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Questions tagged [surreal-analysis]

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4
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0answers
191 views

Algebraic Geometry Over the Surreal and Surrcomplex Numbers

I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?
9
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0answers
86 views

Genetic construction of roots of surreal polynomials

In On Numbers And Games, Conway uses the term "genetic" for definitions of operations on surreal numbers that are inductive in terms of their options. His definitions of addition and multiplication ...
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0answers
240 views

surreal form star is a xor cipher [closed]

i have been doing a lot of surreal analysis lately & it dawn'd on me that star in the surreal numbers has the same properties of a simple xor cipher. star operations are given as: ...
5
votes
1answer
381 views

Can a game be an option of itself?

My question is, can a game contain itself as an option? and can it be a surreal number? For example $A=\{A|\}$ or $B=\{C|B\}$ where $C$ is a surreal number. from the point of view of games, it is ...
12
votes
1answer
546 views

The surreal version of $e$

For a sequence $(x_{\alpha})$ of surreal numbers indexed by the set of all ordinal numbers, we say that $\lim x_{\alpha}=l$ ($l$ is a surreal number) if for each surreal $\epsilon>0$, there exists ...
6
votes
1answer
407 views

Is $\omega^\frac{1}{\omega} > n \forall n \in \mathbb{N}$?

I was thinking about $log(\omega)$ which appears to be $\{\mathbb{N}|\omega^{\frac{1}{n}}\}_{n\in\mathbb{N}}\stackrel{?}{=}\omega^\frac{1}{\omega}$. Intuitively, there's the idea that, if the highest ...
9
votes
1answer
428 views

Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...
12
votes
1answer
438 views

Theory of C* algebras over other fields than the complex numbers

How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...
6
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0answers
452 views

More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
16
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2answers
1k views

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig identities) will still ...
23
votes
3answers
3k views

Surreal numbers vs. non-standard analysis

What is the relationship between the surreal numbers and non-standard analysis? In particular, is there a transfer principle for surreal numbers they way there is for NSA? A specific situation in ...
13
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0answers
1k views

Characterizing the surcomplex numbers

Conway showed that the Field of surreal numbers ("${\bf No}$") is the maximal totally ordered Field. Later Jacob Lurie showed that the Group of all partizan games ${\bf Pg}$ is the universally ...