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2 views

Combinatorics in a dynamic system?

On page 44 of his article "An Imitation of Life," William Grey Walter claims that "six elements would be more than enough to form a system which would provide a new pattern every tenth ...
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0answers
9 views

Trouble undestanding regular expression

I am reading a book example on regular expressions and I have a trouble to get why the asnwer is correct. "Write a regular expression for the regular language that contains all the strings by 0's ...
-1
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0answers
7 views

Decomposition of $f$

Let $f$ be a diffeomorphismus of $\Bbb R^m$. Why we can writte $f(x)=\sum_{i=1}^{n} x_ig_i(x)$, with some $g_i$ which are smooths? Many thanks for your help.
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0answers
4 views

Probability to obtained result by chance

Researchers wanted to test if girl babies have a preference to a doll or a ball. They observed 7 girl babies and found that 5 preferred the dolls. a. How likely is that they obtained this result by ...
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0answers
5 views

How to diagonalize a symmetric square matrix with only one extra row and column?

$$ \left[ \begin{array}{ccccc} a_{11}&a_{12}&a_{13}&\cdots&a_{1N}\\ a_{21}&a_{22}&0 &\cdots &0\\ a_{31}&0&a_{33} &\cdots &0\\ a_{41}&0&0 ...
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0answers
7 views

calculate the probability through area

The random variable X has a probability density function f(x) defined by: fx(x)=x-1 1<= x <=2 , f(x)=3-x 2<= x <=3 a)Find P(|X-2| <= 0.5) I don't understand the 2 I cercled ...
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0answers
6 views

Continuity of linear functionals on $L^1([a,b])$ with the $L^1$ functions composed with $C^1$ functions

I've been thinking about the following idea involving continuity of linear functionals on $L^1([a,b])$. Let $f\in C^1([a,b])$ be fixed. Define $\alpha:=\min f$ and $\beta:=\max f$. Consider the space $...
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0answers
3 views

Moments of Pareto($\alpha$)

I'm trying to show that a random variable $X\sim\text{Pareto}(\alpha)$ for some $\alpha>1$, i.e. $$P(X>x)=x^{-\alpha},\quad x>1$$ has infinit moment generating function $$\mathbb{E}[e^{hX}]=\...
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0answers
4 views

Writing T (m) for the composite trapezium rule and S(2m) for the composite Simpson’s, show that S(2m) = (4/3)T(2m) − (1/3)T(m) .

Given T(m)=h[$\frac{1}{2}$f(x0)+f(x1)+...+f(xm-1)+$\frac{1}{2}$f(xm)] for h=$\frac{b-a}{m}$ and S(2m)=$\frac{h}{3}$[f(x0)+4f(x1)+2f(x2)+...+2f(x2m-2)+4f(x2m-1)+f(x2m)] for h=$\frac{b-a}{2m}$, show S(...
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0answers
21 views

General rule of sequence $1,11,21,1211,…$

I understand that a term is the same as the "reading" of the previous one. The second term is equal to 1 amount of 1 in the first term = 11. The third term is equal to amount of 1 in the ...
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2answers
9 views

For which values is polynomial divisible by a prime number?

Let $f=aX^2+bX+c \in \mathbb{Z}[X]$ be a quadratic polynomial and $p$ a prime number. I need to find all solutions $x \in \mathbb{Z}$, s.t. $p$ divides $f(x)\in \mathbb{Z}$. I'm sure there is an ...
0
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2answers
17 views

How do I prove |z-i|=2 with $z - i = 2cos\theta - 2isin\theta $?

I have the following question. It's basically my first day doing complex numbers, so I am absolutely lost here. I have read that the modulus-arg form is $$ z = r(cos\theta + i sin\theta)$$ Now, in ...
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0answers
10 views

Placing people around table

Find number of ways to place 10 people around circular table, so between two specific persons are exactly other two. For those two specific persons, we have 16 possible combinations, and $8!$ ...
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2answers
18 views

How to Parameterize Hyperbola

I am trying to parametrize general hyperbola (shown below) using $x=t$, $y=1/t$. I tried to factor it, but I didn't get to the correct answer. The hyperbola: The correct answer: thanks a lot.
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0answers
19 views

If A $\bigtriangleup$ B = C $\bigtriangleup$ B, then A=C

I know that the power set P of any set A becomes an abelian group under the operation of symmetric difference so symmetric difference has this property. However, I need another solution. I tried to ...
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0answers
13 views

Prove that if $E(X^n)$ exists then $E(X^i)$ exists for $i=1, …, n-1$

Let $X$ be a discrete random variable and $n$ be a positive integer. Prove that if $E(X^n)$ exists then $E(X^i)$ exists for $i=1, ..., n-1$. This seems like a simple enough question but I'm having ...
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0answers
8 views

Graph created by contraction of edge is planar

Let G be a planar graph and e $\in$ E(G). Prove that graph H, which is created from a graph G by contraction of edge e is also a planar graph. I have to show that H can't induce some $K_{3,3}, K_{5}...
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0answers
11 views

Ring of fractions, localisations, etc. Proving $S^{-1}A \cong A \iff S \subseteq A^{\times}$

I am confused on how to prove the $\Rightarrow$ direction of this implication. The question is in context of http://userpage.fu-berlin.de/aconstant/Alg1/Alg1_2019.pdf p 49, Corollary 3.8. Thank you!
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0answers
9 views

Increments of Brownian bridge are not independent - why?

On the wikipedia of Brownian bridge: https://en.wikipedia.org/wiki/Brownian_bridge it says that the increments are not independent. Why is this true?
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0answers
18 views

Galois Theory and automorphism.

Let $K$ be a field and $L/K$ and algebraic extension of $K$ $(L \subset \overline{K})$. Assume that for all $\sigma \in Gal(\overline{K}/K)$ we have $\sigma/_L: L \to L \in Gal(L/K)$. Show that $G(L/K)...
1
vote
1answer
15 views

Maximising volume of a cylinder when surface area fixed

I know how to start off this problem but get bogged down when it comes to differentiating at the end. A right circular cylinder is of radius r cm. and height pr cm. The total surface area of the ...
1
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0answers
14 views

For what exactly we need that the submartingale is a finite sequence in theorem 6.7.3 of Ash's book?

I want to check if my understanding of the following proof is correct. The following is a slight rephrasing of a theorem of Ash's probability book: 6.7.3 Theorem. Let $\{X_n:n=1,\ldots ,m\}$ a ...
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2answers
25 views

I have a problem with differential geometry. [closed]

Prove that |a×b|²=|a|²|b|²-|a•b|²
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0answers
19 views

What is this non-sense with just ignoring higher order terms that no one ever explains?

I'm going through the derivative of the Euler-Lagrange equations. To get from the minimal form of a function to the weak functional form, from $P(u) = \frac{1}{2}\int c(u')^2 - \int f(x)u(x)dx $ to $\...
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1answer
21 views

Equivalence of $Ax = b$ and the quadratic form?

I want to prove the following: Given $A \in \mathbf{R}^{n \times n}$ is symmetric positive definite. Prove that $\hat{x}$ solves $Ax = b$ if and only if $\hat{x}$ minimizes the quadratic function $f: \...
1
vote
1answer
9 views

Rotations of a surface of revolution about its axis are diffeomorphisms

I'm currently studying Differential Geometry using Manfredo's book. In chapter 2, section 2, exercise 11, I am asked to prove that the rotations of a surface of revolution S about its axis are ...
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1answer
12 views

Combinatorics - Sorting objects into specific sets with no empty set using a generating function

How can you sort 90 people in 3 different rooms with no room being empty? With no restrictions sorting 90 people in 3 rooms would be 3^90. When it comes to no room empty, I start with (x+x^2/2!+...)^3 ...
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1answer
12 views

Working in modal logic, if $W=\{w\}$ for a model $M=(W, R, V)$, do we have that $wRw$?

If our set of worlds $W$ is a singleton, that is $W=\{w\}$, can we automatically conclude that this world $w$ is accessible from itself?
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2answers
20 views

calculate size of sample to reach wanted probability

You are recording neural activity in a cortical brain region. This brain region is known to contain excitatory and inhibitory neurons randomly distributed in space. In the cortex, the number of ...
3
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2answers
28 views

Minimum value of |z| for the locus of a complex number z

I have a question that involves an Argand diagram. The complex number u = 1 + 1i is the center of that circle, and the radius is one. In other words, $$|z - (1 + 1i)| = 1$$ Now my issue is the ...
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2answers
30 views

How to find a useful variable change for this integral

I would like to find the area of the following region $$ D=\left \{(x,y): -\sqrt{1+y^2}\leq x\leq \sqrt{1+y^2}; -1\leq y\leq (x+1)/2\right \}. $$ I try to calculate the double integral brute force, ...
2
votes
2answers
44 views

Lemma used to prove $\left|HK\right|=\frac{\left|H\right|\left|K\right|}{\left|H \cap K\right|}$

Given a group $G$ and $H,K \le G$,then : $$\left|HK\right|=\frac{\left|H\right|\left|K\right|}{\left|H \cap K\right|}$$ Where $HK:=\left\{hk:h \in H ,k \in K\right\}$ Lemma: For $h_1,h_2 \in H$ $$hK=...
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1answer
21 views

Power series substitution

I'm having some problems in series of this kind: $$\sum_{k=0}^{+\infty} a_kf^k(x) \ \ \ , x\in \text{Dom}(f)$$ ($f^k$ is k-th power, not k-th derivative, or iterated compositin). If I make the ...
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0answers
17 views

Hölder function

let $f$ be the $\alpha $-Hölder function. How to prove that for $\beta \leq \alpha$ $f$ is also $\beta $-Hölder function?
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0answers
12 views

Fit sum of exponentials - excluding intercept

Question Given data points, $$(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$$ Suppose I know $$y_i\sim\mathrm{Laplace}(f(x_i),\sigma)$$ Where $$f(x)=be^{px} + c^{qx}$$ For some $b, c, p, q, \sigma > 0\...
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0answers
12 views

Derivative of exponential tensor with respect to a vector

Suppose you have a rank 4 Tensor $T$ whose coordinates are, in some basis, $T_{ijkl}$. Say the indices take values in some representation of $T$. Now suppose I construct a matrix by dotting this ...
2
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0answers
8 views

Multiplication module which is not cyclic

By the definition from Barnard (1980), a multiplication module is an $R$-module $M$ in which for all submodules $N$, there exists an ideal $I$ of $R$ such that $N=IM$. Note that a cyclic $R$-module (a ...
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3answers
16 views

Find distribution function of a PDF

I'm learning probability and trying to solve the following problem: Find distribution function of: $p(x)=\left\{\begin{matrix} 0 & x<2 \\ \frac{A}{(1-x)^2}&x\geq 2 \end{matrix}\right.$ ...
0
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0answers
11 views

differential equations with brownian motion

I have an equation like $dX(t)/dt = f(X(t),t)+\int_{0}^t dW_s$. I was wondering if there is a way to solve it (even in the simple case like $f(X(t),t) = g(X(t))h(t)$). Any hint will be much ...
0
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2answers
19 views

Euler characteristic of a metrized line bundle

During my readings on the web, I found the Euler characteristic of a metrized line bundle, let $L$ a line bundle on a variety $X \to \operatorname{spec}(K)$. And let $\varphi$ a metric on $L$. Someone ...
0
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0answers
4 views

What is the usefulness of diagonalise a Fisher matrix?

In Statistics, we often talk about diagonalising the Covariance matrix (like in Principal Analysis Components) but what would be the interest and the usefulness of diagonalising the Fisher matrix : ...
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1answer
13 views

A single die is rolled 7 times. What is the probability that a six is rolled exactly once, if it is known that at least one six is rolled?

A single die is rolled 7 times. What is the probability that a six is rolled exactly once, if it is known that at least one six is rolled? Could you walk me through the concepts?
1
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1answer
14 views

[Literature Request]: Counter factual estimation and modelling

I'm hoping that someone may be able to help me. I'm beginning a project in which requires asking "what would have happened if X didn't happen", specifically whilst using Bayesian analysis ...
0
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1answer
39 views

Sum geometrically $\sum_{l=1}^{n} l^9 $

How do I solve the sum: $$\sum_{l=1}^{n} l^9 $$ I was asked to solve this geometrically
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0answers
7 views

Inequality between limits of measures

I have another measure problem that I don't really know how to solve Let $μ$ be a non-negative measure on $σ$-algebra $F$ of subsets of space $X$. Let $A$, $A_n$ $∈$ $F$. Show, that $\mu (\varliminf ...
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0answers
5 views

odds ratio logistic regression

Let us suppose that a 2 x 2 factorial with variable coded to ± 1 is used to fit a logistic regression model in a drug study in which 20 subjects were allocated to each of the 4 treatment combinations. ...
0
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0answers
3 views

$h$-vector of Standard graded Cohen-Macaulay algebra over a field

Let $k$ be a field, and $I$ be a homogeneous ideal of $k[x_1,...,x_n]$ such that $R=k[x_1,...,x_n]/I$ is a Cohen-Macaulay ring of dimension $d$. Let the Hilbert-series of the standard graded $k$-...
1
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0answers
16 views

Regularity of solution of $u_t -\Delta u=f$ where $f$ is a continuous function

Let $f \in C(\bar \Omega \times [0,T])$ and consider the following problem: $\begin{align} \partial_t -\Delta u&=f \quad \text{in} \; \Omega \times (0,T)\\ u(\cdot,0)&=u_0 \quad \text{in} \; \...
-1
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0answers
16 views

Physical economic argument

Obviously if economic theory worked, they would just print money. There is nothing magical about foreign investment, it is just money entering the country, and that can easily be done yourself....
0
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2answers
11 views

Intersection of convex cones

I have the following question: Let $C$ and $C_i$, $i=1,\dots,m$ convex cones in $\mathbb{R}^{n\times n}$ that verify: $\cap_{i=1}^m C_i \neq \emptyset$ and $C\cap C_i \neq \emptyset, \forall i=1,\dots,...

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