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 |
|---|---|---|---|---|
2,794,724 | <blockquote>
<p>Let $F:U\rightarrow W$ be a linear transformation from the vector
space $U$ to the vectorspace $W$. Show that the image space to $F$,</p>
<p>$$V(F)=\{w\in W:w=F(u) \ \ \text{for some} \ \ u\in U\},$$</p>
<p>is a subspace of $W$.</p>
</blockquote>
<p>Okay, I know that in order for $M$ to ... | Aloizio Macedo | 59,234 | <p>The curly bracket says that $w$ is an element of the set $V(F)$ if and only if $w$ is the result of applying $F$ to some element $u \in U$.</p>
<p>For instance, if $F: \mathbb{R}^2 \to \mathbb{R}^3$ is given by $F(x,y)=(x,0,0)$, then $(1,0,0)$ is in $V(F)$, since $(1,0,0)=F(1,3)$, for example. But $(3,2,0)$ is not ... |
3,162,338 | <p>Consider <span class="math-container">$ x_1, x_2, ..., x_n \in \mathbb{R}$</span>.</p>
<p>We have to prove that each <span class="math-container">$\sqrt x $</span> is rational if the sum of <span class="math-container">$\sqrt x_1 + \ldots + \sqrt x_n $</span> is rational. </p>
<p>I think that I could prove it us... | Peter Foreman | 631,494 | <p>Let
<span class="math-container">$$x_1=2$$</span>
<span class="math-container">$$x_2=(2-\sqrt{2})^2=6-4\sqrt{2}$$</span>
Then
<span class="math-container">$$\sqrt{x_1}+\sqrt{x_2}=2$$</span>
But
<span class="math-container">$$x_2\not\in \mathbb{Q}$$</span></p>
|
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | Pieter Geerkens | 64,624 | <p>There is more than one way to count <em>genetics</em>. Suppose that 120 years ago all four pairs of your mother's great grand parents (8 in all) were pregnant with their first child, and both pairs of your father's grandparents (4 in all) were pregnant with their first child. </p>
<p>Thus at that moment in time you... |
97,261 | <p>This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: </p>
<p>-Differentiability.
-Open mapping theorem.
-Implicit function theorem.
-Lagrange multipliers. Submanifolds.
-Integrals.
-Integration on surfaces.
-Stokes ... | Adam | 20,333 | <p><a href="http://www.math.harvard.edu/~shlomo/" rel="nofollow">http://www.math.harvard.edu/~shlomo/</a></p>
|
97,261 | <p>This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: </p>
<p>-Differentiability.
-Open mapping theorem.
-Implicit function theorem.
-Lagrange multipliers. Submanifolds.
-Integrals.
-Integration on surfaces.
-Stokes ... | analysisj | 14,966 | <p>Spivak has a good book in multivariable calculus. Harvard, I think, uses Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Another book which is used is Shifrin's Multivariable Mathematics, which I believe Harvard uses as well.. For Shifrin's book at least, you can buy a solut... |
3,585,271 | <p>Let <span class="math-container">$V$</span> be an affine variety in <span class="math-container">$K^n$</span> with ideal <span class="math-container">$I=I(V)$</span>, where <span class="math-container">$K$</span> is an algebraically closed field. Let <span class="math-container">$V'$</span> be the variety with defin... | Con | 682,304 | <p>As varieties they are the same. Usually one considers varieties defined by radical ideals for the correspondence of Hilbert's Nullstellensatz.</p>
<p>Here you can see one reason why one wants to work with schemes. If for example <span class="math-container">$n = 1$</span>, <span class="math-container">$I = (x^2)$</... |
1,384,752 | <p>I ran across a problem which has stumped me involving existential quantifiers.
Let U, our universe, be the set of all people. Let S(x) be the predicate "x is a student" and I(x) be the predicate "x is intelligent".
I want to write the statement "Some students are intelligent" in the correct logical form. I can see 2... | Greg Martin | 16,078 | <p>The two statements are not equivalent. The second one would be true if there is even one nonstudent $x$ in the universe, regardless of intelligence; it would also be true if there is even one intelligent person $x$ in the universe, regardless of student status.</p>
<p>Understanding why this is the case depends on t... |
3,985,302 | <p>Let <span class="math-container">$L/K$</span> be an extension of local fields. We can find <span class="math-container">$\alpha$</span> such that <span class="math-container">$\mathcal{O}_L=\mathcal{O}_K[\alpha]$</span>. What do we know about this generating element? I think that this <span class="math-container">$\... | KCd | 619 | <p>If you look at a <em>proof</em> of that <span class="math-container">$\mathcal O_L = \mathcal O_K[\alpha]$</span> for some <span class="math-container">$\alpha$</span>, such as Prop. 3, Chapter III, Section 1 in Lang's "Algebraic Number Theory" then you should be able to see that you can take as <span clas... |
1,595,946 | <blockquote>
<p>Let $f:(a,b)\to\mathbb{R}$ be a continuous function such that
$\lim_\limits{x\to a^+}{f(x)}=\lim_\limits{x\to b^-}{f(x)}=-\infty$.
Prove that $f$ has a global maximum.</p>
</blockquote>
<p>Apparently, this is similar to the EVT and I believe the proof would be similar, but I cannot think anything... | Tsemo Aristide | 280,301 | <p>Since $lim_{x\rightarrow a}f(x)=lim_{x\rightarrow b}f(x)=-\infty$, there exists $u>0, u<b-a$ such that $\mid x-a\mid<u, and \mid x-b\mid <u$ implies that $f(x)<-1$. The restriction of $f$ on $[a+u,b-u]$ has a maximum $m$ since $[a+u,b-u]$ is compact. Thus for every $x\in R$, $f(x)\leq sup\{-1,m\}$. Th... |
4,083,697 | <p>I'm thinking about the example <span class="math-container">$f(x)=(x-1)^2$</span> which is clearly symmetric about the line <span class="math-container">$x=1$</span>. The question is really how do you show that it is symmetric about <span class="math-container">$x=1$</span> algebraically? I notice that if you plug i... | Andrei | 331,661 | <p>I would write it like this:
<span class="math-container">$$|z\cdot w|=|r_1|\cdot|r_2|\cdot|\cos\theta_1+i\sin\theta_1|\cdot|\cos\theta_2+i\sin\theta_2|=|r_1|\cdot|r_2|\cdot 1\cdot 1$$</span>
Then, if <span class="math-container">$|z\cdot w|=0$</span>, at least one of the factors must be <span class="math-container">... |
3,518,285 | <p>I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying <a href="https://mathoverflow.net/questions/13089/why-do-so-many-textbooks-have-so-much-technical-detail-and-so-little-enlightenme">backwards</a> as much as possible, but I have been stuck on the concepts of <a href=... | BCLC | 140,308 | <p>As a supplement to the other answers, I'm going to prove (6 or) 7 implies 5 without axiom of choice. This is based on <a href="https://math.stackexchange.com/users/431940/joppy">Joppy</a>'s <a href="https://math.stackexchange.com/a/3559659">answer</a> and <a href="https://math.stackexchange.com/users/686397/wooliert... |
607,264 | <p>Let the directional derivative of a function $f(x,y)$ at a point $P$ in the direction of $(1/\sqrt{5})\mathbf{i}+(2/\sqrt{5})\mathbf{j}$ be $16/\sqrt{5}$ and the partial derivative $\partial f / \partial x$ evaluated at $P$ be $6$. Then what is the directional derivative in the direction of $\mathbf{i}-\mathbf{j}$?<... | Eric Thoma | 35,667 | <p>The directional derivative of $f(x,y)$ along some unit vector $\mathbf{v}$ is $\nabla f(x,y) \cdot \mathbf{v}$.</p>
<p>Applying this to the given in the problem:</p>
<p>$$ \frac{1}{\sqrt{5}}\cdot\frac{\partial f}{\partial x} + \frac{2}{\sqrt{5}}\cdot\frac{\partial f}{\partial y}=\frac{16}{\sqrt{5}}$$</p>
<p>Using... |
1,588,665 | <p>I have been reading up on finding the eigenvectors and eigenvalues of a symmetric matrix lately and I am totally unsure of <strong>how and why</strong> it works. Given a matrix, I can find its eigenvectors and values like a machine but the problem is, I have no intuition of how it works.</p>
<p>1) I understand that... | Pedro | 70,305 | <p>To solve an equation, you have to establish a sequence of logical <strong>equivalences</strong>.</p>
<p>In your first method, you only established a sequence of logical <strong>implications</strong>. This is the reason why you lost the solution $x=0$.</p>
<p><em>Remark:</em> In order to solve an equation, an seque... |
353,658 | <p>Let $g : [0, 1] \rightarrow \mathbb{R}$ be twice differentiable with $g^{\prime \prime}(x) > 0$ for all $x \in [0,1]$. Suppose that $g(0) > 0$ and $g(1) = 1$. Prove if $g$ has a fixed point in $(0,1)$, then $g^{\prime}(1) > 1$.</p>
<p>My attempt: Define a function $h(x)=g(x)-x$. Since $g$ has a fixed point... | R Salimi | 71,371 | <p>We don't use Rolle's theorem,since $g^{\prime\prime}>0$,then $g^{\prime}(x)$ is strictly increasing,also $g$ has a fixed point c in $(0,1)$ and $1$ is fixed point for $g$,we can use only Mvt theorem.($\frac{g(1)-g(c)}{1-c}=g^{\prime}(t)=1,t\in(c,1)$).
Of course I mean $g^{\prime}(1)$ as $\lim_{x\to1^-}g^{\prime}(... |
60,322 | <p>I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.</p>
| Asaf | 8,857 | <p>The connections between Dynamics and Lie Groups (or Algebraic groups) comes mainly in two flavours:</p>
<ol>
<li>Smooth dynamics, like others have stated Hamiltonian dyanmics and differential equations.</li>
<li>Applications of Ergodic theory and Topological dynamics to Lie groups (or more generally, homogeneous spa... |
1,652,701 | <p>Am I on the right track to solving this?</p>
<p>$$e^z=6i$$
Let $w=e^z$</p>
<p>Thus,</p>
<p>$$w=6i$$
$$e^w=e^{6i}$$
$$e^w=\cos(6)+i\sin(6)$$
$$\ln(e^w)=\ln(\cos(6)+i\sin(6))$$
$$w=\ln(\cos(6)+i\sin(6))$$
$$e^z=\ln(\cos(6)+i\sin(6))$$
$$\ln(e^z)=\ln(\ln(\cos(6)+i\sin(6)))$$
$$z=\ln(\ln(\cos(6)+i\sin(6)))$$</p>
<p>... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>$$6i=e^{\log(6)+i\pi/2+i2n\pi}=e^z$$</p>
|
1,652,701 | <p>Am I on the right track to solving this?</p>
<p>$$e^z=6i$$
Let $w=e^z$</p>
<p>Thus,</p>
<p>$$w=6i$$
$$e^w=e^{6i}$$
$$e^w=\cos(6)+i\sin(6)$$
$$\ln(e^w)=\ln(\cos(6)+i\sin(6))$$
$$w=\ln(\cos(6)+i\sin(6))$$
$$e^z=\ln(\cos(6)+i\sin(6))$$
$$\ln(e^z)=\ln(\ln(\cos(6)+i\sin(6)))$$
$$z=\ln(\ln(\cos(6)+i\sin(6)))$$</p>
<p>... | Noble Mushtak | 307,483 | <p>Hopefully, from all of these solutions, you know how to solve this problem. Now, let's try doing it your way. You've done everything right so far:
$$z=\ln(\ln(\cos(6)+i\sin(6)))$$</p>
<p>By <a href="https://en.wikipedia.org/wiki/Euler%27s_identity" rel="nofollow noreferrer">Euler's Identity</a>, we have $\cos(6)+i\... |
2,234,744 | <p>Please help me find the value of the following integral:<br>
$$\frac{(5050)\int^1_0(1-x^{50})^{100} dx}{\int^1_0(1-x^{50})^{101} dx}$$
I tried solving both numerator and denominator via by-parts but it isn't giving me a conclusive solution. Any other suggestions?</p>
| Marios Gretsas | 359,315 | <p>one way to solve it besides the change of variable,is by using the binomial theorem in both denominator and enumerator.</p>
<p>$\sum_{n=0}^{100}\binom{100}{n}(x^{50})^{100-n}(-1)^n=(1-x^{50})^{100}$ </p>
<p>$\binom{n}{k}=\frac{n!}{k!(n-k)!}$</p>
<p>Use the linearity of the integral and integrate each term of the... |
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | Robert Israel | 8,508 | <p>When in doubt, convert trig functions to complex exponentials.</p>
<p>If $w = e^{i\pi/7}$,
$\csc^2(\pi/7) = \dfrac{-4}{(w-1/w)^2}$ and similarly for the others
with $w$ replaced by $w^2$ and $w^4$.
Simplifying,
$$ \csc^2(\pi/7) + \csc^2(2\pi/7) + \csc^2(4\pi/7) - 8 \\=
-4\,{\frac {2\,{w}^{16}+{w}^{14}+3\,{w}^{12}+... |
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | lab bhattacharjee | 33,337 | <p>Let $7x=\pi\implies 4x=\pi-3x$</p>
<p>$$\frac1{\sin2x}+\frac1{\sin4x}=\frac{\sin4x+\sin2x}{\sin4x\sin2x}$$</p>
<p>$$=\frac{2\sin3x\cos x}{\sin(\pi-3x)\sin2x}(\text{ using } \sin2A+\sin2B=2\sin(A+B)\cos(A-B))$$</p>
<p>$$=\frac{2\sin3x\cos x}{\sin(3x)2\sin x\cos x}(\text{ using } \sin2C=2\sin C\cos C \text{ and }\... |
2,849,450 | <p>Let's consider a generic linear programming problem. Is it possible that the decision variables of the objective function assume (at the optimal solution) irrational values?</p>
<p>Also, is it possible that some entries of the $A$ matrix are irrational?</p>
| Siong Thye Goh | 306,553 | <p>Yes.</p>
<p>$$\min x$$ subject to $$\sqrt{2}x=1$$ is a valid linear programming instance.</p>
|
664,349 | <blockquote>
<p>If $G$ is a finite group where every non-identity element is generator of $G$, what is the order of $G$?</p>
</blockquote>
<p>I know that the order of $G$ must be prime, but I'm not sure how to go about proving this from the problem statement. </p>
<p>Any hints on where to start?</p>
| AG. | 80,733 | <p>Since every nonidentity element of $G$ is a generator, $G$ is generated by some element $x$; hence $G$ is cyclic. If the order of $G$ is composite, say $|G|=p r$, where $p, r >1$, then there exists a nonidentity element $x^p$ that does not generate $G$, a contradiction. Thus, the order of $G$ must be a prime. ... |
1,581,756 | <p>Find the general solution of $$z(px-qy)=y^2-x^2$$ Let $F(x,y,z,p,q)=z(px-qy)+x^2-y^2$. This gives $$F_x=zp+2x$$
$$F_y=-zq-2y$$
$$F_z=px-qy$$
$$F_p=zx$$
$$F_q=-zy$$
By Charpit's method we have $$\frac{dx}{zx}=\frac{dy}{-zy}=\frac{dz}{z(px-qy)}=\frac{dp}{-zp-2x-p^2x+pqy}=\frac{dq}{zq+2y-pxy+q^2y}$$</p>
<p>By equating... | MathIsNice1729 | 274,536 | <p>For starters, the $\tau$ function is multiplicative. That means, $\tau(mn)=\tau(m)\tau(n)$ where $(m, n) =1$. But, as you seek intuition, I won't use it in the discussion. Now, your claim is true for $4$. The numbers which have $2$ divisors are prime $p$. Now, if we multiply it by something that the product has more... |
3,559,167 | <p>I know that it is a bounded below set and the infimum is 4, but I'm unsure of going about how to prove that it is indeed bounded. Any help would be greatly appreciated! </p>
| Lee Mosher | 26,501 | <p>Hint #1: If you want to investigate intuitively whether or not it is bounded, try evaluating <span class="math-container">$x + \frac{4}{x}$</span> for values of <span class="math-container">$x$</span> that get larger and larger, such as <span class="math-container">$x=10$</span>, <span class="math-container">$x=100$... |
3,820,929 | <p>When i look at my notes , i realized something i have not realized before.It was as to a modular arithmetic
question.</p>
<p>The question is <span class="math-container">${\sqrt 2} \pmod7$</span></p>
<p>It is very trivial question.The solution is: if <span class="math-container">$x \equiv {\sqrt 2} \pmod7$</span> ,t... | YJT | 731,237 | <p><span class="math-container">$\sqrt{2}$</span> is defined to be "a number that when you multiply it by itself, gives you <span class="math-container">$2$</span>". In each field, the definition of "multiply" might be different and hence this number can be different (or it might not exist). In your... |
697,336 | <p>Let $ABCD$ be a trapezoid, such that $AB$ is parallel to $CD$. Through $O$, the intersection point of the diagonals $AC$ and $BD$ consider a parallel line to the bases. This line meets $AD$ at $M$ and $BC$ at $N$. </p>
<p>Prove that $OM=ON$ and: $$\frac{2}{MN}=\frac1{AB}+\frac1{CD}$$</p>
| kmitov | 84,067 | <p><img src="https://i.stack.imgur.com/iLNJo.png" alt="enter image description here"></p>
<p>The picture support the above solution.</p>
<p>On the other hand ot is possible to go another way.</p>
<p>Trinagles AMO and ACD are similar. Then $\frac{MO}{DC}=\frac{AO}{AC}$</p>
<p>Trinagles BMO and BCD are similar. Then ... |
418,748 | <p>I tried to calculate, but couldn't get out of this:
$$\lim_{x\to1}\frac{x^2+5}{x^2 (\sqrt{x^2 +3}+2)-\sqrt{x^2 +3}}$$</p>
<p>then multiply by the conjugate.</p>
<p>$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$ </p>
<p>Thanks!</p>
| colormegone | 71,645 | <p>This limit problem seems to cooperate <em>rather</em> nicely with the "conjugate factor" method. Multiply the numerator and denominator by the "conjugate" of the numerator to obtain</p>
<p>$$\lim_{x \rightarrow 1} \ \frac{\sqrt{x^2+3}-2}{x^2 - 1} \ \cdot \frac{\sqrt{x^2+3}+2}{\sqrt{x^2+3}+2} $$</p>
<p>$$ = \ \... |
1,390,676 | <p>A quasigroup is a pair $(Q,/)$, where $/$ is a binary operation on $Q$, such that (1) for each $a,b\in Q$ there exists unique solutions to the equations
$a/x=b$ and $y/a=b$.</p>
<p>Now I want to extract a class of quasigroups that captures characteristics from $(Q_+,/)$, where $Q_+$ is the set of positive rational ... | Sean English | 220,739 | <p>3 comes from the first two.</p>
<p>$$(a/b)/(c/d)\\=d/(c/(a/b))\\=d/(b/(a/c))\\=(a/c)/(b/d)\\=(a/(b/d))/c\\=(d/(b/a))/c\\=(d/c)/(b/a)$$</p>
|
1,689,923 | <p>I have a sequence $a_{n} = \binom{2n}{n}$ and I need to check whether this sequence converges to a limit without finding the limit itself. Now I tried to calculate $a_{n+1}$ but it doesn't get me anywhere. I think I can show somehow that $a_{n}$ is always increasing and that it has no upper bound, but I'm not sure i... | Christian Blatter | 1,303 | <p>For each $n$-set $A\subset[2n]$ we can form two different $(n+1)$-sets $$A':=A\cup\{2n+1\},\ A'':=A\cup\{2n+2\}\quad\subset [2(n+1)]\ .$$This implies ${2(n+1)\choose n+1}\geq 2{2n\choose n}$, hence $a_n\to\infty$ $(n\to\infty)$.</p>
|
1,843,274 | <p>Good evening to everyone. So I have this inequality: $$\frac{\left(1-x\right)}{x^2+x} <0 $$ It becomes $$ \frac{\left(1-x\right)}{x^2+x} <0 \rightarrow \left(1-x\right)\left(x^2+x\right)<0 \rightarrow x^3-x>0 \rightarrow x\left(x^2-1\right)>0 $$ Therefore from the first $ x>0 $, from the second $ x... | Zau | 307,565 | <p>$$f(x) = x(x^2-1)= x^3-x$$
$$f'(x)= 3x^2-1$$ then x get local extreme value at point x = $\sqrt 3 /3 $or $-\sqrt 3 /3 $</p>
<p>And it is on the interval ($-\infty$,$-\sqrt 3 /3 $) or ($\sqrt 3 /3 $,$+\infty$) is increasing and the interval ($-\sqrt 3 /3 $,$\sqrt 3 /3$) is decreasing.</p>
<p>$$f(-\sqrt 3 /3... |
4,489,675 | <p>When saying that in a small time interval <span class="math-container">$dt$</span>, the velocity has changed by <span class="math-container">$d\vec v$</span>, and so the acceleration <span class="math-container">$\vec a$</span> is <span class="math-container">$d\vec v/dt$</span>, are we not assuming that <span class... | G Cab | 317,234 | <p>Since you are a physicist (and I am an engineer) to avoid to be entangled with
the centuries long debate on infinitesimals on one side, as well as with uncertainty principle on the other (!), let's
go back to the approach that the same Newton took to justify its motion. law:
<a href="https://en.wikipedia.org/wiki/Fi... |
2,135,717 | <p>Let $G$ be an Abelian group of order $mn$ where $\gcd(m,n)=1$. </p>
<p>Assume that $G$ contains an element of $a$ of order $m$ and an element $b$ of order $n$. </p>
<p>Prove $G$ is cyclic with generator $ab$.</p>
<hr>
<p>The idea is that $(ab)^k$ for $k \in [0, \dots , mn-1]$ will make distinct elements but do ... | Bernard | 202,857 | <p>Suppose $(ab)^k=e$. This means $a^k=b^{-k}\in\langle a\rangle\cap\langle b\rangle$. But since $m\wedge n=1$,$\;\langle a\rangle\cap\langle b\rangle=\{e\}$ by <em>Lagrange's theorem</em>. Thus $a^k=e$ and $b^k=e$, which implies $k $ is a common multiple of $m$ and $n$, i.e. a multiple of $mn$ since $m\wedge n=1$.</p... |
3,197,331 | <p><span class="math-container">$X, Y$</span> are quantities and <span class="math-container">$f : X → Y$</span> a function. Show
the equivalence of the following statements:</p>
<p>(i) <span class="math-container">$f$</span> is injective</p>
<p>(ii) <span class="math-container">$f^{-1}\!\bigl(f(A) \bigr)=A \quad \t... | Dr. Mathva | 588,272 | <p><strong>Hint</strong> <span class="math-container">$\;$</span> Evaluate the expected value <span class="math-container">$E$</span> of the game. We know the game is fair if <span class="math-container">$E=$</span>"Money Paid for entering the game".</p>
<blockquote class="spoiler">
<p> <span class="math-container"... |
507,062 | <p>I need your help with a lifelong problem I have always had. There are two things in life I hate most, 1) weddings and 2) Long division. For the life of me I hate division. I am horrible at it. I can multiple large numbers in my head but I can’t divide if my life depended on it. When the division operation is 2 digi... | RDizzl3 | 77,680 | <p>Yes, that operation is legal. It just comes from multiplying $\frac{(4-x^2)}{5x}$ on both sides.</p>
|
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | Mohammad Riazi-Kermani | 514,496 | <p>You are playing with dividing by <span class="math-container">$0$</span> </p>
<p>To avoid that, notice <span class="math-container">$$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1} = \lim_{x\to -\infty} \frac{x^2-2x +1}{x+1}$$</span></p>
<p><span class="math-container">$$= \lim_{x\to -\infty} x-3+\frac {4}{x+1} = -\infty... |
3,078,707 | <p>The above question is the equation <span class="math-container">$(2.4)$</span> of the following paper:</p>
<p><a href="http://www.jmlr.org/papers/volume6/tsuda05a/tsuda05a.pdf" rel="nofollow noreferrer">MATRIX EXPONENTIATED GRADIENT UPDATES</a>.</p>
<p>Let <span class="math-container">$M$</span> and <span class="m... | Intelligenti pauca | 255,730 | <p>If I understand your question properly, we know a point <span class="math-container">$P=(-c,d)$</span> on the ellipse, with its tangent line, we also know vertex <span class="math-container">$A=(0,0)$</span> of the ellipse and that its major axis lies on the <span class="math-container">$x$</span>-axis, but we don't... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | N. S. | 9,176 | <p>Let $a_k...a_0$ be the digits of $n$. Then</p>
<p>$$n=10^ka_k+\cdots+10a_1+a_0$$
and hence</p>
<p>$$n -\mbox{sum of its digits}=\left( 10^ka_k+\cdots+10a_1+a_0\right)-\left( a_k+\cdots+a_1+a_0\right)\\
=a_k(10^k-1)+\cdots+a_1(10-1)=a_k\cdot 99\ldots9+\cdots+a_1 \cdot 9 \\
=9\left( a_k\cdot 11\ldots1+\cdots+a_2\cdo... |
563,431 | <p>Find the absolute maximum and minimum of $f(x,y)= y^2-2xy+x^3-x$ on the region bounded by the curve $y=x^2$ and the line $y=4$. You must use Lagrange Multipliers to study the function on the curve $y=x^2$.</p>
<p>I'm unsure how to approach this because $y=4$ is given. Is this a trick question?</p>
| Gazi | 131,093 | <p>Let G be any group of order 52. Then by sylow theorem it has only one sylow 13 subgroup and hence is normal in G, let it be H. Then G/H forms a quotient group of order 5, which implies G/H is cyclic and hence G is abelian.</p>
|
3,159,199 | <p>I did a question to find relative extrema for the following function:
<span class="math-container">$f(x)=x^2$</span> on <span class="math-container">$[−2,2].$</span></p>
<p>The answer said that there is no relative maxima for this function because relative extrema cannot occur at the end points of a domain.
Why i... | st.math | 645,735 | <p>There is a relative minimum at <span class="math-container">$0$</span>. But there are also relative <em>maxima</em> at <span class="math-container">$2$</span> and at <span class="math-container">$-2$</span>.</p>
<p>The definition for a relative maximum at a point <span class="math-container">$x_0$</span> is that <s... |
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| chroma | 1,006,726 | <p>I know I am kind of late, but here are a few
<span class="math-container">$$\int_{-\infty}^\infty\frac{\ln{(x^2+1)}}{x^4+x^2+1}\,dx$$</span>
<span class="math-container">$$\int_{-\infty}^\infty\frac{\ln{(x^4+x^2+1)}}{x^4+1}\,dx$$</span>
<span class="math-container">$$\int_{-\infty}^\infty\left(x^2+\frac{1}{x^2}\righ... |
203,967 | <p>I have a tree as below. </p>
<pre><code>Graph[{7 -> 2, 2 -> 3, 3 -> 4, 3 -> 5}, VertexLabels -> "Name"]
</code></pre>
<p><a href="https://i.stack.imgur.com/cs0Ztm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cs0Ztm.jpg" alt="enter image description here"></a></p>
<p>However, I... | kglr | 125 | <p>You can use</p>
<ol>
<li><code>Graph</code> with the option <code>GraphLayout -> "LayeredDigraphEmbedding"</code>, or </li>
<li><a href="https://reference.wolfram.com/language/ref/LayeredGraphPlot.html" rel="nofollow noreferrer"><code>LayeredGraphPlot</code></a>, or</li>
<li><a href="https://reference.wolfram.co... |
203,967 | <p>I have a tree as below. </p>
<pre><code>Graph[{7 -> 2, 2 -> 3, 3 -> 4, 3 -> 5}, VertexLabels -> "Name"]
</code></pre>
<p><a href="https://i.stack.imgur.com/cs0Ztm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cs0Ztm.jpg" alt="enter image description here"></a></p>
<p>However, I... | Szabolcs | 12 | <pre><code>Graph[{7 -> 2, 2 -> 3, 3 -> 4, 3 -> 5}, VertexLabels -> "Name",
GraphLayout -> {"LayeredEmbedding", "RootVertex" -> 7}]
</code></pre>
<p><a href="https://i.stack.imgur.com/t4RL2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/t4RL2.png" alt="enter image description ... |
597,899 | <p>Let $\epsilon > 0$ be given. Suppose we have that $$a - \epsilon < F(x) < a + \epsilon$$</p>
<p>Does it follow that $a - \epsilon < F(x) \leq a $ ??</p>
| Haha | 94,689 | <p>In fact the symbolization $a\leq b$ means that </p>
<p>" $a$ is less <strong>or</strong> equal to $b$." $(1)$ </p>
<p>(A proposition with <strong>or</strong>,let's say $p$= $p_1$or $p_2$ is true iff one at least off $p_i$'s is true)</p>
<p>Now in $(1)$ the two individual propositions cannot be true at the same ti... |
3,527,785 | <p>I'm reading James Anderson's <em>Automata Theory with Modern Applications. Here:</em></p>
<blockquote>
<p><a href="https://i.stack.imgur.com/sFWNh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFWNh.png" alt="enter image description here" /></a></p>
<p><a href="https://i.stack.imgur.com/k9zne.pn... | copper.hat | 27,978 | <p>It might be easier to do an inductive proof on the number of code symbols in two words.</p>
<p>Suppose <span class="math-container">$C$</span> is a prefix code for <span class="math-container">$S$</span>.</p>
<p>I use <span class="math-container">$|s|$</span> for the length (in terms of <span class="math-container... |
3,809,699 | <p>While we define norm on a vector space, we consider only real or complex vector field. But can we generalize this norm on a vector space over an arbitrary field ? I think this can be done, but we have to define a suitable modulus function on the ground field to be meaningful in the property ||cx||=|c|||x||, what |c|... | Physical Mathematics | 592,278 | <p>Let <span class="math-container">$F$</span> be a field and <span class="math-container">$V$</span> a vector space over <span class="math-container">$F$</span>. Define <span class="math-container">$|\cdot |: F \to \mathbb{R}$</span> by <span class="math-container">$|f| = 1$</span> for <span class="math-container">$f ... |
354,213 | <p>This is a similar question to the one I have posted <a href="https://math.stackexchange.com/questions/354124/given-u-2-5-3-how-to-find-unit-vectorsu-w-s-t-uv-is-maximal-and-u">before</a>. The problem
is as in the title:</p>
<blockquote>
<p>Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is
maximal, ... | Dimitris | 37,229 | <p>Let A be an open subset of $X\times Y=[0,1]\times [0,1]$. Then you can write it as a countble union : $$A=\bigcup_{i=1 }^\infty I_i \times J_i$$
in which $I_i,J_i$ are open intervals of [0,1].
Of course, $I_i\times J_i$ are all measurable subsets of $[0.1]^2$, and therefore their union is measurable (since the mea... |
78,341 | <p>I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.</p>
<p>However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wonderin... | Denis Osin | 10,251 | <p>I think what Yves meant in his comment to Andreas' answer is the result of Mal'cev [A. I. Mal'cev, On homomorphisms onto finite groups, Ivanov. Gos. Ped. Inst. Ucen. Zap., 18 (1958), pp. 49-60, in Russian] stating that any split extension of a finitely generated residually finite group by a residually finite group i... |
506,397 | <p>I would like to know the condition for a random variable <span class="math-container">$Y$</span> in order to make <span class="math-container">$\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$</span>, where <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> are ii... | Einar Rødland | 37,974 | <p>If you start off by subtracting $\mathrm{E}[X_1]$ from both sides and let $Z=X_2-X_1$, the desired inequality becomes
$$\mathrm{E}[\max(Y,Z)]>\mathrm{E}[\max(0,Z)].$$</p>
<p>For any distribution $U$ with $F_U(t)=\Pr[U\le t]$ the cumulative distribution, we can write
$$
\mathrm{E}[U]=\int_{-\infty}^\infty \chi(t&... |
985,103 | <p>The set $\{u_{1},u_{2}\cdots,u_{6}\}$
is a basis for a subspace $\mathcal{M}$ of $\mathbb{F}^{m}$ if and
only if $\{u_{1}+u_{2},u_{2}+u_{3}\cdots,u_{6}+u_{1}\}$
is also a basis for $\mathcal{M}$.
So far I have that the two basis are just rearranged sums of each other but don't know where else to go with it.</p>
| marty cohen | 13,079 | <p>If $a_n/b_n \to 0$,
then,
for any $c > 0$,
there is an
$N(c)$
such that
$|a_n/b_n|
< c$
for $n > N(c)$,
or
$|a_n|
< c|b_n|$.</p>
<p>If $|b_n|$
is bounded,
there is a $B > 0$
such that
$|b_n| < B$.</p>
<p>Therefore
$|a_n|
< cB$
for $n > N(c)$.</p>
<p>Finally,
choose
$c = \epsilon/B$.
Then
$... |
3,527,004 | <p>As stated in the title, I want <span class="math-container">$f(x)=\frac{1}{x^2}$</span> to be expanded as a series with powers of <span class="math-container">$(x+2)$</span>. </p>
<p>Let <span class="math-container">$u=x+2$</span>. Then <span class="math-container">$f(x)=\frac{1}{x^2}=\frac{1}{(u-2)^2}$</span></p>
... | almagest | 172,006 | <p>Narrowly on your question: <strong>no</strong>, it is not correct. But it is almost correct.</p>
<p>The error is that just before the end you forgot to get the range for <span class="math-container">$n$</span> correct. When you differentiate a term in <span class="math-container">$x^0$</span> you get 0, not a term ... |
1,694,159 | <p>I am prepping for my mid semester exam, and came across with the following question:</p>
<blockquote>
<p>Find the closed form for the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$, using the assumption that $k = 0, 1,...n$ and $n$ can be any natural number.</p>
</blockquote>
<p>So what I have done is to ... | T.A.Tarbox | 413,002 | <p>Let the integer $k$ be fixed and let $\mathcal C_k$ denote the linear operator acting upon polynomials in $x$ that returns the value of the coefficient of $x^k$.
Then evidently,
\begin{align}
\sum_{j=k}^n {(-1)^{j+k}\binom{n}{j}\binom{j}{k}} &= \sum_{j=k}^n {(-1)^{j}\binom{n}{j}\cdot \mathcal C_k \cdot (1-x)^j} ... |
226,265 | <blockquote>
<p>Suppose <span class="math-container">$(X,d)$</span> is a metric space. Does every open cover of <span class="math-container">$X$</span> have a minimal subcover with respect to inclusion?</p>
</blockquote>
<p>In other words:</p>
<blockquote>
<p>If <span class="math-container">$\mathcal{O}$</span> i... | Cameron Buie | 28,900 | <p>Take any $x\in X$, and consider the covering by the sets $\{y\in X:d(x,y)<n\}$ for all positive integers $n$. If $X$ is unbounded (with respect to the metric $d$), then this open cover has no minimal subcover.</p>
|
226,265 | <blockquote>
<p>Suppose <span class="math-container">$(X,d)$</span> is a metric space. Does every open cover of <span class="math-container">$X$</span> have a minimal subcover with respect to inclusion?</p>
</blockquote>
<p>In other words:</p>
<blockquote>
<p>If <span class="math-container">$\mathcal{O}$</span> i... | Thomas Andrews | 7,933 | <p>Let <span class="math-container">$X=\mathbb R$</span> and <span class="math-container">$U_n=(-n,n)$</span>. Then <span class="math-container">$\{U_n\}$</span> covers <span class="math-container">$X$</span> but it doesn't have a minimal sub-cover.</p>
|
1,349 | <p>In this question here the OP asks for hints for a problem rather than a full proof.</p>
<p><a href="https://math.stackexchange.com/questions/14477">Proof of subfactorial formula $!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$</a></p>
<p>Now, while I would like to respect that request, I also feel that questi... | JDH | 413 | <p>What may be needed is a means by which an answer is visible to everyone except the OP.</p>
|
360,293 | <p>Calculate $\sum_{n=2}^\infty ({n^4+2n^3-3n^2-8n-3\over(n+2)!})$</p>
<p>I thought about maybe breaking the polynomial in two different fractions in order to make the sum more manageable and reduce it to something similar to $\lim_{n\to\infty}(1+{1\over1!}+{1\over2!}+...+{1\over n!})$, but didn't manage</p>
| lab bhattacharjee | 33,337 | <p>Express $$n^4+2n^3-3n^2-8n-3=(n+2)(n+1)n(n-1)+B(n+2)(n+1)n+C(n+2)(n+1)+D(n+2)+E--->(1)$$</p>
<p>So that $$T_n=\frac{n^4+2n^3-3n^2-8n-3}{(n+2)!}=\frac1{(n-2)!}+\frac B{(n-1)!}+\frac C{(n)!}+\frac D{(n+1)!}+\frac E{(n+2)!}$$</p>
<p>Putting $n=-2$ in $(1), E=2^4+2\cdot2^3-3\cdot2^2-8\cdot2-3=1$</p>
<p>Similarly, ... |
770,430 | <p>How to find the value of $X$?</p>
<p>If $X$= $\frac {1}{1001}$+$\frac {1}{1002}$+$
\frac {1}{1003}$. . . . $\frac {1}{3001}$</p>
| Umberto | 92,940 | <p>Mathematica gives an approximation of</p>
<p>1.098612251</p>
<p>Ref.: <a href="http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427egmfbkbu8jv&mail=1" rel="nofollow">http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427egmfbkbu8jv&mail=1</a></p>
<p>Just to note that ... |
572,125 | <p>How to show this function's discontinuity?<br></p>
<p>$ f(n) = \left\{
\begin{array}{l l}
\frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\
0 & \quad , \quad(x,y)=(0,0)
\end{array} \right.$</p>
| Brian M. Scott | 12,042 | <p>If $\kappa$ is a regular cardinal, and $\alpha$ is any ordinal less than $\kappa$, then every function $f:\alpha\to\kappa$ is bounded. In particular, any function $f:\omega\to\omega_2$ is bounded.</p>
<p>More generally, all functions $f:\alpha\to\kappa$ will be bounded if $\alpha<\operatorname{cf}\kappa$, the <a... |
4,134,734 | <p>I'm trying to verify that a certain function of two variables <span class="math-container">$F(x,y)$</span> satisfies the conditions of a joint CDF. Showing that each condition holds has been fairly straightforward except, that is, for the condition that</p>
<p><span class="math-container">$a<b,c<d\implies F(b,... | Vons | 274,987 | <p>The condition is the same as checking <span class="math-container">$\frac{F(b,d)-F(b,c)}{F(a,d)-F(a,c)}\ge 1$</span>. Geometrically this is the slope of any cross sectional slice parallel to the xz-plane is greater if the x-value is greater.</p>
<p>Here are some cases that are worth considering.</p>
<p><span class="... |
19,148 | <p>I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?</p>
<p>I think it comes down to not having a ... | Yuri Bakhtin | 2,968 | <p>Suppose you are in the context of collecting data and estimating the mean.</p>
<p>Imagine a situation where SLLN does not hold. It means that with positive probability accumulating new data is useless.</p>
|
2,227,709 | <p>Find the solution of the $x^2+2x+3 \equiv0\mod{198}$</p>
<p>i have no idea for this problem i have small hint to we going consider $x^2+2x+3 \equiv0\mod{12}$</p>
| Travis Willse | 155,629 | <p>Here's a more-or-less generalizable, manual way of finding all of the solutions:</p>
<p>First, as P. Vanchinathan does, change variable to $a := x + 1$, which transforms the equation into one with zero linear term:
$$a^2 + 2 \equiv 0 \pmod {198} .$$
(This step is option, but reduces the amount of later work.)</p>
... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | Asher2211 | 742,113 | <p><span class="math-container">$$(x-y)^2\ge0$$</span><br />
<span class="math-container">$$⇒ x^2+y^2\ge2xy=(x+y)^2-x^2-y^2=(x+y)^2-1$$</span><br />
<span class="math-container">$$⇒ 1=x^2+y^2\ge(x+y)^2-1$$</span><br />
<span class="math-container">$$⇒ 2\ge(x+y)^2$$</span><br />
<span class="math-container">$$⇒ x+y≤\sqr... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | Community | -1 | <p><span class="math-container">$$x^2+y^2=1$$</span> is the unit circle, and</p>
<p><span class="math-container">$$\cos(\theta)+\sin(\theta)=\sqrt2\cos\left(\theta-\dfrac\pi4\right)\le\sqrt 2.$$</span></p>
|
3,960,549 | <p>There are <span class="math-container">$3$</span> cyclone on average in a year in russia .</p>
<p>What is the probability that there will be cyclone in the next TWO years ?</p>
<p>I just want to know the value of <span class="math-container">$\lambda$</span> in Poisson distribution! As we know <span class="math-c... | tommik | 791,458 | <p><span class="math-container">$\lambda=6$</span> thus your probability is</p>
<p><span class="math-container">$$P[X>0]=1-e^{-6}\approx 99.75\%$$</span></p>
|
3,960,549 | <p>There are <span class="math-container">$3$</span> cyclone on average in a year in russia .</p>
<p>What is the probability that there will be cyclone in the next TWO years ?</p>
<p>I just want to know the value of <span class="math-container">$\lambda$</span> in Poisson distribution! As we know <span class="math-c... | Botnakov N. | 452,350 | <p>I suppose that your question means that there is some misunderstanding in the question about taking <span class="math-container">$2 \lambda$</span>: otherwise there would be no questions.</p>
<p>Let <span class="math-container">$\xi$</span> be the number of cyclones during the first year and <span class="math-contai... |
2,605,546 | <p>Simple question but not sure why for, $ f = \frac{\lambda}{2}\sum_{j=1}^{D} w_j^2$
$$\frac{\partial f}{\partial wj}= \lambda w_j$$</p>
<p>I would have thought the answer would be $\frac{\partial f}{\partial wj}= \lambda \sum_{j=1}^{D} w_j^2$</p>
<p>Since we get the derivative of $w_j^2$ which is $2w_j$, pull out t... | Community | -1 | <p>$\frac {\partial \omega_j^2}{\partial \omega_j}=2\omega _j$
And, $\frac {\partial \omega_i^2}{\partial \omega_j}=0$ for $i\not =j $. ..</p>
|
2,605,546 | <p>Simple question but not sure why for, $ f = \frac{\lambda}{2}\sum_{j=1}^{D} w_j^2$
$$\frac{\partial f}{\partial wj}= \lambda w_j$$</p>
<p>I would have thought the answer would be $\frac{\partial f}{\partial wj}= \lambda \sum_{j=1}^{D} w_j^2$</p>
<p>Since we get the derivative of $w_j^2$ which is $2w_j$, pull out t... | user247327 | 247,327 | <p>What you have written doesn't quite make sense! The given function is a function of the D variables, $\omega_1, \omega_2, \cdot\cdot\cdot, \omega_D$. But then what variable do you want to differentiate with respect to? Having used "j" as the summation index, you should not then use "j" as an index outside that su... |
676,171 | <blockquote>
<p>Prove that if $F$ is a field, every proper prime ideal of $F[X]$ is maximal.</p>
</blockquote>
<p>Should I be using the theorem that says an ideal $M$ of a commutative ring $R$ is maximal iff $R/M$ is a field? Any suggestions on this would be appreciated. </p>
| Alex Youcis | 16,497 | <p>Yes.</p>
<p>Hint: Every prime ideal $P$ of $F[x]$ is of the form $P=(f(x))$ for some polynomial $f$. Use this to show that $F[X]/P$ is a domain which is a finite dimensional vector space, and so a field (why?).</p>
|
3,795,932 | <p><span class="math-container">$\mathbb {Z}G $</span> is not Artinian where <span class="math-container">$ G$</span> is a finite group.</p>
<p>I know that <span class="math-container">$\mathbb {Z} $</span> is not Artinian but <span class="math-container">$\mathbb {Z} $</span> is not an ideal of the group ring. So H... | b00n heT | 119,285 | <p>I am not sure that the fraction simplifies as you say (check that again). But the fact that the two results are equivalent derives from
<span class="math-container">$$\ln(2x-2)=\ln(2\cdot(x-1))=\ln(2)+\ln(x-1),$$</span>
So that the <span class="math-container">$2$</span> gets put into the constant <span class="math-... |
849,038 | <p>Im once again struggling to see the equivalence of two definitions. In my abstract algebra book (abstract algebra by beachy and blair) it says that elements $d_1,d_2,....,d_n \in D$ where $D$ is a UFD is said to be relatively prime in $D$ if there is no irreducible element $p \in D$ such that $p \mid a_i$ for $i=1,2... | Michael Hardy | 11,667 | <p>Here I am thinking I should remember something about this, or maybe I shouldn't. We have a matrix "equation":
$$
\begin{bmatrix} 1 & h_1 & w_1 \\ \vdots & \vdots & \vdots \\ 1 & h_n & w_n \end{bmatrix} \begin{bmatrix} 7 \\ 0.08 \\ 0.06 \end{bmatrix} \overset{\text{?}} = \begin{bmatrix} a_1 \... |
3,738,789 | <p>I know if I stick two pins on a paper, and trace a taut loop around them, I get an ellipse. With one pin, I get a circle. Question is, are there names for shapes I get if I trace a taut loop around 3, 4, 5, ..., k pins, assuming the pins are not collinear, and the polygon formed by joining them is convex i.e. every ... | Community | -1 | <p>If you keep the loop taut, the moving part will form a variable triangle defined by two pins and the pen for a while, then change one pin at a time. During this process, the pen draws arcs of ellipse, forming a continuous curve with continuous tangent.</p>
<p>The endpoints of the arcs will be found by lengthening th... |
2,543,834 | <p>Ok, so in my differential equations class we've been doing problems which more or less amount to solving equations of the form:</p>
<p><span class="math-container">$$\frac{dY}{dt} = AY$$</span></p>
<p>Where <span class="math-container">$A$</span> is just some <span class="math-container">$2\times2$</span> linear tra... | Doug M | 317,162 | <p>$Y' = A Y\\
Y = e^{At}Y_0$</p>
<p>$e^{At} = \sum_\limits{n=0}^\infty \frac {A^nt^n}{n!}$</p>
<p>$A = PDP\\
A^n = PD^nP^{-1}$</p>
<p>$e^{At} = P\left(\sum_\limits{n=0}^\infty \frac {D^nt^n}{n!}\right)P^{-1}\\
Y = Pe^{Dt}P^{-1}Y_0$</p>
<p>$e^{Dt} = \begin{bmatrix} e^{\lambda_1 t}\\&e^{\lambda_2 t}\end{bmatrix... |
673,385 | <p>Question:</p>
<blockquote>
<p>Determine the multiplicative inverse of $x^2 + 1$ in $GF(2^4)$ with $$m(x) = x^4 + x + 1.$$ </p>
</blockquote>
<p>My confusion is over the $GF (2^4)$.</p>
| Vadim | 26,767 | <p><strong>Hint</strong>:</p>
<p>$$I_1(x)=\int_x^1\frac{dt}{\sqrt{1-t^4}}=\frac{1}{\sqrt{2}}\int_{0}^{\arccos x}\frac{du}{\sqrt{1-\frac{1}{2}\sin^2u}}$$</p>
<p>$$I_2(x)=\int_0^x \frac{dt}{\sqrt{1+t^4}}=\frac{1}{2}\int_{0}^{\arccos\frac{1-x^2}{1+x^2}}\frac{du}{\sqrt{1-\frac{1}{2}\sin^2u}}$$</p>
<p>and since $$\arccos... |
673,385 | <p>Question:</p>
<blockquote>
<p>Determine the multiplicative inverse of $x^2 + 1$ in $GF(2^4)$ with $$m(x) = x^4 + x + 1.$$ </p>
</blockquote>
<p>My confusion is over the $GF (2^4)$.</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}... |
673,385 | <p>Question:</p>
<blockquote>
<p>Determine the multiplicative inverse of $x^2 + 1$ in $GF(2^4)$ with $$m(x) = x^4 + x + 1.$$ </p>
</blockquote>
<p>My confusion is over the $GF (2^4)$.</p>
| math110 | 58,742 | <p>The first one:let $t^2=\sin{x}$,then
$$I_{1}=\dfrac{1}{2}\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{\sqrt{\sin{x}}}$$</p>
<p>The second one: let
$t^2=\tan{x}$</p>
<p>then
$$I_{2}=\dfrac{1}{2}\int_{0}^{\frac{\pi}{4}}\dfrac{dx}{\sqrt{\sin{x}\cos{x}}}=\dfrac{1}{2\sqrt{2}}\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{\sqrt{\sin{x}}}$$
s... |
3,195,618 | <p>Prove that topological space <span class="math-container">$ \mathbb{R^2} $</span> with dictionary order topology is first countable, but not second countable.</p>
<p>I am a bit stuck. Some hints would help. For first countability I am having trouble finding a local base for each <span class="math-container">$ (x,y)... | Dr. Sonnhard Graubner | 175,066 | <p>Substituting <span class="math-container">$$\frac{dy(x)}{dx}=\frac{dt}{dx}\frac{dy(t)}{dt}$$</span> we get
<span class="math-container">$$y''(t)+9y(t)=0$$</span> and <span class="math-container">$$y''(x)=\frac{d^2}{dx^2}\frac{dy(t)}{dt}+\left(\frac{dt}{dx}\right)^2\frac{d^2y(t)}{dt^2}$$</span></p>
|
237,446 | <p>I find to difficult to evaluate with $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )$$ I tried to use the fact, that $$\frac{1}{1-n} \geqslant \ln(n)\geqslant 1+n$$
what gives $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right ) \geqslant \lim_{n\rightarrow\inft... | WimC | 25,313 | <p>$\log(x) \leq x - 1$, so $\log(x) = n \log(x^{1/n}) \leq n (x^{1/n} - 1)$ for all integral $n \geq 1$. Take $x = \log(n)$ to get $\log(\log(n)) \leq n(\log(n)^{1/n}-1)$ or $n(1-\log(n)^{1/n}) \leq -\log(\log(n))$. This shows that your limit is $-\infty$.</p>
|
237,446 | <p>I find to difficult to evaluate with $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right )$$ I tried to use the fact, that $$\frac{1}{1-n} \geqslant \ln(n)\geqslant 1+n$$
what gives $$\lim_{n\rightarrow\infty}\left ( n\left(1-\sqrt[n]{\ln(n)} \right) \right ) \geqslant \lim_{n\rightarrow\inft... | lisyarus | 135,314 | <p>Use Taylor!</p>
<p>$$n(1-\sqrt[n]{\log n}) = n (1-e^{\frac{\log\log n}{n}}) \approx n\left(1-\left(1+\frac{\log\log n}{n}\right)\right) = - \log\log n$$</p>
<p>which clearly tends to $-\infty$.</p>
|
1,902,878 | <p>If $a^x=bc$, $b^y=ca$ and $c^z=ab$, prove that: $xyz=x+y+z+2$.</p>
<p>My Approach;
Here,</p>
<p>$$a^x=bc$$
$$a={bc}^{\frac {1}{x}}$$</p>
<p>and,</p>
<p>$$b={ca}^{\frac {1}{y}}$$
$$c={ab}^{\frac {1}{z}}$$</p>
<p>I got stopped from here. Please help me to continue </p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>We have $$abc=a^{x+1}=b^{y+1}=c^{z+1}=K\text{(say)}$$</p>
<p>$\implies a=K^{1/(x+1)}$ etc.</p>
<p>Put the values of $a,b,c$ in one of $a^x=bc, b^y=ca,c^z=ab$</p>
|
3,776,217 | <p>Prove that</p>
<p><span class="math-container">\begin{equation}
y(x) = \sqrt{\dfrac{3x}{2x + 3c}}
\end{equation}</span></p>
<p>is a solution of</p>
<p><span class="math-container">\begin{equation}
\dfrac{dy}{dx} + \dfrac{y}{2x} = -\frac{y^3}{3x}
\end{equation}</span></p>
<p>All the math to resolve this differential ... | user | 505,767 | <p>Your work seems fine and we obtain</p>
<p><span class="math-container">$$\dfrac{dy}{dx} =\dfrac{y}{2x} - \dfrac{y^3}{3x}$$</span></p>
<p>which should be the correct differential equation.</p>
<p>I don't understand why you have added the term <span class="math-container">$P(x)y$</span> in the last step.</p>
|
3,776,217 | <p>Prove that</p>
<p><span class="math-container">\begin{equation}
y(x) = \sqrt{\dfrac{3x}{2x + 3c}}
\end{equation}</span></p>
<p>is a solution of</p>
<p><span class="math-container">\begin{equation}
\dfrac{dy}{dx} + \dfrac{y}{2x} = -\frac{y^3}{3x}
\end{equation}</span></p>
<p>All the math to resolve this differential ... | Lutz Lehmann | 115,115 | <p>You could probably get an easier calculation by going over the logarithmic derivative,
<span class="math-container">$$
\log(y(x))=\frac12(\log(3x)-\log(2x+3c))
\\~\\
\implies
\frac{y'(x)}{y(x)}=\frac12\left(\frac1x-\frac2{2x+3c}\right)
=\frac1{2x}-\frac1{3x}y(x)^2
$$</span>
The last step is obtained by doing the min... |
2,886,675 | <p>I suspect the following is exactly true ( for positive $\alpha$ )</p>
<p>\begin{equation}
\sum_{n=1}^\infty e^{- \alpha n^2 }= \frac{1}{2} \sqrt { \frac{ \pi}{ \alpha} }
\end{equation}</p>
<p>If the above is exactly true, then I would like to know a proof of it.
I accept showing a particular limit is true, may b... | Community | -1 | <p>No doubt that the equality is wrong. For large $\alpha$, the first term dominates and the asymptotic behavior is $e^{-\alpha}$.</p>
<p>No even sure that there exist a value of $\alpha$ such that the expressions are equal.</p>
|
2,660,934 | <p>Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$</p>
<p>We know $-1\le \sin \frac{11}{x} \le 1 $ </p>
<p>Therefore, $x\rightarrow \infty $
And so limit of this function does not exist. </p>
<p>Am I on the right track? Any help is much appreciated.</p>
| E.H.E | 187,799 | <h2>Hint</h2>
<p><span class="math-container">$$\lim\limits_{x \to \infty} x\sin\left(\frac{11}{x}\right)=11\lim\limits_{x \to 0^+} \frac{\sin(11x)}{11x}$$</span></p>
|
2,660,934 | <p>Find $$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\right)$$</p>
<p>We know $-1\le \sin \frac{11}{x} \le 1 $ </p>
<p>Therefore, $x\rightarrow \infty $
And so limit of this function does not exist. </p>
<p>Am I on the right track? Any help is much appreciated.</p>
| user | 505,767 | <p>It is true that</p>
<p>$$-1\le \sin \frac{11}{x} \le 1$$</p>
<p>but since $x\to \infty$ we have that $$\sin \frac{11}{x}\to0$$</p>
<p>thus the limit is in the indeterminate form $0\cdot \infty$.</p>
<p>To solve we can set for example $y=\frac1x\to 0$ then</p>
<p>$$\lim_{x \to \infty} x\sin\left(\frac{11}{x}\rig... |
417,064 | <p>Let T be a totally ordered set that is <strong>finite</strong>. Does it follow that minimum and maximum of T exist?
Since T is finite, I believe there exists a minimal of T. From that it maybe able to be shown that the minimal is the minimum but not quite sure whether it is the right approach. </p>
| Seirios | 36,434 | <p>By induction, you can show that any linearly ordered set $T$ of cardinality $n>0$ has a minimum and a maximum. </p>
<p>If $n=1$, $T=\{p\}$ and $p$ is the minimum and the maximum of $T$. Then, if $\text{card}(T)=n+1$ take a subset of cardinality $n$ ant its minimum $m$ and its maximum $M$. If $p$ is the last elem... |
3,991,351 | <p>As stated in the title.</p>
<p>Any arbitrary function can be expressed as
<span class="math-container">$$f(x)=\frac{a_0}{2}+\sum^{\infty}_{n=1}(a_n\cos(nx)+b_n\sin(nx)) \tag{1}$$</span>
where
<span class="math-container">$$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx \tag{2}$$</span>
<span class="math-container"... | frog | 84,997 | <p>If you allow for complex coefficients, then the Fourier-series of <span class="math-container">$x\mapsto \cos (x) + \mathrm i \sin(x)$</span> is just itself (by uniqueness of the Fourier-series).
You can formally compute the Fourier-coefficients of <span class="math-container">$x\mapsto \mathrm e^{\mathrm i x}$</spa... |
3,246,240 | <p>I have the following problem</p>
<p><span class="math-container">$\frac{d^2y}{dx^2} + \lambda y = 0 , y'(0)=0$</span> and <span class="math-container">$y(3)=0$</span></p>
<p>I'm trying to solve for the eigenvalues <span class="math-container">$\lambda_n$</span> for <span class="math-container">$n=1,2,3...$</span>
... | uniquesolution | 265,735 | <p>Starting with your inequality
<span class="math-container">$$|\hat f(\alpha +h)-\hat f(\alpha )|\leq \int_{\mathbb R}|f(x)||e^{-2i\pi xh}-1|dx,$$</span>
observe that the r.h.s does not depend on <span class="math-container">$\alpha$</span>, hence
<span class="math-container">$$\sup_{\alpha\in\mathbb{R}}|\hat f(\alph... |
88,319 | <p>I've had the same effect in Mathematica 9 and 10.</p>
<p>I'm trying to color a 3D Plot with another function, let's call it colorFun ( it should highlight the areas where the colorFun is above a certain threshold), but ColorFunction seems to use the wrong coordinates.</p>
<p>Horribly colored minimal example</p>
<... | Mr.Wizard | 121 | <p>You'll get a much crisper output if you use Mesh functionality instead:</p>
<pre><code>Plot3D[x^2 + y^2, {x, 0, 1}, {y, 0, 2},
MeshFunctions -> {#/#2 &},
Mesh -> {{1}},
MeshShading -> {Red, Blue}
]
</code></pre>
<p><img src="https://i.stack.imgur.com/YHxOM.png" alt="enter image description here... |
2,596,700 | <p>We can see, for example, that the years 2009 and 2015 have identical calendars. Similarly, 2000 and 2028.</p>
<p>I read once that given any year X at most 28 years later there will be another year Y, with the calendar identical to that of X.</p>
<p>Here I am referring to our usual calendar, the Gregorian.</p>
<p>... | Azlif | 513,870 | <p>Hint :
Write the limit as $$\left[\left(1 - \frac{1}{a_n}\right)^{a_n} \right]^{\frac{n}{a_n}}.$$
The limit inside the bracket is $\frac{1}{e}$; what must the value of $n/{a_n}$, as $n\to \infty$, be, to ensure that the limit converges?</p>
|
786,643 | <p>Considering $$\int\frac{\ln(x+1)}{2(x+1)}dx$$ I first solved it seeing it similar to the derivative of $\ln^2(x+1)$ so multiplying by $\frac22$ the solution is $$\int\frac{\ln(x+1)}{2(x+1)}dx=\frac{\ln^2(x+1)}{4}+const.$$. But then we can solve it using by parts' method and so this is the solution that I found:
$$\f... | RRL | 148,510 | <p>Look at the denominator in the integrand of your final result. The 2 is missing that was present in the original integral. Divide both sides of the equation by 2 and you get the desired result.</p>
|
786,643 | <p>Considering $$\int\frac{\ln(x+1)}{2(x+1)}dx$$ I first solved it seeing it similar to the derivative of $\ln^2(x+1)$ so multiplying by $\frac22$ the solution is $$\int\frac{\ln(x+1)}{2(x+1)}dx=\frac{\ln^2(x+1)}{4}+const.$$. But then we can solve it using by parts' method and so this is the solution that I found:
$$\f... | kmitov | 84,067 | <p>Both answers are the same. Can you see that </p>
<p>$\int\frac{ln(x+1)}{(x+1)}dx=\frac{1}{2}\ln(x+1)ln(x+1)+cost.$</p>
<p>$\frac{1}{2}\int\frac{ln(x+1)}{(x+1)}dx=\frac{1}{4}\ln(x+1)ln(x+1)+cost.$</p>
|
66,314 | <p>This is very similar to my earlier question <a href="https://mathematica.stackexchange.com/questions/60069/one-to-many-lists-merge">One to Many Lists Merge</a> but somehow different. I have two lists, first column in each list represents its key. I want to merge these two lists. The only problem is that these two l... | Mr.Wizard | 121 | <p>Another formulation:</p>
<pre><code>merge1[a_List, b_List, pad_] :=
Module[{rules, keys},
rules = Apply[# -> {##2} &, {a, b}, {2}];
keys = Union @@ Keys @ rules;
Join[List /@ keys, ##, 2] & @@
(Lookup[#, keys, pad & /@ #[[1, 2]] ] & /@ rules)
]
</code></pre>
<p>Test:</p>
<... |
3,970,488 | <p>while solving a differential equation i encounter this derivative : let <span class="math-container">$$z=\frac {dt}{dx} $$</span> i don't understand how they make that <span class="math-container">$$ \frac {dz}{dx}=z^3\frac {d^2x}{dt^2}$$</span></p>
| Ninad Munshi | 698,724 | <p>Equivalently we have the expression</p>
<p><span class="math-container">$$\frac{dx}{dt}=\frac{1}{z}$$</span></p>
<p>then take <span class="math-container">$\frac{d}{dt}$</span> on both sides and apply chain rule</p>
<p><span class="math-container">$$\frac{d^2x}{dt^2} = \frac{d}{dt}\frac{1}{z} = \frac{dx}{dt}\cdot\le... |
115,387 | <p>Have two series, just a quick check of some simple series:</p>
<p>$\sum _{1}^{\infty} \frac {1}{\sqrt {2n^{2}-3}}$</p>
<p>Considering
$\frac {1}{\sqrt {2n^{2}-3}}$ > $\frac {1}{\sqrt {4n^{2}}}$ = $\frac {1}{2n}$</p>
<p>Since
$\sum _{1}^{\infty} \frac {1}{2n}$ $\rightarrow$ Diverges, hence by the camparsion tes... | David Mitra | 18,986 | <p>The first argument is ok (you should start the series at $n=2$ though). </p>
<p>The second is not. As you imply, for the second series, you should be able to compute the <i>value of</i> the limit of (part of) the <i> sequence of terms you are adding</i>: $\lim\limits_{n\rightarrow\infty}(1+{1\over n})^n$. Then, b... |
3,693,735 | <p><span class="math-container">$X \not= \emptyset$</span>,<span class="math-container">$Y \not= \emptyset$</span>,<span class="math-container">$(X,T)$</span> and <span class="math-container">$(Y,V)$</span> are topological space. Let <span class="math-container">$f:X \rightarrow Y$</span> function is a homeomorphizm an... | QuantumSpace | 661,543 | <p>Your answer is correct. Here is a more "standard" approach, using the Sylow-theorems (which should be the first thing you think of given your assumptions):</p>
<p>By Sylow's third theorem we have a unique normal Sylow <span class="math-container">$p$</span>-group <span class="math-container">$P$</span> (use <span c... |
273,619 | <p>How to use a <code>list</code> to specify part of another nested list <code>mat</code>? We don't want to write <code>mat[[list[[1]],list[[2]],list[[3]],...]]</code>.</p>
<pre><code>mat = RandomInteger[10, {5, 6, 7, 8}];
list = {3, 4, 6};
mat[[3, 4, 6]]
</code></pre>
| lericr | 84,894 | <p>Couple of options.</p>
<p>Use Extract instead:</p>
<pre><code>Extract[mat, list]
</code></pre>
<p>Apply Sequence to the list:</p>
<pre><code>mat[[Sequence @@ list]]
</code></pre>
|
273,619 | <p>How to use a <code>list</code> to specify part of another nested list <code>mat</code>? We don't want to write <code>mat[[list[[1]],list[[2]],list[[3]],...]]</code>.</p>
<pre><code>mat = RandomInteger[10, {5, 6, 7, 8}];
list = {3, 4, 6};
mat[[3, 4, 6]]
</code></pre>
| Syed | 81,355 | <p>A variation could be:</p>
<pre><code>Fold[Part, mat, list]
</code></pre>
<blockquote>
<p><code>{9, 0, 9, 2, 6, 5, 4, 3}</code></p>
</blockquote>
|
1,771,920 | <p>Okay so here's the question </p>
<blockquote>
<p>Seventy percent of the light aircraft that disappear while in flight
in a certain country are subsequently discovered. Of the aircraft that
are discovered, 60% have an emergency locator, whereas 90% of the
aircraft not discovered do not have such a locator. S... | JKnecht | 298,619 | <p>$P(D \mid E)$ </p>
<p>$= P(D \cap E)/P(E)$</p>
<p>$= P(D \cap E)/P((E \cap D) \cup (E \cap \overline{D}))$</p>
<p>$= P(D \cap E)/\{P(E \cap D) + P(E \cap \overline{D})\}$</p>
<p>$= \{P(D) \cdot P(E \mid D)\}/\{(P(D) \cdot P(E \mid D)) + P(\overline{D}) \cdot P(E \mid \overline{D})\}$</p>
<p>$= (0.70 \cdot 0.60)... |
202,379 | <p>Suppose for some constants $\alpha,\beta,\gamma$ that we're given the following ODE: $$\alpha y''+\beta xy'+\gamma y=0.$$ Now, I know how to find the general solution for $y(x)$ if any of $\alpha,\beta,\gamma$ should turn out to be $0$, but I've just ended up with the ODE $$2y''+xy'+y=0.$$ Can anybody give me the fi... | Robert Israel | 8,508 | <p>Maple gives the general solution using the Kummer M and U functions.</p>
<p>$$ y \left( x \right) =c_{{1}}{{\rm e}^{-{\frac {\beta\,{x}^{2}}{
2 \alpha}}}}
{{\rm M}\left(-{\frac {-2\,\beta+\gamma}{2\beta}},\frac{3}{2},\,{\frac {\beta\,{x}^{2}}{2\alpha}}\right)}
x+c_{{2}}{{\rm e}^{-{\frac {\beta\,{x}^{2}}{2\alpha}}}}... |
202,379 | <p>Suppose for some constants $\alpha,\beta,\gamma$ that we're given the following ODE: $$\alpha y''+\beta xy'+\gamma y=0.$$ Now, I know how to find the general solution for $y(x)$ if any of $\alpha,\beta,\gamma$ should turn out to be $0$, but I've just ended up with the ODE $$2y''+xy'+y=0.$$ Can anybody give me the fi... | Luis Costa | 39,495 | <p>The general form as you have it, is a hypergeometric differential equation. You can manipulate it into a standard form and then apply the Frobenius method. It's already worked out here for several cases:</p>
<p><a href="http://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation" rel="nofollow">h... |
701,241 | <p><code>¬(p∨q)∧(p∨r)</code> Does this mean the negation of both <code>(p∨q)</code> and <code>(p∨r)</code> or just <code>(p∨q)</code>?
If it was just <code>p∨q</code> it would make more sense to me being inside the brackets like <code>(¬p∨q)</code> but maybe that's just the programmer in me. I have also seen <code>(¬p∨... | fgp | 42,986 | <p>Usually negation is assumed to have rather high precedece, i.e. binds strongly to whatever stand to the right of it. So $\lnot a \land b$ means $(\lnot a)\land b$, $\lnot a \lor b$ similarly means $(\lnot a )\lor b$. Whether or not $a$ and $b$ are variables or themselves expressions is irrelevant, so $\lnot(p\lor q)... |
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