benjamin-rood

$${ (x,y)\in \Bbb R^2: x\in [-a,a],y\in [-b,b]}$$

$$(0, + \infty ] = \{ x \in \Bbb R: x > 0 \}$$

$$\mathrm{Lighter \ boxes \ are \ better.}$$ light-grey is too dark a field background, so I tried #eae9e6 instead... maybe even lighter than that is better?

benjamin-rood

$${ x \in \Bbb{R} : x < 2 }$$

benjamin-rood

For question 2, do you mean:

$$ax \ mod \ n = 1$$

$\forall x \in \mathbb{Z}$ ? (Note, no brackets, as me are talking about the operation)

(I suppose as long as $x$ is a number, it doesn't make much difference....)

benjamin-rood

Suppose $\langle a^{d_{1}} \rangle \leq \langle a^{d_{2}} \rangle$. Then

$$a^{d_{1}} = (a^{d_{2}})^{m} \textrm{, for some } m \in \mathbb{Z}$$

Thus

$$ a^{d_{1}-md_{2}} = e = \textrm{ identity.}$$

benjamin-rood

Re: Fine, Gaglione, Rosenburger, Introduction to Abstract Algebra, (2014), pp.377-9

Dear Dr. Fine,

I am trying to understand the formal description in the chapter on Polynomial Rings in Introduction to Abstract Algebra, and there is one particular part that is confusing me. Specifically, about 3/4 of the way down p.378 :

benjamin-rood

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[TOC]

#Preface & Acknowledgements

###github.com/benjamin-rood/abm-cp

benjamin-rood

#Definition of the ABM Environment

Consider the set of all real numbers within $[\,-1,  \,+1\,]$.$\ $ Let $\,F\ $ be the cartesian product of such a set with itself,  *s.t.* individual elements in $\,F\,$ are two-dimensional points, or a cartesian *pair* within a subset of $R^2$.
$$\therefore \ F\ = \  \{\ (x,y) \in\, \mathbb{R}^{2} \, \lvert\  (\,-1 \leq x \leq 1 \,)\ \wedge\,(\,-1 \leq y \leq 1 \,)\  \}$$

Let $\ E\ $ be the symbolic representation of our modelling environment, such that $E$ encompasses the following: 

* A vector space $\ V\ $ over field $\ F$.
* Variable extensions on the field $F$ which act as "environmental" describers *(to define the properties of the medium the agents exist in)* and modifiers *(to be used to model changes to the world state)*.[^env-mod]