qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,700,246 | <p>Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an element $g\in SO_{3}(q)$ defined by: $g: e\mapsto -e$, $f\mapsto \frac{1}{2}e -f +d$, $d\mapsto e+d$. I would like to de... | christina_g | 307,603 | <p>It is obvious that if $y \leq 0$ there is no such $r$. If $y>0$ and $x <0$ then $r=0$. Now if $x \geq 0$, then because of the density of rational numbers you can find a $r \in \Bbb Q, \; \sqrt{x} < r < \sqrt{y}$ and you have $x<r^2<y$. I suppose that you do not have to prove the density of $\Bbb... |
3,380,081 | <p>Question: Suppose <span class="math-container">$n(S)$</span> is the number of subset of <span class="math-container">$S$</span> and <span class="math-container">$|S|$</span> be the number of elements of <span class="math-container">$S$</span>. If <span class="math-container">$n(A)+n(B)+n(C)=n(A\cup B\cup C)$</span> ... | Arthur | 15,500 | <blockquote>
<p>Is it safe to assume that if I keep going infinitely the answer will be exactly 14/3?</p>
</blockquote>
<p>Strictly speaking, no, it is not. The answer could be some number <em>really</em> close to <span class="math-container">$\frac{14}{3}$</span>, and the only way to tell would be to actually do th... |
621,461 | <p>I'm having trouble understanding division when the divisor is greater than the dividend, for ex 1/4.</p>
<p>I think of division as "how many times can the divisor fit into the dividend evenly". </p>
<p>Intuitively, when I see 1/4 in the context of slices of pizza, I think of it as 1 "out of" 4, but I can't seem to... | Bill Dubuque | 242 | <p>$X$ equals $\,\dfrac{1}4\,$ of the total $\,T$ means $\,X\, =\, \dfrac{1}4 T\, =\, \dfrac{T}4\ \,$ or $\ \color{#c00}{\dfrac{X}T = \dfrac{1}4}\,\ $ or $\,\ 4X = T$</p>
<p>$X$ equals $\dfrac{A}B$ of the total $\,T$ means $\,X = \dfrac{A}B T = \dfrac{AT}B\ $ or $\ \color{#c00}{\dfrac{X}T = \dfrac{A}B}\ $ or $\ BX = A... |
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Cody Rudisill | 279,692 | <p>Your question touches on issues both mathematical and philosophical; in fact, you might enjoy an introductory text in philosophy of mathematics and logic, or on the foundations of mathematics.</p>
<p>Truth is, an infinity is not <em>just</em> an infinity. One reason for this peculiarity is due to the seminal work o... |
3,792,683 | <p>Where <span class="math-container">$\alpha$</span> is a real constant, consider the sequence {<span class="math-container">$z_n$</span>} defined by <span class="math-container">$z_n=\frac{1}{n^\alpha}$</span>. For which value of <span class="math-container">$\alpha$</span> is {<span class="math-container">$z_n$</spa... | Quanto | 686,284 | <p>Integrate by parts directly</p>
<p><span class="math-container">$$\begin{array}\
\int^{\infty}_0 \frac{e^{-x^2}}{(x^2+\frac{1}{2})^2} dx
&= \int^{\infty}_0 \frac{e^{-x^2}}x d\left( \frac{x^2}{x^2+\frac{1}{2}} \right)=
2\int^{\infty}_0e^{-x^2}dx= \sqrt{\pi} \end{array}$$</span></p>
|
2,021,126 | <p>Let $f :[a , \infty)\to \mathbb{R}$ a positive and Monotonic function such that $\int_a^\infty f$ converge
<br>
prove:
$\lim_{x\to\infty}f(x)=0$</p>
| Gono | 384,471 | <p>That's an easy calculation… assume $$\lim_{x\to\infty}f(x)\not=0$$ then because f is monotone and positive there exists an $\varepsilon > 0$ s.t. $f>\varepsilon$ so $$\int_a^\infty f \ge \int_a^\infty \varepsilon = \infty$$</p>
|
2,021,126 | <p>Let $f :[a , \infty)\to \mathbb{R}$ a positive and Monotonic function such that $\int_a^\infty f$ converge
<br>
prove:
$\lim_{x\to\infty}f(x)=0$</p>
| Patrick Stevens | 259,262 | <p>Alternative answer: if $f$ is constant then the result is trivial; wlog $f$ is decreasing, since if it is increasing then the integral can't converge.</p>
<p>By the integral test for convergence, $\sum_{i=a}^{\infty} f(i)$ converges. Therefore $f(i) \to 0$ as $i \to \infty$ over the integers.</p>
<p>But $f$ is mon... |
3,568,230 | <p>My question is: why, in general we cannot write down an formula for the <span class="math-container">$n-$</span>th term, <span class="math-container">$S_{n}$</span>, of the sequence of partial sums?</p>
<p>I will explain better in the following but the question is basically that one above.</p>
<p>Suppose then you ... | hamam_Abdallah | 369,188 | <p>You had to write <span class="math-container">$C_n$</span> instead of <span class="math-container">$ C$</span>.
In fact we have
<span class="math-container">$$C_n=S_{n-1} \text{ and } \; S_n=C_n+\frac 1n$$</span>
and all we can say is</p>
<p>If <span class="math-container">$(C_n)$</span> converges then the series i... |
3,711,744 | <p>Let <span class="math-container">$C_1\geq C_2\geq\dots\geq C_n$</span> be a fixed set of positive numbers. Maximize the linear function <span class="math-container">$L(x_1, x_2, \dots, x_n)=\sum^n_1C_jx_j$</span> in the closed set described by the inequalities <span class="math-container">$0\leq x_j\leq 1, \sum^n_1 ... | Sergio | 731,870 | <p>Let <span class="math-container">$A,B$</span> be matrices associated to <span class="math-container">$U,T$</span>. You could look for two matrices such that <span class="math-container">$AB=0$</span>, <span class="math-container">$BA\ne AB$</span>, e.g. you could look for two diagonalizable, but not simultaneously d... |
3,920,812 | <p>Please excuse if the formatting of this post is wrong.</p>
<p>There's a question that asks for the 2nd derivative of <span class="math-container">$y-2x-3xy=2$</span></p>
<p>From what I know, I have to use implicit differentiation, using which I get: <span class="math-container">$$\frac{12+18y}{(1-3x)^{2}}$$</span>
B... | Raffaele | 83,382 | <p>Sure you can in this case
<span class="math-container">$$y=\frac{2 (x+1)}{1-3 x}$$</span>
and then
<span class="math-container">$$y'=\frac{8}{(1-3 x)^2}$$</span>
and finally
<span class="math-container">$$y''=\frac{48}{(1-3 x)^3}$$</span></p>
|
3,920,812 | <p>Please excuse if the formatting of this post is wrong.</p>
<p>There's a question that asks for the 2nd derivative of <span class="math-container">$y-2x-3xy=2$</span></p>
<p>From what I know, I have to use implicit differentiation, using which I get: <span class="math-container">$$\frac{12+18y}{(1-3x)^{2}}$$</span>
B... | Community | -1 | <p>We differentiate once,</p>
<p><span class="math-container">$$y'-2-3y-3xy'=0$$</span>
and twice,</p>
<p><span class="math-container">$$y''-3y'-3y'-3xy''=0.$$</span></p>
<p>Now we can eliminate <span class="math-container">$y'$</span>, using</p>
<p><span class="math-container">$$(1-3x)y'=3y+2$$</span></p>
<p>and</p>
<... |
1,102,709 | <p>I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.</p>
<p>Thanks.</p>
| Julián Aguirre | 4,791 | <p>What you have proved is that for a fixed $t\in\mathbb{R}$, given $\epsilon>0$ there is a polynomial $p$ such that $|f(t)-p(t)|<\epsilon$. But the Weierstrass approximation theorem is about uniform approximation; $|f(t)-p(t)|<\epsilon$ must hold for all $t$ on a given set.</p>
<p>It is true that given an in... |
238,392 | <p>Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:</p>
<blockquote>
<p>There does not exist any Riemannian metric on the torus whose sectional curvature is $<0$.</p>... | HJRW | 1,463 | <p>I'm still a little uncertain about this question, but I'll try to say something about the Virtual Haken conjecture (discussed above) and in the process explain why I think it's a good example.</p>
<p>The Virtual Haken conjecture (now Agol's theorem) can be stated as follows --</p>
<blockquote>
<p>Every hyperboli... |
401,389 | <p>I worked through some examples of Bayes' Theorem and now was reading the proof.</p>
<p>Bayes' Theorem states the following:</p>
<blockquote>
<p>Suppose that the sample space S is partitioned into disjoint subsets <span class="math-container">$B_1, B_2,...,B_n$</span>. That is, <span class="math-container">$S = B_1 ... | Abhra Abir Kundu | 48,639 | <p>You can easily show this using set theoretic arguement. </p>
<p>$A \cap S= A \cap (B_1 \cup B_2 \cup \cdots \cup B_n)=(A \cap B_1) \cup (A\cap B_2) \cup \cdots \cup(A \cap B_n)$</p>
|
3,387,458 | <p>Show that a bounded sequence having one limit point is convergent. </p>
<p>The converse holds true. The fact that a convergent seq is bounded has been shown in Baby Rudin. The fact that it will have only one limit point can be found <a href="https://math.stackexchange.com/questions/3386703/prove-that-a-convergent-s... | copper.hat | 27,978 | <p>Suppose <span class="math-container">$x_n \in \mathbb{R}$</span> is bounded and has exactly one limit point. Then
<span class="math-container">$x_n$</span> converges.</p>
<p>Suppose <span class="math-container">$x_{n_k} \to x$</span>, and suppose <span class="math-container">$|x_n| \le B$</span>.</p>
<p>Let <span ... |
1,943,840 | <p>I'm trying to find out for square matrices with $n \geq 2$ :
$$ \det(A-B) = \det(A)-\det(B).$$ </p>
<p>I know that $\det(AB) = \det(A)\det(B)$, but I'm unable to find proof on why a subtraction (or addition) is not equal. Thanks.</p>
| M.Hannan | 372,918 | <p>This is true if $A=B$. </p>
<p>In that case $\det(A-A) = 0$</p>
<p>and $\det(A) - \det(A) = 0$</p>
<p>so $\det(A-A) = \det(A) - \det(A)$</p>
|
367,254 | <p>Solve the following quadratic congreunce</p>
<p>$x^2+ 7x + 10 \equiv 0$ (mod $11$).</p>
<p>I want to know a general and easy method how to solve this kind of questions.</p>
| shobon | 73,393 | <p>Use the quadratic formula $$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$</p>
<p>so $b^2-4ac = 7^2 - 40 = 9 = 3^2$</p>
<p>so $\pm \sqrt{b^2-4ac} = \pm 3$</p>
<p>and $-b \pm 3 = -10,-4$</p>
<p>and so the answers are -5 and -2.</p>
|
367,254 | <p>Solve the following quadratic congreunce</p>
<p>$x^2+ 7x + 10 \equiv 0$ (mod $11$).</p>
<p>I want to know a general and easy method how to solve this kind of questions.</p>
| Warren Moore | 63,412 | <p>For $ax^2+bx+c\equiv 0$ (mod $p$), if $p$ is an odd prime not dividing $a$, then this is the same as solving a simpler question about quadratic residues $y^2\equiv b^2-4ac$ (mod $p$), and $y\equiv 2ax+b$ (mod $p$).</p>
|
476,895 | <p>Show that for $u,v \in \mathbb{C}$ with $|u|<1, |v|<1$, and $\bar{u}v\neq u\bar{v}$, we always have
$$\left|\left(1+|u|^2\right)v-\left(1+|v|^2\right)u\right|>\left|u\bar{v}-\bar{u}v\right|.$$</p>
| achille hui | 59,379 | <p>Divide both sides of the inequality we want to prove by $(1+|u|^2)(1+|v|^2)$, we find
it is equivalent to following statement
$$| y - x | \stackrel{?}{>} |x\bar{y} - y\bar{x}|
\quad\text{ where }\quad x = \frac{u}{1+|u|^2}\quad\text{ and }\quad y = \frac{v}{1+|v|^2}
$$
Notice
$$\begin{align}
& \bar{u} v \ne ... |
476,895 | <p>Show that for $u,v \in \mathbb{C}$ with $|u|<1, |v|<1$, and $\bar{u}v\neq u\bar{v}$, we always have
$$\left|\left(1+|u|^2\right)v-\left(1+|v|^2\right)u\right|>\left|u\bar{v}-\bar{u}v\right|.$$</p>
| TheSimpliFire | 471,884 | <p>I have answered here as <a href="https://math.stackexchange.com/q/3896729/471884">this question</a> which shows context has been closed as a duplicate.</p>
<hr />
<p>Let <span class="math-container">$u=re^{i\theta}$</span> and <span class="math-container">$v=\rho e^{i\phi}$</span>. Then the inequality is equivalent ... |
3,602,323 | <p>Let <span class="math-container">$ m $</span>, <span class="math-container">$ m+1 $</span>, <span class="math-container">$ m+2 $</span>, <span class="math-container">$ \dots $</span>, <span class="math-container">$ m+p-1 $</span> be an integers and let <span class="math-container">$ p $</span> be an odd prime. I wan... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Work in the field <span class="math-container">$\,\mathbf F_p=\mathbf Z/p\mathbf Z$</span> and observe that by Fermat, for any <span class="math-container">$k\in[1..\,p-1]$</span>, one has
<span class="math-container">$$k^{p-2}=k^{-1}.$$</span>
Now, in this field, the map
<span class=... |
614,962 | <blockquote>
<p>We have a continuous function <span class="math-container">$f:(a,b)\to \mathbb R$</span></p>
<p>Prove that: <span class="math-container">$\forall n: x_1...x_n\in(a,b):\exists x\in(a,b)$</span> such that:</p>
<p><span class="math-container">$$f(x)=\frac1n ( f(x_1)+...+f(x_n) ) $$</span></p>
</blockquote>... | Suraj M S | 85,213 | <p>$$\lim_{x\to 0} \frac{x}{\sin (2x)\cos (5x)}$$
apply the formula $$\sin A\cos B=\frac{1}{2}(\sin (A+B)+\cos (A-B))$$
$$\lim_{x\to 0} \frac{2x}{\sin (7x)-\sin (3x)}$$
further use L hospital to get the limit as $\frac{1}{2}$.</p>
|
282,050 | <p>I have equation $y = -x^2 + 2x + 7$.
How can I change it to canonical form, which looks like $y^2 = 2px$ ?
($p$ will be parameter)</p>
<p>What i ve tried so far:
$$\begin{align}
y &= -x^2 + 2x + 7\\
y &= -(x^2 - 2x + 1) + 8\\
(y-8) &= -(x-1)^2 \\
(y-8)^2 &= 2*(0.5)*(x-1)^4
\end{align}
$$</p>
<p>... | lab bhattacharjee | 33,337 | <p>$$-y=x^2-2x-7=(x-1)^2-8\implies (x-1)^2=-(y-8)$$
which is of the form $(x-\alpha)^2=-4a(y-\beta)$</p>
<p>Now, if we are allowed to make the transformation of axes, we can set $x-1=Y,y-8=X$</p>
<p>So,$Y^2=-X=2\left(-\frac12\right)X$</p>
|
2,225 | <p>If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant.</p>
<p>My question is suppose $f$ is continuous and it satisfies $f(x)=f(2x+1)$, then can the domain of $f$ be restricted so that $f$ doesn't remain a constant. If yes, then ... | doraemonpaul | 30,938 | <p>$f(x)=f(2x+1)$</p>
<p>$f(2^x-1)=f(2(2^x-1)+1)$</p>
<p>$f(2^x-1)=f(2^{x+1}-1)$</p>
<p>Note that the general solution is $f(x)=\Theta(\log_2(x+1))$ , where $\Theta(x)$ is any periodic function with unit period</p>
<p>So if the domain of $f$ can be restricted to $f:(-1,\infty)\to\mathbb{R}$ , then $f$ can be non-co... |
945,736 | <p>:)</p>
<p>I have this matrix:</p>
<p>B = \begin{bmatrix}
0.626 & 2.56 & 2.15 & \\
0.835 & 6.66 & 5.16 & \\
0 & 0 & -1.65 &
\end{bmatrix}</p>
<p>I was wondering how to find a givens matrix such that I could apply it from the right side of the matrix and eliminate... | Serge | 32,384 | <p>If you're familiar with left-applied Givens rotations, you can take the transpose of your desired equation an end up in more familiar territory, where your goal becomes creating a lower-triangular matrix with a left-applied Givens rotation.</p>
<p><span class="math-container">$$(\begin{bmatrix} c & -s & 0 \\... |
3,746,630 | <p>So I am solving some probability/finance books and I've gone through two similar problems that conflict in their answers.</p>
<h2>Paul Wilmott</h2>
<p>The first book is Paul Wilmott's <a href="https://smile.amazon.com/Frequently-Asked-Questions-Quantitative-Finance/dp/0470748753" rel="nofollow noreferrer">Frequently... | Brian M. Scott | 12,042 | <p>They’re computing two entirely different things. Wilmott is computing the minimum number of days out of <span class="math-container">$260$</span> on which you must make a profit in order to come out ahead; Joshi is computing the expected value of your portfolio. Applying Joshi’s calculation to Wilmott’s setting, we ... |
85,052 | <p>A housemate of mine and I disagree on the following question: </p>
<p>Let's say that we play a game of yahtzee. Of the five dice you throw, two dice obtain the value 1, two other dice obtain the value 2, and one die shows you six dots on the top side. Given the fact that you haven't thrown a "full house" yet, you s... | Ilmari Karonen | 9,602 | <p>We can write the process out as an <a href="http://en.wikipedia.org/wiki/Extensive-form_game" rel="nofollow">extensive-form game tree</a>:</p>
<ul>
<li>First throw:<br><br><ul>
<li>Probability $\frac 13$: rolled 1 or 2, stop.
<li>Probability $\frac 23$: rolled 3, 4, 5 or 6, roll again:<br><br><ul>
<li>Probability $... |
92,867 | <p>Suppose we have some random variable $X$ that ranges over some sample space $S$. We also have two probability models $F$ and $G$. Let $f(x)$ and $g(x)$ be the probability density functions for these distributions. Does the following quantity $$ \log \frac{f(x)}{g(x)} = \log \frac{P(F|x)}{P(G|x)}- \log \frac{P(F)}{P(... | Michael Hardy | 11,667 | <p>I was slightly puzzled by the notation, but I'm assuming that by $P(F)$ you mean the probability that $F$ is the right model, and $P(F\mid x)$ is the conditional probability that $F$ is the right model given the event $X=x$.</p>
<p>The identity you write is then a form of a special case of what is sometimes called ... |
389,675 | <p>I'm trying to use a program to find the largest prime factor of 600851475143.
This is for Project Euler here: <a href="http://projecteuler.net/problem=3">http://projecteuler.net/problem=3</a></p>
<p>I first attempted this with the code that goes through every number up to 600851475143, tests its divisibility, and a... | xisk | 73,012 | <p>The thing with Project Euler is that there is usually an obvious brute-force method to do the problem, which will take just about <em>forever</em>. As the questions become more difficult, you will need to implement clever solutions.</p>
<p>One way you can solve this problem is to use a loop that always finds the sm... |
389,675 | <p>I'm trying to use a program to find the largest prime factor of 600851475143.
This is for Project Euler here: <a href="http://projecteuler.net/problem=3">http://projecteuler.net/problem=3</a></p>
<p>I first attempted this with the code that goes through every number up to 600851475143, tests its divisibility, and a... | hmakholm left over Monica | 14,366 | <p>The number looks small enough to be brute-forced on a computer. Just try every possible factor, starting with 2, 3, 4, ... and keep dividing them out as long as the division comes out even. Then continue looking for factors of the quotient. You don't even need to explicitly restrict to primes, because any composite ... |
3,530,492 | <blockquote>
<p>Evaluate</p>
<p><span class="math-container">$$ \int_0^{e^{\pi}} |\cos\ (\ln x)|dx$$</span></p>
</blockquote>
<p><em>My ideas:</em> I substituted <span class="math-container">$u = \ln x$</span> and tried to evaluate</p>
<p><span class="math-container">$$\int_{-\infty}^\pi |\cos u|\ e^u du$$</sp... | LHF | 744,207 | <p>Here's a solution based on GEdgar's hint.</p>
<p><span class="math-container">$$\begin{aligned}
\int_0^{e^{\pi}} |\cos (\ln x) |\; dx &= \int_{-\infty}^{\pi} |\cos u |\; e^u \; du\\
&= \int_{\pi/2}^{\pi} |\cos u |\; e^u \; du +\sum_{k = 0}^{\infty} \int_{-k\pi -\pi/2}^{-k\pi +\pi/2} |\cos u |\; e^u \; du \\... |
623,819 | <p>I do not understand a remark in Adams' Calculus (page 628 <span class="math-container">$7^{th}$</span> edition). This remark is about the derivative of a determinant whose entries are functions as quoted below.</p>
<blockquote>
<p>Since every term in the expansion of a determinant of any order is a product involving... | Lutz Lehmann | 115,115 | <p>The determinant is like a generalized product of vectors (in fact, it is related to the outer product). So considering the rows as factors in this generalized product, this formula reflects the product rule of differentiation.</p>
<p>If $D(a,b,c)$ is generally a function of vectors that is linear in each argument, ... |
118,873 | <p>I understand that the Mellin transform of a modular form is expected to satisfy RH when it is an eigenform of all Hecke operators, in which case it has an Euler product. Now about when the form is not an eigenform: Is it known a case where the zeros are all in the critical strip?</p>
| Johan Andersson | 10,811 | <p>When it is not a Hecke-Eigen form, the Hecke L-series connected with the modular form does not have an Euler product. However it can still be written as a linear combination of Hecke L-series that have Euler-products. Thus the situation will resemble the case of linear combinations of Dirichlet L-series. In particul... |
2,916,099 | <p>Find a Mobius transformation $T$ from the unit disk to the right half plane with condition $T(0)=3$.</p>
<p>First, the transformation from the unit circle to the upper half plane is $T_1(z)=(1-i)\frac{z-i}{z-1}$.</p>
<p>So from the unit circle to the right half plane, $T_2(z)=-i(1-i)\frac{z-i}{z-1}$</p>
<p>How ca... | dxiv | 291,201 | <p>For a purely algebraic derivation, consider the general form of the Möbius transformation <span class="math-container">$\,T(z)=\dfrac{az+b}{cz+d}\,$</span>. Both <span class="math-container">$\,a\,$</span> and <span class="math-container">$\,c\,$</span> cannot be <span class="math-container">$\,0\,$</span>, otherwis... |
4,298,184 | <p><a href="https://i.stack.imgur.com/f5ny0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/f5ny0.png" alt="enter image description here" /></a>
as we can see, we are supposed to use stars and bars where n = 10 and r = 3. but what i dont understand is why we use stars and bars when stars and bars is ... | Marc van Leeuwen | 18,880 | <p>With new unknowns <span class="math-container">$x_i'=x_i+1\in\Bbb N$</span>, you need to count the number of solutions to <span class="math-container">$x_1'+x_2'+x_3'=10$</span>. Think of writing a solution to that last equation representing in the left hand side each <span class="math-container">$x_i$</span> by tha... |
3,130,877 | <p>Normal geometry concepts, such as parallel, angle, area, triangle, do they still apply in Mobius band?</p>
<p>If not, in which case will they fail to do so?</p>
<p>For example, what would three lines on a Mobius band form? A triangle if not parallel? or it might be totally something else?</p>
| Travis Willse | 155,629 | <p>A priori a Mobius strip <span class="math-container">$S$</span> is a topological manifold, perhaps with boundary, depending on how we define it---for simplicity I'll assume no boundary.</p>
<p>If one endows <span class="math-container">$S$</span> with a Riemannian metric <span class="math-container">$g$</span>---an... |
1,457,838 | <p>I am given position vectors: $\vec{OA} = i - 3j$ and $\vec{OC}=3i-j$.
And asked to find a position vector of the point that divides the line $\vec{AC}$ in the ratio $-2:3$. </p>
<p>So I found the vector $\vec{AC}$, and it is $2i+2j$. Then, if the point of interest is $L$, position vector $\vec{OL} = \vec{OA} + \lam... | R.N | 253,742 | <p>Hint: theorem- Every composite number has a proper factor less than or equal to its square root.</p>
<p>Suppose n is composite. I can write $n = ab$ , where $1 < a, b < n$ . If both $a, b > \sqrt{n}$ , then</p>
<p>$$n = \sqrt{n}\cdot \sqrt{n} < a\cdot b = n.$$</p>
<p>This contradiction shows that at ... |
2,762,715 | <blockquote>
<p>Let $\mathrm a,b$ are positive real numbers such that for $\mathrm a - b = 10$, then the smallest value of the constant $\mathrm k$ for which $\mathrm {\sqrt {x^2 + ax}} - {\sqrt{x^2 + bx}} < k$ for all $\mathrm x>0$, is? </p>
</blockquote>
<p>I don't get how to approach this problem. Any help ... | Ross Millikan | 1,827 | <p>First, replace $a$ in your inequality by $b+10$. Now the left hand side has only one parameter. Take the derivative with respect to $x$, set to zero, and find the $x$ value of the maximum as a function of $b$. Plug that in and find the left hand side maximum as a function of $b$. Take the derivative, set to zero... |
2,762,715 | <blockquote>
<p>Let $\mathrm a,b$ are positive real numbers such that for $\mathrm a - b = 10$, then the smallest value of the constant $\mathrm k$ for which $\mathrm {\sqrt {x^2 + ax}} - {\sqrt{x^2 + bx}} < k$ for all $\mathrm x>0$, is? </p>
</blockquote>
<p>I don't get how to approach this problem. Any help ... | Vishaal Sudarsan | 414,699 | <p>Let $f :\mathbb{R}^+ \to \mathbb{R} \quad $such that$\quad f(x) = \sqrt{x^2 + ax} - \sqrt{x^2+bx}$</p>
<p>Notice that $f(x)$ is increasing when $a>b$ , $f(x) = 0$ when $a=b$ and $f(x)$ is decreasing when $a<b$.</p>
<p>Now when $a>b$ $\lim_{x \to \infty} f(x) = 5$</p>
<p>Hence $\sqrt{x^2 + ax} - \sqrt{x... |
3,203,282 | <p>Given that <span class="math-container">$C[-\pi,\pi]$</span> is complete:
How can we prove, by using the supremum norm, that the space:</p>
<p><span class="math-container">$$C_p[-\pi,\pi]=\{f\in C[-\pi,\pi]\mid f(-\pi)=f(\pi)\}$$</span></p>
<p>is also complete? thank you!</p>
| carlosayam | 49,844 | <p>The sum initially goes from <span class="math-container">$n=0$</span> to <span class="math-container">$n=\infty$</span>. In the second line, the first term (<span class="math-container">$n=0$</span>), which is <span class="math-container">$1$</span>, is expanded apart, and now the sum starts at <span class="math-con... |
1,765,022 | <p>The problem is:</p>
<blockquote>
<p>$Prove$ $that$ $|\sin^2 (x)-\sin^2 (y)|\le |x-y|$ $ for $ $ all $ $ x,y>0$.</p>
</blockquote>
<p><em>$My$ $work$ :</em> $$\sin^2 (x)\le|\sin x|\le|x|\le|x-y|+|y|$$ and so is $$|\sin^ 2 (x)-\sin^2 (y)|\le |x-y|+|y|$$ But this is not the actual result I want. I think I have ... | David Heider | 21,026 | <p>The mistake is in my eyes that you allow the polar angle to be in $\phi\in[0,\pi]$ although just integrating over a quadrant integration domain in cartesian coordinates,$(x,y)\in[0,R]^2$. I'd suggest putting $\phi\in[0,\pi/2]$ in order to account for the integration in the first quadrant. This modification of your c... |
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | Community | -1 | <p>you would have to think about how to get the form desired. in this case the first part of:</p>
<p>$7(7^{k+2}+8^{2k+1})+57⋅8^{2k+1}$</p>
<p>We have made a factor, of the form desired. Assuming it's divisible by 57, that part of the sum is, and the other part shows it is already. So, the sum of both parts, must divi... |
2,358,838 | <p>I can see the answer to this in my textbook; however, I am not quite sure how to solve this for myself . . . the book has the following:</p>
<blockquote>
<p>To take advantage of the inductive hypothesis, we use these steps:</p>
<p>$ 7^{(k+1)+2} + 8^{2(k+1)+1} = 7^{k+3} + 8^{2k+3} $</p>
<p>$$
= 7\cdot7^... | Bram28 | 256,001 | <p>The key is to get <em>very clear</em> on what you <em>have</em>, and what you <em>want</em>.</p>
<p>For the inductive step, you <em>have</em> the inductive assumption:</p>
<p><span class="math-container">$7^{k+2}+8^{2k+1}$</span> is divisible by <span class="math-container">$57$</span></p>
<p>And you <em>want</em> t... |
65,304 | <p>I have a plane curve $C$ described by parametric equations $x(t)$ and $y(t)$ and a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. The line integral of $f$ along $C$ is the area of the "fence" whose path is governed by $C$ and height is governed by $f$.</p>
<p><img src="https://i.stack.imgur.com/4rmZy.png" alt="... | Dr. belisarius | 193 | <pre><code>opts = {MeshFunctions -> (#4 &),
MeshShading -> {{Opacity[#2], #1}, {Opacity[#2/2], #1}},
BoxRatios -> {1, 1, 1/2},
BoundaryStyle -> Directive[Thin, Blue]} &;
Show[
ParametricPlot3D[{Cos[t], Sin[t], z (2 + Sin[t] Cos[t]^2)}, {t, 0, π/2}, {z, 0, 1},
... |
1,889,957 | <p>I'm a bit rusty on my math notations and I'd like to write that:</p>
<blockquote>
<p>It exists a unique element $z$ such that $z$ belongs to the collection of values returned by $f(x,y)$</p>
</blockquote>
<p>Honestly I'm not just rusty I'm also mostly ignorant of math except from basic functions and basic matrix... | S.C.B. | 310,930 | <p>$$\sum_{cyc}\frac{a}{b+c}=\sum_{cyc}\frac{a^2}{ab+ac} \ge \frac{(a+b+c+d)^2}{ab+bc+cd+da+2ac+2bd}=\frac{(a+b+c+d)^2}{(a+c)(b+d)+2ac+2bd}$$</p>
<p>From Cauchy. </p>
<p>$$(a+b+c+d)^2=(a+c)^2+(b+d)^2+2(a+c)(b+d) \ge 4ac +4bd+2(a+c)(b+d)$$</p>
<p>Since $(x+y)^2 \ge 4xy$.</p>
<p>So $$\sum_{cyc}\frac{a}{b+c} \ge 2 >... |
202,041 | <p>I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, and is two dimensional for the torus, but the higher dimensional cases stump me, and I am unable to find the result. An... | Alexandre Eremenko | 25,510 | <p>First on terminology. "Riemannian surface" is a surface already equipped with a Riemannian metric. So the question "how many metrics of constant curvature exist on a Riemannian surface" makes sense only if you state what is the relation between the metric of
constant curvature and the original metric on the Riemanni... |
1,167,880 | <p>Given $a = [-5, 8, 1]$ and $b = [2, -7, -3]$, find a vector $c$ such that $a \cdot (b × c) = 0$</p>
<p>I don't know how to get it, I've been looking for examples, but I don't know..</p>
| Timbuc | 118,527 | <p>Proof for you to understand and check that $\;f(x)=x^2\;$ is not u.c. on $\;[0,\infty)\;$:</p>
<p>$$x_n:=\sqrt n\implies |x_{n+1}-x_n|\xrightarrow[n\to\infty]{}0\;,\;\;\text{yet nevertheless}\;\;|f(x_{n+1})-f(x_n)|\rlap{\;\;\;\;\;/}\xrightarrow[n\to\infty]{}0$$</p>
|
1,167,880 | <p>Given $a = [-5, 8, 1]$ and $b = [2, -7, -3]$, find a vector $c$ such that $a \cdot (b × c) = 0$</p>
<p>I don't know how to get it, I've been looking for examples, but I don't know..</p>
| Andrew D. Hwang | 86,418 | <p>$\DeclareMathOperator{\sgn}{sgn}\newcommand{\Reals}{\mathbf{R}}\newcommand{\eps}{\varepsilon}$Uniform continuity is a "global" property, depending not only on the formula(s) defining a function but on the domain $X$. To see why, compare the definitions of "$f$ is continuous on $X$" and "$f$ is uniformly continuous o... |
390,129 | <p>Let <span class="math-container">$O$</span> be a <span class="math-container">$d$</span>-dimensional rotation matrix (i.e., it has real entries and <span class="math-container">$OO^T = O^TO = I$</span>). Let <span class="math-container">$\mathbf{x}$</span> be a uniformly random bitstring of length <span class="math-... | Guillaume Aubrun | 908 | <p>We prove the weaker bound
<span class="math-container">$$ \mathbf{P} \left[ \|O \mathbb{x}\|_1 \leq \frac{cd}{\sqrt{\log d}} \right] \leq 2^{-Cd} $$</span>
for some constants <span class="math-container">$C, c$</span>.</p>
<p>Define the Gaussian mean width of a compact subset <span class="math-container">$A \subset ... |
3,451,629 | <p>Let <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span> be a continous map then which of the following cannot be the image of </p>
<p><span class="math-container">$[0,1)$</span> under <span class="math-container">$f$</span> ?</p>
<p>(a) <span class="math-container">$0$</span></p>
<p>(b) <span clas... | lonza leggiera | 632,373 | <p>If this were a test question, and "<span class="math-container">$0$</span>" were not a typo for "<span class="math-container">$\{0\}$</span>", I would consider it an extremely unfair question, because there are two contradictory answers to the question, both of which seem to me to be reasonable. The problem is that... |
2,723,585 | <p>If $\textbf{A}$ is a square matrix, how can I prove that, by using the power series of matrices that the above equality holds?
Note that the $x \in \mathbb{N}$ and $\textbf{A}$ is a square matrix.</p>
| Tsemo Aristide | 280,301 | <p>Hint: $exp(A+B)=exp(A)exp(B)$ if $A$ commutes with $B$</p>
|
148,972 | <p>I am working with solving a linear system that becomes a tridiagonal matrix. In order to speed up the process for large matrices, I want to use sparse matrices. My problem is that the values are not constant along the bands, but change based on their horizontal position in the matrix (x position, if you will). For i... | george2079 | 2,079 | <p>another way.</p>
<pre><code>Total@MapIndexed[
DiagonalMatrix[SparseArray@#, #2[[1]] - 2] &,
{func3[0, #] & /@ Range[n - 1],
func1[0, #] & /@ Range[n],
func2[0, #] & /@ Range[n - 1]}
]
</code></pre>
<p>This might be faster than using <code>Band</code> (Even for this simple input... |
2,900 | <p>I saved an <code>InterpolationFunction</code> in a ".mx" files using <code>DumpSave</code> on a variable that was scoped by a <code>Module</code>. Here is a stripped-down example:</p>
<pre><code>Module[{interpolation},
interpolation=Interpolation[Range[10]];
DumpSave["interpolation.mx", interpolation];
]
</co... | Leonid Shifrin | 81 | <p>You seem to be out of luck (although I will be happy to be proven wrong). This is a rather subtle point, related to the <code>Temporary</code> attribute and garbage-collection. I will just share a few observations. First, note that after loading of .mx file on a fresh kernel, the variable is not found anywhere, it i... |
2,416,510 | <p>I have a matrix $A \in R^{n×n}$. I would like to choose two diagonal matrices $D_1,D_2 \in R^{n×n}$ such that $\text{cond}(D_1AD_2)$ should be minimal. How to provide such diagonal matrices? </p>
| Raffaele | 83,382 | <p>Each of the four angles inscribed in the circular segment is half the angle with vertex in the centre of the circle. Each of the $\alpha$ is concave and is $360°$ less the convex part. Adding the $4$ we get $1440°-360°=1080°$ so the sum of the $\beta$ is half that is $540°$</p>
<p>Hope this is useful</p>
<p><a hre... |
48,726 | <p>I'm trying to plot a 3d revolution plot from a set of 2d points. These data points form a 2d curve, then we rotate that curve around y axis and get a 3d surface. @<a href="https://mathematica.stackexchange.com/users/50/50">J. M.</a> has a well explained and very helpful post at <a href="https://mathematica.stackexch... | m_goldberg | 3,066 | <p>You have a lot of points, so generating two interpolation functions from them, one for the bottom of the surface and the other for the top, should give a smooth surface of revolution. If the default plot isn't smooth enough, you can always increase <code>PlotPoints</code>.</p>
<pre><code>max = Max[First /@ points];... |
48,726 | <p>I'm trying to plot a 3d revolution plot from a set of 2d points. These data points form a 2d curve, then we rotate that curve around y axis and get a 3d surface. @<a href="https://mathematica.stackexchange.com/users/50/50">J. M.</a> has a well explained and very helpful post at <a href="https://mathematica.stackexch... | xslittlegrass | 1,364 | <p>The the roughness on the surface is due to the noise in the original data, so that the first derivative is not smooth.</p>
<p>This shows the first derivative of the x and y components using original data, we can see it's very noisy</p>
<pre><code>tvals = parametrizeCurve[points];
m = 3;
knots = Join[ConstantArra... |
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | I am a person | 806,777 | <p>In your second equation, the only way you can get from that equation to <span class="math-container">$7\tan^2{x} = 3$</span>, is by assuming that <span class="math-container">$\tan{x}$</span> and <span class="math-container">$1 - \tan^2{x}$</span> is not equal to <span class="math-container">$0$</span> or infinity b... |
4,092,994 | <p>The question is</p>
<blockquote>
<p>Find the solutions to the equation <span class="math-container">$$2\tan(2x)=3\cot(x) , \space 0<x<180$$</span></p>
</blockquote>
<p>I started by applying the tan double angle formula and recipricoal identity for cot</p>
<p><span class="math-container">$$2* \frac{2\tan(x)}{1-... | Arthur | 15,500 | <p>The moment you substitute <span class="math-container">$\cot x\mapsto \frac1{\tan x}$</span>, you are implicitly assuming that <span class="math-container">$x\neq 90^\circ$</span>, because that's required for that substitution to make sense. So that's a case you have to manually check in the original equation becaus... |
4,300,993 | <p>In this paper <a href="https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf" rel="nofollow noreferrer">Five stages of accepting constructive mathematics</a> on page 484 (shown in the image below) it contrastingly shows the use of the axiom of choice (<span class="math-cont... | HallaSurvivor | 655,547 | <p>Here is a clear way (imo) to see that the first proof uses choice:</p>
<blockquote>
<p>Let <span class="math-container">$\mathcal{A}$</span> be an open cover of <span class="math-container">$X$</span>. Then each <span class="math-container">$x$</span> lies in some <span class="math-container">$A \in \mathcal{A}$</sp... |
3,258,249 | <p><span class="math-container">$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$</span></p>
<p>It's clear that we can simplify the limit a little bit, after which we get:</p>
<p><span class="math-container">$\lim\limits_{n\to\infty}{(\frac{1}{4})^{n}\sum\limits_{... | user600016 | 545,151 | <p>Using identify <span class="math-container">${n\choose r} = \frac{n}{r}\cdot {{n-1} \choose {r-1}}$</span></p>
<p><span class="math-container">$$\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$$</span></p>
<p>= <span class="math-container">$$ \lim\limits_{n\to\... |
77,504 | <p>I'm in the embarrassing situation that I want to ask a question that
was <a href="https://mathoverflow.net/questions/14175/how-to-learn-about-shimura-varieties">already asked</a>, but (for complicated reasons) never answered. I'd
like to try with a blank slate.</p>
<p>Shimura varieties show connections to a lot of... | jvo | 6,121 | <p>I think the general wisdom is that Deligne's <em>Travaux de Shimura</em> and Milne's <em>Introduction to Shimura Varieties</em> are the most comprehensive references, with the latter being somewhat lighter on prerequisites (but heavier on examples). </p>
<p>I've heard it suggested by people who work in the area tha... |
4,459,221 | <p>I'm looking to count ways to make <span class="math-container">$n$</span> pairs if there are two groups of <span class="math-container">$n$</span> people and each pair must consist of one person from each group.</p>
<p>My initial thought is <span class="math-container">$^nP_n = n!$</span>, as we can line the groups ... | Stephen Donovan | 869,084 | <p>I like your reasoning for the case of two separate groups, and I believe that it is correct: we can simply look at each of the <span class="math-container">$n$</span> elements of one group in order, and for the <span class="math-container">$i^{th}$</span> element we will have <span class="math-container">$n - i + 1$... |
4,459,221 | <p>I'm looking to count ways to make <span class="math-container">$n$</span> pairs if there are two groups of <span class="math-container">$n$</span> people and each pair must consist of one person from each group.</p>
<p>My initial thought is <span class="math-container">$^nP_n = n!$</span>, as we can line the groups ... | true blue anil | 22,388 | <p>For the first part, you have already given a nice intuitive explanation.</p>
<p>For an intuitive explanation of the second formula, in the <span class="math-container">$(2n)!$</span> permutations of the items which we can pair serially, neither the order of the pairs nor the order <em>within</em> the pairs matter, t... |
2,807,611 | <p>I know the answer is $n=6$, but can't figure out how to solve.
I tried dividing by $n!$, but didn't work because there isn't one in RHS to simplify... also tried using Gamma function properties, but didn't work either... </p>
<p>Any help would be appreciated.</p>
<p>Thanks.</p>
| N8tron | 32,820 | <p>I don't see a way to solve for n but I do see a way to see the only solution is 6</p>
<p>Multiply both sides by 5! And factor out $n!$ to get the equivalent equation</p>
<p>$$
(n^2+3n+1)n!=55(6!)
$$</p>
<p>The left hand side is a strictly increasing function for $n>0$. So for any value of the right hand side, ... |
21,238 | <p>Would someone be able to point me to a good resource explaining step by step the process for solving inhomogenous recurrence relations? (ie something of the form $ a_n = \sum{{b_i}{a_{n-i}}} + f(n)$ )</p>
| Csar Lozano Huerta | 1,547 | <p>For the case of singular plane curves though $C\subset \mathbb{P}^2$, the result is well known. The blowup of the projective plane itself is non singular yeah, but what about the image of the curve $C$ in the blowup? That is, the proper transform of $C$? Is this proper transform $\tilde{C}$ smooth?. Such an informa... |
21,238 | <p>Would someone be able to point me to a good resource explaining step by step the process for solving inhomogenous recurrence relations? (ie something of the form $ a_n = \sum{{b_i}{a_{n-i}}} + f(n)$ )</p>
| Blup | 5,998 | <p>For monomial ideals there is a combinatorial smoothness criterion, see "Blowups in tame monomial ideals" <a href="https://arxiv.org/abs/0905.4511" rel="nofollow noreferrer">https://arxiv.org/abs/0905.4511</a></p>
|
2,604,844 | <p>I have the folowing induction :</p>
<p>"Every graph with n vertices and zero edges is connected"</p>
<ul>
<li><p>Base:</p>
<p>For $n=1$ graph with one vertice is connected, hence a graph with $1$ vertex and zero edge.</p></li>
<li><p>Assumpution:</p>
<p>Every graph with $n-1$ vertices and zero edges is connected... | Gregory Fenn | 389,331 | <p>The induction step (from $n-1$ to $n$, or from all '$k < n$' to $n$) is missing here. All you've done is proven that </p>
<pre><code>if all graphs of n-1 nodes with no edges are connected, then all graphs with n nodes and zero edges have a connected subgraph of n-1 nodes and no edges.
</code></pre>
<p>That's r... |
872,017 | <p>$$\int_0^1 xe^{\sqrt{x}} dx = ? $$</p>
<p>All I can think of is the integration by parts rule, where
$ u = x $ and $ dv= e^{\sqrt(x)} $ $ \Rightarrow du = 1$ and $ v= e^{\sqrt(x)} $ </p>
<p>The answer I get is $e^{\sqrt(x)}(x-1)$ , which is wrong.</p>
<p>Can anyone please explain in detail?</p>
| 2'5 9'2 | 11,123 | <p>You can do integration by parts like this, without substituting. Of course, substituting is fine and all, but you'll have to use Integration by Parts three times either way.</p>
<p>$$\begin{align}
\int_0^1xe^{\sqrt{x}}\,dx
&=\int_0^1\frac{2x\sqrt{x}}{2\sqrt{x}}e^{\sqrt{x}}\,dx\\
&=\int_0^12x\sqrt{x}\left(\f... |
3,732,648 | <p>So I was reading Dugundji's topology and I found myself in trouble trying to prove the following.</p>
<blockquote>
<p>Prove that the following statemets are equivalent:</p>
<ul>
<li><span class="math-container">$f:X\rightarrow{}Y$</span> is continuous</li>
<li><span class="math-container">$f(A')\subseteq \overline{f... | zkutch | 775,801 | <p>You have <span class="math-container">$y=x$</span> as tangent line in <span class="math-container">$x=1$</span> and function <span class="math-container">$x^x$</span> is convex.</p>
|
3,732,648 | <p>So I was reading Dugundji's topology and I found myself in trouble trying to prove the following.</p>
<blockquote>
<p>Prove that the following statemets are equivalent:</p>
<ul>
<li><span class="math-container">$f:X\rightarrow{}Y$</span> is continuous</li>
<li><span class="math-container">$f(A')\subseteq \overline{f... | Martin R | 42,969 | <p>This can be solved without taking derivatives if you know that the logarithm is a strictly increasing function. We have (for all <span class="math-container">$x > 0$</span>)
<span class="math-container">$$
x^x \ge x \iff x \ln x \ge \ln x \iff (x-1) \ln x \ge 0
$$</span>
and the last inequality is always true b... |
43,282 | <p>This question is somewhat related to Tilmans notorious problem in <a href="https://mathoverflow.net/questions/17532/does-linearization-of-categories-reflect-isomorphism">this post</a>. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 \rbrace.$$
... | Torsten Ekedahl | 4,008 | <p>Let $R$ be a finite dimensional algebra over $\mathbb Z/2$. Then $\{1\}\neq
R^\times$ unless $R=(\mathbb Z/2)^n$. Indeed, if $N$ is the radical of $R$, then
$1+N\subseteq R^\times$ so we may assume $R$ is semi-simple. Then $R=\prod_iR_i$
where the $R_i$ are simple algebras and $R^\times=\prod_iR_i^\times$ so we may
... |
791,719 | <p>I have this inequation:
$$5-3|x-6|\leq 3x -7$$</p>
<p>i solved this this way: </p>
<p>i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : </p>
<p>1) for $x\geq6$ (positive): </p>
<p>$5-3x+6\leq 3x -7\\
6x\geq30\\
x\geq5$</p>
<p>2) for $x<6$ (negative): </... | John Joy | 140,156 | <p>Sometimes drawing a diagram is helpful.</p>
<p><img src="https://i.stack.imgur.com/h5mbF.png" alt="enter image description here"></p>
<p>From the diagram it is clear that
$$\tan A = x$$
$$A = \arctan(x)$$
and that also
$$\tan(\pi/2 - A) = 1/x$$
$$\arctan(\tan(\pi/2 - A)) = \arctan(1/x)$$
$$\pi/2 - A = \arctan(1/x)... |
3,766,042 | <p>I was doing the problem</p>
<blockquote>
<p>Find all real solutions for <span class="math-container">$x$</span> in:</p>
<p><span class="math-container">$$ 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2$$</span></p>
</blockquote>
<p>There was a hint, to prove that <span class="math-container">$2^{x} - 1$</span> has the sa... | Michael Rozenberg | 190,319 | <p>Now, for <span class="math-container">$x\neq1$</span>, <span class="math-container">$x\neq0$</span> and <span class="math-container">$x\neq-1$</span> rewrite our equation in the following form:
<span class="math-container">$$\frac{2^x-1}{x}+\frac{2^{x^2-1}-1}{x^2-1}=0.$$</span>
Can you end it now?</p>
|
2,594,837 | <p>Let $R$ be a Noetherian domain with quotient field $K$,<br>
$I \subseteq R$ a finitely generated ideal, $I \neq (0)$ and<br>
$x \in K$ such that $x \cdot I \subseteq I$.<br>
I want to show that $x$ is <a href="https://en.wikipedia.org/wiki/Integral_element" rel="nofollow noreferrer">integral</a> over $R$.</p>
<p>Su... | Jesko Hüttenhain | 11,653 | <p>Your statement is not quite correct, you need to assume that $I$ is also nonzero. In that case however, there is a very elementary way to see this.</p>
<p>So you have $I=(f_1,\ldots,f_n)$ and $x\cdot I\subseteq I$. In particular,
$$
x\cdot f_i = \sum_{j=1}^n x_{ij} f_j
$$
for a matrix $X:=(x_{ij})\in R^{n\times n}... |
2,594,837 | <p>Let $R$ be a Noetherian domain with quotient field $K$,<br>
$I \subseteq R$ a finitely generated ideal, $I \neq (0)$ and<br>
$x \in K$ such that $x \cdot I \subseteq I$.<br>
I want to show that $x$ is <a href="https://en.wikipedia.org/wiki/Integral_element" rel="nofollow noreferrer">integral</a> over $R$.</p>
<p>Su... | Mohan | 245,104 | <p>Here is a proof which uses Noetherianness (not that it is much simpler). You have inclusions of $R$-algebras, $R\subset R[x]\subset \mathrm{End}_R(I)$. Noetherian property implies the last is a finite type $R$-module and hence again by Noetherian property, so is $R[x]$. This immediately implies $x$ is integral over ... |
3,338,388 | <p>I tried to calculate the expression:
<span class="math-container">$$\lim_{n\to\infty}\prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right)$$</span>
in Wolframalpha, but it does not interpret it correctly. </p>
<p>Could someone help me type it in and get the ans... | David G. Stork | 210,401 | <pre><code>Limit[
Product[1 - n/(((n + Sqrt[n^2 + 4])/2)^k + (n + Sqrt[n^2 + 4])/2),
{k, 1, \[Infinity]}],
n -> \[Infinity]]
</code></pre>
|
3,066,967 | <p>Prove that <span class="math-container">$k^2+k+1$</span> is not divisible by <span class="math-container">$101$</span> for any natural <span class="math-container">$k.$</span></p>
| Seewoo Lee | 350,772 | <p>Hint: If <span class="math-container">$101|k^{2}+k+1$</span>, then <span class="math-container">$101|4(k^{2} + k + 1) = (2k+1)^{2} + 3$</span>, which implies that <span class="math-container">$\left(\frac{-3}{101}\right) = 1$</span>. However, you may show that <span class="math-container">$\left (\frac{-3}{101}\righ... |
187,459 | <p>What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle?</p>
<p>Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one vertex in common, then can we characterize them in some way?</p>
<p>When is it possible to draw such a graph... | Aaron Meyerowitz | 8,008 | <p>There should be lots of these, even with the second condition. So many that I can't imagine a classification. I'll call a $4$-cycle a <em>square.</em></p>
<p>One construction is as follows: Start with an appropriate connected graph such that </p>
<ul>
<li>Each edge is on a unique square </li>
<li>All vertices have... |
2,205,042 | <p>I want to show that there exists some $M$ such that for any $n$ and any $x \in [\epsilon, 2\pi - \epsilon]$ we have $\left |\sum_{m = 1}^n e^{imx} \right|\leq M$. Geometrically, it is like starting at the origin facing east, then turning left by $x$ degrees and moving forward by 1 unit of distance, and repeating thi... | J.-E. Pin | 89,374 | <p>Just use regular expressions instead of automata. Indeed,
<span class="math-container">$$S(L_1,L_2) = (L_1L_2)^* \cup (L_1L_2)^*L_1$$</span></p>
|
510,633 | <p>A independent variable is "the input" and the dependent variable is the "output", atleast thats how it was explained to us.</p>
<p>But if you have some random function can't both variables be seen as "affecting" the other variable?</p>
<p>For example, in $ y = 1/x$, "x" could be seen as an input and "y" the output... | Felix Marin | 85,343 | <p>\begin{align}
\left(1 + {1 \over n}\right)^{n}
&=
\sum_{\ell = 0}^{n}{n \choose \ell}{1 \over n^{\ell}}
=
\sum_{\ell = 0}^{n}
{n\left(n - 1\right)\ldots\left(n - \ell + 1\right) \over \ell!}\,
{1 \over n^{\ell}}
\\[3mm]&<
\sum_{\ell = 0}^{n}
{\left(n + 1\right)n\ldots\left(n - \ell + 2\right) \over \ell!}... |
3,756,970 | <p>I know which step is wrong in the following argument, but would like to have contributors' explanations of <em>why</em> it is wrong.</p>
<p>We assume below that weather forecasts always predict whether or not it is going to rain, so <em>not forecast to rain</em> means the same as <em>forecast not to rain</em>. We sh... | José Carlos Santos | 446,262 | <p>No, there is no such set. You can always define the discrete metric on any set.</p>
|
2,847,419 | <p>I know that <br/>
$\sigma , \delta$ be 2 function then <br/>
$1)$ $\sigma \circ \delta$ is onto or one-one if both $\sigma $ and $\delta$ is onto or one one.<br/>
I can prove this fact .
I wanted to find the counterexample for both cases if the converse is not true.
<br/> Any Help will be appreciated </p>
| Chris2018 | 578,559 | <p>At critical points $f'(x)=0$ so</p>
<p>$0=k(x+e^x)^{k-1} \times (1+e^x)$<br>
k and $(1+e^x)$ are positive so $0=x+e^x$<br>
$x_n=-e^{x_{n+1}}$ so $x \approx -0.56714329$</p>
|
1,822,160 | <p>Why is the "column space" on the vertical in a matrix? In my mind the column space is that space that the vectors in the matrix have created.
I mean, for example take the equations:</p>
<pre><code>3x + 4y = 5
2x + 8y = 6
</code></pre>
<p>Then the matrix will be:</p>
<p>\begin{pmatrix}
3 & 4 \\
2 & 8
\en... | Aloizio Macedo | 59,234 | <p>I think what you are meaning to ask is:</p>
<blockquote>
<p>Why is the collumn space the range of the matrix, in the
appropriate bases?</p>
</blockquote>
<p>Otherwise, the collumn space is just a definition per se.</p>
<p>Therefore, we must see who is the image of the basis. That is, compute</p>
<p>$\begin{p... |
770,538 | <p>Let $\alpha$ be cylindrical helix with unit vector $u$, angle $\theta$, and arc length $s$ (measured from $0$). The only curve $\gamma$ such that $$\alpha(t)=\gamma(t)+s(t)\cos(\theta)u$$ is called the cross-section curve of the cylinder where $\alpha$ lies. </p>
<p><strong>$\bf (a)$</strong> How to show $\gamma(t)... | Mick | 42,351 | <p>Referring to the diagram below:-
<img src="https://i.stack.imgur.com/GR8wR.png" alt="enter image description here"></p>
<p>It is not that difficult to see that:-</p>
<p>(1) The two tangents are equal in length (and = 2)</p>
<p>(2) By properties of tangents, $\angle KHO = \angle PHO = x$)</p>
<p>(3) $H$, $K$, $O$... |
94,134 | <p>I have a feeling that the following inequality should be very easy to prove:</p>
<p>$$
x^n \geq \prod_{i=1}^n{(x+k_i)},\quad\text{where } \sum_{i=1}^{n}{k_i}=0,\quad \text{and } x+k_i>0\text{ for all } i
$$</p>
<p>(and the equality only holds when all the $k_i=0$).</p>
<p>It seems intuitively obvious (when $... | Davide Giraudo | 9,849 | <p>We can show it by induction: for $n=2$, we have for $a+b=0$: $x^2-(x+a)(x+b)=-(a+b)x-ab=-ab\geq 0$, since $ab\leq 0$. We assume that the result is true for $n$. Let $(k_1,\ldots,k_{n+1})$ such that $\sum_{j=1}^{n+1}k_j=0$. We can assume that $k_nk_{n+1}\leq 0$. We put $k_j'=k_j$ if $j\leq n-1$, and $k_n'=k_n... |
1,221,639 | <p>Consider two random variables <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. If X and Y are independent random
variables, then it can be shown that:
<span class="math-container">$$E(XY) = E(X)E(Y).$$</span></p>
<p>Let <span class="math-container">$X$</span> be the random variabl... | megas | 191,170 | <p><strong>Hint</strong>: By the definition of independence,
two discrete random variables $X$ and $Y$ are independent if the joint probability mass function $P(X = x \text{ and } Y = y )$ satisfies
$$
P(X = x \text{ and } Y = y ) = P(X = x) \cdot P(Y = y)
$$
for all $x$ and $y$.</p>
<p>What are the possible values... |
1,790,612 | <p>Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex semisimple Lie algebra ${\frak g}_{\Bbb C}={\frak g}\otimes\Bbb C$?</p>
<p>In particular:</p>
<blockquote>
<p>If $... | Dietrich Burde | 83,966 | <p>Take the compact simple real Lie algebra $\mathfrak{su}(2)$. The Killing form is negative definite. Its complexification is isomorphic to the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, where the Killing form is non-degenerate, but does not satisfy $B(x,x)=0$ implies $x=0$. If $x,y,h$ is the standard basis, then $B(x... |
587,275 | <p>I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) = 2^{x}$, but now I'm stuck.</p>
<p>Here's my final step:
<strong>$\displaystyle{{\rm f}'\left(x\right)
= \lim_{h \to 0}{2^{x}\left(2^{h} - 1\right... | Community | -1 | <p>Write $2^h$ as $(e^{\log(2)})^h$. Hence, $$\dfrac{2^h-1}h = \log(2) \cdot \dfrac{e^{\log(2)h}-1}{h\log 2}$$ Now finish off using the fact that
$$\lim_{x \to 0} \dfrac{e^x-1}{x} = 1$$</p>
|
3,753,819 | <p><span class="math-container">$\textbf{Question:}$</span> Let <span class="math-container">$(X,\mathcal{F},\mu)$</span> be an arbitrary measure space. Let <span class="math-container">$\varphi: \mathbb{R} \rightarrow \mathbb{R}$</span> be continuous and satisfy for some <span class="math-container">$K>0$</span>:</... | Oliver Díaz | 121,671 | <p>Observe that he Converse part is equivalent to the statement</p>
<p><strong>Theorem:</strong> If <span class="math-container">$\phi\in C(\mathbb{R})$</span> and for any measure space <span class="math-container">$(X,\mathscr{F},\mu)$</span>
<span class="math-container">$$\phi\circ f\in L_p(\mu)$$</span>
whenever <sp... |
1,812,914 | <p>What is the correct approach to solving a log equation with more than one non log value? Please demonstrate using the following equation:
$$\log(2x-1)=-x+3$$</p>
| Claude Leibovici | 82,404 | <p>As alex jordan answered, there is an analytical solution involving Lambert function. If you cannot or do not want to use it, numerical methods should be required.</p>
<p>Consider the function $$f(x)=\log(2x-1)+x-3$$ and its first derivative $$f'(x)=\frac 2{2x-1}+1$$ It is always positive since, because of the logar... |
831,618 | <p>Please help me to prove that this integral converges.</p>
<p>$$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$</p>
<p>No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result still.</p>
| Ron Gordon | 53,268 | <p>Using the factorization $a^3-b^3 = (a-b)(a^2+ab +b^2) $, rewrite the integral as </p>
<p>$$\int_0^1 dx \, \frac{(1-x)^{-1/3}}{(1+x+x^2)^{1/3}} $$</p>
<p>Sub $y=1-x$ and observe that</p>
<p>$$\int dy \, y^{-1/3} = \frac{3}{2} y^{2/3} + C$$</p>
|
147,363 | <blockquote>
<p>If $\alpha$ is an algebraic element of $\mathbb{C}$, then there is a unique non-zero polynomial $f \in \mathbb{Q}[x]$ with leading coefficient $1$ such that $f(\alpha) = 0$, and $f$ is irreducible. </p>
</blockquote>
<p>The first part of this proof would be proving that $f$ is not a unit, but what do... | Gerry Myerson | 8,269 | <p>It would, if it were of degree zero. </p>
|
3,129,248 | <p>I am solving ordinary differential equation in <span class="math-container">$S'$</span> (dual to Schwartz space) given as:</p>
<p><span class="math-container">$y' + ay = \delta$</span>, where <span class="math-container">$\delta$</span> is a Dirac delta function.</p>
<p>The general solution of homogenous equation ... | xpaul | 66,420 | <p>Let <span class="math-container">$Y(s)=L\{y(t)\}$</span> be the Laplace transform of <span class="math-container">$y$</span>. Using
<span class="math-container">$$ L\{y'(t)\}=sY(s)-y(0), L\{\delta\}=1, $$</span>
one has
<span class="math-container">$$ sY(s)-y(0)+aY(s)=1 $$</span>
from which,
<span class="math-conta... |
2,526,695 | <p>I've got following sequence formula:
$ a_{n}=2a_{n-1}-a_{n-2}+2^{n}+4$</p>
<p>where $ a_{0}=a_{1}=0$</p>
<p>I know what to do when I deal with sequence in form like this:</p>
<p>$ a_{n}=2a_{n-1}-a_{n-2}$
- when there's no other terms but previous terms of the sequence.
Can You tell me how to deal with this typ... | adfriedman | 153,126 | <p>One approach that works to get to the final form is to take the formal power series
$$f(x) = \sum_{n=0}^{\infty} a_n x^n$$
and try and rewrite it in terms of itself. Applying the initial conditions where necessary:
\begin{align}
f(x) &= \sum_{n=0}^{\infty} a_n x^n\\
&= a_0 + a_1 x +\sum_{n=2}^{\infty} \left(... |
104,195 | <p>Background: This year I'll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica's AbstractAlgebra package but (although visually appealing ) this is no longer sufficient (i.e. listing subgroups of <span class="math-container">$S_4$</span> takes ages). So, I want to learn more ... | Olexandr Konovalov | 70,316 | <p>In addition to some answers given in comments to this question (cf. <a href="http://meta.math.stackexchange.com/questions/1559/dealing-with-answers-in-comments">http://meta.math.stackexchange.com/questions/1559/dealing-with-answers-in-comments</a>?), let me add that the following.</p>
<p>The <a href="http://www.gap... |
1,448,427 | <p>Consider a lot consisting of 3 blue balls and 1 red ball.</p>
<p>Suppose I pick 2 balls one after another without replacement, now the probability of 2 balls being blue:</p>
<p>$$\frac{3\choose 2}{4 \choose 2}$$</p>
<p>Now taking different approach using conditional probability, the solution is also:</p>
<p>$$\f... | got it--thanks | 160,720 | <p>Here's how to draw $$\vec {BC} = -\frac 25\vec{AB}$$</p>
<p><a href="https://i.stack.imgur.com/yxJCc.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yxJCc.jpg" alt="enter image description here"></a></p>
<p>I drew $\vec {AB}$ a little to the side so that it wasn't overtop $\vec{BC}$.</p>
|
1,505,336 | <p>Please can someone explain if this identity is correct:</p>
<p>|a| = $\sqrt{a^2} \ $ <p>
I thought it should be:<p> |a| = $(\sqrt{a})^2\ $</p>
<p>being that the former would produce an answer that is either positive or negative.</p>
<p>Thank you for your help.</p>
<p>PS: The full question was comparing say:</p>
... | Thomas Russell | 32,374 | <p>If we examine your suggestion, we would have:</p>
<p>$$|(-3)|=\left(\sqrt{-3}\right)^{2}=\left(\sqrt{3}i\right)^{2} = 3i^{2} = -3 \neq 3$$</p>
<p>Moreover, the $\sqrt{\cdot}$ operator is defined to be non-negative for all non-negative real arguments, so $\sqrt{x^{2}}$ does give us a well defined value $\forall x \... |
3,415,845 | <p>Using the technique proof by cases, show that</p>
<p><strong>"For integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, if <span class="math-container">$xy$</span> is odd, then <span class="math-container">$x$</span> is odd and <span class="math-container">$y$</span> is odd."... | ironX | 534,898 | <p>If <span class="math-container">$xy$</span> is odd, then:</p>
<p>case 1: <span class="math-container">$x$</span> is even, <span class="math-container">$y$</span> is odd. But <span class="math-container">$xy$</span> would be even since any integer (<span class="math-container">$y$</span>) multiple of an even number ... |
3,415,845 | <p>Using the technique proof by cases, show that</p>
<p><strong>"For integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, if <span class="math-container">$xy$</span> is odd, then <span class="math-container">$x$</span> is odd and <span class="math-container">$y$</span> is odd."... | YiFan | 496,634 | <p>The proof is perfectly fine. I suspect your confusion comes from the idea that to show <span class="math-container">$p\rightarrow q$</span> by breaking into cases, we have to break <span class="math-container">$p$</span> itself up into cases and discuss. That is not really true. In particular, the statement <span cl... |
1,692,346 | <p>I have heard of a statement like this:</p>
<blockquote>
<p>A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time.</p>
</blockquote>
<p>Is this possible (mathematics wise)?</p>
| Count Iblis | 155,436 | <p>"A car can technically never run out of gas (when still moving) if the driver uses half of the gas left each time."</p>
<p>A more practical restatement: If you reduce the speed of the car as a function of time like $v(t) = v(0)\exp(-a t)$, then the car will always keep on moving, yet the fuel consumption will be bo... |
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