Recent Questions - Mathematics Stack Exchange most recent 30 from math.stackexchange.com 2019-05-20T12:38:29Z https://math.stackexchange.com/feeds http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://math.stackexchange.com/q/3232987 0 Is there any way to linearize x-x^2<=0? Nazi https://math.stackexchange.com/users/675715 2019-05-20T12:36:24Z 2019-05-20T12:36:24Z <p>I am trying to solve an optimization problem. The objective function and all constraints of this problem are linear except x-x^2&lt;=0. Is there any way to linearize x-x^2&lt;=0, where x is a continuous variable? Note that x is not in the objective function.</p> https://math.stackexchange.com/q/3232986 -1 Vector Valued Integration Octagonal Monk https://math.stackexchange.com/users/581210 2019-05-20T12:36:04Z 2019-05-20T12:36:04Z <p>Can anybody suggest me books or references for the topic of <strong>VECTOR VALUED INTEGRATION</strong>? I need it asap. I looked in up the google, couldn't find it.</p> https://math.stackexchange.com/q/3232981 0 Example about radical ideal and fraction ideal nilpo10 https://math.stackexchange.com/users/156064 2019-05-20T12:30:36Z 2019-05-20T12:30:36Z <p>(First time studying rings, and I need some help)</p> <p>Let <span class="math-container">$\sqrt{I}$</span> be the radical ideal on the commutative ring <span class="math-container">$R$</span>, defined as <span class="math-container">$\sqrt{I}=\{r\in R: r^n\in I\ \mathrm{for\ some\ } n\in\mathbb{Z}^{+}\}$</span>. Then, for example, <span class="math-container">$\sqrt{180\mathbb Z} = 30\mathbb Z$</span>, since <span class="math-container">$180=2^23^25$</span>, and <span class="math-container">$30 = 2*3*5$</span>.</p> <p>Using this example, I can now solve any similar problem, but I do not understand why this method of prime factorization works.</p> <p>I was also confused about the example regarding a fraction ideal.</p> <p><span class="math-container">$(I:J)=\{x\in R:xJ\subseteq I\}$</span>, where <span class="math-container">$I, J$</span> are ideals in <span class="math-container">$R$</span>. </p> <p>The example that was given to me was this: <span class="math-container">$(180\mathbb Z:700\mathbb Z)= 9\mathbb Z$</span>. The method to obtain this is as follows: <span class="math-container">$180=2^23^25, 700 = 2^25^27$</span>. And we see that <span class="math-container">$180/gcd(180, 700) = 9$</span>.</p> <p>I feel like once I see the answer it'll be clear, but as of now I do not see it.</p> https://math.stackexchange.com/q/3232979 0 Is there a point $H$ such that $\frac{AH \cdot DM}{HD^2} = \frac{BH \cdot EN}{HE^2} = \frac{CH \cdot FP}{HF^2}$? Lê Thành Đạt https://math.stackexchange.com/users/654749 2019-05-20T12:28:53Z 2019-05-20T12:28:53Z <p><a href="https://i.stack.imgur.com/fuh2n.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fuh2n.png" alt="enter image description here"></a></p> <blockquote> <p><span class="math-container">$H$</span> is a point in non-isoceles triangle <span class="math-container">$\triangle ABC$</span>. The intersections of <span class="math-container">$AH$</span> and <span class="math-container">$BC$</span>, <span class="math-container">$BH$</span> and <span class="math-container">$CA$</span>, <span class="math-container">$CH$</span> and <span class="math-container">$AB$</span> are respectively <span class="math-container">$D$</span>, <span class="math-container">$E$</span>, <span class="math-container">$F$</span>. <span class="math-container">$AD$</span>, <span class="math-container">$BE$</span> and <span class="math-container">$CF$</span> cuts <span class="math-container">$(A, B, C)$</span> respectively at <span class="math-container">$M$</span>, <span class="math-container">$N$</span> and <span class="math-container">$P$</span>. Is there a point <span class="math-container">$H$</span> such that the following equality is correct? <span class="math-container">$$\large \frac{AH \cdot DM}{HD^2} = \frac{BH \cdot EN}{HE^2} = \frac{CH \cdot FP}{HF^2}$$</span></p> <ul> <li><p>If there is not, prove why.</p></li> <li><p>If there is, illustrate how to put down point <span class="math-container">$H$</span>.</p></li> </ul> </blockquote> <p>Of course, point <span class="math-container">$H$</span> should be one of the triangle centres identified in the <a href="https://faculty.evansville.edu/ck6/encyclopedia/ETC.html" rel="nofollow noreferrer">Encyclopedia of Triangle Centers</a>. But I don't which one it is.</p> https://math.stackexchange.com/q/3232978 0 Expectation E(XY) of two dependent variables Elkana Tovey https://math.stackexchange.com/users/674936 2019-05-20T12:27:54Z 2019-05-20T12:30:47Z <p>If X and Y are 2 dependent variables, how does their combined expectation look? For example, if flipping a fair coin n times, with X representing the number of heads and Y representing the number of tails. How would I calculate E[XY], and what's the intuition behind it?</p> https://math.stackexchange.com/q/3232976 0 How to find the projection along the following vector subspace? model_checker https://math.stackexchange.com/users/255452 2019-05-20T12:26:50Z 2019-05-20T12:26:50Z <p>I am given with the inner product, <span class="math-container">$$\phi(a,b) = a_1b_1+a_2b_3 + a_3b_2$$</span> where <span class="math-container">$a=(a_1,a_2,a_3)\text{ and } b= (b_1,b_2,b_3)\in \mathbb{R}^{3}.$</span> Consider the vector space <span class="math-container">$F = \text{span}(1,1,1).$</span> Then I want to find the <span class="math-container">$P_{F}$</span> or the projection along <span class="math-container">$F$</span> explicitly. Let <span class="math-container">$u=(1,1,1)$</span> then <span class="math-container">$\phi(u,u)=3.$</span> Then <span class="math-container">$\hat{u}=(1,1,1)/\sqrt{3}.$</span> And so <span class="math-container">$$P_F(x) = \phi(x,u)\hat{u} = (x_1+x_2+x_3)\hat{u}.$$</span> Similarily, <span class="math-container">\begin{align*} P_{F^{\perp}}(x) &amp;= x-P_{F}(x) \\ &amp;= (x_1,x_2,x_3)-(x_1+x_2+x_3)\hat{u}\\ &amp;= \frac{1}{\sqrt{3}}\langle(\sqrt{3}-1)x_1+x_2+x_3,x_1+(\sqrt{3}-1)x_2+x_3,x_1+x_2+(\sqrt{3}-1)x_3\rangle \end{align*} .</span></p> <p>Is this calculation correct?</p> https://math.stackexchange.com/q/3232974 1 Does the series $\sum\limits_{n=2}^{\infty}\left(\sqrt{n+3}-\sqrt{n+2}\right)$ converge? Lasan Manujitha Ukwaththa Liya https://math.stackexchange.com/users/675717 2019-05-20T12:23:02Z 2019-05-20T12:33:34Z <p>The series <span class="math-container">$\sum\limits_{n=2}^{\infty}\left(\sqrt{n+3}-\sqrt{n+2}\right)$</span> does this converge ?</p> <p>I tried by multiplying it with the conjugate , but can I do that as both tends to infinity can we assure that the difference of these two terms are not zero?</p> <p>I don't know how to type the exact thing , but cananyone please be kind enough to edit it?</p> <p>Thank you so much !</p> https://math.stackexchange.com/q/3232973 0 Self-study Graph theory shota kobakhidze https://math.stackexchange.com/users/364394 2019-05-20T12:21:19Z 2019-05-20T12:21:19Z <p>I decided to self-study graph theory. I am not newbie in it as I have experience from programming, like I know graph algorithms and how they work, have some experience with discrete mathematics(well, that includes matching problems, coloring). However, I would like to take a book with nice theorems and proofs that will be interesting for someone like me with previous little experience with graphs. Thank you in advance.</p> https://math.stackexchange.com/q/3232972 0 Best way to visualize queueing theory for a Lecture on Markov Chains SABOY https://math.stackexchange.com/users/512018 2019-05-20T12:20:52Z 2019-05-20T12:20:52Z <p>Let the Markov Chain <span class="math-container">$X:=(X_{n})_{n \in \mathbb N_{0}}$</span> denote, for every <span class="math-container">$X_{n}$</span>, the number of people waiting in a line at time <span class="math-container">$n$</span>. Now note that <span class="math-container">$X$</span> is a Markov Chain living on <span class="math-container">$\mathbb Z_{+}$</span>. Then we denote <span class="math-container">$\rho$</span> as the probability mass function of new customers arriving in one time interval.</p> <p>This leads us to the following definition of our transition matrix <span class="math-container">$\Pi$</span>, namely, if <span class="math-container">$x,y \in \mathbb Z_{+}$</span> then </p> <p><span class="math-container">$\Pi(x,y)=\begin{cases} \rho(y) , \operatorname{ if}x=0\\ \rho(y-x+1) \operatorname{ if}x\geq1 \operatorname{ and} y\geq x-1\\ 0 \operatorname{ else} \end{cases}$</span> </p> <p>I have proven that if <span class="math-container">$\mathbb E[\rho]&gt; 1$</span> then the state <span class="math-container">$0$</span> is transient, and conversely if <span class="math-container">$\mathbb E[\rho ]\leq 1$</span> then the state <span class="math-container">$0$</span> is recurrent.</p> <p>But I want to visualize this, by showing that a graph comparing two queues with varying expectation (e.g. one where <span class="math-container">$\mathbb E[\rho]=0.95$</span> and another where <span class="math-container">$\mathbb E[\rho]=1.05$</span></p> <p>In terms of computation, the trials (or the time intervals would only need to be <span class="math-container">$\pm200$</span> and preferably computing during my presentation, and choosing for example the Poisson distribution as the distribution to be investigated. </p> <p>Those are my ideas, but I have no idea where to start in terms of software, can Mathematica help me out, particularly with regards to the varying expectation?</p> <p>Any advice/steps are greatly appreciated</p> https://math.stackexchange.com/q/3232971 0 asymptotic proportionality Giordano Giambartolomei https://math.stackexchange.com/users/303204 2019-05-20T12:19:09Z 2019-05-20T12:37:59Z <p>Is there a well established notation for asymptotic proportionality by a non-zero and non-unitary constant (for null contant we have little-oh, for unitary constant of asymptotic proportionality we have <span class="math-container">$\sim$</span>)? Is for example the usual proportionality symbol <span class="math-container">$\propto$</span> extended to such use?</p> https://math.stackexchange.com/q/3232967 0 Why isn't $\int \frac{-dx}{\sqrt{1-x^2}}$ equal to $\arccos(x)$? J. Doe https://math.stackexchange.com/users/380473 2019-05-20T12:16:38Z 2019-05-20T12:19:01Z <p>The reason I'm asking this is that because <span class="math-container">$d/dx(\arccos(x))=-\frac{1}{\sqrt{1-x^2}}$</span>, why to textbooks opt to write <span class="math-container">$\int -\frac{1}{\sqrt{1-x^2}} dx=-\arcsin(x)$</span> instead?</p> https://math.stackexchange.com/q/3232962 0 how to apply qr decomposition while keep top left block un-change? Mr.Guo https://math.stackexchange.com/users/342686 2019-05-20T12:12:09Z 2019-05-20T12:29:33Z <p>I have a matrix which is already up-triangular like :</p> <p><span class="math-container">$\begin{bmatrix}A &amp; B\\ 0 &amp; C\end{bmatrix}$</span></p> <p>in which A and C is up-triangular block, now I add a block row then get :</p> <p><span class="math-container">$\begin{bmatrix}A &amp; B\\ 0 &amp; C \\ D &amp; E\end{bmatrix}$</span></p> <p>I want to apply a qr decomposition to keep this matrix up-triangular while keep A is not changed:</p> <p><span class="math-container">$Q*\begin{bmatrix}A &amp; B\\ 0 &amp; C \\ D &amp; E\end{bmatrix} = \begin{bmatrix} A &amp; F \\ 0 &amp; G \\ 0 &amp; 0 \end{bmatrix}$</span></p> <p>I know this could be achieved by gauss elimination method, because A is already up-triangular, we can modify the D col by col according the data from A, this could also keep A not changed. </p> <p>but col by col operation is not computation efficient. I know the givens rotation or householder reflection is the common method for QR, but I think they can't keep A not changed, so any more efficient method than gauss elimination ?</p> <hr> <p>and I have an advance question, I have an up-triangular matrix like, which means a linear solve system :</p> <p><span class="math-container">$\begin{bmatrix} A &amp; B &amp;C \\ 0 &amp; D &amp; E \\ 0 &amp; 0 &amp; F \end{bmatrix}$</span></p> <p>A,D,F is up-triangular block. with some variables permute I can get another linear system:</p> <p><span class="math-container">$\begin{bmatrix} D &amp; 0 &amp; E \\ B &amp; A &amp; C \\ 0 &amp; 0 &amp; F \end{bmatrix}$</span></p> <p>I want apply a qr decomposition while keep A not changed, which mean:</p> <p><span class="math-container">$Q* \begin{bmatrix} D &amp; 0 &amp; E \\ B &amp; A &amp; C \\ 0 &amp; 0 &amp; F \end{bmatrix} = \begin{bmatrix} G &amp; H &amp; I \\ 0 &amp; A &amp; J \\ 0 &amp; 0 &amp; K \end{bmatrix}$</span></p> <p>I think even gauss elimination can't achieve this purpose. is there any solution :)?</p> https://math.stackexchange.com/q/3232949 0 What is the matrix exponential of a quadratic Jacobian? Kutsubato https://math.stackexchange.com/users/517830 2019-05-20T11:55:09Z 2019-05-20T12:22:31Z <p>For <span class="math-container">$J\in\mathbb{R}^{N\times N}$</span>, with <span class="math-container">$$J = \begin{bmatrix} \frac{\partial f_1}{\partial y_1} &amp; ... &amp; \frac{\partial f_1}{\partial y_N} \\ ... &amp; ... &amp; ... \\ \frac{\partial f_N}{\partial y_1} &amp; ... &amp; \frac{\partial f_N}{\partial y_N} \end{bmatrix},$$</span> what is <span class="math-container">$$E = e^{J\cdot t}$$</span> where <span class="math-container">$t$</span> is the time, in general?</p> https://math.stackexchange.com/q/3232918 0 Solving an inequality with the floor operator capadocia https://math.stackexchange.com/users/155984 2019-05-20T11:34:25Z 2019-05-20T12:31:18Z <p>Assume that <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span> are positive real numbers and that <span class="math-container">$I$</span> is a positive integer. How could I isolate <span class="math-container">$A$</span> in the inequality <span class="math-container">$\left \lfloor{AB/C}\right \rfloor \geq I$</span> ? The best I could do:</p> <p><span class="math-container">\begin{equation} AB/C -1 \geq I \Rightarrow \left \lfloor{AB/C}\right \rfloor \geq I \end{equation}</span></p> <p>Thus, <span class="math-container">$A\geq C(1+I)/B$</span> is a sufficient condition for <span class="math-container">$\left \lfloor{AB/C}\right \rfloor \geq I$</span>.</p> https://math.stackexchange.com/q/3232909 0 Demonstrate how determine when two mobile will be closest (dp.dv) / ||dv||² Zeldarck https://math.stackexchange.com/users/531452 2019-05-20T11:25:10Z 2019-05-20T12:34:30Z <p>I have two mobile, I now their position and velocity. I want to be able to determine in how many time they will be the closest if they still move at the same velocity. In a book I read, they present this formula : <span class="math-container">$t_{closest} = \frac{dp.dv}{||dv||²}$</span></p> <p>With : </p> <ul> <li>Considering all in 2D</li> <li><span class="math-container">$t_{closest}$</span> is in how many time will they be the closest possible</li> <li>dp = p2 - p1 </li> <li>p1 = position of mobile 1 at start</li> <li>p2 = position of mobile 2 at start</li> <li>dv = v2 - v1 </li> <li>v1 = velocity of mobile 1 at start</li> <li>v2 = velocity of mobile 2 at start</li> <li>||dv|| is the magnitude of dv</li> </ul> <p>I try it and it work, but I didn't success to demonstrate it</p> <p>Is someone able to explain me how demonstrate this formula?</p> <p>Thank you,</p> https://math.stackexchange.com/q/3232903 0 algebra odd numbers Rishira Fernando https://math.stackexchange.com/users/574844 2019-05-20T11:21:39Z 2019-05-20T12:29:43Z <p>A question states, using algebra, prove that when the square of any odd number is divided by four, the remainder is <span class="math-container">$1$</span></p> <p>I managed to go up to <span class="math-container">$4(n^{2}+n)+1$</span>, from <span class="math-container">$(2n+1)^{2}$</span> but I dont know how to prove it. Please help!</p> https://math.stackexchange.com/q/3232877 0 Kronecker delta from cardinal sine Graz https://math.stackexchange.com/users/554828 2019-05-20T10:50:47Z 2019-05-20T12:29:12Z <p>One possible definition for Dirac's delta function is via a limit of the cardinal sine, according to</p> <p><span class="math-container">\begin{equation} \lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a} \mathrm{sinc}\left(\frac{x}{a}\right)\phi(x)\,dx = \phi(0) \end{equation}</span></p> <p>where <span class="math-container">$\phi$</span> is a smooth distribution. I was wondering whether one could formally define in an analaogous fashion the Kroneckers' delta, e.g. via <span class="math-container">$\mathrm{sinc}(2\pi n)$</span>, <span class="math-container">$n\in \mathbb N$</span>.</p> https://math.stackexchange.com/q/3232869 1 Double Binomial coefficient sum identity Fra https://math.stackexchange.com/users/215418 2019-05-20T10:40:58Z 2019-05-20T12:26:38Z <p>I have a sum of factorials that I managed to put in the following form <span class="math-container">$$S = \sum_{k = 0}^{n} (-1)^k {{4n-1-2k}\choose{2n-2k}}{{2n-1}\choose{k}}$$</span> where <span class="math-container">$n\in\mathbb{N}$</span>. Mathematica can sum this object and it gives <span class="math-container">$S = 2^{-1+2n}$</span>. Despite searching countless identities I haven't found a way to prove this result. I have also tried induction but it doesn't seem to be the right approach.</p> https://math.stackexchange.com/q/3232698 1 Notation for reversed rows and/or columns of a matrix? Jean-Claude Arbaut https://math.stackexchange.com/users/43608 2019-05-20T07:42:23Z 2019-05-20T12:21:51Z <p>In <a href="https://math.stackexchange.com/a/3232638/43608">this answer</a> I am using a transformation of matrices, and I would like to know if there is a notation for this.</p> <p>Given a matrix <span class="math-container">$A$</span>, let <span class="math-container">$B$</span> be the same as <span class="math-container">$A$</span> with rows in reverse order, <span class="math-container">$C$</span> with the columns in reverse order, and <span class="math-container">$D$</span> with both rows and columns in reverse order. Is there a usual notation for any of <span class="math-container">$B,C$</span> or <span class="math-container">$D$</span> ?</p> <p>The <span class="math-container">$D$</span> case has a nice property: if it's called <span class="math-container">$f$</span>, then for any matrices <span class="math-container">$A,B$</span>, <span class="math-container">$f(AB)=f(A)f(B)$</span>.</p> <p>I know one can write for instance <span class="math-container">$B=A[n:1,:]$</span> and it's not uncommon to see such MATLAB-like formulas in numerical analysis articles, but I wonder if there is a shorthand for this, or a usual name.</p> https://math.stackexchange.com/q/3232694 0 $A$ is a square complex matrix and $f\in\mathbb C[t]$ such that $f(A)$ is diagonalizable. If $f'(A)$ is invertible, then $A$ is diagonalizble. user549397 https://math.stackexchange.com/users/549397 2019-05-20T07:33:22Z 2019-05-20T12:34:13Z <blockquote> <p>Suppose that <span class="math-container">$A$</span> is a square complex matrix and <span class="math-container">$f$</span> is a polynomial in <span class="math-container">$\mathbb C[t]$</span> such that <span class="math-container">$f(A)$</span> is diagonalizable. If <span class="math-container">$f'(A)$</span> is invertible, where <span class="math-container">$f'$</span> is the derivative of <span class="math-container">$f$</span>, prove that <span class="math-container">$A$</span> is diagonalizable in <span class="math-container">$\mathbb C$</span>.</p> </blockquote> <hr> <p>My attempt:</p> <p>Since <span class="math-container">$f(A)$</span> is diagonalizable, then the minimal polynomial of <span class="math-container">$f(A)$</span> has no multiple roots. Say <span class="math-container">$g(t)=\prod_{i=1}^n (t-r_i)\in\mathbb C[t]$</span>, with <span class="math-container">$r_i$</span> distinct. And <span class="math-container">$g(f(A))=0$</span>. In order to prove <span class="math-container">$A$</span> is diagonalizable in <span class="math-container">$\mathbb C$</span>, we need to find a polynomial <span class="math-container">$p(t)\in\mathbb C[t]$</span> with no multiple roots such that <span class="math-container">$A$</span> is annihilated by <span class="math-container">$p(t)$</span>. We know that <span class="math-container">$p(t)|g(f(t))$</span>. Now the problem is how to use the fact that <span class="math-container">$f'(A)$</span> is invertible?</p> https://math.stackexchange.com/q/3232673 0 Let $T \in T_{2}^{1}$ be associated with the map given as $Z(A,B) = AB - BA$, where $A,B \in V$, find $[T]_B$. Leif https://math.stackexchange.com/users/413652 2019-05-20T07:15:01Z 2019-05-20T12:37:41Z <p>Let <span class="math-container">$V$</span> be the space of all <span class="math-container">$3 \times 3$</span> antisymmetric matrices. And let <span class="math-container">$T \in T_{2}^{1}$</span> (tensors) be associated with the map given as <span class="math-container">$Z(A,B) = AB - BA$</span>, where <span class="math-container">$A,B \in V$</span>. </p> <p>The basis of <span class="math-container">$V$</span> is given as a classic basis of antisymmetric matrices: <span class="math-container">$\begin{pmatrix} 0 &amp; 1 &amp; 0 \\ -1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \end{pmatrix}\!, \\[2pt]$$\begin{pmatrix} 0 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 0 \\ -1 &amp; 0 &amp; 0 \end{pmatrix}\!, \\[2pt]$$\begin{pmatrix} 0 &amp; 0&amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 0 &amp; -1 &amp; 0 \end{pmatrix}\!, \\[2pt]$</span></p> <p>Find <span class="math-container">$[T]_B$</span>.</p> <p>I just have no idea how to work with tensors given like that. Should I write matrix <span class="math-container">$A$</span> as an arbitrary antisymmetric matrix and then apply the <span class="math-container">$Z$</span> map? But how is this a tensor problem? </p> <p>Thanks for any suggestions.</p> https://math.stackexchange.com/q/3232595 4 QR decomposition with lower triangular matrix using Householder reflection Moreblue https://math.stackexchange.com/users/574517 2019-05-20T04:35:49Z 2019-05-20T12:24:19Z <h2>Problem</h2> <blockquote> <p>Find householder matrices <span class="math-container">$H_1,H_2,\cdots,H_n$</span> such that</p> <p><span class="math-container">$$H_n\cdots H_1 A = L$$</span></p> </blockquote> <p>where <span class="math-container">$A$</span> : <span class="math-container">$n \times n$</span> matrix and <span class="math-container">$L$</span> : <span class="math-container">$n \times n$</span> <strong>lower triangular matrix</strong>. </p> <hr> <h2>Try</h2> <p>By defining <span class="math-container">$v_k:= [\cdots, sgn(x_k) |x_k|, \cdots]$</span> and <span class="math-container">$H_k := I - 2v_kv_k^T/v_k^Tv_k$</span>, we can make</p> <p><span class="math-container">$$H_n\cdots H_1 A = U$$</span></p> <p>where <span class="math-container">$U$</span> : <span class="math-container">$n \times n$</span> <strong>upper triangular matrix</strong></p> <p>But I'm currently stuck at how to define <span class="math-container">$v_k$</span> to make RHS lower triangular.</p> https://math.stackexchange.com/q/3232521 3 Shortest distance between two rectangles in 2D fhm https://math.stackexchange.com/users/248789 2019-05-20T02:20:23Z 2019-05-20T12:35:57Z <p>Given the center points and dimensions of two rectangles and the angles <span class="math-container">$\theta_1$</span> and <span class="math-container">$\theta_2$</span>, how to calculate the shortest distance between two rectangles (<span class="math-container">$d$</span>)? <span class="math-container">$\theta$</span> is the angle that the length of rectangle makes with <span class="math-container">$X$</span> axis on a <span class="math-container">$2D$</span> plane.</p> <p><a href="https://i.stack.imgur.com/nLDmq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nLDmq.png" alt="enter image description here"></a></p> https://math.stackexchange.com/q/3231586 1 How many ways to tile a $2 \times N$ ring using $1 \times1$ and $1 \times 2$ tiles with different colors Hang Wu https://math.stackexchange.com/users/376930 2019-05-19T10:16:56Z 2019-05-20T12:37:24Z <p>Suppose we have a closed ring with size <span class="math-container">$2 \times N$</span>, which is created by joining both ends (short edges) of a <span class="math-container">$2 \times N$</span> grid together without twisting. We have as many <span class="math-container">$1 \times1$</span> and <span class="math-container">$1 \times 2$</span> tiles as possible, and each tile is colored by one of the <span class="math-container">$C$</span> colors. It is required to tile the ring with these tiles such that no two tiles overlap, no four tiles intersect at one common point, and no adjacent tiles have the same color. Note that for two tilings, if one becomes the other after some rotation of the ring, we count it as the same tiling; but reflection is counted differently. I'd like to ask how many ways <span class="math-container">$f(N,C)$</span> are there?</p> <p>For example, <span class="math-container">$f(1,C)=C^2, f(2,C)=C(C-1)(C-2)+\frac{C(C-1)}{2}$</span>.</p> https://math.stackexchange.com/q/3231469 3 Differentiability of $G_k(w)$ user549397 https://math.stackexchange.com/users/549397 2019-05-19T07:26:42Z 2019-05-20T12:34:33Z <blockquote> <p>Suppose that <span class="math-container">$f$</span> is holomorphic in a neighborhood of <span class="math-container">$z_0$</span>, and that all complex derivatives of <span class="math-container">$f$</span> up to order <span class="math-container">$m-1$</span> at <span class="math-container">$z_0$</span> vanish, namely, <span class="math-container">$f^{(j)}(z_0)=0$</span> for all <span class="math-container">$j=0,...,m-1$</span>, but that <span class="math-container">$f^{(m)}(z_0)\ne 0$</span>.</p> <p>(a)Prove that there exist <span class="math-container">$\epsilon&gt;0$</span> and <span class="math-container">$\delta&gt;0$</span> such that, for every <span class="math-container">$k\in\mathbb N$</span>, the equation</p> <p><span class="math-container">$$G_k(w)=\frac{1}{2\pi i}\int_{|\zeta-z_0|=\epsilon}\frac{\zeta^k f'(\zeta)}{f(\zeta)-w}d\zeta$$</span> defines a holomorphic function of <span class="math-container">$w$</span> in the set <span class="math-container">$$D_\delta(f(z_0))=\{ w\in\mathbb C:|w-f(z_0)|&lt;\delta \}.$$</span></p> <p>(b) Prove that, in the context of (a), if <span class="math-container">$w\in D_\delta(f(z_0))$</span> then the equation <span class="math-container">$f(z)-w=0$</span> has <span class="math-container">$m$</span> roots (counted with multiplicity), <span class="math-container">$z_1,...,z_m,$</span> inside <span class="math-container">$|z-z_0|&lt;\epsilon$</span>, and that <span class="math-container">$$G_k(w)=\sum_{j=1}^m z_j^k.$$</span></p> </blockquote> <hr> <p>My attempt:</p> <p>(a) Suppose <span class="math-container">$w\in D_\delta(f(z_0))$</span>, then <span class="math-container">\begin{align} \frac{G_k(w+\Delta w)-G_k(w)}{\Delta w}&amp;=\frac{1}{2\pi i}\int_{|\zeta-z_0|=\epsilon}\frac{\zeta^k f'(\zeta)}{(f(\zeta)-w-\Delta w)(f(\zeta)-w)}d\zeta \end{align}</span> I was about to show that the modulus of integrand is bounded by an integrable function so that I can apply the dominated convergence theorem. However, I cannot find such a function... </p> <h2>Edit:</h2> <p>We know that if <span class="math-container">$\gamma$</span> is a Jordan curve, <span class="math-container">$\varphi(\zeta)$</span> is continuous on <span class="math-container">$\gamma$</span>, then the function <span class="math-container">$$F(z)=\frac{1}{2\pi i}\int_\gamma\frac{\varphi (\zeta)}{\zeta-z}d\zeta$$</span> is analytic on each region of <span class="math-container">$\overline{\mathbb C}\setminus\gamma$</span>. The proof of differentiability of <span class="math-container">$F(z)$</span> depends on the non-vanishment of <span class="math-container">$\zeta-z$</span> on <span class="math-container">$\overline{\mathbb C}\setminus\gamma$</span> which clearly is not the case in this problem. So we have to use different techniques.</p> https://math.stackexchange.com/q/3230748 0 An analytic function $f(z)$ on $U$ can be decomposed as $f(z)=f_1(z)+f_2(z)$ user549397 https://math.stackexchange.com/users/549397 2019-05-18T15:58:21Z 2019-05-20T12:34:48Z <blockquote> <p>Let <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span> be simple closed curves in <span class="math-container">$\mathbb C$</span> and assume that <span class="math-container">$C_2$</span> is in the interior of <span class="math-container">$C_1$</span>. Let <span class="math-container">$U$</span> be the region bounded by <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span>.</p> <p>Prove that an analytic function <span class="math-container">$f(z)$</span> on <span class="math-container">$U$</span> can be decomposed as <span class="math-container">$$f(z)=f_1(z)+f_2(z) ,$$</span> where <span class="math-container">$f_1(z)$</span> is analytic in the interior of <span class="math-container">$C_1$</span> and <span class="math-container">$f_2(z)$</span> is analytic in the exterior of <span class="math-container">$C_2$</span>(including <span class="math-container">$\infty$</span>). Moreover the decomposition is unique up to an additive constant.</p> </blockquote> <hr> <p>My Attempt:</p> <p>If <span class="math-container">$C_1$</span> and <span class="math-container">$C_2$</span> are circles then we know that <span class="math-container">$f_1(z)$</span> are the terms in the Laurent series with non-negative power, and <span class="math-container">$f_2(z)$</span> are those with negative power. Otherwise, we can imitate the important technique to prove the Laurent series locally so that we can still treat it as is the circle(since within a small angle, it preserves the inequalities and we can write <span class="math-container">$\frac{1}{\zeta-z}=\frac{1}{\zeta-z_0+z_0-z}=\frac{1}{1+\frac{z_0-z}{\zeta-z_0}}\cdot\frac{1}{\zeta-z_0}$</span> and use the Talor series of <span class="math-container">$\frac 1{1-z}$</span>). Then since the closed curve is also compact, we can get finitely many expansions of <span class="math-container">$f(z)$</span> at different parts of the curve. Finally, since the Laurent series is unique locally, then we can glue each part to get a Laurent series of <span class="math-container">$f(z)$</span> in <span class="math-container">$U$</span>. Does this work?</p> https://math.stackexchange.com/q/3228109 0 inner product in the algebra of shifted symmetric function, $\Lambda^*$ idriskameni https://math.stackexchange.com/users/510091 2019-05-16T08:54:11Z 2019-05-20T12:32:13Z <p>In the ring of symmetric functions we have an inner product defined by <span class="math-container">$$\big \langle h_{\lambda}, m_{\mu} \big \rangle = \delta_{\lambda,\mu}.$$</span> Where <span class="math-container">$\delta_{\lambda,\mu}$</span> is the Kronecker delta.</p> <p>Do we have a similar thing in the algebra of shifted symmetric functions? <span class="math-container">$$\big \langle h^*_{\lambda}, m^*_{\mu} \big \rangle = \delta_{\lambda,\mu}.$$</span></p> https://math.stackexchange.com/q/3219667 0 Relative Bertini theorem random123 https://math.stackexchange.com/users/172843 2019-05-09T12:06:34Z 2019-05-20T12:24:07Z <p><strong>Setup of the problem :</strong> Let <span class="math-container">$f : X \rightarrow Y$</span> be a flat(smooth) projective morphism with a relative very ample sheaf <span class="math-container">$\mathcal{O}(1)$</span>.</p> <p><strong>Q :</strong> Is it possible to find a section of <span class="math-container">$\mathcal{O}(n) \otimes f^*\mathcal{L}^m$</span> for some <span class="math-container">$n &gt; 0, m &gt; 0$</span> and line bundle <span class="math-container">$\mathcal{L}$</span> on <span class="math-container">$Y$</span>, such that <span class="math-container">$Z(s)$</span>, the zero set of <span class="math-container">$s$</span>, is flat(smooth) over <span class="math-container">$Y$</span>?</p> <p>If <span class="math-container">$Y = k$</span>, and we replace flatness by smooth, we get the standard Bertini theorem. I presume that this is a weaker condition and perhaps it admits some generalization in this form.</p> <p><strong>My thoughts so far :</strong> Let <span class="math-container">$E$</span> be a vector bundle over a smooth projective scheme <span class="math-container">$Y$</span> over <span class="math-container">$k(= \overline{k})$</span>, and let <span class="math-container">$X = \mathbb{P}(E) \xrightarrow{\pi} Y$</span>, the projective bundle associated to <span class="math-container">$E$</span>. For any <span class="math-container">$n$</span> large enough, I can find <span class="math-container">$s \in \Gamma(X, \mathcal{O}(n) \otimes \mathcal{L})$</span> for an ample line bundle <span class="math-container">$\mathcal{L}$</span> on <span class="math-container">$Y$</span>. This can be done as follows : We know that <span class="math-container">$\Gamma(X, \mathcal{O}(n) \otimes \mathcal{L}) = \Gamma(Y, Sym^n E \otimes \mathcal{L})$</span>. For <span class="math-container">$n$</span> large enough the rank of <span class="math-container">$Sym^nE$</span> is greater than <span class="math-container">$dim(Y)$</span> and thus can find a everywhere non-vanishing section of <span class="math-container">$Sym^n E \otimes \mathcal{L}$</span> for some ample line bundle <span class="math-container">$\mathcal{L}$</span>. This section gives us a section of <span class="math-container">$\mathcal{O}(n) \otimes \mathcal{L}$</span> such that <span class="math-container">$dim(Z(s)) &lt; dim(X)$</span> and <span class="math-container">$Z(s) \rightarrow Y$</span> is flat. The issue is we cannot repeat this argument again with <span class="math-container">$X = Z(s)$</span>, since <span class="math-container">$X$</span> maynot be projective bundle of a vector bundle.</p> <p>Hence the reformulation of the above problem in a more general setup. Feel free to put any conditions on the morphisms or the spaces which is not stronger than the ones in the above paragraph.</p> <p>Thanks.</p> https://math.stackexchange.com/q/139262 1 Optimization problem defined by polynomials only always leads to algebraic solutions? Ewan Delanoy https://math.stackexchange.com/users/15381 2012-05-01T05:58:34Z 2019-05-20T12:16:25Z <p>Let $\Omega$ be a non-empty set in ${\mathbb R}^n$ defined by a set of polynomial inequalities with rational coefficients $P_i(x_1, \ldots ,x_n) \gt 0 (1 \leq i\leq m)$ and $Q_j(x_1, \ldots ,x_n) \geq 0 (1 \leq j\leq m')$. Let $F(x_1, \ldots ,x_n)$ be a polynomial with rational coefficients such that $F$ attains its minimum on $\Omega$ at a unique point $(z_1, \ldots ,z_n)$. Does it follow that all the $z_k$ are algebraic over $\mathbb Q$ ?</p> <p>UPDATE (15 mins later) :since an answer by André Nicolas pointed out that this is a standard fact in model theory, I guess I should rephrase my question as, "Where may I learn more about this" (preferably on the internet ?)</p> https://math.stackexchange.com/q/46212 26 Interesting implicit surfaces in $\mathbb{R}^3$ Leonardo Fontoura https://math.stackexchange.com/users/7928 2011-06-19T01:34:14Z 2019-05-20T12:24:03Z <p>I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement!</p> <p>Some that I've implemented are:</p> <ul> <li>Basic surfaces like spheres and cylinders;</li> <li>Nordstrand's Weird Surface;</li> <li>Klein Quartic;</li> <li>Goursat's Surface;</li> <li>Heart Surface;</li> </ul> <p>So, my question is, what are other interesting implicit surfaces in $\mathbb{R}^3?$</p> <p>P.S.: I know this is kind of vague, but anything you find interesting will be of use. (:</p> <p>P.P.S: Turn this into a community wiki, if need be.</p>