Recent Questions - Mathematics Stack Exchange |
- What is the meaning of ideals of the form $I_1(X.Y)$
- Parameterization of plane with sphere intersection, get ellipse then make it a circle
- Describe the set $A(r)=\{z\in\mathbb{C}:z=\exp(1/s),0<|s|<r\}$
- Centroid of the region bounded by y = f(x) and x = f(y) curve
- An inequality of integral on [0,1] relative to Fourier transform.
- Show that A limited by the trajectory between $q$ and $\pi(q)$ is an invariant set
- $G$ has $2k>0$ vertices of odd degree if and only if there exists a partition of the edges of $G$ into $k$ open Eulerian trails.
- $\varprojlim k[t]/(t^n)\simeq k[[t]]$
- Are there any optimized root to tip space-filling fractal trees?
- What does "/" (forward slash) mean in this context?
- Fatou's Lemma without MCT
- How should I solve this simple optimization problem? Is it related to optimal control?
- Curvature tensor in basis $e^j$ and vectors $v=v_je^j$
- Calculus 1: How can I evaluate the following limit for this series?
- Expected value of $\sin(wt)$
- Finding the Remainder of $f(x)=(x-1)^2(x+2)Q(x)+R(x)$
- Can we solve a system of nonlinear equations by echelon form?
- The equation $\tan x = \tan 2x \tan 4x \tan 8x$
- $1^2 + 2^2 + 3^2 ... + n^2$ Is/what is the name of this progression / series?
- Lower bounds on the MGF for a mean zero random variable with variance $\sigma^2$
- Find the parameter $k$ given the equation of a circle [closed]
- Computing the First Derivative of a Sum of a Product
- What are the combination outcomes to the following questions based on information given
- The mean IQ of a sample of 1600 children was 99. Is it likely that this was a sample from a population with mean IQ 100 and standard deviation 15?
- Let $M$ be a locally Noetherian module. Then, either $M$ is uniform or it has a uniform direct summand?
- using vandermonde determinant to express general determinant?
- Introduction to the Theory of Distributions, Friedlander and Joshi, Exercise 2.1
- Computing the exact value of $\sum_{n=1}^\infty \left(\frac{2n+3}{3n+2}\right)^n$
- When fitting a polynomial to data points, how to determine the reasonable degree to use?
- $Sl(2,\mathbb{R})$ and matrix exponential
| What is the meaning of ideals of the form $I_1(X.Y)$ Posted: 23 May 2021 08:55 PM PDT We take the polynomial Ring as $\mathbb{K}[x_i,y_j]$ where $\{x_i |1 \le i \le m\} $ and $\{y_j|1 \le j \le n \}$. We consider $X$ is a vandermonde matrix where the entries come from $\langle \{x_i|1 \le i \le m \}\rangle$ and $Y$ as a $ n \times 1$ matrix. Then the ideal generated by $I_1(X.Y)$ is the ideal generated by the $1 \times 1$ minors or the entries of the $ m \times 1$ matrix $XY$. What does this ideal $I_1$ mean is essentially my doubt?As $X$ is a $m \times m+1$ matrix and $Y$ is a $ n \times 1$ matrix. Is it possible to show with an example?  |
| Parameterization of plane with sphere intersection, get ellipse then make it a circle Posted: 23 May 2021 08:49 PM PDT I had two cartesian equations, one of a unit sphere and other a plane. I got the cartesian intersection which led to a projection of the circle to an ellipse in the Y-Z plane and then I found a parametrization of that, being:
From this link below, the fellow by name of :"Fly By Night" he did the same as I did: https://math.stackexchange.com/questions/305894/parametrization-for-intersection-of-sphere-and-plane#:~:text=Given%20is%20the%20sphere%20x,y%3D2%20in%20R3. Then another fellow by name of @bubba, provided a very nice conversion to the true circle of intersection, but I do not understand how 'bubba' obtained these unit vectors, and how it gave the result. I think this approach by @bubba is really a great way to figure this out. I hope bubba or someone else can help out here. Regards,  |
| Describe the set $A(r)=\{z\in\mathbb{C}:z=\exp(1/s),0<|s|<r\}$ Posted: 23 May 2021 08:46 PM PDT I have the next problem:
What i tried was: \begin{equation*} \begin{split} A(r) &=\{z\in\mathbb{C}:z=\exp(1/w),0<|w|<r\}\\ &=\{z\in\mathbb{C}:z=\exp(1/w),0<|w|<r\}\\ &=\{\exp(z)\in\mathbb{C}:z=1/w,0<|w|<r\}\\ &=\{\exp(z)\in\mathbb{C}:|z|>1/r\}\\ &=\{\exp(z)\in\mathbb{C}:z\in[D(0,1/r)]^c\}\\ &=\text{Im}(\exp:\mathbb{C}\setminus D(0,1/r)\to\mathbb{C}) \end{split} \end{equation*} Then $A(r)=\{z\in\mathbb{C}:\exp(w)=z,w\in[D(0,1/r)]^c\}$. I see that it is the image through the exponential, and that it does not have the origin, then I get that $$A(r) \subseteq \mathbb{C}-\{0\}$$ But I can't see how to get the other inclusion, any hints or ideas?  |
| Centroid of the region bounded by y = f(x) and x = f(y) curve Posted: 23 May 2021 08:37 PM PDT How do i find the centroid of the region if x = f(y) is involved ? The formula i found in the book was A is the area of the region bounded by the curve  |
| An inequality of integral on [0,1] relative to Fourier transform. Posted: 23 May 2021 08:21 PM PDT For $\alpha > 0$, I want to prove that there exists a positive number $C$ s.t. the following inequality holds. $$ \int_0^1 (1-t)^{\alpha} cos(2 \pi xt) dt \leq \frac{C}{(1+|x|)^{min \{ 1+\alpha , 2 \} } }$$ I can't find any relation between Riemann-Lebesgue theorem for fourier transform on [0,1] and this inequality, but there is a hint that using the proof of it. Thank you for your answers.  |
| Show that A limited by the trajectory between $q$ and $\pi(q)$ is an invariant set Posted: 23 May 2021 08:21 PM PDT Let $F:U\rightarrow \mathbb{R}^2$ be a vector space and $\gamma$ an isolated periodic orbit of a point $p\in U$, that is, there is a neighborhood $V$ of $\gamma$ without any other periodic trajectory. Let $\phi(t,q)$ be the flow of $F$ passing by $q\in U$ with $\phi(0,q)=q$ Let $\Sigma$ be an exterior (in the sense that $\gamma$ is a Jordan curve and $\Sigma$ is on the exterior part of the Jordan curve) Poincaré section of $p$. For all purposes we may think $\Sigma$ being a straight line segment spanned by $J F(p)$ where $J$ is the anti-clockwise rotation by $\frac{\pi}{2}$. Let $q\in\Sigma$ and $\pi(q)\in\Sigma$ be the first recurrence map of $q$. Assume that $q>\pi(q)>p$ (here the order relation by looking $\Sigma$ as a segment of the real line, WLOG). I want to show that $\pi(q)>\pi(\pi(q))=\pi^2(q)$, that is, the set $A$ limited by the trajectory from $q$ to $\pi(q)$ and the line segment $[q,\pi(q)]\subseteq\Sigma$ is positively invariant, without any Bony–Brezis, just with basic results of flow, tubular flow theorem and Poincaré map theorem. I'm not finding the way to show this, event though it is clear why. I tried by contradiction to show that $\langle F(\pi^2(q)),F(p)\rangle <0$ if $\pi^2(q)>\pi(q)$, without success so far. If anyone could help me I would be grateful  |
| Posted: 23 May 2021 08:18 PM PDT Let $G$ be a connected graph, show that $G$ has $2k>0$ vertices of odd degree if and only if there exists a partition of the edges of $G$ into $k$ open Eulerian walks. Attempt: $\Rightarrow )$ Let $y_1, y_2, \ldots y_k$ and $z_1, z_2, \ldots , z_k$ be the odd vertices of $G$. We construct a new graph $H$ from $G$ by adding $k$ new vertices $x_1, x_2, \ldots , x_k$ to $G$ and joining $x_i$ to $y_i$ and $z_i$ for $i = 1, 2, \ldots , k$. Thus, $H$ is Eulerian and therefore contains an Eulerian circuit $C$. Since $y_ix_i$ and $x_iz_i$ are consecutive on $C$ for $i = 1, 2, \ldots ,k$, deleting the $k$ vertices $x_i$ from $H$ results in $k$ edge-disjoint trails in $G$ connecting odd vertices such that every edge of $G$ lies on one of these trails, i.e., they are an open Eulerian trail. $\Leftarrow )$ Suppose that $G$ has a partition in edge-disjoint trails. Let $P_1, P_2, \ldots P_l$ be the walks contained in G such that $\cup _{i=1}^{l}E(P_i)=E(G)$. Since $G$ is connected, the union of the open trails gives a closed trail $P$, since $E(P_i)\cap E(P_j)=\emptyset$ for all $i\neq j$. Thus $P$ is a closed Eulerian trail.  |
| $\varprojlim k[t]/(t^n)\simeq k[[t]]$ Posted: 23 May 2021 08:11 PM PDT I want to show
This question is already answered here but the answer is identifying $k[[t]]$ and $\prod_{n\in\Bbb N}k[t]/(t^n)$ which is how we construct the inverse limit. I want to show this by showing the universal property which is the original question in the same post.
How can I get such $\varphi$?  |
| Are there any optimized root to tip space-filling fractal trees? Posted: 23 May 2021 08:09 PM PDT I've been looking for fractals optimized for specific purposes, and one of the simplest is a space-filling tree that minimizes the distance needed to travel from the root to any one of the tips/ends. More specifically, the minimization should be per unit length of a fractal, distance from the root to tip is minimized. One example is the H-fractal vs the Pythagorean tree, which both have a clear root and tip relationship. Finally, what mathematical framework can be applied to determine the efficiency of these space-filling trees? Thank you in advance!  |
| What does "/" (forward slash) mean in this context? Posted: 23 May 2021 08:16 PM PDT I was reading A New Introduction to Modal Logic, written by G.E. Hughes and M.J. Cresswell, book on modal logic and the author uses the forward slash {/}:
but I was unable to understand it. So I ask: what does / mean in this situation?  |
| Posted: 23 May 2021 08:53 PM PDT Let $(\mathbb{R^d},\mathcal{M},\mu)$ be a measure space, $\mu$ be the Lebesgue measure, and $\{f_n\}$ a sequence of non-negative measurable functions. Then $$ \int_{\mathbb{R^d}} \liminf_{n\to\infty}f_n~d\mu \leq \liminf_{n\to\infty}\int_{\mathbb{R^d}} f_n~d\mu.$$ This can be easily proved using MCT. I am trying to prove it without using MCT. Here is a proof that does not use MCT: General Fatou's Lemma. Could you explain why the following equality is true? Is this due to an equivalence of the definition of the integral? Thanks. $$ \sup_m \int_{X_m} min( g, m) = \int_{X} \liminf_n f_n \> $$ (the last equality from the proof in the above link)  |
| How should I solve this simple optimization problem? Is it related to optimal control? Posted: 23 May 2021 07:54 PM PDT I was wondering what are the correct FOCs for this problem: $\max_a \int_{c_1}^{d_1} \int_{c_2}^{d_1} f(a(\theta_1,\theta_2)) d\theta_1 d\theta_2$ with constraint $\frac{\partial a(\theta_1,\theta_2) }{\partial \theta_1}\geq 0$ for all $\theta_1 \in [c_1,d_1],\theta_2 \in [c_2,d_2]$  |
| Curvature tensor in basis $e^j$ and vectors $v=v_je^j$ Posted: 23 May 2021 08:11 PM PDT How to demonstrate the curvature tensor $$R(V,W)=\delta_V \Gamma_W -\delta_W \Gamma_V + \Gamma_{VW} - \Gamma_V\Gamma_W $$ for the basis $e^j$ and vectors $v = v_je^j$? I know how to get the result of $R(V,W)u$ from the form of the basis $R(e_i,e_j)e_k$ , but how to fit the vectors? Any reference would be helpful.  |
| Calculus 1: How can I evaluate the following limit for this series? Posted: 23 May 2021 08:20 PM PDT I was solving practice problems for my upcoming calculus 1 final and came across this problem. I'm honestly still a little lost about the series and ratio tests.
I identified $$a_n = \frac{2\cdot 4\cdot 6\cdot \ldots (2n)}{n!}$$ but I'm not sure if a $$a_{n+1} = \frac{2\cdot 4\cdot 6\cdot \ldots (2n)^{n+1}}{(n+1)!}$$ or something else? Please help evaluate the limit using, $$\left|\frac{a_{n+1}}{a_n}\right|.$$ Thank you in advance!  |
| Posted: 23 May 2021 07:55 PM PDT I am given a function $W = \sin(wt)$ where $w$ is uniformly distributed between $0$ and $2\pi$ When I calculate $E[W]$, I get $\frac{1-cos(2\pi t)}{2\pi t}$ My question is why is there a $t$ in the denominator? It suggests that expected value gets smaller with time, looking at $W$, I can't see it happening. Side note : That $t$ in the denominator comes when I integrate $\sin(wt)dw$  |
| Finding the Remainder of $f(x)=(x-1)^2(x+2)Q(x)+R(x)$ Posted: 23 May 2021 08:55 PM PDT So I'm a high school student and I'm stuck on a question. Please help. $f(x)=(x-1)^2Q(x)+3x+1$ $f(x)=(x+2)Q(x)+4$ $f(x)=(x-1)^2(x+2)Q(x)+R(x)$ My first approach was Making $R(x)=ax^2+bx+c$, I soon found out that there are only two equations not three to find $a, b, c$ I don't understand how to solve this.  |
| Can we solve a system of nonlinear equations by echelon form? Posted: 23 May 2021 08:40 PM PDT I have solved the system of nonlinear equations $$ \begin{cases} x+2x^2+3xy=6 \\ x^2+3xy+y=5 \\ x-x^2+y=7 \end{cases} $$ by substituting $x=a,xy=b,x^2=c$ and $y=d$. Then I reduced it in echelon form and it had no solution. And this system actually has no solution. So my question is, may I do so? If yes, then why? Or is there something that we can deduce from this system?  |
| The equation $\tan x = \tan 2x \tan 4x \tan 8x$ Posted: 23 May 2021 08:38 PM PDT In the question we have the equality $$\tan 6^{\circ} \tan 42^{\circ} = \tan 12^{\circ} \tan 24^{\circ}$$ which is equivalent to $$ \tan 6^{\circ} = \tan 12^{\circ} \tan 24^{\circ} \tan 48^{\circ}$$ This means that the equation $$\tan x = \tan 2x \tan 4x \tan 8x$$ has the solution $x =6^{\circ} = \frac{\pi}{30}$. How to find all the solution of this equation?  |
| $1^2 + 2^2 + 3^2 ... + n^2$ Is/what is the name of this progression / series? Posted: 23 May 2021 08:16 PM PDT $$1^2 + 2^2 + 3^2 ... + n^2$$ Is this a series? or can it be converted to a series? What's its name? Is there a formula? ($F_n$?) Thanks. (I'm pretty novice when it comes to mathematical notation btw)  |
| Lower bounds on the MGF for a mean zero random variable with variance $\sigma^2$ Posted: 23 May 2021 08:27 PM PDT Let $X$ be mean-zero with variance $\sigma^2$. Is there a lower bound on the MGF for $X$ (or even simpler, $E e^X$) in terms of $\sigma^2$: $E[e^X] \ge f(\sigma^2)$? What about the general case where we are given the first $k$ moments?  |
| Find the parameter $k$ given the equation of a circle [closed] Posted: 23 May 2021 08:06 PM PDT Find the parameter $k$ so that the circle $$x^2 +y^2-(5k-1)x + (4-2k)y =5k$$ touches the $x$-axis. I know that if $C(p,q)$ is the center of the circle, then $q=r$ because it touches the $x$-axis. The answer is $k=-\frac{1}{5}$ but I don't know how to solve. Help please!!  |
| Computing the First Derivative of a Sum of a Product Posted: 23 May 2021 08:31 PM PDT Given real numbers $a_0, \dots, a_n,$ consider the function$$f(x) = \sum_{j = 0}^n {\left(\prod_{i \neq j} \frac{x - a_i}{a_j - a_i} \right)}.$$ I would like to find $f'(x),$ but I don't know how to answer this question. I think it involves the Chain Rule, but I don't know how to use it.  |
| What are the combination outcomes to the following questions based on information given Posted: 23 May 2021 08:26 PM PDT Right so got this assignment question that needs to be done soon. We haven't received any material on it yet but I don't want to be only starting it too close to the date. I will attempt these questions myself using decent enough formulas online. But could someone attempt these also with me whether it has explanations or not. I just want to be sure im on the right track. Here we go then 3.c) A travel agent has brochures for 20 different countries: 12 warm and 8 cold. They will choose 4 countries to feature in their monthly advertisement. (i) How many groups of 4 countries are possible? (ii) How many groups of 4 countries are possible if Alaska must be chosen? (iii) How many groups of 4 with 3 warm and 1 cold country featured are possible?  |
| Posted: 23 May 2021 08:21 PM PDT I was solving this question like this: Here level of significance (α) is not given, so I took 5% level.
But since 2.67 gives 99.6% confidence level shouldn't we say that it is likely that the sample has been drawn from this sample ? Although, the answers says it is not drawn from the sample.  |
| Posted: 23 May 2021 08:32 PM PDT Definition: An $R$-module $M$ is called locally Noetherian if, any finitely generated submodule of $M$ is Noetherian. Question: Let $M$ be a locally Noetherian module. Then, either $M$ is uniform or it has a uniform direct summand? I don't have any idea about the existence or non-existence of above statement. Please guide me how to initiate.  |
| using vandermonde determinant to express general determinant? Posted: 23 May 2021 08:15 PM PDT I am reading an introduction paragraph about Vandermonde determinant and the following statement troubles me: [Statement]: For a determinant $$D_n=\begin{vmatrix} x_{1n}&x_{1,n-1}&...&x_{11}\\x_{2n}&x_{2,n-1}&...&x_{21}\\...&...&...&...\\x_{nn}&x_{n,n-1}&...&x_{n1}\end{vmatrix}$$ Suppose that in a purely formal (?) way we replace each element $x_{ik}$ by $x_{i}^{k-1}$. After we have made such a substitution, the determinant above clearly becomes the Vandermonde determinant. This leads at once to the formula $$D_n=\begin{align}(x_1-x_2)(x_1-x_3)\ ...\ (x_1-x_n)\\\times(x_2-x_3)\ ...\ (x_2-x_n)\\............\\\times(x_1-x_n)\end{align}$$ Multiply out the terms in $D_n$ above, and in each of the products so obtained, replace $x_k^{s-1}$ by $x_{ks}$. If a product does not contain some power of the number $x_k$, then supply the factor $x_k^0$, which after substitution becomes $x_{k1}$. We note that this rule can be adopted as the definition of a determinant (?) [Questions]:
 |
| Introduction to the Theory of Distributions, Friedlander and Joshi, Exercise 2.1 Posted: 23 May 2021 08:23 PM PDT The problem reads: Show that $$\frac{1}{\pi}\frac{\epsilon}{x^2+\epsilon^2}\to\delta \ \text{in} \ \mathcal{D}'(\mathbb{R}) \ \text{as} \ \epsilon\to 0^+.$$ If I understand correctly, this means that I am supposed to show that for each $\phi\in C_C^{\infty}(\mathbb{R})$ $$\lim_{\epsilon\to 0^+}\int_{\mathbb{R}}\frac{1}{\pi}\frac{\epsilon}{x^2+\epsilon^2}\phi(x)\text{d}x=\phi(0)=\delta(\phi),$$ correct? If so, any tips on how to calculate this integral (or otherwise, prove this result)?  |
| Computing the exact value of $\sum_{n=1}^\infty \left(\frac{2n+3}{3n+2}\right)^n$ Posted: 23 May 2021 07:55 PM PDT I found this problem in my textbook, and I know that it converges, but I wanted to know if there was a way to find the exact value of the convergence (similar to what Euler did with the sum of reciprocal squares). I tried to rewrite the sum as a power series of sorts, but I don't know if it's correct, or if it made anything more complicated. Steps: $$\lim\limits_{a \to \infty} \sum_{n=1}^a \Big(\frac{2n+3}{3n+2}\Big)^n=\lim\limits_{a \to \infty} \sum_{n=1}^a \Big(\frac{2}{3} + \frac{5/3}{3n+2} \Big)^n =\lim\limits_{a \to \infty} \sum_{n=1}^a \Big(\frac{1}{3}\Big)^n\Big(2+\frac{5}{3n+2}\Big)^n$$ $$=\lim\limits_{a \to \infty} \sum_{n=1}^a \Big(\frac{1}{3}\Big)^n \sum_{m=0}^n 2^{n-m}\Big(\frac{5}{3n+2}\Big)^m \binom{n}{m}= \lim\limits_{a \to \infty} \sum_{n=1}^a \Big(\frac{2}{3}\Big)^n \sum_{m=0}^n \Big(\frac{5}{2(3n+2)}\Big)^m \binom{n}{m}$$ $$=\lim\limits_{a \to \infty}\sum_{n=1}^a \Big(\frac{2}{3}\Big)^n \sum_{m=0}^n\Big(\frac{5}{6n+4}\Big)^m \binom{n}{m}$$ I hit a wall here because I am not sure what do with the double sum part of the problem. (Note: I made the top limit of the outer sum $a$, and took the limit as $\,$$a\to\infty$$\,$ because when I tried to make a table to evaluate the double sum, I wanted to use something finite in order to get a finite answer.) EDIT: Is there an explicit formula if $a$ does not approach $\infty$?  |
| When fitting a polynomial to data points, how to determine the reasonable degree to use? Posted: 23 May 2021 08:04 PM PDT I have wondered the following: Suppose that there is a set of data points $(x_i,y_i)$. Then I would like to know if it is more reasonable to assume if there is a polynomial relation of degree $m$ between them or of degree $n$. Is there way to measure it? I know that Lagrange's polynomial gives the exact relation but for example physics formula $F=ma$ says that sometimes it is correct to choose linear polynomial to model the phenomenon.  |
| $Sl(2,\mathbb{R})$ and matrix exponential Posted: 23 May 2021 08:27 PM PDT I'm trying to prove that every matrix in $Sl(2,\mathbb{R})$ can be written as a product of two exponential matrix. First I noted that every matrix in $Sl(2,\mathbb{R})$ can be written as a product of a orthogonal matrix and a upper triangular matrix, so a orthogonal matrix can be written as exponential of some matrix, but my problem is with the upper triangular matrix. If the diagonal elements were the same my problem will be done, but they are different, I'm stucked here, any help will be welcome. Thank you.  |
| You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now. | Email delivery powered by Google |
| Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States | |
No comments:
Post a Comment