Recent Questions - Mathematics Stack Exchange most recent 30 from math.stackexchange.com 2022-10-04T06:57:45Z https://math.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://math.stackexchange.com/q/4544820 -2 Bijective linear mapping on basis bruh https://math.stackexchange.com/users/1103036 2022-10-04T06:50:05Z 2022-10-04T06:50:05Z <p>If L: V -&gt; W is a bijective linear mapping, then, would the basis of V be mapped onto W as a basis of W?</p> https://math.stackexchange.com/q/4544819 0 How do I show that $sin(x^2+3x+2)$ is not periodic? ThirstForMaths https://math.stackexchange.com/users/919818 2022-10-04T06:47:05Z 2022-10-04T06:47:05Z <p>I know that if <span class="math-container">$g(x)$</span> is periodic then <span class="math-container">$f(g(x))$</span> is periodic.</p> <p>This is a sufficient condition but not necessary as <span class="math-container">$sin(x)=f(g(x))$</span> is periodic where <span class="math-container">$g(x)=x$</span>(non periodic) and <span class="math-container">$f(x)=sin(x)$</span>.</p> <p>Can the above condition be made into a necessary and sufficient condition (assuming that <span class="math-container">$g(x)$</span> is not the identity function) or is there a better way to find the solution.</p> <p>This is from a previous year college entrance exam.Any short trick rather to check for the periodicity of <span class="math-container">$f(g(x))$</span> when <span class="math-container">$g(x)$</span> is a complicated function would be helpful.</p> https://math.stackexchange.com/q/4544818 0 Joining two points on manifold by injective smooth curve Rahul Sarkar https://math.stackexchange.com/users/801621 2022-10-04T06:43:50Z 2022-10-04T06:43:50Z <p>Suppose <span class="math-container">$\mathcal{M}$</span> is a smooth, connected manifold, and let <span class="math-container">$p$</span> and <span class="math-container">$q$</span> be distinct points in <span class="math-container">$\mathcal{M}$</span>. Is it true that there always exists a smooth injective regular map <span class="math-container">$\gamma: [0,1] \rightarrow \mathcal{M}$</span> such that <span class="math-container">$\gamma(0)=p$</span> and <span class="math-container">$\gamma(1)=q$</span>?</p> https://math.stackexchange.com/q/4544817 0 How can I prove that the following sets are metric spaces? Đăng Khải https://math.stackexchange.com/users/998718 2022-10-04T06:40:59Z 2022-10-04T06:40:59Z <ol> <li><p>Let <span class="math-container">$\displaystyle l_2=\left\{x=(x_n)_n \subset \mathbb{R}\ |\ \sum_{n=1}^\infty x_{n}^2 &lt; +\infty \right\}$</span> <span class="math-container">$$d(x,y)= \left[\sum_{n=1}^\infty (x_n-y_n)^2\right]$$</span> Show that <span class="math-container">$d$</span> is a metric on <span class="math-container">$l_2$</span>.</p> </li> <li><p><span class="math-container">$X=\{x:[a,b] \longrightarrow \mathbb{R}\ |\ x$</span> is continuous} <span class="math-container">$$d(x,y)=\left[\int\limits_a^b (x(t)-y(t))^2dt \right]^\frac{1}{2}$$</span> Show that <span class="math-container">$d$</span> is a metric on <span class="math-container">$X$</span></p> </li> </ol> https://math.stackexchange.com/q/4544816 0 The linear system $|E+nF|$ Kelvin Lian https://math.stackexchange.com/users/872975 2022-10-04T06:38:26Z 2022-10-04T06:38:26Z <p>I'm asking this question with reference to an answer by BlaCa to an old MathOverflow question I saw <a href="https://mathoverflow.net/questions/122952/on-a-hirzebruch-surface">here</a>.</p> <p>We first recall that the <span class="math-container">$n$</span>-th Hirzebruch surface <span class="math-container">$\mathbb{F}_n$</span> can be viewed as the projective bundle <span class="math-container">$\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$</span>. It contains a unique irreducible curve <span class="math-container">$E$</span> with self-intersection <span class="math-container">$-n$</span>. Let <span class="math-container">$F$</span> be a fiber of <span class="math-container">$\mathbb{F}_n \rightarrow \mathbb{P}^1$</span>.</p> <p>What I'm confused about is the line &quot;The linear system <span class="math-container">$|E+nF|$</span> has degree zero on <span class="math-container">$E$</span>, degree one on <span class="math-container">$F$</span> and degree <span class="math-container">$n$</span> on <span class="math-container">$E+F$</span>.&quot; What does it mean for a linear system to have some degree over a divisor? Also, how are the degrees zero, one and <span class="math-container">$n$</span> deduced?</p> <p>Moreover, the answer followed up by saying <span class="math-container">$E$</span> contracts to a unique singular point whose <em>hyperplane section is a rational normal curve of degree n</em>. Why is this a consequence?</p> https://math.stackexchange.com/q/4544815 0 Solve the first order PDE using Lagrange method Mittal G https://math.stackexchange.com/users/290972 2022-10-04T06:32:46Z 2022-10-04T06:32:46Z <p>Let <span class="math-container">$u(x, y)$</span> be the solution of first order partial differential equation <span class="math-container">$$xu_x+(x^2+y)u_y=u, \ \text{for all}\ x, y\in \mathbb{R}$$</span>satisfying <span class="math-container">$u(2, y)=y-4$</span>. Then find the value of <span class="math-container">$u(1, 2)$</span>.</p> <p>My attempt: I started with Lagrange equations and write <span class="math-container">$$\frac{dx}{x}=\frac{dy}{x^2+y}=\frac{du}{u}$$</span> Using first and third equations, I get <span class="math-container">$x=au$</span> for some constant <span class="math-container">$a$</span>. But I strucked further, as I wanted to solve second and third by making use of <span class="math-container">$x=au$</span> but unable to do so. Please help how to proceed further.</p> https://math.stackexchange.com/q/4544813 0 $D=U^TAU$ A is symmetric positive semi-definite okzoomer https://math.stackexchange.com/users/985287 2022-10-04T06:26:38Z 2022-10-04T06:40:42Z <p>Question came up on singular value decomposition topic.</p> <p>Prove that for all symmetric positive semi-definite matrices <span class="math-container">$A$</span>, there exists an orthonormal matrix <span class="math-container">$U$</span> such that <span class="math-container">$$D=U^TAU$$</span> is a diagonal matrix such that all diagonal entries are non-negative.</p> <hr /> <p><span class="math-container">$$D = \left[ {\begin{array}{c} u_{1}^T\\ \vdots \\ u_{n}^T\\ \end{array} } \right] A \left[ u_1 \dots u_n \right]$$</span></p> <p><span class="math-container">$u_i$</span> is an orthonormal vector of <span class="math-container">$U$</span>. Expanding I'm getting <span class="math-container">$D_{i,j} = u_i^TAu_j$</span>. This is non-negative if <span class="math-container">$i = j$</span>, from the definition we were given for positive semi-definite matrices. I understand that <span class="math-container">$u_i^Tu_{j\neq i} = 0$</span>, but I'm not sure how to show <span class="math-container">$u_i^TAu_{j\neq i} = 0$</span>. Also I am not sure if I'm on the right track, since if <span class="math-container">$u_i^TAu_{j\neq i} = 0$</span> is true, this proves true for all orthonormal <span class="math-container">$U$</span>, which is a much stronger statement than there exist an <span class="math-container">$U$</span>.</p> <p>Another thought I had was to make the <span class="math-container">$D$</span> the <span class="math-container">$\Sigma$</span> in some singular value decomposition of <span class="math-container">$A$</span>, but it doesn't sound right since <span class="math-container">$A$</span> is now restricted to <span class="math-container">$v^TAv\geq0 \; \forall\;v\neq 0$</span> (the only property of positive semi-definite matrices that was given.)</p> <hr /> <p>Edit: After some search, I understand that for orthonormal matrices, <span class="math-container">$U^T = U^{-1}$</span>. So my second approach yields <span class="math-container">$UDU^T = A$</span>. The problem set has not made use of this property of orthonormal matrices, so I am not sure if we can use that, plus it sounds like circular logic.</p> https://math.stackexchange.com/q/4544812 0 Solution of a Laplace equation in 2 dimension Mittal G https://math.stackexchange.com/users/290972 2022-10-04T06:22:03Z 2022-10-04T06:22:03Z <p>Let <span class="math-container">$u_{xx}+u_{yy}=0$</span> be a given Laplace equatio with <span class="math-container">$1&lt;x&lt;2$</span> and <span class="math-container">$1&lt;y&lt;2$</span>. Suppose the given boundary conditions are <span class="math-container">$$u_x(1,y)=y, \ u_x(2, y)=5, 1&lt;y&lt;2,$$</span> <span class="math-container">$$u_y(x, 1)=\frac{15x^2}{7}, u_y(x, 2)=x, 1&lt;x&lt;2.$$</span> Find the solution <span class="math-container">$u$</span>.</p> <p>I know that we can solve a Laplace equation by method of separation of variables or in terms of polar coordinates. But how to solve it from the given boundary conditions. Please help.</p> https://math.stackexchange.com/q/4544810 0 Equation of angle bisector between 2 planes. TshrD23 https://math.stackexchange.com/users/1071321 2022-10-04T06:14:47Z 2022-10-04T06:56:40Z <p>Find the equation of the angles between the both planes below <span class="math-container">$$2x-y+2z+3=0$$</span> <span class="math-container">$$3x-2y+6z+8=0$$</span> and specify the plane which bisects the acute angle and the plane which bisects the obtuse angle.</p> <p>I am facing problems in determining the plane which bisects the acute angle and the one that bisects the obtuse angle.</p> https://math.stackexchange.com/q/4544809 0 Measurability of uncountable union of measurable sets Turin123 https://math.stackexchange.com/users/1102760 2022-10-04T06:13:28Z 2022-10-04T06:30:14Z <p>Let <span class="math-container">$E \subset (0,1) \times (0,1)$</span> be a set such that <span class="math-container">$E \cap (\{p\}\times (0,1))$</span> is (Lebesgue)-measurable and such that <span class="math-container">$E \cap (\{p\}\times (0,1))$</span> has full one-dimensional Lebesgue measure for every <span class="math-container">$p \in (0,1)$</span> . Is the set <span class="math-container">$E$</span> then also (Lebesgue)-measurable?</p> https://math.stackexchange.com/q/4544805 0 À CRT based question from chezch slovak math olympiad baron jary https://math.stackexchange.com/users/960946 2022-10-04T06:09:16Z 2022-10-04T06:09:16Z <p><a href="https://i.stack.imgur.com/UjMEA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/UjMEA.png" alt="enter image description here" /></a></p> <p>This is the first solution .. But i cannot understand it properly . Its another solution is -</p> <p>Simply take <span class="math-container">$a_n=(n!)^3$</span> , then for each <span class="math-container">$k\ge 2$</span> consider the terms of the sequence <span class="math-container">$\dots , k+a_{n},k+a_{n+1},\dots$</span> where <span class="math-container">$n$</span> is sufficiently large , all of them are divisible by <span class="math-container">$k$</span> . So it remains to check when <span class="math-container">$k=0,1$</span> . When <span class="math-container">$k=0$</span> , we're clearly done . If <span class="math-container">$k=1$</span> , <span class="math-container">$(n!)^3+1=(n!+1)((n!)^2-n!+1)$</span> is clearly composite for all <span class="math-container">$n\neq{1}$</span></p> <p><strong>Doubt</strong></p> <p>Please anyone give me detailed explanation of these two solutions .</p> https://math.stackexchange.com/q/4544804 0 How do I get the volume of this? AFM https://math.stackexchange.com/users/1102413 2022-10-04T06:06:56Z 2022-10-04T06:06:56Z <p>An A4 sheet has the dimensions 210 mm x 297 mm.</p> <p>Each corner is cut a square away as shown in the picture, with the side being x mm.</p> <p>The edges are then folded up and taped together so that a box without a lid is formed.</p> <p>a) Write a function V (x) that gives the volume of the box in mm³, where x mm is the side of the square that is cut off.</p> <p>b) Write the domain of the function V(x).</p> <p>We calculate the volume of a rectangular block/box through the width * height * length, the base is usually a two-dimensional measurement. So we need three measurements to calculate the volume. Width, height and length of the box. Then try to identify the three measurements in the figure.</p> <p>But from there I don't know how to solve the problem.</p> <p><a href="https://i.stack.imgur.com/e9AWW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/e9AWW.png" alt="enter image description here" /></a></p> https://math.stackexchange.com/q/4544802 0 Symmetric square of an $\mathfrak{sl}_2$-representation MathManiac https://math.stackexchange.com/users/245727 2022-10-04T06:03:16Z 2022-10-04T06:03:16Z <p>Consider the Lie algebra <span class="math-container">$\mathfrak{sl}_2$</span> with basis <span class="math-container">$\{ x,y,h\}$</span> satisfying the standard commutation relations.</p> <p>Let <span class="math-container">$V$</span> be an <span class="math-container">$\mathfrak{sl}_2$</span>-representation over <span class="math-container">$\mathbb{C}$</span>, that isn't necessarily simple. Then, the space <span class="math-container">$Sym^2(V)$</span> has a natural <span class="math-container">$\mathfrak{sl}_2$</span>-action. Now, pick some <span class="math-container">$v\in V$</span> and consider <span class="math-container">$v^2\in Sym^2(V)$</span>. Suppose <span class="math-container">$v^2 = x.w$</span> for some <span class="math-container">$w\in Sym^2(V)$</span>. Then, I want to claim that <span class="math-container">$v$</span> must lie in the subspace of <span class="math-container">$V$</span> spanned by the eigenvectors of <span class="math-container">$h$</span> in <span class="math-container">$V$</span> having strictly positive eigenvalues.</p> <p>I can prove this by fixing a basis for <span class="math-container">$V$</span> for which the <span class="math-container">$\mathfrak{sl}_2$</span> action is nice, but the proof eventually gets quite messy. So, I am looking for a nicer proof, that hopefully doesn't require fixing any coordinates.</p> https://math.stackexchange.com/q/4544800 0 Determine the function to which the Fourier series converges for $f(x)=x$ John Infinity https://math.stackexchange.com/users/469000 2022-10-04T06:00:51Z 2022-10-04T06:00:51Z <p>Determine the function to which the Fourier series converges for <span class="math-container">$f(x)$</span> given the following</p> <p><span class="math-container">$$f(x)=x,~~~~~-\pi&lt;x&lt;\pi$$</span></p> <p>Solution: <span class="math-container">\begin{align*} a_0&amp;=\frac{1}{\pi}\int_{-\pi}^{\pi} x dx\\ &amp;=\frac{1}{\pi}\left[\frac{x^2}{2}\right]_{-\pi}^{\pi}\\ &amp;=0. \end{align*}</span> Since <span class="math-container">$f(x)$</span> is odd so <span class="math-container">$a_n=0$</span>.</p> <p><span class="math-container">\begin{align*} b_n&amp;=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)dx\\ &amp;=\frac{1}{\pi}\int_{-\pi}^\pi x\sin(nx)dx\\ &amp;=\frac{1}{\pi}\left[\frac{x\cos (nx)}{n}+\dfrac{\sin(nx)}{n^{2}}\right]_{-\pi}^{\pi}\\ &amp;=-\frac{1}{n\pi}[\pi\cos(nx)-(-\pi)\cos(-n\pi)]\\ &amp;= -\frac{1}{n\pi}\cdot 2\pi\cos(n\pi)\\ &amp; = \frac{2\cdot(-1)^{n+1}}{n} \end{align*}</span></p> <p>Therefore, the required fourier series is <span class="math-container">$$f(x)=\sum_{n=1}^\infty \frac{2\cdot(-1)^{n+1}}{n} \sin nx.$$</span></p> <p>Help me check whether this series converges to <span class="math-container">$f(x)$</span>.</p> <p>Thanks in advance!</p> https://math.stackexchange.com/q/4544799 0 Find the value of $x(t)$ that maximizes $F=a^2(t)x(t)+a(t)x(t)-a(t)x^2(t)+z(t)+a(t)+a^2(t)$ XIAO YU https://math.stackexchange.com/users/1092013 2022-10-04T05:59:55Z 2022-10-04T06:16:56Z <p>For an equation <span class="math-container">$F=a^2(t)x(t)+a(t)x(t)-a(t)x^2(t)+z(t)+a(t)+a^2(t)$</span>. For each t, calculate the maximum value of the formula,and for each t, the value of <span class="math-container">$a(t), z(t)$</span> can be known. So what is the value of <span class="math-container">$x(t)$</span> that maximizes this formula.</p> https://math.stackexchange.com/q/4544792 -5 Does the sequence converge or diverge? And why? [closed] Mohammad T. https://math.stackexchange.com/users/1103019 2022-10-04T05:34:36Z 2022-10-04T06:19:22Z <p>Does the sequence a_n = |(-1)^n| converge or diverge? And why? If it converges what value does it converge to?</p> https://math.stackexchange.com/q/4544785 1 Is there an exact, analytical solution to $y'' = a\sin(y)$? Kalcifer https://math.stackexchange.com/users/834094 2022-10-04T05:18:59Z 2022-10-04T06:40:53Z <p>I came across the differential equation <span class="math-container">$$y'' = a\sin(y)$$</span> when analyzing a problem with a pendulum. I tried checking it out in symbolab, and wolfram; however, both services exceeded computation time. I could think of a way to approximate it by taking the taylor series of <span class="math-container">$\sin(y)$</span> to a desired number of terms, but I wonder if there is an exact analytical solution that exists.</p> https://math.stackexchange.com/q/4544783 -4 0 is a natural number. Prove by some techniques and also justify your answer. [closed] Azka Mehak https://math.stackexchange.com/users/1103001 2022-10-04T05:08:01Z 2022-10-04T06:38:27Z <p>[Is 0 a natural number or not?]</p> https://math.stackexchange.com/q/4544781 -3 A Complex Matrix Equation 897968 https://math.stackexchange.com/users/1103007 2022-10-04T05:05:09Z 2022-10-04T06:14:46Z <p>How do we solve <span class="math-container">$$2\alpha[FF^{T}Z(Z^{T}Z)^{-1}(Z^{T}Z)^{-1}-Z(Z^{T}Z)^{-1}(Z^{T}Z)^{-1}Z^{T}FF^{T}Z(Z^{T}Z)^{-1}-Z(Z^{T}Z)^{-1}Z^{T}FF^{T}Z(Z^{T}Z)^{-1}(Z^{T}Z)^{-1}]+\mu Z=\mu Y+\Lambda$$</span> for Z</p> <p>The shape of <span class="math-container">$F$</span> is <span class="math-container">$n\times l$</span>, <span class="math-container">$Z$</span> is <span class="math-container">$n \times k$</span> and <span class="math-container">$\Lambda$</span> is <span class="math-container">$n \times k$</span></p> https://math.stackexchange.com/q/4544779 -1 Let $S=\{1,2,3,...,2019\}.$ Find the cardinality of largest subset A of S such that for any $a,b∈A, a≠b$ Yash https://math.stackexchange.com/users/1102719 2022-10-04T04:59:47Z 2022-10-04T05:59:52Z <p>Full Question : Let <span class="math-container">$S=\{1,2,3,...,2019\}$</span>. Find the cardinality of largest subset A of S such that for any <span class="math-container">$a,b ∈A, a≠b$</span>, There exists a unique isosceles triangle with sides <span class="math-container">$a,b$</span>.</p> <p>I didn't get the question as it's not given that either <span class="math-container">$a&lt;b$</span> or <span class="math-container">$b&lt;a$</span> so I'm getting zero as the answer</p> https://math.stackexchange.com/q/4544778 1 Draw two numbers, A and B, from a set {1, 2, 3, 4, 5, 6}. A and B are drawn sequentially without replacement. Find the variance Var(3A+B). Advay Mansingka https://math.stackexchange.com/users/946423 2022-10-04T04:59:25Z 2022-10-04T06:28:29Z <p>You draw two numbers, <span class="math-container">$A$</span> and <span class="math-container">$B$</span> from a set of integers <span class="math-container">$\{1,2,3,4,5,6\}$</span>. The numbers are drawn sequentially from the set, <strong>without</strong> replacement. Find the variance <span class="math-container">$Var(3A+B)$</span>.</p> <p>I have tried working on this question for a bit but cannot seem to find a quick way to compute it - perhaps there is some insight that would allow us to skip a lot of the computation?. I do not have a lot of experience with similar questions and so I have just been trying a brute force approach so far. This quickly gets quite convoluted for me, as expanding to <span class="math-container">$Var(3A+B) = Var(3A) + Var(B) + Cov(3A,B)$</span> all require further sub steps. Any help greatly appreciated!</p> <p><a href="https://i.stack.imgur.com/GSrCn.jpg" rel="nofollow noreferrer">My approach so far</a></p> https://math.stackexchange.com/q/4544769 0 Blocks of Kronecker product Rich Hinrichsen https://math.stackexchange.com/users/1071026 2022-10-04T04:35:05Z 2022-10-04T06:17:53Z <p>Let <span class="math-container">$A$</span> be the <span class="math-container">$n\times n$</span> matrix of all <span class="math-container">$1$</span>s. Let <span class="math-container">$I$</span> be the <span class="math-container">$m\times m$</span> identity matrix with <span class="math-container">$m&lt;n$</span>. Prove that any <span class="math-container">$n\times n$</span> block of <span class="math-container">$A\otimes I$</span> contains a <span class="math-container">$1$</span>.</p> <p>My approach is to show that the largest possible block of all zeros in <span class="math-container">$A\otimes I$</span> is <span class="math-container">$k\times k$</span> where <span class="math-container">$k=floor(m/2)$</span>. If true, then no <span class="math-container">$n\times n$</span> block could contain all zeroes.</p> https://math.stackexchange.com/q/4544742 5 Is it possible to prove the derivative of $e^x$ is $e^x$ using the limit definition of $e$ without using binomial expansion? Katie https://math.stackexchange.com/users/783429 2022-10-04T03:34:07Z 2022-10-04T06:11:12Z <p>I am teaching my students about the derivative of <span class="math-container">$e^x$</span>. I have walked through what is in our textbook and they have all happily believed me, but I would like to have a better explanation for why the derivative is <span class="math-container">$e^x$</span>. I know several other proofs exist like those that define <span class="math-container">$e$</span> using the slope at <span class="math-container">$0$</span> and those that use the natural log, but I'd like my proof to closely follow the method used in the textbook. That is why <a href="https://math.stackexchange.com/questions/671281/derivative-of-exponential-function-proof">this</a> question and <a href="https://math.stackexchange.com/questions/359023/using-the-limit-definition-to-find-the-derivative-of-ex">this</a> question have not answered all of my questions. Here is what the textbook says. <a href="https://i.stack.imgur.com/hBh6f.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hBh6f.jpg" alt="enter image description here" /></a></p> <p>I understand everything until they decide to let <span class="math-container">$e^{\Delta x}\approx 1 + \Delta x$</span>. My question is why do they not directly use the limit definition of <span class="math-container">$e$</span>. Is it because they can't without introducing students to binomial expansion?</p> <p>I put <span class="math-container">$e^{x}$</span> outside the limit and then tried to substitute the definition of <span class="math-container">$e$</span>. But I think I made a mistake. Here is my work:</p> <p><span class="math-container">$$e^{ x} \lim_{\Delta x\to\ 0} \frac{(\lim_{\Delta x\to\ 0}( 1+ \Delta x)^{\frac{1}{\Delta x}}) ^{\Delta x}-1}{\Delta x}$$</span></p> <p>Using limit properties I know I can rewrite this as: <span class="math-container">$$e^{x} \lim_{\Delta x\to\ 0} \frac{(\lim_{\Delta x\to\ 0}( 1+ \Delta x))-1}{\Delta x}$$</span></p> <p><strong>This is where I get confused/stuck.</strong></p> <p>If I resolve the inner limit first, I get: <span class="math-container">$$e^{x} \lim_{\Delta x\to\ 0} \frac{1-1}{\Delta x}$$</span> <span class="math-container">$$e^{x}\lim_{\Delta x\to\ 0} \frac{0}{\Delta x}$$</span> <span class="math-container">$$e^{x} \times 0 = 0$$</span></p> <p>Is there a property of limits that would allow me to &quot;get rid&quot; of that inner limit?</p> https://math.stackexchange.com/q/4544690 0 Calculating the "time down" different curves (Brachiostone problem)? aayush https://math.stackexchange.com/users/838661 2022-10-04T01:46:24Z 2022-10-04T06:02:17Z <p>I am interested in calculating the time taken to &quot;go down&quot; a variety of curves, such as quadratics and exponential curves etc. However, I am unsure of how to calculate these values. Off the internet, I have found formulae for the cycloid (The Brachiostone Problem) but I am not interested in this type of curve.</p> <p>I will demonstrate what I am intending to find for the curve which I know how to solve; the linear line going from <span class="math-container">$(100,100)$</span> to the origin <span class="math-container">$(0,0)$</span>.<a href="https://i.stack.imgur.com/8gzYL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8gzYL.png" alt="enter image description here" /></a></p> <p>As acceleration will be constant, we can use:</p> <p><span class="math-container">$$mg\Delta h = \frac{mv^2}{2}$$</span> <span class="math-container">$$v^2=2g\Delta h$$</span> <span class="math-container">$$v = \sqrt{2g\Delta h}$$</span></p> <p>As <span class="math-container">$\Delta h = 100$</span>: <span class="math-container">$$v=\sqrt{200g}=44.3\space m\space s^{-1}$$</span></p> <p>And as <span class="math-container">$time=\frac{s}{v}$</span>: <span class="math-container">$$T=\frac{\sqrt{100^2+100^2}}{44.3}=3.19\space s$$</span></p> <p>Therefore, the question becomes: <strong>How do I solve this problem for different types of curves i.e. when acceleration is not constant</strong></p> https://math.stackexchange.com/q/4544673 0 Find fundamental matrix in Floquet theory Bayesian guy https://math.stackexchange.com/users/594523 2022-10-04T01:01:49Z 2022-10-04T05:58:35Z <p>I have problems finding the fundamental matrix for this exercise of Floquet theory <span class="math-container">\begin{align*} \dot{x} &amp;= − \sin(2t)x + (\cos(2t) − 1)y\\ \dot{y} &amp;= (\cos(2t) + 1)x + \sin(2t)y \end{align*}</span></p> <p>I noticed that if I rearrange this system into the form of <span class="math-container">$\dot{x}=A(t)x$</span> then <span class="math-container">$trace(A)=0$</span> and using the formula of the Wronskian I have <span class="math-container">$$W(t)=1,$$</span> I think the matrix is something like <span class="math-container">$$\Phi(t)=\begin{bmatrix} e^{t}\cos(2t) &amp; -e^{-t}\sin(2t)\\ e^{t}\sin(2t) &amp; e^{-t}\cos(2t) \end{bmatrix},$$</span> but that matrix does not satisfy that <span class="math-container">$$\dot{\Phi}(t)=A(t) \Phi(t).$$</span></p> <p>Any help?</p> https://math.stackexchange.com/q/4544628 2 Explicitly Computing Structure Sheaves For Quotients of Polynomial Rings Mnifldz https://math.stackexchange.com/users/210719 2022-10-03T23:14:50Z 2022-10-04T06:28:02Z <p>I'm working through an introductory problem in scheme theory and just want to make sure I'm understanding the concepts correctly. The problem states to consider the following two rings, specify their Zariski open sets, as well as the restriction maps:</p> <ul> <li><span class="math-container">$A = \mathbb{C}[t]/\left (t^2-t\right) = \{a+bt \; | \; a,b\in\mathbb{C}\}$</span></li> <li><span class="math-container">$B = \mathbb{C}[t]/\left (t^3-t^2\right) = \{a+bt+ct^2 \; | \; a,b,c\in\mathbb{C}\}$</span></li> </ul> <p>For ring <span class="math-container">$A$</span> we can compute <span class="math-container">$X_A = \text{Spec} A = \left \{\widetilde{(t)}, \widetilde{(t-1)}\right \}$</span> where the <span class="math-container">$\sim$</span> is to signify that we are really dealing with the quotients <span class="math-container">$\widetilde{(t)} = (t)/\left (t^2-t\right )$</span>. I arrived at this since prime ideals in a ring <span class="math-container">$R/I$</span> are projections of prime ideals of <span class="math-container">$R$</span> containing <span class="math-container">$I$</span>. Furthermore, I concluded that the Zariski topology on <span class="math-container">$X_A$</span> is discrete since both prime ideals can be written as the zero set of an ideal (namely itself) in <span class="math-container">$A$</span>. At the risk of overdoing the notation I'll call the open sets <span class="math-container">$U_t$</span> and <span class="math-container">$U_{t-1}$</span>, then the structure sheaf elements should be</p> <p><span class="math-container">\begin{eqnarray*} \mathcal{O}(U_t) &amp; = &amp; A_{\widetilde{(t)}} \;\; =\;\; \left \{\left .\frac{a+bt}{c+dt} \;\right | \; c+dt \not \in \widetilde{(t)}\right \} \\ \mathcal{O}(U_{t-1}) &amp; = &amp; A_{\widetilde{(t-1)}} \;\; =\;\; \left \{\left .\frac{a+bt}{c+dt} \;\right | \; c+dt \not \in \widetilde{(t-1)}\right \}. \end{eqnarray*}</span></p> <p>Is my logic more or less correct? We as well have that <span class="math-container">$U_t \cap U_{t-1} = \emptyset$</span> so we need not worry about the sheaf element in that case.</p> <p>For ring <span class="math-container">$B$</span> much of my logic remains the same in that <span class="math-container">$X_B = \text{Spec} B = \left \{\widetilde{(t)}, \widetilde{(t-1)}\right \}$</span>, this is discrete as well for similar reasons, and the only difference is in the forms of the structure sheaf elements:</p> <p><span class="math-container">\begin{eqnarray*} \mathcal{O}(U_t) &amp; = &amp; B_{\widetilde{(t)}} \;\; =\;\; \left \{\left .\frac{a+bt+ct^2}{d+ft+gt^2} \;\right | \; d+ft+gt^2 \not \in \widetilde{(t)}\right \} \\ \mathcal{O}(U_{t-1}) &amp; = &amp; B_{\widetilde{(t-1)}} \;\; =\;\; \left \{\left .\frac{a+bt+ct^2}{d+ft+gt^2} \;\right | \; d+ft+gt^2 \not \in \widetilde{(t-1)}\right \}. \end{eqnarray*}</span></p> <p>I'm studying algebraic geometry for the first time out of Bosch's <em>Algebraic Geometry and Commutative Algebra</em>, and I also don't have much of an algebra background aside from groups, Lie groups, and basic ring theory.</p> <p><strong>Response to Comments</strong></p> <p>I think I caught KReiser's comment about the denominators in ring <span class="math-container">$B$</span> being incorrect. In response to the comments about rationalization, for ring <span class="math-container">$A$</span> we can take</p> <p><span class="math-container">$$\frac{a+bt}{c+dt} \;\; =\;\; \frac{at+bt^2}{ct+dt^2} \;\; =\;\; \frac{at+bt}{ct+dt} \;\; =\;\; \frac{a+b}{c+d}$$</span></p> <p>which means that both sheaf elements in ring <span class="math-container">$A$</span> are just copies of <span class="math-container">$\mathbb{C}$</span>. So can we conclude that <span class="math-container">$\mathcal{O}(U_t) \cong \mathbb{C}$</span> as well as <span class="math-container">$\mathcal{O}(U_{t-1}) \cong \mathbb{C}$</span>?</p> <p>Moving on to ring <span class="math-container">$B$</span> we can similarly compute</p> <p><span class="math-container">$$\frac{a+bt+ct^2}{d+ft+gt^2} \;\; =\;\; \frac{at^2 + bt^3 +gt^4}{dt^2+ft^3+gt^4} \;\; =\;\; \frac{a+b+c}{d+f+g}$$</span></p> <p>so, do we get the same result for ring <span class="math-container">$B$</span>?</p> https://math.stackexchange.com/q/4544614 3 When do Several immersions of codimension-1 imply an immersion of higher co-dimension? R. Rankin https://math.stackexchange.com/users/316713 2022-10-03T22:41:32Z 2022-10-04T06:04:53Z <p>Coming at you guys from a physics point of view, so apologies for any sloppiness or improper use of terminology.</p> <p>Suppose I have an n-dimensional manifold <span class="math-container">$\Sigma$</span> with a codimension-1 immersion into another manifold <span class="math-container">$\Omega$</span>. let's call it a map <span class="math-container">$\rho$</span> such that</p> <p><span class="math-container">$$\rho:\Omega\rightarrow\Sigma$$</span></p> <p>Now let us consider the tangent bundle (or an associated bundle) <span class="math-container">$T\Sigma$</span> into which <span class="math-container">$\Omega$</span> is immersed by an (n-1)codimension immersion <span class="math-container">$\sigma$</span>.</p> <p><span class="math-container">$$\sigma:T(\Sigma)\rightarrow\Omega$$</span></p> <p>Let all manifolds be endowed with a (possibly pseudo) Riemannian structure. Consider the composition of immersions <span class="math-container">$$\sigma(\rho):T\Sigma\rightarrow\Sigma$$</span></p> <p>which would be an immersion of codimension <span class="math-container">$n$</span>. We could find in principle <span class="math-container">$n$</span> such codimension-1 immersions</p> <p><span class="math-container">$$\rho_{k}:\Omega_{k}\rightarrow\Sigma$$</span></p> <p>such that the codimensions of each immersion <span class="math-container">$\rho_{k}$</span> (each orthogonal to one another) all together with <span class="math-container">$\Sigma$</span> are equivalent to the space <span class="math-container">$T(\Sigma)$</span>. Does having <span class="math-container">$n$</span> such immersions imply the map:</p> <p><span class="math-container">$$T(\Sigma)\rightarrow\Sigma$$</span></p> <p>or rather is there some equivalence:</p> <p><span class="math-container">$$\rho_{1},\rho_{2}\cdots\rho_{k}=\sigma(\rho)?$$</span></p> <p>What conditions do I need? Moreover, I need the implied immersion to be an isometric embedding. Not sure if it matters, but the space <span class="math-container">$T\Sigma$</span> will be the tangent bundle <span class="math-container">$T(\Sigma)$</span> or an associated bundle (such as the oriented orthonormal frame bundle of <span class="math-container">$\Sigma$</span>)</p> <p>Please let me know if I can clarify anything, I'm very new to this.</p> https://math.stackexchange.com/q/4544411 6 A Weighted Gaussian Inequality: $E[\frac{\sigma_n^2 x_n^2}{\sum_{i=1}^n \sigma_i^2x_i^2} ] \ge \frac{\sigma_n^2}{\sum_{i=1}^n \sigma_i^2}$ Thomas Ahle https://math.stackexchange.com/users/7072 2022-10-03T17:08:11Z 2022-10-04T06:13:18Z <p>Given <span class="math-container">$\sigma_1 \ge \dots \ge \sigma_n \ge 0$</span>, and independent random gaussian variables <span class="math-container">$x_1, \dots, x_n \sim \mathcal N(0,1)$</span>, I want to show: <span class="math-container">$$\mathbb E\left[ \frac{\sigma_n^2 x_n^2}{\sum_{i=1}^n \sigma_i^2 x_i^2} \right] \ge \frac{\sigma_n^2}{\sum_{i=1}^n \sigma_i^2}.$$</span> Note that this corresponds to taking the expectation of the numerator and denominator individually.</p> <hr /> <p>Using Jensen's inequality I can show <span class="math-container">\begin{align} \mathbb E\left[ \frac{x^2}{x^2 + z} \right] &amp;= \mathbb E\left[ \mathbb E\left[ \frac{x^2}{x^2 + z} \mid x \right] \right] \\&amp;\ge \mathbb E\left[ \frac{x^2}{x^2 + \mathbb E[z]} \right] \\&amp;\approx \frac{1}{1 + \sqrt{\mathbb E[z]} + \mathbb E[z]}. \end{align}</span> However, what I would need to be true is <span class="math-container">$\mathbb E\left[ \frac{x^2}{x^2 + z} \right] \ge \frac{1}{1 + \mathbb E[z]}$</span>, and that certainly doesn't hold in general. In particular it seems I need to somehow use that it's the smallest <span class="math-container">$\sigma_n$</span> that's in the numerator. The equivalent result with an arbitrary <span class="math-container">$\sigma_i$</span> doesn't seem to be true in general.</p> <p>It's also interesting to notice that in the simple case <span class="math-container">$n=2$</span> we get <span class="math-container">$$\mathbb E\left[ \frac{\sigma_2^2 x_2^2}{\sigma_1^2 x_1^2 + \sigma_2^2 x_2^2} \right] = \frac{\sigma_2}{\sigma_1 + \sigma_2}.$$</span> (At least Mathematica says this is true, I'd be interested in knowing a proof.) Though that definitely doesn't hold for <span class="math-container">$n &gt; 2$</span>.</p> <p>I suppose the equation <span class="math-container">$\sum_{i=1}^n \sigma_i^2 x_i^2 = 1$</span> corresponds to integrating over an ellipse, but I haven't found a nice geometric way to make use of that.</p> <hr /> <p>I tried something else. In the case <span class="math-container">$\sigma_1 = 1$</span> and <span class="math-container">$\sigma_2=\sigma_3$</span>, Mathematica can evaluate the expectation as <span class="math-container">$$\frac{x_1^2}{x_1^2 + \sigma_2^2 (x_2^2 + x_3^2)} = \frac{1}{1-\sigma_2^2}+\frac{\sigma_2 \sinh ^{-1}\left(\sqrt{\sigma_2^2-1}\right)}{\left(\sigma_2^2-1\right)^{3/2}}.$$</span> As expected this is below <span class="math-container">$1/(1+2\sigma_2^2)$</span> for <span class="math-container">$\sigma_2 &gt; 1$</span>: <a href="https://i.stack.imgur.com/pQfgs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pQfgs.png" alt="graph in the case sigma1=sigma2" /></a></p> <p>Mathematica even finds an expression for the general case <span class="math-container">$E[\frac{x_n^2}{x_n^2+a \chi^2}]$</span> where <span class="math-container">$\chi^2$</span> is Chi-squared distributed with <span class="math-container">$n-1$</span> degrees of freedom. So maybe there's a proof works by &quot;evening out&quot; the larger <span class="math-container">$\sigma$</span> values... The bound with chi-squared isn't particularly pretty though...</p> <p>A statement equivalent to my inequality is that <span class="math-container">$$\mathbb E\left[ \frac{x_n^2}{\sum_{i=1}^n p_i x_i^2} \right] \ge E\left[ \frac{x_n^2}{\frac{1}{n} \sum_{i=1}^n x_i^2} \right],$$</span> where <span class="math-container">$\sum_i p_i=1$</span>, and <span class="math-container">$p_1 \ge p_2 \ge \dots \ge p_n \ge 0$</span>. Since <span class="math-container">$E\left[ \frac{x_n^2}{\sum_{i=1}^n x_i^2} \right]=\frac1n$</span> by symmetry. It might even be that all of this is true independent of <span class="math-container">$x_i$</span> being Gaussian, as long as they are IID.</p> https://math.stackexchange.com/q/2388890 3 Laplace Transform - Why it works 14tim4 https://math.stackexchange.com/users/400843 2017-08-10T09:39:31Z 2022-10-04T06:08:51Z <p>The definition of the Laplace Transform is $$F(s) = \int^{\infty}_0 f(t)e^{-st} dt$$</p> <p>It is very useful in terms of solving linear, constant coefficient ordinary differential equations, but why exactly does it work? Why does taking the Laplace transform of each term in in the differential equation, using the Laplace Transform's linearity to get each individual term in the differential equation in its own Laplace Transform work in solving differential equations?</p> <p>Why does it work?</p> <p>I am not so sure why I am hesitant to accept this. Having seen generating functions of recursive sequences as a way to describe the sequence of numbers and aids in obtaining a (closed) formula for the recursive sequence - this is a similar analogy, we are essentially "transforming" sequences to a power series, but I am more accepting of this idea than the Laplace transform. </p> https://math.stackexchange.com/q/1415542 11 Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable. Dontknowanything https://math.stackexchange.com/users/240480 2015-08-31T06:50:28Z 2022-10-04T06:21:46Z <blockquote> <p>Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.</p> </blockquote>