# All Questions

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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### I have a problem with differential geometry. [closed]

Prove that |a×b|²=|a|²|b|²-|a•b|²
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### 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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### 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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### 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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### 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 ...
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### 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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### 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, ...
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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 $\mu (\varliminf ... 0answers 5 views ### odds ratio logistic regression Let us suppose that a 2 x 2 factorial with variable coded to ± 1 is used to fit a logistic regression model in a drug study in which 20 subjects were allocated to each of the 4 treatment combinations. ... 0answers 3 views ###$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$-... 0answers 16 views ### Regularity of solution of$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} \; \...
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,...