35,379 research outputs found
Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas
We point out that a sequence of natural numbers is the dimension sequence of
a subproduct system if and only if it is the cardinality sequence of a word
system (or factorial language). Determining such sequences is, therefore,
reduced to a purely combinatorial problem in the combinatorics of words. A
corresponding (and equivalent) result for graded algebras has been known in
abstract algebra, but this connection with pure combinatorics has not yet been
noticed by the product systems community. We also introduce Cartesian systems,
which can be seen either as a set theoretic version of subproduct systems or an
abstract version of word systems. Applying this, we provide several new results
on the cardinality sequences of word systems and the dimension sequences of
subproduct systems.Comment: New title; added references; to appear in Journal of Stochastic
Analysi
Bulking II: Classifications of Cellular Automata
This paper is the second part of a series of two papers dealing with bulking:
a way to define quasi-order on cellular automata by comparing space-time
diagrams up to rescaling. In the present paper, we introduce three notions of
simulation between cellular automata and study the quasi-order structures
induced by these simulation relations on the whole set of cellular automata.
Various aspects of these quasi-orders are considered (induced equivalence
relations, maximum elements, induced orders, etc) providing several formal
tools allowing to classify cellular automata
The game semantics of game theory
We use a reformulation of compositional game theory to reunite game theory
with game semantics, by viewing an open game as the System and its choice of
contexts as the Environment. Specifically, the system is jointly controlled by
noncooperative players, each independently optimising a real-valued
payoff. The goal of the system is to play a Nash equilibrium, and the goal of
the environment is to prevent it. The key to this is the realisation that
lenses (from functional programming) form a dialectica category, which have an
existing game-semantic interpretation.
In the second half of this paper, we apply these ideas to build a compact
closed category of `computable open games' by replacing the underlying
dialectica category with a wave-style geometry of interaction category,
specifically the Int-construction applied to the cartesian monoidal category of
directed-complete partial orders
The variety of reductions for a reductive symmetric pair
We define and study the variety of reductions for a reductive symmetric pair
(G,theta), which is the natural compactification of the set of the Cartan
subspaces of the symmetric pair. These varieties generalize the varieties of
reductions for the Severi varieties studied by Iliev and Manivel, which are
Fano varieties.
We develop a theoretical basis to the study these varieties of reductions,
and relate the geometry of these variety to some problems in representation
theory. A very useful result is the rigidity of semi-simple elements in
deformations of algebraic subalgebras of Lie algebras.
We apply this theory to the study of other varieties of reductions in a
companion paper, which yields two new Fano varieties.Comment: 23 page
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