Recent Questions - Mathematics Stack Exchange most recent 30 from math.stackexchange.com 2020-06-05T16:28:09Z https://math.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://math.stackexchange.com/q/3707147 0 Proof checking $\liminf cx_n = c \limsup x_n$ user2820579 https://math.stackexchange.com/users/141841 2020-06-05T16:23:28Z 2020-06-05T16:23:28Z <p>I just started to work with <span class="math-container">$\limsup$</span>'s and <span class="math-container">$\liminf$</span>'s and I would like to know if my proof of the identity </p> <p><span class="math-container">\begin{equation} \liminf cx_n = c \limsup x_n \end{equation}</span></p> <p>with <span class="math-container">$x_n$</span> a bounded sequence and <span class="math-container">$c\leq0$</span> is correct. </p> <p>Let <span class="math-container">$a = \limsup x_n$</span> and <span class="math-container">$\epsilon&gt;0$</span>. Then </p> <p><span class="math-container">\begin{equation} x_n &lt; a+\epsilon \end{equation}</span></p> <p>for <span class="math-container">$n$</span> sufficiently large. Multiplying by <span class="math-container">$c$</span> we get the inequeality</p> <p><span class="math-container">\begin{equation} c x_n &gt;ca + c\epsilon \end{equation}</span></p> <p>or </p> <p><span class="math-container">\begin{equation} cx_n&gt;ca-|c|\epsilon. \end{equation}</span></p> <p>That is <span class="math-container">$\liminf cx_n = ca$</span> which implies <span class="math-container">$\liminf cx_n = c\limsup x_n$</span>.</p> https://math.stackexchange.com/q/3707144 0 sum of absolute differences $|\sigma(i)-i|$ is even Thesinus https://math.stackexchange.com/users/294735 2020-06-05T16:21:39Z 2020-06-05T16:23:23Z <p>I have to show that for all <span class="math-container">$\sigma \in S_n$</span> following equation holds: <span class="math-container">$\sum_{i=1}^n|\sigma(i)-i|=2k$</span> for <span class="math-container">$k\in \mathbb{Z}$</span>.</p> <p>I have no idea how to show it and would be very grateful for any hint.</p> https://math.stackexchange.com/q/3707140 0 Joint distribution and covariance of poisson process and waiting time kim https://math.stackexchange.com/users/749428 2020-06-05T16:18:43Z 2020-06-05T16:24:17Z <p>Hi I am having a trouble solving for this problem where I have to find </p> <blockquote> <p>1) Joint distribution of <span class="math-container">$W_{1}$</span>, <span class="math-container">$W_{r}$</span> for <span class="math-container">$r\geq2$</span>. </p> <p>2) <span class="math-container">$\operatorname{Cov}(W_{1},W_{r})$</span> for <span class="math-container">$r\geq2$</span>.</p> </blockquote> <p>[Notation explanation: <span class="math-container">$W_{r}=\min(t:N_{t}\geq r)$</span> is waiting time until the <span class="math-container">$r^\text{th}$</span> occurrence. </p> <p>Here, <span class="math-container">$(N_{t})$</span>, <span class="math-container">${t\geq0}$</span> is a Poisson process with occurrence rate <span class="math-container">$\lambda&gt;0$</span>.]</p> <p>How should I solve this? I did solve the case when <span class="math-container">$r=2$</span>. In that case <span class="math-container">$\operatorname{pdf}_{W_{1},W_{2}}(t_{1},t_{2})=\lambda^2e^{-\lambda t_{2}}I(_{0&lt;t_{1}&lt;t_{2}})$</span> </p> <p>But as I tried to generalize it to case <span class="math-container">$r$</span>, it became quite complicated and I got lost. I would really appreciate if someone could help.</p> <p>Thanks.</p> https://math.stackexchange.com/q/3707139 -3 How do I merge two Frobenius norm of matrix? El Kal https://math.stackexchange.com/users/796703 2020-06-05T16:16:09Z 2020-06-05T16:24:01Z <p>Consider a soft thresholding peoblem: <span class="math-container">$$\mathop {\arg \min }\limits_B {\left\| B \right\|_1} + \frac{1}{{2\mu }}\left( {\left\| {Y - AB - Z - \Lambda } \right\|_F^2 + \left\| {L - B - \Gamma } \right\|_F^2} \right) % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbWexLMBbXgBd9gzLbvyNv2CaeHbl7mZLdGeaGqiVu0Je9sqqr % pepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs % 0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaai % aabeqaamaabaabauaakeaadaWfqaqaaiGacggacaGGYbGaai4zaiGa % c2gacaGGPbGaaiOBaaWcbaGaamOqaaqabaGcdaqbdaqaaiaadkeaai % aawMa7caGLkWoadaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaWcaaqa % aiaaigdaaeaacaaIYaGaeqiVd0gaamaabmaabaWaauWaaeaacaWGzb % GaeyOeI0IaamyqaiaadkeacqGHsislcaWGAbGaeyOeI0Iaeu4MdWea % caGLjWUaayPcSdWaa0baaSqaaiaadAeaaeaacaaIYaaaaOGaey4kaS % YaauWaaeaacaWGmbGaeyOeI0IaamOqaiabgkHiTiabfo5ahbGaayzc % SlaawQa7amaaDaaaleaacaWGgbaabaGaaGOmaaaaaOGaayjkaiaawM % caaaaa!6755!$$</span> where Y, A, Z are three known matrixes,<span class="math-container">$\Lambda$</span> and <span class="math-container">$\Gamma$</span> are two known Augmented Lagrange multipliers.</p> <p>The question is how can I merge these two F-norm to form standard soft-thresholding equation as follow？</p> <p>Please help me, your knowledge and support will be remembered. Thanks! My life depends on you now. </p> https://math.stackexchange.com/q/3707137 0 Affine transformation between compact sets juan zaragoza https://math.stackexchange.com/users/713859 2020-06-05T16:14:59Z 2020-06-05T16:14:59Z <p>So, I am starting mysefl on the study of the Finite Element Method for numerical PDE and I have found the deffinition of "affine equivalent methods", that is, that two FEM, <span class="math-container">$(K,P,\Sigma)$</span> and <span class="math-container">$(K^*,P^*,\Sigma ^*)$</span> are equivalent if there exists an affine isomorhpism <span class="math-container">$F$</span> between the compact sets <span class="math-container">$K$</span> and <span class="math-container">$K^*$</span> satisfying some things <span class="math-container">$P^*=\lbrace p^*:K\longrightarrow \mathbb{R} \ / p^*\circ F\equiv p\in P\rbrace$</span> and for every <span class="math-container">$L^*\in \Sigma$</span>, <span class="math-container">$L^*(p^*)=L(p^*\circ F)$</span> for some <span class="math-container">$L\in \Sigma$</span>.</p> <p>The entire deffinition is not of much worth for what it means to my question, which is the next one. </p> <p>Lets say <span class="math-container">$K$</span> is a tetrahedron of vertices <span class="math-container">$v_1=(0,0,0),v_2=(1,0,0),v_3=(0,1,0),v_4=(0,0,1)$</span> and let <span class="math-container">$K^*$</span> be any other tetrahedron of vertices <span class="math-container">$u_1,u_2,u_3,u_4$</span>. Consider <span class="math-container">$$[v_1.v_2,v_3,v_4]=\lbrace \mu_1 v_1+\mu_2v_2+\mu_3v_3+\mu_4v_4 / \mu_1+\mu_2+\mu_3+\mu_4=1,\ |\mu_i|\geq 0\rbrace$$</span> the convex closure of <span class="math-container">$K$</span> and <span class="math-container">$$[u_1,u_2,u_3,u_4]=\lbrace \mu_1u_u+\mu_2u_2+\mu_3u_3+\mu_4u_4\ /\mu_1+\mu_2+\mu_3+\mu_4=1,\ |\mu_i|\geq 0\rbrace$$</span> the convex closure of <span class="math-container">$K^*$</span>.</p> <p>Let then <span class="math-container">$F$</span> be the map <span class="math-container">$F:[v_1,v_2,v_3,v_4]\longrightarrow [u_1,u_2,u_3,u_4]$</span> defined by <span class="math-container">$$F(\mu_1 v_1+\mu_2v_2+\mu_3v_3+\mu_4v_4)=\mu_1u_u+\mu_2u_2+\mu_3u_3+\mu_4u_4$$</span> which clearly identifies both tetrahedrons. But, does it do it in an affine way? I guess it does, because it identifies the edges and sides via a composition of a translation, an homotethy and rotations around the axis, but that is just intuition and I find myself unable to prove it. Also, the notion I've always been given of affine transformations is between affine spaces or subspaces, and I am not sure how to adapt it for compact sets.</p> <p>Thank you very much for your time!!</p> https://math.stackexchange.com/q/3707135 0 Infinitude of prime in the arithmetic progression$4n+1$ math is fun https://math.stackexchange.com/users/443052 2020-06-05T16:13:17Z 2020-06-05T16:20:07Z <p><a href="https://i.stack.imgur.com/an3x2.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/an3x2.jpg" alt=""></a>Is to possible to prove the problem with elementary approach as used to prove the case <span class="math-container">$4n+3$</span>. Most of the proof that proves Infinitude of primes of the form <span class="math-container">$4n+1$</span> uses the some theorem from quadratic reciprocity. </p> <p>So I was curious to know whether this proof can also be done as the same way as of the proof for the case <span class="math-container">$4n+1$</span> without using any special result.</p> <p>I am aware of the proof of this fact available in this site. But I just want the proof in the way this book mentioned.</p> <p>Any help would be appreciated. Thanks in advance.</p> https://math.stackexchange.com/q/3707134 0 How to calculate an angle to rotate a line segment anchored by one point to be within a specific distance to another segment? v010dya https://math.stackexchange.com/users/106053 2020-06-05T16:12:54Z 2020-06-05T16:19:40Z <p>The problem is outlined here:</p> <p><a href="https://i.stack.imgur.com/pTHLJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pTHLJ.png" alt="enter image description here"></a></p> <ul> <li>Coordinates of an anchor (a point that cannot move): <code>ax</code>, <code>ay</code></li> <li>Coordinates of a free point: <code>px</code>, <code>py</code></li> <li>Distance around a free point that constitutes the fact that this line touches something: <code>r</code></li> <li>A starting point where the line onto which the above line is falling by rotation: <code>sx</code>, <code>sy</code></li> <li>Angle of this line to a horizonal: <code>α</code></li> </ul> <p>The angle that I am looking for is <code>β</code>.</p> <p>I am thinking of constructing another line parallel to the line that contains <code>sx</code>, <code>sy</code> at the distance <code>r</code> from it. After that it will be possible to write a formula for a circle with the centre at <code>ax</code>, <code>ay</code> and radius <code>sqrt((ax-px)^2 + (ay-py)^2))</code> and solve for intersection of these, after which it will be possible to use the smaller angle as the answer.</p> <p>However, I cannot shake the feeling that I am missing some less intensive way to do that. Am I going about it in a long-winded way? Is there a way to make the calculation simpler?</p> https://math.stackexchange.com/q/3707133 0 Delta method for asymptotic variance Le_Poisson https://math.stackexchange.com/users/793592 2020-06-05T16:12:50Z 2020-06-05T16:12:50Z <p>Let <span class="math-container">$X_i$</span> be i.i.d. r.v. with <span class="math-container">$\sim P(\lambda)$</span> and <span class="math-container">$$\sqrt{n}*(\hat{\theta}-\theta) \overset{d}{\to} N(0,V(\hat{\theta}))$$</span></p> <p>With <span class="math-container">$\hat{\lambda} = \bar{X_n}$</span> (consistent and asymptotically normal) calculate the variance <strong><span class="math-container">$V(\hat{\lambda}) = ?$</span></strong>.</p> <p>So basically I am taking the expectation of an expectation.</p> <p>I first recognize the general variance formula as <span class="math-container">$$V(\hat{\lambda}) = E[\hat{\lambda}^2]-E[\hat{\lambda}]^2$$</span></p> <p>which I would assume I can write as <span class="math-container">$$V(\bar{X_n}) = E[\bar{X_n}^2]-E[\bar{X_n}]^2$$</span></p> <p>and the general delta method as <span class="math-container">$$\sqrt{n}(g(\hat{\lambda})-g(\lambda)) \overset{d}{\to} N(0, g'(\lambda)^2*V(\hat{\lambda}))$$</span></p> <p>From here on I am a bit lost and struggling to find the next step. Is the information that <span class="math-container">$\hat{\lambda}$</span> is asymptotically normal the clue?</p> <p>(There is a second part to the problem using an indicator function (Bernoulli), but I wanted to get this method straight first.)</p> https://math.stackexchange.com/q/3707132 1 $\sup(a + B) = a + \sup B$ John P. https://math.stackexchange.com/users/749683 2020-06-05T16:11:34Z 2020-06-05T16:23:25Z <p>I believe my proof of this simple fact is fine, but after a few false starts, I was hoping that someone could look this over. In particular, I am interested in whether there is an alternate proof.</p> <blockquote> <p>For a real number <span class="math-container">$a$</span> and non-empty subset of reals <span class="math-container">$B$</span>, define <span class="math-container">$a + B = \{a + b : b \in B\}$</span>. Show that if <span class="math-container">$B$</span> is bounded above, then <span class="math-container">$\sup(a + B) = a + \sup B$</span>.</p> </blockquote> <p>My attempt: </p> <blockquote> <p>Fix <span class="math-container">$a \in \mathbb{R}$</span>, take <span class="math-container">$B \subset \mathbb{R}$</span> to be nonempty and bounded above, and define <span class="math-container">$$a + B = \{a + b : b \in B\}.$$</span> Since <span class="math-container">$B$</span> is nonempty and bounded above, the least-upper-bound axiom guarantees the existence of <span class="math-container">$\sup B$</span>. For any <span class="math-container">$b \in B$</span>, we have <span class="math-container">$$b \leq \sup B,$$</span> which implies <span class="math-container">$$a + b \leq a + \sup B.$$</span> As this is true for any <span class="math-container">$b \in B$</span>, it follows that <span class="math-container">$a + \sup B$</span> is an upper bound of <span class="math-container">$a + B$</span>, and hence <span class="math-container">$\sup(a + B)$</span> exists, by the completeness axiom, since <span class="math-container">$B \neq \emptyset$</span> implies immediately that <span class="math-container">$a + B \neq \emptyset$</span>. I claim that <span class="math-container">$a + \sup B$</span> is in fact the least upper bound of <span class="math-container">$a + B$</span>. As we have already shown it to be an upper bound, it suffices to demonstrate that <span class="math-container">$a + \sup B$</span> is the least of the upper bounds. Let <span class="math-container">$\gamma$</span> be an upper bound of <span class="math-container">$a + B$</span>. Hence, for any <span class="math-container">$b \in B$</span>, <span class="math-container">$$b \leq \gamma,$$</span> which implies that <span class="math-container">$$b \leq a + \gamma.$$</span> As this holds for all <span class="math-container">$b \in B$</span>, <span class="math-container">$a + \gamma$</span> is an upper bound of <span class="math-container">$B$</span>. Hence, by the definition of supremum, <span class="math-container">$$a + \gamma \geq \sup B,$$</span> which implies that <span class="math-container">$$\gamma \geq a + \sup B,$$</span> as desired. </p> </blockquote> <p>I tried to write the proof initially be showing that <span class="math-container">$\sup(a + B) \leq a + \sup B$</span> and <span class="math-container">$\sup(a + B) \geq a + \sup B$</span>, but didn't have any luck. If there is a trick to it, I would be interested in hearing it.</p> https://math.stackexchange.com/q/3707131 -2 Show $(G(n+1)-G(n))/G(n)$ is bounded below where $G(n)$ is the number of unordered Goldbach partitions of $2n$. Goldbug https://math.stackexchange.com/users/622895 2020-06-05T16:10:16Z 2020-06-05T16:27:46Z <p>A scatterplot of the first 10k values of <a href="https://oeis.org/A280008" rel="nofollow noreferrer">A280008</a>/<a href="https://oeis.org/A002375" rel="nofollow noreferrer">A002375</a> shows what appears to be a <a href="https://en.wikipedia.org/wiki/Stationary_sequence" rel="nofollow noreferrer">stationary sequence</a> with a sharp lower bound. Is it possible to show a lower bound for this sequence and would doing this necessarily prove the <a href="https://en.wikipedia.org/wiki/Goldbach%27s_conjecture" rel="nofollow noreferrer">Goldbach Conjecture</a>? Has anyone seen any work related to this ratio that might be insightful?</p> <p><a href="https://i.stack.imgur.com/pf65c.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pf65c.png" alt="enter image description here"></a></p> <p>The emperical distribution is also interesting as it shows three main peaks around -0.5, 0, and 1.</p> <p><a href="https://i.stack.imgur.com/VWkCc.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VWkCc.png" alt="enter image description here"></a></p> https://math.stackexchange.com/q/3707127 0 Can we learn pure maths only by reading, I mean without pen and paper? jasmine https://math.stackexchange.com/users/557708 2020-06-05T16:06:47Z 2020-06-05T16:11:56Z <p>How to study Pure mathematics? In Pure mathematics, there are lots of theorem are there which is almost equivalent to literature subject especially Topology and abstract algebra.</p> <p>Some of my friends say me that they don't write the theorem proof in paper for practice, they only read the theorem like Novel books /literature books because it saves lot of time </p> <p>My question is that </p> <p><span class="math-container">$1.$</span>Can we learn pure maths only by reading i mean without pen and paper ?</p> <p>My <span class="math-container">$2$</span>nd question is that what is the best method /way to learn pure mathematics ?</p> https://math.stackexchange.com/q/3707123 0 Linear programming with approximate arithmetic Sébastien Loisel https://math.stackexchange.com/users/193295 2020-06-05T16:03:41Z 2020-06-05T16:19:06Z <p>One of the big achievements since the 80s is the solution of linear programs, e.g., by barrier method. For example, to solve </p> <p><span class="math-container">$$\begin{array}{ll} \text{minimize} &amp; c^T x\\ \text{subject to} &amp; Ax \leq b\end{array}$$</span></p> <p>one instead minimizes </p> <p><span class="math-container">$$F(x) = tc^Tx - \sum_{k=1}^m \log([b-Ax]_k)$$</span> </p> <p>for given <span class="math-container">$t &gt; 0$</span> to obtain the central path <span class="math-container">$x^*(t)$</span>, and then you follow it as <span class="math-container">$t \to \infty$</span> by damped Newton steps. The computational complexity is well-understood, and loosely speaking, it will take <span class="math-container">$O(\sqrt{m}\log (m\epsilon))$</span> damped Newton steps to reach duality gap <span class="math-container">$O(\epsilon)$</span>.</p> <p>I want to replace all arithmetic operations with (very) approximate ones, because my <span class="math-container">$m$</span>, and the dimension <span class="math-container">$n$</span> of <span class="math-container">$x$</span>, are both mind-bogglingly large. The quantities <span class="math-container">$c^Tx$</span> and <span class="math-container">$\log(b-Ax)$</span> will be computed very roughly, and similarly for the gradient and Hessian of <span class="math-container">$F$</span>. This is for high-dimensional PDEs, and I'm considering using hierarchical Tucker or tensor trains to approximate everything, so the errors are far from machine accuracy.</p> <p>But nevermind the tensor stuff, do we have any idea what happens when arithmetic operations are (very) inexact in linear programming? Specifically, how does it affect convergence, and can we still estimate the number of (now approximate) Newton steps?</p> <p>I've tried to google for various combinations of "linear programming inexact arithmetic" but I find nothing. At first blush, it seems to be a can of worms because even feasibility <span class="math-container">$Ax \leq b$</span> is hard to decide for given <span class="math-container">$x$</span>.</p> <p>Thanks. </p> https://math.stackexchange.com/q/3707115 1 Is it possible to calculate the regression sine function given three points? Kantura https://math.stackexchange.com/users/181004 2020-06-05T15:58:12Z 2020-06-05T16:23:56Z <p>Take the three points <span class="math-container">$(10,52)$</span>, <span class="math-container">$(20,38)$</span> and <span class="math-container">$(50,-53)$</span></p> <p>How would you calculate the sine regression line of the form: <span class="math-container">$$f(x)=A\ \sin{\frac{x+B}{C}}$$</span></p> <p>In other words how would you calculate the constants <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and <span class="math-container">$C$</span> ?</p> <p>Using the first two points I got as far as: <span class="math-container">$52\sin{\frac{20+B}{C}}=38\sin{\frac{10+B}{C}}$</span></p> <p>I can't see what the best approach is.</p> <p>All tips appreciated.</p> https://math.stackexchange.com/q/3707109 0 checking uniform convergence of series $\sum_{n=1}^\infty x^n$ Gitika https://math.stackexchange.com/users/794512 2020-06-05T15:53:56Z 2020-06-05T16:13:37Z <p>I have a doubt in a question in which I need to check the uniform convergence of the series given by:</p> <p><span class="math-container">$$\sum_{n=1}^\infty x^n$$</span> on (-<span class="math-container">$1,1$</span>)</p> <p>Now if the series is uniformly convergent,then its sequence of partial sums (s<span class="math-container">$_n$</span>) is uniformly convergent.</p> <p>I have found that <span class="math-container">$s$$_n</span> = <span class="math-container">\frac{1-x^n}{1-x}</span> which point-wise converges to <span class="math-container">s</span>(<span class="math-container">x</span>) = <span class="math-container">\frac{1}{1-x}</span></p> <p>If (<span class="math-container">s$$_n$</span>) is uniformly convergent ,then <span class="math-container">$\sup$</span>{<span class="math-container">$\lvert s_n(x)-s(x)\rvert$</span>:<span class="math-container">$x$</span> <span class="math-container">$\in$</span>(<span class="math-container">$-1,1$</span>)| should tend to <span class="math-container">$0$</span> as <span class="math-container">$n$</span> tends to infinity.</p> <p>Now how to check whether <span class="math-container">$\sup$</span>{<span class="math-container">\lvert $$\frac{x^n}{1-x}\rvert</span>:<span class="math-container">x</span> <span class="math-container">\in</span>(<span class="math-container">-1,1</span>)} converges to zero as <span class="math-container">n</span> tends to infinity or how can I use the definition here?</p> https://math.stackexchange.com/q/3707104 1 If X is path connected then X has not isolated point. Antonio Maria Di Mauro https://math.stackexchange.com/users/736008 2020-06-05T15:51:47Z 2020-06-05T16:25:19Z <blockquote> <p><strong>Lemma</strong></p> <p>If <span class="math-container">X</span> path connnected then for any <span class="math-container">x_0</span> there exist a path such that connect <span class="math-container">x_0</span> with any other point <span class="math-container">X</span> of <span class="math-container">X</span>.</p> <p><strong>Statement</strong></p> <p>If <span class="math-container">X</span> is path connected then <span class="math-container">X</span> has not isolated point.</p> </blockquote> <p>Unfortunately I can't prove the statement, but I'm sure that it is posisble to prove it showing that if <span class="math-container">x_0\in X</span> is an isolated point for <span class="math-container">X</span> then there aren't continuous path that connect <span class="math-container">x_0</span> with any other <span class="math-container">x\in X</span>. If the statement is generally false then is it true for <span class="math-container">\Bbb{R}^n</span>? So could someone help me, please? </p> https://math.stackexchange.com/q/3707065 0 Orthogonality and Linear Independence | Intuition GENIVI-LEARNER https://math.stackexchange.com/users/676416 2020-06-05T15:14:58Z 2020-06-05T16:18:17Z <p>I just like to conceptually understand linear independence and orthogonality. So in reality do one tests for whether say two vectors are co-linear and the other tests whether two vectors are perpendicular? </p> https://math.stackexchange.com/q/3707050 0 What does it mean by binary operation ( taking gcd and lcm )? Omar Muhammed Aly https://math.stackexchange.com/users/796690 2020-06-05T14:59:05Z 2020-06-05T16:16:03Z <p>I'm less than a rookie so it might seem like an easy question but I want a head start to figure things later on my own. The question wants me to prove that the set <span class="math-container">D(m) = \{ x \in \mathbb N - \{0\} : x\mid m \}</span> where <span class="math-container">m</span> is a positive number, is a lattice under two binary operations of taking : gcd and lcm. So I figured out that first I have to prove that it's a partially ordered set ( reflexive - anti Symmetric - transitive), but the problem here I can't understand what does it mean by these two operations. I need an example on the reflexive part so I could understand. or at least any example. and Sorry for this low Question. </p> https://math.stackexchange.com/q/3706889 -2 Does the series \sum_{n=1}^{\infty}(x/e^x)^n converge? Martin https://math.stackexchange.com/users/795849 2020-06-05T12:37:36Z 2020-06-05T16:27:04Z <p>Does the series converge for <span class="math-container">x\in [0,\infty)</span>? I used the ratio test &amp; get out that the series converges for whatever <span class="math-container">x</span> was.</p> https://math.stackexchange.com/q/3706450 2 How to solve vv''+v'^2-6t^2=0 deenodebt1 https://math.stackexchange.com/users/796523 2020-06-05T04:48:06Z 2020-06-05T16:13:47Z <p><strong>Hi, I have been trying to solve this equation for a week now.</strong></p> <p>However, I keep ending up with the same result everytime, it may be because my knowledge on this chapter is not great.</p> <p>I am unable to figure it out, please help. </p> <p>Below is the question and the conditions.</p> <blockquote> <p><span class="math-container">$$vv''+v'^2-6t^2=0 \\ v&gt;0, v'(0)=0,v(0)=1$$</span> </p> </blockquote> <p>They are asking to use <span class="math-container">z=vv'</span> to solve the equation.</p> <p><a href="https://i.stack.imgur.com/aw8GK.jpg" rel="nofollow noreferrer">This is my attempt, but unsure if it is the correct way.</a></p> <p>Thank you so much, any help is appreciated.</p> https://math.stackexchange.com/q/3706428 3 Determinant of 2\times 2 block matrices Vasting https://math.stackexchange.com/users/278029 2020-06-05T03:53:22Z 2020-06-05T16:17:26Z <p>I am trying to solve the problem <a href="https://math.stackexchange.com/questions/1296257/determinant-of-block-matrix-with-commuting-blocks">here</a>: Let <span class="math-container">A,B,C,D,</span> be commuting <span class="math-container">n\times n</span> matrices over the field <span class="math-container">F</span> (it is not given whether any of these matrices are invertible). Show that the determinant of the <span class="math-container">2n\times 2n</span> matrix <span class="math-container">$$\begin{bmatrix} A&amp;B\\C&amp;D \end{bmatrix}$$</span> is <span class="math-container">\det(AD-BC)</span>.</p> <p>I know that there is an answer given in a paper, which mentions working over <span class="math-container">F[x]</span>. However, I have come up with the following idea:</p> <blockquote> <p>Motivated by the adjoint formula for the inverse, we get that <span class="math-container">$$\begin{bmatrix} A&amp;B\\ C&amp;D \end{bmatrix} \begin{bmatrix} D&amp;-B\\ -C&amp;A \end{bmatrix}= \begin{bmatrix} AD-BC &amp; BA-AB\\ CD-DC&amp;AD-BC \end{bmatrix} = \begin{bmatrix} AD-BC &amp; 0\\ 0&amp;AD-BC \end{bmatrix} $$</span> Hence <span class="math-container">$$\det \left(\begin{bmatrix} A&amp;B\\ C&amp;D \end{bmatrix}\right)\det\left(\begin{bmatrix} D&amp;-B\\ -C&amp;A \end{bmatrix}\right)= \det(AD-BC)^2$$</span> But then <span class="math-container">$$\det\left( \begin{bmatrix} A&amp;B\\ C&amp;D \end{bmatrix}\right)= -\det\left(\begin{bmatrix} B&amp;A\\ D&amp;C \end{bmatrix}\right)= -\det\left(\begin{bmatrix} B&amp;D\\ A&amp;C \end{bmatrix}\right)= \det\left(\begin{bmatrix} D&amp;B\\ C&amp;A \end{bmatrix}\right) $$</span> Now, thinking of <span class="math-container">\det</span> in terms of permutations, suppose for a given permutation in the <span class="math-container">\det</span> sum that we pick <span class="math-container">m</span> elements in the first <span class="math-container">n</span> rows from the <span class="math-container">-B</span> side. Then we must also pick <span class="math-container">m</span> elements from the last <span class="math-container">n</span> rows from the <span class="math-container">-C</span> side; i.e. the sign changes must cancel out. Thus, <span class="math-container">$$\det\left(\begin{bmatrix} D&amp;B\\ C&amp;A \end{bmatrix}\right)=\det\left(\begin{bmatrix} D&amp;-B\\ -C&amp;A \end{bmatrix}\right)$$</span> which shows that <span class="math-container">$$\det \left(\begin{bmatrix} A&amp;B\\ C&amp;D \end{bmatrix}\right)=\pm \det(AD-BC)$$</span></p> </blockquote> <p>Is there a way complete the proof from here? All I need to show is that <span class="math-container">$$\det \left(\begin{bmatrix} A&amp;B\\ C&amp;D \end{bmatrix}\right)\quad\text{ and }\quad \det(AD-BC)</span> have the same sign.</p> https://math.stackexchange.com/q/3705864 0 compute the adjoint operator delta https://math.stackexchange.com/users/792905 2020-06-04T18:04:04Z 2020-06-05T16:17:36Z <p>Suppose there is a bounded linear operator <span class="math-container">T:L^2[-1,1]\to L^2[-1,1]</span> given by <span class="math-container">Tf(t)=\int_{-1}^0f(s)ds+(\int_{-1}^1f(s)ds)t^2</span>. We need to compute the adjoint of the operator <span class="math-container">T</span>. When I calculate the integral, I computed as this. <span class="math-container">\begin{aligned}\langle Tf(t)g(t)\rangle&amp;=\int_{-1}^1[\int_{-1}^0f(s)ds+(\int_{-1}^1f(s)ds)t^2]g(t)dt\\&amp;=\int_{-1}^1\int_{-1}^0f(s)dsg(t)dt+\int_{-1}^1\int_{-1}^1f(s)dst^2g(t)dt\\&amp;=\int_{-1}^0\int_{-1}^1f(s)g(t)dtds+\int_{-1}^1\int_{-1}^1f(s)t^2g(t)dtds.\end{aligned}</span> I don't know whether my calculation is correct and I don't know how to compute the next step. </p> https://math.stackexchange.com/q/3705559 0 Distribution of \frac{X_1X_3+X_2X_4}{X_3^2+X_4^2} user587389 https://math.stackexchange.com/users/587389 2020-06-04T14:47:17Z 2020-06-05T16:12:31Z <blockquote> <p><span class="math-container">X_1, X_2, X_3</span> and <span class="math-container">X_4</span> are independent standard normal random variables. Find the distribution of <span class="math-container">T=\frac{X_1X_3+X_2X_4}{X_3^2+X_4^2}$$</span></p> </blockquote> <p>I have found that <span class="math-container">U=X_1X_3+X_2X_4</span> follows standard Laplace distribution. And <span class="math-container">V=X_3^2+X_4^2</span>, being the sum of squares of two independent standard normal random variables, follows Chi-square distribution with d.f. <span class="math-container">2</span>.</p> <p>But I don't know whether <span class="math-container">U</span> and <span class="math-container">V</span> are independent. Because I only have the marginal distributions of <span class="math-container">U</span> and <span class="math-container">V</span>(i.e. standard Laplace and Chi-square, respectively), but not the joint distribution of <span class="math-container">(U,V)</span>. So I cannot check whether they are independent by checking if their joint distribution is product of their marginal distributions. <strong>How can I know that <span class="math-container">U</span> and <span class="math-container">V</span> are independent, since both have common terms like <span class="math-container">X_3</span> and <span class="math-container">X_4</span>? Is there any other way to check their independence?</strong></p> <p>Since I am not sure about their independence, I cannot proceed to find the distribution of <span class="math-container">\frac{U}{V}</span> by using Jacobian technique, because for that joint distribution of <span class="math-container">(U,V)</span> is necessary.</p> <p>Please anyone help me clear this doubt. Thanks in advance. </p> https://math.stackexchange.com/q/3705539 1 Probability with numbered balls and bins Jack Armstrong https://math.stackexchange.com/users/130948 2020-06-04T14:30:38Z 2020-06-05T16:17:11Z <p>I am trying to learn more about probability and came across an interesting question that I am stuck on and can no longer find online. There are 20 numbered balls and 10 bins. Someone is trying to assign the balls to the bins, but does it with replacement on accident.</p> <p>So they did the following: Place a ball in bin 1, record it, then remove ball (with replacement remember). Place a ball in bin 2, record it, then remove ball. Place a ball in bin 3, record it, then remove ball. So for each bin, you have put in 1 ball. There are ten bins, therefore you do that process once for every bin. Once you have done that the experiment is over.</p> <p>What is the probability exactly 1 ball was assigned to exactly 4 bins? What is the probability at least 2 bins received the same ball?</p> <p>A) 1 Ball in 4 Bins:</p> <p>We have <span class="math-container">{20 \choose 1}</span> being the different ways we can choose the 1 ball that was assigned. Also, we have <span class="math-container">{19 \choose 6}</span> being the different ways the other 19 balls can be picked for assignment. However, what is the sample size? Would it be <span class="math-container">20^{10}</span>? Thus the answer would be <span class="math-container">\frac{{20 \choose 1}{19 \choose 6}}{20^{10}}</span>.</p> <p>B) Probability of at least 2 repeated can be represented as <span class="math-container">1-P(\text{Zero Repeated})- P(\text{One Repeated})</span>. So <span class="math-container">P(0) = {20 \choose 10}/20^{10}</span> and <span class="math-container">P(1) = \frac{{20 \choose 1}{19 \choose 9}}{20^{10}}</span>. Then we can plug and chug.</p> <p>Are these right? Is this how to think about this type of problem?</p> https://math.stackexchange.com/q/3702526 2 Why is electric potential function in free space infinitely differentiable? Joe https://math.stackexchange.com/users/419536 2020-06-02T15:12:23Z 2020-06-05T16:23:16Z <p>Electric potential function in free space of a continuous charge distribution <span class="math-container">\rho'</span> distributed over volume <span class="math-container">V' \subset \mathbb{R}^3</span> is denoted by:</p> <p><span class="math-container">\psi (x,y,z): \mathbb{R}^3 \setminus{V'} \to \mathbb{R}</span></p> <p>and is defined as:</p> <p><span class="math-container">$$\psi (x,y,z)=\int_{V'} \dfrac{\rho'}{R}\ dV'$$</span></p> <p>where <span class="math-container">R</span> is distance between point <span class="math-container">(x,y,z)</span> to a point <span class="math-container"> P \in V'</span></p> <p>I know electric potential function in free space is <em>differentiable once</em> but I don't see why it is <em>infinitely differentiable</em>. Please explain why it is so.</p> <blockquote> <p><strong>EDIT</strong></p> <p><strong>Theorem:</strong> PD of <span class="math-container">\dfrac{1}{R}</span> of all orders are differentiable in domain <span class="math-container">\mathbb{R^3} \setminus (R=0)</span></p> <p><strong>Proof:</strong></p> <p>Let <span class="math-container">P_k</span> denote polynomials of degree <span class="math-container">k</span> in <span class="math-container">x,y,z,x',y',z'</span></p> <p><span class="math-container">$$\dfrac{\partial\frac{1}{R}}{\partial x}=-\dfrac{x-x'}{R^3};\ \dfrac{\partial\frac{1}{R}}{\partial y}=-\dfrac{y-y'}{R^3};\ \dfrac{\partial\frac{1}{R}}{\partial z}=-\dfrac{z-z'}{R^3}</span></p> <p>Therefore:</p> <p>PD of <span class="math-container">\dfrac{1}{R}</span> of <span class="math-container">1^{st}</span> order = <span class="math-container">\dfrac{P_1}{R^{(2 \times 1) + 1}}</span></p> <p><span class="math-container">\begin{align} \text{PD of \dfrac{1}{R} of k^{th} order} &amp;= \dfrac{P_k}{R^{2k + 1}}\\ \implies \text{PD of \dfrac{1}{R} of (k+1)^{th} order} &amp;= \dfrac{\partial}{\partial x} [\text{PD of k^{th} order} ]\\ &amp;= \dfrac{\partial \left[ \dfrac{P_k}{R^{2k + 1}} \right]}{\partial x}\\ &amp;= \dfrac{(P_k)'_x}{R^{2k+1}} - (2k+1) \dfrac{P_k}{R^{2k+2}} \dfrac{x-x'}{R}\\ &amp;= \dfrac{(P_k)'_x\ [(x-x')^2+(y-y')^2+(z-z')^2]}{R^{2k+3}} - \dfrac{(2k+1)\ P_k\ (x-x')}{R^{2k+3}}\\ &amp;=\dfrac{P_{k+1}}{R^{2(k+1)+1}}\\ \end{align}</span></p> <p>Thus by induction:</p> <p>PD of <span class="math-container">\dfrac{1}{R}</span> of <span class="math-container">n^{th}</span> order <span class="math-container">=\dfrac{P_k}{R^{2n+1}}</span></p> <p><span class="math-container">P_k</span>, being a polynomial function is continuous in <span class="math-container">x,y,z</span> in domain <span class="math-container">\mathbb{R^3}</span></p> <p><span class="math-container">\dfrac{1}{R^{2n+1}}</span>, being a radial function is continuous in <span class="math-container">x,y,z</span> in domain <span class="math-container">\mathbb{R^3} \setminus (R=0)</span></p> <p>Thus:</p> <p>PD of <span class="math-container">\dfrac{1}{R}</span> of <span class="math-container">n^{th}</span> order is continuous in <span class="math-container">x,y,z</span> in domain <span class="math-container">\mathbb{R^3} \setminus (R=0)</span></p> <p>PD of <span class="math-container">\dfrac{1}{R}</span> of all orders are continuous in <span class="math-container">x,y,z</span> in domain <span class="math-container">\mathbb{R^3} \setminus (R=0)</span></p> <p>PD of <span class="math-container">\dfrac{1}{R}</span> of all orders are differentiable in <span class="math-container">x,y,z</span> in domain <span class="math-container">\mathbb{R^3} \setminus (R=0)</span></p> </blockquote> <p><strong>Now how shall we prove PD if <span class="math-container">\psi</span> of all orders are differentiable in <span class="math-container">x,y,z</span> in domain <span class="math-container">\mathbb{R^3} \setminus {V'}</span>?</strong></p> https://math.stackexchange.com/q/2970426 1 GRE 9367 #62: Prove X=[0,1] in lower limit topology ([a,b)) is not compact, is Hausdorff and is disconnected. BCLC https://math.stackexchange.com/users/140308 2018-10-25T10:57:15Z 2020-06-05T16:13:33Z <p>GRE9367 #62</p> <blockquote> <p><a href="https://i.stack.imgur.com/cyPVZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cyPVZ.png" alt="enter image description here"></a></p> </blockquote> <p><a href="https://www.math.ucla.edu/~iacoley/gre/Practice%203%20solutions.pdf" rel="nofollow noreferrer">Ian Coley</a>'s solution:</p> <blockquote> <p><a href="https://i.stack.imgur.com/98UlF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/98UlF.png" alt="enter image description here"></a></p> </blockquote> <p><a href="https://drive.google.com/file/d/0B4qQg_AuKUglY0I1bkZ5azl6NDQ/view" rel="nofollow noreferrer">Sean Sovine</a>'s solution:</p> <blockquote> <p><a href="https://i.stack.imgur.com/24MBt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/24MBt.png" alt="enter image description here"></a></p> </blockquote> <hr> <ol> <li>Prove <span class="math-container">X</span> is not compact.</li> </ol> <p>My first proof was similar to Ian Coley's, but I came up with another proof:</p> <blockquote> <p>If <span class="math-container">X</span> is compact, then because <span class="math-container">X</span> is Hausdorff, <span class="math-container">X</span> is compact Hausdorff in both standard and lower limit topologies of <span class="math-container">\mathbb R</span>. This implies that the topologies are equal by (*), a contradiction.</p> </blockquote> <p>Did I go wrong somewhere?</p> <ol start="2"> <li>Prove <span class="math-container">X</span> is Hausdorff.</li> </ol> <p>My proof is similar to Sean Sovine's. For Ian Coley's proof, is my understanding right?</p> <blockquote> <p>If there exists the required open sets in standard topology, then we can choose the same sets as the required open sets in the lower limit topology.</p> </blockquote> <ol start="3"> <li>Prove <span class="math-container">X</span> is disconnected.</li> </ol> <blockquote> <p>My proof is the same as Ian Coley's. Is Ian Coley's proof right?</p> </blockquote> <hr> <p>(*) Munkres Exer26.1 (<a href="https://dbfin.com/topology/munkres/chapter-3/section-26-compact-spaces/problem-1-solution/" rel="nofollow noreferrer">dbfin pf</a>)</p> <blockquote> <p><a href="https://i.stack.imgur.com/6JDFA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6JDFA.png" alt="enter image description here"></a></p> </blockquote> https://math.stackexchange.com/q/2968759 0 GRE multiple choice question: Solving a linear system 4 equations in 4 unknown. user1752323 https://math.stackexchange.com/users/574479 2018-10-24T06:30:18Z 2020-06-05T16:15:15Z <p>What is the quickest yet systematic way to solve this question?</p> <p>Consider the system of linear equations: </p> <p><span class="math-container">w + 3x + 2y +2z = 0$$</span></p> <p><span class="math-container">$$w + 4x + y = 0$$</span></p> <p><span class="math-container">$$3w + 5x + 10y + 4z = 0$$</span></p> <p><span class="math-container">$$2w+ 5x + 5y + 6z = 0$$</span></p> <p>with solutions of the form <span class="math-container">(w,x,y,z)</span>, where <span class="math-container">w,x,y,z</span> are real. Which of the following statements is false:</p> <p>A. The system is consistent.</p> <p>B. The system has infinitely many solutions.</p> <p>C. The sum of any two solutions is a solution.</p> <p>D. <span class="math-container">(-5,1,1,0)</span> is a solution.</p> <p>E. Every solution is a scalar multiple of <span class="math-container">(-5,1,1,0)</span></p> <p>A related question is if we have a large <span class="math-container">(4\times 4</span> or <span class="math-container">5\times 5</span>) matrix, and we can't tell if the rows/columns are linearly independent just by looking at them, how do we tell if the matrix is invertible or not? </p> https://math.stackexchange.com/q/2967097 2 GRE 9768 #60 Boolean non-commutative rings: Prove (-s)^2=s^2 without commutativity. BCLC https://math.stackexchange.com/users/140308 2018-10-23T04:28:27Z 2020-06-05T16:15:40Z <p><a href="https://math.stackexchange.com/questions/2967052/">GRE 9768 #60</a></p> <blockquote> <p><a href="https://i.stack.imgur.com/N6BTJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/N6BTJ.png" alt="enter image description here"></a></p> </blockquote> <p><a href="https://www.math.ucla.edu/~iacoley/gre/Practice%205%20solutions.pdf" rel="nofollow noreferrer">Ian Coley's approach</a> is to prove <span class="math-container">(I)</span> and <span class="math-container">(I) \implies (II) \implies (III)</span></p> <blockquote> <p><a href="https://i.stack.imgur.com/KXhz3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KXhz3.png" alt="enter image description here"></a></p> </blockquote> <p>In proving <span class="math-container">(I)</span>, how do we prove <span class="math-container">$$(-s)^2=s^2$$</span> <a href="https://math.stackexchange.com/questions/2967070">without commutativity</a> (but with <span class="math-container">s=s^2</span>, if need be, and of course without <span class="math-container">s+s=0</span>)?</p> <p>Attempt 1:</p> <p><span class="math-container">$$-s=(-s)^2=(-s)(-s)\stackrel{(*)}{=}(-1)(s)(-1)(s)$$</span></p> <p><span class="math-container">$$s=(s)^2=(s)(s)$$</span></p> <p>I'm stuck. Perhaps <span class="math-container">-1</span> commutes with every element of a ring assuming that the ring has a multiplicative identity <span class="math-container">1</span>, but if the ring doesn't have a <span class="math-container">1</span>, then I guess the ring has <span class="math-container">-1</span> only vacuously and therefore <span class="math-container">(*)</span> is meaningless.</p> <p>Attempt 2:</p> <p><span class="math-container">$$-s=(-s)^2=(-s)(-s)$$</span></p> <p><span class="math-container">$$s=(s)^2=(s)(s)$$</span></p> <p><span class="math-container">$$\implies 0=s-s=s^2 + (-s)^2 \implies s^2 = -(-s)^2$$</span></p> <p>I'm stuck.</p> <p>Attempt 3:</p> <p><span class="math-container">$$-s=(-s)^2=(-s)(-s)$$</span></p> <p><span class="math-container">$$s=(s)^2=(s)(s)$$</span></p> <p><span class="math-container">$$\implies -s = -(s)^2=-(s)(s)$$</span></p> <p><span class="math-container">$$\implies (-s)(-s)=-(s)(s)$$</span></p> <p>I'm stuck.</p> https://math.stackexchange.com/q/2922293 3 How can we define the limit of a constant function? Mason https://math.stackexchange.com/users/587231 2018-09-19T02:17:55Z 2020-06-05T16:26:48Z <p>Wikipedia says:</p> <blockquote> <p>In mathematics, a limit is the value that a function(or sequence) "approaches" as the input (or index) "approaches" some value.</p> </blockquote> <p>What if the function was a constant?! A constant function will not approach anything, so, how would we define the limit of a constant function?</p> https://math.stackexchange.com/q/1730299 5 Eigenvalues of block matrix related kalpeshmpopat https://math.stackexchange.com/users/66342 2016-04-06T11:09:46Z 2020-06-05T16:11:38Z <p>What are the eigenvalues of following block matrix?</p> <p><span class="math-container">$$\begin{bmatrix} A &amp; B \\ B^T &amp; O \end{bmatrix}$$</span></p> <p>Here, <span class="math-container">A</span> and <span class="math-container">B</span> are any square matrices of order <span class="math-container">n</span>, <span class="math-container">O</span> is zero matrix of order <span class="math-container">n</span>.</p> https://math.stackexchange.com/q/1721650 2 Determinant of 2 \times 2 block matrix kalpeshmpopat https://math.stackexchange.com/users/66342 2016-03-31T11:15:53Z 2020-06-05T16:26:40Z <p>How to find determinant of following block matrix?</p> <p><span class="math-container">$$\begin{bmatrix} A &amp; A \\ A &amp; kI \end{bmatrix}$</span></p> <p>Where <span class="math-container">$A$</span> is any square matrix, <span class="math-container">$I$</span> is an identity matrix and <span class="math-container">$k\$</span> is any constant.</p>