,
PlotStyle -> Directive[Orange, Specularity[White, 40]], Filling -> Bottom]
(* {{c -> 3/8}} *)
</code></pre>
<p><img src="https://i.stack... |
33,361 | <p>I have a function like this</p>
<p>$f(x,y)=c\,y\,(y-x),\ \text{for}\ 0<x<2,\;-x<y<x$</p>
<p>and I need to find the value of $c$ such that $f(x,y)$ is a PDF. </p>
<p>How can I do that? </p>
<p>I know that the condition is double integral $f(x,y)=1$, but how can I impose that condition with <em>Mathe... | m_goldberg | 3,066 | <p>The double integral gives a very simple value in terms of $c$</p>
<pre><code>Integrate[c y (y - x), {x, 0, 2}, {y, -x, x}]
</code></pre>
<blockquote>
<pre><code>8 c/3
</code></pre>
</blockquote>
<p>so the equation to be solved is a trivial one</p>
<pre><code>8 c/3 == 1
</code></pre>
<p>that we can solve in our ... |
33,361 | <p>I have a function like this</p>
<p>$f(x,y)=c\,y\,(y-x),\ \text{for}\ 0<x<2,\;-x<y<x$</p>
<p>and I need to find the value of $c$ such that $f(x,y)$ is a PDF. </p>
<p>How can I do that? </p>
<p>I know that the condition is double integral $f(x,y)=1$, but how can I impose that condition with <em>Mathe... | ubpdqn | 1,997 | <p>Or just:</p>
<pre><code>1/Integrate[y (y - x), {x, 0, 2}, {y, -x, x}]
</code></pre>
|
236,308 | <p>I solved the following equation the hard way:
$$\sqrt{x+1} +\sqrt{x+33}=\sqrt{x+6} +\sqrt{x+22}$$
The only solution is $x=3$.
I am wondering if there is some easy observation that solves the equation without squaring both sides?</p>
| Matthew Conroy | 2,937 | <p>In general, if an equation of the form
$$ \sqrt{x+a}+\sqrt{x+b} = \sqrt{x+c} + \sqrt{x+d} $$
has a solution, that solution is
$$ x = \frac{t^2-16s^2ab}{8s(2s(a+b)+t)}$$
where $s=a+b-c-d$ and $t=-s^2-4ab+4cd$. A bit messy, but you can be sure that the equation has, at most, one solution. If you find a small intege... |
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