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 |
|---|---|---|---|---|
162,812 | <p>I have an interesting calc question here but im not sure how to solve it. Can someone perhaps give me a helping hand or guide me through steps?</p>
<blockquote>
<p>A balloon that takes images of the earth is shot up in the sky with
rockets from 0 ft off the ground is given by the height of the function s(t)=
... | damagedgods | 33,561 | <p>The rate of change of the objects height with time is the first derivative of the function <em>s(t)</em> this also (intuitively) would be the vertical velocity of the balloon.</p>
<p>When the balloon has a zero velocity it's reached the top of its "curve" - its maximum height.</p>
<p>So if $s(t)=-18t^2+120t$ then ... |
3,082,337 | <p>Forgive my ignorance.<br>
Is the condition <span class="math-container">$x\in\mathbb{R}$</span> necessary to the set statement <span class="math-container">$\{x \in\mathbb{R} \vert x> 0\}$</span>?<br>
In other words, if <span class="math-container">$x$</span> is greater than zero, then is it not, by definition, a... | pwerth | 148,379 | <p>It's definitely necessary. If it weren't, that would imply that
<span class="math-container">$$\{x : x>0\}$$</span>
would always denote <span class="math-container">$\{x\in \mathbb{R}: x> 0\}$</span> but obviously this won't always be the case since it makes sense to write
<span class="math-container">$$\{x\in... |
3,688,829 | <p>We're currently analyzing the convergence of function sequences.</p>
<p>I need to prove
<span class="math-container">$$
\lim_{n \to \infty} \left(1 + \frac{z}{n}\right)^n
$$</span></p>
<p>is not uniformly convergent on <span class="math-container">$\mathbb{C}$</span>. Can I just use the equivalence <span class... | Kavi Rama Murthy | 142,385 | <p>The point-wise limit is <span class="math-container">$e^{z}$</span>. If the convergence is uniform then there exists <span class="math-container">$n_0$</span> such that <span class="math-container">$|(1+\frac z n)^{n} -e^{z}| <1$</span> for all <span class="math-container">$z$</span> whenever <span class="math-co... |
2,685,822 | <p>How can we prove that $L = \lim_{n \to \infty}\frac{\log\left(\frac{n^n}{n!}\right)}{n} = 1$</p>
<p>This is part of a much bigger question however I have reduced my answer to this, I have to determine the limit of $\log(n^{n}/n!)/n$ when $n$ goes to infinity.</p>
<p>Apparently the answer is 1 by wolfram alpha but ... | user | 505,767 | <p>By Stolz-Cesaro</p>
<p>$$\lim_{n \to \infty}\frac{\log\left(\frac{n^n}{n!}\right)}{n} = \lim_{n \to \infty}\frac{\log\left(\frac{(n+1)^{n+1}}{(n+1)!}\right)-\log\left(\frac{n^n}{n!}\right)}{n+1-n} = \lim_{n \to \infty}\log\left(\frac{(n+1)^{n+1}}{(n+1)!}\frac{n!}{n^n}\right)=\\= \lim_{n \to \infty}\log\left(1+\frac... |
101,384 | <p>Calculate the Lebesgue integral of the function</p>
<p>$$ f(x,y)=\left\lbrace\begin{array}{ccl}[x+y]^{2} &\quad&|x|,|y| <12 ,\quad xy \leq 0\\
0 &\quad&\text{otherwise}\end{array} \right.$$</p>
<p>in $\mathbb{R}^2$.</p>
<p>Can anyone help with this? I can't find a way to make the expression of... | AD - Stop Putin - | 1,154 | <p><strong>Hint:</strong></p>
<ol>
<li><p>The function is non-negative, and hence one may apply Tonelli's theorem (sometimes cited as Fubini-Tonelli's or even Fubini' theorem).</p></li>
<li><p>Draw the domain of integration (that is the set where $f(x,y)\ne0$). Split up the domain in order to adopt step 1. </p></li>
<... |
1,231,781 | <p>Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues.
How do I determine the coefficient of the term $x^2$ of the polynomial given by</p>
<p>$q(x) = \det(I_n + xA)$ </p>
| Robert Lewis | 67,071 | <p>When this question was first posted, if I recall correctly, there was a discussion in the comment thread concerning whether or not it is necessary to additionally assume $A$ is diagonalizable in order to obtain (nice) formulas for the coefficients of $x^k$, $0 \le k \le n$, in the polynomial
$q_n(x) = \det(I_n + xA)... |
2,940,072 | <p><span class="math-container">$$\left(\frac{f}{g}\right)'(x_{0})=\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{g^2(x_0)}$$</span></p>
<p>So, <span class="math-container">$\frac{1}{g}.f=\frac{f}{g}$</span>, then <span class="math-container">$$\frac{f}{g}'(x_0)=\frac{f(x)\frac{1}{g(x)}-f(x_0)\frac{1}{g(x_0)}}{x-x_0}=f(x)\frac{\f... | Singh | 83,768 | <p>$(1/g)'\neq 1/g'$ this is where you made the mistake.</p>
|
35,736 | <p>I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical bundle is defined:</p>
<p>$$\mathcal{K}=\mathcal{O}_{X,-\sum D_i}$$</p>
<p>where the $D_i$ are representatives of ... | JRG | 3,912 | <p>Your formula is not quite right for toric varieties. In particular, the sum is not over "representatives of the class group", but over a set of minimal generators for the free group on torus-invariant divisors. Such a set is furnished by the 1-cones in the fan. More precisely,</p>
<blockquote>
<p>Let $X_\Sigma... |
2,347,892 | <p>Let $f$ be a continuous differentiable function in $\mathbb{R}$. If $f(0)=0$ and $|f'(x)|\leq|f(x)|,\;\forall x \in \mathbb{R}$, then $f$ is the null function.</p>
<p>Following @DougM's idea, I think I figured it out:
Consider $f$ restricted to the interval $[0,1]$. Then, by Weirstrass theorem there exists $x_1,x_2... | JJacquelin | 108,514 | <p>given $u_x-6u_y=y$ then $\frac{dx}{1}=\frac{dy}{-6}=\frac{du}{y}\quad$ not $=\frac{du}{u}\quad$(typo).</p>
<p>$y^2+12u=F(6x+y)\quad$ is OK.</p>
<p>Condition $u(x,y)=e^x$ on the line $y=-6x+2\quad\to\quad y^2+12e^x=F(2)=$constant is impossible. Thus, there is no solution.</p>
<p>Condition $u(x,y)=1$ on the line $... |
847,887 | <p>I'm having a very hard time resolving the system of equations after using the Lagrange Multipliers optimization method. For instance:</p>
<p>The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 + y^2 $ over an ellipse. Find the ellipse points that are nearer and farther from the origin.</p>
<p>I know t... | David | 119,775 | <p><strong>Hint</strong>. From my experience of students doing this kind of question, here is the no.1 tip: <strong>do not miss any potential solutions</strong>. For example, the solution of
$$xL_1=yL_1$$
is <strong>not</strong> $x=y$, it is
$$x=y\quad\hbox{or}\quad L_1=0\ .$$</p>
<p>It's hard to give any other gene... |
1,174,433 | <p>I'm trying to prove $\mathbb Z_p^*$ is a group if and only if $p$ is prime. I know that if $p$ is prime $\mathbb Z_p^*$ is a group, but how can I do the converse? In another words, if the equation $ax\equiv 1 (\mod m)$ has solution for every integer $a$ which is not divisible by $m$, then $m$ is prime.</p>
<p>I'm ... | Bill Dubuque | 242 | <p>If $\ \color{#c00}{m = ab},\ a,b > 1\,$ then $\ {\rm mod}\,\ \color{#c00}m\!:\ \color{#0a0}{bc\equiv 1}\,\Rightarrow\,\color{#c00}{0}\equiv (\color{#c00}{ab})c\equiv a(\color{#0a0}{bc})\equiv a,\,$ contra $\,m\nmid a$</p>
<p>Generally, just like $\,b\,$ above, a zero-divisor is not invertible (except in the triv... |
2,907,378 | <p>In my Math book I'm solving a case where this is the situation:</p>
<p>"The demand curve for good X is linear. At a price (p) of 300 the demand is 600 units. At a price of 680 the demand is 220 units. Also the supply curve for good X is linear. If the price is 400 then the supply equals 200 units, whereas for a pri... | Community | -1 | <p>I suspect that this is because of the strange (to me) way these things are defined. </p>
<p>According to the <a href="https://en.wikipedia.org/wiki/Demand_curve" rel="nofollow noreferrer">Demand curve</a> and <a href="https://en.wikipedia.org/wiki/Supply_(economics)" rel="nofollow noreferrer">Supply curve</a> Wikip... |
4,374,739 | <p>Spivak's chapter 6, question 1 asks: for which of the following functions <span class="math-container">$f$</span> is there a continuous function <span class="math-container">$F$</span> with domain <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$F(x) = f(x)$</span> for all <sp... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$x$</span> is an irrational numbers and if <span class="math-container">$\varepsilon>0$</span>, then, in <span class="math-container">$(x-1,x+1)$</span>, there are only finitely many rational numbers which, when written in lowest terms as <span class="math-container">$\frac pq$</sp... |
1,398,464 | <p>Folks - Please help. What's the gradient for the cost function below?</p>
<p>$ D(Y||AX)=\frac{1}{2} ||Y-AX||^2_F $</p>
<p>Additional info - </p>
<p>-need to get the derivative of that with respect to A.
-Multiplicative NMF Algorithms Based on the Squared Euclidean Distance</p>
| uranix | 200,750 | <p>Let
$$
J = \frac{1}{2}\sum_{ij} \left(\sum_k a_{ik}x_{kj} - y_{ij}\right)^2.
$$
Then
$$
\frac{\partial J}{\partial A} =
\frac{\partial J}{\partial a_{lm}} = \sum_{ij} \left(\sum_k a_{ik}x_{kj} - y_{ij}\right) \frac{\partial \left(\sum_k a_{ik}x_{kj} - y_{ij}\right)}{\partial a_{lm}} =\\
= \sum_{ij} \left(\sum_k a_{i... |
4,610,313 | <p>I am working on AoPS Vol. 2 exercises in Chapter 1 and attempting to solve the below problem:</p>
<blockquote>
<p>Given that <span class="math-container">$\log_{4n} 40\sqrt{3} = \log_{3n} 45$</span>, find <span class="math-container">$n^3$</span> (MA<span class="math-container">$\Theta$</span> 1991).</p>
</blockquot... | user2661923 | 464,411 | <p>Alternative approach:</p>
<p>Let <span class="math-container">$~r~$</span> denote <span class="math-container">$\displaystyle \log_{4n} 40\sqrt{3}.$</span></p>
<p>Then</p>
<p><span class="math-container">$$(4n)^r = 40\sqrt{3}, ~~(3n)^r = 45 \implies $$</span></p>
<p><span class="math-container">$$\left[\frac{4}{3}\r... |
254,236 | <p>If a matrix $A$ is diagonalizable, is $A$ invertible? </p>
<p>I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself.</p>
| JLA | 30,952 | <p>No. For instance, the zero matrix is diagonalizable, but isn't invertible.</p>
|
2,586,501 | <p>I studied the Ebbinghaus's logic textbook, which is done under ZF (sometimes is ZFC) set theory, so the Gödel Completeness Theorem is valid in ZF(C) (in my opinion).</p>
<p>When I studied Jech's set theory, it is going to show that</p>
<blockquote>
<p>If ZF is consistent, then so is ZFC+GCH.</p>
</blockquote>
... | Asaf Karagila | 622 | <p>Well. How can you assume that $\sf PA$ is consistent, if by Gödel's work you can prove its not provable? The answer is that adding assumptions makes your theory <em>stronger</em>. And this is also a way to measure the strength of a theory.</p>
<p>So $\sf ZF$ is stronger than $\sf PA$ because it proves that $\sf PA$... |
2,072,866 | <p>My notes say that if $F$ is conservative (i.e. $F=\nabla f$) then $\text{curl }(F)=0$. But I feel this is not quite right.</p>
<p>There is a theorem that says that if $f(x,y,z)$ has continuous second order partial derivatives then $\text{curl }(\nabla f)=0$. (I'm fine with this theorem, it makes sense.) </p>
<p>O... | Teh Rod | 389,818 | <p>When you take the divergence of a function you doing $\left \langle \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} \right \rangle\cdot \bar{F}$ and then when you take the curl of that you're doing another derivative, the second partial derivative. If the second order derivatives ... |
737,054 | <p>Express the vector $\vec{u}$ below as a sum of two vectors $\vec{u}_1$ and $\vec{u}_2$, where $\vec{u}_1$ is parallel to the vector $\vec{v}$ given below, and $\vec{u}_2$ is perpendicular to $\vec{v}$. Make sure that the first vector in your sum is $\vec{u}_1$ and the second is $\vec{u}_2$.</p>
<p>$\vec{v} = [-3,-1... | Kaster | 49,333 | <p>$$
u = u_1 + u_2,\quad u_1 = \alpha v,\quad u_2 \cdot v = 0 \implies (u - \alpha v) \cdot v = 0\implies \alpha = \frac {u \cdot v}{\|v\|^2} \implies \alpha = \frac {22}{11} = 2
$$
Can you take it from here?</p>
|
3,011,216 | <p>If <span class="math-container">$F(x)$</span> is a continuous function in <span class="math-container">$[0,1]$</span> and <span class="math-container">$F(x) = 1$</span> for all rational numbers then <span class="math-container">$F(1/\sqrt2) = 1$</span>. (True/ False)</p>
<p>I think the statement is true because sin... | Joel Pereira | 590,578 | <p>For every <span class="math-container">$n$</span>, there exists a rational number <span class="math-container">$q_n$</span> such that <span class="math-container">$\mid \frac{1}{\sqrt{2}}-q_n\mid < \frac{1}{n}$</span>. So <span class="math-container">$\displaystyle\lim_{n\rightarrow \infty}q_n$</span> = <span cl... |
3,271,507 | <p>Given a ring <span class="math-container">$R$</span> and a multiplicatively closed system <span class="math-container">$S\subset R$</span>, we define the ring <span class="math-container">$S^{-1}R=\{\frac rs | r\in R, s\in S\}$</span>. However if I have <span class="math-container">$R=\Bbb Z$</span> and <span class=... | Matt Samuel | 187,867 | <p>You are misunderstanding what <span class="math-container">$\mathbb Z\setminus 3\mathbb Z$</span> means. It is an unfortunate notation in algebra. What you are confusing it with is <span class="math-container">$\mathbb Z/3\mathbb Z$</span>, which is a quotient rather than a subset of <span class="math-container">$\m... |
3,636,563 | <p>How to find the values of <span class="math-container">$x,y$</span> and <span class="math-container">$z$</span> if
<span class="math-container">$3x²-3(1126)x=96y²+24(124)y=8z²-4(734)z $</span>?</p>
<p>I dont have any idea!! I think we can have many values of <span class="math-container">$x,y$</span> and <span clas... | GEdgar | 442 | <p>The set <span class="math-container">$(0,1]$</span> is the disjoint union
<span class="math-container">$$
(0,1] = \bigcup_{k=2}^{\infty} \left(\frac{1}{k},\frac{1}{k-1}\right]
$$</span>
Therfore, if <span class="math-container">$f$</span> is Lebesgue integrable on <span class="math-container">$(0,1]$</span> we have
... |
2,278,087 | <p>The polynomial is $p(z)=\sum^n_{k=0} a_kz^k$. And I want to prove the following inequality on the unit disk$$\max_{B_1(0)}|p(z)|\geq |a_n|+|a_0|$$</p>
<p>By the maximum modulus principle, the maximum must be on the unit circle and greater than $|a_0|$ by considering $p(0)$. However, I cannot make further conclusion... | Gagar | 170,291 | <p>One gets that
<span class="math-container">$$\max_{B_1(0)}|p(z)|\geq \sqrt{|a_n|^2+|a_{n-1}|^2+ \cdots+|a_0|^2}$$</span>
by calculating
<span class="math-container">$$\mathbb{E} \left[ |p(e^{i \Theta})|^2\right],$$</span>
where <span class="math-container">$\Theta$</span> is a uniform random variable on <span class=... |
3,424,686 | <p>I have a problem, I am uncertain of how to solve:</p>
<p><a href="https://i.stack.imgur.com/DALSn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DALSn.png" alt="enter image description here"></a></p>
<p>Is it enough to show:</p>
<p><span class="math-container">$L_1$</span>: <span class="math-c... | Community | -1 | <p>You know that <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are linear and so proving <span class="math-container">$gf$</span> is linear can be done without using their matrices. </p>
<p><span class="math-container">$$gf(v_1+v_2)=g(f(v_1)+f(v_2))=g(f(v_1))+g(f(v_2))$$</span>
<... |
2,852,248 | <p><em>There are 5 classes with 30 students each. How many ways can a committee of 10 students be formed if each class has to have at least one student on the committee?</em></p>
<p>I figured that we first have to choose 5 people from each class, so there are $10^5$ options. There remain total of 29*5=145 students to ... | Graham Kemp | 135,106 | <p>You are over counting.</p>
<p>Suppose, for a simpler example, you had two classes of three students $\{a,b,c\},\{d,e,f\}$ and have count ways to form distinct committees of three that includes at least one student from each class.</p>
<p>Your method would have you count ways to pick one from $\{a,b,c\}$, one from ... |
983,923 | <p>Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$,
$$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective.</p>
<p>But why for the inverse, suppose that $E$ and $F$ is the subset of $D$,
$$f^{-1}(E \cap F) = f^{-1}(E) \cap f^{-1}(F)$$
without saying that the inverse functi... | Jacob.Lee | 142,729 | <p>proof: $$x\in f^{-1}(E \cap F)\Leftrightarrow f(x)\in E\cap F\\\Leftrightarrow f(x)\in E \ and \ f(x)\in F\\\Leftrightarrow x\in f^{-1}(E) \ and \ x\in f^{-1}(F) \\
\Leftrightarrow x\in f^{-1}(E)\cap f^{-1}(F)\\$$
so the equality holds.</p>
|
177,643 | <p>Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?</p>
| lab bhattacharjee | 33,337 | <p>We have
$$
\tan(A+B+C)=\tan(A+(B+C))=\frac{\tan A+\tan(B+C)}{1-\tan A \tan(B+C)}=
$$
$$
\frac{\tan A+\frac{\tan B+\tan C}{1-\tan B \tan C}}{1-\tan A\frac{\tan B+\tan C}{1-\tan B\tan C}}=
\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A}
$$
In the last step we multiplied the num... |
2,586,466 | <p>Write the value of $n$ if the sum of n terms of the series $1+3+5+7...n =n^2$.</p>
<p>I'm not getting the right value if I proceed with the general formula for finding sum of n terms of a arithmetic series. The general summation formula for arithmetic series is $\frac{n(2a+(n-1)d)}{2}$, where $a$ is the first term,... | Peter | 82,961 | <p>Since we have $$1+3+5+7+\cdots (2n-1)=n^2$$ and we want to have $$1+3+5+7+\cdots +n=n^2$$ we must have $2n-1=n$ , which implies $n=1$</p>
|
1,595,297 | <p>Ordinary cohomology of topological space $X$ are known to be the cohomology of constant sheaf. </p>
<p><strong>Question</strong> Is there analogous description for equivariant cohomology?</p>
<p>More precisely. Consider category of $G$ equivariant sheaves $\mathcal{Sh}_G (X)$. Denote by $\Gamma_G := Inv \circ \Gam... | quinque | 167,762 | <p>The answer is no.
There is an counterexample. $G = X = U(1)$. More precisely $X$ is principal homogeneous space for $U(1)$.</p>
<p>If group acts freely then equivariant cohomology are equal to ordinary homology of quotient space. In this case the quotient space is just a point. Then $H_G^1 (X)= 0$.</p>
<p>On the o... |
654,089 | <p>The book "A First Course in Algebra" says</p>
<blockquote>
<p>In a finite dimensional vector space, every <strong>finite</strong> set of vectors spanning the space contains a subset that is a basis.</p>
</blockquote>
<p>All that is fine. But what about a span having an infinite number of vectors? Surely that too... | pjs36 | 120,540 | <p>It is certainly true, but not a trivial corollary like I'd originally thought. Typically the development of the notion of "dimension" goes something like this:</p>
<p>Talk about linear combinations, independence, and spanning sets. Show all linearly independent spanning sets have the same cardinality, call this the... |
23,639 | <p>Prove that for every positive integer $x$ of exactly four digits, if the sum of digits is divisible by $3$, then $x$ itself is divisible by 3 (i.e., consider $x = 6132$, the sum of digits of $x$ is $6+1+3+2 = 12$, which is divisible by 3, so $x$ is also divisible by $3$.)</p>
<p>How could I approach this proof? I'm... | yunone | 1,583 | <p>Suppose you have a number whose decimal digits are represented $a$, $b$, $c$, and $d$, so $x=abcd$. </p>
<p>In base $10$, this means
$$
x=abcd=a\cdot 10^3+b\cdot 10^2+c\cdot 10^1+d\cdot 10^0.
$$
Try looking at $x$ modulo $3$, and remember that $10\equiv 1\pmod{3}$. </p>
<p>This concept is easily extended to an i... |
80,432 | <p>I know Mathematica's if format is </p>
<pre><code>If[test, then result, else alternative]
</code></pre>
<p>For example, this</p>
<pre><code>y:=If[RandomReal[]<0.2, 1, 3.14]
</code></pre>
<p>would take a random real number between $0$ and $1$, and evaluate it. If it's less than $0.2$, it'll map <code>y</code> ... | kglr | 125 | <p><strong>Update 2:</strong> Using <code>WeightedData</code>, <code>EmpiricalDistribution</code>, <code>Randomvariate</code></p>
<pre><code>ClearAll[wdF]
wdF[t_, v_, n_: 1] := Module[{d = EmpiricalDistribution[
WeightedData[v, Differences[Join[{0}, t, {1}]]]]},
RandomVariate[d, n]]
</code></pre>
<p>Examples:<... |
1,370 | <p>This is in reference to the recent <a href="https://mathematica.stackexchange.com/q/56829/3066">PlotLegends -> “Expressions”</a> question.</p>
<p>I filed a repot on this behavior with WRI tech support. A little later rcollyer posted an answer explaining it was new (undocumented) feature. This morning I received an... | Szabolcs | 12 | <p><strong>I think we should not use the <a href="https://mathematica.stackexchange.com/questions/tagged/bugs" class="post-tag" title="show questions tagged 'bugs'" rel="tag">bugs</a> tag for inaccuracies in the documentation. It should be reserved for situation when there <em>clearly</em> is a bug in the softw... |
1,687,206 | <p>Do I multiply $4$ to the power of $2$ or do I multiply $8$ to the power of $2$? What is the answer? I'm thinking it's $76$.</p>
| Bobson Dugnutt | 259,085 | <p>You have $$3 \times 4+2 \times 4^2=12+2 \times 16=12+32=44$$</p>
|
2,556,255 | <p>For work I have been doing a lot of calculations which look sort of like summing terms similar to $\frac{A}{1+x}$ and $\frac{B}{1+y}$ for some $A, B$ and small values of $0 \leq x, y \leq 0.1$. In my experience I have found that this is approximately equal to $\frac{A + B}{1 + \frac{A}{A+B}x + \frac{B}{A+B}y }$. The... | Laurens Janssen | 511,396 | <p>Using the method suggested by @Karn worked for me. That is, use the Maclaurin series
$$
\frac{A}{1+x} = A( 1 - x + x^2 - x^3 \cdots)
$$
for $|x| < 1$. Apply this to both elements of the sum and discard all values of order $x^2$ and $y^2$ or higher. Then you can multiply the $x$ and $y$ terms by $\frac{A+B}{A+B}$ ... |
243,849 | <p>I was working on an examples from my textbook concerning transforming formulae into disjunctive-normal form (DNF) until I found an expression that I cannot solve. I hope somebody can help me transform the following statement into DNF:</p>
<p>$$ (\lnot q \lor r) \land ( q \lor \lnot r)$$</p>
| amWhy | 9,003 | <p>$$ (\lnot q \lor r) \land ( q \lor \lnot r)\tag{1}$$
$$[(\lnot q \lor r) \land q] \lor [(\lnot q \lor r) \land \lnot r]\tag{2}$$
$$[(\lnot q \land q) \lor (r \land q)] \lor [(\lnot q \land \lnot r) \lor (r \land \lnot r)]\tag{3}$$
$$ \text{False} \lor (r \land q) \lor (\lnot q \land \lnot r) \lor \text{False}\tag{4}... |
4,578,898 | <blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$f$</span> be the function defined on <span class="math-container">$[0,1]$</span> by
<span class="math-container">$$
f(x)=
\begin{cases}
n(-1)^n & \textrm{if }\frac{1}{n+1}<x\leq \frac{1}{n} \\
0 & \textrm{otherwise}
\en... | Ethan Bolker | 72,858 | <p>The English and mathematics in the question as quoted is close to nonsense, but I think I know what the author meant to say.</p>
<blockquote>
<p>The <span class="math-container">$n$</span>th root of <span class="math-container">$m$</span> is the integer <span class="math-container">$k$</span> that minimizes <span cl... |
145,393 | <p>Let $\Omega \subset R^n$ be a bounded region with Lipschitz boundary. Is the trace operator $T: W^{1,1}(\Omega)\rightarrow L^1(\partial \Omega)$ surjective? If not, what is the image? </p>
| username | 40,120 | <p>It is surjective for a $C^1$ boundary, see <a href="https://link.springer.com/content/pdf/bfm%3A978-1-4471-2807-6%2F1.pdf" rel="nofollow noreferrer">Demengel & Demengel</a>, section 3.3. Following the proof will probably answer your question. </p>
|
145,393 | <p>Let $\Omega \subset R^n$ be a bounded region with Lipschitz boundary. Is the trace operator $T: W^{1,1}(\Omega)\rightarrow L^1(\partial \Omega)$ surjective? If not, what is the image? </p>
| Piotr Hajlasz | 121,665 | <blockquote>
<p>The trace operator $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$ is
surjective.</p>
</blockquote>
<p>This is a classical result of Gagliardo [1], see Theorem 18.13 in [2].</p>
<p>In other words for every function $g\in L^1(\partial\Omega)$ there is a function $Eg\in W^{1,1}(\Omega)$ such that $T(Eg)=... |
2,305,474 | <p>Testing a method with the use of C.-H. theorem for <a href="https://math.stackexchange.com/questions/2303997/square-root-of-matrix-a-beginbmatrix-1-2-34-endbmatrix/2304179#2304179">finding square roots</a> of real $ 2 \times 2 $ matrices I have noticed that some matrices probably don't have their square roots with... | drhab | 75,923 | <p>Let $S$ denote the set of elements that are <strong>not</strong> clusterpoints of sequence $(x_n)_n$ and let $s\in S$. </p>
<p>By definition an open set $U_s$ exists such that $\{x_n\in U_s\mid n\in\mathbb N\}$ is finite. </p>
<p>But actually this shows that <strong>every</strong> element of $U_s$ is not a cluster... |
2,305,474 | <p>Testing a method with the use of C.-H. theorem for <a href="https://math.stackexchange.com/questions/2303997/square-root-of-matrix-a-beginbmatrix-1-2-34-endbmatrix/2304179#2304179">finding square roots</a> of real $ 2 \times 2 $ matrices I have noticed that some matrices probably don't have their square roots with... | Umberto P. | 67,536 | <p>Denote the set of cluster points of a set $E$ by $E'$. </p>
<p>If $\{x_n\}''$ is empty, then $\{x_n\}'$ is closed.</p>
<p>Suppose $x \in \{x_n\}''$. Let $U$ be an open set containing $x$.</p>
<p>By definition, $U$ contains infinitely many points of $\{x_n\}'$. Let $y$ be one such point.</p>
<p>Then $U$ is an op... |
4,306,774 | <blockquote>
<p>There is exactly one <strong>countable</strong> model (upto isomorphism) of the first order theory of <span class="math-container">$(\mathbb Q,<)$</span>.</p>
</blockquote>
<p>I am reading about spectrum of complete theories where the spectrum of various theories are stated (but not proved). I am loo... | Peter Smith | 35,151 | <p>But that initial claim [before it was edited] isn't true. You know from the upward Lowenheim-Skolem theorem that any theory with a countable model also has models of all larger cardinalities (which therefore won't be isomorphic with the countable model).</p>
<p>What <em>is</em> true is that all <em>countable</em> mo... |
4,306,774 | <blockquote>
<p>There is exactly one <strong>countable</strong> model (upto isomorphism) of the first order theory of <span class="math-container">$(\mathbb Q,<)$</span>.</p>
</blockquote>
<p>I am reading about spectrum of complete theories where the spectrum of various theories are stated (but not proved). I am loo... | Sahiba Arora | 266,110 | <p>The statement is proved in Theorem 2.5 <a href="https://www.math.uni-hamburg.de/home/geschke/teaching/ModelTheory.pdf" rel="nofollow noreferrer">here</a>.</p>
|
1,350,030 | <blockquote>
<p>Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal polynomial over $F$? </p>
</blockquote>
<p>For reference, I explain my way. (From here forward, all field iso... | P Vanchinathan | 28,915 | <p>Let <span class="math-container">$\alpha$</span> be an algebraic number (over the rationals) whose degree is a prime number <span class="math-container">$p>2$</span>. Let <span class="math-container">$f(x)$</span> be a polynomial with integer/rational coefficients of degree less than <span class="math-container"... |
1,733,923 | <p>I'm working with the following sum and trying to determine what it converges to: $$\sum_{k=1}^{\infty}\ln\left(\frac{k(k+2)}{(k+1)^2}\right)$$</p>
<p>Numerically I see that it seems to be converging to $-\ln(2)$, however I can't see why that is the case. I have expanded the logarithm and expressed the sum as $\sum_... | robjohn | 13,854 | <p>$$
\begin{align}
\sum_{k=1}^n\log\left(\frac{k(k+2)}{(k+1)^2}\right)
&=\sum_{k=1}^n\log(k)+\sum_{k=1}^n\log(k+2)-2\sum_{k=1}^n\log(k+1)\\
&=\sum_{k=1}^n\log(k)+\sum_{k=3}^{n+2}\log(k)-2\sum_{k=2}^{n+1}\log(k)\\[3pt]
&=\log(1)-\log(2)+\log(n+2)-\log(n+1)\\[6pt]
&=-\log(2)+\log\left(\frac{n+2}{n+1}\rig... |
1,733,923 | <p>I'm working with the following sum and trying to determine what it converges to: $$\sum_{k=1}^{\infty}\ln\left(\frac{k(k+2)}{(k+1)^2}\right)$$</p>
<p>Numerically I see that it seems to be converging to $-\ln(2)$, however I can't see why that is the case. I have expanded the logarithm and expressed the sum as $\sum_... | G Cab | 317,234 | <p>In another way:
$$
\eqalign{
& \sum\limits_{k\, = \,1}^n {\log \left( {{{k\left( {k + 2} \right)} \over {\left( {k + 1} \right)^2 }}} \right)} = \log \prod\limits_{k\, = \,1}^n {{{k\left( {k + 2} \right)} \over {\left( {k + 1} \right)^2 }}} = \cr
& = \log \left( {\prod\limits_{k\, = \,1}^n {{k \ove... |
2,558,646 | <p>Okay, so this is bugging me now.</p>
<p>I know this:</p>
<p>while tg angle = Y / X</p>
<p>for y > 0, angle is an element of < 0, PI ></p>
<p>for y < 0, angle is an element of < PI, 2PI ></p>
<p>for y = 0, angle is 0 or PI or 2PI.</p>
<p>Okay, but how to determine what out of those 3 values angle actua... | user | 505,767 | <p>You have to <strong>show that</strong></p>
<p>$$\frac{\frac{f(x)}{g(x)}}{\frac{a}{b}x^{n-m}} \to 1$$</p>
<p>as $x\to \infty$</p>
|
80,242 | <p>Let $K_0$ be a bounded convex set in $\mathbb{R}^n$ within which lie two sets $K_1$ and $K_2$. $K_0,K_1,K_2$ have nonempty interior. Assume that,</p>
<ol>
<li>$K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$.</li>
<li>The boundary between $K_1$ and $K_2$ is unknown. (To avoid the trivial case, I assume that the bounda... | Michael Biro | 19,029 | <p>Without more constraints on the problem, there is no deterministic algorithm that can decide which is convex with a finite number of queries. For randomized algorithms, I think that you can never get a finite expected number of queries.</p>
<p>Take $K_0$ to be a sphere. Run your algorithm, and let every query respo... |
2,916,600 | <p>So I understand the requirements for an orthonormal basis and everything around it. However, there's one thing I am missing:</p>
<p>Suppose you have two vectors which are orthonormal $u_1$ and $u_2$. According to the answerbook the multiplication of vector $u_1$ and $u_2$ results in another orthonormal vector $u_3$... | mechanodroid | 144,766 | <p>Yes, you can check it by direct calculation.</p>
<p>Assume that $u = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}$ and $v = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$ are orthonormal vectors. Their cross product is defined as
$$u \times v = \begin{bmatrix} u_2v_3 - u_3v_2 \\ u_3v_1-u_1v_3 \\ u_1v_2 - u_2v_... |
163,870 | <p><strong>Bug introduced in or after 10.3, persisting through 11.2.</strong></p>
<hr>
<p>I'm trying to solve following PDE (heat equation):</p>
<p>$$ \begin{cases} u_t = a \, u_{xx} \\ u(x,0) = 0\\ \lim_{x\to \infty}u(x,t) =0\\
\alpha\, (\theta_0-u(0,t))+\dot{q}_0=-\lambda u_x(0,t) \end{cases}$$</p>
<p>Where basi... | Ulrich Neumann | 53,677 | <h1>Analytical solution seems to be wrong!</h1>
<p>Just try</p>
<pre><code>heqn = Derivative[0, 1][u][x, t] == a Derivative[2, 0][u][x, t];
ic1 = u[x, 0] == 0;
ic2 = \[Alpha] (\[Theta]air - u[0, t]) +qp0 == -\[Lambda] Derivative[1, 0][u][0, t];
sol = DSolve [{heqn, ic1, ic2}, u[x, t], {x, t}][[1]]
(*{u[x, t] -> -(... |
163,870 | <p><strong>Bug introduced in or after 10.3, persisting through 11.2.</strong></p>
<hr>
<p>I'm trying to solve following PDE (heat equation):</p>
<p>$$ \begin{cases} u_t = a \, u_{xx} \\ u(x,0) = 0\\ \lim_{x\to \infty}u(x,t) =0\\
\alpha\, (\theta_0-u(0,t))+\dot{q}_0=-\lambda u_x(0,t) \end{cases}$$</p>
<p>Where basi... | xzczd | 1,871 | <h1>Semi-analytic Solution</h1>
<p>The approach in the linked post can be used for solving your problem analytically. We just need an extra step i.e. making Laplace transform:</p>
<pre><code>Clear@"`*"
heqn = D[u[x, t], t] == a D[u[x, t], {x, 2}];
ic1 = u[x, 0] == 0;
ic2 = α (θair - u[0, t]) + q == -λ Derivative[1, 0... |
557,680 | <blockquote>
<p>Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Suppose $A\in M_{k,n}$ is such that $AA^t=I_k$. Let $f:M_{k,n}\rightarrow S_k$ be the map $f(B)=BA^t+AB^t$. Prove that $f$ is onto (surjective). (Not... | Community | -1 | <p>In fact it is sufficient that $A$ has maximal rank, that is, since $n\geq k$, $rank(A)=k$. Consequently $AA^T$ is SPD and we may assume that $AA^T=D=diag((\lambda_i)_i)$ where $\lambda_i>0$. We want to find a solution in $B$ of $C=BA^T+AB^T$. Put $B=EA$. Then we seek a $k\times k$ matrix $E$ s.t. $C=E(AA^T)+(AA^T... |
2,687,375 | <p>Let <span class="math-container">$x_1, x_2, \ldots, x_n$</span> be a random sample from the Bernoulli (<span class="math-container">$\theta$</span>).</p>
<p>The question is to find the UMVUE of <span class="math-container">$\theta^k$</span>.</p>
<p>I know the <span class="math-container">$\sum_1^nx_i$</span> is the ... | P.K | 539,714 | <p><strong>Compactness</strong> : In general, removing a point does not preserve compactness.</p>
<p>See that <span class="math-container">$[0,1]$</span> is compact but <span class="math-container">$(0,1]$</span> is not compact.</p>
<p>You can consider sequence <span class="math-container">$\left(\frac{1}{n}\right)$</s... |
3,866,673 | <p>I’m working on my linear algebra assignment and now struggling to solve this problem, say, on the complex field, find the Jordan canonical form of
<span class="math-container">$$A=\begin{bmatrix}\quad &\quad &\quad & a_1\\ \quad & \quad & a_2 & \quad \\ \quad & \dots & \quad & \qu... | sirous | 346,566 | <p>The <span class="math-container">$(k-1)th$</span> term of LHS is:</p>
<p><span class="math-container">$$4(k-1+1)-3=4k-3$$</span></p>
<p>So LHS can be written as:</p>
<p><span class="math-container">$$[1+5+9+ \cdot \cdot\cdot +4k-3]+4(k+1)-3$$</span></p>
|
4,345,182 | <p>Let <span class="math-container">$f \in L^1(\mathbb{R})$</span> be an integrable function on the real line.</p>
<p>Let <span class="math-container">$ p = x^m + a_1 x^{m-1} + \cdots + a_m \in \mathbb{R}[x] $</span> be a real polynomial of degree <span class="math-container">$m$</span>.</p>
<p>Consider the function <s... | Antonio Maria Di Mauro | 736,008 | <p>First of all we observe that the period <span class="math-container">$T$</span> can be suppose not negative without loss of generality: indeed if <span class="math-container">$T$</span> was not positive then <span class="math-container">$-T$</span> would be not negative and it would be such that
<span class="math-co... |
3,291,975 | <p>Transpose this formula to make <span class="math-container">$y$</span> the subject.</p>
<p><span class="math-container">$$x=\sqrt{x^2y^2+1-y}$$</span></p>
<p>My try:</p>
<p><span class="math-container">$$x^2=x^2y^2+1-y$$</span></p>
<p><span class="math-container">$$x^2-x^2y^2=1-y$$</span></p>
<p><span class="ma... | egreg | 62,967 | <p>If you can't see the factorization, treat the thing as a quadratic in <span class="math-container">$y$</span>:
<span class="math-container">$$
x^2y^2-y+(1-x^2)=0
$$</span>
The discriminant is
<span class="math-container">$$
1-4x^2(1-x^2)=4x^4-4x^2+1=(2x^2-1)^2
$$</span>
Apply the quadratic formula:
<span class="math... |
452,889 | <p>My question relates to the conditions under which the spectral decomposition of a nonnegative definite symmetric matrix can be performed. That is if $A$ is a real $n\times n$ symmetric matrix with eigenvalues $\lambda_{1},...,\lambda_{n}$, $X=(x_{1},...,x_{n})$ where $x_{1},...,x_{n}$ are a set of orthonormal eigenv... | Vedran Šego | 78,926 | <p>Matrix square root can be defined in many ways. If you just want $X$ such that $X^2 = A$, you approach is good.</p>
<p>However, the <em>principal square root</em> is defined only for the matrices with no strictly negative eigenvalues and zero being at most nonderogatory eigenvalue (which is unimportant here, since ... |
169,126 | <p>I was trying to prove that $-(x + y) = -x - y$ and as you can see in the image below, I took the liberty of using the $-$ symbol as a number and applying the associative law with it. Is it kosher in all rigorousness given the axioms professional mathematicians use?
<img src="https://i.stack.imgur.com/SMjqb.png" alt=... | fretty | 25,381 | <p>Well the LHS of your equation is just saying "the additive inverse of $x+y$".</p>
<p>So all you have to show is that the additive inverse of $x+y$ really is the RHS of the equation, i.e. $-x-y$, then by uniqueness of inverses in a group the two must be equal.</p>
|
669,696 | <p>How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we let $a_n = (2n^{1/n}+1)^n.$ This yields $\sum\limits_{n=1}^\infty\sqrt[n]{(2n^{1/n}+1)^n}$, which simplifies to $\sum\l... | Yiorgos S. Smyrlis | 57,021 | <p>Note that
$$
2n^{1/n}+1\ge 3,
$$
and thus
$$
(2n^{1/n}+1)^n\ge 3^n\ge 3,
$$
which implies that the series $\sum_{n=1}^\infty (2n^{1/n}+1)^n$ diverges to infinity, because if a series $\sum_{n=1}^\infty a_n$ converges, then $a_n\to 0$.</p>
<p>Note. If instead we had $\sum_{n=0}^\infty(2n^{1/n}-1)^n$, we would still ... |
710,518 | <p>I'm having a brainfart while trying to solve a problem for differential equations that requires me to recall some Calculus. If I have $y' = f(t, y) = 1 - t + 4y$, what is $y''$? Do I just differentiate with respect to $t$ to get $y'' = -1$?</p>
| Marc | 132,141 | <p>\begin{equation}
y'' = \frac{d^2}{dt^2}y = \frac{d}{dt}y' = \frac{d}{dt} (1-t) + 4\frac{d}{dt} y = -1 + 4y' = -1+4-4t+16y = 3 - 4t + 16y
\end{equation}</p>
|
3,997,321 | <p>Let <span class="math-container">$X$</span> be a set, <span class="math-container">$\tau_1,\tau_2$</span> two topologies on <span class="math-container">$X$</span>, and consider the following statements</p>
<ol>
<li><span class="math-container">$\tau_1\subseteq \tau_2$</span> (i.e <span class="math-container">$\tau_... | Henno Brandsma | 4,280 | <p>Suppose (2) holds. Let <span class="math-container">$O \in \tau_1$</span>. Then <span class="math-container">$X\setminus O$</span> is closed in <span class="math-container">$\tau_1$</span>, and it's also closed in <span class="math-container">$\tau_2$</span>: let <span class="math-container">$x$</span> be in the clo... |
960,880 | <p>Could you help me to explain how to find the solution of this equation
$$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}... | cjferes | 89,603 | <p>Reordering,
$$y'(t)+y(t)=\frac{1}{2}e^{-2t}+\frac{1}{2}$$</p>
<p>This DE is of the form
$$y'(t)+P(t)y(t)=Q(t)$$
with $P(t)=1$ and $Q(t)=\frac{1}{2}e^{-2t}+\frac{1}{2}$.</p>
<p>Then, use the integrating factor
$$M(t)=e^{\int_{t_0}^tP(x)\,dx}=e^{\int_{t_0}^t1\,dx}=e^{t-t_0}=e^{t-0}=e^t$$</p>
<p>So, in our DE,
$$\... |
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 ... | Mundron Schmidt | 448,151 | <p>I suppose you have a measure space $(\Omega,\mathcal A, \mu)$ and you like to prove the monotonicity of the measure $\mu$?<p>
Reading your arguments, I suppose you consider $\Omega=\mathbb R$ and you restrict yourself to intervals $A,B\subset \mathbb R$. But the monotonicity of a measure comes directly from the defi... |
1,671,572 | <p>$\|A\vec{x}\|\leq\|A\|\space\|\vec{x}\|$ where $A$ is a $m\times n$ matrix and $\vec{x}$ is a n-dimensional column vector. Assume that $\|A\|=\sqrt{\Sigma_{i}\Sigma_{j}a_{ij}^{2}}$</p>
| Daniel Akech Thiong | 169,316 | <p>$\sup_{\|y\|\leq 1}\|Ay\| = \|A\|$, which implies that $\|A\| \geq \|Ay\|$ whenever $ \|y\| \leq 1$. This is true for $y= \frac{x}{\|x\|}$. Use one of the properties of a norm and you are done.</p>
|
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... | rschwieb | 29,335 | <p>A ring is called a <strong>left duo ring</strong> if every left ideal is also a right ideal. (It's called a duo ring if every one-sided ideal is two-sided.) There are many rings which are duo on one side but not commutative.</p>
<p>As already discussed, division rings and finite products of them are duo rings. </p>... |
214,832 | <p>Say I have the function: </p>
<pre><code>dep = TextStructure["He wrote a book. I read the book he wrote.",
"DependencyStrings", PerformanceGoal -> "Speed"]
</code></pre>
<p>Which outputs:</p>
<pre><code>{"(wrote, 2)((nsubj, (He, 1)), (dobj, (book, 4)((det, (a, 3)))))", \
"(read, 2)((nsubj, (I, 1)), (dobj, (... | user1066 | 106 | <p>A Regex attempt:</p>
<pre><code>StringCases[#,
RegularExpression["[(]nsubj,\s+[(]([^,]+)"] :> "$1"]&/@dep
</code></pre>
<blockquote>
<p>{{He}, {I, he}}</p>
</blockquote>
<pre><code>StringCases[#,
RegularExpression["^[(]([^,]+)"] :> "$1"]&/@dep
</code></pre>
<blockquote>
<p>{{wrote}, {read... |
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$ ... | jdods | 212,426 | <p>Usually, $\int d(f(x))=\int f'(x) dx=f(x)$ since $df(x)=f'(x)dx$. Your work isn't wrong for when $x>0$, and you would just split it up if the region of integration included both positive and negative $x$ values.</p>
<p>A simpler way to look at it is that $\int x d(x^2)=\int 2x^2 dx =2x^3/3$ (as long as $x\geq0$)... |
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... | Robert Lewis | 67,071 | <p>Our OP El Menduko's proof looks fine to me.</p>
<p>As per his (I assume the masculine is <em>apropos</em> here, based on the appearance of "El" in the OP's user name.) request, here is a shorter proof, as indicated in the comments:</p>
<p>If $A$ is invertible, there is a matrix $B$ with</p>
<p>$AB = I; \tag 1$</p... |
1,457,478 | <p>I'm trying to teach myself some number theory. In my textbook, this proof is given:</p>
<blockquote>
<p><strong>Example (2.3.1)</strong> Show that an integer is divisible by 3 if and only if the sum of its digits is a multiple of 3.</p>
<p>Let <span class="math-container">$n=a_0a_1\ldots a_k$</span> be the decimal r... | fleablood | 280,126 | <p>n = $a_0$ + .... + $a_n$ mod (3) means that n and the sum of the digits will be equivalent to the same number modulo 3. If this number is 0 then n and the sum of the digits will both be divisible by 3. If the number isn't 0 (or any other multiple of 3) neither n nor the sum of the digits will be divisible by 3.</... |
730,357 | <p>A group of order 48 must have a normal subgroup of order 8 or 16 .<br>
Solution:Let G be a group of order n.<br>
Let H be a normal subgroup of G.<br>
Then G/H is a group.<br>
Then by Lagrange's Theorem o(G/H)=o(G)/o(H)<br>
So in this case order of G is 48 and divisors of 48 are 8 and 16.<br>
so a group of order 48 ... | Mark Bennet | 2,906 | <p>Your method is incorrect. For example, the group $A_5$ of order $60$ has no non-trivial normal subgroups, but I could use your argument to show that it must have a normal subgroup of order $2$ or $4$.</p>
|
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 ... | Pavel Čoupek | 82,867 | <p>The statement of the theorem holds even for $k=\mathbb{R}$. See the article</p>
<p>Białynicki-Birula, A., Rosenlicht, M.: <a href="http://www.jstor.org/stable/2034464" rel="nofollow">Injective Morphisms of Real Algebraic Varieties</a>.</p>
|
3,600,868 | <p>Ellipse can be
<a href="https://math.stackexchange.com/q/3594700/122782">perfectly packed with <span class="math-container">$n$</span> circles</a>
if </p>
<p><span class="math-container">\begin{align}
b&=a\,\sin\frac{\pi}{2\,n}
\quad
\text{or equivalently, }\quad
e=\cos\frac{\pi}{2\,n}
,
\end{align}</span> <... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/RiOYS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RiOYS.png" alt="enter image description here"></a></p>
<p>An example of the right triangle, which
<a href="https://en.wikipedia.org/wiki/Mandart_inellipse" rel="nofollow noreferrer">Mandart inellipse</a>
can ... |
207,399 | <p><em>Let $A$ and $B$ be non-empty sets, and let $f\,:\,A\rightarrow B$ be a function.</em>
<br/></p>
<hr/>
<blockquote>
<p>$ \color{darkred}{\bf Theorem}$: The function $f$ is injective if and only if $f\circ g=f\circ h$ implies $g=h$ for all functions $g,h:\,Y\rightarrow A$ for all sets Y. ($f\,:\,A\, \rightarr... | Asaf Karagila | 622 | <p>There is a simple way to prove this by contrapositive.</p>
<p>Assume the function is not injective, and find a counterexample. To find it use the fact that there are $u,w\in A$ such that $f(u)=f(w)$ and create two functions which behave differently on those values.</p>
<blockquote class="spoiler">
<p>Define $h_u... |
1,048,644 | <p>$$\sum_{i=1}^{\infty} \frac {(-1)^{i+1}\cdot 1\cdot 4 \cdot 7 \cdots (3i-2)}{i!2^i}$$</p>
<p>By the alternating series test and the ratio test, I found that this series does not <em>absolutely converge</em>. However, I'm not at all sure how to figure out whether it converges conditionally or diverges.</p>
| MathGod | 101,387 | <p>Let, <span class="math-container">$$\text{I(n)}=\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$$</span> </p>
<p>and</p>
<p><span class="math-container">$\text{J}= \text{I(n) - I(n-1)}=\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\sin^2nx-\sin^2(n-1)x}{\sin^2 x} \text{d}x$</span> </p>... |
396,717 | <p>It's not difficult to see that <span class="math-container">$S^{2n}$</span> doesn't admit a Lie group structure. Since if <span class="math-container">$S^{2n}$</span> admit a Lie group structure, then there exists a left invariant vector field. While the Hairy ball theorem says that there exists no continuous tangen... | Connor Malin | 134,512 | <p>Moishe Kohan's comments (<a href="https://mathoverflow.net/questions/396717/prove-that-s2n-doesnt-admit-topological-group-structure-only-by-hairy-ball#comment1015717_396717">1</a> <a href="https://mathoverflow.net/questions/396717/prove-that-s2n-doesnt-admit-topological-group-structure-only-by-hairy-ball#comment1015... |
39,654 | <p>$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$</p>
<p>I've let $u=x^2+3$ but can't seem to get the right answer.</p>
<p>Really not sure what to do.</p>
| Community | -1 | <p>Or please try putting $x = \sqrt{3} \tan\theta$. Then you have $dx = \sqrt{3}\cdot\sec^{2}\theta \ d\theta$. So you have your integral as:</p>
<p>\begin{align*}
\int\frac{1}{x^{2}+3} \ \text{dx} &= \int \frac{\sqrt{3} \cdot \sec^{2}\theta}{3 (1+\tan^{2}\theta)} \ \text{d}\theta \\ &=\frac{1}{\sqrt{3}} \int ... |
3,877,319 | <p>I have been thinking about finding an explicit formula for the tribonacci numbers, where, namely:</p>
<p><span class="math-container">$$a_n = a_{n-1}+a_{n-2}+a_{n-3}$$</span></p>
<p>and <span class="math-container">$a_1 = 0, a_1 = 1, a_2 = 1.$</span> Obviously, these beginning terms can be shifted, but we'll leave t... | Will Jagy | 10,400 | <p>The real root of <span class="math-container">$x^3 - x^2 - x - 1$</span> is
As in comments, the formula is
<span class="math-container">$$ A \alpha^n + B \beta^n + \bar{B} \bar{\beta}^n $$</span></p>
<p>with</p>
<p><span class="math-container">$$ \alpha = \frac{ 1 + \sqrt[3]{19 + \sqrt{297}}+ \sqrt[3]{19 - \sqrt{297... |
3,193,107 | <p>As the title says, I need to find the explicit form of the recursive sequence defined above, and I am very stuck on this.</p>
| 1123581321 | 444,046 | <p>Consider the difference between consecutive terms and take their sum:</p>
<p><span class="math-container">$b_k - b_{k-1} = 2 \times 3^k$</span></p>
<p>Summing over the left hand side gives:</p>
<p><span class="math-container">$\sum_{k = 2}^n (b_k - b_{k-1}) = (b_2 - b_1) + (b_3 - b_2) + ... + (b_n - b_{n-1}) = b_... |
284,507 | <p>In the textbook "Topology without tears" I found the definition.</p>
<p>$(X, \tau)$ is diconnected iff there exists open sets $A,B$ with $X = A \cup B$ and $A \cap B = \emptyset$.</p>
<p>In Walter Rudin: Principles of Analysis, I found.</p>
<p>$E \subseteq X$ is connected iff it is not the union of two nonempty s... | Hagen von Eitzen | 39,174 | <p>First, note that one should (in both versions) add that $A,B$ should be <em>nonempty</em>.</p>
<p>If $A,B$ are open and disjoint, then also $\overline A$ and $B$ are disjoint as $\overline A$ is the intersection of all closed sets containing $A$, thus $\overline A$ is a subset of the closed set $X\setminus B$.</p>
|
4,200,434 | <p>I am having difficulty with what should be a routine question, Exercise 2.2.2 (b) of <em>Understanding Analysis</em> by Stephen Abbott (2015).</p>
<blockquote>
<p><strong>Exercise 2.2.2.</strong> Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.</... | user2661923 | 464,411 | <p>Alternative</p>
<p>Rewrite <span class="math-container">$~~~\frac{2n^2}{n^3 + 3}~~~$</span> as <span class="math-container">$~~~\displaystyle \frac{2}{n + \frac{3}{n^2}}.$</span></p>
<p>So, the problem reduces to showing that as <br>
<span class="math-container">$\displaystyle n \to \infty, ~\left(\frac{2}{n + \frac... |
3,237,094 | <p>Find all differentiable functions <span class="math-container">$f\colon [0,\infty)\to [0,\infty)$</span> for which <span class="math-container">$f(0)=0$</span> and <span class="math-container">$f^{\prime}(x^2)=f(x)$</span> for any <span class="math-container">$x\in [0,\infty)$</span>. </p>
<p>I have tried to reduce... | MathematicsStudent1122 | 238,417 | <p>We prove this in two steps. First, I prove <span class="math-container">$f$</span> must be identically <span class="math-container">$0$</span> on <span class="math-container">$[0,1]$</span>, then I prove the result for <span class="math-container">$[0, \infty)$</span>. We prove the more general result where <span cl... |
3,428,585 | <p>From " I see a tree in front of me" it seems legitimate to infer that " there is actually a tree in front of me". </p>
<p>But Descartes denies the legitimacy of this inference , saying : </p>
<p>In case there were a Malin Genie manipulating your mind, the fact that you see a tree in front of you would not imply th... | lemontree | 344,246 | <p>Re. your comment:<br>
Okay, now three different levels are starting to get mixed up. (In the following, assume "formula = object language" and "fact = meta language").</p>
<p><span class="math-container">$\newcommand{\fml}[1]{\underbrace{#1}_{\text{formula}}}$</span></p>
<p><span class="math-container">$\newcomman... |
3,428,585 | <p>From " I see a tree in front of me" it seems legitimate to infer that " there is actually a tree in front of me". </p>
<p>But Descartes denies the legitimacy of this inference , saying : </p>
<p>In case there were a Malin Genie manipulating your mind, the fact that you see a tree in front of you would not imply th... | Bram28 | 256,001 | <p>First, let me quickly comment on your:</p>
<blockquote>
<p>Is this strategy really fair? Can I really imagine any supposition, any arbitrary scenario to show an inference is not valid? </p>
</blockquote>
<p>Yes! This is what you <em>always</em> do when considering the validity of an argument: you consider <em>a... |
292,221 | <blockquote>
<p>How to prove that
$$\int_{0}^{1}(1+x^n)^{-1-1/n}dx=2^{-1/n}$$</p>
</blockquote>
<p>I have tried letting $t=x^n$,and then convert it into a beta function, but I failed. Is there any hints or solutions?</p>
| Ron Gordon | 53,268 | <p>Let $x=(\tan{t})^{2/n}$. Then the integral becomes</p>
<p>$$\begin{align} \int_0^1 dx \: (1+x^n)^{-\left ( 1 + \frac{1}{n} \right )} &= \frac{2}{n} \int_0^{\pi/4} dt \: \cot{t} (\tan{t})^{2/n} (\sec{t})^{-2/n} \\ &= \frac{2}{n} \int_0^{\pi/4} dt \: \cot{t} (\sin{t})^{2/n} \\ &= \frac{2}{n} \int_0^{\pi/... |
1,632,990 | <p>I'm having a bit of confusion here.</p>
<p>What are the solutions of</p>
<p>$\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$</p>
<p>Clearly,</p>
<p>$x_2=0$<br>
$x_3=0$<br>
$3x_3=0$</p>
<p>$x_1=s\in\mathbb{R}$, because there are no constraints for $x_1$.<br></p>
<p>then e.g.<... | adjan | 219,722 | <p>There are infinitely many solutions $v = (t, 0, 0)$ with $t \in \mathbb{R}$. The rank of the matrix is smaller than its row size.</p>
|
3,660,825 | <p>I know this question has been asked many times and there is good information out there which has clarified a lot for me but I still do not understand how the addition and multiplication tables for <span class="math-container">$GF(4)$</span> is constructed?</p>
<p>I'm just starting to learn about fields in general, ... | MJD | 25,554 | <p>For any given <span class="math-container">$n$</span>, there is at most one field with <span class="math-container">$n$</span> elements: only one, if <span class="math-container">$n$</span> is a power of a prime number (<span class="math-container">$2, 3, 2^2, 5, 7, 2^3, 3^2, 11, 13, \ldots$</span>) and none otherwi... |
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>
| user137481 | 137,481 | <p>Let (x, y, z) be the numbers showing on the 3 dice.<br>
We want x + y + z = 13.<br>
Assuming the dice are distinguishable, the possibilities are:<br>
(1, 6, 6)
(2, 5, 6), (2, 6, 5)<br>
(3, 4, 6), (3, 5, 5), (3, 6, 4)<br>
(4, 3, 6), (4, 4, 5), (4, 5, 4), (4, 6, 3)
(5, 2, 6), (5, 3, 5), (5, 4, 4), (5, 5, 3), (5, 6, 2)... |
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>
| Pieter21 | 170,149 | <p>A practical solution at High School level:</p>
<p>If I throw 2 dice, I have 36 outcomes.</p>
<p>Throw 7 occurs 6 times, and the other 30 are equally divided in 15 times more than 7 and 15 times less than 7.</p>
<p>The 6 and first set of 15 throws can uniquely be completed to 13. The others can't.</p>
<p>$$6+15 =... |
104,186 | <p>I'm having difficulty understanding how to express text in <code>Epilog</code> that is dynamically updated using <code>Log[b, x]</code>. <em>Mathematica</em> changes this to base $e$, but I would like it to be <code>Log[b, x]</code> in traditional format with base $b$, and I can't seem to make it work. I'm guessin... | B flat | 33,996 | <p>Thanks to Dr. Belisarius,</p>
<pre><code>Epilog -> {Text[Subscript[Log, b] "(x)", {3, -5}]}]
</code></pre>
|
349,147 | <p>What is known regarding which hyperbolic groups are cubulated?</p>
<p>I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.</p>
<p>My impression is that not all of them are, but I didn't manage to find references with a counterexample.</p>
<p>Are there known ways... | AGenevois | 122,026 | <p>If a group <span class="math-container">$G$</span> satisfies Kazhdan's property (T), then any action of <span class="math-container">$G$</span> on a CAT(0) cube complex has a global fixed point. See Niblo and Roller's article <em>Groups acting on cubes and Kazhdan's Property (T)</em>. Examples of hyperbolic groups w... |
2,865,943 | <p>I have an integral surface $z = z(x, y)$.</p>
<p>Writing this integral surface in implicit form, we get</p>
<p>$$F(x, y, z) = z(x, y) - z = 0$$</p>
<p>I am then told that the gradient vector $\nabla F = (z_x, z_y, -1)$ is normal to the integral surface $F(x, y, z) = 0$.</p>
<p>First of all, how was this calculat... | Mohammad Riazi-Kermani | 514,496 | <p>Let us start with an example.</p>
<p>$$ z=x^2+y^2$$
$$ F(x,y,z)=x^2+y^2-z$$
$$\nabla F = (z_x, z_y, -1)=< 2x,2y,-1>$$</p>
<p>If a point is given, for example $P(1,2,5)$ Then at that point you have two normal vector to the surface.</p>
<p>Upward normal $$< -2x,-2y,1> = <-2,-4,1>$$
Downward normal... |
2,865,943 | <p>I have an integral surface $z = z(x, y)$.</p>
<p>Writing this integral surface in implicit form, we get</p>
<p>$$F(x, y, z) = z(x, y) - z = 0$$</p>
<p>I am then told that the gradient vector $\nabla F = (z_x, z_y, -1)$ is normal to the integral surface $F(x, y, z) = 0$.</p>
<p>First of all, how was this calculat... | Calvin Khor | 80,734 | <p>maybe its helpful to give the surface equation a different symbol
$$ S = \{ (x,y,z) : z=Z(x,y) \}$$
then with $F(x,y,z):= Z(x,y) - z$,</p>
<p>$$∇ F (x,y,z) = \begin{pmatrix}\partial_x( Z(x,y) - z)\\\partial_y( Z(x,y) - z)\\\partial_z( Z(x,y) - z)\end{pmatrix}= \begin{pmatrix}\partial_x Z(x,y)\\\partial_yZ(x,y)\\\ -... |
129,993 | <p>Let $p$ and $q$ be relative primes, $n$ positive integer.</p>
<p>Given</p>
<ul>
<li>$n\bmod p$ and</li>
<li>$n\bmod q$</li>
</ul>
<p>how do I calculate $n\bmod (pq)$ ?</p>
| lhf | 589 | <p>Since $p$ and $q$ are relatively prime, there are integers $a$ and $b$ such that $ap+bq=1$. You can find $a$ and $b$ using the <a href="http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm" rel="nofollow noreferrer">Extended Euclidean algorithm</a>. Then $n\equiv aps+bqr \bmod pq$ if $n\equiv r \bmod p$ and $... |
129,993 | <p>Let $p$ and $q$ be relative primes, $n$ positive integer.</p>
<p>Given</p>
<ul>
<li>$n\bmod p$ and</li>
<li>$n\bmod q$</li>
</ul>
<p>how do I calculate $n\bmod (pq)$ ?</p>
| hardmath | 3,111 | <p>From the fact that $p,q$ are relatively prime, we can find coefficients $a,b$ such that:</p>
<p>$$ap + bq = 1$$</p>
<p>With these coefficients we can piece together a solution for n from its residues modulo $p$ and $q$. Say:</p>
<p>$$n \equiv r \mod p$$
$$n \equiv s \mod q$$</p>
<p>Then this works: $ n = sap +... |
1,584,933 | <p>Let's have $f_n(x)$ defined on $\mathbb{R}$ by:</p>
<ul>
<li>$f_n(0)=0$</li>
<li>$f_n(x)=\frac{1-e^{-nx^2x^2}}{x}$if $x\neq 0$</li>
</ul>
<p>$f_n(x)\rightarrow \frac{1}{x}$</p>
<p>therefore, $f_n(x)$ converges weakly to $\frac{1}{x}$
\begin{align}
\lim\limits_{x\rightarrow +\infty}\sup\limits_{n\in I}{|f_n(x)-f(x... | Ron Gordon | 53,268 | <p>$$i 2 \pi = \oint_{\gamma} \frac{dz}{z-w} $$</p>
<p>when $w$ is inside $\gamma$ and $\gamma$ only winds around $w$ once counterclockwise (positive orientation).</p>
|
1,818,260 | <p>I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g.,</p>
<p>$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}+\frac{2}{x}.\frac{\partial u}{\partial x}$</p>
<p>$\frac... | ekkilop | 284,417 | <p>It is a rather big question you are asking and it has undoubtedly many answers. I'll try to give an as condensed answer as I can without going in to mathematical detail. </p>
<p>As to the question "How do we decide which formula we should use [to approximate a derivative]?", first think of what a finite difference ... |
90,342 | <p>Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup</p>
<p>$S = \{x \in \mathbb{T}^n : k \cdot x = 0_{\mathbb{T}^n}\}$</p>
<p>where $\cdot$ is the scalar product. I think that $S$ is iso... | Mikhail Borovoi | 4,149 | <p>Consider the subgroup $N:=\langle k\rangle\subset \mathbb{Z}^n$.
There exists a basis $f_1,\dots,f_n$ of $\mathbb{Z}^n$ such that $uf_1$ is a basis of $N$, where $u\in \mathbb{Z}$, $u>0$,
see Vinberg, A Course in Algebra, Thm. 9.1.5,
or Lang, Algebra, 3d ed., Thm. III.7.8.
Changing, if necessary, $f_1$ to $-f_1$,... |
1,990,033 | <p>Suppose I have a point $P(x_1, y_1$) and a line $ax + by + c = 0$. I draw a perpendicular from the point $P$ to the line. The perpendicular meets the line at point $Q(x_2, y_2)$. I want to find the coordinates of the point $Q$, i.e., $x_2$ and $y_2$.</p>
<p>I searched up for similar questions where the coordinates ... | drhab | 75,923 | <p>The following statements are equivalent:</p>
<ul>
<li><p>$4\sin x\sin2x\sin4x=\sin3x$</p></li>
<li><p>$4\left[\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)\right]\left[\frac{1}{2i}\left(e^{2ix}-e^{-2ix}\right)\right]\left[\frac{1}{2i}\left(e^{4ix}-e^{-4ix}\right)\right]=\frac{1}{2i}\left(e^{3ix}-e^{-3ix}\right)$</p></li>... |
1,482,152 | <p>Show that four non-coplanar points in $\mathbb{R}^3$ determine an unique sphere.</p>
<p>I have no idea how to solve this exercise. Thank you for your help.</p>
| mathcounterexamples.net | 187,663 | <p><strong>Hint.</strong></p>
<p>Consider the perpendicular bisector planes of three segments joining couples of points. Those planes intersect at a point that is the center of a sphere passing through the four points. And this is the only sphere having such property.</p>
|
1,599,843 | <p>I am working out of <em>Mathematical Statistics and Data Analysis by John Rice</em> and ran into the following interesting problem I'm having trouble figuring out.</p>
<blockquote>
<p>Ch 2 (#65)</p>
<p>How could random variables with the following density function be generated from a uniform random number generator?... | BruceET | 221,800 | <p>Comment: Demonstration in R with $\alpha = .2$ of answerbook result.</p>
<pre><code> alpha = .2; m = 10^5; u = runif(m)
x = (-1 + 2*sqrt(1/4 + alpha*(1/2 + alpha/4 - u)))/alpha
hist(x, col="wheat", prob=T)
curve((1 + alpha*x)/2, -1, 1, lwd=2, col="blue", add=T)
</code></pre>
<p><a href="https://i.stack.imgur.... |
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