Recent Questions - Mathematics Stack Exchange |
- If $f \in \Bbb Q[X_1, \dots ,X_n]$, is there a $Q \in \Bbb Q[X]$ such that $Q \circ f$ is a Symmetric Polynomial?
- Stabilizers and topological groups acting over a smooth manifold
- Finding an angle in a kite
- Amazing double series
- Using Generating Functions for solving system of recurrence relations
- Number theory : existence of p & m for all n, where p,n ≡ 5 (mod 6), p doesn't divide n, n≡m^3(mod p)
- How to find a condition for when a multivariable limit exists
- Are radius and diameters lengths or line segments itself?
- How to simplify using conjugates in Maple
- Show that $2x + 1$ is a unit in $\mathbb{Z}_4[x]$ and find a unit in $\mathbb{Z}_{p^2} [x]$
- How to calculate coefficients for cubic least squares regression utilizing equations for summation of x- and y- data points only.
- Integral change of variable from a real to vector
- Find the second real root for cubic $x^3+1-x/b=0$
- Chance letter a next to b in circle with whole alphabet such that no vowels next to each other
- Kummer's hypergeometric negative integer
- Two questions about changing variables in multiple integral
- Integral of sin(x)/(x^2+1) with respect to x from 0 to infinity [closed]
- Information about the quiver if the semi-invariant ring being a polynomial ring or hypersurface
- On computing $\nabla^2 (1/4 \pi |x|)$ in three dimensions using mollifiers.
- Proving a polynomial ring isomorphism
- Confusion in the order of the differentials in integral using differential forms
- Show that if $f$ is continuous at $a$, then if $A \subset X$ and $a \in \overline{A} \implies f(a) \in \overline{fA}$
- Limiting Distribution of Bernoulli to Poisson. ISI-PCB-NC$9$
- Waiting times at a bank - probability
- Why is CDF the only way to randomly select from samples?
- Definition for the graph (cosine wave) when velocity is plotted against displacemnt
- Name of distribution that is the square root of the sum of the squares of normal distributions?
- Probability that a random binary matrix is invertible - what is going on?
- Calculating Ext and Tor for Z/mZ and Z/nZ
- GRE Subject Test - Past Papers, Books, Advice
| Posted: 14 Aug 2021 07:48 PM PDT The Fundamental Theorem on Symmetric Polynomials, together with the Vieta's Formulas, give us the following conclusion
$$a_0+a_1X+\cdots + a_{n-1}X^{n-1}+X^n = (X-x_1) \cdots (X-x_n) \quad \Rightarrow \quad f(x_1,...,x_n)=g(a_0,...,a_{n-1})$$ If our polynomial is not symmetric, there's still hope. For example, if $f \in \Bbb Q[X_1, \dots ,X_n]$ is an anti-symmetric polynomial, then $f \cdot f$ is a symmetric polynomial and therefore there is a polynomial $g \in \Bbb Q[X_1, \dots ,X_n]$ such that $$f(x_1,\dots,x_n) = \pm \sqrt{g(a_0,...,a_{n-1})}$$ Another way to say this is that calculating $f(x_1,...,x_n)$ is as hard as calculating the roots of the polynomial $X^2-g(a_0,...,a_{n-1})$ So my question is, given a polynomial $f \in \Bbb Q[X_1, \dots ,X_n]$, can we determine if there is a polynomial $Q \in \Bbb Q[X]$ such that $Q \circ f \in Q[x_1, \dots ,x_n]$ is a symmetric polynomial? This would mean that calculating $f(x_1,...,x_n)$ is as hard as calculating the roots of $Q$.  |
| Stabilizers and topological groups acting over a smooth manifold Posted: 14 Aug 2021 07:43 PM PDT If $G$ is a topological group such that $G$ is acting as a group of transformation of a smooth manifold $M$. Take a point of $M$ the stabilizer is a subgroup of $G$. Now, how can I prove that the stabilizer is closed? Is it always closed? Can anyone help me with an hint for example? thanks in advance.  |
| Posted: 14 Aug 2021 07:29 PM PDT I feel like this shouldn't be this hard, but nothing on the internet seems to have helped at this point. I'm using a leg stretcher (which has 3 bars, 2 holding my legs, and a center one to pull to my crotch) with a belt to train my splits and wanted to know the angle between my legs (and see how I've improved). All given units are in centimeters. Using my legs and the stretcher, I get a kite: $\overline{AB} = \overline{AD} = a = 76$ (Stretcher Bars) $\overline{BC} = \overline{CD} = b = 80$ (Legs) $\overline{AC} = f = 57 + s$ (Center bar + belt) $s = \{35, 30.5, 27, 25\}$ (Belt lengths) I'm trying to find the angle $\gamma$ (located at C, so between the 2 b sides) and the length of $\overline{BD}$ I'd prefer an explanation or formula as I want to use the formula again later ($s$ - the belt - will get smaller and smaller after all). Any help is appreciated. (in the picture I used $\mu$ and $d$ for the desired angle and side ($\mu$ only since I don't have the greek letters on my keyboard, but I do have this)  |
| Posted: 14 Aug 2021 07:46 PM PDT $$\sum_{j=1}^{\infty}\sum_{n=1}^{\infty}\left(\frac{\sqrt{\frac{j}{n}}}{n^2+x^2j^2}-\frac{\sqrt{\frac{n}{j}}}{j^2+x^2n^2}\right) ,x\gt0$$ The double series was proposed by my friend khalef Ruhemi here https://m.facebook.com/story.php?story_fbid=1481248552228493&id=100010300870398 It looks maybe zero but this closed-form is not a zero. Any ideas to solve this problem?  |
| Using Generating Functions for solving system of recurrence relations Posted: 14 Aug 2021 07:07 PM PDT Using Generating Functions solve the system of recurrence relations \begin{eqnarray*} a_n&=&a_{n-1}+b_{n-1}+c_{n-1}\\ b_n&=&4^{n-1}-c_{n-1}\\ c_n&=&4^{n-1}-b_{n-1} \end{eqnarray*}  |
| Posted: 14 Aug 2021 07:37 PM PDT Prove the existence of prime p and integer m for all n, where a) n ≡ 5 (mod 6) b) p doesn't divide n c) n ≡ m^3(mod p)  |
| How to find a condition for when a multivariable limit exists Posted: 14 Aug 2021 07:50 PM PDT If I have the multivariable limit $$ \lim_{(x,y) \to (0,0)} \frac{x^ay^b}{(x+y)^c} $$ How do I find a general relationship/condition for $a$, $b$, $c$ that results in the limit existing? I've found various specific examples of $a$, $b$, $c$ that make the limit exist, but I don't know what the relationship between them is.  |
| Are radius and diameters lengths or line segments itself? Posted: 14 Aug 2021 06:43 PM PDT For a circle, I generally remember radius and diameters as length of line segments only. Former is the length of line segment connecting center of the circle and any point on the circle and the latter is the length of the line segment (chord) connecting any two points on the circle and passes through the origin of the circle. But, in some math sites, authors refer radius and diameter as to corresponding line segments rather than their lengths. Which sense is correct?  |
| How to simplify using conjugates in Maple Posted: 14 Aug 2021 06:55 PM PDT I think it is fairly easy to see that $$\frac{1}{\sqrt 2 -1} = \sqrt{2} + 1$$ Now given the fraction on the left in Maple, how can we get to the equivalent expression on the right? I apologize in advance, if this is rather obvious. But after trying a few things, I have not been able to make it work. I have used  |
| Show that $2x + 1$ is a unit in $\mathbb{Z}_4[x]$ and find a unit in $\mathbb{Z}_{p^2} [x]$ Posted: 14 Aug 2021 06:40 PM PDT
For the first part, $2x+1$ is a unit in $\mathbb{Z}_4[x]$, since $(2x+1)(2x+1)=4x^2+4x+1=1$.My problem is in the second part. I have tried to use the freshman's dream theorem whit the expression $px+1$, but I have come to nothing. I think a hint would be enough. Thank you  |
| Posted: 14 Aug 2021 06:42 PM PDT For quadratic least squares statistical regression equations, the following is utilized for the calculation of coefficients $(a,b,c)$: $a = \cfrac{∑(x^2y) * ∑(xx)-∑(xy) * ∑(xx^2)}{∑(xx) * ∑(x^2x^2)-∑(xx^2)^2}$ $b = \cfrac{ ∑(xy) * ∑(x^2x^2) - ∑(x^2y) * ∑(xx^2)}{∑(xx) * ∑(x^2x^2)- ∑(xx^2)^2}$ $c = \text{mean}(y) - b * \text{mean}(x) - a * \text{mean}(x^2)$ where, $∑(xx) = ∑(x^2) - \cfrac{∑(x)^2}k$ $∑(xy) = ∑(xy) - \cfrac{∑(x)*∑(y)}k$ $∑(xx^2) = ∑(x^3) - \cfrac{∑(x)*∑(x^2)}k$ $∑(x^2y) = ∑(x^2y) - \cfrac{∑(x^2)*∑(y)}k$ $∑(x^2x^2) = ∑(x^4) - \cfrac{∑(x^2)^2}k$ Can anyone help me identify this same method for finding the coefficients $(a,b,c,d)$ with regard to cubic regression statistical equations? Like this: $a = ???$ $b = ???$ $c = ???$ $d = \text{mean}(y) - c * \text{mean}(x) - b * \text{mean}(x^2) - a * \text{mean}(x^3)$ where, the same identities as above, but possibly including $∑(x^3y)$, $∑(x^2x^3)$ and $∑(x^3x^3)$?  |
| Integral change of variable from a real to vector Posted: 14 Aug 2021 07:46 PM PDT Let $f$ be a function $f:\mathbb{R}^k\to\mathbb{R}^+$, $\mathbb{y},\mathbb{x}\in\mathbb{R}^k$ (vectors) and $\xi\in \mathbb{R}$, then consider the integral $$ I=\int_{-\infty}^{\infty} f(\mathbb{y}-\xi\mathbb{x})d\xi, $$ suppose the integral exists, but it is difficult to compute, I was wondering if I could make a change of variable of the form $\mathbb{u}=\xi\mathbb{x}$, letting $\mathbb{x}=(x_1,\cdots,x_k)^T$ and $\mathbb{v}=(1/x_1,\cdots,1/x_k)^T$, then $\xi=\frac{\mathbb{v}^T\mathbb{u}}{k}$ and $k$ $$ I=\int_{\mathbb{R}^{d}} f(\mathbb{y}-\mathbb{u})\left|\frac{\mathbb{v}^T}{k}\right|d\mathbb{u}. $$ This seems odd, but this new integral I can solve. My doubt is whether this change of variable can be done or not.  |
| Find the second real root for cubic $x^3+1-x/b=0$ Posted: 14 Aug 2021 07:06 PM PDT A cubic of the form $$x^3+1-x/b=0$$ has has three real roots Using the Lagrange inversion theorem one of the roots is given by $$x = \sum_{k=0}^\infty \binom{3k}{k} \frac{b^{3k+1} }{(2)k+1} $$ How do you find the second one? I cannot find any info online. I am looking for a simialr series solution. I suspect is is of the form $$x = \sum_{k=0}^\infty \binom{3k+2}{k} \frac{b^{3k+2} }{(3)k+2} $$  |
| Chance letter a next to b in circle with whole alphabet such that no vowels next to each other Posted: 14 Aug 2021 07:43 PM PDT Here's a question from a book on probability I'm working through:
Here's what I did. Let's fix a. Since b can be immediately to the left or right of a, there's $2$ choices for b. Without loss of generality let's say we have ab, and so we have $4$ vowels remaining and $20$ consonants remaining. With the condition that no $2$ vowels come together: We want to find the number of possible places where we can place a vowel, which is in between consonants. With the $20$ consonants, there's $19$ possible "gaps" between, plus the $2$ on the end, for a total of $21$. However, because our b already occupies one of them, we have to subtract $1$, getting us $20$. So out of these $20$ places we're choosing $4$, so there's $\binom{20}{4}$ ways to place our $4$ vowels among the $20$ consonants subject to the condition no $2$ vowels come together. There's $4!$ ways to order the remaining vowels and $20!$ ways to order the remaining consonants. So our numerator is$${{2 \binom{20}{4} 20!4!}}$$Now let's calculate the denominator. Fix a again. This time we have $21$ consonants and $4$ vowels remaining, but only the $20$ possible "gaps" between the consonants to places our $4$ vowels (there's no $2$ at the end this time around), so our numerator is$$\binom{20}{4}21!4!$$Therefore the chance that a is next to b is$${2\over{21}}$$However, the answer in the back of my book (which is known to be wrong in many places in the answers in the back section) is ${1\over{10}}$. So who's correct? And if I'm wrong, where did I specifically go wrong?  |
| Kummer's hypergeometric negative integer Posted: 14 Aug 2021 07:49 PM PDT It is stated in many texts that when a-b+c=1 that 2F1(a,b;c;-1)=$\frac{\Gamma(\frac{b}2+1)\Gamma(b-a+1)}{\Gamma(\frac{b}2+1-a)\Gamma(b+1)}$ and when b is a negative integer using $\Gamma(z)\Gamma(1\!-\!z)=\frac{\pi}{\sin{\pi z}}$ and taking limit as b approaches a negative integer obtain $2\cos(\frac{\pi b}{2})\frac{\Gamma(|b|)\Gamma(b-a+1)}{\Gamma(\frac{|b|}2-a+1)\Gamma(\frac{|b|}2)}$ It is stated in more than one place but nowhere is it proven and I have tried all I can think of such as $\Gamma(\frac 1 2)=\sqrt{\pi},\Gamma(n+\frac 1 2)=2^{-n}(2n-1)!!,\frac{\Gamma(1+b)}{\Gamma(b)}=2\frac{\Gamma(1+\frac b 2)}{\Gamma(\frac b 2)},\sin(x+\frac\pi 2)=\cos(x),\sqrt{1-\sin(x)^2}=\cos(x)$ etc. For one I can't get anything in the original expression nor any substitutions for z that even remotely resembles $\Gamma(z)\Gamma(1-z)$ Does anyone have any idea how to explicitly and in detail proof ?  |
| Two questions about changing variables in multiple integral Posted: 14 Aug 2021 06:19 PM PDT Suppose we want to evaluate the double integral $\mathbf F(x, y)$ under domain $\mathit D$. And suppose there is a function $\mathbf φ(u, v)$ which transform $\mathbf F(x, y)$ into $\mathbf F \circ \mathbf φ(u, v)$. Then the area of each small rectangle $dxdy$ can be approximated by $|det \mathbf J \mathbf φ|dudv$. And we have neglected some higher order infinitesimal area. And here comes my first question: If we restrict our domain $\mathit D$ to be a bounded region. This works obviously. But what if our region $\mathit D$ is an unbounded region? I suspect the summation of those neglected area may not be neglectable. Are there any examples that the Jacobian transformation fails to work in an unbounded region? My second question is the case of two and three variables in Jacobian transformation can be interpreted geometrically by the area of a curve parallelogram. However, how can we prove that Jacobian transformation is still true in higher dimension since I think the geometric meaning fails in dimension higher than fourth. PS. I am sorry for the possibility of unclear of my question. Since some partial derivative symbol are hard to type so I try my best to express in words.  |
| Integral of sin(x)/(x^2+1) with respect to x from 0 to infinity [closed] Posted: 14 Aug 2021 07:00 PM PDT How would I go about evaluating this integral? Integral of sin(x)/(x^2+1) with respect to 'x' from 0 to infinity so far what I've managed to figure out is:
Thank you for your time  |
| Information about the quiver if the semi-invariant ring being a polynomial ring or hypersurface Posted: 14 Aug 2021 06:33 PM PDT I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)? I have been studying about semi-invariant rings in the context of Quiver representation but I don't really understand that if a semi-invariant ring turns out to be a polynomial ring or a hypersurface (or complete intersection), what "representation-theoretical" properties does it tell us about the quiver? Even if the answer is not particularly in the context of Quiver representation, I would still be glad to know.  |
| On computing $\nabla^2 (1/4 \pi |x|)$ in three dimensions using mollifiers. Posted: 14 Aug 2021 07:41 PM PDT I am going through some notes about how to compute that $\nabla^2 G = \nabla^2 (1/4 \pi |x|) = - \delta(x)$ in a sense of distributions, where $x \in \mathbb{R}^3$. Notes are not expected to be completely rigorous, however, I was not able to figure out all of the details to make the arguments precise. The approach starts by defining function $K : \mathbb{R}^3 \to \mathbb{R}$ such that $K(x) = G(x) = 1/4\pi|x|$ for all $|x| > 1/2$ in a sense of ordinary functions, and continued smoothly on the rest of $\mathbb{R}^3$. I believe the motivation is that the point of singularity of $G(x)$ is $x = 0$, and so we would like to define a function that is the same outside of the singularity, but is smoothly continued near singularity so we can treat it using usual real analysis theorems. Question 1: How do I know that such function $K$ can be indeed found? Is there a constructive formula for such function? I was thinking about defining $K(x) = G(x)\Theta(|x|-1/2)$, where $\Theta$ is a step function, and then to take smooth approximations $\{ f_n \}_{n \in \mathbb{N}}$ for step function, but I believe for all $n \in \mathbb{N}$, I would get that $K(x) \neq G(x)$ for $|x| > 1/2$. If such function $K$ indeed exists, then as it is a smooth extension, all derivatives exist in a pointwise real analysis sense. Then, for example, one might compute that for all $|x| > 1/2$, $\nabla^2 K (x) = 0$ pointwise. Also, one might apply divergence theorem to compute the following, where $B_{1/2}(0)$ is an open ball of radius $1/2$ around $0 \in \mathbb{R}^3$. $$ \int_{\mathbb{R}^3} \nabla^2K (x) \mathrm{d}x = \int_{B_{1/2}(0)} \nabla^2K (x) \mathrm{d}x = -1 $$ Now, one defines one-parameter family of functions (which will create a mollifiers). For each $\varepsilon > 0$, define $K_{\varepsilon} : \mathbb{R}^3 \to \mathbb{R}$ as follows. $$ K_{\varepsilon}(x) = \frac{1}{\varepsilon} K\left(\frac{x}{\varepsilon}\right) $$ I think I understand the intuition, as for each $\varepsilon > 0$, $K_{\varepsilon}(x) $ is again smooth, and this leads to the following scaling on Laplacian, which later can be used to change variables under the integrals. $$ \nabla^2 K_{\varepsilon}(x) = \frac{1}{\varepsilon^3} \nabla^2 K \left( \frac{x}{\varepsilon} \right)$$ Question 2: Notes motivate choice of definition of $K_{\varepsilon}$ with the claim that $K_{\varepsilon} * \phi \to G * \phi$, as $\varepsilon \to 0^+$, where $*$ is convolution operator and $\phi$ is suitable test function, say, smooth with a compact support. However, I don't understand in what sense is this true, as I thought that the whole point not working with $G$ is because it is singular at the origin. For example, naively, I would understand $G * \phi$ is the following. $$ (G * \phi)(x) = \int_{\mathbb{R}^3} G(x-y) \phi(y) \mathrm{d}y $$ However, as $G(x) = 1/4 \pi |x|$ defined on $\mathbb{R}^3 - \{0\}$ is not bounded, I believe (Darboux) integral does not exist. One maybe might define it as follows, but I am not sure if this is the convention here. $$ (G * \phi)(x) = \lim \limits_{\delta \to 0} \int_{\mathbb{R}^3 - B_{\delta}(0)} G(x-y) \phi(y) \mathrm{d}y $$ Question 3: Fixing interpretation of $G * \phi$, how to prove it is well defined (as an integral or as a limit)? Question 4: I tried performing the argument on $K_{\varepsilon}$ as the one that is used to show that $\nabla^2 (K_{\varepsilon} * \phi) \to - \phi(x)$ as $\varepsilon \to 0^+$. However, I do not understand what goes wrong. I provide (intuitive and, I think, wrong) argument below. $$ \lim \limits_{\varepsilon \to 0^+} (K_{\varepsilon} * \phi)(x) = \lim \limits_{\varepsilon \to 0^+} \int \limits_{\mathbb{R}^3} K_{\varepsilon}(x-y)\phi(y)\mathrm{d}y = \lim \limits_{\varepsilon \to 0^+} \int \limits_{\mathbb{R}^3} K_{\varepsilon}(y)\phi(x-y)\mathrm{d}y = \lim \limits_{\varepsilon \to 0^+} \int \limits_{\mathbb{R}^3} \frac{1}{\varepsilon} K\left( \frac{y}{\varepsilon} \right)\phi(x-y)\mathrm{d}y $$ $$= \lim \limits_{\varepsilon \to 0^+} \int \limits_{\mathbb{R}^3} \varepsilon^2 K\left( y \right)\phi(x-\varepsilon y)\mathrm{d}y$$ So, I would think that the following happens. $$ \lim \limits_{\varepsilon \to 0^+} \int \limits_{\mathbb{R}^3} K\left( y \right)\phi(x-\varepsilon y)\mathrm{d}y = \phi(x) \int \limits_{\mathbb{R}^3} K\left( y \right) \mathrm{d}y $$ Here, I think the latter integral does not exist, but I feel that there might a different choice of $K(x)$ such that that integral exists (for example, by also taking large $|x|$ cutoff). In that case, I feel that this limit would be finite but because of the $\varepsilon^2$ factor, $K_{\varepsilon} * \phi \to 0$ as $\varepsilon \to 0^+$.  |
| Proving a polynomial ring isomorphism Posted: 14 Aug 2021 07:08 PM PDT From Leitner - Basic Category Theory 2016 p8 My question is how to solve (b) below. In particular it was my thought that, using the homomorphism property proved in (a), the supplied definition of $\psi$ in (b) has no relevance to the solution. Would someone please tell me how am I mistaken? Denote by $\mathbb{Z}[x]$ the polynomial ring over $\mathbb{Z}$ in one variable. (a) Prove that for all rings $R$ and all $r ∈ R$, there exists a unique ring homomorphism $φ: \mathbb{Z}[x] \rightarrow R$ such that $φ(x) = r.$ (b) Let $A$ be a ring and $a ∈ A$. Suppose that for all rings $R$ and all $r ∈ R$, there exists a unique ring homomorphism $φ: A → R$ such that $φ(a) = r$. Prove that there is a unique isomorphism $ι: \mathbb{Z}[x] → A$ such that $ι(x) = a.$ Thank you  |
| Confusion in the order of the differentials in integral using differential forms Posted: 14 Aug 2021 07:44 PM PDT My doubt is very general but I'm going to give an example so I can explain why I'm struggling with this. I have a 2-form $\alpha$ written in Cartesian coordinates $$ \alpha=\alpha_{ij}\,dx^i\wedge dx^j $$ If I want to integrate this on a 2-sphere $S^2\in\mathbb R^3$, I'm going to change $\alpha$ to spherical coordinates $(x,y,z)\to(r,\theta,\phi)$ and have something like this $$ \int_{S^2}\alpha= \int \alpha'_{\theta\phi} \,d\theta \wedge d\phi $$ But I know that $d\theta \wedge d\phi=-d\phi \wedge d\theta $, so I also have $$ \int_{S^2}\alpha= -\int \alpha'_{\theta\phi} \,d\phi \wedge d\theta $$ Which is the correct way to integrate $\alpha$ over $S^2$? $$ \int_{S^2}\alpha= \int \alpha'_{\theta\phi} \,d\theta \,d\phi \quad\text{or}\quad \int_{S^2}\alpha= -\int \alpha'_{\theta\phi} \,d\theta \,d\phi $$ I can't discern difference between the two integrals, I need a procedure to choose the correct way to integrate. And the thing is worse when I'm trying to integrate things less intuitive and with other manifolds. I don't find any source either with clear examples and where I can understand  |
| Posted: 14 Aug 2021 06:59 PM PDT
I was told that this it is enough to show that if $V \subset Y$ is a neighborhood of $f(a)$ it's enough to show that $V \cap fA$ isn't empty, but how is this enough? What if $f(a) \in V$, but $f(a) \notin fA$? The intersection could still be non-empty?  |
| Limiting Distribution of Bernoulli to Poisson. ISI-PCB-NC$9$ Posted: 14 Aug 2021 07:47 PM PDT Question: Let $X_i\sim (i.i.d.)$, Bernoulli($\frac{\lambda}{n}$), $n\ge \lambda\ge 0$. $Y_i\sim (i.i.d.)$, Poisson($\frac{\lambda}{n}$). $\{X_i\}$ and $\{Y_i\}$ are independent. Define $T_n=\sum_{i=1}^{n^2}X_i$ and $S_n=\sum_{i=1}^{n^2}Y_i$. Find the limiting distribution of $\frac{T_n}{S_n}$ as $n\to\infty$. My solution: Hence, $\lim_{n\to\infty}\frac{T_n}{S_n}=1.$ Is there any mistake in the solution? Please help me to correct it. Thanks in advance.  |
| Waiting times at a bank - probability Posted: 14 Aug 2021 07:48 PM PDT Suppose there are 3 bank tellers, all currently occupied with a customer. You are waiting in line (and there are no other customers that will come in after you). All service times are iid exponential random variables with parameter $\lambda$. You must find the probability that you are the last to leave. I have set up this problem as follows:
With this setup, we must computer $P[X_d + X_- > X_+]$. So far, I have established the following:
I'm a bit stuck on how to move on from here. Any guidance would be awesome. Thanks.  |
| Why is CDF the only way to randomly select from samples? Posted: 14 Aug 2021 07:49 PM PDT I am confused with the concept of cumulative distribution function (CDF).
It seems to me that this algorithm takes into account each element's "weight" in the array, it takes into account duplicates and is random. But I have verified via testing that it is wrong. Update:  |
| Definition for the graph (cosine wave) when velocity is plotted against displacemnt Posted: 14 Aug 2021 06:41 PM PDT When $y = \sin x$ where $y$ is the displacement and $y'= \cos x$. So, when plotted (velocity in $y$ axis and distance in $x$ axis), we get a cosine wave(right?). So, when the velocity goes to negative in the graph, what does it really mean ?  |
| Name of distribution that is the square root of the sum of the squares of normal distributions? Posted: 14 Aug 2021 07:18 PM PDT Imagine a dart board. You throw a dart, and you have two normally distributed variables, $X$ and $Y$, representing the distance in the x and y direction from the bullseye. Obviously then, the radius $R = \sqrt{X^2 + Y^2}$. What is the name for the distribution of $R$? As far as I can tell, it's a Rayleigh distribution, but I don't know enough to be sure.  |
| Probability that a random binary matrix is invertible - what is going on? Posted: 14 Aug 2021 07:01 PM PDT This is a follow-up to this question: Probability that a random binary matrix is invertible? The answer says that the probability of a random $\{0,1\}$, $n \times n$ matrix to be invertible is: $$p(n)=\prod_{k=1}^{n}(1-2^{-k})\;,$$ For a $32\times32$, that's about 0.288. But, when I generate a random matrix in Matlab, and check its rank, it's always 32! The code is: Is the answer wrong? Is Matlab/Octave wrong? Please help me solve the mystery. Thanks!  |
| Calculating Ext and Tor for Z/mZ and Z/nZ Posted: 14 Aug 2021 07:22 PM PDT Looking at $Z/mZ, Z/nZ$ as $Z$ modules, how do we determine what $Ext^i(Z/mZ, Z/nZ)$ and $Tor_i(Z/mZ,Z/nZ)$ are for all $i$? I know that $Z$ is a projective module over itself and therefore, we get the following projective resolution: $$\cdots \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \xrightarrow{\times m} \mathbb{Z} \rightarrow Z/mZ \rightarrow 0$$ I am not sure what to do from here.  |
| GRE Subject Test - Past Papers, Books, Advice Posted: 14 Aug 2021 07:32 PM PDT This is not for the Maths part of the General GRE. This is for the GRE Subject Test in Maths. Feel free to add or comment.
http://www.mathematicsgre.com/ http://www.physicsgre.com/viewtopic.php?f=13&t=1078  |
| 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