59,290 research outputs found
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
A survey of max-type recursive distributional equations
In certain problems in a variety of applied probability settings (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form X =^d
g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are
independent copies of the unknown distribution X. We survey this area,
emphasizing examples where the function g(\cdot) is essentially a ``maximum''
or ``minimum'' function. We draw attention to the theoretical question of
endogeny: in the associated recursive tree process X_i, are the X_i measurable
functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Duality methods for a class of quasilinear systems
Duality methods are used to generate explicit solutions to nonlinear Hodge
systems, demonstrate the well-posedness of boundary value problems, and reveal,
via the Hodge-B\"acklund transformation, underlying symmetries among
superficially different forms of the equations.Comment: 14 page
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